by alexander balk, university of utah
DESCRIPTION
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 , - PowerPoint PPT PresentationTRANSCRIPT
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