= Alfvén wave energy transport
{c}
{title2=$\partial_t\mathcal E+\partial_zF_z=0$}
Dot the two <Linear Alfvén wave equations> with $\mathbf u_\perp$ and $\mathbf b_\perp/\mu_0$, respectively, and add. The result is <conservation of energy> with
$$
\mathcal E=\frac{\rho|\mathbf u_\perp|^2}{2}+\frac{|\mathbf b_\perp|^2}{2\mu_0},
\qquad F_z=-\frac{B_0}{\mu_0}\mathbf u_\perp\cdot\mathbf b_\perp.
$$
A single traveling <Alfvén wave> has equal <kinetic energy> and perturbation <magnetic energy>, and $F_z=\pm v_A\mathcal E$. This equality need not hold instantaneously at each point of a superposed <standing wave>.
As an original ideal-model example, stipulate $B_0=1.00\,\mathrm T$, $\rho=10^4\,\mathrm{kg\,m^{-3}}$ and a uniform path $L=0.20\,\mathrm m$. Using $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{H\,m^{-1}}$, the <Alfvén speed> is $8.92\,\mathrm{m\,s^{-1}}$ and the transit time $L/v_A$ is $22.4\,\mathrm{ms}$. Doubling the <magnetic field> halves that time. A sinusoidal <velocity> with peak <amplitude> $0.010\,\mathrm{m\,s^{-1}}$ has magnetic peak <amplitude> $1.12\,\mathrm{mT}$ and mean <energy> flux $\langle F_z\rangle=\rho v_AU^2/2=4.46\,\mathrm{W\,m^{-2}}$ in the positive direction. These are stipulated ideal parameters, not historical measurements or a prediction including vessel walls.
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