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 isunder 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 mostpronounced 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 wherephase/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 groupvelocity 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 thetranslational motion of the particles, rotational and vibrational energies of the molecules, the chemicalenergy. 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 withcomparable masses, the time required for the Maxwell distribution to be established is of the sameorder 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 isdetermined 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 therelaxation 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 isachieved for the “fast” processes only (at the high-frequency part of the spectrum), and where the full1

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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 thefrequency 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 from260m s-1 at about 10 kHz to 270 m s -1 at about 1 MHz. A theoretical description of this effect can befound 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:

2
k

2

 с~ 2 ,C  i 0CP  k BT0 .
с~ 2  P 0
CV 0  i 0CV  m(1)Here , k are wave frequency and wave number, respectively, k B is the Boltzmann constant, mis the mean particle mass and  0 is the relaxation time of vibrational degree of freedom. Quantities
CP , CV 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 inparticles are fully frozen (for an instance, CP  7k B 2m , CV  5k B 2m for diatomic gas). In turn,notations CP 0 , CV 0 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 evennegative. Using the introduced capacities one may easily define adiabatic indices and, as aconsequence, the sound speed in the corresponding limits:с0   0k BT0 m , с    k BT0 m , 0  CP 0 CV 0 ,    CP CV

(2)

Here с0 ,  0 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 fromequation (1) under the assumption of the weak amplification on the wavelength in the followin g simple
form:

cSnd   Rek 



C c    C c 
C    C 
2 2
V0 0
2
V0

2 2 2 20 V 
2 2 2
0 V

(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 ofa so-called heat–loss function Q(  , T )  L(  , T )  H  , T  defined by the difference between theenergy losses L(  , T ) (the cooling rate) and the energy gains H  , T  (the heating rate), excluding
effects of the thermal conduction.
The occurrence of the heating/cooling misbalance can be considered as an e ffective excitation of anadditional “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 Q0 , T0   0
Therefore, we can estimate the relaxation time for this new degree of freedom using the heating andcooling rates calculated for the unperturbed values of the plasma density  0 and temperature T0 . In
other words, we thus introduce a characteristic heating/cooling time as 0  k BT0 mH 0 , T0   kBT0 mL0 , T0 
(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)7587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:kkT  Q   Q 
CV 0  B Q0T , CP 0  B Q0T  Q0  , Q0T  0 , QL 0   0 

(5)
m
m
H 0  T   0 ,T0
H 0     ,T





0

0

Thus, both media have mathematically similar dispersion relation with the only difference thatcoefficients 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 andrelaxing medium to describe some properties of magnetoacoustic(MA) waves in acoustically active
non-adiabatic plasma.

2. THE MODEL AND ASSUMPTIONSIn 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 H (  , T ) and cooling L(  , T ) processes:

B rotV  B ;t
 divV  0 ;tdivB  0 ;

dV1
 P B  rotB ;dt
dT k BT dCV Q (  , T ) ;dt m dt

k TP B .
m(6)Here, P is the pressure, vectors V, B 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 a nd
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.

B 0  B0  sin   x  B0  cos   z . Here, B0 is an absolute value for the magnetic-field vector,  is an angle with respect to the z-axis and x, z are the unit vectors along the corresponding axis. The
representation of the problem geometry is shown in Figure 1. We also introduce notation s 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 ofvariables upon x and y are ignored ( x   y  0) .

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 a z, t   a0 z, t   a1 z, t , where a1 denotes the perturbed

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quantity and a1 / a0 ~   1 . Subsequent substitution of the harmonic wave solution in form
~ exp( it  ikz ) allows us to obtain dispersion equations for the fast and slow MA waves as follows:



2



 c  с~   4с~ c cos  

 2 k  0.5 ca2  с~ 2 


2 2

2
a

2 2
a

2

(7)



The signs ‘+’ and ‘-’ correspond to fast and slow MA wave, respectively. Notation ca  B02 4 0
is for the so-called Alfven speed. If we exclude non-adiabatic processes or consider strictly
high-frequency waves  0  1 , then the complex term с~ 2  с2 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: c  с    4с  c cos  

2  
c f , s     Rek   0.5 ca2  сSnd


2
a

2

2
Snd

2
Snd

2
a

2

(8)



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

c0 f , s 1 22
 ca  с0 
2c  с   4c с cos   , c
2
a

2 2
0

2 2
a 0

2

f , s 1 2
2
 ca  с 2c  с   4c с cos  
2
a

2 2


2 2
a 

2

(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
 0  1 the speed of both slow and fast MA perturbations is determined by the classical expression
for media without non-adiabatic processes c f , s  0 1  сf , s . In turn, in the low-frequency limit

 0  1 the velocity of MA waves is determined by the influence of non-adiabatic heating andcooling processes c f ,s 0 1  c0 f ,s .
Due to the fact that MA waves (as opposed to acoustic waves) are caused by a change in not only thegas-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, themagnitude 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.

  8P0 B02  2c2  ca2

(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 thesame for any frequency in this medium. However, for media with non-adiabatic processes one may
define three qualitatively different ranges   2   , 2  0    2   and   2  0 . In such media
only one harmonic a may have the phase velocity equal to the phase speed of Alfven waves.

a  CV 0  0CV  c02  ca2

 c  c 
2
a

2


(11)

This result is of significant interest for the problems of nonlinear interaction between MA andAlfven wave as it imposes severe restrictions on resonant conditions. And in particular, such result isof importance for the problem of the solar corona heating, This is due to the fact that heating throughthe 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 ofwaves in the medium. Magnetic field specifies the preferential direction and results in the anisotropyof such parameters as phase velocity and wave increment. In order to show the anisotropy of the phasevelocities it is convenient to use the so-called polar diagram. The phase polar diagram represents thedependence of the phase velocity Vph   Rek  k on the angle between the vector k and themagnetic field B vector in the polar coordinate system.

7589

Figure 2 – Phase polar diagram for different values of plasma beta a)   2  0 , b)   2  0 , c)

2  0    2   , 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 B 0 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 ofplasma beta  (see Figure 2) we assume that all perturbations in the medium are subject toattenuation. In other words, we consider the case when low-frequency waves propagate slower thanhigh-frequency waves c0 f , s  cf , s . In the medium without non-adiabatic processes polar diagrams forwhole wave spectrum consist of single curves defined by expression cf ,s (9). However, in medium
with non-adiabatic processes the situation is different. Due to the fact that each frequency defines thecorresponding 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 u se 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 thespectrum to the high-frequency part of the spectrum the polar diagrams for acoustic active medium isa 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 amonotonic 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 groupvelocity, 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 bothgroup and phase polar diagrams). For weakly dissipating/active medi a in which the perturbation
amplitudes vary slowly over space and time(8) the group velocity can be found as:

VG_f,s   Re  Re k   Re k  Re  1(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: 1 dd  dVG_f,s 
k
θ
(13)
dk
d Re kRe k dwhered  c 3f , s
d Re k 


2
2
2
2


 2

 c 2   c f , s  ca cos     , 1 d  1 cSnd ca cos  sin  ,22 f , s  2c 2f , s  ca2  cSnd  Re k d c f , s 2c 2f , s  ca2  cSnd















    c2  c02  2 02CV2CV2 0



(14)

 C    C 
2V0

2 2 2 20 VThe obtained expression in the case of non-adiabatic processes absence is reduced to the classicalexpression (see (6, 7)). As in the case of the phase velocity, the polar diagram for group velocity is agroup 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 thelow- and high-frequency limits expressions (13) are reduced to c0 f , s and cf , s , respectively.
However, the frequency dependence is not monotonic and it has an extreme value on frequency:

ExtMA  CV 0  0CV  c02  ca2 c  c c  3c  4c  c  c 
2
0

2
a

2
0

2


2
a

2


2
a

(15)

Depending on the dispersion type, namely, negative c0  c or positive c0  c , this extremum
can be the minimum or maximum, respectively. Thus, some harmonics may propagate even faster thanvalue cf , s which do not take into account non-adiabatic processes influence and widely used foranalysis of waves in such medium. In order to show this effect we assume again that all perturbationsin 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 cs and cf (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 themagnetic 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)   2  0 , b)   2  0 , c)

2  0    2   , 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 B 0 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 themedium on the expression for group velocity. In other words, we have investigated whether theimaginary component Im  k  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 (amplifi cation) and to describe thepropagation of whistlers in a collisional plasma (9). The analysis have shown that the imaginary
component Im  k  has no significant effect on the group velocity value, and as a result, theexpression for the group velocity of acoustic and magnetoacoustic waves can be calculated asRe k  (see equation (14)). It has been also shown that the effect of the imaginary component
Im  k  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 ( MAwave 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 dueto the imaginary component Im  k  does not occur.

4. CONCLUSIONSIn 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 groupvelocities for fast and slow magnetoacoustic wave. The polar diagrams demonstrating the anisotropyof phase and group velocities have been shown. It have been demonstrated that group velocity incontrast 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 Im  k  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 andmagnetoacoustic waves can be calculated as Re k  .

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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