3D SIP-CESE MHD Model on Triangular Prism Grids

Abstract

This chapter describes the 3D Solar-Interplanetary space-time Conservation Element and Solution Element (SIP-CESE) MHD model and its application on the solar wind study. SIP-CESE MHD model is established on pentahedral cells with each cell a triangular prism composed of two triangular bases and three rectangular sides. The basic principle as well as the implementation detail of the CESE numerical scheme are presented. Two examples are given to illustrate the validity and capacity of modeling solar wind: (i) two-dimensional coronal dynamical structure with multipole magnetic fields and (ii) three-dimensional coronal dynamical structure, using measured solar surface magnetic fields and the empirical values of the plasma properties on the solar surface as the initial conditions for the set of MHD equations, and then the relaxation method to achieve a quasi-steady state. These examples show that the MHD model possesses the ability to model the Sun-Earth environment. Finally, the evolution of the large-scale coronal magnetic structure during solar cycle 23 is investigated by the SIP-CESE MHD model.

Appendix

In this appendix, the Jacobian matrices for the fluxes $\mathbf {F}, \mathbf {G}, \mathbf {H}$, and $\varvec{\varPhi }$ are calculated [16].

$$ \frac{\partial \mathbf {F}}{\partial \mathbf {U}}= \begin{pmatrix} 0&{} 1&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \\ -u^2+(\gamma -1)\frac{u^2+v^2+w^2}{2}&{} (3-\gamma )u&{} (1-\gamma )v&{} (1-\gamma )w&{} \gamma -1&{} - \gamma B_x&{} (2-\gamma )B_y&{}(2-\gamma )B_z \\ -uv &{} v&{} u&{} 0&{} 0&{} -B_y&{} -B_x&{} 0\\ -uw &{} w&{} 0&{} u&{} 0&{} -B_z&{} 0&{} -B_x \\ F_1&{} F_2&{} F_3&{} F_4&{} F_5&{} F_6&{}F_7&{} F_8 \\ 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \\ -\frac{uB_y-vB_x}{\rho }&{} \frac{B_y}{\rho }&{} -\frac{B_x}{\rho }&{} 0&{} 0&{} -v&{} u&{} 0 \\ -\frac{uB_z-wB_x}{\rho }&{} \frac{B_z}{\rho }&{} 0&{} -\frac{B_x}{\rho }&{} 0&{} -w&{} 0&{} u \end{pmatrix} $$

where

$$ F_1=-\frac{\gamma eu}{\rho }+ (\gamma -1)u\left( u^2+v^2+w^2\right) +\frac{\gamma -2}{2}u\frac{B_y^2+B_z^2}{\rho }+\frac{\gamma }{2}u\frac{B_x^2}{\rho }+B_x \frac{v B_y+w B_z}{\rho }$$

$$ F_2=\gamma \frac{e}{\rho }+\frac{3}{2}(1-\gamma )u^2+\frac{1-\gamma }{2}\left( v^2+w^2\right) -\frac{\gamma -2}{2}\frac{B_y^2+B_z^2}{\rho }-\frac{\gamma }{2}\frac{B_x^2}{\rho } $$

$$ F_3=(1-\gamma )uv-\frac{B_x B_y}{\rho } $$

$$ F_4=(1-\gamma )uw-\frac{B_x B_z}{\rho },F_5=\gamma u $$

$$ F_6=-\gamma uB_x-(vB_y+wB_z) $$

$$ F_7=(2-\gamma )uB_y-vB_x $$

$$ F_8=(2-\gamma )uB_z-wB_x $$

$$ \frac{\partial \mathbf {G}}{\partial \mathbf {U}}=\begin{pmatrix} 0&{} 0&{} 1&{} 0&{} 0&{} 0&{} 0&{} 0 \\ -uv&{} v&{} u&{} 0&{} 0&{} - B_y&{} - B_x&{} 0 \\ -v^2+(\gamma -1)\frac{u^2+v^2+w^2}{2}&{} (1-\gamma )u&{} (3-\gamma )v&{} (1-\gamma )w&{} \gamma -1&{} (2-\gamma )B_x&{} - \gamma B_y&{}(2-\gamma )B_z \\ -vw&{} 0&{} w&{}v&{} 0&{} 0&{} -B_z&{} -B_y \\ G_1&{}G_2&{} G_3&{} G_4&{} G_5&{} G_6&{}G_7&{}G_8 \\ -\frac{vB_x-uB_y}{\rho }&{} -\frac{B_y}{\rho }&{} \frac{B_x}{\rho }&{} 0&{} 0&{} v&{} -u&{} 0 \\ 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \\ -\frac{vB_z-wB_y}{\rho }&{} 0&{} \frac{B_z}{\rho }&{} -\frac{B_y}{\rho }&{} 0&{} 0&{} -w&{} v \end{pmatrix} $$

where

$$ G_1=-\frac{\gamma ev}{\rho }+ (\gamma -1)v\left( u^2+v^2+w^2\right) +\frac{\gamma -2}{2}v\frac{B_x^2+B_z^2}{\rho }+\frac{\gamma }{2}v\frac{B_y^2}{\rho }+B_y \frac{u B_x+w B_z}{\rho }$$

$$ G_2=(1-\gamma )vu-\frac{B_y B_x}{\rho } $$

$$ G_3=\gamma \frac{e}{\rho }+\frac{3}{2}(1-\gamma )v^2+\frac{1-\gamma }{2}\left( u^2+w^2\right) -\frac{\gamma -2}{2}\frac{B_x^2+B_z^2}{\rho }-\frac{\gamma }{2}\frac{B_y^2}{\rho } $$

$$ G_4=(1-\gamma )vw-\frac{B_y B_z}{\rho },G_5=\gamma v $$

$$ G_6=(2-\gamma )vB_x-uB_y $$

$$ G_7=-\gamma vB_y-\left( uB_x+wB_z\right) $$

$$ G_8=(2-\gamma )vB_z-wB_y $$

$$ \frac{\partial \mathbf {H}}{\partial \mathbf {U}}=\begin{pmatrix} 0&{} 0&{} 0&{} 1&{} 0&{} 0&{} 0&{} 0 \\ -uw&{} 0&{} w&{} u&{} 0&{} -B_z&{} 0&{} -B_x \\ -vw&{} 0&{} w&{} v&{} 0&{} 0&{} -B_z&{} -B_y \\ -w^2+(\gamma -1)\frac{u^2+v^2+w^2}{2}&{} (1-\gamma )u&{} (1-\gamma )v&{} (3-\gamma )w&{} \gamma -1&{} (2-\gamma )B_x&{} (2-\gamma )B_y&{} -\gamma B_z\\ H_1&{} H_2&{} H_3&{} H_4&{}H_5&{} H_6&{} H_7&{} H_8 \\ -\frac{wB_x-uB_z}{\rho }&{} -\frac{B_z}{\rho }&{} 0&{} \frac{B_x}{\rho }&{} 0&{} w&{} 0&{} -u \\ -\frac{wB_y-vB_z}{\rho }&{} 0&{} -\frac{B_z}{\rho }&{} \frac{B_y}{\rho }&{} 0&{} 0&{} w&{} -v \\ 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \end{pmatrix} $$

where

$$ H_1=-\frac{\gamma ew}{\rho }+ (\gamma -1)w\left( u^2+v^2+w^2\right) +\frac{\gamma -2}{2}w\frac{B_x^2+B_y^2}{\rho }+\frac{\gamma }{2}w\frac{B_z^2}{\rho }+B_z \frac{u B_x+v B_y}{\rho }$$

$$ H_2=(1-\gamma )wu-\frac{B_z B_x}{\rho } $$

$$ H_3=(1-\gamma )wv-\frac{B_z B_y}{\rho } $$

$$ H_4=\gamma \frac{e}{\rho }+\frac{3}{2}(1-\gamma )w^2+\frac{1-\gamma }{2}\left( u^2+v^2\right) -\frac{\gamma -2}{2}\frac{B_x^2+B_y^2}{\rho }-\frac{\gamma }{2}\frac{B_z^2}{\rho } $$

$$ H_5=\gamma w $$

$$ H_6=(2-\gamma )wB_x-uB_z $$

$$ H_7=(2-\gamma )wB_y-vB_z $$

$$ H_8=-\gamma wB_z-(uB_x+vB_y) $$

$$ \frac{\partial \varvec{\varPhi }}{\partial { \mathbf {U}} } =\mathbf {I}-\frac{\varDelta {t}}{2} \frac{\partial {\varvec{\eta }}}{\partial { \mathbf {U}}} \equiv \begin{pmatrix} 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ a_{21}&{}\quad 1&{}\quad a_{23}&{}\quad 0 &{}\quad 0&{}\quad a_{26}&{}\quad 0&{}\quad 0\\ a_{31} &{}\quad a_{32}&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad a_{37}&{}\quad 0\\ a_{41}&{}\quad 0&{}\quad 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad a_{48} \\ a_{51}&{}\quad a_{52}&{}\quad a_{53}&{}\quad a_{54}&{}\quad a_{55}&{}\quad a_{56}&{}\quad a_{57}&{}\quad a_{58} \\ a_{61}&{}\quad a_{62}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 1&{}\quad 0&{}\quad 0 \\ a_{71}&{}\quad 0&{}\quad a_{73}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 1 &{}\quad 0\\ a_{81}&{}\quad 0 &{}\quad 0&{}\quad a_{84}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 1 \end{pmatrix} $$

where

$$ a_{21}=-{a}_\mathrm{gra}{x}\frac{\varDelta {t}}{2}-\omega ^2 x \frac{\varDelta {t}}{2}, a_{23}=-2\omega \frac{\varDelta {t}}{2}, a_{31}=-{a}_\mathrm{gra}{y}\frac{\varDelta {t}}{2}-\omega ^2 y \frac{\varDelta {t}}{2}, a_{32}=2\omega \frac{\varDelta {t}}{2} $$

$$ a_{41}=-{a}_\mathrm{gra}{z}\frac{\varDelta {t}}{2}, a_{26}= a_{37}= a_{48}=({\nabla }\cdot {B})^{n-1/2}_{Q^*}\frac{\varDelta {t}}{2} $$

$$ a_{51}=-(\gamma -1){q}_e \left( {T_0}-\gamma (\gamma -1)\frac{{u}^2+{v}^2+{w}^2}{2}\right) \frac{\varDelta {t}}{2}-({\nabla }\cdot {B})^{n-1/2}_{Q^*} \frac{{u}{B_x}+{v}{B_y}+{w}{B_z}}{{\rho }} \frac{\varDelta {t}}{2} $$

$$ a_{52}=-\left( {a}_\mathrm{gra}{x}+ \gamma (\gamma -1)^2{q}_e{u} \right) \frac{\varDelta {t}}{2}-\omega ^2 x \frac{\varDelta {t}}{2}+({\nabla }\cdot {B})^{n-1/2}_{Q^*}\frac{{B_x}}{{\rho }}\frac{\varDelta {t}}{2} $$

$$ a_{53}=-\left( {a}_\mathrm{gra}{y}+ \gamma (\gamma -1)^2{q}_e{v} \right) \frac{\varDelta {t}}{2}-\omega ^2 y \frac{\varDelta {t}}{2}+({\nabla }\cdot {B})^{n-1/2}_{Q^*}\frac{{B_y}}{{\rho }}\frac{\varDelta {t}}{2} $$

$$ a_{54}=-\left( {a}_\mathrm{gra}{z}+ \gamma (\gamma -1)^2{q}_e{w} \right) \frac{\varDelta {t}}{2}+({\nabla }\cdot {B})^{n-1/2}_{Q^*}\frac{{B_z}}{{\rho }}\frac{\varDelta {t}}{2} $$

$$ a_{55}=1+\gamma (\gamma -1)^2{q}_e\frac{\varDelta {t}}{2} $$

$$ a_{56}=-\gamma (\gamma -1)^2{q}_e{B_x}\frac{\varDelta {t}}{2}+ \left( ({\nabla }\cdot {B})^{n-1/2}_{Q^*}{u} \right) \frac{\varDelta {t}}{2} $$

$$ a_{57}=-\gamma (\gamma -1)^2{q}_e{B_y}\frac{\varDelta {t}}{2}+ \left( ({\nabla }\cdot {B})^{n-1/2}_{Q^*}{v}\right) \frac{\varDelta {t}}{2} $$

$$ a_{58}=-\gamma (\gamma -1)^2{q}_e{B_z}\frac{\varDelta {t}}{2}+ \left( ({\nabla }\cdot {B})^{n-1/2}_{Q^*}{w} \right) \frac{\varDelta {t}}{2} $$

$$ a_{61}=-\frac{({\nabla }\cdot {B})^{n-1/2}_{Q^*} {u}}{{\rho }}\frac{\varDelta {t}}{2},a_{62}= \frac{({\nabla }\cdot {B})^{n-1/2}_{Q^*} }{{\rho }}\frac{\varDelta {t}}{2} $$

$$ a_{71}=-\frac{({\nabla }\cdot {B})^{n-1/2}_{Q^*} {v}}{{\rho }}\frac{\varDelta {t}}{2},a_{73}= \frac{({\nabla }\cdot {B})^{n-1/2}_{Q^*} }{{\rho }}\frac{\varDelta {t}}{2} $$

$$ a_{81}=-\frac{({\nabla }\cdot {B})^{n-1/2}_{Q^*} {w}}{{\rho }}\frac{\varDelta {t}}{2},a_{84}= \frac{({\nabla }\cdot {B})^{n-1/2}_{Q^*} }{{\rho }}\frac{\varDelta {t}}{2} $$

with ${a}_\mathrm{gra}=-\frac{G M_\mathrm{s}}{{r}^3}$ and ${q}_e={q}_0\exp \left[ -\frac{(r-R_\mathrm{s})^2}{\sigma _0^2}\right]$. Then, $\left( \frac{\partial \varvec{\varPhi }}{\partial { \mathbf {U}} }\right) ^{-1}$ can be easily obtained by Mathematica.