Abstract
In Chaps. 7 and 8 of Volume I, an introduction was given to the mathematical treatment of linear waves in general and to water waves in particular. To isolate the specific properties of water waves with a free surface, the influence of the rotation of the reference frame was not considered. Here, our aim is to elucidate the role played by the rotation of the reference frame – the Earth – in the dynamics of large water masses such as ponds, lakes, and the ocean.
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Notes
- 1.
For a portrait of Coriolis and a short biography, see Volume I, Fig. 4.5, p. 98.
- 2.
At this point of the development it may not become clear to the reader, why the parcel should be able to be replaced by a wave-packet. However, in a first approach, we may accept this as an analogy and investigate its consequences. Later formulas will corroborate this.
- 3.
The classical definition of adiabaticity is that a body is thermally isolated from its environment. This means that in this case heat flux and radiation sources are zero. Our weaker interpretation includes in the definition all those terms which are associated with dissipation, through species, momentum, and heat diffusion.
- 4.
The physical interpretation of p 0 is atmospheric pressure at z = ζ which obviously is a function of x, y and t : p 0 = p 0(x, y, t). However, typical length scales over which atmospheric pressures vary, are, in general, larger than the horizontal extent of common lakes, implying that p 0 may be assumed to be constant. When atmospheric fronts move across a lake, then p 0 should not be treated as constant. In this more general case atmospheric pressure gradients would enter (11.15) and (11.17) as driving forces of the motion.
- 5.
Actually, with the linearized boundary condition (11.6), ∂ζ ∕ ∂t = w, at z = 0 and the full boundary condition (11.7), (11.16) reads
$$w(x,y,z,t) = \frac{(z + H)} {H} \,\,\frac{\partial \zeta (x,y,t)} {\partial t} + \left [ \frac{1} {H} \frac{\partial H} {\partial x} \frac{U} {H} + \frac{1} {H} \frac{\partial H} {\partial y} \frac{U} {H}\right ]z$$in which the second bracketed is dropped in the shallow water approximation.
- 6.
This statement assumes that of the three solutions for the wave frequency all are real. Without explicit analysis this is not guaranteed. We here anticipate this fact.
- 7.
Later, we shall also introduce an internal Rossby radius of deformation. For a two-layer system (upper layer depth H 1, density ρ1; lower layer depth H 2, density ρ2) it is given by
$${R}_{\mathrm{D}} = \frac{\sqrt{g\prime{H}_{\mathrm{E}}}} {{f}_{0}},\,\,\,\,\,g\prime = g\,\frac{{\rho }_{2} - {\rho }_{1}} {{\rho }_{2}},\,\,\,\,\,{H}_{\mathrm{E}} = \frac{{H}_{1}{H}_{2}} {{H}_{1} + {H}_{2}}.$$This internal Rossby radius is exactly what we have introduced in (11.3), (11.4) in the baroclinic case with methods of dimensional analysis; it applies for a stratified fluid in two layers and is much smaller than its external counterpart. So, whereas in certain lake basins the effects of the rotation of the Earth may be ignored for surface waves this is not so for internal waves. We shall come back to this property later on.
- 8.
An exception may be the Caspian Sea which has a very deep ( > 1, 000 m) Southern part for which R D ≃ 1, 000 km and L 0 ≃ 250 km and a shallow middle and Northern part, where R D ≃ 70 km, and the same L 0-value. A thorough analysis of this has, however, not been done.
- 9.
For a portrait of Karl Gustav Rossby and a biographical sketch, see Fig. 19.1 in Chap. 19.
- 10.
The boundary condition is in general Ψ = C α = const along each boundary with vanishing flow of water through it. Each boundary has a different value of C α. If a lake has no island, then only a single C arises which may be set to zero, because the stream function can accordingly be normalized.
- 11.
This fact is stated here without proof.
- 12.
The lines of constant f ∕ H are called isotrophs. We choose here to name isolines of H ∕ f ‘isotroph depths’, but mention that this is not an official denotation.
- 13.
With ei(kx − ωt) a positive ω means propagation into the positive x-direction.
- 14.
For portraits and biographical sketches of Sir David Brunt and Vilho Väisälä see Fig. 8.13 of Volume I.
- 15.
Any model for which the mass balance equation reduces to the continuity equation \(\mathrm{div}\,\vec{v} = 0\) does not permit acoustic waves, which are longitudinal as we have seen.
- 16.
Progressive vector diagrams are ‘displacement’ plots from current meter measurements at fixed positions. If Δt is the temporal increment at which the velocity vector \(\vec{{v}}_{i}\) (i = number of the time increment) is measured, then
$$\begin{array}{rcl} \vec{{s}}_{n} ={ \sum \nolimits }_{i=0}^{n}\vec{{v}}_{ i}\Delta t,\quad (n = 0, 1, 2,\ldots )& & \\ \end{array}$$determines the trajectory of a ‘virtual particle’ that passed the current meter at n = 0, see Volume 3, Chapter ‘Instruments and sensors’. The particle is not real, because the current meter does not follow it (Eulerian description). An alternative is to follow a marker (e.g. freely moving drifter buoy) through time (Lagrangean description).
- 17.
For a portrait and biographical sketch of Boussinesq see Fig. 11.8.
- 18.
Consult also Sects. 4.6 and 4.7 of Chap. 4 in Volume I of this book series for a general discussion of these properties.
- 19.
This separation also works in the β-plane approximation, if the perturbation pressure or the y-component of the velocity field are taken as the independent fields, see [8].
- 20.
This argument shows that in vertical eigenvalue problems for problems not based on the shallow water approximation the eigenvalues gh n depend on the rotation of the Earth since the evaluation of gh n is constrained to the satisfaction of the dispersion relation (11.78).
- 21.
Because of the boundedness of lakes, n-layer equivalent depth models for n > 2 are not very useful, because they require the same domain boundary for each n. This is becoming unrealistic, when n > 2. For an ocean, where boundaries are far away, such models are better suited.
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Hutter, K., Wang, Y., Chubarenko, I.P. (2011). The Role of the Earth’s Rotation: Fundamentals – Rotation and Stratification Influenced Dynamics. 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_11
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