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The Rossby Wave Extra Invariant in the Dynamics of 3-d Fluid Layers and the Generation of Zonal Jets : Volume 21, Issue 1 (10/01/2014)

By Balk, A. M.

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Book Id: WPLBN0003992452
Format Type: PDF Article :
File Size: Pages 11
Reproduction Date: 2015

Title: The Rossby Wave Extra Invariant in the Dynamics of 3-d Fluid Layers and the Generation of Zonal Jets : Volume 21, Issue 1 (10/01/2014)  
Author: Balk, A. M.
Volume: Vol. 21, Issue 1
Language: English
Subject: Science, Nonlinear, Processes
Collections: Periodicals: Journal and Magazine Collection, Copernicus GmbH
Historic
Publication Date:
2014
Publisher: Copernicus Gmbh, Göttingen, Germany
Member Page: copernicus

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Balk, A. M. (2014). The Rossby Wave Extra Invariant in the Dynamics of 3-d Fluid Layers and the Generation of Zonal Jets : Volume 21, Issue 1 (10/01/2014). Retrieved from http://kindle.worldlibrary.net/


Description
Description: Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA. We consider an adiabatic-type (approximate) invariant that was earlier obtained for the quasi-geostrophic equation and the shallow water system; it is an extra invariant, in addition to the standard ones (energy, enstrophy, momentum), and it is based on the Rossby waves. The presence of this invariant implies the energy transfer from small-scale eddies to large-scale zonal jets.

We show that this extra invariant can be extended to the dynamics of a three-dimensional (3-D) fluid layer on the beta plane. Combined with the investigation of other researchers, this 3-D extension implies enhanced generation of zonal jets.

For a general physical system, the presence of an extra invariant (in addition to the energy–momentum and wave action) is extremely rare. We summarize the unique conservation properties of geophysical fluid dynamics (with the beta effect) that allow for the existence of the extra invariant, and argue that its presence in various geophysical systems is a strong indication that the formation of zonal jets is indeed related to the extra invariant.

Also, we develop a new, more direct, way to establish extra invariants (without using cubic corrections). For this, we introduce the small denominator lemma.


Summary
The Rossby wave extra invariant in the dynamics of 3-D fluid layers and the generation of zonal jets

Excerpt
Balk, A. M., Nazarenko, S. V., and Zakharov, V. E.: New invariant for drift turbulence, Phys. Lett. A, 152, 276–280, 1991.; Balk, A. M.: A new invariant for Rossby wave systems, Phys. Lett. A, 155, 20–24, 1991.; Balk, A. M.: New conservation laws for the interaction of nonlinear waves, SIAM Rev., 39, 68–94, 1997.; Balk, A. M.: Angular distribution of Rossby wave energy, Phys. Lett. A, 345, 154–160, 2005.; Balk, A. M. and Ferapontov, E. V.: Invariants of 4-wave interactions, Physica D, 65, 274–288, 1993.; Balk, A. M. and Ferapontov, E. V.: Invariants of wave systems and web geometry, in: Nonlinear waves and weak turbulence, edited by: Zakharov, V. E., T. Am. Math. Soc., Ser. 2, vol. 182, 1–30, 1998.; Balk, A. M. and van Heerden, F.: Conservation style of the extra invariant for Rossby waves, Physica D, 223, 109–120, 2006.; Balk, A. M., van Heerden, F., and Weichman, P. B.: Rotating shallow water dynamics: Extra invariant and the formation of zonal jets, Phys. Rev. E, 83, 046320, doi:10.1103/PhysRevE.83.046320, 2011.; Blaschke, W.: Topological differential geometry, Chicago University Press, USA, 1932.; Boltzmann, L.: Über das warmegleichgewicht von gasen, auf welche äußere kräfte wirken, Wiener Berichte, 72, 427–457, 1875 (in German).; Cambon, C., Rubinstein, R., and Godeferd, F. S.: Advances in wave turbulence: rapidly rotating flows, New J. Phys., 6, 73, 29 pp., 2004.; Duran-Matute, M., Flór, J.-B., Godeferd, F. S., and Jause-Labert, C.: Turbulence and columnar vortex formation through inertial-wave focusing, Phys. Rev. E, 87, 041001(R), doi:10.1103/PhysRevE.87.041001, 2013.; Ferapontov, E. V.: Web geometry and mathematical physics, in: Geometry and Algebra of Multidimensional Three-Webs, authored by: Akivis, M. A. and Shelekhov, A. M., Kluwer Academic Publishers, Norwell, MA, 310–323, 1992.; Gill, A. E.: Atmosphere–Ocean Dynamics, Academic Press, New York, 1982.; Greenspan, H. P.: On the nonlinear interaction of inertial modes, J. Fluid Mech., 36, 257–264, 1969.; Landau, L. D. and Lifshitz, E. M.: Mechanics, Course of theoretical physics, v. 1, 3rd Edn., Butterworth-Heinemann, New York, 1976.; Landau, L. D. and Lifshitz, E. M.: Fluid Mechanics, Course of theoretical physics, v. 6, 2nd Edn., Butterworth-Heinemann, New York, 1987.; Manz, P., Xu, G. S., Wan, B. N., Wang, H. Q., Guo, H. Y., Cziegler, I., Fedorczak, N., Holland, C., Muller, S. H., Thakur, S. C., Xu, M., Miki, K., Diamond, P. H., and Tynan, G. R.: Zonal flow triggers the L-H transition in the Experimental Advanced Superconducting Tokamak, Phys. Plasmas, 19, 072311, doi:10.1063/1.4737612, 2012.; Maximenko, N., Bang, B., and Sasaki, H.: Observational evidence of alternating zonal jets in the world ocean, Geophys. Res. Lett., 32, L12607, doi:10.1029/2005GL022728, 2005.; Nazarenko, S. and Quinn, B.: Triple Cascade Behavior in Quasigeostrophic and Drift Turbulence and Generation of Zonal Jets, Phys. Rev. Lett., 103, 118501, doi:10.1103/PhysRevLett.103.118501, 2009.; Pedlosky, J.: Geophysical Fluid Dynamics, Springer, New York, 1987.; Rhines, P. B.: Waves and turbulence on a beta plane, J. Fluid Mech., 69, 417–443, 1975.; Sercignani, C.: Are there more than 5 linearly-independent collision invariants for the Boltzmann equation?, J. Stat. Phys., 58, 817–823, 1990.; Smith, L. M. and Waleffe, F.: Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence, Phys. Fluids, 11, 1608–1622, 1999.; Soomere, T.: Coupling coefficients and kinetic equation for Rossby waves in multi-layer ocean, Nonlin. Processes Geophys., 10, 385–396, doi:10.5194/npg-10-385-20

 

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