Pizza: an open-source pseudo-spectral code for spherical quasi-geostrophic convection

Abstract

We present a new pseudo-spectral open-source code nicknamed pizza. It is dedicated to the study of rapidly-rotating Boussinesq convection under the 2-D spherical quasi-geostrophic approximation, a physical hypothesis that is appropriate to model the turbulent convection that develops in planetary interiors. The code uses a Fourier decomposition in the azimuthal direction and supports both a Chebyshev collocation method and a sparse Chebyshev integration formulation in the cylindrically-radial direction. It supports several temporal discretisation schemes encompassing multi-step time steppers as well as diagonally-implicit Runge-Kutta schemes. The code has been tested and validated by comparing weakly-nonlinear convection with the eigenmodes from a linear solver. The comparison of the two radial discretisation schemes has revealed the superiority of the Chebyshev integration method over the classical collocation approach both in terms of memory requirements and operation counts. The good parallelisation efficiency enables the computation of large problem sizes with $\mathcal{O}(10^4\times 10^4)$ grid points using several thousands of ranks. This allows the computation of numerical models in the turbulent regime of quasi-geostrophic convection characterised by large Reynolds $Re$ and yet small Rossby numbers $Ro$. A preliminary result obtained for a strongly supercritical numerical model with a small Ekman number of $10^{-9}$ and a Prandtl number of unity yields $Re\simeq 10^5$ and $Ro \simeq 10^{-4}$. pizza is hence an efficient tool to study spherical quasi-geostrophic convection in a parameter regime inaccessible to current global 3-D spherical shell models.