2. Numerical Experiment

(1)

(2)

(3)

(4)

[8]  In equations (1)-(4), Re_m = L₀V₀/n_m0 is the magnetic Reynolds number, n_m0 = c²/4ps₀ is the magnetic viscosity for conductivity s at temperature T₀, and s is the conductivity, s₀/s = T^-3/2. The dimensionless coefficient b₀ = 8pn₀ kT₀/B₀², where n₀ = r₀/m_i. It should be emphasized that b₀ in this form is not the ratio between plasma pressure and pressure of magnetic field at a definite point in space, it is simply a dimensionless coefficient expressed through the dimensionless units used for the calculations. The term including viscosity does not exert a considerable influence on the results; it is important for increasing the stability of the finite difference scheme. Re = r₀ L₀ V₀/h is the Reynolds number, h is the viscosity, G_q = L(T₀) r₀ t₀/T₀, L(T _dimens ) is the radiation function for ionization equilibrium in the corona, T _dimens= T₀T. L'(T) = L(T _dimens)/L(T₀) is the dimensionless radiation function. In the problem being solved, radiation did not have a strong effect; e, e ₁, e ₂ are the orthogonal unit vectors parallel and perpendicular to the magnetic field; k_dl = k/(Pk₀ ) is the dimensionless coefficient of thermal conductivity along the magnetic field; P = r₀ L₀ V₀/k₀ is the Peclet number; k₀ is the thermal conductivity at temperature T₀; k is the thermal conductivity; k/k₀ = T^5/2; k _dl = [(kk₀^-1P^-1) (k_B k_0B^-1P_B^-1)]/ [(kk₀^-1P^-1) +(k_B k_0B^-1P_B^-1)] is the dimensionless coefficient of thermal conductivity in the direction perpendicular to the magnetic field; P_B = r₀ L₀ V₀/k_0B is the Peclet number for the thermal conductivity across a strong magnetic field (when the cyclotron radius is much smaller than the free path). Thermal conductivity across a strong magnetic field is denoted as k_B; and k_0B is its magnitude for temperature T₀, plasma density r₀ and magnetic field B₀; k_B/k_0B = r²B^-2 T^-1/2. G_g G is the dimensionless gravitational acceleration. G_g= t₀²/L₀, G is the gravitational acceleration, and g is the adiabatic constant.

[9]  The parameters used for calculations were: g = 5/3, Re_m=8 10⁴, Re=10⁴, b₀=8 10^-6, P=2, P_B=2 10⁶. (We repeat here, that b₀ is not the ratio between plasma pressure and magnetic pressure at a definite point in space, it is only coefficient in equations (1)-(4), which is determined by dimensionless units of pressure and magnetic field). In numerical and laboratory simulation it is impossible to use very big and very little dimensionless parameters. These parameters are chosen bigger or less than unit, but not precisely by the same order of magnitude. The principles according to which the dimensionless parameters were chosen were described by Podgorny and Podgorny [1995, 1996]. For the numerical calculations, the grid 41 41 41 was used; therefore the magnetic Reynolds number was ~50.

Figure 1

[10]  The computational domain was a cube with sides 8 R, in dimensionless units 0 x 1, 0 y 1, 0 z 1. The positions of the Sun, corona, and computational domain in the y = 0.5 plane are shown in Figure 1. The dipole giving rise to the magnetic field B₀ 0.8 G on the Sun's surface is parallel to the Z axis. The magnetic moment of the dipole in dimensionless units is M₁={M_1x=0, M_1y=0, M_1z=9.6 10^-2}. Its center is at the point R₁ = {x₁ = -0.217, y₁ = 0.5, z₁ = 0.5}. On the face x = 0, at the point y = 0.5 and z = 0.5, the magnetic field is 0.15 G. The lines of the

Figure 2

initial (dipole) magnetic field are presented in Figure 2a.

[11]  The vacuum medium cannot be described in the framework of the MHD approximation. For this reason, at t=0 in the computational domain, an extremely low concentration of 10^-1 cm ^-3, whose influence on the dynamics of the corona plasma that expanded in the computational domain was negligibly small, was specified. At the initial moment of time, the temperature inside the region was taken to be as low as 20 eV.

[12]  At t = 0, a thermal expansion of the corona began. The corona parameters ( r_c/m_i = 2 10⁷ cm ^-3, T_c = 200  eV) were specified on the face x=0 in the circle formed by the intersection of this face and a spherical surface with the center ( -0.217, 0.5, 0.5) and radius 2R (Figure 1). The center of the circle was ( y=0.5, z=0.5 ), and its radius was 0.125. The velocity of plasma outflow from the corona was found from the continuity equation, so that the mass flow in the numerical experiment corresponded to the loss of the Sun's mass carried away by the solar wind ( 10^-14 of the Sun's mass per year). From this condition, the velocity of inflow into the region V_x=2.5 10^-4 was specified at the boundary x=0 in the circle with the center ( y=0.5, z=0.5 ) and radius 0.125. The self-consistent values of r, T, and V automatically established in the process of numerical solution of the MHD equations at thermal expansion of corona plasma. The current of the current sheet could freely flow in and out through the planes y=0 and y=1.

[13]  It should be noted that, formally, the parameter b₀ did not correspond to the ratio between pressures at any point. At the point x=0, y=0.5, and z=0.5 the b parameter is 8. In the vicinity of this point in the computational domain, where the condition approximated the vacuum one, b=4 10^-9 at the initial moment.

[14]  The most delicate problem in specifying the boundary conditions is setting the magnetic field at the rightmost boundary X=1. In the process of calculations, and hence extension of field lines along the X axis, all components ( B_x, B_y, B_z ) undergo significant changes, and the tilt of the field line with respect to the X=1 plane also changes. Therefore it is impossible to specify the magnetic field at the boundary X=1. Setting j=0 is also unacceptable because, in the process of extension of field lines, the current sheet reaches the right boundary. The best approximation is to assume dj/dx=0. To be more certain that the obtained solution corresponds to the conditions in the solar wind, a layer boundary X=1 which is by a factor of 2-3 thicker than the current sheet can be excluded from the consideration.

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