PROCEEDINGS of the 23rd International Congress on Acoustics

9 to 13 September 2019 in Aachen, Germany

On some properties of magnetoacoustic waves in acoustically

active non-adiabatic plasma

Dmitrii ZAVERSHINSKII1,2; Nonna MOLEVICH1,2; Igor ZAVERSHINSKII1;

1 Samara National Research University, Russia,

2 Lebedev Physical Institute, Russia

ABSTRACT

Magnetoacoustic(MA) waves in the plasma with acoustic activity caused by the non-adiabatic processes is

under investigation. Acoustic activity of the medium is a consequence of temperature and density

dependence of heating and cooling process which take place in the medium. It is shown that non-adiabatic

process results in the frequency dependence of group/phase velocity. Effect of frequency dependence is most

pronounced near the frequency defined by the inverted heating/cooling time. In the low-/high-frequency

limits effect of frequency dependence can be neglected. However, in contrast to high-frequency case where

phase/group velocities equal to their value in the equilibrium medium, in the low-frequency limit both

velocities are defined by the non-adiabatic processes only. Furthermore, frequency dependence of group

velocity in contrast to phase velocity has as an extrema which can be maxima or minima depending on type of

dispersion (negative or positive, respectively). This result indicates that some harmonics in non-adiabatic

plasma can propagate faster or slower than all others. The expression for group velocity has been obtained

under the assumption of weak dispersion/dissipation. Effects caused by the weak but finite

dispersion/dissipation on group velocity is analyzed as well.

Keywords: acoustic instability, magnetoacoustic wave, non-adiabatic plasma.

1. INTRODUCTION

Generally, the total internal energy of a gas or plasma includes mainly the energy of the

translational motion of the particles, rotational and vibrational energies of the molecules, the chemical

energy. A certain time is required to excite these degrees of freedom and to establish a thermodynamic

equilibrium in the medium. This time is known as a relaxation time. Depending on the degree of

freedom, this characteristic time can vary significantly. Therefore, the thermodynamic equilibrium

tends to be established on certain degrees of freedom faster than on the others. Namely, the r elaxation

time of the translational degrees of freedom has the shortest value, implying that some initial velocity

distribution of particles becomes Maxwellian after even a few elastic collisions. For particles with

comparable masses, the time required for the Maxwell distribution to be established is of the same

order as the average time between gas kinetic collisions. In the case where different degree s of

freedom are present in particles, a typical time for establishing the full thermodynamic equilibrium is

determined by the slowest relaxation time. The characteristic times of these slow processes can differ

by several orders of magnitude from the translational relaxation time. For instance, in the carbon

dioxide at the room temperature the relaxation time of the translational and vibrational degrees of

freedom is 10-10

s and 10-5

s, respectively (1).

Such a huge difference between various relaxation times allows one to analyze each relaxation

process independently from the others. In other words, one may reckon that the thermodynamic

equilibrium of the degrees of freedom with the shortest relaxation times always exists, while the

relaxation of slower processes would generally depend upon the time scales considered. Thus, the time

characterizing such a “slow” process allows us to subdivide the whole frequency spectrum into two

qualitatively different ranges. Namely, these ranges are where the thermodynamic equilibrium is

achieved for the “fast” processes only (at the high-frequency part of the spectrum), and where the full

1 gf@aaa.com 2 gf@ccc.go.jp [Please note: It is optional to provide the email address(es) of the author(s). Please make sure

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thermodynamic equilibrium is established (at low frequencies). As a consequence, the specific heats

and adiabatic index associated with the degrees of freedom present in the medium may vary with the

frequency of the perturbation, leading to the change in the phase velocity of the acoustic wave. For

example, in the carbon dioxide at the room temperature the sound speed was shown to change from

260m s-1

at about 10 kHz to 270 m s-1

at about 1 MHz. A theoretical description of this effect can be

found in classic works (1-3).

Let us consider as an example a vibrationally excited relaxing gas. The linear analysis of acoustic

perturbations propagating in such medium leads to the dispersion relation shown below:

m

Tk

CiC

CiCсс

k

B

VV

PP 0

00

0022

2

2~,~

. (1)

Here k, are wave frequency and wave number, respectively, Bk is the Boltzmann constant, m

is the mean particle mass and 0 is the relaxation time of vibrational degree of freedom. Quantities

VP CC , denote the so-called high-frequency (frozen) specific heats at constant pressure and volume,

respectively. These capacities can be easily defined by using the assumption that all vibrations in

particles are fully frozen (for an instance, mkCmkC BVBP 25,27 for diatomic gas). In turn,

notations 00, VP CC are introduced for description of the so-called low-frequency (equilibrium)

specific heats at constant pressure and volume, respectively.

In contrast to high-frequency analogues, the low-frequency specific heats take into account the

influence of relaxation process. These specific heats may vary very significantly and become even

negative. Using the introduced capacities one may easily define adiabatic indices and, as a

consequence, the sound speed in the corresponding limits:

VPVPBB CCCCmTkсmTkс ,,, 0000000 (2)

Here 00,с and ,с are low- and high-frequency sound speeds and adiabatic indices,

respectively.

The whole frequency dependence of the acoustic wave phase velocity can be obtained from

equation (1) under the assumption of the weak amplification on the wavelength in the following simple

form:

22

022

0

2220

220

20

Re

VV

VVSnd

CC

cCcC

kc

(3)

In the context of a non-adiabatic gas and plasma (interstellar gas, solar corona, etc.), the effects

caused by the vibrational degrees of freedom can be generally neglected. However, another “slow”

process can play a role in the establishment of the thermodynamic equilibrium, which is an interplay

between the temperature T and density dependent heating and cooling processes.

One of the first comprehensive analytical studies of the effects caused by the processes of heating

and cooling in a non-adiabatic active medium was performed in a seminal work by Field(4). According

to this theory, the influence of heating and cooling processes can be described through the inclusion of

a so-called heat–loss function THTLTQ ,),(),( defined by the difference between the

energy losses ),( TL (the cooling rate) and the energy gains TH , (the heating rate), excluding

effects of the thermal conduction.

The occurrence of the heating/cooling misbalance can be considered as an effective excitation of an

additional “degree of freedom” which achieves the equilibrium in some certain time. For plasma being

in a thermal equilibrium, a balance between the heating and cooling processes should be established.

Hence, the heat transfer equation can be re-written in terms of the heat–loss function as 0, 00 TQ

Therefore, we can estimate the relaxation time for this new degree of freedom using the heating and

cooling rates calculated for the unperturbed values of the plasma density 0 and temperature 0T . In

other words, we thus introduce a characteristic heating/cooling time as

0000000 ,, TmLTkTmHTk BB (4)

Taking in mind the analogy between vibrationally excited relaxing gas and non-adiabatic gas, one

may obtain exactly the same dispersion equation as (1) for acoustic waves in non-adiabatic gas(see (5)

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for details). However, in contrast to the relaxing gas the low-frequency heat capacities will be defined

not by relaxation process but by heating and cooling processes which take place in the medium:

0000 ,0

00

,0

0000000 ,,,

T

L

T

TTB

PTB

V

Q

HQ

T

Q

H

TQQQ

m

kCQ

m

kC

(5)

Thus, both media have mathematically similar dispersion relation with the only difference that

coefficients have a different nature. As a consequence, influence of these media on the dispersive

properties of acoustic waves are similar, namely, frequency dependence of phase velocity and wave

increment. Furthermore, amplification or additional damping of acoustic waves may take place.

In the current research, we apply the approach based on analogy between non-adiabatic and

relaxing medium to describe some properties of magnetoacoustic(MA) waves in acoustically active

non-adiabatic plasma.

2. THE MODEL AND ASSUMPTIONS

In order to analyze MA waves we use full system of magnetohydrodynamic (MHD) equations. The

system of equations (1) is slightly different from standard system of equations for the equilibrium

plasma and takes into account influence of heating ),( TH and cooling ),( TL processes:

BVB

rot

t; 0Bdiv ; BB

VrotP

dt

d

1;

0

V

div

t; ),( TQ

dt

d

m

Tk

dt

dTC BV

;

m

TkP B .

(6)

Here, P is the pressure, vectors BV, correspond to the velocity vector and the magnetic field

vector, respectively. Due to the fact that the main focus of this paper is on the non-adiabatic processes,

we neglect the influence of the possible dissipation processes such as thermal conduction and

viscosity.

In the current research, we will analyze the one-dimensional wave dynamics only. To do this we use

Cartesian coordinate system x, y, z and assume that the wave propagation direction is along z-axis. The

vector of equilibrium magnetic field is assumed to lie in the (x,-z) – plane, i.e.

zxB0 cossin 00 BB . Here, 0B is an absolute value for the magnetic-field vector, is an

angle with respect to the z-axis and zx, are the unit vectors along the corresponding axis. The

representation of the problem geometry is shown in Figure 1. We also introduce notations k and k

for wave vector and unite wave vector, respectively, and θ

for angle unite vector. Hereinafter, index

0 indicates the value of any variable in the equilibrium state of the medium. The dependences of

variables upon x and y are ignored )0( yx .

Figure 1 – The representation of the problem geometry

3. DISPERSION PROPERTIES OF MAGNETOACOUSTIC WAVES

To analyze the dispersion properties of MA waves we conduct the linear analysis of equations (6).

For this purpose we use standard methods of perturbation theory. In other words, we expand the plasma

quantities around their stationary value as ,,,, 10 tzatzatza where 1a denotes the perturbed

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quantity and 1~/ 01 aa . Subsequent substitution of the harmonic wave solution in form

)exp(~ ikzti allows us to obtain dispersion equations for the fast and slow MA waves as follows:

2222222222 cos~4~~5.0 aaa cссcсck (7)

The signs ‘+’ and ‘-’ correspond to fast and slow MA wave, respectively. Notation 020 4Bca

is for the so-called Alfven speed. If we exclude non-adiabatic processes or consider strictly

high-frequency waves 10 , then the complex term 22~ сс and equations (7) are reduced to a

generally accepted expressions for equilibrium MHD media (6, 7).

By using the assumption of the weak amplification on the wavelength we obtain the frequency

dependent phase velocities for fast and slow MA wave:

22222222

, cos45.0Re aSndSndaSndasf cссcсckc (8)

In the low- and high-frequency limits these expressions take the form shown below:

22222222

,22

0222

022

02

,0 cos42

1,cos4

2

1сcсcсccсcсcсcc aaasfaaasf (9)

The non-adiabatic processes in the plasma affect phase velocity of MA disturbances in the same

way as they affect purely acoustic waves in the gaseous medium. Namely, in the high-frequency limit

10 the speed of both slow and fast MA perturbations is determined by the classical expression

for media without non-adiabatic processes sfsf сc ,1, 0 . In turn, in the low-frequency limit

10 the velocity of MA waves is determined by the influence of non-adiabatic heating and

cooling processes sfsf cc ,01, 0 .

Due to the fact that MA waves (as opposed to acoustic waves) are caused by a change in not only the

gas-dynamic pressure, but also the magnetic pressure, it is necessary to characterize the degree of its

influence. In order to characterize the magnitude of the magnetic pressure, and, consequently, the

magnitude of the external magnetic field, a dimensionless parameter, the so-called plasma beta ,

which is a ratio of plasma pressure and magnetic pressure, is usually used.

22200 28 accBP (10)

Considering media without non-adiabatic processes one may subdivide the values of plasma beta

into two qualitatively different ranges 2 and 2 where pure acoustic waves are

faster and slower than Alfven waves. This is because the phase velocity of acoustic waves с is the

same for any frequency in this medium. However, for media with non-adiabatic processes one may

define three qualitatively different ranges 2 , 22 0 and 02 . In such media

only one harmonic a may have the phase velocity equal to the phase speed of Alfven waves.

2222000 ccccCC aaVVa (11)

This result is of significant interest for the problems of nonlinear interaction between MA and

Alfven wave as it imposes severe restrictions on resonant conditions. And in particular, such result is

of importance for the problem of the solar corona heating, This is due to the fact that heating through

the nonlinear energy transport from Alfven to acoustic perturbation (which can be easily damped ) is

considered as a possible mechanism of corona heating up to millions degrees.

Not only magnetic field strength but also its direction significantly affects dispersion properties of

waves in the medium. Magnetic field specifies the preferential direction and results in the anisotropy

of such parameters as phase velocity and wave increment. In order to show the anisotropy of the phase

velocities it is convenient to use the so-called polar diagram. The phase polar diagram represents the

dependence of the phase velocity kVph

kRe on the angle between the vector k and the

magnetic field B vector in the polar coordinate system.

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Figure 2 – Phase polar diagram for different values of plasma beta a) 02 , b) 02 , c)

22 0 , d) 2 , e) 2 . The phase diagram corresponding to a particular frequency

is displayed by a color tone. Notations shown in figure means distance between certain point and zero point.

Vector 0B indicate the direction of the external magnetic field.

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In order to show the phase polar diagrams for the fast and slow MA waves at different values of

plasma beta (see Figure 2) we assume that all perturbations in the medium are subject to

attenuation. In other words, we consider the case when low-frequency waves propagate slower than

high-frequency waves sfsf cc ,,0 . In the medium without non-adiabatic processes polar diagrams for

whole wave spectrum consist of single curves defined by expression sfc , (9). However, in medium

with non-adiabatic processes the situation is different. Due to the fact that each frequency defines the

corresponding phase velocity the polar diagram for the whole wave spectrum becomes a group of

curves. To show the shape change of phase polar diagrams with changing frequency we use general

expression (8). The phase diagram corresponding to a particular frequency is shown by a color tone.

The transition from light to dark tone corresponds to the transition from the low-frequency part of the

spectrum to the high-frequency part of the spectrum the polar diagrams for acoustic active medium is

a group of curves for each frequency. To construct the boundary polar diagrams for the fast and slow

MA waves, expressions (9) has been applied. Since the dependence of the phase velocity is a

monotonic function of frequency(8), the phase diagrams for the MA waves are the regions limited by

the above mentioned boundary polar diagrams.

In application to the analysis of wave evolution more interesting is the polar diagram of the group

velocity, i.e. angular dependence of the absolute value of the group velocity. Typically, the group polar

diagrams is called the Friedrichs diagram (sometimes the Friedrichs diagram is understood as both

group and phase polar diagrams). For weakly dissipating/active media in which the perturbation

amplitudes vary slowly over space and time(8) the group velocity can be found as:

1ReReReRe

kkV sG_f, (12)

In this assumption group velocity completely coincides with the rate of energy transfer by a wave.

As a consequence it can qualitatively describe the linear stage of localized Gaussian perturbation

evolution. Thus, the group velocity in the acoustically active medium can be found as follows:

d

d

kkd

d

d

d

Re

1

Reθk

kV sG_f,

(13)

where

,

2

cos

Re 222,

222,2

,3,

Sndasf

asf

sfsfccc

cccc

kd

d

222,

22

, 2

sincos1

Re

1

Sndasf

aSnd

sf ccc

cc

cd

d

k

,

2220

220

20

220

220

2 VVVV CCCCcc

(14)

The obtained expression in the case of non-adiabatic processes absence is reduced to the classical

expression (see (6, 7)). As in the case of the phase velocity, the polar diagram for group velocity is a

group of curves not one curve as in medium without non-adiabatic processes. The frequency

dependence of group velocity is rather similar to the frequency dependence of phase velocity. In the

low- and high-frequency limits expressions (13) are reduced to sfc ,0 and sfc , , respectively.

However, the frequency dependence is not monotonic and it has an extreme value on frequency:

222220

220

22000 43 aaaaVVExtMA cccccccccCC

(15)

Depending on the dispersion type, namely, negative cc0 or positive cc0 , this extremum

can be the minimum or maximum, respectively. Thus, some harmonics may propagate even faster than

value sfc , which do not take into account non-adiabatic processes influence and widely used for

analysis of waves in such medium. In order to show this effect we assume again that all perturbations

in the medium are subject to attenuation and construct group polar diagrams using the expression (13),

(14) (see Figure 3). It is clearly seen that part of the spectrum of fast and slow MA wave has greater

speed than well known sc and fc (red and blue curves in Figure 3). It also seen that fast MA

waves propagate in all possible direction, the slow MA wave packet, in turn, p ropagate along the

magnetic field as cuspoid structure, which usual for MA waves in uniform plasma.

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Figure 3 – Group polar diagram for different values of plasma beta a) 02 , b) 02 , c)

22 0 , d) 2 , e) 2 . The phase diagram corresponding to a particular frequency

is displayed by a color tone. Notations shown in figure means distance between certain point and zero point.

Vector 0B indicate the direction of the external magnetic field.

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Finally, we have studied the effect of dissipation / activity due to non-adiabatic processes of the

medium on the expression for group velocity. In other words, we have investigated whether the

imaginary component kIm would have a significant effect on the value of the group velocity

of waves in a given medium. To study this problem, we have used the technique that was previously

used to describe waves in media with resonant damping (amplification) and to describe the

propagation of whistlers in a collisional plasma (9). The analysis have shown that the imaginary

component kIm has no significant effect on the group velocity value, and as a result, the

expression for the group velocity of acoustic and magnetoacoustic waves can be calculated as

kRe (see equation (14)). It has been also shown that the effect of the imaginary component

kIm is expressed in the fact that in the low-frequency region there will be an increase in the

wave number characterizing the center of the wave packet over time under isentropic instability ( MA

wave amplification) and, conversely, its decrease in case of isentropic stability(MA wave additional

damping). The shift of the characteristic wave number in the high-frequency part of the spectrum due

to the imaginary component kIm does not occur.

4. CONCLUSIONS

In this paper we have discussed some properties of magnetoacoustic waves in acoustically active

non-adiabatic plasma. We have described the anisotropy and frequency dependence of phase and group

velocities for fast and slow magnetoacoustic wave. The polar diagrams demonstrating the anisotropy

of phase and group velocities have been shown. It have been demonstrated that group velocity in

contrast to phase velocity has as an extrema which can be maxima or minima depending on type of

dispersion (negative or positive, respectively). In other words, some harmonics in non-adiabatic

plasma can propagate faster or slower than all others. Our analysis also have shown that the imaginary

component kIm has no significant effect on the group velocity of MA waves in acoustically

active non-adiabatic plasma, and as a result, the expression for the group velocity of acoustic and

magnetoacoustic waves can be calculated as kRe .

ACKNOWLEDGEMENTS

Analysis presented in the reported study were funded by RFBR according to the research project No.

18-32-00344. The study was supported in part by the Ministry of Education and Science of Russia

under the public contract with educational and research institutions within the project

3.1158.2017/4.6.

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