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Response of a Stratified Alpine Lake to External Wind Fields: Numerical Prediction and Comparison with Field Observations

  • Kolumban HutterEmail author
  • Yongqi Wang
  • Irina P. Chubarenko
Chapter
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)

Abstract

The water motion in lakes during summer stratification is primarily generated by wind. It is chiefly influenced by the vertical density distribution, which leads to a strong interplay between the direct circulation dynamics and internal oscillations. In a storm-type wind set-up from rest the on-setting circulation flow also causes the formation of internal seiches, which, according to observations in numerous mountainous lakes, persist for several periods after wind cessation. On the basis of the hydrostatic and Boussinesq-approximated baroclinic non-linear hydrodynamic model, which is discretized by a multi-layered finite difference description, the baroclinic response of Lake Zurich to such a stormy scenario is predicted and compared with an episode, which was observed in Lake Zurich in a summer field campaign in 1978. A careful parameter study shows that the model appears to predict direct wind driven response in a qualitatively correct manner. The model also correctly predicts the formation and propagation of internal temperature surges characterized by an extremely fast falling thermocline. However, it seems to suffer from an excessive dissipation of energy, which causes the internal seiche to oscillate too slowly and die out too quickly. This loss of energy can be attributed to numerical diffusion of the temperature, which is needed to prevent convective instability in the model and avoid large dissipation of kinetic energy caused by the horizontal smoothing of the velocity field needed to control unrealistically large oscillations of the vertical component of velocity. Decreasing vertical eddy viscosity and increasing the vertical resolution near the thermocline reduces but does not eliminate this energy dissipation, some of which appears necessary to assure model stability.

Keywords

Vertical Velocity Internal Wave Horizontal Velocity Coriolis Force Numerical Diffusion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of Symbols

Roman

Symbols

\(A_0\)

Constant momentum Austausch coefficient, horizontal momentum diffusivity

\(A_1\)

Bennett’s areal diffusivity \(A_1 = \textstyle {\frac{1}{4}}\varDelta x\varDelta y\,f\)

\(b(x, y) = z\)

Equation for the lake bottom

\(C(|{\varvec{v}}|)\)

Basal drag coefficient

\(c\)

Phase velocity

\(c_p\)

Specific heat at constant pressure

\(c_{w}\)

Wind drag coefficient

\({\varvec{D}}\)

Strain rate tensor, stretching tensor \({\varvec{D}} = \textstyle {\frac{1}{2}}(\mathrm {grad}\,{\varvec{v}} + \mathrm {grad}\,^{T}{\varvec{v}})\)

\(\text {div}\)

Divergence operator

\(E\)

Total energy \(E = PE + KE\)

\(f = 2|\varOmega |\sin \phi \)

First Coriolis parameter

\(\tilde{f} = 2|\varOmega |\cos \phi \)

Second Coriolis parameter

\({\varvec{g}}, g\)

Gravity vector, gravity constant

\(\mathrm {grad}\,\)

Gradient operator

\(h\)

Specific enthalpy

\(h_1, h_2\)

Epilimnion thickness, hypolimnion thickness

\(h_k\)

Thickness of the \(k\)-th horizontal layer

\(k\)

Counting index

\(KE\)

Kinetic energy

\(\mathsf{M}\)

Fourth order viscosity tensor

\(p\)

Pressure

\(PE\)

Potential energy

\({\varvec{Q}}\)

Heat flux vector

\(Q^\text {atm}\)

Atmospheric heat flow at the lake surface

\({\varvec{Q}}^\text {lam}\)

Laminar heat flux vector \({\varvec{Q}}^\text {lam} = - \kappa \mathrm {grad}\,\,T\)

\({\varvec{Q}}^\text {turb}\)

Mean turbulent heat flux vector \({\varvec{Q}}^\text {turb} = -\rho \langle \,c_{p}T^{\prime }{\varvec{v}}^{\prime }\,\rangle \)

\((Q_{z})_{k+1/2}\)

\(z\) component of the heat flux vector at the interface between layer \(k\) and \(k+1\)

\(R\)

Rossby radius (of deformation)

\(Ri\)

(Gradient) Richardson number

\(s\)

Salinity, mineralization

\(\hat{S}_k \)

Buoyancy weight of the column above \(z = z_{k+1/2}\) \(\hat{S}_k \approx g\int _{z_{k+1/2}}^{0}\sigma (z)\text{ d }z\)

\(T\)

Temperature (Celsius or Kelvin)

\(T^{*}\)

Reference temperature (\(4\,^\circ \text {C}\))

\(T_{k}\)

Average temperature in layer \(k\)

\(T^\text {atm}\)

Atmospheric temperature at the lake surface

\(T_\text {base}\)

Temperature at the bottom of the lake

\({\varvec{T}}\)

Cauchy stress tensor

\({\varvec{T}}_E\)

Extra Cauchy stress tensor (deviator)

\({\varvec{T}}^{\prime }\)

Fluctuation of \({\varvec{T}}\)

\({\varvec{T}}_{E}^\text {turb} \)

Mean turbulent Cauchy stress deviator \({\varvec{T}}_{E}^\text {turb} \equiv \langle \rho {\varvec{v}}^{\prime }\otimes {\varvec{v}}^{\prime } - \textstyle {\frac{1}{3}}\text {tr}(\rho {\varvec{v}}^{\prime }\otimes {\varvec{v}}^{\prime }){\varvec{1}}\rangle \)

\({\varvec{T}}_{E}^\text {lam}\)

Laminar Cauchy stress

\(T_{ij}\)

Components of \({\varvec{T}}\) (\((i,j = 1,2,3)\))

\(t_{x}^\text {wind}, t_{y}^\text {wind}\)

Shear traction components of the wind stress at the lake surface

\(\text {tr}\)

Trace operator

\({\varvec{t}}_{|}\)

Tangential shear traction at the bottom surface

\( u, v, w\)

Cartesian components of the velocity vector

\(u_k, v_k\)

Average horizontal velocity components in layer \(k\)

\({\varvec{v}}\)

Velocity vector

\({\varvec{v}}^{\prime }\)

Turbulent velocity fluctuation

\({\varvec{v}}_{|}\)

Tangential water velocity vector at the bottom surface

\({\varvec{W}}\)

Wind velocity

\(w_{k+1/2}\)

Vertical velocity at the interface between layers \(k\) and \(k+1\)

\(x, y, z\)

Cartesian coordinates

Greek

symbols

\(\varDelta \rho \)

Density difference \(\rho _2 - \rho _1, \rho _2 > \rho _1\)

\(\varDelta \,x, \varDelta \,y, \varDelta \,z\)

Side lengths of the finite difference grid

\(\varphi \)

Angle of geographical latitude

\({\varvec{\kappa }}, \kappa \)

Tensor of heat conductivity, heat conductivity

\(\kappa _v\)

Vertical heat conductivity

\(\nu \)

Kinematic viscosity

\(\nu _v\)

Vertical kinematic viscosity

\(\rho \)

Density of water \(\rho = \rho (T, p, s)\)

\(\rho _0\)

Reference density (\(\rho = \rho (T \approx 30\,^\circ \text {C}\)))

\(\sigma = (\rho /\rho _0 - 1)\)

Density anomaly

\(\zeta (x,y,t) = z\)

Free surface displacement

\(\varvec{\varOmega }\)

Angular velocity of the Earth

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Kolumban Hutter
    • 1
    Email author
  • Yongqi Wang
    • 2
  • Irina P. Chubarenko
    • 3
  1. 1.c/o Laboratory of Hydraulics, Hydrology and Glaciology at ETHZürichSwitzerland
  2. 2.Chair of Fluid Dynamics, Department of Mechanical EngineeringTU DarmstadtDarmstadtGermany
  3. 3.P.P. Shirshov Institute of OceanologyRussian Academy of SciencesKaliningradRussia

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