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Topographic Waves in Enclosed Basins: Fundamentals and Observations

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Physics of Lakes

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

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Abstract

In Sect. 11.2, the notions of first and second class waves were introduced. The former were said to be due to the action of the gravity force. These waves are therefore also called gravity waves. The latter are due to the rotation of the Earth and cease to exist when the frame of reference is inertial. These waves are alternatively also termed Rossby-, vortex-, geostrophic or gyration-waves, see Fig. 19.1.

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Notes

  1. 1.

    For the derivation of (19.1), see Sect. 11.3 and for a first attempt of interpretation Sect. 9.2 of Chap. 9 in Volume I. An elegant derivation follows from the conservation law of barotropic potential vorticity, which requires (see (5.74)–(5.79) in Chap. 5 in Volume I of this book series)

    $$\begin{array}{rcl} \frac{\mathrm{d}{\Pi }_{bt}} {\mathrm{d}t} = \frac{\mathrm{d}} {\mathrm{d}t}\left (\frac{\zeta + f} {H} \right ) = 0,& & \\ \end{array}$$

    in which ζ = ω z  = ∂v ∕ ∂x − ∂u ∕ ∂y = E[Ψ]. Therefore,

    $$\begin{array}{rcl} \frac{1} {H} \frac{\partial {\omega }_{z}} {\partial t} + \frac{\partial } {\partial x}\left (\frac{{\omega }_{z} + f} {H} \right )u + \frac{\partial } {\partial y}\left (\frac{{\omega }_{z} + f} {H} \right )v = 0,& & \\ \end{array}$$

    which, after linearization, is equivalent to (19.1).

  2. 2.

    For a biographical sketch see Fig. 19.1.

  3. 3.

    There is a vast literature on TWs. A reference text may be LeBlond and Mysak [27] who treat primarily waves in the open ocean. A review, perhaps more adequate to the present topic is by Stocker and Hutter [42] and contains a large number of references pertaining to the propagation of TWs in closed or semi-closed basins. Relevant works are also by Stocker and Hutter [40,  43,  44] and Johnson [20,  21,  22], Willmott and Johnson [50], Johnson and Kaoullas [23] and others.

  4. 4.

    Mysak et al. [33, p. 52] list six arguments in support of the TW-model and only the above mentioned discrepancy in the phase relation against it.

  5. 5.

    With \(f = 1{0}^{-4}\,\mathrm{{s}}^{-1},\;g\prime = 0.02\,\mathrm{{ms}}^{-2},\;{D}_{1} = 10\,\mathrm{m},\;{D}_{2} = 270\,\mathrm{m}\) and \({\psi }_{0} = U \cdot L \cdot ({D}_{1} + {D}_{2}) = 0.03 \cdot 1{0}^{4} \times 270\,\mathrm{{m}}^{3}\,\mathrm{{s}}^{-1}\), where U is a velocity scale (approximately 3 cm s − 1 for Lake of Lugano) and L = 104 m, one obtains \({\zeta }_{0} = 40\,\mathrm{m}\).

  6. 6.

    With f = 10 − 4 s − 1,  g′ = 0. 02 ms − 2,  ψ0 = 7 ×104 m3 s − 1 and D 2 = 270 m, (19.21) yields ζ0 = 1 m. Alternatively, using τ0 = ρair c d U w 2 with ρair = 1. 29 kg m − 3,  c d  = 1. 85 ×10 − 3 (an average value for lakes during summer, see Simons [39], p. 92), and U w  = 4 ms − 1, we find τ0 = 0. 038 Nm − 2 and hence according to (19.21), ζ0 = 1. 5 m and according to (19.20), U = τ0 ∕ (ρ1 fD 1) = 3. 8 cm s − 1. Both values are typical observations in Lake Zurich and Lake of Lugano, see Table 19.3.

  7. 7.

    This solution in an elliptic basin with parabolic bottom is constructed in Chap. 20, Sect. 20.3.3.

  8. 8.

    These equations actually imply a statement regarding \(\mathcal{L}\prime ({D}_{1}\vec{{u}}_{1} + {D}_{2}{h}_{2}\vec{{u}}_{2}) \cdot \hat{\vec{ n}}\) rather than the mass transport itself. However, if \(\mathcal{L}g = 0\) along \(\partial \mathcal{D}\) for all time, then necessarily g = 0 as well.

  9. 9.

    Recall that \((\hat{\vec{k}} \times \nabla \psi ) \cdot \nabla {h}^{-1} = 0\) for all times implies that \({\mathcal{L}}^{-1}\{(\hat{\vec{k}} \times \nabla \psi ) \cdot \nabla {h}^{-1}\} = 0.\)

  10. 10.

    We assume the reader to be familiar with the method of the Principle of Weighted Residuals, see e.g. Finlayson [8]. The principle or method of weighted residuals (MWR) is based on the following mathematical equivalence: Let \(f(\vec{x})\) be a function or functional whose value vanishes for all \(\vec{x} \in \mathcal{D}\subset {\mathbb{R}}^{N}\):

    $$\begin{array}{rcl} f(\vec{x}) = 0\quad \mbox{ for all }\vec{x} \in \mathcal{D}\subset {\mathbb{R}}^{N}.& & \end{array}$$
    (19.53)

    Let, moreover, \(\delta {\phi }_{\alpha }(\vec{x})\) be an arbitrary bounded function from a set {δϕα, α = 1, 2, …}. Obviously, (19.53) implies

    $$\begin{array}{rcl} {\int \nolimits \nolimits }_{\mathcal{D}}\delta {\phi }_{\alpha }(\vec{x})f(\vec{x})\mathrm{d}\vec{x} = 0\qquad (\alpha = 1, 2,\ldots ).& & \end{array}$$
    (19.54)

    If (19.54) holds for any complete set of δϕα, then (19.54) also implies (19.53). This equivalence statement lies at the heart of the MWR.

  11. 11.

    An estimate for [P ∕ ρ ∗ ] is obtained as follows: Under hydrostatic conditions the last of (19.51) suggests that [P ∕ ρ ∗ ] ∼ (Δρ ∕ ρ)g[D], where Δρ ∕ ρ is the density anomaly and [D] a typical metalimnion thickness: Thus, with Δρ ∕ ρ ∼ 10 − 3 and [D] ≤ 10 m this yields [P ∕ ρ ∗ ] ≤ 10 − 1m2 s − 2, implying \(\mathbb{B} \leq 1{0}^{-1}\).

  12. 12.

    This argument can even be made more forceful by recognizing that according to (19.58)3 an estimate for H 1 is 10 m (n = 1) so that λ1 = 10 − 3 ×104 ∕ 102 = 10 − 1. Consequently, one baroclinic–barotropic coupling term is about a factor of 100 larger than the corresponding barotropic–baroclinic term.

  13. 13.

    Here, we use the definition ⟨⟨f, g⟩⟩ =  ∫ − H ζ fgdz.

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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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Correspondence to Kolumban Hutter .

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Hutter, K., Wang, Y., Chubarenko, I.P. (2011). Topographic Waves in Enclosed Basins: Fundamentals and Observations. 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_19

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