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

  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},
  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… 
Dynamically consistent shallow‐atmosphere equations with a complete Coriolis force
Shallow‐atmosphere equations retaining both the vertical and horizontal components of the Coriolis force (the latter being neglected in the traditional approximation) are obtained. The derivation
Dynamically consistent shallow‐water equation sets in non‐spherical geometry with latitudinal variation of gravity
The shallow‐water equations in spherical geometry have proven to be an invaluable prototypical tool to advance geophysical fluid dynamics. Many of the fundamental terms and properties, including
Variational Principles in Geophysical Fluid Dynamics and Approximated Equations
In this chapter, the variational principle of Hamilton is applied to different examples from Geophysical Fluid Dynamics. Hamilton’s principle is extended to uniformly rotating flows and to
A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the Pacific Equatorial Undercurrent and thermocline
Using the salient properties of the flow observed in the equatorial Pacific as a guide, an asymptotic procedure is applied to the Euler equation written in a suitable rotating frame. Starting from
Symmetric and asymmetric inertial instability of zonal jets on the $f$ -plane with complete Coriolis force
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
Equatorial inertial instability with full Coriolis force
The zonally symmetric inertial instability of oceanic near-equatorial flows is studied through high-resolution numerical simulations. In homogeneous upper layers, the instability of surface-confined
DYNAMICO-1.0, an icosahedral hydrostatic dynamical core designed for consistency and versatility
Abstract. The design of the icosahedral dynamical core DYNAMICO is presented. DYNAMICO solves the multi-layer rotating shallow-water equations, a compressible variant of the same equivalent to a
Barotropic instability of a zonal jet on the sphere: from non-divergence through quasi-geostrophy to shallow water
Two common approximations to the full Shallow Water Equations (SWEs) are non-divergence and quasi-geostrophy, and the degree to which these approximations lead to biases in numerical solutions are
A variational formulation of geophysical fluid motion in non‐Eulerian coordinates
Systematic methods to derive geophysical equations of motion possessing conservation laws for energy, momentum and potential vorticity have recently been developed. One approach is based on
Unidirectional Modes Induced by Nontraditional Coriolis Force in Stratified Fluids.
Using topology, we unveil the existence of new unidirectional modes in compressible rotating stratified fluids. We relate their emergence to the breaking of time-reversal symmetry by rotation and


Shallow water equations with a complete Coriolis force and topography
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
Dynamically consistent, quasi-hydrostatic equations for global models with a complete representation of the Coriolis force
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
Derivation of the Equations of Atmospheric Motion in Oblate Spheroidal Coordinates
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
Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow-water model
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
Forced-dissipative shallow water turbulence on the sphere
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
The Importance of the Nontraditional Coriolis Terms in Large-Scale Motions in the Tropics Forced by Prescribed Cumulus Heating
AbstractIn meteorological dynamics, the shallow-atmosphere approximation is generally used in the momentum equation, together with the “traditional approximation.” In the traditional approximation,
An oceanic general circulation model framed in hybrid isopycnic-Cartesian coordinates
The Stability of Short Symmetric Internal Waves on Sloping Fronts: Beyond the Traditional Approximation
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
Nonlinear dynamics of rotating shallow water : methods and advances
Variations on a beta-plane: derivation of non-traditional beta-plane equations from Hamilton's principle on a sphere
  • P. Dellar
  • Physics
    Journal of Fluid Mechanics
  • 2011
Starting from Hamilton's principle on a rotating sphere, we derive a series of successively more accurate β-plane approximations. These are Cartesian approximations to motion in spherical geometry