Three-Dimensional Magnetohydrodynamics of the Solar
Corona and of the Solar Wind with Improved Energy
Transport
Roberto Lionello∗ , Jon A. Linker∗ and Zoran Mikić∗
∗
SAIC, 10260 Campus Point Dr., San Diego, CA 92121-1578, USA
Abstract. We have developed a three-dimensional magnetohydrodynamic (MHD) model of the solar corona and of the solar
wind. We specify a magnetic flux distribution on the solar surface and integrate the time dependent MHD equations to steady
state. The model originally employed a polytropic energy equation. In order to improve the physics in our algorithm, we
have incorporated thermal conduction along the magnetic field, radiation losses, and heating into the energy equation. The
2D version of the model is able to reproduce the contrast in density between the open and closed magnetic structures in the
corona and the fast and slow streams of the solar wind. We now present preliminary results of 3D MHD simulations with
improved thermodynamics. The results can be tested against observations by spacecraft and Earth based observatories, in situ
solar wind and magnetic field measurements, heliospheric current sheet crossings.
INTRODUCTION
κ 1.0e+10
The solar corona is well described as a quasi-neutral,
magnetized fluid, or plasma [1, p. 32], that responds dynamically in response to slowly evolving boundary conditions. The appropriate mathematical model for describing the low-frequency motions of such a fluid is MHD.
Several multidimensional MHD algorithms have been
developed to study the global corona [2, 3, 4, 5, 6, 7, 8, 9,
10, 11]. The 3D MHD, polytropic model of Linker et al.
[12] could reproduce many features of the solar corona
during Whole Sun Month, but the plasma density and
temperature were not in quantitative agreement with the
observations. In order to have a better match between the
model and the observations, a more realistic energy equation is needed [13]. Suess et al. [14] and Wang et al. [15]
used a 2D MHD model of the corona and of the solar
wind with heating and thermal conduction. Suess et al.
[16] included multifluid effects. Mikić et al. [17] developed a 3D MHD model that includes thermal conduction
along the magnetic field, radiation losses, and coronal
heating. The model has been used to study the 2D structure of the corona including the transition region and the
upper chromosphere [18]. Here we present preliminary
results of a study of the 3D structure of the solar corona
during the Whole Sun Month. Using the data produced
by the model we calculate emission images that can be
compared with the observations.
Modified Thermal Conduction
Spitzer Thermal Conduction
1.0e+09
1.0e+08
1.0e+07
1.0e+06
1.0e+05
1.0e+04
1.0e+05
1.0e+06
T (K)
FIGURE 1. Parallel thermal conduction coefficient according to Spitzer and modified coefficient used in the present calculation to broaden the temperature gradient of the transition
region.
THE MHD THERMODYNAMIC MODEL
In our time-dependent three-dimensional resistive and
viscous MHD model in spherical coordinates, we solve
the following set of equations here written in CGS units
[17]:
∇ × A = B,
(1)
2
c η
∂A
= v×B−
∇ × B, (2)
∂t
4π
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
222
FIGURE 2. Results from the 3D MHD model for the Sun on August 15, 1996: (a,c,d) temperature at increasingly smaller scales;
(b) magnetic field lines. Closed field regions, where the plasma is trapped, appear hotter.
∂ρ
+ ∇·(ρ v) = 0,
(3)
∂t
1
∂T
m
+ v · ∇T
= −T ∇ · v − (∇ · q
γ − 1 ∂t
kρ
+ne n p Q(T ) − Hch ), (4)
∂v
∇×B×B
ρ
− ∇p + ρ g
+ v·∇v
=
∂t
4π
+∇ · (νρ ∇v),
(5)
flux q, a radiation losses function Q as in Athay [19],
and a coronal heating source Hch . γ = 5/3 is the specific
heats ratio, m the hydrogen atom mass, k is Boltzmann’s
constant, ne (n p ) is the number density of electrons (protons). p is the plasma pressure, g is the acceleration of
gravity, ν is the viscosity.
THE SIMULATION
where A and B are respectively the magnetic vector potential and field, c is the speed of light, η the resistivity,
and v the velocity. ρ is the plasma density, and T the
temperature. The energy equation, (4), contains the heat
We have used our improved model to investigate the
aspect of the corona during the Whole Sun Month. A
magnetic field distribution obtained from the synoptic
charts of the National Solar Observatory at Kitt Peak is
223
FIGURE 3. On the left: superimposed Lasco C1 (FeXIV) and C2 (polarized brightness) images; the solar surface is colored
according to the magnetic flux. On the right: emission calculated using the MHD model.
FIGURE 4. Comparison of 284 Å(FeXV) image taken by SOHO/EIT with the emission calculated using the MHD model.
specified at r = R . A uniform density (ρ0 = 1011 cm−3 )
and temperature (T0 = 20, 000 K) are assumed at the base
of the domain. Below r ∼ 10 R , thermal conduction is
collisional, in the Spitzer form, and directed along B:
q = −κk b̂b̂ · ∇T,
The values of the parameters are Tmod = 3 × 105 K,
α = 0, and ∆Tmod = 2.5×104 K. A plot of κk is presented
in Fig. 1. It shows that for T < Tmod we have a constant
κk , and for T > Tmod we have κk = κ0 T 5/2 . Heating (Hch )
is non uniform in latitude and follows an exponentially
decaying law with the length scale ranging from λ =
0.7 R at the poles to λ = 0.2 R at the equator. Hch
is chosen such that the heat flux at the base ranges from
q0 = 1 × 105 to q0 = 2 × 105 erg cm−2 s−1 .
As initial condition for A we normally prescribe the
potential field corresponding to the surface magnetic flux
distribution. We then integrate Eqs. (2-5) to steady state.
This process takes more than 2 days of real time. If
high resolution is required to resolve the transition region
temperature gradient, and the improved treatment of the
energy equation is used, this translates into many days
(6)
where b̂ is the unit vector along the magnetic field and
κ0 = 9 × 10−7 in CGS units. In order to broaden the
gradient in the transition region without altering qualitatively the solution, we use a special form of κk [20]:
5/2−α
],
κk = κ0 [sT 5/2 + (1 − s)T α Tmod
(7)
T − Tmod
1
.
1 + tanh
2
∆Tmod
(8)
where
s(T ) =
224
parisons with solar-wind measurements are possible and
will be part of future more comprehensive studies.
of computational time. In order to shorten the relaxation
time, we have used as initial condition of the magnetic
field a relaxed MHD state obtained in the past with
the polytropic model. The vector potential (AInt ) and
the current density (JInt ) from the past simulation have
been interpolated on the present, finer mesh. In order to
eliminate the noise introduced by the interpolation, we
have relaxed A by advancing the following equation to
steady state:
4π
∂A
=
J − ∇ × ∇ × A,
∂τ
c Int
A(τ = 0) = AInt .
REFERENCES
1.
2.
3.
(9)
4.
(10)
5.
The solution, A, is used as initial condition of the magnetic field. For the plasma velocity, temperature, and density we use as initial condition a steady state wind solution obtained with a 1D version of our code.
6.
RESULTS
9.
7.
8.
We here show the results for a 3D simulation of the solar
corona, transition region, and upper chromosphere during Whole Sun Month. In Fig. 2 we present the aspect of
the solar corona on August 15, 1996. Figures 2a, 2c, and
2d show the plasma temperature at increasingly smaller
scales. Figures 2b has a large scale view of selected magnetic field lines; the solar surface is colored according to
the magnetic flux. Open field lines are rooted in the polar
caps, whence the fast wind flows, while in the equatorial
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the temperature plot. The temperature appears higher in
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domain to more than two million degrees in the corona.
Using the data obtained from the simulation, we have
calculated the emission in several lines and compared the
result with observations. In Fig. 3a we show a composite
image obtained on Sep 3, 1996 from the C1 (FeXIV) and
C2 (polarized brightness) Lasco coronographs. The calculated emission is in Fig. 3b. The use of a synoptic magnetogram implies that the magnetic flux distribution does
not correspond exactly to that of the days in which the
observations were taken. That may account for some differences. In Fig. 4a we have an image of the corona taken
by SOHO/EIT in the FeXV line. It can be compared with
our result in Fig. 4b, which was obtained using data from
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