Relating Overreflection and Wave Geometry to the Counterpropagating Rossby Wave Perspective: Toward a Deeper Mechanistic Understanding of Shear Instability

The two major approaches to understanding plane parallel shear instability—the counterpropagating Rossby wave (CRW) and overreflection (OR) 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 OR theory in terms of CRW thinking, using a generalization of the CRW approach to multiple infinitesimal potential vorticity (PV) kernels. First the mechanics 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 overreflection) is 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 nondimensional 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 there is a mutual amplification between two adjacent kernels. A further examination of the sources of energy growth yields a generalized view of the restoking of the Orr mechanism, thought to be central to OR: growth depends explicitly on the shear, as in the Orr mechanism, while mutual PV kernel interactions are responsible for holding the perturbation at a configuration that allows it to contribute to growth (a restoking).