# DrIgorShevchenko

Faculty of Natural SciencesDepartment of Mathematics

Research Associate

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### Location

754Huxley BuildingSouth Kensington Campus

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## Publications

Publication Type
Year
to

24 results found

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

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

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

Cotter C, Crisan D, Holm DD, Pan W, Shevchenko Iet al., 2020, Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model, Publisher: arXiv

The stochastic variational approach for geophysical fluid dynamics wasintroduced by Holm (Proc Roy Soc A, 2015) as a framework for derivingstochastic parameterisations for unresolved scales. This paper applies thevariational stochastic parameterisation in a two-layer quasi-geostrophic modelfor a beta-plane channel flow configuration. We present a new method forestimating the stochastic forcing (used in the parameterisation) to approximateunresolved components using data from the high resolution deterministicsimulation, and describe a procedure for computing physically-consistentinitial conditions for the stochastic model. We also quantify uncertainty ofcoarse grid simulations relative to the fine grid ones in homogeneous (teamedwith small-scale vortices) and heterogeneous (featuring horizontally elongatedlarge-scale jets) flows, and analyse how the spread of stochastic solutionsdepends on different parameters of the model. The parameterisation is tested bycomparing it with the true eddy-resolving solution that has reached somestatistical equilibrium and the deterministic solution modelled on alow-resolution grid. The results show that the proposed parameterisationsignificantly depends on the resolution of the stochastic model and gives goodensemble performance for both homogeneous and heterogeneous flows, and theparameterisation lays solid foundations for data assimilation.

Working paper

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

Cotter C, Crisan D, Holm D, Pan W, Shevchenko Iet al., 2020, Data Assimilation for a Quasi-Geostrophic Model with Circulation-Preserving Stochastic Transport Noise, Publisher: SPRINGER

Working paper

Kaltenbacher B, Shevchenko I, 2019, Well-posedness of the Westervelt equation with higher order absorbing boundary conditions, Journal of Mathematical Analysis and Applications, Vol: 479, Pages: 1595-1617, ISSN: 0022-247X

The focus of this work is on the analysis of the Westervelt equation modeling nonlinear propagation of high intensity ultrasound, in the practically relevant setting of a truncated computational domain with absorbing boundary conditions. We especially consider the zero and first order nonlinear absorbing boundary conditions devised in [38] in one and two space dimensions. As a matter of fact, the energy identities and estimates presented here were crucial for designing these absorbing boundary conditions in such a way that the desired energy dissipation through the boundary is guaranteed. Under the hypothesis of small initial data, we establish local well-posedness and provide higher order energy estimates, that we expect to be of additional use in boundary control and stabilization.

Journal article

Cotter C, Crisan D, Holm DD, Pan W, Shevchenko Iet al., 2019, A Particle Filter for Stochastic Advection by Lie Transport (SALT): A case study for the damped and forced incompressible 2D Euler equation, Publisher: arXiv

In this work, we apply a particle filter with three additional procedures(model reduction, tempering and jittering) to a damped and forcedincompressible 2D Euler dynamics defined on a simply connected bounded domain.We show that using the combined algorithm, we are able to successfullyassimilate data from a reference system state (the truth") modelled by ahighly resolved numerical solution of the flow that has roughly $3.1\times10^6$degrees of freedom for $10$ eddy turnover times, using modest computationalhardware. The model reduction is performed through the introduction of a stochasticadvection by Lie transport (SALT) model as the signal on a coarser resolution.The SALT approach was introduced as a general theory using a geometricmechanics framework from Holm, Proc. Roy. Soc. A (2015). This work follows onthe numerical implementation for SALT presented by Cotter et al, SIAMMultiscale Model. Sim. (2019) for the flow in consideration. The modelreduction is substantial: The reduced SALT model has $4.9\times 10^4$ degreesof freedom. Forecast reliability and estimated asymptotic behaviour of the particlefilter are also presented.

Working paper

Cotter CJ, Crisan D, Holm DD, Pan W, Shevchenko Iet al., 2019, Numerically modelling stochastic lie transport in fluid dynamics, SIAM Journal on Scientific Computing, Vol: 17, Pages: 192-232, ISSN: 1064-8275

We present a numerical investigation of stochastic transport in ideal fluids.According to Holm (Proc Roy Soc, 2015) and Cotter et al. (2017), the principlesof transformation theory and multi-time homogenisation, respectively, imply aphysically meaningful, data-driven approach for decomposing the fluid transportvelocity into its drift and stochastic parts, for a certain class of fluidflows. In the current paper, we develop new methodology to implement thisvelocity decomposition and then numerically integrate the resulting stochasticpartial differential equation using a finite element discretisation forincompressible 2D Euler fluid flows. The new methodology tested here is foundto be suitable for coarse graining in this case. Specifically, we performuncertainty quantification tests of the velocity decomposition of Cotter et al.(2017), by comparing ensembles of coarse-grid realisations of solutions of theresulting stochastic partial differential equation with the "true solutions" ofthe deterministic fluid partial differential equation, computed on a refinedgrid. The time discretization used for approximating the solution of thestochastic partial differential equation is shown to be consistent. We includecomprehensive numerical tests that confirm the non-Gaussianity of the streamfunction, velocity and vorticity fields in the case of incompressible 2D Eulerfluid flows.

Journal article

Govorukhin V, Shevchenko I, 2017, Multiple equilibria, bifurcations and selection scenarios in cosymmetricproblem of thermal convection in porous medium, Physica D: Nonlinear Phenomena, ISSN: 0167-2789

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

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

Shevchenko I, Berloff P, Guerrero-López D, Roman Jet al., 2016, On low-frequency variability of the midlatitude ocean gyres, Journal of Fluid Mechanics, Vol: 795, Pages: 423-442, ISSN: 1469-7645

This paper studies the large-scale low-frequency variability of the wind-driven midlatitude ocean gyres and their western boundary currents, such as the Gulf Stream or Kuroshio, simulated with the eddy-resolving quasi-geostrophic model. We applied empirical orthogonal functions analysis to turbulent flow solutions and statistically extracted robust and significant large-scale decadal variability modes concentrated around the eastward jet extension of the western boundary currents. In order to interpret these statistical modes dynamically, we linearized the governing quasi-geostrophic equations around the time-mean circulation and solved for the corresponding full set of linear eigenmodes with their eigenfrequencies. We then projected the extracted decadal variability on the eigenmodes and found that this variability is a multimodal coherent pattern phenomenon rather than a single mode or a combination of several modes as in the flow regimes preceding developed turbulence.

Journal article

Shevchenko I, Kaltenbacher B, 2015, Absorbing boundary conditions for nonlinear acoustics: The Westervelt equation, Journal of Computational Physics, Vol: 302, Pages: 200-221, ISSN: 1090-2716

We consider the Westervelt equation in an unbounded domain and propose nonlinear absorbing boundary conditions for its efficient and robust numerical simulations. We use the theory of pseudo- and para-differential operators as well as asymptotic expansions to derive local in space and time absorbing boundary conditions of low to high orders in a consistent way. We show that the pseudo- and para-differential theories lead to essentially the same absorbing boundary conditions in terms of computational efficiency and numerical accuracy, whereas the asymptotic expansions result in exactly the same boundary conditions as the ones obtained with the para-differential approach. Moreover, we demonstrate that the use of pseudo- and para-differential operators leads to the same boundary conditions if the nonlinear function to be linearized vanishes at zero. The numerical studies demonstrate both the efficiency and effectiveness of the developed boundary conditions for different regimes of wave propagation in a wide range of excitation frequencies and angles of incidence.

Journal article

Shevchenko IV, Berloff PS, 2015, Multi-layer quasi-geostrophic ocean dynamics in Eddy-resolving regimes, Ocean Modelling, Vol: 94, Pages: 1-14, ISSN: 1463-5011

The multi-layer quasi-geostrophic model of the wind-driven ocean gyres is numerically investigated using a combination of long-time runs (200 years) needed for accurate statistics, spatial resolutions (grid interval of less than one kilometer) needed for accurate representation of mesoscale eddies, and large Reynolds number (Re > 104) needed for more realistic flow regimes. We gradually increased the Reynolds number by lowering the eddy viscosity and analysed the corresponding changes of the large-scale circulation, energetics and eddy fluxes, with the goal to understand how the nonlinear eddy dynamics affects the large-scale ocean circulation, as more and more degrees of freedom become dynamically available. Three- and six-layer configurations of the model are considered in order to understand effects of higher baroclinic modes. A parameter sensitivity study is also carried out to show that the explored flow regime is robust.As Re increases, most properties of the flow show no signs of approaching an asymptote, and the following tendencies are found. The time-mean flow properties tend to an asymptote in the three-layer model but not in the six-layer one, suggesting that higher baroclinic modes are dynamically more active at larger Re. The eddy kinetic and potential energies grow faster in the six-layer case. The intensity of the eddy forcing (eddy flux divergence) increases with Re. The inter-gyre eddy potential vorticity flux is predominantly northward and up-gradient for all Re studied. A comparison of the three- and six-layer model solutions revealed an inhibitory influence of high baroclinic modes on the penetration length of the eastward jet extension of the western boundary currents and on the strength of the adjacent recirculation zones. In large-Re regimes, the population of eddies is mostly sustained by the eddy generation at the eastern end of the eastward jet rather than in its central section. Finally, by studying the numerical convergence of the sol

Journal article

Shevchenko I, Kaltenbacher B, 2015, ABSORBING BOUNDARY CONDITIONS FOR THE WESTERVELT EQUATION, Publisher: AMER INST MATHEMATICAL SCIENCES-AIMS

Working paper

Govorukhin V, Shevchenko I, 2013, Selection of steady regimes of a one-parameter family in the problem of plane convective flow through porous medium, Fluid Dynamics, Vol: 48, Pages: 523-532, ISSN: 1573-8507

Convective flow through a porous medium in a rectangular vessel with a linear temperature profile steadily maintained on the boundary is investigated. On the basis of numerical experiments the realization of steady regimes that belong to a globally stable, one-parameter family is studied at different vessel dimensions and initial temperature distributions. The regime selection is shown to strongly depend on the initial fluid temperature: a vicinity of two regimes is realized from initial data similar with the state of rest; at high initial fluid heating the regimes are selected from a vicinity of two other regimes; and in intermediate situations any of infinite number of steady regimes can be attained.

Journal article

Vaik I, Paal G, Kaltenbacher M, Triebenbacher S, Becker S, Shevchenko Iet al., 2013, Aeroacoustics of the edge tone: 2D-3D coupling between CFD and CAA, Acta. Acust. United Ac., Vol: 99, Pages: 245-259

Journal article

Shevchenko I, Kaltenbacher M, Wohlmuth B, 2013, Absorbing boundary conditions for a wave equation with a temperature dependent speed of sound, J. Comput. Acoust., Vol: 21, Pages: 1-20

Journal article

Shevchenko I, Wohlmuth B, 2012, Self-adapting absorbing boundary conditions for the wave equation, Wave motion, Vol: 49, Pages: 461-473

Journal article

Shevchenko I, Kaltenbacher M, Wohlmuth B, 2012, A multi-time stepping integration method for the ultrasonic heating problem, Z. Angew. Math. Mech., Vol: 92, Pages: 869-881

Journal article

Govorukhin VN, Shevchenko IV, 2003, Numerical Investigation of the Second Transition in the Problem of Plane Convective Flow through a Porous Medium, FLUID DYNAMICS, Vol: 38, Pages: 760-771, ISSN: 0015-4628

Journal article

Shevchenko I, Berloff P, On a minimum set of equations for parameterisations in comprehensive ocean circulation models

The complexity of comprehensive ocean models poses an important question forparameterisations: is there a minimum set of equations that should beparameterised, on the one hand, to reduce the development to a minimum, and, onthe other hand, to ensure an accurate representation of large-scale flowpatterns? This work seeks to answer this to assist modern parameterisations bemore selective in their targets. For this, we considered a North Atlanticconfiguration of the MIT general circulation model and studied contributions ofdifferent model equations to the accuracy of representation of the Gulf Streamat low resolution. Our results suggest that it is enough to parameterise onlythe tracer equations for temperature and salinity and leave the other equationsin the hydrodynamic part, as well as the atmospheric model unmodified.

Journal article

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