NPG Nonlinear Processes in Geophysics NPG Nonlin. Processes Geophys. 1607-7946 Copernicus Publications Göttingen, Germany 10.5194/npg-20-437-2013 Diffusion-affected passive scalar transport in an ellipsoidal vortex in a shear flow Koshel K. V. https://orcid.org/0000-0002-8014-7699 1 2 3 Ryzhov E. A. https://orcid.org/0000-0002-2677-5723 1 Zhmur V. V. 4 5 V.I.Il'ichev Pacific Oceanological Institute, 43, Baltiyskaya Street, Vladivostok, 690041, Russia Far Eastern Federal University, 8, Sukhanova Street, Vladivostok, 690950, Russia Institute of Applied Mathematics, 7, Radio Street, Vladivostok, 690022, Russia P. P. Shirshov Institute of Oceanology, 36, Nahimovski prospect, Moscow, 117997, Russia Moscow Institute of Physics and Technology, 9, Institutskiy Pereulok, Dolgoprudnyi, Moscow region, 141700, Russia 01 07 2013 20 4 437 444 Copyright: © 2013 K. V. Koshel et al. 2013 This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/ This article is available from https://npg.copernicus.org/articles/20/437/2013/npg-20-437-2013.html The full text article is available as a PDF file from https://npg.copernicus.org/articles/20/437/2013/npg-20-437-2013.pdf

By employing an analytical model for a constant-vorticity distributed vortex, namely, the ellipsoidal vortex embedded in a constant buoyancy frequency shear flow, the problem of the passive scalar transport through the vortex's boundary is addressed. Since the model's governing equations do not allow such transition to occur, we implement a low-scale diffusion process into the vortex model. Taking into consideration the diffusion term, we study the passive scalar transport in a steady state (the boundary of the ellipsoidal vortex does not change in time) and in a perturbed state (the boundary of the ellipsoidal vortex changes in time periodically) within the time scope corresponding to the characteristic life cycle of a mesoscale oceanic eddy. An increase of the passive scalar transport through the boundary in the perturbed state in comparison with the steady state due to the irregular dynamics of the surrounding flow is shown. The applicability scopes of the investigation for studying oceanic eddies in nature are discussed.

 References Aref, H.: Stirring by chaotic advection, J. Fluid Mech., 143, 1–21, <a href="http://dx.doi.org/10.1017/S0022112084001233">https://doi.org/10.1017/S0022112084001233</a>, 1984. Aref, H.: The development of chaotic advection, Phys. Fluids, 14, 1315–25, <a href="http://dx.doi.org/10.1063/1.1458932">https://doi.org/10.1063/1.1458932</a>, 2002. Balasuriya, S.: An Analytical Study of General Hyper-diffusivity and Barotropic Eddies, Geo, 91, 39–62, <a href="http://dx.doi.org/10.1080/03091920410001659290">https://doi.org/10.1080/03091920410001659290</a>, 2004. Balasuriya, S. and Jones, C. K. R. T.: Diffusive draining and growth of eddies, Nonlin. Processes Geophys., 8, 241–251, <a href="http://dx.doi.org/10.5194/npg-8-241-2001">https://doi.org/10.5194/npg-8-241-2001</a>, 2001. Brown, M. G. and Samelson, R. M.: Particle motion in vorticity-conserving, two-dimensional incompressible flows, Phys. Fluids, 6, 2875–2876, <a href="http://dx.doi.org/10.1063/1.868112">https://doi.org/10.1063/1.868112</a>, 1994. Chelton, D. B., Schlax, M. G., Samelson, R. M., and de Szoeke, R. A.: Global observations of large oceanic eddies, Geophys. Res. Lett., 34, L15606, <a href="http://dx.doi.org/10.1029/2007GL030812">https://doi.org/10.1029/2007GL030812</a>, 2007. Chelton, D. B., Schlax, M. G., and Samelson, R. M.: Global observations of nonlinear mesoscale eddies, Prog. Oceanogr., 91, 167–216, <a href="http://dx.doi.org/10.1016/j.pocean.2011.01.002">https://doi.org/10.1016/j.pocean.2011.01.002</a>, 2011. Dahleh, M. D.: Exterior flow of the Kida ellipse, Phys. Fluids A-Fluid, 4, 1979–1985, <a href="http://dx.doi.org/10.1063/1.858366">https://doi.org/10.1063/1.858366</a>, 1992. Izrailsky, Y. G., Koshel, K. V., and Stepanov, D. V.: Determination of the optimal excitation frequency range in background flows, CHAOS, 18, 013107, <a href="http://dx.doi.org/10.1063/1.2835349">https://doi.org/10.1063/1.2835349</a>, 2008. Jones, S. W.: Interaction of Chaotic Advection and Diffusion, Chaos Soliton. Fract., 4, 929–940, <a href="http://dx.doi.org/10.1016/0960-0779(94)90132-5">https://doi.org/10.1016/0960-0779(94)90132-5</a>, 1994. Kida, S.: Motion of an elliptic vortex in a uniform shear flow, J. Phys. Soc. Jpn., 50, 3517–3520, <a href="http://dx.doi.org/10.1143/JPSJ.50.3517">https://doi.org/10.1143/JPSJ.50.3517</a>, 1981. Klyatskin, V. I.: Statistical description of the diffusion of a passive tracer in a random velocity field, Phys.-Usp.+, 37, 501–513, <a href="http://dx.doi.org/10.1070/PU1994v037n05ABEH000110">https://doi.org/10.1070/PU1994v037n05ABEH000110</a>, 1994. Klyatskin, V. I.: Stochastic Equations through the Eye of the Physicist (Basic Concepts, Exact Results, and Asymptotic Approximations), Elsevier Science, vii–viii, <a href="http://dx.doi.org/10.1016/B978-044451797-5/50000-9">https://doi.org/10.1016/B978-044451797-5/50000-9</a>, 2005. Koshel, K. V. and Alexandrova, O. V.: Some results of a numerical modeling of the diffusion of passive tracers in a random field of velocities, Izv. Atmos. Ocean. Phys., 35, 578–588, 1999. Koshel, K. V. and Prants, S. V.: Chaotic advection in the ocean, Phys.-Usp.+, 176, 1177–1206, <a href="http://dx.doi.org/10.1070/PU2006v049n11ABEH006066">https://doi.org/10.1070/PU2006v049n11ABEH006066</a>, 2006. Koshel, K. V., Sokolovskiy, M. A., and Davies, P. A.: Chaotic advection and nonlinear resonances in an oceanic flow above submerged obstacle, Fluid Dyn. Res., 40, 695–736, <a href="http://dx.doi.org/10.1016/j.fluiddyn.2008.03.001">https://doi.org/10.1016/j.fluiddyn.2008.03.001</a>, 2008. Lichtenberg, A. and Lieberman, M.: Regular and Stochastic Motion, Springer-Verlag, New York, 1983. Meacham, S., Pankratov, K. K., Shchepetkin, A. F., and Zhmur, V. V.: The interaction of ellipsoidal vortices with background shear flows in a stratified fluid, Dynam. Atmos. Oceans, 21, 167–212, <a href="http://dx.doi.org/10.1016/0377-0265(94)90008-6">https://doi.org/10.1016/0377-0265(94)90008-6</a>, 1994. Mesinger, F.: Numerical integration of the primitive equations with a floating set of computation points: experiments with a barotropic global model, Mon. Weather Rev., 99, 15–29, <a href="http://dx.doi.org/10.1175/1520-0493(1971)099< 0015:NIOTPE> 2.3.CO;2">https://doi.org/10.1175/1520-0493(1971)099< 0015:NIOTPE> 2.3.CO;2</a>, 1971. Monin, A. and Ozmidov, R.: Turbulence in the Ocean, Reidel, Dordrecht, 1985. Okubo, A.: Oceanic diffusion diagrams, Deep-Sea Res., 18, 789–802, 1971. Pedlosky, J.: Geophysical Fluid Dynamics, 2nd Edn., Springer, New York, 1987. Polvani, L. and Wisdom, J.: Chaotic Lagrangian trajectories around an elliptical vortex parch embedded in a constant and uniform background shear flow, Phys. Fluids A-Fluid, 2, 123–126, <a href="http://dx.doi.org/10.1063/1.857814">https://doi.org/10.1063/1.857814</a>, 1990. Rom-Kedar, V. and Poje, A. C.: Universal properties of chaotic transport in the presence of diffusion, Phys. Fluids, 11, 2044–2057, <a href="http://dx.doi.org/10.1063/1.870067">https://doi.org/10.1063/1.870067</a>, 1999. Van Dam, G. C., Ozmidov, R. V., Korotenko, K. A., and Suijlen, J. M.: Spectral structure of horizontal water movement in shallow seas with special reference to the North Sea, as related to the dispersion of dissolved matter, J. Marine Syst., 21, 207–228, <a href="http://dx.doi.org/10.1016/S0924-7963(99)00015-9">https://doi.org/10.1016/S0924-7963(99)00015-9</a>, 1999. Zaslavsky, G.: Physics of Chaos in Hamiltonian Dynamics, Imperial College Press, London, 1998. Zhmur, V. V.: Localized eddy formation in a shear-flow, Oceanology, 28, 536–538, 1988. Zhmur, V. V.: Subsurface mesoscale eddy structures in a stratificated ocean, Oceanology, 29, 28–32, 1989. Zhmur, V. V. and Pankratov, K. K.: The dynamics of the semi-ellipsoid subsurface vortex in the non-uniform flow, Okeanologiya, 29, 205–211, 1989. Zhmur, V. V., Ryzhov, E. A., and Koshel, K. V.: Ellipsoidal vortex in a nonuniform flow: Dynamics and chaotic advections, J. Mar. Res., 69, 435–461, <a href="http://dx.doi.org/10.1357/002224011798765204">https://doi.org/10.1357/002224011798765204</a>, 2011. Zhurbas, V. and Oh, I. S.: Drifter-derived maps of lateral diffusivity in the Pacific and Atlantic Oceans in relation to surface circulation patterns, J. Geophys. Res., 109, C05015, <a href="http://dx.doi.org/10.1029/2003JC002241">https://doi.org/10.1029/2003JC002241</a>, 2004.