zbMATH — the first resource for mathematics

The Rossby wave extra invariant in the physical space. (English) Zbl 1160.37392
Summary: It was found out in 1991 that the Fourier space dynamics of Rossby waves possesses an extra positive-definite quadratic invariant, in addition to the energy and enstrophy. This invariant is similar to the adiabatic invariants in the theory of dynamical systems. For many years, it was unclear if this invariant – known only in the Fourier representation – is physically meaningful at all, and if it is, in what sense it is conserved. Does the extra conservation hold only for a class of solutions satisfying certain constraints (like the conservation in the Kadomtsev-Petviashvili equation)? The extra invariant is especially important because this invariant (provided it is meaningful) has been connected to the formation of zonal jets (like Jupiter’s stripes). In the present paper, we find an explicit expression of the extra invariant in the physical (or coordinate) space and show that the invariant is indeed physically meaningful for any fluid flow. In particular, no constraints are needed. The explicit form also enables us to note several properties of the extra invariant.

37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
76B65 Rossby waves (MSC2010)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Full Text: DOI
[1] Vallis, G.K., Atmospheric and oceanic fluid dynamics, fundamentals and large-scale circulation, (2006), Cambridge · Zbl 1374.86002
[2] McWilliams, J.C., Fundamentals of geophysical fluid dynamics, (2006), Cambridge · Zbl 1233.86003
[3] Hasegawa, A.; Mima, K., Pseudo-three-dimensional turbulence in magnetized nonuniform plasma, Phys. fluids, 21, 87-92, (1978) · Zbl 0374.76046
[4] Rhines, P.B., Waves and turbulence on a beta plane, J. fluid mech., 69, 417-443, (1975) · Zbl 0366.76043
[5] Balk, A.M.; Nazarenko, S.V.; Zakharov, V.E., New invariant for drift turbulence, Phys. lett. A, 152, 276-280, (1991)
[6] Balk, A.M., A new invariant for Rossby wave systems, Phys. lett. A, 155, 20-24, (1991)
[7] Pedlosky, J., Geophysical fluid dynamics, (1987), Springer New York · Zbl 0713.76005
[8] Durran, D., Numerical methods for wave equations in geophysical fluid dynamics, (1999), Springer New York
[9] Balk, A.M.; van Heerden, F., Conservation style of the extra invariant for Rossby waves, Physica D, 223, 109-120, (2006) · Zbl 1200.86003
[10] P. Kushner, Private communication (2005)
[11] Balk, A.M., Angular distribution of Rossby wave energy, Phys. lett. A, 345, 154-160, (2005) · Zbl 1345.76021
[12] Newell, A.C., Rossby wave packet intereactions, J. fluid mech., 35, 255-271, (1969) · Zbl 0176.54704
[13] Rhines, P.B., Geostrophic turbulence, Annu. rev. fluid mech., 11, 401-441, (1979) · Zbl 0474.76054
[14] Reznik, G.M.; Soomere, T.E., The evolution of the Rossby-wave ensemble to the anisotropic equilibrium state, Okeanolgiya, 24, 558-565, (1984)
[15] Vallis, G.K.; Maltrud, M.E., Generation of Mean flows and jets on a beta plane and over topography, J. phys. oceanogr., 23, 1346-1362, (1993)
[16] Condie, S.A.; Rhines, P.B., A convective model for the zonal jets in the atmospheres of Jupiter and saturn, Nature, 367, 711-713, (1994)
[17] Danilov, S.; Gurarie, D., Scaling, spectra and zonal jets in beta-plane turbulence, Phys. fluids, 16, 2592-2603, (2004) · Zbl 1186.76124
[18] Diamond, P.H.; Itoh, S.-I.; Itoh, K.; Hahm, T.S., Zonal flows in plasma — a review, Plasma phys. control. fusion, 47, R35-R161, (2005)
[19] Galperin, B.; Sukoriansky, S.; Dikovskaya, N.; Read, P.L.; Yamazaki, Y.H.; Wordsworth, R., Anisotropic turbulence and zonal jets in rotating flows with a \(\beta\)-effect, Nonlinear processes geophys., 13, 89-96, (2006)
[20] Sercignani, C., Are there more than 5 linearly-independent collision invariants for the Boltzmann equation?, J. stat. phys., 58, 817-823, (1990) · Zbl 0718.60113
[21] Zakharov, V.E.; Schulman, E.I., Degenerative dispersion laws, motion invariants, and kinetic equations, Physica D, 10, 192-202, (1980) · Zbl 1194.37162
[22] Zakharov, V.E., Integrable systems in multidimensional spaces, (), 190-216
[23] Zakharov, V.E.; Schulman, E.I., On additional motion invariants of classical Hamiltonian wave systems, Physica D, 29, 283-320, (1988) · Zbl 0651.35080
[24] Zakharov, V.E.; Schulman, E.I., Integrability of nonlinear systems and perturbation theory, (), 185-250, Springer Series on Nonlinear Dynamics · Zbl 0807.35107
[25] Zakharov, V.E., Hamiltonian formalism in the theory of waves in nonlinear media with dispersion, Izv. vuzov, radiofizika, 17, 431-453, (1974)
[26] Balk, A.M.; Ferapontov, E.V.; Zakharov, V.E., Invariants of wave systems and web geometry, Nonlinear waves and weak turbulence, amer. math. soc. trans. ser. 2, 182, 1-30, (1998) · Zbl 0913.76008
[27] Balk, A.M., New conservation laws for the interaction of nonlinear waves, SIAM rev., 39, 68-94, (1997) · Zbl 0874.35089
[28] Blaschke, W., Topological differential geometry, (1932), Chicago University Press
[29] Longuet-Higgins, M.S.; Gill, A.E., Resonance interactions between planetary waves, Proc. roy. soc. London, ser. A, 299, 120-144, (1967)
[30] Reznik, G.M., On the energy transfer equation for weakly interacting waves, Internat. J. non-linear mech., 19, 95-113, (1984) · Zbl 0546.73026
[31] Monin, A.S.; Piterbarg, L.I., A kinetic equation for Rossby waves, Sov. phys. - doklady, 295, 816-820, (1987) · Zbl 0642.76029
[32] Galeev, A.A.; Sagdeev, R.Z., Nonlinear plasma theory, (1969), Benjamin New York · Zbl 0194.58802
[33] Salmon, R., Lectures on geophysical fluid dynamics, (1998), Oxford University Press
[34] Zakharov, V.E.; Piterbarg, L.I., Canonical variables for Rossby and drift waves in plasma, Sov. phys. doklady, 32, 560-561, (1987) · Zbl 0642.76130
[35] Piterbarg, L.I., Hamiltonian formalism for for Rossby and drift waves in plasma, Nonlinear waves and weak turbulence, amer.math. soc. trans. ser., 2, 182, 131-166, (1998) · Zbl 0913.76019
[36] Lin, J.-E.; Chen, H.H., Constraints and conserved quantities of the Kadomtsev-Petviashvili equations, Phys. lett. A, 89, 164-167, (1982)
[37] Arfken, G.B.; Weber, H.J., Mathematical methods for physicists, (2001), Academic Press Harcour · Zbl 0970.00005
[38] Abramowitz, M.; Stegun, I.A., ()
[39] Stein, E.M.; Shakarchi, R., Fourier analysis. an introduction, (2003), Princeton University Press · Zbl 1026.42001
[40] Weichman, P.B., Equilibrium theory of coherent vortex and zonal jet formation in a system of nonlinear Rossby waves, Phys. rev. E, 73, (2006), 036313—1-5
[41] Smolyakov, A.I.; Diamond, P.H.; Shevchenko, V.I., Zonal flow generation by parametric instability in magnetized plasmas and geostrophic fluids, Phys. plasmas, 7, 1349-1351, (2000)
[42] Shukla, P.K.; Stenflo, L., Generation of zonal flows by Rossby waves, Phys. lett. A, 307, 154-157, (2003) · Zbl 1006.76019
[43] Onishchenko, O.G.; Pokhotelov, O.A.; Sagdeev, R.Z.; Shukla, P.K.; Stenflo, L., Generation of zonal flows by Rossby waves in the atmosphere, Nonlinear processes geophys., 11, 241-244, (2004)
[44] Lee, Y.; Smith, L.M., On the formation of geophysical and planetary zonal flows by near-resonant wave interactions, J. fluid mech., 576, 405-424, (2007) · Zbl 1110.76053
[45] R. Salmon, Private communication (2005)
[46] Holloway, G.; Hendershott, M.C., Stochastic closure for nonlinear Rossby waves, J. fluid mech., 82, 747-765, (1977) · Zbl 0363.76014
[47] Nezlin, M.V.; Snezhkin, E.N., Rossby vortices, spiral structures, solitons: astrophysics and plasma physics in shallow water experiments, (1993), Springer-Verlag
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.