= Solution
In this question, the coefficient multiplying $\nabla^2\mathbf B$ is the <magnetic diffusivity>. Denote it $\eta_m$; it is the paper's $\eta$, with dimensions of length squared per time. If $\eta_{\rm Ohm}$ denotes <electrical resistivity>, then $\eta_m=\eta_{\rm Ohm}/\mu_0$. The two conventions must not be interchanged.
For an imposed real positive $\omega$, retain the momentum equation and use
$$
-i\omega\rho_0\mathbf v=\frac{ikB_0}{\mu_0}\mathbf b,\qquad
(-i\omega+\eta_mk^2)\mathbf b=ikB_0\mathbf v.
$$
Elimination gives the exact <Resistive damping of an Alfvén wave> relation
$$
\boxed{\omega^2+i\eta_m\omega k^2-v_A^2k^2=0,\qquad
k^2=\frac{\omega^2}{v_A^2-i\eta_m\omega}.}
$$
Here $k$ is complex because the driving <frequency> is real. Introduce the dimensionless diffusion parameter $\epsilon=\eta_m\omega/v_A^2\ll1$. For propagation and attenuation along $+z$, choose the square-root branch with $\operatorname{Re}k,\operatorname{Im}k>0$. The <spatial damping of a driven Alfvén wave> is
$$
\boxed{k=\frac\omega{v_A}(1-i\epsilon)^{-1/2}=\frac\omega{v_A}\left(1+\frac{i\epsilon}{2}+O(\epsilon^2)\right).}
$$
With the convention $e^{i(kz-\omega t)}$, the <amplitude> decays as $e^{-\operatorname{Im}k\,z}$. Define its e-folding length $L_A=1/\operatorname{Im}k$ and the wavelength $\lambda=2\pi/\operatorname{Re}k$. Then
$$
\boxed{L_A\sim\frac{2v_A^3}{\eta_m\omega^2},\qquad\frac{L_A}{\lambda}\sim\frac1{\pi\epsilon}.}
$$
If damping length instead refers to wave <energy>, the squared <amplitude> decays twice as fast: $L_E=L_A/2$ and $L_E/\lambda\sim1/(2\pi\epsilon)$. Stating the convention avoids a factor-of-two ambiguity. The temporal damping rate $\eta_mk^2/2$ at fixed real $k$ is a different specification from this imposed-frequency problem.
\Image[/past-exam-of-the-advanced-fluid-dynamics-course-of-the-university-of-oxford/2023-driven-alfven-attenuation.png]
{title=Weak spatial attenuation of a driven Alfvén wave and its amplitude envelope}
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