Perspective Chapter: Numerical Simulation to Study Alfvén Waves

Abstract

This chapter presents the Alfvén waves self-interaction in approximately zero-$\beta$ plasma (cold plasma), which is applicable to coronal heating. We consider thermal to magnetic pressure ratio $\beta \approx 0$ (cold plasma). When we consider magnetic field transverse perturbation in approximately zero-$\beta$ plasma, the model dynamical equation of Alfvén wave self interaction turns out to be the modified Zakharov system equation. Numerical simulation has been carried out to study the effect of Alfvén wave self interaction and transverse perturbation, which results in the formation of magnetic field profile structures (coherent structures) and power spectrum for coronal heating and particle acceleration in space plasma. The investigated results reveals that the system of magnetic field profile structures reaches to quasi-steady state and the power spectral index approaches the $k^{-5/3}$, which is consistent with Kolmogorov scaling.

1. Introduction

Alfvén waves are low-frequency, electromagnetic waves in uniform magnetized plasma with a background magnetic field. It is well known that the mechanical effects of a magnetic field $B_0$ in a magnetized plasma are equivalent to an isotropic magnetic pressure $p_B=B_0^2/2{\mu }_0,$ combined with a magnetic tension $T_B=B_0^2/{\mu }_0$ along the magnetic field lines. Analogy with the theory of stretched strings suggests that this tension may lead to a transverse wave propagating along the field lines, with the Alfven velocity $V_A=\sqrt{T_B/\rho }=B_0^2/\sqrt{{\mu }_0\rho },$ where $\rho$ is the mass density of magnetized plasma. These waves were first predicted in classic work by H. Alfvén [1] and can be derived from magnetrohydrodynamic (MHD) equation. These waves propagate parallel to ambient magnetic field. Alfvén waves were first experimentally demonstrated in laboratory by Lundquist (in mercury) [2] and Lehnert (in sodium) [3]. These waves plays very important role in space plasma. so we should understand the mechanism of Alfvén wave formation. The Waves of any kind in nature are driven by some restoring force which opposes displacements in the system. So the restoring force is responsible for the Alfven wave generation, which can be understood by two physical principles in conducting fluid.

1. Lenz’s law applied to conducting Fluids- ‘Electrical currents induced by the motion of a conducting fluid through a magnetic field give rise to electromagnetic forces acting to oppose that fluids motion.’

2. Newton’s second law for Fluids – ‘A force applied to a fluids will result in a change in the momentum of the fluids proportional to the magnitude of the force and in the same direction.’

In general terms, the AWs are transient electromagnetic-hydromagnetic phenomenon occurring in magnetized plasma. These waves are traveling oscillations of ions and magnetic field. The ion mass density provides the inertia and the magnetic field lines tension provides the restoring force in magnetized plasma. The electron also oscillate in an Alfvén wave along the magnetic field lines, and they are responsible for the field-aligned current in Alfvén wave. Alfvén waves are electromagnetic wave, so they oscillation electric field. Inside Alfvén waves, electric and magnetic field oscillate perpendicular to the ambient magnetic field. Together with the electric and magnetic field yield a pointing vector that explains the energy flow of the wave. The oscillation of electric and magnetic field can be understood via an experiment [Davidson, 2001].

Let us look at a system of uniform magnetic field permeating a perfectly conducting fluid, which are uniformly flow initially to the magnetic field lines (Figure 1). The fluid flow will distort the magnetic field lines, so magnetic field lines become curved as shown in fig. (b). The curved of magnetic field lines produce a Lorentz force on the conducting fluid that opposes the next curvature as explained by Lenz’s Law. According to the Newton’s second law, the Lorentz force changes the momentum of conducting fluid, which pushing it to minimize the magnetic field line distortion and restore the system in its equilibrium state. The restoring force provides the basis for the transverse oscillation of magnetic field in conducting fluid. As the curvature of magnetic field lines increases, so the Lorentz force becomes strong enough to chance the direction of the conducting fluid flow from initial direction. Now the magnetic field lines are pushed back to their undistorted state and the Lorentz force associated with their curvature weakens until the field lines becomes strength again. The sequence of flow of magnetic field lines distortion will repeat, but in opposite direction. This cycle will continue indefinitely, until the dissipation will apply.

Figure 1.

Mechanism of Alfvén wave formation: (a) fluid velocity perpendicular to the uniform magnetic field (b) distorted magnetic lines giving rise to a Lorentz force that retards and eventually reverses the fluid velocity (c) field lines return to undisturbed position (d) reversed distortion and (e) repeats the cycle.

The electromagnetic field oscillation generated in magnetic field lines by fluid flow and the elastic string when it plucked, having same mechanism. For both, the tension as a restoring force, magnetic tension in case of magnetic field lines and elastic tension in case of the string. The both waves resultant oscillations propagating in direction perpendicular to their displacement. The resultant oscillation generated by restoring force is the shear Alfven wave.

The low-frequency $\left(\omega <<{\omega }_{\mathit{ci}}=\mathit{eB}/{\mathit{cm}}_i\right)$ and long wavelength $\left({\lambda }_L>>{\rho }_i=v_{\mathit{Ti}}/{\omega }_{\mathit{ci}}\right)$ perturbations apply in magnetized plasma, single- fluid MHD approach are more appropriate because $m_e\to 0$ and $n_e=n_i=n$. The low-frequency waves classified as the fast, slow and intermediate via phase speed of wave. The MHD analysis gives the following dispersion relation of Alfven waves are

$$v_A^2=\frac{{\omega }^2}{k_{\parallel }^2}=\frac{B_o^2}{4\mathit{\pi \rho }}$$E1

The dispersion relation (Eq. (1)) of Alfven wave phase speed is independent of $k_{\perp }$ and perturbation of magnetic field perpendicular to the $B_o$ and k.

$${\omega }^2=\frac{1}{2}{k}^2\left(v_A^2+c_s^2\right)\pm \frac{k^2}{2}\sqrt{{\left(v_A^2+c_s^2\right)}^2-4\frac{k_{\parallel }^2}{k^2}{c}_s^2{v}_A^2}$$E2

The “+” and “- “sign in Eq. (2) leading to the fast magnetosonic wave $\left({\omega }_f>{\omega }_i\right),$ where ${\omega }_i$ is the shear alfven frequency and slow magnetosonic wave $\left({\omega }_s<{\omega }_i\right)$.

For low-$\beta$ (Here $\beta =(2\pi n_{0}T/B_0^2=c_s^2/v_A^2<<1$ is the thermal to magnetic pressure ratio) limit the dispersion relation reduce as

$${\omega }_f^2=k^2{v}_A^2$$E3

In the low-$\beta$ limit the slow magnetosonic wave reduces to ion-acoustic wave and the dispersion relation can be written as

$${\omega }^2=k_{\parallel }^2{c}_s^2$$E4

The shear Alfvén wave characteristics depend on the ion gyroradius (${\rho }_s$) or the electron skin depth (${\lambda }_e$), which have determined as the dispersive Alfvén wave (Kletzing, 1994). The dispersion property of Alfvén waves is determined by the electron-$\beta$, where $\beta =(2\pi n_0{T}_e/B_0^2=\left(m_e/m_e\right)\left(2v_{\mathit{Te}}^2\right)$ and $v_T{e}^2=T_e/m_e$ is the electron thermal velocity. The Alfvén waves property depend upon the whether the Alfvén velocity is larger than the electron thermal speed ($v_{\mathit{Te}}<<v_A$, which is inirtial limit) or the smaller ($v_{\mathit{Te}}>>v_A,$ which is the kinetic limit). Therefore, for kinetic limit the electrons moves fast are able to respond adiabatically to wave field. For low-$\beta$ plasma ($\beta <<m_e/m_i$), the shear Alfvén wave call as the “Inertial Alfvén wave” whereas for intermediate-$\beta$ plasma ($m_e/m_i<<\beta <<1$), the shear Alfvén wave call as the “Kinetic Alfvén wave”. The inertial limit is applicable for cold plasma (Aurora) and while the kinetic limit applicable to the hot plasma (Solar corona and solar wind) (Vincena et al., 2004).

The Alfvén waves have various application in solar physics. Alfvén waves have been proposed as energy transportation mechanism for corona [4, 5] to the accelerate solar wind [6]. The coronal heating is one of the important unsolved problem in solar physics from decades [7, 8, 9, 10]. The coronal heating energy originated in convection zone to heat the coronal loops, but how this energy transported to coronal loops are not well understood. The one possibility of coronal heating is braided magnetic field dissipation in MHD models of coronal loops [11, 12]. Another one possibility to be the magnetic flux tubes, which generated by convective collapse [13, 14]. The transverse MHD waves can be formed in magnetic flux tube by interaction with granule scale convective flow [15]. These transverse MHD waves can propagate upward into the atmosphere and transport there energy in corona [16, 17, 18, 19]. Alfvén wave are prominent because they can propagate at large distance in corona and transport there energy to coronal loops. Alfvén wave have been observed in corona [20, 21, 22, 23], but the role of such waves are not well demonstrated. The 1.5D MHD simulation of torsional Alfvén waves explained, when the Alfvén waves are energized by random motion into photosphere, then the torsional Alfvén waves are sufficient to heat corona [24, 25]. The Alfvén wave nonlinearity generated by coupling of these waves to other MHD mode and leads to MHD shocks. Such type of shocks assume as coronal heating [26]. The 2.5D MHD model conclude that corona heated by fast and slow mode MHD shocks [27, 28].

The plasma exhibits a turbulent cascade over wide range of spectral scale [29]. The turbulent power spectrum consists several scales, in which various mechanism are possible in the transfer of energy from larger to smaller scales. The spectral power law behavior of the spectral indices steeper between power laws in ranges, leading to confirm different turbulent models [30, 31, 32, 33]. MHD turbulence plays a prominent role to transfer wave energy in corona [34]. MHD turbulence is responsible for the transportation of energy from large scale to small scale where it is dissipated [35]. Compressible MHD turbulence proposed as a possible mechanism for wave energy dissipation to heat corona [36, 37, 38]. The wave energy dissipation process cause by the damping of MHD wave to heat corona by Alfvén waves [39, 40]. The interaction between two Alfvén wave leading to the wave energy cascade form source region to energy dissipation region through the MHD turbulence [41, 42, 43, 44].

Alfvén waves are mixed mode having electric field parallel to the ambient magnetic field [45, 46]. Rosenthal and Bogdan [47, 48], numerically studied the wave propagation in a magneto-sphere and observed the positron generated both in fast and slow MHD wave to accelerate these wave to magnetized plasma. Their study explains the coupling of fast and slow mode waves, where the plasma beta unity. De Moortel et al. [49], studied the slow waves on 2D geometrical boundary with density variation. This author also shows the coupling between slow and fast and phase mixing of slow mode waves.

The coupling of fast wave and Alfvén wave has been studied by Parker [50] when the linear density gradient exist. When the nonlinear excitation exist, Nakariakov et al. [51], investigated fast waves nonlinear excitation by Alfvén wave leading to phase mixing in inhomogeneous background density profile. Their numerical simulation study showed that the Alfvén waves not only heat via phase mixing dissipation method but also through nonlinear coupling of fast wave. Such type of phase mixing have been studied by various authors to investigate the coronal heating [52, 53, 54, 55]. Thurgood and McLaughlim [56], studied the nonlinear damping mechanism for Alfvén wave at $\beta =0$ (cold plasma), when the nonlinear ponderomotive force generated by the Alfvén waves can excite fast wave leading to the shock formation (coronal heating). The direct coupling between Alfvén and slow wave has been studied by Zaqarashvili et al. [57] when $\beta =1$.

In this chapter we will study the Alfvén wave self-interaction with magnetic field transverse perturbation in cold plasma ($\beta \approx 0$). In this paper, we focus on the coronal heating by Alfvén wave self-interaction when consider the magnetic field transverse perturbation for cold plasma. This studies have carried out by numerical simulation method.

The content of chapter is organized as follows. In Section 2 the dynamical equation for Alfvén waves self interaction in cold plasma are derived. Section 3 consists the pseudo-spectral method. Section 4 explaines the numerical simulation and obtained results of our model equation. Finally, Section 5 deals with the conclusion of the work.

2. Model equation

We consider the Alfvén waves propagation in compressible, magnetically formed, and zero $\beta$ plasma. Here we derive the dynamical equation of Alfvén wave self interaction with the help of the MHD set of equation, which can be written as [51].

$$\frac{\partial \rho }{\partial t}+\nabla .\left(\mathit{\rho V}\right)=0$$E5

$$\frac{\partial B}{\partial t}=\nabla \times \left(V\times B\right)$$E6

$$\rho \frac{\partial V}{\partial t}+\rho \left(V.\nabla \right)V=-\frac{\left(\nabla \times B\right)\times B}{4\pi }$$E7

$$\nabla .B=0$$E8

which connect the density $\rho$, magnetic field B and plasma velocity V.

We write Eqs. (5)–(8) in Cartesian coordinate (x, y, z) form. Assume the perturbation in the y direction is zero. The Cartesian form of Eqs. (5)–(8) are

$$\frac{\partial \rho }{\partial t}+\frac{\partial }{\partial x}\left(\rho V_x\right)+\frac{\partial }{\partial z}\left(\rho V_z\right)=0,$$E9

$$\frac{\partial B_x}{\partial t}=-\frac{\partial }{\partial z}\left(V_z{B}_x-V_x{B}_z\right),$$E10

$$\frac{\partial B_y}{\partial t}=\frac{\partial }{\partial z}\left(V_x{B}_z-V_z{B}_y\right)-\frac{\partial }{\partial x}\left(V_x{B}_y-V_y{B}_x\right),$$E11

$$\frac{\partial B_z}{\partial t}=\frac{\partial }{\partial x}\left(V_z{B}_x-V_x{B}_z\right),$$E12

$$\rho \left[\frac{\partial V_x}{\partial t}+\left(V_x\frac{\partial V_x}{\partial x}+V_z\frac{\partial V_x}{\partial z}\right)\right]=-\frac{\left[-B_y\frac{\partial B_y}{\partial x}-B_z\left(\frac{\partial B_x}{\partial z}-\frac{\partial B_z}{\partial x}\right)\right]}{4\pi },$$E13

$$\rho \left[\frac{\partial V_y}{\partial t}+\left(V_x\frac{\partial V_y}{\partial x}+V_z\frac{\partial V_y}{\partial z}\right)\right]=-\frac{\left[-B_z\frac{\partial B_y}{\partial z}-B_x\frac{\partial B_y}{\partial x}\right]}{4\pi },$$E14

$$\rho \left[\frac{\partial V_z}{\partial t}+\left(V_x\frac{\partial V_z}{\partial x}+V_z\frac{\partial V_z}{\partial z}\right)\right]=-\frac{\left[B_y\frac{\partial B_y}{\partial z}+B_x\left(\frac{\partial B_x}{\partial z}-\frac{\partial B_z}{\partial x}\right)\right]}{4\pi }.$$E15

Further, we consider the equilibrium state of MHD system is in compressible plasma state of density ${\rho }_0$ with a uniform magnetic field ${\overrightarrow{B}}_0=B_0\widehat{z}$. Initially, the medium is flow freely. The finite amplitude perturbation are taken as $\overrightarrow{V}\left(x,y,t\right)=0+V\left(x,y,z\right)$ for flow, $\overrightarrow{B}\left(x,y,t\right)=B_0+B\left(x,y,z\right)$ for magnetic field and $\rho ={\rho }_0+{\rho }_1\left(x,y,z\right)$ for density.

Then, the finite amplitude perturbation of inhomogeneous equilibrium state written by set of following equation

$$\frac{\partial \rho }{\partial t}+\frac{\partial }{\partial x}\left({\rho }_0{V}_x\right)+\frac{\partial }{\partial z}\left({\rho }_0{V}_z\right)=L_1,$$E16

$$\frac{\partial B_x}{\partial t}-B_0\frac{\partial V_x}{\partial z}=L_2,$$E17

$$\frac{\partial B_y}{\partial t}-B_0\frac{\partial V_y}{\partial z}=L_3,$$E18

$$\frac{\partial B_z}{\partial t}+B_0\frac{\partial V_x}{\partial x}=L_4,$$E19

$${\rho }_0\frac{\partial V_x}{\partial t}-\frac{B_0\left(\frac{\partial B_x}{\partial z}-\frac{\partial B_z}{\partial x}\right)}{4\pi }=L_5,$$E20

$${\rho }_0\frac{\partial V_y}{\partial t}-\frac{B_0\left(\frac{\partial B_y}{\partial z}\right)}{4\pi }=L_6,$$E21

$${\rho }_0\frac{\partial V_z}{\partial t}=L_7.$$E22

Where the left-hand-side and right-hand–side terms govern the linear behavior and nonlinear behavior. The nonlinear terms $L_1$ to $L_7$, we get

$$L_1=-\frac{\partial }{\partial x}\left({\rho }_1\left(x\right)V_x\right)-\frac{\partial }{\partial z}\left({\rho }_1\left(x\right)V_z\right),$$E23

$$L_2=-\frac{\partial }{\partial z}\left(V_x{B}_x-V_x{B}_z\right),$$E24

$$L_3=-\frac{\partial }{\partial z}\left(V_z{B}_y-V_y{B}_z\right)+\frac{\partial }{\partial x}\left(V_y{B}_x-V_x{B}_y\right)$$E25

$$L_4=\frac{\partial }{\partial x}\left(V_z{B}_x-V_x{B}_z\right),$$E26

$$\begin{array}{l}{L}_5=-{\rho }_1\left(x\right)\left(V_x\frac{\partial V_x}{\partial x}+V_z\frac{\partial V_x}{\partial z}\right)-{\rho }_1\left(x\right)\frac{\partial V_x}{\partial t}-{\rho }_0\left(V_x\frac{\partial V_x}{\partial x}+V_z\frac{\partial V_x}{\partial z}\right)\\ -\frac{1}{4\pi }{B}_y\frac{\partial B_y}{\partial x}+\frac{1}{4\pi }{B}_z\left(\frac{\partial B_x}{\partial z}-\frac{\partial B_z}{\partial x}\right),\end{array}$$E27

$$L_6=-{\rho }_1\left(x\right)\frac{\partial B_y}{\partial t}-{\rho }_1\left(x\right)\left(V_z\frac{\partial V_y}{\partial z}+V_x\frac{\partial V_y}{\partial x}\right)-{\rho }_0\left(V_x\frac{\partial B_y}{\partial x}-V_z\frac{\partial V_y}{\partial z}\right)+\frac{\left(B_x\frac{\partial B_y}{\partial x}+B_z\frac{\partial B_y}{\partial z}\right)}{4\pi }.$$E28

$$\begin{array}{l}{L}_7=-{\rho }_1\left(x\right)\frac{\partial V_z}{\partial t}-{\rho }_0\left(V_x\frac{\partial V_z}{\partial x}+V_z\frac{\partial V_z}{\partial z}\right)-{\rho }_1\left(x\right)\left(V_x\frac{\partial V_z}{\partial x}+V_z\frac{\partial V_z}{\partial z}\right)-\frac{1}{4\pi }{B}_y\frac{\partial B_y}{\partial z}\\ -\frac{B_x\left(\frac{\partial B_x}{\partial z}-\frac{\partial B_z}{\partial x}\right)}{4\pi }.\end{array}$$E29

Eqs. (16)–(22) consist the information about the propagation of Alfvén waves and fast magnetosonic waves in the Cartesian form. We used Eqs. (16)–(22) to obtain the nonlinear dynamical equation of Alfvén waves.

$$\frac{{\partial }^2{B}_y}{\partial t^2}-{V_A}^2\frac{{\partial }^2{B}_y}{\partial z^2}=\frac{1}{4{\mathit{\pi \rho }}_0}\left[\frac{{\partial }^2{B}_y}{\partial x\partial t}+\frac{{\partial }^2{B}_y}{\partial z\partial t}\right]+\frac{B_0}{4\pi }\frac{{\partial }^2{B}_y}{\partial x\partial z}$$E30

where $V_A=B_0/\sqrt{4{\mathit{\pi \rho }}_0}$ is the Alfvén speed. The Eq. (30) describes Alfvén wave self interaction, which propagating in x-z plane, i.e. $\overrightarrow{k}=k_x\widehat{x}+k_z\widehat{z}$.

Consider the solution of Eq. (30) is as

$$B_y={\tilde{B}}_y\left(x,z,t\right)e^{\left(k_{0x}x+k_{0z}z-\mathit{\omega t}\right)}$$E31

Substituting Eq. (31) into Eq. (30) and simplify equation for the case $\partial {\tilde{B}}_y<<k_{0_x}{\tilde{B}}_y$, here we get

$$\left(-2\mathit{i\omega }-{\mathit{ik}}_{0x}-{\mathit{ik}}_{0z}\right)\frac{\partial {\tilde{B}}_y}{\partial t}-{V_A}^2\frac{\partial {\tilde{B}}_y}{\partial z}-i\left(\frac{B_0{k}_{0z}}{4\pi }-\frac{\omega }{4{\mathit{\pi \rho }}_0}\right)\frac{\partial {\tilde{B}}_y}{\partial x}-{\mathrm{V}}_A^2\frac{{\partial }^2{\tilde{B}}_y}{\partial z^2}+\frac{B_0}{4\pi }{k}_{0x}{k}_{0z}{\tilde{B}}_y=0$$E32

Where $\omega$ is the Alfvén wave frequency.

The total magnetic field can be written in the form of field vector and transverse perturbation such that,

$${\tilde{B}}_y={\tilde{B}}_{y1}+{\tilde{B}}_{y2}$$E33

where ${\tilde{B}}_{y1}$ and ${\tilde{B}}_{y2}$ is the field vector and perturbation of Alfvén wave, then

$${\left.{\tilde{B}}_{y2}{\tilde{B}}_{y2}^{\ast }\right|}_{x,z,t\ne 0}={\tilde{B}}_{y21}^2\mathit{exp}\left[i\left(k_{0x}x+k_{0z}z-\mathit{\omega t}\right)\right]$$E34

The perturbed field form of Eq. (28) can be written as

$$\begin{array}{l}\left(-2\mathit{i\omega }-{\mathit{ik}}_{0x}-{\mathit{ik}}_{0z}\right)\frac{\partial {\tilde{B}}_{y21}}{\partial t}-{V_A}^2\frac{\partial {\tilde{B}}_{y21}}{\partial z}-i\left(\frac{B_0{k}_{0z}}{4\pi }-\frac{\omega }{4{\mathit{\pi \rho }}_0}\right)\frac{\partial {\tilde{B}}_{y21}}{\partial x}-{\mathrm{V}}_A^2\frac{{\partial }^2{\tilde{B}}_{y21}}{\partial z^2}\\ +\frac{B_0}{4\pi }{k}_{0x}{k}_{0z}\left[{\tilde{B}}_{y21}{\tilde{B}}_{y21}^{\ast }\right]=0.\end{array}$$E35

Here we considered zero $\beta$ plasma. Then the plasma heated by electron only i.e. $T_e\ne 0,T_i=0.$ The modified density fluctuation due to electron [58] can be written as

$$\frac{\delta n_s}{n_0}\approx -\xi {\left|{\tilde{B}}_y\right|}^2\approx -\xi \left({\tilde{B}}_y{\tilde{B}}_y^{\ast }\right)\approx \left({\tilde{B}}_{y21}{\tilde{B}}_{y21}^{\ast }\right)$$E36

where $\xi =\left(1+8k_{0x}^2{\lambda }_e^2\right)/\left(48\pi n_0{T}_e\right)$.

From Eqs. (35) and (36), we get

$$\begin{array}{l}\left(-2\mathit{i\omega }-{\mathit{ik}}_{0x}-{\mathit{ik}}_{0z}\right)\frac{\partial {\tilde{B}}_{y21}}{\partial t}-{V_A}^2\frac{\partial {\tilde{B}}_{y21}}{\partial z}-i\left(\frac{B_0{k}_{0z}}{4\pi }-\frac{\omega }{4{\mathit{\pi \rho }}_0}\right)\frac{\partial {\tilde{B}}_{y21}}{\partial x}-{V_A}^2\frac{{\partial }^2{\tilde{B}}_{y21}}{\partial z^2}\\ +\frac{B_0}{4\pi }{k}_{0x}{k}_{0z}\left[-\xi \left({\tilde{B}}_y{\tilde{B}}_y^{\ast }\right)\right]=0.\end{array}$$E37

The dimensionless form of (33) can be written as

$$-i\frac{\partial {\tilde{B}}_y}{\partial t}-4\frac{\partial {\tilde{B}}_y}{\partial z}-i\frac{\partial {\tilde{B}}_y}{\partial x}-4\frac{{\partial }^2{\tilde{B}}_y}{\partial z^2}-{\left|\tilde{B}\right|}_y^2{\tilde{B}}_y=0$$E38

We obtain the Eq. (37), which is the modified Zakharov system of equation [59]. The normalizing parameters for $\beta =0$ plasma are

$$t_N\approx \frac{4\pi \left(2\omega +k_{0x}+k_{0z}\right)}{B_0{k}_{0x}{k}_{0z}},x_N\approx \left(\frac{\omega }{B_0\rho k_{0x}{k}_{0z}}+\frac{1}{k_{0x}}\right),z_N\approx \frac{4\pi {V_A}^2}{B_0{k}_{0x}{k}_{0z}},B_N\approx {\left[\frac{1+8k_{0x}^2{\lambda }_e^2}{48\pi n_0{T}_e}\right]}^{-1/2}.$$

3. Numerical method

In this chapter, we have use pseudo-spectral method to solve partial differential Eq. (38).

3.1 Pseudo-spectral method

The pseudo-spectral method approaches the partial differential equation are solved pointwise in the physical space (x − t). The Pseudo-Spectral methods have been successfully applied to the fluid dynamics (turbulence modeling, weather predictions) - non-linear waves - seismic modeling - MHD [60].

We have applied them to plasma turbulence simulations, and to the non-linear interaction of Alfven waves with plasmas. The pseudo-spectral method is applicable only on the spatial types of partial differential equations.

The general spatial partial differential equation is

$$\mathit{Lu}\left(x\right)=s\left(X\right),\mathit{XϵV}$$E39

Boundary condition

$$f\left(u\left(y\right)\right)=0,\mathit{yϵ}\partial V$$E40

Where L is the spatial differential operator $L=\partial /\partial x^2+\partial /\partial y^2$. But here we need numerical soluation of $u^N\left(X\right)$.

The general procedure to find $u^N\left(X\right)$ are as follows.

Initially we choose a finite set of expanction function ${\varphi }_j,j=0,..N-1$, and expanding $u^N$ by using these functions

$$u^N\left(X\right)=\sum _{j=0}^{N-1}{\widehat{u}}_j{\varphi }_j\left(X\right)$$E41

Further we choose a set of test function ${\chi }_n,k=\mathrm{0,1,2},\dots \dots N-1$ and demand that

$$\left({\chi }_n,R\right)=0,n=0,1.\dots N-1$$E42

Spectral methods’ deals with the trial functions ${\varphi }_n$ to form a basis for a certain space of smooth functions (Fourier expension).

The test function according to pseudo-spectral method can be written as

$${\chi }_n\left(X\right)=\delta \left(X-X_n\right),$$E43

Where the $X_n,\left(n=0,1,\dots N-1\right)$ are spatial point or the collection of points.

The smallness condition for the residual can be written as

$$0=\left({\chi }_n,R\right)=\left(\delta \left(X-X_n\right),R\right)=R\left(X_n\right)={\mathit{Lu}}^N\left(X_n\right)-s\left(X_n\right)$$E44

$$\sum _{j=0}^{N-1}{\widehat{u}}_{j}L{\varphi }_j\left(X_n\right)-s\left(X_n\right)=0,n=\mathrm{0,1,2},..\dots N-1.$$E45

Where N equations to determine the unknown N coefficients ${\widehat{u}}_j$.

In this chapter applications, we assume periodic boundary condition and use in Fourier series as

$${\varphi }_j\left(X\right)=e^{-{\mathit{ik}}_{j}X}$$E46

Assume this case for 1-D only, then

$$\frac{{\partial }^{2}u}{\partial z^2}=s\left(X\right)$$E47

The Fourier transform of Eq. (47) is

$${\left(-{\mathit{ik}}_z\right)}^2\widehat{u}\left(k_z\right)=\widehat{s}\left(k_z\right)$$E48

We want to solve this numerically. Then our next aim is to find the expension coefficients ${\widehat{u}}_j$ such that residual

$$\sum _{j=0}^{N-1}{\widehat{u}}_{j}L{\varphi }_j\left(X_N\right)-s\left(X_n\right)=0,n=\mathrm{0,1,2},\dots ,N-1.$$E49

or

$$\sum _{j=0}^{N-1}{\widehat{u}}_j{\mathit{Le}}^{-{\mathit{ik}}_j{X}_n}-s\left(X_n\right)=0,n=\mathrm{0,1,2},\dots ,N-1.$$E50

vanishes.

If L considered as linear, then ${\mathit{Le}}^{-{\mathit{ik}}_K{X}_n}=h\left(k_k\right)e^{-{\mathit{ik}}_K{X}_n}$.

$$\sum _{j=0}^{N-1}{\widehat{u}}_{j}h\left(k_j\right)e^{-{\mathit{ik}}_j{X}_n}-s\left(X_n\right)=0,n=\mathrm{0,1,2},\dots ,N-1.$$E51

For 1 − D simplicity, we choose as

$$z_n=n\mathrm{\Delta },n=\mathrm{0,1,2},\dots N-1$$E52

and

$$k_j=2\mathit{\pi j}/\left(N\mathrm{\Delta }\right),j=-N/2,.\dots ,N/2.$$E53

where $\mathrm{\Delta }$ is the spatial resolution and $z_n$ and $k_j$ are equi-spaced, then the condition on the residual becomes.

$$\sum _{j=0}^N{\widehat{u}}_{j}h\left(k_j\right)e^{-2\mathit{\text{πijn}}/N}-s\left(X_n\right)=0,n=\mathrm{0,1,2},\dots ,N-1.$$E54

Further we define the Discrete Fourier transform (DFT) as

$${\widehat{u}}_j=\frac{1}{\sqrt{N}}\sum _{n=0}^{N-1}{u}_n{e}^{-2\mathit{\text{πijn}}/N}.$$E55

and inverse DFT written as

$$u_n=\frac{1}{\sqrt{N}}\sum _{j=-N/2}^{N/2}{\widehat{u}}_j{e}^{-2\mathit{\text{πijn}}/N}.$$E56

Then the DFT can be written as

$${\widehat{u}}_{j}h\left(k_j\right)-\mathit{DFT}\left[s\left(x_n\right)\right]\left(k_j\right)=0.$$E57

Thus, we can manipulate our equations numerically with the DFT.

Finally, we calculate $u_j$ by following steps.

(i) Give at time t at $u_j.$ (ii) Calculate $u_n={\mathit{DFT}}^{-1}\left(u_j\right)$. (iii) Multiply and store $T_n=u_n$ (iv) And move to F-space.

4. Numerical simulation results

It is very complicated to solve dynamical wave Eq. (38) analytically, hence the solution have carried out by numerical method [61, 62].

The numerical code have been written in FORTRAN using the technique of pseudo-spectral method with Fast Fourier Transform (FFT) for transverse (x-z direction) space integration with periodic length $\left(2\pi /{\alpha }_x\right)\times \left(2\pi /{\alpha }_z\right)$ at ${\alpha }_x,{\alpha }_z=1$, implementing $2^8\times 2^8$ grid points and a finite-difference method with a predictor-corrector scheme have used for the integration in the time domain step $\mathit{dt}=5\times {10}^{-5}$. Before solving Eq. (38), we have applied our test code by analyzing the obtained result for solving the cubic nonlinear Schrödinger equation (CNSE). The precision of computation of nonlinear Schrödinger equation (NLSE) is constant $N={\sum }_k{\left|B_k\right|}^2$. The conserved quantity was preserved up to the ${10}^{-5}$ during simulation.

The initial condition of simulation is

$${\tilde{B}}_y\left(x,z,0\right)=\mid a_0\mid \left[1+0.1\mathit{cosh}\left({\alpha }_{x}x\right)\right]\left[1+0.1\mathit{cosh}\left({\alpha }_{z}z\right)\right]$$E58

Where $\mid a_0\mid =1$ is the initial amplitude of Alfven wave.

Figure 2(a-d) depict the magnetic field intensity profile of Alfvén waves. Here, we consider four instant times namely t = 5,9,11,16. Figure 2(a-b) shows the propagation dynamics of Alfvén waves at t = 5,9 where the system reaches to quasi steady state. Figure 2(c-d) shows the formation of localized structures for Alfvén waves at t = 11, 16 and after few step the system reaches to its quasi steady state. The Alfvén waves spread there intensity peaks when the magnetic field intensity raised [63]. Figure 2(a) shows the Alfvén waves propagated without braking intensity at t = 5. Figure 2(b) shows the Alfvén waves propagated with breaking intensity. In Figure 2(c) shows the global spatial structures, where the perturbation takes energy from the interaction of Alfvén wave and form their complex structures (Figure 2(d)). At the initial time waves are propagated without breaking there intensity. When we take time advancement the waves intensity is broken and propagate small part of waves with different intensity peaks. For all instantaneous time the system reaches to quasi-steady state and formed more chaotic structures. These structures might play a crucial role in zero-$\beta$ plasma for coronal heating and particle acceleration in space plasma.

Figure 3(a-d) depict the spectral contour plots of $B_y$ in the $\left(k_x,k_z\right)$ Fourier space at the same instants of time as in Figure 2(a-d) [64]. The most intense magnetic field structures are found in Figure 3(c). But in Figure 3(d), at t = 16, the more complex and chaotic state is found. At early time (t = 5) the complexity were decrease and at late time (t = 16) it increase. It is evident from contour plot that as we take time advancement the intensity and location of the magnetic field profile structures vary. And, with the advancement of time, the spectra in Fourier space become more chaotic and complex, leading to the energy transportation from larger scale to smaller scale (Figures 3 and 4).

Figure 2.

The magnetic field intensity profile of AW for zero-$\beta$ plasma at (a)t = 5, (b)t = 9, (c) t = 11, (d) t = 16.

Figure 3.

The contour plot of ${\left|B_{\mathit{yk}}\right|}^2$ against Fourier modes of AW at (a) t = 5, (b) t = 9, (c) t = 11, (d) t = 16.

Figure 4.

The variation of ${\left|B_{\mathit{yk}}\right|}^2$ against k at (a)t = 5, (b)t = 9, (c) t = 11, (d) t = 16.

Next, we studied the power spectrum of magnetic field for Alfvén waves self interaction by plotting ${\left|B_{\mathit{yk}}\right|}^2$ against k at time t = 5, 9, 11, 16. We found the spectral index steepening range shows the Kolmogorov scaling at t = 5, i.e. $k^{-5/3}$, which is consistent with the particle acceleration in solar wind [41, 65, 66]. However, our investigated results did not match the following reasons. The Alfvén waves are considered as Self-interaction in zero-$\beta$ plasma. Also, the magnetic field transverse perturbation considered for Alfvén waves interaction in cold plasma. With the advancement of time ($t=\mathrm{9,11,16}$) power spectral index shows $k^{-5/4},k^{-7/4},k^{-9/4}$ scaling. The steepening of spectra with advancement of time govern to the coronal heating by Alfvén waves interaction. The consistent results observed by many authors, when the Alfvén waves interaction considered [34, 67, 68]. When the Alfvén waves interact nonlinearly in MHD turbulent plasma, then their spectra steeper the Kolmogorov like spectra ($k^{-5/3}$) studied by Goldreich and Sridhar [41, 59, 63, 64]; Goldstein et al. [66]. The strong turbulent studied by Schekochihin et al. [69] with slop $k^{-7/3}$ and observed particle asymptotic behavior in space plasma. The weak turbulent steepening of Alfvén waves were observed between kinetic and MHD range with slop −3 to −4 [70].

For application purpose, the typical values for solar Corona [71] are as follows: $B_0\approx 10G,n_0\approx {10}^8,\beta \approx {10}^{-2},v_A\approx 2\times {10}^8,{\lambda }_e\approx 2\times {10}^3.$ Using these values, one find, ${\rho }_s\approx 3.4\times {10}^9\mathit{cm},c_s\approx 2\times {10}^7\mathit{cm}/s,{\omega }_{\mathit{ci}}\approx 5.8\times {10}^3.$ For $k_{0x}{\lambda }_e\approx 0.98$, and $\frac{{\omega }_0}{{\omega }_{\mathit{ce}}}\approx 0.85$, one can calculate $k_{0x}\approx 4.9\times {10}^{-4}{\mathit{cm}}^{-1},{\omega }_0\approx 4.1\times {10}^3{\mathit{sec}}^{-1},k_{0z}\approx 2\times {10}^{-5}{\mathit{cm}}^{-1}.$ The normalizing parameter are $x_n\approx 2\times {10}^3\mathit{cm},z_n\approx 5.1\times {10}^{24}\mathit{cm},t_n\approx 1.1\times {10}^{12},B_n\approx 1.5G.$

5. Conclusion

In this chapter, we have investigated the Alfvén waves self interaction in approximately zero-$\beta$ plasma, when the magnetic field transverse perturbation considered. The numerical simulation of Eq. (38) carried out by pseudo-spectral method, which govern the dynamics of Alfvén waves interaction in cold plasma. The transverse perturbation leads to the steepening of power-spectrum which are $k^{-5/4},k^{-7/4},k^{-9/4}$ at time $t=\mathrm{9,11,16}$. The present chapter determine the Alfvén waves self interaction in approximately zero-$\beta$ plasma for corona heating, when the transverse perturbation considered. We also investigate the relevance of proposed self-interaction to explain the observed power spectrum for the solar corona. Our observed results shows the spectrum steepens, which consequence to transport the energy from large scale to small scale. Observed results also revel that the transverse magnetic field profile structures (coherent structures) are break at highest intensity lead to cascade energy. Hence the transverse perturbation affect the coherent structures and power spectrum, govern to coronal heating and particle acceleration in space plasma.