Beyond Long's solution: a Newton solver for nonlinear mountain waves with rotation

Much of our basic understanding of mountain waves comes from the steady-state theory, with the nonlinear analysis of Long playing a prominent role. However, Long's theory is valid only for a relatively limited range of parameter space---specifically, 2D nonrotating flows with uniform upstream states.

The present study describes a method for obtaining nonlinear steady-state solutions under more general flow conditions, with a particular emphasis on 2D rotating flows. The method is based on a Newton iteration applied to a finite-difference discretization in terrain-following coordinates. Convergence of the method is shown to be robust and computationally efficient for all test cases considered. For sufficiently large domain sizes, the method is insensitive to boundary conditions, thus effectively providing an extension of Long's theory to more general cases.

Calculations are presented to explore the nonlinear, rotating parameter space for the case of 2D flow past an isolated ridge. The governing control parameters are then the nondimensional terrain height (Nh/U) and the Rossby number (Ro = U/fL). The critical Nh/U for steady-state wave overturning is computed as a function of Ro, and the surface drag is mapped as a function of Ro and Nh/U. Comparisons are made to time-dependent model integrations so as to characterize the approach to steady state.

In a companion study, the method is used to provide steady-state background solutions for use in normal-mode instability calculations.