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Taming Jets in magnetized Fluids
Y. Kosuga, N. C. Brummell
Ackn: M. Proctor, D. Hughes, P. H. Diamond, J. Mak, etc etc...
Outline
- Motivation
- Linear theory (quick review)
- Generation, quenching, revenge(?) of jets: nice colorful graphs!
- w/o B field → Howard-Krishnamurti
- w/ B field
- Quenching of Jets
- Revival of Jets
- Conclusion
- Effects on the HK bifurcation
→ Critical Modes: Steady and overstable→ Important parameters?
Motivation- Turbulence + subsequently driven Jets/large scale shear flow → Ubiquitous phenomena
Sun: Convective turbulence + Differential Rotation
- Focus: Jets in 2D convective turbulence
- Neutral Fluids: Krishnamurti + Howard ’81 (Exp.), H+K ’86 (Theory)Brummell + Julien (Sim. unpublished)
- What happens with B-field (horizontal)?
- Controlling parameter?
Planets: Geostrophic turbulence + Zonal Jets
Tokamak: Drift wave turbulence + Intrinsic Rotation, Zonal Flow
- Jets: emerge? quench? enforced?
conduction convection tilted cell + Jets chaotic motion
mess
Oscillation
eg)
- Tobias, Hughes, Diamond ’07 → with arbitrary forcing: here, with convectively driven turbulence
Set up- 2D Boussinesq + Uniform Horizontal Magnetic Field
z
xHot
Cold
B
∂t∇2ψ + J(ψ,∇2ψ) = σR∂xθ + σζQ∂xA+ σζQJ(A,∇A2) + σ∇2∇2ψ
∂tθ + J(ψ, θ) = ∂xψ +∇2θ
∂tA+ J(ψ, A) = ∂xψ + ζ∇2A
Vorticity
Temperature
Magnetic Potential
- periodic in x
- Stress free + Perfect conductor (thermal & electric)
σ ≡ ν/κ ζ ≡ η/κ R ≡ gα∆Td3/(νκ) Q ≡ B20d
2/(4πρ0νη)Dim. less. #
Magnetic Potential
B.C.s :
λ ≡ d/L = 1/4
d
L
Linear Instability
- Normal mode solution:
- 2 possibilities
Magnetic Potential
ψ =�
k
ψke−iωt+ikxx sin kzz
ω = 0 ω2 =1− ζ
1 + σσζQk2x − ζ2k4⊥
R =k6⊥k2x
+Qk2⊥R =
(σ + ζ)(ζ + 1)
σ
�k6⊥k2x
+ζ
1 + ζ
σ
1 + σQk2⊥
�
steady cell overstability (only when )
→ Rayleigh-Benard + B field
ζ < 1
→ reduces to Alfven waves for inviscid case
→ more or less close to Rayleigh-Benard
→ unique in convection with magnetic field
→ First focus on this branch → More Later
k2⊥ ≡ k2x + k2z
Linear theory
- Critical Rayleigh number v.s. Q
Rc
Rs =k6⊥k2x
+Qk2⊥
Ro = (1 +ζ
σ)(1 + σ)
k6⊥k2x
+σ + ζ
σ + 1ζQk2⊥
- Keep criticality constant as varying Q
Imposing R at fixed supercriticalityfor steady modes
only for ζ < 1
- Linear stability becomes harder to occur when Q becomes large
→ large Field strength, bending of field lines
→ small magnetic diffusivity, freezing-in law
Q ≡ B20d
2/(4πρ0νη)
Sim. Results
Magnetic Potential
- Unmagnetized case → Howard & Krishnamurti
ζ = 1
Q = 0
- No stationary pattern of cells
- Flow reversal occurs
- Exclude overstable modes (more later)
R = 660000
σ = 10
- Fairly supercritical: for instabilityR = 417136
- Fairly chaotic
Sim. Results
Magnetic Potential
ζ = 1
- Flow and Cell patterns oscillate
- ‘spike’ in kinetic energy corresponds to the flow reversal
Q = 100.5
→ flow flips sign
→ instability of flow, i.e. Kelvin-Helmholtz type instability?
σ = 10 R = 7.23× 106 - Criticality fixed
- starts behaving ‘well’ → B field makes flow pattern more organized
Sim. Results
Magnetic Potential
ζ = 1
- Stationary state achieved! → B field makes flow pattern stationary
- Cell and flow patterns fixed, well behaving jets
Q = 150σ = 10 R = 7.56× 106
Sim. Results
Magnetic Potential
ζ = 1 Q = 200
- Jets quenched! → B field quench Jets
- Still linearly unstable: convection
σ = 10 R = 7.88× 106
Little summary- As Q increases...
chaotic motion
mess
convectiontilted cell + Jets
- ‘Reverse’ the bifurcation series in Howard + Krishnamurti even at constant criticality!
Oscillation
- Why symmetric convection cell without Jets is preferred?
Q = 0 Q = 100.5
Q = 150
Q = 200
- Lantz et al confirmed the ‘reversing’ of the bifurcation series with constant Rayleigh #
→ Simply aiming at linearly stable state
→ Why Jets are quenched?
∂t�U(z)�+ ∂z�w�u� − σζQB�zB
�x� = σ∂zz�U(z)�
Behind the VGs
- Mean Flow:
→ Reynolds v.s. Maxwell Stresses!
- Quenching, why?
→ Max. increases ‘relative’ amplitude, ultimately becomes comparable with
Rey. stress
→ Cell starts standing ‘still’
Q = 130 Q = 150 Q = 170
mag
netic
pot
entia
lst
ream
func
tion
→ Chunk of CurrentAmpere’s law!
→ Lorentz Force cancels for
symmetric cell
Quenching
- Schematically...
|Total Stress|
Q
mess?
170110 130 150
Rey > Max
Rey = Max
- Jet is quenched when Rey = Max → Why Q=170?
- when Jets emerge, always |Rey| > |Max|: Jets driven by Rey. stress
Rey = Max=0
- Once Jet is quenched Rey = Max = 0
- Back of an envelope estimate:
|v2| ∼ σζQ|B2| ζ
�|B2| =
�∂xψA A ∼ τc∂xψ ∼ (kz|vrms|)−1∂xψ
Equipartition of energy + Zeldovich theorem + Mixing of magnetic potential
Q ∼ k2⊥k2x
kz|vrms|σ
∼ 30 → need larger envelope... future work, left as an exercise for... speaker?
There’s the other guy...
→ How magnetic diffusivity affects Jets dynamics?
ζ- Up to this point, varied with fixedQ
→ Relax freezing-in constraint, expect becomes harder to affect Jets with field
- For each different vary Qζ
No Jets Stationary Jets
- Similar behavior as Tobias et al ’07
- Not same ‘power’ law
ζ > 1
Chaotic motion (?)
- may have additional
transition lines
Real fun begins...
- What of overstable modes, i.e. ζ < 1 ?
- Critical Rayleigh number for stationary and overstable cell depends on Q differently
Rc
Q
Rs =k6⊥k2x
+Qk2⊥
Ro = (1 +ζ
σ)(1 + σ)
k6⊥k2x
+σ + ζ
σ + 1ζQk2⊥
- Cross over at
Imposed R: keep same criticality
Qcross =1 + σ
σ
ζ
1− ζ
k4⊥k2x
→ here, Q ∼ 700 (715.976)
σ = 10
ζ = 0.5 kx = 2π/λ = 8π
kz = π
- Expect a different behavior at Q > Qcross
Sim. Results
- Before the crossover: same behavior as before
→ Cell starts standing ‘still’
Q = 150 Q = 200 Q = 500
→ oscillating flow pattern, stationary flow pattern
→ Reynolds v.s. Maxwell, Rey>Max: flow is Rey. Stress driven
→ overstable mode!ζ = 0.5
Sim. Results
- Maxwell stress exceeds Reynolds stress! → Max. Driven flow!
Q = 600 Q = 800 Q = 1000
Sim. Results- Cell/Flow change its tilting direction
Q = 1100 Q = 1300 Q = 1500
- Maxwell > Reynolds → Max. Driven flow!
- Finally Jets quenched
Sim. Results
- Flow Pattern
→ Cell starts standing ‘still’
quenching of Jets
→ Rey. Driven→ Max. exceeds Rey.
Max. Driven
→ Max. DrivenFlow reversal
- ‘Squeeze Jets’, fail and results in flow reversal
Diagram
- Schematically...
|Total Stress|
Q
mess?
Rey > Max
Rey = Max
Rey = Max=0
Rey = MaxMax > Rey
1000 1100200 1500 2000500 600
Max. exceeds Rey. Flow Flips signsSame as steady cell
Q ∼ 700 (715.976)- Note the crossover of stationary and overstable cell at
→ even before the crossover, relative importance of overstable modes increases (diagram for critical Rayleigh)
→ Ball park??? transition from Rey. driven to Max. driven before the crossover
Transition at Q = 1000 to 1100 and Q=1500: Why? Again, left as an exercise...
Conclusion
- Extended Howard-Krishnamurti problem to problem with horizontal B-field
- Confirmed Jets behavior with magnetic field as Tobias et al ’07 with natural forcing
→ Magnetic field can quench Jets by Rey. and Max. stress cancellation, while criticality is kept constant
- With overstable modes, Maxwell stress can dominate Reynolds stress and support flows
- Transition from Rey. dominated flow to Max. dominated flow around the crossover of steady and overstable modes