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 Energy, Office of Science, Office of Fusion Energy Sciences, under Award Numbers DE-FG02-04ER54738 1

Outline • Motivations Mesoscale temperature profile corrugation and nonlinear drive • • Bistable spreading of the turbulence intensity: —subcritical excitation —propagation 2

Motivations How turbulent fluctuations penetrates stable domains? → Anomalous transport → Collapse of H-mode Most previous works treat turbulence spreading as a Fisher front. 3

Conventional wisdom: Fisher front with a nonlinear diffusivity A ≡ −∂ x T Generic structure of Fisher spreading equation: ∂ ∂ x I ∂ ∂ ⎛ ⎞ ) I − γ nl I 2 + D 1 ( ∂ t I = γ 0 〈 A 〉 − A c ⎜ ⎟ ∂ x I ⎝ ⎠ linear excitation nonlinear propagation Nontrivial solution requires: 〈 A 〉 > A c ⇒ I ∝ 〈 A 〉 − A c Suffering from two serious drawbacks: *insufficient near marginal state **can be strongly damped in subcritical region 4

Motivations The turbulence intensity is unistable in the Fisher model. However: A hysteretic relation between turbulence intensity and temperature gradient also observed: Inagaki2013 ⇒ An indication of bistability of the turbulence intensity! 5

We missed something here?? ∂ ∂ x I ∂ ∂ ⎛ ⎞ ) I − γ nl I 2 + D 1 ( ∂ t I = γ 0 〈 A 〉 − A c + ⎜ ⎟ ∂ x I ?? ⎝ ⎠ 6

In this talk, we propose the missed piece is the nonlinear drive induced by the corrugation of the temperature field . 7

The key for nonlinear turbulence excitation: temperature corrugation by inhomogeneous turbulent mixing . ⇑ Inevitable consequence of potential enstrophy conservation A consistent treatment of multi-scale , multi-field couplings is required… 8

An example from Rayleigh-Bernard convection T Toppaladodi et al 2017 Roughened temperature profile enhances turbulent heat flux. 9

How mesoscale fields impact evolution of turbulence intensity? ( ) meso Drive: ∇ T ( ) meso V ′ Dissipation: V ′ ZF ∝ ∇ 2 T local force balance ZF ⇒ Generally, drive&dissipation act in different regions. How the turbulence intensity is excited and spreads i n the presence of a corrugated temperature profile? 10

The Model: bistable turbulence Intensity The basic structure of 𝐽 ’s evolution is ∂ ∂ x I ∂ ∂ ( ) I − γ nl I 2 + D 1 ⎛ ⎞ ! m ) A ! m ∂ t I = γ 0 〈 A 〉 − A c + Θ ( A ⎜ ⎟ ∂ x I ⎝ ⎠ nonlinear drive ! ! Θ ( ! For a mean field approximation, Θ ( ! A m ) ! A m = Θ ( ! A m ) ! A m + Θ ( ! A m ) ! A m ) ! A m A m ⇒ ∂ ∂ x I ∂ ∂ ( ) I − γ nl I 2 + D 1 ⎛ ⎞ ! m ) A ! m 〉 − A c ∂ t I = γ 0 〈 A 〉 + 〈Θ ( A ⎜ ⎟ ∂ x I ⎝ ⎠ 11

Relation between ! m ) A ! m 〉 and I ? 〈Θ ( A 12

Strength of Mesoscopic ∇ 𝑈 Fluctuations ∂ ∂ 2 ∂ t T + ∇⋅ Q T = χ neo ∂ x 2 T + S δ ( x ) (*) ! m + ! ! m :meso scale; ! T = T + ! , Q T = ! T = T + T vT , T T s :micro scale T s Define two types of average: .. s − micro timescale; .. m − meso timescale ⇒ m = T m ≡ T T s ( ) and carrying out a double average Multiplying T on both sides of ∗ .. s m yields Entropy balance of the turbulence: ! v ! ! m + m " χ neo A m ! 2 A 0 Q T T A m s m entropy dissipation due to neoclassical diffusion entropy production due The essential process triple coupling between to turbulent mixing of the for subcritical micro and meso scales mean temperature field turbulence excitation 13

A closure on the triple coupling: ‘negative’ thermal conductivity ! v ! s = ! v ! A m ! A m ! T T s v ! s = χ m ! A m = − χ m ! ! T A m up gradient heat flux on mesoscale χ m < 0 the negative diffusivity. ∂ x T m Q m The underlying physics: roll-over of vs duo to ZF shear. ⇒ Zeldovich relation in multi-cale coupling system: 𝑈 ’s profile gets corrugated by the inhomogeneous turbulent mixing. 14

The closed loop ‘negative’ heat flux on mesoscale inhomogeneous mixing T ’s corrugation enhancing local excitation of turbulence 15

Bistable spreading of the turbulence intensity: subcritical excitation I ’s evolution with subcritical drive( Fitzhugh-Nagumo type, not Fisher! ) s γ 2 0 D 0 h A i 2 ∂ ✓ ◆ ∂ I ∂ χ neo + | χ m | I 3 / 2 � γ nl I 2 ∂ tI = γ 0 ( h A i � A C ) I + + D 1 ∂ xI ∂ x ✓ ◆ On the subcritical excitation ∂ tI = � δ F ( I ) ∂ , δ I s γ 2 0 D 0 h A i 2 F ( I ) = � 1 2 γ 0 ( h A i � A C ) I 2 � 2 χ neo + | χ m | I 5 / 2 +1 3 γ nl I 3 . 5 I = 0 and I = I + , Two stable solutions: I = I − One unstable solution: 16

F excitation threshold metastable I = I + I = I − I = 0 I absolutely stable 17

How the bistable spreading happens? Looking for wave-like solution: I ( x , t ) = I ( z )with z = x − c * t ✓ d ◆ 2 δ I + D 1 I d 2 � c ⇤ d dz I = � δ F dz 2 I + D 1 dz I 1 Propagation speed of the bistable front: R + 1 �1 I 0 3 dz � D 1 c ⇤ = F (+ 1 ) � F ( �1 ) 2 + , R + 1 R + 1 �1 I 0 2 dz �1 I 0 2 dz I ( z = −∞ ) = I + at t 1 & t 2 : C * > 0 c ( t 2 ) = c ( t 1 ) (B) c ( t 1 ) I ( z = + ∞ ) = 0 at t 1 & t 2 18

How the bistable spreading happens? inhomogeneous mixing of the temperature gradient T turbulence pulse driven an initial turbulence pulse by temperature corrugation x 19

Summary&future work 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? 20