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Budapest University of Technology and Economics

Department of Hydraulic and Water Resources Engineering

SHALLOW LAKE HYDRODYNAMICS

Theory, measurement and numerical model

applications

A Hungarian-Finnish experience

Prof. Jnos Jzsa

Budapest, 2006


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SHALLOW LAKE HYDRODYNAMICS Theory, measurement and numerical

model applications

Author: Jnos Jzsa

Budapest, 2006

Mundus-Euroaquae lecture notes

Keywords

Shallow lake, air-water interaction, wind-induced circulation, water exchange, numericalmodelling, field measurements

Contact details

Budapest University of Technology and Economics

Department of Hydraulic and Water Resources EngineeringMegyetem rkp 3., K mf. 4.

H-1111 Budapest

HUNGARY

Tel: +36 1 463-1164 Fax: +36 1 463-1879

Web: www.vit.bme.hu

E-mail: [email protected]


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Shallow Lake Hydrodynamics

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Table of contents1 Introduction.............................................................................................................................1

1.1 Characteristic water motions in shallow lakes ................................................................. 11.2 On the role of wind-induced circulatory flows ................................................................ 21.3 The Hungarian shallow lakes as case studies ................................................................... 21.4 Main research tools........................................................................................................... 3

1.4.1 Field measurements .................................................................................................. 31.4.2 Analysis of the field data.......................................................................................... 41.4.3 Mathematical description and numerical modelling of the processes...................... 41.4.4 Selecting prevailing events....................................................................................... 41.4.5 Model calibration and verification for prevailing events ......................................... 41.4.6 Modelling experiments............................................................................................. 4

1.5 Outline of the present Notes ............................................................................................. 4

2 Air-water interaction in lake environment..............................................................................5

2.1 Near-surface wind and wind shear stress conditions........................................................ 62.1.1 Effect of the lake size, surrounding topography and land roughness....................... 6

2.1.2 Roughness conditions on the lake surface................................................................ 62.1.3 Near-surface wind profile and the aerodynamic drag .............................................. 72.1.4 Momentum exchange at the surface: the wind shear stress...................................... 82.1.5 Effect of an abrupt change in surface roughness: development of the internal

boundary layer.........................................................................................................82.2 Coupled modelling of the internal boundary layer and the surface wind shear stress ..... 9

3 Mathematical description of wind-induced lake flows......................................................... 13

3.1 Three-dimensional governing equations ........................................................................ 133.2 Identifying the shallow water conditions ....................................................................... 153.3 Depth-integrated approach ............................................................................................. 163.4 Three-dimensional approach with analytical solution of the velocity profile................ 17

3.5 Governing equation in vorticity form............................................................................. 173.6 Analysis of the vorticity balance .................................................................................... 18

3.6.1 Vorticity advection and dispersion......................................................................... 183.6.2 Vorticity sources..................................................................................................... 183.6.3 Vorticity sinks ........................................................................................................193.6.4 Relative weight of the source terms and their role in inducing circulations .......... 19

3.7 Depth-integrated governing equations ........................................................................... 193.8 Transport modelling of water exchange processes......................................................... 20

3.8.1 Water exchange as water mass advection-dispersion............................................. 203.8.2 Water exchange characterised as the evolution of the mean residence time field 203.8.3 Residence time field decomposition into contributions of sub-domains................ 20

3.8.4 Applied numerical transport model ........................................................................ 214 Field measurement of wind-induced flows........................................................................... 21

4.1 Applied tools, measuring and deploying principles .......................................................214.2 Data processing and analysis.......................................................................................... 24

4.2.1 Vector-time series representation ........................................................................... 244.2.2 Directional statistics ............................................................................................... 244.2.3 Digital filtering of the time series........................................................................... 24

5 Wind-induced circulations in Lake Fert (Neusiedl) ...........................................................25


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5.1 Progress in field measurements ...................................................................................... 265.2 Wind conditions.............................................................................................................. 27

5.2.1 Prevailing winds ..................................................................................................... 275.2.2 Investigation of the near-surface wind conditions in a bay.................................... 275.2.3 Calibration and verification of the surface wind shear stress model...................... 28

5.3 Wind-induced flows ....................................................................................................... 30

5.3.1 Prevailing circulation patterns in the study bay .....................................................305.3.2 Model calibration for prevailing circulation patterns............................................. 365.3.3 Hydrodynamic exploration of the Northern pelagic area ....................................... 395.3.4 Water exchange mechanisms as a combination of the seiche and circulations...... 44

6 Model investigations in a fictitious sample lake................................................................... 44

6.1 Aims of the investigations .............................................................................................. 446.2 Joint effect of the depth and wind shear stress distribution............................................ 446.3 Effect of emergent vegetation cover............................................................................... 506.4 Effect of large scale dredging......................................................................................... 51

7 Wind-induced circulations in Lake Velence......................................................................... 53

7.1 Local wind conditions ....................................................................................................537.2 Analysis of the measurements ........................................................................................ 547.3 Flow model calibration and verification......................................................................... 60

8 Wind-induced circulations in the Western part of Lake Balaton ......................................... 64

8.1 Field measurement campaigns in the late nineties .........................................................658.2 Wind conditions in the study region............................................................................... 688.3 Flow conditions .............................................................................................................. 68

8.3.1 Analysis of the simultaneous wind and flow measurements.................................. 698.3.2 Modelling the prevailing circulation patterns......................................................... 75

9 Suspended sediment transport modelling in lake Balaton.................................................... 77

10 On the applicability for other shallow lakes ......................................................................... 80

11 Acknowledgements...............................................................................................................80

12 References and relevant literature ........................................................................................ 80


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1 IntroductionShallow lakes have recently received enhanced attention all over the world. Their unique

value and multi-purpose utility have been more and more recognised which have led then tomisusing a number of them, thus worsening their ecological state even to an alarming extent atplaces. Furthermore, the recent changes in the global climate or, at least the fact that extremesseem to grow, changed also the boundary conditions for these vulnerable water bodies. In spiteif this, lake studies are still quite moderately financed compared to maritime research, andoften focuse on deep lakes, only. When trying to adapt the result obtained in deep water lakeor shallow coastal seas, one has to cope, nevertheless, with a number of problems due to thedifferences in the prevailing time and space scales found in shallow lakes. In fact, shallowlakes have got their own features and need special research and management methodology.

In Hungary, situated in the middle of the rather flat Carpathian Basin with no direct accessto the sea, besides the large rivers shallow lakes, first of all Lake Balaton, the largest shallowlake in Central-Europe, and Lake Fert (Neusiedler See in German) the West-most largesteppe lake in the continent are far the most important surface waters. In addition to thetraditional hydro-meteorological observations, hydrodynamic explorations started as early asthe sixties, first investigating the wind-induced seiche motion and looking for possible reasonsfor the unfavourable silting up in some parts of the lakes. Field measurement campaignscarried out in the sixties, seventies and early eighties provided significant results except for thewater currents on which data with only very limited value could be collected due to lackinginstrumentation. This was an obstacle for a while also for validating numerical flow modelsdeveloped since the mid-eighties.

Parallel to this in Finland, the country of thousand lakes, in the late seventies recordingcurrent meters, meant originally for sea conditions, were successfully applied also in shallowlakes, which gave a significant impulse to the lake hydrodynamics research there, including itsmultidimensional numerical modelling. In fact, good quality calibration data proved vital forreliable model parameterisation.

In order to combine shallow lake research efforts of the two countries a bilateralcooperation framework was established in the mid-eighties, leading to a number of fruitfuljoint research programs and methodological development in both countries. The present Notesattempt to give an insight into the achievements in theory, measurements and numericalmodelling related to the Hungarian lakes studied in the framework of the cooperation,discussing more in details the ones with applicability for other shallow lake hydrodynamic,sediment transport and water quality studies.

1.1 Characteristic water motions in shallow lakes

Except rather few cases, water motions in shallow lakes are primarily induced by the wind.Part of the momentum of the wind over the lake is transferred into the water at the lake surface

generating then e.g. waves, turbulence, drift currents, Langmuir circulations, as well as largescale circulations and seiche. This momentum flux drives then indirectly the exchangeprocesses at the lake bottom, mixing in the water body, and the interaction between the littoraland the pelagic zones. In fact, in general the more shallow, the more efficient the influence ofthe external surface forces on the bottom. As one of the main features of shallow lakes, surfacewaves generated even by moderate winds can induce shear stress at the bottom and stir up theuppermost sediment layer of the lake bed (see e.g. Sheng and Lick, 1979; Dyer, 1986; Luettichet al., 1990; Rkczi and Jzsa, 1999). While kept in resuspension by turbulence, particles canbe transported by advective currents over large distances and settle if local conditions promote


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it. As another consequence of shallowness, wind-induced turbulence can easily destroytemperature stratification thus making the lake homogenous.

1.2 On the role of wind-induced circulatory flows

The largest scale wind-induced hydrodynamic processes are the oscillatory (seiche) andcirculatory currents. Oscillations in themselves usually would not result in significant netexchange, but when accompanied wind-induced circulations, their combined influence issignificantly enhanced. While seiche can be generated either by the wind or horizontalatmospheric pressure gradient, circulations are induced primarily by the surface wind shearstress field and shaped by various factors.

Circulations, though always three-dimensional in the most general sense, very often showorganised horizontal pattern consisting of large-scale gyres. In fact, circulation needs rotationsources to be generated and maintained. Such sources can be originated e.g. from bottomslopes relative to the depth (Simons, 1980), irregular wind field (Jzsa et al., 1990; Jzsa et al.,1998; Jinxiu et al, 1999), irregular lake surface exposure to the wind, due to lakeshore forest(Podsetchine and Schernewski, 1999), emergent littoral vegetation (Sarkkula et al., 1991;Jzsa et al., 1999) or indirectly from the earth rotation itself (Simons, 1980).

Windstorm with 10-15 m/s speed can easily generate lake-wide circulations with 10-15cm/s flow velocity which if sustained e.g. for half day may result in significant, even 5-6 kmnet advection. In most shallow lakes it is comparable with their horizontal size, consequentlysuch circulations with sufficient duration and stability can often induce lake-wide exchange ofwater masses and all the substances carried by them.

Though hydrodynamics is challenging in itself, it has been also recognised that thedescription of sediment transport, morphological changes, water quality or even ecologicalprocesses have to be built on sound lake circulation bases. In order to do that it is essential toexplore the most important features of the external forces driving the hydrodynamics.

1.3 The Hungarian shallow lakes as case studies

The protection and restoration of the large shallow lakes has become one of the main tasksin water management in Hungary (Fig. 1). As was mentioned earlier, the largest lakes havelong been investigated both in domestic and international framework. As far the mostimportant, Lake Balaton, nearly 80 km long, 1-10 km wide and 3 m deep on average, is one ofthe most visited, in summer time often overcrowded lake in Europe, offering a great variety ofleisure time activities. High nutrient supply from stirred up sediment grains as internal loadmay, however, result fro time to time in poor water quality. One of the ways to improve theconditions is to remove the upper-most polluted bed layer by dredging. Detailed informationon the hydro- and sediment dynamics as the lake response to the prevailing wind is mostessential when planning and performing such a dredging.

Lake Fert, the second largest, extremely shallow lake with its 1.5 m mean depth and large

reed belt covering more than half of its surface also needs protection and restoration measuresat places. There mainly the sediment and reed quality has been in the focus due to very lowdissolved oxygen content especially late summer as a result of insufficient water exchange between the pelagic and littoral zones. To find the way for proper measures to preserve oreven improve the existing conditions has initiated long-term hydrodynamic investigations thefirst phase of which was to explore wind-induced circulations in the pelagic areas.


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Much smaller in total surface but covered by reed even in relatively greater extent, LakeVelence with its 2 m average depth is the third in the series that has been thoroughlyinvestigated from hydrodynamic point of view. As the lakebed and the reed cover weresignificantly reshaped by large scale dredging in the late seventies, it has recently becomenecessary to identify the main hydrodynamic features of the lake after 20 years. The

occasional adverse effects, e.g. unfavourable siltation at places, were also considered in orderto support possible future remedial works.

The above mentioned three lakes have served as excellent case studies providing both lake-specific features and general conclusions on shallow lakes. In fact, the conditions made it possible to carry out useful cross-verifications especially as far as the new findings wereconcerned. Comparisons with similar cases from abroad (Finland, China and Germany) furtherenhanced the validation of the general applicability (Sarkkula et al., 1991; Jinxiu et al, 1999;Podsetchine and Schernewski, 1999).

1.4 Main research tools

In the research projects a fruitful coupling of theory, field measurements and numerical

modelling have been performed. As the projects were carried out in several subsequent phasesthe experience could be recycled and research tools as well as methodologies could beupgraded accordingly.

1.4.1 Field measurements

In all the research campaigns field measurements consisted of simultaneous, long term,automated recording type of observation of the wind, flow and recently turbidity at a numberof representative sites. In most cases the campaigns focused only on a particular part of thelake at once. In some phases special emphasis was put on special issues such as e.g. the wind

Figure 1. Hungary in the Carpathian Basin. Lake Balaton in Mid-West, Lake Fert in North-west,Lake Velence somewhat to the East-Northeast from Balaton.


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speed variation along the fetch, water exchange at the interface zone of adjacent bays, or thecirculation structure in near-shore regions.

In all the research projects the up-to-date instrumentation provided by AanderaaInstruments, Norway was utilised. In fact, their wind station, propeller as well as Acoustic-Doppler type current meters, and optical turbidity sensors have proved very robust tools inmost situations. Nevertheless, in order to apply them in the given shallow (typically from 1 to

4 m) conditions, special deploying techniques had to be developed.1.4.2 Analysis of the field data

The simultaneous wind and flow velocity vector time series collected in the measurementprograms were then analysed with special attention to the nature of the processes. Both globalcharacteristics and particular features of prevailing storm events were looked at. Vector timeseries display, low-, high- and band-pass filtering as well as directional statistics served as themain analysis and evaluation techniques.

1.4.3 Mathematical description and numerical modelling of the processes

The mathematical description of the hydrodynamic processes is based on the generalconservation laws applied to volume and momentum. Starting from the general three-

dimensional time-dependent approach, reasonable simplifications can be introduced due to theshallow water conditions. As will be seen, special forms of the equations facilitate to highlightthe main driving mechanisms and understand the importance of the improved representation ofexternal forcing fields.

In order to describe the lake-wide water mass transport in most cases the depth-integratedapproach of the wind-driven circulation patterns proved sufficient.

1.4.4 Selecting prevailing events

When selecting prevailing storm events and corresponding flow patterns, to identifyperiods and situations for numerical model calibration and verification was of primary interest.Particular events with clear wind direction, various wind speed range and duration were first

classified and subject to inter-comparison.1.4.5 Model calibration and verification for prevailing events

Model calibration and verification covered the proper tuning of wind, flow and sedimenttransport models in order to obtain a kind of best fit of the model result to the measuredvalues. Once having calibrated, the reasonable approximation of data in further selected casescould then verify the model parameters. Cases similar to each other could confirm thereproduction ability in given specific situations, whereas the acceptable fitting of calculatedand measured quantities in substantially different conditions or lakes could validate therobustness of the approach.

1.4.6 Modelling experiments

Apart from case studies of existing lakes, model experiments in simple, fictitious lakegeometry, nevertheless, representing realistic scales, were also performed. The simplifiedbathymetry and shape made it possible to carry out a number of sensitivity tests efficiently.

1.5 Outline of the present Notes

In the remaining part of the present Notes first an overview of the main lines of the air-water interaction in lake environment, including a model coupling the internal boundary layerdevelopment over the lake to the wind shear stress at the lake surface is provided.


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Once the main characteristics of the external driving force are given, the governingequations of the water flows are presented. Special interest is paid to the vorticity form of theshallow water equations and the analysis of the various terms playing the role of vorticitysources and sinks in the system. This approach facilitates to identify the main factors andestimate their weight in generating advective circulation patterns.

As information gained from field measurements has been so vital in all the research

projects, separate chapters are devoted to present the main findings in the three case studylakes. However, a special chapter on the most relevant results of a fictitious lake is insertedafter the one on Lake Fert, in order to give a systematic presentation of the most importantshallow lake features in simplified conditions.

In order to give a typical example on the utility of hydrodynamics research, at last the main principles of the suspended sediment transport modelling of Lake Balaton are given alongwith the application to planning thin-layer dredging in one part of the lake.

Finally a set of recommendations on possible utilisation of the achievements in othershallow lake studies is given, along with the need for field data and numerical models forreasonable adaptation in the local conditions.

The Notes are supplemented with a representative, though not complete selection ofliterature relevant to wind-driven shallow lake hydrodynamics in general and to the varioussub-topics in particular.

2 Air-water interaction in lake environmentThe air masses moving over the lake surface transfer part of their momentum to the water

whereas the wavy water surface, first of all by its characteristic aerodynamic roughnesssignificantly influences the near surface wind profile. In fact, momentum is transferred as aresult of complicated air-water interaction mechanism, the proper parameterisation of which isstill subject to vast research activities.

In the subsequent case studies it has become more and more evident that the theoreticalexplanation and related numerical reproduction of the measured wind-induced circulationpatterns can hardly be done if spatial irregularities inherent in the surface wind shear stressfield are poorly represented. The zero-order approach, namely running a flow model withspatially constant wind stress distribution often resulted in flow velocities opposite to themeasured ones. In order to explore the main features, relevant space scales, fetch dependentvariation and limitation of the wind shear stress, appropriate measurement campaign withspecial, multiple wind measurement set up had to be performed. Detailed analysis of thesimultaneous wind data and thorough parameter calibration of the relationships resulted in asubstantially improved estimation of the wind shear stress field over lakes. The essential of theapproach, well known in boundary layer meteorology but more or less ignored so far in lakecirculations, lays in the development of a so-called internal boundary layer (IBL hereafter)downwind of an abrupt change in the surface roughness conditions, such as the one at theland-water interface.

The improved wind stress field could provide good fitting of calculated and measured flowpatterns in a wide range of situations. Flow models upgraded according to that proved then inall the other case study lakes the general applicability, provided proper value of the upwindland surface roughness parameter, dependent first of all on the vegetation cover, wasintroduced.


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2.1 Near-surface wind and wind shear stress conditions

Before discussing the coupled near-surface air-water model, an introduction to thetheoretical and semi-empirical relationships leading to a reasonable estimation of the strengthof the momentum flux from the wind to the lake at one particular site is given. It is also meantto provide a better insight to the way of approximation of the surface wind shear stress basedon single point wind measurements at a given height above the surface.

2.1.1 Effect of the lake size, surrounding topography and land roughness

Some of the lakes are horizontally large enough to give room to the development of lightand moderate winds or short duration storms with significant local differences. There are,however, few lakes large enough to feel the synoptic-scale spatial structure of atmosphericcyclones.

As another source of irregularity, large scale topographic features upstream of a lake canalso result in spatially varying wind field over the lake and can have occasionally significantimpact on shaping the circulation patterns, as have been recognised in several cases.(Shanahan et al., 1986; Jzsa et al., 1990; Jinxiu et al., 1999). Its reasonable estimation needseither dense enough wind measurement network, preferably coupled then with some sort of

mezo-scale atmospheric boundary layer model of appropriate vertical and horizontalresolution. However, measurements are seldom dense enough in space to form a firm basis forwind field reconstruction in themselves, moreover, there still seems to be a gap to fill inbetween numerical modelling capability of small scale computational fluid dynamics problemsand large, synoptic scale atmospheric modelling.

In any case, near the surface it is primarily the upwind land roughness conditions thatdetermine the vertical structure of the horizontal wind speed. It is of course strongly related tothe land use in the surroundings. In case of e.g. more or less homogeneous vegetation coverthe wind profile is in equilibrium, which is then severely disturbed by the abrupt change in thesurface roughness conditions at the shoreline. In fact, even the most rough-looking watersurface is several orders of magnitude smoother in an aerodynamic sense than a typical land

surface. This sudden change in the lower boundary conditions for the airflow results in thedevelopment of a new, so-called internal boundary layer, the height of which grows along thefetch following rather simple semi-empirical relationships. Within the IBL the well-knownlogarithmic distribution of the wind speed still holds, whereas outside this layer the profile isundisturbed and preserves its over-land features. However, the momentum flux from air towater is governed by the shear stress at the bottom of the IBL that is at the lake surface.

It will be shown that provided the size of the lake in the prevailing wind direction is in theorder of less than say 20 km, the IBL-based estimation in thermally neutral, nearly stabilised,moderate and strong winds can explain quite a large part of the wind shear stress distribution.

2.1.2 Roughness conditions on the lake surface

In the formulation and parameterisation of the wavy water surface roughness Charnocks(1955) paper has been proved a milestone. Based on data collected in a moderate sizereservoir, it was probably first him to relate the effective height of the roughness elements ofthe wavy surface to the so-called friction velocity W and the acceleration of gravity g. In

average conditions this estimation leads to a value much larger than the typical thickness ofthe so-called viscous sub-layer, therefore the lake surface is in aerodynamically roughconditions in general. It was much later concluded by Wu (1994) by analysing a combined setof data, that the water surface reaches its least rough state at about 5 m/s, a transition rangefrom capillary to gravity waves, and becomes rougher when either increasing or decreasing in


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speed. In the upper range, the so-called roughness height 0z due mainly to the shorter waves

on the lake surface can be considered simply as

g

Wz

2

0= , (1)

providing the well-known, widely used Charnock-formula, with as a coupling parameter.

Though considered as a parameter to be calibrated, reasonable results have been widelyobtained with a value of 0.0185 given by Wu (1982). Recently, a lot of effort have been also put into linking the aerodynamic roughness height more directly to the wave conditions,furthermore, identifying the role of wave stress and turbulent stress in the total stress, but anoverall formula has not come out to date (see e.g. Donelan 1998; Taylor and Yelland, 2001).

2.1.3 Near-surface wind profile and the aerodynamic drag

Once the roughness height is estimated, and there are no significant differences betweenthe air and water temperature, that is the thermal stratification is practically neutral at theinterface, the vertical distribution of the near-surface, horizontal wind speed W(z) in terms ofthe heightzabove the lake surface obeys the well-known logarithmic law as follows:

( )0

lnz

zWzW

= , (2)

where is the von Krmn constant equal to 0.4. In fact, it starts with zero speed at theroughness height with sharp initial increase, which becomes then more and more gentle withheight. Note that this relationship makes it possible to determine both the roughness heightand the friction velocity from wind speed measurements taken in one vertical at two differentheights.

The ratio of the square of the friction velocity and that of the wind speed at zis known asthe aerodynamic drag zc corresponding to the given height (see e.g. Plate and Wengefeld,

1979; Graf and Prost, 1979; Stull, 1991). Using (1) and (2) it is expressed as

2

2

2

2

0

2

2

2

lnln

=

==

W

zg

z

zW

Wc

z

z

. (3)

The customary wind measurement height is usually 10 m for which the drag formula,combining (1), (2) and (3) is

2

21010

2

2

2

2

2

0

2

210

2

10

10ln

10ln

10ln

=

=

==

Wc

g

W

g

z

W

Wc

, (4)

Collecting and analysing a number of data from various sources, Wu managed to derivethe following linear relationship between the wind at 10 m and the corresponding dragcoefficient, proved reasonable from breeze to hurricane (see e.g. Wu, 1982):

( ) 31010 10065,08,0+= Wc . (5)

Similar formulae have been derived by a number of researchers with slight differences inthe empirical coefficients.


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2.1.4 Momentum exchange at the surface: the wind shear stress

Now using the relationship

a

sW

= , (6)

which by definition links the friction velocity to the surface wind shear stress s and air

density a , a series of formulae can be set up, all expressing the shear stress in various

combination as follows:

2102

2

22

102

0

22

10102

10ln

10ln

W

W

gW

z

WcW laaas

=

===

. (7)

Note that by using (1), (2), (5) and (6) the whole set of parameters (friction velocity,roughness height, drag coefficient), finally the wind shear stress and the wind profile itself can be determined in an iterative way, based on wind speed measurements taken at one single,arbitrary, though representative height, provided the boundary layer structure above the lake isdeveloped at least as high as the measurement height. In other words, we are in principle freeto choose the anemometer height, which is a great help in most field conditions, when facingsevere deploying constraints, or if significant, though slow water level changes occur.

Once the wind speed and the drag coefficients are determined, the local specific value ofthe rate of work of the wind on unit water surface area can be given as (West et al., 2000)

31010WcE as = . (8)

If these parameters are known not only locally but all over the lake based on measurementsor some functional expansion, the total available work per unit time can be obtained byintegrating the specific values over the entire lake surfaceA as

( )=A

sts AEE d, . (9)

In case the lake is partly vegetated and the emergent vegetation covers significant part ofthe lake the above mentioned work input is reduced accordingly. Of course the vegetation,either patchy or not, floating on the surface or even if somewhat submerged, can modify the bulk surface roughness conditions, consequently the wind profile and finally the effectivewind shear stress acting on the water surface as well.

2.1.5 Effect of an abrupt change in surface roughness: development ofthe internal boundary layer

As was described earlier, the abrupt change in the aerodynamic roughness conditions at thelakeshore result in an IBL development downwind on the lake. Such phenomena are wellknown and often detected in the lower part of the atmospheric boundary layer. The name IBLis usually retained for boundary layer developing due to mechanical turbulence whereas theone developing due to temperature changes is called thermal boundary layer.

Till very recently, the use of IBL-based wind, wind stress and energy estimations in lakeenvironment have been restricted to determine turbulent energy production rate for verticaltemperature stratification calculations in deep lakes, improve humidity and evaporation fluxcalculations, or to improve wind input for surface wave modelling, only.


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In the Finnish-Hungarian lake research studies reported here, even if wind was measuredoften at one single site, detailed flow measurements could indirectly indicate that there spatialstructure must be significantly influenced by fetch-dependent wind shear stress variation. Infact, at the typical horizontal length scale of the study lakes fetch-unlimited conditions canhardly be found. It is the case with the IBL development over the lake too in stabilized, neutralconditions, resulting in a continuous speed-up of the wind at given height. As will be seen,

however, the the strongest growth rate is found at the initial part, which becomes thenasymptotically more and more gentle. As will be also made clear, for a given wind directionthis fetch-aligned feature depends mainly on the characteristic land surface roughness and thewind speed, the former being far the most important parameter to calibrate.

Though thermal effects can significantly modify the picture, in most of our selected eventsthe conditions were (maybe due to the efficient mixing capacity of the high wind speed) nearlyneutral and the mechanically generated IBL alone could explain most of the observed changes.

Following the general description given by Taylor and Lee (1984), the IBL height b startsto develop at the shoreline in the section of the abrupt roughness change with an initialthickness

( ) 2,00 z

b

= , (10)

equal to that of the roughness height 2,0z of the lake water surface. Its development is

governed mainly by two parameters, the fetchFand the lake surface roughness height as

( )8,0

2,02,075,0

=

z

FzFb , (11)

the roughness being also a function of the fetch.

As a result of this gradual change, the vertical profile of the horizontal wind speed consistsof two parts: outside the IBL the profile is still assumed to be identical with that given at theshoreline at heightzover the land, simply expressed by

( ) ( ) ( )0: zzb WFWzF = , (12)

whereas within the IBL the wind speed is determined by the formula

( ) ( ) ( )

( )

( )

2,01,0

1,02,0

lnln

lnln

0:

z

F

z

z

z

F

z

z

WFWzFb

b

zzb

=> . (13)

Thus the profile remains logarithmic by nature, however, its actual shape is calculated bytwo subsequent transformations.

2.2 Coupled modelling of the internal boundary layer and thesurface wind shear stress

A coupling between the IBL development and surface wind shear stress estimation can beestablished if the Charnock-formula for estimating the local lake surface roughness isintroduced in the algorithm in an iterative way. It means to solve the IBL and surfaceroughness height development along the fetch given the wind speed, its measurement height(the so-called anemometer height) over the land and the land roughness height. Since the lake


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roughness calculation requires the wind speed at 10 m, an initial transformation is needed ifwind data happen to be available at a height different from that as follows:

( ) ( )

1,0

1,010

ln

10ln

00

z

z

zWW

aza

= . (14)

Using (1), (5) and (6) tailored to the IBL formulae (11) and (13) a wind speed equal to

( ) ( )( )

( )

( )( )FzF

z

z

F

FzWFW

b

b

2,01,0

1,02,01010

ln10

ln

ln10

ln

0

= (15)

can be obtained iteratively, which gives then the actual speed in case

( ) ( ) ( )FWFWFb 1010:10 => , (16)

whereas no change occurs in case

( ) ( ) ( )0:10 1010 WFWFb = . (17)

A set of longitudinal profiles is given below on the fetch- and wind speed-dependentbehaviour of the parameters. A typical range of wind speed capable to induce significant watermotion was used and the first 2 km of the fetch was considered. The roughness height of theupwind land was set to 0.15 m, a verified characteristic value for Lake Fert, as will be seenlater. Note especially the longitudinal wind and shear stress profile, moreover its significantincrease due to enhanced 0.4 m roughness, representing high (but not densely packed) forest.

0

10

20

30

40

50

60

70

80

90

100

0 200 400 600 800 1000 1200 1400 1600 1800 2000

F, [m]

b,

[m] 12 m/s

10 m/s

8 m/s

Figure 2. Development of the internal boundary layer over the water surface along the fetch forvarious offshore wind speeds at the shoreline, land roughness equal to 0.15 m.


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0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.001

0 200 400 600 800 1000 1200 1400 1600 1800 2000

F, [m]

z0,2,

[m] 12 m/s

10 m/s

8 m/s

Figure 3. Roughness height of the water surface along the fetch for various offshore wind speeds atthe shoreline, land roughness equal to 0.15 m.

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0.002

0 200 400 600 800 1000 1200 1400 1600 1800 2000

F, [m]

C10,

[m] 12 m/s

10 m/s

8 m/s

Figure 4. Aerodynamic drag coefficient along the fetch for various offshore wind speeds at theshoreline, land roughness equal to 0.15 m.


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5

6

7

8

9

10

11

12

13

14

15

0 200 400 600 800 1000 1200 1400 1600 1800 2000

F, [m]

W10,

[m/s]

12 m/s

10 m/s

8 m/s

Figure 5. Wind speed at 10 m height above the water surface along the fetch for various offshorewind speeds at the shoreline, land roughness equal to 0.15 m.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 200 400 600 800 1000 1200 1400 1600 1800 2000

F, [m]

f,

[Pa] 12 m/s

10 m/s

8 m/s

Figure 6. Wind shear stress at the water surface along the fetch for various offshore wind speeds atthe shoreline, land roughness equal to 0.15m.


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 200 400 600 800 1000 1200 1400 1600 1800 2000

F, [m]

f,

[Pa] 12 m/s

10 m/s

8 m/s

Figure 7. Wind shear stress at the water surface along the fetch for various offshore wind speeds atthe shoreline, land roughness equal to 0.4 m.

3 Mathematical description of wind-induced lake flowsIn the overview of the mathematical description of wind-induced lake hydrodynamics we

first follow the derivation by Simons (1980) and Falconer (1994) but then pay particularattention on the so-called vorticity form of the depth-integrated governing equations. By usingthat, the role of the various terms played in the generation and shaping of the circulation

patterns becomes certainly more visible. Vorticity input first of all due to the curl of the windshear stress field (absent in case of uniform distribution), and that of the relative bottomgradient will be discussed as their joint effect seem to play the main role in the process.

In order to quantify water mass exchange mechanisms in complex conditions, a novelsimulation method of the so-called mean residence time distribution will be also presented,and later on applied when investigating of the sensitivity of the wind-induced processes onvarious parameters.

3.1 Three-dimensional governing equations

The conservation of water volume and momentum in space and time expressed in velocity-pressure form, assuming constant density and hydrostatic pressure distribution, often valid in

shallow water bodies under the action of wind can be written as follows:

0=

+

+

z

w

y

v

x

u, (18)

x

pfv

z

uw

y

uv

x

uu

t

u

=

+

+

+

1, (19)

y

pfu

z

vw

y

vv

x

vu

t

v

=

+

+

+

1, (20)


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z

pg

=

1. (21)

where x, y and zare Cartesian space coordinates, u, v and w are velocity components in x, yand z-direction, p is pressure, is water density and f is the Coriolis coefficient due to the

earth rotation.

Given that there is at least an order of magnitude difference between the horizontal andvertical velocity magnitudes, after appropriate time-averaging, adopting the Boussinesqapproximation for turbulent momentum exchange and introducing the divergence operator

=

yx, (22)

in the horizontal plane only, the momentum equations from now on for the time-averagedvariables can be written as

( )

++

=+

z

u

zu

x

pfvu

t

uvtht ,,

1

v , (23)

( )

++

=+

z

v

zv

y

pfuv

t

vvtht ,,

1

v (24)

where v stands for the (u, v) horizontal velocity vector, ht, and vt, are the horizontal and

vertical eddy viscosity coefficient, respectively, and from this point on all the variables aremeant to be the so-called Reynolds-averaged ones. In free surface flows for these equations atthe free surface the

:=z

+

= v

tw (25)

so-called kinematic condition, whereas at the bottom

:b

zz = bzw = v (26)

boundary condition applies. To make the description complete, one has to specify themomentum flux at both boundaries, expressed in the form of shear stresses as follows:

:=z

xs

vtz

u ,, =

, (27)

ys

vtz

v ,, =

, (28)

:bzz=

xb

vtz

u ,, =

, (29)

ys

vtz

v ,, =

, (30)

where xs, and ys, are the surface, xb, and yb, are the bottom shear stress components,

respectively.


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3.2 Identifying the shallow water conditions

In certain conditions further simplifications can be introduced to the above description,the validity of which can be determined by various characteristic dimensionless anddimensional quantities (Simons, 1980; Shanahan et al., 1981; Hutter, 1984). In the followingparagraphs the case study lakes will be analysed from this point of view, in order to defineshallowness in the light of these quantities.

First, it is important to estimate the rate of irregularity in the vertical profile of thehorizontal velocities, whether the flow can be reasonably approximated by depth-averagedvalues. For this purpose Hutter (1984) introduced the

L4

VH

T

T

,

2

adv vt

h

==

formula, where H is the characteristic depth, V is the characteristic horizontal velocity and L isthe characteristic horizontal length scale over which the velocity presents a significant change.It expresses the ratio of the Th and Tadv characteristic timescales of momentum transportgenerated by the vt, vertical eddy viscosity and the V horizontal advection, respectively. In

case this ration is much smaller than unity (say < 0.1), it indicates that the irregularities in thevertical profile of the horizontal advective velocity are efficiently damped by the vertical eddyviscosity, as a result of which the profile is quite uniform except of course the highly shearednear boundary regions. Assuming 3102 m2/s for the eddy viscosity (see e.g. Simons, 1980;Shanahan et al., 1981; Hutter, 1984; Signell et al., 1990), is estimated for the study lakes asfollows:

Lake Balaton: H=3m; V= 110 m/s; L= 3105 m 02.0 ;

Lake Fert, inner bays: H=1,5m; V= 110 m/s; L= 310 m 03.0 ;

Lake Fert, large northern pelagic region: H=2m; V= 110 m/s; L= 3105 m 01.0 ;

Lake Velence, middle pelagic region: H=2m; V=

2

105

m/s; L=

3

102

m

01.0

.As can be seen the criteria are met in every case, which means in other words that the flowfield can be approximated in depth-averaged way, without significant loss of information inthe description of the resulting horizontal water exchange, compared to direct three-dimensional simulations.

In order to determine if the Coriolis-effect is able to exert its influence in the verticalresulting in the Ekman-spiral in the velocity field, Simons (1980) and Heaps (1984) suggestedto use the

f

vt,2

H

=

dimensionless depth, as the ratio of the characteristic depth and the so-called Ekman-, orfriction-depth. In case this quantity is much smaller than unity, the velocity distribution alongthe vertical is dominated only by the shear stresses at the surface and the bottom, otherwisethere is room for the Ekman-spiral to develop. It is to be noted that in moderate windconditions accompanied with moderate vertical eddy viscosity even a small depth lake canbehave as if it was deep in the sense of the above parameters, as was shown by several field


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measurements data. Assuming 14 s10 =f at the latitude of the case study lakes, can be

estimated as follows:

Lake Balaton: 15.0 ;

Lake Fert, inner bays: 08.0 ;

Lake Fert, large northern pelagic region: 1.0 ;

Lake Velence, middle pelagic region: 1.0 .

3.3 Depth-integrated approach

Though losing the vertical structure, nevertheless, in shallow conditions preserving most ofthe horizontal water mass transport features, the governing equations can be integrating alongthe vertical resulting in a set of equations for the

( ) zzuqbz

x d=

, ( ) zzvqbz

y d=

(31)

specific discharges (or volume fluxes), and for thebzh = (32)

water depth. Dividing the fluxes by the depth results in the depth-averaged velocities

h

qU x= ,

h

qV

y= . (33)

Applying the boundary conditions, considering that pressure gradients over large domainscan be originated also from the ap atmospheric pressure, and introducing a so-called

momentum dispersion coefficient accounting for momentum correction in averaged non-

uniform velocity profile, the following set of equations hold:

0=+

q

t

(34)

x

p

xg

x

p a

+

=

(35)

y

p

yg

y

p a

+

=

(36)

( ) ( )xhexbxsa

yxx q

x

ph

xghfqq

t

q++

=+

,

,,V

(37)

( ) ( )yhe

ybysaxy

yq

y

ph

yghfqq

t

q++

=+

,

,,V

(38)


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3.4 Three-dimensional approach with analytical solution of thevelocity profile

Not discussed here in details, we mention that there are approximate analytical ways toreconstruct the vertical distribution of the horizontal velocities. We refer to the work of Cruz(1997), who managed to set up an analytical profile with sharp, logarithmic transition both atthe surface and at the bottom, often found in nature and lab conditions. It can then serve to

quantify the dispersion correction coefficient and to obtain proper bottom shear stresses morerealistic than the ones estimated on depth-averaged velocity basis.

3.5 Governing equation in vorticity form

Assuming steady-state flow conditions as a result of long enough constant external forcing,smooth spatial variation of the momentum correction coefficient and introducing a depth-averaged horizontal momentum exchange coefficient, the depth averaged momentumequations can be written as

( )Uhhhhx

gfVU hexbxs ++

= ,

,, 1V

, (39)

( )Vhhhhy

gfUV heybys ++

= ,

,, 1V

. (40)

A different form of the above set of equations can be established for the velocity and the

V=

=

y

U

x

V (41)

curl of the depth-averaged velocity by taking the curl of the equation set resulting in

( ) ( )

+++= V

1VV , h

hhhf he

bs

, (42)

representing the overall transport of vorticity in a velocity field influenced largely of course bythe vorticity itself in a domain small enough for the Coriolis coefficient to be constant. Furthersimplifications are introduced assuming that horizontal vorticity exchange is governed mainlyby an apparent local viscosity coefficient giving the form

( )

2VV +++= h

bs

hhf . (43)

Expending the two first terms on the right hand side, applying the continuity relationshipand multiplying by the depth provides the final form of the vorticity balance as follows:

( )

211

VV+

++=

h

b

ss hh

h

hhhfh (44)


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3.6 Analysis of the vorticity balance

Below a short interpretation of the various terms in the vorticity balance is given, withspecial attention on the sources and their dependence on the main lake characteristics. One hasto keep in mind that in the simplified system vorticity with vertical axis that is rotating in thehorizontal plain can only occur.

3.6.1 Vorticity advection and dispersionThe terms

Vh (45)

and

2 hh (46)

express the advective transport and horizontal redistribution (with a tendency towardsmoothing) of the vorticity once introduced to the computational domain.

3.6.2 Vorticity sources

The termh V (47)

expresses the effect of the depth variation on strengthening or weakening vorticity. Imaginevertical vortex tubes attached both to the surface and the bottom according to Helmholtzs principle, which then stretch or shorten, consequently speed up or slow down in angularvelocity when transported toward deeper or shallower zones, respectively. Thus advectivetransport along isobaths results in no change in vorticity.

The term

hf V (48)

represents the vorticity source due to the Earth rotation itself. Note, however, that this termdiffers from zero only if flows are not parallel to the isobaths. Consequently long-shore flowsare not affected, but so are e.g. the end-zones of elongated-shape semi-or totally enclosedbasins.

From the point of view of shallow lake circulation certainly the most important terms are

s

1

(49)

and

hh s

1, (50)

(49) representing the vorticity introduced to the system by the curl of the wind shear stressfield resulting from its spatial irregularity governed largely by the IBL evolution discussedearlier, and the combined effect of the local wind shear stress vector and relative bottomgradient in the form of a vector-product (50). Note first that in case of uniform wind shearstress field (49) vanishes and the bottom slope related term dominates the circulation resultingin the well-known barotropic topographic gyres. Second, bottom gradient are activated asvorticity sources by wind shear stresses inclined to them, that is winds perpendicular to


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isobaths do not do that, but winds blowing parallel to them are most efficient in that. Third, ascan be seen (50) scales inversely with the depth whereas (49) is independent of it,consequently in given bottom topography conditions lake-wide water level rising results inweakening, in turn water level lowering in strengthening of (50) against (49). These featureswill be referred to both in the case studies and in the analysis of the simple model lakebehaviour.

3.6.3 Vorticity sinksVorticity sinks in the system are linked to dissipation terms acting at the bottom as

b

1

(51)

hh

b

(52)

their primary influence is to suck vorticity out of the system via rotation damping due tofriction.

3.6.4 Relative weight of the source terms and their role in inducingcirculations

A rigorous order of magnitude analysis of the different terms should be carried out indimensionless form by introducing characteristic length and velocity scales. Here we skipgoing into the details, merely underline that in our study cases the source terms related to thewind shear stress and the bottom gradient dominated the patterns. We will discuss it includingalso some deviatory cases and locations in the modelling chapters.

3.7 Depth-integrated governing equations

The numerical model applied to describe the wind-induced seiche motion and thecirculatory water mass transport pattern is based on the following, somewhat further

simplified form of the governing equations as follows:

0=

+

+

y

q

x

q

t

yx (53)

( )xhe

xsxsby

yxxx qx

zhghfq

h

qq

yh

q

xt

q 2,

,,2

++

+=

+

+

, (54)

( )yhe

ysysbx

yxyyq

y

zhghfq

h

qq

xh

q

yt

q2

,,,

2

++

+=

+

+

, (55)

supplied with the IBL-based surface wind shear stress model outlined earlier, and

xyxxb qqqhk

g 22372,

+=

, (56)

yyxyb qqqhk

g 22372,

+=

, (57)

conventional Manning-type quadratic bottom shear stress laws.


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The discretised numerical solution of the equation set is performed by standard finitedifference method on equidistant Cartesian grid.

3.8 Transport modelling of water exchange processes

Though the flow model provides the space-time evolution of the wind-induced lake

motion, further modelling techniques are needed for quantifying more in details e.g. itsexchange mechanism.

3.8.1 Water exchange as water mass advection-dispersion

Focusing on quantifying the water mass exchange only, the process in depth integratedform can be considered as conventional advection-dispersion, where this time the transportedsubstance is the water itself. Residence time (in some other interpretation flushing time) is oneof the representative indicators of water exchange efficiency. There are various ways toestimate it either on an Eulerian solute transport or Lagrangian particle-tracking basis. Below,however, a novel technique will be briefly outlined making it possible to calculate even so-called mean residence time fields and their dynamics on a water age simulation basis (Jzsaand Krmer, 2000; Jzsa et al., 2001).

3.8.2 Water exchange characterised as the evolution of the meanresidence time field

In the approach, in a way identical to the conventional advection-dispersion description theage of water masses is considered as transport variable. Water masses with various age in thestudy domain are advected without age modification, undergo mixing to each other and getolder following a zero-order reaction kinetics law, expressing the fact that ageing is linear intime and independent from the actual age. The one-dimensional form of the equation with pure advection and unit rate of growth of the residence timeR with boundary conditionrepresenting zero age at the inflow section can be written as

( ) 0,,1 0 =+

=

txRx

Rut

R , (58)

providing linearly increasing longitudinal residence time distribution at constant flow velocity.

Including turbulent diffusion (and assuming mixing of water particles is identical to that ofany other substance dissolved in it will give the following concise vector form of the process:

( ) 1=+

RR

t

RtDv . (59)

In the investigations to be presented later on here the depth integrated version of theapproach is applied written as

( ) ( ) ( ) hRhhRthR hhh =+

dDV . (60)

3.8.3 Residence time field decomposition into contributions of sub-domains

The basically Eulerian description can get some Lagrangian character if the study domainis divided into characteristic sub-regions (e.g. zone with emergent or submerged vegetation,areas with polluted bottom sediment etc.) and the residence time is decomposed to the


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contributions of the sub-regions by means of a kind of multi-species version of the approachas follows:

)( ) ( ) hRhhR

t

hRjhjh

jh =+

,,

,d

DV , (61)

based on which the overall mean residence time field can be reconstructed by simple

summation as

=

=n

j

jhh RR1

, . (62)

3.8.4 Applied numerical transport model

The two-dimensional equation given above is solved by finite differences on a grididentical with that of the flow model. In order to avoid numerical diffusion and oscillations,the advective terms are approximated by appropriate, higher order robust upwind schemes(Koren, 1993).

4 Field measurement of wind-induced flowsAs was already mentioned in the introduction, making field measurements successful

required reliable, up-to-date instrumentation, proper deploying methodology, as well asadvanced analysis of the enormous amount of data provided by the recording instruments.

4.1 Applied tools, measuring and deploying principles

The wind, flow and turbidity measurements carried out in the joint research projects in thecase study lakes made use of the development efforts of Aanderaa Instruments, Norway. Infact the company, specialised to sea conditions mainly, has put more and more attention tomake its instruments robust enough to use also in shallow lake conditions. Even some of their products were used as one of their first applications in Hungarian lakes, occasionally inextreme wind, flow or turbidity conditions. The first 10 years of application in Hungarian

surface waters was overviewed and evaluated in the framework of an international workshopheld at Lake Balaton in 1997 (see Fig. 8).

Fig. 9 shows the standard wind direction and speed sensors, to be attached to data scanningand storing units, whereas one version of the traditional propeller-type current meter RCM7and the most recently developed Acoustic-Doppler current meter RCM 9 are seen in Fig. 10.The two latter is mounted onto a data logger and RCM9 is supplied also with e.g. turbidity,conductivity, temperature, pressure and dissolved oxygen sensors. Measurements are madewith high time resolution and data at pre-selected regular intervals are stored as integralaverages. The interval can be set from 1 minute to 2 hours upon demand, however, givenlimited data storing capacity it determines the time the instrument is able to collect datacontinuously. The customary interval applied all over the projects has been 10 minutes.


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The acoustic-Doppler current and optical turbidity sensors are available also separately,which requires then a data logger to connect.

Several deployment techniques have been tested. The most recently developed methodshown in Fig. 11 has been applied either in very shallow water or in investigating currents inthe lower part of the vertical. The current meters have been used both in single- and multi-layer measurements. Due to the size, to the disturbances of the acoustic signal near theboundaries as well as to the presence of surface waves, in very shallow conditions it is oftenone single set-up in a vertical that can be perform, only. Nevertheless, the currents have beenin all cases captured in representative amount containing a number of pronounced events idealfor analysis as well as numerical model calibration.

Figure 8. Workshop on the 10 years anniversary of Aanderaa Instruments operating in LakeBalaton.


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(a) (b)

Figure 9. Standard Aanderaa wind speed (a) and direction (b) sensors.

(a) (b)

Figure 10. Aanderaa RCM7 and 9 recording current meters.

Figure 11. Customary bottom deployment set-up for RCM9 in shallow water.


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4.2 Data processing and analysis

Wind and flow data collected simultaneously at a number of sites in long term campaignshave to be jointly analysed in order to explore the relationships between the external drivingforce and the lake response to that. Both overall and separate event-based analyses are neededto identify at all important scales. Prevailing events are then utilised also in model calibrationand verification. In fact, apart from statistical forecasting, the only tool for making prediction

on the future fate of a lake is a numerical model validated against measured field data. Such atool is useful then to investigate e.g. the impact of any natural or man-made change in the lake,and also to evaluate measures in order to maintain existing or restore favourable pastconditions of the lake.

When analysing either the entire period of a particular campaign or selected events, thereare basically three main tools to draw lake-specific information from the data.

4.2.1 Vector-time series representation

An informative displaying method of the raw or filtered data is their vector time seriesrepresentation along the time axis. In the version adopted here the vectors lay with their footpoint on the axis and point away from it, but no arrowhead is used in order to avoid disturbing

graphical overlapping. At the beginning of the axis a reference scale is given in m/s and incm/s for the wind and water current, respectively. Note that vectors represent values valid atthe time at their foot point, which needs some care to find at places.

4.2.2 Directional statistics

The most traditional way to present wind and lake current statistics is the so-called windand current rose, which gives the simple directional distribution of the velocity vectors. Notethat by convention for the wind the frequency values in percentage are put in the directionwhere the wind was blowing from, whereas for the currents they are put in the direction theflow is pointing to. Once weighting the calculus by the speed and its square, one obtains theroute and energy distribution, respectively. The first is applied usually in air pollution studiesfor the wind and e.g. for suspended sediment advection characterisation for the currents, whereit is important to know the directions to propagate at the largest distances. For lakehydrodynamics, however, the wind energy distribution (a term inherited from the generalkinetic energy formula), proportional in fact with the surface wind shear stress is relevant.Similar to that, the current energy directional distribution represents the directions in whichthe sediment stirring up capability of the flow is the most energetic. In the present study thetotal circle is split into 16 sectors providing 22.5o resolution.

4.2.3 Digital filtering of the time series

In order to decompose the vector time series to characteristic elements representing anypre-selected range of their kinetic energy spectra over the time period, appropriate numericalfiltering techniques are applied. In the present study a Gaussian-type filtering algorithm is

used for the so-called low and band-pass filtering of the data. Low pass filtering stands forremoving variability with characteristic frequency higher, or corresponding time periodshorter than a prescribed cut-off value, thus preserving the parts varying slower than that.Band-pass filtering aims at preserving elements with periodicity falling into the range withintwo chosen cut-off values. Using subsequent ranges the time series can be decomposed intotheir harmonics, representing different portion of the total kinetic energy content.

This technique makes it possible to find e.g. the compounds of the wind most relevantfrom the energetic point of view for a lake with characteristic response time scale toturbulence, wave, seiche and circulation generation (see e.g. Findikakis and Law, 1999;


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Luettich et al., 1990; Jzsa et al., 2000). As to the currents, compounds due to seiche can beseparated from the ones representing circulatory motion. In doing so, in general low-passfiltering removing the compounds with less than 1 hour periodicity is applied, along with anappropriate band-pass filtering to capture the first mode of the local seiche if there is any.

For demonstration the simple algorithm of the Gaussian-type low-pass filtering techniqueis given below in which represents half of the so-called cut-off time period:

( )+

= =

c

c

c

j

jj

cjj jjGxx 2,~ (63)

( )( )

2

2

e1

=

cjj

cjjG

(64)

= 2,2,~

ccc jjjxxx (65)

2121 2,2,22,~~~

= ccc jjj xxx (66)

5 Wind-induced circulations in Lake Fert (Neusiedl)The first case study lake to be presented is Lake Fert situated at the Austro-Hungarian

water, with a special character of being largely covered by reed (Fig. 12). The average depthof the pelagic areas is less than 2 m whereas in the read covered zones, occupying more thanhalf of the lake surface, the average depth is below half a meter.

Figure 12. Layout of Lake Fert (Neusiedl). Note the large reed belt.

Both in the pelagic and littoral zones the bottom is very flat and mild-slope, which expectsto give a pronounced role to the spatial irregularities in the surface wind shear stress field. Thelakebed is muddy with strong siltation tendencies at places. Water exchange between thepelagic and littoral zones is often poor resulting then in occasional reed quality deterioration.


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5.2 Wind conditions

This part of the Carpathian Basin is especially windy making lake hydrodynamics veryactive all over the year. Both long-term observations and shorter, month long high resolutiondata, collected during the recent measurement projects are available for sound characterisation.

5.2.1 Prevailing winds

As seen in Fig. 15, there exists a particular axis with NNW-SSE orientation, which givemost of the wind directions. NW-NNW is far the most energetic, thus considered prevailingfrom the point of view of generating circulations as well as seiche motion.

Figure 15. Typical directional distribution of the wind (left), wind route (middle) and wind energy(right), as measured over the lake between 28 October 6 December, 1995.

5.2.2 Investigation of the near-surface wind conditions in a bay

Fertrkos Bay, situated in the Western side of the Hungarian part of the lake is the mostinvestigated area of the lake. Several measurement campaigns have been carried out there inorder to explore the details of the water and related sediment motion, along with the waterexchange across the reed - open water interface zone. Strong clock-wise bay-wide circulationshave been very often found to develop despite the fact the bay bottom is nearly horizontal.Namely, the depth in the middle part do not exceed 1.5 m whereas drops to 1 m next to the

reed border, where a half a meter high sudden step is observed and prevails in the littoral zone.This has been proven suitable to carry out the field measurements focusing on the IBLdevelopment and related circulation pattern. In order to do so, a special measurement set-upwas conceived as seen in Fig.16, measuring the wind speed along the prevailing winddirection axis at two sites, some 1.5 km far from each other and a bit more than 100 m farfrom the reed border each. Reed stems on average were less then 3 m high. Upwind from thebay the reed belt, though patchy at places, is more or less continuous, expected to give room tofull wind profile development over those characteristic roughness conditions. The sensorswere set at 3.3 m high, to be outside the wave-disturbed layer. At both sites RCM9 currentmeters were deployed too to measure the currents induced simultaneously.

2001 Spring

Site No. Water depth, m Sensor depth, m Instrument type

1. flow 1.1 0.6 RCM9

2. flow 1.2 0.7 RCM9

1. wind 1.1 Sensor height: 3.3 Standard Aanderaa

2. wind 1.2 Sensor height: 3.3 Standard Aanderaa


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Figure 16. Wind and flow measurement sites (indicated with dots) in 2001.

5.2.3 Calibration and verification of the surface wind shear stress model

The simultaneous wind data collected over a month have been used then for calibratingand verifying the IBL-model for the area. The IBL development in the study area has been

proven a very well defined process for which the IBL-model outlined earlier could becalibrated with high confidence. Some 6 prevailing storm events coming basically from NNWwere detected and used for the overall validation. The technique adopted here was to carry outan inverse transformation of the speeds to the reed border from both wind measurement sitesand check the agreement there. As can be seen in Fig. 17, raw data at Site 2 has got strongtrend to give substantially higher speed due to the fetch generally 1.5 km longer. In the figureall the data corresponding to the prevailing wind sector and not deviating too much from eachother (which was most often the case) are displayed.

The coupled speeds corresponding to 0.15 m reed roughness height, obtained by a leastsquare fitting basis, are shown in Fig. 18. Frankly, the determinism found in the system isstrong enough, at the same time the points scatter enough to make it even more convincing.

Performing the parameter fitting for the selected individual storm events the roughnessheight varied between 0.1 and 0.18 m, a very narrow range compared to very wide ones givenin relevant boundary layer meteorology textbooks for various canopy types.

Apart from the wind data, Fig. 19 shows also the current velocity vectors, measuredsimultaneously at the two sites. It comes out clearly that in the absence of significant bottomgradients the strong imbalance in the surface wind shear stress dominates the bay-widecirculations, in which water masses go with the wind in the strongly exposed downwind zone,whereas return currents occur in the upwind-zone relatively poor in wind shear stress.


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0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16

Mrt Wza az 1. pontban, [m/s]

MrtWzaa2.pontban,

[m/s

]

Teljes egyezs

Figure 17. Simultaneous wind speeds measured at Site 1 (horizontal axis) and Site 2 (vertical axis)

with 300-350 direction and less than 30 difference. Perfect agreement indicated by continuous line.

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16

W10(0) az 1. pontbl transzformlva, [m/s]

W10(0)a2.pontbltranszformlva,

[m/s]

Teljes egyezs

Figure 18. Simultaneous wind speeds transformed from Site 1 (horizontal axis) and Site 2 (verticalaxis) to the upwind reed border. Perfect agreement indicated by continuous line.


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1. pont

-15

0

15

-15 0 15

u, [cm/s]

v,

[cm/s]

2. pont

-15

0

15

-15 0 15

u, [cm/s]

v,

[cm/s]

1. pont

-15

0

15

-15 0 15

Wx, [m/s]

Wy,

[m/s]

2. pont

-15

0

15

-15 0 15

Wx, [m/s]

Wy,

[m/s]

Figure 19. Simultaneous wind (left) and flow (right) at Site 1 (upper) and Site 2 (lower), indicated bytheir end point only.

5.3 Wind-induced flows

In this section the simultaneous wind and flow vector time series of the strongest stormevent of the measurement period discussed above will be displayed followed by anotherrepresentative prevailing event selected from an earlier measurement campaign, carried out indifferent set-up in the bay. Model calibration will be then shortly presented for the prevailingwinds and circulation patterns. Similar procedure will be done for the large, Northern part ofthe lake for which both measured and modelled patterns will be overviewed. Finallyconclusions about the water exchange as a combined effect of circulations and seiche will bedawn.

5.3.1 Prevailing circulation patterns in the study bay

Fig. 20-21 shows the low-pass filtered time series of the event selected from 2001. Basedon spectral analysis, compounds with periodicity up to 5 hours considered to contain most ofthe seiche, consequently variation slower than that (seen in Fig. 21) were taken representativeto find the circulatory pattern. The differences in wind speed between the two sites, moreover,

the stability of the flow in direction, and closely following the large-scale variation of the windin magnitude are all evident. Fig. 22 gives a typical snapshot of the pattern prevailing in itsstructure for nearly two day, resulting certainly in efficient, bay-wide water exchange.

Fig. 23-27 further confirm the development of the above mentioned pattern in NNW wind,and the behaviour found at Site 7 suggests similar structure also outside the bay. Fig. 27 givesan insight to the directional distribution, sufficiently stable to prove indirectly that the patternis primarily two-dimensional, that is the water mass transport occurs mainly in lateral sense,rather than in the form of an overturning flow, for which the water is unfavourably shallow todevelop at low momentum loss level.


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Figure 20. Vector time series of a prevailing storm event. From top to bottom: wind 1, wind 2, flow 1,flow 2. Harmonics with less than 1 hour time period removed.

Figure 21. Same as Fig. 20, but harmonics with time period lass than 5 hours removed.


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Figure 22. Typical snapshot of the wind and flow pattern on 6/4/2001 14:00. Wind vectors shifted tothe circles to be distinguishable.

Figure 23. Measurement sites in 1995.


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1995 Autumn


1. 1.5 0.8 RCM7

2. 1.4 0.7 DCS 3500

3. 1.3 0.7 RCM4S4. 1.3 0.7 RCM7

5. 1.3 0.7 RCM4S

7. 1.4 0.7 RCM4S

Wind: 2. 1.4 Height: 3.0 Standard Aanderaa

Figure 24. Vector time series of a prevailing storm event. From top to bottom: wind 2; flow 1, 2, 3, 4, 5and 7. Fluctuations with less than 1 hour time period removed.


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Figure 25. Same as Fig. 24, but here fluctuations with less than 5 hours time period removed.

Figure 26. Typical snapshot of the wind and flow pattern on 11/18/1995 14:00. Wind vector shifted to2W to be distinguishable.


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Figure 27. Directional distributions in the period seen in Fig. 25-26. Left box: fluctuations with lessthan 1 hour periodicity removed. Right box: 1-5 hours periodicity band retained.


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5.3.2 Model calibration for prevailing circulation patterns

A 200 m cell size grid was used to apply the depth-integrated finite difference flow modelfor the lake. The wind shear stress field was provided by the IBL-based model, for which nofurther calibration was needed. Instead, the smoothness coefficients of the pelagic areas andinside the reed were subject to tuning. As to the wind effect on the reed-covered zones, it wasentirely disregarded due to the high reed density. Reaching steady-state conditions, the

difference between the flow patterns corresponding to constant (Fig. 30) and IBL-dependent(Fig. 31) shear stress field is striking, the latter being very close to what was measured insimilar, stabilised conditions. Note that the fully opposite, SSE wind direction results in thesame sense of rotation of the water in the bay due to the changes in the wind stress field. Forthat direction, the bulk reed roughness was taken somewhat lower, expressing the effect on thewind profile by crossing a number of bays and reed islands along the long fetch.

Water residence time simulations focusing on the estimation of the time that water massesspend in the Northern part of the bay in half a day in prevailing wind conditions are shown inFig. 33. The large differences between applying constant and IBL-dependent wind shear stressforcing can be clearly seen.

Figure 28. Bottom topography and reed cover of the lake displayed on the 200 m cell size finitedifference grid.

Figure 29. Zooming in on the study bay.


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Figure 30. Modelled steady-state flow pattern, induced by a wind shear stress field uniform over thepelagic areas, corresponding to W10=13m/s NNW wind. Highest velocities within the study bay arearound 6 cm/s.

Figure 31. Modelled steady-state flow pattern, induced by a wind shear stress field according to theIBL development, corresponding to W10=12m/s NNW wind and 0.15 m roughness height at the upwindreed border. Highest velocities in the study bay around 12 cm/s.


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Figure 32. Modelled steady-state flow pattern, induced by a wind shear stress field according to theIBL development, corresponding to W10=10m/s SSE wind and 0.06 m roughness height at the upwindreed border. Highest velocities in the study bay around 8 cm/s.

Figure 33. Half a day state of the distribution of the mean residence time (in hours) that water massesspent in the Northern half of the study bay, induced by modelled flow fields seen in Fig. 30 (left) andFig. 31 (right).


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5.3.3 Hydrodynamic exploration of the Northern pelagic area

As to the orientation, more or less the same exposure to the wind is found in the large, Northern pelagic area of the lake, so as one would expect it results then rather similarcirculation patterns, too. The extensive measurement campaign carried out in 1992 gaveseveral events in which the expected pattern holds. In fact, though fetches are much longer, thevorticity introduced by the wind shear stress vector field seems sufficient to exceed the one

due to the rather mild depth gradients, of course somewhat enhanced by shallowness.Fig. 35-36 shows the time series of one typical event during which lake-wide, stable

circulation was induced by the two days long, strong wind. It can be also seen, however, howintensive back-flow could be generated by the seemingly harmless change in the wind speed atabout half of the period. Observe that for Site 8 the velocity scaling is significantly reduced, tomake it suitable for plotting together with the other data. At that particular site the straitssubstantially speed up the currents which are then locally mainly gravity- rather than wind-driven there. A typical snapshot of the pattern and the directional statistics are given in Fig. 37and 38, respectively.

Modelled flow field with the simplistic constant wind forcing is seen in Fig. 39, whereasthe calibrated pattern with 0.15 m upwind reed belt roughness is given in Fig 40, representing

the significant differences particularly in the West of the lake. Both here and in the previousstudy bay Manning k equal to 40 in the pelagic, and equal to 6 inside the reed provedreasonable.

Figure 34. Measurement sites in 1992. Wind measured at Site 3.


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1992 Spring


1. 1.4 0.7 RCM7

2. 1.5 0.7 RCM4S

3. 1.8 0.9 RCM4S

4. 1.6 0.8 RCM7

7. 1.4 0.7 RCM4S

8. 1.5 0.8 RCM4S

Wind: 3. 1.8 Height: 3.0 Standard Aanderaa

Figure 35. Vector time series of a prevailing storm event. From top to bottom: wind 3; flow 1, 2, 3, 4, 7and 8. Fluctuations with less than 1 hour time period removed.


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Figure 36. The same as Fig. 35, but fluctuations with less than 5 hours time period removed.

Figure 37. Typical snapshot of the wind and flow pattern on 5/4/1992 11:00. Wind vector shiftedoutside the lake to be distinguishable.


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Figure 38. Directional distributions in the period seen in Fig. 36-37. Left box: fluctuations with lessthan 1 hour periodicity removed. Right box: 1-5 hours periodicity band retained.


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Figure 39. Modelled steady-state flow pattern, induced by a wind shear stress field uniform over thepelagic areas, corresponding to W10=13m/s NNW wind. Highest velocities around 6 cm/s.

Figure 40. Modelled steady-state flow pattern, induced by a wind shear stress field according to theIBL development, corresponding to W10=12m/s NNW wind and 0.15 m roughness height at the upwindreed border. Highest velocities around 12 cm/s.


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5.3.4 Water exchange mechanisms as a combination of the seiche andcirculations

Analysing the combined effect of circulations and seiche, it seems to play a key role ingenerating w

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