arXiv subjects

Ian Grooms

Publications and source records attributed to Ian Grooms.

At least 19 recordsLinked to original sources

arXiv2026

Demographics of Mesoscale Eddies in an Eddy-Permitting Ocean Model and Reanalysis

Ocean mesoscale eddies can be thought of as the "weather" of the ocean and strongly influence the ocean's physics, chemistry, and biology; they influence other components of the Earth system via air-sea and sea-ice interactions, and are crucial drivers of marine heat waves. Thus, proper modeling of eddies in both historical and future climates is crucial to accurately capturing the Earth system. Climate projections using global coupled models with eddying ocean components are only recently starting to be more widely used. Despite their critical role in understanding and forecasting climate characteristics, these so-called eddy-permitting models have not been explored to verify that resolved eddies are realistic, and thus any downstream scientific testing of hypotheses in biogeochemistry, ocean physics or other associated Earth systems impacted by eddies hinge on this critical assumption. This paper compares observed eddies with lifetimes longer than 6 weeks present in $1/4^\circ$ satellite altimetry data with observed eddies in $1/4^\circ$ reanalysis data and ocean model output. When compared to eddies observed in satellite altimetry data, eddies in reanalysis data and ocean model output are missing almost 30% of the number of eddy trajectories. In addition to missing eddy trajectories, the characteristics of eddies in reanalysis data and ocean model output differ from eddies observed in satellite altimetry data. At a high level, eddies in reanalysis data and ocean model output tend to live longer, are larger, and are weaker than eddies in observed altimetry data. This paper presents a variety of statistics describing these differences both spatially and in global aggregate.

physics.ao-ph↗

arXiv2025

Data Assimilation With An Integral-Form Ensemble Square-Root Filter

Geoscientific applications of ensemble Kalman filters face several computational challenges arising from the high dimensionality of the forecast covariance matrix, particularly when this matrix incorporates localization. For square-root filters, updating the perturbations of the ensemble members from their mean is an especially challenging step, one which generally requires approximations that introduce a trade-off between accuracy and computational cost. This paper describes an ensemble square-root filter which achieves a favorable trade-off between these factors by discretizing an integral representation of the Kalman filter update equations, and in doing so, avoids a direct evaluation of the matrix square-root in the perturbation update stage. This algorithm, which we call InFo-ESRF ("Integral-Form Ensemble Square-Root Filter"), is parallelizable and uses a preconditioned Krylov method to update perturbations to a high degree of accuracy. Through numerical experiments with both a Gaussian forecast model and a multi-layer Lorenz-type system, we demonstrate that InFo-ESRF is competitive or superior to several existing localized square-root filters in terms of accuracy and cost.

physics.comp-ph↗

arXiv2025

Asymptotic approximations for convection onset with Ekman pumping at low wavenumbers

Ekman pumping is a phenomenon induced by no-slip boundary conditions in rotating fluids. In the context of Rayleigh-B\'enard convection, Ekman pumping causes a significant change in the linear stability of the system compared to when it is not present (that is, stress-free). Motivated by numerical solutions to the marginal stability problem of the incompressible Navier-Stokes (iNSE) system, we seek analytical asymptotic solutions which describe the departure of the no-slip solution from the stress-free. The substitution of normal modes into a reduced asymptotic model yields a linear system for which we explore analytical solutions for various scalings of wavenumber. We find very good agreement between the analytical asymptotic solutions and the numerical solutions to the iNSE linear stability problem with no-slip boundary conditions.

physics.flu-dyn↗

arXiv2024

Parameterized Ekman boundary layers on the tilted $f$-plane

Rotating convection is considered on the tilted $f$-plane where gravity and rotation are not aligned. For sufficiently large rotation rates, $Ω$, the Taylor-Proudman effect results in the gyroscopic alignment of anisotropic columnar structures with the rotation axis giving rise to rapidly varying radial length scales that vanishes as $Ω^{-1/3}$ for $Ω\rightarrow\infty$. Compounding this phenomenon is the existence of viscous (Ekman) layers adjacent to the impenetrable bounding surfaces that diminish in scale as $Ω^{-1/2}$. In this investigation, these constraints are relaxed upon utilizing a non-orthogonal coordinate representation of the fluid equations where the upright coordinate aligns with rotation axis. This exposes the problem to asymptotic perturbation methods that permit: (i) relaxation of the constraints of gyroscopic alignment; (ii) the filtering of Ekman layers through the uncovering of parameterized velocity pumping boundary conditions; and (iii) the development of reduced quasi-geostrophic systems valid in the limit $Ω\rightarrow\infty$. Linear stability investigations reveal excellent quantitative agreement between results from parameterized or unapproximated mechanical boundary conditions. For no-slip boundaries, it is demonstrated that the associated Ekman pumping dramatically alters convective onset through an enhanced destabilization of large spatial scales. The range of unstable modes at a fixed thermal forcing is thus significantly extended with a direct dependence on $Ω$. This holds true even for geophysical and astrophysical regimes characterized by extreme values of the non-dimensional Ekman number $E$. The nonlinear regime is explored via the global heat and momentum transport of single-mode solutions to the quasi-geostrophic systems which indicate $\mathcal{O}(1)$ changes irrespective of the smallness of $E$.

physics.flu-dyn↗

arXiv2022

A comparison of nonlinear extensions to the ensemble Kalman filter: Gaussian Anamorphosis and Two-Step Ensemble Filters

Ensemble Kalman filters are based on a Gaussian assumption, which can limit their performance in some non-Gaussian settings. This paper reviews two nonlinear, non-Gaussian extensions of the Ensemble Kalman Filter: Gaussian anamorphosis (GA) methods and two-step updates, of which the rank histogram filter (RHF) is a prototypical example. GA-EnKF methods apply univariate transforms to the state and observation variables to make their distribution more Gaussian before applying an EnKF. The two-step methods use a scalar Bayesian update for the first step, followed by linear regression for the second step. The connection of the two-step framework to the full Bayesian problem is made, which opens the door to more advanced two-step methods in the full Bayesian setting. A new method for the first part of the two-step framework is proposed, with a similar form to the RHF but a different motivation, called the `improved RHF' (iRHF). A suite of experiments with the Lorenz-`96 model demonstrate situations where the GA-EnKF methods are similar to EnKF, and where they outperform EnKF. The experiments also strongly support the accuracy of the RHF and iRHF filters for nonlinear and non-Gaussian observations; these methods uniformly beat the EnKF and GA-EnKF methods in the experiments reported here. The new iRHF method is only more accurate than RHF at small ensemble sizes in the experiments reported here.

stat.CO↗

arXiv2021

Machine Learning Techniques to Construct Patched Analog Ensembles for Data Assimilation

Using generative models from the machine learning literature to create artificial ensemble members for use within data assimilation schemes has been introduced in [Grooms QJRMS, 2020] as constructed analog ensemble optimal interpolation (cAnEnOI). Specifically, we study general and variational autoencoders for the machine learning component of this method, and combine the ideas of constructed analogs and ensemble optimal interpolation in the data assimilation piece. To extend the scalability of cAnEnOI for use in data assimilation on complex dynamical models, we propose using patching schemes to divide the global spatial domain into digestible chunks. Using patches makes training the generative models possible and has the added benefit of being able to exploit parallelism during the generative step. Testing this new algorithm on a 1D toy model, we find that larger patch sizes make it harder to train an accurate generative model (i.e. a model whose reconstruction error is small), while conversely the data assimilation performance improves at larger patch sizes. There is thus a sweet spot where the patch size is large enough to enable good data assimilation performance, but not so large that it becomes difficult to train an accurate generative model. In our tests the new patched cAnEnOI method outperforms the original (unpatched) cAnEnOI, as well as the ensemble square root filter results from [Grooms QJRMS, 2020].

physics.comp-ph↗

arXiv2021

A hybrid particle-ensemble Kalman filter for problems with medium nonlinearity

A hybrid particle ensemble Kalman filter is developed for problems with medium non-Gaussianity, i.e. problems where the prior is very non-Gaussian but the posterior is approximately Gaussian. Such situations arise, e.g., when nonlinear dynamics produce a non-Gaussian forecast but a tight Gaussian likelihood leads to a nearly-Gaussian posterior. The hybrid filter starts by factoring the likelihood. First the particle filter assimilates the observations with one factor of the likelihood to produce an intermediate prior that is close to Gaussian, and then the ensemble Kalman filter completes the assimilation with the remaining factor. How the likelihood gets split between the two stages is determined in such a way to ensure that the particle filter avoids collapse, and particle degeneracy is broken by a mean-preserving random orthogonal transformation. The hybrid is tested in a simple two-dimensional (2D) problem and a multiscale system of ODEs motivated by the Lorenz-`96 model. In the 2D problem it outperforms both a pure particle filter and a pure ensemble Kalman filter, and in the multiscale Lorenz-`96 model it is shown to outperform a pure ensemble Kalman filter, provided that the ensemble size is large enough.

stat.ME↗

arXiv2020

A note on the formulation of the Ensemble Adjustment Kalman Filter

The ensemble adjustment Kalman filter (EAKF; Anderson, 2001) is one of the earliest ensemble square root filters. This note clarifies the correct formulation of the EAKF, which depends on a careful treatment of an eigen-decomposition of one of the matrices involved in the formulation.

stat.ME↗

arXiv2020

Analog ensemble data assimilation and a method for constructing analogs with variational autoencoders

It is proposed to use analogs of the forecast mean to generate an ensemble of perturbations for use in ensemble optimal interpolation (EnOI) or ensemble variational (EnVar) methods. A new method of constructing analogs using variational autoencoders (VAEs; a machine learning method) is proposed. The resulting analog methods using analogs from a catalog (AnEnOI), and using constructed analogs (cAnEnOI), are tested in the context of a multiscale Lorenz-`96 model, with standard EnOI and an ensemble square root filter for comparison. The use of analogs from a modestly-sized catalog is shown to improve the performance of EnOI, with limited marginal improvements resulting from increases in the catalog size. The method using constructed analogs (cAnEnOI) is found to perform as well as a full ensemble square root filter, and to be robust over a wide range of tuning parameters.

physics.comp-ph↗

arXiv2020

A fast tunable blurring algorithm for scattered data

A blurring algorithm with linear time complexity can reduce the small-scale content of data observed at scattered locations in a spatially extended domain of arbitrary dimension. The method works by forming a Gaussian interpolant of the input data, and then convolving the interpolant with a multiresolution Gaussian approximation of the Green's function to a differential operator whose spectrum can be tuned for problem-specific considerations. Like conventional blurring algorithms, which the new algorithm generalizes to data measured at locations other than a uniform grid, applications include deblurring and separation of spatial scales. An example illustrates a possible application toward enabling importance sampling approaches to data assimilation of geophysical observations, which are often scattered over a spatial domain, since blurring observations can make particle filters more effective at state estimation of large scales. Another example, motivated by data analysis of dynamics like ocean eddies that have strong separation of spatial scales, uses the algorithm to decompose scattered oceanographic float measurements into large-scale and small-scale components.

stat.CO↗

arXiv2019

Energy-conserving Galerkin approximations for quasigeostrophic dynamics

A method is presented for constructing energy-conserving Galerkin approximations in the vertical coordinate of the full quasigeostrophic model with active surface buoyancy. The derivation generalizes the approach of Rocha \emph{et al.} (2016) to allow for general bases. Details are then presented for a specific set of bases: Legendre polynomials for potential vorticity and a recombined Legendre basis from Shen (1994) for the streamfunction. The method is tested in the context of linear baroclinic instability calculations, where it is compared to the standard second-order finite-difference method and to a Chebyshev collocation method. The Galerkin scheme is quite accurate even for a small number of degrees of freedom $N$, and growth rates converge much more quickly with increasing $N$ for the Galerkin scheme than for the finite-difference scheme. The Galerkin scheme is at least as accurate as finite differences and can in some cases achieve the same accuracy as the finite difference scheme with ten times fewer degrees of freedom. The energy-conserving Galerkin scheme is of comparable accuracy to the Chebyshev collocation scheme in most linear stability calculations, but not in the Eady problem where the Chebyshev scheme is significantly more accurate. Finally the three methods are compared in the context of a simplified version of the nonlinear equations: the two-surface model with zero potential vorticity. The Chebyshev scheme is the most accurate, followed by the Galerkin scheme and then the finite difference scheme. All three methods conserve energy with similar accuracy, despite not having any a priori guarantee of energy conservation for the Chebyshev scheme. Further nonlinear tests with non-zero potential vorticity to assess the merits of the methods will be performed in a future work.

physics.ao-ph↗

arXiv2018

Exact instantaneous optimals in the non-geostrophic Eady problem and the detrimental effects of discretization

We derive exact analytical expressions for flow configurations that optimize the instantaneous growth rate of energy in the linear Eady problem, along with the associated growth rates. These optimal perturbations are relevant linear stability analysis, but, more importantly, they are relevant for understanding the energetics of fully nonlinear baroclinic turbulence. The optimal perturbations and their growth rates are independent of the Richardson number. The growth rates of the optimal perturbations grow linearly as the horizontal wavelength of the perturbation decreases. Perturbation energy growth at large scales is driven by extraction of potential energy from the mean flow, while at small scales it is driven by extraction of kinetic energy from the mean shear. We also analyze the effect of spatial discretization on the optimal perturbations and their growth rates. A second order energy-conserving discretization on the Arakawa B grid generally has too-weak growth rates at small scales and is less accurate than two second order discretizations on the Arakawa C grid. The two C grid discretizations, one that conserves energy and another that conserves both energy and enstrophy, yield very similar optimal perturbation growth rates that are significantly more accurate than the B grid discretization at small scales.

physics.ao-ph↗

arXiv2018

Improving particle filter performance by smoothing observations

This article shows that increasing the observation variance at small scales can reduce the ensemble size required to avoid collapse in particle filtering of spatially-extended dynamics and improve the resulting uncertainty quantification at large scales. Particle filter weights depend on how well ensemble members agree with observations, and collapse occurs when a few ensemble members receive most of the weight. Collapse causes catastrophic variance underestimation. Increasing small-scale variance in the observation error model reduces the incidence of collapse by de-emphasizing small-scale differences between the ensemble members and the observations. Doing so smooths the posterior mean, though it does not smooth the individual ensemble members. Two options for implementing the proposed observation error model are described. Taking discretized elliptic differential operators as an observation error covariance matrix provides the desired property of a spectrum that grows in the approach to small scales. This choice also introduces structure exploitable by scalable computation techniques, including multigrid solvers and multiresolution approximations to the corresponding integral operator. Alternatively the observations can be smoothed and then assimilated under the assumption of independent errors, which is equivalent to assuming large errors at small scales. The method is demonstrated on a linear stochastic partial differential equation, where it significantly reduces the occurrence of particle filter collapse while maintaining accuracy. It also improves continuous ranked probability scores by as much as 25%, indicating that the weighted ensemble more accurately represents the true distribution. The method is compatible with other techniques for improving the performance of particle filters.

stat.AP↗

arXiv2017

An eddifying Stommel model: Fast eddy effects in a two-box ocean

A system of stochastic differential equations is formulated describing the heat and salt content of a two-box ocean. Variability in the heat and salt content and in the thermohaline circulation between the boxes is driven by fast Gaussian atmospheric forcing and by ocean-intrinsic, eddy-driven variability. The eddy forcing of the slow dynamics takes the form of a colored, non-Gaussian noise. The qualitative effects of this non-Gaussianity are investigated by comparing to two approximate models: one that includes only the mean eddy effects (the `averaged model'), and one that includes an additional Gaussian white-noise approximation of the eddy effects (the `Gaussian model'). Both of these approximate models are derived using the methods of fast averaging and homogenization. In the parameter regime where the dynamics has a single stable equilibrium the averaged model has too little variability. The Gaussian model has accurate second-order statistics, but incorrect skew and rare-event probabilities. In the parameter regime where the dynamics has two stable equilibria the eddy noise is much smaller than the atmospheric noise. The averaged, Gaussian, and non-Gaussian models all have similar stationary distributions, but the jump rates between equilibria are too small for the averaged and Gaussian models.

math.DS↗

arXiv2016

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

We present an investigation of rapidly rotating (small Rossby number $Ro\ll 1$) and stratified turbulence where the stratification strength is varied from weak (large Froude number $Fr\gg1$) to strong ($Fr\ll1$). The investigation is set in the context of a reduced model derived from the Boussinesq equations that efficiently retains anisotropic inertia-gravity waves with order-one frequencies and highlights a regime of wave-eddy interactions. Numerical simulations of the reduced model are performed where energy is injected by a stochastic forcing of vertical velocity, which forces wave modes only. The simulations reveal two regimes characterized by the presence of well-formed, persistent and thin turbulent layers of locally-weakened stratification at small Froude numbers, and by the absence of layers at large Froude numbers. Both regimes are characterized by a large-scale barotropic dipole enclosed by small-scale turbulence. When the Reynolds number is not too large a direct cascade of barotropic kinetic energy is observed, leading to total energy equilibration. We examine net energy exchanges that occur through vortex stretching and vertical buoyancy flux and diagnose the horizontal scales active in these exchanges. We find that the baroclinic motions inject energy directly to the largest scales of the barotropic mode, implying that the large-scale barotropic dipole is not the end result of an inverse cascade within the barotropic mode.

physics.flu-dyn↗

arXiv2015

On Gakerkin approximations for the surface-active quasigeostrophic equations

We study 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. We extend two existing Galerkin approaches (A and B) and develop a new Galerkin approximation (C). Approximation A, due to \cite{flierl1978}, represents the streamfunction as a truncated Galerkin series and defines the potential vorticity (PV) that satisfies the inversion problem exactly. Approximation B, due to \cite{tulloch_smith2009b}, 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. We discuss the advantages and limitations of approximations A, B, and C in terms of mathematical rigor and conservation laws, and illustrate their relative efficiency by solving linear stability problems with nonzero surface buoyancy. With moderate number of modes, B and C have have superior accuracy than A at high wavenumbers. Because B lacks conservation of energy, we recommend approximation C for constructing solutions to the surface-active QG equations using Galerkin series with standard vertical modes.

physics.ao-ph↗

arXiv2014

Asymptotic Behavior of Heat Transport for a Class of Exact Solutions in Rotating Rayleigh-Bénard Convection

The non-hydrostatic, quasigeostrophic approximation for rapidly rotating Rayleigh-Bénard convection admits a class of exact `single mode' solutions. These solutions correspond to steady laminar convection with a separable structure consisting of a horizontal planform characterized by a single wavenumber multiplied by a vertical amplitude profile, with the latter given as the solution of a nonlinear boundary value problem. The heat transport associated with these solutions is studied in the regime of strong thermal forcing (large reduced Rayleigh number $\widetilde{Ra}$). It is shown that the Nusselt number $Nu$, a nondimensional measure of the efficiency of heat transport by convection, for this class of solutions is bounded below by $Nu\gtrsim \widetilde{Ra}^{3/2}$, independent of the Prandtl number, in the limit of large reduced Rayleigh number. Matching upper bounds include only logarithmic corrections, showing the accuracy of the estimate. Numerical solutions of the nonlinear boundary value problem for the vertical structure are consistent with the analytical bounds.

physics.flu-dyn↗

arXiv2014

Bounds on Heat Transport in Rapidly Rotating Rayleigh-Bénard Convection

The heat transport in rotating Rayleigh-Bénard convection is considered in the limit of rapid rotation (small Ekman number $E$) and strong thermal forcing (large Rayleigh number $Ra$). The analysis proceeds from a set of asymptotically reduced equations appropriate for rotationally constrained dynamics; the conjectured range of validity for these equations is $Ra \lesssim E^{-8/5}$. A rigorous bound on heat transport of $Nu \le 20.56Ra^3E^4$ is derived in the limit of infinite Prandtl number using the background method. We demonstrate that the exponent in this bound cannot be improved on using a piece-wise monotonic background temperature profile like the one used here. This is true for finite Prandtl numbers as well, i.e. $Nu \lesssim Ra^3$ is the best upper bound for this particular setup of the background method. The feature that obstructs the availability of a better bound in this case is the appearance of small-scale thermal plumes emanating from (or entering) the thermal boundary layer.

physics.flu-dyn↗