Numerical MHD modelling of waves in solar coronal loops

1
Numerical MHD modelling of waves in solar coronal
loops
Petr Jelínek1,2 and Marian Karlický2
1University of South Bohemia, Department of
PhysicsCeské Budejovice, Czech
Republic2Astronomical Institute, Academy of
Sciences of the Czech Republic Ondrejov, Czech
Republic
2
Ceské Budejovice

• Small university town in the South of Bohemia
  with 100 000 inhabi-tants
• Approximately 200 km from Wien and 100 km from
  Linz

http//www.c-budejovice.cz/cz

• University of South Bohemia 7 faculties and 2
  research institutes, about 12 000 students

http//www.jcu.cz/
3
Ondrejov

• Small village outlying 30 km from Prague
  http//www.obecondrejov.cz/

• Astronomical Institute, Academy of Sciences of
  the Czech Republic
• Founded in 1898, 4 main scientific departments

http//www.asu.cas.cz/
4
Outline

• Motivation of numerical studies
• Equations of magnetohydrodynamics (MHD)
• Numerical methods solutions
• Results
• 1D model impulsively generated acoustic waves
• 1D model gravitational stratification
• 2D model impulsively generated acoustic waves
• 2D model modelling of wavetrains
• Conclusions

5
Motivation of numerical studies I.

• Oscillations in solar coronal loops have been
  observed for a few decades
• The importance of such oscillations lies in their
  potential for the diagnostics of solar coronal
  structure (magnetic field, gas density, etc.)

• The various oscillation modes in coronal loops
  were observed with highly sensitive instruments
  such as SUMER (SoHO), TRACE
• The observed oscillations include propagating and
  slow magnetosonic waves. There are also
  observations of fast magnetosonic waves, kink and
  sausage modes of waves

6
Motivation of numerical studies II.

• Coronal loop oscillations were studied
  analytically but these studies are unfortunately
  applicable only onto highly idealised situations
• The numerical simulations are often used for
  solutions of more complex problems these
  studies are based on numerical solution of the
  full set of MHD equations

• Mentioned studies of coronal loop oscilla-tions
  are very important in connection with the problem
  of coronal heating, solar wind acceleration and
  many unsolved problems in solar physics
• Magnetohydrodynamic coronal seismology is one of
  the main reasons for studying wa-ves in solar
  corona

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MHD equations

• In our models we describe plasma dynamics in a
  coronal loop by the ideal magnetohydrodynamic
  equations

• The plasma energy density

• The flux vector

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Numerical solution of MHD equations

• The MHD equations (1) (4) are transformed into
  a conservation form

• For the solution of the equations in conservation
  form exist many numerical algorithms including
  professional software such as NIRVANA, ATHENA,
  FLASH, .... (www.astro-sim.org)

9
Numerical methods I.

• There exist a lot of numerical methods used for
  the solution of equations in conservation form in
  numerical mathematics
• Generally we can use the two types of numerical
  methods
• explicit methods calculate the state of a
  system at a later time from the state of the
  system at the current time
• ? easy to programming
• ? unstable in many cases
• implicit methods find the solution by solving
  an equation involving both the current state of
  the system and the later one
• ? unconditionally stable
• ? difficult to programming (tridiagonal matrix
  solution by Thomas algorithm)

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Numerical methods II.

• We use only explicit methods in our calculations
  for this reason we must use the artificial
  smoothing for the stabilisation of the numerical
  scheme
• Some mathematical definitions of numerical
  methods for PDEs
• Consistency the numerical scheme is called
  consistent if
• Convergence the numerical method is called
  convergent if

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Numerical methods III.

• For the solution of the MHD equations in a
  conservation form the methods of so-called flux
  limiters are used
• These numerical methods are able to jump down the
  oscillations near sharp discontinuities and jumps
• Generally, for the solution of PDE in
  conservation form in 1D we can write

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Numerical methods IV.

• Many authors often use the linear methods
• upwind scheme
• Lax-Wendroff scheme (downwind slope)
• Beam-Warming scheme (upwind slope)
• Fromm scheme (centered slope)

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Numerical methods V.
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Numerical methods VI.

• To avoid the overshoots we limit the slope by
  flux limiter methods
• minmod
• superbee
• MC
• van Leer

• And many others van Albada, OSPRE, UMIST, MUSCL
  schemes

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Numerical methods VII.
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1D model of acoustic standing waves

• There exists a lot of types of oscillations in
  solar coronal loop
• acoustic oscillations
• kink and sausage oscillations
• fast and slow propagating waves, ...
• Acoustic oscillations are easy to simulate, they
  can be modelled in 1D, without magnetic field,
  etc.
• Kink and sausage oscillations were directly
  observed (SOHO, TRACE) and there are many
  unanswered questions excitation and damping
  mechanisms, etc.
• We focused on the impulsively generated acoustic
  standing waves in coronal loops

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1D model initial conditions

• Initial condtitions

• The length of the coronal loop was L 50 Mm
  which corresponds to loop radius about 16 Mm.
• The loop footpoints were settled at positions x
  0 and x L

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1D model perturbations

• Perturbations
• In the view of our interest to study impulsively
  generated waves in the solar coronal loops, we
  have launched a pulse in the pressure and mass
  density
• The pulse had the following form

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1D model numerical solution

• The numerical region was covered by a uniform
  grid with 2 500 cells and open boundary
  conditions that allow a wave signal freely leave
  the region were applied
• The time step used in our calculations satisfied
  the Courant-Friedrichs-Levy stability condition
  in the form

• In order to stabilize of numerical methods we
  have used the artificial smoothing as the
  replacing all the variables at each grid point
  and after each full time step as

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Results 1D model
Time evolution of velocity v(x L/4,t), mass
density r(x L/4,t) (top panels) and spatial
profiles of velocity v(x,Dt), v(x,7.12T1) (bottom
panels) all for mass density contrast d 108,
pulse width w L/40, and initial pulse position
x0 L/4.
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Results 1D model
Time evolution of velocity v(x L/4,t), mass
density r(x L/4,t) (top panels) and spatial
profiles of velocity v(x,Dt), v(x,7.89T2) (bottom
panels) all for mass density contrast d 108,
pulse width w L/40, and initial pulse position
x0L/2.
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Results 1D model
Time evolution of velocity v(x L/4,t), mass
density r(x L/4,t) (top panels) and spatial
profiles of velocity v(x,Dt), v(x,11.00T1)
(bottom panels) all for mass density contrast d
108, pulse width w L/40, and initial pulse
position x0 L/50.
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Results 1D model
Time evolution of average pressure, increased by
the factor 103, initial pulse position x0 L/4
(left top panel), x0 L/2 (right top panel) and
x0 L/50 (bottom panel), mass density contrast d
108 and pulse width w L/40 note that x-axis
is in the logarithmic scale.
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Results 1D model
Fourier power spectra of velocities v for initial
pulse position x0 L/2 (left) and x0 L/4
(right), mass density contrast d 105 (top
panels) and d 108 (bottom panels) and pulse
width w L/40. The amplitude of the power
spectrum A(P) is normalized to 1.
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Results 1D model
Time evolution of total (red), pressure (blue)
and kinetic (green) energies for various
positions in numerical box. Left upper panel
whole simulation region, left upper panel
transition region, bottom panel coronal
region. The initial pulse position x0 L/4, d
108 and pulse width w L/40.
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Results 1D model
Time evolution of total (red), pressure (blue)
and kinetic (green) energies for various
positions in numerical box. Left upper panel
whole simulation region, left upper panel
transition region, bottom panel coronal
region. The initial pulse position x0 L/2, d
108 and pulse width w L/40.
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1D gravitational stratification

• To create more realistic model the gravitational
  stratification was added
• We consider a semi-circular loop with the
  curvature radius RL, in this model we incorporate
  the effect of loop plane inclination the shift of
  circular loop centre from the baseline was omitted

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1D gravitational stratification I.

• The MHD equation of motion has the following form

• The gravitational acceleration at a distance s
  measured from the footpoint along the loop, is

• For the plasma pressure in the loop we can write

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1D gravitational stratification II.

• The temperature profile was calculated by means
  of this formula

• The mass density was calculated from

• The length of the coronal loop was L 100 Mm in
  this case which corresponds to loop radius about
  32 Mm.

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Gravitational stratification first results in 1D
Time evolution of velocity v(x L/4,t), mass
density contrast d 102, pulse width w L/80,
and initial pulse position x0 L/4 and x0 L/2,
inclination angle a 0 (blue line) and a 45
(red line).
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2D modelling of magnetoacoustic standing waves

• We consider a coronal slab with a width w 1Mm
  and mass density ri, embedded in a environment of
  mass density re

• The pressure, mass density, temperature and
  initial pulses in pressure and mass density are
  calculated similarly as in 1D model

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Numerical solution in 2D

• For the solution of 2D MHD equations the
  Lax-Wendroff numerical scheme was used, this
  method is often used for the solutions of MHD by
  many authors

• Step 1

• Step 2

• The stability condition

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Results 2D model
Time evolution of velocity v(x L/4, y 0, t)
(left top panel). Spatial profile of x-component
of velocity vx at time t 8.17 T1 (right top
panel) and the corresponding slices of vx along y
H/2 (x L/2) bottom left (right) panel all
for mass density contrast d 108, pulse width w
L/40, and initial pulse position x0 L/2.
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Results 2D model
Time evolution of velocity v(x L/4, y 0, t)
(left top panel). Spatial profile of x-component
of velocity vx at time t 6.15 T2 (right top
panel) and the corresponding slices of vx along y
H/2 (x L/4) bottom left (right) panel all
for mass density contrast d 108, pulse width w
L/40, and initial pulse position x0 L/4.
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Modelling of wave trains in 2D

• The wave trains were directly observed and
  discovered by SECIS (Solar Eclipse Coronal
  Imaging System)
• Observed in Ondrejov in radio waves
• The theoretical description is needed the
  comparison of observed and modelled tadpoles ?
  what type of waves are present

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Modelling of wave trains in 2D

• We study impulsively generated magnetoacoustic
  wave trains propagating along a coronal loop
• The problem is modelled by means of 2D model
  presented before, but magnetic field is parallel
  to the y axis
• The equilibrium is perturbed by a pulse in
  velocity, situated at L/4 of the numerical domain

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Wave trains first results I.
The spatial profile of the velocity vx at time t
30 s from initial pulse (left upper panel), and
corresponding slices of vx along x axis (y H/2)
(right upper panel) and along y axis (x L/4).
Initial pulse position x0 L/4, mass density
contrast d 108, pulse width w L/40
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Wave trains first results II.
Time evolution of mass density r(x L/2,t), (top
panel) and corresponding wavelet analysis (bottom
panel) all for mass density contrast d 108,
pulse width w L/40, and initial pulse position
x0 L/4.
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Conclusions I.

• Computer modelling seems to be very useful tool
  for the understan-ding of processes in solar
  coronal loops
• The next step in our research will be the
  extension of current model to three dimensions
  (by means of mentioned software Athena,
  Nirvana, FLASH...), including the source terms
  such as cooling term, heating term, gravitational
  stratification, etc.
• By means of this model we could investigate
  effects like attenuation of waves in coronal
  loops, plasma energy leakage by the dissipation
  into solar atmosphere and more very interesting
  problems in solar coronal physics...

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Conclusions II.

• More informations about 1D or 2D models can be
  found in
• Jelínek P., Karlický, M. Numerical Modelling of
  Slow Standing Waves in a Solar Coronal Loop,
  Proc. 12th ESPM, Freiburg, Germany, 2008
• Jelínek, P., Karlický, M. Computational Study of
  Implusively Generated Standing Slow Acoustic
  Waves in a Solar Coronal Loop, Eur. Phys. J. D,
  after revisions.

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References

• 1 M. Aschwanden, Physics of the Solar Corona
  (Springer, Praxis Publ., Chichester UK 2004).
• 2 T. J. Chung, Computational Fluid Dynamics
  (Cambridge University Press, New York USA 2002).
• 3 E. R. Priest, Solar Magnetohydrodynamics (D.
  Reidel Publishing Company, London England 1982).
• 4 M. Selwa, K. Murawski, S. K. Solanki, AA
  436, 701 (2005).
• 5 Tsiklauri, D., Nakariakov, V. M., AA, 379,
  1106 (2001).
• 6 Nakariakov, V. M. et al. Mon. Not. R.
  Astron. Soc., 349, 705 (2004).

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Thank you for your attention