76 results found
Shevchenko I, Berloff P, 2022, A method for preserving nominally-resolved flow patterns in low-resolution ocean simulations: Constrained dynamics, Ocean Modelling, Vol: 178, Pages: 1-6, ISSN: 1463-5003
Inability of low-resolution ocean models to simulate many important aspects of the large-scale general circulation is a common problem. In the view of physics, the main reason for this failure are the missed dynamical effects of the unresolved small scales of motion on the explicitly resolved large-scale circulation. Complimentary to this mainstream physics-based perspective, we propose to address this failure from the dynamical systems point of view, namely, as the persistent tendency of phase space trajectories representing the low-resolution solution to escape the right region of the corresponding phase space, which is occupied by the reference eddy-resolving solution. Based on this concept, we propose to use methods of constrained optimization to confine the low-resolution solution to remain within the correct phase space region, without attempting to amend the eddy physics by introducing a process-based parameterization. This approach is advocated as a novel framework for data-driven hyper-parameterizations of mesoscale oceanic eddies in non-eddy-resolving models. We tested the idea in the context of classical, baroclinic beta-plane turbulence model and showed that non-eddy-resolving solution can be substantially improved towards the reference eddy-resolving benchmark.
Ryzhov EA, Berloff P, 2022, On transport tensor of dynamically unresolved oceanic mesoscale eddies, Journal of Fluid Mechanics, Vol: 939, ISSN: 0022-1120
Parameterizing mesoscale eddies in ocean circulation models remains an open problem due to the ambiguity with separating the eddies from large-scale flow, so that their interplay is consistent with the resolving skill of the employed non-eddy-resolving model. One way to address the issue is by using recently formulated dynamically filtered eddies. These eddies are obtained as the field errors of fitting some given reference ocean circulation into the employed coarse-grid ocean model. The main strengths are (i) no explicit spatio-temporal filter is needed for separating the large-scale and eddy flow components, (ii) the eddies are dynamically translated into the error-correcting forcing that perfectly augments the coarse-grid model towards reproducing the reference circulation. We uncovered physical properties of the eddies by interpreting involved nonlinear eddy/large-scale interactions via the classical flux-gradient relation. We described the eddies in terms of their full, space–time dependent transport tensor, which was made unique by constraining it to be the same for the potential vorticity, momentum and buoyancy fluxes. Both diffusive and advective parts of the transport tensor were found to be significant. The diffusive tensor component is characterised by polar eigenvalues and is further decomposed into isotropic and filamentation components. The latter component completely dominates, therefore, it should be taken into account by eddy parameterizations, which is not yet the case. We also showed that spatial inhomogeneities of the transport tensor components are important. Comparing these properties with those obtained for more common, locally filtered eddies revealed that they are distinctly different.
Haigh M, Berloff P, 2022, On the stability of tracer simulations with opposite-signed diffusivities, Journal of Fluid Mechanics, Vol: 937, Pages: 1-11, ISSN: 0022-1120
Many recent studies have diagnosed opposite-signed diffusion eigenvalues to be a prevalent feature of the transfer tensor for diffusive tracer transport by oceanic mesoscale eddies. This diagnosed tensor, which we refer to as the diffusion tensor, therefore accounts for tracer filamentation effects. The preferential orientation of this filamentation is quantified by the principal axis of the diffusion tensor, namely the diffusion axis. Parameterisations of eddy diffusion commonly invoke a diffusion tensor, typically one with non-negative eigenvalues to avoid numerical issues. Motivated by the need to parameterise tracer filamentation, in this study we examine diffusion of a Gaussian tracer patch with imposed opposite-signed diffusion eigenvalues, and in particular we focus on the time scale for the onset of instability. For a fixed diffusion axis, numerical instability is an inevitable consequence of persistent up-gradient fluxes associated with the negative eigenvalue. For typical oceanic scales and diffusion magnitudes, this time scale is of the order of 100 days, but is shorter for larger negative eigenvalues or for finer grid resolutions. We show that imposing a time-dependent diffusion axis can lead to simulations with no onset of instability after 100 000 days of tracer evolution. Although motivated by oceanographic fluid dynamics, our results have much broader applications since diffusive processes are present in a wide range of fluid flows.
Shevchenko I, Berloff P, 2022, A method for preserving nominally-resolved flow patterns in low-resolution ocean simulations: Dynamical system reconstruction, Ocean Modelling, Vol: 170, Pages: 101939-101939, ISSN: 1463-5003
Accurate representation of large-scale flow patterns in low-resolution ocean simulations is one of the most challenging problems in ocean modelling. The main difficulty is to correctly reproduce effects of unresolved small scales on the resolved large scales. For this purpose, most of current research is focused on development of parameterizations directly accounting for the small scales. In this work we propose an alternative to the mainstream ideas by showing how to reconstruct a dynamical system from the available reference solution data (our proxy for observations) and, then, how to use this system for modelling not only large-scale but also nominally-resolved flow patterns at low resolutions. This approach is advocated as a part of the novel framework for data-driven hyper-parameterization of mesoscale oceanic eddies in non-eddy-resolving models. The main characteristic of this framework is that it does not require to know the physics behind large–small scale interactions to reproduce both large and small scales in low-resolution ocean simulations. We tested it in the context of a three-layer, statistically equilibrated, steadily forced quasigeostrophic model for the beta-plane configuration and showed that non-eddy-resolving solution can be substantially improved towards the reference eddy-resolving benchmark. The proposed methodology robustly allows to retrieve a system of equations governing reduced dynamics of the observed data, while the additional adaptive nudging counteracts numerical instabilities by keeping solutions in the region of phase space occupied by the reference fields. Remarkably, its solutions simulate not only large-scale but also small-scale flow features, which can be nominally resolved by the low-resolution grid. In addition, the proposed method reproduces realistic vortex trajectories. One of the important and general conclusions that can be drawn from our results is that not only mesoscale eddy parameterization is possible in pri
Shevchenko I, Berloff P, 2021, On a minimum set of equations for parameterisations in comprehensive ocean circulation models, Ocean Modelling, Vol: 168, Pages: 1-7, ISSN: 1463-5003
The complexity of comprehensive ocean models poses an important question for parameterisations: is there a minimum set of equations that should be parameterised, on the one hand, to reduce the development to a minimum, and, on the other hand, to ensure an accurate representation of large-scale flow patterns? This work seeks to answer the question to assist modern parameterisations be more selective in their targets. For this, we considered a model of the North Atlantic and studied contributions of different model equations to the accuracy of representation of the Gulf Stream at low resolution. Our results suggest that one should focus on parameterising the tracer equations for temperature and salinity, and may leave the other equations in the hydrodynamic part, as well as the atmospheric model unmodified. They also suggest that parameterisations representing only Kinetic Energy Backscatter cannot be fully efficient and the main focus should be shifted towards developing parameterisations of combined Potential/Kinetic Energy Backscatters.
Haigh M, Berloff P, 2021, On co-existing diffusive and anti-diffusive tracer transport by oceanic mesoscale eddies, Ocean Modelling, Vol: 168, ISSN: 1463-5003
A common approach for parameterising eddy transport of passive tracers by mesoscale eddies in the ocean is by invoking a transport tensor. The symmetric part of this tensor, the diffusion tensor, quantifies diffusive eddy tracer transport. Recent studies have diagnosed opposite-signed eigenvalues (diffusivities) of the diffusion tensor from eddy-resolving simulations, while all current parameterisations implement only positive diffusivities. For opposite-signed eigenvalues the associated diffusive eddy tracer flux is not necessarily down-gradient and therefore may not mix the tracer by transferring variance to the small scales. In this study we explore such diffusive eddy fluxes by using an eddy-resolving simulation of passive tracers with a relaxation (source/sink) forcing. After confirming that the diffusion tensors for different tracer pairs have opposite-signed eigenvalues, we show that the corresponding diffusive eddy tracer flux drives a net down-gradient transfer of variance, as would be guaranteed when the diffusion tensor eigenvalues are both positive. Locally up-gradient fluxes are common, with their frequency strongly dependent on the relaxation profile. The effects of weakening/strengthening the relaxation on the frequency of down-gradient fluxes is different for each tracer. However, for all tracers considered the amplitude of the net down-gradient transfer weakens as the relaxation strengthens, a consequence of the homogeneous diffusion dissipating less eddy variance. Our results indicate that for oceanic tracers with sources/sinks the parameterised diffusive eddy tracer fluxes should not be globally down-gradient.
Agarwal N, Ryzhov E, Kondrashov D, et al., 2021, Correlation-based flow decomposition and statistical analysis of the eddy forcing, Journal of Fluid Mechanics, Vol: 924, Pages: 1-30, ISSN: 0022-1120
We present a comprehensive study of the mesoscale eddy forcing in the ocean by proposing spatially local filtering of the high-resolution double-gyre ocean circulation solution into its large- and small-scale (eddy) components. The large-scale component is dominated by the mid-latitude gyres, the western boundary currents and their highly transient eastward jet extension; the eddy component is concentrated around the eastward jet and strongly interacts with it. The proposed decomposition method achieves flow filtering based on the local spatial correlations. This is different from the existing decomposition methods, e.g. classical Reynolds decomposition and moving-average (spatial) filtering with a constant filter size based on the first baroclinic Rossby deformation radius. Next, we characterize the dynamical impacts of the resulting eddy forcing on the large-scale flow in terms of their mutual time-lagged spatial correlations, formulated as product integral characteristics. Its temporal statistics uncover robust causality between the eddy forcing and the induced large-scale potential vorticity anomalies – referred to as the eddy backscatter. The results also prove the significance of the transient eddy forcing and the time lag dependence of the eddy backscatter. We argue that these properties are to be considered by eddy parametrization schemes. We further used the decomposed eddy fields to augment a coarse-resolution ocean model. The augmented solution statistically reproduces the missing eastward jet extension, enhances the eddy activities around it and recovers the essential large-scale low-frequency variability. This justifies a reduced-order statistical emulation of the eddies – an emerging methodology for including eddy effects in non-eddy-resolving ocean models.
Agarwal N, Kondrashov D, Dueben P, et al., 2021, A Comparison of Data-Driven Approaches to Build Low-Dimensional Ocean Models, JOURNAL OF ADVANCES IN MODELING EARTH SYSTEMS, Vol: 13
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Haigh M, Sun L, McWilliams JC, et al., 2021, On eddy transport in the ocean. Part II: The advection tensor, Ocean Modelling, Vol: 165, Pages: 1-17, ISSN: 1463-5003
This study considers the isopycnal eddy transport of mass and passive tracers in eddy-resolving doublegyre quasigeostrophic oceanic circulation. Here we focus on advective transport, whereas a companion paperfocuses on eddy-induced diffusive tracer transport. To work towards parameterising eddy tracer transport wequantify the eddy tracer flux using a transport tensor with eddies defined using a spatial filter, which leadsto results distinct from those obtained via a temporal Reynolds eddy decomposition. The advection tensoris the antisymmetric part of the transport tensor, and is so named since the associated tracer transport canbe expressed as advection of the large-scale tracer field by a rotational eddy-induced velocity (EIV) 𝒖𝑐∗ withstreamfunction 𝐴. The EIV 𝒖𝑐∗is fastest (∼ 1 m s−1) where eddy activity is strongest, e.g., in the upper layer,near the eastward jet and western boundary current. Our results suggest that a stochastic closure for the eddytransport would be most suitable since 𝐴 exhibits a probabilistic distribution when conditioned on, for example,the large-scale relative vorticity. Consistent with closures in ocean circulation models, we quantify eddy mass(isopycnal layer thickness) fluxes as eddy-induced advection by the thickness EIV 𝒖ℎ∗. The divergent part of𝒖ℎ∗– the only part relevant for mass transport in the quasigeostrophic limit – tends to be oriented down thethickness gradient suggesting it quantifies some baroclinic eddy effects similar to those parameterised by theGent & McWilliams (GM90) EIV. Although 𝒖ℎ∗has some qualitative similarities to 𝒖𝑐∗, our results suggest thateddy-induced tracer advection is driven by more than just the thickness-determined EIV and, in turn, morethan just the GM90 EIV.
Berloff P, Ryzhov E, Shevchenko I, 2021, On dynamically unresolved oceanic mesoscale motions, Journal of Fluid Mechanics, Vol: 920, Pages: 1-24, ISSN: 0022-1120
The problem of defining oceanic mesoscale eddies remains generally unresolved because there is no unique local spatio-temporal filter that can be used for extracting the eddies, and it is unclear what part of the eddy field cannot be actually resolved and needs to be parameterized in a coarse-grid model. We propose using of the coarse-grid model itself for reconstructing dynamically unresolved eddies, which are actually field errors on the top of the dynamically resolved, large-scale reference flow. The novelties and strengths of the approach are that (i) no spatio-temporal filtering is ever needed, (ii) field errors are dynamically translated into the error-correcting forcing and (iii) the latter exactly augments the coarse-grid model solution towards the reference flow. After implementation of the proposed approach, we study statistical properties of the field errors, show their robustness and reveal their significant differences from the locally filtered eddies. We argue that dynamical effects of unresolved eddies can be ultimately parameterized by emulating field errors and closing them on the dynamically resolved flow. So far, our results are limited to the quasigeostrophic approximation, but this serves as a proof of concept and starting point for the follow-up extension into the primitive equations, which are used routinely in the comprehensive oceanic general circulation models.
Sun L, Haigh M, Shevchenko I, et al., 2021, On non-uniqueness of the mesoscale eddy diffusivity, Journal of Fluid Mechanics, Vol: 920, Pages: 1-27, ISSN: 0022-1120
Oceanic mesoscale currents (‘eddies’) can have significant effects on the distributions of passive tracers. The associated inhomogeneous and anisotropic eddy fluxes are traditionally parametrised using a transport tensor (K-tensor), which contains both diffusive and advective components. In this study, we analyse the eddy transport tensor in a quasigeostrophic double-gyre flow. First, the flow and passive tracer fields are decomposed into large- and small-scale (eddy) components by spatial filtering, and the resulting eddy forcing includes an eddy tracer flux representing advection by eddies and non-advective terms. Second, we use the flux-gradient relation between the eddy fluxes and the large-scale tracer gradient to estimate the associated K-tensors in their entire structural, spatial and temporal complexity, without making any additional assumptions or simplifications. The divergent components of the eddy tracer fluxes are extracted via the Helmholtz decomposition, which yields a divergent tensor. The remaining rotational flux does not affect the tracer evolution, but dominates the total tracer flux, affecting both its magnitude and spatial structure. However, in terms of estimating the eddy forcing, the transport tensor prevails over its divergent counterpart because of the significant numerical errors induced by the Helmholtz decomposition. Our analyses demonstrate that, in general, the K-tensor for the eddy forcing is not unique, that is, it is tracer-dependent. Our study raises serious questions on how to interpret and use various estimates of K-tensors obtained from either observations or eddy-resolving model solutions.
Haigh M, Sun L, McWilliams JC, et al., 2021, On eddy transport in the ocean. Part I: The diffusion tensor, Ocean Modelling, Vol: 164, Pages: 1-15, ISSN: 1463-5003
This study provides an interpretation of isopycnal eddy transport for mass and passive tracers in double-gyreeddy-resolving oceanic circulation. This paper focuses on a transport/diffusion tensor representation of theeddy tracer flux, and a companion paper will focus on advective eddy-induced tracer and mass transports. Weuse a spatial filter to separate the large and small scales, which leads to results distinct from those obtained viaa temporal Reynolds eddy decomposition. To work towards a parameterisation, we relate the eddy tracer fluxto the large-scale tracer gradient via the transport tensor 𝑲. The symmetric part of 𝑲 is the diffusion tensor,𝑺, which parameterises diffusive fluxes and whose mixing properties are determined by the signs of its eigenvalues. The eigenvalues of 𝑺 are robustly of opposite sign (polar) and thus quantify filamentation of the tracervia both up- and down-gradient fluxes. Given the prevalence of polar eigenvalues – which are also obtained forReynolds eddy fluxes – representing their associated effects should be a target of future eddy tracer transportclosures. Given the inherent inhomogeneity and anisotropy of the eddy-induced transport, we argue that a fulltransport tensor is better suited to this task than scalar coefficients or diagonal tensors. The diffusion axis,which represents the direction of preferential mixing, tends to align with the large-scale velocity vector andcontours of large-scale relative vorticity and layer thickness. Strong shears can inhibit this alignment. We showthat the large-scale velocity gradient matrix may be suitable for parameterising the transport tensor, in particular at depth. Furthermore, since entries of 𝑲 and 𝑺 exhibit probabilistic distributions when conditioned on certain large-scale flow features, we suggest that a stochastic closure for the eddy transport would be most suitable.
Kurashina R, Berloff P, Shevchenko I, 2021, Western boundary layer nonlinear control of the oceanic gyres, Journal of Fluid Mechanics, Vol: 918, Pages: 1-26, ISSN: 0022-1120
This study examines the influence of flow nonlinearity in western boundary layers upon the turbulent wind-driven ocean gyres. Our analysis involves comparisons between large-scale circulation properties of the linear and nonlinear states, as well as a Lagrangian particle analysis of relevant flow features. We find that the so-called counter-rotating gyre anomalies, which are nonlinear circulation features embedded in the gyres, are consistent in shape with the linear, weakened, wind-curl response created by the geometric wind effect. However, the linear response is far too weak without considering nonlinear effects. Within the western boundary layer lobe of these features, the nonlinear boundary layer has a pivotal impact upon the global circulation. Effects of potential vorticity advection inhibit viscous relative vorticity fluxes through the western boundary. This creates a significant potential vorticity imbalance between the gyres. Consequently, this generates an accumulation of enstrophy downstream in the inertial recirculation zones, which in turn supports the eastward jet. However, within the ocean basin, the growing imbalance is eventually rectified by inter-gyre potential vorticity exchanges owing to nonlinear fluxes. The Lagrangian particle analysis reveals the inter-gyre exchange mechanism, where particles seeded within the western boundary layer migrate between the gyres and weaken the eastward jet extension.
Shevchenko I, Berloff P, 2021, A method for preserving large-scale flow patterns in low-resolution ocean simulations, Ocean Modelling, Vol: 161, Pages: 1-6, ISSN: 1463-5003
It is typical for low-resolution simulations of the ocean to miss not only small- but also large-scale patterns of the flow dynamics compared with their high-resolution analogues. It is usually attributed to the inability of coarse-grid models to properly reproduce the effects of the unresolved small-scale dynamics on the resolved large scales. In part, the reason for that is that coarse-grid models fail to at least keep the coarse-grid solution within the region of phase space occupied by the true solution (the high-resolution solution projected onto the coarse grid). In this paper we offer a solution to this problem by computing the image point in the phase space restricted to the region of the true flow dynamics. The proposed method shows encouraging results for both low- and high-dimensional phase spaces, it takes a near-zero effort to implement into existing numerical codes and has ample room for further improvements.
Kamenkovich I, Berloff P, Haigh M, et al., 2021, Complexity of mesoscale eddy diffusivity in the ocean, Geophysical Research Letters, Vol: 48, Pages: 1-12, ISSN: 0094-8276
Stirring of water by mesoscale currents (“eddies”) leads to large‐scale transport of many important oceanic properties (“tracers”). These eddy‐induced transports can be related to the large‐scale tracer gradients, using the concept of turbulent diffusion. The concept is widely used to describe these transports in the real ocean and to represent them in climate models. This study focuses on the inherent complexity of the corresponding coefficient tensor (“K‐tensor”) and its components, defined here in all its spatio‐temporal complexity. Results demonstrate that this comprehensive K‐tensor is space‐, time‐, direction‐ and tracer‐dependent. Using numerical simulations with both idealized and comprehensive models of the Atlantic circulation, we show that these properties lead to upgradient eddy fluxes and the potential importance of all tensor components. The uncovered complexity of the eddy transports calls for reconsideration of how they are estimated in practice, included in the general circulation models and theoretically interpreted.
Davies J, Khatri H, Berloff P, 2021, Linear stability analysis for flows over sinusoidal bottom topography, Journal of Fluid Mechanics, Vol: 911, Pages: 1-25, ISSN: 0022-1120
This is an ocean motivated study which investigates the impacts of sinusoidal bottom topography on baroclinic instability of zonal vertically sheared flows in the two-layer quasigeostrophic model. The corresponding linear stability problem is solved by assuming Fourier-mode solutions in both the zonal and meridional directions. In the presence of variable topographic features, the Fourier modes become coupled due to phase shifts in the wavevectors. The spectral discretisation method used in this work retains the primary relationship between different Fourier modes; thus, the linear stability eigenproblem can be solved for any periodic topography. Moreover, this method does not need any additional assumptions, such as considering small-amplitude or large-scale bottom irregularities, as in some previous studies. In this work, the eigenproblem is solved for a range of topographic amplitudes and wavenumbers, and their effects on the growth rates and shapes of the most unstable eigenmodes are discussed. In general, both the zonal and meridional variations in topography tend to suppress the baroclinic instability. However, it is found that only meridionally varying topography affects the magnitudes of the fastest growth rates. In this instance, unstable modes appear to form two clusters well separated in the zonal wavenumber axis and growth rate maxima occur at two distinct zonal wavenumbers. Dependencies of the characteristics of these clusters on the values of topography amplitude and ridge width are reviewed. Finally, doubly periodic numerical simulations are used to verify the results from the linear stability analysis.
Ryzhov EA, Kondrashov D, Agarwal N, et al., 2020, On data-driven induction of the low-frequency variability in a coarse-resolution ocean model, Ocean Modelling, Vol: 153, Pages: 1-11, ISSN: 1463-5003
This study makes progress towards a data-driven parameterization for mesoscale oceanic eddies. To demonstrate the concept and reveal accompanying caveats, we aimed at replacing a computationally expensive, standard high-resolution ocean model with its inexpensive low-resolution analogue augmented by the parameterization. We considered eddy-resolving and non-eddy-resolving double-gyre ocean circulation models characterized by drastically different solutions due to the nonlinear mesoscale eddy effects. The key step of the proposed approach is to extract from the high-resolution reference solution its eddy field varying in space and time, and then to use this information to improve the low-resolution analogue model.By interactively coupling both the continuously supplied history of the eddy field and the explicitly modeled low-resolution large-scale flow, we obtained the additional eddy forcing term which modified the low-resolution model and significantly augmented its solutions. This eddy forcing term represents the action of the eddy field, its coupling with the large-scale flow and is a key dynamical constraint imposed on the augmentation procedure.Although the augmentation drastically improved the low-resolution circulation patterns, it did not recover the robust, intrinsic, large-scale low-frequency variability (LFV), which is an important feature of the high-resolution solution. This is by itself an important (negative) result that has significant implication for any data-driven eddy parameterization, especially, given the fact that we used the most complete information about the space–time history of the eddy fields. Note, when we supplied the reference (true) eddy forcing, rather than just the eddy field, the LFV was recovered. This suggests that the LFV is crucially dependent on the details of the space–time eddy forcing/large-scale flow correlations, which are not fully respected by the proposed augmentation procedure.In order to overcome the def
Stepanov D, Ryzhov EA, Berloff P, et al., 2020, Floating tracer clustering in divergent random flows modulated by an unsteady mesoscale ocean field, Geophysical and Astrophysical Fluid Dynamics, Vol: 114, Pages: 690-714, ISSN: 0309-1929
Clustering of tracers floating on the ocean surface and evolving due to combined velocity fields consisting of a deterministic mesoscale component and a kinematic random component is analysed. The random component represents the influence of submesoscale motions. A theory of exponential clustering in random velocity fields is applied to characterise the obtained clustering scenarios in both steady and unsteady time-dependent mesoscale flows, as simulated by a comprehensive realistic, eddy-resolving, general circulation model for the Japan/East Sea. The mesoscale flow field abounds in transient eddy-like patterns modulating and branching the main currents, and the underlying time-mean flow component features closed recirculation zones that can entrap the tracer. The submesoscale flow component is modelled kinematically, as a divergent random velocity field with a prescribed correlation radius and variance. The combined flow induces tracer clustering, that is, the exponential growth of tracer density in patches with vanishing areas. The statistical topography methodology, which provides integral characteristics to quantify the emerging clusters, uncovers drastic dependence of the clustering rates on whether the mesoscale flow component is taken to be steady or time-dependent. The former situation favours robust exponential clustering, similar to the theoretically understood case of purely divergent and zero-mean random velocity. The latter situation, on the contrary, hinders exponential clustering due to significant advection of the tracer out of the nearly enclosed eddies, at the rate faster than the clustering rate.
Kondrashov D, Ryzhov EA, Berloff P, 2020, Data-adaptive harmonic analysis of oceanic waves and turbulent flows, Chaos: an interdisciplinary journal of nonlinear science, Vol: 30, Pages: 1-12, ISSN: 1054-1500
We introduce new features of data-adaptive harmonic decomposition (DAHD) that are showcased to characterize spatiotemporal variability in high-dimensional datasets of complex and mutsicale oceanic flows, offering new perspectives and novel insights. First, we present a didactic example with synthetic data for identification of coherent oceanic waves embedded in high amplitude noise. Then, DAHD is applied to analyze turbulent oceanic flows simulated by the Regional Oceanic Modeling System and an eddy-resolving three-layer quasigeostrophic ocean model, where resulting spectra exhibit a thin line capturing nearly all the energy at a given temporal frequency and showing well-defined scaling behavior across frequencies. DAHD thus permits sparse representation of complex, multiscale, and chaotic dynamics by a relatively few data-inferred spatial patterns evolving with simple temporal dynamics, namely, oscillating harmonically in time at a given single frequency. The detection of this low-rank behavior is facilitated by an eigendecomposition of the Hermitian cross-spectral matrix and resulting eigenvectors that represent an orthonormal set of global spatiotemporal modes associated with a specific temporal frequency, which in turn allows to rank these modes by their captured energy and across frequencies, and allow accurate space-time reconstruction. Furthermore, by using a correlogram estimator of the Hermitian cross-spectral density matrix, DAHD is both closely related and distinctly different from the spectral proper orthogonal decomposition that relies on Welch’s periodogram as its estimator method.The turbulent oceanic flows consist of ubiquitous complex motions—jets, vortices, and waves—that co-exist on very different spatiotemporal scales but also without a clear scale separation, and it brings natural challenge to characterize the whole complexity across the scales. In particular, the study of temporal scales has got less attention than of spatial
Haigh M, Sun L, Shevchenko I, et al., 2020, Tracer-based estimates of eddy-induced diffusivities, Deep Sea Research Part I: Oceanographic Research Papers, Vol: 160, Pages: 1-8, ISSN: 0967-0637
This study provides estimates of the mean eddy-induced diffusivities of passive tracers in a three-layer, double-gyre quasigeostrophic (QG) simulation. A key aspect of this study is the use of a spatial filter to separate the flow and tracer fields into small-scale and large-scale components, and we compare results with those obtained using Reynolds temporal averaging. The eddy tracer flux is related to a rank-2 diffusivity tensor via the flux-gradient relation, which is solved for a pair of tracers with misaligned large-scale gradients. We concentrate on the symmetric part of the resulting diffusivity tensor which represents irreversible mixing processes. The eigenvalues of the symmetric tensor exhibit complicated behaviour, but a particularly dominant and robust feature is the positive/negative eigenvalue pairs, which physically represent filamentation of the tracer concentration. The large off-diagonal diffusivity tensor component is the primary contributor to the eigenvalue polarity, and since this is such a prevalent feature we argue that the (horizontal) eddy-induced diffusivity should always be treated as a full tensor. Diffusivity magnitudes are largest in the upper layer and in the eastward jet region, where the eddying flow is strongest. After removing the rotational part of the eddy tracer flux, typical mean diffusivities (eigenvalues) in the upper-layer are on the order of m2 s−1 in the jet region and m2 s−1 elsewhere. We also confirm that the time-mean of the diffusivity calculated from instantaneous fluxes is not the same as the diffusivity associated with the time-mean fluxes.
Haigh M, Berloff P, 2020, Rossby waves and zonal momentum redistribution induced by localised forcing in the rotating shallow-water model, Journal of Fluid Mechanics, Vol: 885, Pages: A43-1-A43-26, ISSN: 0022-1120
The aim of this study is to understand the dynamics of Rossby waves induced by a localised and periodic ‘plunger’ forcing – imposed on a background flow – which is intended as an elementary representation of transient mesoscale eddy forcing in the ocean. We consider linearised dynamics and its quasi-nonlinear extension, and focus on the rotating shallow-water model. The plunger induces a spectrum of Rossby waves that drive zonal momentum flux convergence at the forced latitudes. This behaviour has a robust and significant dependence on the background flow, which we treat as zonal and uniform. We systematically analyse this dependence using two methods. First, we use the eddy geometry formulation, in which Reynolds stresses are expressed in terms of eddy elongation and eddy tilt parameters, and consider the relationship between eddy geometry and zonal momentum redistribution. Second, we implement decompositions of flow responses into linear dynamical eigenmodes and compare with expectations from linear Rossby wave theory. Both methods compliment each other and aid the understanding of zonal momentum redistribution and its dependence on uniform background flow. We find that this dependence is determined by two factors: (i) dispersion-constrained resonance with the plunger forcing and (ii) efficiency of nonlinear eddy self-interactions. These results significantly improve our understanding of shallow-water Rossby waves, and may also be applied towards the development of parameterisations of oceanic mesoscale eddies.
Stepanov DV, Ryzhov EA, Zagumennov AA, et al., 2020, Clustering of floating tracer due to mesoscale vortex and submesoscale fields, Geophysical Research Letters, Vol: 47, Pages: 1-10, ISSN: 0094-8276
American Geophysical Union. All Rights Reserved. Floating tracer clustering is studied in oceanic flows that combine both a field of coherent mesoscale vortices, as simulated by a regional, comprehensive, eddy-resolving general circulation model, and kinematic random submesoscale velocity fields. Both fields have rotational and divergent velocity components, and depending on their relative contributions, as well as on the local characteristics of the mesoscale vortices, we identified different clustering scenarios. We found that the mesoscale vortices do not prevent clustering but significantly modify its rate and spatial pattern. We also demonstrated that even weak surface-velocity divergence has to be taken into account to avoid significant errors in model predictions of the floating tracer patterns. Our approach combining dynamically constrained and random velocity fields, and the applied diagnostic methods, are proposed as standard tools for analyses and predictions of floating tracer distributions, in both observational data and general circulation models.
Berloff P, 2019, Clustering of floating tracers in weakly divergent velocity fields, Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, Vol: 100, Pages: 1-15, ISSN: 1539-3755
This work deals with buoyant tracers floating at the ocean surface, where the geostrophic velocitycomponent is 2D and rotational (non-divergent), and the ageostrophic component can containcomparable in size rotational and potential (divergent) contributions. We consider a randomkinematic flow model and study the process of clustering, that is, aggregation of the floatingtracer in localized spatial patches. In the large-time limit, and in the cases of strongly and weaklydivergent flows, the existing analytical theory predicts the process of exponential clustering, whichis the emergence of spatial singularities containing all the available tracer. Here, we confirm thisanalytical prediction, in numerical model solutions spanning different combinations of rotationaland potential surface velocity components, and report that exponential clustering persists evenin weakly divergent flows, however, at significantly slower rates. For a wide range of parameters,we analyzed not only the exponential clustering, but also the other type of tracer aggregation,referred to as fragmentation clustering, as well as the coarse-graining effects on clustering. For thepresented analyses we considered ensembles of Lagrangian particles, and introduced and appliedthe statistical topography methodology.
Berloff P, 2019, Rossby waves and momentum fluxes induced by transient localised forcing in the shallow-water model, Journal of Fluid Mechanics, ISSN: 0022-1120
The aim of this study is to understand the dynamics of eddies induced by a localised and periodic‘plunger’ forcing - imposed on a background flow - which is intended as an elementary representation oftransient mesoscale eddy forcing in the ocean. We consider linearised dynamics and its quasi-nonlinearextension, and focus on the rotating shallow-water model. The plunger induces a spectrum of Rossbywaves that drive zonal momentum flux convergence at the forced latitudes. This behaviour has arobust and significant dependence on the background flow, which we treat as zonal and uniform. Wesystematically analyse this dependence using two methods. First, we use the eddy geometry formulation, in which Reynolds stresses are expressed in terms of eddy elongation and eddy tilt parameters,and consider the relationship between eddy geometry and zonal momentum redistribution. Second,we implement decompositions of flow responses into linear dynamical eigenmodes and compare withexpectations from linear Rossby wave theory. Both methods compliment each other and aid the understanding of zonal momentum redistribution and its dependence on uniform background flow. Wefind that this dependence is determined by two factors: (1) dispersion-constrained resonance with theplunger forcing and (2) efficiency of nonlinear eddy self-interactions. These results significantly improveour understanding of shallow-water Rossby waves, and may also be applied towards the developmentof parameterisations of oceanic mesoscale eddies.
Ryzhov EA, Kondrashov D, Agarwal N, et al., 2019, On data-driven augmentation of low-resolution ocean model dynamics, Ocean Modelling, Vol: 142, ISSN: 1463-5003
The problem of augmenting low-resolution ocean circulation models with the information extracted from the data relevant to the unresolved subgrid processes is addressed. A highly nonlinear model of eddy-resolving oceanic circulation – quasigeostrophic wind-driven double gyres – is considered. The model solutions are characterized by a vigorous dynamic coupling between the resolved large-scale and small-scale (eddy) flow features. This solution provides the data for augmenting the low-resolution model with the same configuration. The eddy forcing field, which contains the essential information about coupling between the large and eddy scales, is obtained, modified, coarse-grained and added to augment the low-resolution model. The implemented modification involves novel data-adaptive harmonic decomposition analysis and dynamical constraining based on the low-resolution nonlinear advection operator. The resulting augmentation of the low-resolution model significantly improves the solution, including its time-mean circulation and low-frequency variability. This result also paves the way for a systematic data-driven emulation of unresolved and under-resolved scales of motion.
Berloff P, Khatri H, 2019, Tilted, drifting jets over a zonally sloped topography: Effects of vanishing eddy viscosity, Journal of Fluid Mechanics, Vol: 876, Pages: 939-961, ISSN: 0022-1120
Oceanic multiple jets are seen to possess spatiotemporal variability imposed by varyingbottom topography resulting in jets that can drift and merge. The dynamics of multiplejets over a topographic zonal slope is studied in a two-layer quasi-geostrophic model.The jets tilt from the zonal direction and drift meridionally. In addition to the tilted jets,other large-scale spatial patterns are observed, which are extracted using the principalcomponent analysis. The variances of these patterns are strongly influenced by the valuesof eddy viscosity and bottom friction parameters. The contribution of the tilted jets tothe full flow field decreases with decreasing friction and viscosity parameters, and purelyzonal large-scale modes, propagating in the meridional direction, populate the flow field.Linear stability analysis and two-dimensional kinetic-energy spectrum analysis suggestthat the zonal modes gain energy from ambient eddies as well as from the tilted jetsthrough nonlinear interactions. However, viscous dissipation and bottom friction tendto suppress the nonlinear interactions, which results in the inhibition of the upscaleenergy transfer from eddies to the zonal modes. These simulations suggest that, in thepresence of topography, alternating jet patterns may be sustained through interactionsamong various large-scale modes. This is different from the classical zonal jet formationarguments, in which direct eddy forcing maintains the jets.
Khatri H, Berloff P, 2018, Role of eddies in the maintenance of multiple jets embedded in eastward and westward baroclinic shears, Fluids, Vol: 3, ISSN: 2311-5521
Multiple zonal jets observed in many parts of the global ocean are often embedded in large-scale eastward and westward vertically sheared background flows. Properties of the jets and ambient eddies, as well as their dynamic interactions, are found to be different between eastward and westward shears. However, the impact of these differences on overall eddy dynamics remains poorly understood and is the main subject of this study. The roles of eddy relative vorticity and buoyancy fluxes in the maintenance of oceanic zonal jets are studied in a two-layer quasigeostrophic model. Both eastward and westward uniform, zonal vertically sheared cases are considered in the study. It is shown that, despite the differences in eddy structure and local characteristics, the fundamental dynamics are essentially the same in both cases: the relative-vorticity fluxes force the jets in the entire fluid column, and the eddy-buoyancy fluxes transfer momentum from the top to the bottom layer, where it is balanced by bottom friction. It is also observed that the jets gain more energy via Reynolds stress work in the layer having a positive gradient in the background potential vorticity, and this is qualitatively explained by a simple reasoning based on Rossby wave group velocity.
Haigh MC, Berloff PS, 2018, Potential vorticity redistribution by localised transient forcing in the shallow-water model, Journal of Fluid Mechanics, Vol: 852, Pages: 199-225, ISSN: 0022-1120
This study is motivated by the need to develop stochastic parameterisations for representing the effects of mesoscale oceanic eddies in non-eddy-resolving and eddy-permitting ocean circulation models. A necessary logical step on the way to such parameterisations is the understanding of flow responses to spatially stationary and localised, time-dependent ‘plunger’ forcings intended to represent transient eddy flux divergences. Specifically, this study develops an understanding of the plunger-induced convergence of potential vorticity (PV) fluxes using the linearised single-layer shallow-water model. Time-periodic solutions are obtained and the ‘footprint’, defined as the time-mean, quasi-linear PV flux convergence, quantifies the cumulative PV redistribution induced by the plunger. Using the footprint, the equivalent eddy flux (EEF) is defined such that it succinctly quantifies the extent of the PV redistribution, and its dependencies on the forcing latitude and the background flow are examined in detail. For a uniform background flow the EEF is positive for all forcing latitudes, corresponding to net-poleward PV flux convergence, as expected by current theory of -plane Rossby waves. The EEF also has a robust dependence on the direction and magnitude of a uniform background flow, which is a useful quality for the EEF to provide a basis for a parameterisation of eddy PV fluxes. We also examine the PV redistribution due to forcing on top of a Gaussian jet background flow and find that forcing proximity to the jet core is the primary factor in determining whether the jet is sharpened or broadened.
Khatri H, Berloff P, 2018, A mechanism for jet drift over topography, Journal of Fluid Mechanics, Vol: 845, Pages: 392-416, ISSN: 0022-1120
The dynamics of multiple alternating oceanic jets has been studied in the presence of a simple bottom topography with constant slope in the zonal direction. A baroclinic quasi-geostrophic model forced with a horizontally uniform and vertically sheared background flow generates mesoscale eddies and jets that are tilted from the zonal direction and drift with constant speed. The governing dynamical equations are rewritten in a tilted frame of reference moving with the jets, and the cross-jet time-mean profiles of the linear and nonlinear stress terms are analysed. Here, the linear stress terms are present because of the zonally asymmetric topography. It is demonstrated that the linear dynamics controls the drift mechanism. Also, it is found that the drifting jets are directly forced by the imposed vertical shear, whereas the eddies oppose the jets, although this is limited to continuously forced dissipative systems. This role of the eddies is opposite to the one in the classical baroclinic model of stationary, zonally symmetric multiple jets. This is expected to be more generic in the ocean, which is zonally asymmetric nearly everywhere.
Berloff P, 2018, Dynamically consistent parameterization of mesoscale eddies. Part III: deterministic approach., Ocean Modelling, Vol: 127, Pages: 15-15, ISSN: 1463-5003
This work continues development of dynamically consistent parameterizations for representing mesoscale eddy effects in non-eddy-resolving and eddy-permitting ocean circulation models and focuses on the classical double-gyre problem, in which the main dynamic eddy effects maintain eastward jet extension of the western boundary currents and its adjacent recirculation zones via eddy backscatter mechanism. Despite its fundamental importance, this mechanism remains poorly understood, and in this paper we, first, study it and, then, propose and test its novel parameterization.We start by decomposing the reference eddy-resolving flow solution into the large-scale and eddy components defined by spatial filtering, rather than by the Reynolds decomposition. Next, we find that the eastward jet and its recirculations are robustly present not only in the large-scale flow itself, but also in the rectified time-mean eddies, and in the transient rectified eddy component, which consists of highly anisotropic ribbons of the opposite-sign potential vorticity anomalies straddling the instantaneous eastward jet core and being responsible for its continuous amplification. The transient rectified component is separated from the flow by a novel remapping method. We hypothesize that the above three components of the eastward jet are ultimately driven by the small-scale transient eddy forcing via the eddy backscatter mechanism, rather than by the mean eddy forcing and large-scale nonlinearities. We verify this hypothesis by progressively turning down the backscatter and observing the induced flow anomalies.The backscatter analysis leads us to formulating the key eddy parameterization hypothesis: in an eddy-permitting model at least partially resolved eddy backscatter can be significantly amplified to improve the flow solution. Such amplification is a simple and novel eddy parameterization framework implemented here in terms of local, deterministic flow roughening controlled by single paramet
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