Ocean Circulation and Pollution Control ­A Mathematical and Numerical Investigation

Springer-Verlag Berlin Heidelberg GmbH

J esus Ildefonso Dfaz (Editor)

Ocean Circulation and Pollution Control­A Mathematical and Numericallnvestigation A Diderot Mathematical Forum

t Springer

Editor: Jesus Ildefonso Diaz Departamento de Matematica Aplicada Facultad de Matematicas Universidad Complutense de Madrid 28040 Madrid Spain e-mail: ildejonso_diaz@mat.ucm.es

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ISBN 978-3-642-62289-2 ISBN 978-3-642-18780-3 (eBook) DOI 10.1007/978-3-642-18780-3

Mathematics Subject Classification (2000): 35Qxx, 86A05, 65-XX, 91-XX

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Preface

In the framework of the Diderot Mathematical Forum (DMF) of the EuropeanMathematic al Society (EMS) , a videoconference linking three teams of specialistswas held in Amsterd am , Madrid and Venice on Decemb er 19th and 20th, 1997.The general subject of this videoconference ---which was the second one of theDMF series- was Mathetratics and Environm ent: Problems related to Water.The large number of common problems that were treated in the three sit eshighlights the global nature of environmental studies.

This book contains the written cont ribut ions assembled in Madrid. It is notsur prising that a large part of this mat eri al focuses on questions related withOceanography, since Spain is one of t he European countries with longest coas t.

In the cont ribution by R. Bermejo , ocean circulati on is considered from t hepoint of view of a new algorithm for the numerical approach to the so-calledocean primitive equat ions. A different perspect ive is followed in the pap er byJ. Macias, C. Pares and M. J. Castro. They consider a "local problem" whosegreat relevanc e goes beyond Spain : the Strait of Gibraltar . Important concre teaspect s must be correct ly mod elled by means of a mod el obtained under somesimplifi cations: the multilayer shallow seas model. The numerical exp er iencespresented in this article allow to und erst and cert ain phenomena that have beenknown empirically since long time ago. A related question was considered byB. Sommeijer in Amsterdam. The use of sophist icated numerical algorit hms wasalso the aim of one of the contributions presented in Venice (by A. Quarteroni) .

One of the main environment al problems related with water concerns con­t amination and its many different aspects. Groundwater pollution is conside redin the pap er by M. W. Saaltink and J . Carrera (and was also discussed in thelect ure by F . J . Elorza) . T his type of problem was, in fact , the main subjectdevelop ed in the presentations made in Amsterdam by C. J . van Duijn, M. deGee, B. H. Gilding and A. Stein.

The contaminat ion of wat ers has many common points with the dispersion ofair pollulan ts from combust ion of fuel, as presented in the art icle by G. W inter ,J . Betan cor and G. Montero (and also by L. de Haan in the set of Amsterd amlectures) .

The mod elling of the interaction between physical and biological aspect s incoastal ecosystems has a crucial relevance near cit ies and beach es. Such problemsare considered in t he paper by A. Bermudez, C. Rodriguez, M. E. Vazquez­Mendez and A. Martinez. Since this problem is especially important in placessuch as Venice, it is underst andab le that the subject was extensively t aken intoaccount by Italian lecturers (A. Bergamasco, V. Casulli and G. Gambolati) .It was also considered by B. Sommeijer in Amst erd am .

VI Preface

The above-mentioned pro blems are mod elled in terms of systems of non-linearpartial differenti al equat ions. Nevertheless, t he st udy of environment very oft enrequires methods that come from statist ics. T he use (and abuse) of st atisticsin environmental issues was the main content of severa l lectures given in Am ­sterdam (by L. de Haan , R. D. Gill and A. St ein). Finally, economical asp ectsand policies related with pollution limit ations were t aken into considerat ion byJ . 1. Diaz and J . L. Lions by st udying the optimisat ion and control of multi­criteri a under a deterministic state law. A different optimal cont rol problem isstudied in the pap er by A. Bermudez, C. Rodriguez, M. E. Vazquez-Mendez andA. Martinez.

Let us mention that severa l results contained in this book have not beenpublished anywhere else. It is t he case , for instance, of the cont ributio n signedby t he late J acques-Louis Lions. Thanks to his dedication to this field in thelast t en yea rs of his life, the study of environmental problems reached a greatpop ularity among mathemati cians of many count ries.

The videoconference was a complete success. The linking between the threesit es worked perfectl y and according to the pro gramme. In fact , a video of morethan 10 hours was recorded . As everybody knows, the organisation of an inter­national meeting is always complicate d. But t his one was singularly difficult ,due to the technical aspects requi red for a corre ct linking. The job of the t hreemain organisers (E. Canest relli, M. Keane and myself) was possib le thanks tothe help of a large number of people and institutions. Following a chronologicalorder , we start by acknowledging the efficient work made by the EMS Com­mittee on Special Events. T hanks to its President , Jean-Pierre Bourguignon- who was also President of the EMS at that time- and the Secret ary, MireilleCh aleyat-Maurel, a joint preparatory meeting with t he local organisers, held inParis in May 1997, mad e easier the coordination and scient ific design of thevideoconference.

Green light was given by funds received from the DGXII of the EuropeanCommission . Support also came from other sources, such as the Unione Matem­atica It alian a and local institutions, especially the three universities linked bythe event . We convey our indebtedness to all of them.

After the scienti fic design of the videoconference, the preparations requiredthe help of many techni cians and colleagues at each site. It is impossible to listall of them , but a special mention must be made to the high efficiency of BenSchouten in Amsterd am , who helped not only wit h tec hnical questions but alsowith many other aspects, including the scient ific ones.

I cannot finish this shor t introduction without mentioning the importanthelp received from Juan Francisco Padi al , Carles Casacuberta (EMS Publica­tions Officer) , and the editors and st aff of Springer-Verlag for the preparation ofthis book.

Jesus Ildefonso Diez

The DMF Series

In 1996, the European Mathematical Society launched an original series of meet­ings named "Diderot Mathematical Forums" . Here is how this choice came about:

- Since the ambition was to create a series of events during which mathe­maticians of different origins and specialists coming from other fields couldexcha nge knowledge and discuss their views around a definite topic, why notcall them forums?

- In the periode des Lumieres, Denis Diderot was one of the driving forcesbehind the writing and edit ing of L 'Encuclopedie, an unb elievably ambitiousent reprise exhibiting a new concept ion of the links between science, techniqueand society. He was also a man who always insisted in his writings on thehuman dimensions of things, and, in doing so, he was , in his own way, warningagainst "scient ism", that was yet form ally to be born. On top of that , byvisiting different count ries and working in them, he exp erienced Europe ata time where it was not so common; this makes of course ano ther point forthe relevance of his name in this context.

The main purpose of this series was (and still is) twofold . First, to providemathematicians with a tool to change the image that their community projectsoutside in the wider society, by showing their interest in engaging themselves inexchanges with other professional groups. Second, to have an internal effect bygiving more visibility to new avenues of thought for mathematicians, in particularyoung ones, with the definite purpose of showing how the interaction wit h othercommuniti es brings in new problems and sheds a new light on the mathematicalpracti ce and on the needs of other people.

This led the EMS to tryout a rather unusual format for these Forums,namely having two-d ay meetings held in three different European cities linkedthrough videotransmissions for a part of the event . The purpose at each locationwas to have man ageabl e size audiences, say up to 100 participants, making thelocal organisation light enough while creating condit ions for the thought forconfront at ion.

Very early, the theme "Mathemat ics and Environment" was identified as atopic to which a Diderot Mathematical Forum should be devoted. Indeed , thisdomain is exemplary of interdisciplinary act ions that one will need to develop inthe near future on a much larger scale if one is serious about coming to grips withcomplex systems such as this one. But it soon appeared that the domain wastoo wide to allow fruitful exchanges. This led the EMS commit tee in char ge ofthe Forums to propose and concentrate attention on a narrower subject, namelyenvironmental problems related to water . In t his context, Amsterdam and Veniceappeared as two emb lematic European cit ies to discuss such questions.

The theme "wate r" speaks to every citi zen, and it is especially appropriatefor confront ing the views of mathematicians with those of many other scientists

VIII The DMF Series

(chemists, physicists , eart h scient ists, biologists , etc. ). It touches upon serioussocietal issues, in the solution of which scient ist s of all sorts have yet to findtheir true place. This dim ension , namely making the relevance of present daymathematics plain to a wider public, is part of the challenge that the Did erotMathematical Forums want to face.

It is by now well recognised that t here is still a lot to be done in the de­velopment of mathematical mod els allowing a better knowledge and predictionof water presence in the soil, together with pollution risks. In many Europeancount ries, and also in other cont inent s, improving the production of usabl e wateris very high in the priority list of problems to be solved. Many other issues re­lat ed to water were addressed in t he Forum, in part icular those connecte d to seabehaviour. This topic was dominantly dealt with in the third city participatingin the Forum, namely Madrid.

It is my pleasure to end this foreword by acknowledging the hard work don eby Professor Ildefonso Diaz from the Universidad Complutense in Madrid, Pro­fessor Michael Keane from the Centrum voor Wiskunde en Informatica in Ams­terdam , and Professor Elio Canest relli from t he Universita di Venezia, to set upthis Diderot Mathemati cal Forum. We are grateful to them for their hard work.

Jean-Pierre BourguignonEMS President 1995-1998

President of the EMS Committee on Sp ecial Events

Second Diderot Mathematical Forum

Mathematics and Environment: Problems Related to Water

Amsterdam , Madrid, Venice, December 19-20, 1997

Amsterdam

Organisers: MICHAEL K EANE AND BEN SCHOUTEN

R . M. COOKE (Technische Universi teit Delft)

Expert judgement and the theory of dry water

C . J . VAN D UIJN (Centrum voor Wiskunde en Informati ca, Amst erdam)

Salt water in trusion in coastal regions

M . DE G EE (Landbouw Universiteit Wageningen)

Semi-numerical methods for groundwater contaminant transport

B. H . GILDING (Universiteit van Twente)

On the wetting front - transport of moisture in soil

R. D . GILL (Rijksuniversiteit Utrecht)

Lies, damn ed lies, or statistics of the environment; use and abuse ofstatisti cs in environmental issues

L. DE HAAN (Erasmus Universiteit Rotterdam)

Sea and wind: mult ivariate extremes at work

B . SOMMEIJER (Centrum voor W iskunde en Informati ca , Amst erdam)

Num erical modelling of three-dim ensional bio-chemical transport inshallow seas

A. STEIN (Landbouw Universit eit Wageningen)

Point processes, random sets and geostatistics for analyzing patternsof methylene blue coloured soil

Madrid

Organiser: J ESUS ILDEFONSO DfAZ

R . BERMEJO (Universidad Complutense de Madrid)

Eulerian versus semi-Lagrangian schemes in some ocean circulationproblems: a preliminary study

X Second Diderot Mathematical Forum

A. BERMUDEZ, C. RODRiGUEZ, M. E. VAZQUEZ-MENDEZ AND A. MARTiNEZ(Universidad de Santiago de Compostela)

Mathematical modelling and optimal control methods in wastewater discharges

J.1. DiAZ (Univ. Complutense de Madrid) AND J. L. LIONS (College de France)

On the approximate controllability of Stackelberg-Nash strategies

F . J . ELORZA (Universidad Politecnica de Madrid)

Transport of pollutants in ground water and low permeability rocks

J. MAciAS, C . PARES AND M. J . CASTRO (Universidad de Malaga)

Numerical simulation in Oceanography. Applications to the Alboran Seaand the Strait of Gibraltar

M. W. SAALTINK AND J . CARRERA (Univ. Politecnica de Catalunya, Barcelona)

Simulation of reactive transport in groundwater. A comparison oftwo calculation methods

G. WINTER, J . BETANCOR AND G. MONTERO (Universidad de Las Palmas)

3D Simulation in the lower troposphere: wind field adjustment toobservational data and dispersion of air pollutants from combustionof sulfur-containing fuel

Venice

Organiser: Ei.ro CANESTRELLI

A. BERGAMASCO (Ist. per 10 Studio della Dinamica delle Grandi Masse, Venice)

Coupling physical and biological modelling in coastal ecosystems:the Venice lagoon example

V . CASULLI (Universita di Trento)

A mathematical model of the Venice lagoon

G. GAMBOLATI (Universita degli Studi, Padova)

The mathematical model of the Venice subsurface system

A. QUARTERONI (Istituto Politecnico di Milano and CRS4 Cagliari)

Physical-numerical modelling of environmental processes

Contents

Part A: Oceanic pollution control

A. BERMUDEZ, C. RODRiGU EZ, M. E. VAZQUEZ-MENDEZ

AND A. MARTiNEZ. ..... . ..... . . . . . . . . ... .. ... . ..... . . . . . . . . . ....... .. .. 3

Math ematical modelling and optimal control methods in wastewater discharges

J . 1. DiAZ AND J. L. LIONS 17

On the approximate controllability of Stackelberg-Nash strategies

G. WINTER, J . BETANCOR AND G. MONTERO 29

3D Simulation in the lower troposphere: wind field adjustment toobservational data and dispersion of air pollutants from combustionof sulfur-containing fuel

Part B: Numerical methods in oceanic circulation

R. BERMEJO 55

Eulerian versus semi-Lagrangian schemes in some oceancirculation problems: a preliminary study

J . MAciAS , C . PARES AND M . J . CASTRO 75

Numerical simulation in Oceanography. Applications to theAlboran Sea and the Strait of Gibraltar

M. W . SAALTINK AND J . CARRERA 99

Simulation of reactive transport in groundwater . A comparison oftwo calculation methods

Part A

Oceanic pollution control

Mathematical modelling and optimal controlmethods in waste water discharges

A. Bermudez"; C . Rodriguez." , M. E. Vazquez-Mend ea" , and A. Martinez 2

1 Departam ento de Maternatica Aplicada , Facultad de Matematicas,Universidad de Santiago de Compostela, 15706 Santiago, Spain

bermudezOzmat.usc.es, carmenOzmat .usc.es , ernestoOlugo.usc.es2 Departamento de Mat ernatica Aplicada , E.T .S.I. Telecomunicaciones,

Universidad de Vigo, 36200 Vigo, SpainaureaOdma.uvigo .es

1 Introduction

In this work we show how mathematical models and optimal control techniquescan help to solve certain problems of environmental engineering - more precisely,water po llution problems arising from discharges into coastal ar eas or rivers.

Usually, waste waters originated from urban areas or industry undergo aphysico-chemi cal and biolog ica l tr eatment in a plant. Then th ey ar e dischargedthrough an outfall into an aquat ic medium like a lake, a river or a coastal area,at an adequa te distance from protect ed areas .

Before bu ild ing such a system, st udies of environmental impact ar e necessaryin order to ensure that pollu t ion does not reach swimming areas or marine cultureareas. At this stage, mathem atical models can be very useful , becau se they arecheaper and less aggress ive than experiment al methods. Fur th ermore, the answercan be ob tained in a shorter t ime (see for instan ce [12], [24], [3], [13J , [5], [6], [7]).

There are two main ty pes of models, corresponding to the two followingstages of effluent flow:

Buoyant flow from th e point of discharge towards th e surface (jet models).- Hori zontal t ra nsport by cur rent action from the fina l level in th e previous

stage (fa rfield models).

The first ones are systems of ordinary differential equat ions a long th e axis ofth e jet , while farfield models invo lve partial differential equations. In the presentpaper we deal with the second ones.

Very oft en, treatment plants discharge waste waters through outfalls wh ichare placed in the same ar ea (estuary, lake, etc.) . Thus, a ll of th em contribute towater po llution. In these circumst ances, th e problem of design and managementof th e who le sytem of treatment plants and outfalls arises. Optim ization methodscan help decision makers in formulating rational po licies in order to minim izecosts while keeping th e prescribed levels of water quality (see for instance [14],[20], [4], [8], [11], [7]). An exam ple is considered in Section 3. The problemis formula ted as a pointwise op timal contr ol problem with sta te and controlconstraints (see [21], [22], [23]) . We present numerical resu lts for a real problemposed in th e ria of Vigo (Spain) .

J. I. Díaz (ed.), Ocean Circulation and Pollution Control - A Mathematical and Numerical Investigation© Springer-Verlag Berlin Heidelberg 2004

4 A. Bermudez et al.

2 Mathematical models

First of a ll, it is convenient to notice that , since the volume of discharges issmall compared with that of receiving waters, hydrodynamical equ ations maydecouple from pollution dispersion equations. Therefore, the first ste p is to setand solve a model for simula ting flows in the area under consideration. Thenmathematical inodels can be used to simulate the dispersion of pollutants. Asan importat example, we will present a system of partial differential equationsgoverning the evolution of the Biological Oxigen Demand (BOD) and the Dis­solved Oxigen (DO) .

2. 1 Hydrodynamic models

Currents are very oft en the main factor for dispersion of po llutants in farfie ld .In this section we reca ll the Saint Venan t equat ions, which yield a useful math­em at ical model for hydrodynamic flows in shallow regions.

Consider an incompressible viscous newtonian fluid in a shallow domain de­fined as follows (see figure 1):

where:

Q is t he Xl , Xz projection of the domain filled by the fluid ,h(xl, Xz, t) is th e height of the fluid layer at a point (Xl, xz) and a t a time t,X3 = b(Xl' xz ) is the equation of the bottom surface,IJ( Xl' xz) = A - b(Xl, xz ) is th e depth from a fixed reference level A,T](Xl, Xz , t) = h(Xl' Xz, t) - H(xl, xz) the surface elevation from the referencelevel A .

Assuming that pressure is hydrostatic and integrating the incompressibleNav ier-Stokes equations, t he following sys tem of partial differential equationscan be obtained:

1. Mass conservation equation:

2. Momentum conservation equations:

a(hut} a(hui) a(hU1UZ) a lb+h u' z d- --+-- -+ + - 1 X3at aXl axz aXl b

a lb+h, , aPa aT]

+~ U1Uz dx3 +~ +gh~ = 2wsin<1ihuzUXz b UX l UXl

1 I I gUl';ui + u§+pllOVl V - C? '

(2)

(3)

Mathematical modelling and optimal control in waste water discharges 5

Fig. 1. The shallow domain.

(4)

where:

111 and 112, the horizontal mean velocities, are defined bYUl = *fbb+h

U1 dX3,

1 rb+h112 = t: Jb U2 dX3,

c. = U1 - 111, {h = U2 - 112,

g is gravity,p is density,Pa is atmospheric pressure,v is the velocity of wind 10 m above water surface,w is the angular velocity of the earth,C is the Chezy coefficient,<Jj is the north latitude,il0vlvl = Tw (wind stress),pgulul/C2 = Tl (bottom friction stress).

The four terms involving 01 and O2 are called the Reynolds stresses and rep­resent the dispersive effects due to velocity fluctuations from the mean. Some-

6 A. Bermudez et ai.

t imes they can be neg lected , as we will do in t his paper. W hen this is not done,a "closur e" givi ng these terms as functions of Ul and Uz might be used .

The shallow water equations (2, 3 and 4) can be written in terms of theconservative variables 7) and Q (the flow rate vector, given by Ql = Ulh =r b+h rb+hJb U1dX3 a nd Qz = uzh = Jb Us dX3), as follows:

107)-+'\7 · Q=Oot

oQ; ~ ( o (uj Q;)) h~ oPa - F . ~( _ )"'t + L.J '" + g '" + '" - , + Tw Tf'U j =1 U X j U X ; U X ; P

i = 1,2,(5)

where F = 2w sen4i( Qz , - Qt} accounts for t he Co riolis effect .Furthermore, boundary a nd initial conditions have to be specified:

(i) Coast or effluent (To ):

Q ·v = ! , v unit normal vector (f = 0 on the coast). (6)

(ii) Open sea (rt}:h = ¢J + II (¢J a given function) . (7)

(iii) Init ial conditions:h(x, 0) = ho(x ) (8)

u (X, 0) = Uo (x ) (9)

2.2 Pollutant d ispersion: the BOD IDa model

In order to control water pollution, some paramete rs a re used whic h indicate thequality level of liqu id media and t hei r capacity to hold aquatic life. Among t heseindicators a re d issolved oxygen, heavy metals, temperature, pH , radioactivity,etc.

Oxygen plays a major role in all kinds of life. In particular, it is used bybact erias to decompose organic matter. If the oxygen demand is not satisfied,pla nct on and other higher forms of animal life disappear. However, the decom po­sition of organic matter goes on by anaerobic processes which do not use oxygenbut produce sulfure of hydrogen and methane, both having nauseous smell. Thelevel of dissolved oxygen depends on two categories of factors:

1. The quantity of organic matter in t he water to be decomposed a nd themechanism for this decompositi on.

2. Some physica l conditions as temperature, depth, flow rate, turbulence, etc.

T he organic matter can be measured in terms of t he oxygen needed to decom­pose it . This is the so-called biological oxygen demand (BOD) . If t he po llutionlevel is not too hig h, th is demand can be satisfied by the dissolved oxygen. Noticet hat the dissolved oxygen is very sensitive to wastewater discharges, namely to

Mathematical modelling and optimal contro l in waste water discharges 7

the thermal ones. Indeed, at high temperatures the solubily of oxyge n decreaseswhile activity of microorganisms - which is oxygen consuming- increases.

Let us denote by P1 (respectively P2) the density of BOD (respectively DO).Then they sa t isfy th e following system of partial differ ential equations (see[7], [30]):

(10)

(11)

where:

h is the height of water ,u is t he horizonta l mean velocity ,N E is the number of ou tfalls,mj is th e discharge of BOD at a point Pj,<5(x - Pj ) denotes the Dirac measure a t a po int Pj ,

/31 and /32 are hori zontal viscosity coefficients,"1 is a kinetic parameter related to temperature,"2 is the interface transfer rate for oxygen,ds is th e oxygen saturation density in water,I B is t he intensity of sunlight on th e bo ttom ,M is th e surface population density of algae,r'p is the respiration coefficient of algae,a, b and c are empirica l coefficients.

3 Application of optim al control methods: Optimalmanagement of a wastewater treatment system

3 .1 Statement of the p roblem

In this section we suppose that severa l (NE) t rea tme nt plants disch arge wastewaters through outfalls which are placed in the same domain [2 C ]R2 occupiedby shallow waters (estuary, lake, etc. ) and, thus, all of them contribute to waterpollution. Moreover, each plant has an associa ted cost ot t reatment (Ii) whichis a function of the final pollutant (BOD) concentrations. These functions canbe different from one to another by several reasons, such as the technology, thena ture of pollution content of a rriving waste waters, etc.

Finally, we assume that th ere a re N z areas in [2 (beaches, fish nurseri es,etc.), where the pollution level has to be below the maxirriun value permitted byofficial regulations.

Then the following problem arises: What is the reduction of pollutant concen­tration to be made at each plant in order to minimize the tota l cost of the whole

8 A. Bermudez et al.

system, while keeping the prescribed levels of water qua lity in the protectedareas?

Taking BOD/DO as water quality indicators and assuming that the depura­tion cost in a particular depuration plant is a funct ion of the BOD dischargedat the end of th e process through the corr esponding outfall, th e optimal controlproblem can be written as follows:

Find the values (after depuration) of BOD concentrations mj(t) 2: 0, j =1, . .. , NE , which verify the state system (non-conservative form of the model(10)-(11 ) without algae effects and with boundar y and initial conditions) :

apl 1 NE7ft: + uV'PI - f31 iJ. P l = - K I P I + h ?= mjJ(x - Pj ) in D x (0,T )

J= l

apl = 0 on r x (0, T)anPl(X, 0) = 0 in D;

apz 17ft:+ uV' pz - f3ziJ. pz = - K I P I + hKZ(ds - pz) in D x (0,T)

apz = 0 on r x (0, T)anpz(x, 0) = pzo(x) in D;

satisfy the constraints

P1 IA iX(O,1') ~ ai, i = 1, . .. , Ne ,

PZIAiX(O,l') 2: ( i , i = 1, . .. , Nz:

and minimize the cost function

NE t"J (m ) = 2: l.. fj (mj(t)) dt .

j = l 0

3 .2 Discretization

(12)

(13)

(14)

(15)

The state equations are discretized by using finite element and characteristicmethods (see [30]). Then we get a numerical approximation of the state variables(BOD and DO) at some grid points an d time steps (pij(x) R:! p;(Xj, tn)) , and wedefine the function g, which is putting together the discretized constraints:

where:

g: rn:N x N E ----+ rn:N x N vz x rn:N x N vz x rn:N x N E

m t-----tg(m) =(pl -<T,(- pz, _ m )t ," v J "-v--"

=gt(m) = g2(m )

(16)

Mathematical modelling and optimal control in waste water discharges 9

m is the vector consisting of all of the discharges a t all times,Nv z is the number of vertices in the protected areas,Pi is a vector of valu es of Pi a t vertices included in the protected areas andfor all times.

The cost fun ctional is also discretized by using a quad rature formula, and wedefine the new cost:

J: jR N xNE --t jR

mN E N - l

I-----t J(m) = L1t.E .E Qjn!i(mj n),j=l n=O

(17)

'V'J(m) + 'V'gl>" - Ie = 0,

G1(m)>.. = 0, G2(m)O = 0,

where mjn is the amount of BOD discharged in Pj at a time tn, and Qj n arethe weights of the quadrature formula.

Then the discrete optimal control problem has the following form:

{

min J(m)(PF) m Ellt N X N E

such that g(m) S; O.

3.3 Solving the discrete optimization problem: an interiorpoint method

We have solved th e pr oblem (PF) by means of an admissible points methodwhich is based on a globally convergent algorithm introduced by Herskovits [17]an d Panier et al [25] for nonlinear constraints.

If we denote the vector of the dual variabl es by (>.. , e) , then we can writethe first ord er Karush-Kuhn-Tucker optimality conditions for our problem asfollows:

(18)

(19)

(20)

gI(m) S; 0, g2(m) S; 0, (21)

where G1(m) and G2(m) are diagonal matrices, with ((Gi) (m))j j = (gi (m))j.The basic idea of the algorithm of ad missible points consists of solving th e

system of equations (18)-(19) in (m,>.. ,e) by using an iterative method, in sucha way that the conditions (20)-(21) hold at each iteration.

For a given point (mk,>..k,ek) t, th e Newton method applied to th e previoussystem computes th e next ite ra tion (m~+l, A~+l, e~+l)t by solving

( m~+ 1 ) (m

k)

,\~+1 ,\k

o~+1 Ok(

H(mk, ,\k,Ok) "Vg1 _I) -1 ("Vj(mk) + "Vg1,\k _ 10k

)

Ak("Vg1)t G~ 0 G~,\k

_8k 0 G~ G~Ok

10 A. Bermudez et a l.

where :

- H(m ,).., 0) = 'VZj(m) + 'L-{=l )..;'VZgli(m) + 'L-f=l O;'VZgz;(m) is the hessi anof th e lagrangian,

- G1 = G1(mk) , G~ = Gz(m k

) ,

- Ak, ek are diagonal matrices, with (Ak) ii = ()..k); and (ek );; = (Ok) ;.

In general, the point (m~+l , )..~ +l , O~+l) t is not feasible (since the equations(20) and (21) do not hold) . Then, we define dk = m~+l _ m k as a search directionin m and rewrite the previous equality by computing (dk ,)..~+l,O~+l)t as thesolution of the followi ng linear system:

[ ll~:;~::')~k ) :' ~l ) [A;:') [- V'~(mk )) (22)

_ek 0 G~ O~+l 0

Now, in order to determine the new primal point m k +1 , we perform a linesearch along dk to obtain a step t k which leads us to a new admissible pointm k+1 = m k + tkdk where th e cost reduction is satisfactory. Finally, the newvalue of the dual variable ()..k+l, Ok+l)t can be computed from ()..~+l, O~+l)t byseveral updat ing methods.

According to this, the general sketch of the algorithm is the following:

A D M ISSI B LE POINTS A LGO R I T H M

Previous 1. Compute gl (0) , 'VsnInformation 2. Choose (mo,)..0,8o)t such that g(mO)::; 0, >..0 ~ 0, 8° ~ 0

STEP 1 Compute the descent dir ecti on d" by solving the linear syst em (22)

STEP 2 Compute the ste p len gth t" by employing a line search techniqueand define m k+1 = m k + t kd k

STEP 3 Update the dual vari able: Defin e (>""T ,8"Tl) from (>,,~T , 8~Tl)

ST EP 4 Test of Convergence:

I. If it is OK --+ stop algorithm and acce p t mk+1 as solution ofthe problem (PF)

11. If it is not OK --+ go back to STEP 1

3.4 Numerical results

The problem (PF) has been solved for a study case corresponding to th e Riaof Vigo (Spain) . First of all , we have calculated the velocit y and the height of

Mathematical modelling and optimal control in waste water discharges 11

the water by solving the shallow water equations (5) (figure 2 shows the velocityfield a t high tide). Then we have considered two discharge points (NE = 2) aswell as two protected areas (Nz = 2), and we assume that the pollution level inarea 1 must be lower than in area 2:

maximum BOD

(}l = 5.810- 2 Kg/m3,

(}2 = 6.610- 2 Kg/rn",

minimum DO

81 = 7.186410-3 Kg/m3

82 = 7.035410- 3 Kg/m3

Moreover, we suppose that the cost of the depuration is the same for thetwo purifying plants, and that the mass flow rate of BOD arriving to both is150 Kg/s, so the cost function above this value is constant (see figure 5).

In the figure 3 we show the isolines for concentration of BOD at high tide.State constraints hold everywhere in the protected areas and saturate at onevertex in zone 1. At low tide, after a tidal cycle , the BOD concentrations can beseen in th e figure 4. Now saturation takes place at one vertex of area 2.

The optimal values of discharges are given in the figure 6. One can observethat during rising tide the discharge rate is greater at point 2 than at point1. However, during ebb tide (after t = 60) the flow rate decreases at P2 andincreases at Pl. This is an obvious consequence of th e position ofthe two outfalls.

Fig. 2. The velocity field at high tide.

12 A. Bermudez ct al.

Zone 1

Zone 2

[ill] Isle

MOOULEF :1"BHPO 6015/07/98mailvigo c

coorvigo::::cvel.sdb

738 POIWI'S738 NOEUDS

1241 ELEMENTS1241 TRIANGLES

INCONNUE: 1 MNEMO:VN

5.2916&-04

5.0000E-04 =024.6734&-04

4.0552E:-04

3.7460E-04

3.4B40E-04 = (OJ 1

3.1278&-04

2.5096E-04,10

11

12 9.6398E-05

13 6.5486&-05

14 3.45HZ-OS

15 3.662!Z-06

738 PO:IN'TS738 NORunS

:1.241 ELEMlI:NTS:1.241 ~QLES

MODULZF •'rXMPO :1.2016/07/98

ma.11v.t90_'"

:~:::_a

Fig. 3. BOD concentration at high tide.

[ill] Isle

Fig. 4. BOD concentration at low tide.

1

aa.s,,e,ac11aa

"U"

:l..1321B'-03

1.020"'1':-03

9.0313B-04

8.4449:&:-04

7.2720&-04

6.0991B-04

5.5:1.26:11:-04

::::~~:=~: =(Qr23.7532l!:-04

::~::~:=~: = ([ff12.5803ll:-04

1..9S139:S:-04

1.4Q74E_04

Mathematical mod elling and opt imal control in waste water discharges 13

BOO

600

400

200

{

(100)(150)'

f(x)= x' - 3(150)x2+ 3(150)2X

100

si x s 150

si x > 150

50 100 150 2 00

Fig . 5. Cost function .

D. 8. 0

93. 1

0 . 0 40. 0 8 0. 0 120 . 0

CLJ

R eferences

Fig. 6 . Optimal disch arges during a tidalcycle.

1. Abbot [1985]: Computa tio nal Hydraulics, Pi tman, Boston .2. Alcrudo, F ., Garcia-Navarro , P. and Savir6n , J . M. [1993]: Flux-differenc e splitting

for ID op en channel flow equa tions, Int. J. Num. Methods in Eng . 14 , 1009-1018.3. Ames , W . F . [1988]: Analysis of mathem atical models for po llutant transport and

diss ipation, Comput . Mat h. with Appl. 16 , 939-985.4. Antonios, M. N. [1989]: Optimization problems rela ted to water quality control in

aquatic ecosys te ms, Compu t. Math . with A ppl. 18, 851-870.

14 A. Bermudez et al.

5. Bermudez, A. [1993]: Mathematical techniques for some environmental problemsrelated to water pollution cont rol, in Math em at ics , Climate and Env ironme n t,Diaz, J. I. and Lions, J. L. eds., Masson , Paris.

6. Bermudez, A. [1994]: Numerical modelling of water poll ution problems, Environ­me nt, Eco nomics and their Math emat ical Models, Diaz, J. 1. and Lions, J. L. eds.,Masson, Pari s.

7. Bermudez, A. [1997]: Mathematical modelling an d op timal control methods inwater pollut ion problems, The Math em at ics of Mod els for Climatology and En vi­ronme nt, Na to ASI Se ries I 48, Dfaz, J . 1. ed ., Springer Verlag, Berlin, Heidelb erg ,New York .

8. Bermud ez, A., Martinez, A., Rodriguez, C. [1991]: Un problerne de controleponctuel lie a l'em placement op timal d 'emissaires d 'evacuation sous-marins,C. R . Acad. Sc i. Paris t. 313 , Serie I, 515-518.

9. Bermudez, A., Rodriguez, C., Vilar, M. A. [1991]: Solving shallow water equat ionsby a mixed implicit finite element method, IMA J. of N um. Analysis 11 , 79-97.

10. Bermudez, A., Vazqu ez, M. E . [1994]: Upwind methods for hyp erbolic conservationlaws with source te r ms, Computers and Flu ids 23 , n . 8 1049-1071.

11. Bogobowicz, A. [1991]: T heoretical aspe cts of modeling and control of water qualityin river sect ion , Appl. Math. and Compo 41 , 35-60.

12. Br ebbia, C . A. (Ed .) [1976]: Math ematical Model for Environme ntal Probl ems,Pentech Press, London.

13. Ga mbola ti, G ., Rinaldo , A., Bre bbia , C. A., Gray, W . G., Pinder, G. F . [1990]:Computatio nal Methods in Su rface Hyd rology, Springer Verlag , Berlin.

14. Haimes, Y. Y. [1976]: Hierarchical analysis of water ressources systems , McGrawHill , New York.

15. Herskovits, J ., Santos, G. [1982]: A two-stage feasible directi on algori thm includ­ing variable metric techniques for nonlinear optimization problems, Rapports deRecherche, INRlA.

16. Herskovits, J . [1992]: An int erior point technique for nonli near optimization, Rap­ports de Recherche, INRIA.

17. Herskovits, J ., Santos , G. [1997]: On the computer implemetat ion offeasible Direc­t ion Point Algori thms for nonlinear optimization . COP PE - federal University ofRio de Janeiro , Mechanical Engineering Program, Caixa Postal 68503, 21945-970,Rio de .Ian eiro, Brazil.

18. Lions , J. 1. [1968]: Controle Opt imal des S ys te me s (l ouoernes par des Equatio nsaux Deriu ees Partiell es , Dunod, Paris.

19. Lions, J . 1. [1979]: Nouveaux espaces fonctionnels en theorie du controls dessysternes distribues, C. R . A cad. Sc i. Pari s t. 289, Serie I, 315-319.

20. Loucks, D. P., St edinger , J . R., Haith, D. A. [1978]: Wat er Resou rces Sy st emsPlanni ng and Analys is, Prentice Hall , New York .

21. Mar tinez , A., Rodriguez, C., Vazqu ez Mendez, M. E . [1998]: Resoluci6n numeric ade un problem a de control relativo ala depuraci6n de aguas residuales , Aetas delXV CEDYA/ X CMA, Universidad de Vigo, Spain .

22. Martinez , A., Rodriguez, C ., Vazqu ez-Mendez, M. E. [1998]: Theore tical and nu­meri cal analysis of an optimal control probl em related to wast ewater treatmen t,P reprint , Dept . Matematica Aplicada , Univ. Santiago de Compostela.

23. Mar tinez , A., Rodriguez, C., Vazqu ez-Mendez, M. E. [1998]: A con trol probl emrelated to wast ewater treatment . C. R. Acad. Sci. Pari s , Serie I, in pres s.

24. Nihoul, .J. (Ed .) [1975]: Modelling of Marine Sy st em s, Elsevier , Amst erdam .

Mathematical modelling and optimal control in waste water d ischarges 15

25. Panier, E . R. , Tits, A.-L., Herskovits , J. [1988]: A QP-free, globally convergent ,locally sup erlinearly convergent algorithm for inequality constrained optimization,SIAM Journal of Control and Optimization 26 , 788-810.

26. Quetin, B. , De Rouville, M. [1986]: Submarine sewer outfalls - A design manua l,Marine Pollu tion B ullet in 17, 133- 183.

27. Rahman , M. [1988]: Th e Hydrodynamics of Waves and Tides, with Applications,Computational Mechanics Publications, Southampton.

28. Stoker, J . J . [1957]: Water Wa ves , lnterscience, New York.29. Thomann, R. V. [1972]: Systems Analysis and Wat er Quality Management, Envi ­

ronm ental Research and Applications Inc, New York .30. Vazquez-Mendez, M. E . [1992]: Contribuci6n a la resoluci6n numerica de mode­

los para el estudio de la contaminaci6n de aguas , Master T hesis, Publ. Depart.Mat ernati ca Aplicada, Univ ersity of Santiago de Compostela .

On the approximate controllability ofStackelberg-Nash strategies

J . I. DiazI an d J. 1. Lions /

I Departamento de Matematica Aplicada, Facultad de Matematicas,Universidad Complu tense de Madrid , 28040 Madrid , Spain, j i_diazOmat . ucm . as

2 College de France, 3 ru e d 'Ul m, 75231, Paris Cedex 05, France

1 Introduction

Let us consider a distributed sys tem, i.e. a sys te m whos e sta te is defined by thesolut ion of a Partial Differential Equa tion (PD E). We ass ume that we can acton this system by a hierarchy of controls. There is a "global" control v, which isth e leader, a nd th ere a re N "loca l" controls, denoted by WI, ••• , W N , which areth e follow ers. The followers, assuming th at the lead er has mad e a choice v of itspolicy, look for a Nash equilibrium of th eir cost functions (the criteria th ey areinterested in). Then t he leader makes its final choice for the whole system . Th isis the Sta ckelberg- Nash strategy.

Such situations arise in very many fields of Enviro nment an d of Engineering(and, by the way, for sys tems not necessarily described by PDE's). In order toexplain mo re precisely our mo ti va tion , let us choo se here an example taken fromEnvironment: let us consider a resort lake, represented by a domain [2 of ~3.

T he state of the syste m is denoted by y . It is a vector function y = {YI , ... , YN },

each Yi being a function of x and t , x E [2, t = time. The Yi 'S correspond toconcent rations of various chemicals in the lake [2 or of living organisms. The Yi'Sare therefore given by th e solution of a set of diffu sion equations. In the resort,there a re local agents or local plants, PI , . . . ,P N . Eac h plant P i can decide (withsome constraints) it s poli cy uu , There is also a genera l manager of th e resort. He(or she) has th e choic e of th e poli cy deno ted by v. Therefore the state equationsare given by

C;; + A (y ) = sources + sinks + global cont rol v + local cont rol {WI , • .• , W N } , (1)

where th e initial state is su pposed to be given ,

y( x ,0)= yo(x), (2)

and where there are appropria te boundary conditions (of course this is mad emore pr ecise in the next section of this paper) . The general goal of the managerv is to maintain the lak e as "clean" as possible. In oth er word s, if t he sit uationat t = 0 is not ent irely sa tis factory, he (or she) wants to "drive the system" ata chosen time horizon T as close as possible to an ideal state, denoted by s" .Each plant P i has essent ia lly th e same goal , bu t of course, P i will be parti cularly

J. I. Díaz (ed.), Ocean Circulation and Pollution Control - A Mathematical and Numerical Investigation© Springer-Verlag Berlin Heidelberg 2004

18 J . I. Diaz and J . L. Lions

careful to the state y near its location. Let Pi be a smooth function given in Qsuch that

Pi(X) 2: 0, Pi = 1 near the loca tion of Pi. (3)

Then Pi will try to choose ui, such that the state at time T, y(x, T ), be "close"to PST , and to achieve this at minimum cost . This leads to the in troduction of

(4)

where Illwilll represents t he cost of Wi, O'i is a given positive constant andII Pi (y (., T) - yT) II is a measure of the "localized distance" between the actualst ate at time T and the desired state yT.

Remark 1.1 We have assumed here that the system (1), (2) (together with ap­propriate boundary conditions) admits a unique solution y(x, t; V; WI, .. • , WN) .

In (4) , y (.,T) denotes the function x 1-7 y (x,T ; Vi11'1, • •• , W N).

T he "loca l" controls wI, • • • , 11'N ass ume that th e leader has made a choiceV and th ey try to find a Na sh equilibrium of their cost Ji, i.e. they look for11'1, • • • , 11'N (as functions of v) such that

Ji(V ; WI , • • • , Wi-I, Wi , W i+l , ... ., WN) ~ Ji(V ; WI , . .. , Wi -I , Wi ,Wi+l , ... , WN) ,

for all Wi , for i = 1, .. . , N.(5)

(6)

If w = {W1 , • • • , WN} satisfies (5), one says it is a N ash equilibrium.The leader v wants now that the global state (i.e. the sta te y( ., T) in the

whole domain Q) to be as clos e as poss ible to yT . This will be possible, for anygiven function yT, if the problem is approximately controllable, i.e. if

y(x , t ; V; WI, .• • , W N) describes a dense subset of the given statespace when v spans the set of all controls available to the leader.

Remark 1.2 We emphasize again that in (6) the controls Wi are chosen sothat (5) is satisfied. Therefore they are functions of v.

Remark 1. 3 T he above st rategy is of t he Stackelberq's typ e. This strategy hasbeen introduced by Stackelberg [12] in 1934 for problems arising in Economics. Ithas been used in problems of distributed systems in Lions [7], without referenceto controllability questions and in Lions [8] in a different setting without usingNash equilibria.

Remark 1.4 We have explained the family of problems we are interested in forenvironment questions, but problems of this type arise in many other questions,such as the control of large engineering systems.

R emark 1.5 It is clea r that yT is not going to be an arbitrary function inthe state space. Therefore t he reso rt cou ld be maintained in a satisfactory state

Approximate controllability of Stackelberg-Nash st rategies 19

even without t he system being approximately controllable (in th e sense of (6)) .Bu t if there is a seri ous degrad a tion followin g, for instance, an accident, t henthe in itial state can be "anything" so that it is certainly preferable to live in a"controllable resor t" . . .

Remark 1.6 Of course , t he Stackelberg's type strategy is not the only possible!One could also rep lace the Nash equilibrium by a Pareto equilibrium for thefollowers WI, . .. ,WN (see, for instance, Lions [9]). Here a ll the controls Wi agreeto work in a strategy where v is the lead er, an d they agree to work in th e contextof a Nash equilibrium. Their personal (selfish) interests a re expressed in the costfunctions J; as we shall see in the next section. .

Remark 1.7 In the above context there does not always exist a Nash equi­librium . We prove in Section 4 some sufficient conditions for the existence anduniqu eness of a Nash equilibrium. We also present a general counterexampleshowing that those conditions are, in some sense, necessary. W hat we (essen­t ia lly) show in this paper (t he first of a series ) is that for linear systems, if thereis existence and uniqueness of a Nash equilibrium for the followers, then thelead er can cont rol the system (in the sense of app roximate controlla bility). T hestudy of the case of nonlinear systems is the main su bject of Dfaz and Lions [2].

The content of the rest of this paper is the following: In the next section wemake precise the statement of our main result by taking one state equation, i.e .y is a scalar funct ion y instead of a vect or function {Yl' . . . , YN}. T his is just forthe sa ke of simplicity of the exposit ion. It is by no means a serious restriction.But we shall make a very strong assumption, namely that the state equat ion islinear. The proof of the approximate controlla bility will be given in Section 3.The st udy of su itable assumptions (and their optimality) implying the existenceand uniqueness of a Nas h equilibrium is ca rried out in Sect ion 4. Finally, somefurther remarks are presented in Sect ion 5.

2 Statement of the approximate controllability theorem

Let A be a second order elliptic operator in Q :

N O( 0 ) N 0A<p = - L - ai ,j(x )~ + L ai(x)-.:£. + ao(x )<p,.. 1 OXi OXj . 1 OXiz,J = z=

where a ll coefficients are smooth eno ugh and where

N N

L ai,j (x )t.it.j 2:: Q' Lt.;, Q' > 0, x E Q.i ,j = 1 i = 1

We assume that the state equation is given by

o N

o~ + Ay= vx+ L WiXi, =1

(7)

(8)

(9)

20 J .1. Diaz and J. L. Lions

whereX is the characteristic function of 0 C Q, and

Xi is the characteristic function of O, C Q.(10)

Remark 2 .1 The control function v(x, t ) of the leader is distributed in 0 andthe control function Wi (x, t) of the follower "i" is distributed in O],

Remark 2 .2 All the results to follow are a lso valid for boundary controls. T hecase of distributed controls permits to avoid some difficulties of a purely technicalty pe .

We assume that the initial state is

y(x, 0) = 0, x E Q. (11)

Remark 2.3 Since the system is linear, there is no restriction in assuming theinitial state to be zero, in the same way as there is no restriction in assumingin (9) that sources + sin ks are zero (com pare to (1)).

We assume that the boundary conditions are

y = 0 on aQ x (0, T ). (12)

Remark 2.4 Again (12) is not at all a serious restriction. We could consideras well y to be nonzero and that the following resu lts app ly for other boundaryconditions.

We introduce now functions Pi such that

Pi E Loo(Q), n : 0, }

Pi = 1 in a domain 9i C Q,

and we define the cost function J, (compare to (4))

T

111 2 (Xi II T 112

J i (Vi Wl , . . . ,WN) = - Wi dxdt + -2 Piy(TiV, W) - PiY ,2 0 0;

where 11 ·11 is the norm in L 2 (Q).

(13)

(14)

Remark 2.5 In the case of the example presented in the Introduction, 9i isthe regio n of the lake the plant Pi is part icularly interested in (the place nearPi for instance!). If Pi is selfish, then Pi = 0 outside 9i ·

Approximat e controllability of Stackelberg-Nash strategies 21

Remark 2.6 From a mathematical view point, th e only hypothesis needed onPi is that Pi E LCO(Q) (one could even take Pi in a sui table LP(Q) space , butthis is irrelevant here) .

Remark 2 .7 We assume that

v E L 2(O X (0, 1')) , Wi E L 2(Oi X (0,1'))

and th at y( x, t ; v, w) is th e solution of (9), (11) , (12) .

Given v E L 2(O X (0, 1')) , we now define (cf. (5))

w ={WI , . .. , WN}, a Nash equilibrium for th e cost, }(15)

and fun ctions h, ... , I N given by (14).

We will show in Section 3 how (under hypotheses which are presented in Sec­tion 4) that this Na sh equilibrium can be defined as a funct ion of v :

w = w( v) or Wi = Wi(V), i = 1, .. . , N .

We th en replace in (9) Wi by Wi(V) :

8 N8

Y + A y = vx+ L ui, (V)x it i = 1

(16)

(17)

subject to (11) and (12) . T he system (17) , (11) and (12) admit s a unique solut iony(x , t ;v, w(v)) . In Section 3 we prove th e following resul t.

Theorem 2.1 Assume that

th e set of in equalities (5) admits a unique solution (a N ash equilibrium) . (18)

Th en , when v spans L 2(O x (0, 1')) , th e fun ct ion s y(., 1'; v, w(v) ) describe adense subset of L 2 (Q). In oth er words,

th ere is approximate controllability of th e syst emwhen a strategy of th e Stackelberg- Nash type is followed.

3 Proof of the main theorem

3.1 Nash equilibrium

We have (5) iff

(19)

iT1 ui ;Wi dxdt + Q'i1pr(y(1'; v , w) - yT) fj; (1') dx = 0, VWi, (20)o 0 , n

22 J .1. Diaz and J . L. Lions

where fj; is defined by

ofj; A~ - }7ft + Yi = WiXi,

fj;(0) = °in D, fj; = °in oD X (0,T).(21)

(22)

In order to express (20) in a conve nient form, we introduce the adjoint state Pidefined by

- O~i + A*Pi =°in D x (O,T), }

Pi(X, T) = p;(x)(y~x , T; v, w ) - yT (x)) in D,

Pi =°in oD x (0, T) ,

where A* stands for the adjoint of A. If we mult iply (22) by fj; and if we integrateby parts, we find

LP7 (y(T;v , w) - yT) fj; (T) dx = iT LpiWixi dxdt ,

so that (20) becomes

rT r(Wi + Cl:ip;)Wi dxdt = 0, VWi,lo i.

i.e.

Wi + Cl:iPiXi = 0.

Then, if w = {WI, . .. , WN} is a Nash equilibrium, we have

oy Not + Ay + L Cl:iPiXi =VX,

i = l

OPi A* N- 7ft+ Pi =O , i =I, .. . , ,

y(O) = 0, Pi( X,T) = p;(x)(y(x , T; v, w ) - yT (x)) in D,

y = 0, Pi =°in oD x (0,T).

(23)

(24)

We recall that here we are assuming the existence and uniqueness of a Nashequilibrium (hypothesis (18)) . We ret urn to that in Section 4.

3.2 Approximate controllability : Proof of Theorem 2.1

We want to show that the set described by y(.,T; v) is dense in £2(D), where yis the solut ion given by (24) and when v spans £2(0 x (0, T)) . We do not restrictthe problem by assuming th at

yT == °(it suffices to use a t ranslation argument). Let f be given in £2(D) and let usassume that

(y(.,T ;v),J) = 0, Vv E £2(D) . (25)

(26)

Approximate controllability of Stackelberg-Nash strategies 23

We want to show that f == O. Let us introduce the solution {ip , 'l/J l' . .. , 'l/JN} ofthe adjoin t system

Oip A* 0-7it+ ip =,

o'l/J ;---rft + A'l/J; = -Ct;ipX;,

ip (T ) = f + L;'l/J ;(T )PT,

'l/J; (O) = 0,

ip= 0, 'l/J; = 0 in o[l x (0, T ).

We multiply the first (resp. the second) equation in (26) by y (resp. Pi) ' Weobtain

-(f + I: 'l/J;(T)p;,y(T)) +iTL ip(~~+AY) dxdt+,

I:( 'l/J ;(T) ,p;(T)) + (27)i

+I: rTl 'l/Ji (- 0;:;+ A*Pi) dxdt = - I:Ct; rTl ipPi Xi dx dt.i Jo n ot i Jo n

Using (24) (where yT == 0), (27) redu ces to

-(f, y(T )) +iTL ipVXdxdt = O. (28)

Therefore, if (25) holds, th en

ip = 0 on 0 x (0, T ). (29)

Using Mizohata's Uniqueness Theorem (see Mizohata [5] or Saut and Scheurer[10]) - t his is the only place where some smoothness on the coefficients of A isneeded- it follows fro m (26h an d (29) tha t

ip= 0 on [lx(O ,T ).

Then (26h , (26) 4 and 'l/Ji = 0 in o[l x (0, T ) im ply that

'l/Ji = 0 in [l x (0, T) , i = 1, ... , N,

so th at (26h gives f == O.

(30)

(31)

4 On the existence and uniqueness of Nash equilibrium

4 .1 A criterion of existence and uniqueness

We consider th e funct ionals (14) . Let us define

11. ; = L 2(O; X (0,T)) , }

11. = TI~l li i ,

L;VJ; = fj;(T) (cf. (21)) , which defines L; E L(li ;;L2 ([l )).

(32)

24 J . 1. Dfaz and J . L. Lions

Since v is fixed , one can writ e

N

y(T;v, w) =L LiWi + zT, zT fixed .i = l

W ith these no ta t ions (14) can be rew ritten

J;(v;w) = ~ II w;II~ . + ~; p; (~ Lj Wj _"T) ,where 1JT = yT - zT . Then w E 1l is a N ash equilibrium iff

(33)

(34)

or

Wi + aiL; (prt Lj Wj) = aiL; (pt1JT) , i = 1, . . . , N

J = l

(where L; E £( L 2 (Q);1l i ) is the adjoint of Li ), or equivalent ly

Lw = given in 1l , }

L E £ (1l ;1l) ,

[Lw}, = W, + ai L; (pt2:f=l Lj Wj) .

T he n we have

Proposition 4 .1 Assume that

a , = a, for a ll i,

and that

(36)

(37)

(38)

a Ilpi - Pj t OO (D) IlpiIILOO (D) is small en ough, for any i, j = 1, . . . , N. (39)

Then L is invertible. In particular there is a unique Nash equilibrium of (14).

Remark 4.1 Of course, if N = 1 the situation is much simpler. In that case,

hence L is coercive and so the existence and uniqu en ess of a minim um W ofh (v;w) , when v is fixed , is a classical result .

Approximate controllability of Stackelberg-Nash strategies 25

Proof of Proposition 4.1 : In t he genera l case N> 1, one has

T hen one can wri te

Applying Young's inequ ali ty, it follows that, und er hypothesis (39) , L is coercive,i.e .

(Lw , w) 2: } Ilwll~ , for some } > O.

T he conclusion is now a consequence of the Lax-Milgram theorem .

(42)

Remark 4.2 The hypothesis (39) is certainly sa ti sfied if Pi = P for a ll i, inwh ich case th ere is only one fun ction J, = J1 for a ll i, and we are back toRemark 4.1 (with w = {WI,"" WN}).

4.2 Some non-existence and non-uniqueness results

We begin this subsect ion by some general considera t ions on th e existence , ornon-exist ence, of Nash equilibr ium solutions.

Let Hi , Kj be two families of N real Hilbert spaces (i , j = 1, . .. , N ), th escalar product (or norm) in a space H being denoted by ( , )1£ (or 11111£) '

We consid er linea r cont inuous ope rators ai,j

an d we ass ume tha t

ai,j is compact , Vi, j.

We define w = {WI , ... , WN} , w E H = IT~ 1 H i = IT~1 Ki'

2N

1 2 a i '"""'Ji(w) = "2llwill1£i dxdt + 2 L.J ai,jWj - TJij =1

where a i is a positive given constant, and where

N

11= {TJl ,· · · ,TJN} is given in ilKi.i =1

(43)

(44)

(45)

(46)

We are looking for the N ash equilibrium poin ts of the [un ctionals h , ... , I N.We are going to show that "in general" with respect to a = {ad E ~";: , there

26 J . I. Dfaz and J . L. Lions

exists a uniq ue N ash equilibrium fo r th e fun ct ionals Ji , Wh en (Y is "excep tional"in lR ~, th en "in gen eral" with respect to TJ = {ryd ETI~1 J(i, the re is no solu­ti on. Wh en (Y and TJ are "exceptional", th ere is a finit e dimen sional subspace ofsolutions in TI~1 c;

Of course, this "result" has to be mad e prec ise. An element w = {WI , ... , W N }

is a Nash equilibr ium iff

(Wi , Wih ii + (Yi (L aijWj - TJi' aiiWi) = 0, i = 1, ... , N, 'VWi E t:J x,

i.e.

N

* L 1 *.aii aijWj + -Wi = aii TJi, 1 = 1, . . . , N,(Y oj=l '

where aij E £(J(i' 1£j) denotes the adjoint of aij .Let us define

A E c (TI~1 1£i , TI~1 1£i) , }

Aw = {aii I:f=l aij Wj},

( ~) = diagonal operator {Wi} t---t {~i Wi } ,

( = aii TJi , (= {(d·

Then (47) is equivalent to

Aw+ (~) w = (, in 1£ =IT 1£i,,=1

(47)

(48)

(49)

(50)

(51)

where, by virtue of (44), A is com pact in £(1£,1£). Then th e "res ult" st a tedabove is a trivial consequence of th e classical Fredholm alternative. Ind eed , letus consi der th e a 's such that

1- = Ii >'" I i fixed,(Yi

(52)

all th ese numbers being posit ive. Then, according to the Fredho lm alte rnative,(51) and (52) admits a unique solution except for a counta ble set of >" ' s. Thismakes precise th e fact th at there is, "in general" with resp ect to a , a uniquesolut ion. If >.. belongs to th e spect ru m of A + ,>.., then there is a solu tion iff (is or thogonal to the null space of A" +" a conclusion which is "in general" notsa tisfied by ( , i.e. by TJ = {TJd . If it is satisfied, then there is a finit e dim ensionalspace of solu tions.

Approximate cont rollability of Stackelberg-Nash strat egies 27

Remark 4.3 Of course, the formula (51) does not use the hypothesis (44) .Th erefore, on e has tha t without th e hypotesis (44) th ere exists a unique Nashequilibrium if

IlaAII£ (1l ,1l) < 1 (53)

(where (aA)w = {O:iaii 2:j aijWj }).

All the a bove remarks apply to (32), (33) if we take

Qij = PiLj , 17i = Pi17T, K , = L 2 (Q), Vi (54)

(then (53) amounts to 0: Ilpi - pJIIL OO( J?) IlpiIILOO(J?) being small enough ) if one

verifies that Lj , as defin ed by

L iWi = Yi(T) , Yi solut ion of (20) (with Wi replaced by w;), (55)

is compact from L2(Oi x (0 , T)) = 'H, into L 2 (Q).If the co efficien ts of the opera tor A a re smooth enoug h , then the solution Yi

of (20) satisfies

2 2 1 {)Yi 2 ( 2 ( ))Yi E L (0, T: II (Q) n lIo(Q)) , at E L 0, T: L Q

(r ecall t ha t Yi(O) = 0) , so that L; E £(1ii; lI6(Q)) , hen ce L; is compact from 1iiinto L2(Q) (since the injection HJ(Q) '---t L 2(Q) is compact wh en Q is bounded).

References

1. Brezi s, H., 1973, Operat eur s maximaux monotones et sem igroupes de contrac tionsdan s les espaces de Hilbert , North-Holland, Amsterd am .

2. Diaz, J . I. and Lions, J. L., 1998, article in pr eparat ion.3. Gabay, D. and Lions, J. L., 1994, Decisions strategiques amoindres regrets, C. R.

A cad. Sc i. Par is, t, 319 , Serie I, 1049- 1056.4. Gilbarg , D. and Trudinger , N. S. , 1977, Elliptic Partial Differential Equati ons of

Second Order, Springer , Berlin.5. Mizohata, S., 1958, Unicite du prolongement des solut ions pour quelques operat eurs

differenti els paraboliques, Me m. Coli . Sc i. Univ . Kyoto, Ser. A31, 3 , 219-239.6. Lebeau , G. and Robbi ano , L., 1995, Cont role exact de l'equation de la chaleur,

Communications in PD E, 20 , 335-356.7. Lions , J . L., 1981, Some Methods in the Math ematical Analysis of S yste ms an d

Their Control, Science Press and Gordon and Breach.8. Lions, J . L., 1994, Some Remarks on Stackelberg 's Optimization , Mathematical

Models and Methods in Applied Sciences, 4 , no. 4, 477-487.9. Lions, J . L., 1986, Controle de Pareto de syst ernes distribues: Le cas d 'evolution,

C. R . A cad. Sci. Paris, t , 302 , Serie I, 413-417.10. Saut , J . C. and Scheurer , B. , 1987, Unique Continuation for Some Evolu tion Equa­

tions, J. Differential Equati ons, 66 , 118-139.11. Simon , J. , 1987, Compact Set s in th e Space LP(O,T ; B) , Annali di Matematica

Pura ed Applicata (IV) , CXLVI, 65- 96.12. Stackelberg , H. von, 1934, Marktfo rm und Gleichgewicht , Springer, Berlin .

3D Simulation in the lower troposphere: windfield adjustment to observational data and

dispersion of air pollutants from combustion ofsulfur-containing fuel

G. Winter , J . Bet ancor , and G. Montero

Escuela Tecni ca Superior de Ingenieros Industriales, Universidad de Las Palmas ,Edificio de Ingenierias , Campus Uni versitario de Tafir a Baja ,

35017 Las Palmas de Gran Can ari a , Spain

1 Introduction

Combinations of mathematical models with data at particular points from ob­servat ional networks are required in ord er to generate physically consistent windfields and at mospheric pollutant distributions . We describ e a methodology usedto evaluate the modifications to wind flow and pollutant dispersion, mainlycaused by the interaction of the air flow with the terrain.

The troposhere extends from the ground until an average alt it ude of 11 Km.We focus our attention on the part of the troposphere that is directly influ­enced by the presence of the eart h's sur face, a region where sur face-at mosphereturbulent exchange processes t ake place, the so-called planetary boundary layer(PBL) . Often this layer corresponds to t he lowest 500-1500 m of the at mosphere,which is the most important region from many viewpoints, as pollutant emis­sion , frictional drag or t errain induced flow modifi cation. Indirectly, the wholetroposphere can change in response to sur face char acteristic s, but this responseis relatively slow outside of the boundary layer . The t hickness of this region isquite variable in space and t ime due to the thermal st ability condit ions, rang­ing between 100 m at night time with light wind and turbulence to 1-2 Km onsunny days with surface heating. It is usually assumed that t he boundary layerincludes a state ment about one-hour or less timescales. On the ot her hand , if theground sur face is not spat ially homogeneous (as it is t he case in general), thisinhomogeneity is reflected in t he P BL. For this reason, the context of numeric alsimulation in 3D is of interest . Mean wind is responsible for very rapid horizon­t al t ransport or advect ion. Wind velocity increases from 0 to about 70% of it smaximum PBL valu e, whil e the wind direction is nearly constant with height.

W ithin the PBL, in particular near the ground sur face , typi cally up to 10-100m, we have the sur face layer (SF). It is usua lly assumed that the SL cover s thebottom 10% of the PBL. In this layer, where the wind is influenced by the pre­vailing high-level flows and the effect of the surface is well felt , the wind is mainlydetermined by the nat ure of the sur face and t he vertic al temperature gradient.Effect s of dens ity st ratification are small and the wind sp eed follows a nearl y

J. I. Díaz (ed.), Ocean Circulation and Pollution Control - A Mathematical and Numerical Investigation© Springer-Verlag Berlin Heidelberg 2004

30 G. Winter et al.

logarithmic vert ica l profile, even under stable and unstable condition forms ofthe velocity profiles. These profiles are very useful because of th e stratified­boundary-layer conservation equations, which are difficult to solve due to theclosur e problem in the turbulence models.

Within the SF layer , in the vicinity of the ground surface, turbulence isstrongly affected by roughness a nd viscous effects may become significative. Im­mediately adjacent to the surface, a laminar sublayer (also call ed interfaciallayer, or microlayer) is identified, in which strong mol ecular viscosities becomeimportant. However , the thickness of this layer is typically less than a centimet erand for all purposes it can be ignored. Above the SL, the role of the Coriolis forcebecomes relevant with respect to th e friction forces. The wind velocity changesslowly while the wind direction veers describing the so-call ed Ekman spiral. Thispart of the PBL is ca lled Ekman layer. In these layers, some important simpli­fications in the equa t ions of continuity, motion and energy can be made, as thecontinuity equation for an incompressible fluid.

At th e top of the PBL, the flow is nearly independent of the nature of thesurface, and above of this top the region is call ed free atmosphere. This layer isusually called the geostrophic layer, where motion of ai r approximates that ofan inviscid fluid in laminar flow and the direction of winds is mainly determinedby hori zontal pressure gradients and Coriolis forces.

An accurate est imat ion of wind and atmospheric pollutant distributions re­quires to use mathem atical models linked with met eorological observations. Wetreat a model of wind field modelisation on complex terrain, e.g. adjusting me­teorology and topography data with sm all computational effort in 3D. This ad­justment model is a so-call ed mass-consistent model (MMC) , which satisfies thefollowing: the vorticity of the observed wind field is conserved by the adjustme ntrotational; the flow velocit y field is nondivergent (under incompressible condi­tions corresponds to th e continuity equa t ion); and an impermeability condit ionholds at the ground. This velocity field adjustment model is characterized bya mi xed vari ational formulation as a result of the corresponding op timi zationproblem, defined and solved by looking for a saddle point of the associa ted La­grangian function. This model fits the available experimental measu rements, andtheir mixed variational formulation is very suitable, since the numerical solutionexactly sa t isfies divergence-free condit ions pointwise.

The purpose is to provide a realistic methodology for simulation of winds,where the boundary conditions and initial velocity field a re constru cted in a con­sistent way from experimental data with the use of different sources, some groundst ations, geostrophic wind, atmospheric st ability class, roughness parameter andth e dependence of wind speed on height given by appropriate logarithmic pro­files for each of the above-mentioned atmospheric layers. Numerical results areshowed with the MMC model in a region of Can ary Islands with real data.

The 3D Navier-Stokes (NS) equation needs boundary conditions, and aninitial velocity field has to be specified . We highlight that a possible alternativemethodology to provide the initial velocity field and boundary conditions to th e

(1)

3D Simulat ion in th e lower troposphere 31

3D NS formulation can be established from resul ts obtained with th e wind fieldadj ustment model.

In what follows we consider some asp ects on two numerical me thods to solvecoupled convection-difussion equat ions for modelling of air pollutants: one ofthem based on cha racte rist ic lines and another one on a Taylor-Galerkin pro­cedure. Some results of numeric al stability and consistence are compared . Weespecially focus on modelling oxid ation and hydrolysis of sulfur and nitrogen ox­ides released to the su rface layer , which, once oxid ated, are major contributorsto acid ra in in geograph ical regions , and producing ae rosols with proved climaticimplications . Nit rogen oxid es play an important role in th e atmospheric photo­chem ist ry of other greenhouse gases. The dry deposition process is representedby t he so-call ed deposition velocity, which is proportional to the degree of ab­sorptivity of the surface, and it is assumed to be a proportional constant betweenvertical flow and concentration , and thus it is trea ted as a boundary condition .The wet deposition is considered as a source term in the convection-diffusionequation using the washou t coefficient . A numerical applica t ion considering thesame topogra phy and wind field and relative to calculate the distribution ofconcentrations of sulfur e oxide and sulfate is presented.

2 Wind field adjustment model

For a given bounded op en three-d imensional domain with boundary r = r1urZ,we look for a field u t ha t adjusts, in a least square sense, to a velocity fieldUo, obtained from the interpolation of experime ntal meas ure me nts and verticalextrapolation by suitable profiles for each atmospheric layer within th e PBL,and verifying

V·u = 0 on f?u· n = 0 at r 1 ,

where r 1 = r t u r u , with r u the upper altitude of the PBL, and r t th e surfaceof terrain. A zero flux boundary condition is used for th e remaining boundariesr z. T he least square functional to be minimized is:

J(u)=.!. {(u-uo)t.p.(u-uo)dD+ fi {n .(u-uo)Zdr, (2)2 In 2 Jr,

where P denotes a diagonal matrix. Different values of thei r entr ies (the Gausspr ecision moduli) allow ponderation between horizontal and vertical velocitycomponents (usu all y less value with relation to the vertical component). Then ,the wind field will be a solution of th e followin g problem:

"Fi nd u E K that ver ifies

J (u) = min J (v)vEK

with K = {v; V· v = 0, v . nlr 1 = O} ."

32 G. Winter et al.

The problem can be formulated as a saddle point problem for the Lagrangian:

L(v , q) = J(v) +Ls :V . v dQ.

More precisely, if L 2 (Q) is t he space of square integrable functions and H1(Q)the subspace of L 2 (Q) with square integrable first der ivatives, we denote:

H6,r2(Q)

= { <p E H1(Q); <P lr2

= O} ,

H(V , Q) = {v E (L 2(Q)) d; V· v E L2(Q)} ,

and, by introd ucing the space of vector functions such that v . n = 0 on r1 in ameaningful way, say,

HO,r, (V, Q) = {v E H(V ,v) ; t <pv . n sr = 0 V<p E H6,r2(Q)}

, (3)

we search for the couple (u ,>.) E Ho,r,(V,Q) x L 2(Q) such that

L(u , q) :S L(u, >.) :S L(v , >.)

for all q E L 2(Q) and all v E HO,r, (V, Q) , which is characterized by

8L(u, >.) 8L(v , >.)8v = 0 an d 8q = 0 for a ll v E Ho,r, (V, Q) , (4)

Lq. V· u «a= 0 for all q E L2 (Q), (5)

ob taining

Lvt .p.(u-uo) «a: L>'V 'VdQ+ f3t2v 'n(n'(u-uo)) dr=O. (6)

The vari ational formulation given by (5) and (6) can be solved with mixedfinite elements (see [6] for more details) , with great advantages. However , onemo re classical formulation, known as matrix mass-consistent model (MMC), isused instead . It can be derived from (6) with the assumption that the Lagrangemultiplier be sufficiently regular (see [2]), and then we obtain

Now, the problem to solve is

u = Uo + P-1V>.. (7)

(8)

(9)on Tz.

on 1\

3D Simulat ion in the lower troposphere 33

subject to the following boundary conditions:

- 1 0>"-p . - = n 'uoon_p- 1 . 0>" = ~

on (3

The standard finite eleme nt method can be used to obtain the Lagrangemultiplier from (8) subject to the boundary conditions (9), and then the veloci tyfield u from (7). This traditional procedure is usually considered, bu t the fieldthus obtained is discontinuous through the faces of the finite elements and doesnot satisfy th e incompressibility condition pointwise.

2 .1 Initial wind field

The construction of the initial wind field is the most important and criticalstep in mass-consistent models, since it introduces the experimental data intothe model at each node of the computational mesh . Usu ally, observational dataare available at 10 m above th e terrain at different locations, from sensors. Wepropose a two-step procedure with th e interpolation of a velocity field a t x 3 e

over terrain as a first step using the following exp ression (see [5] and [8]):

(10)

n n

L~ L~i= l di i= l I~hi I

UO( X 3 J = E n + (1 - E) n ,1 1

~ di ~I~hi lwhere di represents the horizontal distance from the ith station to th e pointconsidered , I~h i I corresponds to the height differences between them , and n isthe number of observation st ations. The use of th e parameter E, such that 0 ~E ~ 1, will allow us to balan ce the contribution of both weights of interpolation:horizontal distance and height differences. In practical applications, good resultswere obtained with m = 2.

Once th e velocity field has been interpolated a t X 3 e over ter rain , it is verticallyextrapolated using different profi les in every layer where PBL is considered tobe subdivided by similari ty theory, taking into account st ability data obtainedfrom vertical soundings. T hus, within th e surface layer the velocity is computeda t different heights by (see [14]):

X 3 4.7In - + - ( X 3 - X 3o)

X 3 0 LIn _X_3

X3 0

In _X_3 + 2 (_1 1_)+X 30 tan "1r tan "10

I("15 + 1)("10 + 1)2+ n 7'-' ':----:--:-:---7"''"("1; + l)("1r + 1)2

( > 0 stable

( = 0 neutral

( < 0 unstable,

(11)

34 G. Winter et al.

with

(X 3 ) 1/4 ( X3 0 ) 1/4

77r = 1 - 15r: 770 = 1 - 15£ (12)

where K, is the Von Karman parameter (usually equal to 0.4), L the Monin­Obukhov length, and X 3 0 the characteristic length of rugosity, which representsthe depth of the laminar sublayer adjacent to the ear th's surface. Its value willdepend on the surface itself.

In the Ekman layer , the velocity vector is computed using a linear interpola­tion formula between the wind field at the top of the Surface Layer , com putedby (11) , and the geostrophic wind, as suggested by Troen (see [5]) . A third-orderweight function is used :

Over the PBL, in the geostrophic ar ea , th e velocity vector is consider to beconstant with height u g , computed from soundings data.

The atmospheric stability will be characterized by the Monin-Obukhov lengthL , which is rela ted to the Richardson flux dimensionless number Rf. Thus, pos­itive values for L denote a stable atmosphere while negative values representan unstable atmosphere. For L -+ 00 , th e atmosphere is said to be in neutralcondition. Several methods have been developed to compute L from directlyobservable data. We use the Golder method (see [3]) to compute L from

1L = a + blog x3 0 ' (13)

where a and b a re two coefficient s whos e values are indicated in Table 1 asa function of the corresponding Pasquill stability classes (see [4]) , which canbe determined from observational data of incoming solar radiation, mean windsp eed , and night-time cloud-cover fraction.

Table 1. Coefficients for (13) as a function of Pasquill stabi lity classes (see [10]).

Atmospheric Pasquill Coefficientscondition stability class a b

Extremely unstable A -0.096 0.029Moderately unstabl e B -0.037 0.029Slightly unstable C - 0.002 0.018Neutral D 0 0Slightly stab le E 0.004 -0.018Moderate ly stable F 0.035

3D Simulation in the lower troposphere 35

As scale parameter, the modulus of the friction veloci ty vector lu*I is used.Thus th e height of th e PBL can be evaluated by the expression

where f is the Coriolis par ameter, , is a constant and its value depends onatmospheric stability, usu all y between 0.15 and 0.35.

The maximum height of the mi xed layer can be evalua ted by

{ ,~h = 'V f L stable

X3PB L non stable,

where " is a parameter whose value is 0.4 . The height of the surface laye r ishere assumed to be

2.2 Parameters of ponderation

Atmospheric stability is also considered in order to evaluate the Gauss pr ecisionmoduli ai, i = 1,2, 3. Identical Gauss precision moduli are generally consideredfor the horizontal directions and a unique parameter is then considered:

2 a~ Tva = 2 = - =T,

a h Th

which is call ed transmissivity coefficient . Man y authors have proposed differentmethods for th e calculation of the optimal value of this parameter , which is, infact, very case-dependent. For our model, we have choo sen the method proposedby Moussiopoulos [13], which cons iders the Strouhal dimensionless number Str,as the most representa tiv e par amet er which takes into account the effects ofthe atmospheric st ability through the buoyancy frequency N, and the orogra­phy th rough the characteristic height difference 1£ . The Strouhal dimensionlessnumber is defined by

where

stable and neutral

unstabl e,

and ll i=

36 G. Winte r et al.

Thus, a must be computed in stable and neutral conditions by

Str" ( )a Z = 1 - - 2- VI+ 4 Str- 4 - 1 . (14)

For unstable conditions, th e following rela tionship must be satisfied. How­ever, it is observed that the parametrization for unstable stratification has nosign ificant influence on the resulting wind field .

(15)

3 Pollution transport model

We cons ider th e at mospheric convect ion-diffusion equation, which provides amore appropriated model than the Gaussian models, because of the ability to in­clude changes in wind speed and variable eddy diffusivities. Dealing wit h gaseouspollutants, it is assumed that th eir concentration does not affect the meteorologya t some extent , and the equation of continuity can be solved ind ep end ently of thecoupled momentum and energy equat ions. Under appropriate restrictions, theflux of pollutants is proportional to the gradient of its mean concent ra t ion . Thus,in a three-dimensional, inhomogeneous environment, we will have th e simplifiedequation:

BCifit + V . (u . Ci) - V . (K i . V . Ci) = Ii (16)

(17)

where c; = Ci (;v , t ) is the mean concentration of the ith atmospheric pollutantspecies and K i is a diagonal matrix with difussion coefficients. The set of equa­tions (16) (one for each considered pollutant) are to be solved subject to thefollowing initial and boundary conditions:

Ci=Cio(;V ,O) att =O

Ci = Ci, (;V ) onrz

- Ki(;V), Vci· n =vdi(;v)·Ci onrt

V Ci ·n= O

where Cio (;v, 0) is the initial concentration field of th e ith pollutant, Ci,(;V) itsconcentration a long the frontier r z, and V d i the deposition velocity parameter,which models the vertica l flux downward of the ith species above the surface,known as dry deposition.

3.1 D ry deposition

It refers to the vertical flux downward the ea rth 's surface of both gaseous andparticulate pollutants, which are absorbed by soil , water or veget ation. It s com­plex m echanism can be viewed as consist ing of three consecuti ve steps, each withits own resistance to the flow: Transport from the surface layer to the vicinity of

3D Simulation in the lower troposphere 37

the earth's surface, diffusion through laminar sublayer, and absorption or trans­fer to surface recipients. This flux can be modelled by the concentration of thepollutant at some height just above the surface multiplied by the depositionvelocity Vd i' which is a function of the species to be removed, meteorologicalproperties on the surface layer, and the degree of absorptivity of the surfaceitself. Functionally, this flux can be expressed as

and, thus, will be included as a boundary condition at the earth's surface.

3.2 Source term

We assume that it is composed of three terms for every species i:

Ii = E, + Hi + Pi,

where E; (;r, t) includes continuous and instantaneous emission of the ith speciesto the atmosphere, Hi (CI' . . . , Cn , t) represents the rate of generation or elimina­tion of the ith species through chemical reactions, and Pi (;r, t) accounts for therate of elimination of the ith species through wet deposition because of water 'absorption of pollutants during rain or other cloud processes.

To model the emission rate E, of the ith pollutant to the atmosphere froma power plant , a point source function is assumed:

(18)

where (Xl, X2, X 3)O are the location coordinates of the power plant. This expres­sion leads us to consider an instantaneous emission at i = to with

and, thus, by the application of the superposition principle, any combination ofcontinous arid instantaneous sources. The adopted models for the wet depositionand the chemical reactions are now discussed in detail.

3.3 Wet removal processes

Wet deposition accounts for the removal processes of pollutants that take placein the atmosphere because of their absorption by clouds, rainfall water, etc.

We assume wet deposition as a first-order process. It is assumed that the rateof removal depends linearly on the airborne concentration of the material andis independent of the quantity of material previously scavenged. Thus, the localrate of removal of the ith gas is given by

Pi = -Ai (X3' t) c, (Xl, X2, X 3, t)

where Ai (X3 ' t) is the washout coefficient of the ith gas, which, in general, willdepend on the height over the earth's surface. With the assumption that the

38 G. Winter et al.

material being scavenged is uniformly distributed vertically in the layer of depthh or the mixed layer height, a wet deposition velocity V w is defined by analogywith dry deposition (see [11]) as '

V W i = Ah,

which can be expressed in terms of more usual parameters such as the washoutratio W r i , and the precipitation intensity Po, whose value is affected by the dom­inant atmospheric condition ranging from 0.5 to 25 mm· h- 1 (see [10]).

3.4 Chemical transformations

We focus our attention on the oxides of sulfur and nitrogen which are typicallyreleased to the atmosphere in the combustion process of any industrial fuel.These species, once released, are oxidized to their corresponding acids, whichare largely responsible for the acid rain. The set of reactions that involve nitro­gen and sulfur oxides are summarized in Fig. 1. It shows the coupling betweenmechanisms of reaction that involve families of both sulfur and nitrogen corn­pounds, through shared radicals OH· and H02·. Most of these reactions arephotochemically excited and involve a great number of intermediate species ofshort lifetime, such as S02' or HOS02·.

J ••••••••••••••••••••••••••••••••••••••••••••••.••••• • • • • • • • • • • • •

NO, + HO· --7 HN03:

h·v

NO+lIO,'---7 NO, «on

Regeneraci6n de OH·

Fig. 1. Set of chemical reactions involving OH· and HO z"

Two relevant mechanisms are remarked in the dotted boxes of Fig. 1: Oneof them represents the chemical equilibrium of the oxides of nitrogen, and theother one leads to the regeneration of radicals OH· and H02·, which will allow usto uncouple the mechanisms of reaction of both families of pollutants. Certainly,the oxidation of S02 by H02· is so slow when compared with the reactions thatinvolve the chemical equilibrium of nitrogen oxides that it is assumed as first

3D Simulation in the lower troposphere 39

approximation that its concentra tion remains nearly constant because of thesereactions . With these assumptions, t he rates of reaction of the fina l species are:

d[NO x ] = -2k k [NO]dt 1 S x

d[HN03 ] = 2k k [NO ]dt 1 S x

d[HzS04 ] = 2 k1kg [SO]dt ks Z

(19)

(20)

(21)

(22)

which are linear respect to the concentration of the pollutants from its corre­sponding family. In a more general representation,

N

n; = L <Xi j Cj

j =l

with i, j = 1,2, . . . , N, where <Xij are the specific rates of gene ration or elimina­tion of the ith speci es in the different reaction mechanisms.

In the following, we will collect the te rms related to the chem ical t ransfor­ma tions and wet deposition because of their functional similarity in a term

N

R; = n; + Pi = L <X; j Cj ,

j =l

so the sour ce term will be referr ed as Ii = E, + R;.

3.5 Numerical solu t ion method

About the numerical met hod to solve th e problem (16), we describe two proce­dures: One related with the characteristic lines of flow, and the another one isa Taylor-Galerkin scheme. We summarize the establishment of both numericalschemes particular ized to our a ir pollution problem. We have implem ented bothschemes with a finite element method in 3D to carry out the applica t ions.

Characteristic line s sch em e: From the transient equation

de,dt - V . (K i . V . cd = Ii (23)

we can establish a discrete treatment of the material derivative in the implicitscheme

40 G. Winter et al.

where a: is the position of a fluid particle on the characteristic line at a timetn' Now we can expand in a Taylor series the concentration ci (a:) at a locationa: from th e location x with concentration ci (x) through a characteristic line offlow, which results in

(24)

Expanding the position vector at a: so that every coordinate p of the positionvector is

(25)

and substituting (25) in (24), we can evaluate all its terms a t a location x atdifferent times by

cn+! - 6.tV . (tc, . V . c,:, +l ) - 6.1" cl ·c~+l = 6.tE:>+l + c':' -, , ~ ' J J , ,

j=l

6.t 2 3 acr:' 6.t2 3 3 a2cr:' (x)-6.tu· Vc? + -2- L u· VU pax' + -2- L L UpUq ax 'ax + 0(t

3),

p=l P p=lq=l P q

whic h can be treated numerically by standar d finite elem ent techniques. A similartreatment can be given to (27) in an explicit or even semiimplicit sch eme.

Taylor-Galerkin scheme: The Taylor-Galerkin scheme proposed here is avariant that introduces an additional term respect to other classical sch emes ofth is kind. We consider the following Taylor a pproxim at ion :

(26)

where e is a number such that 0 ::; e ::; 1. The first derivative of the pollutantconcentration ca n be obtained from (16), whil e for t he second derivative weevaluate

a2Cii a [aci] aal2 = at at = - at [V · (u · Ci) - V· (Ki . V· Ci) - fdn+en+e n+e

(27)whose terms can be evaluated as follows:

aat [V . (u . Ci )]= - V . (u . (V . (u . Ci) - V . (K i . V . c;) - f;))

3D Simulation in the lower troposphere

Nafi aEi '" I7ft = 7ft - LJ aij (V . (u · Ci) - V· (Ki . V · Ci) - f; )

j = l

a [ aCi ] [ at. ]- . [V ·(K · ·V · c·)]=v, K · ·V·- =v · K · ·V·-at " 'at ' at '

41

(28)

where third or higher degree deriva t ives have been ignored.By substitution of (28) in (27) , a final expression for the numerical scheme

is obtained:

c7 +l = c7 - ~t (V · (u · Ci) - fi)n - ~t (v. (Ki . v ·Ci))n +<9~t2

+-2- (1 - 28 ) {V· iK, . V · f;)}n+<9

~t2+-2- {V . (u . (V . (u· Ci) - f;))

N

- '" a~ · (V . (u . c ·) - V . (K · . V . c·) - f ·)}LJ ' J ' '" n+<9j = l

whose te rms evaluated at time n + 8 will be approximated by weighted interpo­lat ion between t ime n and t ime n + 1. The expression obtained can be treateddirectly wit h standard finite eleme nt techniques .

W hen considering explicit schemes, 8 = 0, identical results will be ob tainedwith both charact erist ic lines an d the Tay lor-Galerkin method, which means thatthe Taylor-Galerkin method allows us to establish a scheme to solve numericall ythe convection-diffusion problem without considering the nature of the fluid ormedia in which the process takes place, extendi ng its validity to pr oblems inwhi ch the concept of fluid particle is meaningless.

3 .6 Stability and accuracy

In this section, we analyse the stability and accuracy of the adopted scheme forsolving convection-diffusion equation. The analyt ical solution of the pr oblem

aC- + V . (u . c) - V . (K . V . c) = acat

(29)

subject to initial and bou ndary conditions (17) may be expanded m Fourierser ies, each mode m hav ing the expression

where Fm is the amplitude of t he mode, em t he wave vector , Qm = v . em t hefrequency of the mode m, and 15m= J( lem12 - a is the damping. We will use alinear finite element approximation in the regula r mesh represented in Fig. 2.

42 G. Winter et al.

2

4 o

6

Fig. 2. Hegular 2D mesh used for the study of accuracy and stability.

Characteristic lines scheme: Considering an implicit formulation , t he nu­merical schem e through characteristic lines for (29) may be expressed by

2 2 2 2 n

(1 + Cl' At)en +1 -AtV · (K . V · en +1) = en -AtV · (u . c

n)+ A; L L U iUj 8~i~Xi = l j=l J

(30)

Applying th e finite element me thod to (30) with consistent integration in a ll oftheir terms, we obtain th e following equivalent finite difference scheme

h2

(1 + At) (6n +l +~ n + l ) + AtK (4 n +l n + l n + l n + l n + l ) _12 CI' Co L...J c , Co - C1 - Ca - C4 - C6 -

i=l

= ~~ (6C~ +t C;) - At~ [U l (2 (cr' - c~ ) + c~ - c; + c~ - c~ ) ++U2 (2 (c; - c~ ) + c~ - c~ + c~ - c~)] -

At2

[ 2 (2 n n n) 2 (2 n n n)- - 2- Ul Co - Cl - C4 + U 2 Co - Ca - C6 +-l-ui U2 ( c~ - c~ + c; + e~ - c~ + c~ - 2c~)],

3D Simulation in the lower troposphere 43

where c; is the numerical solut ion at the ith node and h t he parameter of thediscretization . Let II be the angle between the velocity vector and the x-axis and

(}1 = cos II

(i = cos (~im h)

Xi = sin (~imh)

(}2 = sin II

(12 = cos (6m + 6 m) hXi = sin (6rn + 6m) h.

Then the am plification factor for the numerica l solut ion becom es

1 2C(; (1 - a .6.t) (3 + ( I + (2 + (12) + Pe (2 - (I - (2)

. C (h (2Xl + Xl2 - X2 ) + B2 (2X2 + Xl2 - xI)- z · "3 1 2C

(; (1 - a .6.t) (3 + (I + (2 + (12) + Pe (2 - (I - (2)

where C and Pe are the local Courant and Peclet dimensionless numbers, ex­pressed by

c _ lul.0.t P _ lui h- h e - J( ,

where h is t he characteristic di mension of the element in the flow directi on wit hvelocity u .

We use the Von Neumann stability criter ion to establish the stability limit onthe Courant number. T he most restrictive situation corresponds to the particularcase in which II = - rr/ 4, as shown in Table 2, where stability limits for twoimpor tant values of II are compared.

Table 2 . Stability limit on Courant number for various wave vector directions aridconsistent integration in all terms.

Wave vector Stability limit on Courant number

1 [ 1 1 cu",2 V ]~ = Ci -- -+ - 2 +--(2 + a .6.t)cos?v Pe Pe 6

~= C i- ~ ·j1 [~ J~ 1 - sin 2v (2 .6. )]

1 - sin 2v Pe + Pe2 + 6 + a t

For 0: > 0, there is an ad ditional condition for stability: .0.[ :::; 2/0:. Theseresults agree with the conclusions of Pera ire, Zienk iewicz an d Morgan [9], in the1D problem and also wit h the result obtained in 2D by Montenegro et al. [15].

For im plicit formulation with red uced integration on the cn +1 and c" terms,the application of the Von Neumann criterion leads us to Table 3.

44 G. Winter et al.

Table 3. Stability limit on Courant num ber for various wave vector directions an dreduced integration in all terms.

Wave vector Stability limit on Courant nu mber

1 [1-- -+cos- v P e

1 cos- V ]- 2 + --(2+ a D-t)Pe 2

~= ~ ·i- ~·j ~1 - sm 2v [

2- J~ 1 - sin 2v (2 A ) ]

P + 2 + 2 + a u te P e

(31)

From the poi nt of view of acc uracy, this scheme is first order of numericalconsistency and second order of consistency for the pure convection problemwit h J( = O.

Taylor-Galerkin scheme: The stability analysis of the Taylor-Galerkin schemeis made for the uni di mensional problem, which is formula ted by the implicit ex­pression

(1 _ a 2 flt

2e) en+1 _ (flt J( + flt

2u2 _ a flt 2J( (1 _ e) ) e {)Z e

n+! _

2 2 ax 2

aen+! ( flt

2)- aflt 2ue-- = 1 - a flt + a 2_ (1 - e) en +

ax 2

(fl t 2 ) a2en

+ fl tJ( + -2-U2 - aflt2J( (1 - e) (1 - 0) ax 2 -

aen

- (u flt - au fl t 2 (1 - e)) ax .

In the case of considering reduced integration on the en+1 and en te rms, thenumerical amplification factor is

1 - A(1 - e) sin 2 ¥ -i· B sin emh

1 + Ae sin 2 e~h - i· C sin emh

where

4 (flt2

) fl t2

A = h2 J( flt + - 2-v2 +««fl t 2 + a 2-2-

1B = h (v fl t + va fl t2(1 - e))

1C = hav flt2e

(32)

(33)

(34)

3D Simulation in the lower troposphere 45

and the most restrictive case corres ponds to emh = 11" , and then the aboveexpression yields

G(e ) = 1 - A(1 - B) .m 1+AB

(35)

When the Von Neumann criterion is applied to (35), t he above schem e isun conditionall y stable (and hence convergent ) for 0 2: 1/2.

T he same condition is ob tained when consistent integration is considered ,which corres ponds to a numerical amplification factor given by

(AI( ll) 1). 2emh . . h1 - 1 - u + 6 sm -2- - z . B sm em

G(em) = ( 1) e h '1 + A'B - 6 sin2

~ - i · C sin emh(36)

where A' = A + /20:2~t2, with A defined by (32). The most restrict ive condi­tion occurs when emh = 11" . In thi s sit uation, th e scheme becomes stable if th efollowing cond itions (37) and (38) are satisfied.

A' > 0

A'(1 - 20) :::; 5/ 3.

(37)

(38)

For values 0 2: 1/2, the pr evious cond itions a re always satisfied, so we canconclude th at thi s schem e is unconditionally stable for B 2: 1/ 2. From th e pointof view of the accuracy, t his scheme is second order of numerical consistency andthird order of consistency for the semiimplicit formulation with 0 = 1/3.

4 Numerical application

A numerical application of our model in a region of Lanzarote (Canary Islands)is presented. The computational domain has 12 Km from west to east, 17 Kmfrom south to north and 2500m over th e maximum alti t ude on te rrain, as shownin Fig. 3, and it has been discretized using tetrahedral eleme nts with higher levelsof discretization near the ground surface and around th e emission source. TheDelaunay t riangulat ion algorit hm (see [12]) has been used for generating th ecom putat ional mesh, shown in Fig. 4.

4.1 Wind field adjustment

Five stations of measurement of the hor izontal velocities (see Table 4) were avai l­able for this application together with sounding data , which led us to consid era slightly un st ability atmospheric condi ti on according to Pasq uill st ability classth eor y. In th e upp er atmosphere, t he dominant direction of the wind is observedfrom soundings, together with th e corresponding geost rophic wind speed . Dataavai lable for th e sim ulation include:

46 G. W inter et a l.

LANZAROTE

Fig. 3. Location of the computational domain in Lanzarot e island.

Fig.4. Computational mesh for numerical application .

3D Simulation in the lower troposphere 47

Geostrophic wind: u g = (-28.0 , 28.0 ) m - S-l .

Parameters of ponderation: 7h = 1; Tv =0.17.Other data: e = 0.75 ; X 3 0 = 0.1 m .

Table 4. Data from measurement stations.

Station X(Km) Y(Km) Vx(m ,s") Vy(m . s )Famara I 7,674 13,245 - 3.33 -8.05Famara II 6,956 12,053 - 6.48 - 6.48Famara III 5,973 12,684 - 3.30 - 9.23Famara IV 7,583 10,813 - 0.84 -6.85Tao 3,057 4,097 - 0.85 - 9.76

Fig. 5 shows the wind field obtained with a n MMC m odel at different heights:200, 270, 350, and 800 m over the sea level. From its observation, we can see :

Dominant effect of the orography on the wind field near the ground .Trend of the velocity vector to raise up the orography obstacles, instead ofrounding t hem up horizontally, because of the slight unstability consideredin the computation .

4.2 Dispersion of atmospheric pollutants

We consider an emission source corresponding to a power plant , loca ted at8.0 Km from east side and 10.5 Km from the south limit, with a height of emissionof 70 m over the te rrain surface, which emit s S02 at a uniform rate of 2.0 kg-s" ".Table 5 includes the remaining data for this applica t ion.

Table 5. Data used for the application of dispersion of pollutants.

Parameter S02 H2SO4

co(g . m -3) 0.0 0.0C2 (g . m -3) 0.0 0.0Vd rn v s 1) 0.0044 0.0026V w m . s 1) 0.28 0.14

J(h m2 . s 1) 25 25Ie m2 . s 1) 50 50

o (s 1) -0.0012 0.0012

In Figs . 6, 7, and 8, isolines corresponding to dis t ribution of concentrationlevels of S02 and H2S04 at steady state are shown at 200, 270, and 350 mrespectively over the sea leve l.

48 G. Winter et a l.

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t t t tIII

(a) Velocity vect or a t 200 III overthe sea level

(b) Velocity vector a t 270 III overthe sea level

//////////////// / / / / / // / / / / / / / / / / / / / / / / / / / III/III///////// ///////////// ///// /////////////////////1////////////// /////// // //// ///////////// ////////1/ 1/1/ / /1/ 1/1/1/ / /1/ / 1/ /11/1/1/ / 1//1/1/ / 1///1//1/ / / / II/II / ///////////// /////////////////////////1/1/1/ 1/1/1/1/ / 1/ 1/ / I/I/I/I/I! 1///1/1/1/1/1///1/1/ 1/ / 1/ 1// 1/ 1/ / 1/ 1/ II 1/1//1/ 1/ 1/1/1/1/ / 1////// I! / 1/ 1/ II / / 1/ 1/ / 1/ / II 1/ / 1/1/111//1//11/////11'I/I! I! / II / / / / 1/ 1/ / / II / 1/ / II 1/1/11111/11/11///111/ 1/ / II 1/ / I! / II / / / 1/ 1/ 1/1/1//////1/1/11/////11/ I! 1// / 1/ /I /I / / / / / /I / /1//1/1///1/1/1//1//// IJ Ill / / / / / / // / / / / / / / / / / / / / / / / III/III / //////11/ / 1// / 1/ /1/ 1/ / / / / /I /I / / / JIll / / / / /////////'1/ /I /I / /1/1// / / / / / / / / / / / III / ////////1////11///// / /I / 1/ / / 1/ / / // / / Ji ll / / / / II ///1//// J/ / / III / ///1///////////// / / / / II / /1///1// J1111//1/1/11//// / / /////////1/ /11111/// I I1/1/1//////1//////1//1/1///1/ I / I // I I fill/////// /// //////// ////////1//111/1// / 11/ /1 1

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:" /iiiIHIH!HII iiHHHHiiiHiiiiiiiiiii;i;i/ //11 I I I I //11 I I I I I I I I /////////////////1////1/// /11 I I II 1////1 I 1I I I I 11111/////// //////1//1/1//I //1 I I 1111//1 I I I I I I I 11/////////////////1///1//I /1// I I I I 1III // I I I 111///1///1///////////111///I //1111 I I I // 11/1111 I /111/1//11// I ///// /I /I /1//I 1/11/1 I I I 1/11 I II II/ I 11/////////////1 ///11/1//I 1//1/11 1I 1/ I I II I I I I I 1/1/ /////////////////////I //11/1111///11/1 111//////1//11////1/1////1/1I 1111//11 I I 1///// 1 // 1/ 1/ / / / / / 1/////1//11//1///I /11 11// I /1////11//1 /////// I ///11/11//1/111111I 1/11/111 I III/I II I 111// I /1 I I 1///111' 1///1/ I 1//t 1IftIIIIJI/II IIIJJJJlllll!l!!I I/'I III I I III I/!

(c) Velocity vector at 350 III overthe sea level

(d) Velocity vector a t 800 III overt he sea level

Fig. 5. Velocity field at different heights over the sea level.

(a) Isolines of concentra­tion of S02

3D Simulation in t he lower troposphere 49

(b) Isolin es of conce nt ra­tion of H2S04

Fig.6. Isolines of concentration of pollutants at 200 m over the sea level.

(a) Isolines of concentra­tion of S02

o (

~ f~

(b) Isolines of concentra­tion of H2S04

Fig. 7. Isolin es of conce ntration of po llutants at 270 ill over the sea level.

50 G. Winter et a l.

(a) Isolines of concentra­tion of S02

(b) Isolines of conce ntra­tion of H2S04

Fig. 8 . Isolines of concentration of po llutants at 350 m over the sea level.

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3D Simulation in the lower troposphere 51

11. Mata L. J ., Ga rcia R , and Santana R Simulating acid deposition in tropicalregions. In Baldasano J . M., Brebbia C. A., Power H., and P. Zannetti, editors,Air Pollution II: Pollution Control and Monitoring, pages 59-67, Boston, 1984.Computational Mechanics Publications.

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13. Moussiopoulos N., Flassak Th., and Knittel G. A refined diagnosti c wind model.En viron. Software, 3:85-94, 1988.

14. Benoit R On the integral of the surface layer profile-gradient func tions . J. Appl,Meteorol. , 16:859-860, 1977.

15. Montenegro R. Aplicaci6n de Metodos de Eleme ntos Finitos Adaptativos a Pro­blem as de Con vecci6n-Difusi6n. PhD thesis, Universidad Politecnica de Canarias ,1989.

Part B

Numerical methods in oceanic circulation

Eulerian versus semi-Lagrangian schemes insome ocean circulation problems:

a preliminary study

Rodolfo Bermejo

Universidad Complutense de MadridDepartamen to de Matem atica Aplicada

28040 Madrid, SpainrbermejoOmat.ucm.es

Abstract . A comparative study of explicit semi-Lagrangian and Eu­lerian schemes is carried ou t in the cont ex t of ocean circ ulation prob­lem s. We prop ose the explicit semi-Lagrangian sche mes to overcomesome computational difficulties p ossessed by the st andard implicit semi­Lagrangian ones when they are used in ocean general circulation mo de lsformulat ed in spherical coordinates. The numeric al compara t ive studyof the new semi-Lagrangian schem es with Euler ian schem es, whi ch areused in many ocean models , is p erformed on problems whose solutionsare represen tative of relevant ocean circulation features.

1 Introduction

The general circula tion of the ocean is a blend of mo tions that vary on a widerange of space and time sca les. Based on th e magnitude of such sca les we canestablish the follow ing rou gh hierarchy of ocean motions which are of interestto the general circula tion of the ocean : (i) Large scale motion s. This categoryis charact eri zed by motions with space scales of 0 (103 km) and time scales ex­tending from years to mill enia. Clear examples of such ocean ph enomena arethe ocean gyres driven by wind , heat and mat ter exchanges with the a tmo­sphere; t he equatorial current sys tem ; and th e western bound ary currents, suchas Kuroshio and the Gulf st ream, whos e influences on th eir respecti ve basin cir­culations are so profound. Also, we should includ e the th ermohaline circula t ion.(ii) The m esoscale motions. T his category is char acterized by a range of spacescales of 0(10 km - 102 km) . Much of th e energy of the ocean is concentratedin this ran ge, because th e t ra nsient marine eddies as well as th e ph enomenaassociated with the inte nse boundary currents occur in this range.

As for time scales, the ocean has man y dynamically important time scalesexte ndi ng from millenia variations of th e te rmohaline circula t ion down to th erapid varia tions in t ime of sur face gravity waves.

This variety of sca les an d dynamics is so complex and rich th at it representsa difficult chall enge for physicists and applied mathematicians to interpret, sim­ulate and analyse. In this paper, we shall focu s on some issues concern ing with

J. I. Díaz (ed.), Ocean Circulation and Pollution Control - A Mathematical and Numerical Investigation© Springer-Verlag Berlin Heidelberg 2004

56 R. Bermejo

the design and efficiency of the construction of numerical models to simulatethe general circulation of the ocean. Specifically, we explore some applicationsin ocean models of the so called semi-Lagrangian schemes whose success in nu­merical weather prediction and atm osphere circulation models is nowadays wellestablished.

2 The primitive equations of the general circulation ofthe ocean

The design of a numerical ocean general circulation model, hereafter NOGCM,is a long process that consists of a compromise among several fields of scientificknowledge. First, the physics of the ocean motions; the second one is numericalanalysis that contributes to properly formulate the numerical model; the thirdone, but not the less important, is computational science because the computeris the tool to carry out the physical and numerical formulation of the model. Thespectacular progress made in computing power during the last decade has madepossible to carry out better and more complete simulations with old numericalmodels such as the Cox-Bryan model. Its is only through interactions of theabove three fields that efficient and accurate models can be constructed andexecuted.

The physics is mathematically represented by the so called primitive equa­tions (PEs) in a spherical coordinate system on an ocean domain. The oceanis assumed to be a slightly compressible Newtonian fluid under the influenceof Coriolis force . The quantities that describe the ocean circulation are velocity,pressure, temperature, salinity and density of sea water. The governing equationsare formulated in a spherical coordinate system and consist of the Navier-Stokesequations for velocity and pressure, transport-diffusion equations for tempera­ture and salinity plus one equation of state. On the basis of relevant features ofthe ocean circulation, the formulation of the PEs can be simplified through thefollowing assumptions:

Al Boussinesq approximation. The density of sea water is assumed to be con­stant except in the buoyancy terms and in the equation of state .

A2 Thin layer approximation. H ja « 1, where a and H are the radius of theEarth and the maximum depth of the ocean respectively. Let ((), 'P,r) bethe spherical coordinate system with the origin at the center of the Earth,where () is the colatitude, 0 ~ () ~ 1i; 'P is the longitude, 0 ~ 'P ~ 21i; r isthe radial component and z = r - a is the vertical component with respectto the sea level. By virtue of A2 we have z « a, so we can replace r by ain the equations and substitute ajar by ajaz. Hence the coordinate system((),'P,r) becomes ((),'P,z) with scale factor hI = a, hz = c sin e' and h3 = 1.The unit vectors in the ()-, 'P-, and z-directions are:

1 aee = ~ a()'

1 ae =---,

If a sin () a'Pa

ez =-.az

Eulerian versus semi-Lagrangian schemes in ocean circulation problems 57

A3 Hydrostatic approximation. The gravity force is in balanc e with th e verticalcomponent of the pressure gradient in th e momentum equations.

Let M be the domain occupied by the ocean with boundary r = ru U rb Un,where ru denotes the sea surface; n means the ocean bottomf], == -H(O, iP) ,such that we assume that there is Hi, > 0 sa tis fying -H(O, iP) < <H«: and Ti isthe lateral boundary.

We decompose th e velocity vecto r Va as

Va = V + wez , (1)

with

V = ueo + ve<.p, (2)

wheredO . 0diP dz

(3)u = a dt ' v = c sm --;]i' w= - ·dt

Using AI -A3, the PEs for the gen eral circulation of the ocean are:

DV 1-- + f ez /\ V = --'VrP+ F ,Dt Po

{}P- = - pg,{}z

{}wfu + 'Vr . V = o.

p = p(T, S) ,

(4)

(5)

(6)

plus boundary and initial conditions for V , w, T and S.In th e above equ ations, Po is a constant reference density, f denotes the

Coriolis paramet er given as f = 21.01 sin 0, where .0 is th e angular velocity ofthe rotating Earth. F are frictional terms, whereas fiT , tis and kT , ks are thehorizontal and vertical eddy diffusion coefficients for temperature and salinityrespectively. The material derivatives for V , T and S have the following expres­sions:

DV {}V {}V(i) - = - + V . 'V V + w -

Dt {}t r {} z '

(·1·1) D {} V {}" d S- = - + . 'V + w - lor T an ,Dt {}t r {}z'

58 R. Bermejo

where-~- au _ 1::. cot 0]a Sin B a<p a

,_1_ &v !£ cot 0asinB &<p + a

for T and S scalars,1aT 1 er

V'rT =- J'1() + -.-ll~'au asmu u<p

and an analogous expression for V'rS , Here, V'r denotes the gradient operatorin the ((),<p, z) system with respect to () and ip , A commonly used formulationfor F in ocean general circulation models is the following:

F=V'·a (7)

where a is the symmetric stress tensor. By virtue of A3 and the transverseisotropy of the ocean motions [8] the components of a are given as:

au = A(~ au __~_ av - ::: cot 0) = -0'22,a aO asmO a<p a

(1 au 1 av v )

0'12 = 0'21 = A -.-- + - - - - cot 0 ,a sin () a<p a a() a

au av0'13 = 0'31 = k az ' 0'23 = 0'32 = k az '

where A and k are horizontal and vertical eddy viscosity coefficients respectively.Note that in the PEs the component w appears as a diagnostic variable that

has to be calculated from the values of V via the divergence equation. ManyNOGCMs take w =°at z = 0, this is the so called rigid lid boundary condition,so that by vertical integration of the hydrostatic and divergence equations weget:

and

W(V)(t , 0, <p, z) = v, '1°V(t, (), v.z) dz,

v, . fO V dz' = 0,-H

pet, 0, v,z) = ps(t, 0, <p) + 1°gp dz' ·

(8)

(9)

(10)

Equation (8) is the prognostic equation for wand (9) represents a non-localconstraint of the PEs whose Lagrange multiplier is Ps (t, 0, <p). This is a distinctivefeature of the PEs in relation with the conventional three-dimensional Navier­Stokes equations for which the divergence equation is a local constraint.

The aproach to find the numerical solution of the PEs consists of discretizingin time and in space the equations. The discretization in space can be carriedout either by finite elements or by finite differences. Although many numericalmodels use finite differences, however an advantage of finite elements is that one

Eulerian versus semi-Lagrangian schemes in ocean circulation problems 59

Fig. 1. A triangular finite element mesh for the North Atlantic

can refine selectively the mesh based on the dynamics and numerical propertiesof the solution. We show in Fig. 1 a triangular mesh for the North Atlantic Oceangenerated by our finite element code still under development.

As for the time discretization, most of the NOGCMs use explicit schemes,mainly a combination of the leap-frog scheme for the advective terms with afirst order explicit Euler scheme for the diffusion and friction terms. The rea­sons for doing so are historical and also computational. To explain the latter,let us consider a relatively small ocean domain such as the North Atlantic, onwhich we generate a coarse grid of about 10 x 10 size with an average of 15points in the vertical. Such a grid has 0(104

) points. If we use either an im­plicit or semi-implicit scheme (explicit for the advective terms and implicit forthe viscous and diffusion terms) to perform a long-term computation, say 0(10)years, we would require large amount of both computer storage and CPU time,because, in addition to the large number of unknowns, the equations for the uand v components are coupled through the non-linear and friction terms (seethe expression for V . V'T V and a"). So that ocean modelers have favored the useof explicit schemes that allow to uncoupling the u- and v- equations and requiremuch less computer storage; however, explicit schemes have a restriction overthe time step f::...t since they must satisfy a stability criterium. A possible way to

60 R. Bermejo

overcome such time step constraint , in particular if the restriction comes fromthe non-linear terms, is the use of semi-Lagrangian schemes.

3 The schemes

The structure of the PEs as regards time discretization can be represented bythe advection-diffusion equ ation. Let C be a physical magnitude, where C standsfor V , Tor S . The equation we ar e concerne d with is:

DCDt = V· (vV C) in (0, t) x M,

C(O, x) = Co(x),

BC = G on the boundary r,

(11)

where ~~ = ~~ + V . VC, V is a velocity vector, v is an eddy coefficient, Bdenotes the boundary operator, x is a point of the domain M , and V· and Vare the divergence and gradient operators respectively.

Since mo st of th e operating NOGCMs use finite differences with a uniformgrid and a constant time step 6.t , we restrict our pr esentation of Eul erian andsemi-Lagrangian schem es to this sit uat ion. Furthermore, in order to simplify theexposition, and without loss of generality of the points we wish to make in thecomparisons of these schem es, we shall assume tha t M is a bounded domain in lEt 2

and that (11) is formulated in a Cartesian coordinate system. Thus, to computethe numerical solution of (11) we divide the internal [O,t] in to N subintervals[tn' tn+ 1] , n = 0,1 , ... , N - 1, of equal length 6.t and define a regul ar gridover M with grid points X ij such that i = 1, .. . , I and j = 1, . . . , J. We use th enotation X ij == (X i, Yj) and aij == a (Xij, tn) unless otherwise stated. We define anEulerian scheme as a scheme that for all n and all indices i, j approximates ~~

a t each grid point Xi j by a time discretization scheme. In contrast, we define asemi-Lagrangian scheme as one that for each subinterval [tn, tn+l] approximates(11) along the trajectories of the fluid particles that will reach the grid pointsX ij at time tn + 1 [6] .

3.1 An Eulerian scheme

A very much used explicit Euler scheme in NOGCMs is the following [1]:For all n = 1,2, ... , N - 1, i = 1, ... , I, and j = 1, ... , J , compute

C :,.+1 _ C:,.- 1IJ 26.t IJ + (V,Vh C )ij = (V h·(vVC))7j -

1• (12)

Her e the operators Vh' and Vh are th e fini te difference discretizations for V· andV , resp ectively. The scheme (12) combines the leap-frog scheme for advection .with the explicit Euler scheme for diffusion/dissip ation. Relevant properties ofth is scheme are:

Eulerian versus semi-Lagrangian schemes in ocean circulation problems 61

(i) T he scheme is eas y to implement and well adapted to an y type of machine,i.e., scalar, vector or parall el machines.

(ii) T he schem e keeps the storage requirements reasonably low.(iii) It has a trunca tion error O(~x2 ) + O(~i) .

(iv) It is conditionall y stable and has to sat isfy th e following restrict ive crit erium :

2 (~i maxlVI) 2 (4dJ-l~i)~x + ~x2 < 1,

where d is t he dime nsion of space (d = 1, 2 or 3). Consequent ly, for finegrids or grid s with refinement the time step of this scheme is very sm all. Theschem e is used for most of the NOGCMs based on the Cox- Bryan model.

3.2 An explicit semi-Lagrangian Runge-Kutta scheme

Semi-Lagrangian schemes have shown to be very efficien t in atmospher ic mod­els; however , th ey have no t been used very much in ocean modeling. The waythese schemes are implemented in met eorology or atmospheric po llution modelscombines the integra tion of th e advecti ve terms along th e trajectories of thefluid particles with an implicit formulation for the diffusive/viscous te rms . Aswe saw in Section 2, the viscous terms of th e u and v com ponents are coupled,so the use of implicit formula tion of these terms, even within a semi -Lagrangianframework , may not be affordable. Therefore , we shall describe and test semi­Lag rangian schemes that deal with the diffusion terms of (10) in the same wayas explicit Runge-Kutta methods do.

In ord er to facilitate the presentation of our explicit semi-Lagrangian schem e,let us bri efly recall the formula tion of the first and second ord er explicit Runge­Ku t ta schem es as they are used to find th e numerical solut ion to ini tial valueproblems. To this end, let us consider the system:

{~~ = f(i , z) , to < i :::; T,

z(io) = zo,

(13)

where z and f are in ~m, m 2: 1. We approximate th e solut ion to (13) by thefollowing second ord er Runge-Kutta method:

Let zO = zoo For n = 0, 1, ... , N - 1, compute

K 1 = ~if(in , zn ) ,

K 2 = ~i f (i n + !~i , zn + !K1 ) ,

and set

If in (14) we pu t

(14)

(15)

62 R. Bermejo

then we obtain the first ord er explicit Eul er scheme.To const ruct the semi-Lagrangian Runge-Kutta schemes for computing the

solution of (11) we need to calculate, in each subinte rval [tn' tn+l], the traj ecto­ries X ij (Xij, tn+l; t) of the particles th a t reach the grid points Xij at tn+l . Thisis don e by solving for each pai r i, j th e system of equations:

(16)

For t E [tn , tn+1], the differential of arc dSij of the trajectory X ij (Xij, tn+l; t)is given by

dSij = Jdt 2 + dX~ + dY;j = dt 1/Jij (t ),

where, by vir tu e of (16) , 1/Jij (t ) is expressed as

Thus,

Not e tha t Sij (tn) = 0, and Sij (tn+l) is th e arc length from the departurepoint X ij (Xij , tn+l;tn) to the grid point Xi j at tn+ 1•

Let us denote by Cij (s(t)) the valu e of C (x , t) associa ted with the particlethat describes the trajectory X ij (Xij , t n+ 1 ; t) . Then it follows that , for tn ::; t ::;t n +1,

dCij __1_ DCij .ds 1/Jij (t ) Dt

Hence, we approximate (11) by the semi-discrete system of ordinary differ­ential equations:

(17)

B Cij (s(t)) = gij on the boundary r.

Based on th e formulation of the convent iona l Rung e-Kutta explicit schem esintroduced above, we are now in a position to form ulate the explicit semi­Lagrangian Runge-Kutta schemes to int egrate (17) as follows:

Eulerian versus semi-Lagrangian schemes in ocean circulation problems 63

(i) For all grid points X i j find X i j ( Xij, t n +l;tn ) by solving (16) using, for in­stance, scheme 2 of Temperton and Staniforth [7].

(ii) Evaluate CtT == Ci]" (Xi j ( X ij , t n+ l ; t n ) ) by an interpolation procedure fromth e values Cij" , T he interpolation procedure we have employed in the nu­me rical tests of this paper are piecewise bicubic or biquadratic Lagrangeinterpolation.

For each pair i, i . DO:

(iii) Set

and th en evaluate

(iv) Set

C err I}'i j = ij + '2 \.lij

and compute K 2i j as

(v) Then compute

Cn +1 c *n + }"i j = i j \.2 i j

to obtain an explicit second order semi-Lagrangian Runge-Kutta scheme(SLRK2), or

Cn +1 _ c*n + Ki j - ij \.lij

to obtain the explicit first order semi-Lagrangian Runge-Kutta scheme(SLRK1).

T he relevant properties of these schemes are:

(i) They are exp licit, so th ey are well adapted to run on any type of computerarchitecture.

(ii) SLRK2 has a tru ncat ion error O(~x2) + O(~y2) + O(~t2), while the trun­cation error of SLRK1 is O(~x2) + O(~y2) + O(~t).

(iii) It is well known that semi-Lagrangian schemes are un conditionally stable forth e advect ive terms [4]; however, SLRK1 and SLRK2 are both exp licit forthe diffusion terms, because of this they are conditionally stable. It is easy toprove by Von Neuman analysis of linear stability that a necessary conditionfor SLRK1 and SLRK2 to be stable is:

l/~t 1~x2 :'S 2d '

where d is the dimension of space.

64 R. Bermejo

NUMf'.RICAL SOLUJ1o.'l" CUADRATICIN1ERPOLATION CUBIClN1ERPOLATION

T=O.3.

'~.,.-- --1.-j

_~·L..

~h-

'~'f ~"' 1 \

IIit

I!

~

Fig. 2. Semi-Lagrangian solution at different times

4 Numerical tests

4.1 The two-fronts problem

Our first benchmark problem consists of th e advection-diffusion of two fron tsthat t rave l a t different sp eeds and eventually coalesce into one fron t. The maxi­mum velocity of both fronts is along the main diagonal of th e square domain D .This problem has been pr eviously solved by Krishnamachari et al. [3] and repre­sents an interesting test to illu st rate th e behavi our of the numerical schem es inthe numerical simu la tion of oceanic and atmospheric fronts. T he mathematic alformulation of the problem is:

ae ae aeat + u(x , t) ax + v(y,t) ay = vD.e in D x (0, T ],

elr = u(x , t)v(y,t)lr,

e(x, 0) = u(x , O)v(y,0),

(18)

(19)

(20)

Eulerian versus semi-Lagrangian schemes in ocean circulation problems 65

NUMERICAL SOLUTION

'f=(J.3 s

'f=(JA s

MAlNDIAGONALCROSS-SECTION

Fig. 3. Eulerian solution at different times

whereO.le- A i + 0.5e-Bi + e-Ci

Vi(~, t) = e- A i + e-Bi + e-Ci ' i = 1,2, (21)

0.05( ) 0.25 ( ) 0.5 ( )Ai = - ~ - 0.5 + 4.95t , Bi = - ~ - 0.5 + 0.75t , Ci = - ~ - 0.375 ,v v v

with ~ = x for i = 1, and ~ = y for i = 2. The analytical solution is given as

c(x, t) = u(x, t)v(y, t). (22)

The parameters used in this example are ~x = ~y = II' V = 5 X 10-4 m2s- I ,~t = 5 x 10-2 s, and T = 0.6 s. The numerical solution obtained by the SLRK2scheme with biquadratic and bicubic Lagrange interpolation is displayed in Fig­ure 2, where we have included three dimensional plots together with cross­sections along the main diagonal. The results of a fully implicit Crank-NicolsonEulerian scheme are shown in Figure 3. From a simple inspection of these figures ,it is clear that the semi-Lagrangian solutions are more accurate than the Eule­rian ones. In order to further illustrate this fact, we compute pointwise errors

66 R. Bermejo

Advection: EulerStreamline function at timestep n == 10000 (T == 5000 h )

0 .600 10

0 .600

0.400 105

F ig. 4.

along the main diagonal an d a longit udinal cross-section for the SLRK2 withbiquadratic interpolation and Eul erian schem es, respectively. We notice that inthe semi-Lagrangian schemes, large errors are concentrated in narrow regionsaround the two st eep fronts , whereas outside such regions the errors are verysmall. In contrast , we observe that in the Eulerian scheme, there are wakes ofla rge errors, left behind the steep gradients, which are distributed over wideregions.

4.2 The wind driven homogeneus ocean

Our second example is the wind driven circulation of a ,a-plane homogeneusocean. T here are various issues we wish to address in this numerical experi­ment. The first one is referred to the numerical diffusion of semi-Lagrangianschemes. The reason for this is to test the validity of the extended idea in theocean modeling community that semi-Lagrangian schemes ar e more diffusive

Eulerian versus semi-Lagran gian schemes in ocean circulat ion problems 67

Advection : EulerUp to timestep n = 10000 (T = 5000 h )

0.35E-l.,------------------------,

V>Q>~ 0.30E-l

<:;>

>.!5 0.25E-lc:Q)

0.20E-l

0.15E-l

0.10E-l

0.SOE-2

O+'---r----,---y--...,...---r---;r--,---,---.,..---{o 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

ENERGY n

Fig. 5.

than the Euleria n schemes being curre n t ly used in ocean models; hence, semi­Lagrangian schemes are not suita ble for the type of long-term com putat ionswihich are need ed in ocean studies. Of course, t his is t rue if sem i-Lagrangia nschemes em ploy linear int erpola ti on; however , if high er order interpolation isused then things m ay not be th at way. So that , we want to examine t he effect sof the numerical di ffusion of semi -Lagrang ia n schemes with high er order int erpo­lation on lon g term simulation of a wind dri ven homogeneus ocean circulat ion.The second issue is concerne d with stability, accuracy and com putat ional costof sem i-Lagrangian sche mes versus Eulerian ones in long term computat ions ofocean models. Our t hird concern has to do with the interpolation procedure usedin the ca lculati ons done a t the depar tur e points. Bicubic spline interpolation isadm it te d as a very good interpolator, albeit expensive, but not suitable when­the domain has an ir regula r boundary. Other interpolator freq uently used inat m osphere models is piecewise bicubic Lagrange interpolation, and this couldal so be implemented in ocean models. However , if one is in terest ed in modelingwith finite eleme nts with triangula r gr ids, t hen piecewis e biquadratic Lagrangeinterpola t ion is a natural choice compatible with the finite element formulation.Because of these arguments, we use piecewise biquadratic Lagrange interpolationin the semi-Lagrangi an schemes for the numerica l simulations of t his exa m ple.We must say, in advance, tha t the conclusions that eme rge from this expe rim entshould be conside red as orientative rather than conclusive. However , it is ourhope that they m ay cont ribute to clarify some mi sunderstandings about sem i­Lagrangian sch em es that ar e so exte nded in th e ocean m odeling com munit y.

(27)

68 R. Bermejo

Our barotropic ocean model is mathematically formulated by the followingequations [5]:

ow o'l/J 17ft + J(w, 'l/J) + j3 ox = II~W -,w + If curl . r in M x (0, T], (23)

wlr = Wb(X, y, t) and w(x, y, 0) = 0 in M, (24)

~'l/J = w in M x (O,T], (25)

'l/Jlr = 0, and 'l/J(x, y, 0) = 0 in M, (26)

where w = ~~ - ~~ is the relative vorticity, 'l/J is the stream function which is

related to the velocity components by u = - ~~ and v = ~~. J is the Jacobian

operator given by ~; *-~~ ~~, M is a square domain of side L = 1000 Km,j3 is the first order latitudinal variation of the planetary vorticity 1 = 10 +{3 (y - %L), 10 being the value of 1 at the mid-latitude of the domain. If denotesthe constant depth of the ocean and T is the steady wind stress acting at theupper surface with components T1 = 0, TZ = -Tocos(7fy/L). II is a lateral eddyviscosity coefficient and, is the bottom friction coefficient. We shall compare thesemi-Lagrangian solutions of (23)-(26) with the one obtained by the Eulerianscheme previously employed by many researchers such as Holland [2] and othersto compute the numerical solutions of (23)-(26). Thus, the Eulerian model isgiven by the following equations:

n+1 At A n+1 _ . n-1 At A n-1CX 1Wij - U IIUhWij - CXZWij + U IIUhWij -

2~t (Jh (wi}, 'l/Ji}) + ss, 'l/Ji}) + 2~t curl, T,

~h'l/Jij = wi}, (28)

with homogeneus boundary conditions for both 'l/J and w. In equations (27)and (28), ~h denotes the 5-point discrete Laplace operator, Jx is the centereddifference operator in the x-direction, CX1 = (1 + ~lf), CXz = (1 - ~lf) and theJacobian operator h in (27) is the discrete Arakawa Jacobian. This Eulerianmethod is known to be O(~XZ) + O(~yZ) + O(~tZ) and has to satisfy the CFLcondition because the nonlinear terms are explicit in time.

The semi-Lagrangian scheme that we use to compute the solutions of (23) ­(26) is the SLRK2. By virtue of the relationship between the stream function andthe velocity components, we can write (23) as an advection-diffusion equationsuitable for the application of semi-Lagrangian schemes. Thus, we apply SLRK2to the following equation:

Dw o'l/J 1 .Dt = II~W -,w - j3 ox + If CUrlzT III M x (0,T] . (29)

The velocity components, which are needed to calculate the departure points,are evaluated by the approximations Urj = -Jy'l/Ji} and vi} = Jx'l/Ji}, where Jx , Jy

are centered difference operators in the x- and y- directions respectively.

Eulerian versus semi-Lagrangian schemes in ocean circulation problems 69

Advection: Semi-LagrangianStreamline function at timestep n =10000 (T = 5000 h )

Fig. 6.

The discrete ex plicit SLRK 2 scheme is formulated as follows:

kli j = b..t (vb.. hw;p- , W;jn ) + b..t (~ curl, Til - (3e5x r/J;P )

- on + lkWi j = Wij 2" l i j

(30)

wi/ l = w;p + k2i j

where we have int rod uced t he nota ti on x7/ t to denote X ij ( X ij ' tn+l ; tn+t) ,

0.25E-l

0.30E-l

70 R . Bermejo

Advection: Semi-LagrangianUp to limestep n = 10000 (T = 5000 h )

OAOE-l,---------'-----------'--,

(i)Q) 0.35E-j:;,ge>Q)cQ)

· O.20E- l

0.15E-l

0.10E-l

O.50E-2

o¥--..,.--..,.--,--,--..,.--..,.--..,.--.,...--.,...----lo 1000 2000 3000 4000 5000 6000 7000 BOOO 9000 10000

ENERGY n

Fig. 7.

and

with01.*n+~ = ~ (Sol.n. _ 01.,,:.-1)'f/'J 2 'f/'J 'f/'J .

The Eulerian scheme is run with parameter values: ~t1 = 1800s, ~x =6.y = 20 Krn, j3 = 2.0 X 10- 11 m-1s-\ TO = 2.0 X 10-4 ms:", H = 800 m,u = 340 m2s-1, 'Y = 10-7 S-1 . Notice that the values for 'Y and u were previouslyused by Holland [2] in some of his eddy resolving models. In the semi-Lagrangianruns we change the value of 6.t1 and maintain the values of the other parameters.

Figure 4 displays a snapshot of the streamfunction after 10000 time stepsof integration (T = 208.33 days) of the Eulerian scheme. Figure 5 representsthe total kinetic energy curve of this experiment. Figure 6 shows a streamfunc­tion plot after 10000 steps of semi-Lagrangian integration with 6.t1 = 1800s.The total kinetic energy curve of this semi-Lagrangian experiment is shown inFigure 7.

A simple visual comparison of these figures reveals that the semi-Lagrangianscheme, with biquadratic interpolation, gives a flow that is qualitatively similarto the one obtained by the Eulerian scheme, but with a ore intense recircula­tion and with a silghtly higher energy than the Eulerian model. The interestingthing to remark about these figures is that if the numerical diffusion of the semi­Lagrangian schemes had a strong influence on the long term computations, thenthe energy curve of the semi-Lagrangian results would have been different (with

Eulerian versus semi-Lagrangian schemes in ocean circulation problems 71

Advection: Semi-LagrangianStreamline function at timestep n = 500 (T = 5000 h )

0 .600

0 .4 00 lOs

F ig. 8.

lower values) from th e Eul erian curve. Sinc e this is not th e case in our exp er­iment , we may say that for sufficiently fine grids th e semi-Lagrangian schem eswith biquadratic interpolati on are not mo re diffusive than Eulerian schemes.

Fig ures 8 and 9 show stream funct ion plots for T = 208.33 days an d theto tal kinetic energy curve of the nume rical results obtained by semi-Lagra ng ianschemes with D..t = 20D..tl' We observe in these figures that up to D..t = 20D..t 1

the semi-Lagrangian results ar e basica lly the same as the Eulerian ones, and itis a t D..t = 40D..t 1 when the differences are appreciable.

In this experime nt , the CPU time spe nt to perform a semi-Lagrangian st epis about four times la rger than for the Eu lerian step; however , since the semi­Lagran gian schemes are able to yield numerica l resul ts as acc ur ate as the Eule­rian scheme wit h D..t twenty t imes larger, then the use of semi-Lagragian schemescan rep resent sign ificant savings in comput ing time.

72 R. Bermejo

Advection : Semi-LagrangianUp to timestep n = 500 (T = 5000 h )

O.40E-l

0;-Q) O.3SE-l:;0'='>- O.30E-le>Q)cQ)

02SE-l

020E-l

O.lSE-l

O.t OE· l

0.50E-2

0

5 Conclusions

100 200 300

ENERGY

Fig. 9.

400 saon

T he pri ncipal conclus ions reac hed thro ugh this preliminary com parative studyare the following.

(i) For ocean models wit h moderate or low eddy viscos ity (or diffusion ) coef­ficients, as those used in ocean eddy resolving mo dels, the explicit semi­Lagrangia n schemes with quadra tic Lagrange interpola tion may be a valida lte rnative in the design of new numerical ocean circulation models.

(ii) For models with la rge eddy viscosity (or diffusion) coefficients, t he restric­tion imposed on /!;.t by th e stability criterium that explicit sem i-Lagrang ianschemes have to satisfy, may somet imes requ ires a /!;.t unreasoneable small.

(iii) For models formu la ted in Cartesian coordinates, or wit h no viscous cou pling,semi-Lag ra ng ian Crank-Nicolson schemes are a good choice becau se t hey areboth acc ur a te and uncondit ionally stable.

T he above conclusions should be taken as orienta ti ve, ra th er than defini­t ive. However , it is clear that under suitable conditions explicit sem i-Lagrangianschemes show a considera ble promise in ocean gene ra l circulation mode ls.

Acknowledgements

T he author was partially su pporte d by gra nt CLI95-1823 from Com isi6n Inter­min ist erial de Ciencia y Tecnologia .

Eulerian vers us semi-Lagrangian schemes in ocean circulat ion problems 73

References

1. Cox, M. D., 1984: GFDL Ocean Group Techni cal Repor t No.1 (unpublished) .2. Hollan d , W . R. , 1978: The role of mesoscale eddies in the general circulat ion of the

ocean -numerical experiments using a wind-driven quasi-geostrophic model. J . Phys.Oceanogr . 10, 1010-1031.

3. Krisnamachari, S. V., L. J . Hayes and T . F . Ru ssell, 1989: A finit e elem en talt erna t ing-direction method combined with a mo dified method of characte rist icsfor convecti on-diffusion pr oblems. SIAM J . Numer. Ana l. 26, 6, 1462-1473.

4. McDonald , A., 1984: Accuracy of mul tiple-upstream semi-Lagrangia n advectiveschemes. Mon. Wea . Rev. 112, 1267-1275.

5. Pedlosky, J ., 1979: Geop hysical Fluid Dyna mics. Springer-Verlag, 624 pp.6. Stani for th, A. and J . Co te, 1991: Semi-Lagrangian int egration schemes for the a t­

mospheric mod els: a review. Mon . Wea . Rev. 113, 1050-1065.7. Temperton , C. and A. Stanifor th, 1987: An efficient two-time-level semi-Lagrangian

semi-implicit int egra tion scheme. Quart . J . Roy. Met eor. Soc. 113, 1025-1039.8. Waj sowicz, C . R ., 1993: A consistent formulation of the ani sotropic st ress tensor for

use in models of large-scale ocean circulation . J . CompoPhys. 105, 333-338.

Numerical simulation in oceanography.applications to the Alboran Sea

and the Strait of Gibraltar

J. Macias, C. Pares, and M. J. Castro

Departamento de Analisis Maternatico, Universidad de Malaga,Campus de Teatinos sin, 29080 Malaga, Spain

grupoOanamat. cie. uma.eshttp ://alboran. cie .uma .es

1 Introduction

Mathematical and numerical models are now fundamental tools in Marine Sci­ence, as in many other fields of scientific research. Nevertheless, the models,however complex, may provide at best a reasonable estimate of the behaviour ofthe system under study. We must be aware of this, and avoid overly relying andblindly believing in model predictions. In t he task of better understanding therole that models can play in different scientific disciplines, it appears suitable toclassify them according to their scope and objectives. Following Nihoul (1994),three kinds of models can be listed depending on their objectives:

Test-oriented models are used to test mathematical and numerical mo delsand their implementat ion. These models are based on a reduced set of equa­tions and, hence, are not intended to realistically simulate reality.

Process-oriented models, which focus on a few dominant processes. This oftenimplies sacrificing some of the realism of the results, but allows understand­ing the main mechanisms driving the system under study.

System-oriented models, used to understand and/or predict a who le system.The results of these models must be as realistic as possible, which usuallyrequires an appropriate data assimilation pr ocedure to operate together wit hthe prognostic model. Encompassing a large number of processes, system­oriented models are generally very complex, so that they are probably notthe tool best suited for invest iga ting or trying to explain the processes ormechanisms.

The long-term objective of our research focuses on the development andimplementation of different numerical models for the simulation of the dynamicsin the most western part of the Med iterranean Sea (the Alboran Sea and theStrait of Gibraltar), at var ious temporal and spatial sca les. In order to do this,the initia l step cons ists in developing some test-oriented models to be the base ofmore complex models. In this contribution we present a further stage, a process­oriented model that tries to repr esent the main characteristics of the la rge-scale

J. I. Díaz (ed.), Ocean Circulation and Pollution Control - A Mathematical and Numerical Investigation© Springer-Verlag Berlin Heidelberg 2004

76 J . Macias et al.

dy namics in this reg ion of the Mediterranean . In futur e research, we plan todeve lop another process-orient ed model for the study of the generation of internaltides in the Strait of Gibraltar. The final goal in th is wider project is to dev elopa sys tem-oriented model for the Strait of Gibraltar an d the adjacent basins.

The numerical techniques proposed take advantage of methods allowingnon-structured mes hes as finit e volume and finite element methods. They ap­pear to be suitable for the spatial discretization of problems that need to besolved in geometrically complex domains. T hey also permit the use of anisotropicmesh adaptation techniques. T hese techniques a llow the automatic generation ofmeshes well adapted to the flow characterist ics without dramatically increasingthe computational cost . The use of ad apted meshes makes it possible to cap­ture phenomena along a wider spa tial scale range by increasing th e density ofdiscret iza ti on points only on certain regions of the com putational do main (see[11)).

This work has been undertaken in collaboration wit h the Universities of San­tiago de Compostela and Seville (Spain), t he Department of Applied Physics ofthe University of Ma laga and the lnstituto Esp afiol de Oceanograffa.

The possible applications of this an d ot her modelling and numerical simu­lat ion efforts may be:

a better un derstanding of the hydrodynam ical processes,

helping in navigation,

operational forecasting: marine accidents and contaminant spills, etc.,

understanding the marine ecosystem,

applications in Civi l Engineering: pipelines, high tension cables, etc.

In the next section we describe some of the most im portant features thatcharacterize the large scale dynamics in the Alboran Sea. The understandingof the physical problem and the main phenomena that must be reproduced bya model is the first necessary step in any modelling effort . This is extremelynecessary in order to know which are the requirements the numerical modelmust possess to be able to represent these basic features . Section 3 is devotedto obtaining the numerica l model and we briefly describe how the model isnumerica lly solved. This model has been obtained by generalizing to mu lt i-layersystems the shallow-water solver introduced by Bermudez et al. (1991), basedon a mixed finite element method . To our knowledge, this is the first time amu lti-layer finite element model has been used to study the water flow in theAlboran Sea. Besides this gene rali zation, we have improved, in several ways, theperformance of the shallow water solver in order to reduce t he computationalcost (see [13], [26], [31] for furt her details) . In section 4 we show some numerica lresult s. These results are bri efly compared , from a qualitative point of view,with the main characteristics of the observed large sca le dynamics described insection 2. Last section concludes wit h some final remarks.

Numerical simulat ion in Oceanography. Applications 77

2 The oceanographical problem

T he Alboran Sea is, by its dynamics and by its economical and ecologicalimportance, a very interesting area of the marine ecosystem. Being the westernmost part of the Mediterranean, it is th e first basin to receive the Atlantic oceanwater coming from the Strait of Gibraltar. It is also the last basin from whichflows the Mediterranean water leaving this "sea between earths", feeding thedeep current of the Strait. The Alboran Sea is, therefore, a transition basincharacterized by interesting and sp ecific dynamics.

The Mediterranean Sea is subjected to a particularly dry continental cli­mate , even in wintertime, which causes intense evaporat ion throughout the year(Tchernia, 1978). The losses of water by evaporation exceed the gains due toprecipitation and river contributions. This balance translates into an annual lossbetween 0.5 and 1 meters of water in the whole of the Mediterranean (B ithoux,1979; IIarzall ah, 1990). Nevertheless, the characteristics of the Mediterraneanwaters do not appear to have cha nged over the previous centuries. Is the connec­tion of the Mediterranean Sea with the Atlantic Ocean by the Strait of Gibraltarwhich produces an exchange that counteracts this deficit of water and therebyequilibrates the sa lt bal an ce (Lacombe et al., 1981).

The mean exchange through the Strait is constituted by a superposed two­layer fluid. At the surface, Atl an tic-origin less dense water flows eastwards, while,at depth, a much denser Mediterranean wat er penetrates into the Atlantic Ocean(Lacombe and Richez, 1982). This circula tion satisfies a balance (Knudsen's re­lations) which ensur es that the amount of water and salt in th e Mediterraneanremain constant (see Bryden and Stommel, 1984 for further details). Marsigli(J681) showed, by performing an ingenious laboratory study of the interact ionbetween two masses of water of different densities initially separated by a wall ,that the water could return undern ea th as a dense underflowing current. Twocenturies later, Carpenter and Jeffreys (J870) confirmed this hypothesis whenthey submerged a drag line 450 m down in the Strait of Gibraltar, and observedthat the drag drifted to the west, in the opposite dir ect ion to the surface current.

Now it is common knowledge that the surface flux in the Strait is a conse­quence of the increase in density of the water masses produced in the Mediter­ranean basin under the effect of the ocean-atmosphere interactions. In effect,th e Med iterranean Sea is a concentration basin that transforms the inflowingAtlantic water through th e Strait into Mediter ranean water (Lacombe and Tch ­ernia, 1972). The Atlanti c water, th at is characterized by temperatures around16°C and salinities about 36.5 1, penetrates in the Alboran Sea where it flowsforming two large anticyclonic gyres (Lano ix, 1974; Cheney and Dobler, 1982;Gascard and Ri chez, 1985; Kinder and Parrilla, 1991). Afterwards, it continuesalong the Algerian coast (Millot, 1985; Millot, 1991). Figure 1 schemati zes thesurface circulation described. A portion of these waters remains in the west­ern Mediterr an ean where they accomplish a large cyclonic circuit, forming tothe north the Liguro-Provenzal-Catalan current. Another portion of this water

1 salinities will be expressed in the practical salinity scale (pss) .

78 J. Macias et al.

3go r--"""'T""----r--.,....-..,....--.,..-~r__-~-"""'T"I'...,...__..,.--,.....,

N

SPAIN

ALGERIA

ZOE

Fig. 1. Characteristic scheme of the dynamical st ructure of the circulation in the Al­boran Sea . (From Arnone et al., 1987.)

makes its way to th e eastern basin. In th ese two basins, the Atlantic-origin wateris subj ect to transformation mechanisms by convective mixing, in response towinter atmospherical forcing.

Recip rocally , it is also well known th e deep influence Medit erranean waterhas in th e general circulation of the Atl an tic Ocean and , in particular, in theformation of th e Nor th Atl antic Deep Waters (Reid, 1979). The Mediterraneanwater that penetrates in the Northern Atl antic Ocean creates a maximum insalinity at around 1000 meters depth. This saline contribution helps make theAtl antic Ocean water saltier than the waters of the Pacific a nd Indian Oceans.

Therefore, the Strait of Gibraltar plays an important role in the dynamics ofthe two basins that it conn ects and, especially, in the dynamics of the AlboranSea. This sea is especially interesting as transition basin between the dynamicsin th e Strait and those of the rest of the Mediterranean .

2.1 The Alb ora n Sea: a t ran sition basi n

Numerous field studies (Lan oix, 1974; Cano, 1977; Heburn and La Violett e,1990; Tiniore et al., 1991; Viudez et al., 1996; .. .) show a quasi permanent cir­culation pattern in the western basin of the Alboran Sea dominated by th e pres­ence of an anticyclonic gyre approximately cent ered at 4° 10' west longitude, 35°50' north la t it ude. This gyre has also been reproduced in both labora tory mod­els (Whitehead and Mill er, 1979; Gleizon, 1994), and numerical mo dels (Preller,1986; Werner et al., 1988; Speich, 1992 among others).

Numerical simulation in Oceanography. Applications 79

....SURFACE DYNAMICAL TOPOGRAPHYRELATIVE TO 30008 ( IN CMO)

AUGUST 1976

MAL4"'04_ ..-- - --.....

2'

,.

Fig. 2. Geostrophic currents obtained during the oceanographic cruise carried out bythe O. V. Cornide de Saavedra in August 1976. From Cano (1978 a).

The main characteristics of the surface circul ation in the Alboran Sea aredepicted in figure 2 representing the geostrophic circulation in the surface layerof the Alboran Sea obtained from data of a campaign of the lEO (SpanishInsti tu te for Oceanography) in August 1976 (Gano, 1978b). The Atlantic jetfeeds two large ant i-cyclones th at occupy, respect ively, th e west ern and easternpart of the Alboran Sea. The Atlantic water enters the Alboran Sea as a veryintense surface current afte r having suffered an acceleration and a reductionin its thickness at its exit from the Strait (Farm er and Armi, 1988; Perkinset al., 1990). This Atlantic jet constitutes the northern border of th e WesternGyre, creating a saline front at its entrance into the Alboran Sea. A portionof the At lantic water that penetrates through the Strait forms , in most of theobservations, a first anticyclonic gyre located just to the south of th e incomingjet . This jet may extend up to the African coasts (Kinder and Parrilla , 1997).The rest of this mass of water follows the Moroccan coast , moving to the eas t as acoastal current and engendering, sometimes, a large anticyclonic meander beforecontinuing its way along the Afric an coast in the Algerian basin (La Violett e,1985; Tintore et al., 1988; Heburn and La Violett e, 1990).

A schema tic picture of the dynamics described can be seen in figure 1. Nev­ertheless , a remarkable difference between th e two gyres represented in figure 1has to be noted: while the Western Gyre is quasi-permanent , th e Eastern Gyrepossesses a higher variabi lity . Herburn and La Violette {1990j , from satellite im­agery, statistically study the presence of the two anticyclonic gyr es of th e Albo­ran Sea, concluding th at both gyres can disappear. These authors find situations

80 J . Macias et al.

where any of the gyres may not be present, but never where the simultaneousabsence of both gyres were found. T hus, for example, an analysis of data ofAugust 1976 (Gano, 1978b) shows the presence of the Eastern Gyre (figu r e 2) ,this appears reduced to a small ant icyclonic gyre leewards Cape Tres Forcas inthe observations of May- J une 1973 (figure 4(a) , from Cano, 1971) and it evendisappears in the data of July- August 1962 (figu r e 4(b) , from Lanoix, 1974).This variability of the Eastern Gyre opposes to t he continuous presence, in allthese studies, of a well deve loped Western Gyre. In the other ha nd , Herburn andLa Violette (1990) show satellite images where only the Eastern Gyre is present.T he eastern branch of the second of this two gyre forms what its known in theliterature as the Almerfa- Oran fro nt i Tin tor e et al. , 1988). This front, whenit is present , defines an eastern limit for the Alboran basin (see figure 1). Insitu st udies dur ing t he campaigns of the "Western Mediterra nea n CirculationExperiment" (WMCE) showed that the Almeria-Oran front was characterizedby a strong density gradient, confined to the up per 300 m an d was associatedwit h an intense baroclinic jet in the first 50 to 75 meters wit h surface cur rentsof the order of U.fi m s"! (Arnone et al., 1987; Tintore et al., 1988).

ar-

,.SU1FACE DYNAMICAL TOPOGRAPHYRELATIVE TO 300 08 (IN eMO)

MAY-JUNE 1973

S~ACE DYNAMICAL TOPOGRAPHYRELATI VE TO 300 DB (IN CMD)

JULY-AUGUST 19 62

37 '

(a) In May - J une 1973. From Gano(1977) .

(b) In J uly- August 1962. From Gano(1978a) .

Fig.3. Geostrophic currents obtained during the oceanographical campaign by theO. V. Cornide de Saavedra.

Numerical simulat ion in Oceanography. Applications 81

3 The numerical model

3 .1 Primitive equations

In Oceanography, it is usually accepted tha t t he circulation of water mass isgovern ed by th e following syst em of P.D. E.:

OtU h + [u · V'uh + ~V'hP+ fk X U h - YU(u) = 0,Po

ozp = -pg,

V' . u = 0,

OtT + V'. (Tu) = FT (T ),

OtS + V'. (Su) = FS (S ),

p = p(T , S ,p) .

(1)

(2)

(3)(4)

(5)

(6)

In this sys tem the unkn owns are: U h , t he horizontal component of the three-di­mensional velocity U; the temperature, T ; the sa linity, S; t he vert ica l componentofthe velocity, W; t he density, p and pressure, p. Other no ta t ions in these equa­tions are: k the local vertical axis, f th e Cor iolis parameter, 9 t he accelerationdue to gravity, Po a refere nce value for the density and F U

, FT an d F S that rep­resent parameterizatio ns of the effects of dissipa t ion due to mol ecular viscosityor su b-grid mi xing processes.

T hese equations, the so-called primitive equations, a re derived from theincompressible Navier-Stokes equations using the Boussinesq approxi­mation, in which density variations are neglect ed everywhere bu t in th e gravityterm. Density is related to the temperature, salinity an d pr essure th rough anequation of state (6).

Another hyp oth esis made in Oceanography is the hydrostatic approxi­mation, which cons iders fluid vertical acce leration negligib le compar ed wit hgravity-buoyancy effects. T herefore, ver tical pressure gra dient equilibrates withBoussin esq force . This is represented by equation (2). T his hypothesis elim­ina tes convective processes from the primary Navier- Stokes equ a tions, whichmeans that in a three-dimensional mo del they mu st be parameter ized , and in a2D mode l these processes are not represented. Besides , using this approximat ionallows onl y long waves (the so-called shallow water waves) to be simulated.

Equation (3) represents the incompressi bility hypo th esis (the three-dimen­sional divergence of the velocity vect or is assumed to be zero ), an d the four thand fifth equations are convect ion-diffusion equations for the te mperature an dsa linity. The domain where the former system of P.D.E. has to be solved isdefined by

o, = { (Xl , X2, X3) I (X l , X2 ) E st, b( X t , X2 ) < X3 < S(X I' X2, t)},

that repr esents the three- dimensional region between the sea bo t tom an d thefree sea surface .

82 J . Macias et al.

In order to have a complete problem, some initial and boundary conditionsmust be considered depending on the particular problem. Remark that the at­mospheric effects input the ocean model as boundary conditions through the seasurface.

The difficulties encountered in these equations, both from th e mathem aticaland th e numerical point of view, a re manifold: non linearities, geometricallycomplex three-dimensional domains, free surfaces, turbulent regimes, boundarylayers, anisotropy between horizontal and vertical movements or the wide rangeof space-time scal es involved. Due to all th ese difficulties, it is a common pr acticein Oceanography (and other branches of Science) to turn to simplified models.

A usual simplification made in Oceanography and Climatology is the so­ca lled shallow water approximation, th at consi sts in obtaining from theprimitive equat ions a bidimensional system by means of a procedure of verticalintegra tion of the equat ions. This is the approximation that we have followed inthe present study. More pr ecisely, we suppose that the water column is composedof several inmiscible layers of fluid with different constant densities and considera shallow-water approximation at each of these layers of water . Therefo re, amulti-layer model is dedu ced. For this approximat ion to be valid, wavelengthsof th e phenomena to be simulated must be, roughly speaking, much larger thanth e thi ckness of the water layer .

In th e next section , we introduce th e one-layer mo del. Then th e model isgeneralized to a multi-layer configura t ion and used in its two-layer version tosimulate the large-scale dynamics in the Alboran Sea.

3.2 The one-layer model equations

Before introducing model equati ons, let us consider the notations given in thefollowing figur e.

h(xi , x z,t) = s(Xl,X z,t) - b(Xi,XZ) represents the thickness of the waterlayer, 'fJ is the elevation of the free sea surface above a chosen reference level A(for example the mean sea surface height) and H is the bathymetry of th e basin(depth from the referen ce level to the sea bottom), if the reference level has beenchosen to be the mean sea surface.

As the three-dimension al primitive equations must be vertically integrated todeduce the two-dimensional sha llow-water model, we introduce the mean velocityvert icall y integrated ii , defined as:

Aft er the vertical integration pro cess and some fu rther simplification hy­potheses (see [3], [12], [25] for details on th e deduction of model equations) , thesystem of equations obtain ed can be written in conservative form (Q = hii) as:

at]at + \7 . Q = 0 in [l , (7)

Numerical simulation in Oceanography. Applicat ions 83

A

hp

H

Fig. 4. Notations for the one-layer model

in a, (8)

where Q represents the horizontal projection of the three-dimensional domain011 t5 is the unit tensor, F = I (Qz - Qd the te rm du e to Coriolis effects, Tw =I I . h . . . d a- d g lu l - .I v V IS t e term paramet erizing win stress effects an Tb = Cz u paramet erizes

bottom dr ag effects.This model was originally used by B ermudez, Rodriguez, Vilar (1991) to

simulate tidal effects in the Galician Rias .

Boundar y conditions

The boundary conditions permitted by the model are:

(a) N or-mal comp o n ent of t he flux gi ven,

Q·n =1,

where n is the normal outward unit vector to the boundary of Q , f) Q. T hiscondit ion is useful on th e inflow/outflow boundaries when the flux is known,and on coast al boundaries where 1 = o.

(b) Elevation give n,TJ = 'P.

It is used on the inflow/outflow boundaries, when the elevation is known.This condition is useful , for instance, to simulate the effects due to tidefluctuations.

84 J . Macias et al.

Initial conditions

As initial conditions, the elevation and mean integrated velocity initial statemust be given :

1](X, 0) = 1]o(x) in il ,

u(x , 0) = uo(x) in a.

3.3 The numerical scheme

In this section we outline the numerical tech niques used to numerically solve(7)-(8). The equat ions to be solved form a bidimension al system of non-linearfirst-orde r hyp erbolic equations. Therefore, their properties are simila r to thoseof the compressibl e Euler equations. As it is well know, schemes based on explicitEuler discretization in ti me together with cente red differences for flux ter ms areunconditionally unstable for hyperbolic equations even in the linear case. Theupwind discreti za tion should be used. In [3] a method combining the charac­teristics met hod to discretize convective terms with first-order Ravi art.-Thomasfinit e eleme nts for th e space discretization is given. This is the method we haveused and extended to multi-l ayer sys tems.

3.3.1 Time discretization

T he mass conservation equa t ion (7) is discreti zed using an Euler implicitmethod, obtaining the di scretized equation,

where we use th e standard notations: b..t is th e time step an d 1]n, Qn are theapproximat ions of the eleva tion 1] and the flux Q a t t ime nb..t .

To discretize th e conv ective term in th e momentum equa t ion (8) the methodof characteristics (cf. [33]) is used. This me thod is based on th e fact th at thefirst two adden ds in the momentum equa tion (8) coin cide with the materialder iva tiv e of the product J Q, i.e.,

D 8QDt (JQ) = at + V'. (u 0 Q),

where gt = tt + U • V' is the material or convective derivative, i.e., thederiva tive following part icle trajectories and J represents the evolut ion of thevolume eleme nt, i.e., is the J acobian of th e tr an sformation associa ted to th e flux,that is characte rized as being the solution of t he differential equation

{

'£.1(x , t ;r) = V· u (X(x , t ;T),r) .1(x , t; r)dr

J( x , l ; t) = 1.

Numerical simulat ion in Oceanography. Applications 85

Here the function T --+ X(x, t; T) represents the trajecto ry of the particle that islocated at a point x at time t, and therefore X(x, t; T) is the solution of

{

:T X(x , t; T) = U(X(x, t ;T), T)

X(x , t; t) = x.

The idea behind the method consists of applying a scheme of type backward finitedifferences following the trajectories. Observe that while at t ime n + 1 functionsare evaluated at a point x, at the previous time step n they are evaluated atXn(x), which rep resents the position occupied at time n by a particle locatedat th e point x at time n + 1. This method has good proper ties of stability andeliminates the non -linearity of th e convective term.

3.3. 2 Discretized equations

Using the two t ime discretization schemes described above, the sem i-discretiz edequations ar e written as follows:

Given Qn, rt :Qn+l + !f... \7 ((1]n+l)2 + 21]n+lH) _ 91]n+l\7H = rQn [x

n].

ti.t 2 ti.t '1]n+l = 1]n - ti.t \7 . Qn+l in Q.

In order to simplify the notation (dropping the n' s), the problem to be solvedat each time iteration can be rewritten as

;t + ~ \7 (1]2 + 21]H) - 91]\7H = F in Q ,

1] = 1]0 - ti.t \7 . Q in Q,

Q . n = 0 at {)Q ,

where 1]0 denotes th e elevat ion at th e previous t ime step and F represents theterm given by the method of characteristics.

Although the method of characteristics eliminates th e non-lineari ty of th econvective term, the non-linearity due to the pressure term remains. In fact th edifficulty for numerically solving this system of equations comes from this non­linearity (1]2 + 21]H) , that is a maximal-monotonous operator , which allows theuse of a numerical algorithm proposed by Bermudez and Moreno (1981) (see [3]or [25] for further details on the numerical resolution of model equation s).

The space discretization is performed using the first -order Raviart-ThomasFinite Element. The degrees of freedom for th e flux Q a re the values on th emiddle points of the edges of the elements and th e elevation 1] is constant bytriangle. The resu lting linear systems (with non-symmet ric matrices) are solvedby a Stabilized Conjugate Bigradient preconditioned with an LU incompletetyp e factorization (see [25] and [13] for fur ther details).

86 J. Macias et al.

The main drawbacks of the chosen algorithm derive from the implicit na­ture of the discretization performed, and from the restrictions on the type ofboundary conditions that can be imposed. On the one hand, it is known thatimplicit schemes introduce additional damping effects on the numerical solutionand increase the computational cost when compared with explicit schemes. Theadvantage of the chosen discretization is that it allows dealing with the appear­ance of regions where the thickness of the water layer vanishes (see [25], [26]).Moreover, at this first stage, we only look for obtaining steady states or solutionswithout sharp gradient regions, and damping effects are assumed not to be im­portant. Nevertheless, this algorithm provides good results in more general casesif small enough time steps are chosen. Therefore, in order to have an efficientalgorithm, the computational cost of solving the non-linear problems appearingat each time step must be small. In order to have this, we have accelerated theoriginal algorithm by implementing an automatic choice of parameters (see [25]or [31]). Concerning the boundary conditions, the algorithm used to solve thenon-linear problems include an elliptic regularization of the problem. Due to thisregularisation, only one condition can be imposed on each part of the boundary.Therefore, supercritical flows close to the input/output boundaries cannot behandled. Currently we are working on the adaptation of the algorithm to theseconditions.

3.4 The multilayer model

Once the one-layer model has been introduced, we undertake the descriptionof the multilayer model. For that case we consider the notations given in thefollowing figure representing, for the sake of simplicity, a two-layer configuration:

pz

1771----

Two-layer model scheme and notations

Numerical simulation in Oceanography. Applications 87

As it was pointed out in the section devoted to the introduction of thephys­ical problem, a one-layer model cannot represent the observed dynamics in theAlboran Sea and the Strait of Gibraltar (basically composed by a two-layer fluidwith different densities flowing in opposite directions). Therefore , a one-layermod el is not admissible in that case. The multilayer model will suppose the wa­ter column composed of several layers of water. At each layer , a shallow waterapproximation is considered, i.e.: the primitive equations are written at eachlayer with suitable boundary condit ions at the interface and are vertically inte­grated at each of the layers. Doing so, a coupled shallow water system is obtainedwhere the coupling takes place through the pressure and friction terms. In theformulation used in this study, this system of equations writes as follows:

First layer

Lower layer

8hn----at + V' . Qn = 0,

8~n + V'. [(un @ Qn) + ~g(iJ~ + 2iJnHn)J]

- PI h Pn-l h• TIn = TIn + - I + ... + -- n- 1,Pn Pn

- PI h Pn-l h• H n =H n - - 1-·· · - -- n-l·Pn Pn

The new variables Hand iJ ar e introduced in order to formally obtain th esame equations at each layer and, therefore, to enable us to apply the same

88 J. Macias et al.

algorithm of resolution in all the layers. The unknowns to be computed in thissyst em of equations are Qk , the fluxes at each layer, and 11k, th e elevat ions atth e different interfaces.

In practice, we will apply the two-layer model for simulat ing th e dynamicsin th e Alboran Sea. In th e next section som e num erical results ar e presented.

4 Some numerical results

In this section we depict some figures to illustrate model results. The experimentsperformed were aimed to study th e two Alboran Sea gyres and th eir variationsto different wind conditions. A complete set of figures and comments can befound in [25], here we restrict ourselves to a single numerical example where nowind conditions were imposed.

In th e experiment shown here the two layer model have been used. The upperlayer (initially at 80 ill depth) represents the inflowing Atlantic wat er that entersthrough the Strait of Gibraltar and exits into the western Mediterranean basin.In this layer the constant value for the water density was taken to be equal to1027 kg mr", The lower layer repr esents th e Mediterranean water pouring fromth e eastern Mediterranean into the Atlantic. Th e constant density value takenin this layer was 1029 kgm- 3 •

Fig. 5. Spatial domain considered and it s finit e element discr et ization . The mesh con­tains 4,792 triangles and 7,317 nodes.

Numerical simulation in Oceanography. Applications 89

The physical domain considered presented 4 different boundaries. Two "nat­ural" boundaries, corresponding to the Spanish and Moroccan coasts and two"artificial" boundaries limiting the computational domain on the east and west.The western boundary near Tarifa, in the Strait of Gibraltar, and the easternlimit consisted of the sides of a large rectangle. These sides followed the orien­tation of the Spanish and African coasts at that part of the Mediterranean (seefigure 5). The meshes were generated from digitalized cartographic data pro­vided by the I.E.O. (Instituto Espafiol de Oceanografia), using HYPACK code.When real bottom bathymetry was considered, the bathymetry function H wascomputed from digital cartographic data by means of an automatic interpolationprocess over the mesh vertices. From this discrete function and the correspondingfirst layer mesh, the second, and subsequent, layer meshes were automaticallyconstructed suppressing the spare triangles, i.e., the elements, k in the first layer,with H (k) > -80 m were suppressed. Figure 6 depicts the second layer meshobtained by this procedure from the first layer triangularization shown in fig. 5.

Fig. 6. Second layer finite element discretization, when real bottom bathymetry istaken into account. Otherwise, when constant depth is considered the second layerspatial discretization is the same as for the first layer. The mesh contains 4,094 trianglesand 6,283 nodes.

As initial conditions, elevation and fluxes equal to zero have been imposed,i.e., start from a resting sea. At the first layer, the input flux was imposed overthe Strait of Gibraltar. A total flux of 1 Sv (1 Sv = 106 m3s- 1 ) was taken, which

90 J . Macias et al.

is the estimate corresponding to the annual mean of the Atlantic input flux (see[5]). The profile considered for this input flow was designed to fulfil a criterion .of conservation of the potential vorticity as described in [25]. On the sides ofthe rectangle, coastal conditions were imposed at both layers . For the secondlayer at the Strait, an output symmetric to the first layer input was imposed. Toconserve mass at each layer it was necessary to consider an exchange betweenlayers of 1Sv. In the example presented here, this exchange was imposed totake place in the most eastern part of the domain, in the region limited bythe rectangle. As external forcing, the wind can be imposed. In the examplepresented here no wind conditions are considered. Therefore, the energy neededfor the system to move was exclusively provided by the input flow through theStrait of Gibraltar. Numerical experiments showing the effects of different windconditions and starting from other initial conditions can be found in [25]. In thesimulation shown here the time step is 15 min.

Figures 7 and 9 depict the time evolution of the first layer velocity field.Initially, an evolving "coastal mode" is developed. Figure 7 shows the velocityfield after 10 (left panel) and 20 days (right panel) of integration. The elevationsof the sea surface and interface reproduced by the model after 20 days of integra­tion are shown in figure 8. It can be observed that , when an Atlantic jet flowingclose to the African coast is simulated, an accumulation of water in this coastis produced by effect of Coriolis force. This reflects in a rising of the sea surfaceand a deepening of the interface (up to 37 m under the reference level) due tothe pressure exerted by the thicker Atlantic layer. In the other hand, it can beobserved that in regions with cyclonic circulation the free surface is depressed byeffect of the Coriolis force and at the same time, the interface becomes shallower.This translates into a sea surface up to 13 cm below the mean sea level and aninterface up to 15m above its reference level. '

After a month (not shown) , the Atlantic jet is separating from the Africancoast, producing the formation of a Western Gyre of reduced dimensions, whileat the eastern part of the Alboran Sea, the evolution and growing of anotherlarge anticyclonic structure also starts to be evident. This structure ~as alreadypresent after 10 days of integration (figure 7, left panel), although reduced to onehalf of its maximal size. Ten days latter, it has reduced its size and is confinedleewards of Cape Tres Forcas (figure 7, right panel) and after a month it is againincreasing in size. Figure 9 left panel, shows model results after 40 days. Atthat time, the eastern gyre was well developed and the cyclonic structure appearsmore confined to the north of the Atlantic jet. The eastern gyre was also formed.This figure and next one (figure 9, right panel) show a two-gyre configurationvery similar to that depicted in figures 1 and 2 in the section devoted to theintroduction of the physical problem.

Performing further in the time integration, the Western Gyre continues grow­ing and the eastern one disappears. A one-gyre configuration is obtained, withthe single gyre located to the west of Cape Tres Forcas (not shown) . Figure 10shows the elevations simulated after 50 days. It can be observed the clear sig-

Numerical simulation in Oceanography. Applications 91

ITt UTIOIl \000

JUt ",IJrnnn ...." U 11.eQJIT'

...1 . "' U I -O. 1 .140

""""'.J .ONOC . \. 1010

"I~r '_

' Ian "",1:1 nuxl" PlltD

JU, JIOlwr.nu ...." 'J ~

.,.' .OIUI ·"

""""'.

Fig. 7. First layer mean velocity field at time iterations 1,000 and 2,000.

92 J. Macias et al.

JSU l'OurN112' ....• 7' 1 ~

n:u POIWTInIt .....

'1f' a.-.rn

Fig. 8. Sea surface and interface elevation at iteration 2,000.

' .Utl.,U"' .111'' .nlu·n• •unl-OJ,."" ..-.2l .lCS&I·U1."411-12J ."U,.·IJ..........

. ... . . u • . 1]

- \. ' U "-I).) . UI "~."'J

· .... ng·n·' .n n l· fa- 't . 'tI"....J· t .nU I - I .". 1' 61_•.u. "-' . U t )

n .n11 ,70' .In\ .."I . "U

. .... n.·n- J . \ "· I. I U-' . t H-lI .n-U. n·n."-lI.'.· :u .u-:n."- )6 I I

.:n."-111. "· .. .n-n .n

Numerical simulation in Oceanography. Applications 93

nature of the anticyclonic gyre on the sea surface elevation. On the other hand,the opposite effect is evident in the regions with cyclonic circulation.

The differences on the simulated sea level are of about 30em between higherand lower sea surface regions. Again it can be observed the effect of the ris­ing/sinking of the sea surface on the location of the interface: below the an­ticyclonic gyres water accumulates and the interface sinks (up to 35m belowthe reference level in this simulation); under the cyclonic gyres the interface isshallower (15 m above the reference level) . It can be observed that the maximumsinking is not located exactly below the center of the anticyclonic gyres, as occurswith the sea surface elevat ions, but .this is displaced southwards. This situationhas also been reproduced by other models (see, for example, [36]) although inobservational data this seems not to be the case. In what respect' the sea sur­face, it would be interesting to dispose of suitable satellite data to compare andvalidate model results. Nevertheless, satellite data currently available are of verylarge spatial resolution to be used in regions of th e dimensions of the AlboranSea (reduced at global scale). In the other hand, for such a comparison it mightbe taken into account that the sea surface elevations simulated by the modelcorrespond to a dynamical topography of the sea surface in which the effects ofthe atmospheric pressure are not included.

5 Final remarks

The development of the model presented fits into a wider project whose objectiveis the modelling and numerical simulation of the Alboran Sea and the Strait ofGibraltar dynamics at different spatial and temporal scales, by means of the useof various models, among them the shallow-water multi-layer model presentedhere. At an initial stage, the aim of this project was to find solutions that qual­itatively approximate the large-scale structures characterizing the dynamics inthe Alboran Sea. In view of the all numerical results obtained, some of whichare outlined in the text, it seems that the model developed can be used as atool to better understand the physical problem set by the Strait-of Gibraltarand the Alboran Sea environment. Nevertheless, it must be pointed out thatit seems suitable, for a deeper study of the dynamics simulated by the model,as well as to understand the observed dynamics, to perform a set of numericalexperiments, which would include testing the effects of different winds, variablein time and/or space, or the effect of time variations in the input Atlantic flow,together with other characteristics of this flow, such as its vorticity.

In order to improve the model, we are carrying out the following researchactivities:

Dealing with vanishing thickness regions and using penalisation techniquesto improve convergence.Generalizing the choice of boundary conditions. Study the possibility of usingdata assimilation techniques.Implementation and comparison of another time and/or space discretization,including second-order schemes.

94 J. Macias et al.

IT'IUoTIl» . 4000

JU' JIOl.nlHal .xaa" " ~

"I. ""'"In••• · • • 1"&..,..., .... Of • 1. ...

.., ........,--

ITnATU* . ...

HH POlWTO'Ull acr.-a" ., ~

.,.1 UUI ."

, .tot I. U'

Fig. 9. First layer mean velocity field at time iterations 4,000 and 5,000.

Numerical simulation in Oceanography. Applications 95

~"aI. ,...

' .lU'• •11'2' .UII' .IK4

'.'UlI -'J7 . 41)7& -'2' .'U11-1:a• • 'flU ·OJ) . 'UlI -12I.JUU·.J

-) .Uln-II· 1. ' 'Jl n·OJ·l .UUI-'J.c. n. '.·u· I .'UJ.-U-'. u n a·t J" ..,..,a ·1l· ' . lI U·o,u ..."un

J I.:U I'OnrttJIlt .xaa'7'2 c.-n

lJ .n11 .)\I . U '

' .'044J . U)

••• n-I.'"' • • 211... .. ].,....-u."- 11 . 61-n.u· 19 . '1'7-u. n· U. II

- J 'L 'U· lO ,"-u. n·n .u

Fig. 10. Sea surface and interface elevation at iteration 5,000.

96 J . Macias et al.

Testing the model under more realistic wind and boundary conditions. Com­parison with the results with previous works and experimental measure­ments.Application of the model to the study of the variability of the Alboran gyres.Analysing the mathematical model.

Acknowledgments: This research was partially supported by the C.I.C.Y.T.(project MAR97-1 055-C02-0 1) .

References

1. R. A. Arnone, D. A. Wiesenburg, and K. D. Saunders. Origins and characteristicsof the Algerian Current. EOS, Trans . Americ.Geophys. Union , 68:1725, 1987.

2. A. Bermudez and C. Moreno. Duality methods for solving variational inequalities.Compo and Maths . with Appls., 7:43-58, 1981. Pergamon Press Ltd.

3. A. Bermudez, C. Rodriguez, and M. A. Vilar. Solving shallow water equationsby a mixed implicit finite element method. IMA Journal of Numerical Analysis,11:79-97, 1991.

4. J . P. Bethoux, Budgets of the Mediterranean Sea : Their dependence on the localclimate and on the characteristics of the Atlantic waters. Oceonol. Acta, 7(3) :289­296, 1979.

5. H. Bryden, J . Candela, and T. H. Kinder. Exchange through the Strait of Gibraltar.Prog. Oceanogr ., 33:201-248, 1994.

6. H. 1. Bryden and H. M. Stommel. Limiting processes that determine basic featuresof the circulation in the Mediterranean Sea . Oceanol . Acta, 7(3):289-296, 1984.

7. N. Cano. Resultados de la carnpafia "Alboran 73" . Boletfn 1 (230) :103-177, Inst .Esp. Oceanogr., 1977.

8. N. Cano. Hidrologia del Mar de Alboran en Primavera-Verano. Boletin 4(248) :51­66, Inst . Esp. Oceanogr., 1978a.

9. N. Cano. Resultados de la campafia "Alboran 76" . Boletfn 4 (247) :3-49, Inst. Esp.Oceanogr., 1978b .

10. W. B. Carpenter and J . G. Jeffreys. Report on deep-sea researches carried- onduring the months of July, August, and September 1870 in HM surveying-shipPorcupine. Proceedings of the Royal Soc iety, London, 19:146-221, 1870.

11. M. J. Castro. Generacion y Adaptacion Anisotropa de Mallados de ElementosFinitos para la Hesolucion Numerica de E.D.P. PhD thesis, Universidad de Malaga,Noviembre 1996. 216 pp.

12. M. J . Castro and J . Macias. Modelo Matemdtico de las Corrientes Forzadas porel Viento en el Mar de Alboran, volume 5. Publicaciones del Grupo de AnalisisMaternatico Aplicado de la Universidad de Malaga, 1994. 350pp, ISBN :84-7496­252-8.

13. M. J . Castro, J . Macias, and C. Pares. An incomplete LU-based family ofprecondi­tioners for numerical resolution of a shallow water system using a duality method.Applications. Appl. Math . Lett., 14:651-656 , 2001.

14. R. E . Cheney and R. A. Dobler, Structure and variability of the Alboran Seafrontal system. J. Geophys. Res ., 87(Cl) , 1982.

15. J . C . Gascard and C. Richez . Water masses and circulation in the western AlboranSea and in the Strait of Gibraltar. Progr. Oceanogr., 15:157-215, 1985.

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16. P. Gleizon . Etude Exp er im entel de la Formation et de l'Estab ilite de des Tourb illonsAnticyc/oniques Engendres par un Courant Baroclin e l ssu d 'un Detroit. Applicat iona la Mer d 'Alboran. PhD thesis , Universite Joseph Fourier , Grenoble I, 1994.240 pp.

17. A. Harzallah. Etude aeroloqique et oceanique de l'hydrologie du bassin medi­terran een. PhD thesis, Universite Paris VI, Paris, 1990. 212pp.

18. G. W. Heburn and P. E. La Violette. Variations in the structure of the anticyclonicgyres found in the Alboran Sea . J. of Geophys . Res., 95(C2) :1599-1613, February1990.

19. T . H. Kinder and G. Parrilla. The summer 1982 Alboran Sea gyre. J. Geophys .Res ., Accepted, 1997.

20. P. E. La Violette. Western Mediterranean Circulation Experiment Op eration Plan.WMCE Newsletter 5, 1985. 48pp.

21. H. Lacombe, J .-C. Gascard, J . Gonella, and J. Bethoux. Response of the Mediter­ranean Sea to water and energy fluxes across its surface, on a seasonal and inter­annual scales. Oceanol. Acta, 4(2):247-255, 1981.

22. H. Lacombe and C. Richez. The regimen of the Strait of Gibraltar. In J . C.J . Nihoul, editor, Hydrodynamics of Semi-Enclosed Seas, pages 13-73. Elsevier,Amsterdam, 1982.

23. H. Lacombe and P. Tchernia. Caracteres hydrologiques et circulation des eaux enMediterranee. In D. J. Stanley, editor, The Medit erran ean Sea: A Natural Sedi­mentation Laboratory. Dowden, Hutchinson and Ross, Stroudsburg, Pennsylvania,1972. 765 pp .

24. F. Lanoix. Project Alboran. Etude hydrologique et dynarnique de la Merd 'Alboran. Rapport Technique 66, OTAN, Brussels , ],974.

25" J . Macias. Some Topics in Numeri cal Modell ing in Ocean ography. PhD thesis ,University of Paris VI, Paris , November 1998.

26. J . Macias, C. Pares, and M. J . Castro. Improvement and generalization of ashallow-wa ter solver to multilayer syste ms. Int. J. Nu me r. Meth ods Flu ids, 31:1037­1059, 1999.

27. 1. F . Marsigli . Internal observation of the Thracian Bosphorus, or true channel ofConstantinople, represented in letters to her majesty , Queen Christina of Sweden(Translated by E. Hudson) . In M.B. Deacon, editor, Oceanography: concepts an dhist ory, 394 pp. Dowden, Hutchinson and Ross , Stroudsburg, Pensylvania, 1681.

28. C. Millot. Some features of the Algerian Current . J. Geophys. Res.;.g0:7169-7176,1985.

29. C. Millot . Mesoscale and seasonal variabilities of the circulation in the WesternMediterranean. Dynamics of Atmospheres and Oceans , 15:179-214, 1991.

30. J . C. J. Nihoul. Do not use a simple model when a complex one will do. J. Mar .Syst ., 5:401-406, 1994.

31. C. Pares, J . Macias, and M. J. Castro. Duality methods with an au tomatic choice ofparameters . Application to shallow-water equations in conservative form. Num er.Math ., 89(1) :161-189, 2001.

32. H. Perkins, T . H. Kinder, and P. E. La Violette. The Atlantic inflow in the westernAlboran Sea . J. Phys . Oceanogr. , 20:242-263, 1990.

33. O. Pironneau. Methodes des Elements Fin is pour les Fluides, volum e 7 of RMA .Masson, 1988.

34. R. H. Preller . A numerical model st udy of the Alboran Sea gyre. Progr. inOceanogr., 16:113-146,'1986.

35. J. 1. Reid . On the contribution of the Mediterranean Sea outflow to the NorwegianGreenland Sea . Deep Sea Res., 26:1199-1223, 1977.

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36. S. Speich. Etude du Forcaqe de la Circulation Oceanique par les Detroits: Cas dela Mer d'Alboran. PhD thesis, Universite Paris VI, November 1992. 245 pp .

37. P. Tchernia. Oceanographie regionale, description physique des oceans et des mers.Centre d'edition et de documentation, ENSTA, 1978. 277 pp.

38. J . Tintore, D. Gomis, S. Alonso, and G. Parrilla. Mesoscale dynamics and verticalmotion in the Alboran Sea. J. Phys. Oceanogr., 21(6):811-823, June 1991.

39. J . Tintore, P. E . La Violette, I. Blade, and A. Cruzado. A study of an intense den­sity front in the eastern Alboran Sea. The Almeria-Oran front . J. Phys . Oceanogr.,18(10):1384-1397, 1988.

40. A. Viudez, J . Tintore, and R. L. Haney. Circulation in the Alboran Sea as de­termined by quasi-synoptic hydrographic observations. Part I : three-dimentionalstructure of the two anticyclonic gyres. J. Phys . Oceanogr., 26:684-705, 1996.

41. F . E. Werner, A. Cantos-Figuerola, and G. Parrilla. A sensitivity study of reduced­gravity channel flows with applications to the Alboran Sea. J. Phys . Oceanogr.,18:373-383, 1988.

42. J . A. Whitehead and A. R. Miller. Laboratory simulation of the gyre in the AlboranSea . J. Geophys. Res., 84:3733-3742 , 1979.

Simulation of reactive transport in groundwater.A comparison of two calculation methods

Maarten W. Saaltink and Jesus Carrera

Dep. d'Enginyeria del Terreny i Cartografica, ETSECCPB, Universitat Politecnica deCatalunya, c/Jordi Girona 1-3, Modul D-2, 08034 Barcelona, Spain.

{dsaaltink,carrera} @etseccpb.upc.es

Abstract. Numerical simulation of reactive transport in groundwater (that is,transport of solutes undergoing chemical reactions) requires the solution of alarge number of mathematical equations, which can be highly non linear. Thechoice of a method to solve these equations may effect significantly bothcomputation time and numerical behavior of the solution. Two types ofmethods exist: The Direct Substitution Approach (DSA), based on Newton­Raphson, and the Picard or Sequential Iteration Approach (SIA). Theadvantage of the DSA is that it converges faster and is more robust than theSIA. Its disadvantage is that one has to solve simultaneously a much larger setof equations than for the SIA. We applied both methods to several examplesand compared computational behavior. Results showed that, for chemicallydifficult, cases, the SIA may require very small time steps leading to excessivecomputation times. The DSA displays a much more robust behavior, withcomputation times much less sensitive to the value of chemical parametersandgenerally smaller than the SIA.

1 Introduction

The use of numerical models can greatly help the performance assessment of wastedisposal facilities, the study of groundwater contamination and the understanding ofgroundwater quality in natural systems and the processes undergone by rocks. Thesemodels should consider the concentrations of several species and should be able tosimulate both solute transport processes, such as advection and dispersion, andchemical reactions, such as complexation, adsorption and precipitation. This requiresthe solution of a large number of mathematical equations, which can be highly nonlinear. For complex problems this may easily lead to excessive computation times .Therefore, the choice of an approach to solve these mathematical equations isimportant. Several approaches are available. However, one can consider them to bevariants of two .

The first one is the Picard method that includes the Sequential Iteration Approach(SIA) and the Sequential Non Iteration Approach (SNIA). It consists of separatelysolving the chemical equations and the transport equations. The difference betweenthe SIA and the SNIA is that the first iterates between these two types of equations,whereas the second does not. The SIA has been used by, among others, Kinzelbach

J. I. Díaz (ed.), Ocean Circulation and Pollution Control - A Mathematical and Numerical Investigation© Springer-Verlag Berlin Heidelberg 2004

100 M. W . Saaltink and J. Carrera

[1991], Yeh and Tripathi [1991], Engesgaard and Kipp [1992], Simunek and Suarez[1994], Zysset et al. [1994], Morrison et al. [1995], Schafer and Therrien [1995] andStollenwerk [1995] . The SNIA has been used by, among others, Liu and Narasimhan[1989a], McNab and Narasimhan [1994], Walter et al. [1994], Engesgaard andTraberg [1996]. Valocchi and Malmstead [1992], Miller and Rabideau [1993] andBarry et al. [1996] discussed some limitations of the SNIA and proposed solutions.

The second approach is the Newton-Raphson method, also called one-step, globalimplicit or Direct Substitution Approach (DSA). It consists of substituting thechemical equations into the transport equations and solving them simultaneously,applying Newton-Raphson. It has been used by, among others, Valocchi et al. [1981],Steefel and Lasaga [1994], White [1995], Grindrod and Takase [1996] and Saaltinket al. [1998].

The main disadvantage of the DSA is the large set of equations that one has tosolve simultaneously, leading to high computational costs per iteration. We shouldalso mention that programming the DSA is significantly more difficult than the SIA.On the other hand, the SIA and SNIA generally show slower convergence and areless robust and more stiff. This may require finer temporal discretisations than theDSA, leading to a larger number of iterations. Reeves and Kirkner [1988] and Steefeland MacQuarrie [1996] compared the different approaches by applying them to anumber of cases of small one-dimensional grids. Both reported more numericalproblems for the SIA and/or SNIA than for the DSA. The first found generallysmaller computation times for the DSA, whereas the latter for the SIA and SNIA.Nevertheless, in both articles the computation times for the different approaches werealways of the same order of magnitude. In an article which has had great impact, Yehand Tripathi [1989] compared the different approaches for larger grids of one, twoand three dimensions. They concluded that the DSA leads to excessive CPU memoryand CPU times of realistic two- and three-dimensional cases, due to the very large setof equations that one has to solve for the DSA in these cases. However, they madetheir comparison on a theoretical basis without in fact applying them and measuringCPU times. Therefore, they could not take into account the fact that the DSA mayrequire fewer iterations. We conjecture that for some cases this may be important andthat hence the DSA might become more advisable than stated by Yeh and Tripathi[1989].

The objective of our work is precisely to test this conjecture. To do so, we firstformulate. several cases of varying chemical complexity, second, solve them with theSIA and DSA and, third, compare required temporal discretisation, number ofiteration and CPU time.

We start by explaining the mathematical formulation for reactive transport. Thenext section treats the implementation of SIA and DSA. Then, we give a shortdescription of the cases that we used for the comparison . The next section discussesthe results of the comparison between the SIA and DSA. Finally, the last sectioncontains some conclusions.

Simulation of reactive transport in groundwater 101

2 Basic Equations

In this chapter we briefly explain the mathematical formulation for reactive transport.For a more detailed explanation we refer to Sao/rink et at. [1998]. There are two typesof equations that one has to solve: equations that express the chemical reactions andthose that express mass balances and transport processes such as advection,dispersion and diffusion.

2.1 Chemical Reactions

If a system is in chemical equilibrium one can apply the mass action law that relatesthe concentration of reactant and products of a chemical reac tion. This can be writtenfor the whole system in the following form:

S.(log e+ logy(e)) =logk (1)

where S, is a N, x N, matrix (N; being the number of reactions and Ns the number ofchemical species) containing the stoichiometric constants of the reactions (i.e., thenumber of moles supplied/consumed in each reaction s), c is a vector of theconcentrations of all chemical species, k is a vector of equ ilibr ium constants and 'Y avector of activity coefficients which are a function of all concentratio ns. A specialcase are the mineral s. One normally assumes that their activity (the product of theactivity coefficient and concentration) always equals one.

The mass action law only applies at equilibrium. In other cases, slow chemica lreactions are characterized by the reaction rate (rk), which is defined as the amount ofreactants evolving to products of a chemical reaction per unit time. It depends on theconcentrations of specie s involved in the reaction but it may also depend on theconcentration of catalysts, on the reactive surface (e.g., for precipitation/dissolution ofminerals), on the amount of bacteria (for biological reactions), etc. In this work, wesimply state that the reaction rate is a function of all concentrations:

2.2 Transport Equations

The basic equations for reacti ve transport can be written as follows:

de I Iat=ML(e) +Sere+Skrk(e)

(2)

(3)

where M is a diagonal matrix that specifies whether a species is immobile or not, Sk isthe stoichiometric matri x for kinetic reactions, r , is the vector of reaction rates forequilibrium reactions and L is a linear operator for the convec tion, dispersion andprescribed sink/source terms. Notice that equation 3 simply expresses the contributionto the change in the concentration of all species (ocld t) caused by transport processes

102 M. W . Saaltink and J . Carrera

(ML(c)) , all equilibrium reactions (Se're) and all slow reactions (Sk'rk). It is worthmentioning also that, while rk can be written explicitly as a function of concentrations(equation 2), equilibrium reaction rates, re, cannot. They can only be expressedimplicitly by coupling equations 1 and 3. For the sake of simplicity, we will consideronly transport in a single, aqueous phase. Then, L(e) is given by:

(4),

where q is the water flux, rf> the volumetric water content, D the dispersion tensor andm represent s sources and sinks.

As there are N, concentrations per node, there are also N, transport equations pernode. We can reduce the number of transport equations by eliminating reoIn order todo so, one should recall that Se is a full-ranked N, x N, matrix. Full rank is assuredbecause all equilibrium reactions must be independent. Therefore, it is possible toobtain a full ranked (N, - Nt) x N, kernel matrix (U) such that:

(5)

Multiplying equation 4 by U allows us to eliminate the equilibrium reaction ratesterm re:

(6)

Since the dimensions of U are (N, - Nt) x Nt> the number of transport equations pernode reduces from N, in equation 4 to N, - N, in equation 6. We will call U thecomponent matrix, because it adds up the total amount of a component, distributedover the various chemical species . Components are defined in such a way, that everyspecies can be uniquely represented as a combination of one or more components[Yeh and Tripathi ,1989] . In addition , equation 5 ensures that the global mass of acomponent is independent of equilibrium reactions [Rubin, 1983]. In a closed systemthe global mass only depends on kinetic reactions, whereas in an open system theglobal mass depends on mass fluxes as well.

Due to the assumption that the activity of minerals equals one, we can alsoeliminate the concentrations of these species . To do this, we write equation 6 in thefollowing form:

(7)

where vector c and matrices U have been split up into parts referring to the aqueousand therefore mobile species (with subscript a), sorbed and therefore immobilespecies (with subscript s) and minerals (with subscript m):

Simul ation of reactive transport in groundwater 103

(8)

(9)

In the same way as for the elimination of equilibrium reaction rates, we elimin ate .Umckn/at by multiplying equation 7 by an elimination matrix E defined in such a waythat:

dC mEU =O=::}EU -=0m m dt

Then , equation 7 become s:

dCa dC s tEUa-at+ EUsi)t= EUaL(ca)+EUSkrk(ca)

(10)

(11)

Multiplying by E reduces the number of transport equations from N, - N; inequation 7 to N, - N, - Nm in equat ion II (Nm being the number of mineral s inequilibrium).

3 Numerical Approaches

3.1 Sequential Iteration Approach (SIA)

We use equation 7 but written in a slightly different form :

dUa dUs dUm t----atrs:+----at = L(u a ) + USkrk(ua)(12)

where Ua , Us and Urn are vectors containing the total concentrations of the componentin aqueous, sorbed and mineral form respectively. They are defined as:

(13)

(14)

(15)

The SIA consists of first solving the transport equations 12 with the total aqueousconcentrations of every chemic al component (vector ua) as unknowns. It treats the

104 M. W. Saaltink and J . Carrera

concentrations of sorbed species, minerals and kinetic reactions as source-sink term(vector 1), computed by the previous iteration :

dUi. . I

_a = L(u' )+f'-dt a

(16)

where the superscript i refers to the iteration number. Note that 16 has the same formas the transport equation without chemical reactions . We used finite elements forspatial and finite differences for the temporal discretization. This leads to linearequations that one can solve for every component separately. We used LUdecomposition of a banded matrix to solve these linear equations. As the matrix ofthis system only changes if the time increment changes, one can take advantage of thedecomposed matrix to solve the systems of all components and even of previous timesteps as long as the time increment is not changed .

In the second step one updates the source-sink terms. As there is no explicitexpression for f as a function of U a, one has to calculate first the concentrations (c)from the total aqueous concentrations (ua) by means of the chemical equations:

(17)

One has to solve these equations together with equations 14 to 16 and those forchemical equilibrium (equation I). One usually substitutes the chemical equationsinto 17 with the exception of the equations for reactions that involve minerals. Thisleads to N, - N, + Nm number of equations. Because these equations are non linear, weapplied a Newton-Raphson scheme for its solution (not to be confused with theNewton -Raphson applied to the DSA to solve the whole set of equations). One can dothis for every node separately. From c one calculates new source-sink terms:

(18)

This term can now be substituted in (16) for the next iteration and the whole processis repeated until convergence.

3.2 Direct Substitution Approach (DSA)

For the DSA we substitute all chemical equations (1) into the transport equation (11)and apply Newton-Raphson as follows. We define a vector of transport equations (g)and of chemical equations (h) :

dCa dCs t (19)g = EUa-at+EUs ---:;;r-EUaL(ca)-EUSkrk(Ca) = 0

h = S, (logc+ logy(C)) -logk = 0 (20) -

Simulation of reactive transport in groundwater 105

We also define a vector of N, - N; - Nm concentrations (CI) which we will link to thetransport equations and another of N, concentrations (cz) linked to the chemicalequations. Then the Newton-Raphson scheme becomes:

dg ( , I ' ) dg ( , I ' ) ,-- e+-c' +-- e+ -c' =-g'dC I 1 dC z ZI Z

db ( , I ' ) db ( , I ' ) ,-- e+ -C' +-- c'+ -C' =-b'dC I 1 dC z ZI Z

(21)

(22)

where hi and gi stand for htc') and g(c\ respectively. If we ensure concentrations Cz

to be in equilibrium with c), chemical equations (h) equal zero. To fulfill thiscondition we applied the following Newton-Raphson scheme to calculateconcentrations Cz for given concentrations CI :

db ( , I ,) ,-- cJ+ -cJ =-bJ

dC z zz

We substitute 22 into 21 with chemical equations (h) being zero:

(dg + dg dC z ) ( i+l i) i- -- C -C --gdC

IdC z dC

tI i r:

where oczle)cl is defined by:

db dCz _ dbdC z dCt dCt

(23)

(24)

(25)

The approach consist of first calculating Cz by means of equation 23. Then wecalculate oCz/OCt from 25. Note, that equation 23 and 25 have the same jacobianmatrix (oh/ocz) and that they are local, that is, they represent equilibrium conditionsat every point, so that they can be solved for every node separately . Moreover, as fornormal situations in groundwater y generally varies only slightly with concentrations,we can write the chemical equations (h) almost as an explicit function of Ct (cz =f(ct» [Saaltink et at., 1998], which leads a jacobian matrix equal to the unity matrix ,Therefore, calculation of 23 and 25 is not extraordinarily costly . After calculation ofCz and OCt/ocz we calculate a new value of CI by means of equation 24, This equationhas N, - N, - Nm unknowns that must be solved for all nodes simultaneously,

As for the SIA, we used finite elements for spatial and finite difference for thetemporal discretization.

3.3 Time increment control

Both the SIA and DSA can fail to converge if one uses a too large time increment. Onthe other hand, CPU times can become unnecessarily large if one uses a too small

106 M. W . Saaltink and J. Carrera

time increment. As we conjecture, that the SIA requires smaller time increments thanthe DSA, it is important to work with the optimal time increments for bothapproaches . However, it is difficult to estimate a priori an optimal time increment.Therefore, we developed an algorithm that changes automatically the time incrementduring the simulation, according to the following scheme:

IF (failed to converge) THENDecrease time step DT by a factor FD

ELSEIF (number of iter. < min . threshold THRMIN) THEN

Increase DT by a factor FIIF (DT > maximum time increment DTMAX) THEN

DT = DTMAXENDIF

ELSEIF (number of iter. > max. threshold THRMAX) THENdecrease DT by factor FD

ELSEmaintain DT

ENDIFENDIFDo next time step

If convergence cannot not be reached, it repeats the calculations with a smaller timestep. In case of a successful convergence, it reduces the time increment for the nexttime step, if the number of iterations is large, whereas the time increment is increased ,if this number is small.

4 Case descriptions

Table I shows a summary of the cases used for the comparison. The first set ofexamples (CAL) is the most simple one. It treats the dissolution of calcite in a onedimensional domain. Initially the water is saturated with calcite. Infiltrating water,that is subsaturated to calcite, dissolves the calcite . This case consists of severalsubcases: one assuming equilibrium dissolution of calcite (CAL-E) and four withkinetic dissolution , with various calcite dissolution rates (CAL-l to CAL-4, the firsthaving the slowest rate and the last the fastest) . We also calculated a case withoutcalcite (CAL-O).

The next set (WAD) contains cases of the flushing of saline water by fresh water inthe Waddenzee (the Netherlands) in a one dimensional domain described by Appeloand Postma [1994]. They include dissociation of water, carbonate reactions, cationexchange and dissolution of calcite . Likewise the calcite dissolution cases, wecalculated cases of equilibrium, kinetic and no calcite dissolution.

The third set (DEDO) simulates the replacement of dolomite with calcite, which isdriven by the infiltration of Ca rich water, called dedolomitization [Ayora et al.,1998]. We used a two dimensional domain, which includes a fracture with a 100times higher water velocity than in the surrounding rock. Note that a high number of

Simulation of react ive transport in groundwater 107

pore volumes are flushed for this case. The number of flushed pore volumes is thevolume of water that enters the domain during the simulated time divided by thevolume of water in the domain.

The last case (OSA) is chemically the most complex one. It models the depositionof uranium resulting from infiltration of oxygenated, uranium bearing groundwaterthrough a hydrothermally altered phonolitical host rock at the Osamu Utsumi uraniummine, POl;OSde Caldas, Brazil [Lichtner and Waber, 1992]. As for the DEDO cases , ahigh number of pore volumes are flushed.

Table 1. Characteristics of the casesused for comparison.

Case name No. of No. of No. of No. of No. of No. of Flushednodes primary second. adorbed minerals minera ls pore

species species species (eguil.) (kin.) volumesCAL-O 21 3 5 1.0CAL- l 21 3 5 1.0CAL-2 21 3 5 1.0CAL-3 21 3 5 1.0CAL-4 21 3 5 1.0CAL-E 21 3 5 1.0WAD-O 21 6 3 3 37.5WAD- l 21 6 3 3 37.5WAD -2 21 6 3 3 37.5WAD-3 21 6 3 3 37.5WAD-4 21 6 3 3 37.5WAD-E 21 6 3 3 1 37.5DEDO-E 15x 15 7 8 2 - 22704.6DEDO-K 15x l5 7 8 2 22704.6OSA 101 13 29 8 - 80000 .0

5 Comparison

We measured the number of required time steps, the number of required iterationsand the CPU times are shown in figures 1 through 3, respectively, for the casesdescribed above. The numbers of time steps and iterations may depend on theparameters that control the time increment and converge nce criteria and the CPU timeon the programming style. So one should interpret these figures with care.Nevertheless, one can observe some clear differences between the two approac hes.For the cases with a small number of flushed pore volumes (CAL and WAD), the SIAand the DSA behave similarly, when the mineral is in equilibrium or absent. Thenumber of time steps and iterations and consequently the CPU time of the SIA risewith higher dissolution rates, when one assumes a kinetic dissolution. On the otherhand, dissolution rates do not seem to have much influence on the numericalbehaviour of the DSA. A bigger kinetic rate makes the non-linear source-sink term f

108 M. W. Saaltink and J. Carrera

in equation 10 to be more important, causing more numerical problems for the SIA.The DSA does not show these problems thanks to its robustness.

For the cases with a high number of t1ushed pore volumes (DEDO, OSA) the SIArequires really excessive number of time steps, iteration and CPU time (centuries)also for cases that assume equilibrium dissolution-precipitation (in fact, we haveestimated these figures by extrapolating from runs that took about one day). The highnumber of f1ushed pore volumes makes that fulfilling the Courant eondition wouldlead to a very high number of time step. This condition states that the solute cannotgo over an element during a simple time step:

(26)

where ,ix and M are the element size and the time increment. It seems that theCourant condition is important for the SIA, whereas it is not for the DSA.

'" 10Q,<l)...'" 8<l)

.s...6'-

0

'"'<l) 4,.Q

S:: 2;::OJ)0

0....la :'; ') J 7 N ~ 7 "' ~ ;:J s~ '-' ~ ;i ~ Co Co Co Co ~< -c < -c < -c «: -c < 0 6 0u u u u u u

'" '" '" '" '" '" 0 0

"' cc0 0

Case

Fig. 1. Number of time steps for all cases. The values for the DEDO and OSA cases calculatedby the SIA have been derived by extrapolating those for a small number of time steps. Note thelog scale.

Simulation of reactive transport in groundwater 109

2

o ,:3-cu

Case

;;jo

Fig. 2. Number of iterations for all cases. The values for the DEDO and OSA cases calculatedby the SIA have been derived by extrapolating those for a small number of time steps. Note thelog scale.

10,.-.,

'" .OSA-e 8=0 SIA"'" 6(J)

'"'-'~ 4~

UeJ) 20

....l

00 "i -r '7 0 N ~ ., rc ;;i:;: J

"""'J

"""' :;;! '" '" '" ~ ;;l '"-c ~ -c < ~ ~ ~ -c 0u u u u u u " " " " " " '""'cCase

Fig. 3. CPU times for all cases. The values for the DEDO and OSA cases calculated by the SIAhave been derived by extrapolating those for a small number of time steps. Note the log scale.

110 M. W . Saaltink and J . Carrera

6 Conclusions and Discussion

The results show that indeed the SIA requires generally more iterations than the DSA.The SIA particularly gives problems for two types of cases: cases with high kineticrates and cases with a high number of flushed pore volumes. The DSA does not showthese problems thanks to its robustness. However, all presented cases have grids of asmall number of nodes. For 2 and 3 dimensional grids with large number of nodes thesolution of the linear system may give more problems for the DSA than for the SIA.More research is required on this issue. At the very least, one can conclude that thechoice between the two methods is not so strongly in favor of the SIA as stated byYeh and Tripathi [1989] .

Acknowledgments

This work was funded by ENRESA (Spanish Nuclear Waste Management Company).

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