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Mathematical modeling
of thermal bar in Lake Teletskoye
O.F.VASILIEV, V.I.KVON, D.V.KVON
Institute
for Water and Environmental Problems, SB RAS
Morskoy
Prospekt 2, Novosibirsk 630090, Russia
Phone/Fax:
+7 (3832) 356005, E-mail:vasiliev@ad-sbras.nsc.ru
ABSTRACT
The
formation and development of longitudinal and nearshore thermal bars are considered in the Lake Teletskoye (Altai
Mountains, west Siberia). This deep lake has an oblong form and the maximum
depth of 325 m. The 2-D vertical hydrodynamic models were developed to
simulate both types of thermal bars: a longitudinal one with lateral averaging
and a cross-sectional one, for the first and second cases respectively. The
model contains the dynamic equation, continuity equation, heat transfer equation
and the equation of state for water. The model of turbulence is based on the equations for kinetic energy of turbulence and rate of its
dissipation. The model takes into account of water compressibility,
meteorological factors and the through - flow along the lake due to river
inflow and outflow. The account of water compressibility allows more precisely
to describe thermal stratification in a deeper part of lake. Numerical
calculations were performed to portray the thermal structure of the lake in a
spring-summer period. Comparison of the numerical results with the field
measurement data has shown that the mathematical model describes well the main
features of the hydrothermal regime of the lake including the formation of
specific longitudinal thermal bar. The peculiarities
of development longitudinal and nearshore
thermal bars are compared.
Keywords:
lake hydrology, lake hydrodynamics, lake hydrothermal behavior, numerical
modeling of lakes
INTRODUCTION
The
Lake Teletskoye is one of the largest water bodies of Siberia situated in the
Altai Mountains. The lake has an oblong form, his length is 77.8 km, the
maximum width is 5.2 km and the
maximum depth of 325 m [1]. It has a through-flow along it length. The
main part of inflow comes to the lake from the Chulyshman river in the southern
part of the lake and the outflow takes place through the Biya river in the
opposite northwestern part. Because of the significant length and depth of the
lake and the relatively small width the main changes in the season dynamics of
its thermal structure are related to the longitudinal and vertical directions
(longitudinal thermal bar) ), though the development
of nearshore thermal bar is of interest too. Twice a year in spring-summer and
autumn-winter periods thermal bars are observed In lake, the front of which
being moved to the central part of lake from both ends in longitudinal
direction [1]. The duration of spring-summer thermal bar is approximately from 1.5 to 2 months. In theoretical studies
of thermal bar (see, for example, [2]) the water density is usually accepted as
dependent only upon temperature.
MATHEMATICAL MODEL
The
basic hydrodynamic equations averaged along a width of lake is used here, for
simulation of a longitudinal thermal bar in the Lake Teletskoye. The
compressibility of water is taken into account because of the significant depth
of lake. The system of basic equations includes the dynamic equation,
continuity equation, of heat transfer equation and the equation of state with
taking account of compressibility of water are as following [3,4]:
, (1)
, (2)
, (3)
, (4)
where 
= longitudinal coordinate, directed from the Biya
River to the Chulyshman River; 
= vertical coordinate; 
= longitudinal flow velocity; 
= vertical flow velocity; 
= hydrostatic pressure; 
= lateral width; 
= the fluid density; 
= a referential fluid density; 
= rate of lateral inflow; 
= lateral inflow temperature; 
= deviation of the water surface elevation from
its undisturbed level 
; 
= parameter chracterised the shape of shore slopes
and 
= shear stress factor for these surfaces. The
equation of state for water in the form (4) is recommended by the UNESCO joint
panel on oceanographic tables and standards [5,6]. In this equation 
= density of water at standard atmospheric
pressure; 
= the volumetric module of compressibility of
water; 
= salinity of water, accepted here as constant and
equal 75 mg/l.
The
initial and boundary conditions must be adjoined to the set of equations (1) -
(4). On water surface the usual kinematics conditions, wind shear stress and
heat flux are specified. At the bottom the shear stress is assumed to be
proportional to value of near-bottom velocity squared and zeroth heat flux is
settled. At the inlet section where the River Chulyshman inflows the water
discharge, velocity and the temperature inflowing water are given. At the
outlet section of the lake where the River Biya outflows from it, the water
discharge (or the relationship between discharge and water level known as a
rating curve) and some other conditions are set.
The lateral
inflows q was defined as the delta - functions in the inlet points. The wind
shear stress and the heat flux though the water surface were calculated by the
known formulas, with using of the flux of solar radiation and other
meteorological data (wind velocity, air temperature and humidity, atmospheric
pressure and cloud cover).
The
vertical eddy viscosity and eddy diffusivity are defined with using the model
of turbulence based upon the equations of kinetic energy of turbulence and rate
of its dissipation[7]. The horizontal eddy viscosity and eddy diffusivity are
defined using the Richardson formula [8].
THE NUMERICAL RESULTS
The model
is realized using semi-implicit finite difference scheme based on splitting of
the physical processes: at the first fractional step the momentum transport is
performed by advection and diffusion and at the second fractional step the
hydrodynamic fields adaptation is simulated [4].
The
simulations of the thermal structure of Lake Teletskoye were performed for the
time interval since May 20 till August 30 1968 because that year measurements
are the most completely presented in the literature. The initial conditions
correspond to state of rest and to an
uniform distribution of temperature in the lake (its value assumed to be equal
to 2.3 oC according to observations [1]).
According to
the observation data [1], the characteristic feature of the hydrothermal
behavior of Lake Teletskoye is the formation of thermal bars arising at its
ends. Then their fronts are moving along the lake to its central part
(longitudinal thermal bar). This phenomenon is conditioned by faster heating of
water in the areas with smaller depth at the ends of oblong lake and by the
temperature differences between these areas and the central part of lake where
at that period the water temperature is less than 4 oC. In the
northwestern part of the lake the water is warmed more rapidly because of the
relatively small depth in this part, and in the southern part the water warming
is aided also the warmer water inflow from the Chulyshman River. In the zones
of cold and warm water mixing (in the vicinities of the thermal bar fronts)
where the water temperature reaches the value of maximal density 4 oC,
the thermal gravity convection with the down-flows is developed.
The
results of the simulation of longitudinal thermal bar are in accordance with
the measurements data [1]. The thermal gravity convection flows which arise
near the fronts of thermal bar can carry oxygen-rich surface waters to the
deeper layers of the lake. Both the measurements data and the simulations
results shows that though the Biya outflow current carries out the warm water
from northwestern part of the lake, the thermal bar front moves there in the
opposite direction to the central part of the lake. The simulation is
described the thermal bars formation at
the both ends of the lake and moving their fronts to the central part of the
lake with the following confluence of those there in accordance with observed
data. According to the measurements data «by the July 15 the fronts of the
thermal bar are closed up and the lake becomes uniform» [1] in the longitudinal
direction. The time of the thermal bars closing up is not detected precisely in
the measurements data. According to the numerical results, the closing up takes
place at July 11, and the uniform stratification along the length of the lake
(or, at least, along a significant part
of lake) establishes somewhat near July 13 - 15.
Figure 1: Temperature distribution in the longitudinal - vertical section of lake for the moments of shaping of the thermal bar fronts (June 20), before the closing up of fronts (July 8) and after it (July 15).
In the
Figure 1 the locations of thermal bar fronts much earlier the closing up (June
20), in the moment near the closing up (July 11) and after it (July 15) are
shown. We remind, at the figure the longitudinal-vertical section of the lake
is shown. The Chulyshman River inflows to the lake at the right (the southern
part) and the Biya River outflows at the left (the north-western part).
According
to the observations [1] in formation of Lake Teletskoye thermal structure the
main role belongs to the development of longitudinal thermal bar. The nearshore
thermal bar plays a less significant
role.
Figure 2: Development the nearshore thermal bar in a cross - section area of lake (the stages on July 10, 15 and 20 are shown).
In
Figure 2 the development of the nearshore thermal bar in a cross-section area
of lake in the deepest central part of lake is shown. The closing up of the
fronts of the thermal bar, moving from both shores to the center of a
cross-section could happen much later, than the closing up of the longitudinal
thermal bar fronts. The distribution of temperature in this section is almost
homogeneous for a long period of time, its value in the major part of the
section being changes from 2.3 C on May 20 (after the turn-over of waters
in the lake) to 3.6 C on July 15 ( at the moment of closing up of the
longitudinal thermal bar fronts). Thus, though the transverse distance is much
smaller than the longitudinal to cover by the fronts of two types of thermal
bar respectively, the front of the longitudinal one arrives to the meeting
point much earlier rather than the front of transverse one.
CONCLUSIONS
(1)
The numerical results and its comparison
with the data of observations in the lake had shown. The mathematical model
describes the main features of a thermal regime of Lake Teletskoye, including
the formation of specific longitudinal thermal bar, the propagation of its
fronts and their closing up.
(2)
A preliminary analysis of the formation
and propagation of the nearshore thermal bar is carried out. The comparison of
time-scale for both types of thermal bars are given.
ACKNOLEGEMENT
The
authors acknowledge with thanks the financial support of the Russian Foundation
for Basic Research given to carry out this study (Project N 96-01-01940).
REFERENCES
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