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Topographic RossbyWaves in Basins of Simple Geometry

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Physics of Lakes

Part of the book series: Advances in Geophysical and Environmental Mechanics and Mathematics ((AGEM,volume 2))

Abstract

In the last chapter, topographic Rossby-waves on the f-plane were studied with emphasis of their mathematical description as extracted from the governing equations of fluid mechanics. Their possible observation by synoptic measurements was also discussed: they pertain to horizontal velocity and temperature-time series from moored thermistor chains and current recorders. It was shown by appropriately scaling the adiabatic Boussinesq approximated equations that in lakes with shallow epilimnion and deep hypolimnion – more specifically lakes which satisfy the so-called Gratton-scaling – the barotropic-baroclinic coupling is one-sided from the barotropic to the baroclinic TWs but not vice versa. In other words, if a topographic wave or a free or wind-induced oscillation in a lake, whose spectral component can be associated with a barotropic topographic wave mode, is acting in a lake, then this spectral component exerts a sizeable effect on the vertical baroclinic water movement which is (in principle) measurable in isotherm–depth–time series. Conversely, a baroclinic wave signal has a negligible influence on the barotropic TW response. This implies that for all those lakes whose geometry and stratification falls into the range of Gratton’s scaling – most Alpine lakes satisfy this scaling – the spectral structure can be found from the spectral analysis of the TW-operator, yet observational inferences can be drawn not only from cross-correlation analyses of moored current meters but equally also from such analyses involving isotherm–depth or temperature–time series.

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Notes

  1. 1.

    The thalweg of an elongated lake is the line which follows the deepest points of the basin cross sections. For simple shapes and simple bathymetries (ellipses with parabolic bottom surface, rectangles, etc.) the thalweg can readily be defined. For arbitrary basins the thalweg cannot be defined this way. In those cases, the lake axis is a line roughly defining the middle between opposite shores.

  2. 2.

    With \({\psi }_{ij} = {\psi }_{ij}^{(0)}{\mathrm{e}}^{-\mbox{ i}\sigma t}\) (i, j = 0, 1), (20.36) can be transformed to a homogeneous linear system for ψ ij (0) which possesses a solution provided that its determinant vanishes. Equation (20.37) makes this determinant to vanish.

  3. 3.

    Equation (20.48) implies that \(B = -A\tan (\lambda {\xi }_{S})\), and then (20.49) yields \((\tan (\lambda {\xi }_{I}) -\tan (\lambda {\xi }_{S})) = 0\), or tan(λξ S ) = tan(λξ I ) so, \(\lambda {\xi }_{S} = (\lambda {\xi }_{I} + n\pi ),\,n = 1, 2,\ldots \).

  4. 4.

    Clearly, in elliptical coordinates ξ ≥ 0. Continuity of a quantity ϕ(ξ, η) ‘across’ ξ = 0 means \({\lim }_{\xi \downarrow 0}\phi (\xi , 2\pi - \eta ) {=\lim }_{\xi \downarrow 0}\phi (\xi ,\eta ),\; 0 < \eta < 2\pi \).

  5. 5.

    A source where second class waves in the ocean are studied is LeBlond and Mysak [19]. Additional works are e.g. by Allen [2], Brink [5,  6], Djurfeldt [9], Gratton [10], Gratton and LeBlond [11], Huthnance [13], Koutitonsky [16], Lie [20], Lie and El-Sabh [21], Mysak [25,  26], Ou [29], Takeda [40] and others.

  6. 6.

    [​[ϕ(y)]​] at y = s denotes the jump of the quantity ϕ defined by

    $$[\![\phi (s)]\!] {=\lim }_{\epsilon \downarrow 0}(\phi (s + \epsilon ) - \phi (s - \epsilon )).$$
  7. 7.

    Right-bounded means that the shallower region is to the right when looking into the direction of phase propagation.

  8. 8.

    Because of its significance h′ ∕ h is often referred to as slope parameter S ≡ h′ ∕ h.

  9. 9.

    See the transformation formulae (20.12) and Fig. 20.1 and compare the TW-equation in Cartesian, (20.5) and (20.6), and in elliptical (20.12) coordinates.

  10. 10.

    An identical equation as that for F can also be obtained for G if

    $$\begin{array}{rcl} \chi = - \frac{1} {{\eta }_{R} - {\eta }_{L}}\eta - \frac{{\eta }_{R} + {\eta }_{L}} {2({\eta }_{R} - {\eta }_{L})} + \frac{2\pi } {{\eta }_{R} - {\eta }_{L}}& & \\ \end{array}$$

    is chosen.

  11. 11.

    The text below follows Stocker [38] with some minor changes.

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Authors and Affiliations

  1. c/o Versuchsanstalt für Wasserbau Hydrologie und Glaziologie ETH-Zentrum, ETH Zürich, Gloriastr. 37/39, 8092, Zürich, Switzerland

    Prof. Dr. Kolumban Hutter

  2. Department of Mechanical Engineering, Darmstadt University of Technology, Petersenstr. 30, 64287, Darmstadt, Germany

    PD. Dr. Yongqi Wang (Chair of Fluid Dynamics)

  3. P.P. Shirshov Institute of Oceanology, Russian Academy of Sciences, prospect Mira 1, 236000, Kaliningrad, Russia

    Dr. Irina P. Chubarenko

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  1. Prof. Dr. Kolumban Hutter
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  2. PD. Dr. Yongqi Wang
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  3. Dr. Irina P. Chubarenko
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Correspondence to Kolumban Hutter .

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Hutter, K., Wang, Y., Chubarenko, I.P. (2011). Topographic RossbyWaves in Basins of Simple Geometry. In: Physics of Lakes. Advances in Geophysical and Environmental Mechanics and Mathematics, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19112-1_20

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