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
ANISOTROPIC MHD CASCADE
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.
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 λdiss ≈ (ν 3 /)1/4 , ν the kinematic viscosity and the dissipation rate of energy per unit mass. A key
approximate connection between turbulence energy
per unit mass u2 , the energy-containing scale, and
the dissipation rate is ≈ 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-
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
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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.
500
400
y
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.
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.)
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 ≈ Z 3 /λ⊥ , except that
the lengthscale that appears, λ⊥ , is a perpendicular
energy-containing scale. Here Z = (u 2 + b2)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 <
428
Transmission
B0
Reflection
Counter-prop waves
drive 2D turbulence.
Energy cascades to
small perp scales.
--> heating
Waves
launched
upwards
FIGURE 2.
FIGURE 3.
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.
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 ∼ Ωci /B0 , 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
OPEN FIELD CORONA AT 2 RS
Magnetic fluctuations in some form are a likely conduit that transports the required 6 × 10 5 erg/cm2sec [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 > 106 K within 2 Rs 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—1000s,
the waves might be of much lower frequency.
429
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].
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 10 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 patches2 of electric
field, which last
for several Alfvén times (10 —103 s in the corona).
/
R
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.
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
430
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.
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-
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-
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about nonlinear kinetic heating in the corona, and
perhaps in other space environments as well. If so,
it would be quite gratifying that plasma physics discovered in one space venue might elucidate fundamental physics in other applications.
ACKNOWLEDGMENTS
Supported by NASA (NAG5—8134), NSF (ATM0105254), and the UK PPARC.
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