Instabilities and waves on a columnar vortex in a strongly-stratified and rotating fluid

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Stratorotational instability

Figure 1.17: (a) Two boundary modes. Vectors represent the horizontal velocity and contour plots in color represent the vertical velocity (Molemaker et al., 2001). (b) Kalliroscope visual-ization of the stratorotational instability in the vertically stratified salt water (Le Bars & Le Gal, 2007). The stratorotational instability (SRI) is an instability of the Taylor-Couette flow which appears if the fluid is stratified. It occurs outside the regime of the centrifugal instability: 2 where = o= i and = ri=ro are the ratios of the angular velocities and the radii of the inner and outer cylinders. Molemaker et al. (2001); Yavneh et al. (2001) have shown that the SRI exists in the regime 2 < 1 in inviscid fluids. The instability is due to a resonance between two boundary waves which exist in the presence of density stratification. These waves are trapped near the cylinders by the shear of the mean flow (figure 1.17a). The SRI has been observed in viscous fluids in both numerical studies (Shalybkov & Rüdiger, 2005) and experiments (Le Bars & Le Gal, 2007). As shown in figure 1.17(b) of the experiment by Le Bars & Le Gal (2007), the SRI is non-axisymmetric unlike the centrifugal instability which forms axisymmetric vortices known as the Taylor vortices (Andereck et al., 1986). The stratorotational instability of the stratified Taylor-Couette flow has been shown to be related to the radiative instability by Le Dizès & Riedinger (2010). In this thesis, we will investigate the stability of the Taylor-Couette flow in the regime > 1. We shall show that it is unstable to the SRI.

Structure of the thesis

In chapter 2, the numerical and asymptotic methods used to investigate the linear stability of the different basic flows are presented. In chapter 3, the stability of the Rankine vortex in a strongly stratified and rotating fluid is investigated. This chapter focuses on the effect of a cyclonic rotation but the effect of a weak anticyclonic rotation is also presented to describe the competition between the radiative instability and the centrifugal instability. The stability of a smoothed Rankine vortex and the Lamb-Oseen vortex are also investigated briefly. In chapter 4, the stability of the Rankine vortex in the regime 1 Ro < 0 is investigated. We show that the radiative instability also occurs in this range. The stability of the stratified Taylor-Couette flow in the regime > 1 is also investigated in this chapter since this regime is analogous to the regime 1 Ro < 0 for a vortex. In chapter 5, conclusions and discussions are established. The appendix presents some validations of the numerical codes.

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Chebyshev spectral collocation method

In the presence of viscosity, we use a spectral method. One of the advantages of the spectral method is that we do not need a guess value for the eigenfrequency since the global spectrum is computed (Schmid & Henningson, 2001). The collocation points are distributed along radial direction r using the Chebyshev discretization. The MATLAB differentiation matrix suite for the Chebyshev polynomials written by Weideman & Reddy (2000) is used. The Chebyshev collocation points sj 2 [ 1; 1] are defined as sj = cos j 1) ; j = 1; : : : ; N; ( (2.31) N 1. where N is the number of collocation points (Weideman & Reddy, 2000). For a vortex, we use the algebraic mappings from s to r as r(s) = p s ; or (2.32a) 1 s2 r(s) = s ; (2.32b) 1 s2. where is the stretching factor. These mappings have been used previously in various works (Antkowiak, 2005; Fabre & Jacquin, 2004; Fabre et al., 2006; Riedinger et al., 2010b). Note that r varies in the interval r 2 [ ; 1] and not in the interval [0; 1]. The advantage of these mappings is that using the parity properties [ur( r); u ( r)] = ( 1)jmj [ur(r); u (r)] ; (2.33) [uz( r); p( r); ( r)] = ( 1)jmj [uz(r); p(r); (r)] ; (2.34) we do not need to impose any boundary conditions at the vortex center r = 0. The Cheby-shev expansions are of order 2N but there are N collocation points (Fabre & Jacquin, 2004; Antkowiak, 2005). Additionally, if we introduce an angle , we can avoid the singularities on the real axis by integrating in the complex plane r0 = rei and eigenvalues can be computed more efficiently (Fabre et al., 2006; Riedinger et al., 2010b).

Table of contents :

1 Introduction 
1 Introduction to Geophysical Fluid Dynamics
2 Objectives of the thesis
3 Structure of the thesis
2 Problem formulation 
1 Equations of motions under the Boussinesq approximation
2 Linear stability equations
3 Boundary conditions
4 Numerical methods
5 The WKBJ approximation
3 Effect of rotation on the radiative instability 
1 Instabilities and waves on a columnar vortex in a strongly-stratified and rotating fluid
2 Stability of the cyclonic Lamb-Oseen vortex
4 Effect of strong anticyclonic rotation with 􀀀1 < Ro < 0 71
1 Radiative instability of an anticyclonic vortex in a stratified rotating fluid
2 Stability of the stratified Taylor-Couette flow
5 Conclusions and perspectives 
Appendix

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