Simulating convection at extreme parameters on a logarithmic Fourier lattice

Our ability to study numerically the scaling properties of turbulent convection is limited by the high cost of direct numerical simulations (DNS) as the Rayleigh number (Ra) increases. This motivates the exploration of alternatives to DNS which enable faster computation by using reduced models of the full dynamics. Here we explore the use of logarithmic Fourier lattices (LFL) to capture extreme dynamic ranges of spatial scales in Rayleigh-Benard convection (RBC) at high Ra. The LFL scheme uses a Fourier series with logarithmically rather than linearly distributed wavenumbers. This scheme exactly captures the dynamics of constant-coefficient linear operators, but approximates nonlinear operators with a finite number of lattice-supported triads. By combining an LFL horizontal discretization with a sparse Chebyshev method in the vertical, we can simulate RBC at a substantially reduced cost compared to DNS. We will discuss ongoing work to implement efficient mixed LFL-Chebyshev solvers in 2D and 3D, along with results from RBC simulations in various parameter regimes. We compare these simulations to DNS results and assess their suitability for extrapolation beyond the current capabilities of DNS.