Bistable Dynamics of Turbulence Intensity in a Corrugated - - PowerPoint PPT Presentation
Bistable Dynamics of Turbulence Intensity in a Corrugated Temperature Profile Zhibin Guo University of California, San Diego Collaborator: P.H. Diamond, UCSD Chengdu, 2017 This material is based upon work supported by the U.S. Department of
Bistable Dynamics of Turbulence Intensity in a Corrugated Temperature Profile
Zhibin Guo University of California, San Diego Chengdu, 2017
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Collaborator: P.H. Diamond, UCSD
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, under Award Numbers DE-FG02-04ER54738
Outline
—subcritical excitation —propagation
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Motivations
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How turbulent fluctuations penetrates stable domains?
Collapse of H-mode Most previous works treat turbulence spreading as a Fisher front.
Conventional wisdom: Fisher front with a nonlinear diffusivity
∂ ∂t I = γ 0 〈A〉 − Ac
( )I −γ nlI 2 + D1
∂ ∂x I ∂ ∂x I ⎛ ⎝ ⎜ ⎞ ⎠ ⎟
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A ≡ −∂xT
Nontrivial solution requires:
〈A〉 > Ac ⇒ I ∝ 〈A〉 − Ac
*insufficient near marginal state **can be strongly damped in subcritical region
linear excitation nonlinear propagation Generic structure of Fisher spreading equation: Suffering from two serious drawbacks:
Motivations
Inagaki2013
A hysteretic relation between turbulence intensity and temperature gradient also observed:
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The turbulence intensity is unistable in the Fisher model. However:
An indication of bistability of the turbulence intensity!
∂ ∂t I = γ 0 〈A〉 − Ac +
( )I −γ nlI 2 + D1
∂ ∂x I ∂ ∂x I ⎛ ⎝ ⎜ ⎞ ⎠ ⎟
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??
We missed something here??
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In this talk, we propose the missed piece is the nonlinear drive induced by the corrugation of the temperature field.
The key for nonlinear turbulence excitation: temperature corrugation by inhomogeneous turbulent mixing.
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Inevitable consequence of potential enstrophy conservation
A consistent treatment of multi-scale, multi-field couplings is required…
Toppaladodi et al 2017
T
Roughened temperature profile enhances turbulent heat flux. An example from Rayleigh-Bernard convection
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How mesoscale fields impact evolution of turbulence intensity?
Drive: ∇T
( )meso
Dissipation: V ′
ZF
local force balance
V ′
ZF ∝ ∇2T
( )meso
Generally, drive&dissipation act in different regions.
How the turbulence intensity is excited and spreads in the presence of a corrugated temperature profile?
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The Model: bistable turbulence Intensity The basic structure of 𝐽’s evolution is For a mean field approximation, Θ( !
Am) ! Am = Θ( ! Am) ! Am + Θ( ! Am) ! Am
! ! Θ( !
Am) ! Am
∂ ∂t I = γ 0 〈A〉 + 〈Θ(A
! m)A ! m〉 − Ac
( )I −γ nlI 2 + D1
∂ ∂x I ∂ ∂x I ⎛ ⎝ ⎜ ⎞ ⎠ ⎟
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nonlinear drive
∂ ∂t I = γ 0 〈A〉 − Ac + Θ(A
! m)A ! m
( )I −γ nlI 2 + D1
∂ ∂x I ∂ ∂x I ⎛ ⎝ ⎜ ⎞ ⎠ ⎟
Relation between
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Strength of Mesoscopic ∇𝑈 Fluctuations
∂ ∂t T + ∇⋅QT = χneo ∂2 ∂x2 T + Sδ(x)
T = T + ! T = T +T
! m + !
Ts
T
! m :meso scale; !
Ts :micro scale
, QT = ! vT,
Multiplying T on both sides of ∗
( ) and carrying out a double average
.. s
m yields
A0 QT
m +
! Am ! v ! T
s m " χneo
! Am
2 m
Define two types of average: .. s − micro timescale; .. m − meso timescale
(*)
Entropy balance of the turbulence:
entropy production due to turbulent mixing of the mean temperature field entropy dissipation due to neoclassical diffusion triple coupling between micro and meso scales
⇒ T
s m = T m ≡ T
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The essential process for subcritical turbulence excitation
A closure on the triple coupling: ‘negative’ thermal conductivity
! Am ! v ! T
s = !
Am ! v ! T
s
up gradient heat flux on mesoscale
! v ! T
s = χm !
Am = − χm ! Am
χm < 0 the negative diffusivity.
𝑈’s profile gets corrugated by the inhomogeneous turbulent mixing.
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Zeldovich relation in multi-cale coupling system: The underlying physics: roll-over of vs duo to ZF shear.
Qm ∂xTm
The closed loop
inhomogeneous mixing
‘negative’ heat flux
T’s corrugation local excitation of turbulence
enhancing
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Bistable spreading of the turbulence intensity: subcritical excitation
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∂ ∂tI = γ0 (hAi AC) I + s γ2
0D0hAi2
χneo + |χm|I3/2 γnlI2 ✓ ◆
+ D1 ∂ ∂x ✓ I ∂ ∂xI ◆
I’s evolution with subcritical drive(Fitzhugh-Nagumo type, not Fisher!)
On the subcritical excitation ∂ ∂tI = δF(I) δI ,
F(I) = 1 2γ0 (hAi AC) I22 5 s γ2
0D0hAi2
χneo + |χm|I5/2+1 3γnlI3.
I = 0 and I = I+,
Two stable solutions: One unstable solution:
I = I−
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F
metastable
I = I− I = I+
absolutely stable
I = 0
excitation threshold
I
How the bistable spreading happens?
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c⇤ d dz I = δF δI + D1I d2 dz2 I + D1 ✓ d dz I ◆2
Looking for wave-like solution: I(x,t) = I(z)with z = x − c*t
1 c⇤ = F(+1) F(1) R +1
1 I02dz
+ D1
2
R +1
1 I03dz
R +1
1 I02dz
,
Propagation speed of the bistable front:
I(z = −∞) = I+ at t1 &t2
I(z = +∞) = 0 at t1 & t2
c(t1) c(t2) = c(t1)
(B)
:C* > 0
How the bistable spreading happens?
inhomogeneous mixing of the temperature gradient turbulence pulse driven by temperature corrugation an initial turbulence pulse
T
x
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Summary&future work
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Mesoscale temperature profile corrugations provide a natural way for subcritical turbulence excitation, and the following spreading. Next: Temperature profile evolution needs to be treated in a more consistent way; Stability problem of the front, i.e., can the front be splitter by any external/internal noise?