Frontiers in Nonlinear WavesFrontiers in Nonlinear Wavesin honor of V. E. Zakharov in honor of V. E. Zakharov

birthdaybirthdayMarch 26–29, 2010March 26–29, 2010

University of Arizona, Tucson, AZUniversity of Arizona, Tucson, AZ

Extra Invariant Extra Invariant and Zonal Jetsand Zonal Jets

by Alexander Balk, by Alexander Balk, University of UtahUniversity of Utah

Francois van Heerden,Francois van Heerden,Nuclear Energy Corporation of S.AfricaNuclear Energy Corporation of S.Africa,,

and Peter Weichman, and Peter Weichman, British Aerospace, MassachusettsBritish Aerospace, Massachusetts

(submitted to J. Fluid Mech.)(submitted to J. Fluid Mech.)

Zonal jetsZonal jets The famous example – stripes on The famous example – stripes on

JupiterJupiter

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

Magnetized Plasma:Magnetized Plasma:

ZonalZonal Jets Jets areare

Transport Transport BarriersBarriers

Another situation Another situation

Rotation Rotation (of a planet)(of a planet) ~ Magnetic Field ~ Magnetic Field (in plasma)(in plasma)

In this talk:In this talk:1.1. 3 adiabatic-type invariants: 3 adiabatic-type invariants:

Energy, EnstrophyEnergy, Enstrophy, , Extra InvariantExtra Invariant (started in B., Nazarenko, Zakharov (started in B., Nazarenko, Zakharov

1991)1991)

2.2. Well known: Energy and Well known: Energy and Enstrophy => Inverse Cascade.Enstrophy => Inverse Cascade.

Extra invariant Extra invariant =>=> Anisotropy of the Inv. Cascade:Anisotropy of the Inv. Cascade: Energy accumulates in the Zonal Energy accumulates in the Zonal

JetsJets

3.3. Zonal jets more pronounced at Zonal jets more pronounced at the the EquatorEquator

Rotating Shallow Rotating Shallow WaterWater

0)()(

yxt

yyxt

xyxt

HvHuH

Hguyfvvvuv

Hgvyfuvuuu β-plane approx.:f=f₀+βy+O(y²)

Two Modes:

22 k

p

kradius deform.

inverse

Filtering out inertia-gravity modeGeostrophic Balance (impossible Near Equator)

1. Inertia-Gravity waves ω²=k²+α²+O(β)2. Rossby waves

3 3 approximate, approximate, adiabatic-type, adiabatic-type,

invariants: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

ConservationConservation Style 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 Weakly nonlinear dynamics conserves:conserves:

• Extra invariantExtra invariant

kkk

k dtF

I

),( component;Rossby theof

spectrumenergy theis

qp

tF

kk

pk

• Enstrophy (east-west momentum)Enstrophy (east-west momentum)

• EnergyEnergy

22225

323arctan

3arctan

1

k

p

k

pq

k

pq

Balance argument for Balance argument for the the formation of zonal jetsformation of zonal jets

),( vs.

density spectralenergy the

density spectralinvariant extra the

plot

qp

kk

kk

20

4

k

What forcing is better What forcing is better for generation of for generation of Zonal JetsZonal Jets

(B. & Zakharov 2009)(B. & Zakharov 2009)

Important for fusion plasmas,Important for fusion plasmas,

as Zonal Jets prove to be the transport as Zonal Jets prove to be the transport barriersbarriers

Energy accumulates in the sector of polar angles θ> 60˚.

Agrees with the analysis of energy spectraof very long Rossby waves

[with periods of several years](Glazman & Weichman, 2005)

Not always zonal jets.Not always zonal jets.Long wave limit: k/Long wave limit: k/αα→→00

Nonlinearity taken into Nonlinearity taken into account:account:

• Balance argument works for waves with Balance argument works for waves with Rossby dispersion 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 for special forcing the energy can still

concentrate in zonal jets, even in the long concentrate in zonal jets, even in the long wavewave

situation situation (Balk & Zakharov 2009)(Balk & Zakharov 2009)

• In the short wave case specially arranged In the short wave case specially arranged forcing can accelerate the formation of forcing can accelerate the formation of Zonal Jets (Applications to Nuclear Fusion).Zonal Jets (Applications to Nuclear Fusion).

w

v

u

gz

y

x

vgr velocity 0

0

gravity

Up

North

East

scoordinate Local

:planeTangent

West East

x

y z

Ω

H(x,y,t)

Ωz

g

Coriolis parameter f=2Ωz

Conserves:Conserves: 1. 1. EnergyEnergy 2.2.Space averaged fluid Space averaged fluid depth depth H₀ (mass conservation) x-momentum (translational symmetry in zonal direction) infinite series of potential vorticity integrals

radius deform. inverse

/ 0Hgf