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, and Peter Weichman, British Aerospace, Massachusetts and Peter Weichman, British Aerospace, Massachusetts (submitted to J. Fluid Mech.)

Zonal jets The famous example – stripes on Jupiter

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

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

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

Rotating Shallow Water β-plane approx.: f=f ₀ +βy+O(y²) Two Modes: Filtering out inertia-gravity mode Geostrophic Balance (impossible Near Equator) 1. Inertia-Gravity waves ω²=k²+ α ²+O(β) 2. Rossby waves

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

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

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

Balance argument for the formation of zonal jets

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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

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) Not always zonal jets. Long wave limit: k/ α→0

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

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

Conserves: 1. Energy 2.Space averaged fluid depth Conserves: 1. Energy 2.Space averaged fluid depth H ₀ (mass conservation) x-momentum (translational symmetry in zonal direction) infinite series of potential vorticity integrals