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by Alexander Balk, University of Utah

Frontiers in Nonlinear Waves in honor of V. E. Zakharov birthday March 26–29, 2010 University of Arizona, Tucson, AZ Extra Invariant and Zonal Jets. by Alexander Balk, University of Utah Francois van Heerden, Nuclear Energy Corporation of S.Africa ,

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by Alexander Balk, University of Utah

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  1. Frontiers in Nonlinear Wavesin honor of V. E. Zakharov birthdayMarch 26–29, 2010University of Arizona, Tucson, AZExtra Invariant and Zonal Jets by Alexander Balk, University of Utah Francois van Heerden,Nuclear Energy Corporation of S.Africa, and Peter Weichman, British Aerospace, Massachusetts (submitted to J. Fluid Mech.)

  2. Zonal jetsThe famous example – stripes on Jupiter

  3. O. G. Onishchenko, O. A. Pokhotelov, R. Z. Sagdeev, P. K. Shukla, and L. Stenflo 2004

  4. Another situation Magnetized Plasma: Rotation (of a planet) ~ Magnetic Field (in plasma) Zonal Jets are Transport Barriers

  5. In this talk: 1. 3 adiabatic-type invariants: Energy, Enstrophy, Extra Invariant (started in B., Nazarenko, Zakharov 1991) 2. Well known: Energy and Enstrophy => Inverse Cascade. Extra invariant => Anisotropy of the Inv. Cascade: Energy accumulates in the Zonal Jets 3. Zonal jets more pronounced at the Equator

  6. Rotating Shallow Water β-plane approx.: f=f₀+βy+O(y²) Two Modes: 1. Inertia-Gravity waves ω²=k²+α²+O(β)2. Rossby waves Filtering out inertia-gravity mode Geostrophic Balance (impossible Near Equator)

  7. 3 approximate, adiabatic-type, invariants: (1) Energy and (2) Enstrophy of the Rossby component => inverse energy cascade (3) Extra invariant => anisotropy of the inverse cascade Energy accumulates in Zonal Jets

  8. Conservation Style Similar to adiabatic conservation in Dynamical Systems But instead of slow parameter change, small nonlinearity • Conserved similar to: • Manley-Rowe relations in optics • => balance of photon fluxes • Wave action for surface gravity waves • => inverse cascade (Zakharov, 1985)

  9. Weakly nonlinear dynamics conserves: • Extra invariant • Energy • Enstrophy (east-west momentum)

  10. Balance argument for the formation of zonal jets

  11. 20

  12. What forcing is better for generation of Zonal Jets (B. & Zakharov 2009) Important for fusion plasmas, as Zonal Jets prove to be the transport barriers

  13. Not always zonal jets.Long wave limit: k/α→0 Energy accumulates in the sector of polar angles θ> 60˚. Agrees with the analysis of energy spectra of very long Rossby waves [with periods of several years] (Glazman & Weichman, 2005)

  14. Nonlinearity taken into account: • Balance argument works for waves with Rossby dispersion Nonlinearity can be different • If nonlinearity is taken into account, for special forcing the energy can still concentrate in zonal jets, even in the long wave situation (Balk & Zakharov 2009) • In the short wave case specially arranged forcing can accelerate the formation of Zonal Jets (Applications to Nuclear Fusion).

  15. Ω Ωz z y x H(x,y,t) g West East Coriolis parameter f=2Ωz

  16. Conserves: 1. Energy2.Space averaged fluid depth H₀(mass conservation)x-momentum (translational symmetry in zonal direction) infinite series of potential vorticity integrals

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