2. Dynamics of the Inertial Alfvén Wave

The nonlinear governing equation of the finite-frequency (ω ₀ < ω_ci) IAW is assumed to be propagating in the x − y − z plane, i.e., $\stackrel{⟶}{k}=k_x\hat{x}+k_y\hat{y}+k_z\hat{z}$ having background magnetic field B ₀ along the z-direction, i.e., ${\stackrel{⟶}{B}}_0=B_0\hat{z}$ , and the dynamical equation for the 3D-IAW can be obtained as follows:

(1) $\begin{array}{}\frac{{\partial }^2{\tilde{A}}_z}{\partial t^2}={\lambda }_e{}^2\left(\frac{{\partial }^4{\tilde{A}}_z}{\partial t^2\partial x^2}+\frac{{\partial }^4{\tilde{A}}_z}{\partial t^2\partial y^2}\right)+\frac{V_A{}^2}{{\omega }_{ci}{}^2}\left({\omega }_{ci}{}^2+\frac{{\partial }^2}{\partial t^2}\right)\left(1-\frac{n}{n_0}\right)\frac{{\partial }^2{\tilde{A}}_z}{\partial z^2}.\end{array}$

Here, $n\approx {\tilde{n}}_e\approx {\tilde{n}}_i$ and ${\lambda }_e=c/{\omega }_{pe}$ is the electron inertial length. Equation (1) gives the dispersion relation as follows:

(2) $\begin{array}{}\frac{{\omega }_0{}^2}{V_A{}^2{k}_{0z}{}^2}=\frac{1}{\left(1+{\lambda }_e{}^2\left(k_{0x}{}^2+k_{0y}{}^2\right)+{\lambda }_i{}^2{k}_{0z}{}^2\right)}.\end{array}$

Here, ${\lambda }_i=c/{\omega }_{pi}$ is an ion inertial length. The term ${\lambda }_i{}^2{k}_{0z}{}^2$ in equation (2) appears due to the finite frequency correction (ω₀/ω_ci).

We seek the solution of equation (1) for vector potential A_z as follows:

(3) $\begin{array}{}{\tilde{A}}_z=A_z\left(x,y,z,t\right)e^{i\left(k_{0x}x+k_{0y}y+k_{0z}z-{\omega }_{0}t\right)}.\end{array}$

Using equation (3) in equation (1), the following equation has been obtained for the case when ${\partial }_z{A}_z\ll k_{0z}{A}_z,$

(4) $\begin{array}{c}i\frac{2{\omega }_0\left(1+{\lambda }_e{}^2{k}_{0\perp }{}^2+\tau \right)}{V_A{}^2{k}_{0z}{}^2\left(1-\alpha \right)}\frac{\partial A_z}{\partial t}-2i{k}_{0\perp }\frac{{\omega }_0{}^2{\lambda }_e{}^2}{V_A{}^2{k}_{0z}{}^2\left(1-\alpha \right)}\left(\frac{\partial A_z}{\partial x}+\frac{\partial A_z}{\partial y}\right)-\frac{{\omega }_0{}^2{\lambda }_e{}^2}{V_A{}^2{k}_{0z}{}^2\left(1-\alpha \right)}\left(\frac{{\partial }^2{A}_z}{\partial x^2}+\frac{{\partial }^2{A}_z}{\partial y^2}\right)+i\frac{2}{k_{0z}}\frac{\partial A_z}{\partial z}+\frac{n}{n_0}{A}_z=0.\end{array}$

Here, $\alpha ={\omega }_0{}^2/{\omega }_{ci}^2,\tau =V_A{}^2{k}_{0z}^2/{\omega }_{ci}^2$ , k_0x, k_0y (k_0z) is the component of the wave vector perpendicular (parallel) to $B_0\hat{z}$ , and ω ₀ is the frequency of the 3D-IAW.

3. Dynamics of the Magnetosonic Wave

Let us assume that the low-frequency magnetosonic wave is propagating perpendicularly with respect to the background magnetic field along the x− axis, i.e., $\stackrel{⟶}{k}=k_x\hat{x}$ . The dynamical equation for the PMSW can be obtained by the standard approach using the basic plasma equations mentioned in the following:

1. (i) The equation of motion:

   (5) $\begin{array}{}\frac{\partial {\stackrel{⟶}{v}}_j}{\partial t}=\frac{q_j}{m_j}{}_{\stackrel{⟶}{E}}+\frac{q_j}{c{m}_j}\left({\stackrel{⟶}{v}}_j\times {\stackrel{⟶}{B}}_0\right)-\frac{{\gamma }_{j}k{T}_j}{m_j}\stackrel{⟶}{\nabla }\frac{n_j}{n_0}+{\stackrel{⟶}{F}}_j.\end{array}$

2. (ii) The continuity equation:

   (6) $\begin{array}{}\frac{\partial n}{\partial t}+\stackrel{⟶}{\nabla }.\left(n{\stackrel{⟶}{v}}_j\right)=0.\end{array}$

3. (iii) Faraday’s law:

   (7) $\begin{array}{}\left(\stackrel{⟶}{\nabla }\times \stackrel{⟶}{E}\right)=-\frac{1}{c}\frac{\partial \stackrel{⟶}{B}}{\partial t}.\end{array}$

Here, the index j = e or i accounts for the electrons and ions, respectively, q _i = −q _e = e is the charge, $j=e{n}_0\left(v_i-v_e\right)$ is the current density, v_j is the velocity, n₀ is the background number density, m_j and T_j are the masses and temperature of jth species, and ${\stackrel{⟶}{F}}_j=-\left(m_j\left({\stackrel{⟶}{\upsilon }}_j.\stackrel{⟶}{\nabla }\right){\stackrel{⟶}{\upsilon }}_j-q_j/c\left({\stackrel{⟶}{\upsilon }}_j\times {\stackrel{⟶}{B}}_0\right)\right)$ is the ponderomotive force due to 3D-IAW. Putting the values of ${\stackrel{⟶}{v}}_j$ in the wave equation and taking the y component of that, one can have

(8) $\begin{array}{}\frac{{\partial }^2{E}_y}{\partial x^2}-\frac{1}{c^2}\frac{{\partial }^2{E}_y}{\partial t^2}-\frac{1}{v_A^2}\frac{{\partial }^2{E}_y}{\partial t^2}=\frac{4\pi n_0{T}_e}{c{B}_0}\frac{{\partial }^2}{\partial x\partial t}\left(\frac{n_e}{n_0}\right)+\frac{4\pi n_{0}e}{c^2}\left(-\frac{\omega }{i{\omega }_{cj}}\frac{F_{jx}}{m_j}-\frac{{\omega }^2}{i{\omega }_{cj}^2}\frac{F_{jy}}{m_j}\right).\end{array}$

Here, ions are assumed to be cold and $\omega \ll {\omega }_{ci}$ . The electron continuity equation yields

(9) $\begin{array}{}\frac{\partial }{\partial t}\left(\frac{n_e}{n_0}\right)=-\frac{\partial }{\partial x}\left(\frac{c{E}_y}{B_0}\right).\end{array}$

Components of ponderomotive force are given as

(10) $\begin{array}{}{F}_{ex}={\left(\frac{{\omega }_o}{B_0}\right)}^2\left(-\frac{m_e}{4}\frac{{\left(1+\epsilon \right)}^2{k}_{\perp 0}^2}{2k_{z0}^2}+\frac{m_i}{8}\frac{\left(1+\epsilon \right)k_{\perp 0}^2}{\left(1-\alpha \right)k_{z0}^2}\right)\frac{\partial }{\partial x}{\left(A_{0z}\right)}^2,\end{array}$

(11) $\begin{array}{}{F}_{ey}=\frac{m_e}{4}{\left(\frac{{\omega }_0}{B_0}\right)}^2\frac{{\left(1+\epsilon \right)}^2{k}_{\perp 0}^2}{2k_{z0}^2}\frac{\partial }{\partial x}{\left(A_{0z}\right)}^2,\end{array}$

(12) $\begin{array}{}{F}_{ix}=\frac{m_i}{4}{\left(\frac{{\omega }_o}{B_0}\right)}^2\left(-\frac{{\left(1+\epsilon \right)}^2\left(1+\alpha \right)k_{\perp 0}^2}{{\left(1-\alpha \right)}^2{k}_{z0}^2}+\frac{\epsilon }{\alpha }\right)\frac{\partial }{\partial x}{\left(A_{0z}\right)}^2,\end{array}$

(13) $\begin{array}{}{F}_{iy}=\frac{m_i}{4}{\left(\frac{{\omega }_o}{B_0}\right)}^2\left(\frac{{\left(1+\epsilon \right)}^2{k}_{\perp 0}^2}{2{\left(1-\alpha \right)}^2{k}_{z0}^2}\left(1+\left(\frac{2i}{1-\alpha }\right)\left(\frac{{\omega }_0}{{\omega }_{ci}}\right)\right)\right)\frac{\partial }{\partial x}{\left(A_{0z}\right)}^2.\end{array}$

Substituting equations (9), (11), and (12) into equation (8), one obtains

(14) $\begin{array}{c}\left(\left(1+\beta \right)\frac{{\partial }^2}{\partial x^2}-\frac{1}{V_A{}^2}\frac{{\partial }^2}{\partial t^2}\right)\left(\frac{n_e}{n_0}\right)=\frac{{\omega }_0^2}{4V_A{}^2{B}_0{}^2}\left(-\frac{\left({\left(1+\epsilon \right)}^2{k}_{0\perp }{}^2\right)}{2k_{0z}^2}\left(\frac{\left(1+\alpha \right)}{{\left(1-\alpha \right)}^2}+\frac{m_e}{m_i}\right)+\left(\frac{\epsilon }{\alpha }+\frac{k_{0\perp }{}^2}{k_{0z}^2}\right)\right)\left(\frac{{\partial }^2\left({\left(A_z\right)}^2\right)}{\partial x^2}\right).\end{array}$

Here, $k_{0\perp }{}^2=k_{0x}{}^2+k_{0y}{}^2,$ $\beta =C_s^2/V_A^2$ , $\epsilon =k_{0\perp }{}^2{\lambda }_e{}^2$ , and $v_A={\left(B_0^2/4\pi n_0{m}_i\right)}^{1/2}$ .

Equation (13) represents the dynamical equation of the PMSW, and its right-hand side represents the ponderomotive force due to 3D-IAW.

After normalization of equations (4) and (13), these can be written in the dimensionless form as

(15) $\begin{array}{}i\frac{\partial A_z}{\partial t}-i{\xi }_1\frac{\partial A_z}{\partial x}-{\xi }_1\frac{\partial A_z}{\partial y}-{\xi }_2\frac{{\partial }^2{A}_z}{\partial x^2}-{\xi }_2\frac{{\partial }^2{A}_z}{\partial y^2}+i\frac{\partial A_z}{\partial z}+n{A}_z=0,\end{array}$

(16) $\begin{array}{}\left(\frac{{\partial }^2}{\partial x^2}-{\xi }_3\frac{{\partial }^2}{\partial t^2}\right)n=-\frac{{\partial }^2\left({\left(A_z\right)}^2\right)}{\partial x^2}.\end{array}$

Here, ${\xi }_1=2{\omega }_0{}^2/V_A{}^2{k}_{0z}{}^2{k}_{0\perp }{\lambda }_e\left(1-\alpha \right)$ , ${\xi }_2={\omega }_0^2/V_A{}^2{k}_{0\perp }{}^2{\lambda }_e{}^2{k}_{0z}{}^2\left(1-\alpha \right)$ , and ${\xi }_3\approx V_A^2{\lambda }_e^2/4\left(1+\beta \right){\left(k_{0z}^2\left(1-\alpha \right)/{\omega }_0\left(1+{\lambda }_e^2{k}_{0\perp }^2+\tau \right)\right)}^2$ . The normalizing parameters are $x_n\approx y_n\approx {\lambda }_e,$ $z_n\approx 1/2k_{0\perp }^2{\lambda }_e^2{k}_{0z},$ $t_n\approx 2{\omega }_0\left(1+{\lambda }_e{}^2{k}_{0\perp }^2+\tau \right)/V_A{}^2{k}_{0z}^2\left(1-\alpha \right),$ $n_n\approx k_{0\perp }{}^2{\lambda }_e{}^2{n}_0$ , and ${\left(A_z\right)}_n\approx {\left(4{\lambda }_e^2{V}_A^2\left(1+\beta \right)/{\omega }_0^2\left(-\left({\left(1+\epsilon \right)}^2\left(k_{0x}^2+k_{0y}^2\right)\right)/2k_{0z}^2\right)\left(\left(1+\alpha \right)/{\left(1-\alpha \right)}^2+m_e/m_i\right)+\left(\epsilon /\alpha +\left(k_{0x}^2+k_{0y}^2\right)/k_{0z}^2\right)\right)}^{1/2}{B}_0.$

4. Numerical Simulation

Numerical simulation has been performed using the pseudo-spectral method for equations (13) and (14). The wave numbers of perturbation are taken as α_x = α_y = α_z = 0.2 which is normalized by $x_n^{-1}$ , $y_n^{-1}$ , and $z_n^{-1}$ , respectively. Simulation has been carried out with (64)³ grid points, and a periodic spatial domain of $\left(2\pi /{\alpha }_x\right)\times \left(2\pi /{\alpha }_y\right)\times \left(2\pi /{\alpha }_z\right)$ with the initial conditions of simulation is as follows:

(17) $\begin{array}{}{A}_z\left(x,y,z,0\right)=A_{z0}\left(1+0.1\cos \left({\alpha }_{x}x\right)\right)\left(1+0.1\cos \left({\alpha }_{y}y\right)\right)\left(1+0.1\cos \left({\alpha }_{z}z\right)\right),\end{array}$

(18) $\begin{array}{}n\left(x,y,z,0\right)=-{\left(A_z\left(x,y,z,0\right)\right)}^2.\end{array}$

Here, |A_z0| = 1 is the amplitude of the homogenous pump 3D-IAW. An algorithm for the nonlinear Schrödinger (NLS) equation has been studied and tested against its invariant in order to carry out the simulation of equations (13) and (14).

The accuracy was determined by consistency of the number $N={\sum }_k{\left(A_{zk}\right)}^2$ in the case of cubic NLS equation. During computation, the conserved quantity was preserved to the order of 10⁻⁵. After testing algorithm of cubic NLS, it has been modified for solving dimensionless equations (13) and (14), which is used to study nonlinear coupling of 3D-IAW with PMSW and the resulting formation of localized structures, turbulent scaling, and density cavities.

The values of ξ ₁ and ξ ₂ can be evaluated from the low-β plasma parameters applicable to aurora. For the application purpose in low-β plasma, the typical parameters for the auroral altitude of 1700 km [Reference Wu, Huang and Wang11] are as follows: $B_0\approx 0.3G,$ $n_0\approx 5\times {10}^{3}c{m}^{-3},$ and $T_e\approx 1.16\times {10}^{4}K.$ Using these parameters, one can find $V_A\approx 9.25\times {10}^8\text{cm}/\text{s},$ ${\lambda }_e\approx 7.52\times {10}^3\text{cm},$ ${\omega }_{ci}\approx 2.87\times {10}^3{\sec }^{-1},{\rho }_s\approx 481.80\text{cm},\text{and}{c}_s=1.38\times {10}^6\text{cm}/\text{s}.$ For $k_{0\perp }{\lambda }_e\approx 0.30$ and ${\omega }_0/{\omega }_{ci}\approx 0.30$ , one can calculate $k_{0x}\approx 1.19\times {10}^{-5}{\text{cm}}^{-1},$ $k_{0y}\approx 1.19\times {10}^{-5}{\text{cm}}^{-1},$ $k_{0z}\approx 6.38\times {10}^{-7}{\text{cm}}^{-1}$ , and ${\omega }_0\approx 0.574\times {10}^3{\sec }^{-1}$ . The normalizing parameter values are $x_n\approx 7.52\times {10}^3\text{cm},$ $y_n\approx 7.52\times {10}^3\text{cm},$ $z_n\approx 9.66\times {10}^7\text{cm},t_n\approx 0.11\sec ,n_n\approx 0.16\times {10}^3{\text{cm}}^{-3}$ , and ${\left(A_z\right)}_n\approx 5.36\times {10}^3\text{G}-\text{cm}$ .