arXiv subjects

Beth A. Wingate

Publications and source records attributed to Beth A. Wingate.

7 recordsLinked to original sources

arXiv2025

Quasi-geostrophic limiting dynamics and energetics of the LANS-$α$ model

The Lagrangian-Averaged Navier-Stokes-$α$ (LANS-$α$) model, a turbulence closure scheme based on energy-conserving modifications to nonlinear advection, can produce more energetic simulations than standard models, leading to improved fidelity (e.g., in ocean models). However, comprehensive understanding of the mechanism driving this energetic enhancement has proven elusive. To address this gap, we derive the fast quasi-geostrophic limit of the three-dimensional, stably-stratified LANS-$α$ equations. This provides both the slow, balanced flow and the leading-order fast wave dynamics. Analysis of these wave dynamics suggests that an explanation for the energetic enhancement lies in the dual role of the smoothing parameter itself: increasing $α$ regularizes the dynamics and simultaneously generates a robust landscape of wave-wave resonant interactions. Direct numerical simulations show that $α$ plays an analogous role to that of the Burger number ($Bu$) in governing the partition of energy between slow and fast modes -- and consequently, the timescale of geostrophic adjustment -- but with key differences. Increasing $α$, regardless of the relative strengths of rotation and stratification, extends the lifetime of wave energy by delaying the dominance of the slow modes. We find that the creation of an energy pathway only involving fast waves is a universal outcome of the regularization across all values of $Bu$, contrasting with a disruption of slow-fast interactions that is most impactful only in the $Bu=1$ case. These insights unify the LANS-$α$ model's characteristic energetic enhancement with, in some cases, its known numerical stiffness, identifying potential pathways to mitigate stability issues hindering the broader application of LANS-$α$-type models.

physics.flu-dyn↗

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.

math.NA↗

arXiv2012

Spontaneous Formation of Columnar Vortices

A fluid dynamics video of the rotating, weakly stratified Boussinesq equations is presented that illustrates the spontaneous formation of columnar vortices in the presence of stochastic, white noise forcing.

physics.flu-dyn↗

arXiv2011

Low Rossby limiting dynamics for stably stratified flow with Finite Froude number

In this paper we explore the fast rotation, nonhydrostatic limit of the rotating and stratified Boussinesq equations. We derive new reduced equations for the slow dynamics that describe Taylor-Proudman flows. One new aspect of the dynamics is a decoupling of the horizontal kinetic energy, described by 2D Navier-Stokes, from new dynamics that describe the coupling of vertical kinetic energy and buoyancy. We support the theory with high resolution numerical simulations of the full Boussinesq equations that, in this limit, reveal the spontaneous formation of Taylor-Proudman columns and their dynamics.

nlin.CD↗

arXiv2007

Implementation of the LANS-alpha turbulence model in a primitive equation ocean model

This paper presents the first numerical implementation and tests of the Lagrangian-averaged Navier-Stokes-alpha (LANS-alpha) turbulence model in a primitive equation ocean model. The ocean model in which we work is the Los Alamos Parallel Ocean Program (POP); we refer to POP and our implementation of LANS-alpha as POP-alpha. Two versions of POP-alpha are presented: the full POP-alpha algorithm is derived from the LANS-alpha primitive equations, but requires a nested iteration that makes it too slow for practical simulations; a reduced POP-alpha algorithm is proposed, which lacks the nested iteration and is two to three times faster than the full algorithm. The reduced algorithm does not follow from a formal derivation of the LANS-alpha model equations. Despite this, simulations of the reduced algorithm are nearly identical to the full algorithm, as judged by globally averaged temperature and kinetic energy, and snapshots of temperature and velocity fields. Both POP-alpha algorithms can run stably with longer timesteps than standard POP. Comparison of implementations of full and reduced POP-alpha algorithms are made within an idealized test problem that captures some aspects of the Antarctic Circumpolar Current, a problem in which baroclinic instability is prominent. Both POP-alpha algorithms produce statistics that resemble higher-resolution simulations of standard POP. A linear stability analysis shows that both the full and reduced POP-alpha algorithms benefit from the way the LANS-alpha equations take into account the effects of the small scales on the large. Both algorithms (1) are stable; (2) make the Rossby Radius effectively larger; and (3) slow down Rossby and gravity waves.

physics.ao-ph↗

arXiv2005

Several new quadrature formulas for polynomial integration in the triangle

We present several new quadrature formulas in the triangle for exact integration of polynomials. The points were computed numerically with a cardinal function algorithm which imposes that the number of quadrature points $N$ be equal to the dimension of a lower dimensional polynomial space. Quadrature forumulas are presented for up to degree $d=25$, all which have positive weights and contain no points outside the triangle. Seven of these quadrature formulas improve on previously known results.

math.NA↗