MHD instability: $\boldsymbol{k}\cdot\boldsymbol{B}=0$? In this paper, on page 3, it states that to trigger an instability, a disturbance needs to have a wave vector, $\boldsymbol{k}$, which satisfies,
$$\boldsymbol{k}\cdot\boldsymbol{B}=0,$$
where $\boldsymbol{B}$ gives the magnetic induction. In astrophysical contexts, $\boldsymbol{B}$ is referred to as the magnetic field. Do you know where the equation above comes from? Im guessing this formula only applies to instabilities in a linearised system?
 A: I am by no means an expert in plasma physics, but I think the equation of your interest comes from the linear perturbation analysis of the equations of ideal, inviscid magnetohydrodynamics, and studying the perturbations plus linearising the resulting perturbation equations, one gets1
\begin{align}
    \vec{k} \cdot \delta\vec{B} &= 0 \quad,\\
    \\
    \omega \cdot \delta \vec{B} - \vec{k} \times \left(\vec{B}_0 \times \delta\vec{v} \right) &= 0 \quad,
\end{align}
as conditions for the magnetic field instability respectively.
I hope this helps somewhat.
$$ **\quad Edit \; 01 \quad** $$
Although not open access, I found2 those specific types of MHD instabilites are apparently referred to as Interchange instabilities and in the book by Boyd & Sanderson elaborates on this specific condition in more depth through the following two paragraphs,

By introducing magnetic shear we can make sure that for any given mode with propagation vector $\vec{k}$ this least stable condition, $\vec{k} \cdot \vec{B} = 0$, is restricted to specific layers and does not occur throughout the plasma. Thus, in our analysis we want to allow for arbitrary orientation of $\vec{k}$ to $\vec{B}$ and vertical variation of the direction of $\vec{B}$. Without loss of generality, we may choose coordinates such that the $y$-axis is vertical and the $z$-axis is parallel to $\vec{k}$, the direction of propagation of the mode under investigation.

Furthermore,

Surfaces where $\vec{k} \cdot \vec{B}_0 = 0$ play a crucial role in stability analysis quite generally and are called resonant surfaces.


Footnotes & References:
1 Bartelmann, M.: Theoretical Astrophysics - An Introduction. Wiley-VCH Verlag, 2013, p. 266ff.
2 Boyd, T. J. M. & Sanderson, J. J.: The Physics of Plasmas. University of Cambridge Press, 2010, p. 124ff.
A: All magnetic fields satisfy $$i\mathbf{k} \cdot \mathbf{B}(\mathbf{k}) = 0 $$ at all times, since this is the Fourier transform of $$\mathbf{\nabla} \cdot \mathbf{B}(\mathbf{x}) = 0 $$
