Bistable Dynamics of Turbulence Intensity in a Corrugated Temperature Profile

Transcription

1 Bistable Dynamics of Turbulence Intensity in a Corrugated Temperature Profile Zhibin Guo University of California, San Diego Collaborator: P.H. Diamond, UCSD Chengdu, 07 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-FG0-04ER54738

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

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

4 Conventional wisdom: Fisher front with a nonlinear diffusivity Generic structure of Fisher spreading equation: A x T t I = γ 0 Nontrivial solution requires: ( A A )I γ I + D c nl x linear excitation A > A c I A A c I x I nonlinear propagation Suffering from two serious drawbacks: *insufficient near marginal state **can be strongly damped in subcritical region 4

5 Motivations The turbulence intensity is unistable in the Fisher model. However: A hysteretic relation between turbulence intensity and temperature gradient also observed: Inagaki03 An indication of bistability of the turbulence intensity! 5

6 We missed something here?? t I = γ ( 0 A A c + )I γ I?? + D nl x I x I 6

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

8 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

9 An example from Rayleigh-Bernard convection PRL 8, (07) PHYSICAL REVIEW LETTERS week end 7 FEBRUAR T et al 07 FIG. 4. A snapshot of the temperature field for λ ¼ 0. and Ra ¼ 09. ToToppaladodi see the effects of roughness, the flow field here contrasted with that in the smooth case studied by Johnston & Doering [8]. See also Fig. 3 of [40], which shows the transition fr planar to the rough flow field in the case of one rough wall. achieved, and in the limits λ λopt and λ λopt, the maximum value of β was found to be β 0.36 [40], but at a Roughened temperature profile enhances turbulent heat flux. slightly larger λ. This highlights the role played by the value of β is recovered, which may underlie why c 9 second rough wall in further decreasing the role of the BLs experiments found no effect of periodic roughness

10 How mesoscale fields impact evolution of turbulence intensity? Drive: ( T ) meso Dissipation: V ZF local force balance V ZF ( T ) meso Generally, drive&dissipation act in different regions. How the turbulence intensity is excited and spreads in the presence of a corrugated temperature profile? 0

11 The Model: bistable turbulence Intensity The basic structure of I s evolution is t I = γ 0 ( A A c + Θ(A! m )A! m )I γ I + D nl x nonlinear drive I x I or a mean field approximation, Θ(! A m )! A m = Θ(! A m )! A m + Θ(! A m ) A m! Θ( Am! ) A! m t I = γ 0 ( A + Θ(A! m )A! m A )I γ I + D c nl x I x I

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

13 Strength of Mesoscopic T Fluctuations t T + Q T = χ neo T + Sδ (x) x (*) T = T +!T = T + T! m +!T s, Q T =!vt, T! m :meso scale;! T s :micro scale Define two types of average:.. s micro timescale;.. m meso timescale T s m = T m T Multiplying T on both sides of ( ) and carrying out a double average.. yields s m Entropy balance of the turbulence: A 0 Q T m +! Am!v! T s m " χ neo! Am m entropy dissipation due to neoclassical diffusion entropy production due to turbulent mixing of the mean temperature field triple coupling between micro and meso scales 3 The essential process for subcritical turbulence excitation

14 A closure on the triple coupling: negative thermal conductivity!a m!v!t s =!A m!v!t s up gradient heat flux on mesoscale!v!t s = χ m!a m = χ m!a m χ m < 0 the negative diffusivity. he underlying physics: roll-over of Q m vs x T m duo to ZF shear. Zeldovich relation in multi-cale coupling system: T s profile gets corrugated by the inhomogeneous turbulent mixing. 4

15 The closed loop inhomogeneous mixing negative heat flux on mesoscale T s corrugation enhancing local excitation of turbulence 5

16 Bistable spreading of the turbulence intensity: subcritical excitation q I s evolution with subcritical drive(fitzhugh-nagumo type, not Fisher!) h I = 0 0 (hai A C ) I + D 0hAi neo + m I3/ + @x I On the subcritical excitation F (I) = 0 (hai A C ) I I = 0 D 0hAi s F (I) I s neo + m I5/ + 3 nl I 3., Two stable solutions: One unstable solution: I = 0 and I = I +, I = I 6

17 F excitation threshold metastable I = 0 I = I I = I + I absolutely stable 7

18 derstand how these domains are connected, one needs fies the local gradient and stability as it passes. This future. Th front(i.e., the inner solution), one has I(x, t) = I(z) with incorporate theturbulence, spatial coupling term inhappens? Eqn.(6). For a the bistable spreading induces bistability of How the in contrast to the can b z = x c t(c speed ofshow the I(x, frontt)propagation). Fisherfront(i.e., model, which is unistable. In this work,one we The linear the inner solution), has = I(z) with Eqn.(6) sprea that temperature profile becomes corrugation can provide such a * can be incorp z = x c t(c speed of the front propagation). Eqn.(6) Lookingdrive, for and wave-like I(x,t)of=turI(z)with z = x c t nonlinear so sustainsolution: a finite amplitude ture spreading mo becomes bulence intensity in the subcritical region. This enables d F d d gion c I =l < 0 with + Dthat D I. () I I + turbulence spreading, and reconciles ture gradient dz I dz dz ation phenomenon. d F d d c I = + D I I + D I. () d gion is enlarg I+ is dz c is alsoi the eigenvalue dz dz of Eqn.(). Multiplying dz Iations on then b trans both sides of Enq.() and integrating from z = to I is decreas + d and e c is also the eigenvalue of Eqn.(). Multiplying I on I (z = ) = γ / γ atspeed t &t zof=the + yieldsfront: dz Propagation bistable transport bab can both sides of Enq.() and integrating from zr = to and excited s D absol I dz (A) z = + yieldsc(t ) F (+) F ( ) c(t ) < c(t ) can c = + R +, (3) become to a R + 0 dz 0 dz R I I + 3 D 0 at t 0 absolutely s bista I (z = + ) I dz F (+) F ( ) I(z = + ) = 0 at t c = +, (3) to a laminar also p R R I (z = ) = I at t & t where II 0 di/dz. The propagation of the front is 0 dz dz I trans bistable mod driven by two e ects: () the free energy di erence of resid also provides 0 two homogeneous states, and () the c(t ) = c(t ) where I thedi/dz. The propagation of the front is steepc(t ) (B) * spatial turbu transition pr ening of the turbulence intensity due to nonlinear difdriven by two e ects: () the free energy di erence of expec residual term fusion. For the type of front sketched in Fig., one = + ) = 0() at t & t the spatial steepthe two homogeneous states, I (zand lineac turbulence F (+) intensity F ( ) =due F (0)to nonlinear F (I+ ) > 0(Fig.3) and ening of thehas turbulence difthe a 0 expect that I < 0, so that both e ects make a positive contribufusion. Forfront thein Fisher type of front sketched in Fig., one speed equation FIG. : (A) Turbulencetion based model linear satura to c. Since Eqn.() is invariant under the trans8 hasfront F (+) F (local) = (0)turbulence and and the is modified by growth; (B) 0 F 0 F (I+ ) > 0(Fig.3) l nl + :C > 0

19 How the bistable spreading happens? T inhomogeneous mixing of the temperature gradient an initial turbulence pulse turbulence pulse driven by temperature corrugation x 9

20 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? 0