Methods of substructuring in lake circulation dynamics
Yongqi Wang
*, Kolumban Hutter
1Institute of Mechanics, Darmstadt University of Technology, Hochschulstr. 1, D-64289 Darmstadt, Germany
Received 27 February 1998; received in revised form 19 January 1999; accepted 15 April 1999
Abstract
A semi-implicit semi-spectral hydrodynamic primitive equation model is used in combination with a substructuring technique to study wind-induced motions and tracer diusion in the homogeneous and strati®ed Lake Constance. An impulsively applied spatially uniform wind is applied in the long direction of the lake (305°from True North) lasting inde®nitely. Tracer mass is released at various locations of the free surface over a ®nite area for 24 h starting together with the imposed wind.
We compute for a given wind and tracer scenario the ¯ow and concentration ®elds for the entire lake using a coarse grid res-olution. Within three dierent subregions, where improvements of the obtained results are sought, computations are repeated by using a ®ner grid and employing the results of the global computations at the open boundaries of the subregion. We demonstrate that this substructuring technique can be used: (i) to improve results where subgrid processes are signi®cant as e.g., near shore, at early times in the neighbourhood of tracer sources and in complex geometric areas (bays); (ii) to better resolve and graphically display velocity and tracer concentration distributions on larger scales. We employ the technique for the homogeneous and the strati®ed Lake Constance. The technique is seen to be an economically ecient procedure in improving computational results when implementation of ®ne resolutions is not feasible. Ó 2000 Elsevier Science Ltd. All rights reserved.
AMS:65M06; 65M12; 65M70; 76B15; 86A05
Keywords:Substructuring; Lake circulation; Tracer diusion; Limnology
1. Introduction
Lakes and the ocean are physical systems which re-spond to the input of solar radiation and wind. Classical models that describe this response are based on the Boussinesq approximated shallow water equations, paired with diusion equations for the transport and dispersion of materials and/or pollutants. These equa-tions were discretized by various methods, ®nite dier-ences, ®nite elements and spectral methods, to name a few, and with the emerging codes many lakes were studied when typical meteorological scenarios were ap-plied, both for homogeneous as well as strati®ed water masses.
In earlier works we employed a semi-spectral method to the rotating shallow water equations by altering the Haidvogel et al. [2] SPEModel to account for an implicit temporal integration (in the vertical direction), Wang [11] Wang and Hutter [12]. Such a semi-implicit
inte-gration technique was necessary because with the ex-plicit integration in time used in the original SPEM the total CPU-times for integration of a wind-induced sce-nario in a realistic lake (say Lake Constance) was un-duly large. This is so, because the grid sizes in lakes must be much smaller than in the ocean. Because of the conditional stability of the explicit integration scheme through the CLF-condition an implicit temporal inte-gration was compelling, if computations over realistic times should be possible. In Hutter and Wang [5] the extension of the semi-implicit SPEM to diusion prob-lems was studied and it was demonstrated that the in-ertial wave dynamics in homogeneous water, and the Kelvin-and Poincare-type wave dynamics in strati®ed
water could be seen in the computed tracer concentra-tion data. This, among other things, was partial cor-roboration for the correctness of the applied numerical code determining the velocity, temperature and tracer concentration ®elds.
Despite the use of the semi-implicit integration tech-niques and the associated larger time steps that could be employed, the horizontal discretizations that could so far be applied in realistic basins, were still rather coarse
Advances in Water Resources 23 (2000) 399±425
*Corresponding author.
E-mail addresses: wang@mechanik.th-darmstadt.de (Y. Wang), hutter@mechanik.th-darmstadt.de. (K. Hutter)
± an average horizontal grid size of approximately 1 km2
and 12 and 30 Chebyshev polynomials were used; ex-isting computer capacity made ®ner resolutions uneco-nomical. For global analyses of the circulation pattern this was sucient, however to describe detailed pro-cesses in special regions, i.e., in the vicinity of a shore region, a drinking water intake or the source region of a contaminant, a considerably ®ner grid size is necessary. As computation of the entire lake with a ®ne grid is uneconomical, we propose here to use the method of
substructuring. According to this method, the subregion,
in which more accurate computation of the velocity, temperature and tracer concentration ®elds is requested, is discretized with a ®ner net and recomputed for the same external scenario by using the ®eld quantities along the boundaries of the subregion as computed with the coarser grid. In this process interpolations are nec-essary, and it is tacitly assumed that in the computations performed with the coarser grid no essential physical processes were lost that are signi®cant in the subregion. To implement such a program on a computer is a for-midable endeavour by itself. We describe here how it was done. However our intention is also to delineate the suitability of computational results obtained with a particular grid by comparing some of its results with the corresponding results of a substructuring. This will provide automatically certain ``thumb'' rules when substructuring is necessary and under which circum-stances it can be avoided.
In what follows we shall present in Section 2 the governing equations. Section 3 deals with the method of substructuring. Section 4 applies the method of sub-structuring to the homgeneous Lake Constance while Section 5 does so for the strati®ed lake. In Section 6 a summary is given.
2. Governing equations and selected parameterizations
2.1. Governing equations
These comprise of the ®eld equations valid in the domain occupied by the water and the boundary con-ditions along the free surface and the lake bottom that bound the lake domain.
2.1.1. Balance laws of mass, momentum and energy We assume that the water is ``contaminated'' by a tracer or a number of tracers, but that their concentra-tion is so minute that the density of the mixture, i.e., water plus tracers is not aected by the presence of the latter. Thus, the balance laws of mass for the mixture and each tracer and the balances of linear momentum and energy for the mixture together with the thermal equation of state form the ®eld equations for the con-sidered ¯uid system. We impose the Boussinesq
as-sumption (which states that density variations only aect the buoyancy force) and also employ the shallow water assumption (which asserts that physical variables change much slower over horizontal distances than over vertical ones). Thus, the ®eld equations read (see, e.g. [4]) used; (x, y) are horizontal, and zis vertically upwards, against the direction of gravity. The ®eld variables and parameters arising in Eqs. (1)±(7) are de®ned in Table 1. Diusion is accounted for by postulating Fick's ®rst law for the constituent mass ¯ux
ja ÿ Dca
which introduces orthotropic diusive behaviour, equal in both horizontal directions but dierent in the vertical direction. Eqs. (3) and (4) account for momentum dif-fusion dierent in the vertical and the two horizontal directions through the turbulent viscosities mV and mH, respectively. The thermal equation of state (6) is used in the form
qÿq0 q0
ÿb ÿT T02; 9
Eq. (7) is the balance of internal energy; it accounts
for orthotropic thermal turbulent diusion, but not for radiation; so no seasonal variations of the thermocline are in focus. Nevertheless, changes of the temperature distribution, and therefore density distribution, can be accounted for, but not if they are due to solar irradia-tion. In principle, incorporation of radiation is, how-ever, straightforward.
2.1.2. Boundary conditions for mass, momentum and energy
The above laws comprise 5m®eld equations for the ®eldsv;ca;/ andT which are also 5munknown ®elds.
They must be subjected to boundary conditions of both kinematic and dynamic nature. Let the undeformed free surface be described by z0 and let the base be de-scribed by z ÿh(x, y). We shall formulate the boun-dary conditions on these surfaces, and thus impose the
rigid lid assumption.
(a) Mechanical and thermal conditions. With the
above restrictions the kinematic conditions read
ÿw0; at z0;
ÿvH rhÿw0; at z ÿh x;y:
11
Here vH u;v; and r is the (horizontal) gradient
operator.
The dynamic boundary conditions are taken in the forms
In these equations,T0
x;yare the horizontal components
of the shear tractions exerted by the wind on the water surface,qathe density of air,c0the drag coecient, and
(U, V) the horizontal velocity components of the wind 10 m above the free surface. Similarly,Th
x;yare the shear
tractions exerted by the bottom on the water,ch('10ÿ3/
10ÿ4 m sÿ1) the frictional coecient, which depends on
many dierent factors, but mainly on the bottom roughness, and uh;vh the horizontal water current components at the bottom. (13a) is a sliding law for the bottom boundary layer. Qgeoth is the geothermal
tem-perature gradient. The conventional method of relating the surface wind stress to the wind velocity is by the quadratic relationship (12a). Direct measurements of momentum ¯uxes over water have indicated that the non-dimensional value of the drag coecientc0depends on wind velocity. If the wind velocity is less than 10 m sÿ1, c
0 can be regarded as a constant c0'1:810ÿ3.
Correspondingly, the bottom stress will have to be re-lated to the water velocity at the bottom. One commonly assumes that the bottom stress is linearly or quadratic-ally dependent on velocity. The dierence of the sliding
Table 1
Variables and parameters arising in the ®eld equations and boundary conditions
ca(mg mÿ3) Mass density of constituenta(concentration)
cv(J kgÿ1°Cÿ1) Speci®c heat of water at constant volume c0'0:0018ÿif jUj<10 m sÿ1 Drag coecient for evaluation of wind stress cw
a mg mÿ
3 Concentration of constituentaat the boundary whenvn>0 Dca
H;D ca
V m2sÿ1 Turbulent horizontal and vertical mass diusivities of constituenta DT
H;D T
V m
2sÿ1 Turbulent horizontal and vertical thermal diusivities
f2Xsin U sÿ1 Coriolis parameter, whereXis the angular velocity of the Earth andUthe geographical
latitude
g9.81 (m sÿ2) Gravity constant
ja
w mgmÿ2sÿ1 Concentration ¯ow of constituentaat the boundary whenvn<0
Qgeoth(°C mÿ1) Geothermal temperature gradient
T(°C) Temperature
U;V (m sÿ1) Horizontal components of the wind 10 m above water surface u;v;w(m sÿ1) Components of the water current in thexÿ,yÿ,zÿdirections uh,vh(m sÿ1) The horizontal water current components at the bottom
b'6:810ÿ6 Cÿ2
Quadratic coecient of thermal expansion
ch'10ÿ
310ÿ4 m sÿ1 bottom drag coecient
/p/q(kg mÿ1sÿ2) Dynamic normalized pressure
mH,mV(m2sÿ1) Kinematic turbulent horizontal and vertical momentum diusivities (viscosities)
q(kg mÿ3) Density of water
q0(kg mÿ3) Density of water at 4°C
qa1.225 (kg mÿ3) Density of air
laws has an obvious in¯uence only on the ®eld near the bottom. In our computations we choose the linear relation (13a) and an appropriate value of the bottom drag coecientch'510ÿ4 m sÿ1. One reason of the
choice of the linear relation is the larger has more ex-periences with the bottom drag coecient in the linear relation. The other reason is in maintaining the linearity of the vertical diusive term, which is important if one uses implicit integration in time for this term. This will be the case in our numerical method.
(b) Boundary conditions for the tracer mass ¯ux. It
shall be assumed that any tracer mass can only be brought into the water body via the river in¯ows and by a source at the free surface of the lake. By the same token it can leave the lake domain only through its exit rivers and by sedimentation at the bottom surface. Thus one prescribes an in¯ow
janjaw; at inflow coordinates whenvn<0; 14
whilst out¯ow conditions are given by
jancwavn; at outflow coordinates whenvn>0:
15 At all other boundary points no trace mass leaves the domain, i.e.
jan0; whenvn0: 16
In the above Eqs. (14)±(16),ja
wdenotes the in¯ow tracer
mass ¯ux at the boundary (wall), and cw
a is the tracer
concentration at the boundary. If sedimentation of trac-ers or their back solution into the water mass must be modelled, thenja
wmust also be prescribed along the
bot-tom boundary. This however will not be our concern here. This completes the formulation of the boundary conditions.
2.2. Numerics
The above system of dierential equations for the velocity, temperature and tracer mass ®elds, paired with the corresponding boundary conditions has been nu-merically programmed. Since the tracer concentrations are supposed to be so small that the density of the mixture is only negligibly aected by the presence of the tracer, and because also boundary conditions for the tracers are decoupled from those for the velocity and temperature ®elds the system can be decoupled and consecutively solved: ®rst the equations for the velocity and temperature ®elds are solved and subsequently with their knowledge the tracer ®elds are determined.
To this end a semi-spectral model was designed with implicit integration in time. The model is the semi-spectral model SPEM developed by Haidvogel et al. [2], and was extended by us to account for implicit tem-poral integration. The variation of the ®eld variables in the vertical direction is accounted for by a superposi-tion of Chebyshev polynomials, but ®nite dierence
discretization is used in the horizontal direction. By using the so-calledr-transformation, the lake domain is transformated to a new domain with constant depth and this cylindrical region is once again transformed in the horizontal coordinates by using conformal mapping which maps the shore as far as possible onto a rectangle. Theoretically such a mapping always exists, but when the bounding line deviates in some segments from the actual shore line deviations occur. This will be the case for Lake Constance.
Because of the small water depths of lakes in com-parison to the ocean the original SPEM model had to be altered to permit economically justi®able time steps. It is well known that in the computation of the circulation of a lake, very ®ne grids need to be used near the free surface in the vertical direction. Moreover, in turbulent ¯ows, the eddy viscosity may be several orders of mag-nitude larger than the molecular viscosity. This makes the implicit treatment of the viscous terms imperative because the viscous stability limit is much more restric-tive than the inviscid Courant-Friedrichs-Lewy (CFL) condition near the free surface. In Wang and Hutter [12] several ®nite dierence schemes, implicit in time, were introduced; that scheme which used implicit integration in time for the viscous terms in the vertical direction was the most successful one. The eectiveness of the pro-posed method is demonstrated by Wang [11] and Wang and Hutter [12] and its workability for diusion prob-lems was demonstrated in Hutter and Wang [5].
Here we consider only the tracer diusion (2) and show how the discretization is implemented. This is done by using a leap frog procedure for integration in time (more precisely, this leap frog procedure is only used for the advective terms), upstream dierencing for the advective terms and central dierences for the hor-izontal diusive terms as follows
17
indicate the time step. The terms indicated by the curly brackets are evaluated at the new, unknown time step. Because of the spectral expansion in the vertical, these terms with the vertical dierentiations need special handling when being discretized, see [12]. The use of an actually forward scheme in time for the horizontal dif-fusive terms is because of the fact that for a diusive equation a leap frog procedure for integration in time tends always to numerical instability. We show in the mentioned paper that for each water column only a linear system of equations must be solved to advance the computation in time, a step that can quickly and un-problematically be solved.
2.3. Parameter selection
Computations were performed for lake constance under homogeneous, barotropic and strati®ed, baro-clinic situations and exposed to external wind forcings.
LM 6517 mesh points were chosen in the hori-zontal direction amounting to an average grid size of DxDy1 km.
In ensuing developments values of the diusivities will be prescribed even though they ought to be com-puted according to the turbulence intensity present at a certain location of the water body. This is done so here since the model is still in a phase of development where its proper performance is tested. Later applications ought to use algebraic Reynolds stress parameterization.
The diusivities will be taken as follows: (i) for homogeneous water:
Except for the tracer diusivities these choices were motivated and extensively discussed by Wang and Hutter [12]. The expressions of the diusivities indicate the fact that for homogeneous waters the diusivities
can be approximately assumed as constants, while for strati®ed waters non-constant vertical distributions of the vertical diusivities (19) are more realistic as they account for smaller diusivities (viscosities) in the met-alimnion than in the epi- and hypolimnion. We must point out that their choice is not entirely free, as it de-pends to a certain extent also on the numerical stability of the code. At ®xed spatial resolution the turbulent mass, momentum and thermal diusivities must be suciently large to guarantee that numerical oscillations (noise) are attenuated, and computations can stably be executed. Should the numerical values of the austausch coecients needed according to these requirements be greater than physically permitted, then physically im-portant phenomena might be damped away to such an extent that they are no longer recognizable or not as persistent as in nature. In such cases an increase of the spatial resolution and a simultaneous reduction of the values of the diusivities might help and yield better and more stable results. Thus, with the above choices of the diusivities (18) and (19) the number of Chebyshev polynomials must for homogeneous water be at least 12 and for strati®ed water 30 to perform stable computa-tions. Compared with physically realistic values the diusivities (18) and (19) are still somewhat large (see, e.g. [3,6±8]). They can be further reduced and better adjusted to values closer to physical reality, if the number of polynomials (vertical resolution) is enlarged. For lake constance, the above parameterizations Eq. (19) led to stable computations, if the initial tem-perature pro®le
was chosen. It mimics a typical summer strati®cation of an alpine lake, but has its maximum vertical slope at 10± 20 m thus above the depth where the turbulent diu-sivities are largest. This is not ideal but was needed to obtain stable computations. Numerical stability could have been reached also with smaller vertical diusivities, but they would require a larger number of Chebyshev polynomials and make computation time unduly long.
3. Substructuring
In Hutter and Wang [5] and Wang and Hutter [12] computations with the semi-implicit SPEM were per-formed in which the resolution in the horizontal plane was relatively coarse. For an arti®cial lake of 65´ 17 km2and 100 m depth and for Lake Constance a horizontal grid with 65´17 nodal points was chosen and led in Lake Constance ± because of the curvilinear
coordinate system used ± to grid lengths between 200 and 2700 m in the x-direction and 100±2200 m in the y-direction. This non-uniformity is the result of the conformal transformation used to obtain a grid system that naturally follows the lake shore line. Uniformity in grid size distribution is intended, because the numerical oscillations (instabilities) preferably occur on the small scales; however, it is dicult to achieve in complex ge-ometries. In such cases, to attain a uniform grid as far as possible, a bounding line, which deviates in some seg-ments from the actual lake boundaries, is used for conformal mapping. In these segments the actual boundaries can only be approximated by a step func-tion. This will be the case for Lake Constance. Obvi-ously, the best computational results for the velocity, temperature and tracer concentration ®elds are obtained with a grid as ®ne as possible and with a number of Chebyshev polynomials as large as possible; however CPU-times of our workstations of up to 100±200 h set a natural limitation to such intentions. 65´17 grid points in the horizontal directions and 12 and 30 Chebyshev polynomials for homogeneous and strati®ed water bodies, respectively, could not be surpassed without making CPU-times longer than a week. The global, i.e., basin wide circulation dynamics could be obtained with satisfactory accuracy, but for closer examination of particular processes or areas, as for instance the intake location of a drinking water catchment site or the source region of a contaminant, it is compelling to introduce an increased gird resolution in these regions.
This goal can be reached by two dierent methods. Either one uses the domain decomposition method. In this approach, the physical domain or grid is decom-posed into a number of overlapping or non-overlapping subdomains on each of which an independent incom-plete factorization can be computed and applied in parallel. In the subdomain where higher resolution is needed a particularly ®ne grid is selected, while a suc-cessively coarser grid size is employed as one moves out of this region. Usually, the interfaces or overlapping regions between the subdomains must be treated in a special manner. The advantage of this approach is that it is quite general and can be used with dierent methods within dierent subdomains. This method has been successfully applied to solve many physical problems (see, e.g. [1,9,10,13,14]). Alternatively, one may perform a ®rst integration with a coarse grid and large time step for the entire domain. This may yield suciently accu-rate results over most part of the integration domain, but if a subregion exists where this is not so, a subdo-main containing this subregion can be selected. Two kinds of boundaries that exist in the subdomain con-taining this subregion can be selected. Two kinds of boundaries may exist in the subdomain, namely the
physical boundary which encloses the physical domain
and the inter-grid boundarywhich lies in the interior of
the global domain. Then, the same computation can be repeated only for this subdomain with a ®ner resolution of the grid and a smaller time step, using the results of the global calculations with the coarser gird along its inter-grid boundaries. This procedure, known as
sub-structuring, requires interpolation of the coarse-grid
data obtained from the global calculations to the ®ner meshes on the inter-grid boundaries of the subdomain, while on the physical boundaries of the subdomain the same boundary conditions as in the global calculations are applied, and it assumes that the ®elds at the boundaries of the subdomain and within its complement are suciently accurate when being imposed from the coarse-gird computations.
One essential step when applying substructuring techniques is the selection of the sub-domain within which the grid resolution must be increased. In this paper we shall report experiences gained when trying to use the above described substructuring technique to focus on details of the velocity and tracer concentration ®elds in three distinct areas of Lake Constance, the two basins consisting of the Obersee andUberlinger See, see Fig. 1(a). Since the Untersee is dynamically uncoupled from the remaining two basins we shall not deal with this latter lake basin in this study. However, we shall show results obtained with sub-structuring techniques for theUberlinger See as a whole, the middle portion of the Obersee and a near shore region between Roman-shorn and Rorschach.
In principle, each attempt to estimate the discretiza-tion error in a numerical method is a comparison of results obtained with dierent grid sizes. We performed such comparisons in our computations of the global dynamics of Lake Constance by varying the number of Chebyshev polynomials, Wang and Hutter [12]. It turned out that a representation of the vertical distri-bution of the ®eld variables with 12 and 30 polynomials for homogeneous and strati®ed water was sucient to obtain results with a satisfactory degree of convergence. For this reason we shall here not enlarge the number of Chebyshev polynomials used and implement the in-crease of resolution in the horizontal direction only. The subdomains will be selected as some domains, de®ned by the coarse resolution using 65´17 grid points for the global analysis of the entire lake. The procedure will be as follows.
· In the ®rst step the velocity, temperature and tracer
concentration ®elds will be calculated with the coarse resolution in the entire lake basin. The values (i.e., time series) of the ®elds at the grid points of the in-ter-grid boundary of the subdomain within which computations are being repeated with the ®ner reso-lution must for each time step be stored.
· In step two a new and ®ner net of grid points is
curve de®ning the boundary of this coarse grid pro-jection and using the conformal Schwarz±Chrystoel transformation to generate a new and ®ner net of or-thogonal curvilinear coordinates.
· In step three the computations are to be performed
with this new discretization within the subdomain only. In doing so, the inter-grid boundary values of the ®eld variables at the boundary of the grid points of the ®ne net at all time steps of the ®ne integration process must be interpolated from the boundary data generated with the coarse net. At the physical bound-aries, e.g., at the basal, free surface and along the shore line, the same boundary conditions as in the global analysis in the ®rst step and prescribed at the grid points with the ®ne resolution.
These procedures will ®rst be illustrated using a typ-ical meteorologtyp-ical scenario for homogeneous Lake Constance.
4. Homogeneous Lake Constance
Fig. 1(a) shows a map of Lake Constance with a few of the larger towns situated along its shore and bathy-metric lines indicating its depth. This Alpine lake bor-ders Austria, Germany and Switzerland and consists of three basins: the larger Obersee, Uberlinger See and Untersee. The latter is dynamically disconnected from the others by the 5 km long ``Seerhein'', the exit river of the two other basins. Obersee and Uberlinger See are 64 km long and have a mean width of about 10 km. The
Uberlinger See is relatively shallow (147 m deep) when
compared with the 252 m deep Obersee and somewhat separated from the latter by a sill north of the island Mainau. The mean depths are 79 and 101 m, respec-tively. Fig. 1(b) also shows the curvilinear coordinate system employed by us with the mesh of 65´17 grid points. Notice that the boundary from which the coor-dinate net was constructed by conformal mapping does not everywhere coincide with the shoreline. The island Mainau and the segment Rorschach±Bregenz are spared out, and the shorelines are indicated by thick lines in these regions. As mentioned before we have done it so that the grids are uniformly distributed as far as possi-ble.
4.1. Global results
Consider Lake Constance under homogeneous con-ditions as they prevail from November through March, approximately. Let the wind scenario be an impulsively applied wind uniformly distributed over the entire basin, acting in the long direction from 305°True NW (north west wind) with a strength of 0.05 N mÿ2(corresponding
to 4.7 m sÿ1) and lasting for ever. The set-up of the
current within the lake is then expected to be accom-panied with the formation and (slow) attenuation of inertial waves.
Fig. 2 displays the horizontal velocity distribution four days after the onset of the wind in 0, 10, 20 and 40 m depth. Time series also show that steady-state conditions have practically been reached at this time. At o-shore positions the turning of the horizontal velocity towards the right, that increases with depth and is due to
Fig. 1. (a) Map of Lake Constance with bathymetry and a few towns indicated along its shore. Three basins characterize the lake: Obersee,
the eects of the rotation of the Earth, can clearly be recognized. The near-shore currents are generally very strong, parallel to the shore and with the wind, while the o-shore currents are mixed, with an angle of re¯ection to the right of the wind at the surface. This angle in-creases with depth below the surface so that the ¯ow is clearly against the wind at 40 m depth.
Time series of the horizontal componentsu;v, of the water velocity at two mid-basin positions of the Uber- linger See and the Obersee, respectively, are displayed in Figs. 3 and 4 for the depths from 0 to 100 m below the free surface. The Coriolis-force-induced-inertial oscilla-tions with a period of approximately 16.3 h are
recongnizable in all time series at all depths, but these oscillations are much faster attenuated in theUberlinger See than in the Obersee, the reason obviously being the increased boundary friction due to the narrowness of the shores.
At near-shore positions (at most a few hundred meters o-shore), as for instance the three positions in Fig. 5, the inertial oscillations can no longer be seen but at most guessed; the frictional eects of the boundaries prevent the development of these oscillations. At the western most point in theUberlinger See the current pattern is built-up within the ®rst few hours and then levels o at constant values which are reached after approximately
20 h (Fig. 5(a) and (b)). At the position east of Ror-schach (Fig. 5(c) and (d)), 500 m from shore steady conditions have not been attained within the ®rst four days as they-components perform a long-periodic swing that is not equilibrated after 100 h. At the position of Fig. 5(e) and (f), 720 m from the southern shore between Romanshorn and Rorschach, small amplitude inertial oscillations can be discerned that are superposed upon the almost monotonous approach into a steady state that here seems to be reached after 100 h. To see the eects of the inertial motion, one only needs to analyse the data at an o-shore position as indicated in Fig. 6, which dis-plays the time series of the horizontal velocity compo-nents at a location approximately 3 km o-shore from Arbon (between Rorschach and Romanshorn), where the water depth is again larger than 100 m.
These results will henceforth be used for comparison when the substructuring techniques are used.
4.2. Substructuring in the middle part of the Obersee
It is to be expected that the central portion of the large lake basin is satisfactorily modelled with the res-olution of the 65 ´17 grid points of the global analysis as presented in Section 4.1. This is indeed so; therefore, the purpose of the application of the substructuring technique in this area must primarily be a device to ``better represent the results'' rather than to improve upon them.
Fig. 7(a) shows the location of the subdomain within the bathymetric chart of Lake Constance and Fig. 7(b) displays the applied orthogonal curvilinear coordinate system which was constructed with the four boundary segments via the application of the conformal transfor-mation; in computing this net, the boundary points of this domain as obtained from the coarse discretization were interpolated to de®ne the ®ner net. The boundary
Fig. 3.Homogeneous Lake Constance. Time series of the horizontal velocity componentsu;vat two midlake positions in the Obersee as indicated in the insets for an impulsively applied spatially uniform wind from 305°NW. The oscillations have an approximate period of 16.3 h. The labels (1;2;3;. . .;11) indicate depths at (0;10;20;. . .;100) m; panels (a, b) are for the position shown in the inset of Fig(a), panels (c, d) for that shown in the inset of Fig(c).
conditions at the free and at the bottom surfaces need be prescribed in the same way they were described for the global analysis.
Fig. 8 shows the distribution of the horizontal ve-locities four days after the onset of wind for horizontal sections at the 0, 10, 20 and 40 m depth level. When comparing this ®gure with Fig. 2, which displays the corresponding results for the same but global analysis the same features can be discerned, the substructuring process, however, allows us to better see and more easily interpret the current structure.
A better proof to see whether the grid resolution used in the global analysis is sucient to accurately model barotropic wind-induced currents is to compare time series of the horizontal velocity components for loca-tions within the subdomain. We have done this for the two positions shown in the insets of Fig. 3; these points lie in the western and eastern portions of the subdomain as indicated in Fig. 7, and the time series for the hori-zontal velocity components are virtually identical with
those of Fig. 3 so there is no need to show them sepa-rately. The results prove that the global analysis with the 65´17 grid points accurately reproduces results at midlake positions of the Obersee.
4.3. Substructuring of theUberlinger See
Uberlinger See is particularly signi®cant because the ``Bodensee-Wasserversorgung'' in Stuttgart operates at a midlake position ``Sipplingen'' a water intake site for drinking water. Our coarse grid distribution has already been selected with emphasis on a ®ner resolution within
Uberlinger See. Despite this, we now repeat computa-tions in theUberlinger See west of the sill at Mainau by employing substructuring with a yet ®ner resolution. Fig. 9 displays the subdomain in the bathymetric chart of Lake Constance and the selected orthogonal curvili-near coordinates used for it. This domain is special in-sofar as the ®eld variables of the global analysis must only be taken over at grid points of the eastern
boundary sector in which the velocity components, the stream functions and the tracer concentration must be prescribed; apart from the boundary data at the free and
bottom surfaces these are the only boundary data that are needed in the computations at the substructure level. Along the other three shore lines the same boundary
Fig. 5.Homogeneous Lake Constance.Time series of the horizontal velocity components in thex-direction (left) andy-direction (right) for an impulsively applied spatially uniform wind from 305°NW. Panels (a, b) are for a position 540 m from the western end of theUberlinger See, panels (c, d) are for a position 560 m east of the bight of Rorschach and (e, f) are for a position between Romanshorn and Arbon 720 m from the southern shore, all locations being indicated in the inset maps. The labels (1 2, 3, 4, 5) denote depths at (0, 10, 20, 30, 40) m.
conditions as in the global analysis are applied as are the boundary conditions at the free and bottom surfaces for wind stress and ¯ux density of tracer.
The advantages of the method of substructuring are now clearly seen in Fig. 10 which shows the vector plots of the horizontal velocity for the Uberlinger See four days after the wind set-up (i.e., for nearly steady con-ditions) for horizontal sections in 0, 10, 20 and 40 m depth and for the same wind input as before. When comparing this ®gure with Fig. 2 which illustrates the
same results it is clear that the current patterns can much better be seen than with the global analysis. In particular, Fig. 10 discloses interesting details how the ¯ow changes from a general orientation with the wind at the free surface to one against the wind at the 40 m depth. The Ekman type rotation of the horizontal cur-rents is particularly strong in the middle portion of this basin. Furthermore, the currents along the shores are generally with the wind and perhaps also somewhat stronger than o-shore.
Fig. 7. Position of the subdomain indicated by a thick solid line within the bathymetric chart of Lake Constance (a), and distribution of the grid points within this subdomain of area 67.27 km2(b). In (b) the symbolsindicates the positions at which time series of the horizontal velocity
components are plotted in Fig. 3.
If time series of the horizontal velocity components at the positions indicated in Fig. 9 are plotted, then the results at o-shore points are practically identical with those of Fig. 4 (the positions are the same); dierences are hardly visible and so the corresponding ®gures will not be repeated. At the near-shore position (the western most point in Fig. 9) results, however, dier
quantita-tively, but not qualitaquantita-tively, compare Fig. 5(a) and (b) with Fig. 11(a) and (b).
So far our focus has been the velocity ®elds. The method of substructuring was also tested in computa-tions of tracer diusion. To this end consider the same meteorological scenario with an impulsively applied spatially uniform and temporally constant wind from
Fig. 8.Homogeneous Lake Constance. Horizontal velocity vector plots for steady conditions, four days after the onset of a spatially uniform, temporally constant wind from 305°NW at the depths 0, 10, 20, 40 m. Each panel has its own velocity scale in m sÿ1(see insets, and compare also
with Fig. 2).
305°NW. Consider that a tracer is released at the free surface at a position 5 km from the western end on an area of 730´ 140 m2with a ¯ow rate of 20 mg mÿ2sÿ1. Let this ¯ow rate be applied for 24 h at the beginning and then be shut-o. The position and the area over which the tracer is released is shown in Fig. 9(b). In that ®gure we also show neighbouring points (indicated by symbol +) at which time series are shown in Figs. 12 and 13. The former shows the time series of the tracer concentration at various depths at these locations as computed with the coarse-grid-global analysis, the lat-ter repeats these results as obtained with the substruc-turing technique. Details can also be obtained from the ®gure captions. A super®cial glance at these ®gures seems to indicate that the two sets of graphs are not dierent from one another, however a closer look dis-closes signi®cant dierences in particular close to the location of tracer input. For instance, while the tem-poral evolution of the tracer concentration at the center point (Figs. 12(a) and 13(a)) and east as well as south of it (Figs. 12(c),(d) and 13(c),(d)) are very similar at all depths that are shown their absolute values dier somewhat. At the locations south and south-east of the central source point dierences between the two com-putations are larger (Fig. 12(b), (e) and (f) and Fig. 13(b), (e) and (f)). In particular the surface-near evolution of the tracer concentration is dierent at early times while at depth such dierences set in somewhat later. This indicates that substructuring techniques are physically important and not simply ``cosmetic''. The later time evolution of the tracer concentration depends upon how early time dispersion is achieved.
4.4. Substructuring in the near-shore zone of the south-east Obersee
In the global discretization that we employed, the shore line between Rorschach and Bregenz was not identical with a bounding segment from which the cur-vilinear coordinate system was constructed; the result was a step-wise approximation of the shore line in this region. One may therefore justly suspect that results obtained with the coarse grid should be improvable with the substructuring technique. To demonstrate this, we now select the subdomain indicated in Fig. 14; it is not free of a step-wise approximation of the shore east of Rorschach, however as seen in Fig. 14(b) the approxi-mation with the new orthogonal curvilinear coordinates is much improved. Boundary data along the northern and the western boundary of the subdomain must be provided by the global coarse-grid computations and interpolations; boundary data along the southern and eastern shore are prescribed the same way as they are prescribed in the global analysis, and so are the boun-dary conditions at the free and bottom surfaces.
Comparing the results obtained for the horizontal velocity ®eld four days after the wind set-up indicates how useful and signi®cant the substructuring technique turns out to be in this near-shore region. Fig. 15 displays the velocity ®eld for horizontal sections 0, 10, 20 and 40 m below the free surface and Fig. 2 gives the same re-sults as obtained with the global coarse-grid computa-tions. Details of the ¯ow structure with the strong near-shore current are now very clearly visible as is the strong northward current at the eastern shore.
Fig. 9. Position of the subdomain indicated by a thick solid line within the bathymetric chart of Lake Constance (a), and distribution of the grid points within this subdomain of area 32.77 km2(b). In (b) the shadow shows the area over which the tracer is released. The symbol + indicates the
Fig. 11.Homogeneous Lake Constance. Time series of horizontal velocity components at a position 540 m east of the western end of theUberlinger See. Conditions are the same as those in Fig. 5(a) and (b) but results have been obtained using substructuring for the subdomain.
Fig. 10.Homogeneous Lake Constance. Vector plots the horizontal velocity, four days the wind set-up for an impulsively applied spatially uniform wind from 305°NW in 0, 10, 20 and 40 m depth. Conditions are very nearly steady state. Note, every panel has its own velocity scale in m sÿ1as
indicated in the inset.
Fig. 16 displays graphs of the time series of the hor-izontal velocity components at 10 m depth intervals at the positions, indicated in the insets and also shown in
Fig. 14(b); the results are obtained by using the sub-structuring technique. The corresponding results ob-tained with the global coarse-grid analysis are displayed
Fig. 12.Homogeneous Lake Constance. Time series of the tracer concentration for the positions shown in the inset in the middle of theUberlinger See (a) 720 m towards west, (b) 740 m towards east, (c) 280 m towards south, (d) 1.5 km towards south-east, (e) and 3.2 km towards south-east, (f) of it (the positions indicated in the insets are also shown in Fig. 9) for various depths. At timet0 an impulsively applied temporally constant spatially uniform wind sets in from 305°NW lasting for ever and a tracer is released on the free surface on an area of 730´140 m2situated 5 km east of the
western end with a ¯ow rate of 20 mg mÿ2sÿ1, (see also inset in Fig. 9). The labels (1, 2, 3;. . .;11) indicate the depths (0, 10, 20;. . .;100) m.
in Fig. 5(c)±(f) and Fig. 6(a) and (b). At the o-shore position (3 km from shore) the time series for the hori-zontal velocity components are very similar in the two computations (see Fig. 6(a) and (b) and Fig. 16(e) and
(f)). Closer to the shore, but still 720 m from it, the evolution of the horizontal velocity components is very similar in the two cases and absolute values dier only slightly (see Fig. 5(e) and (f) and Fig. 16(c) and (d)). At
Fig. 13.Homogeneous Lake Constance. Same as Fig. 12, but computations were now performed within the subdomain ``Uberlinger See'' employing substructuring techniques.
the third location that is closest to the shore (560 m from shore, Fig. 5(c) and (d) and Fig. 16(a) and (b)) dier-ences in the results for the velocity components are largest. The results obtained within the coarse-grid global analysis seem to indicate that steady conditions have not been reached within the ®rst 100 h, while the more accurate computations hint at steady conditions at 50 h. The probable reason of this dierence is likely the poor step-like approximation of the global discretiza-tion.
Tracer concentrations were also computed with the substructuring technique for a tracer release about 3 km o-shore the southern shore between Romanshorn and Arbon (see Fig. 14) with a ¯ow rate of 20 mg mÿ2 sÿ1
over an area of 1200 ´520 m2at the surface lasting 24 h.
Results are shown in Fig. 17 as time series of the con-centration at certain depths for positions displayed in the inset maps; computations were also performed for the coarse grid. They dier in all positions from those of Fig. 17 except for the central point of tracer release (panel a). The dierences are quantitative and moderate, but not qualitative.
A better overall impression about the tracer disper-sion is obtained if snapshots of the tracer concentration are shown at the free surface for the times 1, 2, 3, 4 days after initiation of the tracer release. Fig. 18 dis-plays the concentration distribution at these instances in the subdomain between Romanshorn±Rorschach. The eects of the advection by the velocity ®eld are very clearly seen in this ®gure. The tracer masses follow
the south-eastern surface currents until they reach the shore and then turn northward very close to the shore with the highest concentrations at the shore. Superim-posed are the diusive processes that contribute to a further dispersion of tracer mass in the horizontal di-rection.
5. Strati®ed lake constance
5.1. Global results
Strati®cation was implemented for the temperature pro®le (20) and turbulent diusivities are selected as given in (19). It turned out that for the uniform con-stant wind from 305° True NW with strength 4.7 m sÿ1 30 Chebyshev polynomials were required to
achieve stable numerical integration. Corresponding CPU times became very large, so that we limit henceforth integration to three days (the limitation comes from the substructuring and less from the global analysis).
Fig. 19 displays the horizontal velocity distribution two days after the onset of the application of the spatially uniform, temporally constant wind from 305° at the depths 0, 10, 20, 40 m. In the upper layer (0±20 m), the currents are exclusively directed with the wind. This ¯ow structure continues in the Obersee, but east of approximately the Romanshorn±Friedrichsha-fen cross section a large counterclockwise gyre is
formed that ®lls the entire width of the basin. This gyre continues to exist in the hypolimnion as is seen from the 40 m-depth panel of Fig. 19. It can be traced even at 100 m depth. By contract, in theUberlinger See and the western part of the Obersee at 40 m depth, the baroclinic return current has already been formed. Of
interest are also the two smaller clockwise rotating epilimnetic gyres at the bay of Konstanz and west of the Alpenrhein in¯ow.
Time series of the horizontal velocity components in the x- and y-directions at the three positions in the south-eastern part of the Obersee as indicated in Fig. 14
Fig. 15.Homogeneous Lake Constance. Vector plots of the horizontal velocity, four days after the wind set-up at the 0, 10, 20, 40 m depths for an impulsively applied spatially uniform wind from 305°NW and nearly steady-state conditions. The subdomain is a shore region near Romanshorn± Rorschach. Note, every panel has its own velocity in m sÿ1as indicated in the insets.
already are displayed in Fig. 20. They are displayed for 72 h (3 days) at 10 m intervals from the surface (curve label 1) to 100 m depth (curve label 11). Besides the
general trend of a fast increase of the speeds in the ®rst approximately 20 h and a subsequent moderate to small relaxation to a suspected steady state, the time series
also show traces of oscillations with an estimated period of 14 h. The amplitudes of these oscillations are largest at the position that is farthest from shore and can hardly
be discerned in the other two positions. It can be shown that these oscillations likely correspond to the lowest order Poincare mode.
Fig. 17.Homogeneous Lake Constance. Time series of the tracer concentration for the positions indicated in the inset maps, in the middle of the tracer release area of 1200´520 m2(a), 1.2 km towards west (b), 1.2 km towards east (c), 700 m towards south (d), 3.5 km towards south-east (e) and 5.3
5.2. Substructuring in the near-shore zone of the south-east Obersee
The very interesting current structure exhibited in Fig. 19 calls for a detailed study of the substructuring technique in baroclinic circulation dynamics. Here we restrict considerations to demonstrating its usefulness in analyses of physical limnology. Because of the projected 40 Mio. $; output drinking water site we choose the
near-shore zone of the south-east Obersee in which it lies.
Fig. 21 displays vector plots of the horizontal velocity for the same conditions and time as those of the global analysis in Fig. 19. Whereas the graphs of this latter ®gure allow in the upper 20 m to infer a fairly strong eastward ¯ow with a suspected ``coastal jet'' out of the region towards north at ``Alpenrhein'', and as one sup-poses a clear clockwise rotating gyre at 40 m depth west of Alpenrhein, these features can now be better quanti-®ed in Fig. 21. The strong eastward current in the up-permost 20 m and the equally strong northward jet at the north-east corner are now clearly detailed. The velocities
Fig. 18. Homogeneous Lake Constance. Isolines of tracer concentration on the free surface 1 2, 3, 4 days after the commencement of the tracer in¯ow of 20 mg mÿ2 sÿ1 mass over the area of 1200´520 m2during 24 h. The subregion is the near-shore zone between Romanshorn
and Rorschach and the wind is impulsively applied and uniform over the entire lake blowing from 305°True North (in the long direction of the
at the 40 m-depth seem to be smaller than in Fig. 19 and the clockwise gyre in the bay of Rorschach is not formed. Fig. 22 displays time series of the horizontal velocity components for the same conditions as Fig. 20, but now computed with substructuring. A ®rst glance at both ®gures shows that qualitatively the results are very similar, but the maximum values of the velocities ob-tained with substructuring are up to twice as large as in the global analysis. The Poincare-type oscillations are
however reduced in the graphs of Fig. 22 (as compared to those of Fig. 20) and can now only be clearly discerned at the o-shore position; interestingly the x-components show these oscillations now very vaguely (compare Fig. 20(e) and Fig. 22(e)). At the other two
positions (Fig. 20(a)±(d) and Fig. 22(a)±(d)) dierences are particularly visible at the early time response.
These results should indicate that near-shore behaviour of the baroclinic motion should always be studied in more detail by subjecting such regions to substructuring procedures. Moreover, these results, and especially Fig. 19 draws attention to a wealth of further interesting questions. For instance the region involving the gyre in the bay of Konstanz and the eastern half of the Obersee should be studied with sub-structuring techniques, perhaps by employing repeated nestings to resolve the proper current pattern in the bay of Bregenz. Such further analyses as well as the study of the tracer diusion are deferred to a later paper.
Fig. 19.Strati®ed Lake Constance. Horizontal velocity vector plots, two days after the onset of a spatially uniform, temporally constant wind from 305°NW (in the long direction of the basin) at the depths 0, 10, 20, 40 m. Each panel has its own velocity scale in m sÿ1(see insets).
6. Concluding remarks
In this paper we studied wind-induced circulation and tracer diusion in Lake Constance by using a
semi-im-plicit semi-spectral numerical model based on SPEM ([2]), in which a semi-implicit temporal integration routine was implemented ([12]). We employed the model at the global level using the entire lake basin and
Fig. 20.Strati®ed Lake Constance. Time series of the horizontal velocity components in thex-direction (left) andy-direction (right) for an impulsively applied spatially unform wind from 305°NW. Panels (a, b) are for a position 560 m east of the bight of Rorschach, panels (c, d) are for a position
covering it with a coarse grid. With this discretization, the velocity and tracer mass ®elds to an impulsively applied wind uniform in space and to a surfacial in¯ow of tracer mass were determined. In a more local second analysis i.e., in subregions where the thus computed
®elds are likely to be inaccurate the computations were repeated with a substantially increased number of mesh points. Along the open boundaries of these subregions the ®eld quantities obtained in the global analysis must be prescribed and interploated, while on outer
bound-Fig. 21.Strati®ed Lake Constance. Vector plots of the horizontal velocity, two days after the wind set-up at the 0, 10, 20, 40 m depths for an im-pulsively applied spatially uniform wind from 305°NW. The subdomain is a shore region near Romanshorn±Rorschach. Note, every panel has its
own velocity scale in m sÿ1as indicated in the insets. Compare also with Fig. 20.
aries they are prescribed in the same way as in the global analysis. This sub-structuring technique allowed im-proved computation of the ®eld variables where subgrid processes could not be resolved with the coarser grid of the global analysis.
This technique of substructuring was demonstrated for three subregions of the homogeneous Lake Con-stance and a subregion of the strati®ed Lake ConCon-stance. To generate the subregions, new curvilinear, orthogonal coordinates were constructed by using the closed
Fig. 22.Strati®ed Lake Constance. Time series of the horizontal velocity components at the locations shown in the inset-maps for various depths and an impulsively applied temporally constant spatially uniform wind from 305°NW. The positions are also shown in Fig. 14(b). Results are obtained
polygon de®ning the subregion in the coarse grid of the global analysis. The ®eld variables at interior grid points along this boundary are at any given time known from the global analysis and can be interpolated to de®ne corresponding ®eld quantities at intermediate boundary points. Boundary conditions at all other (external) boundary points of the subregion are prescribed as for the global analysis.
We showed that for barotropic and baroclinic pro-cesses the global results dier from those obtained via substructuring provided the region necessitating closer analysis is close to the shore. At o-shore subregions a global analysis may well be sucient. Moreover, at early times of tracer diusion problem and close to the source of the trance mass in¯ow, an increased accuracy is equally necessary if an accurate estimation of the tracer concentration at a ®eld position close to the source is looked for.
In summary, an ecient workable substructuring scheme is now available with the aid of which realistic scenarios of wind-induced currents and tracer dispersion in lakes can be computed.
Acknowledgements
We acknowledge ®nancial support from the A.v. Humboldt Foundation and the Max Planck Society through K. Hutter's Max Planck Prize and the Deutsche Forschungsgemeinschaft. We thank Ms. Danner for her help with typing the text.
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