Heat Transport in a Stochastic Magnetic Field. John Sarff Physics Dept, UW-Madison

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1 Heat Transport in a Stochastic Magnetic Field John Sarff Physics Dept, UW-Madison CMPD & CMSO Winter School UCLA Jan 5-10, 2009

2 Magnetic perturbations can destroy the nested-surface topology desired for magnetic confinement. Stochastic instability occurs when magnetic islands overlap, causing the field lines to wander randomly throughout the plasma volume. Parallel streaming along the stochastic field leads to radial transport. Astrophysical plasmas have weak ordered field (naturally tangled ) nested magnetic surfaces (ideal) magnetic island formation if islands overlap, stochastic field (B perturbations from instability or error components)

3 Projection of radial field yields intuitive estimate of stochastic transport. Recall parallel heat transport T t = χ ( ˆ b ) 2 T where ˆ b = B/ B If B = B 0 + B r r ˆ where B 0 = well-ordered field, forming nested magnetic surfaces T t = χ ( b ˆ ) 2 B T = χ r 2 T B 0 r 2 2 effective perpendicular transport (not quite rigorous, ok for fluid limit)

4 Small fluctuation amplitudes can yield large transport. Recall for classical electron transport χ 2 ν c ~ λ mfp χ ρ ν c Small magnetic fluctuation amplitude yields substantial transport χ B r B 0 2 ~ χ for B r B 0 ~ 10 3

5 Outline. Model for stochastic transport Comparisons with experimental measurements (mostly from the RFP)

6 Fluctuation-induced transport fluxes. Linearizing the drift kinetic equation f t + v f = 0 f = f 0 + f f 0 t = v E f = B 0 2 B 0 B + v B 0 f drift associated with electrostatic fluctuations streaming associated with magnetic fluctuations

7 Fluctuation-induced transport fluxes. Moments of the d.k.e. lead to the fluctuation-induced transport fluxes particle Γ r = dv f E B 0 2 B 0 B + v B 0 r ˆ = n E / B 0 + J B r /eb 0 electrostatic magnetic energy Q r = dv v 2 f E B 0 2 B 0 B + v B 0 r ˆ = p E / B 0 + Q B r / B 0 where... denotes an appropriate average, e.g., over an unperturbed magnetic flux surface

8 Model for stochastic magnetic transport. Very few self-consistent models for magnetic fluctuation induced transport have been developed. Most analysis has been for a static, imposed set of magnetic fluctuations Error fields from misaligned magnets and other stray fields Low frequency turbulence Stochastic magnetic transport is described by a double diffusion process 1. Random walk of the magnetic field lines 2. Collisional or other cross-field transport process is required for particles to lose memory of which field line they follow

9 Magnetic diffusion. Divergence of neighboring field lines: flux tube r 0 distance, s, along unperturbed field B 0 r(s) = r 0 e s/lk δ δ(s) = r 0 e s/l K L K = Kolmogorov-Lyaponov length

10 Magnetic diffusion. Magnetic diffusion coefficient: D m = (Δr)2 Δs = 0 B r (0) B r (s)ds B 0 2 (units of length) = L ac B r 2 / B 0 2 L ac = auto-correlation length for B L ac is related to the width of the k spectrum, L ac π /Δk ( L K ) in general

11 Stochastic transport in the collisionless limit. Consider a test particle streaming along the magnetic field flux tube distance, s, along unperturbed field B 0 (Δr) 2 = D m Δs average radial displacement associated with field line diffusion For λ mfp >> L ac χ st = (Δr)2 Δt = D mλ mfp τ c = D m v T v T = T / m (thermal velocity) τ c = λ mfp /v T (collision time)

12 Stochastic transport in the collisional limit. For λ mfp << L ac Δs ~ L ac, test particle must first diffuse along the field The parallel diffusion is given by: χ = (Δs)2 Δt = λ 2 mfp τ c χ st = (Δr)2 Δt = D m Δs Δt = D m L ac L 2 ac / χ = D m v λ mfp T L ac B 2 = χ r B 0 2

13 Stochastic transport in the collisional limit. For λ mfp << L ac Δs ~ L ac, test particle must first diffuse along the field The parallel diffusion is given by: χ = (Δs)2 Δt = λ 2 mfp τ c χ st = (Δr)2 Δt = D m Δs Δt = D m L ac L 2 ac / χ = D m v λ mfp T L ac B 2 = χ r B 0 2 Smooth transitional form: χ st = v T L eff B r 2 B 0 2 with L 1 eff = L 1 1 ac + λ mfp Krommes et al. provided a unifying discussion of various collisional limits with respect to characteristic scale lengths.

14 How well does the static field model work? Few direct measurements of stochastic transport. Inferences via energetic particles in tokamak plasmas, exploiting expected velocity dependence. Self-organizing plasmas like the RFP and spheromak provide good opportunity to test expectations, because they exhibit a broad spectrum of low frequency magnetic fluctuations.

15 The Reversed Field Pinch plasma configuration. RFP Magnetic Geometry MST parameters: n ~ cm 3 T e < 2 kev T ion ~ T e B < 0.5 T ρ ion ~ 1 cm

16 The MST at UW-Madison. R = 1.5 m a = 0.5 m Ip < 0.6 MA

17 Main source of symmetry breaking magnetic field in the RFP is MHD tearing instability. Linear stability analysis using force balance J B = p B k = m θ ˆ n r R ˆ r F + φ F + k 2 B r 0 F = k B 0 F F ~ B 0 J ~ r B 0 B yields Growth rate depends on r J and the plasmaʼs resistivity Mode resonance appears at the minor radius where k B 0 0 k B 0 = 0 m n = rb φ RB θ = q(r) B r B 0 (see 2008 Winter School lectures) rdl θ Rdl φ

18 Tearing permits the creation of magnetic islands. resonant layer k B 0 0 r = r s r = r s Tearing reconnection magnetic island forms to B 0 island width w m,n = 4 B r,m,n (r s )L s B 0 (r s )k

19 Chirikov threshold condition for stochastic instability. If neighboring magnetic islands overlap, the field lines are allowed to wander from island-to-island randomly. s = 1 2 w n+1 + w n r s,n+1 r s,n stochasticity parameter (crudely the number of islands overlapping a given radial location) s < 1 : islands do not overlap, no stochastic transport (but transport across the island is typically enhanced by its topology) s ~ 1 : weakly stochastic, magnetic diffusion and transport are transitional (e.g., as discussed by Boozer and White) s >> 1 : magnetic field line wandering is well approximated as a random-walk diffusion process

20 Many possible tearing resonances occur across the radius of the RFP configuration. q(r) = rb φ RB θ B n B 0 1% Observed Spectrum Toroidal Mode, n

21 Chirikov threshold is exceeded, particularly in the mid-radius region where the density of rational magnetic surfaces is large q(r) s

22 Magnetic puncture plot indicates widespread magnetic stochasticity. B r,m,n (r) Eigenfunctions from nonlinear resistive MHD computation, normalized to measured B m,n (r = a). Field is modeled using B m,n (r) eigenfunctions, combined with equilibrium reconstruction that provides B 0 (r).

23 Direct measurement of magnetic fluctuation-induced stochastic transport. Measurements were made in MST (RFP), CCT (tokamak), and TJ-II (stellarator)

24 Measured electron heat flux in the edge of MST plasmas.

25 Measured island-induced heat flux in CCT (tokamak at UCLA). Heat flux in the magnetic island scales as if stochastic

26 The amplitude of the tearing fluctuations in the RFP can be reduced using current profile control (PPCD). ~5X reduction of most modes B n B 0 allows tests of χ st ~ B 2 r scaling and dependence on spectral features

27 Region of stochastic field shrinks with current profile control. Standard PPCD Toroidal, φ Toroidal, φ r / a r / a

28 Power balance measurements provide the experimental electron heat conductivity profile. Electron heat flux Q e = χ e n r T e Te 0.6 (KeV) PPCD-Improved PPCD Standard r/a χe (m2/s) Standard PPCD-Improved PPCD PPCD r/a

29 Measured heat diffusivity consistent with collisionless stochastic transport model (where the field is stochastic). Standard PPCD 1000 Magnetic diffusivity is evaluated directly from an ensemble of magnetic field lines. 100 L ac << λ mfp χ e 10 χ st ~1 m ~10ʼs m χ st χ st = D m v T r/a r/a 1

30 Magnetic diffusivity as expressed by Rechester-Rosenbluth, PRL ʼ78. D m = πr m,n B r,m,n (r) 2 B z 2 δ[ m /n q(r) ] auto-correlation length, L ac RMS fluctuation amplitude^2 but only k = 0 modes resonant nearby r

31 Estimate of the auto-correlation length from the spectral width. For a tokamak B φ >> B θ k = k B B 1 m B φ r B θ + n R B φ = 1 m R q + n Δk ~ Δr k = n r rs R Δr 1 dq q dr rs ~ 1 mode radial width ~ 1 R (n=1 typically dominant)

32 Rechester-Rosenbluth magnetic diffusivity overestimates χ st for regions with low s.

33 Electron temperature gradient correlates with amplitude of tearing modes resonant at mid-radius T e (0) 0.8 (kev) 0.6 PPCD 0.4 Standard B rms = 15 n=8 B 2 n (a) mid-radius modes m =1, n 8

34 Electron temperature gradient does not correlate with largest mode, resonant in the core T e (0) 0.8 (kev) 0.6 PPCD T e (0) 0.8 (kev) Standard B rms = 15 n=8 B 2 n (a) m = 1, n = Dominant Mode B 1,6

35 Though parallel streaming transport is nonlocal, the tearing reconnection process is local B r B 2 1,6 linear eigenmodes D m ~ m,n B r,m,n (r) 2 B z 2 δ[ m /n q(r) ] RMS m =1, n =8-15 illustrates importance of k = r/a 1,6

36 Stochastic particle transport is affected by its inherent non-ambipolar character. Since the thermal velocity is mass-dependent, electron and ion stochastic diffusion are not automatically ambipolar (unlike E B 0 motion). Harvey derived from the drift kinetic equation (collisionless limit) 1 Γ r ~ D m v T n 1 Q r ~ D m v T n n r + 1 2T n r + 3 2T T r + ee A T T r + ee A T n nt Setting and Γ r,e 0 Q r,e ~ D m v T n T r yields the ambipolar electric field E A = T e e r ln(nt 1/2 e )

37 Non-ambipolar transport predicts a radially outward directed electrostatic field due to the high mobility of electrons. Heavy ion beam probe observes the positive potential in the core. Lei et al. MST plasma

38 In astrophysical plasmas, stochastic field can reduce heat transport. Reflects large transport anisotropy in a magnetized plasma. Consider collisionless limit L ac << λ mfp : D χ st = D m v T = χ m L = χ ac B r λ mfp λ mfp B 0 2 < 1, even for B ~ B 0 Has been applied to cooling flows in galactic clusters to argue small heat conduction.

39 References 1. Rosenbluth, Sagdeev, Taylor, Nucl. Fusion 6, 297 (1966) 2. Jokipii and Parker, Ap. J. 155, 777 (1969) 3. Rechester and Rosenbluth, Phys. Rev. Lett. 40, 38 (1978) 4. Harvey, McCoy, Hsu, Mirin, Phys. Rev. Lett. 47, 102 (1981) 5. Boozer and White, Phys. Rev. Lett. 49, 786 (1982) 6. Krommes, Oberman, Kleva, J. Plasma Physics 30, 11 (1983) 7. Liewer, Nucl. Fusion 25, 543 (1985) 8. Prager, Plasma Phys. Control. Fusion 32, 903 (1990) 9. Stoneking et al., Phys. Rev. Lett. 73, 549 (1994) 10. Fiksel et al., Plasma Phys. Control. Fusion 38, A213 ( 1996) 11. Chandran and Cowley, Phys. Rev. Lett 80, 3077 (1998) 12. Biewer et al., Phys. Rev. Lett 91, (2003) 13. Fiksel et al, Phys. Rev. Lett 95, (2005)

40 Homework problem a) Consider the RFP magnetic equilibrium. Using the Chirikov stochasticity parameter, derive from the threshold condition, s=1, the recursion relation below for the width of the magnetic island associated with toroidal mode, n, so that it just touches its nearest neighbors (assume m=1 for all modes): w n = 1 1 q n n(n +1) where q n = dq and r n is the minor radius of the dr r=rn resonant surface b) Estimate the stochastic heat diffusivity, χ st, for a fluctuation spectrum described by the recursion relation above. c) For fusion parameters, discuss the magnitude of χ st relative to other transport mechanisms, such as classical (or neoclassical) transport and anomalous transport as observed in tokamak plasmas. For what n is χ e,st < 1 m 2 /s?

41 Homework problem (illustration and partial answer) n=6 w n = 4 B r,n (r n )r n B θ (r n )n q n (a=plasma minor radius)

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