A new model for heating of Solar North Polar Coronal Hole
This paper presents a new model of North Polar Coronal Hole (NPCH) to study dissipation/propagation of MHD waves. We investigate the effects of the isotropic viscosity and heat conduction on the propagation characteristics of the MHD waves in NPCH. We first model NPCH by considering the differences in radial as well as in the direction perpendicular to the line of sight (los) in temperature, particle number density and non-thermal velocities between plumes and interplume lanes for the specific case of \ionOVI ions. This model includes parallel and perpendicular (to the magnetic field) heat conduction and viscous dissipation. Next, we derive the dispersion relations for the MHD waves in the case of absence and presence of parallel heat conduction. In the case of absence of parallel heat conduction, we find that MHD wave dissipation strongly depends on the viscosity for modified acoustic and Alfven waves. The energy flux density of acoustic waves varies between and while the energy flux density of Alfven waves turned out to be between . But, solutions of the magnetoacustic waves show that the parallel heat conduction introduce anomalous dispersion to the NPCH plasma wherein the group velocity of waves exceeds the speed of light in vacuum. Our results suggests all these waves may provide significant source for the observed preferential accelerating and heating of \ionOVI ions, in turn coronal plasma heating and an extra accelerating agent for fast solar wind in NPCH.
keywords:magnetohydrodynamics (MHD)–Sun:corona–solar wind–waves
Observations as well as theoretical studies on coronal heating and solar wind acceleration had long since drawn attention to the dissipation of MHD waves through various processes. Particularly, North Polar Coronal Hole (NPCH) have been observed by various satellites, for instance, Yohkoh, Wind, Soho, Hinode, TRACE, etc. The effective temperature of NPCH has been found as high as K. Since the surface temperature of the Sun is much lower than the NPCH one, two questions are immediately required to be answered: 1) what is the continuous source of energy for NPCH, since the plasma with that much high temperature is bound to lose energy to the transition region below through heat conduction and optically thin emission? and 2) what kind of force causes solar wind acceleration? Observations revealed that NPCH may be regarded as collisional up to the radial distance 1.7 - 2.1 R range, where . Within this domain, MHD waves may donate their energies into random thermal motions of particles through microscopic mechanisms, i.e., viscosity, thermal conductivity and resistivity.
Braginskii (1965) pointed out that dissipative processes, we listed above, cause damping of MHD waves and thus heat the plasma. In the extensive review articles by Ofman (2005) and Cranmer (2009) it was shown that Alfven waves, fast and slow magnetosonic waves are the most likely candidates to heat the coronal plasma and accelerate the solar wind. Energy fluxes of acoustic waves are so low and the damping length scales are so short that they are excluded from the list of energy supplier candidates (Athay & White, 1978; Mein & Schmieder, 1981). Suzuki (2002) claimed that acoustic waves with period and energy flux density may heat the plasma to the temperatures of K by dissipation of N-waves after the formation of the shocks. He also noted that other collective processes with larger length scales may heat the coronal plasma and accelerate the solar wind.
Presence of MHD waves in NPCH is revealed by EUV observations carried out by SOHO, Hinode and TRACE satellites. Ofman et al. (1997), De Moortel et al. (2002), Banerjee et al. (2009a) and Gupta et al. (2010) reported the presence of propagating and standing compressional waves in corona. Alfven waves are also observed (Tomczyk et al., 2007; Banerjee et al., 1998, 2009b; Landi & Cranmer, 2009; Gupta et al., 2010; Bemporad & Abbo, 2012).
In the heliocentric distance range 1.4-2.6 the coronal heating and solar wind acceleration are shown by UVCS observations to be more effective (Fisher & Guhathakurta, 1995). In earlier studies by Hollweg (1986) and Hollweg & Johnson (1988), the required energy flux density to heat the corona is evaluated as . When electron number density, magnetic field strength and the turbulent velocity amplitude are taking as , and respectively, it was shown that energy flux density of an MHD wave becomes at (Banerjee et al., 1998). Tomczyk et al. (2007) carried out multichannel polarimetric observations and detected Alfven waves with a phase speed . They concluded that the observed Alfven waves were not energetic enough to heat the corona. However, McIntosh et al. (2011) managed to observe the Alfven waves and measured their amplitudes, phase speeds, the periods (of order of ) and then calculated the energy flux densities of these waves which were found to be high enough to heat the NPCH and accelerate the solar wind.
Banerjee et al. (2011) reviewed the observational evidences of propagating MHD waves in coronal holes. Periods of the MHD waves observed in coronal hole via remote sensing are listed in Table 1 of this paper. Periods of waves detected in plumes and interplume lanes vary between and . Waves observed other regions of coronal holes (on-disk and off-limb) have a period range of to .
Ruderman et al. (1998) developed a model which is specifically applicable to the coronal hole. In this model, the authors considered the Alfven wave phase mixing in a magnetic field which consists of open field lines and has two dimensions wherein the vertical scale height is greater than that of the horizontal. In their model they considered viscosity as a free parameter since the kinematic viscosity coefficient in the solar corona has no certain value. Their model predicts that Alfven waves with a period falling in a range are not damped near the plume boundary if the vertical height and . On the other hand, when the viscosity coefficient is raised to the value and the other parameters are kept unchanged, Alfven waves are damped and transferred their energy to the magnetised plasma.
Surface wave heating is also one of the possibilities for heating coronal hole (Narain & Sharma, 1998). The authors evaluated that in a radial magnetic field configuration with a strength , waves go through a strong dissipation within a radial distance range (in a nonlinear regime) and (in a linear regime). Their model predicts that in a magnetic field showing distortion in radial direction (“strong spreading” in their jargon) surface waves are strongly dissipated by the coronal hole plasma.
The presence of linearly polarized spherical Alfven waves in coronal hole is also considered as a possible agent to heat the plasma and accelerate the solar wind (Nakariakov et al., 2000). The authors investigated the weakly non-linear dynamics of the waves in a medium wherein only the shear viscosity plays an important role in dissipation. They remind the reader that there is no reliable information about the radial dependence of the viscosity coefficient in a coronal hole and assume constant viscosity. Their conclusion is that in a plasma with normalized viscosity , linearly polarized Alfven waves with periods are damped within a radial distance less than .
The upper parts of the chromosphere and the corona are believed to be characterized by strong resistivity, anisotropic viscosity and thermal conductivity (Ruderman et al., 2000). The authors considered slow surface waves propagating in the above-defined plasma threaded by equilibrium magnetic field. In the fluid description, they assumed that away from the dissipative region only the parallel (to the magnetic field) thermal conductivity and viscosity are important. However, at the layer where waves are dissipated they adopted the full Braginskii expressions for viscosity, electrical resistivity and the heat flux. In their conclusion, it is the resonant absorption that causes the surface wave damping.
Ultraviolet Coronograph Spectrometer white-light channel (UVCS/WLC) and Extreme Ultraviolet Imaging Telescope (EIT) observations revealed the presence of slow magnetoacoustic waves in polar plumes. Using these data (Ofman et al., 2000) investigated the damping of waves propagating in a gravitationally stratified medium threaded by radially divergent magnetic field. Two-dimensional MHD model of the authors contains the non-linear effects of the solar wind and viscosity. Dissipation length scale of the waves with amplitude turns out to be . If the viscosity is taken into account for the waves with amplitude and with a period , damping length scale becomes . Close to the Sun, dissipated waves may contribute to the acceleration of the solar wind either by increasing the thermal pressure of the coronal plasma or by undergoing Landau damping and thus transferring momentum at the collisionless part of the hole which is further away from the Sun.
Devlen & Pekünlü (2010), in an MHD approximation, considered the wave propagation in NPCH wherein temperature and number density of \ionOVI ions show gradients both in the parallel and the perpendicular directions to the magnetic field. They also took into account the heat conduction in both directions to the magnetic field, but neglected viscosity. Even the fluid description of NPCH revealed the resonance absorption of the waves.
In this paper, we assume that the parallel and the perpendicular (to the magnetic field) heat conduction and viscosity affect the propagation characteristics of MHD waves in NPCH and these waves, in turn, interact with \ionOVI ions and preferentially heat them. SOHO/UVCS data on \ionMgX and \ionOVI ions showed that NPCH plasma becomes collisionless beyond the radial distance (Doyle et al., 1999). Therefore, we considered NPCH plasma collisional up to radial distance and used MHD approximation. This study investigates the effects of the isotropic viscosity and heat conduction on the propagation characteristics of the MHD waves in NPCH.
This paper is organized as follows. In the next section we give details of our model of NPCH. In Section 3 we give basic MHD equations and linearized MHD equations. We obtain the dispersion relations of the driven waves in the case of absence and presence of parallel heat conduction, separately. We estimate the damping length scales and energy flux densities of these waves. Finally Section 4 concludes this paper with a brief summary and discussion of our major results.
2 Model of NPCH
SOHO/UVCS data revealed that NPCH was structured with so-called plumes and interplume lanes (Wilhelm et al., 1998). In their Figure 1, Wilhelm et al. (1998) drew attention to the fact that plumes are brighter, higher in density but cooler than interplume lanes. Temperature anisotropy for \ionOVI ions in NPCH were revealed by SOHO/UVCS data, i.e., , where subscripts refer to the perpendicular and the parallel directions to the magnetic field, respectively (Kohl et al., 1997; Cranmer et al., 2008). Spectral line widths of \ionOVI ions taken from interplume lanes turned out to be wider than the ones taken from plumes. This, of course, indicates that perpendicular (to the magnetic field lines) heating in interplume lanes is more effective than that of plumes (Cranmer, 2002; Wilhelm et al., 2011). Besides, minor ion species He, \ionSiVIII, \ionMgX, etc. also confirmed the temperature anisotropy in NPCH.
Many authors considered NPCH as electrically quasi-neutral, i.e., (e.g. Marsch, 1999; Endeve & Leer, 2001; Voitenko & Goossens, 2002). In Figure 2 of Cranmer et al. (1999) electron number densities in plumes and interplume lanes are plotted. This figure reveals that in plumes is about 10% higher than that of interplume lanes. One can find electron number density plots with respect to heliocentric distance in Wilhelm et al. (2011)’s study. The empirical fit by Doyle et al. (1999) seems to be the most consistent one with data points,
where is the heliocentric distance. Following the authors who consider NPCH as quasi-neutral, we assume that and try to model NPCH in two dimensions: one is the heliocentric distance, i.e., and the other, , is the direction perpendicular to both the magnetic field and the line of sight (); is measured in arc seconds. Thus, proton number density may be expressed in space as below:
where superscripts IPL ad PL stand for interplume lanes and plumes, respectively.
Magnetic field in NPCH is assumed to be in the radial direction (Hollweg, 1999),
where . Because of the absence of data as to how the magnetic filed varies in plumes and interplume lanes, and besides in direction, we assume that the equation (3) above is applied to both PL and IPL.
OVI line width is shown to vary both in the and the direction (Kohl et al., 1997). Wilhelm et al. (1998) formulated the total effective temperatures of minor ion species as the sum of ion temperature and the contribution from non-thermal processes,
where is the most probable speed of ion along the and is the most probable speed of an isotropic, Gaussian distributed, turbulent velocity field. Dolla & Solomon (2008) argued that the origin of non-thermal part of the line width is either Alfven wave dissipation or ion cyclotron resonance (ICR) after a turbulent cascade. It is almost impossible to separate these contributions to line widths.
The effective temperature of the \ionOVI ions is given by Devlen & Pekünlü (2010),
It was shown by Esser et al. (1999) that the value of for \ionOVI and \ionMgX ions are quite close to each other and vary in the same manner with . Taking this fact into account and by using \ionMgX data for , Devlen & Pekünlü (2010) derived a polynomial for the non-thermal part of the effective temperature of \ionOVI ions.
Wilhelm et al. (1998) report that \ionOVI ion effective temperature in the interplume lanes is about 30% higher than that of plumes (see Fig. 1). With this observational fact we may express the effective temperature in two dimensions as below:
The thermal conductivity and viscosity have anisotropic character due to the presence of magnetic field in NPCH. The parallel (to the magnetic field) thermal diffusivity for \ionOVI ions is given by (Banks, 1966)
For the plumes and interplume lanes under the typical coronal conditions . In NPCH the ratio is unknown, therefore we take it as a free parameter. For solar corona, van Hoven & Mok (1984) took it as . It may be argued that for solar coronal conditions . But it is not negligible when . van Hoven & Mok (1984) argued that even the ratio assumes the value it should be taken into consideration because it removes unphysical singularity in the case of for thermal instability.
The coefficient of kinematic viscosity in the isotropic case is given by
where is the Coulomb logarithm (Spitzer, 1962). Ruderman et al. (1998) argued that the viscosity coefficient for solar corona is quite uncertain. There is a same uncertainty on the coefficient of the conduction. Bearing this fact in mind, in this study we use both coefficients as free parameters remaining around theirs estimated values from Equation (8) and (9) for the model of NPCH.
3 Basic MHD Equations
The basic equations for the investigation of wave equation and dissipation in a plasma are the continuity of mass, momentum, energy and magnetic induction, together with the magnetic flux equations, in the forms given as (e.g. Priest, 1984):
where is the mass density, is the fluid velocity, is the scalar pressure, is the magnetic field, is the coefficient of kinematic viscosity (assumed uniform), is the heat flux, where , and which is unit vector along the magnetic field, and the is adiabatic index (its value is taken as for coronal plasma) and is a Lagrangian derivative.
We apply a standart Wentzel-Kramers-Brillouin (WKB) perturbation analysis on the equilibrium state. In this analysis, all the variables in the MHD equations are denoted by sums of equilibrium (denoted with a “0” subscript) and a small perturbed quantity (denoted with a “1” subscript), i.e. etc.
The quasi-linear equations derived from equations (10)-(14) are
where is the sound speed. In the linear approximation, perturbed values are usually assumed to be negligible in comparison with the equilibrium values. In sec. 3.3 we will investigate the effects of the isotropic viscosity and heat conduction on the propagation characteristics of the MHD waves in NPCH in linear approximation. But, in the framework of quasi-linear MHD, the lowest order non-linear term in rms velocity is retained, i.e. we must not neglect the term. Because the observed perturbed velocity of \ionMgX falls within the range 95 and 140 in (Esser et al., 1999). Esser et al. (1999) showed that the non-thermal velocities of \ionOVI and \ionMgX are quite close to each other and vary in the same manner with . Hence we kept the term in our first approach. But we neglect term. Although we try very hard to find the magnitude of the perturbed magnetic field, i.e. in the literature, but in vain. Therefore, we assume that the waves perturbing the magnetic field and giving it a radius of curvature so huge that becomes almost zero, where is the radius of curvature imposed by the propagation of waves!
The set of equations (15)-(19) is reduced to a single equation by differentiating equation (16) with respect to time and substituting for (), (), () from equations (15), (17) and (18) respectively. After time differentiation, equation (16) reads as
Remembering that the perturbation values vary as we make the below substitutions: and and we put , . There is no harm in repeating that if and does not vanish, two coupled (in and ) equations may be constructed and they may be eliminated so that the dispersion relation is found (Priest, 1984). Then we obtain
where and . We make dot product the equation (21) with , , and , respectively and reach the coupled equations in and :
From equations (22) and (23), we find the equation (25) depending only on :
where , , , and .
In NPCH the effective temperature increases by 30% in -direction where from the midpoint of a plume to the midpoint of neighbouring interplume lane. Distance in x-direction between from the midpoint of a plume and the midpoint of neighbouring interplume lane at = 1.7 is about 27 885 . The effective temperature change between these two points is about 30%. Measuring from = 1.7, the increase of the effective temperature by 30% in -direction takes place in 374 120 . Thus, the temperature gradient height scale is about . After , the temperature change in direction will occur at . We mentioned that isn’t negligible when (van Hoven & Mok, 1984) in Section 2. If these height scales are used, one can easily see that is satisfied. Therefore, especially to see the effect of on the evolution of wave propagation in the direction, we may assume that temperature gradient does not depend on in the interval (i.e., ) and thus and . After that we will search solutions of equation (25) in two steps; by assuming is negligible, i.e., , in subsection 3.1, and by retaining it in equation (25) in subsection 3.2.
3.1 In the absence of parallel heat conduction
If = 0 then equation (25) becomes
Equation (26) have two roots: and . By substituting and into Equation (22) or (23), we obtain a pair of ( , ):
3.1.1 Acoustic Waves in NPCH
First, let us substitute (,) into equation (24) and try to find the dispersion relation of the perturbation:
This mode may be labelled as “acoustic wave modified by viscosity and perpendicular heat conduction”, nevertheless, for the sake of brevity we call it acoustic wave. Figure 2 shows the variation of the damping length scales plotted versus logarithms of perpendicular diffusivity and frequency for acoustic wave propagating in the plume () at the fiducial radius and different viscosity values. Due to the fact that the coefficients of heat conduction and viscosiy in the solar corona are not well-established, we adopted these coefficients as free parameters. Thus, we determined the necessary values of these parameters by considering the damping length scales which would cause the solar wind acceleration and heating corona. The damping length scale () of this wave derived from equation (27) is given in terms of solar radius. In all the graphics showing the damping length scales between we used false colours. Distances greater than are indicated by grey colour and the ones shorter than by white colour. We chose wave angular frequency range of to which corresponds to period interval .
As shown the Figure 2, for the values of greater than , it is seen that the value of the perpendicular diffusivity coefficient increases linearly with frequency for all the values of viscosity and all the damping length scales. For the values of smaller than 100, the viscosity should have values higher than if the higher frequency waves are to be damped at the required distances. When the viscosity is about even the low frequency waves get damped at distances wherein coronal heating and solar wind acceleration take place.
Mechanical energy flux density of waves is given by Priest (1984):
where is the group velocity of the wave and is the non-thermal velocity which is related to the small scale unresolved turbulent motions in Equation (4).
Figure 3 shows the variation of the damping length scales of the waves with angular frequency propagating in plumes and interplume lanes (PL and IPL; upper graphics) and the variation of the energy flux density of the same waves with viscosity and perpendicular diffusivity (lower graphics). angular frequency waves correspond to frequency waves with a period of . Within plumes, when the perpendicular diffusivity has a value between , damping length scales are independent of viscosity. When the viscosity has higher values within the range , damping occurs independent of perpendicular diffusivity. Within IPLs damping of the waves occurs for the values of viscosity between and , otherwise the damping occurs at much greater distances.
When the damping length scales within PLs and IPLs are between and , the energy flux density of the acoustic wave begins and increases with increasing values of the diffusivity and viscosity(lower graphics in the Fig. 3).
In the literature, the energies of acoustic waves are found to be inadequate for heating the corona (see Athay & White (1978); Mein & Schmieder (1981)). Withbroe (1988) pointed out that most of the heat supply to the corona should be within . Besides, in order to produce fast solar wind, a little more energy flux density () should be deposited at the sound point (a few solar radii). Strachan et al. (2012) studied the velocity maps of the solar wind and concluded that both the slow and the fast solar wind reveals the existence of an extra accelerating agent beyond .
Our study shows that when the variations of heat, density and temperature perpendicular to the magnetic field are taken into, the sound wave fluxes reach levels required for heating of the corona and accelerating the solar wind. The energy flux density of acoustic waves varies between and . The damping length scale of these waves varies between and . In even the negligible value range of perpendicular diffusivity (i.e. ) wave damping occurs as clearly seen in graphics. We may claim that these waves may be responsible for perpendicular heating of \ionOVI ions and thus contribute to the heating of the coronal base in NPCH. Besides, our solution for the acoustic waves show that these waves may supply the observed extra acceleration for the fast solar wind because of its larger damping length scale ().This result is consistent with observations (Cranmer, 2002; Wilhelm et al., 2011).
3.1.2 Alfven waves in NPCH
If we substitute (,) into equation (24) we get the dispersion relation for Alfven waves:
In Figure 4, it is shown how frequency and viscosity cause variation of the damping length scale (left) in terms of solar radius and the energy flux density (right) of Alfven waves along the plume at the fiducial radius . There is a turning point of viscosity for each contours of the damping length scales. Up to the value of turning point firstly viscosity increases linearly with decreasing frequency and later increases with increasing frequency. For example the viscosity value of turning point is about for the damping length scales between . Wave flux is about for the required damping length scales. When the frequency is set to a constant value, it is seen that wave flux increases with increasing viscosity.
The damping length scales and the energy flux densities of this waves are similar for both the plumes and interplume lanes.
The total Alfven wave energy flux required to heat the quiet corona is (Withbroe & Noyes, 1977). We showed above that the energy flux density of Alfven waves falls within a range about . This amount is believed to replace the energy loss in NPCH by optically thin emission and by heat conduction to the transition region. The damping length scale and energy flux density are and for wave with frequency at around Spitzer value of viscosity coeffcient (), respectively. Alfven waves may contribute to the heating of upper corona in NPCH and supply the observed extra acceleration for the fast solar wind.
In the second part of the investigation, we searched the wave propagation characteristics of the acoustic and Alfven waves along the radial direction . Taking as the distance of the midpoint of plume we found the damping length scales and energy flux densities along the radial direction. Table 1 shows these values of acoustic and Alfven waves having frequencies and at some radial distances assuming viscosity has a value of . Fluxes decrease with radial distance and this decrease is steeper at higher frequencies. For smaller viscosity values (e.g. of ) a similar variation occurs at frequencies and , but fluxes are smaller and variation along the radial distance is smoother for .
3.2 In the presence of parallel heat conduction
If we take then equation (25) yields three values and thus three (,) pairs as given below
where , .
The square root in the F and G terms is rewritten as . Since the value of the second term under square root is less than one, we apply the Taylor series expansion, i.e., if the below condition is fulfilled
This condition is fulfilled in our model.
3.2.1 Solution for ()
Let us substitute (,) into equation (24) using Taylor series expansion and after a lengthy effort we reach the dispersion relation as Equation (A1) in the Appendix.
Figures 5 and 6 show the solutions of ten degree dispersion relation (Equation A1). The variation of damping length scales of Mode 1 (upper graphics) and Mode 2 (lower graphics) versus frequency and perpendicular diffusivity is shown in Figure 5 wherein viscosity and three parallel diffusivity values are assumed to be and , respectively. For Mode 1 in case , there appears damping length scales only dependent on frequency at the values smaller than for perpendicular diffusivity. At values greater than for perpendicular diffusivity, the value of perpendicular diffusivity increases linearly with increasing frequency for all the damping length scales. At extremely high value of , there appears a linear increase of perpendicular diffusivity with increasing frequency for all the damping length scales within the range for perpendicular diffusivity. For other values, wave damping occurs at either very far or very close distances.
On the other hand, it is seen that the damping length scales of Mode 2 is independent of the value of . Waves with lower frequencies () get damped at very far distances. Similiarly, there appears damping length scales independent of perpendicular diffusivity and only dependent on frequency at the values smaller than for perpendicular diffusivity while there appears a linear increase of perpendicular diffusivity with increasing frequency for all damping length scales at the values greater than for perpendicular diffusivity.
Mode 3 shows a similar tendency with Mode 2. While Mode 4 gets damped very near to the sun, Mode 5 propagates to very far distances before damping.
If we assume in the dispersion relation (A1), the solution gives an acoustic wave given by the equation (27).
When only is taken into consideration and other effects are assumed to be negligible in Equation (A1), damping length scale increases with increasing for a fixed frequency. At lower frequencies, required damping length scales for heating corona and acceleration of solar wind obtained only for smaller values of .
When only and viscosity are taken into consideration, Mode 1 gets damped at distances . Mode 2 waves with frequencies within the range get damped within the required radial distances. Damping is independent of the value of . Mode 3 has required damping length scales when values fall within a narrow range, i.e., . Waves with frequencies lesser than get damped at very far distances (i.e. further than ).
Figure 6 (top and middle) shows the variation of damping length scales with viscosity and perpendicular diffusivity of Mode 1 and Mode 2 having a frequency when values are assumed to be and . Plots in the bottom of Figure 6 show the variation of damping length scale of Mode 3 having a frequency and and the variation of damping length scale of the same mode (Mode 3) having a frequency and values.
For Mode 1 when , there is a region wherein waves get damped independent of viscosity for values of perpendicular diffusivity between and . At smaller values of perpendicular diffusivity and within viscosity value range there appears the second damping region. First damping region disappears while values increase and the second region enlarges towards smaller viscosity values.
Mode 2 has only the first damping region like the Mode 1. This mode has not been significantly affected by the variation of .
When perpendicular diffusivity value is smaller than , Mode 3 with gets damped within the radial distance range . When assumes higher values, wave gets damped at further radial distances. For a wave with a frequency at the value of damping occurs, on the average, at for smaller values of perpendicular diffusivity. But when , the wave with a same frequency gets damped, on the average, at .
The energy flux densities of the three modes are calculated as high as . The “inflationary” values of these energy flux densities result from the group velocities of the waves which are three orders of magnitude higher than the speed of light in vacuum! Therefore, energy flux densities of these modes have no physical meaning since they are calculated by equation (28) containing the group velocity of the waves.
Jackson (1975) noted that the group velocity often becomes larger than the speed of light in a dispersive medium, but ideas of special relativity does not violated, because approximations caused that the transport of energy occurs with the group velocity are no longer valid. Besides, Venutelli (2010) referring to Lombardi (2002) conclude that physical properties of a wave packet should not be described by group velocity in a medium with anomalous dispersion.
3.2.2 Solution for ()
If we substitute into Equation (24) we obtain dispersion relation given as
When assumed that in this equation, Alfven wave given with equation (29) is obtained.
Figure 7 shows the variation of the damping length scale of a magnetosonic wave with frequency and perpendicular diffusivity, assuming the value of viscosity as and values as and . For , while the waves with frequencies greater than get damped at the required distances for the heating of corona and solar wind acceleration, waves with smaller frequencies get damped at greater radial distances. When , while the lesser frequency waves get damped at the required distances for perpendicular diffusivity smaller than , higher frequency waves get damped at diffusivity values greater than . When the , waves get damped generally at smaller radial distances. There is a small region wherein required damping occurs for greater perpendicular diffusivity values.
In Figure 8, by assuming different values (i.e. ) we plotted the variation of the damping length scale of the wave having a frequency with viscosity and perpendicular diffusivity. While damping occurs independent of diffusivity when the viscosity has a value within the range for , in case have greater values, damping of waves occurs within the larger range of viscosity. When damping of waves occurs independent of diffusivity for small viscosity () and perpendicular diffusivity values within the range . For the viscosity values within the range it is seen that the value of the perpendicular diffusion coefficient increases linearly with viscosity for all the damping length scales.
Although the parameter ranges giving required damping length scales for these waves have been determined, it is seen that the calculated wave fluxes are inadequate for heating the corona and solar wind acceleration. These waves can not replace the loss energy in NPCH.
3.2.3 Solution for ()
Let us substitute (,) given into equation (24) using Taylor series expansion and after a lengthy effort we reach the dispersion relation as Equation (A2) in the Appendix.
The solution of the dispersion relation (A2) yields five modes. Solutions of three modes are shown in Figures 9 and 10. Other two mode have very long damping length scales, therefore we didn’t give their graphs.
In Figure 9, assuming that viscosity has a constant value (), variations of the damping length scales of Mode 1 (top), Mode 2 (middle) and Mode 3 (bottom) with wave frequency and perpendicular diffusivity are shown for values.
In case , the damping length scale of Mode 1 have required values for heating corona and solar wind acceleration. When , in order the waves with frequencies smaller than to be damped within required distances, perpendicular diffusivity should have values greater than . For , the region of the damping length scale with range disappears. Waves get damped very close to the sun.
Damping length scale of Mode 2 is more or less independent of perpendicular diffusivity. However, as increases, waves with lower frequencies (i.e. ) get damped at further radial distances instead of required ones for perpendicular diffusivity value .
Mode 3 shows similar variation to Mode 2. But, in any case, waves with frequencies smaller than get damped at very far radial distances.
In Figure 10, assuming , the variation of damping length scales of Mode 1 (top), Mode 2 (middle) and Mode 3 (bottom) having frequency with viscosity and perpendicular diffusivity are shown.
When viscosity values are taken within the range and the value of perpendicular diffusivity is greater than , damping of Mode 1 occurs at the required distances. For this wave, larger viscosity values are required in order the wave damping occuring at required distances when smaller values of perpendicular diffusivity are assumed. When reaches a value , waves get damped at the required distances only when higher perpendicular diffusivity values are assumed.
For Mode 2 in cases of and , when the perpendicular diffusivity is less than , there is a region around wherein waves get damped. In this region, damping is independent of viscosity values. For greater perpendicular diffusivity values () a secondary damping region emerges. This region disappears as the value of increases and the first damping region takes over.
Mode 3 displays a similar structure as the Mode 2 but does not show a secondary damping region.
3.3 Linear Approximation
In order to compare the results of quasi-linear and linear approximations in this sub-section, we neglect nonlinear term in the momentum equation (16) and then we obtain dispersion relation as given:
The dispersion relation has two roots: one root gives damping Alfven waves. We have given the solutions of this wave already in subsection 3.1.2. The