Fast and slow MHD waves in thermally active plasma slab

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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:

$\frac{{\rho }^{\gamma }}{\gamma -1}\frac{\text{D}}{\text{D}t}\left(\frac{P}{{\rho }^{\gamma }}\right)=-\rho W\left(\rho \mathrm{,}T\right)=L\left(\rho \mathrm{,}T\right)-\Gamma \left(\rho \mathrm{,}T\right)\mathrm{,}$ (1.1)

Here, $\rho$, $T$, and $P$ are the plasma density, temperature and pressure, respectively. The parameter $\gamma$, is for the adiabatic index. In the equation (1) the function $W\text{}\left(\rho \mathrm{,}T\right)\text{}$ describes the net heat-loss function which is the difference between radiative cooling $\text{}L\text{}\left(\rho \mathrm{,}T\right)$ and some heating $\Gamma \text{}\left(\rho \mathrm{,}T\right)$ processes. These rates balance each other in the case of the stationary conditions $\text{}L\text{}\left({\rho }_0\mathrm{,}{T}_0\right)\text{=}\Gamma \text{}\left({\rho }_0\mathrm{,}{T}_0\right)$, or $W\left({\rho }_0\mathrm{,}{T}_0\right)=0$.

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: $a=a_0+a_1\mathrm{,}{a}_1/a_0~\epsilon <<1$, 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:

$B_0\text{}\left(x\right)\text{=}\left\{\begin{array}{cc}{B}_i\mathrm{,}& \left|x\right|\le x_0\mathrm{,}\\ B_e\mathrm{,}& \left|x\right|x_0\mathrm{.}\end{array}\right.$ (1.2)

Here, $B_0$ and $x_0$ are the stationary value of magnetic field strength and the slab half-width, respectively. Searching the solution of the linearized equation in the form $a_1={\tilde{a}}_1\left(x\right)e^{i\left(\omega t+k_{z}z\right)}$, 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:

$(k_z^2{c_{\text{A}}^2}_i-{\omega }^2)\frac{k_{x_e}}{k_{x_i}}=-\frac{{\rho }_{0_e}}{{\rho }_{0_i}}\left(k_z^2{c_{\text{A}}^2}_e\text{}-{\omega }^2\right)\text{}\left(\begin{array}{c}coth\left(k_{x_i}{x}_0\right)\\ tanh\left(k_{x_i}{x}_0\right)\end{array}\right)\text{}\mathrm{.}$ (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:

${\tau }_V=C_V/W_{0T}\mathrm{,} {\tau }_P=C_P{T}_0/\left(W_{0T}{T}_0-W_{0\rho }{\rho }_0\right)\mathrm{,}$ (1.4)

where $C_V$ and $C_P$ are the specific heat capacities at constant volume and constant pressure, respectively. We also use the derivatives $W_{0T}=\partial Q/\partial T{|}_{{\rho }_0\mathrm{,}{T}_0}\mathrm{,}{W}_{0\rho }=\partial Q/\partial \rho {|}_{{\rho }_0\mathrm{,}{T}_0}$

Some characteristic speeds of fast and slow MHD waves are shown below:

$c_{\text{A}}=\sqrt{\frac{B_0^2}{4\pi {\rho }_0}}\mathrm{,} c_{\text{S}}=\sqrt{\gamma \frac{k_{\text{B}}{T}_0}{m}}\mathrm{,} c_{\text{SQ}}=\sqrt{\frac{{\tau }_V}{{\tau }_P}\gamma \frac{k_{\text{B}}{T}_0}{m}}\mathrm{,} c_{\text{T}}=\sqrt{\frac{c_{\text{S}}^2{c}_{\text{A}}^2}{\left(c_{\text{A}}^2+c_{\text{S}}^2\right)}}\mathrm{,} c_{\text{TQ}}=\sqrt{\frac{c_{\text{A}}^2{c}_{\text{SQ}}^2}{\left(c_{\text{A}}^2+c_{\text{SQ}}^2\right)}}\mathrm{.}$ (1.5)

Here, speed $c_{\text{A}}$ is for the Alfven speed. The speed $c_{\text{S}}$ is the usual speed of sound for ideal plasma. The so-called tube speed $c_{\text{T}}$ is a result of the pure wave-guide dispersion effect. The speed $c_{\text{SQ}}$ 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 $c_{\text{TQ}}$, which is a result of the both wave-guide and thermal activity dispersion effects.

In the dispersion relation (3) we also use following notations:

$k_{x_{i\mathrm{,}e}}^2=\frac{\left(A_{{\text{Q}}_{\text{i,e}}}^2\text{}{m}_{{\text{Q}}_{\text{i,e}}}^2+i\omega {{\tau }_V}_{i\mathrm{,}e}{A}_{i\mathrm{,}e}^2{m}_{i\mathrm{,}e}^2\right)}{\left(A_{{\text{Q}}_{\text{i,e}}}^2+i\omega {{\tau }_V}_{i\mathrm{,}e}{A}_{i\mathrm{,}e}^2\right)}\mathrm{,}$

$m^2=\frac{\left(k_z^2{c}_{\text{A}}^2-{\omega }^2\right)\left(k_z^2{c}_{\text{S}}^2-{\omega }^2\right)}{\left(c_{\text{A}}^2+c_{\text{S}}^2\right)\left(k_z^2{c}_{\text{T}}^2-{\omega }^2\right)}\mathrm{,} m_{\text{Q}}^2=\frac{\left(k_z^2{c}_{\text{A}}^2-{\omega }^2\right)\left(k_z^2{c}_{\text{SQ}}^2-{\omega }^2\right)}{\left(c_{\text{A}}^2+c_{\text{SQ}}^2\right)\left(k_z^2{c}_{\text{TQ}}^2-{\omega }^2\right)}\mathrm{,}$

$A^2=\left(c_{\text{A}}^2+c_{\text{S}}^2\right)\left(k_z^2{c}_{\text{T}}^2-{\omega }^2\right)\text{}\mathrm{,} A_{\text{Q}}^2=\left(c_{\text{A}}^2+c_{\text{SQ}}^2\right)\left(k_z^2{c}_{\text{TQ}}^2-{\omega }^2\right)\text{}\mathrm{.}$

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.

 

Table

Slab parameters used for calculations

Таблица

Параметры слоя, используемые для расчетов

Parameter	Value
Magnetic field strength inside the slab $\left(B_{0_i}\right)$	100 G
Temperature $\left(T_{0_i}\right)$	6 MK
Number density inside the slab $\left(n_{0_i}\right)$	1011 cm
Density contrast $\left(n_{0_i}/n_{0_e}\right)$	10
Slab width $\left(2x_0\right)$	2 Mm

 

The parameterization of the radiative cooling function $L\left(\rho \mathrm{,}T\right)=\chi \rho T^{\alpha }\mathrm{,}$ has been calculated using the CHIANTI Version 10.0 database [9]. The heating rate has been modeled as $\Gamma \left(\rho \mathrm{,}T\right)=h{\rho }^{0.5}{T}^{-3.5}$. 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.

 

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

 

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

 

It can be easily seen that the slow modes in the thermally active plasma can be found between sound speed $c_{\text{Si}}$ and the modified tube speed $c_{\text{TQi}}$. In the ideal plasma case, the long-wavelength limit is $c_{\text{Ti}}$. The fast modes in the plasma with the thermal misbalance range between $c_{\text{Ae}}$ and $c_{\text{Ai}}$. 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.

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 long-wavelength becomes equal to $c_{\text{TQ}}$, 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 $c_{\text{T}}$ 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).

About the authors

Daria 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, Samara; Samara

Sergey 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, Samara; Samara

Nonna 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, chief researcher of the Theoretical Department; Professor of the Department of Physics

Russian Federation, Samara; Samara

Dmitrii I. Zavershinskii

Samara National Research Universit; 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, Samara; Samara

References

Supplementary files