Konstantin V. Koshel

We consider the effects of passive tracer clustering (e.g., the magnetic field energy in stellar atmospheres) in a random acoustic velocity field. A method for numerical modeling of a two-dimensional random acoustic field is proposed. The field is described by a correlation tensor defined by traveling isotropic waves, taking into account dissipation. Two metrics for measuring the clustering effects are used: concentration and density. Using numerical modeling, we show that the tracer concentration is almost always clustered. The situation with the density is different; as the dissipation tends to zero, the time to reach the clustered states increases significantly. In addition, due to the tracer transport out of the density clustering regions, only a part of the tracer is clustered. For the presented analyses, we considered ensembles of Lagrangian particles and introduced and applied the statistical topography methodology.

The influence of a background vortex flow on the clustering of floating tracers is addressed. The vortex flow considered is induced by an ellipsoidal vortex evolving in a deformation. The system exhibits various vortex motion regimes: (1) a steady state, (2) oscillation and (3) rotation of the ellipsoidal vortex core. The latter two induce an unsteady velocity field for the tracer, thus leading to irregular (chaotic) tracer motion. Superimposing a stochastic divergent velocity field onto the deterministic vortex flow allows us to observe significantly different tracer evolution. An ellipsoidal vortex has ellipsoidal symmetry, and the tracer’s trajectories exhibit the same symmetry inside the vortex. Outside the vortex, the external deformation flow symmetry dominates. Diffusion scattering and chaotic advection give tracers the opportunity to leave the region of ellipsoidal symmetry and form a picture of shear flow symmetry. We use the method of characteristics to integrate the floating tracer density evolution equation and the Euler Ito scheme for obtaining the floating tracer trajectories with a random velocity field. The cluster area and cluster mass from the statistical topography are used as the quantitative diagnostics of a floating tracer’s clustering. For the case of a steady ellipsoidal vortex embedded into the deformation flow with a random velocity field component, we found that the clustering characteristics were weakened by the steady vortex. For the cases of an unsteady ellipsoidal vortex, we observed clustering in the floating tracer density field if the contribution of the divergent component was greater than or equal to that of the rotational (nondivergent) component. Even when the initial floating tracer patch was set on the boundary of the oscillating ellipsoidal vortex, we observed the formation of clusters. In the case of a rotating ellipsoidal vortex, we also observed pronounced clustering. Thus, we argue that unsteady ellipsoidal vortex regimes (oscillation and rotation), which induce chaotic motion of the nearby passive tracer’s trajectories, are still conducive to clustering of floating tracers observed in the density field, despite the intense deformation introduced by strain and shear.

The influence of a background vortex flow on clustering of floating tracer is addressed. The vortex flow considered is induced by an ellipsoidal vortex evolving in a deformation. The system exhibits various vortex motion regimes: (i) steady state, (ii) oscillation and (iii) rotation of the ellipsoidal vortex core. The latter two induce unsteady velocity field for the tracer thus leading to irregular (chaotic) tracer motion. Superimposing a stochastic divergent velocity field onto the deterministic vortex flow allows us to observe significantly different tracer evolution. We use the method of characteristics to integrate the floating tracer density evolution equation and the Euler-Ito scheme for obtaining the floating tracer trajectories with the random velocity field. The cluster area and cluster mass from the statistical topography were used as the quantitative diagnostics of floating-tracer’s clustering. For the case of the steady ellipsoidal vortex embedded into the deformation flow with a random velocity field component we found that clustering characteristics are weakened by the steady vortex. For the cases of an unsteady ellipsoidal vortex, we observed clustering in the floating-tracer density field if the contribution of the divergent component is higher or equal to that of the rotational (nondivergent) component. Even when the initial floating tracer patch was set on the boundary of oscillating ellipsoidal vortex, we observed formation of clusters. In the case of a rotating ellipsoidal vortex, we also observed pronounced clustering. Thus, we argue that unsteady ellipsoidal vortex regimes (oscillation and rotation), which induce chaotic motion of the nearby passive tracer’s trajectories, are still conducive to clustering of floating tracer observed in the density field despite the intense deformation introduced by strain and shear.

A generalized form of the matrix-invariant imbedding method was developed to solve boundary-value problems for coupled systems of Helmholtz-type equations. Within this approach, a boundary-value problem solution can be obtained by solving evolutionary first-order imbedding equations for a matrix-valued function. The proposed method is applied to the solution of coupled equations for mode amplitudes describing the propagation of acoustic waves in a range-dependent shallow-water waveguide. The back-scattering of modes by bathymetry features is investigated, and the coefficients of the modal expansion of the wave reflected by an inhomogeneity in the bottom relief are computed. It is demonstrated that back-scattering is strongly connected with the modal interactions and that the back-scattered field consists of modes with numbers different from the number of the incident mode.

При моделировании распространения звука в мелком море с проницаемым дном, как правило, пренебрегают эффектом обратного рассеяния акустических волн. В частности, метод параболического уравнения имеет в своей основе именно приближение однонаправленного распространения. Тем не менее в литературе практически отсутствуют оценки амплитуды обратного рассеяния. В данной работе такие оценки получены с использованием модового представления поля в адиабатическом приближении. Данное приближение позволяет отделить обратное рассеяние от взаимодействия мод - другого важного эффекта, наблюдаемого при рассеянии акустических волн на неоднородностях батиметрии. Для случая падения нормальной волны на локализованную неоднородность дна исследована зависимость амплитуды волны, рассеянной в обратном направлении, от размеров неоднородности и номера моды. Для расчета коэффициента отражения используется численное решение уравнения Риккати, которое получается при переходе от краевой задачи для уравнения Гельмгольца для модовой амплитуды к соответствующим уравнениям погружения. Обсуждаются возможности дальнейшего использования модовых уравнений погружения при решении задач распространения звука в...

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.

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.

This work deals with buoyant tracers floating at the ocean surface, where the geostrophic velocity component is two dimensional and rotational (nondivergent) and the ageostrophic component can contain rotational and potential (divergent) contributions that are comparable in size. We consider a random kinematic flow model and study the process of clustering, that is, aggregation of the floating tracer in localized spatial patches. In the long-time limit and in the cases of strongly and weakly divergent flows, the existing analytical theory predicts the process of exponential clustering, which is the emergence of spatial singularities containing all the available tracer. Here we confirm this analytical prediction, in numerical model solutions spanning different combinations of rotational and potential surface velocity components, and report that exponential clustering persists even in weakly divergent flows, however at significantly slower rates. For a wide range of parameters, we analyze 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 the presented analysis we consider ensembles of Lagrangian particles and introduce and apply the statistical topography methodology.

Floating tracer clustering is studied in oceanic flows that combine both a field of coherent mesoscale vortices simulated by a regional, comprehensive, eddy-resolving general circulation model and randomly modeled 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 reported different clustering scenarios. We found that inclusion of the mesoscale vortices does not prevent clustering, but the rates and patterns of clustering become significantly modified. We also demonstrated that even when the surface velocity divergence is weak, it 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, both in observational data and general circulation models.