On short-wave instability of the stratified Kolmogorov flow

Abstract

The problem of the linear stability of the stratified Kolmogorov flow driven by a sinusoidal in space force in a viscous and diffusive Boussinesq fluid is re-visited using the Floquet theory, Galerkin approximations and the method of (generalized) continued fractions. Numerical and analytical arguments are provided in favor of a conjecture that an ideal stratified Kolmogorov flow is prone to short-wave instability for Richardson numbers markedly greater than the critical Richardson number Ri $=$ ¼ that appears in the Miles–Howard theorem. The short-wave instability of the stratified Kolmogorov flow is conjectured to be due to a resonance amplification of the Doppler-shifted internal gravity wave modes, in the presence of critical levels of the main flow that are ignored in the proof of the Miles–Howard theorem, but it is emphasized that the complete resolution of the above paradox is a task for future research.

Graphical abstract

Appendices

Appendix A: comments on the Miles–Howard theorem application

When the solution of (5) is sought in the form $\left( {\psi ,\xi } \right) =\left( {\hat{{\psi }}\left( z \right) ,\hat{{\xi }}\left( z \right) } \right) \exp \left\{ {\text{ i }k\left( {x-ct} \right) } \right\}$, then Eqs. (5a, 5b) read

$$\begin{aligned} \left( {\varepsilon \cos z+c} \right) \left( {\frac{\text{ d}^{2}\hat{{\psi }}}{\text{ d }z^{2}}-k^{2}\hat{{\psi }}} \right) +\varepsilon \cos z{}{}\hat{{\psi }}=-\hat{{\xi }},\quad \left( {\varepsilon \cos z+c} \right) \hat{{\xi }}{}=\hat{{\psi }}, \end{aligned}$$

(A.1)

By eliminating $\hat{{\xi }}$ between (A.1) it follows that

$$\begin{aligned} \left( {\varepsilon \cos z+c} \right) \left( {\frac{\text{ d}^{2}\hat{{\psi }}}{\text{ d }z^{2}}-k^{2}\hat{{\psi }}} \right) +\left( {\varepsilon \cos z+\frac{1}{\varepsilon \cos z+c}} \right) \hat{{\psi }}=0. \end{aligned}$$

(A.2)

Equation (A.2) is the Taylor–Goldstein equation. We denote $W=\varepsilon \cos z+c$ and by dividing both sides of (A.2) by W rewrite (A.2) as

$$\begin{aligned} \frac{\text{ d}^{2}\hat{{\psi }}}{\text{ d }z^{2}}-k^{2}\hat{{\psi }}-\frac{{W}''}{W}\hat{{\psi }}+\frac{1}{W^{2}}\hat{{\psi }}=0. \end{aligned}$$

(A.3)

Primes denote differentiation with respect to z. We note that passing from (A.1) to (A.3), by dividing by W, we disregard the continuous spectrum of eigenvalues, which correspond to the critical levels $\varepsilon \cos z+c=0$ in the problem (see more below). Therefore, the subsequent stability analysis due to [10] may be incomplete in this regard.

Following [10], assume that c has a nonzero imaginary part $c_{i}$ and introduce a new dependent variable $G={\hat{{\psi }}} / {\sqrt{W} }$. After some algebra, we obtain

$$\begin{aligned} \left( {W{G}'} \right) ^{\prime }-\left[ {\textstyle {1 \over 2}{W}''+k^{2}W+W^{-1}\left( {\textstyle {1 \over 4}{W}'^{2}-1} \right) } \right] G=0. \end{aligned}$$

(A.4)

Then, multiplying both sides of (A.4) by $\bar{{G}}$, where the overbar denotes the complex conjugate, integrating over z, integrating $\left( {W{G}'} \right) ^{\prime }$ by parts and invoking the periodic boundary conditions, we obtain

$$\begin{aligned} \int {\left\{ {W\left| {{G}'} \right| ^{2}+k^{2}W\left| G \right| ^{2}+\textstyle {1 \over 2}{W}''\left| G \right| ^{2}+W^{-1}\left( {\textstyle {1 \over 4}{W}'^{2}-1} \right) \left| G \right| ^{2}} \right\} } {}\text{ d }z=0. \end{aligned}$$

(A.5)

Taking the imaginary part of (A.5), we have

$$\begin{aligned} c_{i} \int {\left\{ {\left| {{G}'} \right| ^{2}+k^{2}\left| G \right| ^{2}+\left| W \right| ^{-2}\left( {1-\textstyle {1 \over 4}{W}'^{2}} \right) \left| G \right| ^{2}} \right\} } {}\text{ d }z=0. \end{aligned}$$

Because $c_{i} \ne 0$, this equality is possible only when $\textstyle {1 \over 4}{W}'^{2}>1$. Recalling that ${W}'^{2}=\varepsilon ^{2}\sin ^{2}z$, it shows that this necessary condition for instability is fulfilled when $\varepsilon >2$, that is, $\text{ Ri }<\textstyle {1 \over 4}$ [10].

Alternatively, by eliminating $\hat{{\psi }}$ between (A.1), we obtain, see also [9]

$$\begin{aligned} \frac{\text{ d }}{\text{ d }z}\left( {W^{2}\frac{\text{ d }\hat{{\xi }}}{\text{ d }z}} \right) -k^{2}W^{2}\hat{{\xi }}=-\hat{{\xi }}. \end{aligned}$$

(A.6)

Then, multiplying (A.6) by $\bar{{\hat{{\xi }}}}$ and integrating over z with periodic boundary conditions, we have

$$\begin{aligned} \int {W^{2}\left( {\left| {{\hat{{\xi }}}'} \right| ^{2}+k^{2}\left| {\hat{{\xi }}} \right| ^{2}} \right) {}{}} {}\text{ d }z=\int {\left| {\hat{{\xi }}} \right| ^{2}} \text{ d }z. \end{aligned}$$

(A.7)

Taking the imaginary part of (A.7), we obtain (cf. [9])

$$\begin{aligned} c_{i} \int {\left( {\varepsilon \cos z+c_{r} } \right) \left( {\left| {{\hat{{\xi }}}'} \right| ^{2}+k^{2}\left| {\hat{{\xi }}} \right| ^{2}} \right) {}{}} {}\text{ d }z=0. \end{aligned}$$

(A.8)

If $c_{i} \ne 0$ then it follows from (A.8) that $c_{r} \in \left[ {-\varepsilon ,\,\varepsilon } \right]$. Due to symmetry with respect to mirror reflections about the z-axis, we have

$$\begin{aligned} \int {\left( {\varepsilon \cos z+c_{r} } \right) \left( {\left| {{\hat{{\xi }}}'} \right| ^{2}+k^{2}\left| {\hat{{\xi }}} \right| ^{2}} \right) {}{}} {}\text{ d }z=\int {\left( {\varepsilon \cos z-c_{r} } \right) \left( {\left| {{\hat{{\xi }}}'} \right| ^{2}+k^{2}\left| {\hat{{\xi }}} \right| ^{2}} \right) {}{}} {}\text{ d }z, \end{aligned}$$

(A.9)

and (A.9) shows that $c_{r} =0$, that is, the principle of exchange of stabilities is in operation. Now, the equation $\varepsilon \cos z+c_{r} =0$, which generally determines the continuum spectrum of eigenvalues in the problem, shows that $\cos z_{*} =0$ and $z_{*} ={{\pi }} /2+n{\pi }$, $n=0,\,\pm 1,\,\pm 2,\ldots$.Therefore, the spectrum quantization occurs and it reduces to the eigenvalue $c_{r} =0$, corresponding to a countable infinite set of critical levels responsible for instability via the resonance amplification mechanism (see the main text). Next, taking the real part of (A.7) we have

$$\begin{aligned} \int {\left( {\varepsilon ^{2}\cos ^{2}z-c_{i}^{2} } \right) \left( {\left| {{\hat{{\xi }}}'} \right| ^{2}+k^{2}\left| {\hat{{\xi }}} \right| ^{2}} \right) {}{}} {}\text{ d }z=\int {\left| {\hat{{\xi }}} \right| ^{2}} \text{ d }z>0, \end{aligned}$$

(A.10)

that is, $c_{i}$ cannot exceed $\varepsilon$.This is the Howard’s [10] semicircle theorem as applied to the stratified Kolmogorov flow. This theorem does not exclude the possibility of instability at $\varepsilon <2$.

Finally, it is worth mentioning that solution (11), (12), which supports the deductions of the Miles–Howard theorem, does not exhaust all the solutions of (6) that are relevant to the neutral stability conditions. When $c=0$, then the substitution $\eta =\tan z$ transforms (A.6) to

$$\begin{aligned} \left( {\eta ^{2}+1} \right) ^{2}\frac{\text{ d}^{2}\hat{{\xi }}}{\text{ d }\eta ^{2}}-k^{2}\hat{{\xi }}+\frac{1}{\varepsilon ^{2}}\left( {\eta ^{2}+1} \right) =0. \end{aligned}$$

(A.11)

For $k^{2}=3/ 4$ and $\varepsilon <2$ this equation has the exact solution (see, e.g., [28], eq. 2.385)

$$\begin{aligned} \hat{{\xi }}=\left( {\eta ^{2}+1} \right) ^{1 / 4}\left( {C_{1} \cos I+C_{2} \sin I} \right) ,\quad I=\textstyle {1 \over 2}\sqrt{4 / {\varepsilon ^{2}}-1} \,\,\log \left( {\eta +\sqrt{\eta ^{2}+1} } \right) , \end{aligned}$$

(A.12)

Here, $C_{1}$ and $C_{2}$ are arbitrary constants. For $\varepsilon >2$ it is necessary to replace $4 / {\varepsilon ^{2}}-1$, $\cos$, $\sin$ with, respectively, $1-4 /{\varepsilon ^{2}}$, $\cosh$, $\sinh$. Interesting is the case of $\varepsilon =2$, that is, $\text{ Ri }=\textstyle {1 \over 4}$. Now the solution of (A.11) is $\hat{{\xi }}=C_{1} \left( {\eta ^{2}+1} \right) ^{1 / 4}$, which exactly matches (11) and (12). However, in the case of $\varepsilon <2$, that is, $\text{ Ri }>\textstyle {1 \over 4}$ relevant to this study, we have the continuum of solutions belonging to candidate neutral stability curves, and the arguments of Sect. 6 virtually distinguish that solution which corresponds to $\varepsilon =2\sqrt{2 /3}$, that is, $\text{ Ri }=\textstyle {3 \over 8}>\textstyle {1 \over 4}$. There is evidence based on Maplesoft application that for $k^{2}\ne 3 / 4$, including $k^{2}>1$, the solutions of (A.11) for $\varepsilon <2$ are given by a complex analytical expression containing associated Legendre functions of the first and second kind.

Appendix B: comparison with Balmforth and Young [7]

For the comparison with [7] it is convenient to change to the Richardson number $\text{ Ri }={Y^{2}k^{2}} / 4$ and write $\gamma _{n} =2-{4\text{ Ri }} / {\left( {n^{2}+k^{2}-1} \right) }$. Now, solvability conditions (14) and (15), the latter taken with the plus sign on its right-hand side, have a ”trivial” solution $\text{ Ri }=\text{0 }$. Indeed, in this case the resulting infinite continued fraction

$$\begin{aligned} S_{\infty } =2-\frac{1}{2-\frac{1}{2-\cdots }} \end{aligned}$$

is summed up, $S_{\infty } =1$, see the main text. The only “non-trivial” solution to be compared with curve (7) originating from [7] is supplied by Eq. (16) with the minus sign on its right-hand side. At very small $k^{2}$-values we have approximately from (16) by setting $\text{ Ri }=\text{0 }$ in the second term on the left-hand side of (16)

$$\begin{aligned} 2-\frac{4\text{ Ri }}{k^{2}}-1\cong -1. \end{aligned}$$

This equation has a solution $\text{ Ri }\cong {k^{2}}/ 2$ consistently with (11). To find a correction to this asymptotic solution we proceed as follows. We start from (18), make a replacement $Y^{2}k^{2}=4\text{ Ri }$, and find that expressions (19) assume the form

$$\begin{aligned} f\left( M \right) =1-4\text{ Ri }\sum \limits _{n=2}^M {\frac{\left( {n-1} \right) }{\left( {2n+1} \right) ^{2}+k^{2}-1}} ,\quad g\left( M \right) =1-4\text{ Ri }\sum \limits _{n=1}^M {\frac{\,n}{\left( {2n+1} \right) ^{2}+k^{2}-1}} . \end{aligned}$$

It is readily seen that

$$\begin{aligned} f\left( M \right) \equiv 1-4\text{ Ri }\sum \limits _{n=1}^M {\frac{\left( {n-1} \right) }{\left( {2n+1} \right) ^{2}+k^{2}-1}} =g\left( M \right) +4\text{ Ri }\sum \limits _{n=1}^M {\frac{1}{\left( {2n+1} \right) ^{2}+k^{2}-1}} \end{aligned}$$

and therefore

$$\begin{aligned} \frac{f\left( M \right) }{g\left( M \right) }=1+4\text{ Ri }\sum \limits _{n=1}^M {\frac{1}{\left( {2n+1} \right) ^{2}+k^{2}-1}} \cdot \left( {1-4\text{ Ri }\sum \limits _{n=1}^M {\frac{\,n}{\left( {2n+1} \right) ^{2}+k^{2}-1}} } \right) ^{-1}. \end{aligned}$$

(B.1)

By assuming $\text{ Ri }<<1$, $k^{2}<<1$, thus neglecting the contribution from $k^{2}$ everywhere in (B.1) and taking the limit $M\rightarrow \infty$, such that the resulting infinite set before the parentheses is readily summed up, we obtain

$$\begin{aligned} \lim \limits _{M\rightarrow \infty } \frac{f\left( M \right) }{g\left( M \right) }\cong 1+4\text{ Ri }\cdot \frac{1}{4}\left( {1-4\text{ Ri }\cdot \lim \limits _{M\rightarrow \infty } \sum \limits _{n=1}^M {\frac{\,n}{\left( {2n+1} \right) ^{2}-1}} } \right) ^{-1}. \end{aligned}$$

(B.2)

We reached a somewhat paradoxical situation. If the Richardson number $\text{ Ri }$ in (B.2) assumes a very small but finite value, then the second logarithmically divergent term in parentheses dominates (cf. Sect. 6) and we obtain from (B.2)

$$\begin{aligned} \lim \limits _{M\rightarrow \infty } \frac{f\left( M \right) }{g\left( M \right) }\cong 1. \end{aligned}$$

(B.3)

On the other hand, if we rigorously set $\text{ Ri }=\text{0 }$ in parentheses in (B.2), then

$$\begin{aligned} \lim \limits _{M\rightarrow \infty } \frac{f\left( M \right) }{g\left( M \right) }\cong 1+4\text{ Ri }\cdot \frac{1}{4}. \end{aligned}$$

(B.4)

From (B.4) we would have, cf. (18) in Sect. 6,

$$\begin{aligned} 2-\frac{4\text{ Ri }}{k^{2}}-4\text{ Ri }\cdot \frac{1}{4}\cong 0, \end{aligned}$$

which in the first approximation yields $\text{ Ri }\cong {k^{2}} / 2-{k^{4}} / 8$. This coincides exactly with the two first terms of the binominal expansion of the equation $\text{ Ri }=\sqrt{1-k^{2}} -\left( {1-k^{2}} \right) \cong {k^{2}} / 2-{k^{4}} / 8+\cdots$ for the neutral stability curve in [7], cf. Sect. 3. This coincidence may not be accidental, but correlate with a singular character of solution (8a, 8b) in [7]. The two neutral stability curves, (i) $\text{ Ri }=\sqrt{1-k^{2}} -\left( {1-k^{2}} \right)$ of [7] and (ii) $\text{ Ri }={k^{2}} / 2$ obtained from (A.1) and (A.3), diverge significantly afterward. In particular, this can be seen from the fact that while for the first curve $\text{ Ri }=0$ at $k^{2}=1$ it is not the case for the second curve, as for example follows from (A.1), the arguments of Sect. 6 show explicitly that $\text{ Ri }\approx \textstyle {1 \over 2}$ for $k^{2}=1$.

At $k^{2}\rightarrow 0$ irregular solution (12) $\psi _{1} =C_{1} \left( {\cos z} \right) ^{\sqrt{1-k^{2}} }$ reduces to a regular, spatially periodic solution $\psi _{1} =C_{1} \cos z$ that fits the Galerkin first approximation in [5, 12]. It can be hypothesized, although cannot be proved at this point, that $k^{2}=0$ represents a bifurcation point in the parameter space at which two independent solutions branch off from the solution $\psi _{1} =C_{1} \cos z$. The first solution corresponds to $\psi _{1} =C_{1} \left( {\cos z} \right) ^{\sqrt{1-k^{2}} }$, see [7], and the second one meets our solution obtained by the method of continued fractions, while the Miles−Howard theorem is relevant only to the first of them.

Appendix C: on the Lyapunov stability of the stratified Kolmogorov flow

We will show that (14) corresponds to what follows from Arnol’d’s energy-Casimir method [18, 19]. We start from the nonlinear equations of ideal fluid motion derived from Eq. (1), taken in a dimensionless form using a transition to dimensionless variables as described in the main text

$$\begin{aligned} \frac{\partial }{\partial t}\nabla ^{2}\psi +J\left( {\psi ,\nabla ^{2}\psi } \right) =\frac{\partial \xi }{\partial x},\quad \frac{\partial }{\partial t}\xi +J\left( {\psi ,\xi } \right) =-\frac{\partial \psi }{\partial x}. \end{aligned}$$

(C.1)

The asterisks on the dimensionless variables are omitted. The energy conservation law

$$\begin{aligned} \frac{\text{ d }}{\text{ d }t}E=0,\quad E=\frac{1}{2}\int \!\!\!\int {\left\{ {\left( {\nabla \psi } \right) ^{2}+\xi ^{2}} \right\} } {}{}\text{ d }x\text{ d }z, \end{aligned}$$

and the invariance of the Casimir functional (cf. [19])

$$\begin{aligned} \frac{\text{ d }}{\text{ d }t}F=0,\quad F=\int \!\!\!\int {\left\{ {\Phi \left( {z+\xi } \right) +\Gamma \left( {x+\xi } \right) \nabla ^{2}\psi } \right\} } {}{}\text{ d }x\text{ d }z, \end{aligned}$$

with $\Phi$ and $\Gamma$ as arbitrary differentiable functions, follow from (C.1) given the periodic boundary conditions. We study the nonlinear Lyapunov stability of the solution $\psi _{0} =\varepsilon \sin z$ (a non-essential additive constant is omitted), $\xi _{0} =0$, with respect to small but finite variations $\psi =\psi _{0} +\delta \psi$, $\xi =\delta \xi$. We introduce the functional $H=E+F$ and expand $H\left[ {\psi ,\xi } \right] -H\left[ {\psi _{0} ,\xi _{0} } \right]$ into a series of variations of increasing order $H\left[ {\psi ,\xi } \right] -H\left[ {\psi _{0} ,\xi _{0} } \right] =\delta H+\textstyle {1 \over 2}\delta ^{2}H+\cdots$. We have

$$\begin{aligned} \begin{array}{l} \delta H=\int \!\!\!\int {\left\{ {\nabla \psi _{0} \cdot \nabla \delta \psi +{\Phi }'\delta \xi +{\Gamma }'\delta \xi {}{}\nabla ^{2}\psi _{0} +\Gamma \nabla ^{2}\delta \psi } \right\} } {}{}\text{ d }x\text{ d }z \\ \quad \;\;\,=\int \!\!\!\int {\left\{ {\left( {-\psi _{0} +\Gamma } \right) \nabla ^{2}\delta \psi +\left( {{\Phi }'+{\Gamma }'\nabla ^{2}\psi _{0} } \right) \delta \xi {}{}} \right\} } {}{}\text{ d }x\text{ d }z \\ \end{array}, \end{aligned}$$

Here, the prime denotes differentiation. The conditions for vanishing of the first variation read as $\Gamma \left( z \right) =\varepsilon \sin z$, ${\Phi }'\left( z \right) =\varepsilon ^{2}\sin z\cos z\equiv \textstyle {1 \over 2}\varepsilon ^{2}\sin 2z$. Next, we calculate the second variation

$$\begin{aligned} \delta ^{2}H=\int \!\!\!\int {\left\{ {\left( {\nabla \delta \psi } \right) ^{2}+\left( {1+{\Phi }''+{\Gamma }''\nabla ^{2}\psi _{0} } \right) \left( {\delta \xi } \right) ^{2}{}+2{\Gamma }'\nabla ^{2}\delta \psi \delta \xi {}{}} \right\} } {}{}\text{ d }x\text{ d }z. \end{aligned}$$

Because, ${\Phi }''\left( z \right) =\varepsilon ^{2}\cos 2z\equiv \varepsilon ^{2}\left( {\cos ^{2}z-\sin ^{2}z} \right)$, ${\Gamma }'=\varepsilon \cos z$ and ${\Gamma }''\nabla ^{2}\psi _{0} =\varepsilon ^{2}\sin ^{2}z$, we obtain

$$\begin{aligned} \delta ^{2}H=\int \!\!\!\int {\left\{ {\left( {\nabla \delta \psi } \right) ^{2}+\left( {1+\varepsilon ^{2}\cos ^{2}z} \right) \left( {\delta \xi } \right) ^{2}{}+2\varepsilon \cos z{}{}\nabla ^{2}\delta \psi \delta \xi } \right\} } {}{}\text{ d }x\text{ d }z, \end{aligned}$$

(C.2)

which up to “1/2” matches (14). If the integrand in (C.2) were positive definite for some $\varepsilon$ value, then the stratified Kolmogorov flow would be stable according to Lyapunov for this $\varepsilon$ value. However, this is impossible in principle due to the last term in (C.2) which describes the destabilizing effect of short-wave perturbations (cf. [19]). The indefiniteness of such a second variation implies either the existence of negative-energy modes (which by definition are neutrally stable, but can be unstable (a) in the presence of dissipation or (b) with the inclusion of nonlinearity) or linear instability, and there is no way in general of distinguishing from the expressions for $\delta ^{2}H$ alone which is the case [29]. It is conjectured in [26] that (a) and (b) are generic; i.e., although there exist cases where the second variation is indefinite and the system is stable, and there exist special types of dissipation that do not result in instability, these are conjectured to be exceptional. The authors of [26] conclude that if we accept the conjecture and ignore these possibilities because of their rarity, we obtain a sense in which the second variation provides a "necessary" and sufficient condition for stability. It is worthwhile mentioning in this context how it is stated in [19] that “Consistent with indefiniteness of the second variation we conjecture that all stratified, incompressible, ideal fluid flows for sufficiently high wavenumbers of the density variation are nonlinearly unstable” (italics is as in [19]). We also have a feeling that the second variation $\delta ^{2}H$ provides a "necessary" and sufficient condition for stability of the stratified Kolmogorov flow, but because $\delta ^{2}H$ exactly matches invariant (14) of linearized equations, we attribute indefiniteness of $\delta ^{2}H$ to the linear instability through resonance amplification of internal gravity wave modes in the presence of critical levels of the main flow, although in view of the short-wave nature of the emerging instability, the role of dissipative factors may be crucial.

Furthermore, following [19] to assume the upper cutoff for the spectrum of short-wave disturbances, we have

$$\begin{aligned} \int \!\!\!\int {\left( {\nabla ^{2}\delta \psi } \right) ^{2}} {}{}\text{ d }x\text{ d }z<K^{2}\int \!\!\!\int {\left( {\nabla \delta \psi } \right) ^{2}} {}{}\text{ d }x\text{ d }z, \end{aligned}$$

where $K^{2}$ is a (large) positive constant having the meaning of dimensionless wavenumber squared. Now, second variation (C.2) admits a lower estimate

$$\begin{aligned} \delta ^{2}H>\delta ^{2}\tilde{{H}}=\int \!\!\!\int {\left\{ {K^{-2}\left( {\nabla ^{2}\delta \psi } \right) ^{2}+\left( {1+\varepsilon ^{2}\cos ^{2}z} \right) \left( {\delta \xi } \right) ^{2}{}+2\varepsilon \cos z{}{}\nabla ^{2}\delta \psi \delta \xi } \right\} } {}{}\text{ d }x\text{ d }z \end{aligned}$$

(C.3)

and the binary form in (C.3) is positive definite, that is, the stratified Kolmogorov flow is stable according to Lyapunov, if $\varepsilon ^{2}\cos ^{2}z<\left( {K^{2}-1} \right) ^{-1}$ everywhere in the flow, that is, if $\varepsilon ^{2}<\left( {K^{2}-1} \right) ^{-1}$. At $K^{2}\rightarrow \infty$ the zone of guaranteed Lyapunov stability shrinks to zero.

Appendix D: Band-like structure of instability

Table 1 The band-like structure of the possible instability zones as revealed by application of the Galerkin approximations corresponding to $k=3$ and $M=1,2,\ldots ,7$ in the non-viscous and non-diffusive case $R=S=\infty$

Table 2 The band-like structure of the possible instability zones as revealed by application of the Galerkin approximations corresponding to $k=3$ and $M=1,\ldots ,7$ for a slightly viscous and diffusive case when $R=S=4000$

Table 3 The roots of the equation $S_{M} =-1$, see (17), calculated for $k=3$ and $M=1,\ldots ,41$. Underlined are the numbers, which coincide exactly with those calculated for the Galerkin first, third, fifth and seventh approximations (see Table 1)