Parameterized Ekman boundary layers on the tilted $f$-plane

Abstract

Rotating convection is considered on the tilted $f$-plane where gravity and rotation are not aligned. For sufficiently large rotation rates, $\Omega$, the Taylor-Proudman effect results in the gyroscopic alignment of anisotropic columnar structures with the rotation axis giving rise to rapidly varying radial length scales that vanishes as $\Omega^{-1/3}$ for $\Omega\rightarrow\infty$. Compounding this phenomenon is the existence of viscous (Ekman) layers adjacent to the impenetrable bounding surfaces that diminish in scale as $\Omega^{-1/2}$. In this investigation, these constraints are relaxed upon utilizing a non-orthogonal coordinate representation of the fluid equations where the upright coordinate aligns with rotation axis. This exposes the problem to asymptotic perturbation methods that permit: (i) relaxation of the constraints of gyroscopic alignment; (ii) the filtering of Ekman layers through the uncovering of parameterized velocity pumping boundary conditions; and (iii) the development of reduced quasi-geostrophic systems valid in the limit $\Omega\rightarrow\infty$. Linear stability investigations reveal excellent quantitative agreement between results from parameterized or unapproximated mechanical boundary conditions. For no-slip boundaries, it is demonstrated that the associated Ekman pumping dramatically alters convective onset through an enhanced destabilization of large spatial scales. The range of unstable modes at a fixed thermal forcing is thus significantly extended with a direct dependence on $\Omega$. This holds true even for geophysical and astrophysical regimes characterized by extreme values of the non-dimensional Ekman number $E$. The nonlinear regime is explored via the global heat and momentum transport of single-mode solutions to the quasi-geostrophic systems which indicate $\mathcal{O}(1)$ changes irrespective of the smallness of $E$.