# Consistent shallow-water equations on the rotating sphere with complete Coriolis force and topography

@article{Tort2014ConsistentSE, title={Consistent shallow-water equations on the rotating sphere with complete Coriolis force and topography}, author={Marine Tort and Thomas Dubos and François Bouchut and Vladimir Zeitlin}, journal={Journal of Fluid Mechanics}, year={2014}, volume={748}, pages={789 - 821} }

Abstract Consistent shallow-water equations are derived on the rotating sphere with topography retaining the Coriolis force due to the horizontal component of the planetary angular velocity. Unlike the traditional approximation, this ‘non-traditional’ approximation captures the increase with height of the solid-body velocity due to planetary rotation. The conservation of energy, angular momentum and potential vorticity are ensured in the system. The caveats in extending the standard shallow… Expand

#### 15 Citations

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#### References

SHOWING 1-10 OF 35 REFERENCES

Shallow water equations with a complete Coriolis force and topography

- Physics
- 2005

This paper derives a set of two-dimensional equations describing a thin inviscid fluid layer flowing over topography in a frame rotating about an arbitrary axis. These equations retain various terms… Expand

Dynamically consistent, quasi-hydrostatic equations for global models with a complete representation of the Coriolis force

- Mathematics
- 1995

The spherical polar components of the Coriolis force consist of terms in sin ϕ and terms in cos ϕ, where ϕ is latitude (referred to the frame-rotation vector as polar axis). The cos ϕ Coriolis terms… Expand

Derivation of the Equations of Atmospheric Motion in Oblate Spheroidal Coordinates

- Physics
- 2004

Since Earth is more nearly an oblate spheroid than a sphere, it is of at least theoretical interest to develop the atmospheric equations of motion in spheroidal coordinates. In this system the… Expand

Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow-water model

- Physics
- Journal of Fluid Mechanics
- 2001

We develop a theory of nonlinear geostrophic adjustment of arbitrary localized (i.e. finite-energy) disturbances in the framework of the non-dissipative rotating shallow-water dynamics. The only… Expand

Forced-dissipative shallow water turbulence on the sphere

- Geology
- 2006

Although possibly the simplest model for the atmospheres of the giant planets, the turbulent forceddissipative shallow-water system in spherical geometry has not, to date, been investigated; the… Expand

The Importance of the Nontraditional Coriolis Terms in Large-Scale Motions in the Tropics Forced by Prescribed Cumulus Heating

- Physics
- 2012

AbstractIn meteorological dynamics, the shallow-atmosphere approximation is generally used in the momentum equation, together with the “traditional approximation.” In the traditional approximation,… Expand

An oceanic general circulation model framed in hybrid isopycnic-Cartesian coordinates

- Computer Science
- 2002

The intent of this approach, some of whose features are reminiscent of the Arbitrary Lagrangian–Eulerian (ALE) technique, is to combine the best features of isopycnic-coordinate and fixed-grid circulation models within a single framework. Expand

The Stability of Short Symmetric Internal Waves on Sloping Fronts: Beyond the Traditional Approximation

- Physics
- 2012

AbstractThe interaction of internal waves with geostrophic flows is found to be strongly dependent upon the background stratification. Under the traditional approximation of neglecting the horizontal… Expand

Nonlinear dynamics of rotating shallow water : methods and advances

- Physics
- 2007

The rotating shallow water (RSW) model is of wide use as a conceptual tool in geophysical fluid dynamics (GFD), because, in spite of its simplicity, it contains all essential ingredients of… Expand

Consistent approximate models of the global atmosphere: shallow, deep, hydrostatic, quasi-hydrostatic and non-hydrostatic

- Physics
- 2005

We study global atmosphere models that are at least as accurate as the hydrostatic primitive equations (HPEs), reviewing known results and reporting some new ones. The HPEs make spherical… Expand