🧲 MHD#

MHD (Magnetohydrodynamics) (also called magneto-fluid dynamics or hydromagnetics) is a model of electrically conducting fluids that treats all interpenetrating particle species together as a single continuous medium. It is primarily concerned with the low-frequency, large-scale, magnetic behavior in plasmas and liquid metals and has applications in numerous fields including geophysics, astrophysics, and engineering.


Incompressible MHD#

In a connected, bounded domain \(\Omega \subset \mathbb{R}^{d}\), \(d\in\left\lbrace2,3\right\rbrace\) with Lipschitz boundary \(\partial \Omega\), the incompressible constant density magnetohydrodynamic (or simply incompressible MHD) equations are given as

(1)#\[\begin{split}\begin{equation} \begin{aligned} \rho \left[ \partial_t\boldsymbol{u}^* + \left(\boldsymbol{u}^* \cdot \nabla\right)\boldsymbol{u}^* \right] - \tilde{\mu} \Delta \boldsymbol{u}^* - \boldsymbol{j}^* \times \boldsymbol{B}^* + \nabla p^* &= \rho \boldsymbol{f}^*, \\ \nabla\cdot \boldsymbol{u}^* &= 0 ,\\ \partial_t \boldsymbol{B}^* + \nabla\times \boldsymbol{E}^* &= \boldsymbol{0} ,\\ \boldsymbol{j}^* - \sigma \left(\boldsymbol{E}^* + \boldsymbol{u}^*\times\boldsymbol{B}^*\right) &= \boldsymbol{0} , \\ \boldsymbol{j}^* - \nabla\times \boldsymbol{H}^* &= \boldsymbol{0} ,\\ \boldsymbol{B}^* &= \mu \boldsymbol{H}^*, \end{aligned} \end{equation}\end{split}\]


  • \(\boldsymbol{u}^*\) fluid velocity

  • \(\boldsymbol{j}^*\) electric current density

  • \(\boldsymbol{B}^*\) magnetic flux density

  • \(p^*\) hydrodynamic pressure

  • \(\boldsymbol{f}^*\) body force

  • \(\boldsymbol{E}^*\) electric field strength

  • \(\boldsymbol{H}^*\) magnetic field strength

subject to material parameters the fluid density \(\rho\), the dynamic viscosity \(\tilde{\mu}\), the electric conductivity \(\sigma\), and the magnetic permeability \(\mu\).

By selecting the characteristic quantities of length \(L\), velocity \(U\), and magnetic flux density \(B\), a non-dimensional formulation of (1) is

(2)#\[\begin{split}\begin{equation} \begin{aligned} \partial_t\boldsymbol{u} + \left(\boldsymbol{u} \cdot \nabla\right)\boldsymbol{u} - \mathrm{R}_f^{-1} \Delta \boldsymbol{u} - \mathrm{A}_l^{-2}\boldsymbol{j} \times \boldsymbol{B} + \nabla p &= \boldsymbol{f}, \\ \nabla\cdot \boldsymbol{u} &= 0 ,\\ \partial_t \boldsymbol{B} + \nabla\times \boldsymbol{E} &= \boldsymbol{0} ,\\ \mathrm{R}_m^{-1}\boldsymbol{j} - \left(\boldsymbol{E} + \boldsymbol{u}\times\boldsymbol{B}\right) &= \boldsymbol{0} , \\ \boldsymbol{j} - \nabla\times \boldsymbol{B} &= \boldsymbol{0} ,\\ \end{aligned} \end{equation}\end{split}\]

where \(\boldsymbol{u}\), \(\boldsymbol{j}\), \(\boldsymbol{B}\), \(p\), \(\boldsymbol{f}\), and \(\boldsymbol{E}\) are the non-dimensional variables, and \(\mathrm{R}_f = \dfrac{\rho U L}{\tilde{\mu}} = \dfrac{U L}{\nu}\) (with \(\nu=\dfrac{\tilde{\mu}}{\rho}\) being the kinematic viscosity) is the fluid Reynolds number, \(\mathrm{A}_l = \dfrac{U\sqrt{\rho \mu}}{B} = \dfrac{U}{U_A}\) (with \(U_A = \dfrac{B}{\sqrt{\rho\mu}}\) being the Alfvén speed), and \(\mathrm{R}_m = \mu\sigma U L\) is the magnetic Reynolds number.

If we further introduce \(\boldsymbol{\omega}:=\nabla\times\boldsymbol{u}\) and \(P:=p+\frac{1}{2}\boldsymbol{u}\cdot \boldsymbol{u}\), (2) can be written into the rotational form:

(3)#\[\begin{split}\begin{equation} \begin{aligned} \partial_t\boldsymbol{u} + \boldsymbol{\omega}\times\boldsymbol{u} - \mathrm{R}_f^{-1} \Delta \boldsymbol{u} - \mathrm{A}_l^{-2}\boldsymbol{j} \times \boldsymbol{B} + \nabla P &= \boldsymbol{f}, \\ \boldsymbol{\omega} - \nabla\times\boldsymbol{u} &= \boldsymbol{0} ,\\ \nabla\cdot \boldsymbol{u} &= 0 ,\\ \partial_t \boldsymbol{B} + \nabla\times \boldsymbol{E} &= \boldsymbol{0} ,\\ \mathrm{R}_m^{-1}\boldsymbol{j} - \left(\boldsymbol{E} + \boldsymbol{u}\times\boldsymbol{B}\right) &= \boldsymbol{0} , \\ \boldsymbol{j} - \nabla\times \boldsymbol{B} &= \boldsymbol{0} .\\ \end{aligned} \end{equation}\end{split}\]

Numerical Examples#

For numerical examples of MHD, see

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