Hydrodynamic simulation with mike21 of Mele Bay and Port

HYDRODYNAMIC SIMULATION WITHMIKE21 OF MELE BAY AND

PORT VILA, VANUATU

Robert KleinSOPAC Secretariat

SOPAC Technical Report 263

August 1998

[2]

[SOPAC Technical Report 263 – Klein]

ACKNOWLEDGEMENTS

The French Embassy for the funds that enabled me to have a placement at SOPAC.

The Director of SOPAC for granting the permission for my placement at the SOPAC Secretariat.

The SOPAC Staff for their support, especially Robert Smith my supervisor, thank you verymuch.

The National Tidal Facility of The Flinders University of South Australia is acknowledged for theprovision of tidal data.

[3]


TABLE OF CONTENTS

ACKNOWLEDGEMENTS ........................................................................................................ 2

SUMMARY............................................................................................................................... 5

INTRODUCTION...................................................................................................................... 6

1. GENERALITIES ................................................................................................................ 7

Port Vila, Vanuatu ............................................................................................................. 7

Hazards............................................................................................................................. 7

OBJECTIVES OF THE STUDY......................................................................................... 9

THEORETICAL MODEL ................................................................................................. 11

Mike 21 Package............................................................................................................. 11

Numerical Modelling........................................................................................................ 13

Shallow Water Equations ................................................................................................ 14

Numerical Formulation .................................................................................................... 20

2. SET UP OF THE BATHYMETRY MODEL OF PORT VILA-MELE BAY .......................... 22

Brief report of the bathymetric surveys ............................................................................ 22

Digitization of the maps under AutoCAD ......................................................................... 23

Geographic Transformation............................................................................................. 23

Correction of the bathymetric model................................................................................ 24

Set up of a 100-meter interpolation grid........................................................................... 24

3. SET UP OF THE HYDRODYNAMIC MODEL FOR PORT VILA-MELE BAY................... 26

The bathymetry ............................................................................................................... 26

Selecting the model area................................................................................................. 28

Grid orientation................................................................................................................ 28

Model parameters ........................................................................................................... 29

Boundary conditions........................................................................................................ 30

Model calibration ............................................................................................................. 35

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4. RESULTS AND VALIDATION OF THE MODEL OF PORT VILA-MELE BAY.................. 36

Simulations using the Admiralty Chart data ..................................................................... 36

Simulations of Mele Bay and Port Vila run with the measurements ................................. 42

5. MODEL OF PORT VILA.................................................................................................. 49

Set up of the bathymetry model with AutoCAD................................................................ 49

Set up of the hydrodynamic model under Mike 21........................................................... 51

Advection-dispersion modelling....................................................................................... 57

RECOMMENDATIONS.......................................................................................................... 59

CONCLUSION ....................................................................................................................... 59

REFERENCES ...................................................................................................................... 60

ANNEX

Introduction to SOPAC .................................................................................................... 61

[5]


SUMMARY

In recent years with the development of powerful computers, numerical models are increasinglybeing used to simulate the processes of nature. Mike 21 is but one example of professionalmodelling software for 2D free surface flows and comes in modular form with four mainapplication areas, coastal hydraulics and oceanography, waves, sediment processes andenvironmental hydraulics.

As harbours and lagoons are all subject to some, or all, pollution, wave action, storm surge,seiching, tsunami, erosion and sedimentation, and sea-level rise studies linked to coastalmanagement are often limited by the data sets available, seasonal variations and cost.Numerical modelling provides an opportunity to view and analyse coastal problems and riskswith minimal penalties for error as we are able to change the input parameters in the model andobserve the response, a valuable symbiosis between development and application.

Port Vila Harbour is but one example of many in the region that suffers from a number ofsources of pollution, is vulnerable to storm surges and tsunamis. In particular, seiching wasnoted in the harbour during one earthquake event. Coastal areas are under pressure fromdevelopment and today’s decision maker requires all the help he/she can get if developmentand preservation of the environment are to be sustainable. The development of the hydraulicmodels for Mele Bay and Port Vila provides the foundation from which quantitative predictivemodels for many environmental issues can be addressed in particular the water quality of theharbour.

Two hydrodynamic models were developed, one model encompassing both Mele Bay and PortVila with a grid resolution of 100 m and a smaller but more detailed model of Port Vila with agrid resolution of 15 m. Good results were obtained from both models for the comparison of themodel water elevation with the National Tidal Facility recorded data. Although no calibrationdata was available for currents the periodicity in the model current velocities and directioncorrelate well with tide providing reasonable if not good evidence for the models correctsimulation of the hydraulic behaviour of Mele Bay and Port Vila.

Animated simulations produced from advection and dispersion modelling and based on thehydrodynamic model clearly demonstrate the potential impacts that can be expected from aninner harbour oil spill or with city growth the impacts of pollutants entering the harbour viadrainage and storm water runoff. With an economy dependent on tourism, protecting theenvironment can only but contribute towards and promote positive growth.

[6]


INTRODUCTION

SOPAC (South Pacific Applied Geoscience Commission), based in Suva (Fiji Islands), is in theearly stages of a project which is aimed at predicting the effects of natural disasters and thedegree of risk to life and property on Port Vila and Mele Bay (Vanuatu). The hazards that will beconsidered range through earthquake, tsunami, cyclone, storm surge flooding, sea level rise,foundation problems and harbour pollution. The development of numerical models is a tool inwhich to assess the likely hazards. These predictions are essential to the efficiency of post-disaster impacts and operations and, in the longer term, for the assessment of sustainableurban planning of the city and surrounds.

The work carried out at the Coastal Unit and presented in this report has been mainly focusedon the development of a hydrodynamic model for Mele Bay and Port Vila (Vanuatu).

Developed by the DHI (Danish Institute of Technology), Mike 21 is two-dimensional depthaveraged hydrodynamic software. To perform all the different models like a water quality, asedimentary transport or a tsunami model for instance, it is essential to first set up the basichydrodynamic model where all the others will be based on. Then two hydrodynamic modelshave been set up; one for the whole area of both Port Vila and Mele Bay and the other is asmaller one containing Port Vila only.

Following a brief summary of the area studied here and the objectives of this project, theassumptions used by Mike 21 and basic equations that form the basis of numerical modellingare described. In all hydrodynamics situations, geographical configuration and bathymetry arethe principal controlling factors that determine the hydraulics.

Compilation of the bathymetry model was based on two bathymetric surveys 1990, and 1997with the model compiled in AutoCAD 14. Boundary conditions are based on National TidalFacility (Flinders University of South Australia) tide data. Further calibration data is required tovalidate the model.

The parameters specified in Mike 21 to run the simulations are described as well as theparameters for the calibrations. The numerical results obtained and the measurements arecompared for the validation of both models.

[7]


1. GENERALITIES

Port Vila, Vanuatu

Vanuatu, formerly the New Hebrides, is a archipelago of about 80 islands. The total landarea is reported to be 11 880 km² of which 75 percent comprises the four main islands ofEspiritu Santo, Malekula, Pentecost and Efate. In mid-1991 the population was estimated at150864. The population of Port Vila, 17°45’S-168°19’E, the administrative capital, growingat a faster rate than the rest of the country, is about 21000. The main island Efatecontaining the capital is about 2250 km northeast of Sydney and about 800 km from Suva,Fiji Islands.

The islands have a tropical climate with warm humid period from November to April and acooler drier period under the influence of SE trade winds between May and October. Theaverage maximum and minimum temperatures are 29°C and 22,5°C in summer, and 26°Cand 20°C in winter with extremes of 12,6°C and 33,3°C.

The thirty year mean annual rainfall at Port Vila was 2366mm with a maximum in March(381) and a minimum in October (94mm), and the rainfall is generally in excess of theevaporation.

Principal industry, is canning and copra oil production, centred on the larger Island of Efate.Agricultural livestock, copra, fisheries, and tourism remain the largest sector of theeconomy. Copra, tourism, and fishing are the principal export earnings respectively.

Tourism commenced around 1972 and has been described by Vanuatu’s Reserve Bank as“the locomotive of the economy”. By late 1993 an additional 11 hotels in the greater PortVila area, offering some 536 rooms and 1220 beds have since become established with anadditional 100 beds elsewhere in the country.

Hazards

Port Vila now suffers from pollution from a number of sources, and remains vulnerable tostorm surge seiching and tsunamis and cyclones. Mele Bay is fronted by a high-energybeach system that has been subjected to erosion, which can be attributed to both humanand natural causes. Bathymetry and high resolution seismic offshore surveys haveidentified areas in Mele Bay where slumping has occurred (Ref [8]). Tourism is a majorindustry, with Mele Bay and Port Vila having significant tourist amenities. Their attraction isdependent on maintaining a pristine and healthy environment. Impacts from developinginfrastructure, the siting of sewerage outfall, city storm drainage, are important issues to beconsidered in preserving this environment.

[8]


Earthquakes and Tsunamis

Like volcanic activity, earthquakes in the region occur largely in the countries which liewithin “The Pacific Rim of Fire”, associated with the plate boundary zone between thePacific and Tonga.

Damaging large earthquakes are not uncommon (Ref. [6]). Fortunately a large part of theseismically active area of the region is rural and therefore effects are often minimal.

However earthquakes often trigger other events, like tsunamis, evident in historical recordsof Port Vila. An earthquake on January 24th 1927 with a recorded magnitude greater than7.1 centered in south Malekula produced the worst tsunami. In 1985 an interview with ChiefGraham Kalsakou of Ifira Island clearly indicated that the seiche effect produced by thetsunami entering the almost enclosed harbour resulted in flooding of the coastline to severalmeters above normal tide. A more recent example is the tsunami that occurred in July 1998in Papua New Guinea (same rim as Vanuatu) where 3000 people died.

An advantage of a tsunami model is developing scenarios from which predictions might bemade of the areas most likely to be affected, so that local authorities can use theinformation in their building codes, or to make decisions about relocating highly vulnerablecommunities to safer locations.

Erosion

Mele Bay is a large embayment in the southern of Efate (Figure 1) just adjacent to Port Vila.The coastal plain at the head of Mele Bay is an extensive coastal alluvial fan bounded bytwo main headlands, Devil’s Point to the west and Pongo point to the southwest.

Coastal erosion has been a recurring problem at Mele Bay, and there is evidence tosuggest that has been linked to the unregulated mining of beach sand at two sites in thebay at an average rate of 450-500m≥ per month (Ref. [9]).

Following the survey done by Robert Smith in 1990 (Ref [9]), the slope of the beach isconsistent for the entire bay, but near the beach mining sites, the beach is significantlylower and narrower than at other parts of the bay, reflecting the damage to the beach byremoval of large quantities of sand. Coastal erosion during major storms increased, ashappened in 1986, because the buffering effect of a wider beach is reduced. There is nofringing reef to protect the bay.

Seismic data show a variety of seafloor landforms which indicate significant instability ofbottom sediments on the shelf slopes. Factors contributing to this instability are very narrowshelf, steep shelf-slopes gradients, active seismicity which generation of large quantities ofunstable sediment offshore by tropical cyclones. One such result event is a slide on theshelf slopes off LaColle River delta interpreted from seismic data is estimated to haveresulted in between 90 000 and 180 000 m≥ of sediment moved in one. Such sedimentslumping may generate a localized tsunami affecting the Mele Bay plain.

Studies of the coastal zone, show that the sediment input from the rivers is unable to keeppace with the total amount of sand being removed by beach mining or by removal ofsediment offshore and natural process occurring.

[9]


Based in the hydrodynamic model, a sediment transport model can be developed fromwhich accretion and erosion rates may be determined as well as the areas where themining can be done.

Water quality

There has been little work done on the water quality of Port Vila, and the last water qualitysurveys done show an increase of pollution levels in the coastal waters around Port Vila(Ref.[2]). The principal results are:

• The ship discharge within Port Vila Harbour reduces water quality there.• The water quality within the harbour tends to deteriorate with time due in a major part to

the human activity within the area.

Now, tourism both Mele Bay and Port Vila have significant tourist amenities, which aredependent on maintaining a pristine and healthy environment. Then, the development of awater quality model is essential to assess the circulation of pollutants and likely impacts onthe environment.

OBJECTIVES OF THE STUDY

The first task in order to perform all these study, that will be done when all the essentialdata will be available, is the set up of a hydrodynamic model, and it is mainly this workwhich is presented in this report.

So that this report may be useful for any future Mike 21 user, the parameters specified forthe simulations are detailed as well as the problems that have occurred during this work.

Then two hydrodynamic model have been set up:

• A model of Port Vila and Mele Bay together, with a 100-meter grid

• A model of Port Vila only, with a 15-meter grid

The process is more detailed for the first model as we may see in the following twochapters, whereas in the one of Port Vila, we present mainly the results.

At the end of the last chapter, an example of a water quality model is shown to give an ideaof the circulation of pollutants.

[10]


[11]


THEORETICAL MODEL

Sedimentology, long-term morphodynamics, water quality and contaminant dispersion, aresome of the many topics relying on a clear understanding of hydrodynamics. The latterdiscipline is thus the corner stone of many studies involving floodplain processes. To predictor explain the flow patterns in a river, to design bridges and dykes, engineers have longbeen doomed to resort to empirical or simplified formulas, or to build scale models. Despitethe basic flaw of similitude, scale models have been widely used up to now. Their rangeapplications still very important, though shrinking every day. Scale models are nowgradually replaced by numerical simulation, but will still thrive for a long time, at least forvery complex problems like the assessment of dyke stability, or simply for validatingnumerical models.

Mike 21 package

Among the numerous Hydrodynamic Modeling Software available today in the market, Mike21, developed by DHI (Danish Hydrodynamic Institute), is a modelling system for 2-dimentional, free-surface flows, and is used by the Coastal Section at SOPAC.

Application areas

Mike 21 can be applied to a wide range of hydraulic and related phenomena. These can bedivided into four main application areas:

• Coastal hydraulics and Oceanography. This includes modeling of tidal hydraulics, windand wave generated currents, storm surges and flood waves.

• Environmental Hydraulics. This encompasses everything from normal advection-dispersion simulation of conservative pollutants to complex water quality simulationsincluding chemical reactions.

This means that we can investigate the impact on the marine environment from varioussources such as sewage, storm water and cooling water outfalls. The environmentparameters which could be studied are bacterial concentrations, eutrophication, algalblooms, BOD-OD (bacteriological oxygen demand – dissolved oxygen) and others.Heavy metal dispersion and its influence on marine flora and fauna can also beinvestigated.

• Waves. This covers wave agitation in harbours, harbour seiche, hindcast and forecast,design wave parameters, non-linear transformation and ship motions.

We can therefore use Mike 21 as a tool in the design of harbours and offshorestructures. It can be used to test the effects of new breakwater alignments, navigationchannels, wharf areas etc., and to produce design wave parameters for offshoreinstallations.

• Sediment Processes on coast, in estuaries and rivers. These include sediment transportinvestigations of navigation channels, harbour entrances, coasts, river ports, etc.

[12]


The computational Modules

The 13 computational modules of which Mike 21 consists have the following properties:

Hydrodynamic Module• Full non-linear equations of continuity and conservation of momentum• ADI finite difference solution of second-order accuracy• Smagorinsky eddy formulation

Advection-dispersion module• Conservation of mass equation• Transport of conservative, heat dissipating and linear decaying matter• Third-order explicit finite difference solution

Water quality module• BOD-DO, oxygen depletion and bacterial decay balances• Organic nitrogen, ammonia and nitrate balances

Eutrophication module• Phytoplancton, benthic algae, zooplancton, oxygen balances and mineralisation

estimates• Dependence on nutrient availability, light and temperature

Heavy Metals Module• Combined physical, chemical and biological processes• Uptake in organisms

Mud Transport module• Cohesive sediment erosion, transport and deposition

Sand transport• Non-cohesive sediment transport using Engelund-Fredsoe’s formulation under the

action of currents and waves, both breaking and non-breaking

Particle Module• Transport and fate of solutes or suspended matter• Pollutants are considered as particles• Lagrangian approach

Boussinesq Wave Module• Boussinesq equations with improved linear dispersion characteristics• Irregular, directional waves from deep to shallow water• Non-linear wave-wave interactions• Diffraction, partial reflection

Elliptic Mild-Slope Wave Module• Linear refraction-diffraction in sheltered areas• Harbour resonance and seiching• Radiation stress include diffraction and reflection• Monochromatic waves, wave breaking, reflection

Parabolic Mild-Slope Wave Module• Linear refraction-diffraction in larger coastal areas

[13]


• No diffraction behind obstacles and no reflection• Superposition over frequency and direction

Nearshore Spectral Wind-Wave module• Discrete directions, parametric in frequency• Stationary wind fields• Wave breaking, wave-current interaction, radiation stresses• No diffraction, no reflection

Offshore Spectral Wind-Wave module• Discrete directions and frequency• Time varying wind fields• Second generation wind-wave model• Parametric wave-wave interaction

All these modules require at least the Hydrodynamic Module to be run. For this reason theset up of the hydrodynamic model is by far the most important task for any study. Therefore,the result of an advection-dispersion model or a sediment transport model, for instant, arereliant on the quality of the hydrodynamic model.

In the study presented here, the body of the work was to set up two hydrodynamic models,one for Mele Bay and Port Vila together and one for Port Vila only, in order to assess waterquality and erosion of the beaches for these areas. A number of simulations for a waterquality model will also be run to have an idea of the circulation of the pollutants within thebay and harbour.

Therefore, the scientific background we will discuss here concerns only the hydrodynamicand advection-dispersion part.

Numerical Modelling

Unlike other related disciplines, such as sedimentology and water quality, the basicequations are well known in hydrodynamics. The only and difficult problem is to solve them.All equations in Hydrodynamics stem from the three-dimensional Navier-Stokes equationsthat read:

Continuity

Momentum

)1(0=∂∂+

∂∂+

∂∂

zw

yv

xu

)4(1

)3(1

)2(1

2

2

2

2

2

22

2

2

2

2

2

22

2

2

2

2

2

22

gzw

yw

xw

yp

zw

yvw

xuw

tw

zv

yv

xv

yp

zvw

yv

xuv

tv

zu

yu

xu

xp

zuw

yuv

xu

tu

−

∂∂+

∂∂+

∂∂+

∂∂−=

∂∂+

∂∂+

∂∂+

∂∂

∂∂+

∂∂+

∂∂+

∂∂−=

∂∂+

∂∂+

∂∂+

∂∂

∂∂+

∂∂+

∂∂+

∂∂−=

∂∂+

∂∂+

∂∂+

∂∂

υρ

υρ

υρ

[14]


Boundary conditions:

Free surface condition

Bottom conditionu=v=w=0 pour z=-h (6)

Where x, y, z denote the space coordinates, u, v, w represent the three components of thevelocity, ρ is the density and p is the pressure.

The continuity equation expresses the mass conservation, while the momentum equation isactually the fundamental law of dynamics, written for fluids. The only assumption in theseequations is that the fluid should be Newtonian and this is indeed an excellentapproximation for water. The main difficulty of the Navier-Stokes equations stems in thenon-linear terms that challenge the numerical algorithms and are also responsible for theflow turbulence. As a matter of fact, it is important to note here that turbulence, with all itscomplexity, is contained in the Navier Stokes equations.

For the time being, no industrial tool for directly solving the 3D Navier-Stokes equations forfree surface flows is available. One of the greater difficulties is the free surface itself, whichcauses the computational domain to vary in time. Many kinds of simplifications have beenproposed, the most popular being the Shallow Water equations given by Barre de SaintVenant one century ago, and Mike21 is based on it.

Shallow Water Equations

The 2D Shallow Water equations are obtained by means of averaging of the 3D Navier-Stokes Equations over the depth. The new variables obtained are mean values over thedepth. Let’s call:

Where ζ is the free surface and h is the bottom elevation. Solving the equations will consistof finding the values of U, V and h everywhere in a domain, during a given lapse of time, asfunctions of initial conditions and the boundary conditions.

Full derivation of the equations

The derivation of the shallow water equations is mainly based on the Leibnitz rule, thatreads:

)5(),,(for zyxzyx

ut

w ζζζτ =∂∂+

∂∂+

∂∂=

∫ ∫− −

==ζ ζ

h h

vdzzyxVudztyxU )7(),,(),,(

)8()()( 11

22

2

1

2

1xz

zuxz

zudxxuudz

t

z

z

z

z ∂∂−

∂∂+

∂∂=

∂∂ ∫∫

[15]


• Integration of the continuity equation:

Following the bottom and the free surface conditions, it remains:

Taking the average we finally obtain:

• Integration of the vertical movement equation:

in order to obtain the pressure in z, the following is:

From the Leibnitz rule, we obtain the expressions:

),,,(~),,,(t)z,y,(x,:~ termfluctuant a and termaveragean with divided becan function Any

tzyxtzyx ϕϕϕϕϕ

+=

)11(0)()()()()()(

)10(0

)9(0

=−−+∂∂−−

∂∂−

∂∂+

∂∂−−

∂∂−

∂∂

=∂∂+

∂∂+

∂∂

=∂∂+

∂∂+

∂∂

∫ ∫

∫∫

− −

−−

hwwyhhv

yuvdz

xxhhu

xuudz

x

dzzwdz

yv

xu

zw

yv

xu

h h

hh

ζζζζζζ ζ

ζζ

)12(0=∂∂+

∂∂+

∂∂ ∫∫

−−

ζζζhh

vdzy

udzxt

)13(0=∂∂+

∂∂+

∂∂

tV

tU

tζ

)14(1

2

2

2

2

2

22

∫∫∫∫ −

∂∂+

∂∂+

∂∂+

∂∂−=

∂∂+

∂∂+

∂∂+

∂∂ ζζζζ

υρ zzzz

gdzdzzw

yw

xw

dzyp

dzz

wy

vwx

uwtw

∫ ∫−∂

∂=∂∂ζ ζ

z h

wdzt

dztw )15(

[16]


The free surface condition shows that:

Moreover, the pressure at the free surface is taken to be equal to zero, i.e. p(z=ζ )=0.Summing up all the terms, we obtain:

)20()(

)19()()(

)18()()(

)17()()(

)16()()(

222

2

∫

∫

∫

∫ ∫

∫ ∫

−=

−=∂∂

−=∂∂

∂∂−

∂∂=

∂∂

∂∂−

∂∂=

∂∂

ζ

ζ

ζ

ζ ζ

ζ ζ

ζ

ζ

ζ

ζζζ

ζζζ

z

z

z

z z

z z

zggdz

zppdzzp

zwwdzxw

ywvvwdz

ydz

yvw

xwuuwdz

xdz

xuw

)21()()()()()(2 ζζζζζζζζ wy

vx

uwt

zw

∂∂+

∂∂+

∂∂==

)22()()(1

)(2 ∫∫∫∫ ∆++−=−∂∂+

∂∂+

∂∂ ζζζζ

ζρ zzzz

wdzvhgzpzwvwdzy

uwdzx

wdzt

)23(0=∂∂=

∂∂ ∫∫

ζζ

zz

dzwt

wdzt

flow. thefacing are when theyavoided be should slopes steep that is econsequenc overlookedoften an :0w that assume then wesmall,

remains velocity vertical that theassumption themaking then equation; above theofmean theTaking=

)28()()(

)27()()(

)26(~~~)~(

)25(~~~)~(

)24(0

zgzg

zPzp

dzwvy

dzwvvy

vwdzy

dzwux

dzwuux

uwdzx

dzwvwdzv

zzz

zzz

zz

−=−

=

∂∂=+

∂∂=

∂∂

∂∂=+

∂∂=

∂∂

=∆=∆

∫∫∫

∫∫∫

∫∫

ζζρρ

ζζζ

ζζζ

ζζ

[17]


This gives the following result:

• Integration of the horizontal momentum equations:

The derivation is detailed term by term for the first momentum equation:

When summing all these terms, it results in the following term that is equal to zero becauseof the free surface condition:

The pressure at the free surface and the speed at the bottom are taken as zero. It remains:

)29()(~~~~)()( 2 zwdzwv

ydzwu

xzg

zP

zz

−∂∂+

∂∂+−= ∫∫

ζζ

ζρ

)35()()(

)34()()(

)33()()()()(

)32()()(

)31()(

)30(1

2222

2

2

2

2

2

22

xp

th

hppdzt

dzxp

wudzz

uw

yhhvhu

yvuuvdz

ydz

yuv

xh

hux

udzux

dzx

u

tuudz

tdz

tu

dzzu

yu

xudz

xpdz

zuw

yuv

xu

tu

hh

h

hh

hh

hh

hhh

∂∂+

∂∂−−

∂∂=

∂∂

=∂

∂

∂∂−−−

∂∂−

∂∂=

∂∂

∂∂−−

∂∂−

∂∂=

∂∂

∂∂−

∂∂=

∂∂

∂∂+

∂∂+

∂∂+

∂∂−=

∂∂+

∂∂+

∂∂+

∂∂

∫∫

∫

∫∫

∫∫

∫∫

∫∫∫

−−

−

−−

−−

−−

−−−

ζζ

ζζ

ζζζ

ζζ

ζζ

υρ

ζζ

ζ

ζζ

ζζ

ζζ

ζζζ

)36(0)()()()( =

∂∂−

∂∂−

∂∂−

tyv

xuwu

ζζζζζζζ

)39()(2 ∫∫∫∫∫

−−−−−

∆=∂∂−−

∂∂+

∂∂+

∂∂+

∂∂ ζζζζζ

υρρ hhhhh

udzxhhp

dzp

xuvdz

ydzu

xudz

t

[18]


Taking the mean of each term:

We obtain the movement equation integrated on the vertical and averaged on a period

Two terms depend on the speed fluctuations:

Or for the second momentum equation:

)43()()(21p(-h)

:Hence(42)h)g(-P(-h)p

:bottom at the pressure dynamic average theas defined is

)41(

)40(~~)~)(~(

)39(~)~(

)38()~(

2

2222

xhghg

xxhp

xh

p

dzuudz

dzvuy

dzvuy

dzvvuuy

dzyuv

dzux

dzux

dzuux

dzux

tUdzuu

tdz

tu

hh

h hhh

h hhh

hh

∂∂+−

+

∂∂+

∂∂=

∂∂

+=

∆=∆

∂∂+

∂∂=++

∂∂=

∂∂

∂∂+

∂∂=+

∂∂=

∂∂

∂∂=+

∂∂=

∂∂

∫∫

∫ ∫∫∫

∫ ∫∫∫

∫∫

−−

− −−−

− −−−

−−

ζζζρ

ζρ

υυζζ

ζ ζζζ

ζ ζζζ

ζζ

)45()(

)44(~~)(21~ 22

2

∫

∫ ∫ ∫∫

−

− − −−

∆+∂∂+

∂∂+−

∂∂−

+−+

∂∂=

∂∂+

∂∂

∂∂+

∂∂

ζ

ζ ζ ζζ

υζξ

ζρ

h

h h hh

udzxh

px

hg

dzvuy

hgdzp

ux

dzvuy

dzt

uxt

U

)49(~~

)48(h)(2g 22

∫

∫

−

−

=

+−+=

ζ

ζ

ρ

ζρρ

hyx

hyy

dzuvS

dzvpS

)47(~~

)46(h)g( 22

∫

∫

−

−

=

+−+=

ζ

ζ

ρ

ζρρ

hyy

hxx

dzvuS

dzupS

[19]


A last assumption is made assuming that the mean horizontal velocities are constant alongthe vertical.

This last assumption is clearly a limitation of the Shallow Water Equations: if the horizontalvelocity varies too much along the vertical, the average value will have no physicalmeaning. For example, a contaminant will have the same velocity if it travels near thesurface or near the bottom. It is a property of long wave to have a constant velocity alongthe vertical. For this reason the Shallow Water Equations are well suited for thecomputation of floods, tides, and tsunamis.

Then we obtain the Shallow Water equations expressed in averaged velocity. Introducingthe wind and bottom friction stress components, the system of equations becomes:

stress.shear effective theof components theare termsThe ijj

ij

x

S

∂∂

=τ

)51((

(50)Uuh)(22

vUVuh)vudzvu

udzu

h

h

==+=

=+=

∫

∫

−

−

ζ

ζ

ζ

ζ

parameter Coriolis :

viscosityturbulent :stressfriction wind:

stressfriction bottom:

stressshear effective ofcomponent :with

)54(

h)(

)53(

h)(

)52(0

f

UfyV

y

xV

xxg

yV

UxV

Ut

V

VfyU

y

xU

xxg

yUU

xUU

tU

tV

tU

t

ij

wi

bi

ij

yy

xywybyyyyx

xy

xxwxbxxyxx

υτττ

υ

υρ

τρ

τρ

τρ

τζζ

υ

υρ

τρ

τρ

τρ

τζζ

ζ

−

∂∂

∂∂+

∂∂

∂∂+++++

∂∂+−=

∂∂+

∂∂+

∂∂

+

∂∂

∂∂+

∂∂

∂∂+++++

∂∂+−=

∂∂+

∂∂+

∂∂

=∂∂+

∂∂+

∂∂

[20]


The values of the bottom friction stress and of the turbulent viscosity terms are veryimportant in the stability of the numerical model. The bottom friction stress term is obtainedintroducing the Chézy coefficient written C.

Numerical Formulation

The numerical model is a finite-difference schema using an A.D.I (Alternating Direction-Implicit) technique to integrate the equation for mass and momentum conservation in thespace-time domain. The equation matrices are resolved by a Double Sweep (DS) algorithm.

Mike 21 HD has the following properties:

• Zero numerical mass and momentum falsification and negligible numerical energyfalsification, over the range of practical applications, through centering of all differenceterms and dominant coefficients, achieved without resort to iteration.

• Second to third-order accurate convective momentum terms, i.e. “second and thirdorder” respectively in terms of discretization error in a Taylor series expansion.

The difference terms are expressed on a staggered grid in x, y-space as shown below.

Where p and q are flux densities in x- and y-direction (m≥/s/m)=(uh,vh); (u,v)=depthaveraged velocities in x- and y-direction.

VC

VUgUC

VUgC

VUg bybxb 2

22

2

22

2

22 +=+=+= ρτρτρτ

[21]


Time centering of three equations in Mike 21 HD is achieved as sketched below.

The equations are solved in one-dimensional sweep, alternating between x and ydirections. In the x-sweep the continuity and x-momentum equations are solved, taking ζfrom n to n+1/2 and p from n to n+1. n-1 and n+1/2. For the terms involving q, the two levels ofold, known values are used, i.e. n-1/2 and n+1/2.

In the y-sweep the continuity and y-momentum equations are solved, taking ζ from n+1/2 ton+1 and q from n from n+1/2 to n+3, while terms in p use the value just calculated in the x-sweep at n and n+1.

Adding the two sweeps together gives time centering at n+1/2.

[22]


2. SET UP OF THE BATHYMETRY MODEL OF PORTVILA-MELE BAY

The first step and by far the most important task in a modelling process was setting up thebathymetry model. Data used for the compilation was sourced from two surveys completedrespectively in Mele Bay and Port Vila in 1990 and 1997.

Instead of setting up the model directly in Mike 21, we prefer using the software Quicksurfthat is a surface modelling system running inside AutoCAD 14. Quicksurf is a package usedfor generation of contour maps, profiles, and interpolation grids.

Thus, after having generated an interpolation grid, the next step consists in exporting it intoMike 21after having converted the grid data into a Mike 21 format using a Fortran program.

Brief report of the bathymetric surveys

Survey of 1990

In the nearshore zone of Mele Bay, thirty-two geophysical line profiles totalingapproximately 50 km were surveyed between 7 March and 19 March 1990. The base mapfor navigation control was digitized from 1:2500 series DOS 065. A Del Norte microwave-positioning system with all data logged and processed on laptop PCs provided vesselposition. A Raytheon De719e echo sounder with a digital depth output was recorded by alaptop and files created were merged with the navigation data to produce computergenerated bathymetric maps. Admiralty tide table was used to reduce bathymetric data to acommon datum, in this case Lowest Astronomical Tide (LAT). For the seabed morphologyand sub-bottom information a Data Sonic SBT220 3.5 kHz sub-bottom profiler was used toproduce analogue records for interpretation. A detailed account of survey techniques anddata processing can be found in Smith and Saphore (1990) and Smith (1991).

Survey of 1997

Bathymetric mapping was completed in Port Vila harbour and Mele Bay areas to a water depthof 200 meters.

All navigation control was accomplished with a Del Norte1009+ Differential GPS in real time.The reference station was sited at a number of known control points based on BellevueTransverse Mercator1977, local grid. An Echotrak precision echo sounder was used for profilingthe seabed, and the digital output logged by the 1009. All bathymetric data collected are to bereduced to zero of the NFT tide station sited at Port Vila wharf.

A detailed account of survey techniques and data processing can be found in SmithShorten and Young (1997).

[23]


Digitization of the maps under AutoCAD

Due to the unavailability of all the 2-meter contour 1:2500 series DOS 065 maps for thewhole area, we have digitized with AutoCAD only about 50% of Port Vila-Mele Bay with themaps available. The rest has been completed with a 1/50000 Ile Efate S.O map.

Due to the lack of a complete set of maps with 2-meter contour, the simulation of harborseiching, tsunamis or storm surge was not done. In representing flooding brought about bysuch an event, detailed topography contour maps are necessary. A new digitization of thetopography will be made when maps with a 2-meter contour covering the whole areabecome available.

Geographic transformation

To maintain geodetic correctness for all the maps and the bathymetric data files andconverting data sets to one common datum before digitalization and before importing theASCII point files within the model, a geographic calculator was used. The geographicconversions were done using the software Geographic Calculator (version 3) under theprojection of the D.O.S maps. The following figures (Figure 1) shows an example of ageographic conversion.

Figure 1. Efate Conversion with Geographic Calculator from the Projection Geodetic39 Bellevue to EfateBellevue

Figure 2. Result of the bathymetry surveys of Port Vila and Mele Bay.

[24]


Correction of the bathymetric model

Some corrections have been necessary to avoid the superposition of some data and tohave a coherent bathymetry according to the maps. These corrections have been donemainly on the reef where data was difficult to collect during the survey because it was veryshallow. We have added some points near the open boundary where the bathymetry is verydeep (around 300 meters). Moreover the model has been reduced to the area of interest i.e.to the area of Mele Bay and Port Vila.

Set up of a 100-meter interpolation grid

The choice of a 100-meter grid will be explained in the next chapter.

Mike 21 uses a finite difference method to solve the depth-integrated equations ofconservation of mass and momentum. This implies that the model must work with a discreterepresentation of the bathymetry in the simulation area, i.e. with the depth values given in anumber of points. These points must be placed in a rectangular matrix, or in acomputational grid, with constant spacing in the two directions.

That’s why a 100-meter spacing interpolation grid is made under AutoCAD (Figure 3 and 4)taking the precaution of writing down the grid origin and the number of cells that areessential for the conversion into a Mike 21 format through a Fortran program.

The process is not detailed here, but requires a lot of manipulations in order to obtained themost accurate grid.

Figure 3. Representation of the 100 meters grid spacing.

[25]


Figure 4. 3-D representation of the grid under AutoCAD.

[26]


3. SET UP OF THE HYDRODYNAMIC MODEL FORPORT VILA-MELE BAY

After having imported the bathymetry grid from AutoCAD into Mike 21, the next stepinvolves rectifying the grid following some rules, which will be given below. To run thesimulation, some parameters have to be specified, and often changed in order to obtain agood calibration according to certain measures. Then, the purpose of this chapter is to givethe different steps followed for the set up and the calibration of the model giving theproperty of the parameters that have to be specified.

For the description of the area, we have to remember that the basic assumptions on whichMike 21 is built:

• Two-dimensional flow• Vertically homogeneous• Slowly varying bathymetry

In this chapter, only the case of the model of Port Vila-Mele Bay will be discussed, knowingthat for the study of Port Vila only, the same steps will be followed.

The bathymetry

As mentioned before, describing the water depths in the model area for the hydrodynamicmodel is without doubt the most important task in the modelling process A few hours lessspent in setting up the model bathymetry, might later mean extra days spent in thecalibration process.

• Some water depth values interpolated through AutoCAD are aberrant (Figure 1). Then,the first process is “to clean” the bathymetry under the Mike 21 grid in order to have acoherent depth water (Figures 2 and 3). It can be done point by point into the grid editor.

• The fact that the grid is 100-meter spacing, makes it difficult to represent the areas ofdimensions less than 100 meters, like a 50-meter offshore fringing reef, or a 30-meterlong wharf

[27]


Figure 1. Bathymetry imported from Autocad.

Figure 2. Bathymetry after rectification.

[28]


Figure 3. 3D-representation of the bathymetry.

Selecting the model area

For this model, the area of interest is both Port Vila and Mele Bay. To decide where to placethe open boundaries, some modelling guideline are followed in order to avoid many pitfallsin the modelling process. For this particular model only, one open boundary was requiredand the logical place was between Devil’s Point and Pongo Point.

• The area or point of interest should lie well inside the model area, say at least 10 gridpoints from the boundary but preferably more. As the area of interest is Port Vila and theNorth coast of Mele Bay, the placement of the open boundary was more than 10 gridpoints far from the study area. Therefore, not only the area immediately surrounding thearea or point of interest is included, but larger area is needed; for example, to compute awind surge properly.

• In order to have a "well behaved" flow at the open boundary, the bathymetry issmoothed close to it.

Grid orientation

After having determined the areas of interest and influence, we turned our attention to theorientation of the model

Because of the way Mike 21 solves the equations, better results are normally obtained withflow parallel to one of the coordinate axes. Therefore the model should be placed so thatthe main flow direction is aligned with the grid.

[29]


From the bathymetry it must be assumed that the tide will generate a flow that is goingdirectly in and out of the bay, i.e. parallel with the coast. It can be seen from the Figure 4that the x-axis is much better aligned with the dominant flow direction than in the Figure 2.

An other advantage in model rotation as in the case of Mele Bay resulted in one openboundary.

Figure 4. Bathymetry model after a 62 degrees rotation.

Model Parameters

The grid spacing and the courrent number

In Mike 21 ngrid cell size and time step are predetermined to ensure a low courant numberis obtained.

The grid spacing is linked with the current number as follows:

where c is the celerity, ∆t the time step and ∆x the grid spacing. For a tidal wave the celerityis

where g is gravity and h is the water depth.

xtcRc

∆∆×=

hgc ×=

[30]


In the same way Mike 21 calculates the water level and flow in a number of discrete pointsin the computational grid. It also calculates them at a number of discrete time steps. Andjust as the grid point should be placed equidistant so should the time step, i.e. thecomputation should progress with a constant time increment.

As the information (for water levels and fluxes) in the computational grid travel at a speedcorresponding to the celerity, the Courant number is an expression of how many grid pointsthe information moves in one time step.

Normally we can have a maximum Courant number in the model of up to 5. The maximumvalue, which can be used without having stability problems does however depend on thebathymetry. For very smooth bathymetry, Mike 21 allows Courant numbers up to about 20.

Therefore, the grid spacing becomes a compromise between the purpose of the study, thetime step and the Courant number. For the model of Mele Bay, a grid spacing of 100 metersproduced a Courant number of 5.6.

Boundary conditions

If the description of the bathymetry is the most important task in the modelling process thenthe description of the water levels and flow at the open boundaries (i.e. the "boundaryconditions") is the second most important task. The better the boundary conditions thebetter the results and the fewer the instability problems.

MIKE 21 solves the partial differential equations that govern nearly horizontal flow and, likeall other differential equations, these need boundary conditions. As the unknown variablesare surface elevation and flux densities in the x-direction and y-direction one of these twovariables must be, in principle, specified in all grid points along the open boundary at eachtime step. In this application, we will use the surface elevation through the boundary givingthe same elevation to each point of the grid.

For this, we use the file data containing the water level recorded during the last surveybetween the 14th and the 31st of March 1997 (Figure 5). Even if the recordings have beenmade in front of the main wharf in Port Vila, i.e. 10km from the open boundary, we will usethese measures as a boundary condition. According to the fact that the depth between theopen boundary and the main wharf varies between 300 m and 50 m, tides can be expectingto propagate at speeds of the order of 20-50 m/s (formula above), thus traversing the bay inless than 10 minutes. Period of which tidal levels used for the open boundary wererecorded at 6 minute intervals.

Boundary conditions can be either derived theoretically from tidal constituents or bemeasured values.

[31]


Figure 5. Graph of the measured water elevation.

In Mike 21 using tidal constituents (Figure 6), a time series of water levels can be derived togenerate a boundary file.

Harmonicconstants

Phase[deg]

Amplitude[m]

M2 163 0.37S2 182 0.11K1 205 0.16O1 171 0.09

Figure 6. Tidal constituents of Port Vila from the Admiralty Chart Table

Figure 7. Water elevation calculated from the Admiralty Tide Table constituents

Using both computed and measured time series for boundary conditions a comparison ofthe two models simulation can be made.

[32]


Bed Resistance

A very important term in the equations, especially in river flows, is the bottom friction. Thegeneral 1D expression of friction is given by the formula:

with units in kg/m/s″, where ρ is the density, u the velocity of the flow and Cf is the frictioncoefficient (without dimension). The 2D equivalent of this formula is

The friction coefficient Cf is not frequently used in hydraulics and is generally replaced bythe Chézy coefficient which is defined as

Then (2) becomes

According to the Chézy lay, the friction force will then appear in the momentum equation asa source term having the following form:

Another formulation for 2D flows is given by the Manning empirical law, which expresses Cas a function of the water depth h, that is

As in Mike 21 we specify the bed resistance as 1/m, we will call M the inverse of theManning number: M=1/mAccording to Nikuradse, the Chézy coefficient may related to the grain size ks on thebottom:

Mostly, the value of the friction coefficients will be unknown and will have to be estimated bymeans of measurements or by using catalogues.

As Mike 21, there are two ways of specifying the bed resistance: either as a Chézy numberor as the inverse of a Manning number. One value is given to all grid points, or a mapsimilar to that of the bathymetry with a resistance value for each grid point can be made.

As there is not enough information about the bottom of the area (grain size of thesediments, vegetation), the manning number will be specified following the recommendedvalues given under Mike 21. A default value of M=32 m↓/s will be applied for the first

,)/( 2uC21 fρ=τ

)()/( 2uuC21 fρ=τ

).// sm(inCg2C 1/2f=

2C

uug ..=τ

222

222 vuv

hCgFandvuu

hCgF yx +−=+−=

).s/m(in number Manning theis Where

1

31

6/1

m

hm

C ⋅=

=

skhC 11ln63.7

[33]


simulations. A user-defined map of bed resistance will be made as well to compare if betterresults are obtained

From a depth –10m to 2m: 1/M=15From –60 to –10m: 1/M=20For a water level up to 60 1/M=32.

Turbulence and eddy viscosity

Turbulence remains one of the fundamental unresolved problems of physics, the complexityof the problem being due essentially to the non-linearity of the convective terms of theNavier-Stokes equations. But as far as we are concerned here, it is more of practicalproblem, that is to say of how to model the influence of the flow turbulence on the meanstructure of the flow.

The effective shear stresses in the momentum equations contain momentum fluxes due toturbulence and vertical integration. The terms are included using an eddy viscosityformulation.

The formulation of the eddy viscosity in the equations has been implemented in two ways:

Flux based formulation

Velocity based formulation

Where u is the velocity in the x-direction and h the water depth.

Strictly speaking the first formulation is only correct at a constant depth and should beapplied with great care in order to avoid falsification of the flow pattern.

The velocity based formulation, which is more correct, is unfortunately also more difficult toimplement in the numerical algorithm. This is because the system uses the fluxes as theunknown parameters and not the velocities. Therefore the velocity-based formulation isimplemented by using the velocities from the previous time step. This can, however, lead tostability problems when the eddy viscosity coefficient E becoming large. The coefficientmust fulfil the criterion:

Specifying the eddy viscosityThe eddy viscosity coefficient E can be specified in three different ways:

As a constant value for the entire computational domain

momentum)-(x

∂∂

∂∂+

∂∂

∂∂

yP

Eyx

PE

x

∂∂⋅

∂∂+

∂∂⋅

∂∂

yu

Ehyx

uEh

x

2/1. ≤∆∆xtE

[34]


From a type 2 data file (grid matrix file) giving the value at each grid pointA time-varying function of the local gradients in the velocity field. This formulation is basedon the so-called Smagorinski concept, which yields

The Smagorinski facility is combined with the following formulation of the shear tresses:

For our studies, it was be sufficient to specify a constant turbulent viscosity for the wholeflow field. However, when turbulence becomes important, in more complex flow situations, aconstant eddy viscosity model is too simple.

Initial Surface Elevation

Since the model initializes the fluxes or current velocities to zero, it must be specified aninitial surface elevation that is in agreement with these conditions. This means that to avoidinstability at the first time steps of the simulation, we specify a value that matches theboundary condition at the first time step. This allows the model to obtain stable solutionsfaster.

Flooding and drying

A problem of numerical algorithms is drying zones where some terms in the equations havedivisions with h which tend to infinity, when h tends to 0.There are two solutions to avoidthese problems, but only the second one is implemented in Mike21:

• Solving the equations everywhere and coping with spurious terms,• Removing the dry zones from the computational domain.

The first one is the simplest, but correction must be applied in the wetting and drying zones,to avoid infinite terms and spurious values of the free surface gradient. As a matter of fact,in dry areas, the free surface gradient is equal to the gradient of the bottom topography andin that case must not act as a driving force in the momentum equation.

The second option, often referred to as the “moving boundary technique”, consists ofremoving the dry zone from the domain, but this task increases the computational time, andthe disk space required.

MIKE 21 is capable of including and excluding computational areas dynamically during thesimulation or, in other words, compute the flow in an area, which sometimes dries out and issometimes flooded.

1.0. to0.25 of interval in thechosen be oconstant t a is and spacing grid theis direction,-y and-in x components velocity averagedepth are V U,Where

21

22222

s

s

c

yV

xV

yU

xU

cE

∆

∂∂+

∂∂+

∂∂+

∂∂∆=

∂∂+

∂∂⋅⋅

∂∂+

∂

∂⋅∂∂

xV

yU

Ehyx

UEh

x2/1

[35]


Continuity is then preserved during the flooding and drying process. The water depths atthe points that are dried out are saved and then reused when the point becomes floodedagain.

As our model is driven by tidal forces we have included flooding and drying to both models.

Specifying flooding and dryingTo enable the possibility of flooding and drying areas, we have to specify at what depth thecomputational points should be taken out or reentered into the computations.

The minimum water depth before drying can normally be specified in the range 0.1 - 0.2 mand the minimum water depth before flooding in the range 0.2 - 0.4 m. A difference betweenthe two depths of 0.1 m is recommended. If the water level variations occur very rapidly(compared to the time step) we can increase the difference to 0.2 m or even more.

Model Calibration

The purpose of the calibration is to tune the model in order to reproduce satisfactorilyresults which compare well with measured conditions for a particular period known as thecalibration period. It is rare that the first few simulations will provide good results as modelinstability can result in the simulation ending prematurely, commonly called a blow up.These usually become unstable and end the simulation prematurely, what we call usually ablow up.

The main parameters to adjust during the calibration phase are:

• Bed resistanceThe bed resistance can be used to stabilize and calibrate the model. The shallower thearea, the more effective it is to change the bed resistance. Increasing/decreasing bedresistance may be used to change amplitude and phase of a tidal wave..

• Eddy viscosityThe eddy viscosity is mainly used to stabilize the solution. If the results are spurious withhigh frequent oscillations in the water levels or wiggles in the flow field -zigzaggingcurrent vectors in an area of the model, then the result can be smoothen by increasingthe eddy viscosity in the area.

• BathymetryThe bathymetry is far the most important calibration parameter. During the first numbersof calibration simulations, the bathymetry should be inspected and possibly changedbefore trying out the other calibration parameters. Sometimes, we have to revise thebathymetry area with schematization many times before obtaining good results.

• Boundary conditionsThe boundary conditions can also be used to calibrate the model. Particularly theselection of a tilt point can change the result quite dramatically.

Once these parameters specified, some simulations can be run. In the next chapter, theexact values of the parameters will be given for each simulation, and calibration.

[36]


4. RESULTS AND VALIDATION OF THE MODEL OFPORT VILA-MELE BAY

The purpose of the following simulations is to produce a basic hydrodynamic model (HD) thatcan be used for the set-up of a water quality, sedimentary transport or tsunami model. It canalso give an idea of the current velocities and the flux in the model. Comparing the results withmeasured sea level, assesses the validity of the hydrodynamic model. But, the fact that nocurrent or flux data are available does not allow us to completely validate the current velocity orflux of the model.

As we will see later, some instability has occurred in the simulation using the measurements.That’s one of the reasons why some simulations using the Admiralty tide table have beenperformed. Moreover it is also interesting to make comparisons between the two models.

This chapter also gives the values of the parameters specified for each simulation and tocalibrate the model. In order to be helpful for another Mike 21-user, the blow-ups and thecalibration problems that have occurred during the simulations are discussed.

Simulations using the admiralty Chart data

Definition of the model

Here are the parameters specified for the first simulations:

• The bathymetryThe bathymetry model selected is the one containing both Port Vila and Mele Bay. Thewater depth at the open boundary is about 300 meters and is well smoothed.(see figurechapter).

The flood and dry checking is done as a default value for a drying depth of 0.2 meter and aflooding depth of 0.3 meter.

• Area DescriptionThe grid spacing is 100 meters, and the dimension of the grid is 1203150.The grid is rotated of 628 in order to have a flow perpendicular to the open boundary.The origin of the grid is given as 17,748S and 168.38E.

• Simulation periodThe simulation period is from the 14th to the 29th of March of 1997.The calculated time step interval is 10 second, with the number of time step being 138240.This produces a courant number of 5.1 which is a good indicator for model stability. Some20 hours of computation have been necessary to complete the simulation.

• Eddy viscosityFor the first simulations, no Eddy viscosity was specified. Later simulations, differentnumbers for eddy viscosity were experienced with.

• Open boundaryThe only open boundary defined was a line between Pongo and Devil’s Point, at the head ofMele Bay, grid points (0,48) and (0,132).

[37]


A file representing the water elevation as a function of time calculated under Mike 21 withfrom Harmonic Constants given in the Admiralty Tide Table is taken for the open boundary.The initial surface elevation for the whole model was given as the same value as the initialsurface elevation at the open boundary, i.e. 0.05m to avoid instability at the first time stepsand obtain stable results faster.

• WindThe Monthly Data Report of March 1997, edited by AusAID, gives a wind speed less than10 knots for the whole period of the survey. The direction was very variable. For this reasonwind was not included in the simulations.

Obtaining results

In the first simulation, many blow-ups occurred ending the simulations prematurely. These blow-ups came from certain grid points where bathymetry was not well smoothed, or where the slopebetween two grid-points was too steep.

Then, after a few rectifications and schematizations in the bathymetry, the simulation can end,and the model can now be verified.

The first process in verifying the model was to compare the computed water level at a numberof points in the model to the water level derived from the Admiralty Chart Data (Figure 1).

[38]


Figure 1. Example of difference in water level (in meter) between computed (in red)and Admiralty Tidedata (in black) at the passage of Port Vila for a high tide phase.

Figure 2. Aberrant flux vectors at the passage between Port Vila and Mele Bay.

[39]


We can see that the difference between the 2 graphs (Figure 1) representing the waterelevation at the point (60,33) compare well meaning that the model behaves well for the waterlevel used. When examining the flow in these areas especially in the passage between Port Vilaand Mele Bay (Figure 2), we see some instabilities in the flux (zigzagging), and some vectors ofthe flux seem to be very often aberrant.

Without current and flux measurements, it’s very difficult to make a good interpretation of theseparameters. Meanwhile, in other areas, where the bathymetry is well smoothed, the vectors arestill confused where they should appear more linear.

Calibration

A way to obtain a more linear flow is to use the Eddy viscosity (E). If we had had measurementsof the velocities and flux, we could have changed it in order to obtain computational valuesclose to the measurements. As that is not the case, the Eddy viscosity has been chosen toprovide the minimum of instabilities.

Following a number of simulations with different Eddy viscosity values, a value of 5m″/sproduced the best results.

Final results

Analysis of the water elevationAfter the last calibration of Eddy Viscosity, there was almost no difference between thecomputed water elevation, and the one from the Admiralty Tide Table (Figure 3). Therefore, wecan say that the model is well calibrated for the water elevation.

Figure 3. Comparison between the water elevation at the passage (grid point (56,22)) fromthe simulation and the Admiralty tide table.

[40]


Analysis of the speeds and velocitiesThe graphs of the velocities within the passage between Port Vila and Mele Bay with E=5 m″/sand E=0 m″/s (Figure 4) show that there are less instabilities when adding some viscosity. It canbe observed that the periodicity of the velocities corresponds to the tide periodicity.

The 3D-speed map (Figure 5) shows that the speed of the flow is higher near the shore then inthe passages. The difference of the amplitude of the speeds in the Figure 6 between the flowclose to the open boundary and the one in the passage is important. Moreover, the graph of thespeed near the open boundary is really better smoothed than in the passage. This observationcan be explained by the fact that the tidal waves have undergone transformations (diffraction,reflection) when spreading into Port Vila Bay.

Figure 4. Velocities at the passage (grid point 60,33) between Port Vila and Mele Bay following an axisparallel to the flow (in red E=0 m″/s, in green E=5 m″/s).

[41]


Figure 5. Distribution of the flow speed for the whole model at mid tide.

Figure 6. Speeds (in m/s) at the open boundary (in black) at the grid point (10,90), and in red in thepassage between Port Vila and Mele Bay (grid point 60,33).

[42]


Conclusion

The water level computed fit well with the one taken at the open boundary from the Admiraltyconstituents with speeds corresponding to reality with a periodicity of the speed graphcorresponding to the periodicity of the tide.

We have presented here the results found in the main passage of Port Vila-Mele Bay becausethe instabilities were more common to occur close to this area. Several other areas for thewhole model have been verified and produced acceptable results.

Simulations of Mele Bay and Port Vila run with the measurements

After performing the transformations necessary on the original file of the measurements in orderto be imported in Mike 21, the first simulations are run using the same specifications as in theprevious model, except at the open boundary. Considering the fact that both graphs of the waterlevels (Figure 7) from the measurements and from the ones computed with the HarmonicConstants are very similar (i.e. same mean value, amplitude and phase), a good calibration ofthe model can be obtained with the same parameters.

Figure 7.Comparison between measured and calculated data from the Admiralty Tide Tablewater elevation.

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Obtaining results

Analysis of the water levelThe Figure 8 shows that the computed water level are very close to the measured values.Meanwhile, on examining the results in more detail (Figure 9), we can see that the differencebetween the curves for NTF are larger than those for the simulation based on the Admiraltyderived boundary conditions.

Figure 8.Water elevation measured and computed at the grid point (60,33).

Figure 9. Zoom on the water elevation instabilities of the graph above.

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Analysis of flux and velocitiesThe analysis of the 2D map representing the current vectors an anomaly in the model. Here, wecan observe that the direction of the vectors are changing direction almost at every time steps(Figure 10), while these should be kept at nearly the same direction during a flooding or ebbingtide.

This observation can be confirmed in the graph of the velocities (Figure 11), which appearsconfused. The graph corresponds to the velocity of a grid point taken in the main passage ofPort Vila-Mele Bay, following an axis parallel to the main flow. Modelled velocity changes frompositive to negative almost at each time step, with no periodicity able to be identified.

Figure 10. Direction of the flow for two consecutive times steps during an ebbing tide.

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Figure 11. Velocities at the grid point (60,33) following an axis parallel to the flow.

Calibration

The plots of the graphs of velocities for different grid points within the model presents also showsimilar characteristics seen above, and even close to the open boundary. Though, whenviewing at the 2D map plots of velocity vectors, the flow appears linear with no “zigzagging”vector as seen in the model using ATT as boundary driver with no viscosity. It proves that theinstabilities of the velocity graphs can’t be solved with the Eddy Viscosity. But the fact that weobtained bad results compared to the previous model run with the same bathymetry, viscosity,bed resistance… proves that the problem comes from the water elevation measurements usedat the open boundary. It seemed also surprising at first that small differences between the ATTand the measurements could give such aberrant velocities.

A number of variables were experimented with to improve velocity results. The first idea was tochange the bed resistance (values given in the pervious chapter) in order to obtain a computedwater level close to the one using the ATT. The bed resistance can be used to change theamplitude and phase, but the resulting graph of velocities shoed little variation from the firstplots.

In fact, when we do a zooming of the NTF (National tide Facilities) water elevationmeasurements, we can see that the graph is not smoothed (Figure 12), whereas the waterelevation calculated with Mike 21 (ATT) shows the curve to be more smoothed.

Then the next idea was to smooth the NTF water elevation measurement file used for the openboundary. Actually, the very small imperfections of the curve must be amplified during thepropagation of the flow. This may therefore explain aberrations in the results in particular withthe velocity plots.

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Figure 12. Comparisons between smoothed and non-smoothed water elevation measurements graphs.

Results with a smoothed NTF boundary condition

Analysis of the water elevationFrom a plot of water elevation at several points of the model, there is now almost no differencebetween the numerical water elevation and the measured, the model is calibrated for waterelevation

Analysis of the speeds and velocitiesHaving taken the smoothed NTF boundary condition, there is no fluctuation of the flow at eachtime step anymore, plots of the 2D map. Flow appears to behave correctly.

From the plots of speeds, good correlation with the periodicity of the tide is evident, and thecurve of the graph of the speed is clearly more smoothed (Figure 13). In contrast the model runwith the ATT for boundary, it was found that the velocities are higher around Port Vila and in themain passage than in the middle of Mele Bay.

Having no speed measurements does not allow us to interpret the amplitudes of the speeds.Model results show speeds that are of the magnitude lower than expected, with speeds as lowas 1/100 m/s in the main passage of Port Vila in comparison with other studies of the samenature (Majuro), where velocities are of the order of 1/10 (m/s) in a passage. We can observeagain that the velocities are greater in the main passage of Port Vila-Mele Bay than at the openboundary (Figure 14).

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Figure 13. Comparison of the computed speeds at the main passage with the smoothed and non-smoothed tide.

Figure 14. Comparison of the speed close the open boundary (10,90) and between the main passage

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Conclusion

The instabilities observed mainly in the velocities appear to have come from the “imperfections”of the water elevation measurement curve used for the open boundary. It is good to keep inmind that the quality of these data greatly influences the accuracy of the results, and thatsmoothing of the data improves model stability.

An Eddy Viscosity of 5 m″/s limits the numerical instabilities due to the turbulence which appearmainly within the passages and close to the shore.

The model seems to be coherent if we consider the results of water level and velocity. But wecan’t verify the velocity model especially concerning the amplitude since we don’t have anymeasurement of the velocity at any points of the area.

Apart from calibrate the model against measured current information, the hydrodynamic modelcan be used to explore further water quality, eutrophication and sediment transport in Port Vilaand Mele Bay.

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5. MODEL OF PORT VILA

The steps followed to set up the model of Port Vila follows the same procedure used for the oneof Port Vila-Mele Bay. An advantage of produced a model of Port Vila only, is a higher grid cellresolution permits the inclusion with a smaller grid which would permit the inclusion in the modelof elements that are of the same dimension as a grid cell. The resolution of the model istherefore improved. For instance, it was not possible with the previous model when the grid sizewas 100 meters to study the influence of a 30-meter long dyke or wharf on the area.

The objectives for the setting up of a higher resolution hydrodynamic model for Port Vila properare at first the impacts of a number of coastal structures yet to be implemented could beassessed.

Set up of the bathymetry model with AutoCAD

To set up the bathymetry model in AutoCAD, we make use of the previous model removing thearea of Mele Bay. As the grid is chosen at 15 meters spacing, to make the model moreaccurate, some details in the bathymetry are added, like fringing reefs.

In developing the Port Vila Model, the number of open boundaries was taken into consideration.With two open boundaries (which one drying during a tidal cycle) it may increase the risk ofinstability, and moreover, in that case, no analysis may be done in an area close to thepassage. The influence of the two passages in the behavior of the flow is important. That’s why,the open boundaries have to be far from the passages. Then, choosing an open boundary at 40grid cells from the passages into Mele Bay reduces its number to one, and any study close tothe passages can now be done.

The Figures 1 and 2 respectively show the bathymetry and the grid of the model of Port Vilaunder AutoCAD. The grid is later exported as ASCII points from AutoCAD and converted to theMike 21 format with the help of a FORTRAN program.

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Figure 1. Bathymetry of Port Vila under AutoCAD

Figure 2. 15-meter grid represented the bathymetry of Port Vila under AutoCAD

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Set up of the hydrodynamic model under Mike 21

The first step in building the bathymetry model imported from AutoCAD to Mike 21 requiresediting, cleaning and orientation to a flow the most perpendicular to the open boundary.

Having completed this task the model parameters required are defined as follows:

Parameters specified

• The bathymetryThe bathymetry model selected is the one of Port Vila. The highest water depth at the openboundary is about 50 meters. The open boundary is smooth in order to prevent instability ofthe flow.

For flooding and drying the default values for a drying depth of 0.2 meter and a floodingdepth of 0.3meter are used.

• Area descriptionA grid spacing is 15 and the dimension of the grid being 250⋅350. A rotation of 18° anticlockwise of the final grid is applied.

The origin of the grid is given as 17,74°S and 168,3°E

• Simulation periodThe simulation period is from the 14th to the 29th of March of 1997 which is the period of thetide measurements.For a time step interval of 7 seconds, it gives a total of time steps of about 180000 alltogether. The courant number is then 10,4.

The computation takes about 35 hours to completion. But the first simulations are run for ashort period until the results are acceptable. Indeed so many simulations have to be runbefore finding the right parameters that running each of them for the whole period could takemonths. That’s why a period of a few days is taken for each simulation until the calibration isdone, and then the model is verified for the whole period.

• Eddy viscosityAn Eddy viscosity of 5 was selected based on the good results obtained in the previousmodel.

• Open boundaryThere is only one open boundary situated in Mele Bay at about 40 grid cells from the twopassages, from the point (0,8) to the point (0,213).

The water elevation measurement file is taken for the open boundary. An initial water elevationwas set to –0.14m for the whole model to limit the instabilities at the first time steps (effect ofwaterfalls). This value corresponds to the first water elevation measured value

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Figure 3. 2D-Bathymetry of Port Vila under Mike 21.

Figure 4. 3D Bathymetry of Port Vila under Mike 21.

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Results of simulations

After having run several simulations and corrected the bathymetry in order to avoid blow upswhich end the simulations prematurely, output graphs of water elevation and velocities can beanalyzed.

Water elevationThe following graph compares the measured and the computed water elevation (Fig.5). Itclearly shows very little difference between the two curves. Additional points from the modelwere examined and similar if not better correlation were produced between the measured andthe computed water elevation.

Figure 5. Correlation of measured and computed water elevations of point 60,33 located in the passageleading into Mele Bay from Port Vila.

VelocitiesAn analysis of the 2D velocity maps shows that the flow is linear at almost every time steps. Thebest results were obtained using an eddy viscosity value of 5 m″/s. For no eddy viscosity thevector in the 2D plots were confuse while without eddy viscosity, the vectors remain very messy.

In examining the velocities at different grid points in the model in the model, it can be observedthat the current follows the tide well. The graphs of the velocities at any point of the area arewell smoothed except in the small passage between Port Vila and Mele Bay where we observesome ‘peaks’ of velocity (Fig 7)

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It is difficult to give an interpretation of the graph of the velocities in this small passage (Figure7). The curve is in the phase opposite to the one in the main passage, and its amplitude issometimes much higher.

The fact that the velocities are more important in the small passage is not surprising because itis narrower than the big passage. Concerning the shape of the curve, the large peaks observedjust before low tide or just after high tide may be due to the effect of flooding and drying in themodel. The point where velocity is analyzed is close to a flooding and drying area; then, justbefore being dry (0.2m) some points of this area are removed from the calculation and replacedwhen the water depth reaches a certain value (0.3m). Perhaps it is the effect of theseoperations that creates the spurious results at this point.

Recommendations: A way to verify these assumptions would be to place a current meter at thispoint in Port Vila to compare the velocity measurements to the computed results.

Apart from the anomalous results in this particular area, the whole model seems to reproducevelocities that are coherent..

Figure 6. Flow through the small passage between Port Vila and Mele Bay during a flooding tide .

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Figure 7. A comparison between the velocities from within the main ( 60,180) and the small passage(61,188) between Port Vila and Mele Bay.

Comparison with the results of the model of Port Vila-Mele Bay

It is interesting to compare the results from the model of both Port Vila-Mele Bay, and the one ofPort Vila only.

Concerning the water elevations, it’s clear that the results are the same for both models, asthese are both close to the measurements.

When comparing the velocities of both models (Figure 8), amplitude and phase correlate well.The most significant difference being that the curve of the graph of the model of Port Vila issmoother.

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Figure 8.Comparison of computed velocities of the model of Port Vila-Mele Bay and the one of Port Vila,at the same point within the main passage of Port Vila to Mele Bay.

Figure 9. A 3D perspective showing velocity in the Port Vila area during a flooding tide.

Conclusion

Considering the results of the computed water elevation and velocities, the model appearscalibrated. An eddy viscosity of 5m/s seems to give the best results.

The distribution of the velocities seems to correspond to the reality since these are the mostimportant within the passages, and near the coast (Figure 9).

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In comparison with the previous model representing Port Vila and Mele Bay with a 100-metergrid spacing, we obtain more accurate results with a 15-meter grid spacing.

Graphs of velocity in a flooding and drying area are not very reliable.

Further measurements of velocity would however permit to verify the computed velocities andespecially the amplitude.

Advection-dispersion modelling

Next, we run a simulation of a catastrophic scenario of pollution close to the main wharf of PortVila (Figures 10 and 11). The decay of the pollutant is very weak. It may be a black tide causedby spill from a tanker, for instance.

There is no recent data of the pollutant within Port Vila available at the moment, so fictitiousparameters have been taken for this simulation.

Figure 10. Propagation of oil after 5 hours.

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Figure 11. Propagation of the oil at 3 hours interval.

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RECOMMENDATIONS

In order to improve the accuracy of the computed velocities of the model, a survey recording thewater level, velocity and direction of the flow should be performed at several areas of Port Vila,and especially, close to the passages.

If other studies don’t require a 15-meter grid spacing, the grid spacing could be increase toreduce the computational time and the memory requirement, because it was found that for the15-meter grid CPU time was high.

In representing flooding brought by tsunami or flooding, detailed topography contour mapsare necessary.

Water quality survey should be done within Port Vila Harbour to assess the concentration ofpollutants to set up a water quality model.

CONCLUSION

Two hydrodynamics models have been set up through Mike 21. The first is covering the area ofPort Vial and Mele Bay together with 100-meter grid spacing and the other includes only PortVila with 15-meters grid spacing.

A major part of this study has been also concentrated on the set up the bathymetry of themodels under AutoCAD using the data of the two surveys done in 1990 and 1997 and severalmaps.

The hydrodynamics simulations have been run using NTF water level measurements at theopen boundary and the calibration have been performed comparing the computed to themeasured water elevation.

Good results were easily obtained for the comparison of water elevation. Good results wereobtained for velocity after having taken an eddy viscosity of 5 m″/s and after having smoothedthe water elevation graph used for the open boundary. Indeed, the few “imperfections” in thecurve of its graph could be amplified during the propagation of the flow giving aberrantcomputed velocity results. The periodicity of computed velocities obtained correlate well with thetide, and their intensity is higher in the different passages and close to the coast. Moreover,when comparing the two models at a point in Port Vila, the results are more accurate with the15-meter grid spacing.

The numerical models described herein are therefore an efficient tool for obtaining an overviewof the instantaneous currents and can be a good way to determine the long-term transport ofparticles under tidal influence.

The next task is to collect velocity data to complete the calibration of the model and to includethe topography within the models to assess the impact of tsunami.

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REFERENCES

[1] Bonnefille René, 1992. Cours d’hydraulique Maritime, Ed. Masson.

[2] Carter R., 1990. Hydraulic and water quality studies, Erakor lagoons and Port VilaHarbour, Vanuatu. Sopac Technical Report 117.

[3] Douillet P., 1997. Tidal dynamics of the South-West lagoon of News Caledonia:observation and 2D numerical modelling. Orstom, Centre de Noumea.

[4] Hardisty J., 1990. Beaches Form & Process. Ed. Chapman & Hall.

[5] Hervouet J.-M. and L. Van Haren, 1996. Recent Advances in Numerical Methods forFluid Flows. Laboratoire National D’Hydraulique, Chatou, Paris, France.

[6] Howorth, R., 1985. Baseline coastal studies, Port Vila, Vanuatu. Holocene uplift recordand evidence for recurrence of large earthquakes. CCOP/SOPAC Technical Report116.

[7] Molin B., 1997. Introduction a l’hydronynamique. Esim.

[8] Smith, R. 1991. Nearshore bathymetry and sea bed morphology, Mele Bay, Efate,Vanuatu. Sopac Technical Report 126.

[9] Smith, R. and Saphore, E. 1990. Geophysical cruise report for Mele Bay and PortHavanah, Vanuatu. Sopac Preliminary Report 22.

[10] Svendsen Ib A. and Ivar G. Jonsson, 1980. Hydronynamic of Coastal Region,Technical University of Danemark.

[61]


ANNEX

SOUTH PACIFIC APPLIED GEOSCIENCE COMMISSION (SOPAC)

An Introduction

BACKGROUND

The South Pacific Applied Geoscience Commission (SOPAC) is an independent, inter-governmental,regional organisation established by a few South Pacific nations in 1972, originally as CCOP/SOPAC. ItsSecretariat is located in Suva, Fiji, and has about 40 professional and support staff.

Member countries are currently Australia, Cook Islands, Federated States of Micronesia, Fiji Islands,Guam, Kiribati, Marshall Islands, New Zealand, Niue, Papua New Guinea, Samoa, Solomon Islands,Kingdom of Tonga, Tuvalu and Vanuatu. French Polynesia and New Caledonia are Associate Members.Twelve members are developing Pacific Islands Countries. Australia and New Zealand do not receiveWork Program assistance but are major donors.

MISSION STATEMENT

To improve the well being of the peoples of Pacific island member countries through the application ofgeoscience to the management and sustainable development of their non-living resources.

WORK PROGRAM

"It is difficult to think of any development of any country which hasn't benefited in some way from basicgeoscientific knowledge."

Through its Secretariat, SOPAC carries out a wide range of geoscience activities in the region. TheSecretariat's primary roles are to gather new data to assist member countries to assess their naturalresources, and to build national capacities in the geosciences towards self sufficiency in the long term.Not all of the activities listed below are carried out at any one time, the balance of the Work Programdepending on member country priorities and on the level of funding available to SOPAC at the time.

Resource Development Program

Mineral Resources

• mineral and aggregate assessments/surveys• deepsea mineral assessments/surveys• promotion of mineral potential• minerals legislation and policy development• hydrocarbons promotion

Water Resources

• sector strategy and action planning• technological and equipment support• surface and groundwater resource assessment• water supply and waste disposal advice• sanitation (with SPC)• solid waste management (with SPREP)

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Energy

• policy and planning• small energy projects• renewable energy projects• monitoring alternative energy developments (OTEC, Wave & Geothermal)

Environmental Science Program

Coastal

• physical environment monitoring• physical processes studies• coastal and shallow water seabed mapping• field equipment operation and maintenance

Hazard Assessment

• geohazard studies• vulnerability assessments• protection and engineering advice• lagoon/shallow water circulation studies

Ocean

• ocean environment monitoring• deep water and seabed mapping• cruise coordination• physical oceanography• LOS/EEZ issues (MSR, ISBA, Continental Shelf)

National Capacity Development Program

Human Resource Development

• ESMG Certificate• fellowships• workshops and seminars• HRD advisory assistance• distance education development

Information Technology

• computing services• communications, PEACESAT and data transfer• remote sensing and resource monitoring• database and GIS development

Disaster Reduction

• regional coordination• research information

Publications and Library

• editing• publications and reporting• library• public awareness

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FUNDING

An annual budget of about F$5 million supports the implementation of the Work Program and theoperation of the Secretariat. SOPAC funding consists of Core Funding, for Corporate Services includingcertain key technical support services derived primarily from member country contributions, and ProjectFunding, derived from supporting donor governments and agencies for project activities which focus onfield work; data collection, processing and interpretation; promotion of results; technical advice andassistance; and education and training.

In addition to the member countries, supporting governments include Canada, France, Japan, Korea,Peoples' Republic of China, Taiwan and United States of America. The European Union, theCommonwealth Fund for Technical Cooperation and the United Nations Development Program are theprincipal multilateral supporting agencies. In addition to levied contributions, member countries provideconsiderable support during survey work. Research ship time is contributed by other countries from timeto time. Linkages are maintained with other regional organisations and with research institutes worldwide.

PUBLICATIONS

SOPAC publishes numerous technical reports and bulletins, an annual report, a Corporate Plan,Proceedings of the Annual Session, a newsletter (SOPAC News), non-technical summaries of its work(SOPAC Projects), and a publications list.

FURTHER INFORMATION

To find out more about SOPACs work, or to request any publications, contact:

The DirectorSOPAC SecretariatPrivate Mail BagSuva, FIJI

Street Address : Mead Road, Nabua, Suva

Telephone : +(679) 381377 or +(679) 381139Fax : +(679) 370040E-mail : [email protected] : http://www.sopac.org.fj

Documents

Hydrodynamic simulation with mike21 of Mele Bay and Port