Revisiting the Charney Baroclinic Instability Problem and Point-jet Barotropic Instability Problem, Part II: Matched Asymptotic Expansions & Overreflection Without Delta-Functions

Abstract

Abstract Baroclinic instability generates the cyclones and anticyclones of midlatitude weather. Charney developed the first effective theory for the infancy of this cyclogenesis in 1947. His linear eigenproblem is analytically solvable by confluent hypergeometric functions. It is also, with extension of the domain of the coordinate from [0,∞] to [−∞,∞] by reflection about the origin, the point-jet model of barotropic instability, important for tropical cyclogenesis. (Note that the coordinate is height z in the Charney model, but latitude y for the point-jet bartropic instability. It is a great simplification that the Charney and point-jet instability problems are mathematically identical, but it also is confusing that the mathematical analysis in y also applies to the Charney problem with the substitution of z for y.) Unfortunately, the theory is full of distributions like the Dirac delta-function and the reflected Charney eigenfunction has a discontinuous first derivative at y = 0. Here we regularize the Charney problem by replacing a linear mean current, U = |y|, by either U = є log(cosh(y/є)) or U = є y erf(y/є), followed by matched asymptotic perturbation expansions in powers of the small regularization parameter є. The series is carried to third order because the lowest nonzero correction to the phase speed is O(є2) and this correction is determined simultaneously with the third order approximation to the eigenfunction. The result is both an explicit, analytic regularization of a problem important in atmospheric and ocean dynamics, but also a good school problem because the series is explicit with nothing worse than polylogarithms and confluent hypergeometric functions. The primary meteorological conclusion is that the delta functions in the Charney problem are harmless as demonstrated both by third order perturbation theory and by spectrally-accurate numerical solutions. The physics of the regularized Charney problem is not significantly changed from that of the original Charney problem.