= Elsässer energy invariant
{c}
{title2=$I_\pm$}
For smooth <ideal magnetohydrodynamics> of constant <mass density>, define $I_\pm=\int_V|\mathbf z^\pm|^2dV$. Taking the <dot product> of the <Elsässer variable> equation with $2\mathbf z^\pm$ gives
$$
\partial_t|\mathbf z^\pm|^2+\nabla\cdot\left(|\mathbf z^\pm|^2\mathbf z^\mp+2\psi\mathbf z^\pm\right)=0.
$$
Therefore each integral is constant when its outward boundary flux vanishes, for example when both <velocity> and <magnetic field> are tangent to the fixed boundary, or for <periodic boundary conditions>. The <kinetic energy> plus <magnetic energy> is $\rho(I_++I_-)/4$, while the <cross-helicity> is $\sqrt{\mu_0\rho}(I_+-I_-)/4$.
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