arXiv:astro-ph/9705050v1 7 May 1997
Turbulent coronal heating and the distribution of nanoflares
Pablo Dmitruk
1
and Daniel O. G´omez
2,3
Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires,
Ciudad Universitaria, Pabell´on I, 1428 Buenos Aires, Argentina
Received
; accepted
1
Fellow of Universidad de Buenos Aires
2
Also at the Instituto de Astronom´ıa y F´ısica del Espacio, Buenos Aires, Argentina
3
Member of the Carrera del Investigador, CONICET, Argentina
– 2 –
ABSTRACT
We perfo r m direct numerical simulations of an externally driven two-
dimensional magnetohydrodynamic system over extended periods o f time
to simulate the dynamics of a tr ansverse section of a solar coronal loop.
A stationary and large-scale magnetic forcing was imposed, to model the
photospheric motions at the magnetic loop footpoints. A turbulent stationary
regime is reached, which corresponds to energy dissipation rates consistent with
the heating requirements of coronal loops.
The temp oral behavior o f quantities such as the energy dissipation rate
show clear indications of intermittency, which are exclusively due to the
strong nonlinearity of the system. We tenta tively associate these impulsive
events of magnetic energy dissipation (from 5 × 10
24
erg t o 10
26
erg) to the
so-called nanoflares. A statistical analysis of these events yields a power law
distribution as a function of their energies with a negative slope of 1.5, which
is consistent with those obtained for flare energy distributions reported f r om
X-ray observations.
Subject headings: Sun: flares — MHD — turbulence
– 3 –
1. Introduction
Coronal loops in active regions are likely to be heated by Joule dissipation of highly
structured electric currents. The formation o f small scales in the spatia l distribution of
electric currents is necessary t o enhance magnetic energy dissipation and therefore provide
sufficient heating to the plasma confined in these loops. Various scenarios of how these fine
scale structures might be g enerated have been proposed, such as the spontaneous formation
of tangent ial discontinuities (Parker 1972, Parker 1983), the development of an energy
cascade driven by random footpoint motions on a force-fr ee configuration (van Ballegooijen
1986), or the direct energy cascade a ssociated to a fully turbulent magnetohydrodynamic
(MHD) regime (Heyvaerts & Priest 1992, G´omez & Ferro Fo nt´an 1992). These rather
different models share in common the dominant r ole of nonlinearities in generating fine
spatial structure.
In this paper we assume that the dynamics of a corona l loop driven by footpo int
motions is described by t he MHD equations. Since the kinetic (R) and magnetic (S)
Reynolds numbers in coronal active regions are extremely large (R ∼ S ∼ 1 0
10−12
), we
exp ect footpoint motions to drive the loop into a strongly turbulent MHD regime.
Footpoint motions whose lengthscales are much smaller than the loop length cause
the coronal plasma to move in planes perpendicular to the axial magnetic field, generating
a small transverse magnetic field component . In §2 we model this coupling to simulate
the driving action of footpoint motions on a generic transverse section of the loop. The
numerical technique used for the integration of the two-dimensional MHD equations is
described in §3. In §4 we report the energy dissipation rate that we obtain, and a statistical
analysis of dissipation events is presented in §5. Finally, the relevant results of this paper
are summarized in §6.
– 4 –
2. Forced two-dimensional magnetohydrodynamics
The dynamics of a coronal loop with a uniform magnetic field B = B
0
z, length L and
transverse section ( 2πl) × (2πl), can be modeled by the RMHD equations (Strauss 1976):
∂
t
a = v
A
∂
z
ψ + [ψ, a] + η∇
2
a (1)
∂
t
w = v
A
∂
z
j + [ψ, w] − [a, j] + ν∇
2
w (2)
where v
A
= B
0
/
√
4πρ is the Alfven speed, ν is the kinematic viscosity, η is the plasma
resistivity, ψ is the stream function, a is the vector potential, w = −∇
2
ψ is the fluid
vorticity, j = −∇
2
a is the electric current density and [u, v] = z · ∇u × ∇v. For given
photospheric motio ns applied at the footpoints (plates z = 0 and z = L) horizontal velocity
and magnetic field components develop in the interior of the loop, given by v = ∇ × (zψ)
and b = ∇ × (za).
The RMHD equations can be regarded as describing a set of two- dimensional MHD
systems stacked along the loop axis and interacting among themselves t hr ough the v
A
∂
z
terms. For simplicity, hereafter we study the evolution of a generic two-dimensional slice of
a loop t o which end we model the v
A
∂
z
terms as external forces (see Einaudi et al. 1996
for a similar approach). We a ssume the vector potential t o be independent of z and the
stream function to interpola t e linearly between ψ(z = 0) = 0 and ψ(z = L) = Ψ, where
Ψ(x, y, t) is the stream function for the photospheric velocity field. These assumptions yield
v
A
∂
z
ψ = v
A
Ψ/L (in Eqn (1)) and v
A
∂
z
j = 0 (in Eqn (2)) and correspond to an idealized
scenario where the magnetic stress distributes uniformly throughout the loop. The 2D
equations for a generic transverse slice of a loop become,
∂
t
a = [ψ, a] + η∇
2
a + f (3)
∂
t
w = [ψ, w] − [a, j] + ν∇
2
w (4)
where f = (v
A
/L)Ψ.
– 5 –
3. Numerical pro cedures
We performed numerical simula tions of Eqs (3)-(4) on a square box of sides 2π, with
periodic boundary conditions. The magnetic vector potential and the stream f unction
are expanded in Fourier series. To be able to perform long-time integrations, we worked
with a moderate resolution version of the code (96 × 9 6 grid points). The code is of the
pseudo-sp ectral type, with 2/3 de-aliasing (Canuto et al. 1988). The temporal integration
scheme is a fifth order predictor-corrector, to achieve almost exact energy balance over our
extended time simulations.
We turn Eqs (3)-(4) int o a dimensionless version, choosing l and L as the units
for transverse and longitudinal distances, and v
0
=
√
f
0
as the unit for velocities
(f
0
= v
A
u
p
(l/L), u
p
: typical photospheric velocity), since the field intensities are determined
by the forcing strength. The dimensionless dissipation coefficients are ν
0
= ν/(l
√
f
0
) and
η
0
= η/(l
√
f
0
). The forcing is constant in time and non zero only in a narrow band in
k-space cor responding to 3 ≤ k l ≤ 4. In spite of t he narrow forcing and even though
velocity and magnetic fields are initially zero, non-linear terms quickly populate all the
modes across the spectrum and a stationary turbulent state is reached.
4. Energy dissipation rate
To restore the dimensions to o ur numerical results, we used typical values for the solar
corona: L ∼ 5 × 10
9
cm, l ∼ 10
8
cm, v
p
∼ 10
5
cm/s, B
0
∼ 100 G, n ∼ 5 × 10
9
cm
−3
and
ν
0
= η
0
= 3 × 10
−2
.
Fig. 1 shows mag netic and kinetic energy vs time. The behavior of both energies is
highly intermittent despite the fact that the forcing is constant and coherent. This kind
of behavior is usually called internal intermittency, to emphasize the fact that the ra pid