arXiv subjects

Timothy C. Andrews

Publications and source records attributed to Timothy C. Andrews.

5 recordsLinked to original sources

arXiv2026

A perfectly matched layer for damping vertically propagating waves in the compressible Boussinesq equations

This paper introduces a new application of the perfectly matched layer (PML) for mitigating model top wave reflections in geophysical fluid models. Typically, a strong Laplacian or Rayleigh damping sponge layer is used near the upper boundary, but these often need many vertical levels or a high model top to be sufficiently effective. An advantage of the PML is that, at the continuous level, it is free of wave reflection at the onset of the damping layer. This enables the PML to be effective even with a thin damping layer. We derive PMLs for the linear and nonlinear versions of the Boussinesq equations, which are a simplified model for vertical dynamics in the atmosphere. In the nonlinear system, we define a novel PML that damps perturbations from a hydrostatically balanced reference state. We approximate the PML equations using the compatible finite element method for numerical experiments. First, tests with the linear Boussinesq system show that the PML is more effective than a typical sponge layer in absorbing acoustic waves near the model top. Next, tests in the nonlinear system show that i) the PML can damp acoustic waves even when they are under-resolved by the time discretisation, and ii) the PML can avoid the standing wave pattern caused by model top reflection of orographic gravity waves. We propose that the PML is worth further development and investigation as a sponge layer alternative in dynamical cores for atmospheric modelling.

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arXiv2026

A conservative, discontinuous Galerkin, tracer transport scheme using compatible finite elements

This paper outlines a conservative transport scheme for scalar tracers within a compatible finite element model for geophysical fluid equations. Instead of using the advective transport equation for a mixing ratio, a conservative transport equation is solved for the tracer density of the mixing ratio multiplied by the dry density. This ensures mass conservation in the continuous equations, which can be preserved in the discrete equations with a discontinuous Galerkin transport scheme. Our method is designed to work for two placements of the mixing ratio in a Charney-Phillips vertical staggering: either co-located with the dry density or vertically staggered from it. The new scheme is designed to conserve the tracer density and ensure consistency by maintaining a constant mixing ratio. Additionally, a mass-conserving limiter is developed to ensure non-negativity in the co-located configuration. Tests with terminator toy chemistry and a moist rising bubble show the use of the new transport scheme with physics terms and its ability to accurately model mass conservation of moisture species in a dynamical core setup.

math.NA↗

arXiv2025

Stability analyses of divergence and vorticity damping on gnomonic cubed-sphere grids

Divergence and vorticity damping, which operate upon horizontal divergence and relative vorticity, are explicit diffusion mechanisms used in dynamical cores to ensure stability. To avoid numerical blow-up from excessively strong diffusion, there are mesh-dependent upper bounds on the coefficients of the diffusion operators. This work considers such stability limits for three gnomonic cubed-sphere meshes: the 1) equidistant, 2) equiangular, and 3) equi-edge mappings. Stability limits are derived from a von Neumann analysis of damping with a simplified pseudo-Laplacian operator, as used in NOAA GFDL's finite-volume dynamical core on the cubed-sphere (FV3), and with the full curvilinear Laplacian. The resulting stability limits depend on the gnomonic mapping through the cubed-sphere cell areas, aspect ratios, and grid nonorthogonality. The analytical stability limits are compared to practical divergence and vorticity damping upper bounds in FV3, using idealised tests and the equiangular and equi-edge grids. For divergence damping, both the magnitude of maximum stable coefficients and the locations of instability agree with linear theory. Due to implicit vorticity diffusion in the FV3 transport scheme, practical limits for vorticity damping are lower than the explicit stability limits and depend on the choice of horizontal transport scheme.

math.NA↗

arXiv2024

A mean correction for improved phase-averaging accuracy in oscillatory, multiscale, differential equations

This paper introduces a new algorithm to improve the accuracy of numerical phase-averaging in oscillatory, multiscale, differential equations. Phase-averaging is a timestepping method which averages a mapped variable to remove highly oscillatory linear terms from the differential equation. This retains the main contribution of fast waves on the low frequencies without explicitly resolving the rapid oscillations. However, this comes at the cost of introducing an averaging error. To offset this, we propose a modified mapping that includes a mean correction term encoding an average measure of the nonlinear interactions. This mapping was introduced in Tao (2019) for weak nonlinearity and relied on classical time-averaging, which leaves only the zero frequencies. Our algorithm instead considers mean corrected phase-averaging when 1) the nonlinearity is not weak but the linear oscillations are fast and 2) finite averaging windows are applied via a smooth kernel, which has the advantage of retaining low frequencies whilst still eliminating the fastest oscillations. In particular, we introduce a local mean correction that combines the concepts of a mean correction and finite averaging; this retains low-frequency components in the mean correction that are removed with classical time-averaging. We show that the new timestepping algorithm reduces phase errors in the mapped variable for the swinging spring ODE in various dynamical configurations. We also show accuracy improvements with a local mean correction compared to standard phase-averaging in the one-dimensional rotating shallow water equations, a useful test case for weather and climate applications.

math.NA↗

arXiv2024

The effect of linear dispersive errors on nonlinear timestepping accuracy in the f-plane rotating shallow water equations

For simulations of time evolution problems, such as weather and climate models, taking the largest stable timestep is advantageous for reducing wall-clock time. A drawback of doing so is the potential reduction in nonlinear accuracy of the numerical solution - we investigate this for the Rotating Shallow Water Equations (RSWEs) on an f-plane. First, we examine how linear dispersion errors can impact the nonlinear dynamics. By deriving an alternate time evolution equation for the RSWEs, the dynamics can be expressed through interactions of three linear waves in triads. Linear dispersion errors may appear in the numerical representation of the frequency of each triad, which will impact the timestepped nonlinear dynamics. A new triadic error quantifies this by composing three stability polynomials from the oscillatory Dahlquist test equation. Second, we design two new test cases to examine the effect of timestep size in a numerical model. These tests investigate how well a timestepper replicates slow nonlinear dynamics amidst fast linear oscillations. The first test case of a Gaussian height perturbation contains a nonlinear phase shift that can be missed with a large timestep. The second set of triadic test cases excite a few linear waves to instigate specific triadic interactions. Two triadic cases are examined: one with a dominant directly resonant triad and another with near-resonances that redistribute fast mode energy into rings in spectral space. Three numerical models, including LFRic from the Met Office, are examined in these test cases with different timesteppers.

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