= Solution
In <ideal magnetohydrodynamics>, the first-order <electric field> is $\mathbf E_1=-\mathbf v\times B_0\mathbf e_z$. Retaining its product with both the background and first-order <magnetic field> gives the <Poynting vector>
$$
\mathbf S=\frac1{\mu_0}\mathbf E_1\times(B_0\mathbf e_z+\mathbf b)
=\frac{B_0^2}{\mu_0}\mathbf v-\frac{B_0}{\mu_0}(\mathbf v\cdot\mathbf b)\mathbf e_z.
$$
The first term is transverse and oscillatory, with zero period average. The second is quadratic and transports wave <energy>. A potential quadratic ideal electric-field term $-\mathbf v\times\mathbf b$ vanishes here because the real traveling-wave fields are parallel or antiparallel.
For the $+z$ branch, $\mathbf b=-\sqrt{\mu_0\rho_0}\mathbf v$, so
$$
\boxed{\langle\mathbf S\rangle=\rho_0v_A\langle|\mathbf v|^2\rangle\mathbf e_z=v_A\langle\mathcal E\rangle\mathbf e_z.}
$$
For the opposite branch, its sign reverses. The averaged <Poynting flux> is the total wave <energy density>, including equal kinetic and magnetic shares, multiplied by the <group velocity>. This agrees with <Alfvén wave energy transport>. Although the flux in this expression is electromagnetic, it supplies the transport of both components of the coupled fluid-field wave.
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