MHD Turbulence driven by low frequency waves and reflection from inhomogeneities: Theory, simulation and application to coronal heating W H Matthaeus Bartol Research Institute, University of Delaware Collaborators: P. Dmitruk, L. Milano, D. Mullan, G. Zank and S Oughton MHD turbulence and heating in the open field line corona: quasi-2D cascade model driven by low frequency waves Reflection and sustainment of turbulence driven by waves Turbulence and the origin of the coronal heat function Q(r) Fourth International Workshop on Nonlinear Waves and Chaos in Space Plasmas, Tromsø, Norway, June 17-22, 2001.

Corona and Solar Wind: Open and Closed Field Line Regions

Parker (1972), Priest et al (1998), Einaudi et al (1996)... Parker (1991), Axford and McKenzie (1995)

[McKenzie et al, 1995; Axford and McKenzie, 1997]

Model: - Low frequency waves propagate upwards at the coronal base - Inhomogeneity cause REFLECTION - Counterpropagating interact, drive a low frequency “Reduced MHD” cascade - Turbulent dissipation is sustained; Efficiency = Turbulent Dissipation/Flux Supplied Rate of transmission: Alfven Speed / parallel “box” length Rate of Reflection: Rate of turbulent dissipation

Reduced MHD: Strong B, low frequency limit RMHD regime High frequency Low frequency

RMHD model - Must add reflection terms for inhomogeneous cases - Fluctuation energy injected/removed at boundaries or by volume force

Feasibility of the model: RMHD + Waves +Reflection + Transmission = Heating ??

Physical structure of the “box” model Phenomenology: –one point homogeneous closure –ad hoc parameterization of T: transmission, R: Reflection, F: supplied upwards fluctuation energy Simulation: –RMHD –periodic spectral method –R = Rm = 200 –F is body force –ad hoc R, T

RMHD simulation, “box model”, F=1, R=1/2, T=0.3 Steady energy nonpropagating modes nonzero mixed cross helicity statistically steady turbulent dissipation, NB nonpropagating contribution Efficiency = Diss/F around 1/4 to 1/2

RMHD (Box): Broadband spectra, random transient reconnection/current sheets T=100 Magnetic field and current density Velocity field and vorticity PDF of Electric current density: Intermittency

Box model with R and T Efficiency of 10-50% or moderate to high R, fixed T=1 Dissipation --> 0 if R=0 But Steady dissipation is insensitive to initial seed turbulence level RMHD simulation Phenomenology

Box models show –waves can drive turbulence -- if there is Reflection –efficiencies 10%-50% easily attainable –insensitivity to I.C.s –intermittent fully developed turbulence

Conditions for sustainment of turbulence: open boundaries and “real” reflection/coronal profiles

Under what conditions is MHD turbulent sustained by low frequency wave driving in open boundaries? RMHD model (incompressible) Open boundaries Upward propagating fluctuations injected at base Seed level of broad band turbulence Runs I and II: No reflection; Two boundary conditions (+/- suppression of nonpropagating structures) Runs III and IV: Reflection due to Va(z); 2 boundary conditions

Run I: no R, no “structures” Cross helicity: becomes unidirectional Dissipation efficiency goes --> 0 Turbulent dissipation goes away

Run II: no R, allow structures Cross helicity becomes unidirectional Dissipation efficiency --> 0 Oscillatory but transient turbulent dissipation “Dynamic alignment” turns off the turbulence

Run III: Reflection, but no structures Cross helicity oscillates Very low periodic dissipiation efficiency Turbulent dissipation very small Not real turbulence

Run IV: Reflection and nonpropagating structures “turned on” Cross helicity goes to a statistically steady, mixed value Dissipation efficiency oscillates around value ~40% Almost all the dissipation is turbulent dissipation (spectral transfer dominant) “Real turbulence” broadband “-5/3” spectrum

Cannot sustain MHD turbulence driven by unidirectionally propagating waves alone To sustain turbulence, must –have some source of downward fluctuations, e.g., REFLECTION –permit very low frequency “nonpropagating modes” Compressible and kinetic effects have not been included! Open boundaries

Low frequency waves + reflection = RMHD cascade = sustained turbulent heating Turbulent heating is insensitive to –initial conditions –many details of the fluctuations Turbulent heating is sensitive to –reflection –boundary conditions --- “non-propagating “ modes Efficiencies of 10% - 50 % are easily attainable Kinetic mechanisms to absorb energy at high k perp are not yet identified Conclusions: low frequency wave-driven MHD turbulence as a candidate mechanism for heating the open field line corona