Topographic drag in rapidly rotating fluids

Flows over topography in stratified fluids have long been studied in oceanography and atmospheric sciences, where internal gravity waves play a major role in energy dissipation, mixing and angular-momentum transfer. By contrast, the impact of topography in rapidly rotating homogeneous fluids remains poorly understood, despite its potential importance in well-mixed rotating fluid layers (e.g. planetary liquid cores or vigorously convecting subsurface oceans). In particular, predictive scaling laws for the drag generated by finite-amplitude topography are still lacking, even in the simplest case of a uniform steady flow past a topography. Here, we investigate the pressure drag produced by a uniform flow over a three-dimensional periodic topography in a rapidly rotating Cartesian fluid layer using asymptotic analysis and direct numerical simulations. We show that the topography excites inertial waves, which radiate energy into the fluid interior and generate a wave-driven pressure drag upper D D D . In the shallow-slope limit left parenthesis h 0 k much less than 1 right parenthesis ( h 0 k ≪ 1 ) (h₀k 1) , we recover the classical scaling law upper D proportional to h 0 squared k D ∝ h 0 2 k D ∝ h₀² k . We then extend this theory to finite-slope topographies left parenthesis h 0 k greater than or equivalent to 1 right parenthesis ( h 0 k ≳ 1 ) (h₀k 1) in the quasi-geostrophic regime left parenthesis upper U 0 k much less than upper Omega right parenthesis ( U 0 k ≪ Ω ) (U₀k ) , deriving a correction that quantitatively agrees with direct numerical simulations. In the asymptotic steep limit left parenthesis h 0 k much greater than 1 right parenthesis ( h 0 k ≫ 1 ) (h₀k 1) , the drag becomes independent of topographic height and scales as upper D proportional to k Superscript negative 1 <m