three-dimensional wind-induced baroclinic circulation in rectangular basins

Three-dimensional wind-induced baroclinic circulation in rectangularbasins

Yongqi Wang a,*, Kolumban Hutter a, Erich B�auerle b

a Institute of Mechanics, Darmstadt University of Technology, Hochschulstr. 1, D-64289 Darmstadt, Germanyb Limnological Institute, University of Constance, D-78457 Constance, Germany

Received 29 July 1999; received in revised form 14 June 2000; accepted 15 June 2000

Dedicated to Professor K. Wilmanski on occasion of his 60th birthday

Abstract

We present results of various circulation scenarios for the wind-induced currents in two vertically strati®ed rectangular basins of

constant depth with di�erent sizes; these are obtained with the aid of a semi-spectral semi-implicit ®nite di�erence code developed in

Haidvogel DB, Wilkin JL, Young R. J. Comput. Phys. 94 (1991) 151±185 and Wang Y, Hutter K. J. Comput. Phys. 139 (1998) 209±

241. Our focus is to see whether the code allows reproduction of the many well-known processes exhibited in strati®ed waters of a

lake basin on the rotating Earth. Often, the internal dynamics exhibits Kelvin- and Poincar�e-type oscillations, whose periods depend

upon the strati®cation and the geometry of the basin and which persist for a long time, the attenuation being the result of the

turbulent dissipation mechanisms. It is shown that the numerical dissipation of our code can be su�ciently restricted that such wave

dynamics obtained with it is realistically persistent for typical time scales of physical limnology. Direct responses to wind forcing and

the oscillating behaviour after wind secession are studied and numerical results are illustrated for longitudinal and transverse winds,

respectively. By solving the eigenvalue problem of the linearized shallow water equations of two-layered closed rectangular basins,

the interpretation of the oscillations as Kelvin- and Poincar�e-type waves is corroborated. Ó 2000 Elsevier Science Ltd. All rights

reserved.

Keywords: Lake circulation; Limnology; Kelvin- and Poincar�e-type oscillations

1. Introduction

The strati®cation in lakes is almost exclusively es-tablished by the heat input due to solar radiation.During the summer season, in temperate climate zones,a strati®cation given at one particular instant generallypersists for a time span that is long in comparison to thetime scales of the dynamic circulation which is estab-lished by the wind shear traction applied at the watersurface. Only during short-lived episodes, when theshearing in the surface layer is very large (i.e., typicallyduring strong storms), Kelvin±Helmholtz instabilitiescan form and generate internal bores that destroy thestrati®cation or may erode the thermocline. During theshort periods in-between the change of the strati®cation

caused by the dynamic circulation is relatively small.Thus, a stable underlying strati®cation with no motionmay be assumed as an initial condition from which thetemperature and velocity ®eld may develop withoutforming internal instabilities.

The underlying thermo-mechanical processes canmathematically be described by the shallow waterequations in the Boussinesq approximation: in theseequations the vertical momentum balance reduces to theforce balance between pressure gradient and gravity(buoyancy) force, and density variations are onlyaccounted for in the buoyancy term. The equationsdescribe both, barotropic and baroclinic processes. Theformer are those that develop when density variationsdo not exist and, in temperate zones, arise in lateautumn and during winter. They also exist in a trulystrati®ed lake or ocean basin, but are then over-shadowed by the baroclinic processes that are typical forthe summer seasons when the underlying strati®cationis strong. We studied barotropic processes for idealgeometries and Lake Constance in [13]. Here we focus

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Advances in Water Resources 24 (2001) 11±27

* Corresponding author. Tel.: +49-6151-163196; fax: +49-6151-

164120.

E-mail addresses: [email protected] (Y. Wang),

[email protected] (K. Hutter), Erich.Baeuerle@uni-

konstanz.de (E. BaÈuerle).

0309-1708/01/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 3 4 - 8

upon baroclinic processes for ideal basins of rectangulargeometry in order to eliminate the complicated e�ects oftopography and irregular shore of a real lake from thebaroclinic processes, reserving a detailed study forstrati®ed Lake Constance to a further article [14].

The highly non-linear initial boundary value problemthat describes wind induced circulation dynamics inenclosed basins will not be described in this paper; thereader is referred to [12]. Neither shall we describe indetail the semi-spectral primitive equation model(SPEM) with the aid of which the shallow water equa-tions in the Boussinesq approximation are numericallysolved for given external atmospheric driving. To thatend the reader is directed to [12]. Here our aim is thepresentation of explicit solutions to given strati®cationand wind stress scenarios. The reasons for our detailedinvestigation are as follows: classical ®nite di�erence and(less frequent) ®nite element implementations of thehydrodynamic equations in the shallow water andBoussinesq approximation with a Cartesian coordinatesystem are generally fraught with numerical di�usionthat overshadows the physical di�usion of momentumand energy in magnitude. The probable reasons lie in theuse of conditionally stable (explicit) integration tech-niques and in the low order accuracy of the discretiza-tion techniques. For usual spatial and temporal timesteps (that obey the Courant±Friedrichs±Levy criterion)unduly large momentum and energy di�usivities areneeded to make integrations possible. As a result, timedependent processes that are persistently created by thetime-varying winds die out relatively quickly. In extremecases, e.g., when an oscillating process should developfrom a wind signal, the computed motion dies out be-fore it even has had a chance to fully develop.

Because of the small size of lakes in comparison tothe ocean, their boundaries transform the time depen-dent (oscillating) driving mechanisms into a response ofcurrent and temperature distributions which manifestthemselves both in quantized components, characteristicof the geometry of the enclosed basins and the strati®-cation plus a direct signal. The quantized component isreminiscent of the Kelvin and Poincar�e wave dynamicsand, during summer, often persists for days and weeksas conspicuous signals of water displacement in themetalimnion. Naturally, computational codes ought toreproduce this oscillatory motion, which explains whythree-dimensional codes must adequately account for aslittle numerical di�usion as possible.

We believe that the version of SPEM with semi-im-plicit numerical integration as sketched below is partic-ularly suitable to cope with such conditions, sincenumerical di�usion can be kept su�ciently low in ordernot to overshadow its physical counterpart. In Section 2we brie¯y outline how the model works. In Section 3various scenarios of wind-driven baroclinic processes ina rectangular basin with constant depth are studied and

the predicted Kelvin- and Poincar�e-type oscillations canbe displayed in the results. In Section 4 by solving theeigenvalue problem of the linearized shallow waterequations of a two-layered closed rectangular basin, theinterpretation of the oscillations as Kelvin- and Poin-car�e-type waves can be ascertained. To study the in¯u-ence of basin dimension on the baroclinic processes,the same computations as in Sections 3 and 4 are per-formed in Section 5 for a smaller rectangular basin.We conclude with a summary and some remarks inSection 6.

2. SPEM with curvilinear orthogonal coordinates

We brie¯y describe here the numerical model asconstructed by Haidvogel et al. [5] and changed by us toallow large-time-temporal integration [12].

2.1. Basic equations

The basic hydrodynamic equations consist of thebalance equations of mass, momentum and energy aswell as a thermal equation of state; here the followingsimplifying assumptions are invoked:· Boussinesq assumption, i.e., density variations are on-

ly accounted for in the buoyancy term, and balanceof mass reduces to the continuity equation�divv � 0�.

· Shallow water assumption, i.e., the vertical momen-tum balance reduces to the hydrostatic equation bal-ancing the vertical pressure gradient with thebuoyancy force, and the e�ects of the eccentric rota-tion (second Coriolis parameter) are ignored.

· Rigid lid assumption, i.e., in the vertically integratedmass balance the in¯uence of the motion of the freesurface is ignored. This amounts to assuming the vol-ume transport to be solenoidal and derivable froma stream function that can be determined indepen-dently of the baroclinic response.Thus, under these assumptions the ®eld equations

read:

ouox� ov

oy� ow

oz� 0;

ouot� v � grad uÿ fv

� ÿ o/ox� o

oxmH

ouox

� �� o

oymH

ouoy

� �� o

ozmV

ouoz

� �;

ovot� v � grad v� fu

� ÿ o/oy� o

oxmH

ovox

� �� o

oymH

ovoy

� �� o

ozmV

ovoz

� �;

12 Y. Wang et al. / Advances in Water Resources 24 (2001) 11±27

0 � ÿ o/ozÿ qg

q0

;

oTot� v � grad T

� oox

DTH

oTox

� �� o

oyDT

H

oToy

� �� o

ozDT

V

oToz

� �;

qÿ q0

q0

� ÿ6:8� 10ÿ6 � �T ÿ T0�2 �T in °C�:

�1�

Here a Cartesian coordinate system �x; y; z� has beenused; �x; y� are horizontal, and z is vertically upwards,against the direction of gravity; v � �u; v;w�; f ; q; q0;/; g; T are, respectively, the velocity vector, Coriolisparameter, density, reference density (q0 � 1000 kg mÿ3

at temperature T0 � 4°C), dynamic pressure (/ � p=q0;p is pressure), gravity force �g � 9:8 m sÿ2�, tempera-ture. Furthermore, mH; mV are horizontal and verticalmomentum, DT

H, DTV, horizontal and vertical heat di�u-

sivities. A derivation of these equations from a properscaling analysis is, e.g., given by Hutter [6]. We note thatdispersion in integrated layer models is due to the in-clusion of the acceleration terms in the vertical momen-tum balance and necessarily stabilizes those models bothphysically and computationally. This is one reason whyhydrostatic models generally require a large amount ofnumerical di�usion and is no di�erent in SPEM.

2.2. Original version of SPEM

Haidvogel et al. [5] developed a SPEM from the hy-drodynamic equations (1). In SPEM, smoothness in therepresentation of the discretization is the principle tominimize the necessary numerical di�usion; hydrosta-ticity is maintained. To this end the r-transformation inthe vertical direction, mapping irregular lake topogra-phy into constant depth (in the r-coordinate), and theSchwarz±Chrysto�el transformation in the horizontal,mapping the irregular lake shoreline onto a rectangle,are implemented, viz.,

n � n�x; y�; g � g�x; y�; r � 1� 2z

h�x; y� ; �2�

where h�x; y� is the lake depth. The computational do-main becomes now a cube in �n; g; r�-space. Thistransformation avoids step-like discontinuous approxi-mations of the lake boundary in FD-approximationswhich could also be achieved by using FE-approxi-mations. Of course, for a rectangular basin such atransformation is unnecessary. In SPEM, the verticalr-dependence of the model variables is represented as anexpansion in a ®nite polynomial basis set fTk�r�g, thatis,

b�n; g; r; t� �XN

k�0

Tk�r�b̂k�n; g; t�; �3�

where b is any unknown ®eld variable, fTk�r�g is a setof orthogonal polynomials de®ned in �ÿ1; 1�, the onlyrestriction placed on their form is thatZ 1

ÿ1

Tk�r�dr � 2dk0;

where dk0 is the Kronecker delta ± i.e., only the lowestorder polynomial �k � 0� has a non-zero vertical inte-gral. This isolates the depth averaged barotropic com-ponent of the ®eld. fTk�r�g are chosen to be modi®edChebyshev polynomials, and the expansion guarantees avery smooth representation of the variables in ther-direction if only N is su�ciently large. In thehorizontal, i.e., the n; g-directions, ®nite di�erencerepresentations with an Arakawa C-grid are used.

SPEM in its original version employs an explicitscheme for temporal integration and thus is numericallyonly conditionally stable, i.e., the allowable time step isrestricted by the horizontal resolution and the number Nof polynomials taken into account in (3). This con-ditional stability also dictates which numerical valuesthe horizontal and vertical di�usivities of momentumand energy can take. Because of the smoothness pre-cautions taken in (2) and (3) these values are of the orderof what would be physically suggested (or somewhatlarger) but the temporal steps of integration are stillprohibitively small for many problems.

2.3. Implementation of semi-implicit integration in time

A fully implicit integration in time is equally pro-hibitive because of the large non-linear systems ofequations that emerge; so we considered alternatedirection implicit (ADI) procedures. Accordingly, theintegration step from time level n to n� 1 is subdividedas follows:1. tn ! tn�1=3, implicit in n-direction, explicit in g- and

r-directions,2. tn�1=3 ! tn�2=3, implicit in g-direction, explicit in

n- and r-directions,3. tn�2=3 ! tn�1, implicit in r-direction, explicit in n- and

g-directions.It turned out that for usual lakes tripling the time step Dtaccording to this procedure is less e�ective than to em-ploy a semi-implicit integration in the r-direction alonewith the full time step Dt. Furthermore, computationsfor barotropic wind induced processes in Lake Con-stance showed that time steps could be 200±400 timeslarger than with the explicit integration of the originalSPEM while the computational time could be reducedby a factor of 60. For baroclinic processes the gain ismuch less but still a factor of approximately 20, or morefor time steps and a factor of 6 for the computationaltime, respectively.

Y. Wang et al. / Advances in Water Resources 24 (2001) 11±27 13

A detailed analysis of this, including a study of se-lection of the optimal horizontal and vertical di�usivitiesis given by Wang and Hutter [12].

3. Wind-induced baroclinic response in a rectangular basin

3.1. Parameter and scenario setting

This model is driven by prescribing at each nodalpoint on the free surface of the discretized domain timeseries of the shear-traction vector (its n- and g-compo-nents) and the surface temperature or surface heat low.In principle, therefore, arbitrary spatial distributions ofthe wind forcing can be prescribed if so desired, how-ever, we will here restrict attention to relatively simplescenarios: uniformly distributed wind in a preferred di-rection and Heaviside-type in time, e.g., uniform inspace and constant in time wind only lasting the ®rst twodays. A wind speed of Vwind � 5 m sÿ1, 10 m above thewater surface, is used, corresponding to a wind stress of0.055 N mÿ2 at the water surface. Similarly, basal sheartractions obey a viscous sliding law. As for otherboundary conditions, the rigid lid assumption is appliedat the free surface and the low is tangential to the bed.Furthermore, it will be assumed that no heat lows acrossthe free or basal surfaces: dT=dn � 0 , where n is the unitnormal vector of the boundary surface.

We shall now apply our model to a rectangular basinwith constant depth and subject it to an initial strati®-cation as prescribed below. Length, width and depth ofthe rectangle are 65 km, 17 km and 100 m, respectively.The mesh size will be Dx � Dy � 1 km, whereas in thevertical 30 Chebyshev-polynomials will be employed.

The strati®cation varies seasonally following that thesolar radiation heats the upper most layers of the lake,and turbulence di�uses this heat to greater depths. Bylate summer these processes will have established a dis-tinct strati®cation, that essentially divides the watermass into a warm upper layer (called epilimnion), a colddeep layer (hypolimnion) which are separated by atransition zone (metalimnion), in which the epilimniontemperatures above it are transferred to the hypolim-nion temperatures below it. Vertical temperaturegradients in this layer are larger than in the epi- andhypolimnion with a maximum at the so-called thermo-cline. A vertical temperature pro®le, typical of a latesummer situation for Alpine lakes (see, e.g., [2,7]), is

T �t � 0�

� 17ÿ 2 exp�ÿ�z� 20�=5�; z P ÿ 20 m5� 10 exp��z� 20�=20�; z < ÿ20 m

��°C�;

�4�it will be chosen as the initial temperature pro®le fromwhich also the initial density distribution can be com-

puted according to the thermal equation of state, thesixth equation of (1). For computational and physicalpurposes this representation is su�ciently accurate. Weshall later present results for the internal eigen-oscilla-tions also on the basis that the continuous strati®cationis replaced by a two-layer model, consisting of anepilimnion with constant density and a hypolimnionequally with constant but larger density separated at thethermocline by a sharp material interface.

The computations of the baroclinic currents is per-formed for the following choices of di�usivities vH; vV ofmomentum, and DT

H;DTV of heat, respectively:

vV �0:016; z > ÿ20 m;

0:0016; ÿ20 m P z Pÿ 40 m;

0:008; z < ÿ40 m

8>><>>: �m2 sÿ1�;

vH � 1:0 m2 sÿ1;

DTV �

0:0002; z > ÿ20 m;

0:00002; ÿ20 m P z Pÿ 40 m;

0:00004; z < ÿ40 m

8>><>>: �m2 sÿ1�;

DTH � 1:0 m2 sÿ1; �5�

in which the horizontal di�usivities are simply chosen asconstants. For the non-constant vertical di�usivities,this choice accounts for the fact that the Austauschcoe�cients are considerably smaller in the metalimnionregion than in the epilimnion above and the hypolim-nion below. This is so because the thermocline preventsdi�usion through this `interface'. The di�usivities in thehypolimnion are also smaller than in the epilimnionbecause of its smaller ambient turbulence. Comparedwith realistic measured values they are basically in therange that is thought to be physically acceptable, eventhough they are still somewhat larger than measuredvalues [6,8,11]. Computations with even smaller di�u-sivities can also be performed if for numerical stableconsiderations higher vertical resolution (larger numberof Chebyshev polynomials) is used, of course, thereforelonger computational time is needed.

In the following two subsections the simulated resultsare displayed and discussed for winds in the longitudinaland transverse direction, respectively, lasting only the®rst two days.

3.2. Uniform wind in the longitudinal direction

In this subsection we consider the response of therectangular basin to an impulsively applied constantwind force in the longitudinal direction from west with aduration of two days. The model is started at rest andintegrated over a period of 12 days.

Fig. 1 shows time series of the vertical velocity com-ponent w in 30 and 60 m depth, respectively, at the four


near-shore midpoints as sketched in panel (a) of Fig. 1.Immediately after the westerly wind set up, the motionstarts with an upwelling at the western end and adownwelling at the eastern end due to the direct e�ect ofwind stress at the surface. In the northern (southern)mid boundary points there occurs initially an upwelling(downwelling). This behavior is, clearly, due to theCoriolis force that causes a velocity drift to the right andtherefore is responsible for the initial downwelling (up-welling) at the southern (northern) shore. The motion ischaracterized by oscillation, of which two conspicuouscomponents can be clearly identi®ed. The longerperiodic oscillation, barely visible in the overall behav-

iour but conspicuously indicated by the strong down-stroke signals, can be identi®ed as an internal Kelvin-type wave, the shorter periodic one as a Poincar�e-typewave. We will in the next section provide convincingdetails for this interpretation. In Fig. 1(a) for the west-ern shore point the downstroke after the wind secession(48 h) is conspicuously seen at �73 h. If we follow thisdownwelling signal around the lake, then it is seen inpanels (a)±(d) of Fig. 1 at �74, 104, 133, 158 h. As thenext downwelling signal it emerges again at �186, 213,240, 264 h, respectively, the four points are encounteredcounterclockwise around the basin, with a travel time ofapproximately 105±110 h around the basin. Fig. 1 also

Fig. 1. Time series of the vertical velocity component w at the four near-shore midpoints, counterclockwise around the basin (western (a), southern

(b), eastern (c) and northern shore (d) indicated in the inset) at 30 and 60 m depths in the strati®ed rectangular basin of constant depth subject to

constant wind from west (in the long direction) lasting two days. The labels correspond to the depths (30 and 60 m).


discloses very clearly the short periodic oscillations thatare superimposed on the long term trend in the signals.This oscillating signal may be interpreted as a standingPoincar�e-type wave. These waves persist for a very longtime, for more than 10 days, as one can also observe inreal lakes.

The Kelvin-type oscillations can be seen even moreclearly in the isotherm-depth time series at the same fournear-shore midpoints displayed in Fig. 2. This ®gureclearly shows that the metalimnion experiences a strongdownward motion whenever the downwelling strokesindicated in Fig. 1 pass the respective position empha-sized by the arrows in Fig. 2. After the rapid downwardmotion the isotherms rise again, much more slowlythough, until the second rapid downward motion isinitiated by the second passage of the downwelling sig-nal. If instead of isotherm-depth-time series tempera-ture-time series at a certain depth are drawn, the graphsof Fig. 3 are obtained. These graphs are better suited toshow the epilimnetic and hypolimnetic behaviour, butthey also disclose the conspicuous motion in the meta-limnion. This motion circulates counterclockwisearound the basin with a period of Tk � 110 h. In theepilimnion (above �15 m depth) and in the hypolimnion(below �45 m depth) the temperature change is rela-tively small. It is mainly caused by the vertical motion ofthe thermocline (metalimnion); the changes associatedwith direct thermal di�usion are relatively small. Con-trary to this Kelvin-type behaviour, the Poincar�e-typeoscillation cannot be identi®ed in the isotherm-depth-

and temperature-time series of Figs. 2 and 3. Neithercan they be discerned in corresponding isotherm depthor temperature-time series at mid-lake positions. This isso because of the very small vertical velocities occurringat mid-lake positions; except for a small temperaturedrop at the beginning which is due to turbulent mixingthese temperature results give no conclusive inferences(not shown here for brevity).

Fig. 4 shows the longshore velocity at the mid-pointsof the western (a), southern (b), eastern (c) and northernshores (d) as functions of time at various verticallocations. As expected, the Kelvin-type wave may beclearly identi®ed. The vertical structure of the current®eld exhibits ®rst baroclinic mode behaviour in that thehorizontal velocity oscillates above the metalimnionwith the opposite phase from that below the metalim-nion; this is very similar to the result of a two-layermodel. In the metalimnion the amplitudes of theoscillation of the horizontal velocity is relatively small.Furthermore, the times of vanishing horizontal motionseparate a 110 h interval, the suspected Kelvin waveperiod; these times occur simultaneously at the mid-points of western and eastern shores and a quarterperiod (28 h) later at southern and northern shore, asexpected. In other words, this position of vanishinghorizontal motion travels counterclockwise around thebasin.

In Fig. 5 the time series of the horizontal velocitycomponents u (a) and v (b) in the center of the basin at0 m and 60 m depths are displayed. It is known that

Fig. 2. Isotherm-depth time series at the four near-shore midpoints, counterclockwise around the basin (western (a), southern (b), eastern (c) and

northern shore (d)) in the strati®ed rectangular basin of constant depth subject to a constant wind from west (in the longitudinal direction) lasting

two days. The labels correspond to the temperatures in °C.


rotational e�ects on gravitational waves (e.g., Kelvinand Poincar�e waves) can be detected in basins which arewide as compared, or at least comparable to the Rossbyradius R � cint=f , where cint is the phase speed of inter-nal gravitational waves. The Rossby radius measures acritical length scale over which rotation is important forthe dynamics of the motion. For baroclinic waves R isjust a few kilometers. We will see in the next section thatthe phase speed of the internal baroclinic waves for thestrati®cation (4) is nearly cint ' 0:39 msÿ1 , which cor-responds to a Rossby radius of R ' 3:7 km for a Cori-olis parameter f � 1:07� 10ÿ4 sÿ1. The amplitude of aKelvin-type wave decays exponentially as one moves

away from the boundary, the decay rate being given bythe Rossby radius. Even though Kelvin-type waves de-cay with the distance from shore, this kind of oscillationcan still be recognized in the middle of the lake if thelake width does not extend over more than three or fourRossby radii. This is the case here. Obviously, theamplitude of Kelvin-type waves in the center of thebasin is much smaller than near-shore. As opposed tothe temperature signal, for which an amphidromic pointresults at the centre of the rectangle, the remainingcontributions of the shore-parallel currents which areequally oriented at opposite shores, are added together.For R � 3:7 and B � 17 km one would have the long

Fig. 3. Time series of the temperature at the four near-shore midpoints, counterclockwise around the basin (western (a), southern (b), eastern (c) and

northern shore (d)) at various depths in the strati®ed rectangular basin of constant depth subject to a constant wind from west (in the longitudinal

direction) lasting two days. The labels correspond to the depths in meters.


velocity component in the middle, if friction can be ex-cluded,

u�y � B=2� � �u�y � 0� � u�y � B�� exp

�ÿ B

2R

�� 2ushore exp

�ÿ 8:5

3:7

�� 0:2ushore;

which agrees well with the computational results (com-pare u�z � 0� at the time t � 90 h in Figs. 4(a) and (c)(u � �14� 6�=2 cm sÿ1� with u�z � 0� at t � 90 h inFig. 5, once the Poincar�e contributions are subtracted.When the amplitude near the shore amounts approxi-

mately to 10±12 cm sÿ1, it will be reduced to 2 cm sÿ1 inthe middle of the lake.

Evidence of the second, transverse Poincar�e-typewaves, can be more clearly found in the middle of thelake, especially in the transverse component of the ve-locity. From the component of the velocity u, a super-position of Kelvin- and Poincar�e-type waves can beidenti®ed, but in the transverse component v nearly onlyPoincar�e-type waves exist with a period of approxi-mately Tp � 13:3 h. One typical feature of Poincar�ewaves is that the horizontal projection of the velocityvector rotates clockwise (on the northern hemisphere).This behaviour can be inferred from the two panels in

Fig. 4. Time series of the shore-parallel velocity component at the four near-shore midpoints, counterclockwise around the basin (western (a),

southern (b), eastern (c) and northern shore (d) indicated in the inset) at various depths in the strati®ed rectangular basin of constant depth subject to

a constant wind from west (in the longitudinal direction) lasting two days. The labels correspond to the depths in meters.


Fig. 5, where the diamond symbols mark times at whichv reaches a maximum and u lies between a relativeminimum and maximum. This is reminiscent of a 45°phase shift of the y-component behind the x-component,exactly what is expected by a Poincar�e-type wave.

For a horizontally in®nitely extended basin with aconstant depth the vertical component of the velocityremains always zero under the action of a horizontal,spatially uniform wind-forcing. If the basin is horizon-tally bounded, the in¯uence of the horizontal boundarypropagates from the boundary toward the center of thebasin. In a bounded domain at a given point away fromthe boundary of the basin the vertical component of thevelocity vector is zero as long as no signal from an en-counter with a horizontal boundary reaches this point.The perturbation induced by such an encounter propa-gates with the phase speed of the wave. Fig. 6 shows thedistribution of the vertical component w of the velocityin the cross-section through the basin center for the ®rst6 h after the wind set-up. It can be clearly seen bycomparison of the curves for various times that theperturbation propagates away from the boundary to-ward the lake center. Immediately after the onset of thewind from the West, due to the e�ect of the Coriolisforce, the vertical motion in this mid-lake cross sectioncommences with a downwelling (negative w) at thesouthern shore and an upwelling (positive w) near thenorthern shore. As time proceeds the regions of down-and upwelling become larger, as they move towardsthe mid-point of the lake. It takes nearly 6 h until the

perturbation reaches the center of the basin, 8.5 km fromshore. This propagation speed amounts to approxi-mately 0:4 m sÿ1. We will see in Section 4 that this speedis almost exactly equal to the Kelvin-type phase speed ofinternal oscillations with strati®cation (4), obtained bysolving an eigenvalue problem of the vertical velocitycomponent. If we observe the vertical velocity compo-nent at the western and eastern ends for short times afterthe onset of the wind, a similar behaviour is observed.

As we have showed, for the chosen di�usivities (5),the Kelvin- and Poincar�e-type waves could be easilyidenti®ed. These waves persisted for a long time. If werepeat the same computation but using the ten timeslarger vertical di�usivities (which are obviously muchlarger than physically acceptable) than listed in (5), theoscillations can no longer be detected in the wind-deduced motion. In Fig. 7 the time series are displayedof the horizontal velocity components u and v computedwith these larger di�usivities in the center of the basin atthe free surface and 60 m depth, respectively. In this

Fig. 5. Temporal evolution of the horizontal velocity components u (a)

and v (b) for a midlake position in the strati®ed rectangular lake of

constant depth subject to a constant wind from west (in the longitu-

dinal direction) lasting two days. The labels correspond to the depths

in meters. The diamond symbols mark times at which v reaches a

maximum and u lies between a relative minimum and maximum. Two

types of oscillations can be clearly seen with periods of approximately

13.3 and 110 h, respectively.

Fig. 6. Distribution of the vertical velocity component w at the cross-

section through the midlake (indicated in the inset) at 30 m depth for the

®rst 6 h after the wind set-up. The labels correspond to the time in hours.

Fig. 7. Same as Fig. 5 but now computed with the 10 times larger

vertical di�usivities than in Fig. 5.


case, transient Kelvin-type and Poincar�e waves developonly within a short time after the wind set-up and se-cession while they are largely damped away beforethey are fully developed. After the wind secession thevelocities die out very rapidly basically without anysuperimposed oscillations.

3.3. Uniform wind in the transverse direction

It is known that for wind-forcing in the transversedirection the Poincar�e-type waves are more easily ex-cited. We also performed computations for an uniformtransverse wind (from South) lasting two days. Fig. 8

shows the longshore velocity components at the fourmid-points of the shore around the basin. From thesetime series the superposition of the Poincar�e-type os-cillations on the long periodic Kelvin-type wave can beclearly identi®ed. On the other hand, as we have seenin Fig. 4, for a wind in the longitudinal direction of thebasin, this kind of transverse wave could not at all beidenti®ed from the horizontal velocity near the shore inthat case. Strong Poincar�e-type oscillations generatedby a transverse wind can still be more easily identi®edin time series of the horizontal velocity components inthe center of the basin (Fig. 9). Here no sign of aKelvin-type response is discernable, quite contrary to


southern (b), eastern (c) and northern shore (d) indicated in the inset) at various depths in the strati®ed rectangular basin of constant depth subject to

constant wind from south (in the transverse direction) lasting two days. The labels correspond to the depths in meters.


what was seen in panel (a) of Fig. 5 for a longshorewind.

4. Internal seiches: an eigenvalue problem of the two-layer

model

To corroborate the interpretation of the oscillationsemerging from the wind-induced motion, an eigenvalueproblem is now solved. The ensuing analysis of freeoscillations in this rectangular basin will be based upona two-layer ¯uid system on the rotating Earth. We em-ploy the shallow water equations of a two-layer ¯uid,ignore frictional e�ects and thus may suppose the layervelocity vectors to be independent of the depth coordi-nate. The depth integrated linearized momentum andconservation of mass equations then take the form (see,e.g., [8])

oV1

ot� f k� V1 � gh1 grad f1 � 0;

of1

ot� div V1 � div V2 � 0;

oV2

ot� f k� V2 � g�1ÿ ��h1 grad f1 � g�h2 grad f2 � 0;

of2

ot� div V2 � 0 �6�

with the boundary conditions for a closed water basin

V1 � n � 0 and V2 � n � 0: �7�

Here, f is the Coriolis parameter, k a unit vector in the z-direction and g the gravity constant. V i �i � 1; 2� are thehorizontal components of the volume transport vectorin the ith layer; fi is the displacement of the free surfaceand interface, hi the layer depth and � � �q2 ÿ q1�=q2,where qi are the constant densities of the two layers,respectively. n is the unit normal vector along theboundary.

For constant depth basins, h � h1 � h2 � const:, thetwo-layer equations (6) and (7) can be decoupled intotwo one-layer models governing the free barotropic andbaroclinic oscillations separately. In these new equationsthe horizontal motion is either unidirectional in the en-tire water column and then models the barotropic re-sponse, or back and forth in the epi- and hypolimnionwith compensating mass transport and then models thebaroclinic internal response. The equations for theselatter free oscillations are:

oV int

ot� f k� V int � ghint grad fint � 0:

ofint

ot� div V int � 0;

V int � n � 0 along the boundary

�8�

with

hint � � h1h2

h�O��2�; V int � V1 ÿ h1

h2

V2 �O��;

fint � f1 ÿhh2

f2 �O��9�

and

c2int � ghint � g

q2 ÿ q1

q2

h1h2

h1 � h2

: �10�

cint is the phase speed of the internal gravity wave of thetwo-layer model and hint the corresponding reduceddepth. Using the harmonic ansatz

�V int�x; y; t�; f�x; y; t�� V�int�x; y�; f��x; y��eixt; �11�

with frequency x and the amplitude V� and f� in Eq. (8)an eigenvalue problem emerges for the eigenvalue x andthe mode functions V� and f� which can be routinelydetermined [1].

For constant depth basins the same decompositionalso exists for continuously strati®ed water bodieswith the di�erence that now a countable spectrum ofbaroclinic modes exists of which the highest frequencycorresponds to the above baroclinic solution of the two-layer model. Incorporating the rigid lid assumption onemust solve in this case the problem [9]

Fig. 9. Temporal evolution of the horizontal velocity components u (a)

and v (b) for a midlake position in the strati®ed rectangular lake of

constant depth subject to constant wind from south (in the transverse

direction) lasting two days. The labels correspond to the depths in

meters.


N 2�z�div grad w� o2

ot2

�� f 2

�o2woz2� 0

with

N 2�z� � ÿ gq�

oqoz; w � 0 at z � 0; z � ÿh; �12�

for the vertical component of the velocity ®eld, w; q� andq�z� are a reference density and the vertical densitypro®le and N is the so-called buoyancy frequency. Withthe separation solution w�x; y; z; t� � Zn�z�wn�x; y; t� theSturm±Liouville eigenvalue problem

d2Zn

dz2� N 2�z�

ghnZn � 0; Zn � 0 at z � 0 and z � ÿh

�13�emerges which possesses a countably in®nite set of ei-genvalues c2

n � ghn and associated eigenfunctions. Thelargest of these eigenvalues is identi®ed with c2

int � ghint

of the two-layer problem (8)±(11). With the temperaturedistribution (4) the solution of this vertical mode iscint �

��ghint

p � 0:391 m sÿ1, corresponding to a two-layer situation with h1 � 20 m, h2 � 80 m and� � 9:741� 10ÿ4. Using this value of cint in Eqs. (8)±(11)allows evaluation of the horizontal structure of the cor-responding oscillations. Some of the frequencies ob-tained this way are listed in Table 1. All these modespossess the same vertical structure, but only di�er intheir horizontal distribution.

The horizontal structures of the 1. and 9. mode of theoscillations, which are identi®ed by means of the lowellipses, and the corresponding horizontal velocity dis-tributions for four successive phases (xt � 0; p=4; p=2;3p=4� are displayed in Figs. 10 and 11. In Figs. 10 and11 a each ellipse represents a trajectory of a particle,which moves periodically around the center of the re-spective ellipse. If a particle traverses its ellipse in theanticlockwise direction the wave motion is dominatedby Kelvin-type behaviour, and ellipses are ®lled; if aparticle traverses it in the clockwise direction this mo-tion is primarily Poincar�e-type and the ellipses are not®lled. The size of the ellipses gives an indication of theexcursion a particle encounters.

The motion of the 1. mode of oscillations is the ro-tation-modi®ed fundamental longitudinal oscillation ofthe basin (Kelvin-type wave). For small lakes, this seiche

period (®rst mode) can be calculated by a simple one-dimensional method as a function of the length of thebasin, Lx, and of the phase speed cint with Merian'sequation

Tp � 2Lx

cint

: �14�

As the size of a lake increases or strati®cation decreasesthe e�ects of the rotation of the Earth become more andmore important, the one-dimensional seiches with theirstanding waves sloshing back and forth are increasinglydistorted and modi®ed to a rotation around the basin.This is evidenced in the case treated here. The intensityof the velocity decays exponentially with the distancefrom shore with a strong Kelvin-type motion along theshores. The periods are, however, not signi®cantly al-tered by the rotation, indeed with (14) one obtainsTp � 92:6 h compared to 103.96 h in Table 1. With aninternal Rossby radius of deformation of Rint � 3:8 kmin this case, which is less than a fourth of the basinwidth, it is clear that the e�ects of the rotation of theEarth must be important. The velocity ®eld of the lowestmode is shore bound and counterclockwise around thebasin (Figs. 10(b)±(f) as expected for Kelvin-type waves.The next higher (2.±6.) modes whose frequencies also liebelow the inertial frequency f are likewise Kelvin-typeoscillations with increasing number of amphidromicsystems and are not likely to be excited by an uniformwind in the longitudinal direction of the basin. The nexttwo modes (7. and 8., which are the ®rst two superin-ertial ones) possess complex structure and are hardlyexcited. Mode 9., the third superinertial mode, has asimple structure (see Fig. 11); it may be interpreted asthe superposition of two Poincar�e waves propagating inopposite directions (see, e.g., [10]). The next, 10th and11th modes have again intricate structures and arehardly excited.

Therefore, it seems that only the fundamental sub-inertial Kelvin-type oscillation (1. mode) and the 3.superinertial Poincar�e-type oscillation (9. mode) can beeasily excited by a spacially uniform wind. The othermodes may occur under very complex wind distribu-tions. Which of superinertial Poincar�e-type oscillationsactually does possess relatively simple structure dependssomewhat on the in¯uence of the Coriolis forces (on thevalue of the internal Rossby radius compared to the

Table 1

Eigen frequencies and periods of the fundamental oscillation (1.mode) and the ®rst ®ve superinertial oscillations (7., 8., 9., 10. and 11. modes) subject

to the strati®cation (4) (cint � 39:1 cm sÿ1), which corresponds to a two-layer model of h1 � 20 m; h2 � 80 m, and � � 9:741� 10ÿ4

Mode

1. . . . 7. 8. 9. 10. 11. . . .

Frequency

��10ÿ4 sÿ1�0.16787 . . . 1.1052 1.2421 1.3022 1.3487 1.3702 . . .

Period [h] 103.96 . . . 15.78 14.05 13.40 12.94 12.74 . . .


width of the basin). In the calculated wind-induced cir-culation in Section 3 we indeed saw these two kinds ofwaves. Their behaviour is the same as in the eigen os-cillations. Kelvin-type oscillations circulate counter-clockwise around the basin and possess large amplitudesnear-shore, Poincar�e-type oscillations occur mainly inthe center of the basin and rotate in the clockwise di-rection. The periods of the oscillations estimated fromthe wind-induced circulations and the eigen oscillationsbasically coincide.

5. Baroclinic oscillations in a smaller rectangular basin

In lakes, the periods of the oscillations depend onlake size, topography, and on the vertical densitystructure of the water body. To demonstrate the in-¯uence of basin size on the periods of baroclinic waves,the same computations as in Sections 3 and 4 areperformed in this section for a smaller basin with dif-ferent aspect ratio. Consider a rectangular basin of20 km � 10 km extent and 100 m depth, initially at

Fig. 10. Elliptical ¯uid particle trajectories (a) and the corresponding horizontal velocities for four successive phases (b)±(e) for the ®rst internal

oscillation mode (Kelvin-type oscillation).


rest, and subject to external wind-forcing in thelongitudinal direction lasting two days. All otherconditions and parameters are kept the same as inSection 3.

5.1. Wind-induced baroclinic response

The wind-induced circulation for this smaller basinis displayed in Fig. 12 for the longshore velocity at themid-points of the shore around the basin. Similarly,

Fig. 13 displays the two horizontal velocity compo-nents at the basin center. It is seen that two nearlyperiodic `components' form the dominant signals ofthe computed water motion. The ®rst (Fig. 12) pos-sesses a period of approximately 32±33 h with am-plitudes much larger near-shore than at mid-lakepositions. As indicated in Fig. 12 the maxima andminima at the western/eastern near-shore mid-pointsare a quarter period out of phase from those at thenorthern/southern near-shore mid-points, facts that

Fig. 11. Elliptical ¯uid particle trajectories (a) and the corresponding horizontal velocities for four successive phases (b)±(e) for the ninth internal

oscillation mode (Poincar�e-type oscillation).


are reminiscent of Kelvin-type wave dynamics. Thesecond oscillation, well visible in Fig. 13(b), has aperiod of approximately 10 h, is best seen in thetransverse velocity components at the mid lake posi-tion, it is also seen in the mid lake longitudinal ve-locity components of Fig. 13(a), but they are almostcompletely overshadowed there by the Kelvin-typefundamental mode. Because the basin width is onlyabout 2.5 internal Rossby radii it is also understand-able why the Kelvin-type wave signal is less attenu-ated at the basin center than in the 65 km � 17 kmbasin of the last section.

5.2. Eigen oscillation

An eigenvalue problem as in Section 4 has also beensolved for this smaller basin. The eigen frequencies andperiods of the ®rst ®ve oscillations are listed in Table 2.Only the ®rst and ®fth mode, of which elliptical ¯uid-particle trajectories are exhibited in Fig. 14, possessstructures, which are thought to be easily excited. Theoscillation of the ®rst mode propagates along the shorein the counterclockwise direction, marking Kelvin-typewave; the oscillation of the ®fth mode rotates in theclockwise direction and `lives' mainly o� shore,


southern (b), eastern (c) and northern shore (d)) at various depths in a strati®ed smaller rectangular basin of constant depth subject to a constant

wind from west (in the longitudinal direction) lasting two days. The labels correspond to the depths in meters.


indicating Poincar�e-type behaviour. The correspondingperiods, 31.9 and 9.9 h, respectively, are very close tothose obtained from the wind-induced circulation.

6. Concluding remarks

The internal response of medium size lakes (as theyexist in the Alpine regions and other mountainous areas)

to wind forcings is dominated by Kelvin- and Poincar�e-type wave dynamics. The circulation pattern also con-tains a long periodic topography induced component,but it is less signi®cant than in the ocean and oftenso much damped by the bottom friction that the longperiodic signals are attenuated before they have had thetime to form. It follows that numerical codes must beable to reproduce the dynamics of long internal wave.

Classical codes of the shallow water equations withexplicit temporal integration require large numericaldi�usion to the extent that long internal waves are soquickly attenuated that the excited waves die out muchfaster than in reality. It is therefore important to employsemi-implicit time integration so that numerical di�u-sion is minimized and restricted to physically realisticvalues. The demonstration that our semi-implicit codeSPEM enjoys these features, has been the principal goalof this paper.

We reported about the three-dimensional wind-induced baroclinic response of a rectangular basin

Fig. 13. Temporal evolution of the horizontal velocity components u (a) and v (b) for a midlake position in a strati®ed rectangular lake of constant

depth �20 km� 10 km� 100 m� subject to a constant wind from west (in the across direction) lasting two days. The labels correspond to the depths

in meters.

Table 2

Eigen frequencies and periods of the ®rst ®ve oscillations subject to the strati®cation (4)

Mode

1. 2. 3. 4. 5. . . .

Frequency [�10ÿ4 sÿ1] 0.54773 1.0589 1.4587 1.7361 1.7676 . . .Period [h] 31.9 16.5 11.5 10.05 9.9 . . .

Fig. 14. Elliptical ¯uid particle trajectories for the 1. mode ((a), Kel-

vin-type oscillation) and the 5. mode ((b), Poincar�e-type oscillation).

Open ellipses express motions in the clockwise direction, ®lled ellipses

mark motions in the counterclockwise direction.


with constant depth. The sizes of the basins werechosen to be reminiscent of Alpine lakes. Direct re-sponse to (simple) wind forcing and the oscillatingbehaviour after wind secession were studied. It wasdemonstrated that this behaviour was well predicted,in particular, the `life time' of the dominant Kelvin-and Poincar�e-type waves were reasonably repro-duced, despite the fact that the attenuation isstill somewhat larger than in real lakes of comparablesize.

The performance of the model can still be amended,i.e., di�usive mechanisms reduced, by using a largernumber of Chebyshev polynomials, computations,however, may become unduly long. The choice of thehorizontal and (more so) vertical di�usivities is verycrucial, in fact the velocity ®eld depends on the selec-tion of the di�usivities. However, while our programallows to choose the values in the range that is thoughtto be physically acceptable, the vertical distributionand temporal variation can still not be chosen withsu�cient assurance of physical reliability. This dictatesthat the Reynolds-closure conditions must be comput-ed along with the balances of mass, momentum andenergy.

A ®rst step towards an improvement of this situationwould be to add to these balance laws a closure con-dition of higher order, say using an algebraic formulafor the Reynolds stresses or a two-parameter-closurescheme, such as the k±e model. This has been done [3,4],however this code still faces problems of instability un-der very strong wind forcing.

Nevertheless, our computations over nearly 15 daysindicate that non-linear wave dynamics in strati®edlakes of realistic dimension can be performed, and thelake behaviour be studied. In a subsequent paper weshall do this for Lake Constance.

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three-dimensional wind-induced baroclinic circulation in rectangular basins

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