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# Advances in Rapid Distortion Theory: From Rotating Shear Flows to the Baroclinic Instability

[+] Author and Article Information
Aziz Salhi

Département de Physique, Campus Universitaire de Tunis, 1060 Tunis, Tunisie

Claude Cambon

Laboratoire de Mécanique des Fluides et d’Acoustique, U.M.R. CNRS 5509, Ecole Centrale de Lyon, 69134 Ecully Cedex, Francecambon@mecaflu.ec-lyon.fr

In fact, both nonlinearity and nonlocality have to be considered, particularly the nonlocal relationship of pressure to velocity.

$Gij$ for fixed $j$ satisfy the same equations as $ûi$. A different initialization $Gij=δij$ was prescribed by Townsend. Eq. 5 presents some advantages, since $kiGij=0$ can be satisfied at any time, and the RDT Green’s function can be more easily related to Kraichnan’s response function.

Given $R̂ij$, for instance, obtained by RDT started with isotropic initial data, evaluation of the integral over 3D Fourier space can be a nontrivial task. For example, the RDT Green’s function is determined analytically in the case of simple shear, but the integrals in Eq. 2 are not straightforward and must be evaluated numerically or asymptotically (40).

J. Appl. Mech 73(3), 449-460 (Oct 07, 2005) (12 pages) doi:10.1115/1.2150234 History: Received December 10, 2003; Revised October 07, 2005

## Abstract

The essentials of rapid distortion theory (RDT) are briefly recalled for homogeneous turbulence subjected to rotational mean flows, including its linkage to stability analysis. The latter “linkage” is of particular importance from our viewpoint, since it also attracted the attention of Charles Speziale, resulting in at least two papers [Speziale, C. G., Abid, R., and Blaisdell, G. A., 1996, “On the Consistency of Reynolds Stress Turbulence Closures With Hydrodynamic Stability Theory  ,” Phys. Fluids, 8, pp. 781–788 and Salhi, A., Cambon, C., and Speziale, C. G., 1997, “Linear Stability Analysis of Plane Quadratic Flows in a Rotating Frame  ,” Phys. Fluids, 9(8), pp. 2300–2309] with particular emphasis on rotating flows. New analytical solutions and related RDT results are presented for shear flows including buoyancy forces, with system rotation or mean density stratification. Finally, combining shear, rotation and stratification, RDT is shown to be pertinent to revisiting the baroclinic instability. This instability results from the tilting of mean isopycnal surfaces under combined effects of vertical shear and system rotation, in a vertically (stably) stratified medium rotating around the vertical direction. In addition, the challenge of reproducing RDT dynamics in single-point closure models is briefly discussed, from the viewpoint of structure-based modeling [Cambon C., Jacquin, L., and Lubrano, J.-L., 1992, “Towards a New Reynolds Stress Model for Rotating Turbulent Flows  ,” Phys. Fluids A, 4, pp. 812–824 and Kassinos, S. C., Reynolds, W. C., and Rogers, M. M., 2000, “One-Point Turbulence Structure Tensors  ,” J. Fluid Mech., 428, pp. 213–248.

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## Figures

Figure 1

Decomposition onto four subdomains of the circular area k12+k22⩽K2sin2θ in the (k1,k2) plane; μ (or γ) is complex in the dashed areas

Figure 7

Neutral curves ω0=0 in the (Ri,θ) plane for the k1=0 mode, at several values of ϵ

Figure 8

Polar-spherical system of coordinates for k and related Craya–Herring frame of reference

Figure 2

Time evolution of the (symmetrized) components T11(lin)T22(lin) and T13(lin) in the stratified shear case (lines) and in the rotating shear case (symbols) at B=1∕4 and Ri=1∕8 (corresponding to φc=π∕4)

Figure 3

Time evolution of the shear stress component u1′u3′¯∕q02 (“total”) and the parts u1′u3′¯(c)∕Sq02 (complex) and u1′u3′¯(r)∕Sq02 (real) in the stratified shear case at (Ri=1∕8)

Figure 4

Time evolution of the (symmetrized) component T13(lin)∕(Sq02) (“total”) and the parts T13(lin)(c)∕(Sq02) (complex) and T13(lin)(r)∕(Sq02) (real) at B=1∕4

Figure 5

Sketch of the mean flow including system rotation (a), mean vertical density gradient (b) and mean shear (c), with possible tilting of isopycnal lines

Figure 6

Neutral curves ω0=0 in the (Ri,θ) plane for the k1=0 mode, at several values of Ro=S∕f

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