scholarly journals Ellipticity Conditions of the Shallow Water Balance Equations for Atmospheric Data

2006 ◽  
Vol 63 (5) ◽  
pp. 1559-1566 ◽  
Author(s):  
Andrei Bourchtein
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Well Posedness ◽  
Nonlinear Normal Mode ◽  
Balance Equations ◽  
Pressure Fields ◽  
Atmospheric Data ◽  
Nonlinear Normal ◽  
Normal Mode Initialization ◽  
Orthogonal Coordinates

Abstract Nonlinear normal mode initialization equations, which provide required balance relations for atmospheric data, are considered in the generalized case of shallow water equations in arbitrary orthogonal coordinates. Using the concept of ellipticity in the sense of Douglis–Nirenberg, the conditions of well posedness of boundary value problems for balance equations are derived in the cases of constrained streamfunction, constrained potential vorticity, and constrained pressure fields.

The equatorial counterpart of the quasi-geostrophic model

2009 ◽  
Vol 637 ◽  
pp. 327-356 ◽  
Author(s):  
JÜRGEN THEISS ◽  
ALI R. MOHEBALHOJEH
Keyword(s):  
Shallow Water Equations ◽  
Analytic Form ◽  
Equatorial Region ◽  
Rossby Number ◽  
Nonlinear Normal Mode ◽  
Geometric Framework ◽  
Balanced Model ◽  
Nonlinear Normal ◽  
Normal Mode Initialization ◽  
Standard Derivation

A uniformly valid balanced model that represents the quasi-geostrophic model's counterpart in the equatorial region is derived. The quasi-geostrophic model itself fails in the equatorial region because it is only valid where the dominant balance is geostrophic, i.e. where the Rossby number is small. The smallness of the Rossby number is assumed in the quasi-geostrophic model's standard derivation and therefore this derivation cannot be repeated for the equatorial region. An alternative derivation of the quasi-geostrophic model that is independent of the Rossby number was presented by Leith in 1980, using the geometric framework of nonlinear normal mode initialization. Its independence of the Rossby number allows it to be repeated for the equatorial region, leading to an equatorial balanced model that thus represents the equatorial counterpart of the quasi-geostrophic model. As such it also coincides with the quasi-geostrophic model sufficiently far away from the equator. Its dispersion relation can be expressed in an explicit analytic form and, compared to that of other balanced models of similar simplicity, approximates that of the shallow water equations strikingly well.

scholarly journals IMPLICIT NONLINEAR NORMAL MODE INITIALIZATION FOR A BAROTROPIC PRIMITIVE EQUATION LIMITED AREA MODEL

MAUSAM ◽  
2021 ◽  
Vol 44 (1) ◽  
pp. 1-8
Author(s):  
D. R. C. NAIR ◽  
B. CHAKRAVARTY ◽  
P. NIYOGI
Keyword(s):  
Normal Mode ◽  
Shallow Water Equations ◽  
Equation Model ◽  
Limited Area ◽  
Nonlinear Normal Mode ◽  
Model Formulation ◽  
Primitive Equation ◽  
Level Model ◽  
Nonlinear Normal ◽  
Normal Mode Initialization

 A simple version of implicit nonlinear normal mode initialization is applied to a limited area one-level primitive equation model over a tropical domain. The model formulation is based on shallow water equations in spherical co-ordinate and potential enstrophy conserving finite difference scheme is employed. The model is used for predicting the movement of a typical monsoon depression formed over the Bay of Bengal. The above scheme is found to be very effective as it requires only three iterations for attaining balance between the mass and wind tields. However this model is not able to predict the movement of the depression very ac-curately due to the limitations of such a one-level model.

scholarly journals A Quasi-Variational Algorithm for Nonlinear Normal-Mode Initialization

2010 ◽  
Vol 138 (3) ◽  
pp. 951-961
Author(s):  
Andrei Bourchtein
Keyword(s):  
Partial Differential Equations ◽  
Normal Mode ◽  
Iterative Algorithms ◽  
Nonlinear Normal Mode ◽  
Balance Equations ◽  
Good Balance ◽  
Variational Approaches ◽  
Nonlinear Normal ◽  
The Given ◽  
Normal Mode Initialization

Abstract Balance equations of normal-mode initialization are nonlinear time-independent partial differential equations solved by iterative methods. For the given geopotential, there are regions where these equations are not elliptic, which is reflected in the divergence of iterative algorithms. Variational approaches used to minimize the geopotential changes are more expensive than conventional methods. In this study a simple quasi-variational algorithm is proposed based on different forms of normal-mode initialization equations, which achieves a good balance of atmospheric fields and ensures small changes of geopotential analysis values.

scholarly journals On the Well-Posedness for the Viscous Shallow Water Equations

2008 ◽  
Vol 40 (2) ◽  
pp. 443-474 ◽  
Author(s):  
Qionglei Chen ◽  
Changxing Miao ◽  
Zhifei Zhang
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Well Posedness

Balance dynamics in rotating stratified turbulence

2016 ◽  
Vol 795 ◽  
pp. 914-949 ◽  
Author(s):  
Hossein A. Kafiabad ◽  
Peter Bartello
Keyword(s):  
Energy Spectrum ◽  
Rossby Number ◽  
Stratified Turbulence ◽  
Nonlinear Normal Mode ◽  
Initial Flow ◽  
First Order ◽  
Initialization Scheme ◽  
Nonlinear Normal ◽  
Balance Dynamics ◽  
Normal Mode Initialization

If classical quasigeostrophic (QG) flow breaks down at smaller scales, it gives rise to questions of whether higher-order nonlinear balance can be maintained, to what scale and for how long. These are naturally followed by asking how this is affected by stratification and rotation. To address these questions, we perform non-hydrostatic Boussinesq simulations where the initial data is balanced using the Baer–Tribbia nonlinear normal mode initialization scheme (NNMI), which is accurate to second order in the Rossby number, as the next-order improvement to first-order QG theory. The NNMI procedure yields an ageostrophic contribution to the energy spectrum that has a very steep slope. However, as time passes, a shallow range emerges in the ageostrophic spectrum when the Rossby number is large enough for a given Reynolds number. It is argued that this shallow range is the unbalanced part of the motion that develops spontaneously in time and eventually dominates the energy at small scales. If the initial flow is not nonlinearly balanced, the shallow range emerges at even lower Rossby number and it appears at larger scales. Through numerous simulations at different rotation and stratification, this study gives a clear picture of how energy is cascaded in different initially balanced regimes of rotating stratified flow. We find that at low Rossby number the flow mainly consists of a geostrophic part and a balanced ageostrophic part with a steep spectrum. As the Rossby number increases, the unbalanced part of the ageostrophic energy increases at a rate faster than the balanced part. Hence, the total energy spectrum displays a shallow range above a transition wavenumber. This wavenumber evolves to smaller values as rotation weakens.

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