Effect of Magnetic Field Diffusion on the Structure of Extended Envelopes of Hot Jupiters

Abstract

The paper presents a modification of the numerical MHD model of extended envelopes of hot Jupiters, in which the magnetic viscosity is considered. In the new version of the numerical model, we included the processes of magnetic field diffusion due to the ohmic (turbulent) viscosity, as well as the Bohm-type viscosity. In the region of the planet’s magnetosphere, Bohm diffusion can play a significant role in the vicinity of neutral points of the magnetic field and near current sheets. The magnetic field diffusion equation is solved using an implicit absolutely stable scheme. The structure of a quasi-open super-Alfvén envelope for a typical hot Jupiter HD 209458b was calculated with and without magnetic viscosity. An analysis of the calculation results leads to the conclusion that, at short times on the order of the orbital period, diffusion effects are insignificant. However, in the quasi-stationary envelope of a hot Jupiter, magnetic field diffusion will effectively work for longer times on the order of several orbital periods. In addition, the effectiveness of Bohm diffusion can increase significantly even at short times in spatial regions of sufficiently small size. In the calculations presented, the physical diffusion of the magnetic field turns out to be comparable with the effect of numerical diffusion.

Appendix

1.1 DIFFERENCE SCHEME FOR THE EQUATION OF MAGNETIC FIELD DIFFUSION

The equation describing magnetic field diffusion can be written as follows:

$$\frac{{\partial {\mathbf{b}}}}{{\partial t}} = - \nabla \times \left( {\eta {\kern 1pt} \nabla \times {\mathbf{b}}} \right).$$

(A.1)

In Cartesian coordinates, this equation can be conveniently rewritten as

$$\frac{{\partial {\mathbf{b}}}}{{\partial t}} = \frac{\partial }{{\partial {{x}^{k}}}}\left( {\eta \frac{{\partial {\mathbf{b}}}}{{\partial {{x}^{k}}}}} \right) - \frac{{\partial \eta }}{{\partial {{x}^{k}}}}\nabla {{b}_{k}},$$

(A.2)

where the summation over the index $k$ from 1 to 3 is assumed. Since the diffusion coefficient $\eta$ depends, generally speaking, on the magnitude of the magnetic field, the use of explicit schemes for solving Eq. (A.2) leads to too strong restrictions on the time step. Therefore, to approximate this equation on an interval ${{t}^{n}} \leqslant \Delta t \leqslant {{t}^{{n + 1}}}$, our numerical code uses the absolutely stable completely implicit scheme [59]

$$\frac{{{{{\mathbf{b}}}^{{n + 1}}} - {{{\mathbf{b}}}^{n}}}}{{\Delta t}} = {{\hat {D}}^{{n + 1}}}{{{\mathbf{b}}}^{{n + 1}}} - {{{\mathbf{R}}}^{{n + 1}}}.$$

(A.3)

Here, the diffusion operator $\hat {D}$ approximates the first term on the right side of (A.2) and ${\mathbf{R}}$ approximates the source term.

The diffusion operator and the source term can be conveniently split into spatial directions:

$$\hat {D} = {{\hat {D}}_{x}} + {{\hat {D}}_{y}} + {{\hat {D}}_{z}},\quad {\mathbf{R}} = {{{\mathbf{R}}}_{x}} + {{{\mathbf{R}}}_{y}} + {{{\mathbf{R}}}_{z}},$$

(A.4)

where, for example, the operator ${{\hat {D}}_{x}}$ approximates the expression $\partial {\text{/}}\partial x(\eta \partial {\mathbf{b}}{\text{/}}\partial x)$ and ${{{\mathbf{R}}}_{x}}$ approximates the expression $(\partial \eta {\text{/}}\partial x)\nabla {{b}_{x}}$. To implement scheme (A.3), we organize an iterative process

$$\frac{{{{{\mathbf{b}}}^{{(p + 1/3)}}} - {{{\mathbf{b}}}^{n}}}}{{\Delta t}} = {{\hat {D}}^{{(p)}}}{{{\mathbf{b}}}^{{(p + 1/3)}}} - {\mathbf{R}}_{x}^{{(p)}},$$

(A.5)

$$\frac{{{{{\mathbf{b}}}^{{(p + 2/3)}}} - {{{\mathbf{b}}}^{{(p + 1/3)}}}}}{{\Delta t}} = {{\hat {D}}^{{(p)}}}{{{\mathbf{b}}}^{{(p + 2/3)}}} - {\mathbf{R}}_{y}^{{(p)}},$$

(A.6)

$$\frac{{{{{\mathbf{b}}}^{{(p + 1)}}} - {{{\mathbf{b}}}^{{(p + 2/3)}}}}}{{\Delta t}} = {{\hat {D}}^{{(p)}}}{{{\mathbf{b}}}^{{(p + 1)}}} - {\mathbf{R}}_{z}^{{(p)}},$$

(A.7)

where $p$ is the iteration number. In this case, we assume that ${{{\mathbf{b}}}^{{(0)}}} = {{{\mathbf{b}}}^{n}}$ and, in the limit at $p \to \infty$, the process converges to the sought solution, ${{{\mathbf{b}}}^{{(p)}}} \to {{{\mathbf{b}}}^{{n + 1}}}$. The convergence of the process is determined by the condition

$$\left\| {{{{\mathbf{b}}}^{{(p + 1)}}} - {{{\mathbf{b}}}^{{(p)}}}} \right\| \leqslant {{\varepsilon }_{1}}\left\| {{{{\vec {b}}}^{{(p)}}}} \right\| + {{\varepsilon }_{2}},$$

(A.8)

where ${{\varepsilon }_{1}}$ and ${{\varepsilon }_{2}}$ are convergence parameters.

The calculation of the magnetic field value on a new iterative layer $p + 1$ from the known value on the layer $p$ consists of three stages. At the first stage, we obtain the value ${{{\mathbf{b}}}^{{(p + 1/3)}}}$; then, from this value, we find ${{{\mathbf{b}}}^{{(p + 2/3)}}}$; and, finally, we obtain ${{{\mathbf{b}}}^{{(p + 1)}}}$. At each stage, we have a difference scheme of the form

$$\frac{{{{{\mathbf{b}}}_{C}} - {\mathbf{b}}_{C}^{0}}}{{\Delta t}} = \frac{1}{{{{h}_{C}}}}\left( {{{\eta }_{2}}\frac{{{{{\mathbf{b}}}_{R}} - {{{\mathbf{b}}}_{C}}}}{{{{h}_{2}}}} - {{\eta }_{1}}\frac{{{{{\mathbf{b}}}_{C}} - {{{\mathbf{b}}}_{L}}}}{{{{h}_{1}}}}} \right) - {{{\mathbf{R}}}_{C}},$$

(A.9)

where the indices $L$, $C$, and $R$ define three adjacent cells of the computational grid in a given coordinate direction and the corresponding nodes are denoted 1 (between cells $L$ and $C$) and 2 (between cells $C$ and $R$). The size of the cell $C$ is denoted ${{h}_{C}}$, and the distances between the corresponding centers of the cells are denoted ${{h}_{1}}$ and ${{h}_{2}}$. This scheme can be rewritten as follows:

$$ - {{a}_{1}}{{{\mathbf{b}}}_{L}} + ({{d}_{1}} + {{a}_{1}} + {{a}_{2}}){{{\mathbf{b}}}_{C}} - {{a}_{2}}{{{\mathbf{b}}}_{R}} = {{{\mathbf{f}}}_{1}},$$

(A.10)

where the coefficients are

$${{a}_{{1,2}}} = \frac{{\Delta t{\kern 1pt} {{\eta }_{{1,2}}}}}{{{{h}_{{1,2}}}}},\quad {{d}_{1}} = {{h}_{C}},\quad {{{\mathbf{f}}}_{1}} = {{h}_{C}}({{{\mathbf{b}}}^{0}} - \Delta t{{{\mathbf{R}}}_{C}}).$$

(A.11)

Thus, the systems of linear algebraic equations that arise in the implementation of the scheme can be solved by the tridiagonal scalar algorithm in each spatial direction. In this case, we use the stream version of the algorithm [60]. The stream tridiagonal algorithm gives a more accurate solution in the case of stiff diffusion problems, when the diffusion coefficient in the computational domain varies over a wide range. Therefore, when considering, e.g., Bohm diffusion of the magnetic field, this version of the tridiagonal algorithm seems to be preferable. The described scheme is implicit, absolutely stable, and satisfies the positivity condition.