Turbulent dissipation in the solar wind and corona

Turbulent dissipation in the solar wind and corona W. H. Matthaeus, P. Dmitruk, S. Oughton and D. Mullan Bartol Research Institute, University of Delaware, Newark, DE 19716 USA Department of Mathematics, University of Waikato, Hamilton NZ Abstract. Models based upon anisotropic magnetohydrodynamic (MHD) cascade offer promising explanations for observations of both interplanetary and coronal turbulence and heating, which are reviewed here. In the standard picture the cascade proceeds by driving at the energy-containing scales, transfers through the inertial range, and into small scales where it drives small-scale random turbulent reconnection events. In order to understand more fully the heating and dissipation processes, one also needs to understand how small-scale MHD-driven reconnection involving current sheets and filaments induces kinetic plasma processes that thermalize the fluid energy. Here we suggest that in these reconnection sites MHD electric fields drive ion beam instabilities and nonlinear electron dynamics involving electron solitary wave structures, in analogy with the kinetic physics observed near parallel electric field auroral regions by the FAST spacecraft. INTRODUCTION A familiar signature of turbulence, whether it be hydrodynamic [1] or magnetohydrodynamic [2], heliospheric [3], solar [4] or astrophysical [5], is a powerlaw wavenumber fluctuation spectrum that extends over a wide range of spatial scales. Powerlaw spectra are indicative of scale invariant processes that distribute the fluctuation energy across a wide range of scales, but which on balance transfer energy towards smaller scales The cascade leads to small scale dissipation, which in a collisionless plasma may be anisotropic and dependent upon detailed plasma physics processes. An example is the role of the Hall effect in collisionless reconnection [6, 7]. For some purposes, the inertial range itself can drop out of the picture, acting as a lossless pipeline that connects the large-scale energy sources to the smallscale dissipation. In the solar wind this pipeline spans four orders of magnitude, from the correlation scale at 5 10 11 cm down to around the ion inertial scale at 2 10 7 cm. Important physics lies in the identification of the energy sources for the cascade, as well as identification of dissipation mechanisms. Here we will focus on three heliospheric problems; namely to explain why (1) the solar wind is too hot at 1 AU; (2) the solar wind is too hot at 30 AU; and (3) the corona is too hot at 2 R s. Each of these puzzles can be addressed using simple energycontaining range treatments of MHD turbulence in which large scale dynamics govern both the cascade rate and the rate of dissipation. ANISOTROPIC MHD CASCADE MHD turbulence in regimes of incompressibility or near-incompressibility gives rise to anisotropic spectral transfer relative to an externally imposed dc magnetic field direction [8, 9]. There is an equivalent type of anisotropy relative to a local mean magnetic field direction [10, 11]. The preferred spectral transfer generates strong gradients across the mean magnetic field, and greater excitation in perpendicular wavevectors than in parallel wavevectors. Frequently, perpendicular transfer leads to lowfrequency, quasi-2d turbulence for which Reduced MHD (RMHD) [48] is a good approximation. The perpendicular cascade ends when it reaches the dissipation range of wavenumbers. In hydrodynamics the large scale nonlinear time is λ/u for turbulent velocity fluctuation amplitude u, and energycontaining scale λ. The estimated dissipation scale is λ diss (ν 3 /ɛ) 1/4, ν the kinematic viscosity and ɛ the dissipation rate of energy per unit mass. A key approximate connection between turbulence energy per unit mass u 2, the energy-containing scale, and the dissipation rate is ɛ u 3 /λ [12]. Simple MHD models include both a viscosity and a resistivity µ. Provided that these are not too dissimilar in value, it is well established that the anisotropic MHD cascade will terminate in randomly distributed reconnection sites. This is illustrated in Fig. 1. When there is a high degree of anisotropy caused by a (locally) uniform mean magnetic field, the reconnection sites tend to be highly oblique, with elec-

y 500 400 300 200 100 0 0 100 200 300 400 500 x FIGURE 1. Cross section of a spectral method simulation of decaying 2D MHD turbulence showing magnetic field-lines and electric current density (gray scale) in the out-of-page direction. Current is concentrated in sheets or filaments between interacting magnetic flux structures. The cascade process is characterized by random driven reconnection events, involving magnetic island merger and generation of strong current channels. tric current density tending to be aligned with the mean field. Because of anisotropy, the extension of the decay phenomenology [13] for low cross helicity is hydrodynamic-like with ɛ Z 3 /λ, except that the lengthscale that appears, λ,isaperpendicular energy-containing scale. Here Z =( u 2 + b 2 ) 1/2, u the turbulent velocity, b the turbulent magnetic field in Alfvén speed units, and... denotes an ensemble or volume average. SOLAR WIND AT 1 AU From the days of early spacecraft exploration of the solar wind it has been known that the proton temperature at 1 AU is much higher than what would be expected based upon reasonable coronal base temperatures and an adiabatic expansion [14]. Some local heating appears to be required. The first suggestion that a strong hydrodynamic-like cascade [12, 15] might be responsible for this heating was due to Coleman [16]. Influential papers by Tu et al. [17] and Hollweg [18] formulated spectral and energycontaining range theories (respectively) that would quantitatively account for this turbulent heating of the wind. There are complications in the inner heliosphere, such as nonzero cross helicity effects, and proximity to the initial data at distances < 1 AU [19 22]; however, apart from details, it seems fairly certain that a nonlinear, turbulent cascade contributes to the heating of the solar wind at 1 AU. Recent studies by Leamon et al. [23 26] examine turbulent dissipation at 1 AU in WIND data by examining the spectral breakpoints in magnetic spectra near 1 Hz. Some main conclusions are: (1) simple parallel cyclotron resonance does not provide a consistent explanation; (2) association of the breakpoint with oblique structures of size equal to the ion inertial scale provides a better fit than either the cyclotron frequency itself, or the parallel resonant wavenumber; (3) there is evidence for about equal mixture of processes with cyclotron signatures and other processes that lack cyclotron signatures (e.g., Landau damping). These results are consistent with involvement of current sheets having wavevectors with average orientation of about 70 80 to the mean magnetic field. SOLAR WIND AT 30 AU Interpretation of solar wind observations beyond 1 AU have also presented difficulties. On the one hand [27] a transport theory ( WKB ) for noninteracting waves seems to give approximately the correct radial variation of fluctuation amplitude. However, persistence of powerlaw spectra [3] and the highly non-adiabatic radial proton temperature profile [28, 29] point to an ongoing deposition of energy in the extended solar wind plasma. A plausible explanation is that the turbulence cascade is maintained at near steady conditions, so that driving balances dissipation [27], although development of an appropriate transport theory [30] was needed to demonstrate that a near-wkb radial energy density variation of r 3 would be maintained to 10 AU or so. A source of driving for the turbulence is needed, and simple estimates of shear-driving due to instability of interfaces between high and low speed solar wind appears to work well [30]. Finally, turbulence theory [31, 32] accounts well for the observed radial variation of proton temperature. (See paper by J. Richardson, these proceedings.) The structure of this theory includes a one-point closure model of the turbulence decay rate, which in accordance with the discussion of the previous section, is chosen to vary as Z 3 /λ where λ is the transverse energycontaining lengthscale, which controls the cascade rate. The assumption of an anisotropic cascade is central in the turbulence theory explanation of heating of the outer heliospheric solar wind. (See paper by Oughton, these Proceedings.)

Transmission B 0 Reflection Counter-prop waves drive 2D turbulence. Energy cascades to small perp scales. --> heating Waves launched upwards FIGURE 2. Diagram of an open field-line coronal heating model that is powered by low-frequency turbulence. Incident waves periods of 1000 s and transverse scales of 30,000 km, enter through the coronal base. Inhomogeneity of the Alfvén speed induces some reflection. Counterpropagating wave trains interact through nonlinear couplings, driving a quasi-2d low-frequency cascade. Some wave energy exits through the top and is lost to the system. If the cascade is sustained, then some fraction of the total energy flux supplied is converted into heat by turbulent dissipation. OPEN FIELD CORONA AT 2 R S Magnetic fluctuations in some form are a likely conduit that transports the required 6 10 5 erg/cm 2 - sec [33] into the corona. The search for a heating process is constrained by Spartan [36] and UVCS [37] observations of proton temperatures that reach T >10 6 K within 2 R s of the photosphere. Accordingly observations by UVCS indicate a solar wind speed > 200 km/s at r R s. The wind speed reaches 500 km/s or more by r 6 R s [38]. Theories of solar wind acceleration [39, 40] confirm that deposition of heat close to the coronal base can accelerate this fast polar wind. Low compressibility upwards propagating Alfvén waves are candidates for transporting energy into the corona. These might be high-frequency waves generated in a furnace region in the chromospheric network [33] which has a typical transverse scale of 30,000 km. High-frequency waves near the cyclotron frequency, perhaps 1 khz, may be readily transmitted from the network to the corona and dissipated by cyclotron absorption [33 35, 40]. Alternatively, in view of the fact that the photospheric motions have convective time scales of 100 1000 s, the waves might be of much lower frequency. FIGURE 3. Profiles of heating (per unit volume) from an analytical phenomenology that describes coronal heating by waves, reflection, and cascade. The various curves correspond to different values of the perpendicular energy-containing scale. All resemble an exponential, with e-folding length a fraction of a solar radius, similar to the ad hoc heat functions used in models of the acceleration of the solar wind. There are several reasons to consider driving by low-frequency waves, apart from the better match with the convective photospheric timescales. One reason is that cyclotron resonance may be, in a sense, too efficient [41 43]. The cumulative effect of many minor ion resonances [41] may be to leave insufficient wave energy to heat the protons. This would then require that the wave spectrum be regenerated within the corona. However to resupply the coronal highfrequency wave spectrum through an MHD cascade requires cascade to k Ω ci /B 0, i.e., to high parallel wavenumbers. But MHD cascade [8, 9] mainly generates high perpendicular wavenumber excitations, and so it may be very difficult to use a nearly incompressible [44] cascade to resupply the high-frequency spectrum. The above reasons (See paper by Isenberg, these proceedings.) provide ample motivation to examine an alternative scenario for coronal heating, illustrated in Fig. 2. Low-frequency (period 10 3 s) Alfvénic fluctuations enter the corona and propagate upwards. These represent excitations of only one (say, Z ) of the two (Z ± ) Elsässer amplitudes. However, to excite strong incompressible wave-wave couplings and turbulence, both Elsässer amplitudes are needed, as upwards-type waves interact nonlinearly only with downwards-type waves, and vice versa. Consequently there needs to be a source of

downwards waves, and in this regard the inhomogeneity of the medium enters in a crucial way. Density changes, or more precisely Alfvén speed gradients [45 47], produce refraction and reflection of Alfvén waves. Downward waves produced in this way engage in nonlinear interactions with the upwards waves. With a strong vertical magnetic field, and low plasma beta, this results in a quasi-2d nearly incompressible cascade that can be appropriately described by the equations of RMHD [48] The efficacy of the above described model can be evaluated using a variety of analytical and numerical modeling approaches. Numerical studies have established some of the general conditions for maintaining a cascade driven by waves [49], confirming the general features of the above picture. Analytical phenomenologies can be developed that include transport, reflection, and simple cascade models [50], and these have suggested that reasonable efficiencies (10% to 30%) can be maintained. Numerical RMHD simulations [51] confirm that these are attainable efficiencies. Most recently, numerical RMHD simulations have been extended to employ reasonable models of coronal density and magnetic field profiles [52]. Results show that the deposition of heat by this mechanism is, on a per-unit-volume basis, concentrated near the coronal base, while the per-unit-mass heating is extended to at least several solar radii (see Fig. 3). The model has the interesting property that the deposition of heat by the cascade follows a radial profile that is very similar in shape to the assumed coronal density profile. For a simple isothermal gravitationally stratified coronal model [45] the density behaves similarly to an exponential in altitude. In general, in the wave reflection-driven turbulence model the macroscopic density profile can control the cascade [52]. RECONNECTION & DISSIPATION The physical picture remains incomplete until we understand how the kinetic plasma responds to the cascade and generates heat. As discussed above, most of the energy in a nearly incompressible MHD cascade is expected to dissipate through small-scale reconnection processes. This points to a suggestion: the key to understand the dissipation and heating associated with MHD turbulence in collisionless plasmas is to be found in the kinetic plasma response to turbulent reconnection. In particular, turbulence generates patches of strong electric field surrounding the reconnection sites; see Fig. (4). Furthermore, when there is a large-scale magnetic field, anisotropy of the FIGURE 4. Cross-section of the vertical MHD electric field from an RMHD simulation with coronal density profile, and wave driving at the bottom boundary. Large scale Reynolds number and magnetic Reynolds number are 2000. The snapshot is 1/10R s above the base, after a statistically steady state has been achieved. Regions of intense electric field (intense white or black shading) are concentrated around sites of magnetic reconnection. Kinetic response of ions and electrons will be strongly influenced by these patches of electric field, which last for several Alfvén times (10 2 10 3 s in the corona). cascade implies that magnetic reconnection sites and their associated kinetic responses will be distributed anisotropically. What kind of kinetic response do we expect from driven, turbulent reconnection? This is a difficult problem that appears not to have been completely solved as yet. There are several areas in which we can seek some guidance however. Test particle response can be viewed as an upper limit to the impact that reconnection fields can have on the kinetic plasma. Pending newer results, we anticipate some of the physics by looking back at the test particle results [53] from a simulation of decaying 2D reconnection in the presence of a finite-amplitude broadband turbulence. The reconnection dynamics used in these simulations admits strong fluctuations in space and time. There are multiple and highly dynamic reconnection zones, and a growth of reconnected magnetic islands faster than either tearing mode theory or laminar Sweet Parker theory (see, e.g., [54]). Accordingly, the test particle response is complex. The highest energy particles tend to be those that are temporarily trapped in the turbulent reconnection zones, in which typically there are several highly dynamic X-point regions. Fig. 5 illustrates this feature.

What we can conclude based upon test particle results is fairly limited. However, they confirm the expectation that the electric fields appearing in and near turbulent reconnection sites excite beam-like distributions of particles. Being much lighter, electrons would absorb energy faster than protons. Very rapid electron dynamical timescales imply that self-consistent modifications of the distributions would be manifest quickly. Vlasov and PIC simulations (e.g., [55]) provide some insight into electron response to low-frequency electric fields. Electron plasma oscillations and low-frequency ion acoustic waves are generated, followed by damping of the high-frequency waves and formation of suprathermal tails. Implications for proton distribution are not yet clear from these studies, presumably due to computational limitations. Another view of the response of kinetic plasma to reconnection is obtained in the simulations of the smooth, nondriven, non-turbulent reconnection simulations that have been carried out in the GEM collaborations [6]. For example, it has been observed [7] that whistler turbulence is excited by kinetic effects in laminar reconnection, and electron Hall currents give rise to distinctive magnetic field signatures near the boundaries of the ion reconnection layer. Numerical modeling of reconnection and turbulence is moving rapidly, and we expect additional findings will pave the way to advancement in our understanding of dissipation processes in a variety of space plasma environments in which reconnection and turbulence occur. On the observational side, further breakthroughs in understanding solar microphysics will await results of proposed missions, including SDO, RAM, ASCE, and Solar Probe. In the interplanetary environment, important results about microphysics and dissipation will come from an operational multi-spacecraft cluster, such as a Multispacecraft Heliospheric Mission, or an L1 Cluster of existing missions such as ACE, WIND, Genesis, SOHO, and Triana. Geospace missions, especially in the auroral acceleration regions, are providing important insights now. CLUES IN AURORAL PHYSICS The accuracy and time resolution of current spacecraft experiments makes it possible to explore previously unobservable kinetic phenomena. An excellent example is the FAST mission [56] study of the auroral acceleration region in which there is now clear evidence for electric field parallel to the local magnetic field [57]. FAST shows distinct signatures de- FIGURE 5. Test particles speeds vs. position along the nominal current sheet in a turbulent reconnection simulation, 2.9 eddy turnover times from the initial state, a 2D current sheet extended in the x-direction. Turbulence triggers highly dynamic reconnection. Higher energy particles are produced in small regions along the current sheet, the positions corresponding to small transient X-point regions. At this time particles are collimated in the z-direction (not shown); later the particles isotropize. Adapted from Ambrosiano et al [1988]. pending upon the electric field direction. Upwards electric field drives electrons downwards, generating the aurora. In this case protons stream upwards, forming beams and associated cyclotron instabilities. For downwards electric field, there are upwards electron beams, and associated solitary structures (highly nonlinear phase space holes ) that propagate upwards. In the latter case protons are observed to have high perpendicular temperature, presumably due to very fast time scale encounters with propagating solitary structures [58, 59]. Moreover, in this case there also appears to be a deficit of power at harmonics of the proton cyclotron frequency. There are great differences between the origins of the parallel electric field in the aurora and in the corona. However from auroral studies and analogy with the corona, we may be able to learn something about the favored mode of response of a collisionless plasma to a parallel electric field. Production of nonlinear electron beams, copious solitary structures and associated broadband noise, and perpendicular heating of protons, may well be the favored response when electrons are the preferred charge carriers. When proton beams are formed, cyclotron beam instabilities and production of cyclotron waves may be preferred. Since the corona, too, is gravitationally stratified, and the vertical electric fields associated with random reconnections are alternately upwards or downwards directed, both types of phe-

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