= Elsässer variable
{c}
{title2=$\mathbf z^\pm$}
For an <incompressible flow> with constant <mass density>, the <Elsässer variables> are $\mathbf z^\pm=\mathbf u\pm\mathbf v_a$, where $\mathbf v_a$ is the <Alfvén velocity>. Adding and subtracting the <Euler momentum equation> and <ideal magnetohydrodynamic induction equation> yields $\partial_t\mathbf z^\pm+(\mathbf z^\mp\cdot\nabla)\mathbf z^\pm=-\nabla\psi$, with $\psi=p/\rho+\Phi+|\mathbf v_a|^2/2$ and $\nabla\cdot\mathbf z^\pm=0$. Thus each field is transported by the other, a useful way to expose interactions of oppositely directed <Alfvén waves>.
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