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.

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.

Incremental Nonlinear Normal-Mode Initialization

1992 ◽  
Vol 120 (8) ◽  
pp. 1723-1734 ◽  
Author(s):  
Bradley Ballish ◽  
Xianhe Cao ◽  
Eugenia Kalnay ◽  
Masao Kanamitsu
Keyword(s):  
Normal Mode ◽  
Nonlinear Normal Mode ◽  
Nonlinear Normal ◽  
Normal Mode Initialization

Initialization Procedures

1998 ◽  
Author(s):  
T. N. Krishnamurti ◽  
H. S. Bedi ◽  
V. M. Hardiker
Keyword(s):  
Normal Mode ◽  
High Frequency ◽  
Normal Modes ◽  
Nonlinear Normal Mode ◽  
Model Equations ◽  
Nonlinear Normal ◽  
Number Of Cycles ◽  
Physical Initialization ◽  
Normal Mode Initialization ◽  
Initialization Procedure

In this chapter we describe two of the most commonly used initialization procedures. These are the dynamic normal mode initialization and the physical initialization methods. Historically, initialization for primitive equation models started from a hierarchy of static initialization methods. These include balancing the mass and the wind fields using a linear or nonlinear balance equation (Charney 1955; Phillips 1960), variational techniques for such adjustments satisfying the constraints of the model equations (Sasaki 1958), and dynamic initialization involving forward and backward integration of the model over a number of cycles to suppress high frequency gravity oscillations before the start of the integration (Miyakoda and Moyer 1968; Nitta and Hovermale 1969; Temperton 1976). A description of these classical methods can be found in textbooks such as Haltiner and Williams (1980). Basically, these methods invoke a balanced relationship between the mass and motion fields. However, it was soon realized that significant departures from the balance laws do occur over the tropics and the upperlevel jet stream region. It was also noted that such departures can be functions of the heat sources and sinks and dynamic instabilities of the atmosphere. The procedure called nonlinear normal mode initialization with physics overcomes some of these difficulties. Physical initialization is a powerful method that permits the incorporation of realistic rainfall distribution in the model’s initial state. This is an elegant and successful initialization procedure based on selective damping of the normal modes of the atmosphere, where the high-frequency gravity modes are suppressed while the slow-moving Rossby modes are left untouched. Williamson (1976) used the normal modes of a shallow water model for initialization by setting the initial amplitudes of the high frequency gravity modes equal to zero. Machenhauer (1977) and Baer (1977) developed the procedure for nonlinear normal mode initialization (NMI), which takes into account the nonlinearities in the model equations. Kitade (1983) incorporated the effect of physical processes in this initialization procedure. We describe here the normal mode initialization procedure. Essentially following Kasahara and Puri (1981), we first derive the equations for vertical and horizontal modes of the linearized form of the model equations.

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