Tuesday, 13 May 2014

The Solar Global Eigenmodes of Oscillation

This post presents work we have undertaken to understand the dynamics generated in the solar atmosphere by the solar global oscillating eigenmode. We describe a model for studying dynamics in the solar atmosphere and describe simulations performed using this model. The results provide evidence of induced dynamics in the solar atmosphere.

The objectives of this work are to;
  • Demonstrate code validity by repeating earlier hydrodynamic 2D models of Malins and Erdelyi,
  • demonstrate cavity modes in the Chromosphere and surface modes in the transition region and
  • understand the dynamics allowing mode tunneling and energy leakage into the corona.
The Earliest observations of dynamical motions of the solar surface dates back to around 1918, (St.John C.E. et al, from Doppler Shift when studying solar rotation). In 1960 Leighton, reported observed Vertical motions of 300-400m/s, these were explained by Ulrich 1970. The solar p modes are generated by global resonant oscillations and turbulent motions just beneath the photosphere.  Modes which trapped below the photosphere are refracted by a sharp change in density. There is  a power peak at 5 – minutes these are p mode oscillations.  The picture below shows an example of a power spectrum obtained from 144 days the MDI Medium-l data for the modes averaged over the azimuthal order m. The power concentrates in ridges corresponding to solar acoustic (p) modes. The lowest weak ridge corresponds to the fundamental (f) mode.


It is found that  modes are evanescent around the photosphere, propagation into the corona is inhibited. However, modes can tunnel through and propagate into the solar atmosphere. Earlier work of Malins and Erdelyi using 2D hydrodynamical models to study dynamics with point drivers demonstrated cavity modes in the Chromosphere. There are also surface modes at the transition region. It is interesting to understand dynamics which enable tunneling.  Local acoustic cut-off is a natural period, disturbances at the cut off can cause dynamic responses. Propagation occurs below the cutoff, there is evanescence above the cut-off.  As demonstrated by Schmitz, 1998, the Cutoff can be calculated in different ways and for an Isothermal atmosphere or a highly stratified atmosphere. For the simulations undertaken here we represent oscillation modes as a vibrating membrane driver located at temperature minimum. We ensure that the simulation is constructed such that the driver delivers the same amount of energy for different modes.

The plots below display the computed cut-off for atmosphere models  


In the context of oscillating systems, the eigenmodes (or normal modes) of that system refers to the sinusoidal motion with which the  system oscillates. In particular all parts of the system move with the same frequency and phase relation. This motion of the normal mode is termed a resonance. Consider a solution for an oscillating membrane


where
Since the solution satisfies the wave equation we have the following relation for the modes of oscillation

The total membrane energy is obtained by integrating over its surface, this may be expressed as

Here we have made the wave speed the same as the speed of sound and we use the expression for an adiabatic gas

If we assume that the total membrane energy for all modes is the same as the fundamental 00 mode i.e.
then evaluation of the surface integral above the relationship between the mode amplitudes

Expressing the membrane solution using the wave number

The wave number can be expressed as
Here we have used
Using c=8.4km/s (speed of sound at the photosphere) this can be calculated from the VALIIIc data

We can calculate the frequency of oscillation using
For each mode of oscillation we keep the ratio
we also keep the energy fixed by using the relationship for Anm above

A visualisation of such a membrane with oscillations described by the above relations is shown in the animation below illustrating the 1,1 mode

This was obtained using the scilab script membrane_modes.sce which is available on github.

Simulations were performed using the stratified MHD code enabled for GPUs (SMAUG).  For the computational Model we used a Model of the stratified solar atmosphere using data sets from VALIIIC and McWhirter.  The model height is a 6Mm through the atmosphere with a cross section of 4Mmx4Mm. The computational domain was divided into 128x128x128 equal computational elements. Simulations were run with membrane drivers, with periods of 30s,180s,300s these were run for the 00,01 and 02 modes.

To study the energy leakage and the evolution of the membrane oscillations we present simulation results showing a view through the simulation box for the vertical component of the velocity. We also present plots showing the integral of the total energy for each time step through the solar atmosphere. The integration of the total energy is performed over the model cross section and at each height of the computational domain. Since most of the energy is trapped near the photosphere, we plot the energies for the transition region and into the Corona.

Results for the 30s driver with 0,1 mode are shown below




Results for the 180s driver with 0,1 mode are shown below




Results for the 300s driver with 0,1 mode are shown below





The results above show that the 180s and 300s modes provide energy leakage  with period the same as that of the driver, the 30s driver is the least efficient at coronal  energy leakage. The results also provide evidence for non linear behaviour. Using the formula

we compute the frequencies for the normal modes, for the 00, 01 and 03 mode. For the 00 mode a 670s driver was used, for the 01 mode a 430s driver was used and for the 03 mode a driver with a period of 230s was used. For these normal modes of oscillation we obtain.





The image above shows distance time Plots for 0,1 mode from left to right 30s, 180s, 300s for the vertical component of the velocity. It is clearly seen that the 30s mode does not support oscillations in the corona. The 180s and 300s modes support standing modes in the transition region and the corona. It is informative to summarise this collection of results by presenting plots which show the energy deposited into the corona and transition zone. The energies have been integrated over these regions of height and also over time, an average was taken over the times. For computing the integration the chromosphere is from the base of the simulation box to a height of 1.78Mm,  the transition region is between 1.78 and 2.16Mm and the corona is the remainder of the box above 2.16Mm. For the integrated energies we compute the difference between the total energy and the background energy, i.e. the perturbed energy,  these integrated energies are shown in the plots below for the different modes and different driver frequencies.




The 180s fundamental is effective at leaking energy into the transition region but energy leaks from the atmosphere  and back to the transition region. The low values for the 30s driver are related to the cutoff? For the fundamental mode the 300s driver appears to be the most effective at enabling energy leakage into the atmosphere. Not all wave energy of the driver will go to the corona. The drivers can excite surface waves in the TR region, waves can be reflected from the TR region.

Conclusions

  • The GPU code performs well 
  • The results provide evidence for energy leakage into the corona
  • The fundamental mode 5 min driver (300s) is effective for energy supply to the corona region and the 3 min driver is effective for TR
  • 01 mode  5 min nothing for corona and 3 min again effective for TR
  • 02 mode 5 min nothing for corona and 3 min effective for corona
  • The distance time plots illustrate cavity modes in the chromosphere
  • For the 300s and 180s drivers there is a clear indication  of induced dynamics in the corona
  • There are unexplained resonances e.g. for 30s driver probably resulting from non-linear behaviour
  • Further characterisation of the normal modes and run models with a greater range of modes of oscillation
Further work will consider genuine MHD examples for a vertical B field, a horizontal B field and for flux tubes.

References


We also consider magnetic effects referiing to 


Magnetohydrodynamic waves driven by p-modes Elena Khomenko, Irantzu Calvo Santamaria

 Simulations of the Dynamics Generated in the Solar Atmosphere by Solar Global Oscillating Eigenmodes (slides for NAM2014, June 2014, Portsmouth Univesity)

F.Schmitz and B. Fleck Astronomy and Astrophysics, v.337, p.487-494 (1998)





Course Notes on Oscillations and Waves




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