Barotropic response in a lake to wind-forcing Y. Wang, K. Hutter, E. Bäuerle

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Y. Wang, K. Hutter, E. Bäuerle. Barotropic response in a lake to wind-forcing. Annales Geophysicae, European Geosciences Union, 2001, 19 (3), pp.367-388. ￿hal-00316778￿

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Barotropic response in a lake to wind-forcing

Y. Wang1, K. Hutter1, and E. Bauerle¨ 2 1Institute of Mechanics, Darmstadt University of Technology, Hochschulstr. 1, D-64289 Darmstadt, Germany 2Limnological Institute, University of Constance, D-78457 Konstanz, Germany Received: 25 July 2000 – Revised: 21 November 2000 – Accepted: 15 January 2001

Abstract. We report results gained with a three-dimensional, restricted to the barotropic component for which the internal semi-implicit, semi-spectral model of the shallow water hydrostatic pressure gradient is exclusively due to the varia- equations on the rotating Earth that allowed one to compute tion of the free surface over its equilibrium value. It is com- the wind-induced motion in lakes. The barotropic response mon knowledge that numerical codes integrating these spa- to unidirectional, uniform winds, Heaviside in time, is de- tially three-dimensional shallow water equations, explicitly termined in a rectangular basin with constant depth, and in in time, are conditionally stable; due to the explicit treatment Lake Constance, for different values and vertical distribu- of the friction (or diffusion) terms, the time step is limited by tions of the vertical eddy viscosities. It is computationally the smallest mesh size. For most problems of ocean dynam- demonstrated that both the transitory oscillating, as well as ics, since space scales are usually large, the maximum us- the steady state current distribution, depends strongly upon able time steps can equally be relatively large, and thus, time the absolute value and vertical shape of the vertical eddy vis- integration over physically relevant intervals is often possi- cosity. In particular, the excitation and attenuation in time ble. Since lakes are small, sufficient spatial resolution re- of the inertial waves, the structure of the Ekman spiral, the quires small mesh sizes limiting the economically justifiable thickness of the Ekman layer, and the exact distribution and integration time to values below the physically relevant inter- magnitude of the upwelling and downwelling zones are all vals. Thus, implicit or semi-implicit integration routines are significantly affected by the eddy viscosities. Observations needed. For the latter, time steps are still limited by the value indicate that the eddy viscosities must be sufficiently small of the eddy viscosity and the space steps in the horizontal so that the oscillatory behaviour can be adequately modelled. direction; but this restriction is not severe because the hori- Comparison of the measured current-time series at depth in zontal mesh sizes are usually fairly large. That is why, often, one position of Lake Constance with those computed on the only semi-implicit schemes are applied instead of fully im- basis of the measured wind demonstrates fair agreement, in- plicit or ADI schemes. There are also many ocean models cluding the rotation-induced inertial oscillation. which include implicit schemes for good reasons, especially in coastal ocean models or when a time- and space-dependent Key words. Oceanography: general (limnology) – eddy viscosity is computed. Various implicit models have Oceanography: physical (Coriolis effects; general cir- been developed and used by many researchers (e.g. Back- culation) haus, 1983, 1985, 1987; Blumberg and Mellor, 1987; Davies, 1987; Bleck et al., 1992; Davies and Lawrence, 1994; Ip and Lynch, 1994; Song and Haidvogel, 1994). 1 Introduction On the other hand, with regard to non-linear advection terms, if the second-order centered finite difference scheme Knowledge of water movements is a prerequisite for the study is used (or other traditional high-order schemes, e.g. the spec- of a multitude of water quality problems of natural and artifi- tral method used here), numerical oscillations always occur. cial lakes. The computation of the current distribution in ho- The smaller the eddy viscosity is, the smaller the mesh size mogeneous lakes is commonly performed with the shallow and time step for stable integration must be. Thus, less nu- water equations on the rotating Earth. In the present study, merical oscillations are provoked, so that the small eddy vis- density variations are ignored (that is the case for most of the cosity can still assure stable numerical simulations. The at- Alpine lakes in winter), so that the wind-induced motion is tenuation with time of any oscillating motion increases with Correspondence to: Y. Wang growing eddy viscosity. It follows that amplitude and persis- ([email protected]) tence of oscillations are directly tied to the eddy viscosities. 368 Y. Wang et al.: Barotropic response in a lake to wind-forcing

Values needed for stable integration are often larger than is ered fluid system. We apply these hydrodynamic equations physically justified, i.e. computed physical oscillations are in the shallow water approximation, with the Coriolis term faster attenuated than in reality. A study how the vertical and the hydrostatic pressure equation implemented. Thus, eddy viscosity affects the time dependent and steady state the field equations read wind-induced velocities is, therefore, urgent. ∂u ∂v ∂w Wind-induced barotropic responses in lakes or the ocean + + = 0 (1) have been studied by many researchers (e.g. Mortimer, 1974, ∂x ∂y ∂z ∂u 1980; Simons, 1980; Hutter, 1982; Heaps, 1984; Sunder-¨ + v·∇ u − f v mann, 1984). However, for most of these works, if the three- ∂t dimensional field equations including the nonlinear advec- ∂φ ∂ ∂u ∂ ∂u ∂ ∂u = − + (νh ) + (νh ) + (νv ), (2) tion terms are numerically solved, the inertial waves seem to ∂x ∂x ∂x ∂y ∂y ∂z ∂z be hardly observable in the numerical results, or they remain ∂v + v·∇ v + f u only for a fairly short time and then are rapidly damped out. ∂t This fact is due to the much larger eddy viscosities used in ∂φ ∂ ∂v ∂ ∂v ∂ ∂v = − + (ν ) + (ν ) + (ν ), (3) the models in order to restrain the numerical (not physical) ∂y ∂x h ∂x ∂y h ∂y ∂z v ∂z oscillations and thus, assure numerical stability. ∂φ We employ the three-dimensional, hydrodynamic, semi- 0 = − − g . (4) ∂z implicit, semi-spectral model SPEM, as originally developed by Haidvogel et al. (1991) and extended for semi-implicit Here a Cartesian coordinate system (x, y, z) has been used; integration in time by Wang and Hutter (1998). This soft- (x, y) are horizontal, and z is vertically upwards, against the ware is able to cope with the above questions in a reasonable direction of gravity. The field variables (u, v, w) are the way. We propose three functional relations for the vertical components of the velocity vector v in the (x, y, z) direc- distribution of the vertical eddy viscosity: (i) constant, (ii) tions, φ is the dynamic pressure φ = p/ρ with pressure p sinusoidal half wave in an upper turbulently active layer ex- and density ρ, f is the Coriolis parameter and g is the grav- tended to a constant value below and (iii) a linear increase ity acceleration. Eddy viscosities are taken into account by with depth in the upper layer to a maximum continued by a differentiating between the horizontal and vertical directions, constant value at this maximum below it. All these distribu- νh and νv, respectively; thus, the anisotropy effects of the tur- tions are based on well justified arguments; here we use these bulent intensity are considered accordingly. distributions and vary their intensity. Wind stress forcing at the water surface, free-slip lateral We show that the offset of the surface current to the right boundary conditions, and linear bottom friction are used. of the wind, the Ekman spiral and the thickness of the Ek- Along all boundaries the normal component of the current man layer depend strongly upon both the absolute value and is set equal to zero, which, at the free surface, corresponds the distribution of the vertical eddy viscosity, as do the wind- to the rigid-lid approximation. With this method, however, a induced inertial oscillations, the stored kinetic energy and the Poisson equation must be solved at every time step to obtain details in the up- and downwelling distributions of the mo- a pressure field which ensures ∇ ·(hv¯) = 0, where h indi- tion. It is further demonstrated that the numerical values of cates water depth and v¯ is the depth-averaged velocity (see the eddy viscosities need to be sufficiently small, i.e., at lev- Haidvogel et al. (1991) and Wang (1996) for details). els of their physical counterparts if the observed dynamics A semi-spectral model was designed with semi-implicit is to be reproduced by the computations. This demonstrates integration in time to solve the system of differential equa- that robust software is needed if the barotropic circulation tions numerically. The model is based on the semi-spectral dynamics is to be adequately modelled. This paper, together model SPEM developed by Haidvogel et al. (1991), in which with the previous work relating to the baroclinic circulation the variation of the field variables in the vertical direction dynamics in lakes (Wang et al., 2000), makes up a complete is accounted for by a superposition of Chebyshev polyno- pattern of forced motion response in enclosed lakes. mials, whereas a centered finite difference discretization on The paper is structured as follows: In Sect. 2, the numer- a staggered Arakawa grid is employed in the horizontal di- ical method is briefly introduced. Results are discussed in rection; it was extended by Wang and Hutter (1998) to ac- Sect. 3 for a rectangular basin with a constant depth, and count for implicit temporal integration. Due to the small wa- in Sect. 4, for Lake Constance. Comparison of a series of ter depths of lakes, in comparison to the ocean, the original measured results with the numerical simulation of that case SPEM model had to be altered to permit economically justifi- is performed in Sect. 5. We conclude with a summary and able time steps in the computation of the circulation of a lake. some remarks in Sect. 6. In Wang and Hutter (1998) several finite difference schemes, implicit in time, were introduced; that scheme which uses implicit integration in time for the viscous terms in the verti- 2 Numerical method cal direction was the most successful one. For a realistic irregular lake, a vertical topography follow- The balance law of mass and the balances of linear momen- ing the σ -coordinate, and a horizontal shore following the tum form the hydrodynamic field equations for the consid- curvilinear coordinate system is employed. By using the so- Y. Wang et al.: Barotropic response in a lake to wind-forcing 369 16

(iii) ]

1 Wind − Φ cm s [ (ii) Vsurf [degrees] surf Φ V

(i)

(a) (b)

2 1 2 1 νv,A,A0 [m s− ] νv,A,A0 [m s− ] FIG. 1: Rotation angle Φ between wind and surface current (a) and absolute fluid velocity Vsurf at the Fig. 1. Rotation angle 8 between wind and surface current (a) and absolute fluid velocity Vsurf at the free surface (b). Here 8 and Vsurf are free surface (b). Here Φ and Vsurf are plotted against νv (solid curves), A (broken curves), A0 (dotted plotted against νv (solid curves), A (broken curves), A0 (dotted curves), respectively, corresponding to the cases (i), (ii) and (iii). curves),respectively,corresponding to the cases (i),(ii) and (iii) called σ-transformation,(a) a lake near domain the western with varying shore topog- homogeneous water,(b) near initially the eastern at rest, shore and subject to external raphy is transformed to a new domain with constant depth, wind forcing. This wind blows in the long direction of the and this region is once again transformed in the horizontal rectangle, uniformly in space, Heaviside in time, and with ] ] (c)

1 − coordinates by using conformal mapping, which maps the strength1 4.5 m s 1 (10(a) m above the water(b) surface), corre- − − (d)r shore, as far as possible, onto a rectangle. Uniformity in grid sponding to a wind stressr of approximately 0.0447r Pa at the mms

size distribution is intended because numerical oscillations watermms surface. This is the wind-forcing we apply throughout

[ r [ w (instabilities) occur preferably on the small scales; however, thew paper, except for Sect. 5. Integration starts at rest until a it is difficult to achieve it in complex geometries. In such steady state is reached. We shall implement the discretization cases, to attain an uniform grid as far as possible, a bounding with 10–30 Chebyshev polynomials; the exact number de- line, which deviates in some segments from the actual lake pends on the magnitude of the vertical eddy viscosities (the boundaries, is used for the conformal mapping. In these seg- smaller the vertical eddy viscosities, the more Chebyshev ments the actual boundaries can only be approximatedupwelling by a polynomials are needed to achievedownwelling stable and convergent nu- Vertical velocity step function, and the land areas must be masked with a spe- mericalVertical velocity computations). We choose 1x = 1y = 1 km and let cial technique (Wilkin et al., 1995). For the computation in a the numerical value of the horizontal eddy viscosity of mo- rectangular basin, such a conformal mapping in the horizon- mentum be constant ν = 1 m2s−1, because it turned out that Time [h] h Time [h] tal direction is not needed; however, for the Lake Constance, the numerical values of ν are not very crucial (Wang, 1996), (c) near the northern shore h it is required, but not unique. In this case, a conformal map- and this value(d) is generally near the reasonable southern shore (Csanady, 1978; Hut- ping is performed for a bordering line which deviates in some ter, 1984). The insensitivity of the horizontal eddy viscosity ]

segments from the actual shore line, so that a more uniform can] be expected and easily recognized because the horizontal 1

1 downwelling − grid system is obtained. Then, in the numerical computa- variation− of the velocity field is relatively small, in compari- tions, the land areas within the grid system must be excluded son with its vertical variation, except in the vicinity of shores mms [ mms by the aforementioned masking technique. and[ when tracer diffusion is considered. w This numerical code has proved its suitability in several w lake applications, including diffusion problems (Hutter and Wang, 1998), substructuring procedures (Wang and Hutter, 3.1 Surface steady current in relation to distributions and

2000) and wave dynamics in baroclinic wind-driven circula- amplitudes of the vertical eddy viscosity2 1 ...... N=30,νv =0.001m s− tion dynamics (Wang et al., 2000). Here, the application of 2 1 upwelling ---- N=20,νv =0.005m s− 2 1 this code toVertical velocity the barotropic motions is performed. It will be Three chosen vertical——– eddyN=10 viscosities,νv =0.02 m ares− prescribed as fol- Vertical velocity seen in the following numerical results that almost all typical lows: features in a homogeneous lake, e.g. inertial waves, Ekman ∈ [ ] 2 −1 spirals, can be reproduced by thisTime model. [h] – case (i): νv 0.005Time, 1 m [h]s , FIG. 2: Time series of the vertical velocity component w at– thecase four (ii): indicated (inset in panel b) near-shore 3 Barotropicmidpoints motions at 30 min deptha rectangular in the homogeneous basin rectangular basin of( constant depthπ z subject to constant) wind A · sin − , z > −45m 2 −1 from West. Results were obtained with three different verticalνv = eddy viscosities2νv30. N denotes the numberm s , A rectangularof Chebyshev basin of 65polynomials×17 km2 extent needed with to aachieve 100 m depthstable and convergent numerical0.01 , computations. z ≤ −45m corresponds to the space scale of Lake Constance; we assume where A ∈ [0.01, 1] m2s−1 , 16

(iii) ]

1 Wind − Φ cm s [ (ii) Vsurf [degrees] surf Φ V

(i)

(a) (b)

2 1 2 1 νv,A,A0 [m s− ] νv,A,A0 [m s− ] FIG. 1: Rotation angle Φ between wind and surface current (a) and absolute fluid velocity Vsurf at the free surface (b). Here Φ and Vsurf are plotted against νv (solid curves), A (broken curves), A0 (dotted 370curves),respectively,corresponding to the cases (i),(ii) and Y. Wang (iii) et al.: Barotropic response in a lake to wind-forcing

(a) near the western shore (b) near the eastern shore ] ] (c) 1 1 (a) (b) − − (d)r r r mms mms

[ r [ w w

upwelling downwelling Vertical velocity Vertical velocity

Time [h] Time [h] (c) near the northern shore (d) near the southern shore ] ] 1

1 downwelling − − mms [ mms [ w w

2 1 ...... N=30,νv =0.001m s− 2 1 upwelling ---- N=20,νv =0.005m s− 2 1 Vertical velocity ——– N=10,νv =0.02 m s− Vertical velocity

Time [h] Time [h] FIG. 2: Time series of the vertical velocity component w at the four indicated (inset in panel b) near-shore Fig. 2.midpointsTime series at of 30the m vertical depth velocity in the component homogeneousw at the rectangular four indicated basin (inset of in constant panel b) depth near-shore subject midpoints to constant at 30 m depth wind in the homogeneous rectangular basin of constant depth, subject to constant wind from the West. Results were obtained with three different vertical from West. Results were obtained with three different vertical eddy viscosities ν . N denotes the number eddy viscosities νv. N denotes the number of Chebyshev polynomials needed to achieve stable and convergentv numerical computations. of Chebyshev polynomials needed to achieve stable and convergent numerical computations.

– case (iii): varying the free parameters A and A0. Choices like these are ( z ) considered realistic and have, to some extent, been analysed A0 + (A0 − 0.03) , z > −10m 2 −1 νv = 10 m s , with a linear equation system, in which the nonlinear advec- 0.03, z ≤ −10m tion terms are not included in finite depth oceans (see Heaps, 2 −1 where A0 ∈ [0.01, 1] m s . 1984).

In case (i), the vertical eddy viscosities are spatially con- One typical result of the Ekman problem is the angle 8 stant, but their values are varied. Case (ii) corresponds to between the direction of the wind and the surface current in a z-dependence of νv, where νv is large in the upper layer, steady state. In the original problem, an infinite ocean with growing first with depth until it reaches a maximum in 30 m constant vertical eddy viscosity exists, and the surface cur- 2 −1 ◦ and then rapidly decreases to the value 0.01 m s at 45 m rent vsurf is 45 to the right of the wind. Results of our non- depth and below. This form corresponds with the vertical linear calculations are summarized in Fig. 1. In case (i), the ◦ 2 −1 eddy viscosity distribution obtained by Svensson employing maximum value of 8 is 42.5 for νv = 0.005 m s , and 8 an one-dimensional k − ε model (Svensson, 1979) and by decreases rapidly with increasing vertical eddy viscosity to ◦ 2 −1 Guting,¨ using a three-dimensional k − ε closure condition a value as low as 4 for νv = 1 m s . It can never reach ◦ (Guting,¨ 1998). Case (iii) assumes a linear variation of νv 45 as it does (independent of the value of νv) in the infinite from the free surface to 10 m depth below, in which νv is ocean, because of the finiteness of the rectangular basin that 2 −1 kept constant at the value νv = 0.03 m s (Csanady, 1980; induces a geostrophic flow which has the tendency to reduce Madsen, 1977). The maximum values of νv can be varied by 8. This simply indicates how significantly the circulating Y. Wang et al.: Barotropic response in a lake to wind-forcing 371 17 17 ] ] ] 1 ] 1 1 1 − − − − cms cms [ [ cm s cm s [ [ v v u u Horizontal velocity

Horizontal velocity (b) Horizontal velocity (b) Horizontal velocity (a)(a)

Time [h] TimeTime [h] [h] Time [h] FIG. 3: FIG. 3: TimeTime series series of of the the horizontal horizontal velocity velocity components componentsu (a)u (a) and andv (b)v (b) in the in the rectangular rectangular basin basin subject subject Fig. 3. Time series of the horizontal velocity components u (a) and v (b) in the rectangular basin, subject to constant2 1 wind from the West, to constant wind from West computed with constant− vertical eddy viscosity νv =0.005m s−2. The1 labels computedto constant with constant wind vertical from eddy West viscosity computedν = 0 with.005 m constant2s 1. The vertical labels on eddy the curves viscosity stand forνv depth=0.005m in meters.s− . The labels on curves stand for depth in meters. v on curves stand for depth in meters. motion in a finite basin can depend on the absolute values viscosity is, the stronger the superimposing oscillations will of the eddy viscosities. Even more surprising1 are the results be, and the longer the oscillations will persist. For the larger 1.4 cm s− ✲ 1 = 2 −1 of case (ii). Here 8 varies non-monotonously1.4 cm s−✲ with A: For value νv 0.02 m s , only one day after the wind starts, 20 m 60 m 1 A ≤ 0.12 it decreases with growing A, while it grows with the inertial oscillations are damped out,0.42 whereas cm s− the oscilla- 20 m 60 m ✲ 1 2 −1 0.42 cm s− A when A ≥ 0.12. Case (iii) presumes, for most values of tions for νv = 0.001 m s can still be clearly identified✲ af- A0, a very active turbulent near-surface layer and illustrates ter six days. For this small eddy viscosity, due to the emerg- that for such a stiff situation, the typical dependence of 8 ing strong oscillation, even a short-time downwelling (up- can even be reversed. In Fig. 1b the moduli of the surface ve- welling) at the western (eastern) shore can be observed. locities vsurf are shown as functions of the eddy viscosities. In Fig. 3, the time series of the horizontal velocity compo- As expected, they rapidly decrease with increasing eddy vis- nents u (a) and v (b) for a medium value of ν = 0.005 m2s−1 20 m (b) v cosities. These results demonstrate how critical the distribu- in the center20 m of the basin, at different(b) depths, are displayed. tion and absolute values of the eddy viscosities are; reducing SteadyFIG. motion 4: Hodographs is reached of approximately the horizontal after velocity five days, but the numerical10 m diffusion to levels below the physical values initially,vectorsFIG. an inoscillating 4: theHodographs middle motion of ofthe can the rectangular be horizontal seen at all basin watervelocity depths, is, therefore, compelling, if results are not to be physically 10 m however,invectors 0,5,10,20 with in decreasing and the 60 middle m depthamplitude of the (a). rectangular as In the order depth to basin increases. falsified. It is also interesting that the horizontal velocity that oscil- seein the 0,5,10,20 spiral at and 20 60 m m depth depth more (a). clearly In order the to lates above 30–40 m, with the opposite phase from below 20see m thehodograph spiral at is 20 shown m depth separately more clearly in (b). the 3.2 Time series of the5m velocity components this depth, corresponds approximately to the Ekman depth. ThisThe can20 motion be m explained hodograph starts by from theis shown use a state of a separately linear at rest. equation The in (b). system. (a) 5m 0m It issmall knownThe circles motion that mark for starts a time linearized from intervals a equation state of 1/4 at system, inertial rest. Thethe hori- Figure 2 shows the(a) time series of the vertical0m velocity com- ponent w in 30 m depth at the four nearshore midpoints, as zontalperiodsmall motion circles consists mark of a time drift intervals current directly of 1/4 due inertial to wind- sketched in panel (b) of Fig. 2, subject to three different mag- stress,period and a geostrophic current due to the surface slope (the nitudes of the constant vertical eddy viscosity, respectively. surface pressure gradient) that can be described as follows: Immediately after the west wind sets in, the motion starts Neglecting the horizontal friction terms, the linearized hor- rapidly with an upwelling at the western end (Fig. 2a) and a izontal momentum equations (2) and (3) then take the form downwelling at the eastern end (Fig. 2b), due to the direct 2 effect of wind stress at the surface. In the northern mid- ∂u ∂φ ∂ u − f v = − + νv boundary point, there occurs an upwelling (Fig. 2c), while ∂t ∂x ∂z2 in the southern shore, a downwelling emerges (Fig. 2d). This ∂v ∂φ ∂2v + f u = − + ν (5) behaviour is, clearly, due to the Coriolis force that causes ∂t ∂y v ∂z2 a horizontal velocity drift to the right and therefore, is re- sponsible for the downwelling (upwelling) at the southern with νv = const. For the homogeneous case the dynamic (northern) shore on the northern hemisphere. The motion is pressure is independent of z. The local solution u(z, t), v(z, t) characterized by oscillation, with a period of approximately of (5) (depending only parametrically on x, y) may be writ- 16.3 hours, which can obviously be identified as the inertial ten as period using a Coriolis parameter approach for Lake Con- −4 −1 stance, f = 1.07 × 10 s . The smaller the vertical eddy u = u1(t) + u2(z, t), v = v1(t) + v2(z, t), (6) 17 ] ] 1 1 − − cms [ cm s [ v

u 17 ] ] 1 1 − − cms Horizontal velocity [

cm s (b) Horizontal velocity

[ (a) v u

Time [h] Time [h] FIG. 3: Time series of the horizontal velocity components u (a) and v (b) in the rectangular basin subject 2 1 372to constant wind from West computed with constant vertical Y. Wang eddy viscosity et al.: Barotropicνv =0.005m responses− . The in labels a lake to wind-forcing on curves stand for depth in meters.

Horizontal velocity (b) Horizontal velocity (a)

1 1.4 cm s− Time [h] ✲ Time [h] 20 m 60 m 1 FIG. 3: Time series of the horizontal velocity components u (a) and v (b) in the rectangular0.42 cm basin s✲− subject 2 1 to constant wind from West computed with constant vertical eddy viscosity νv =0.005m s− . The labels on curves stand for depth in meters.

1 1.4 cm s−✲ 20 m (b) 20 m 60 m 1 FIG. 4: Hodographs of the horizontal velocity0.42 cm s✲− 10 m vectors in the middle of the rectangular basin in 0,5,10,20 and 60 m depth (a). In order to see the spiral at 20 m depth more clearly the 20 m hodograph is shown separately in (b). 5m The motion starts from a state at rest. The 0m (a) small circles20 m mark time intervals(b) of 1/4 inertial period FIG. 4: Hodographs of the horizontal velocity Fig. 4. Hodographs10 of m the horizontal velocity vectors in the middle of the rectangularvectors inbasin the in middle 0, 5, 10, of the 20 andrectangular 60 m depths basin(a). In order to see more clearly the spiral at 20 m depth, the 20 m hodograph is shown separatelyin 0,5,10,20 in (b). and The 60 motion m depth starts (a). from Inorder a state to at rest. The small circles mark time intervals of 1/4 inertial period. see the spiral at 20 m depth more clearly the 20 m hodograph is shown separately in (b). 5m The motion starts from a state at rest. The 0m where u1(t), v1(t) is the geostrophic(a) current which is a so- the Ekmansmall circles depth, mark the timedrift intervals motion (u of 1/42, v inertial2), induced directly lution of by windstress,period is dominant, whilst below this depth, the Ek- man drift motion dies out and only the geostrophic current ∂u1 ∂φ ∂v1 ∂φ − f v1 = − , + f u1 = − , (7) (u , v ) remains, which is independent of depth under ho- ∂t ∂x ∂t ∂y 1 1 mogeneous, hydrostatic conditions. From the solutions of due to the surface pressure gradient, whilst u2(z, t), v2(z, t) such a simplified linear, horizontal momentum equation sys- is the depth-dependent or frictionally induced velocity com- tem, one can easily see that these two contributions to the ponents current oscillate with opposite phase; at 30 to 40 m depth, the two oscillations possess comparably large amplitudes, and 2 2 ∂u2 ∂ u2 ∂v2 ∂ v2 hence, the superimposing oscillation is hardly visible at this − f v2 = νv , + f u2 = νv . (8) ∂t ∂z2 ∂t ∂z2 depth (see the 40m-depth curves in Fig. 3). Such oscillations have been discussed by Krauss (1979) and Csanady (1984). With the suddenly imposed windstress τ 0 at the surface in the positive x-direction, and with a surface pressure gradient (a The same time series of the velocity components u and surface slope) ∂φ/∂x in the positive x direction and bottom v, as shown in Fig. 3, are displayed again in the form of stresses neglected, (7) and (8) have the local solutions hodographs in Fig. 4 for 0, 5, 10, 20 and 60 m depths. The small circles along the curves mark time intervals of 1/4 of 1 ∂φ 1 ∂φ the inertial oscillations, which is approximately 4 hours. The u1 = − sin f t, v1 = (1 − cos f t) (9) f ∂x f ∂x arrows represent the velocity vectors in the steady state. This of the geostrophic current and graph is very similar to the result obtained by Krauss (1979) in an infinite channel of rectangular cross-section. In the up- ! τ 0 Z t cos f s z2 per layer (0, 5, 10, 20 m), the wind-induced drift current, due u = − s, 2 1/2 1/2 exp d to sudden wind, approximates the steady state in the form π (νvs) 4νvs 0 of inertial oscillations. The spiral in 60 m depth reflects ! τ 0 Z t sin f s z2 the adaptation of the current field to the geostrophic current. v = − exp − ds (10) 2 1/2 1/2 The reflection of the water surface, due to the sudden wind, π 0 (νvs) 4νvs produces a sudden change of the pressure field, in the en- of the drift motion, respectively. The solution (10) was first tire water column. The adaptation of the current field, which given by Fredholm (Ekman, 1905). It is obvious that the two was zero, to the new condition, occurs in the form of inertial motions have opposite phase. It is of interest to note that waves. Similar to those in the surface layer, these waves start such a combination of the currents can give a satisfactory together with the wind because the pressure gradient is felt approximation for the vertical current profile in regions far immediately. In the hodograph for 20 m depth, one can see, away from the lateral boundaries, even in those cases where at first, (perhaps during the first two hours) the same north- nonlinear effects cannot be neglected in solving the horizon- westward current as in 60 m depth; at this initial time, only tal circulation problem (Nihoul and Ronday, 1976). Above the geostrophic current exists in 20 m, but when the surface 18 Y. Wang et al.: Barotropic response in a lake to wind-forcing 373

2 1 2 1 a) νv=0.02 m s− b) νv=0.02 m s− in an infinitely large basin at the center of the rectangular basin without bottom friction

2 1 2 1 c) νv=0.02 m s− d) νv=0.02 m s− at the center of at the midpoint of the rectangular basin the southern shore of with bottom friction the rectangular basin with bottom friction

2 1 e) νv=0.005 m s− 2 1 f) νv=0.005 m s− at the center of at the midpoint of the rectangular basin the southern shore of with bottom friction the rectangular basin with bottom friction

× Fig. 5. EkmanFIG. spirals 5: Ekman in an infinite spirals ocean in an(a) infiniteand in aocean rectangular (a) and basin in with a rectangular 65 km 17 basin km side with lengths 65 km and 10017 m km depth side(b-f) lengths, respectively. The graphs show the horizontal velocities in a vertical profile at the indicated positions. The drawn arrows× are for positions 5 m apart from and 100 m depth (b-f),respectively. The graphs show the horizontal velocities2 −1 in a vertical profile at the one another, from the free surface to the bottom. (a) in an infinite ocean with νv = 0.02 m s ; (b) at the midlake position of the rectangular indicated positions.2 −1 The drawn arrows are for positions 5 m apart from one another from the free surface2 −1 basin with νv = 0.02 m s , no bottom friction; (c) at the midlake position and (d) near the southern shore with νv = 0.02 m s and 2 1 2 −1 bottom friction;to the bottom.(e) at the midlake a) in an position infinite and ocean(f) near with theν southernv =0.02 shore m s− with; b)νv at= the0.005 midlake m s positionand bottom of friction. the rectangular Note the different 2 1 scales usedbasin in each with graph.νv =0.02 m s− ,no bottom friction; c) at the midlake position and d) near the southern shore 2 1 with νv =0.02 m s− and bottom friction; e) at the midlake position and f) near the southern shore with 2 1 νv =0.005 m s− and bottom friction. Note the different scales used in each graph. drift current reaches this depth, an abrupt change in direction 3.3 Ekman spirals occurs. It can be explicitly detected that the wind-induced current in the upper layer (in 0, 5, 10 and 20 m depth) os- In Fig. 5 we display vertical structures of the horizontal ve- cillates with opposite phase of the geostrophic current in the locities in the form of Ekman spirals for several different lower layer. cases in steady state. For a wind-induced motion, where 19

374 Y. Wang et al.: Barotropic response in a lake to wind-forcing

2 1 ...... N=30,νv =0.001m s− ] ] 2 1 ---- N=20,νv =0.005m s−

Nm 2 1 Nm ——– N=10,νv =0.02 m s− 9 9 10 10 × × [ [ Total kinetic energy (a) Total kinetic energy (b)

Time [h] Time [h] FIG. 6: Time series of the total kinetic energy in the rectangular basin subject to constant wind from Fig. 6. WestTime seriesin the of long the total direction kinetic (a)energy and in from the rectangular South in basin the subjectcross direction to constant (b) wind with from three the West different in the long vertical direction eddy(a) and = from theviscosities South in the (see cross inset, directionN =(b) numberwith three of Chebyshev different vertical polynomials eddy viscosities used). (see inset, N number of Chebyshev polynomials used). the vertical velocity is negligible, for example, in the mid- cal eddy viscosities. Here too, the value of the vertical eddy dle of a lake, the motion under homogeneous conditions can viscosity considerably affects the distribution of the velocity. be considered a linear combination of (i) the geostrophic cur- rent, constant in the vertical, (ii) a bottom current deviation 3.4 Total kinetic energy in relation to the vertical eddy vis- indicating the bottom Ekman layer,N decaying exponentially cosity and the wind-fetchN away from the bottom which, together with the geostrophic current, satisfies the bottom boundary condition, and (iii) a In Fig. 6, the time series of the total kinetic energy in the surface-drift current (the Ekman layer) decreasing rapidly homogeneous rectangular basin of constant depth, subject to with depth and satisfying the surface dynamic boundary con- constant western (a) and southern (b) wind, respectively, are dition. In an infinite ocean with infinite depth, there exists displayed, as previously computed with three vertical eddy only the surface-drift current. As is evident from Fig. 5a, viscosities, as indicated. As has been seen in the time series for a constant vertical eddy viscosity, the drift current at the for the horizontal velocity, the inertial motions of the tran- ◦ 2 −1 surface,WE in steady state, is directed 45 to the right of the sientWE energy persist much longer, when νv = 0.005 m s 2 −1 windstress (on the northern hemisphere). In a bounded rect- than when νv = 0.02 m s , and this difference is even 2 −1 angular basin, due to the surface slope under windstress, a more distinct between νv = 0.005 m s and νv = 0.001 2 −1 geostrophic current shares the wind-induced motion. Sub- m s . The total kinetic energy stored in the basin for νv = 2 −1 2 −1 ject to a western wind, a surface pressure gradient towards 0.001 m s is much larger than for νv = 0.005 m s or 2 −1 the east is built up, and hence, the caused geostrophic cur- νv = 0.02 m s , due to the much larger energy input from rent points towards the north, as displayed in Fig. 5b; the windstress because of the much larger water velocity at the current below the Ekman depth is approximately 60 m for free surface (or more precisely, its component in the wind 2 −1 νv = 0.02 m s . If the northward geostrophic current is direction), and the likely smaller dissipation for the smaller eliminated from the motion in Fig. 5b,S a velocity field is ob- vertical eddy viscosity. ComparisonS of Figs. 6a,b shows that (a) (b) tained which is almost identical with that of the infinitely the kinetic energy subject to the longitudinal, western wind large oceanFIG. (Fig. 7: Total 5a). kinetic If bottom energy friction computed cannot be three ignored, days afteris the only wind slightly set-up larger (steady than that state) which subject is subject to constant to the trans- the bottom Ekman layer occurs (Fig. 5c). The surface drift wind from different directions with the vertical eddyverse, viscosity southernν =0 wind..02 m2s 1. The computations are current, as well as the bottom current deviation, decreases v − performed for two case: (a) the Coriolis force effect is considered;The dependency (b) the of Coriolis the total force kinetic is energy neglected. on the The wind di- rapidly with distance from the boundaries and a decay rate rection is worked out in terms of a polar diagram, which plots directions of the drawn arrows are the wind directions (for an interval of 10 ) around the basin,their depending on the magnitude of the vertical eddy viscosity, as the variation of the total kinetic◦ energy stored in the station- can belengths seen from indicate a comparison the magnitudes of Fig. 5c,e. of the We total also kinetic note energy.ary circulation The dashed of the circles rectangular show basin the maximum as a function value of the di- that theof northward the total geostrophic kinetic energy current in all automatically directions,which reduces is 5 .24 109N m for (a),but 4 .49 1010Nm for(b). rection× of the wind. We compute× the water motion subject to the off-set angle of the surface current relative to that of the a constant wind from different directions (in intervals of 10◦) 2 −1 Ekman drift, a fact visible when comparing Figs. 5a and 5b. for a fixed vertical eddy viscosity νv = 0.02 m s . The po- Finally, Figs. 5d,f display the Ekman spirals for a position lar diagram of the total kinetic energy is displayed in Fig. 7a. near the midpoint of the southern shore (the distance from It is seen that the kinetic energy in the rectangular basin with shore is 500 m) that is computed for two values of the verti- constant depth depends only marginally on the wind direc- 19

2 1 ...... N=30,νv =0.001m s− ] ] 2 1 ---- N=20,νv =0.005m s−

Nm 2 1 Nm ——– N=10,νv =0.02 m s− 9 9 10 10 × × [ [ Total kinetic energy (a) Total kinetic energy (b)

Time [h] Time [h] FIG. 6: Time series of the total kinetic energy in the rectangular basin subject to constant wind from West in the long direction (a) and from South in the cross direction (b) with three different vertical eddy viscosities (see inset, N = number of Chebyshev polynomials used).

Y. Wang et al.: Barotropic response in a lake to wind-forcing 375

N N

WEWE

S S (a) (b)

FIG. 7: Total kinetic energy computed three days after the wind set-up (steady state) subject to constant Fig. 7. Total kinetic energy computed three days after the wind set-up (steady state) subject to constant2 1 wind from different directions20 with wind from different directions2 −1 with the vertical eddy viscosity νv =0.02 m s− . The computations are the vertical eddy viscosity νv = 0.02 m s . The computations are performed for two case: (a) the Coriolis force effect is considered; (b) the Coriolisperformed force is neglected. for two case: The directions (a) the Coriolis of the drawn force arrows effect are isconsidered; the wind directions (b) the (for Coriolis an interval force of is 10 neglected.◦) around the The basin, their lengths indicatedirections the magnitudes of the drawn of the arrows total kinetic are the energy. wind The directions dashed circles (for showan interval the maximum of 10◦) value around of the the total basin,their kinetic energy in all directions,lengths which is indicate 5.24 × 10 the9 N magnitudes m for (a), but of 4.49 the× total1010 N kinetic m for (b). energy. The dashed circles show the maximum value of the total kinetic energy in all directions,which is 5 .24 109N m for (a),but 4 .49 1010Nm for(b). × × N N

WEWE

S S (a) (b) FIG. 8: Same as Fig. 7 but for a rectangular basin with larger aspect ratio. Here the basin dimension is × × Fig. 8. Same60 km as Fig.5km 7, but100 for m a rectangular instead of basin the dimension with larger of aspect 65 km ratio.17 Here, km the100 basin m in dimension Fig. 7. The is 60 dashed km 5 circles km show100 m instead × × of the dimension× of 65 km× 17 km 100 m in Fig. 7. The dashed circles× show the× 8 maximum value of the total kinetic energy, which is 8the maximum value of the total kinetic energy,which is 9 .71 10 9N m for the case with consideration of 9.71 × 10 N m for the case with consideration of the Coriolis force (a), but 8.57×× 10 N m for the case without the Coriolis effect (b). the Coriolis force (a),but 8 .57 109N m for the case without the Coriolis effect (b). × tion. The kinetic energy attains its maximum for longitu- southeast wind. This loss of symmetry is due to the effect of dinal, western wind, while for a transverse southern wind, the Coriolis force, which can be demonstrated by means of the kinetic energy is minimum, although the maximum and repeating the computations, but under the neglect of the Cori- the minimum differ from each other only by approximately olis force. The analogous polar diagram of the total kinetic 15%. As expected, the dependence of the kinetic energy on energy, without consideration of the Coriolis force, is dis- the wind direction is anti-symmetric about the E-W or S-N played in Fig. 7b. In this case, the dependence of the kinetic axis. It is also seen that the kinetic energy subject to south- energy on the wind direction is symmetric about the E-W or west or northeast wind is larger than that for northwest or S-N axes. When the Coriolis force is ignored, the kinetic

FIG. 9: Map of Lake Constance. Untersee, Uberlinger¨ See and Obersee. Also shown are the 50 m und 200 m isobaths and a few main towns along the shore. 20

N N

WEWE

S S (a) (b) FIG. 8: Same as Fig. 7 but for a rectangular basin with larger aspect ratio. Here the basin dimension is 60 km 5km 100 m instead of the dimension of 65 km 17 km 100 m in Fig. 7. The dashed circles show × × × × the maximum value of the total kinetic energy,which is 9 .71 108N m for the case with consideration of × the Coriolis force (a),but 8 .57 109N m for the case without the Coriolis effect (b). ×

376 Y. Wang et al.: Barotropic response in a lake to wind-forcing

FIG. 9: Map of Lake Constance. Untersee, Uberlinger¨ See and Obersee. Also shown are the 50 m und Fig. 9. Map of Lake Constance. Untersee, Uberlinger¨ See and Obersee. Also shown are the 50 m und 200 m isobaths and a few main towns along200 the m shore. isobaths and a few main towns along the shore. energy is almost an order of magnitude larger. One likely 4 Barotropic circulation in Lake Constance reason may be that the water motion decays exponentially with increasing depth when the Coriolis force is accounted Lake Constance consists of three basins: Obersee, Uberlinger¨ for, while the velocity decreases only linearly with depth, if See and Untersee, but the Untersee is separated from the the Coriolis force is neglected. The other reason may be that other two basins by a 5 km long channel; we shall be con- in the absence of the Coriolis force, the water motion at the cerned here with the ensemble Obersee+Uberlinger¨ See, for free surface is mainly along the wind direction, hence, there brevity, also referred to as Lake Constance. It is approxi- is more energy input from the windstress. When the Coriolis mately 64 km long and 16 km wide with has a maximum force is neglected, the maximum kinetic energy occurs for a depth of 253 m and an approximate mean depth of 100 m, as wind blowing in the diagonal direction of the basin. In fact, shown in Fig. 9. the difference in the total kinetic energy depends primarily on A study analogous to that above was also performed for the wind-fetch, i.e. the integrated distance over which wind- Lake Constance with the same number of grid points (65 × action takes place, taken along the wind direction. For diago- 17) as for the rectangle. Computations were also done for 2 −1 2 −1 2 −1 nal winds, the wind-fetch is at maximum. In the presence of νh = 1.0 m s , νv = 0.02 m s and νv = 0.005 m s , rotation, the integral for the fetch would probably have to be respectively. In the second case, the larger number of poly- taken over the Ekman layer. This may be why the maximum nomials, namely N = 25 instead of N = 10, was needed of the kinetic energy is not reached for diagonal winds. to achieve stable and convergent numerical integration. Let a spatially uniform, temporally constant wind with a wind speed of 4.5 m s−1 blow from Northwest (305◦ (NW) in the longitudinal direction of the lake) until steady state condi- The stronger dependence of the total kinetic energy on the tions are established. wind direction can be expected for a basin with a larger as- pect ratio (e.g. a larger variation of the wind-fetch). In or- 4.1 Horizontal distribution of the steady currents der to demonstrate this point, we repeat the computations for a basin with a larger aspect ratio. A basin dimension of The horizontal variation of the currents at the surface and at 60 km × 5 km × 100 m is used instead of 65 km × 17 km × depth levels 10, 20 and 40 m is depicted in Fig. 10. The hori- 100 m. The results are displayed in Fig. 8. In this case, the to- zontal motion is displayed by arrow-diagrams. In the central tal kinetic energy depends on the wind direction much more part of the lake, the surface current is deflected by about 50– intensively than in Fig. 7. Obviously, the basin must be very 60◦ to the right of the downwind direction and amounts to 5 long and narrow before a directional dependence of the total cm s−1, on average. The boundary currents somewhat off the kinetic energy is appreciable. northern and southern shore increase to a magnitude of 8–10 cm s−1 and are directed parallel to the boundary. As is typical for drift currents, the direction of the motion In the next section we will see that the response of Lake in the open lake continuously turns to the right on with in- Constance, subject to a wind, is much more conspicuously creasing depth and decreasing velocities. At 10 m depth and dependent on the wind direction with regard to the total ki- far away from the shores, the current deflection to the right of netic energy, even though its geometry is close to the rectan- the wind exceeds 90◦, and the average velocity decreases to gular basin in Fig. 7. This means that such a dependence on 3–4 cm s−1. At the 40 m depth, the motion in the middle of the wind direction depends not only on the wind-fetch, but the lake is nearly reflected to the upwind direction. However, also on the bathymetry of the lake basin. above the 40 m depth, the shore currents are still basically Y. Wang et al.: Barotropic response in a lake to wind-forcing 377 21

Velocity (Time = 6.0 days)

Depth = 0 m

10 m

20 m

40 m

WIND

✲

N

FIG. 10: Velocity vectors,6 days after a constant northwest wind (305 NW in the longitudinal direction ◦ ◦ Fig. 10. Velocityof the basin) vectors, sets 6 days up inafter 0 m,10 a constant m,20 northwest m and 40 wind m (305depth.NW Each in the panel longitudinal has its directionown velocity of the scale. basin) started, in 0 m, 10 m, 20 m and 40 m depth. Each panel has its own velocity scale. 22 378 Y. Wang et al.: Barotropic response in a lake to wind-forcing22

anticyclonicanticyclonic gyre gyre PPP P PP PPPqPq ✏✶ ✏✏ ✏✶ ✏✏ ✏✏ ✏✏ ✏ ✏✏ ✏✏ ✏ ✏✏ cyclonic✏ gyre✏ cyclonic gyre

FIG. 11: Stream function of the vertically integrated volume transport in the steady state (6 days after a FIG. 11: Fig. 11. Streamconstant functionStream longshore of function the vertically northwest of the integrated wind vertically sets volume up). integrated Solid transport (dashed) in volume the streamlinessteady transport state (6 belong days in the after to steady cyclonic a constant state and longshore anticyclonic (6 days northwest after a wind started).constant Solidrotations. (dashed) longshore streamlines northwest belong wind to cyclonic sets up). and anticyclonicSolid (dashed) rotations. streamlines belong to cyclonic and anticyclonic rotations.

(a) (b) (a) (b)

Wind = ⇒ Wind = ⇒

(c) (d)

(c) (d)

Wind = ⇒

Wind = 2 1 2 1 νv =0.02 m s⇒− νv =0.005 m s−

FIG. 12: Ekman spirals for a wind in the longitudinal direction of Lake Constance at the positions shown Fig. 12. Ekman spirals for a wind in the longitudinal direction of Lake Constance at the positions shown in the insets. The graphs show the horizontalin velocitiesthe insets. in two The vertical graphs profiles show the computed horizontal for large velocities (left) and in small two vertical (right) vertical profiles eddy computed viscosities, for as large indicated. (left) The drawn arrows are forand positions small (right) 5 m apart vertical from oneeddy2 another,1 viscosities from as the indicated. free surface The to the drawn bottom. arrows Each are panel for has positions its2 own1 5 velocity m apart scale, from as indicated. one another fromνv =0 the.02 free m surfaces− to the bottom. Each panel has its ownνv velocity=0.005 scale m ass− indicated. FIG. 12: Ekman spirals for a wind in the longitudinal direction of Lake Constance at the positions shown parallel to the shores with the exception of the northeastern In order to show the general influence of the depth con- in the insets. The graphs show the horizontal velocities in two vertical profiles computed for large (left) shore, where the motion below the 20 m depth is almost in figuration on the wind-driven circulation, the vertically inte- and small (right) vertical eddy viscosities as indicated. The drawn arrows are for positions 5 m apart from the upwind direction. Near the northeastern corner, a cy- grated volume transport is displayed in Fig. 11 for the same clonicone gyre another is visible. from the free surface to the bottom. Each panellongitudinal has its constant own velocity wind forcing; scale as the indicated. figure displays vol- Y. Wang et al.: Barotropic response in a lake to wind-forcing 379 23

60 m 60 m

1 1 1.6 cm s−✲ 1.8 cm s−✲

10 m 10 m

5m 5m 0m 0m (a) (b)

1 1 0.37 cm s✲− 0.61 cm s✲−

20 m 20 m (c) (d)

FIG. 13: Hodographs,i.e.,time series of the horizontal velocities in the middles of the Uberlinger¨ See (a) Fig. 13. Hodographs, i.e., time series of the horizontal velocities in the middles of the Uberlinger¨ See (a) and the Obersee (b) in 0, 5, 10 and and the Obersee (b) in 0,5,10 and 60 m depths. The motion is set up from rest. The small circles mark 60 m depths. The motion is set up from rest. The small circles mark time intervals of approximately four hours. In order not to confuse the time intervals of approximately four hours. In order not to confuse the spirals at 20 m and 60 m depth the spirals at 20 m and 60 m depths, the 20 m hodographs are shown separately in the panels (c) and (d) for the two positions. 20 m hodographs are shown separately in the panels (c) and (d) for the two positions. ume transport streamlines. The circulation consists, in prin- and the Obersee (below), as they form, for an impulsively ciple, of two cells which rotate in such a way that the trans- applied uniform wind from 305◦ (NW) (approximately in the ] port is downwind] along the shores, orientated roughly in the long direction), and as obtained with the two different indi- Nm Nm 9

direction of the9 wind. The central part of the circulation cated eddy viscosities. Those Ekman spirals are considerably 10 10 shows a net transport with an upwind component which re- affected× by the ν -values. The turning of the arrows which [

× v sults from the[ geographic current (caused and bounded by make up the spirals, also indicates that the surface Ekman the bottom topography); with the bottom current prevailing boundary layer is thinner for the smaller value of the eddy over the surface drift current. In contrast, the drift current viscosities (right panel) than for the larger value (left panel). dominates the other currents in the shallow, nearshore region. Equally interesting is the comparison of the time series Due to the Coriolis force, subject to a western wind forcing,2 1 ---- N=25,νv =0.005m s− of the horizontal velocity components u and v for various the sloping bottom causes an intensification of the northward2 1

——– N=10,νv =0.02 m s− Total kinetic energy

Total kinetic energy ¨ flow component at the(a) northwestern shore as well as at the depths(b) at the midlake positions of the Uberlinger See and the Obersee, as displayed by the hodographs in Fig. 13 and southeastern shore (Serruya et al., 1984; Wang, 1996). If 2 −1 obtained with νv = 0.005 m s . At both positions, tran- there would be no Coriolis effect, theTime circulation [h] would con- Time [h] sist of two gyres, which are located almost exactly to the sient oscillations can be discerned with the inertial period of FIG. 14: Time series of the total kinetic energy in Lake Constance subject to constant wind from 305◦ north and south of the Talweg, rotating in the clockwise and approximately 16 h. The oscillations can be seen at all wa- NW approximately in the long direction (a) and from 215◦ SW approximately in the cross direction (b) anticlockwise directions, respectively. ter depths, however, with decreasing amplitude as the depth with two different values of the vertical eddy viscosity. increases. Furthermore, they die out before two inertial pe- 4.2 Ekman spirals and time series of the horizontal velocity riods in the Uberlinger¨ See, but only after 5–6 inertial pe- riods in the Obersee. The reason is the smaller size of the The vertical variation of the current depends strongly on the Uberlinger¨ See and, therefore, the enhanced frictional re- eddy viscosity. We display in Fig. 12 two steady Ekman spi- sistance due to the lake bottom and the side shores. In the rals at the midlake positions of the Uberlinger¨ See (above) Obersee, the transient velocities oscillate around a slowly 23 23

60 m 60 m 60 m 60 m

1.6 cm s 1 1.8 cm s 1 1 −✲ 1 −✲ 1.6 cm s−✲ 1.8 cm s−✲

10 m 10 m 10 m 10 m

5m5m 5m5m 0m0m 0m0m (a)(a) (b)(b)

1 1 0.37 cm1 s− 0.61 cm1 s− 0.37 cm s✲− ✲ 0.61 cm s✲− ✲

20 m20 m 20 m20 m (c)(c) (d)(d)

FIG.FIG. 13: 13:Hodographs,i.e.,timeHodographs,i.e.,time series series of the of the horizontal horizontal velocities velocities in the in the middles middles of the of theUberlinger¨ Uberlinger¨ See See(a) (a) andand the the Obersee Obersee (b) (b) in in0,5,10 0,5,10 and and 60 60m depths.m depths. The The motion motion is set is set up fromup from rest. rest. The The small small circles circles mark mark timetime intervals intervals of approximatelyof approximately four four hours. hours. In order In order not not to confuse to confuse the the spirals spirals at 20 at m 20 and m and 60 m 60 depth m depth the the 3802020 m m hodographs hodographs are are shown shown separately separately in the in the panels panels (c) (c)Y. and Wang and (d) (d) for et al.: forthe Barotropicthe two two positions. positions. response in a lake to wind-forcing ] ] ] ] Nm Nm Nm Nm

9 24 9 9 9 10 10 10 10 × × [ × [ × [ [

2 1 N=25,ν =0.005 −2 1 ------N=25v,νv =0.m005sm s− 2 21 1

——– N=10,νv =0.02 m s− Total kinetic energy

——– N=10,ν =0.02 m s− Total kinetic energy Total kinetic energy v Total kinetic energy (a)(a) (b)(b)

Time [h] TimeTime [h] [h] Time [h] FIG. 14: Time series of the total kinetic energy in Lake Constance subject to constant wind from 305 FIG. 14: Time series of the total kinetic energy in Lake Constance subject to constant wind from◦ 305◦ Fig. 14. NWTime approximately series of the total in thekinetic long energy direction in Lake (a) Constance, and from subject 215 SW to constant approximately wind from in 305the◦ crossNW approximately direction (b) in the long NW approximately in the long direction (a) and from 215◦ ◦ SW approximately in the cross direction (b) direction (a) and from 215◦ SW, approximately in the cross direction (b) with two different values of the vertical eddy viscosity. withwith two two different different values values of theof the vertical vertical eddy eddy viscosity. viscosity.

N N

WEWE

S S (a) (b)

FIG. 15: Total kinetic energy computed three days after the wind set-up (steady state) subject to constant Fig. 15. Total kinetic energy computed three days after the wind set-up (steady state) subject to constant2 1 wind from different directions with wind from different directions with the vertical eddy viscosity νv =0.02 m s− . The computations are the vertical eddy viscosity ν = 0.02 m2s−1. The computations are performed for two cases: (a) the Coriolis effect is considered; (b) the performed for twov cases: (a) the Coriolis effect is considered; (b) the Coriolis force is neglected. The Coriolis force is neglected. The directions of the drawn arrows are the wind directions (for an interval of 10◦) around the basin, their lengths indicate thedirections magnitudes of of the the drawn total kinetic arrows energy. are The the dashed wind directions circles show (for the maximum an interval value of of 10 the◦) aroundtotal kinetic the energy basin,their in all directions, which is 1lengths.11 × 1010 indicateN m for the (a), magnitudes but 4.65 × 10 of10 N the m fortotal (b). kinetic energy. The dashed circles show the maximum value of the total kinetic energy in all directions,which is 1 .11 1010N m for (a),but 4 .65 1010Nm for(b). × × varying mid-velocity at each depth; this is different from in which the geostrophic motion is basically perpendicular to the corresponding behaviour in the rectangular basin where the wind direction or surface pressure gradient (90◦ to the left the motion oscillates approximately around the steady state on the northern hemisphere). (compare with Fig. 4). This difference is due to the complex topography of Lake Constance. From the spirals at larger 4.3 Total kinetic energy in relation to the vertical eddy depth (e.g. 20 m), during an initial time interval after the viscosity and the wind direction wind started, a sudden change in the direction of the mo- tion can be seen when the wind-induced surface drift flow The time series of the stored, total kinetic energy in Lake 2 −1 reaches this depth. Before this time, only the geostrophic Constance for NW and SW wind, νv = 0.02 m s and 2 −1 current exists at this depth which, due to the restriction of νv = 0.005 m s (Fig. 14) show that the inertial oscilla- the bottom topography, is along the Talweg. This is different tions persist longer, and the value of the total kinetic energy from the behaviour of the flow in a basin with constant depth, is much larger for the smaller νv-value (as shown before for Y. Wang et al.: Barotropic response in a lake to wind-forcing 381 the rectangular basin). More interesting is the comparison of of the y-coordinates in the cross-section, with the wind di- the kinetic energies for the case of longitudinal NW wind and rection (positive, broken lines) and against the wind (nega- transverse SW wind (Fig. 14a,b). For the transverse wind, the tive, dotted lines), in cross-sections through the centers of the temporal inertial oscillations persist much longer than those Uberlinger¨ See and the Obersee, respectively. From Fig. 18, associated with the longitudinal wind, whereas the magni- the nearshore coastal jet in the direction of the wind can be tude of the kinetic energy subject to the longitudinal wind is seen more clearly, not only in the Uberlinger¨ See, but also in much larger. the Obersee. Close to the shore, the transport is almost only The variable response of the lake to constant wind forcing in the direction of the wind; far away from shore, where the from different directions at 10◦ interval is represented in the motion is dominantly in the upwind direction, the transport polar diagram of the kinetic energy uptake in Fig. 15a, com- is against it. This feature is due to the approximate parabolic 2 −1 puted for νv = 0.02 m s . The stored, total energy depends shape of the cross direction. very strongly on the wind direction. The geographical direc- If the positive and the negative transports are added, the tion of maximum exposure coincides approximately with the net volume fluxes per unit width (the solid lines in Fig. 18) longitudinal direction of the lake. The kinetic energy under are obtained. These lines would be obtained with a depth a transverse wind is only one fifth of thekinetic energy for integrated model. Obviously, the horizontal integration of a longitudinal wind. The strong dependence of the total ki- the net volume flux along the cross-section must vanish in netic energy on the direction of the wind should be mainly steady state. due to the topography, and less to the dimension of the lake, The vertical distributions of the horizontally integrated since Lake Constance has a comparable aspect ratio to the transport at the centers of the Uberlinger¨ See and the Obersee rectangular basin in Fig. 7, in which the dependence on the can be extracted from Fig. 19. It can be seen that the transport direction of the wind is much weaker. This dependence of the occurs primarly in the top 80 m, especially in the top 40 m. total kinetic energy on the direction of the wind is similar, to The transport in the wind direction (positive, broken lines) some extent, to that obtained by Serruya et al. (1984), where assumes its maximum at the surface, while the maximum of only the kinetic energy of the two-dimensional vertically in- the transport against the wind (negative, dotted lines) occurs tegrated net transport was calculated. The dependence of the at approximately 20–30 m depth. Below 100 m there exists kinetic energy on the wind direction, without consideration only the transport against the wind with very small values. of the Earth’s rotation, is also displayed in Fig. 15, in panel The sum of the two fluxes for a fixed depth yields the net b. In this case, the total kinetic energy is much larger than the volume flux per unit depth, which is also displayed as solid kinetic energy with the Coriolis force, while the dependence lines in Fig. 19. Obviously, its vertically integrated total flux of the kinetic energy on the direction of the wind is somewhat through a cross-section must vanish in a steady state. It is weaker; here, the kinetic energy under a transverse wind is also obvious that these results cannot be obtained with the nearly half as large as the kinetic energy for a longitudinal use of a vertically integrated model. wind. In Figs. 17, 20 and 21, graphs are displayed for the iso- tachs of the three velocity components in a cross-section of 4.4 Vertical distributions of the steady currents the Uberlinger¨ See (Fig. 17) as well as the horizontal and vertical distributions of the longshore transport in the two in- The isotachs of the three velocity components in a steady dicated cross-sections of the Uberlinger¨ See (Fig. 20) and the state on the cross-section through the center of the Uberlinger¨ Obersee (Fig. 21), respectively. This is also the case in Figs. See are presented in Fig. 16. The wind is blowing in the di- 16, 18 and 20, but now with a smaller vertical eddy viscos- 2 −1 rection of the longitudinal axis of the lake, perpendicular to ity νv = 0.005 m s . Principally, they are very similar to 2 −1 the plane of the graphs upwards. The most distinguished fea- those subject to νv = 0.02 m s . The most distinguishing ture of the motion is a nearshore coastal jet in the direction differences is that for the smaller vertical eddy viscosity, the of wind (Fig. 16a). The change of the horizontal velocity surface layers of positive x-velocity are much smaller (Figs. with depth happens primarly in the upper layer and in the 17 and 21) than in Figs. 16 and 19, while the absolute val- middle of the cross-section (Fig. 16a,b). At lower depths, ues of the velocity are much larger, and the two-dimensional where the geostrophic current plays an important part in the depth integrated model is less justified (compare Figs. 18 and motion, the velocity components do not significantly change 20). with depth. Due to the effect of the Coriolis force, for a west wind, there occurs a downwelling along the southern shore (left side in Fig. 16c) and an upwelling along the northern 5 Comparison of measured values and computational shore (right in Fig. 16c), however, the vertical velocity com- results in Lake Constance ponent is much smaller than the horizontal components. In the zone far away from shore, the vertical velocity compo- Unfortunately, there are insufficient data available which nent is practically negligible. would allow validation of the model. Nevertheless, a partial Figure 18 displays the transverse variations of the long- check of the reliability of the numerical method was possible shore transports. These are the depth integrals of the posi- with data from 16 February 1993 (0 MET) to 22 February tive and negative velocities in the x-direction, as functions 1993 (24 MET), a period of seven days. Wind velocity and 26

SN SN

N N S Depth [m]

Depth [m] S

26 26 (a) (b)

SN 2 1 2 1 SNVolume flux per unit depth [m s− ] Volume flux per unit depth [m s− ] SNSN FIG. 18: Horizontally integrated volume flux per unit depth through the cross-sections in the Uberlinger¨

See (a) and in the Obersee (b) as indicated in the insets computed with the vertical eddy viscosity νv = 2 1 25 0.02m s− . Broken curves indicate the transport part with the wind,dotted curves against the wind direction. The solid lines show the net volume flux 382 Y. Wang et al.: Barotropic response in a lake to wind-forcing

SNN SNSNN SN 0 0 0 N 0 N S Depth [m] S Depth [m]

Depth [m] Z S Z

Z Z Depth [m] S [m] [m] [m] [m] -50 -50 -50 -50 (a) (b) (a) (b)

2 1 2 1 Volume flux per unit depth [m s− ] Volume flux per unit depth [m s− 2] 1 2 1 -100 -100 -100 Volume flux per unit depth [m s− ] -100 Volume flux per unit depth [m s− ] FIG. 18: ¨ Horizontally integrated volume flux per unit depthFIG. through 18: Horizontally the cross-sections integrated in the volumeUberlinger flux per unit depth through the cross-sections in the Uberlinger¨ See (a) and in the Obersee (b) as indicated in the insets computed with the vertical eddy viscosity ν = See (a) and in the Obersee (b) as indicated inv the insets computed with the vertical eddy viscosity νv = 0.02m2s 1. Broken curves indicate the25 transport part with the2 1 wind,dotted curves against the25 wind − 0.02m s− . Broken curves indicate the(a) transport part with the wind,dotted curves against the(b) wind direction. The solid lines show the(a) net volume flux (b) -150 -150 -150direction. The solid lines show the net volume flux -150 0y[km]1.0 2.0 0y[km]0y[km]1.0 1.0 2.0 2.0 0y[km]1.0 2.0 SNSNSNSNSNSN SN0 SN0 0 00 0 0 0 Z Z ZZ Z Z Z [m] [m] [m][m] [m] [m][m] [m] -50 -50-50 -50 -50-50 -50 -50 -50 FIG. 19: FIG. 16: Isotachs of the two horizontal and ver- Isotachs of the two horizontal and verti- tical velocity components u (a), v (b) and w (c) cal velocity components u (a), v (b) and w (c) in a ¨ in a cross-section of the middle of the Uberlinger¨ cross-section of the middle of the Uberlinger See in -100 -100 steady state subject to constant wind from 305◦ NW -100 -100-100 See-100 in steady-100-100 state subject to constant wind from -100 approximately in the longitudinal direction (out of 305◦ NW approximately in the longitudinal direc- tion (out of the page) with the vertical eddy viscos- the page) computed with the vertical eddy viscosity 2 1 2 1 νv =0.005m s− (a) ity νv =0.02m s− (c)(a)(b) (b) (a) (a)(c)(b) -150 (b) -150 -150-150-150 -150-150 -150 -150 -150 0y[km]1.0 2.0 0y[km]1.0 2.0 0y[km]0y[km]0y[km]1.01.0 1.0 2.0 2.0 0y[km]0y[km]1.0 1.0 2.0 2.0 0y[km]1.0 2.0 SNSNSNSNSNSN 0 0 0 0 ] ] N N 1 Z 1 Z Z Z − − s s S 2 [m] [m]2 [m] [m] S -50 -50 -50 -50 FIG. 19: FIG. 16: Isotachs of the two horizontal and ver- FIG. 16: IsotachsIsotachs of the of two the horizontaltwo horizontal and and ver- verti- FIG. 19: Isotachs of the two horizontal and verti- tical velocity components u (a), v (b) and w (c) tical velocitycal velocity components componentsu (a),u (a),v v(b)(b) and andww(c)(c) in a cal velocity components u (a), v (b) and w (c) in a ¨ ¨ in a cross-section of the middle of the Uberlinger¨ in a cross-sectioncross-section of the middle middle of of the theUberlingerUberlinger¨ See in cross-section of the middle of the Uberlinger See in steady state subject to constant wind from 305 NW steady state subject to constant wind from 305 NW -100 See-100 in steady-100 state subject to constant wind from See in steady-100 state subject to constant wind from◦ ◦ approximately in the longitudinal direction (out of approximately in the longitudinal direction (out of 305◦ NW approximately in the longitudinal direc- 305◦ NW approximately in the longitudinal direc- tion (out of the page) with the vertical eddy viscos- tion (outthe page) of the computed page) with with the thevertical vertical eddy eddy viscos- viscosity the page) computed with the vertical eddy viscosity 2 1 ν =0.005m2 21s 1 ν =0.005m2s 1 ity νv =0.02m s− (a) ity νv =0v .02m s− − (b) v − (c) (c) (c) (c) Volume flux per unit width [m -150 -150 -150 Volume flow per unit width [m -150 0y[km]1.0 2.0 0y[km]0y[km]1.0 1.0 2.0 2.0 0y[km]1.0 2.0 Distance from the southern shore [km] Distance from the southern shore [km] SNSNSNSN Fig.FIG. 16. Isotachs 17: Vertically of the two horizontal integrated and volume vertical velocity flux per compo- unit widthFig. in17. theIsotachs cross-sections of the two horizontal of the and center vertical of velocitythe compo- ]

] nents (a) u, (b)v and (c) w in a cross-section of2 the1 middle of the

N ¨ N ] nents ] (a) u, (b) v and (c) w in a cross-section of the middle of the

1 N N 1 Uberlinger See (a) and in the Obersee (b) computed with the vertical eddy viscosity ν =0.02m s− .

1 v 1 ◦ ◦ − ¨ −

¨ − Uberlinger See in steady state, subject to constant wind from 305 − s

s Uberlinger See in steady state, subject to constant wind from 305 s S s 2 2 Broken curves indicate the transportS part with the wind direction,dotted curves against the wind direc- 2 NW2 approximately in the longitudinal direction (perpendicular to NW approximately in the longitudinal direction (perpendicular to tion. The solid lines showS the sum of the two. The location of the cross-sections is shownS as inset. the plane of the graphs upwards) with the vertical eddy viscosity the plane of the graphs upwards) computed with the vertical eddy 2 −1 2 −1 νv = 0.02 m s . viscosity νv = 0.005 m s .

direction were measured at “Boje Mitte”, a buoy with me- Uberlinger¨ See. These data are displayed in Fig. 22. Dur- teorological instruments approximately in the center of the ing the first three days (16–18 February), the wind was very

(a) (b) (a) (b) Volume flux per unit width [m Volume flow per unit width [m Volume flux per unit width [m Volume flow per unit width [m

Distance from the southern shore [km] DistanceDistance from from the the southern southern shore shore [km] [km] Distance from the southern shore [km] FIG. 17: Vertically integrated volume flux per unit widthFIG. in 17: theVertically cross-sections integrated of the volume center flux of the per unit width in the cross-sections of the center of the Uberlinger¨ See (a) and in the Obersee (b) computed with¨ the vertical eddy viscosity ν =0.02m2s 1. 2 1 Uberlinger See (a) and in the Oberseev (b) computed− with the vertical eddy viscosity νv =0.02m s− . Broken curves indicate the transport part with the windBroken direction,dotted curves indicate curves the against transport the wind part direc- with the wind direction,dotted curves against the wind direc- tion. The solid lines show the sum of the two. The locationtion. of Thethe cross-sections solid lines show is shown the sum as of inset. the two. The location of the cross-sections is shown as inset. 25

SNSN 0 0

Z Z [m] [m] -50 -50

-100 -100

(a) (b) -150 -150 0y[km]1.0 2.0 0y[km]1.0 2.0 SN 0

Z [m] -50 FIG. 16: Isotachs of the two horizontal and ver- tical velocity components u (a), v (b) and w (c) in a cross-section of the middle of the Uberlinger¨ -100 See in steady state subject to constant wind from 305◦ NW approximately in the longitudinal direc- tion (out of the page) with the vertical eddy viscos- 2 1 ity νv =0.02m s− Y. Wang et al.: Barotropic response in a lake to wind-forcing(c) 383 -150 0y[km]1.0 2.0 SNSN ] ] N N 1 1 − − s s S 2 2 S

(a) (b) Volume flux per unit width [m Volume flow per unit width [m

Distance from the southern shore [km] Distance from the southern shore [km] FIG. 17: Vertically integrated volume flux per unit width in the cross-sections of the center of the ¨ Fig. 18.¨Vertically integrated volume flux per unit width in the cross-sections of the center of the Uberlinger See (a) and in the Obersee2 261 (b) Uberlinger See (a) and in the Obersee2 (b)−1 computed with the vertical eddy viscosity νv =0.02m s− . computed with the vertical eddy viscosity νv = 0.02 m s . Broken curves indicate the transport part with the wind direction, dotted curves against theBroken wind direction.curves indicate The solid the lines transport show the sum part of with the two. the The wind location direction,dotted of the cross-sections curves is shown against as inset.the wind direc- tion. The solid lines show the sum of the two. The location of the cross-sections is shown as inset. SN SN

N N S Depth [m]

Depth [m] S

(a) (b)

2 1 2 1 Volume flux per unit depth [m s− ] Volume flux per unit depth [m s− ] FIG. 18: Horizontally integrated volume flux per unit depth through the cross-sections in the Uberlinger¨ ¨ Fig. 19.SeeHorizontally (a) and in integrated the Obersee volume (b) flux as per indicated unit depth inthrough the insets the cross-sections computed in with the Uberlinger the vertical See eddy(a) and viscosity in the Oberseeνv =(b) as 2 −1 indicated in the2 insets,1 computed with the vertical eddy viscosity νv = 0.02 m s . Broken curves indicate the transport part with the wind, 0.02m s− . Broken curves indicate the transport part with the wind,dotted curves against the wind dotted curves against the wind direction. The solid lines show the net volume flux. direction. The solid lines show the net volume flux weak. On 19SN February, a strong wind was initiated; its am- HereSN we simulate, numerically, the corresponding water plitude reached0 8 ms−1 and it persisted for the remainder of motion0 with the measured wind input and check if the mea- the period. For the same period, the water velocity and its di- sured water velocity can be reproduced by the computed ve- Z Z rection[m] at the “Mainauschwelle” (near the island Mainau) at locity[m] field. In winter (here, in February), the water den- 80 m depth were also measured; they are displayed as solid sity in Lake Constance can be considered to be homogenous curves-50 in Fig. 23a,b. In the first 80 hours, the water velocity (Bauerle¨-50 et al., 1998). For this simulation the measured wind was less than 1.1 cm s−1, which was below the threshold of at the station “Boje Mitte” is used and extrapolated to the en- the current meter. At the fourth day, strong currents started tire lake. This uniformity is certainly unlikely to be realistic, with superimposed oscillations of a period of approximately but we apply it due to lack of better knowledge. The shear 16 hours,-100 which can be interpreted as inertial oscillations. stress-100 at the water surface can be calculated with the aid of The flow direction superimposed by oscillations is approxi- the classical drag formulas ◦ mately 300–340 (NW). 0 τ = ρac0 | V wind | V wind, (a) (b) -150 -150 0y[km]1.0 2.0 0y[km]1.0 2.0 SN 0

Z [m] -50 FIG. 19: Isotachs of the two horizontal and verti- cal velocity components u (a), v (b) and w (c) in a cross-section of the middle of the Uberlinger¨ See in -100 steady state subject to constant wind from 305◦ NW approximately in the longitudinal direction (out of the page) computed with the vertical eddy viscosity ν =0.005m2s 1 (c) v − -150 0y[km]1.0 2.0 27

SNSN ] ] N 1 1 −

N − s s 2 2 27 S S

384 Y. Wang et al.: Barotropic response in a lake to wind-forcing

SNSN ] ] N 1 1 −

N − s s 2 S 2 (a) S (b) Volume flux per unit width [m Volume flux per unit width [m

Distance from the southern shore [km] Distance from the southern shore [km]

FIG. 20: Vertically integrated volume flux per unit width in the Uberlinger¨ See (a) and in the Obersee 2 1 (b) computed with the vertical eddy viscosity νv =0.005m s− . Broken curves indicate the transport part along the wind direction,dotted curves against the wind. The solid lines show the sum of the two. The location of the cross-sections is shown as insets.(a) (b) Volume flux per unit width [m Volume flux per unit width [m

Distance from the southern shore [km] Distance from the southern shore [km]

FIG. 20: Vertically integrated volume flux per unit width in the Uberlinger¨ See (a) and in the Obersee ¨ Fig. 20. Vertically integrated volume flux per unit width in the Uberlinger See2 (a)1 and in the Obersee (b) computed with the vertical eddy (b) computed2 with−1 the vertical eddy viscosity νv =0.005m s− . Broken curves indicate the transport part viscosity νv = 0.005 m s . Broken curves indicate the transport part along the wind direction, dotted curves against the wind. The solid lines showalong the sum the of wind the two. direction,dotted The location of thecurves cross-sections against the is shown wind. as insets.The solid lines show the sum of the two. The location of the cross-sections is shown as insets. SNSN

N SNN SN

Depth [m] S S Depth [m]

(a) (b)

N N 2 1 Volume flux per unit depth [m s− ] 2 1

Depth [m] S Volume flux per unit depthS [m s− ] FIG. 21: Horizontally integrated volume flux per unit depthDepth [m] through the cross-sections in the Uberlinger¨ Fig. 21. Horizontally integrated volume flux per unit depth through the cross-sections in the Uberlinger¨ See (a) and in the Obersee (b) as See (a) and in the Obersee (b) as indicated in the insets computed with the vertical eddy viscosity νv = indicated in the insets, computed with the vertical eddy viscosity ν = 0.005 m2s−1. Broken curves indicate the transport part with the 0.005m2s 1. Broken curves indicate the transportv part with the wind,dotted curves against the(b) wind wind, dotted curves− against the wind direction. The solid lines show(a) the net volume flux. direction. The solid lines show the net volume flux

2 1 Volume flux per unit depth [m s− ] 2 1 where V wind is the wind speed 10 m above the water sur- face (roughnessVolume length). flux per In unit the depth computations, [m s− ] we choose = −3 = × −4 face, ρaFIG.the air 21: density (ρa 1.225 kg m ) and c0 a di- z0 1.0 10 m. The vertical eddy viscosity¨ is assumed Horizontally integrated volume flux per unit depth through the cross-sections2 − in1 the Uberlinger mensionless friction coefficient. The specification of c is constant with value νv = 0.02 m s and the simulation is See (a) and in the Obersee (b) as indicated in0 the insets computed with the vertical eddy viscosity ν = not unique and varies from author to author. A typical value started from rest, although at the initial time, a smallv motion 2 1 0.005m s− . Broken curves indicate the transport−1 part with the wind,dotted curves against the wind for a weak or medium wind strength (Vwind < 10 m s ) is must exist in nature. direction.−3 The solid lines show the net volume flux c0 = 1.8×10 (Lehmann, 1992). The wind speed measured at 4.4 m above the lake surface (Fig. 22) must be converted The computed water velocity and its direction in 80 meters to the value at 10 m. Assuming a logarithmic wind profile, depth, at the position “Mainauschwelle” where the measured yields time series are plotted in Fig. 23, are also displayed in Fig. 23 as dashed lines. During the first three days, when a weak V ln 10 − ln z 10 = 0 , wind prevailed, the measured and computed current speeds V4.4 ln 4.4 − ln z0 could not be compared, but their directions could (Fig. 23b), where z0 is a measure for the corrugation of the water sur- and, they deviated considerably from one another. This is 28

29

Y. Wang et al.: Barotropic response in a lake to wind-forcing 385 ] 1 − ] 1 [m s − [cm s

Wind velocity (a) Flow velocity Time [h] (a)

Time [h] ] ◦ ] ◦

(b) [ Wind direction Φ [

Time [h] FIG. 22: Measured wind speed (a) 4.4 m above the lake surface at the station “Boje Mitte” (approximately at the centerFig. of the 22.Uberlinger¨ Measured See) from wind 16.2.’93 speed 000(a)MET4.4 until m above 22.2.’93 the 2400 lakeMET,and surface direction (b)Flow direction of (b) which the meanat valuethe station is approximately “Boje Mitte” 270◦ (West) (approximately at the center of the Uberlinger¨ See) from 16 February 1993 (0 MET) until 22 Febru- ary 1993 (24 MET), and direction (b) of which the mean value is Time [h] ◦ approximately 270 (West). FIG. 23: a) Flow velocity at 80 m depth near the Mainauschwelle subject to the wind forcing displayed in Fig. 22. TheFig. solid 23. curve (a) indicatesFlow velocity the measured at 80 values,the m depth broken near curve the corresponds Mainauschwelle to the computational result. b) Samesubject as a) to but the for wind the flow forcing direction. displayed Its value in after Fig. the 22. wind The set-up solid is approximately curve toward 320 (northwest). ◦ indicates the measured values, the broken curve corresponds to the most likely due to the fact that the computations started from computational result. (b) Same as (a) but for the flow direction. Its a state of rest, but there was some (small) motion in na- value after the wind set-up is approximately toward 320◦ (north- ture (however, not caught by the current meter) which was west). not considered in the simulation. A relatively strong wind is needed to bring the computed and the measured veloci- ties together. This strong wind occurs after 80 hours when computation but without consideration of the Coriolis force both measured and computed current speed and direction co- was repeated. For this case, the computed water velocities incide quite well with one another. Only at the seventh day at the Mainauschwelle, at 80 m depth, and at the center of do the current speeds of the computations differ from the re- the Obersee, at several depths, are shown in Figs. 25 and spective measured values. The computed current orientation 26. Comparison of Figs. 23 and 25 shows that without the at this depth lies between 300–340◦ (towards Northwest), Coriolis term, the computed velocity at the Mainauschwelle which is in fair agreement with the measured values (com- does not exhibit the oscillations shown by the measurements. ◦ pare the solid and dashed lines of Fig. 23b), which are basi- Moreover, the direction of the current is toward 260 , which cally around 320◦, superimposed by oscillations. Given the is substantially different when one accounts for the Coriolis measuring technique and the bold extrapolation of the wind effects. Similar differences are also seen when Figs. 24 and over the entire basin, a better agreement can hardly be ex- 26 are compared. The computed current oscillations at the pected. midpoint position of the Obersee are not established without the Coriolis force. These facts should be sufficient demon- In Fig. 24a,b we have plotted time series of the two hor- stration that the measured oscillations are indeed rotational izontal velocity components at several depths in the mid- effects due to the motion of the Earth. lake position in the Obersee, as indicated in the inset. Ev- idently, the strong wind commencing after 80 hours gener- ates equally strong oscillations; with a period of almost ex- 6 Concluding remarks actly 16.3 hours, they are most likely, inertial oscillations. On the other hand, comparison of the water velocity at 80 m In this paper, wind induced barotropic circulation in lakes depth at the Mainauschwelle (Fig. 23) with the driving wind was studied, first, from a more fundamental point of view, suggests that these oscillations might simply be the direct using a rectangular basin with constant depth, but later, with response to the wind forcing. In order to confirm that the Lake Constance, a medium-size Alpine lake. The tenor of the oscillations exhibited by the time series of water velocity are investigation was to see how the absolute values of the eddy indeed inertial oscillations due to the Coriolis force, the same viscosities affect the flow within such basins. Vertical eddy 386 Y. Wang et al.: Barotropic response in a lake to wind-forcing3030

(a) (b) ] ]

(a) 1 (b) 1 ] ] − 1 − 1 − − cms cm s [ [ cms v cm s [ u [ v u Horizontal velocity Horizontal velocity Horizontal velocity Horizontal velocity

Time [h] Time [h] Time [h] Time [h] FIG. 24: Time series of the horizontal velocity components u (a) and v (b) at the center of the Obersee Fig. 24.FIG.Time(see inset)series 24: Time of subject the serieshorizontal to theof thevelocity wind horizontal forcing components of velocity Fig.(a) 22.u and components The(b) labelsv at the indicate centeru (a) of and thev Obersee depths(b) at (see in the meters inset) center subject of the to the Obersee wind forcing of Fig.(see 22. The inset) labels subject indicate to the the depths wind in meters. forcing of Fig. 22. The labels indicate the depths in meters (a) (b) (a) (b) ] 1 ] − 1 ] 1 − ] − 1 cm s [ − v [cm s cm s [ v [cm s Flow velocity Flow direction Flow velocity Flow direction

Time [h] Time [h]

FIG. 25: Time series ofTime the [h] horizontal velocity (a) and its direction (b) at 80Time m depth [h] near the Main- auschwelle subject to the wind forcing displayed in Fig. 22. In this computation the Coriolis force is not FIG. 25: Time series of the horizontal velocity (a) and its direction (b) at 80 m depth near the Main- considered Fig. 25.auschwelleTime series subjectof the horizontal to the velocity wind forcing components displayed(a) u and in(b) Fig.v at 22. the centerIn this of computation the Obersee subject the to Coriolis the wind force forcing is of not Fig. 22, where the Coriolis force is not considered. The labels indicate the depths in meters. considered (a) (b) ] ] 1 1 − viscosities were varied in three different ways, as suggested(a) − cosity (but less on its vertical distribution).(b) Its transient ] ] 1 1 cm s cms [ by other− studies, and the steady Ekman problem was solved. behaviour from a state of rest to a steady state is char- [ − u The following results were obtained, each demonstrating the v acterized by oscillations that are attenuated in time, but cm s cms [ significance of the eddy viscosity: [ equally, the smaller the vertical eddy viscosity is, the u v – The direction of the surface water velocity at the mid- larger the oscillations. This is true for the rectangle point of a rectangular basin relative to that of the wind, (Fig. 6), as well as Lake Constance (Fig. 14). the so-called wind set-off, depends strongly on the ab- solute value and vertical distribution of the vertical eddy – The wind-directional dependence of the total kinetic en- Horizontal velocity viscosity. It cannot be concluded that the current set-off Horizontal velocity ergy depends strongly on the wind-fetch and the lake (to the right on the northern hemisphere) decreases with bathymetry. For a rectangle with constant depth and a Horizontal velocity increasing eddy viscosity. Depending upon the vertical Horizontal velocity width to length ratio of 0.25, this dependence is less distribution of the eddy viscosity,Time opposite [h] or even non- than approximatelyTime 10% [h] (Fig. 7); for Lake Constance, monotonicFIG. 26: behaviourTime series may occurof the (Fig. horizontal 1a). velocity componentsitu is(a) substantial and v (b) (Fig. at 15). the center of the Obersee Time [h] Time [h] – Thesubject surface to current the wind speed, forcing however, of Fig. decreases 22,where mono- the Coriolis force is not considered. The labels indicate the – The Coriolis force implies a significant reduction of the tonicallyFIG.depths 26: with inTime meters an increase series in of the the eddy horizontal viscosity velocity (Fig. 1b). components u (a) and v (b) at the center of the Obersee total kinetic energy that can be established in a basin subject to the wind forcing of Fig. 22,where the Coriolis force is not considered. The labels indicate the – The amount of kinetic energy stored in the water de- due to wind forces, when compared to the case when pendsdepths equally in meters upon the absolute value of the eddy vis- rotational effects are neglected (Figs. 7, 8 and 15). 30

(a) (b) ] ] 1 1 − − cms cm s [ [ v u Horizontal velocity Horizontal velocity

Time [h] Time [h] FIG. 24: Time series of the horizontal velocity components u (a) and v (b) at the center of the Obersee (see inset) subject to the wind forcing of Fig. 22. The labels indicate the depths in meters

(a) (b) ] 1 ] − 1 − cm s [ v [cm s Flow velocity Flow direction

Time [h] Time [h] FIG. 25: Time series of the horizontal velocity (a) and its direction (b) at 80 m depth near the Main- auschwelle subject to the wind forcing displayed in Fig. 22. In this computation the Coriolis force is not Y. Wang et al.: Barotropic response in a lake to wind-forcing 387 considered

(a) (b) ] ] 1 1 − − cm s cms [ [ u v Horizontal velocity Horizontal velocity

Time [h] Time [h] FIG. 26: Time series of the horizontal velocity components u (a) and v (b) at the center of the Obersee Fig. 26.subjectTime series to the of the wind horizontal forcing velocity of Fig. components 22,where(a) theu and Coriolis(b) v at force the center is not of the considered. Obersee subject The to labels the wind indicate forcing the of Fig. 22, where the Coriolis force is not considered. The labels indicate the depths in meters. depths in meters

– The vertical distribution of the water velocity, i.e., the sults that can be compared with the measured ones. This Ekman spiral, depends considerably upon the absolute wave dynamic, in turn, determines the advective properties values of the eddy viscosities. Not only the Ekman and thus, the transports of nutrients and pollutants both hori- depth is affected by these values, but equally so, the zontally and vertically. orientational distribution of the horizontal current with depth (Figs. 5, 12). Acknowledgement. Topical Editor N. Pinardi thanks S. Pier`ıni and J. O. Backhaus for their help in evaluating this paper. – Inertial oscillations are more easily established in open water than bounded channels; this demonstrates the significance of the frictional effects due to boundaries References (Fig. 13). Backhaus, J. O., A semi-implicit scheme for the shallow water All these results support the conjecture that it is important equations for application to shelf sea modelling, Continental for adequate reproduction or prediction of observed current Shelf Research, 2(4), 243–254, 1983. fields in lakes to use the correct orders of magnitudes of the Backhaus, J. O., A three-dimensional model for simulation of shelf numerical eddy viscosities. On the other hand, the eddy vis- sea dynamics, Deutsche Hydrographische Zeitschrift, 38(4), cosity should be chosen according to physical considerations, 165–187, 1985. and simultaneously, ensure numerical stability of a simula- Backhaus, J. O. and Hainbucher, D., A finite difference general cir- tion. On the other hand, if the eddy viscosity desired by nu- culation model for shelf seas and its application to low frequency merical stability is much larger than the values suggested by variablity on the North European shelf, in Three-dimensional measurements, the numerical code needs to be adjusted, es- models of marine and estuarine dynamics, Eds. J. C. J. Nihoul and B. M. Jamart, Elsevier Oceanography Series, No. 45, 221– pecially with some modern numerical treatments of the non- 244, 1987. linear advection terms. Inappropriate numerical schemes of Bauerle,¨ E., Ollinger, D., and Ilmberger, J., Some meteorological, these advection terms are often the reason for the production hydrological, and hydrodynamical aspects of Upper Lake Con- of numerical oscillations, which require large eddy viscosi- stance, in Lake Constance: Characterisation of an ecosystem in ties in order to prevent their development and hence, assure transition, Eds. E. Bauerle,¨ and U. Gaedke, Arch. Hydrobiol. numerical stability. At present, we are making an effort in Spec. Issues Advanc. Limnol. 53, E. Schweizerbart’sche Verlags- this aspect. It is exactly the problem in many existing three- buchhandlung, Stuttgart, pp. 31–83, 1998. dimensional lake circulation models, that in order to reach Bleck, R., Rooth, C., Hu, C., and Smith, L., Salinity-driven ther- numerical stability, the eddy viscosities must be chosen to be mocline transients in a wind- and thermohaline-forced isopycnic much larger than physically acceptable so that the physical coordinate model of the North Atlantic, J. Phys. Oceanogr., 22, oscillations are also rapidly damped away or are even indis- 1486–1505, 1992. Blumberg, A. F. and Mellor, G. L., A description of a cernible. It has been clearly demonstrated by this model that three-dimensional coastal ocean circulation model, in Three- observed inertial oscillations due to the rotation of the Earth dimensional Coastal Ocean Models, Ed. N. Heaps, Coastal and persist long and are slowly attenuated. That these inertial Estuarine Sciences, 4, 1–16, 1987. waves are indeed a dominant effect is shown in a restricted Csanady, G. T., Water circulation and dispersal mechanisms, in comparison of the measured current at 80 m depth for Lake Lakes: Chemistry, Geology, Physics, Ed. A. Lerman, Springer, Constance. Neglecting the Coriolis effects does not yield re- New York-Heidelberg-Berlin, 1978. 388 Y. Wang et al.: Barotropic response in a lake to wind-forcing

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