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Modelling the radiative signature of turbulent
heating in coronal loops. S. Parenti1, E.
Buchlin2, S. Galtier1 and J-C. Vial1, P. J.
Cargill31. IAS, Université Paris Sud - CNRS,
FR 2. University of Florence, IT 3. Space and
Atmospheric Physics, Blackett Laboratory,
Imperial College, UK
Introduction In this work we investigate the
statistical properties of a cooling coronal loop
subject to a turbulent heating along the lines of
Cargill (1994) work. We are interested to see if
the statistical properties of the injected energy
are conserved by the radiation produced during
the cooling phase. The potentiality of this kind
of study is in the direct connection between the
theory adopted for the coronal heating and the
real data available from the observations. We
model a coronal loop as composed by n unresolved
identical threads. Each elemental thread has a
half length L, a section area A and at each time
t it is described by one temperature T and one
density N. We simulate the heating-cooling cycle
of each thread and we calculate its radiative
losses. We use this quantity to built synthetic
spectra that we will compare with instrumental
observations, such as SOHO/EIT and SolarB/EIS.
The heating model We use the Buchlin et al
(2003) model to simulate the injection of energy
in the loop system in the corona. This energy
originates from turbulent fluctuations in the
photosphere and propagates in the corona through
Alfvén waves.The dissipation is done through
instantaneous nanoflare events (ltQgt 1.2 1024
ergs) during a period of about 105 s. The
Probability Distribution Function (PDF) of this
energy (Q) is a power law function with index ?
- 1.6. Individual events with energy Q are
randomly distributed in space.
At each instant t, if ?R gt ?c conduction is the
dominant process, whereas if ?R lt ?C radiation
dominates.
?C ? N L2/ T 5/2 ?R ? T 1-? / N ?
?(T) ? T ?
T is the plasma temperature, N is the density,
L is the half length of the loop, ?(T) is the
loss function calculated assuming Mazzotta et al.
98 ionisation equilibrium and coronal abundances
(CHIANTI v. 4.2). The histories of the plasma
parameters T, N during the time of the
simulations are then used for statistical
studies. Some general properties of the radiative
emission of a loop subject to a power law heating
function have been anticipated in Cargill
Klimchuk 04
The cooling model The elemental strand that is
instantaneously heated, cools through conduction
and radiation (Cargill 94). These processes are
governed by their characteristic time scales ?C
and ?R.
Some results
FIGURE 3
FIGURE 4
FIGURE 5
Figure 4 shows the Probability Distribution
Function for the intensities of Figure 3. The
simulations show that this distribution follows a
power law similar to the heating function. This
means that part of the information on the heating
is still present. However, the intensities are
distributed over only about one decade, contrary
to the distribution of the heating.
Parameter L We tested the intensity behaviour
when changing the geometrical parameters of the
elemental strands. These parameters affect the
way the strands cool down and imply a change in
the DEM profile. Figure 5 shows an example where
we have decreased the loop length L with respect
to Figure 4 (L2108 cm, A 81013 cm2). The
index of the power law decreases.
Fit -3.9
The index of the power law has changed. For this
simulation we have assumed A 81013 cm2 and L
109 cm.
FIGURE 6
Parameter A In Figure 6 we have increased the
strand section A with respect to Figure 4 (L
1109 cm and A 81014 cm2). In this case a
power law does not represent any more the
intensity distribution.

• Conclusions
• The results of this work show that the properties
  of the PDF of the energy given as input for our
  loop model are partially conserved during the
  radiation phase.
• The power law distribution of the intensity is
  better conserved when we assume small and thin
  strands.
• The index of the power law changes with the
  assumed fine structure of the loop. Smaller and
  thinner are the strands, more the index gets
  closer to that of the heating distribution.
• Higher is the temperature of the line we
  synthesize, more similar to the heating function
  is the PDF of the intensity.
• We have shown here that the coronal plasma
  response to the heating depends on the unresolved
  fine structure. To understand the plasma
  behavior is an important issue in order to
  properly associate the plasma heating properties
  with those derived from the observations (i.e.
  Aschwanden et al, 2000, Aletti et al. 2000).
• Our work shows that it is important to direct the
  attention to the very hot lines. These are
  emitted just after the heating injection. Here we
  may find direct information on the heating.
• The SolarB/EIS instrument, with its high
  resolution and large wavelength bands, which
  include high temperature lines, will be an
  important source of information for studying the
  nanoflare case.
• References
• Aletti, V et al., 2000, ApJ, 544,
  550 Cargill, P.J. Klimchuk, J.A., 2004,
  ApJ, 605, 911
• Buchlin, E. et al., 2003, AA, 406,
  1061 Mazzotta, P. et al., 1998, AAS, 133, 403
• Aschwanden, M.J. et al., 2000, ApJ, 541,
  1059 Young, P.R. et al., 2003, ApJS, 144, 135
• Cargill, P.J., 1994, ApJ, 422, 381

The temperature parameter Because we are
interested in the heating signature, we have
calculated the intensity PDF for lines emitted at
higher temperatures. Figure 7 shows the example
for Fe XIX 1118 Å (SOHO/SUMER, logT 7) and
Figure 8 for Fe XXI 188 Å (SolarB/EIS, logT7.1).
Note the change in the index of the power law
that gets closer to that of the heating function
for the hottest line.
FIGURE 7
FIGURE 8
Fit -2.0
Fit -1.53