Solar Wind Heating by MHD Turbulence

Solar Wind Heating by MHD Turbulence C. S. Ng, A. Bhattacharjee, and D. Munsi Space Science Center University of New Hampshire Acknowledgment: P. A. Isenberg Work partially supported by NSF, NASA CMSO General Meeting, June 4, 2007 University of New Hampshire, Durham, NH

Hydrodynamic turbulence Kolmogorov (1941): can get energy spectrum by dimensional analysis Energy cascade rate:!(k) " k # E(k) $! " = 5/ 2, # = 3/ 2 Kolmogorov spectrum: { Energy spectrum: Kolmogorov constant: C K ~ 1.4! 2 = constant E(k) ~ C K! 2 / 3 k "5/ 3! E(k) dk = total energy

MHD turbulence Additional dependence on the Alfvén speed V A!(k) " k # E(k) $ % Energy cascade rate: V A! " = (5 #$ ) / 2, % = (3 #$ ) / 2 Spectral index: ν = α / β = (5 γ ) (3 γ ) = constant! = 0 : Kolmogorov spectrum E(k) ~ C K! 2 / 3 k "5/ 3 E(k) ~ C IK ε 1/ 2 1/ 2 3/ 2 γ = 1 : IK spectrum V A k Iroshnikov (1963), Kraichnan (1965) C IK ~ 1.8 2.2 Dimensionless parameter:! " k1/ 2 E k 1/ 2 V A #1 = v k / V A << 1

Weak MHD turbulence Energy cascade possible only if two Alfvén wave packets propagating in opposite directions collide Weak turbulence: χ << 1 Time scales: Eddy turn-over time! N ~ 1/ kv k Alfvén time! A ~ 1/ kv A = "! N <<! N Energy cascade time! E ~! A / " 2 =! N / " >>! N Kolmogorov cascade rate:! K ~ v k 2 / " N ~ k 5/ 2 E k 3/ 2 IK cascade rate:! IK ~ v k 2 / " E ~! K # ~ k 3 E k 2 V A $1 <<! K

simulations of 2D MHD turbulence no B 0 k - 3/2 energy spectrum [Ng et al. 2003] 1D energy spectrum decaying periodic in x,y 2048 2 consistent with IK spectrum magnetic energy kinetic energy

Energy cascade rate! Fit numerical values of or #3 / " = C 2 K " K = C #2 IK " IK where ε IK = k 3 E 2 1 k V! A by IK theory ε K = k 5 / 2 E k 3/ 2 by Kolmogorov theory E(k) = C K " 2 / 3 k #5 / 3 = C IK " 1/ 2 V A 1/ 2 k #3 / 2 C IK 1.8 C K 2048 2 1024 2 4.6 1.8 4.0 More consistent with IK theory c.f. usually found values: C K ~ 1.4 2, C IK ~ 1.8 2.2

Energy cascade rate 2048 2 1024 2! E energy cascade rate ε IK = k 3 E k 2 VA 1 ε A amplitude changing rate ε K = k 5/ 2 E k 3/2 More consistent with IK theory

Energy cascade rate -- observations From [Vasquez et al. 2007, preprint], scatter plot of the Kolmogorov cascade rate (red diamonds - left), the IK cascade rate (red diamonds -right), and expected heating rate (blue crosses) as a function of proton temperature T pr. The Kolmogorov rate exceeds the expected one by a factor of 10 or more, while the IK rate is roughly in agreement.

Solar wind turbulence model The steady state solar wind turbulence model developed by [Matthaeus et al. 1994, 1996] and later developments: Assumptions: Steady state Radially expanding solar wind with uniform speed V sw 1D (radial position r) Turbulence characterized by two fields: fluctuation velocity (Z) and correlation length (λ) Kolmogorov type cascade rate Solar wind (proton) temperature can be calculated passively

Solar wind turbulence model The steady state solar wind turbulence model developed by [Matthaeus et al. 1994, 1996] and later developments: dz 2 dr = " AZ 2 r " #Z 3 $V SW + Q V SW d" dr = # C" r + $Z V SW # $"Q V SW Z 2! dt dr = " 4T 3r + m#z 3 3k B $V SW Z 2 :! average turbulence energy with Z 2 2 + = Z " λ: turbulence correlation length T: solar! wind proton temperature Q: turbulence generation rate due to pickup ions (interstellar neutral particles entering the! heliosphere and get ionized)

Solar wind turbulence model vs observations From [Smith et al. 2001]

Temperature comparison with pickup ions From [Isenberg et al. 2003]

Solar wind model with IK cascade The solar wind turbulence model changed to [Matthaeus et al. 1994, Hossain et al. 1995]: dz 2 dr = " AZ 2 r " #Z 4 $V SW V A + Q V SW d" dr = # C" r + $ % Z 4 ( ' * # $"Q V SW & V A ) V SW Z 2! dt dr = " 4T 3r + m#z 4 3k B $V SW V A Z 2!: average turbulence energy with Z 2 2 + = Z " λ: turbulence correlation length T:! solar wind temperature Q: turbulence generation rate due to pickup ions! 1/ 3

Comparisons with observations IK without Q IK with Q Kolmogorov with Q Kolmogorov without Q c.f. [Isenberg et al. 2003]

Comparisons with observations Kolmogorov without Q Kolmogorov with Q C.f. [Smith et al. 2001], with corrected temperature equation.

Comparisons with observations Kolmogorov without Q Kolmogorov with Q IK without Q C.f. [Smith et al. 2001], with corrected temperature equation.

Comparisons with observations Kolmogorov without Q Kolmogorov with Q IK without Q IK with Q C.f. [Smith et al. 2001], with corrected temperature equation.

Conclusion 2D MHD turbulence is found in simulations to be weak 3 /2 with IK spectrum E(k) k and the energy cascade rate is found to be more consistent with IK scaling. Although a solar wind model based on Kolmogorov cascade compares well with proton temperature observations with contributions from pickup ions, it is also consistent with IK cascade with or without that. The predication of correlation length based on IK cascade with or without pickup ions agrees better with data than Kolmogorov cascade with pickup ions. Ion heating based on IK cascade is a possible alternative to Kolmogorov based theories and should be investigated more extensively.