×

zbMATH — the first resource for mathematics

Conservation style of the extra invariant for Rossby waves. (English) Zbl 1200.86003
Summary: We consider the dynamics of a system of Rossby waves with nonlinear interaction. It has been shown that such a system–besides the energy and momentum (enstrophy)–has an extra invariant, which is conserved approximately in the limit of small wave amplitudes. This invariant implies the anisotropy of the inverse cascade, when the energy is transferred from small scale eddies to the large scale zonal jets. The invariant has a quadratic (in wave amplitudes) term, which has a universal form, and a cubic term, whose form depends on the form of nonlinear interaction between Rossby waves.In this paper, we show that it is impossible to extend the extra invariant by higher order nonlinear terms (fourth order and higher) to obtain an exact conservation law. This fact holds irrespective of the form of nonlinearity (including the three-wave interaction). The extra invariant is similar to the adiabatic invariants in the theory of dynamical systems.We also show that on a “long time scale”, the cubic part of the invariant can be dropped (without sacrificing the accuracy of conservation), so that the extra invariant is given by the universal quadratic part.Finally, we show that the extra invariant mostly represents large scale modes, even to a greater extent than the energy. The presence of a small dissipation destroys the conservation of the enstrophy, which is based on small scale modes. At the same time, the extra invariant, along with the energy, remains almost conserved.

MSC:
86A05 Hydrology, hydrography, oceanography
76B55 Internal waves for incompressible inviscid fluids
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Gill, A.E., Atmosphere — Ocean dynamics, (1982), Academic Press New York
[2] Pedlosky, J., Geophysical fluid dynamics, (1987), Springer New York · Zbl 0713.76005
[3] Salmon, R., Lectures on geophysical fluid dynamics, (1987), Springer New York
[4] Newell, A.C., Rossby wave packet interactions, J. fluid mech., 35, 255-271, (1969) · Zbl 0176.54704
[5] Rhines, P.B., Waves and turbulence on a beta plane, J. fluid mech., 69, 417-443, (1975) · Zbl 0366.76043
[7] Rhines, P.B., Jets, Chaos, 4, 313-339, (1994)
[8] 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)
[9] Balk, A.M., A new invariant for Rossby wave systems, Phys. lett. A, 155, 20-24, (1991)
[10] Balk, A.M., Angular distribution of Rossby wave energy, Phys. lett. A, 345, 154-160, (2005) · Zbl 1345.76021
[11] Zakharov, V.E., Stability of periodic waves of finite amplitude on the surface of deep fluid, J. appl. mech. tech. phys., 2, 190, (1968)
[12] Zakharov, V.E.; Schulman, E.I., On additional motion invariants of classical Hamiltonian wave systems, Physica D, 29, 283-320, (1988) · Zbl 0651.35080
[13] Kukharkin, N.; Orszag, S.A., Generation and structure of Rossby vortices in rotating fluids, Phys. rev. E, 54, R4524-R4527, (1996)
[14] Kaladze, T.D.; Wu, D.J.; Pokhotelov, O.A.; Sagdeev, R.Z.; Stenflo, L.; Shukla, P.K., Drift wave driven zonal flows in plasmas, Phys. plasmas, 12, 122311, (2005)
[15] A.M. Balk, E.V. Ferapontov, Invariants of wave systems and web geometry, in: V.E. Zakharov, (Ed.), Nonlinear Waves and Weak Turbulence, in: Amer. Math. Soc. Trans. Ser. 2, vol. 182, Providence, RI, 1998, pp. 1-30 · Zbl 0913.76008
[16] Zakharov, V.E., Hamiltonian formalism in the theory of waves in nonlinear media with dispersion, Izv. VUZ. radiofiz., 17, 431-453, (1974)
[17] Blaschke, W., Einführung in die geometrie der waben, (1955), Birkhäuser-Verlag Basel-Stuttgart
[18] Lin, J.-E.; Chen, H.H., Constraints and conserved quantities of the kadomtsev – petviashvili equations, Phys. lett. A, 89, 164-167, (1982)
[19] Walker, R.J., Algebraic curves, (1950), Princeton University Press Princeton, NJ · Zbl 0039.37701
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.