1. Introduction
In 1958, Eugene Parker proposed a physical mechanism to explain how the Sun's outer atmosphere, primarily composed of hydrogen plasma, could be accelerated to supersonic speeds. This acceleration would cause a continuous outward flow of plasma from the Sun, known as the solar wind [1]. The existence of the solar wind was later confirmed through direct observations by NASA's Mariner 2 spacecraft passing by Venus in 1962 [2]. One potential mechanism for transferring energy from the lower solar atmosphere to the solar wind and aiding in its acceleration is through the presence of hydromagnetic waves [3]. These waves can originate either in the solar corona [4] or the lower solar atmosphere and subsequently propagate into the corona [3].
Swift [5] has shown that the existence of a wave electric field becomes apparent when the perpendicular wavelength of shear Alfvén waves aligns with the electron inertial length, and this alignment enables inertial scale waves to facilitate the acceleration of electrons parallel to the magnetic field. Inertial Alfvén waves have been suggested to be participating in particle acceleration in various space plasma systems such as in the Earth's ionosphere [6], Jovian flux tubes [7], Ganymede's footprint tail aurora [8] etc. Thompson and Lysak in 1996 [6] incorporated conservation of energy into their model by allowing particles (electrons in their case) to influence the wave via Landau damping. Here we have utilized a similar approach in which inertial Alfvén waves deliver their energy to resonant solar plasma particles via Landau damping. We have particularly focused on how the charged particles speed up after receiving energy from the waves. We used a kinetic model and varying parameters such as the parallel wavenumber (k∥), temperature (T), ambient magnetic field (B0), and particle number density (n0) to understand the behavior of these waves across different solar regions. The chosen parameters fall within the reported range in observations [9–11].
2. Mathematical model
2.1. Inertial Alfvén waves
The dispersion relation, derived from the general Vlasov–Maxwell system, for inertial Alfvén waves is given by [12]
$\begin{eqnarray}D=\left|\begin{array}{ll}{\epsilon }_{{xx}}-{n}_{\parallel }^{2} & {n}_{\parallel }{n}_{\perp }\\ {n}_{\parallel }{n}_{\perp } & {\epsilon }_{{zz}}-{n}_{\perp }^{2}\end{array}\right|,\end{eqnarray}$
where n∥ (=k∥c/ω) and n⊥ (=k⊥c/ω) are the parallel and perpendicular refractive indices, respectively. The expressions for εxx and εxx are given by [13] $\begin{eqnarray*}{\epsilon }_{{xx}}=\frac{{c}^{2}}{{V}_{{\rm{A}}}^{2}}\left(1-\frac{3}{4}{k}_{\perp }^{2}{\rho }_{{\rm{i}}}^{2}\right),\end{eqnarray*}$
and $\begin{eqnarray*}{\epsilon }_{{zz}}=-\displaystyle \frac{{\omega }_{{\rm{pe}}}^{2}}{{\omega }^{2}}+2{\rm{i}}\sqrt{\pi }\left(\displaystyle \frac{{\xi }_{{\rm{oe}}}{\omega }_{{\rm{pe}}}^{2}}{{k}_{\parallel }^{2}{v}_{{\rm{Te}}}^{2}}\right)\exp (-{\xi }_{{\rm{oe}}}^{2}),\end{eqnarray*}$
where VA is the Alfvén speed, ρi is the ion gyroradius, ωpe is the plasma frequency, vTe is the electron thermal velocity, and ξoe = ω/k∥vTe. All other symbols have their usual meanings. Solving for the real and imaginary parts, respectively, we get [12] $\begin{eqnarray}{\omega }_{{\rm{r}}}^{2}=\displaystyle \frac{{k}_{\parallel }^{2}{V}_{{\rm{A}}}^{2}}{1+\tfrac{{k}_{\perp }^{2}{c}^{2}}{{\omega }_{{\rm{pe}}}^{2}}},\end{eqnarray}$
and $\begin{eqnarray}{\omega }_{{\rm{i}}}=-\displaystyle \frac{{k}_{\perp }^{2}{c}^{2}{k}_{\parallel }{V}_{{\rm{A}}}^{4}\sqrt{\pi }\exp (-{\xi }_{{\rm{oe}}}^{2})}{{\omega }_{{\rm{pe}}}^{2}{v}_{{\rm{Te}}}^{3}{\left(1+\tfrac{{k}_{\perp }^{2}{c}^{2}}{{\omega }_{{\rm{pe}}}^{2}}\right)}^{3}}.\end{eqnarray}$
2.2. Poynting flux for Alfvén waves
When the electromagnetic perturbations are sinusoidal, the time derivative term, averaging over a complete cycle, in the differential form of Poynting's theorem ($\tfrac{\partial u}{\partial t}=\ -{\boldsymbol{J}}\cdot {\boldsymbol{E}}-{\rm{\nabla }}\cdot {\boldsymbol{S}}$) is zero [13]. The terms S, E, and J can be represented as phasors, expressed as5 ) to get7 ) with ω = ωr + iωi, we get8 ) in Equation (6 ) we have9 ) is given by [11]10 ) is the expression for the Poynting flux associated with inertial Alfvén waves. It represents the energy transport of the waves as a function of distance and allows us to quantify the amount of electromagnetic energy that is transferred from the waves to the plasma as they propagate spatially away from the reference point z = 0. Initially, at z = 0 the waves are excited and exhibit a Poynting flux with a magnitude of S(0). However, as the waves progress, the Poynting flux gradually diminishes.
$\begin{eqnarray*}{\boldsymbol{S}}=\displaystyle \frac{1}{2{\mu }_{0}}\mathrm{Re}({{\boldsymbol{E}}}^{* }\times {\boldsymbol{B}}),\end{eqnarray*}$
and $\begin{eqnarray*}{\boldsymbol{J}}\cdot {\boldsymbol{E}}=-\frac{1}{2}{\rm{Re}}({{\boldsymbol{J}}}^{\ast }\cdot {\boldsymbol{E}}).\end{eqnarray*}$
Hence $\begin{eqnarray}{\rm{\nabla }}\cdot {\boldsymbol{S}}=-\frac{1}{2}\mathrm{Re}({{\boldsymbol{J}}}^{* }\cdot {\boldsymbol{E}}).\end{eqnarray}$
The wave is considered to propagate in the x–z plane, therefore Sy is zero. Furthermore, Sx ≪ Sz [13], as Alfvén waves generally carry most of the energy along the magnetic field lines. For simplicity, we will use S for Sz. Following the steady-state Poynting's theorem we can write [13] $\begin{eqnarray}\displaystyle \frac{\partial S}{\partial z}=-\displaystyle \frac{1}{2}\mathrm{Re}({J}_{z}^{* }{E}_{z}).\end{eqnarray}$
The term Jz can be found using Ampere's law ${J}_{z}=\tfrac{1}{{\mu }_{0}}\tfrac{\partial {B}_{y}}{\partial x}$, and eliminating the operator we have $\begin{eqnarray*}{J}_{z}=\displaystyle \frac{{\text{i}k}_{x}{B}_{y}}{{\mu }_{0}}.\end{eqnarray*}$
Substituting Jz in Equation ( $\begin{eqnarray*}\displaystyle \frac{{\rm{\partial }}S}{{\rm{\partial }}z}=\displaystyle \frac{1}{2{\mu }_{0}}{\rm{Re}}({\text{i}k}_{x}{B}_{y}^{\ast }{E}_{z}),\end{eqnarray*}$
$\begin{eqnarray}\displaystyle \frac{{\rm{\partial }}S}{{\rm{\partial }}z}={k}_{x}{\rm{Re}}\left(\text{i}\displaystyle \frac{{E}_{z}}{{E}_{x}}\right)S.\end{eqnarray}$
The ratio $\tfrac{{E}_{z}}{{E}_{x}}$ can be obtained by solving row one of the following matrix [13]: $\begin{eqnarray*}\left(\begin{array}{cc}{\epsilon }_{{xx}}-{n}_{\parallel }^{2} & {n}_{\parallel }{n}_{\perp }\\ {n}_{\parallel }{n}_{\perp } & {\epsilon }_{{zz}}-{n}_{\perp }^{2}\end{array}\right)\left(\begin{array}{c}{E}_{x}\\ {E}_{z}\end{array}\right)=0.\end{eqnarray*}$
Hence $\begin{eqnarray}\displaystyle \frac{{E}_{z}}{{E}_{x}}=\displaystyle \frac{{c}^{2}{k}_{z}^{2}-{\omega }^{2}{\epsilon }_{{xx}}}{{c}^{2}{k}_{z}{k}_{x}}.\end{eqnarray}$
Rewriting Equation ( $\begin{eqnarray*}\frac{{E}_{z}}{{E}_{x}}=\frac{{c}^{2}{k}_{z}^{2}-({\omega }_{\text{r}}^{2}-{\omega }_{\text{i}}^{2}+2\text{i}{\omega }_{\text{r}}{\omega }_{\text{i}}){\epsilon }_{{xx}}}{{c}^{2}{k}_{z}{k}_{x}}.\end{eqnarray*}$
Neglecting ωi in comparison with ωr as it is small and considering the imaginary part $\begin{eqnarray}{\rm{Re}}\left(\text{i}\displaystyle \frac{{E}_{z}}{{E}_{x}}\right)=\displaystyle \frac{2{\epsilon }_{{xx}}{\omega }_{\text{r}}{\omega }_{\text{i}}}{{c}^{2}{k}_{x}{k}_{z}}.\end{eqnarray}$
Substituting Equation ( $\begin{eqnarray}\displaystyle \frac{{\rm{\partial }}S}{{\rm{\partial }}z}=\displaystyle \frac{2{\epsilon }_{{xx}}{\omega }_{\text{r}}{\omega }_{\text{i}}}{{c}^{2}{k}_{z}}S.\end{eqnarray}$
The solution of Equation ( $\begin{eqnarray}S(z)=S(0)\exp \left(\displaystyle \frac{2{\epsilon }_{{xx}}{\omega }_{\text{r}}{\omega }_{\text{i}}}{{c}^{2}{k}_{z}}z\right).\end{eqnarray}$
Equation (2.3. Net velocity of resonant particles
The kinetic energy of the particles passing through unit surface area per unit time (kinetic energy flux) can be written as [14]
$\begin{eqnarray*}\mathrm{Kinetic}\ \mathrm{energy}\ \mathrm{flux}=\displaystyle \frac{1}{2}\rho {v}^{3},\end{eqnarray*}$
where ρ is the mass density and v is the particle velocity.The wave energy per unit area per unit time is given by the Poynting flux expression S. According to the law of conservation of energy
$\begin{eqnarray*}\displaystyle \frac{1}{2}\rho {v}^{3}={S}_{z}.\end{eqnarray*}$
Solving for v, we obtain $\begin{eqnarray*}v={\left(\displaystyle \frac{2{S}_{z}}{\rho }\right)}^{\tfrac{1}{3}},\end{eqnarray*}$
which is the speed the particles gain from the wave.The particles that fulfill the resonant condition, interacting with the wave and receiving energy from the wave, have velocity that is equal to the sum of the particles' initial velocity (which must be approximately equal to the wave phase velocity for particles to resonate with the wave) and the energy the particles gain from the wave as it damps. Hence
$\begin{eqnarray}{v}_{\mathrm{net}}=\displaystyle \frac{{\omega }_{{\rm{r}}}}{{k}_{\parallel }}+{\left(\displaystyle \frac{2{S}_{z}}{\rho }\right)}^{\tfrac{1}{3}},\end{eqnarray}$
where vnet is the net speed of resonant particles. The parallel component of the wave vector is considered in the equation because these waves mostly deliver their energy along the magnetic field lines.3. Results and discussion
In our analysis, we have made an assumption regarding the solar environment being studied, with particle number density n0 estimated to be of the order of 1011 cm−3 [10]. The background magnetic field, B0, is approximately 250 G [9]. Based on these assumptions, the ratio of thermal pressure to magnetic pressure ($4\pi {n}_{0}{T}_{\text{e}}/{B}_{0}^{2}$) is lower than the ratio of electron mass to ion mass (me/mi). As a result, the waves under consideration in this study are classified as inertial Alfvén waves.
In Figure 1 we see that the Poynting flux decays faster with the increase in parallel wavenumber (Figure 1(a)), and the wave delivers its energy to the particles at a faster rate as the parallel wavenumber increases (Figure 1(b)). The analysis of the expressions for the dispersion (Equation (2 )) and damping rate (Equation (3 )) of inertial Alfvén waves reveals a direct relationship with the parallel wavenumber. However, when examining the expressions for the Poynting flux (Equation (10 )) and the net velocity of resonant particles (Equation (11 )), an exponential decrease with the parallel wavenumber becomes evident, which is consistent with Figures 1(a) and (b). Figure 2 shows that the regions with higher temperatures have slower decay of Poynting flux (2(a)). The resonant particles have large velocities in regions where the temperature is high (2(b)).
Figure 1. Variation in (a) Poynting flux and (b) net velocity of resonant particles with distance in solar radii at different values of parallel wavenumber when perpendicular wavenumber (k⊥) is 5 cm−1, temperature (T) is 7000 K, ambient magnetic field (B0) has a value of 250 G, and particle number density (n0) is 1 × 1011 cm−3. |
Figure 2. Variation in (a) Poynting flux and (b) net velocity of resonant particles with distance in solar radii at different values of temperature when perpendicular wavenumber (k⊥) is 5 cm−1, parallel wavenumber (k∥) is 8 × 10−10 cm−1, ambient magnetic field (B0) has a value of 250 G, and particle number density (n0) is 1 × 1011 cm−3. |
In the equation for Poynting flux (Equation (10 )), the term in brackets has a direct relationship with the Alfvén speed (VA), which in turn is directly related to the magnetic field. Consequently, regions with a stronger magnetic field demonstrate a faster decay of the flux, as depicted in Figure 3(a). In addition to depending on the Poynting flux, the net velocity of resonant particles displays a direct relationship with the frequency (ωr). Based on the graph in Figure 3(b), it can be inferred that the initial velocity term in Equation (11 ) becomes more dominant in regions characterized by a stronger magnetic field.
Figure 3. Variation in (a) Poynting flux and (b) net velocity of resonant particles with distance in solar radii at different values of ambient magnetic field when perpendicular wavenumber (k⊥) is 5 cm−1, parallel wavenumber (k∥) is 8 × 10−10 cm−1, temperature (T) is 7000 K, and particle number density (n0) is 1 × 1011 cm−3. |
Figure 4. Variation in (a) Poynting flux and (b) net velocity of resonant particles with distance in solar radii at different values of particle number density when perpendicular wavenumber (k⊥) is 5 cm−1, parallel wavenumber (k∥) is 8 × 10−10 cm−1, temperature (T) is 7000 K, and ambient magnetic field (B0) has a value of 250 G. |
To conclude, the analysis of the results reveals that inertial Alfvén waves experience more rapid damping as the parallel wavenumber, strength of ambient magnetic field, and particle number density rise, and the temperature decreases. Consequently, the rate of energy transfer to resonant particles intensifies as the temperature drops and the parallel wavenumber and particle number density increase. Additionally, particles with higher initial velocities actively participate in Landau damping primarily in regions where the ambient magnetic field is more intense. This model, despite being simple, provides us with a good understanding of how inertial Alfvén waves behave in environments with different physical parameters in plasma (i.e., parallel wavenumber, temperature, magnetic field, and particle number density). Further work may be done using a more advanced and realistic model simulating the actual solar atmosphere. This model may be used for non-Maxwellian distributions as well.