The concept of negative energy (NE) waves is very useful in studying stability. For long time only the NE wave instability of propagating waves was studied. Some time ago, Ruderman (Reference Ruderman2018) (Paper I below) studied for the fist time this type of instability for standing waves. He used a model problem of stability of a tangential magnetohydrodynamic (MHD) discontinuity in an incompressible plasma. The main conclusion that he made is that the NE wave instability of standing waves occurs when the flow velocity exceeds a critical velocity. This condition is the same as the condition of the NE wave instability for propagating waves. While this conclusion is perfectly correct, the expression for the instability increment obtained by Paper I is wrong. We aim to correct the error made in that paper and derive the correct expression for the instability increment.

We start from a brief reminder of the problem set-up and its solution given in by Paper I. The equilibrium state is an MHD tangential discontinuity in an incompressible plasma. The equation of this discontinuity is $z = 0$ in Cartesian coordinates $x,y,z$ . The plasma below the discontinuity is immovable and viscous, while the plasma above the discontinuity is moving with the velocity $U$ in the positive $x$ -direction. It was assumed that the Reynolds number is large, so dissipation in the viscous plasma only takes place in a narrow dissipative layer near the discontinuity. The perturbed discontinuity is defined by $z = \zeta (t,x)$ . In the linear approximation the evolution of $\zeta (t,x)$ is described by the equation derived by Ruderman & Goossens (Reference Ruderman and Goossens1995). It was assumed that the magnetic field lines are frozen in a dense plasma at $x = 0$ and $x = L$ , so $\zeta (t,0) = \zeta (t,L) = 0$ . The wave propagating in the direction opposite to the direction of flow becomes an NE wave when $U \gt U_c$ , where

(1) \begin{equation} U_c^2 = \frac {\rho _1 V_1^2 + \rho _2 V_2^2}{\rho _2} = \frac {\rho _1 V_{KH}^2} {\rho _1 + \rho _2}, \quad V_{1,2} = \frac {B_1^2}{\mu _0\rho _{1,2}}, \end{equation}

where $\rho$ is the density, $B$ the magnetic field, $\mu _0$ magnetic permeability of free space, the indices 1 and 2 refer to the equilibrium quantities below and above the discontinuity and $V_{KH}^2$ is the Kelvin-Helmholtz threshold velocity.

The NE wave instability increment is proportional to the coefficient of kinematic viscosity $\nu$ . It was assumed in Paper I that the instability growth time is much higher than the oscillation period of a standing wave. In accordance with this the ‘slow’ time $T = \epsilon t, \epsilon \ll 1,$ and the scaled coefficient of kinematic viscosity $\bar \nu = \epsilon ^{-1}\nu$ were introduced. Then the solution to the problem was looked for in the form of expansion $\zeta = \zeta _1 + \epsilon \zeta _2 + \ldots$ In the first-order approximation the expression for $\zeta _1$ was obtained. It reads

(2) \begin{equation} \zeta _1 = A(T)[\cos (\omega t - k_+ x) - \cos (\omega t - k_+ x)], \end{equation}

where $A(T)$ is a function to be determined and the frequency $\omega$ is given by equation (3.12) in Paper I. The expression for $\omega$ is incorrect. The correct expression is

(3) \begin{equation} \omega = \frac {kn \rho _2(U^2 - U_c^2)}{\sqrt {\rho _1\rho _2(V_{KH}^2 - U_c^2)}}, \quad k = \frac \pi L, \quad n = 1,2,\ldots \end{equation}

The wavenumbers $k_\pm$ are defined by

(4) \begin{equation} k_\pm = \omega \frac {\rho _2 U \mp \sqrt {\rho _1\rho _2(V_{KH}^2 - U^2)}} {\rho _2(U^2 - U_c^2)}. \end{equation}

This quantities are related by $k_- - k_+ = 2kn$ .

In the second-order approximation the equation for $\zeta _2$ was derived (see equation (3.12) in Paper I. It is convenient to transform this equation to

(5) \begin{eqnarray} (\rho _1 + \rho _2)\frac {\partial ^2\zeta _2}{\partial t^2} &+& 2\rho _2 U \frac {\partial ^2\zeta _2}{\partial t\partial x} + \rho _2(U^2 - U_c^2) \frac {\partial ^2\zeta _2}{\partial x^2} \nonumber \\ &=& -\textrm { ie}^{\textrm { i}\omega t}\left (a_+ \textrm { e}^{-\textrm { i}k_+ x} + a_- \textrm { e}^{-\textrm { i}k_- x}\right ) + \mbox {c.c.}, \end{eqnarray}

where c.c. indicates complex conjugate and

(6) \begin{equation} a_\pm = k_\pm \sqrt {\rho _1\rho _2(V_{KH}^2 - U^2)}\frac {dA}{dT} \pm 2\bar \nu \omega \rho _1(k_\pm ^2 + k_y^2)A. \end{equation}

We note that $\zeta _2 = 0$ at $x = 0,L$ . The condition of existence of solution to equation (5) bounded with respect to time results in the equation determining $A(T)$ . This equation was incorrectly derived in Paper I. This resulted in a wrong expression for the instability increment $\gamma$ . We now derive the correct expression for this quantity. We look for the solution to equation (5) satisfying the zero boundary conditions in the form $\zeta _2 = \zeta _p + \zeta _h$ , where $\zeta _p$ is a particular solution to equation (5) and $\zeta _h$ is the general solution to its homogeneous counterpart. It can be shown that the solution bounded with respect to time to the homogeneous counterpart satisfying the zero boundary conditions and arbitrary initial conditions always exists. We do not give this proof here. We look for $\zeta _p$ in the form $\zeta _p = f(x)\textrm { e}^{\textrm { i}\omega t} + \mbox {c.c.}$ Substituting this expression in equation (5) yields

(7) \begin{equation} \rho _2(U^2 - U_c^2)f'' + 2\textrm { i}\rho _2\omega Uf' - (\rho _1 + \rho _2)\omega ^2 f = -\textrm { i}\left (a_+ \textrm { e}^{-\textrm { i}k_+ x} + a_- \textrm { e}^{-\textrm { i}k_- x}\right ), \end{equation}

where the prime indicates the derivative with respect to $x$ . We look for the solution to this equation in the form

(8) \begin{equation} f(x) = x\left (H_+ \textrm { e}^{-\textrm { i}k_+ x} + H_- \textrm { e}^{-\textrm { i}k_- x}\right ). \end{equation}

Substituting this expression in equation (7) and collecting terms proportional to $\textrm { e}^{-\textrm { i}k_+ x}$ and $\textrm { e}^{-\textrm { i}k_- x}$ we obtain

(9) \begin{equation} H_\pm = \mp \frac {a_\pm }{2\omega }{\sqrt {\rho _1\rho _2(V_{KH}^2 - U^2)}}. \end{equation}

The function $f(x)$ must satisfy the conditions $f(0) = f(L) = 0$ . Obviously $f(0) = 0$ . Using equations (3), (4), (6), (9) and the relation $k_- - k_+ = 2kn$ we obtain from the condition $f(L) = 0$ is $H_+ + H_- = 0$ . Using equations (6) and (9) we obtain from this relation

(10) \begin{equation} \sqrt {\rho _1\rho _2(V_{KH}^2 - U^2)}\frac {dA}{dT} = \frac {\bar \nu \rho _1\rho _2 (U^2 - U_c^2)(k_+^2 + k_-^2)}{\sqrt {\rho _1\rho _2(V_{KH}^2 - U^2)}}A. \end{equation}

It follows from this equation that $A = A_0\textrm { e}^{\gamma T}$ . Using equations (3) and (4) we obtain that the expression for $\gamma$ is given by

(11) \begin{equation} \gamma = 2\bar \nu \frac {U^2 - U_c^2}{V_{KH}^2 - U^2}\left (k^2 n^2 \frac {\rho _2 U^2 + \rho _1(V_{KH}^2 - U^2)}{\rho _1(V_{KH}^2 - U^2)} + k_y^2\right ). \end{equation}