Ian Grooms

Statistical and physical balances in low Rossby number Rayleigh–Bénard convection

Rapidly rotating Rayleigh–Bénard convection is studied using an asymptotically reduced equation set valid in the limit of low Rossby numbers. Four distinct dynamical regimes are identified: a disordered cellular regime near threshold, a regime of weakly interacting convective Taylor columns at larger Rayleigh numbers, followed for yet larger Rayleigh numbers by a breakdown of the convective Taylor columns into a disordered plume regime characterized by reduced efficiency and finally by geostrophic turbulence. The transitions are quantified by examining the properties of the horizontally and temporally averaged temperature and thermal dissipation rate. The maximum of the thermal dissipation rate is used to define the width of the thermal boundary layer. In contrast to the non-rotating Rayleigh–Bénard convection, the temperature drop across this layer decreases monotonically with increasing Rayleigh number and does not saturate. The breakdown of the convective Taylor column regime is attributed to the onset of convective instability of the thermal boundary layer and confirmed using the explicit linear stability analysis. Horizontal spectra of the vorticity, vertical velocity and temperature fluctuations are computed and their evolution with time is elucidated. A large-scale barotropic mode evolves from random initial conditions on an extremely long time scale and leads to continued evolution of the nominally saturated Nusselt number and its variance over very long times. The results are used to provide insights into the dynamics of rapidly rotating convection outside the asymptotic regime described by the reduced equations.

Efficient stochastic superparameterization for geophysical turbulence

Efficient computation of geophysical turbulence, such as occurs in the atmosphere and ocean, is a formidable challenge for the following reasons: the complex combination of waves, jets, and vortices; significant energetic backscatter from unresolved small scales to resolved large scales; a lack of dynamical scale separation between large and small scales; and small-scale instabilities, conditional on the large scales, which do not saturate. Nevertheless, efficient methods are needed to allow large ensemble simulations of sufficient size to provide meaningful quantifications of uncertainty in future predictions and past reanalyses through data assimilation and filtering. Here, a class of efficient stochastic superparameterization algorithms is introduced. In contrast to conventional superparameterization, the method here (i) does not require the simulation of nonlinear eddy dynamics on periodic embedded domains, (ii) includes a better representation of unresolved small-scale instabilities, and (iii) allows efficient representation of a much wider range of unresolved scales. The simplest algorithm implemented here radically improves efficiency by representing small-scale eddies at and below the limit of computational resolution by a suitable one-dimensional stochastic model of random-direction plane waves. In contrast to heterogeneous multiscale methods, the methods developed here do not require strong scale separation or conditional equilibration of local statistics. The simplest algorithm introduced here shows excellent performance on a difficult test suite of prototype problems for geophysical turbulence with waves, jets, and vortices, with a speedup of several orders of magnitude compared with direct simulation.

Linearly implicit methods for nonlinear PDEs with linear dispersion and dissipation

The linear stability of IMEX (IMplicit-EXplicit) methods and exponential integrators for stiff systems of ODEs arising in the discrete solution of PDEs is examined for nonlinear PDEs with both linear dispersion and dissipation, and a clear method of visualization of the linear stability regions is proposed. Predictions are made based on these visualizations and are supported by a series of experiments on five PDEs including quasigeostrophic equations and stratified Boussinesq equations. The experiments, involving 24 IMEX and exponential methods of third and fourth order, confirm the predictions of the linear stability analysis, that the methods are typically limited by small eigenvalues of the linear term and by eigenvalues on or near the imaginary axis rather than by large eigenvalues near the negative real axis. The experiments also demonstrate that IMEX methods achieve comparable stability to exponential methods, and that exponential methods are significantly more accurate only when the problem is nearly linear. Novel IMEX predictor-corrector methods are also derived.

Mesoscale Eddy Energy Locality in an Idealized Ocean Model

This paper investigates the energy budget of mesoscale eddies in wind-driven two-layer quasigeostrophic simulations. Intuitively, eddy energy can be generated, dissipated, and fluxed from place to place; regions where the budget balances generation and dissipation are ‘‘local’’ and regions that export or import large amounts of eddy energy are ‘‘nonlocal.’’ Many mesoscale parameterizations assume that statistics of the unresolved eddies behave as local functions of the resolved large scales, and studies that relate doubly periodic simulations to ocean patches must assume that the ocean patches have local energetics. This study derives and diagnoses the eddy energy budget in simulations of wind-driven gyres. To more closely approximate the ideas of subgrid-scale parameterization, the authors define the mean and eddies using a spatial filter rather than the more common time average. The eddy energy budget is strongly nonlocal over nearly half the domain in the simulations. In particular, in the intergyre region the eddies lose energy through interactions with the mean, and this energy loss can only be compensated by nonlocal flux of energy from elsewhere in the domain. This study also runs doubly periodic simulations corresponding to ocean patches from basin simulations. The eddy energy level of ocean patches in the basin simulations matches the level in the periodic simulations only in regions with local eddy energy budgets.

Molecular Embedding via a Second Order Dissimilarity Parameterized Approach

We describe a computational approach to the embedding problem in structural molecular biology. The approach is based on a dissimilarity parameterization of the problem that leads to a large-scale nonconvex bound constrained matrix optimization problem. The underlying idea is that an increased number of independent variables decouples the complicated effects of varying the location of individual atoms in coordinate-based formulations. Numerical tests support this hypothesis and indicate that the optimization problem that results is relatively benign and easy to solve, despite being large and nonconvex. We can solve problems with millions of independent variables in a few dozen to a few score optimization iterations. The nonconvexity arises due to matrix rank constraints in the problem, and we focus on their efficient computational treatment. We present numerical results for a number of synthetic and real protein data sets and comment on features of real experimental data that can cause computational difficulties.

Numerical Schemes for Stochastic Backscatter in the Inverse Cascade of Quasigeostrophic Turbulence

Backscatter is the process of energy transfer from small to large scales in turbulence; it is crucially important in the inverse energy cascades of geophysical turbulence, where the net transfer of energy is from small to large scales. One approach to modeling backscatter in underresolved simulations is to add a stochastic forcing term. This study, set in the idealized context of the inverse cascade of two-layer quasigeostrophic turbulence, focuses on the importance of spatial and temporal correlation in numerical stochastic backscatter schemes when used with low-order finite-difference spatial discretizations. A minimal stochastic backscatter scheme is developed as a stripped-down version of stochastic superparameterization [Grooms and Majda, J. Comput. Phys., 271, (2014), pp. 78--98]. This simplified scheme allows detailed numerical analysis of the spatial and temporal correlation structure of the modeled backscatter. Its essential properties include a local formulation amenable to implementation in finite difference codes and nonperiodic domains, and tunable spatial and temporal correlations. Experiments with this scheme in the idealized context of homogeneous two-layer quasigeostrophic turbulence demonstrate the need for stochastic backscatter to be smooth at the coarse grid scale when used with low-order finite-difference schemes in an inverse-cascade regime. In contrast, temporal correlation of the backscatter is much less important for achieving realistic energy spectra. It is expected that the spatial and temporal correlation properties of the simplified backscatter schemes examined here will inform the development of stochastic backscatter schemes in more realistic models.

Ensemble Filtering and Low-Resolution Model Error: Covariance Inflation, Stochastic Parameterization, and Model Numerics

AbstractThe use of under-resolved models in ensemble data assimilation schemes leads to two kinds of model errors: truncation errors associated with discretization of the large-scale dynamics and errors associated with interactions with subgrid scales. Multiplicative and additive covariance inflation can be used to account for model errors in ensemble Kalman filters, but they do not reduce the model error. Truncation errors can be reduced by increasing the accuracy of the numerical discretization of the large-scale dynamics, and subgrid-scale parameterizations can reduce errors associated with subgrid-scale interactions. Stochastic subgrid-scale parameterizations both reduce the model error and inflate the ensemble spread, so their effectiveness in ensemble assimilation schemes can be gauged by comparing with covariance inflation techniques. The effects of covariance inflation, stochastic parameterizations, and model numerics in two-layer periodic quasigeostrophic turbulence are compared on an f plane and...

On Galerkin Approximations of the Surface Active Quasigeostrophic Equations

AbstractThis study investigates the representation of solutions of the three-dimensional quasigeostrophic (QG) equations using Galerkin series with standard vertical modes, with particular attention to the incorporation of active surface buoyancy dynamics. This study extends two existing Galerkin approaches (A and B) and develops a new Galerkin approximation (C). Approximation A, due to Flierl, represents the streamfunction as a truncated Galerkin series and defines the potential vorticity (PV) that satisfies the inversion problem exactly. Approximation B, due to Tulloch and Smith, represents the PV as a truncated Galerkin series and calculates the streamfunction that satisfies the inversion problem exactly. Approximation C, the true Galerkin approximation for the QG equations, represents both streamfunction and PV as truncated Galerkin series but does not satisfy the inversion equation exactly. The three approximations are fundamentally different unless the boundaries are isopycnal surfaces. The authors ...

Assimilation of ocean sea-surface height observations of mesoscale eddies

Mesoscale eddies are one of the dominant sources of variability in the worlds oceans. With eddy-resolving global ocean models, it becomes important to assimilate observations of mesoscale eddies to correctly represent the state of the mesoscale. Here, we investigate strategies for assimilating a reduced number of sea-surface height observations by focusing on the coherent mesoscale eddies. The study is carried out in an idealized perfect-model framework using two-layer forced quasigeostrophic dynamics, which captures the dominant dynamics of ocean mesoscale eddies. We study errors in state-estimation as well as error growth in forecasts and find that as fewer observations are assimilated, assimilating at vortex locations results in reduced state estimation and forecast errors.

Investigations of non-hydrostatic, stably stratified and rapidly rotating flows

We present an investigation of rapidly rotating (small Rossby number