Shallow Water Waves on the Rotating Earth

Paldor, Nathan

doi:10.1007/978-3-319-20261-7

Cited by 20 publications

(20 citation statements)

References 5 publications

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“…This limitation of harmonic waves to narrow channels only is also clearly evident a priori when one realizes that in large meridional domains the changes in f ( y ) cannot be neglected in comparison to f 0 , which is the basis for obtaining the harmonic waves as solutions of the constant‐coefficient ordinary differential equation. This applicability of the harmonic theory to narrow channels and that of the trapped wave theory to (infinitely) wide channels is also encountered in channels on the equatorial β ‐plane (see Erlick et al, ; and chapter 3 in Paldor, ).…”

Section: Discussionmentioning

confidence: 81%

“…In recent years a non‐harmonic, trapped wave, theory was developed for inertia‐gravity waves and planetary waves in a wide channel in midlatitudes (Paldor et al, ; Paldor and Sigalov, ; Paldor, ). The amplitudes of the eigenfunctions in this theory vary meridionally as the regular (at +infinity) Airy function, Ai , while the zonal and temporal variations are identical to those of the harmonic waves since the waves propagate zonally in both cases so their zonal and temporal variations are given by Re{e i( kx − ωt ) }.…”

Section: Introductionmentioning

confidence: 99%

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Numerical simulation of harmonic, and trapped, Rossby waves in a channel on the midlatitude β‐plane

Gildor

Paldor

Ben-Shushan

2016

Quart J Royal Meteoro Soc

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The applicability of two competing wave theories of the rotating shallow-water equations, the harmonic planetary waves and the trapped planetary waves, is examined by solving the equations numerically in a zonal channel on the midlatitude β-plane. The examination is carried out by initializing the numerical shallow-water solver by harmonic and trapped waves and comparing the temporal evolution to that expected from the corresponding wave theory. The simulations confirm that harmonic waves provide accurate approximations for the temporal evolution in narrow channels while trapped waves do so in wide channels. The transition from a narrow channel to a wide channel occurs when the non-dimensional number L(where L is the channel width, R d is the radius of deformation, a is Earth's radius and φ 0 is the latitude of the centre of the channel) is increased above a threshold value that larger than 4 and that depends on the meridional mode number of the trapped wave. Numerical solutions of the nonlinear equations show that, for the same wave amplitude, trapped waves approximate the nonlinear solutions in a wide channel for much longer times than harmonic waves in a narrow channel.

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Section: Discussionmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 99%

Numerical simulation of harmonic, and trapped, Rossby waves in a channel on the midlatitude β‐plane

Gildor

Paldor

Ben-Shushan

2016

Quart J Royal Meteoro Soc

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“…In contrast, solutions of system are not known to form a complete set and no relationship can be established between the values of C and the latitudinal structure of V cos ϕ and η . The formulation of an eigenvalue equation associated with waves proved to be highly informative in several set‐ups including the midlatitude β ‐plane (Paldor et al ., ; Paldor and Sigalov, ), an equatorial channel on a sphere (De‐Leon et al ., ) and the entire sphere (De‐Leon and Paldor, ; Paldor et al ., ; Paldor, ). In contrast to these channel problems where the boundary conditions are ‘no normal flow’ at the channel walls, the boundary conditions that solutions of must satisfy on a sphere are their regularity at the singular poles.…”

Section: A Schrödinger Eigenvalue Equation Of the Lrswementioning

confidence: 98%

“…Recently, Schrödinger eigenvalue problems were formulated for zonally propagating wave solutions of the LRSWE in midlatitudes (Paldor et al, 2007;Paldor and Sigalov, 2008) and approximate Schrödinger eigenvalue problems were also formulated on a sphere (De-Leon and Paldor, 2011;Paldor et al, 2013;Paldor, 2015). These formulations provide a clear and concise way of identifying the various waves and modes that solve the LRSWE.…”

Section: Introductionmentioning

confidence: 99%

Classification of eastward propagating waves on the spherical Earth

Garfinkel

Fouxon

Shamir

et al. 2017

Quart J Royal Meteoro Soc

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Observational evidence for an equatorial non-dispersive mode propagating at the speed of gravity waves is strong, and while the structure and dispersion relation of such a mode can be accurately described by a wave theory on the equatorial β-plane, prior theories on the sphere were unable to find such a mode except for particular asymptotic limits of gravity wave phase speeds and/or certain zonal wave numbers. Here, an ad hoc solution of the linearized rotating shallow-water equations (LRSWE) on a sphere is developed, which propagates eastward with phase speed that nearly equals the speed of gravity waves at all zonal wave numbers. The physical interpretation of this mode in the context of other modes that solve the LRSWE is clarified through numerical calculations and through eigenvalue analysis of a Schrödinger eigenvalue equation that approximates the LRSWE. By comparing the meridional amplitude structure and phase speed of the ad hoc mode with those of the lowest gravity mode on a non-rotating sphere we show that at large zonal wave number the former is a rotation-modified counterpart of the latter. We also find that the dispersion relation of the ad hoc mode is identical to the n = 0 eastward propagating inertia-gravity (EIG0) wave on a rotating sphere which is also nearly non-dispersive, so this solution could be classified as both a Kelvin wave and as the EIG0 wave. This is in contrast to Cartesian coordinates where Kelvin waves are a distinct wave solution that supplements the EIG0 mode. Furthermore, the eigenvalue equation for the meridional velocity on the β-plane can be formally derived as an asymptotic limit (for small (Lamb Number) -1/4 ) of the corresponding second order equation on a sphere, but this expansion is invalid when the phase speed equals that of gravity waves i.e. for Kelvin waves. Various expressions found in the literature for both Kelvin waves and inertia-gravity waves and which are valid only in certain asymptotic limits (e.g. slow and fast rotation) are compared with the expressions found here for the two wave types.

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“…That gyres exist as a natural consequence of the general equations of motion (in spherical geometry), with clear and well-defined properties, is, we suggest, an important observation, even though a number of less critical physical characteristics may have to be suppressed in the development. We note that there is a growing interest in the study of ocean flows beyond the planar setting of the f - or β -plane approximation, within the framework of spherical geometry—see the discussion in [3] for a survey of the relevant research literature.…”

Section: Introductionmentioning

confidence: 99%

Large gyres as a shallow-water asymptotic solution of Euler’s equation in spherical coordinates

Constantin

Johnson

2017

Proc. R. Soc. A.

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Starting from the Euler equation expressed in a rotating frame in spherical coordinates, coupled with the equation of mass conservation and the appropriate boundary conditions, a thin-layer (i.e. shallow water) asymptotic approximation is developed. The analysis is driven by a single, overarching assumption based on the smallness of one parameter: the ratio of the average depth of the oceans to the radius of the Earth. Consistent with this, the magnitude of the vertical velocity component through the layer is necessarily much smaller than the horizontal components along the layer. A choice of the size of this speed ratio is made, which corresponds, roughly, to the observational data for gyres; thus the problem is characterized by, and reduced to an analysis based on, a single small parameter. The nonlinear leading-order problem retains all the rotational contributions of the moving frame, describing motion in a thin spherical shell. There are many solutions of this system, corresponding to different vorticities, all described by a novel vorticity equation: this couples the vorticity generated by the spin of the Earth with the underlying vorticity due to the movement of the oceans. Some explicit solutions are obtained, which exhibit gyre-like flows of any size; indeed, the technique developed here allows for many different choices of the flow field and of any suitable free-surface profile. We comment briefly on the next order problem, which provides the structure through the layer. Some observations about the new vorticity equation are given, and a brief indication of how these results can be extended is offered.

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Shallow Water Waves on the Rotating Earth

Cited by 20 publications

References 5 publications

Numerical simulation of harmonic, and trapped, Rossby waves in a channel on the midlatitude β‐plane

Numerical simulation of harmonic, and trapped, Rossby waves in a channel on the midlatitude β‐plane

Classification of eastward propagating waves on the spherical Earth

Large gyres as a shallow-water asymptotic solution of Euler’s equation in spherical coordinates

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