\frac{D B}{D t} = (B \cdot \nabla) u - B \nabla \cdot u,

\frac{D u}{D t} = - \nabla (h + Φ) + T \nabla s + \frac{( \nabla \times B ) \times B}{4 π ρ} .

The magnetic force is perpendicular to

B

. For

h_{c} = u \cdot B

, the two equations therefore give

\frac{D h _{c}}{D t} = - B \cdot \nabla (h + Φ - \frac{u ^{2}}{2}) + T B \cdot \nabla s - h_{c} \nabla \cdot u .

Since

\nabla \cdot B = 0

\frac{\partial h _{c}}{\partial t} + \nabla \cdot F_{c} = S_{c},

with

F_{c} = u h_{c} + B (h + Φ - \frac{u ^{2}}{2}), S_{c} = T B \cdot \nabla s .

\nabla s = 0

, so the source vanishes.

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