THEORY AND SIMULATION OF ROTATIONAL SHEAR STABILIZATION OF TURBULENCE

Transcription

1 THEORY AND SIMULATION OF ROTATIONAL SHEAR STABILIZATION OF TURBULENCE R.E. WALTZ, R.L.DEWAR*, and X. GARBET** *The Australian National University, **Association Euratom-CEA Cadarache GENERAL ATOMICS San Diego, California Division of Plasma Physics APS Meeting, Pittsburgh, PA, November 17-21, 1997

2 WHAT THIS TALK IS ABOUT This talk is about the quench rule for rotational shear stabilization of turbulence in tokamaks: when the ExB shear rate γ E =(r/q) (qv ExB /r)/ r exceeds the maximum local ballooning mode growth rate γ max turbulent diffusion is quenched. This rule is the basis for transport models χ=(1 -γ E/ γ max )χ describing core transport barriers (Waltz, Staebler, Dorland, Hammett, Kotschenreuther & Konings 1997) This talk is not about the experimental fidelity of the rule. For this see Burrell (APS review talk 1996 ); Synakowski et al & Greenfield et al (APS invited papers 1996 ) Lao et al (APS invited paper 1995) This talk is about the numerical and theoretical evidence and limitations for the rule.

3 BACKGROUND TO ANALYTIC THEORIES ExB shear rate : γ E =(r/q) (qv ExB /r)/ r complex ballooning mode growth rate: γ nonlinear 2 point renormalization linear eigenmode stability ρ Connor,Taylor,&Wilson 1993 V ExB >> V EXB shear only x-variation only Im (dγ /dx) = ky γ E Biglari, Diamond, &Terry 199 Shaing, Crume,& Houlberg 199 Zhang & Mahajan 1992 turbulence suppression γ E _norm ω( kx/ ky) ω D kx 2 no distinction slab / torus toroidal process at smallest γ E reduces eigenmode rate to small ballooning mode angle average rate transport follows eigenmode rate resulting in slab-like levels ρ finite V ExB =O ( V ) x-variation (dγ /dx) = γ ' profile shear x2-variation (d2γ dx2) =γ '' profile curvature Rominelli & Zonca 1993 Dewar 1996 profile curvature Re(γ '') works against profile shear stabilizationim(γ ')

4 PREVIOUS NONLINEAR SIMULATIONS ExB shear rate : γ E =(r/q) (qv ExB /r)/ r complex ballooning mode growth rate: γ gyrofluid ITG ballooning mode or "flux tube" code ρ V ExB >> V EXB shear only x-variation only Im (dγ /dx) = ky γ E Waltz, Kerbel, & Milovich 1994 turbulence quenched γ E _crit γ max = max {Re (γ )} χ /[cs ρ s 2 /a γ P =18γ E γ 3 3 P =12γ 3 E.5 3 γ P = γ E /[c s /a] γ E γ max χ=(1 -γ E/ γ max )χ basis of models for core transport barriers Get quench for fixed γ P but for pure toroidal rotation, parallel shear rate γ P =(Rq/r)γ E can increase γ max (γ P ) faster than γ E avoiding quench. Superficial resemblance to Biglari, Diamond,&Terry rule since ω ( kx/ ky) tracks γ max for isotropic turbulence Appears not to follow eigenmode stability byconnor, Taylor,& Wilson rule since eigenmodes are stable at very small γ E here.

5 OUTLINE TO NEW WORK Review of the ExB shear in ballooning mode representations Coupling in ballooning angle θ versus convection in ballooning angle θ How a new mode centered Floquet ballooning representation avoids numerical "box mode" instabilities. Gyrofluid ITG ballooning mode or ρ "flux tube" numerical illustrations showing how the γ E _crit γ max quench rule results from a nonlinear convective amplification process unique to toroidal geometry. Likely modifications and limitations on the γ E _crit γ max quench rule for finite ρ and general profile shear and profile curvature: lessons from the ballooning- Schrodinger eigenmode equation.

6 BALLOONING MODE AND REPRESENTATIONS df/dt = F/ t + v~ ExB F ExB nonlinear coupling x'-space radial direction V ExB = γ E x' γ E =(r/q) (qv ExB /r)/ r df/dt = -(γ E x') i ky F + L F L linear operations transform kx'-space or ballooning mode -space F(θ,θ) x'f = -i F/ kx' kx' kys^θ ky nq / r s^ ( r/q ) ( d q/dr ) df(θ,θ)/dt = -(γ E /s^) F(θ,θ)/ θ + L[θ -θ, cos(θ), / θ] F(θ,θ) _each F(θ,θ) centered at θ θ _For finite γ E, θ is no longer a "good quantum number" _ExB shear γ E linearly "couples"θ 's _θ discrete: F/ θ F/ θ

7 BALLOONING MODE AND REPRESENTATIONS continued Cooper transformation θ θ+(γ E /s^)t "Floquet ballooning mode" F'(θ,θ) df'(θ,θ)/dt = L [θ - (θ+(γ E /s^)t), cos(θ), / θ ] F'(θ,θ) _ExB shear γ E linearly convects each θ _Easy to see Connor,Taylor, & Wilson rule for eigenmode or time average growth rate at small γ E /s^ when modes not distorted or broken up: γ = γ ( θ ) d θ / π 2 << γ ()

8 BALLOONING MODE AND REPRESENTATIONS continued Rotating frame transformation θ θ+(γ E /s^)t "centered Floquet ballooning mode" F''(θ,θ) df''(θ,θ)/dt=(γ E /s^) F"(θ,θ)/ θ+l[θ-θ,cos(θ+(γ E /s^)t), / θ] F''(θ,θ) _Explicitly displays "poloidal breakup term " which stabilizes modes in the slab when cos( ) dependence is dropped or stabilizes toroidal modes at large γ E /s^ even when γ = γ ( θ ) d θ / π 2 >. _Leaves nonlinear terms invariant, modes stay centered at θ θ and do not rotate out of the finite θ numerical box

9 Why the centered Floquet ballooning mode representation is best for numerical simulations _All represenations are equivalent if θ is a continuous variable! _But to do ( ky, kx') nonlinear coupling we must treat [kx'=kys^θ] θ as a discreet variable. Discreteness in θ means we have a "cyclic box" in x-space Two "visions" of "homogeneous" ExB shear. "box modes" live in corners of shear θ continuum _Miller and Waltz (1994) showed that discrete θ could lead to "box modes" which require a finite γ E to stabilize, even when "true" continuum eigenmodes were stable v E γ = γ ( θ ) d θ / π 2 < cyclic boxes x θ grid _The quench rule γ E_crit γ max in the discrete θ nonlinear ballooning mode code was later found to be associated with linear stabilization of "box modes".

10 Numerical illustrations with new centered Floquet ballooning mode representation which avoids numerical "box modes" but still recovers approximate quench rule γ E_crit γ max ITG with adiabatic electrons R / a = 3, a/ln=1, a/lt=3, q=2, s^=1, α=, γ P = At γ E =.3, least stable mode kyρs=.3 time average growth rate or eigenmode rate same as ballooning mode angle average rate γ = γ ( θ ) d θ / π 2 = -.33 which is negative. Convective amplification of 316x

11 At small γ E turbulence persists from convective amplification and nonlinear coupling even with stable eigenmodes. ITG with adiabatic electrons At γ E =.3 diffusion is only slightly suppressed. After multiplying all amplitudes by.1x, the turbulence is able recover but it does not recover after.1x. Implies that Floquet modes passing bad curvature region are convectively amplified and nonlinearly couple energy to convectively decaying modes passing good curvature region preventing eigenmode decay.

12 At higher shear rates turbulence quenched at near γ E _crit γ max R / a = 3, a/ln=1, s^=1, α=, γ P = χ i / [c s ρ s2 /a] =4 =3 γ max s^=1. q=2 γ max χ i / [c s ρ s2 /a] q=3 q=2 q=4 γ max γ max s^=1. =3 γ max γ E γ E

13 kyρs-spectrum as a function of γe R / a = 3, a/ln=1, a/lt=3, q=2, s^=1, α=, γ P = Depletion at low-ky suggests local rule γ E _crit γ (ky) may be justified. (n~/n ) rms = ( Σ kx [(n~/n ) rms ]2 )1/ γ E = ρ s k y s^=1. =3 q=3

14 Quench rule γ E _crit γ max persists at low s^ = (r/q ) (d q/ dr) Even though the Floquet θ rotation rate is γ E/ s^, there is no evidence that the critical γ E s^ γ max. Hence no evidence here that s^= is the "seat" of core reversed shear transport barrier. χ i /χ i () C G K O s^=.25 s^=.5 s^=1. s^=1.5 =3 χ i /χ i () C G K O S s^=.25 s^=.5 s^=1. s^=1.5 s^=2. = γ E / γ max γ E / γ max (γ E /s^)/γ max scaling does not give a good invariance with s^.

15 At low s^ convection speed γe/s^ goes up, but most unstable mode at θ = must convect 1/s^ times farther in θ to reach stability. θ π/2 θ 4π Heuristic argument: Balancing growth time with convection time implies γ E _crit s^ θ γ max Ο(1) γ max

16 γ E_crit does not follow eigenmode stability in toroidal geometry R / a = 3, a/ln=1, q=2, α=, γ P = γ E_ <γ> is the ExB shear rate for eigenmode stability <γ> =. Due to the poloidal breakup effect, actual rate <γ> <γ> Connor,Taylor,& Wilson rate.2.4 γ E_crit.15 γ max =3.3 γ max.1 γ E_crit.2.5 γ E_<γ>=.1 γ E_<γ>= = s^ s^

17 γ E_crit does follow eigenmode stability in slab geometry γ E _crit >> γ max R / a = 3, a/ln=1, a/lt=3, q=2, s^=1, α=, γ P = toroidal curvature terms "turned off".8.7 slab no curvature γ max =3 s^=1. poloidal breakup rate χ i / [c s ρ s2 /a] =3 s^=1. γ max = γ E γ E

18 Summary of ITG simulation cases for quench rule R / a = 3, a/ln=1, α=, γ P = γ max B D F a/l =4 T s^ =2.->.25 =3 s^ =1.5->.25 a/l =3 T q =2->4 In summary, for the vanishing ρ limit, it appears that γ E_crit 4/3 γ max is the best approximate description of the critical ExB shear rate for quenching transport in toroidal geometry..1 slab γ E_crit

19 Quantitative test of 2-point nonlinear renormalization theories _Theory: nonvanishing suppression factor χ/χ = 1/[1 + (γ E /γ E_norm )2 ] γ E_norm ω t ( k x / k y ) ω t = D k x 2 used interchangeably ω t ( k x / k y ) B E H =4 s^ =2.->.25 =3 s^ =1.->.5 =3 q =2->4 slab 2 ω t =D k x γ max χ i /χ i () =4 s^=1.5 1 / [1+(γ E /γ norm ) 2 ] γ max 2-pt. renorm. theory toroidal ITG simulation γ E ω t ( k x / k y ) does tracks γ max using D = χ But supression formula gives less than 1% reduction at the actual quench point. _Most importantly, this (ρ ->) theory does not account for the difference between simulations in slab and toroidal geometry.

20 Likely modifications and limitations on the γ E _crit γ max quench rule : lessons from the ballooning-schrodinger eigenmode equation. complex ballooning mode growth rate: γ (x,θ) expand in x=r-r ρ finite V ExB = O ( V ) x-variation (dγ /dx) = γ ' profile shear Im(dγ /dx)=ky γ mode x2-variation (d2γ dx2) =γ '' profile curvature γ = γ + γ' x + γ''/2 x2 + [Re(γ ) -<γ> ][cos(θ)-1] x i ss / θ θ i ss / x generalize γ E γ mode =(r/q) (qv mode /r)/ r to include shear in intrinsic mode phase velocity not just ExB Doppler shift γ mode _crit γ max

21 Likely modifications and limitations on the γ E _crit γ max quench rule : lessons from the ballooning-schrodinger eigenmode equation...continued eigenmode growth rate o ballooning mode at γ mode = x with 1/n radial stabilization "bound" harmonic oscillator solution <γ> "passing" CTW solution jump < γ > with poloidal breakup Im(γ ) / k y = γ mode γ bound =γ -1/2 (-γ '' ss 2γ )1/2-1/2(γ '2/ γ '' ) γ pass = γ = γ ( θ ) dθ / π 2 Re (γ) = max {Re ( γ bound ), Re ( γ pass ) } 1/n "radial correction" is small Ballooning-Schrodinger equation does not contain poloidal breakup Jump point likely criterion for minimum profile shear γ ' or γ mode to ignore profile curvature γ '' which prevents mode circulation and convective amplifcation: Re [1/2 (γ'2/ γ'' )] > Re [ γ -<γ>] otherwise quench rule may not hold

22 CONCLUSIONS From ITG adiabatic electron simulations the quench rule for rotational shear stabilization γ E =(r/q) (qvexb/r)/ r γ max results from a nonlinear convective amplification process unique to toroidal geometry. The rule holds over a wide parameter range including low s^= (r/q ) (d q/dr) at vanishing ρ ExB shear stabilization does not follow eigenmode stability in toroidal geometry but does follow eigenmode stability in slab geometry. From the ballooning-schrodinger eigenmode equation at finite ρ we speculate: The rule should be modified to include the total mode velocity not just the EXB Dopper shift. The quench rule is likely to hold only if the profile shear γ '= dγ /dx is large enough to overcome profile curvature γ '' = d2γ /dx2 which prevents full circulation in ballooning mode angle and convective amplifcation. Studies with 3D full radius nonlinear simulations are needed at finite but small ρ