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 unpublished)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