2. A plane diffusion wave with regard to the Hall effect

Let the components of magnetic field depend on coordinate z only, i.e., H ¼ Hxð Þ t; z ; Hy ð Þ t; z ; Hz0Þ. We neglect the medium motion. Then, the magnetic field equation (for components) is written in the following form:

$$\frac{dH\_x}{dt} = \nu \frac{\partial^2 H\_x}{\partial z^2} + \beta \frac{\partial^2 H\_y}{\partial z^2}, \quad \frac{dH\_y}{dt} = \nu \frac{\partial^2 H\_y}{\partial z^2} - \beta \frac{\partial^2 H\_x}{\partial z^2}, \quad \frac{dH\_z}{dt} = 0, \quad \beta = bH\_{z0}. \tag{2}$$

Consider the problem of a diffusion wave propagating in an unbounded medium with the given boundary and initial conditions:

Magnetohydrodynamic problems are naturally divided into two groups: problems for an ideal infinitely conducting plasma and problems with dissipative processes in the form of heat

Numerous publications on the construction of difference methods for ideal magnetohydrodynamics use a standard set of test problems. These include propagation of one-dimensional Alfven waves at various angles to grid lines [3–5], Riemann problem for MHD equations [6–9], and various two-dimensional problems accounting for the presence of a uniform magnetic field [3, 5, 10]. In [11], a number of additional ideal MHD benchmarks are presented, which are basically shock-wave problems. A special class of tests includes problems with a weak magnetic field not affecting the medium motion. If there is an exact solution for a given hydrodynamic problem, the magnetic field "freezing-in" principle allows finding components of the

at any time with the known medium displacements X <sup>¼</sup> X Xð Þ <sup>0</sup>; <sup>t</sup> .

The representation in publications of the problem of testing the dissipative stage of MHD equations is much the worse. Possibly, this is owing to complexity problems that require accounting the interaction of the shock-wave processes, heat conduction, diffusion of magnetic

Numerical simulations of some of the tests presented here have been done using the Lagrangian-Eulerian code EGIDA developed at VNIIEF [12, 13] for multi-material compress-

The magnetohydrodynamic equation system in one-temperature approximation modified by

where <sup>ν</sup> <sup>¼</sup> <sup>c</sup><sup>2</sup>=ð Þ <sup>4</sup>πσ is the magnetic viscosity coefficient, <sup>κ</sup> is the heat conduction factor, b ¼ c=ð Þ 4πene is a local exchange (Hall) parameter [2], and e and ne are charge and density of electrons. When writing Eq. (18), we assume that bias currents and electron inertia are negligibly small [2]. Equation system (1) differs from equation system for ideal MHD owing to

Let the components of magnetic field depend on coordinate z only, i.e., H ¼ Hxð Þ t; z ; Hy

ð Þ t; z ; Hz0Þ. We neglect the medium motion. Then, the magnetic field equation (for components)

dt <sup>þ</sup> divðr<sup>u</sup> <sup>⊗</sup> <sup>u</sup> <sup>þ</sup> ð Þ <sup>P</sup> <sup>þ</sup> PH <sup>I</sup> � <sup>0</sup>:5<sup>H</sup> <sup>⊗</sup> <sup>H</sup>Þ ¼ <sup>0</sup>, PH <sup>¼</sup> <sup>0</sup>:5j j <sup>H</sup> <sup>2</sup>

,

(1)

<sup>∂</sup><sup>t</sup> <sup>þ</sup> divð Þ¼ <sup>r</sup>nε<sup>u</sup> <sup>0</sup>,

þ PH,

the Hall effect can be written in the following conservative form [2]:

<sup>∂</sup><sup>t</sup> <sup>þ</sup> divð<sup>u</sup> <sup>⊗</sup> <sup>H</sup> � <sup>H</sup> <sup>⊗</sup> <sup>u</sup>޼�rotð Þ <sup>ν</sup> � rot<sup>H</sup> <sup>þ</sup> <sup>b</sup>½ � <sup>H</sup> � rot<sup>H</sup> , <sup>∂</sup>rn<sup>ε</sup>

<sup>∂</sup><sup>t</sup> <sup>þ</sup> divðð Þ <sup>Ξ</sup> <sup>þ</sup> <sup>P</sup> <sup>þ</sup> PH <sup>u</sup> � H uð Þ� � <sup>H</sup> <sup>κ</sup>gradTÞ ¼ <sup>0</sup>, <sup>Ξ</sup> <sup>¼</sup> <sup>r</sup> <sup>e</sup> <sup>þ</sup> <sup>0</sup>:5j j <sup>u</sup> <sup>2</sup>

diffusion terms present in the equations of energy and inductance of magnetic field.

2. A plane diffusion wave with regard to the Hall effect

conduction and magnetic viscosity.

218 Numerical Simulations in Engineering and Science

field H Hx; Hy; Hz

field, and Joule heating.

ible flow simulations.

<sup>þ</sup> divr<sup>u</sup> <sup>¼</sup> <sup>0</sup>, <sup>∂</sup>r<sup>u</sup>

is written in the following form:

P ¼ Pð Þ r; T , ε ¼ εð Þ r; T :

∂r ∂t

∂H

∂Ξ

$$\mathbf{H}(t, z=0) = \mathbf{H}\_{\mathsf{I}}, \quad \mathbf{H}(t, z \to \ast) = \mathbf{H}\_{\mathsf{I}}, \quad \mathbf{H}(t=0, z) = \mathbf{H}\_{\mathsf{I}}, \quad \mathbf{H}\_{\mathsf{I}} = (0, 0, H\_{\mathsf{II}}), \quad \mathbf{H}\_{\mathsf{I}} = \begin{pmatrix} H\_{\mathsf{II}}, H\_{\mathsf{II}}, H\_{\mathsf{II}} \end{pmatrix}. \tag{3}$$

Let γ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>ν</sup><sup>2</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup> q . A general solution to Eq. (2) for the self-similar variable <sup>ξ</sup> <sup>¼</sup> <sup>z</sup><sup>=</sup> ffiffiffiffiffiffiffi 4γt p looks like

$$H\_{\mathbf{x}} = H\_{\mathbf{x}0} + \mathsf{C}\_1 \Phi(\xi) + \mathsf{C}\_2 \Psi(\xi), \\ H\_y = H\_{y0} + \mathsf{C}\_1 \Psi(\xi) - \mathsf{C}\_2 \Phi(\xi), \\ H\_z = H\_{z0}$$

where <sup>Φ</sup>ð Þ¼ <sup>ξ</sup> <sup>Ð</sup> ξ 0 exp �νx<sup>2</sup>=<sup>γ</sup> � � sin <sup>β</sup>x<sup>2</sup>=<sup>γ</sup> � �dx, <sup>Ψ</sup>ð Þ¼ <sup>ξ</sup> <sup>Ð</sup> ξ 0 exp �νx<sup>2</sup>=<sup>γ</sup> � � cos <sup>β</sup>x<sup>2</sup>=<sup>γ</sup> � �dx:

Since <sup>Φ</sup>ð Þ¼ <sup>∞</sup> <sup>Γ</sup> <sup>1</sup> 2 ffiffiffiffiffiffi γ�ν 2γ q ¼ ffiffiffiffiffiffiffiffiffiffiffi π γð Þ �ν 2γ q , <sup>Ψ</sup>ð Þ¼ <sup>∞</sup> <sup>Γ</sup> <sup>1</sup> 2 ffiffiffiffiffiffi γþν 2γ q ¼ ffiffiffiffiffiffiffiffiffiffiffi π γð Þ þν 2γ q constants C1, C<sup>2</sup> with regard to boundary conditions can be found from equations

$$\mathcal{C}\_{1} = \frac{-H\_{x0}\Phi(\circ\circ) - H\_{y0}\Psi(\circ\circ)}{\Phi(\circ\circ) + \Psi(\circ\circ)} = -\frac{2}{\sqrt{\pi}} \left( H\_{x0} \sqrt{\frac{\gamma - \nu}{2\gamma}} + H\_{y0} \sqrt{\frac{\gamma + \nu}{2\gamma}} \right).$$

$$\mathcal{C}\_{2} = \frac{-H\_{x0}\Psi(\circ\circ) + H\_{y0}\Phi(\circ\circ)}{\Phi(\circ\circ) + \Psi(\circ\circ)} = -\frac{2}{\sqrt{\pi}} \left( H\_{x0} \sqrt{\frac{\gamma + \nu}{2\gamma}} - H\_{y0} \sqrt{\frac{\gamma - \nu}{2\gamma}} \right).$$

Simulation setup: the initial data is described by Eq. (3). A bounded computational domain 0 < z < L, L = 1 is considered. For this reason, the magnetic field value taken from the analytical solution Hxð Þ¼ <sup>t</sup>; <sup>z</sup> <sup>¼</sup> <sup>L</sup> <sup>1</sup> <sup>þ</sup> ð Þ <sup>C</sup>1<sup>Φ</sup> <sup>þ</sup> <sup>C</sup>2<sup>Ψ</sup> <sup>L</sup>ffiffiffiffiffi <sup>4</sup>γ<sup>t</sup> <sup>p</sup> , Hyð Þ¼ <sup>t</sup>; <sup>z</sup> <sup>¼</sup> <sup>L</sup> <sup>1</sup> <sup>þ</sup> ð Þ <sup>C</sup>1<sup>Ψ</sup> � <sup>C</sup>2<sup>Φ</sup> <sup>L</sup>ffiffiffiffiffi <sup>4</sup>γ<sup>t</sup> <sup>p</sup> , Hzðt; <sup>z</sup> <sup>¼</sup>

Figure 1. Profiles of field components at time t = 0.01: (a) Hx and (b) Hy.

With such assumptions, the problem is reduced to solving equations

dT

Here, r is the density of plasma, γ is the heat capacity ratio (adiabatic index), σ<sup>0</sup> is the conductivity at T = Ry (it is expressed via atomic constants), and Ry is the Rydberg constant.

At initial time t=0, all quantities depend on one space coordinate. It is assumed that the magnetic field has only one component, H ¼ ð Þ 0; 0; Hz . The solution is considered for the

dt <sup>¼</sup> ð Þ <sup>γ</sup> � <sup>1</sup> <sup>ν</sup>ð Þ rot<sup>H</sup> � rot<sup>H</sup> , (4)

Benchmarks for Non-Ideal Magnetohydrodynamics http://dx.doi.org/10.5772/intechopen.75713

(5)

221

, α ¼ 3=2, γ � 1 ¼ R=CV, R ¼ 1, CV ¼ 1:5

,T xð Þ¼ ; t ¼ 0 T0, rð Þ¼ x; t ¼ 0 r<sup>0</sup>

4πσ<sup>0</sup>

dξ � �<sup>2</sup>

4π

Ei �ξ<sup>2</sup>

T0 Ry � �<sup>α</sup>

, <sup>η</sup> <sup>¼</sup> ð Þ <sup>γ</sup> � <sup>1</sup> <sup>H</sup><sup>2</sup>

¼ 0: (8)

=4 � �: (10)

~0 for the linear

0 rT<sup>0</sup>

: (7)

Hzð Þ¼ x ! �∞; t 0, Hzð Þ¼ x ! ∞; t H0,T xð Þ¼ ! �∞; t T0: (6)

<sup>ν</sup>0<sup>t</sup> <sup>p</sup> . The solution can be obtained by integrating the system of ordinary

<sup>d</sup><sup>ξ</sup> <sup>þ</sup> ητ�<sup>α</sup> dhz

hzð Þ¼ ξ ! �∞ 0, hzð Þ¼ ξ ! ∞ 1, τ ξð Þ¼ ! �∞ 1: (9)

<sup>i</sup>�i! , temperature in the vicinity of interface <sup>ξ</sup><sup>2</sup>

=4π.

~0, because of no integral curves of Eq. (8) satisfying the boundary conditions at

, ν τð Þ¼ <sup>ν</sup>0τ�<sup>α</sup>, <sup>ν</sup><sup>0</sup> <sup>¼</sup> <sup>c</sup><sup>2</sup>

In an infinite region (�∞ < x < ∞), the problem has a self-similar solution depending on the

¼ 0, ξ 2 dτ

Note that for a linear case, α = 0, the solution of Eqs. (8), (9) can be found in quadratures

In general, if α > 0, one does not manage to establish the asymptotic law for temperature in the

hzð Þ¼ <sup>ξ</sup> <sup>0</sup>:5 1ð Þ <sup>þ</sup> signð Þ <sup>ξ</sup> erfð Þ <sup>ξ</sup>=<sup>2</sup> , τ ξð Þ¼ <sup>1</sup> � <sup>η</sup>

dt ¼ �rotð Þ <sup>ν</sup> � rot<sup>H</sup> , <sup>r</sup>

problem with initial conditions Eq. (5) and boundary conditions Eq. (6):

�

dt <sup>¼</sup> ην τð Þ <sup>∂</sup>hz

d <sup>d</sup><sup>ξ</sup> <sup>τ</sup>�<sup>α</sup> dhz dξ � �

i ð Þ �<sup>1</sup> <sup>i</sup> xi

case <sup>α</sup> = 0 has the logarithmic profile τ ξð Þ��<sup>η</sup> ln <sup>ξ</sup><sup>2</sup>

∂x � �<sup>2</sup>

0 if x < 0 H<sup>0</sup> if x > 0

For dimensionless variables, hz ¼ Hz=H0, τ ¼ T=T<sup>0</sup> Eq. (4) are reduced to the form:

dH

Energy units have been chosen for temperature.

Hzð Þ¼ x; t ¼ 0

ξ 2 dhz dξ þ

ν τð Þ <sup>∂</sup>hz ∂x , dτ

variable <sup>ξ</sup> <sup>¼</sup> <sup>x</sup><sup>=</sup> ffiffiffiffiffiffi

differential equations:

with boundary conditions

Since Eið Þ¼ �<sup>x</sup> <sup>C</sup> <sup>þ</sup> ln <sup>x</sup> <sup>þ</sup> <sup>P</sup>

vicinity of ξ<sup>2</sup>

infinity Eq. (9).

dhz dt <sup>¼</sup> <sup>∂</sup> ∂x

where <sup>ν</sup> <sup>¼</sup> <sup>c</sup><sup>2</sup>=4πσ, <sup>σ</sup> <sup>¼</sup> <sup>σ</sup>0ð Þ <sup>T</sup>=Ry <sup>α</sup>

Figure 2. Profiles of field components at time t = 0.1: (a) Hx and (b) Hy.

LÞ ¼ 1 is imposed on the right boundary z = L. On the left boundary z=0, the field components take constant values according to Eq. (3). In simulations with 2D and 3D codes, boundary conditions ∂H=∂n ¼ 0 (n is a normal vector to a face) are imposed on lateral faces. By varying parameters ν and β, we can study the effect of the diffusion and Hall terms in Eq. (2) on the diffusion wave parameters. Consider an option with the Hall effect dominating over the diffusion effect: ν = 0, β = 1, Hx0 = Hy0 = Hz0 = 1. Profiles of magnetic field's components Hx, Hy at time t = 0.01, 0.1 are shown in Figures 1 and 2. With the grid refinement, convergence to the reference solution takes place.
