Two-Layer Shallow Water Equations with Complete Coriolis Force and Topography

Stewart, Andrew L.; Dellar, Paul J.

doi:10.1007/978-3-642-12110-4_164

Cited by 22 publications

(3 citation statements)

References 9 publications

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“…We consider in this section the two-layer rotating shallow-water equations on the NT f -plane which were derived by Stewart & Dellar (2010). The flow is a barotropic Bickley jet (in the sense of absence of vertical shear) evolving in mid-latitudes in the northern hemisphere, such that the traditional Coriolis parameter is positive f > 0 and the NT Coriolis parameter is F = O( f ).…”

Section: Linear Stability Analysis In the Two-layer Modelmentioning

confidence: 99%

“…In particular, there are two materially conserved potential vorticities (Stewart & Dellar 2010): Inertial instability requires that the product of the (traditional) Coriolis parameter and the absolute vorticity be negative. As NT q i={1,2} differs by terms proportional to F from the traditional potential vorticities for which F = 0 in (2.8)-(2.9), the stability bounds may be altered.…”

Section: Equations Of Motion and Relevant Parametersmentioning

confidence: 99%

“…The two-layer shallow-water equations with free upper surface and flat bottom on the NT f -plane (Stewart & Dellar 2010) are…”

Section: Equations Of Motion and Relevant Parametersmentioning

confidence: 99%

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Symmetric and asymmetric inertial instability of zonal jets on the -plane with complete Coriolis force

2016

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Symmetric and asymmetric inertial instability of the westerly mid-latitude barotropic Bickley jet is analysed without the traditional approximation which neglects the vertical component of the Coriolis force, as well as the contribution of the vertical velocity to the latter. A detailed linear stability analysis of the jet at large Rossby numbers on the non-traditional f -plane is performed for long waves in both the two-layer rotating shallow-water and continuously stratified Boussinesq models. The dependence of the instability on both the Rossby and Burger numbers of the jet is investigated and compared to the traditional case. It is shown that non-traditional effects significantly increase the growth rate of the instability at small enough Burger numbers (weak stratifications) for realistic aspect ratios of the jet. The main results are as follows. (i) Two-layer shallow-water model. In the parameter regimes where the jet is inertially stable on the traditional f -plane, the symmetric inertial instability with respect to perturbations with zero along-jet wavenumber arises on the non-traditional f -plane. Both non-traditional symmetric and asymmetric (small but non-zero wavenumbers) inertial instabilities have higher growth rates than their traditional counterparts. (ii) Continuously stratified model. It is shown that by a proper change of variables the linear stability problem for the barotropic jet, on the non-traditional f -plane, can be rendered separable and analysed along the same lines as in the traditional approximation. Neutral, weak and strong background stratifications are considered. For the neutral stratification the jet is inertially unstable if the traditional approximation is relaxed, while its traditional counterpart is not. For a sufficiently weak stratification, both symmetric and asymmetric inertial instabilities have substantially higher growth rates than in the traditional approximation. The across-jet structure of non-traditional unstable modes is strikingly different, as compared to those under the traditional approximation. No discernible differences between the two approximations are observed for strong enough stratifications. The influence of dissipation and non-hydrostatic effects upon the instability is quantified.

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Section: Linear Stability Analysis In the Two-layer Modelmentioning

confidence: 99%

Section: Equations Of Motion and Relevant Parametersmentioning

confidence: 99%

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Symmetric and asymmetric inertial instability of zonal jets on the -plane with complete Coriolis force

2016

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Well-balanced schemes for the shallow water equations with Coriolis forces

et al. 2017

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Geophysical Fluid Dynamics

Zeitlin

2018

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The book explains the key notions and fundamental processes in the dynamics of the fluid envelopes of the Earth (transposable to other planets), and methods of their analysis, from the unifying viewpoint of rotating shallow-water model (RSW). The model, in its one- or two-layer versions, plays a distinguished role in geophysical fluid dynamics, having been used for around a century for conceptual understanding of various phenomena, for elaboration of approaches and methods, to be applied later in more complete models, for development and testing of numerical codes and schemes of data assimilations, and many other purposes. Principles of modelling of large-scale atmospheric and oceanic flows, and corresponding approximations, are explained and it is shown how single- and multi-layer versions of RSW arise from the primitive equations by vertical averaging, and how further time-averaging produces celebrated quasi-geostrophic reductions of the model. Key concepts of geophysical fluid dynamics are exposed and interpreted in RSW terms, and fundamentals of vortex and wave dynamics are explained in Part 1 of the book, which is supplied with exercises and can be used as a textbook. Solutions of the problems are available at Editorial Office by request. In-depth treatment of dynamical processes, with special accent on the primordial process of geostrophic adjustment, on instabilities in geophysical flows, vortex and wave turbulence and on nonlinear wave interactions follows in Part 2. Recently arisen new approaches in, and applications of RSW, including moist-convective processes constitute Part 3.

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Two-Layer Shallow Water Equations with Complete Coriolis Force and Topography

Cited by 22 publications

References 9 publications

Symmetric and asymmetric inertial instability of zonal jets on the -plane with complete Coriolis force

Symmetric and asymmetric inertial instability of zonal jets on the -plane with complete Coriolis force

Well-balanced schemes for the shallow water equations with Coriolis forces

Geophysical Fluid Dynamics

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