A Balanced Approximation of the One-Layer Shallow-Water Equations on a Sphere

2009 ◽  
Vol 66 (6) ◽  
pp. 1735-1748 ◽  
Author(s):  
W. T. M. Verkley
Keyword(s):  
Shallow Water ◽  
Potential Vorticity ◽  
Shallow Water Equations ◽  
Rossby Waves ◽  
Coriolis Parameter ◽  
Beta Plane ◽  
Propagating Waves ◽  
Barotropic Vorticity Equation ◽  
Barotropic Vorticity ◽  
The One

Abstract A global version of the equivalent barotropic vorticity equation is derived for the one-layer shallow-water equations on a sphere. The equation has the same form as the corresponding beta plane version, but with one important difference: the stretching (Cressman) term in the expression of the potential vorticity retains its full dependence on f 2, where f is the Coriolis parameter. As a check of the resulting system, the dynamics of linear Rossby waves are considered. It is shown that these waves are rather accurate approximations of the westward-propagating waves of the second class of the original shallow-water equations. It is also concluded that for Rossby waves with short meridional wavelengths the factor f 2 in the stretching term can be replaced by the constant value f02, where f0 is the Coriolis parameter at ±45° latitude.

Introduction to Geophysical Fluid Dynamics

1998 ◽  
Author(s):  
Rick Salmon
Keyword(s):  
Fluid Dynamics ◽  
Shallow Water ◽  
Potential Vorticity ◽  
Shallow Water Equations ◽  
Coordinate Frame ◽  
Geophysical Fluid Dynamics ◽  
Homogeneous Fluid ◽  
The One ◽  
Quasigeostrophic Equations ◽  
Inertia Gravity Waves

This second chapter offers a brief introduction to geophysical fluid dynamics—the dynamics of rotating, stratified flows. We start with the shallow water equations, which govern columnar motion in a thin layer of homogeneous fluid. Roughly speaking, the solutions of the shallow-water equations comprise two types of motion: ageostrophic motions, including inertia-gravity waves, on the one hand, and nearly geostrophic motions on the other. In rapidly rotating flow, these two types of motion may, in some sense, decouple. We seek simpler equations that describe only the nearly geostrophic motion. The simplest such equations are the quasigeostrophic equations. In the quasigcostrophic equations, potential vorticity plays the key role: The potential vorticity completely determines the velocity field that transports it, thereby controlling the whole dynamics. We begin by generalizing our previously derived fluid equations to a rotating coordinate frame.

On the stability of elliptical vortex solutions of the shallow-water equations

1987 ◽  
Vol 183 ◽  
pp. 343-363 ◽  
Author(s):  
P. Ripa
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Normal Modes ◽  
Finite Amplitude ◽  
Rossby Number ◽  
Coriolis Parameter ◽  
Vortex Solutions ◽  
The Stability ◽  
The One ◽  
Two Parameters

The one-layer reduced gravity (or ‘shallow water’) equations in the f-plane have solutions such that the active layer is horizontally bounded by an ellipse that rotates steadily. In a frame where the height contours are stationary, fluid particles move along similar ellipses with the same revolution period. Both motions (translation along an elliptical path and precession of that orbit) are anticyclonic and their frequencies are not independent; a Rossby number (R0) based on the combination of both of them is bounded by unity. These solutions may be taken, with some optimism, as a model of ocean warm eddies; their stability is studied here for all values of R0 and of the ellipse eccentricity (these two parameters determine uniquely the properties of the solution).Sufficient stability conditions are derived from the integrals of motion; f-plane flows that satisfy them must be either axisymmetric or parallel. For the model vortex, the circular case simply corresponds to a solid-body rotation, and is found to be stable to finite-amplitude perturbations for all values of R0. This includes R0 > ½, which implies an anticyclonic absolute vorticity.The stability of the truly elliptical cases are studied in the normal modes sense. The height perturbation is an n-order polynomial of the horizontal coordinates; the cases for 0 ≤ n ≤ 6 are analysed, for all possible values of the Rossby number and of the eccentricity. All eddies are stable to perturbations with n ≤ 2. (A property of the shallow-water equations, probably related to the last result, is that a general finite-amplitude n-order field is an exact nonlinear solution for n ≤ 2.) Many vortices - noticeably the more eccentric ones - are unstable to perturbations with n ≥ 3; growth rates are O(R02f) where f is the Coriolis parameter.

scholarly journals Propagation of a finite-amplitude potential vorticity front along the wall of a stratified fluid

2002 ◽  
Vol 468 ◽  
pp. 179-204 ◽  
Author(s):  
MELVIN E. STERN ◽  
KARL R. HELFRICH
Keyword(s):  
Laboratory Experiment ◽  
Shallow Water ◽  
Potential Vorticity ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Similarity Theory ◽  
Wave Structure ◽  
Small Scale ◽  
Reduced Gravity ◽  
Coriolis Parameter

A similarity solution to the long-wave shallow-water equations is obtained for a density current (reduced gravity = g′, Coriolis parameter = f) propagating alongshore (y = 0). The potential vorticity q = f/H1 is uniform in −∞ < x [les ] xnose(t), 0 < y [les ] L(x, t), and the nose of this advancing potential vorticity front displaces fluid of greater q = f/H0, which is located at L < y < ∞. If L0 = L(−∞, t), the nose point with L(xnose(t), t) = 0 moves with velocity Unose = √g′H0 φ, where φ is a function of H1/H0, f2L20/g′H0. The assumptions made in the similarity theory are verified by an initial value solution of the complete reduced-gravity shallow-water equations. The latter also reveal the new effect of a Kelvin shock wave colliding with a potential vorticity front, as is confirmed by a laboratory experiment. Also confirmed is the expansion wave structure of the intrusion, but the observed values of Unose are only in qualitative agreement; the difference is attributed to the presence of small-scale (non-hydrostatic) turbulence in the laboratory experiment but not in the numerical solutions.

scholarly journals Instability of coupled gravity-inertial-Rossby waves on a β-plane in solar system atmospheres

2009 ◽  
Vol 27 (11) ◽  
pp. 4221-4227 ◽  
Author(s):  
J. F. McKenzie
Keyword(s):  
Rossby Waves ◽  
Narrow Band ◽  
Boussinesq Approximation ◽  
The Other ◽  
The Earth ◽  
Propagating Waves ◽  
Inertial Wave ◽  
The One ◽  
Fifth Order ◽  
Critical Latitude

Abstract. This paper provides an analysis of the combined theory of gravity-inertial-Rossby waves on a β-plane in the Boussinesq approximation. The wave equation for the system is fifth order in space and time and demonstrates how gravity-inertial waves on the one hand are coupled to Rossby waves on the other through the combined effects of β, the stratification characterized by the Väisälä-Brunt frequency N, the Coriolis frequency f at a given latitude, and vertical propagation which permits buoyancy modes to interact with westward propagating Rossby waves. The corresponding dispersion equation shows that the frequency of a westward propagating gravity-inertial wave is reduced by the coupling, whereas the frequency of a Rossby wave is increased. If the coupling is sufficiently strong these two modes coalesce giving rise to an instability. The instability condition translates into a curve of critical latitude Θc versus effective equatorial rotational Mach number M, with the region below this curve exhibiting instability. "Supersonic" fast rotators are unstable in a narrow band of latitudes around the equator. For example Θc~12° for Jupiter. On the other hand slow "subsonic" rotators (e.g. Mercury, Venus and the Sun's Corona) are unstable at all latitudes except very close to the poles where the β effect vanishes. "Transonic" rotators, such as the Earth and Mars, exhibit instability within latitudes of 34° and 39°, respectively, around the Equator. Similar results pertain to Oceans. In the case of an Earth's Ocean of depth 4km say, purely westward propagating waves are unstable up to 26° about the Equator. The nonlinear evolution of this instability which feeds off rotational energy and gravitational buoyancy may play an important role in atmospheric dynamics.

scholarly journals Propagation properties of Rossby waves for latitudinal β-plane variations of <I>f</I> and zonal variations of the shallow water speed

2012 ◽  
Vol 30 (5) ◽  
pp. 849-855 ◽  
Author(s):  
C. T. Duba ◽  
J. F. McKenzie
Keyword(s):  
Shallow Water ◽  
Group Velocity ◽  
Rossby Wave ◽  
Rossby Waves ◽  
Low Frequency ◽  
Wave Number ◽  
Coriolis Parameter ◽  
Propagation Properties ◽  
Wave Normal ◽  
Water Speed

Abstract. Using the shallow water equations for a rotating layer of fluid, the wave and dispersion equations for Rossby waves are developed for the cases of both the standard β-plane approximation for the latitudinal variation of the Coriolis parameter f and a zonal variation of the shallow water speed. It is well known that the wave normal diagram for the standard (mid-latitude) Rossby wave on a β-plane is a circle in wave number (ky,kx) space, whose centre is displaced −β/2 ω units along the negative kx axis, and whose radius is less than this displacement, which means that phase propagation is entirely westward. This form of anisotropy (arising from the latitudinal y variation of f), combined with the highly dispersive nature of the wave, gives rise to a group velocity diagram which permits eastward as well as westward propagation. It is shown that the group velocity diagram is an ellipse, whose centre is displaced westward, and whose major and minor axes give the maximum westward, eastward and northward (southward) group speeds as functions of the frequency and a parameter m which measures the ratio of the low frequency-long wavelength Rossby wave speed to the shallow water speed. We believe these properties of group velocity diagram have not been elucidated in this way before. We present a similar derivation of the wave normal diagram and its associated group velocity curve for the case of a zonal (x) variation of the shallow water speed, which may arise when the depth of an ocean varies zonally from a continental shelf.

Convection in a rotating cylindrical annulus. Part 2. Transitions to asymmetric and vacillating flow

1987 ◽  
Vol 174 ◽  
pp. 313-326 ◽  
Author(s):  
A. C. Or ◽  
F. H. Busse
Keyword(s):  
Rayleigh Number ◽  
Rossby Waves ◽  
Period Doubling ◽  
Critical Value ◽  
Coriolis Parameter ◽  
Onset Of Convection ◽  
Cylindrical Annulus ◽  
Propagating Waves ◽  
Annular Layer ◽  
Doubling Sequence

The instabilities of convection columns (also called thermal Rossby waves) in a cylindrical annulus rotating about its axis and heated from the outside are investigated as a function of the Prandtl number P and the Coriolis parameter η*. When this latter parameter is sufficiently large, it is found that the primary solution observed at the onset of convection becomes unstable when the Rayleigh number exceeds its critical value by a relatively small amount. Transitions occur to columnar convection which is non-symmetric with respect to the mid-plane of the small-gap annular layer. Further transitions introduce convection flows that vacillate in time or tend to split the row of columns into an inner and an outer row of separately propagating waves. Of special interest is the regime of non-symmetric convection, which exhibits decreasing Nusselt number with increasing Rayleigh number, and the indication of a period doubling sequence associated with vacillating convection.

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