Existence of a strong solution for the 2D four-field RMHD equations

Abstract

In this paper, we investigated the two-dimensional, four-field reduced magnetohydrodynamics equations that are used to examine the tearing instability and magnetic reconnection of phenomena in plasmas. The generalized Ohm’s law is used in the derivation of the 2D four-field RMHD equations. We established the existence and uniqueness of a strong solution for the 2D four-field RMHD equations under periodic boundary conditions during some time interval in Sobolev-Slobodetskiĭ space with the help of successive approximations and $a\ priori$ estimates.

Appendices

Appendix A1. Derivation of the reduced MHD equations (2)

Let us consider the following MHD equations:

$$\begin{aligned} v_t + (v \cdot \nabla )v = - \nabla p + J \times B + D \Delta v, \end{aligned}$$

(A.1)

$$\begin{aligned} \nabla \cdot v =0, \end{aligned}$$

(A.2)

$$\begin{aligned} B_t = \nabla \times ( v \times B - \eta J), \end{aligned}$$

(A.3)

$$\begin{aligned} \nabla \cdot B =0, \end{aligned}$$

(A.4)

$$\begin{aligned} \nabla \times B =J, \end{aligned}$$

(A.5)

where $x = (x_1, x_2)$, $(x, x_3 ) \in \mathbf{R^3}$, $\nabla = \left( \partial _1, \partial _2, \partial _3 \right)$, and $\Delta = \partial _{11}^2 + \partial _{22}^2 + \partial _{33}^2$. Here $v = v(x,t) = (v_1(x,t), v_2(x,t), 0)$, $B = B(x,t) = (B_1(x,t), B_2(x,t), 0)$, $J=J(x,t)$, and $p=p(x,t)$ denote the unknown velocity of the fluid, the magnetic field, the current density, and the pressure of the fluid. The density of the fluid is $\rho _0=1$. Here, (A.3) comes from Faraday’s law $\displaystyle B_t = - \nabla \times E$ and Ohm’s law $E + v \times B = \eta J$. From (A.5) and $B = (B_1(x,t), B_2(x,t), 0)$, J can be written as $J=(0,0, j(x,t))$.

By substituting Ampère’s law (A.5) into (A.3), and then using (A.4) and the equality $\nabla \times ( \nabla \times A ) = \nabla ( \nabla \cdot A ) - \nabla ^2 A$ ( [17]), we have $B_t = \nabla \times (v \times B ) + \eta \Delta B.$ Using the equality $\nabla \times (A \times B) = A (\nabla \cdot B) - B ( \nabla \cdot A) + (B \cdot \nabla ) A - (A \cdot \nabla ) B$ ( [17]), we have

$$\begin{aligned} B_t = - ( v \cdot \nabla ) B + ( B \cdot \nabla ) v + \eta \Delta B. \end{aligned}$$

(A.6)

From (A.2) and (A.4), B and v can be written as follows:

$$\begin{aligned} B = \nabla \psi \times {\textbf{e}} = \left( \partial _2 \psi , - \partial _1 \psi , 0 \right) , \qquad v = \nabla \phi \times {\textbf{e}}, \end{aligned}$$

where ${\textbf{e}}=(0,0,1)$. Hence, from (A.6), we obtain

$$\begin{aligned} \nabla \times \left( 0, 0, \psi _t \right) = \nabla \times (0, 0, - \left\{ \phi , \psi \right\} ) + \eta \nabla \times (0,0, \Delta _\perp \psi ), \end{aligned}$$

(A.7)

where $\Delta _\perp = \partial _{11}^2 + \partial _{22}^2$. In Appendix A1, two symbols $\Delta$ and $\Delta _\perp$ are used, but in the text, we use only the symbol $\Delta$. Because all unknown functions are independent of $x_3$, from (A.7), we have $\psi _t + \left\{ \phi , \psi \right\} = \eta \Delta _\perp \psi .$ If we consider the external force of the electric fields, we have

$$\begin{aligned} \psi _t + \left\{ \phi , \psi \right\} = \eta \Delta _\perp (\psi - \psi _*). \end{aligned}$$

Notice that from Faraday’s law $\displaystyle B_t = - \nabla \times E$ and $B = \nabla \psi \times {\textbf{e}}$, we have $\displaystyle E = - \nabla \xi + \psi _t$ for a suitable function $\xi$. From (A.5) and $B = \nabla \psi \times {\textbf{e}}$, we have $J = (0,0,j) = (0,0, - \Delta _\perp \psi )$.

By substituting (A.5) into (A.1) and using the equality $\nabla \mid B \mid ^2 = 2 (B \cdot \nabla ) B + 2 B \times \left( \nabla \times B \right)$ ([17]), we obtain

$$\begin{aligned} v_t + (v \cdot \nabla )v = - \nabla p - \frac{1}{2} \nabla \mid B \mid ^2 + ( B \cdot \nabla ) B + D \Delta v. \end{aligned}$$

(A.8)

By taking $\nabla \times$ for (A.8) and using the equality $\nabla \times \nabla \psi =0$ ( [17]), we have

$$\begin{aligned} \nabla \times \left( v_t + (v \cdot \nabla )v \right) = \nabla \times \left( ( B \cdot \nabla ) B + D \Delta _\perp v \right) . \end{aligned}$$

Let us introduce the vorticity $\omega = (\nabla \times v) \cdot {\textbf{e}}$. Then, using $v = \nabla \phi \times {\textbf{e}}$, we have

$$\begin{aligned} \left( 0,0, \omega _t + \left\{ \phi , \omega \right\} \right) = (0,0, \left\{ \psi , j \right\} + D \Delta _\perp \omega ). \end{aligned}$$

Hence we have $\omega _t + \left\{ \phi , \omega \right\} = \left\{ \psi , j \right\} + D \Delta _\perp \omega .$

Appendix A2. Numerical algorithm and numerical simulation for (2)

To obtain Fig. 1, we use a numerical scheme that is similar to the methods described in [10] and [31]. We let $\mu = \eta$, and then, we solve (2) with periodic boundary conditions by the finite differential method. We apply the third-order Karniadakis time integration scheme to the first and second equations of (2), and we apply the successive over-relaxation (SOR) method to the third and fourth equations of (2). The SOR method is well known, so we will discuss only the third-order Karniadakis time integration scheme [19].

When we use the third-order Karniadakis time integration scheme, we use the central difference for the spatial difference ${\partial \phi }/{\partial x_2}$ and $\Delta$, and Arakawa’s method for the Poisson bracket $\{ \cdot , \cdot \}$. We set $L_1 =10$ and $L_2 = 7$, we use a grid width of $d x_1= d x_2=h$, and we use $256 \times 256$ grid points. Let us write the grid number as $N=256$ and the time step width of the numerical calculation as dt. Furthermore, we write $t_k = k dt$ for $k=0, 1, 2, \ldots$. At the grid point (i, j) and time step k, we write the unknown function f as $f_{i,j}^k$. We choose the following as the given function and the initial value:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \psi _* = F(x_2) \psi ^0(x_2) - 0.00001 \cos \left( 2 \pi x_1 L_1^{-1} - \pi \right) , \\ \displaystyle \psi _0 = \psi _*, \quad \phi _0 = 0, \end{array} \right. \end{aligned}$$

(A.9)

where

$$\begin{aligned} \psi ^0(x_2) = \left( \cosh ^2 \left( 2 \pi x_2 L_1^{-1} - \pi \right) \right) ^{-1}, \\ F(x_2) = -1 + \tanh ^2 \left( 2 \pi x_2 L_2^{-1} \right) + \tanh ^2 \left( 2 \pi - 2 \pi x_2 L_2^{-1} \right) . \end{aligned}$$

Note that $\psi ^0(x_2)$ and $\phi ^0 =0$ is a stationary solution for (2) when $\eta =0, \ \mu =0, \ \psi _* =0$, and $(\psi ^0(x_2), \phi ^0)$ is tearing unstable when $L_2 / L_1 < \sqrt{5}$ ( [27, 33]). When $\psi = \psi ^0(x_2)$, then $B= ( \partial _2 \psi ^0(x_2),0, 0)$, where $\partial _2 \psi ^0(0)=0$, $\partial _2 \psi ^0(x_2) <0$ for $x_2 >0$, and $\partial _2 \psi ^0(x_2) >0$ for $x_2 <0$. Hence, magnetic field lines with different directions are adjacent to each other, with a straight line $x_2=0$ as a boundary. If we take large $L_1$ and $L_2$, we can impose periodic boundary conditions because $\psi ^0(x_2) \rightarrow 0$ and $d \psi ^0(x_2)/ dx_2 \rightarrow 0$ as $\mid \, x_2 \mid \rightarrow \infty$ ( [33]). However, $L_1 =10$, and $L_2 = 7$ are not so large. Thus, we use the cutoff function $F(x_2)$, which satisfies $F(x_2) \approx 0, \ \partial _2 F(x_2) \approx 0$ at $x_2 = 0, L_2$. In Fig. 1, we take $\mu$ and $\eta$ to be small ($\mu = \eta =0.0002$).

Arakawa [4] introduced the following discretization of $\{ g, f \}_{i,j}^k$ at the grid point (i, j) and time step k:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \{ g, f \}_{i,j}^k = - \frac{1}{12 h^2} [ ( f_{i, j-1}^k + f_{i+1,j-1}^k - f_{i,j+1}^k - f_{i+1,j+1}^k) ( g_{i+1,j}^k + g_{i,j}^k )\\ \displaystyle - ( f_{i-1, j-1}^k + f_{i,j-1}^k - f_{i-1,j+1}^k - f_{i,j+1}^k) ( g_{i,j}^k + g_{i-1,j}^k ) \\ \displaystyle + ( f_{i+1, j}^k + f_{i+1,j+1}^k - f_{i-1,j}^k - f_{i-1,j+1}^k) ( g_{i,j+1}^k + g_{i,j}^k ) \\ \displaystyle - ( f_{i+1, j-1}^k + f_{i+1,j}^k - f_{i-1,j-1}^k - f_{i-1,j}^k) ( g_{i,j}^k + g_{i,j-1}^k ) \\ \displaystyle + ( f_{i+1,j}^k - f_{i,j+1}^k ) ( g_{i+1,j+1}^k + g_{i,j}^k ) - ( f_{i,j-1}^k - f_{i-1,j}^k ) ( g_{i,j}^k + g_{i-1,j-1}^k ) \\ \displaystyle + ( f_{i,j+1}^k - f_{i-1,j}^k ) ( g_{i-1,j+1}^k + g_{i,j}^k ) - ( f_{i+1,j}^k - f_{i,j-1}^k ) ( g_{i,j}^k + g_{i+1,j-1}^k ) ]. \end{array} \right. \end{aligned}$$

(A.10)

This discretization of the convection term exactly conserves energy, enstrophy, and circulation. Arakawa’s method is compared with the spectral method in [30]. It is important to note that in [21], the left-hand side of (A.10) is written as $\{ f, g \}_{i,j}^k$, but that was an error.

Let us denote the discretization of the right-hand sides of the first and second equations of (2) at the grid point (i, j) and time step k as $F_{i,j}^k$ and $G_{i,j}^k$, as follows:

$$\begin{aligned}{} & {} F_{i,j}^k = - \left\{ \phi , \omega \right\} _{i,j}^k + \left\{ \psi , j \right\} _{i,j}^k \\{} & {} \qquad + D h^{-2} \left( ( \omega _{i-1,j}^k - 2 \omega _{i,j}^k + \omega _{i+1,j}^k ) + ( \omega _{i,j-1}^k - 2 \omega _{i,j}^k + \omega _{i,j+1}^k ) \right) , \\{} & {} G_{i,j}^k = - \left\{ \phi , \psi \right\} _{i,j}^k \\{} & {} \qquad + \eta h^{-2} \left( ( \psi _{i-1,j}^k - 2 \psi _{i,j}^k + \psi _{i+1,j}^k ) + ( \psi _{i,j-1}^k - 2 \psi _{i,j}^k + \psi _{i,j+1}^k ) \right) \\{} & {} \qquad - \eta h^{-2} \left( ( \psi _{* i-1,j}^k - 2 \psi _{* i,j}^k + \psi _{* i+1,j}^k ) + ( \psi _{* i,j-1}^k - 2 \psi _{* i,j}^k + \psi _{* i,j+1}^k ) \right) . \end{aligned}$$

We apply the third-order Karniadakis time-integration scheme [19] to the first and second equations of (2):

$$\begin{aligned}{} & {} \omega _{i,j}^{k+1} = \frac{6}{11} \left( 3 \omega _{i,j}^{k} - \frac{3}{2} \omega _{i,j}^{k-1} + \frac{1}{3} \omega _{i,j}^{k-2} + dt \left( 3 F_{i,j}^{k} - 3 F_{i,j}^{k-1} + F_{i,j}^{k-2} \right) \right) , \\{} & {} \psi _{i,j}^{k+1} = \frac{6}{11} \left( 3 \psi _{i,j}^{k} - \frac{3}{2} \psi _{i,j}^{k-1} + \frac{1}{3} \psi _{i,j}^{k-2} + dt \left( 3 G_{i,j}^{k} - 3 G_{i,j}^{k-1} + G_{i,j}^{k-2} \right) \right) . \end{aligned}$$

When we solve the first and second equations of (2) using the third-order Karniadakis time-integration scheme, we define dt the same way that [21] defined it (see, [18]).

As shown in Fig. 1(b), a shape similar to the letter “x” is created around $(x_1, x_2) =(5, 3.5)$ because of tearing instability. Note that a periodic boundary condition is imposed, and we can confirm that an island shape is formed around $(x_1, x_2) =(0, 3.5)$ in Fig. 1(b).