## Estudio de las propiedades de las ondas en simulaciones de magneto-convección

##### Author

Larrodera Baca, Carlos##### Date

2018##### Abstract

The study of the sun was based on the analysis of images obtained using ground
based telescopes and, later, using space based telescopes. Through these images we
could analyze the observed structures (for example, sunspots), or also the polarization
of the sun light using spectropolarimeters. Over the years, the telescopes and the tools
for the analysis have been improving looking for a better spatial resolution and magnetic
field sensitivity, allowing to solve the smallest structures observed to better determine
their characteristics.
The theoretical framework was developed, leading to the equations which define the
movement, and its properties, of a plasma with a magnetic field, to result in the magnetohydrodynamics
equations or MHD equations. This theoretical framework showed the
existence of unknown features, like Alfv´en waves or the transformation between waves
when passing through the equipartition layer with β=1. It was this frame, together
with the computer development, which allowed the growing of the numerical simulations,
which are able to solve the MHD equations to simulate the behavior of the sun
and reproduce the observed structures.(Sec:2)
In this work we have analyzed the result of a simulation made with the MANCHA
3D code (Khomenko & Collados, 2006), who solved the MHD equations with a realistic
equation of state and radiative transfer in a grey atmosphere. The domain of the simulation
covers 5, 8 × 5, 8 × 1, 6 Mm3
, located in such a way that the z axis covers from a
depth z=0,95 Mm below the surface to z=0,62 Mm above it.
In the article that describes the simulation, Khomenko et al. (2017) analyze a model
based on the action of the Biermann battery effect. This effect generates a magnetic
field due to the local imbalance of the electronic pressure in the partially ionized solar
plasma. It is shown that the battery effect by itself is able, to create a seed magnetic
field with a strength around µG, and together with the amplification of the dynamo
mechanism, they allow the generation of a magnetic field with an average strength
similar to that obtained from the observations. The generated magnetic field represents,
the distribution of the quiet sun with a average value of < B >= 100 G. During the
simulation, one 3D snapshot has been saved each 0,4 s, with a total time of 79 s. The
grid has 20 km horizontally and 14 km vertically, with dimensions of 5, 8 × 5, 8 × 1, 6
Mm3
, creating 199 data cubes with a size of 180 GB.
The importance of this type of simulations is the high frequency of saving the snapshots,
in this case each 0,4 s, which allow to study the behavior of the waves in the
interval between 100-200 mHz, while observations currently only reach up to 10-20
mHz. This type of waves have the property that they are not affected by the acoustic
cut-off frequency, so they can spread to the upper layers of the atmosphere. Because
they have a shorter wavelength, the theoretical models that are generally based on
homogeneous or slightly stratified situations serve better to explain their behavior.
The analysis of the data has been based on the study of the compressible and incompressible
waves at a given height of z=0,31 Mm above the surface to avoid the
contamination by the convection, through the Fourier temporal transform of ∇ × ~ ~v
and ∇ · ~ ~v as representative magnitudes of the compressible and incompressible waves
respectively. We have also analyzed the relation between these magnitudes and the physical
magnitudes such as the magnetic field strength (B~ ), temperature (T) and azimuth
(φ) in order to establish possible phase differences, useful to localized these kind of
waves.(Sec:3)
The simulation represents a region below and above the surface. With the data, we
have calculated the value of β for the height of study and also in the surface, and in
both layers, using a histogram we can see that the value of β is much greater than
1.(Sec:2.4)
All the analysis has been carried out in two different intervals of frequency, low and
high, (0,013-0,1 Hz y 0,11-0,2 Hz), chosen in such a way that there is a complementarity
in temporal power, so that the distribution of power in the low frequency range is
anticorrelated with the power in the high frequency range.(Sec:4.1)
In the study of power maps, we have shown that the temporal evolution of the
incompressible waves is defined by the fluctuation of the magnetic tension, but for
the compressible waves the representative magnitude is the fluctuation of the total
pressure.(Sec:4.2)
In the last part, we have studied the differences between the phases of the Fourier
transform of ∇ × ~ ~v and ∇ · ~ ~v, and those of B~ , T and φ. With this analysis, we want
to determine which phase difference may be most useful to determine the existence of
one or other kind of waves. So, for example if we want to study the differences between
the compressible waves and the temperature, we begin by analyzing the difference in
all the points of the maps of power distribution (from now on we call it “Total”case).
Then we select those points where the power is larger than a threshold in both maps
to obtain the most representative points. We call this the “Mask”case.
For the low frequencies, the magnitude that can serve as an indicator of waves is the
temperature. For the phase shift ΦI − ΦT , the histograms in both cases, “Mask”and
“Total”, appear to be most different. For the high frequencies, this method seems more
effective and we obtained results which are different from zero, not only with T, but
also with B~ and φ. In the “Mask”case, for the phase shift ΦI − ΦB, the histogram has
two peaks around -30o and 40o
, and in the one of ΦC − ΦB, 3 peaks appear at -80o
,0o
and 70o
. When comparing with φ, is representative the histogram of the phase shift
ΦC − Φφ in the “Total”case, where a peak appears in the positive part (around 20o
)
and a tail in the negative part, and when we study the “Mask”case, both characteristics
are reinforced. And for the comparison with T, we obtained significant differences in the
“Total”case for both phase shift histograms, ΦI −ΦT and ΦC −ΦT . In the “Mask”case,
the peak for the histogram of ΦI − ΦT is reinforced around 80o
, while the phase shift
ΦC − ΦT unfolds into two peaks, one near 0o and the other around 110o
(Sec:4.3).