Imperial College London

ProfessorPavelBerloff

Faculty of Natural SciencesDepartment of Mathematics

Professor in Applied Mathematics
 
 
 
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Contact

 

+44 (0)20 7594 9662p.berloff Website

 
 
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Location

 

745Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Publication Type
Year
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70 results found

Agarwal N, Ryzhov E, Kondrashov D, Berloff Pet 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.

Journal article

Haigh M, Sun L, McWilliams JC, Berloff Pet 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.

Journal article

Agarwal N, Kondrashov D, Dueben P, Ryzhov E, Berloff Pet al., 2021, A Comparison of Data-Driven Approaches to Build Low-Dimensional Ocean Models, JOURNAL OF ADVANCES IN MODELING EARTH SYSTEMS, Vol: 13

Journal article

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.

Journal article

Sun L, Haigh M, Shevchenko I, Berloff P, Kamenkovich Iet 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.

Journal article

Haigh M, Sun L, McWilliams JC, Berloff Pet 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.

Journal article

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.

Journal article

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.

Journal article

Kamenkovich I, Berloff P, Haigh M, Sun L, Lu Yet 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.

Journal article

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.

Journal article

Ryzhov EA, Kondrashov D, Agarwal N, McWilliams JC, Berloff Pet 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

Journal article

Stepanov D, Ryzhov EA, Berloff P, Koshel Ket 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.

Journal article

Haigh M, Sun L, Shevchenko I, Berloff Pet 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.

Journal article

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

Journal article

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.

Journal article

Stepanov DV, Ryzhov EA, Zagumennov AA, Berloff P, Koshel KVet 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.

Journal article

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.

Journal article

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.

Journal article

Ryzhov EA, Kondrashov D, Agarwal N, Berloff PSet 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.

Journal article

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.

Journal article

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.

Journal article

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.

Journal article

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.

Journal article

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

Journal article

Kondrashov D, Chekroun MD, Berloff P, 2018, Multiscale Stuart-Landau emulators: application to wind-driven ocean gyres, Fluids, Vol: 3, ISSN: 2311-5521

The multiscale variability of the ocean circulation due to its nonlinear dynamics remains a big challenge for theoretical understanding and practical ocean modeling. This paper demonstrates how the data-adaptive harmonic (DAH) decomposition and inverse stochastic modeling techniques introduced in (Chekroun and Kondrashov, (2017), Chaos, 27), allow for reproducing with high fidelity the main statistical properties of multiscale variability in a coarse-grained eddy-resolving ocean flow. This fully-data-driven approach relies on extraction of frequency-ranked time-dependent coefficients describing the evolution of spatio-temporal DAH modes (DAHMs) in the oceanic flow data. In turn, the time series of these coefficients are efficiently modeled by a family of low-order stochastic differential equations (SDEs) stacked per frequency, involving a fixed set of predictor functions and a small number of model coefficients. These SDEs take the form of stochastic oscillators, identified as multilayer Stuart–Landau models (MSLMs), and their use is justified by relying on the theory of Ruelle–Pollicott resonances. The good modeling skills shown by the resulting DAH-MSLM emulators demonstrates the feasibility of using a network of stochastic oscillators for the modeling of geophysical turbulence. In a certain sense, the original quasiperiodic Landau view of turbulence, with the amendment of the inclusion of stochasticity, may be well suited to describe turbulence.

Journal article

van Sebille E, Griffies SM, Abernathey R, Adams T, Berloff P, Biastoch A, Blanke B, Chassignet E, Cheng Y, Cotter C, Deleersnijder E, Doos K, Drake H, Drijfhout S, Gary S, Heemink A, Kjellsson J, Koszalka I, Lange M, Lique C, MacGilchrist G, Marsh R, Mayorga Adame G, McAdam R, Nencioli F, Paris C, Piggott M, Polton J, Ruhs S, Shah S, Thomas M, Wang J, Wolfram P, Zanna L, Zika Jet al., 2017, Lagrangian ocean analysis: fundamentals and practices, Ocean Modelling, Vol: 121, Pages: 49-75, ISSN: 1463-5003

Lagrangian analysis is a powerful way to analyse the output of ocean circulation models and other ocean velocity data such as from altimetry. In the Lagrangian approach, large sets of virtual particles are integrated within the three-dimensional, time-evolving velocity fields. Over several decades, a variety of tools and methods for this purpose have emerged. Here, we review the state of the art in the field of Lagrangian analysis of ocean velocity data, starting from a fundamental kinematic framework and with a focus on large-scale open ocean applications. Beyond the use of explicit velocity fields, we consider the influence of unresolved physics and dynamics on particle trajectories. We comprehensively list and discuss the tools currently available for tracking virtual particles. We then showcase some of the innovative applications of trajectory data, and conclude with some open questions and an outlook. The overall goal of this review paper is to reconcile some of the different techniques and methods in Lagrangian ocean analysis, while recognising the rich diversity of codes that have and continue to emerge, and the challenges of the coming age of petascale computing.

Journal article

Shevchenko I, Berloff P, 2017, On the roles of baroclinic modes in eddy-resolving midlatitude ocean dynamics, Ocean Modelling, Vol: 111, Pages: 55-65, ISSN: 1463-5011

This work concerns how different baroclinic modes interact and influence solutions of the midlatitude oceandynamics described by the eddy-resolving quasi-geostrophic model of wind-driven gyres. We developedmulti-modal energetics analysis to illuminate dynamical roles of the vertical modes, carried out a systematicanalysis of modal energetics and found that the eddy-resolving dynamics of the eastward jet extension of thewestern boundary currents, such as the Gulf Stream or Kuroshio, is dominated by the barotropic, and thefirst and second baroclinic modes, which become more energized with smaller eddy viscosity. In the absenceof high baroclinic modes, the energy input from the wind is more efficiently focused onto the lower modes,therefore, the eddy backscatter maintaining the eastward jet and its adjacent recirculation zones is thestrongest and overestimated with respect to cases including higher baroclinic modes. In the presence of highbaroclinic modes, the eddy backscatter effect on the eastward jet is much weaker. Thus, the higher baroclinicmodes play effectively the inhibiting role in the backscatter, which is opposite to what has been previouslythought. The higher baroclinic modes are less energetic and have progressively decreasing effect on the flowdynamics; nevertheless, they still play important roles in inter-mode energy transfers (by injecting energyinto the region of the most intensive eddy forcing, in the neighborhood of the eastward jet) that have to betaken into account for correct representation of the backscatter and, thus, for determining the eastward jetextension.

Journal article

Chen C, Kamenkovich I, Berloff P, 2016, Eddy trains and striations in quasigeostrophic simulations and the ocean, Journal of Physical Oceanography, Vol: 46, Pages: 2807-2825, ISSN: 1520-0485

This study explores the relationship between coherent eddies and zonally elongated striations. The investigation involves an analysis of two baroclinic quasigeostrophic models of a zonal and double-gyre flow and a set of altimetry sea level anomaly data in the North Pacific. Striations are defined by either spatiotemporal filtering or empirical orthogonal functions (EOFs), with both approaches leading to consistent results. Coherent eddies, identified here by the modified Okubo–Weiss parameter, tend to propagate along well-defined paths, thus forming “eddy trains” that coincide with striations. The striations and eddy trains tend to drift away from the intergyre boundary at the same speed in both the model and observations. The EOF analysis further confirms that these striations in model simulations and altimetry are not an artifact of temporal averaging of random, spatially uncorrelated vortices. This study suggests instead that eddies organize into eddy trains, which manifest themselves as striations in low-pass filtered data and EOF modes.

Journal article

Shevchenko I, Berloff P, 2016, Eddy Backscatter and Counter-Rotating Gyre Anomalies of Midlatitude Ocean Dynamics, Fluids, Vol: 1, ISSN: 2311-5521

This work concerns how two competing mechanisms – eddy backscatter and2 counter-rotating gyre anomalies – influence the midlatitude ocean dynamics, as described by3 the eddy-resolving quasi-geostrophic (QG) model of wind-driven gyres. We analysed dynamical4 balances and effects of different eddy forcing components, as well as their dependencies on5 increasing vertical resolution and decreasing eddy viscosity and found that the eastward jet6 and its adjacent recirculation zones are maintained mostly by the eddy forcing via the eddy7 backscatter mechanism, whereas the time-mean eddy-forcing component plays not only direct8 jet-supporting but also indirect jet-inhibiting role. The latter is achieved by inducing zonally9 elongated anticyclonic/cyclonic Counter-rotating Gyre Anomaly (CGA) in the subpolar/subtropical10 gyre. The indirect effect of CGAs on the eastward jet is found to be moderate relative to the dominant11 eddy backscatter mechanism. We also found that the higher is the vertical baroclinic mode, the12 weaker is its backscatter role and the stronger is its CGA-driving role. Although the barotropic and13 first baroclinic modes are the most efficient ones in maintaining the backscatter, the higher, up to the14 fifth baroclinic modes also have significant but reverse impact that reduces the backscatter

Journal article

Berloff P, 2016, Dynamically Consistent Parameterization of Mesoscale Eddies-Part II: Eddy Fluxes and Diffusivity from Transient Impulses, Fluids, Vol: 1, ISSN: 2311-5521

This work continues development of the framework for dynamically consistent parameterization of mesoscale eddy effects in non-eddy-resolving ocean circulation models. Here, we refine and extend the previous results obtained for the double gyres and aim to account for the eddy backscatter mechanism that maintains eastward jet extension of the western boundary currents. We start by overcoming the local homogeneity assumption and by taking into account full large-scale circulation. We achieve this by considering linearized-dynamic responses to finite-time transient impulses. Feedback from these impulses on the large-scale circulation are referred to as footprints. We find that the local homogeneity assumption yields only quantitative errors in most of the gyres but breaks down in the eastward jet region, which is characterized by the most significant eddy effects. The approach taken provides new insights into the eddy/mean interactions and framework for parameterization of unresolved eddy effects. Footprints provide us with maps of potential vorticity anomalies expected to be induced by transient eddy forcing. This information is used to calculate the equivalent eddy potential vorticity fluxes and their divergences that partition the double-gyre circulation into distinct geographical regions with specific eddy effects. In particular, this allows approximation of the real eddy effects that maintain the eastward jet extension of the western boundary currents and its adjacent recirculation zones. Next, from footprints and their equivalent eddy fluxes and from underlying large-scale flow gradients, we calculate spatially inhomogeneous and anisotropic eddy diffusivity tensor. Its properties suggest that imposing parameterized source terms, that is, equivalent eddy flux divergences, is a better parameterization strategy than implementation of the eddy diffusion.

Journal article

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