Lecture 3
Conservative (Divergence) Form of MHD
Overview
Conservative form is the bookkeeping language of MHD.
-
1.
- It rewrites the equations in terms of fluxes through surfaces.
-
2.
- It makes momentum transport by the magnetic field look as concrete as momentum
transport by the fluid itself.
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3.
- It is the natural form for finite-volume numerics, jump conditions, and integral
conservation laws.
The point of conservative form is not aesthetic. It is the form in which one can integrate over a control
volume and read off exactly what crosses the boundary. Starting from the continuity equation (1.7), the
explicit momentum equation (2.9), and the induction equation (2.12), we now reorganize the same physics
into flux form.
3.1 Mass Conservation
\[\boxed { \pp {\rho }{t} + \divergence (\rho \uvec ) = 0. } \tag{3.1}\]
This is already conservative. It says that mass changes inside a volume only because the mass flux \(\rho \uvec \) crosses
its boundary.
3.2 Momentum Flux
Using (2.9) together with (3.1), one obtains
\[\boxed { \pp {(\rho \uvec )}{t} + \divergence \left ( \rho \uvec \uvec + p\,\tens {I} - \tens {\Pi } \right ) = \J \times \B . } \tag{3.2}\]
The dyadic term \(\rho \uvec \uvec \) is advective momentum transport; \(p\,\tens {I}-\tens {\Pi }\) is the mechanical stress from Lecture 2.
3.3 Lorentz Force as a Divergence
Using Ampère’s law (1.11) and \(\divergence \B =0\) from (1.12),
\[\J \times \B = \divergence \left [ \left ( \frac {B^2}{2\muo } \right )\tens {I} - \frac {\B \B }{\muo } \right ]. \tag{3.3}\]
Substituting (3.3) into (3.2) gives the full conservative momentum equation \[\boxed { \pp {(\rho \uvec )}{t} + \divergence \tens {T} = 0, } \tag{3.4}\]
with total momentum-flux tensor \[\boxed { \tens {T} = \rho \uvec \uvec + \left ( p+\frac {B^2}{2\muo } \right )\tens {I} - \frac {\B \B }{\muo } - \tens {\Pi }. } \tag{3.5}\]
Caution
The isotropic term \((B^2/2\muo )\tens {I}\) looks like a pressure and \(-\B \B /\muo \) looks like a tension, but they should really
be interpreted together through (3.3) and (3.5). The field transports momentum as a
tensor, not as a scalar pressure with an optional correction.
3.4 Energy Densities
Define the total energy density
\[\mathcal {E} = \frac {1}{2}\rho u^2 + \rho e + \frac {B^2}{2\muo }. \tag{3.6}\]
Its three pieces are kinetic, internal, and magnetic energy.
3.5 Kinetic Energy Equation
Dotting (2.9) with \(\uvec \) and using (3.1) gives
\[\pp {}{t} \left ( \frac {1}{2}\rho u^2 \right ) + \divergence \left [ \left ( \frac {1}{2}\rho u^2 \right )\uvec + \left ( p\,\tens {I} - \tens {\Pi } \right )\cdot \uvec \right ] = p\,\divergence \uvec - \tens {\Pi }:\grad \uvec + \J \cdot \E - \eta J^2. \tag{3.7}\]
3.6 Magnetic Energy Equation
Starting from Faraday’s law (1.10) and Ampère’s law (1.11),
\[\pp {}{t} \left ( \frac {B^2}{2\muo } \right ) + \divergence \left ( \frac {\E \times \B }{\muo } \right ) = - \J \cdot \E . \tag{3.8}\]
The flux \(\E \times \B /\muo \) is the Poynting flux.
3.7 Internal Energy Equation
A convenient conservative form for the internal energy is
\[\pp {(\rho e)}{t} + \divergence (\rho e\,\uvec ) = - p\,\divergence \uvec - \divergence \vect {q} + \tens {\Pi }:\grad \uvec + \eta J^2. \tag{3.9}\]
For an ideal gas, \[e=\frac {p}{(\gamma -1)\rho }. \tag{3.10}\]
If, in addition, the flow is adiabatic and we neglect \(\tens {\Pi }\), \(\vect {q}\), and \(\eta \), then (3.9) reduces to \[\boxed { \frac {Dp}{Dt} = -\gamma p\,\divergence \uvec , } \tag{3.11}\]
which is equivalent to the adiabatic law (1.14).
3.8 Total Energy Conservation
Adding (3.7), (3.8), and (3.9) gives
\[\boxed { \pp {\mathcal {E}}{t} + \divergence \left [ \left ( \frac {1}{2}\rho u^2 + \rho e + p \right )\uvec - \tens {\Pi }\cdot \uvec + \vect {q} + \frac {\E \times \B }{\muo } \right ] = 0. } \tag{3.12}\]
3.9 Why This Form Matters
For shocks, discontinuities, and numerical algorithms, (3.1), (3.4), and (3.12) are the natural starting point
because they survive volume integration without hidden integration-by-parts steps. Conservative form is
where the geometry of fluxes becomes algebraically explicit.
Takeaways
- Magnetic stresses belong in the same flux bookkeeping as fluid stresses.
- The Maxwell stress in (3.3) is the cleanest way to see how magnetic fields carry
momentum.
- The conservative pressure law (3.11) is not fundamental by itself; it is the reduced
form that survives after a closure has been chosen.