Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue4.2000:

Advances in Water Resources 23 (2000) 399±425

Methods of substructuring in lake circulation dynamics
Yongqi Wang *, Kolumban Hutter

1

Institute 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 di€usion 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 resolution. Within three di€erent 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 ecient 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 di€usion; Limnology

1. Introduction
Lakes and the ocean are physical systems which respond 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 di€usion equations for the transport and
dispersion of materials and/or pollutants. These equations were discretized by various methods, ®nite di€erences, ®nite elements and spectral methods, to name a
few, and with the emerging codes many lakes were
studied when typical meteorological scenarios were applied, 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*

Corresponding author.

E-mail addresses: wang@mechanik.th-darmstadt.de (Y. Wang),
hutter@mechanik.th-darmstadt.de. (K. Hutter)

gration technique was necessary because with the explicit integration in time used in the original SPEM the
total CPU-times for integration of a wind-induced scenario in a realistic lake (say Lake Constance) was unduly 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 integration was compelling, if computations over realistic
times should be possible. In Hutter and Wang [5] the
extension of the semi-implicit SPEM to di€usion problems was studied and it was demonstrated that the inertial wave dynamics in homogeneous water, and the
Kelvin-and Poincare-type wave dynamics in strati®ed
water could be seen in the computed tracer concentration data. This, among other things, was partial corroboration for the correctness of the applied numerical
code determining the velocity, temperature and tracer
concentration ®elds.
Despite the use of the semi-implicit integration techniques 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

0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 9 9 ) 0 0 0 2 4 - X

400

Y. Wang, K. Hutter / Advances in Water Resources 23 (2000) 399±425

± an average horizontal grid size of approximately 1 km2
and 12 and 30 Chebyshev polynomials were used; existing computer capacity made ®ner resolutions uneconomical. For global analyses of the circulation pattern
this was sucient, however to describe detailed processes 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 necessary, 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 formidable 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 circumstances 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 substructuring 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 conditions 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 concentration is so minute that the density of the mixture, i.e.,
water plus tracers is not a€ected 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 considered ¯uid system. We impose the Boussinesq as-

sumption (which states that density variations only
a€ect 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])
ou ov ow
‡ ‡
ˆ 0;
ox oy oz





oca
o
o
ca oca
ca oca
DH
DH
‡
‡ v
grad ca ˆ
ox
oy
ot
ox
oy


o
oca

; …a ˆ 1; . . . ; m†;
DcHa
‡
oz
oz

…1†

…2†

ou
‡ v
grad u ÿ fv
ot






o/ o

ou
o
ou
o
ou
‡
‡
;
‡
mH
mH
mV
ˆÿ
ox ox
ox
oy
oy
oz
oz
…3†

ov
‡ v
grad v ‡ fu
ot






o/ o
ou
o
ov
o
ov
‡
‡
;
‡
mH

mH
mV
ˆÿ
oy ox
ox
oy
oy
oz
oz
…4†
0ˆÿ

o/ qg
ÿ ;
oz q0

q ˆ q…T †;
oT
‡ v
grad T
ot







o
oT
o
oT
o
oT
‡
‡
:
DTH
DTH
DTV
ˆ
ox
ox
oy
oy
oz
oz

…5†
…6†

…7†

Here a Cartesian coordinate system (x, y, z) has been
used; (x, y) are horizontal, and z is vertically upwards,
against the direction of gravity. The ®eld variables and
parameters arising in Eqs. (1)±(7) are de®ned in Table 1.
Di€usion is accounted for by postulating Fick's ®rst law
for the constituent mass ¯ux


a
ca oca
ca oca
ca oca
;
…8†
; DH
; DV
j ˆ ÿ DH
ox
oy
oz
which introduces orthotropic di€usive behaviour, equal
in both horizontal directions but di€erent in the vertical
direction. Eqs. (3) and (4) account for momentum diffusion di€erent 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
q ÿ q0
2
ˆ ÿb…T ÿ T0 † ;
…9†
q0
valid for T 2 ‰0; 40Š°C. The fact that dependencies on
impurities (salt) and pressure are ignored restricts considerations to fresh water lakes with moderate depths.

Y. Wang, K. Hutter / Advances in Water Resources 23 (2000) 399±425

401

Table 1
Variables and parameters arising in the ®eld equations and boundary conditions
ca (mg mÿ3 )
cv (J kgÿ1 °Cÿ1 )
c0 ' 0:0018‰ÿŠ if jU j < 10 m sÿ1
cwa …mg mÿ3 †
DcHa ; DcVa …m2 sÿ1 †
DTH ; DTV …m2 sÿ1 †
f ˆ 2X sin U …sÿ1 †
g ˆ 9.81 (m sÿ2 )
jaw …mgmÿ2 sÿ1 †
Qgeoth (°C mÿ1 )
T (°C)
U ; V (m sÿ1 )
u; v; w (m sÿ1 )
uh , vh (m sÿ1 )
b ' 6:8  10ÿ6 … Cÿ2 †
ch ' 10ÿ3  10ÿ4 …m sÿ1 †
/ ˆ p/q (kg mÿ1 sÿ2 )
mH , mV (m2 sÿ1 )
q (kg mÿ3 )
q0 (kg mÿ3 )
qa ˆ 1.225 (kg mÿ3 )

Mass density of constituent a (concentration)
Speci®c heat of water at constant volume
Drag coecient for evaluation of wind stress
Concentration of constituent a at the boundary when v
n > 0
Turbulent horizontal and vertical mass di€usivities of constituent a
Turbulent horizontal and vertical thermal di€usivities
Coriolis parameter, where X is the angular velocity of the Earth and U the geographical
latitude
Gravity constant
Concentration ¯ow of constituent a at the boundary when v
n < 0
Geothermal temperature gradient
Temperature
Horizontal components of the wind 10 m above water surface
Components of the water current in the xÿ, yÿ, zÿdirections
The horizontal water current components at the bottom
Quadratic coecient of thermal expansion
bottom drag coecient
Dynamic normalized pressure
Kinematic turbulent horizontal and vertical momentum di€usivities (viscosities)
Density of water
Density of water at 4°C
Density of air

Eq. (7) is the balance of internal energy; it accounts
through Fourier's heat law


T oT
T oT
T oT
;
…10†
; DH
; DV
q ˆ ÿqcv DH
ox
oy
oz
for orthotropic thermal turbulent di€usion, 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 irradiation. In principle, incorporation of radiation is, however, straightforward.
2.1.2. Boundary conditions for mass, momentum and
energy
The above laws comprise 5 ‡ m ®eld equations for the
®elds v; ca ; / and T which are also 5 ‡ m unknown ®elds.
They must be subjected to boundary conditions of both
kinematic and dynamic nature. Let the undeformed free
surface be described by z ˆ 0 and let the base be described by z ˆ ÿh(x, y). We shall formulate the boundary conditions on these surfaces, and thus impose the
rigid lid assumption.
(a) Mechanical and thermal conditions. With the
above restrictions the kinematic conditions read
ÿ w ˆ 0; at z ˆ 0;
ÿ vH
rh ÿ w ˆ 0; at z ˆ ÿh…x; y†:

…11†

Here vH ˆ …u; v†; and r is the (horizontal) gradient
operator.
The dynamic boundary conditions are taken in the
forms

· at the free surface z ˆ 0:


p
s0x ; s0y ˆ qa c0 U 2 ‡ V 2 …U ; V †;

oT
ˆ 0;
oz
· at the bottom surface z ˆ ÿh…x; y†:


shx ; shy ˆ q0 ch …uh ; vh †;

…12a†
…12b†

…13a†

oT
ˆ 0:
…13b†
oz
0
In these equations, Tx;y are the horizontal components
of the shear tractions exerted by the wind on the water
surface, qa the density of air, c0 the drag coecient, and
(U, V) the horizontal velocity components of the wind
h
are the shear
10 m above the free surface. Similarly, Tx;y
tractions exerted by the bottom on the water, ch (' 10ÿ3 /
10ÿ4 m sÿ1 ) the frictional coecient, which depends on
many di€erent 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 temperature 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 coecient c0 depends
on wind velocity. If the wind velocity is less than 10 m
sÿ1 , c0 can be regarded as a constant c0 ' 1:8  10ÿ3 .
Correspondingly, the bottom stress will have to be related to the water velocity at the bottom. One commonly
assumes that the bottom stress is linearly or quadratically dependent on velocity. The di€erence of the sliding
Qgeoth ÿ

402

Y. Wang, K. Hutter / Advances in Water Resources 23 (2000) 399±425

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 coecient ch ' 5  10ÿ4 m sÿ1 . One reason of the
choice of the linear relation is the larger has more experiences with the bottom drag coecient in the linear
relation. The other reason is in maintaining the linearity
of the vertical di€usive 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
j a
n ˆ jaw ; at inflow coordinates when v
n < 0;
whilst out¯ow conditions are given by
j a
n ˆ cwa v
n;

…14†

at outflow coordinates when v
n > 0:

…15†
At all other boundary points no trace mass leaves the
domain, i.e.
…16†
j a
n ˆ 0; when v
n ˆ 0:
In the above Eqs. (14)±(16), jaw denotes the in¯ow tracer
mass ¯ux at the boundary (wall), and cwa is the tracer
concentration at the boundary. If sedimentation of tracers or their back solution into the water mass must be
modelled, then jaw must also be prescribed along the bottom boundary. This however will not be our concern here.
This completes the formulation of the boundary
conditions.
2.2. Numerics
The above system of di€erential equations for the
velocity, temperature and tracer mass ®elds, paired with
the corresponding boundary conditions has been numerically programmed. Since the tracer concentrations
are supposed to be so small that the density of the
mixture is only negligibly a€ected 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
semi-implicit integration in time. The model is the semispectral model SPEM developed by Haidvogel et al. [2],
and was extended by us to account for implicit temporal integration. The variation of the ®eld variables in
the vertical direction is accounted for by a superposition of Chebyshev polynomials, but ®nite di€erence

discretization is used in the horizontal direction. By
using the so-called r-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 comparison 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 magnitude larger than the molecular viscosity. This makes
the implicit treatment of the viscous terms imperative
because the viscous stability limit is much more restrictive than the inviscid Courant-Friedrichs-Lewy (CFL)
condition near the free surface. In Wang and Hutter [12]
several ®nite di€erence 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 e€ectiveness of the proposed method is demonstrated by Wang [11] and Wang
and Hutter [12] and its workability for di€usion problems was demonstrated in Hutter and Wang [5].
Here we consider only the tracer di€usion (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 di€erencing for
the advective terms and central di€erences for the horizontal di€usive terms as follows

…17†

In this equation lower case Latin subscripts …i; j† denote
horizontal mesh points while upper case superscripts …n†

Y. Wang, K. Hutter / Advances in Water Resources 23 (2000) 399±425

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 di€erentiations need special
handling when being discretized, see [12]. The use of an
actually forward scheme in time for the horizontal diffusive terms is because of the fact that for a di€usive
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 unproblematically be solved.
2.3. Parameter selection
Computations were performed for lake constance
under homogeneous, barotropic and strati®ed, baroclinic situations and exposed to external wind forcings.
L  M ˆ 65  17 mesh points were chosen in the horizontal direction amounting to an average grid size of
Dx ˆ Dy ˆ 1 km.
In ensuing developments values of the di€usivities
will be prescribed even though they ought to be computed 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 di€usivities will be taken as follows:
(i) for homogeneous water:
mH ˆ 1:0 m2 sÿ1 ;

mV ˆ 0:02 m2 sÿ1 ;

2 ÿ1
Dca
V ˆ 0:005 m s ;
(ii) for strati®ed water:
8
0:02; z > ÿ20 m;
>
<
mV ˆ 0:002; ÿ20 m P z P ÿ 40 m;
>
:
0:005; z < ÿ40 m;

mH ˆ 1:0 m2 sÿ1 ;
8
0:0005;
z > ÿ20 m;
>
<
T
DV ˆ 0:00005; ÿ20 m P z P ÿ 40 m;
>
:
0:0001; z < ÿ40 m;

DTH ˆ 1:0 m2 sÿ1 ;
8
0:01; z > ÿ20 m;
>
<
ca
DV ˆ 0:001; ÿ20 m P z P ÿ 40 m;
>
:
0:003; z < ÿ40 m;

DcHa

2

ÿ1

ˆ 1:0 m s ;

…18†

…m2 sÿ1 †;

…m2 sÿ1 †;

403

can be approximately assumed as constants, while for
strati®ed waters non-constant vertical distributions of
the vertical di€usivities (19) are more realistic as they
account for smaller di€usivities (viscosities) in the metalimnion than in the epi- and hypolimnion. We must
point out that their choice is not entirely free, as it depends to a certain extent also on the numerical stability
of the code. At ®xed spatial resolution the turbulent
mass, momentum and thermal di€usivities must be
suciently large to guarantee that numerical oscillations
(noise) are attenuated, and computations can stably be
executed. Should the numerical values of the austausch
coecients needed according to these requirements be
greater than physically permitted, then physically important 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 di€usivities might help and yield better and
more stable results. Thus, with the above choices of the
di€usivities (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 computations. Compared with physically realistic values the
di€usivities (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 temperature pro®le
T …t ˆ 0†

17 ÿ 2 exp…ÿ…z ‡ 10†=5†; z P ÿ 10 m; 
ˆ
… C†
5 ‡ 10 exp……z ‡ 10†=20†; z < ÿ10 m;
…20†
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 di€usivities are largest. This is not ideal but was needed to
obtain stable computations. Numerical stability could
have been reached also with smaller vertical di€usivities,
but they would require a larger number of Chebyshev
polynomials and make computation time unduly long.

3. Substructuring

…m2 sÿ1 †;
…19†

Except for the tracer di€usivities these choices were
motivated and extensively discussed by Wang and
Hutter [12]. The expressions of the di€usivities indicate
the fact that for homogeneous waters the di€usivities

In Hutter and Wang [5] and Wang and Hutter [12]
computations with the semi-implicit SPEM were performed in which the resolution in the horizontal plane
was relatively coarse. For an arti®cial lake of
65 ´ 17 km2 and 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

404

Y. Wang, K. Hutter / Advances in Water Resources 23 (2000) 399±425

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 dicult to achieve in complex geometries. In such cases, to attain a uniform grid as far as
possible, a bounding line, which deviates in some segments from the actual lake boundaries, is used for
conformal mapping. In these segments the actual
boundaries can only be approximated by a step function. This will be the case for Lake Constance. Obviously, 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 di€erent methods.
Either one uses the domain decomposition method. In
this approach, the physical domain or grid is decomposed into a number of overlapping or non-overlapping
subdomains on each of which an independent incomplete factorization can be computed and applied in
parallel. In the subdomain where higher resolution is
needed a particularly ®ne grid is selected, while a successively 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 di€erent methods
within di€erent 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 suciently accurate results over most part of the integration domain,
but if a subregion exists where this is not so, a subdomain containing this subregion can be selected. Two
kinds of boundaries that exist in the subdomain containing 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 boundary which 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 substructuring, 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 suciently 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 and Uberlinger
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 the Uberlinger
See as a whole, the middle portion of
the Obersee and a near shore region between Romanshorn and Rorschach.
In principle, each attempt to estimate the discretization error in a numerical method is a comparison of
results obtained with di€erent 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 distribution of the ®eld variables with 12 and 30 polynomials
for homogeneous and strati®ed water was sucient 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 increase 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 inter-grid boundary of the subdomain within which
computations are being repeated with the ®ner resolution must for each time step be stored.
· In step two a new and ®ner net of grid points is generated in the horizontal projection of the subdomain.
This is done by considering the closed polygonal

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405

Fig. 1. (a) Map of Lake Constance with bathymetry and a few towns indicated along its shore. Three basins characterize the lake: Obersee,


Uberlinger
See and Untersee. (b) Coordinate net for the Obersee and the Uberlinger
See with the ``computational'' shore line indicated as thick line.

curve de®ning the boundary of this coarse grid projection and using the conformal Schwarz±Chrysto€el
transformation to generate a new and ®ner net of orthogonal 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 boundaries, 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 typical meteorological 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 bathymetric lines indicating its depth. This Alpine lake borders 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, respectively. 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 coordinate 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 possible.
4.1. Global results
Consider Lake Constance under homogeneous conditions 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 accompanied 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

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Fig. 2. 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 (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).

the e€ects 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 increases with depth below the surface so that the ¯ow is
clearly against the wind at 40 m depth.
Time series of the horizontal components u; v, of the

water velocity at two mid-basin positions of the Uberlinger 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 oscillations 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 the Uberlinger
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 e€ects of the boundaries prevent
the development of these oscillations. At the western

most point in the Uberlinger
See the current pattern is
built-up within the ®rst few hours and then levels o€ at
constant values which are reached after approximately

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407

Fig. 3. Homogeneous Lake Constance. Time series of the horizontal velocity components u; v at 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).

20 h (Fig. 5(a) and (b)). At the position east of Rorschach (Fig. 5(c) and (d)), 500 m from shore steady
conditions have not been attained within the ®rst four
days as the y-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 e€ects of
the inertial motion, one only needs to analyse the data at
an o€-shore position as indicated in Fig. 6, which displays the time series of the horizontal velocity components 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 resolution 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 transformation; 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

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Y. Wang, K. Hutter / Advances in Water Resources 23 (2000) 399±425


Fig. 4. Homogeneous Lake Constance. Same as Fig. 3 but for two midbasin positions in the Uberlinger
see, 9 km (panels a, b) and 5 km (panels c, d)
from the western end, respectively.

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 velocities 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 sucient to accurately model
barotropic wind-induced currents is to compare time
series of the horizontal velocity components for locations 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 horizontal velocity components are virtually identical with

those of Fig. 3 so there is no need to show them separately. The results prove that the global analysis with the
65 ´ 17 grid points accurately reproduces results at
midlake positions of the Obersee.
 berlinger See
4.3. Substructuring of the U

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 the Uberlinger
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 curvilinear coordinates used for it. This domain is special insofar as the ®eld variables of the global analysis must
only be taken over at grid points of the eastern

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409

Fig. 5. Homogeneous Lake Constance. Time series of the horizontal velocity components in the x-direction (left) and y-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 the Uberlinger
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.

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

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Y. Wang, K. Hutter / Advances in Water Resources 23 (2000) 399±425

Fig. 6. Homogeneous Lake Constance. Time series of the horizontal velocity components u; v at a position 3 km o€-shore from Arbon as indicated in
the inset map for various depths and an impulsively applied spatially uniform wind from 305° NW. The labels (1, 2, 3,. . ., 11) denote depths at (0, 10,
20; . . . ; 100) m.

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 symbol s indicates the positions at which time series of the horizontal velocity
components are plotted in Fig. 3.

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 conditions) 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 currents 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.

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411

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).

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); di€erences
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, di€er quantita-

tively, but not qualitatively, 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 computations of tracer di€usion. To this end consider the same
meteorological scenario with an impulsively applied
spatially uniform and temporally constant wind from

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Y. Wang, K. Hutter / Advances in Water Resources 23 (2000) 399±425

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
positions at which time series of the tracer concentration are displayed in Figs. 12 and 13, whereas the symbol s indicates the positions at which time
series of the horizontal velocity components are plotted in Figs. 4 and 5ab and Fig. 11ab.

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 m2 with a ¯ow rate of 20 mg mÿ2 sÿ1 .
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 latter repeats these results as obtained with the substructuring 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
di€erent from one another, however a closer look discloses signi®cant di€erences in particular close to the
location of tracer input. For instance, while the temporal 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 di€er
somewhat. At the locations south and south-east of the
central source point di€erences between the two computations 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 di€erent at early
times while at depth such di€erences 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 southeast 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 curvilinear 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 approximation 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 boundary 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 results as obtained with the global coarse-grid computations. Details of the ¯ow structure with the strong nearshore current are now very clearly visible as is the strong
northward current at the eastern shore.

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413

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 sÿ1 as
indicated in the inset.


Fig. 11. Homogeneous Lake Constance. Time series of horizontal velocity components at a position 540 m east of the western end of the Uberlinger
See. Conditions are the same as those in Fig. 5(a) and (b) but results have been obtained using substructuring for the subdomain.

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Y. Wang, K. Hutter / Advances in Water Resources 23 (2000) 399±425


Fig. 12. Homogeneous Lake Constance. Time series of the tracer concentration for the positions shown in the inset in the middle of the Uberlinger
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 time t ˆ 0 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 m2 situated 5 km east of the
western end with a ¯ow rate of 20 mg mÿ2 sÿ1 , (see also inset in Fig. 9). The labels (1, 2, 3; . . . ; 11) indicate the depths (0, 10, 20; . . . ; 100) m.
Computations were performed globally with the coarse grid shown in Fig. 1.

Fig. 16 displays graphs of the time series of the horizontal 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 substructuring technique. The corresponding results obtained with the global coarse-grid analysis are displayed

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415


Fig. 13. Homogeneous Lake Constance. Same as Fig. 12, but computations were now performed within the subdomain ``Uberlinger
See'' employing
substructuring techniques.

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 horizontal 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 di€er only
slightly (see Fig. 5(e) and (f) and Fig. 16(c) and (d)). At

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Y. Wang, K. Hutter / Advances in Water Resources 23 (2000) 399±425

Fig. 14. Location of the subdomain indicated by a thick solid line within the bathymetric chart of Lake Constance (a), and selection of the curvilinear
coordinates leading to the grid points as indicated in (b). In (b) the shadow shows the area over which the tracer is released. The symbol + indicates
the positions at which time series of the tracer concentration are displayed in Fig. 17, whereas the symbol s indicates the positions at which time
series of the horizontal velocity components are plotted in Fig. 5(c)±(f), Fig. 6(a) and (b) and Fig. 16.

the third location that is closest to the shore (560 m from
shore, Fig. 5(c) and (d) and Fig. 16(a) and (b)) di€erences 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 pro