FAST AND SLOW MHD WAVES IN THERMALLY ACTIVE PLASMA SLAB


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Abstract

We considered the combined influence of the thermal activity and the magnetic structuring on properties of the compressional magnetohydrodynamic (MHD) waves. To model MHD waves we use the single magnetic slab geometry. To derive dispersion equations for the symmetric (sausage) and anti-symmetric (kink) waves, we apply the assumption of strong magnetic structuring. In our calculations we use parameters corresponding to the highly magnetized coronal loop. The thermal activity leads to the changes in the phase velocity and in the wave increment/decrement. We show that the spatial scales where the dispersion effects caused by the thermal activity is most pronounced are longer than the geometry dispersion spatial scale. The thermal activity and wave-guide geometry have comparable effect on the slow-waves phase velocity dispersion. However, the main source of the phase velocity dispersion for the fast MHD waves remains the wave-guide geometry. We also show that the damping of slow MHD waves caused by the thermal activity is greater than the damping of fast MHD waves.

Full Text

Introduction
The solar atmosphere is the rarified and highly-magnetized plasma. Due to this fact, there are numerous
magnetic structures, which exist in the solar corona including coronal loops, prominences, polar plumes and
etc. In fact, these magnetic structures play the role of the wave-guides for the compressional and compressional
MHD waves, which routinely observed in the solar corona [1; 2]. The geometry of these wave-guides is
prescribed by the balance between the gas-dynamic and magnetic forces and lead to the dispersion of the
compressional modes.
However, the magnetic structures owe their existence not only to mechanical but also to the thermal
balance. The coronal cooling/heating rates are temperature and density dependent. As a consequence, some
compressional perturbation can disturb balance between these processes, and can be amplified or damped by
the additional impact from the non-adiabatic processes. Thus, there are some feedback between waves and
plasma. In other word, the coronal plasma is the thermally active one [3; 4]. Furthermore, the non-adiabatic
processes affect propagation speed of the compressional perturbation [5].
Further, we analyze the combined influence of the thermal activity and the magnetic structuring on
properties of the compressional magnetohydrodynamic (MHD) waves propagating in the highly magnetized
plasma. Most similar studies use strong limitations on the wavelength and the magnitude of the external
magnetic field [6; 7]. In our study, we will not use such strong assumption and apply the most general
approach.
The paper is organized as follows. In the Section 2, we introduce the initial equations and obtained
dispersion relation. In the Section 3, we show calculated dispersion curves. In the Section 4, we discuss the
obtained results.
1. Model and dispersion relation
We analyze the waves in the fully ionized optically thin coronal plasma. We neglect the gravitational
forces and the effects of viscosity, thermal conductivity and finite magnetic conductivity. The difference of
basic equations from equations describing the ideal plasma is in the energy transport equation:
122
Agapova D.V. et al. Fast and slow MHD waves in thermally active plasma slab
Агапова Д.В. и др. Быстрые и медленные МГД-волны в термически активном плазменном слое

− 1
D
Dt
(
P

)
= −W(; T) =L(; T) − ????(; T) ; (1.1)
Here, , T, and P are the plasma density, temperature and pressure, respectively. The parameter , is
for the adiabatic index. In the equation (1) the function W(; T) describes the net heat-loss function which
is the difference between radiative cooling L(; T) and some heating ????(; T) processes. These rates balance
each other in the case of the stationary conditions L(0; T0) = ????(0; T0), or W (0; T0) = 0 .
To analyze the properties of the MHD waves we apply the standard methods of the perturbation theory. In
other words, we linearize basic equations using substitution of the following form: a = a0+a1; a1=a0 ∼ " << 1 ,
where a is some plasma parameter. We model the coronal wave-guide by the magnetic slab with the magnetic
field directed along z -axis as follows:
B0(x) =
{
Bi; |x| 6 x0;
Be; |x| > x0:
(1.2)
Here, B0 and x0 are the stationary value of magnetic field strength and the slab half-width, respectively.
Searching the solution of the linearized equation in the form a1 = ea1 (x) ei(!t+kzz), one may obtain dispersion
relation for the set (fast/slow and body/surface) of the sausage/kink magnetoacoustic waves in the magnetic
slab as follows:
(k2
zc2
Ai
− !2)
kxe
kxi
=−0e
0i
(
k2
zc2
Ae
− !2)(
coth (kxix0)
tanh (kxix0)
)
: (1.3)
The complete derivations steps can be found in our previous study (see [8]). We use the and hyperbolic
functions for the kink and sausage modes, respectively. We introduce the following characteristic temporal
scale to describe the thermal activity influence:
V = CV =W0T ; P = CP T0= (W0T T0 −W00) ; (1.4)
where CV and CP are the specific heat capacities at constant volume and constant pressure, respectively.
We also use the derivatives W0T = @Q=@T|0;T0 ;W0 = @Q=@|0;T0
Some characteristic speeds of fast and slow MHD waves are shown below:
cA =

B2
0
40
; cS =


kBT0
m
; cSQ =

V
P

kBT0
m
; cT =

c2
Sc2
A
(c2
A + c2
S)
; cTQ =
vuut
c2
Ac2
( SQ
c2
A + c2
SQ
): (1.5)
Here, speed cA is for the Alfven speed. The speed cS is the usual speed of sound for ideal plasma. The
so-called tube speed cT is a result of the pure wave-guide dispersion effect. The speed cSQ is long-wavelength
limit value of the slow-wave or acoustic perturbation in the case the thermally active uniform plasma. And
the last but not the least is the speed cTQ, which is a result of the both wave-guide and thermal activity
dispersion effects.
In the dispersion relation (3) we also use following notations:
k2x
i;e =
(
A2
Qi;e
m2
Qi;e
+ i!V i;eA2
i;em2
i;e
)
(
A2
Qi;e
+ i!V i;eA2
i;e
) ;
m2 =
(
k2
zc2
A
− !2
) (
k2
zc2
S
− !2
)
(c2
A + c2
S) (k2
zc2
T
− !2)
;m2
Q =
(
k2
zc2
A
− !2
) (
k2
zc2
SQ
− !2
)
(
c2
A + c2
SQ
) (
k2
zc2
TQ
− !2
);
A2 =
(
c2
A + c2
S
) (
k2
zc2
T
− !2)
;A2
Q =
(
c2
A + c2
SQ
) (
k2
zc2
TQ
− !2)
:
Further, we will use dispersion relation (3) to calculate the dependencies of phase velocities and
increment/decrement of MA waves on the wavenumbers in the solar atmosphere conditions.
2. Results
Let us apply the obtained dispersion relation to the coronal conditions. In the current study we will focus
on properties of waves in the highly magnetized coronal loop. The magnetic slab parameters corresponding
to the mentioned loop are shown in Table.
The parameterization of the radiative cooling function L(; T) = T; has been calculated using the
CHIANTI Version 10.0 database [9]. The heating rate has been modeled as ????(; T) = h0:5T????3:5. Such heating
scenario has been seismologically proposed by [10] using the observations of the damped slow magnetoacoustic
waves in the long-lived coronal plasma structures. The dispersion curves for body fast/slow sausage/kink
magnetoacoustic waves are shown in Figures 1, 2.
Вестник Самарского университета. Естественнонаучная серия. 2022. Том 28, № 1–2. С. 120–127
Vestnik of Samara University. Natural Science Series. 2022, vol. 28, no. 1–2, pp. 120–127 123
Table
Slab parameters used for calculations
Таблица
Параметры слоя, используемые для расчетов
Parameter Value
Magnetic field strength inside the slab (B0i ) 100G
Temperature (T0i ) 6MK
Number density inside the slab (n0i ) 1011 cm????3
Density contrast (n0i=n0e ) 10
Slab width (2x0) 2Mm
Fig. 2.1. Phase velocities of sausage and kink waves in the highly magnetized coronal plasma (see Table). Fast
and slow waves are shown on different spatial scales. The range where the scale is changing is indicated by saw
teeth. The top and bottom panels are for the sausage and the kink modes, respectively. We use different colours
for different modes. The approximate position where the dispersion effect of the slow waves is the most
pronounce is indicated by star. The range of speeds where is no roots corresponding to MHD waves can be
found are shown by grey dashing
Рис. 2.1. Фазовые скорости волн перетяжек и изгибных волн в сильно замагниченной корональной плазме
(см. таблицу). Быстрые и медленные волны показаны в разных пространственных масштабах. Диапазон
изменения шкалы обозначен пилообразным символом. Верхняя и нижняя панели предназначены для волн
перетяжек и изгибных волн соответственно. Мы используем разные цвета для разных мод. Приблизительное
положение, в котором дисперсионный эффект медленных волн наиболее заметен, обозначено звездой. Серым
пунктиром показан диапазон скоростей, в котором не могут быть найдены корни,
соответствующие МГД-волнам
It can be easily seen that the slow modes in the thermally active plasma can be found between sound
speed cSi and the modified tube speed cTQi. In the ideal plasma case, the long-wavelength limit is cTi. The
fast modes in the plasma with the thermal misbalance range between cAe and cAi. The slow waves are
highly affected by both thermal activity and wave-guide dispersion. The impact of the thermal activity on
the fast-wave dispersion is negligible.
124
Agapova D.V. et al. Fast and slow MHD waves in thermally active plasma slab
Агапова Д.В. и др. Быстрые и медленные МГД-волны в термически активном плазменном слое
Fig. 2.2. Decrement of sausage and kink waves in the highly magnetized coronal plasma (see Table). The top
panel corresponds to the sausage modes. The bottom panel is the for kink modes. Different colours correspond to
different modes
Рис. 2.2. Декремент волн перетяжек и изгибных волн в сильно замагниченной корональной плазме
(см. таблицу). Верхняя панель соответствует волнам перетяжек. Нижняя панель предназначена для изгибных
волн. Разные цвета соответствуют различным модам
One may notice that decrement of both fast sausage and kink modes are lower than decrement of slow
waves. This effect is in agreement with result obtained for the uniform plasma [4]. However, the magnetic
structuring leads to non-monotonic behavior of wave decrement, with maximum in the long-wavelength part
of the spectrum. In uniform plasma behavior was shown to be monotonic [4]. The calculated decrement
of rather weak and cannot explain observed fast wave damping. The slow-wave decrement is comparable
with the observed decay time. Moreover, in highly magnetic plasma decrement of slow-waves become greater
(compare with results obtained for the hot loop in [8]. This is due to the fact that with decrease of plasma
beta/increase of magnetic field the slow wave becomes more acoustic than magnetic mode.
3. Discussion and conclusions
In the presented study we analyzed the combined influence of the thermal activity and the magnetic structuring on properties of the compressional magnetohydrodynamic (MHD) waves. Using perturbation theory, assumption of strong magnetic structuring and the slab geometry, we obtain the dispersion relation for the set (fast/slow and body/surface) of the sausage/kink magnetoacoustic waves. We solve the obtained dispersion relation numerically and use to the higly-magnetized solar corona conditions. We showed that slow-waves are higly affected by both thermal activity and wave-guide dispersion. In particular the longwavelength becomes equal to cTQ, which is defined not only by geometry of the wave-guide but also by the acting non-adiabatic processes. As a result, the usage of the value cT for the helioseismological needs may cause mistakes. On the contrary, the oscillation of the fundamental modes may be used for phenomenological determination of unknown coronal heating function. We also showed the phase velocity dispersion for the fast MHD waves remains the wave-guide geometry. In the magnetically structured plasma the wave decrement becomes non-monotonic with maximum in the long-wavelength part of the spectrum. The calculated slow wave decrements are comparable with the observed decay times.
The study was supported in part by the Ministry of Education and Science of Russia by State assignment to educational and research institutions under Project No. FSSS-2020-0014 and No. 0023-2019-0003, and by RFBR, project number 20-32-90018. CHIANTI is a collaborative project involving George Mason University, the University of Michigan (USA), University of Cambridge (UK), and NASA Goddard Space Flight Center (USA).

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About the authors

D. V. Agapova

Samara National Research University;
Samara branch of P.N. Lebedev Physical Institute of the Russian Academy of Sciences

Author for correspondence.
Email: agapovadaria2019@gmail.com
ORCID iD: 0000-0002-3957-7339

Master degree student of the Department of Physics; Engineer of the Theoretical Department

Russian Federation, 34, Moskovskoye shosse, Samara, 443086, Russian Federation; 221, Novo-Sadovaya Street, Samara, 443011, Russian Federation

S. A. Belov

Samara National Research University; Samara branch of P.N. Lebedev Physical Institute of the Russian Academy of Sciences

Email: mr_beloff@mail.ru
ORCID iD: 0000-0002-3505-9542

postgraduate student of the Department of Physics; Research associate of the Theoretical Department

Russian Federation, 34, Moskovskoye shosse, Samara, 443086, Russian Federation; 221, Novo-Sadovaya Street, Samara, 443011, Russian Federation

N. E. Molevich

Samara National Research University; Samara branch of P.N. Lebedev Physical Institute of the Russian Academy of Sciences

Email: nonna.molevich@mail.ru
ORCID iD: 0000-0001-5950-5394

Doctor of Physical and Mathematical Sciences, professor of the Department of Physics; chief researcher of the Theoretical
Department 

Russian Federation, 34, Moskovskoye shosse, Samara, 443086, Russian Federation; 221, Novo-Sadovaya Street, Samara, 443011, Russian Federation

D. I. Zavershinskii

Samara National Research University; Samara branch of P.N. Lebedev Physical Institute of the Russian Academy of Sciences

Email: d.zavershinskii@gmail.com
ORCID iD: 0000-0002-3746-7064

Candidate of Physical and Mathematical Sciences, associate professor of the Department of Physics; Research associate of the Theoretical Department

Russian Federation, 34, Moskovskoye shosse, Samara, 443086, Russian Federation; 221, Novo-Sadovaya Street, Samara, 443011, Russian Federation

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Copyright (c) 2022 Agapova D.V., Belov S.A., Molevich N.E., Zavershinskii D.I.

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