# Barotropic Wind-Induced Motions in a Shallow Lake

Chapter
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)

## Abstract

In this chapter the intention is to describe the horizontal velocity distribution in a homogeneous lake by the spatially three-dimensional dynamical equations, based on the hydrostatic pressure assumption on the one hand, and their spatially two dimensional depth integrated reduction on the other hand. Comparison of the two sets of solutions for wind forcing, uniform in space and Heaviside in time, from various directions discloses the conditions when the depth averaged equations likely yield valid approximations of the three dimensional situation. Lake Zurich is used as an example. The extensive computations reveal that the problem of approximate determination of the barotropic velocity distribution in a homogeneous lake needs careful scrutiny. We shall analyze this problem by applying layered versions of the equations of motion in the hydrostatic pressure assumption to Lake Zurich and comparing the wind-induced current distribution obtained for a number of wind scenarios of a one layer and an eight-layer model. It may be deduced that depth integrated models deliver horizontal currents in homogeneous lakes of extremely shall depths (ca 5 m) only.

## Keywords

Advective Term Shallow Part Shear Traction Horizontal Velocity Component Shallow Basin
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

$$A, A_0$$

Mean horizontal momentum Austausch coefficient

$$\mathcal {C}(x,y)$$

Boundary loop of the lake surface

$$c_D$$

Wind drag coefficient

$$f$$

Coriolis parameter

$$g$$

Gravity constant

$$H$$

Water depth of the lake at rest

$$h_k$$

(Constant) thickness of layer $$k$$

$$k$$

Identifier of layers

$$p^\mathrm {atm}$$

Atmospheric pressure at lake surface

$$Q_{\mathcal C}^{k}(x,y,t)$$

Volume flux through the side boundary of layer $$k$$

$$R = \frac{\varDelta _{\mathrm {Diff}}}{\varDelta _{\mathrm {Adv}}} = \frac{H_{0}^{2}V_{0}}{4DL_{0}}$$

Parameter estimating, whether depth integrated barotropic models can be applied

$$r \approx 2\times 10^{-3}$$

Basal friction drag coefficient

$$R^{(x)}, R^{(y)}$$

Basal friction stress components in $$(x,y)$$ directions

$$(T_{xy}, T_{yz})_{k+1/2}$$

Horizontal shear stresses at the interface between layers $$k$$ and $$k+1$$

$$t_{x}^{b}, t_{y}^{b}$$

Bottom friction tractions in $$(x,y)$$ directions

$$U, V$$

Horizontal volume flux in $$(x,y)$$ directions

$$u, v$$

Depth averaged horizontal velocity components in $$(x,y)$$ directions

$$u^{\mathrm {wind}}, v^{\mathrm {wind}}$$

Horizontal wind velocity components in $$(x,y)$$ directions

$$u_k, v_k, w_k$$

Cartesian velocity components in layer $$k$$

$$W^{(x)}, W^{(y)}$$

Wind shear stress components in $$(x,y)$$ directions at the lake surface

Greek

Symbols

$$\varDelta _{\mathrm {Adv}} = \frac{L}{V_0}$$

$$\varDelta _{\mathrm {Diff}} = \frac{H_{0}^{2}}{4D}$$

Diffusive time scale

$$\varDelta _H$$

Horizontal Laplace operator

$$\zeta$$

Free surface displacement

$$\zeta = \frac{z}{H}$$

Dimensionless vertical coordinate

$$\nu _{k+1/2}$$

Vertical kinematic viscosity at the interface between layers $$k$$ and $$k+1$$

$$\rho$$

(Constant) water density

$$\varPsi$$

Volume transport stream function

## References

1. 1.
Hutter, K.:(Ed.) Hydrodynamics of Lakes, Springer, New York, NY (1984)Google Scholar
2. 2.
Hutter, K.: Mathematische Vorhersage von barotropen und baroklinen Prozessen im Zürich und Luganersee. Vierteljahresschrift der Naturforschenden Gesellschaft in Zürich, 129(1), 51–92 (1984)Google Scholar
3. 3.
Hutter, K., Oman, G. and Ramming, G.: Windbedingte Strömungen des homogenen Zürichsees. Mitteilung Nr 61 der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie an der ETH Zürich, pp. 1–123 (1982)Google Scholar
4. 4.
Hutter, K., Wang, Y. and Chubarenko, I.: Physics of Lakes Volume 1: Foundation of the Mathematical and Physical Background, Springer Verlag, Berlin etc. pp. 434 (2011)Google Scholar
5. 5.
Oman, G.: Das Verhalten des geschichteten Zürichsees unter äusseren Windlasten - Resultate eines numerischen Modells. Mitteilung Nr 60 der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie an der ETH Zürich, pp. 1–185 (1982)Google Scholar
6. 6.
Platzman, G. W.: The dynamic prediction of wind tides on Lake Erie. Meteorolog. Monographs, Am. Meteorol. Soc. 4(26), 44 p. (1963)Google Scholar
7. 7.
Ramming, H. G. and Kowalik, Z.: Numerical Modeling of Marine Hydrodynamics, Elsevier, Amsterdam, Oxford, New York, (1980)Google Scholar
8. 8.
Simons, T. J.: Circulation models of lakes and inland seas. Canadian Bulletin of Fisheries and Aquatic Sciences, Bulletin 203, Ottawa (1980)Google Scholar
9. 9.
Taylor, G. I.: Tidal friction in the Irish Sea. Phil. Trans. Royal. Soc. London, A220, pp. 1–33 (1919)Google Scholar

© 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 