Geostrophic vs magneto-geostrophic adjustment and nonlinear magneto-inertia-gravity waves in rotating shallow water magnetohydrodynamics

2015 ◽  
Vol 109 (5) ◽  
pp. 497-523 ◽  
Author(s):  
Vladimir Zeitlin ◽  
Christelle Lusso ◽  
François Bouchut
Keyword(s):  
Shallow Water ◽  
Gravity Waves ◽  
Geostrophic Adjustment ◽  
Rotating Shallow Water ◽  
Shallow Water Magnetohydrodynamics ◽  
Inertia Gravity Waves

Existence and properties of ageostrophic modons and coherent tripoles in the two-layer rotating shallow water model on the -plane

2012 ◽  
Vol 706 ◽  
pp. 71-107 ◽  
Author(s):  
Noé Lahaye ◽  
Vladimir Zeitlin
Keyword(s):  
Shallow Water ◽  
Gravity Waves ◽  
Coherent Structures ◽  
Water Model ◽  
Shallow Water Model ◽  
Scatter Plot ◽  
Weak Stratification ◽  
Rotating Shallow Water ◽  
Frontal Collisions ◽  
Inertia Gravity Waves

AbstractWe study formation and properties of new coherent structures: ageostrophic modons in the two-layer rotating shallow water model. The ageostrophic modons are obtained by ‘ageostrophic adjustment’ of the exact modon solutions of the two-layer quasi-geostrophic equations with the free surface, which are used to initialize the full two-layer shallow water model. Numerical simulations are performed using a well-balanced high-resolution finite volume numerical scheme. For large enough Rossby numbers, the initial configurations undergo ageostrophic adjustment towards asymmetric ageostrophic quasi-stationary coherent dipoles. This process is accompanied by substantial emission of inertia–gravity waves. The resulting dipole is shown to be robust and survives frontal collisions. It contains captured inertia–gravity waves and, for higher Rossby numbers and weak stratification, carries a (baroclinic) hydraulic jump at its axis. For stronger stratifications and high enough Rossby numbers, ‘rider’ coherent structures appear as a result of adjustment, with a monopole in one layer and a dipole in another. Other ageostrophic coherent structures, such as two-layer tripoles and two-layer modons with nonlinear scatter plot, result from the collisions of ageostrophic modons. They are shown to be long-living and robust, and to capture waves.

Semi-Lagrangian exponential time-integration method for the shallow water equations on the cubed sphere grid

2020 ◽  
Vol 35 (6) ◽  
pp. 355-366
Author(s):  
Vladimir V. Shashkin ◽  
Gordey S. Goyman
Keyword(s):  
Shallow Water ◽  
Gravity Waves ◽  
Shallow Water Equations ◽  
Time Integration ◽  
Standard Test ◽  
Integration Scheme ◽  
Test Problems ◽  
Cubed Sphere ◽  
Lagrangian Scheme ◽  
Inertia Gravity Waves

AbstractThis paper proposes the combination of matrix exponential method with the semi-Lagrangian approach for the time integration of shallow water equations on the sphere. The second order accuracy of the developed scheme is shown. Exponential semi-Lagrangian scheme in the combination with spatial approximation on the cubed-sphere grid is verified using the standard test problems for shallow water models. The developed scheme is as good as the conventional semi-implicit semi-Lagrangian scheme in accuracy of slowly varying flow component reproduction and significantly better in the reproduction of the fast inertia-gravity waves. The accuracy of inertia-gravity waves reproduction is close to that of the explicit time-integration scheme. The computational efficiency of the proposed exponential semi-Lagrangian scheme is somewhat lower than the efficiency of semi-implicit semi-Lagrangian scheme, but significantly higher than the efficiency of explicit, semi-implicit, and exponential Eulerian schemes.

scholarly journals Analogy between electro-magnetic waves in cold unmagnetized plasma and shallow water inertio-gravity waves in geophysical systems

2021 ◽  
Author(s):  
E. Heifetz ◽  
L. R. M. Maas ◽  
J. Mak ◽  
I. Pomerantz
Keyword(s):  
Dispersion Relation ◽  
Shallow Water ◽  
Gravity Waves ◽  
Collisionless Plasma ◽  
Water System ◽  
Phase Speed ◽  
Rotating Shallow Water ◽  
Unmagnetized Plasma ◽  
Electro Magnetic ◽  
Magnetic Waves

Abstract The fundamental dispersion relation of transverse electro-magnetic waves in a cold collisionless plasma is formally equivalent to the two-dimensional dispersion relation of inertio-gravity waves in a rotating shallow water system, where the Coriolis frequency can be identified with the plasma frequency, and the shallow water gravity wave phase speed plays the role of the speed of light. Here we examine this equivalence and compare between the propagation wave mechanisms in these seemingly unrelated physical systems.

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

2001 ◽  
Vol 445 ◽  
pp. 93-120 ◽  
Author(s):  
G. M. REZNIK ◽  
V. ZEITLIN ◽  
M. BEN JELLOUL
Keyword(s):  
Shallow Water ◽  
Slow Component ◽  
Fast Component ◽  
Water Dynamics ◽  
Water Model ◽  
Rossby Number ◽  
Geostrophic Adjustment ◽  
Perturbation Expansions ◽  
Modulation Equation ◽  
Rotating Shallow Water

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 assumptions made are the well-defined scale of disturbance and the smallness of the Rossby number Ro. By systematically using the multi-time-scale perturbation expansions in Rossby number it is shown that the resulting field is split in a unique way into slow and fast components evolving with characteristic time scales f−10 and (f0Ro)−1 respectively, where f0 is the Coriolis parameter. The slow component is not influenced by the fast one and remains close to the geostrophic balance. The algorithm of its initialization readily follows by construction.The scenario of adjustment depends on the characteristic scale and/or initial relative elevation of the free surface ΔH/H0, where ΔH and H0 are typical values of the initial elevation and the mean depth, respectively. For small relative elevations (ΔH/H0 = O(Ro)) the evolution of the slow motion is governed by the well-known quasi-geostrophic potential vorticity equation for times t [les ] (f0Ro)−1. We find modifications to this equation for longer times t [les ] (f0Ro2)−1. The fast component consists mainly of linear inertia–gravity waves rapidly propagating outward from the initial disturbance.For large relative elevations (ΔH/H0 [Gt ] Ro) the slow field is governed by the frontal geostrophic dynamics equation. The fast component in this case is a spatially localized packet of inertial oscillations coupled to the slow component of the flow. Its envelope experiences slow modulation and obeys a Schrödinger-type modulation equation describing advection and dispersion of the packet. A case of intermediate elevation is also considered.

Spontaneous Imbalance and Hybrid Vortex–Gravity Structures

2009 ◽  
Vol 66 (5) ◽  
pp. 1315-1326 ◽  
Author(s):  
Michael E. McIntyre
Keyword(s):  
Shallow Water ◽  
Gravity Waves ◽  
Large Scale ◽  
General Rule ◽  
Radiation Reaction ◽  
Water Dynamics ◽  
Small Scale ◽  
Wave Source ◽  
Vortical Motion ◽  
Inertia Gravity Waves

Abstract After reviewing the background, this article discusses the recently discovered examples of hybrid propagating structures consisting of vortex dipoles and comoving gravity waves undergoing wave capture. It is shown how these examples fall outside the scope of the Lighthill theory of spontaneous imbalance and, concomitantly, outside the scope of shallow-water dynamics. Besides the fact that going from shallow-water to continuous stratification allows disparate vertical scales—small for inertia–gravity waves and large for vortical motion—the key points are 1) that by contrast with cases covered by the Lighthill theory, the wave source feels a substantial radiation reaction when Rossby numbers R ≳ 1, so that the source cannot be prescribed in advance; 2) that examples of this sort may supply exceptions to the general rule that spontaneous imbalance is exponentially small in R; and 3) that unsteady vortical motion in continuous stratification can stay close to balance thanks to three quite separate mechanisms. These are as follows: first, the near-suppression, by the Lighthill mechanism, of large-scale imbalance (inertia–gravity waves of large horizontal scale), where “large” means large relative to a Rossby deformation length LD characterizing the vortical motion; second, the flaccidity, and hence near-steadiness, of LD-wide jets that meander and form loops, Gulf-Stream-like, on streamwise scales ≫ LD; and third, the dissipation of small-scale imbalance by wave capture leading to wave breaking, which is generically probable in an environment of random shear and straining. Shallow-water models include the first two mechanisms but exclude the third.

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