Planetary (Rossby) waves and inertia–gravity (Poincaré) waves in a barotropic ocean over a sphere

Paldor, Nathan; De-Leon, Yair; Shamir, Ofer

doi:10.1017/jfm.2013.219

Cited by 25 publications

(52 citation statements)

References 15 publications

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“…6.3 (the figures in this subsection were reproduced from Paldor et al (2013)) clearly demonstrate the accuracy of the analytical estimates of E n that were formally derived in (6.16) only for n → ∞ but seem to be accurate even for n = 5, i.e., when αn 2 = 1.25, which is only marginally larger than 1. 6.3 (the figures in this subsection were reproduced from Paldor et al (2013)) clearly demonstrate the accuracy of the analytical estimates of E n that were formally derived in (6.16) only for n → ∞ but seem to be accurate even for n = 5, i.e., when αn 2 = 1.25, which is only marginally larger than 1.…”

Section: Dispersion Relation Of Planetary Wavessupporting

confidence: 53%

“…6.1 for α = 0.01 are very close Fig. (6.2) when αn 2 is not small follows the method described in Paldor et al (2013). The approximations of U 1 ð/Þ þ aU 2 ð/Þ (red dashed line) by U 1 ð/Þ (thin solid blue line) are not accurate near the poles.…”

Section: ð6:8þmentioning

confidence: 56%

“…Similarly, when V cos / is eliminated from (4.4) the definition of wð/Þ and the expression for U 2 ð/Þ in (6.1), calculated in the same way as the expressions in (6.3), turn out to be (see Paldor et al 2013 for more details):…”

Section: ð6:3þmentioning

confidence: 99%

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Shallow Water Waves on the Rotating Earth

Paldor¹

2015

SpringerBriefs in Earth System Sciences

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Section: Dispersion Relation Of Planetary Wavessupporting

confidence: 53%

Section: ð6:8þmentioning

confidence: 56%

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Shallow Water Waves on the Rotating Earth

Paldor¹

2015

SpringerBriefs in Earth System Sciences

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“…For comparison, the frequencies predicted by Haurwitz () (for the value of gH used here) are superimposed on the figure (dashed‐lines). Note that, while the frequencies predicted by Haurwitz () are indistinguishable from those predicted for Rossby waves by Paldor et al () (except for n = 1, k = 1,2), the eigenfunctions predicted by the two theories are fundamentally different. In particular, the solutions obtained by Paldor et al () include the small but significant divergence field which is completely missing from the Haurwitz solutions.…”

Section: The Analytic Solutionsmentioning

confidence: 99%

“…First, as is clearly demonstrated in this work, initial wave modes based on the analytic solutions obtained by Paldor et al () are accurately preserved for at least 100 wave periods, compared to only O (1) for initial Rossby–Haurwitz waves. In particular, except for small round‐off errors in the initial conditions, the error in the solution in the first 100 wave periods results mainly from the numerical scheme being used.…”

Section: Introductionmentioning

confidence: 99%

A quantitative test case for global‐scale dynamical cores based on analytic wave solutions of the shallow‐water equations

Shamir

Paldor

2016

Quart J Royal Meteoro Soc

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Recently derived analytic wave solutions of the shallow‐water equations (SWEs) on the rotating spherical Earth are employed to construct a test case for hydrostatic dynamical cores of global‐scale general circulation models (GCMs). The proposed test case is more relevant to the SWEs than the frequently used Rossby–Haurwitz test case which is based on wave solutions of the non‐divergent barotropic vorticity equation and not the SWEs. The applicability of the proposed test case to operational GCMs is demonstrated by using the spectral Eulerian dynamical core of the atmospheric component of NCAR's Community Earth System Model to simulate the analytic solutions. An initial slowly propagating Rossby wave and a fast eastward propagating inertia–gravity wave are both accurately simulated for 100 wave periods. In order to quantify the accuracy of the simulations, two error‐measures are suggested which complement the conservation of global energy and, unlike the frequently used L2 error‐measure, provide independent assessments of the errors in the phase speeds and the meridional structures of the simulated waves and are therefore more relevant to periodic wave solutions.

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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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Planetary (Rossby) waves and inertia–gravity (Poincaré) waves in a barotropic ocean over a sphere

Cited by 25 publications

References 15 publications

Shallow Water Waves on the Rotating Earth

Shallow Water Waves on the Rotating Earth

A quantitative test case for global‐scale dynamical cores based on analytic wave solutions of the shallow‐water equations

Classification of eastward propagating waves on the spherical Earth

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