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.
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Notes
- 1.
For a parabolic velocity profile in the interval \(0\leqslant z \leqslant H\) of the form
$$\begin{aligned} u = {\frac{3}{2}}\bar{u}(2\zeta - \zeta ^{2}) \quad \zeta := z/H \end{aligned}$$we have \(u(0) = 0, u^{\prime }(H) = 0\) and \(\bar{u} = {\frac{1}{H}}\int _{0}^{H}u(z){\text {d}}z\) and obtain after an easy computation
$$\begin{aligned} \frac{1}{H}\int _{0}^{H}\left( \frac{\partial u}{\partial x}u\right) (z){\text {d}}z = 1.20\frac{\partial \bar{u}}{\partial x}\bar{u} > \frac{\partial \bar{u}}{\partial x}\bar{u}, \end{aligned}$$which shows that the convective term for the parabolic velocity profile is larger by a factor of 1.20 than the corresponding term for a uniform velocity profile.
- 2.
Substitute (23.4) into (23.1)\(_{2,3}\), differentiate (23.1)\(_{2}\) with respect to \(y\) and (23.1)\(_{3}\) with respect to \(x\) and subtract the emerging equations. The resulting equation will be a nonlinear partial differential equation for \(\varPsi \) subject to the no-flux condition along the shore,
$$\begin{aligned} \varPsi = 0\quad {\text {along}}\quad {{\mathcal {C}}}(x, y) = 0. \end{aligned}$$ - 3.
Note that in (23.5)\({}_1\) \(u_{\text {wind}}\) should actually be \((u_{\text {wind}}- u_{\text {water}})_{\text {surface}}\), etc., but since the wind velocity is generally much larger than the water velocity at the surface, the latter is ignored.
- 4.
Note that the mean layer thicknesses \({\frac{1}{2}}(h_{k+1}^{x,y} + h_{k}^{x,y})\) are in the lake interior the same constant. Only close to the shore \(h_{k+1}^{x,y}\) and/or \(h_{k}^{x,y}\) may be smaller than their corresponding layer thicknesses, so that a non-constant mean layer thickness results.
- 5.
Recall that the steady solutions of the topographic wave operator lead to stream lines which follow lines of constant \(H/f\), where \(H\) is the water depth and \(f\) is the constant Coriolis parameter. We shall not show graphs here; for details see [3].
- 6.
Abbreviations
- 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
- \(\varDelta _{\mathrm {Adv}} = \frac{L}{V_0}\) :
-
Advective time scale
- \(\varDelta _{\mathrm {Diff}} = \frac{H_{0}^{2}}{4D}\) :
-
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
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Hutter, K., Wang, Y., Chubarenko, I.P. (2014). Barotropic Wind-Induced Motions in a Shallow Lake. In: Physics of Lakes. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-00473-0_23
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