The timescales and heating efficiency of MHD wave-driven
turbulence in an open magnetic region
Pablo Dmitruk and William H. Matthaeus
Bartol Research Institute, University of Delaware, Newark, DE 19716, USA
Abstract. Incompressible MHD waves propagating in a non-homogeneous background, with Alfven speed parallel gradients,
can interact with their reflections producing a strong turbulent energy cascade to small perpendicular length scales. In an open
magnetic region, with a strong background magnetic field, this nonlinear process can be described by the reduced MHD
equations, in which incompressible or weakly compressible fluctuations are transverse to the mean field and their gradients
in the parallel direction are much weaker than in the perpendicular direction. The system is forced by Alfven waves injected
through the bottom boundary, with characteristic perpendicular length scale, amplitude and frequency. We identify the relevant
timescales of the system and propose an ordering which favors the turbulent dissipation efficiency. We have applied this
mechanism for the heating of the lower corona required in models of the origin of the solar wind.
The presence of magnetohydrodynamic (MHD) turbulence in coronal holes, as regions of the origin of the
fast solar wind, is an attractive idea, since it can provide a mechanism by which the energy found in lowfrequency motions and large scales can be cascaded (i.e.,
transferred) to small scales and higher frequencies where
it can be more efficiently dissipated. Provided a kinetic
mechanism can be identified for the dissipation of the
cascaded energy, the proper amount of heating can be
obtained for the requirements imposed in fluid models
of the acceleration of the solar wind. From the point of
view of MHD turbulence theory however, a number of
important considerations have to be taken into account to
properly address this turbulent cascade in an open magnetic region. First, the presence of a strong radially directed magnetic field B 0 impose an anisotropy of the turbulent cascade: the energy transfer in the direction of
the magnetic field is suppressed in favor of a transfer
of energy to small perpendicular (to B 0 ) length scales
[1, 2, 3] This is important if a dissipation mechanism is
invoked, since it has to take into account that anisotropy
in the fluctuations scale dependence. Second, the same
presence of the strong magnetic field also induces the
fast propagation of almost non-dissipative Alfven waves
along B0 . As a consequence, the injected energy would
leave the open region considered if no other process is
identified to allow the nonlinear transfer to more dissipative small scales. Both considerations have been addressed in [4] by doing numerical simulations of the reduced MHD equations in a basic model of an open magnetic region. The reduced MHD approximation [5, 6, 7]
describes the anisotropy found in the turbulent cascade.
The presence of Alfven speed parallel gradients produce
reflections of the injected waves [8, 9, 10, 11] and the
coexistence of counter-propagating waves is a necessary
condition to allow nonlinearities to activate and sustain
the turbulence [12, 13, 14, 15, 16]. A more subtle issue is that non-propagating structures favor turbulence
sustainment [1, 2, 17] while the presence of such structures is controlled by the nature of the boundary conditions [4] We further analyze this system here, by identifying the relevant timescales and proposing an ordering of those timescales that favors the presence of turbulence and hence the enhancement of dissipation. The
present report offers a brief discussion of the basic physical issues and associated timescales that affect the sustainment and efficiency of MHD turbulence in an open
region. We have addressed elsewhere [18] the application of this model to more realistic density and magnetic
field radial profiles and their effect on the heating in a
coronal open magnetic region.
The reduced MHD approximation [5, 6, 7] describes
an incompressible or weakly compressible plasma in
the presence of a strong background magnetic field
B0 . A condition in RMHD is that gradients of fluctuations in the direction parallel to B 0 are much weaker
than gradients in the perpendicular direction, thus, the
anisotropy of the turbulent cascade is inherent to the description. Transverse fluctuations of the magnetic b and
velocity fields v are conveniently described by the upward/downward propagating fluctuations z
v b. In
an inhomogeneous background, with an Alfven speed
4πρ s,
varying in the parallel direction VA s B0
where ρ s is the smoothly varying background density,
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
347
the RMHD equations are:
at different length scales 1 k. On the bottom boundary,
the amplitude of the waves is given by a typical imposed
fluctuation amplitude, δ z 0 and at a typical perpendicular
length scale l0 1 k0, where k1 k0 k2 is a wavenumber included in the narrow band forcing. The timescale
introduced by this forcing is t 0 l0 δ z0 . If there is a
turbulent cascade, the ordering is t NL k t0 and it is actually t0 which determines the typical timescale of nonlinearities.
Both the RMHD approximation and the efficiency
of the turbulence on transferring the energy to smaller
scales is favored if the ordering t 0 tA is satisfied.
If Alfven waves travel too fast through the box (i.e,
if tA t0 ) the energy of the fluctuations is mostly
transmitted and lost through the top boundary. The condition t0 tA also means that the correlation scale of the
forcing fluctuations (given by the forcing length scale
l0 ) is small enough to favor nonlinearities to develop,
l0 Lδ z0 B0 . The “competition” between the emptying behavior given by the Alfven wave crossing time t A
and the turbulent cascade development given by the nonlinear timescale t0 is key to the system and determines
the efficiency on dissipating the energy injected through
the bottom.
Another important timescale is given by the frequency
of the forcing, t f
1 f , since it is well known from
linear studies of waves in inhomogeneous media [8, 9,
10, 11] that high frequency waves can travel almost unaffected by reflections and hence nonlinearities can not
survive in that case, as stated before. So, the condition
which favors the dissipation efficiency regarding that issue is tR t f . Finally, the dissipative term in Eq. (1)
contains the timescale tη l02 η for the dissipation of
the large scale forcing structures. The actual dissipation
mechanism of the system is in general more complicated,
and it should be described considering kinetic theory.
For the macroscopic MHD description considered here,
it suffices to say that the dissipation timescale associated
with the large scale energy containing structures is very
large. This is to say that the Reynolds number of the system (comparison of the nonlinear term to the dissipative
term) is very large.
From the previous considerations, we can conjecture
that the timescale ordering which would favor the development of a strong turbulent cascade and sustained heating efficiency is given by:
∂ z
∂z
1 dVA
1 dVA
z z
VA ∂t
∂s
2 ds 2 ds
2
∇ p z ∇ z η ∇z
where η is the resistivity (assumed equal to the viscosity) and p is the (magnetic plus thermal) pressure. The
injection of waves from the bottom, which allows nonpropagating structures to be present in the system (see
[4]), is described by the boundary condition:
∂ z bot
k ∂s
Ak cos2π f t if k1 k k2 (1)
where f is the frequency of the forcing and A controls the
amplitude of the injected waves. The forcing is narrow in
Fourier space, with a k-band given by k 1 and k2 . At the
top boundary, no waves are injected, so, the condition
imposed is:
∂ z top
k ∂s
0
k
(2)
The dynamical RMHD equations and the boundary conditions, contain several timescales and it is of relevance
for the dissipation efficiency of the turbulent cascade to
identify them. We will proceed now to describe those
timescales.
Ignoring the right hand side Eq. (1) leaves us with a
description of the linear propagation of waves in the system. For constant VA , the solutions z expik s VAt correspond to downward (+) and upward (-) propagating
Alfven waves, with speed VA . The timescale in which
a linear fluctuation would propagate out of the box of
height L is tA L VA , which can be computed using the
average VA when there are gradients.
The wave period (given by the forcing frequency of the
boundary conditions at the bottom) and the wave crossing time are the fundamental scales of the problem until
the additional effects contained in the r.h.s. of Eq. (1) are
considered, namely, reflection, nonlinear couplings and
dissipation. The first two terms on the r.h.s. of Eq. (1)
describe the reflection of the waves due to the presence
of parallel gradients of the Alfven speed V A s. Those reflections introduce another timescale, t R L ∆VA , due
to the variation of the Alfven speed within the box. If
tR tA a moderate amount of downward fluctuations can
be generated from a population of only upward injected
fluctuations. In that case, the nonlinear term in the r.h.s.
of Eq (1) z ∇ z is not trivial and the system can
develop turbulence. If, on the other hand, there are no
Alfven speed gradients, t R ∞, and it can be seen [4] that
the turbulence can not be sustained. Even a very small
amount of reflected waves (less than 5%) can be enough
for sustaining the turbulence. The nonlinear term also introduce the timescales t NL k lk zk for fluctuations zk
tNL k t0 tA tR t f tη
(3)
This relationship embodies a balance among numerous
effects – wave propagation speed, the spectral transfer
rate of turbulence, the strength of driving, the large scale
Reynolds number, the self-similarity of the cascade, the
degree of inhomogeneity and strength of reflections, the
ratio of the system size to the turbulence energy contain-
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100
1.0
t0/tA = 0.1
−2
10
t0/tA = 1
0.6
Ek⊥
Efficiency
0.8
0.4
−4
10
t0/tA = 10
−6
0.2
10
0.0
0
20
40
60
Time / t0
80
10−8
100
1
10
100
k⊥
FIGURE 1. Turbulent dissipation efficiency for different
timescale ratios, t0 tA 01 (dashed), t0 tA 1 (continuous)
and t0 tA 10 (dotted, but of such low value it is almost invisible)
FIGURE 2. Perpendicular spectra for different timescale ratios, t0 tA 01 (dashed), t0 tA 1 (continuous) and t0 tA 10
(dotted)
open magnetic region forced through the boundaries.
The conditions which favor the development of turbulence and the enhancement of heating efficiency
have been described. In general terms, high dissipation
efficiency can be obtained when the nonlinear timescale
is less than the Alfven wave crossing time, even if the
population of the necessary reflected waves is low. The
considered problem is of relevance for the heating in
coronal holes and the consequent acceleration of the
solar wind. We have discussed such heating application
of the present model in [18].
ing scale – each of which enters into the overall balance that determines whether a strong cascade driven
by waves can persist in an inhomogeneous medium with
open field line boundary conditions. Extensive study using numerical simulations for a variety of conditions
and parameters will be required to fully investigate the
validity and consequences of the orderings implied by
this inequality. This substantial effort will be deferred
for the present. However to explore some simple consequences of the above discussion, we present a comparison of numerical simulations of Eqs (1)-(3), using
a pseudo-spectral type code (see [4] for a previous use
of this code), for three cases corresponding to different
timescale ratios t0 tA 01 1 10 and fixed forcing frequency, t0 t f 005, dissipative timescale t 0 tη 0001
and tR tA in each case. The dissipation efficiency, defined as the turbulent energy transfer rate divided by the
energy injection rate is shown in Fig. 1. Clearly, the
t0 tA 1 case is the most efficient, while in the case
with t0 tA 1 the dissipation efficiency is very low.
The ratio of downward propagating fluctuations to the
upward propagating fluctuations is in all these cases less
than 5% which indicates that (for the t 0 tA 01 1 cases)
turbulence is still efficient even for low reflected waves
population.
In Fig. 2 we show the perpendicular energy spectra
for the same cases. For the cases t 0 tA 01 1 the spectrum is broad band and consistent with a Kolmogorov
cascade description. The spectrum for the t 0 tA 10 is
much steeper, indicating the fact that the emptying property of fast Alfven waves is dominating over the nonlinear turbulent transfer. The spectral properties of the system under different timescale conditions deserves further
study.
In conclusion, we have identified the relevant
timescales in a wave-driven turbulent system in an
ACKNOWLEDGMENTS
This research is supported in part by National Aeronautics and Space Administration Sun-Earth Connection
Theory Program grant NAG5-8134, NASA grant NAG57164 and National Science Foundation grant ATM0105254 to the University of Delaware Bartol Research
Institute.
REFERENCES
1. J. V. Shebalin, W. H. Matthaeus and D. C. Montgomery, J.
Plasma Phys. 29, 525 (1983).
2. S. Oughton, E. R. Priest and W. H. Matthaeus, J. Fluid
Mech. 280, 95 (1994)
3. W. H. Matthaeus, S. Ghosh, S. Oughton and D. A. Roberts,
J. Geophys. Res. 101, 7619 (1996)
4. P. Dmitruk, W. H. Matthaeus, L. J. Milano and S. Oughton,
Phys. Plasmas 8, 2377 (2001).
5. H. R. Strauss, Phys. Fluids 19, 134 (1976)
6. D.C. Montgomery, Physica Scripta T2/1, 83 (1982)
7. G. P. Zank and W. H. Matthaeus, J. Plasma Phys. 48, 85
(1992)
349
8. M. Heinemann and S. Olbert, J. Geophys. Res. 85, 1311
(1980)
9. J. V. Hollweg, Solar Phys. 70, 25 (1981)
10. C.-H. An, Z. E. Musielak, R. L. Moore and S. T. Suess,
Astrophys. J. 345, 597 (1989)
11. M. Velli, Astron. Astrophys. 270, 304 (1993)
12. R. H. Kraichnan, Phys. Fluids 8, 138 (1965)
13. M. Dobrowolny, A. Mangeney and P. Veltri, Phys. Rev.
Lett. 45, 144 (1980)
14. R. Grappin, U. Frisch, J. Léorat and A. Pouquet, Astron.
Astrophys. 105, 6 (1982)
15. A. Pouquet, U. Frisch and M. Meneguzzi, Phys. Rev. A 33
4266 (1986)
16. S. Ghosh, W. H. Matthaeus and D.C. Montgomery, Phys.
Fluids 31, 2171 (1988)
17. D. C. Montgomery and W. H. Matthaeus, Astrophs. J.
447, 706 (1995)
18. P. Dmitruk, W. H. Matthaeus, L. J. Milano, S. Oughton,
G. P. Zank and D. J. Mullan, Astrophys. J., in press (2002).
350