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) Simulations of magnetohydrodynamic turbulence in astrophysics: recent achievements and perspectives Paris, France, July 2001

Observations of the Solar Atmosphere show indications of highly dynamic activity involving a wide range of spatial and time scales Turbulent Dynamo interaction of waves and non- propagating structures Transverse structure nonlinear MHD effects, turbulent reconnection, cascade... EIT/SOHO Lasco/SOHO TRACE

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

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

Physical structure of the 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 -- if there is Reflection –efficiencies 10%-50% easily attainable –insensitivity to I.C.s –intermittent fully developed turbulence Need to look at –coronal profiles of density, field –consistent treatment of propagation, reflection, transmission, boundary effects –I.E., a coronal model

Step II 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/ Coronal Profiles

Step III Try to explain properties of the corona: the heat function Q(r)

Well-known but Ad hoc Heat Function Q(r) Used in a variety of studies of solar and stellar winds (Holzer and Axford (1970), Koppand Orrall (1976), Hammer (1982) Can provide an ad hoc explanation for many observed properties of the fast wind and polar coronal holes (McKenzie et al (1995), Habbal et al (1995), Axford and McKenzie (1997) As yet no accepted theoretical basis r 0 =Solar radius L ~ 1/4 - 1/2

Wave driven Coronal Model Adopt Coronal profile RMHD model of a section with open boundaries Phenomenology: wave amplitudes with turbulent drag and realistic reflection rates

RMHD simulation with coronal profile Radial expansion R=Rm=600 Forcing at bottom –single low frequency 0.1/T A –broadband in Kperp Waves escape from top and bottom

Dissipation per unit volume is concentrated near the coronal base Q(r) = energy per unit mass per unit time dissipated by the turbulence at an altitude r

Coronal phenomenology Detail of transverse structure is suppressed -- modeled by turbulence phenomenology Perpendicular energy containing (correlation) scale controls turbulence Propagation, reflection and boundary effects are like 1D waves

Parameter scan using coronal phenomenological model shows that Q is approximately exponential with scale height L ~ 1/4 solar radius

Conclusions: Turbulence and Heat function RMHD Simulations with coronal profiles show that – turbulent heating is sustained –dissipation per unit volume is concentrated near coronal base Phenomenological model supports the above conclusions –allows parameter studies –shows approximately exponential behavior –provides a high R e limit an asymptotic treatment of the phenomenology shows –Q(R) is related to coronal density and magnetic field profile

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 Heat function –determined by coronal density profiles –approximately exponential with L= R sun /3 for isothermal gravitationally stratified atmosphere 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

Efficiency studied in the phenomenological model Higher efficiency for lower frequency (quasi-static!) smaller transverse length scale of the turbulence (< 30,000 km) Large Alfven speed gradients

Can we develop a model that converts low frequency wave energy into a turbulent cascade that produces rapid and sustained heating? Plenty of energy: usually, power spectra are peaked at low frequency, long wavelength Crucial role of refection Important role of “non- propagating” modes Similar approach seems to work well for solar wind Typical k^(-5/3) magnetic field spectrum at 1 AU in SW (Voyager 2, 1978)

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) Simulations of magnetohydrodynamic turbulence in astrophysics: recent achievements and perspectives Paris, France, July 2001