Parameterized Ekman boundary layers on the tilted <i>f</i>-plane

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 scale as $\Omega ^{-1/2}$ . In this investigation, these constraints are relaxed upon utilising 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 parameterised 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 parameterised or unapproximated mechanical boundary conditions. For no-slip boundaries, it is demonstrated that the associated Ekman pumping alters convective onset through an enhanced destabilisation 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 characterised 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 $O(1)$ changes which do not scale with the size of $E$ .