Boundary Conditions

If Bn ≠ 0, B n must also be specified. t
Aerospace & Energetics Research Program
Plasma Dynamics Group
Boundary Conditions for Insulators
Consider the case of an insulating, permeable wall. The integrated induction equation becomes
Expressing the MHD equations in compact form,
Q·F t

Sp
Integrating over the domain,
tdVQdSn· FdVSp
For ideal MHD, RHS vanishes.
Consistent boundary values must be specified for the normal fluxes in the surface integral.
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Plasma Dynamics Group
Boundary Conditions for Ideal MHD
For the energy equation, the boundary conditions give
n ? e p B 22 v B ? v B e p B 22 v n B t? v tB n
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Plasma Dynamics Group
Boundary Conditions for Ideal MHD
For the momentum equation, the boundary conditions give
n 贩 v v B B p B B /2 I n 贩 v v n B B n 稩 p B ? B2
If Bn ≠ 0, B n must also be specified. t
Aerospace & Energetics Research Program
Plasma Dynamics Group
Boundary Conditions for Resistive MHD
If the resistive MHD equations are considered, the RHS terms modify the required boundary conditions.
Boundary Conditions for MHD
Resistive MHD equations in weakly conservative form (balance form, divergence form, flux/source form) are

v

0

tB ev· e vvpB B B v ?B B 2pB v vB ?B B?v 2B I? ( 0 B )B )B
If no initial field penetrates boundary, Bn = 0, vt is unconstrained.
Typically, extrapolate from domain, or simply
vt 0 n If initial field penetrates boundary, Bn ≠ 0, it must be specified and vt 0
For the energy equation, the integral of the RHS gives
dV?B)BdSn?B)B


Need to specify ηjt and Bt.
dSn?j)B
If
Bn
=
0,
B
t
,
B t n
must be specified.
equation. Bt is unconstrained.
Aerospace & Energetics Research Program
Plasma Dynamics Group
Boundary Conditions for Resistive MHD
If the resistive MHD equations are considered, the RHS terms modify the required boundary conditions.
dV E
dSBnvt vnBt n(jt )
A tangential electric field can be applied by any combination of the terms:
Bnvt , vnBt , and ηjt.
0 B n B n 稩 pB n 2/2B t2/2
If Bn = 0, Bt is unconstrained, but net pressure at boundary must be
specified.
p Bt2 / 2
If Bn ≠ 0, B and p must be specified at boundary.
Aerospace & Energetics Research Program
Plasma Dynamics Group
谢谢！
where the total energy is given by
ep1v2贩 vB2B
Aerospace & Energetics Research Program
Plasma Dynamics Group
Deriving Consistent Boundary Conditions for MHD*
For the continuity equation, this gives
n?vvn
Density is unconstrained. Typically, extrapolate from domain, or simply
0 n
Aerospace & Energetics Research Program
For the induction equation, the integral of the RHS gives
dV(B)dSn(B)


Need to specify ηjt.
dSn(j)
If Bn = 0,
B t n
must be specified.
n ·F
*The discussion is limited to Cartesian coordinates. Additional terms
arise in other coordinate systems, e.g. cylindrical.
Aerospace & Energetics Research Program
B t· v tB n
No additional constraints are derived.
The first term is unconstrained since vn = 0. If Bn = 0, Bt ·vt is unconstrained. If Bn ≠ 0, Bt ·vt must be specified. However, vt = 0 from induction
Plasma Dynamics Group
Boundary Conditions for Ideal MHD
Consider the case of perfectly conducting, impermeable wall
nE0Et
Bv t
0
t
Bn
0
n?v vn0
Plasma Dynamics Group
Boundary Conditions for Ideal MHD
For the induction equation, the boundary conditions give
n 贩 v B B v n v B t n ? B v t v n B t B n v t B n v t