Relating Over-Reflection and Wave Geometry to the Counter Propagating Rossby Wave Perspective: Toward a Deeper Mechanistic Understanding of Shear Instability

The two major approaches to understanding plane parallel shear instability - the Counter propagating Rossby Wave (CRW) and Over-Reflection (O-R) approaches, illuminate very different fundamental aspects. This work examines how such seemingly different views can explain the same phenomena and why these differences arise, with the overall goal of deepening our understanding of shear instability. This is done by rationalizing the O-R theory in terms of CRW thinking, using a generalization of the CRW approach to multiple infinitesimal PV kernels - a Green function approach. First the mechanistics of cross-shear wave propagation and the exponential decay in an evanescent region are qualitatively explained. Then the cross-shear behavior in different wave geometries (full, partial and over reflection) are examined. It is found that overreflection is basically a mutual amplification between PV kernels, with the main kernels being at the two sides of the evanescent region, which forms on one side of the critical level. The nondimentional overreflection coefficient is obtained by assuming a balance between the mutual amplification on the two sides of the evanescent region, and the rate at which this amplification is carried away. The critical level is found to be the only point at which we have a mutual interaction between two adjacent kernels. We will also examine the sources of energy growth and the role of the Orr mechanism, thought to be central to O-R.