1. Plasma Dynamics Group Aerospace & Energetics Research Program Boundary Conditions for MHD Resistive MHD equations in weakly conservative form (balance form, divergence form, flux/source form) are where the total energy is given by

2. Plasma Dynamics Group Aerospace & Energetics Research Program Deriving Consistent Boundary Conditions for MHD* Expressing the MHD equations in compact form, Integrating over the domain, For ideal MHD, RHS vanishes. Consistent boundary values must be specified for the normal fluxes in the surface integral. *The discussion is limited to Cartesian coordinates. Additional terms arise in other coordinate systems, e.g. cylindrical.

3. Plasma Dynamics Group Aerospace & Energetics Research Program Boundary Conditions for Ideal MHD Consider the case of perfectly conducting, impermeable wall For the continuity equation, this gives Density is unconstrained. Typically, extrapolate from domain, or simply

4. Plasma Dynamics Group Aerospace & Energetics Research Program Boundary Conditions for Ideal MHD For the induction equation, the boundary conditions give If no initial field penetrates boundary, B n = 0, v t is unconstrained. Typically, extrapolate from domain, or simply If initial field penetrates boundary, B n ≠ 0, it must be specified and

5. Plasma Dynamics Group Aerospace & Energetics Research Program Boundary Conditions for Ideal MHD For the momentum equation, the boundary conditions give If B n = 0, B t is unconstrained, but net pressure at boundary must be specified. If B n ≠ 0, B and p must be specified at boundary.

6. Plasma Dynamics Group Aerospace & Energetics Research Program Boundary Conditions for Ideal MHD For the energy equation, the boundary conditions give No additional constraints are derived. The first term is unconstrained since v n = 0. If B n = 0, B t ·v t is unconstrained. If B n ≠ 0, B t ·v t must be specified. However, v t = 0 from induction equation. B t is unconstrained.

7. Plasma Dynamics Group Aerospace & Energetics Research Program Boundary Conditions for Resistive MHD If the resistive MHD equations are considered, the RHS terms modify the required boundary conditions. For the induction equation, the integral of the RHS gives Need to specify ηj t. If B n = 0,must be specified. If B n ≠ 0,must also be specified.

8. Plasma Dynamics Group Aerospace & Energetics Research Program Boundary Conditions for Resistive MHD If the resistive MHD equations are considered, the RHS terms modify the required boundary conditions. For the energy equation, the integral of the RHS gives Need to specify ηj t and B t. If B n = 0, must be specified. If B n ≠ 0,must also be specified.

9. Plasma Dynamics Group Aerospace & Energetics Research Program Boundary Conditions for Insulators Consider the case of an insulating, permeable wall. The integrated induction equation becomes A tangential electric field can be applied by any combination of the terms: B n v t, v n B t, and ηj t.