tokens: RWI is an analysis of a configuration with only one local pressure extremum). This analysis of the barotropic shear layer model gets simplified if we implement the strategy used in Umurhan ( 2010 ), wherein is assumed. In this limit, the continuous shear layer flow profile becomes an analytically tractable problem comprised of three piecewise linear shear profiles. The solution procedure follows the steps detailed in the Appendix relating to the same type of analysis done on the asymmetric jet (examined in the next section). 12 12 It ought to be noted that there are subtleties introduced because of the use of piecewise linear profiles. For instance, it is sometimes the case that piecewise linear profiles predict modes that do necessarily have counterparts in analysis of flow profiles that are infinitely differentiable. The normal modes generated herein, through the use of these piecewise linear representations, all have counterpart modes in their corresponding infinitely differentiable flow profiles. For the shear layer considered here it is a slight generalization of Rayleigh’s shear layer analysis (Rayleigh 1880 ) with the additional superposition of the background Keplerian shear profile. The resulting frequency response is given byInspection shows that this system becomes unstable when ω goes through zero. The physical condition for instability is the phase locking of two counter-propagating Rossby waves (Baines Mitsudera 1994 ; Heifetz et al. 1999 ). By setting we find the condition for marginal growth to be given by where is the shear layer amplitude corresponding to marginality. We do not consider the positive roots, , because these generally correspond to composite flows that are strongly cyclonic, i.e., flows in which while the negative root, on the other hand, often corresponds to anti-cyclonic shear profiles. 13 13 This is generally true on examination of the marginal condition in Equation ( 53 ). If we consider the setting examined in M13 , we suppose that the smallest azimuthal wavenumber is the one fitting the box considered therein. The corresponding azimuthal length scale in their box units is approximately . This corresponds to a fundamental azimuthal wavenumber . Similarly, the width of the shear line charge is approximately 0.28 (also in their code units). Translating this into our setup, the total width is , and thus we adopt a value of . In this case, the corresponding critical value of for the R-mode is given from Equation ( 53 ) by , with a corresponding critical Rossby number of . Values of should then be stable to the RWI. We find that the value of predicted in Equation ( 53 ) generally predicts more negative values of that are borne out in our models. The reason for this is that we adopt non-zero values of () for all the models reported in this section. The discrepency is generally 2%–5% of the predicted values. 14 14 We have separately checked (but have not included in this manuscript) and verified that as the model
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