Collisional filtration model in the solar transition region
A. Greco and P. Veltri
Dipartimento di fisica, Università della Calabria (Italy)
Abstract. In this work we will try to show the physical reasons for the steep temperature increase in the solar transition
region, trying to look for a solution of the Boltzmann equation, which seems to be the most suitable tool for describing the
plasma of the solar transition region. To solve the Boltzmann equation, we have used a Monte Carlo method; as a solution of
this equation we have obtained a temperature enhancement and a density drop; besides, we have computed an electron heat
flux towards the innersphere, showing a departure from the classical transport theory.
INTRODUCTION
atmosphere while heat flux is zero. To overcome the difficulty to reproduce the temperature jump, in the collisionless approach, Scudder [3, 4] proposed a new mechanism
called the velocity filtration effect: if the lower boundary is overpopulated with high-velocity particles, this excess is recovered in the higher regions, giving rise to a
temperature enhancement (see figure (3) and (4) of [3])
and this result is obtained analitically when using a generalized Lorentian (or Kappa) distribution for particles
at the lower boundary. Scudder strategy was to baypass
the need for a collisional calculation since the suprathermal tails which overcome the potential barrier and enter in corona, are essentially non collisional (the free
v4 ). Collisionless exospheric modpath scales as λ
els are limeted because collisions are not negligible and
because they require non-thermal distribution functions
(like Kappa-functions) to reproduce a temperature gradient.
For the above mentioned limitations, a complete description of the solar TR requires a kinetic model taking collisions into account; so, it is natural to consider the Boltzmann equation. Due to complexity of the physical problem and of the mentioned mathematical equation, we
have made some simplifying assumptions, described in
the next section.
A challenging problem in solar physics is represented
by the attempt to understand the reasons for the steep
temperature increase and density drop over a very short
(with respect to solar scale lengths) distance (some thousand kilometers) in the transition region (TR). The electron mean free path increases from some hundred centimeters in chromosphere to several thousand kilometers in corona [2] and this means that in the transition region a matching is realized between a collisional
medium, where a fluid description is adequate, and a
medium where collisionless transport processes are expected to be dominant, so the plasma should be described
in term of the kinetic Vlasov equation. For the above
mentioned reasons two opposite approaches to this problem, have been pursued: the classical hydrodynamic approach [7, 8] yielding to the fluid models and the collisionless one yielding to the exospheric models [5]. The
former is based on the assumption that the plasma is
collision-dominated and on the assumption of weak external gradients. In this case there are only small departures from Maxwellian distribution functions and the
Fourier’s law for the heat flux holds. Collisional fluid descriptions for modelling the solar transition region are
limited because of the presence of huge temperature and
density gradients and because the thermal Knudsen numbers KT are large enough for the classical transport coefficients to become modified [12]. On the other hand, the
collisionless approach assumes that the effects of collisions are neglegible, so the Boltzmann equation for the
particle flow is reduced to the Vlasov equation and any
function of the total energy is a solution. If we start
with a Maxwellian distribution function (DF) at the lower
boundary, at each height a Maxwellian is found with the
same temperature and the density is that of an isothermal
NUMERICAL SIMULATION
Let us assume a stationary case, a planar onedimensional geometry in the physical space and a
cylindrical one around the vertical direction in the
velocity space. In this case the DF depends on three
variables, f x u u , where x is the vertical coordinate
(the distance from the solar surface), u ux is the
velocity parallel to the vertical direction and u is the
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
273
modulus of the velocity perpendicular to this direction;
then, the stationary Boltzmann equation for the electrons
is written
ux
d f x u eE d f x u
dx
me dux
of the distribution function through a shock wave profile.
We will briefly recall here the main features of this technique. This method is based on five main points: (i) about
104 electrons are injected in the simulation box at the
lower boundary with the assumed DF for positive parallel
velocities; (ii) electrons propagate in the simulation box
under the effect of the above electric field and are subjected to Coulomb collisions whose rate is determined
by the DF of the target electrons; (iii) when particles arrive at the upper boundary we reverse their radial velocity if it is lower than the escape velocity; (iv) we stop to
follow the particles trajectory when they arrive either at
the lower boundary or to the upper one, but with a radial
velocity higher than the escape one; (v) to compute the
DF we have divided the phase space in cells and the DF
for the incident electrons is computed by accumulating
the time spent by the particles in each cell of the phase
space.
Clearly, the knowledge of the target DF is needed, so we
use an iterative procedure, in which the DF for the incident electrons obtained in the previous iteration is used
as the DF of the target electrons in the following iteration. When the distribution functions of two following
iterations are almost the same, the convergence has been
obtained and we have a solution to the Boltzmann equation and the two populations (incident and target ones)
can be considered identical. At the first iteration step, we
started with a guess target DF and we have chosen for
this a Maxwellian one; for the density and temperature
profiles of the zero-order distribution, we have used those
obtained as a solution of the Fourier’s law with constant
heat flux qo [6]: T x To7 2 7qo x xo 2k2 7 and
from the guess assumption of constant pressure p o , the
density profile is given by n x p o kB T x. To , qo and
po are guess parameters, referred to their value at the
lower boundary of the simulation box.
(1)
bmax
dv
bdb
0
2π
0
u vd ε
f x u f x v f x u f x v
where e and m e are the electron charge and mass, respectively.
The right hand-side term of (1) is the collisional integral,
in which two populations of particles are considered: the
target ones with velocity v and the incident ones with velocity u; v and u are their velocities after a collision,
respectively; b is the impact parameter and, since we describe Coulomb collisions between electrons, the maximum impact parameter b max is the Debye radius; ε is the
angle made by the collision plane and an arbitrary plane,
measured in a plane normal to the relative velocity vector
u v (see figure 2 of [1]).
To avoid solving the Boltzmann equation for the protons, we have made the hypotheses of charge neutrality,
equal bulk velocity V and temperature T in the momentum equations of the two kinds of particles; from these,
we can obtain the expression of the electric field, where
GMS
m m
the electrons move E i 2e e V dV
dx x2 , obtained in
1972 by Lemaire and Scherer.
We studied the solution of (1) in a region which starts
at 2000 Km above the photosphere and extends up to
about 7 10 4 Km in the corona. At the lower boundary we supposed that distribution function for the incoming electrons is Maxwellian; at the upper boundary, we
assumed that the motion outside the region under study
is ballistic, so that electrons which arrive with a velocity
less than the escape one, come back to this boundary with
a reversed sign of velocity (the escape velocity is considered as a parameter of the problem). Solving (1) with the
above reported boundary conditions, we obtain electron
distribution function at each level of the simulation box,
which in turn allows us to determine the density, temperature, bulk velocity and heat flux profiles.
To solve the Boltzmann equation we use a Montecarlo
method widely described in [1]; it is defined as a test
particle method based on an iterative process, where
the "random walk" of a particle (an electron) is studied while it interacts (through Coulomb collisions) with
the field (target electrons). This tecnique shows the distinct advantage that is not necessary to introduce simplifying assumption to evaluate the collisional integral. The
same method has been developed to study the Boltzmann
equation in inhomogeneous situations, like the evolution
NUMERICAL RESULTS
Each length in the code is normalized to the Debye radius at the lower boundary r Do , velocities to the elctron
thermal velocity at the lower boundary v tho and times
to the inverse of the electron plasma frequency at the
lower boundary ω peo vtho rDo . With this choice, den3 , temperatures to
sity profiles are normalized to n rDo
2 2K (T and T have the same value) and
T me vtho
o
b
3 .
the heat fluxes to q n me vtho
The figure 1 on the left shows the 6 iterations needed to
reach the convergence of the moments of the distribution
function; the plot on the top left and on the top right are
the density and the average temperature, respectively and
the plots on the bottom left and on the bottom right are
the electron bulk velocity and the electron heat flux, re-
274
spectively. The guess parameters for this simulation are
To 104 K, po 69 103 erg cm3 (no 5 109
cm3 ) and qo 5 105 erg s1 cm2 (see [6]). Besides,
the density normalization n is equal to 106 cm3 and
heat flux normalization q is about 150 erg s 1 cm2 .
From the two plots on the top, we can note that density
decreases by a factor of about 100 and temperature increases by the same factor, reaching values of 1 million
of degrees in corona. The electron temperature is an average between the parallel and perpendicular one and we
have found that the perpendicular heating is much more
effective than the parallel heating, but this result is not
shown. As previously said, the escape velocity is an input parameter which determines the bulk velocity of the
simulation; we have used a value of 10v tho to obtain these
results; the reader can note that the value of the obtained
electron bulk velocity is very high and unrealistic, but it
is a consequence of the very low value of the escape velocity, therefore many electrons have a parallel velocity
greater than the escape one and these are the particles
which provide such a high bulk velocity in corona. We
would stress the fact that using an higher and more realstic value of escape velocity, the bulk velocity would
be determined by the more energetic particles in the tails
of the distribution function, where the statistic is poorer,
so much more particles are needed in order to find some
electron which has a parallel velocity geater than the escape one. Our efforts are going in this direction, but a
very high computing power is required. As we can see
from the fourth panel of the figure 1 on the left, we obtain
an heat flux peak value of the order of 2000q 3 105
erg/cm2 sec. If we compare with the free-streaming value
of the heat flux coming from corona with an electron
3
thermal velocity vth1 , q f s n1 me vth1
7 106 erg/cm2
sec (n1 is electron density at the upper boundary), we
find that the electron heat flux is limeted by the value
for streaming particles in a neutral gas [13]. Another upper limit fort the electron transported heat flux is the non
collisional value q NC KnL To TL1 2 TL To1 2 (K is a
constant), where To and TL are the temperatures at the
boundaries of the simulation box and n L is the density
at the upper side [14]; if we suppose that the plasma is
non collisional and that there are two Maxwellian distribution functions at the boundaries, the above expression
gives 43 10 5 erg/cm2 sec in each point, which is
greater than the obtained value. So, the general behaviour
of the system is not that of a collisionless plasma, because our heat flux is spatially variable and smaller than
the collisionless value. Finally, the profile of the classical Spitzer-Ḧarm heat flux [9] given by the expression
qSH ke T 5 2 dT
dx (calculated with the obtained temperature and density profiles) is very close to the obtained one
in corona, where temperature and density gradients are
very small, but near the transition region, where the hypoteses for the classical heat conduction don’t hold, the
profile shows a significant departure from the obtained
one.
The same figure 1 on the right shows the evolution of the
computed electron DF, of the last iteration for the case
no 5 109 cm3 , versus parallel velocity normalized
to vtho , at four distances from the solar surface integrated
on perpendicular velocities: at 7 10 4 Km (the upper
boundary) the DF is not symmetric, as we can see from
the comparison with a Maxwellian distribution function
with the same temperature and density (dotted line). It is
probably due to the lost particles having a parallel velocity greater than the escape one. As the altitude decreases (104 Km, 21 10 3 Km, 2 103 Km) the electron DF becomes more non-Maxwellian, in fact it exhibits an anisotropic, high-velocity tail going downward
from corona to chromosphere, carring a heat flux, as it is
possible to note in the fourth panel of the figure 1 on the
left. In the same time an enthalpic flux 32 nV Kb T pV appears in order to balance the heat flux. Future works are
going in the direction to check the energy balance for all
the forms of energy in the system. At the lower boundary (2 10 3 Km) the distribution for positive velocities
is consistent with the assumed Maxwellian conditions at
the injection layer, whereas for negative velocities the
distribution deviates from a Maxwellian at 2 3v tho.
DISCUSSION AND CONCLUSIONS
Concerning the broadening of the electron DF as the altitude increases, we think that a filtration process occurs
in our system and this effect is mainly carried out by
the collisions (besides the attractive potential), therefore
in our case we do not need to inject any energy density in the tails at the base of TR: an electron which is
going towards the corona has a great probability to collide with a hotter target electron, so it could gain energy
and its free path could became very large (the free path
λ v4 ). In this way it will suffer few collisions during its
path and can have a quasi-ballistic motion up to corona,
where the rate of collisions is decreased; so, if this electron has enough energy to overcome the potential well, it
arrives to the upper boundary with large velocities. Particles which do not gain energy with collisions display
a diffusive motion which keeps them for longer times in
the lower regions. As result, in higher regions we mainly
find high energy particles which in turn gives rise an enhancement of temperature.
The figure 2 illustrates very clearly this concept of filter
due to collisions: the two temperature profiles depicted
on the left are obtained from the simulation with two different density values at the lower boundary n o 1010
cm3 (solid line), and n o 5 109 cm3 (dashed line);
the temperature jump is less for n o 5 109 cm3 ; in-
275
ACKNOWLEDGMENTS
We gratefully acknowledge the lectures and notes of André Mangney, the helpfull discussions with Francesco
Malara and the help of Fedele Stabile given for the parallelization of the Monte Carlo code on the AlphaServerSC
cluster.
This work is part of a research programme which is financially supported by the Ministero dell’Università e
della Ricerca Scientifica e Tecnologica (MURST), the
Consiglio Nazionale delle Ricerche (CNR), contract no.
98.00148.CT02 and no. 98.00129.CT02, the Agenzia
Spaziale Italiana (ASI), contract no. ARS98-82.
FIGURE 1. On the left: iterations for the moments of the
distribution function, normalized at their value at the lower
boundary of the simulation box. On the right: the distribution
function of the last itaration, displayed for 4 distances above
the solar surface. The dot line is explained in the text.
1*100
6*10-1
6
REFERENCES
1*10
3*10-1
6*105
-1
1*10
1. Haviland J. K., The Solution of Two Molecular flow
Problems by Monte Carlo Method in Methods in
computational Physics, Academic Press, 4 (1965)
2. Mariska J. T., The Solar Transition Region, Cambridge
Astrophysics Series (1992)
3. Scudder J. D., On the causes of temperature change in
inhomogeneus low-density astrophysical plasmas, ApJ,
398 (1992)
4. Scudder J. D., Why all stars should posses circumstellar
temperature inversion, ApJ, 398 (1992)
5. Maksimovic M., Pierrard, V., Lemaire, J. F., A kinetic
model of the solar wind with Kappa distribution functions
in the corona, A&A, 324 (1997)
6. Owocki S. P., Canfield, R. C., The role of non-classical
electron transport in the lower transition region, ApJ, 300
(1986)
7. Spitzer L., Ḧarm, R., Phys. Rev, 89 (1953)
8. Braginskii, S., Transport Processes in a plasma in Reviews
in Plasma Physics, 1 (1965)
9. Lie-Svendsen O., Holzer, E., Leer, E., Electron heat flux
in the solar transition region: validity of the classical
description, ApJ, 525 (1999)
10. Lie-Svendsen O., Leer, E., The electron velocity
distribution in high-speed solar wind: Modelling the
effects of protons, JGR, 105 (2000)
11. Shoub E. C., Invalidity of local thermodynamic
equilibrium for electrons in the solar transition region. I.
Fokker-Planck results, ApJ, 266 (1983)
12. Gray D. R., Kilkenny J. D., Plasma Phys., 22 (1980)
13. Campbell P. M., Transport phenomena in a completely
ionized gas with large temperature gradients, Physical
Review A, 30 (1984)
14. Landi S., Pantellini F. G. E., On the temperature profile
and heat flux in the solar corona: Kinetic simulations,
A&A, 372 (2001)
6*10-2
n
T (K)
3*105
5
1*10
3*10-2
4
6*10
3*104
-2
1*10
6*10-3
1*104
6*103
0*100
3*10-3
2*104
4*104
x (Km)
6*104
8*104
0*100
2*104
4*104
6*104
8*104
x (Km)
FIGURE 2. On the left: two temperature profiles for two values of density no . On the right: two density profiles normalized
to their value no . For both panels solid lines refers to no 1010
cm 3 and dashed ones to no 5 109 cm 3
deed in this case the filter due to collisions is less effective because density is lower (so the medium is less
collisional), then the "cold"electrons from the bottom
have not many possibilities to gain energy, so they reach
the corona with relatively low velocities, and because of
"cold" electrons are more numerous (they represent the
core of the distribution), their density is higher in corona,
as it is displayed in figure 2 on the right.
Clearly, we have made several simplyfing assumptions;
on of these is to not consider electron-proton collisions.
Because of the small mass ratio between electrons and
protons, these collisions are not very effective for trasferring energy between the two species; however, even
if protons do not matter for the energy distribution, they
could be important for the angular distribution and for the
isotropy of the electrons [10]. We have already said that
the presence of protons is taken into account through the
electric field in which electrons move; however, computing the proton distribution function in such a simulation
where particle move in fixed external fields, requires that
the charge neutrality has to be checked at each iteration
step otherwise very huge electric fields could appear in
the system.
276