2. Exact solution

We start from the linearized hydrostatic-Boussinesq equations for the propagation of a three-dimensional disturbance in a uniformly stratified sheared flow $\overline{\boldsymbol{u}}_{0}=({\it\Lambda}z,0,0)$ , where the shear ${\it\Lambda}>0$ and the Brunt–Väisälä frequency $N$ are constant. When the Coriolis parameter $f$ is also constant, a stationary monochromatic disturbance with vertical velocity of the form

(2.1) $$\begin{eqnarray}w(x,y,z,t)=\text{Re}\,\{W(z)\text{e}^{\text{i}(kx+ly-{\it\omega}t)}\},\end{eqnarray}$$

with ${\it\omega}$ the absolute frequency, satisfies the vertical structure equation

(2.2) $$\begin{eqnarray}\frac{1-{\it\xi}^{2}}{{\it\xi}^{2}}W_{{\it\xi}{\it\xi}}-\left(\frac{2}{{\it\xi}^{3}}-\frac{2\text{i}{\it\nu}}{{\it\xi}^{2}}\right)W_{{\it\xi}}-\left[\frac{(1+{\it\nu}^{2})J}{{\it\xi}^{2}}+\frac{2\text{i}{\it\nu}}{{\it\xi}^{3}}\right]W=0,\end{eqnarray}$$

where

(2.3a−c ) $$\begin{eqnarray}J=\frac{N^{2}}{{\it\Lambda}^{2}},\quad {\it\nu}=\frac{l}{k}\quad \text{and}\quad {\it\xi}=\frac{k{\it\Lambda}z}{f}-\frac{{\it\omega}}{f}.\end{eqnarray}$$

This is equation (6) in Yamanaka & Tanaka (Reference Yamanaka and Tanaka1984), but see also Inverarity & Shutts (Reference Inverarity and Shutts2000). Note that in the absence of horizontal boundaries, as assumed here, and for $k\neq 0$ , there are no unstable modes so we can take $\text{Im}\,{\it\omega}=0$ ; for $k=0$ , inertial instability occurs for $J<1$ , leading to growing modes with $\text{Im}\,{\it\omega}\neq 0$ which we discuss further in § 4.

Equation (2.2) has two singularities at the inertial levels ${\it\xi}=\pm 1$ that require a special treatment; the critical level ${\it\xi}=0$ is an apparent singularity. Following LPV12, we write the exact solutions of (2.2) as

(2.4a ) $$\begin{eqnarray}\displaystyle & W_{u}^{(III)}({\it\xi})\quad \text{for}~{\it\xi}>1, & \displaystyle\end{eqnarray}$$

(2.4b ) $$\begin{eqnarray}\displaystyle & AW_{d}^{(II)}({\it\xi})+BW_{u}^{(II)}({\it\xi})\quad \text{for}~-1<{\it\xi}<1, & \displaystyle\end{eqnarray}$$

(2.4c ) $$\begin{eqnarray}\displaystyle & CW_{d}^{(I)}({\it\xi})+DW_{u}^{(I)}({\it\xi})\quad \text{for}~{\it\xi}<-1. & \displaystyle\end{eqnarray}$$

$A$

$B$

$C$

$D$

$W$

(2.5) $$\begin{eqnarray}W_{u}^{(III)}({\it\xi})\sim {\it\xi}^{1/2+\text{i}{\it\mu}}\quad \text{as}~{\it\xi}\rightarrow \infty ,\end{eqnarray}$$

corresponding to an upward-propagating gravity wave (Booker & Bretherton Reference Booker and Bretherton1967). In (2.4c ) the two functions are such that the asymptotic form is given by

(2.6) $$\begin{eqnarray}W^{(I)}\sim C|{\it\xi}|^{1/2-\text{i}{\it\mu}}+D|{\it\xi}|^{1/2+\text{i}{\it\mu}}\quad \text{as}~{\it\xi}\rightarrow -\infty .\end{eqnarray}$$

To evaluate $A$ , $B$ , $C$ , and $D$ , we first use asymptotic expansions of (2.4a )–(2.4b ) and (2.4b )–(2.4c ) in the vicinity of ${\it\xi}=1$ and ${\it\xi}=-1$ , respectively (see appendix A). Next we follow Booker & Bretherton (Reference Booker and Bretherton1967) and introduce an infinitesimally small linear damping to determine the physically relevant branches of multivalued functions (see figure 1). This is equivalent to shifting the path of integration of (2.2) into the lower half of the complex plane so that

(2.7) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}ll@{}}{\it\xi}-1=(1-{\it\xi})\text{e}^{-\text{i}{\rm\pi}} & \text{for}~{\it\xi}<1,\\ {\it\xi}+1=|{\it\xi}+1|\text{e}^{-\text{i}{\rm\pi}} & \text{for}~{\it\xi}<-1.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Using this, the asymptotic expansions of (2.4a ) and (2.4b ) can be continued for ${\it\xi}\rightarrow 1^{-}$ and ${\it\xi}\rightarrow -1^{-}$ and compared with the corresponding expansions of (2.4b ) and (2.4c ) (see appendix A for an illustration of the procedure in the simpler context of the WKB solution). Matching expansions near ${\it\xi}=1$ gives

(2.8) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}{\it\alpha}^{\prime }=\text{e}^{-{\it\nu}{\rm\pi}}({\it\alpha}A+{\it\alpha}^{\prime \prime }B),\\ {\it\beta}^{\prime }={\it\beta}A+{\it\beta}^{\prime \prime }B,\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where the coefficients ${\it\alpha},\,{\it\beta},{\it\alpha}^{\prime }$ , etc. are known functions of ${\it\mu}$ and ${\it\nu}$ given in appendix A. Similarly, matching near ${\it\xi}=-1$ gives

(2.9) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}rcl@{}}{\it\beta}A-{\it\beta}^{\prime \prime }B\hspace{-7.0pt} & = & \hspace{-7.0pt}\text{e}^{{\it\nu}{\rm\pi}}({\it\beta}^{\prime \prime \prime }C+{\it\beta}^{\prime }D),\\ {\it\alpha}^{\prime \prime \prime }C+{\it\alpha}^{\prime }D\hspace{-7.0pt} & = & \hspace{-7.0pt}{\it\alpha}A-{\it\alpha}^{\prime \prime }B.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The constants $A,B,C$ and $D$ are then deduced by solving the linear system (2.8) and (2.9). Note that the valve effect, involving an exponential dependence on ${\it\nu}$ , can be traced to the factors $\text{e}^{\pm {\it\nu}{\rm\pi}}$ in (2.8) and (2.9), with the amplification across the lower inertial level associated with the factor in (2.9) and the balancing attenuation across the upper critical level associated with that in (2.8).

After involved manipulations it can be shown that the reflection and transmission coefficients are given by

(2.10a,b ) $$\begin{eqnarray}R=\frac{C}{D}=0\quad \text{and}\quad T=\frac{1}{D}=\text{i}\text{e}^{-{\it\mu}{\rm\pi}},\end{eqnarray}$$

where ${\it\mu}^{2}=J(1+{\it\nu}^{2})-1/4$ (see § A.1). This is the result of Jones (Reference Jones1967), which can be obtained much more directly by integration along a semi-circle with $|{\it\xi}|\gg 1$ , as recalled in the introduction (see figure 1). Our solution provides the details of the solution between the inertial levels that this result ignores. In particular, it makes it possible to compute the EP flux, proportional to

(2.11) $$\begin{eqnarray}F^{z}=\text{Re}\,\left(\text{i}\frac{1-{\it\xi}^{2}}{{\it\xi}^{2}}W_{{\it\xi}}W^{\ast }-{\it\nu}\frac{WW^{\ast }}{{\it\xi}^{2}}\right),\end{eqnarray}$$

and constant apart from jumps at the inertial levels (this conservation law is derived directly from (2.2) by multiplying by $W^{\ast }$ , the complex conjugate of $W$ , and integrating by parts). We can obtain the three distinct values of the EP flux on either side of and between the inertial levels using the first terms in the expansion of the hypergeometric functions for ${\it\xi}\rightarrow \pm \infty$ and ${\it\xi}\rightarrow 0$ . Computations detailed in § A.2 show that the ratios of the EP flux between and above the inertial levels to the EP flux below are

(2.12a,b ) $$\begin{eqnarray}\frac{F^{z(II)}}{|F^{z(I)}|}=\frac{\text{e}^{2{\rm\pi}{\it\nu}}+1}{\text{e}^{2{\rm\pi}{\it\mu}}-1}\quad \text{and}\quad \frac{F^{z(III)}}{|F^{z(I)}|}=\text{e}^{-2{\rm\pi}{\it\mu}},\end{eqnarray}$$

where the absolute value in the denominators is required because the incoming EP flux $F^{z(I)}<0$ whereas the other two fluxes are positive (see appendix A). The first of these ratios describes the crossing of the lower inertial level by the incident waves and displays the strong dependence on the direction ${\it\nu}$ of the wavevector that characterizes the valve effect. It is shown as a function of ${\it\nu}$ and $J$ in figure 2. The figure indicates that, for $J<1$ , the ratio can exceed 1 and can take very large values for ${\it\nu}>0$ . We discuss this phenomenon further in § 4.

Figure 2.

Ratio of the EP flux between the inertial levels to the incident EP flux, $\text{log}_{10}(F^{z(II)}/F^{z(I)})$ , as a function of the Richardson number $J$ and wave aspect ratio ${\it\nu}=l/k$ .

3. WKB approximation and tunnelling

A WKB solution of (2.2), strictly valid for ${\it\mu}\gg 1$ but qualitatively correct for ${\it\mu}\gtrsim 1$ , sheds light on the behaviour of the solution and on the valve effect in particular. To derive this solution, it is convenient to follow Miyahara (Reference Miyahara1981) and recast (2.2) in the canonical form

(3.1a ) $$\begin{eqnarray}{\it\Psi}_{{\it\xi}{\it\xi}}+\left[\frac{{\it\mu}^{2}+1/4}{{\it\xi}^{2}-1}+\frac{(3+{\it\nu}^{2}){\it\xi}^{2}-2}{{\it\xi}^{2}({\it\xi}^{2}-1)^{2}}\right]{\it\Psi}=0,\end{eqnarray}$$

where

(3.1b ) $$\begin{eqnarray}W({\it\xi})={\it\xi}({\it\xi}-1)^{-1/2+\text{i}{\it\nu}/2}({\it\xi}+1)^{-1/2-\text{i}{\it\nu}/2}{\it\Psi}({\it\xi}).\end{eqnarray}$$

Assuming ${\it\mu}\gg 1$ , we then introduce the expansion

(3.2) $$\begin{eqnarray}{\it\Psi}({\it\xi})=({\it\Psi}_{0}({\it\xi})+{\it\mu}^{-1}{\it\Psi}_{1}({\it\xi})+\cdots \,)\exp \left({\it\mu}\int ^{{\it\xi}}{\it\phi}({\it\xi}^{\prime })\,\text{d}{\it\xi}^{\prime }\right)\end{eqnarray}$$

and find at orders ${\it\mu}^{2}$ , ${\it\mu}$ and 1 that

(3.3a−c ) $$\begin{eqnarray}{\it\phi}=\frac{{\it\epsilon}\text{i}}{\sqrt{{\it\xi}^{2}-1}},\quad {\it\Psi}_{0}=({\it\xi}^{2}-1)^{1/4}\quad \text{and}\quad {\it\Psi}_{1}=-{\it\epsilon}\text{i}\frac{1+({\it\nu}^{2}/2-7/8){\it\xi}^{2}}{{\it\xi}({\it\xi}^{2}-1)^{1/4}},\end{eqnarray}$$

where ${\it\epsilon}=\pm 1$ selects the two possible solutions. The expansion (3.2) breaks down in the regions $||{\it\xi}|-1|=O({\it\mu}^{-2})$ surrounding the inertial levels where an approximation in terms of Hankel functions can be constructed (see LPV12 for the rescaling leading to this conclusion). Here we avoid this complication by integrating (3.1a ) along a path that remains at a distance larger than ${\it\mu}^{-1}$ from the inertial levels and avoids the branch cut (see figure 1). This path does not cross the only Stokes line (where $\text{Im}\,{\it\phi}=0$ , e.g. Ablowitz & Fokas Reference Ablowitz and Fokas1997), namely ${\it\xi}\in [-1,1]$ . Therefore, a single WKB solution, determined by the radiation condition to correspond to ${\it\epsilon}=1$ , can be continued from ${\it\xi}\rightarrow \infty$ to ${\it\xi}\rightarrow -\infty$ . To make the absorptive properties more transparent, we express the approximations at order ${\it\mu}$ for a unit-amplitude incident wave (rather than for a unit-amplitude transmitted wave in the exact case, see § A.3 for details). This gives

(3.4a ) $$\begin{eqnarray}\displaystyle & W_{WKB}^{(III)}=-\text{i}g({\it\xi})~\text{e}^{-{\it\mu}{\rm\pi}}~\exp \left(\text{i}{\it\mu}\ln ({\it\xi}+\sqrt{{\it\xi}^{2}-1})\right)\quad \text{for}\;{\it\xi}>1, & \displaystyle\end{eqnarray}$$

(3.4b ) $$\begin{eqnarray}\displaystyle & \displaystyle W_{WKB}^{(II)}=-\text{e}^{{\it\nu}{\rm\pi}/2}~\frac{1-\text{i}}{\sqrt{2}}g({\it\xi})~\exp (-{\it\mu}({\rm\pi}/2+\sin ^{-1}{\it\xi}))\quad \text{for}\;-1<{\it\xi}<1, & \displaystyle\end{eqnarray}$$

(3.4c ) $$\begin{eqnarray}\displaystyle & W_{WKB}^{(I)}=g({\it\xi})~\exp \left(\text{i}{\it\mu}\ln (|{\it\xi}|+\sqrt{{\it\xi}^{2}-1})\right)\quad \text{for}\;{\it\xi}<-1. & \displaystyle\end{eqnarray}$$

$g({\it\xi})={\it\xi}|{\it\xi}-1|^{-(1/4)+\text{i}({\it\nu}/2)}|{\it\xi}+1|^{-(1/4)-\text{i}({\it\nu}/2)}$

${\it\xi}\pm 1$

$-1/4\pm \text{i}{\it\nu}/2$

$|{\it\xi}|\gg 1$

$\exp (\text{i}{\it\mu}\ln (|{\it\xi}|+\sqrt{{\it\xi}^{2}-1}))\approx 2^{\text{i}{\it\mu}}|{\it\xi}|^{\text{i}{\it\mu}}$

$g({\it\xi})\approx \text{sign}({\it\xi})|{\it\xi}|^{1/2}$

$W_{WKB}^{(I)}=|{\it\xi}|^{1/2+\text{i}{\it\mu}}$

The WKB solution (3.4) is instructive in several respects. First, as ${\it\xi}\rightarrow \pm \infty$ , it reproduces exactly the ratio between the amplitudes of the upward-propagating IGWs (2.5) and (2.6) found from the exact solution. In particular, the reflection and transmission coefficients are the exact ones in (2.10). The WKB solution also shows that the amplitude of the solution is dominated by the exponential term $\exp (-{\it\mu}\sin ^{-1}{\it\xi})$ in (3.4b ) that rapidly decays monotonically between the inertial levels. This decay is strongly reminiscent of the exponential decay that characterizes the neutral solution in the quasi-geostrophic approximation. This can be made more transparent by noting that near the critical level ${\it\xi}=0$ , the $O({\it\mu}^{-1})$ term (retaining ${\it\Psi}_{1}$ in (3.2)) is proportional to the quasi-geostrophic solution $(1+{\it\mu}{\it\xi})\text{e}^{-{\it\mu}{\it\xi}}$ (see equation (2.16) in LPV12). Away from the immediate vicinity of ${\it\xi}=0$ , the decay is in fact faster than predicted by the quasi-geostrophic approximation since $|\text{sin}^{-1}{\it\xi}|\geqslant {\it\xi}$ . The valve effect is also evident in the WKB solution, through the factor $\text{e}^{{\it\nu}{\rm\pi}/2}$ in (3.4b ). The dependence of the amplitude on ${\it\nu}$ associated with this factor is, however, weak compared to the dependence on $J$ and $|{\it\nu}|$ stemming from the factor $\exp (-{\it\mu}({\rm\pi}/2+\sin ^{-1}{\it\xi}))\sim \exp (-\sqrt{J(1+{\it\nu}^{2})}({\rm\pi}/2+\sin ^{-1}{\it\xi}))$ in (3.4b ) since $J\gg 1$ . This factor represents a very rapid amplitude decay of the solution above ${\it\xi}=-1$ , and corresponds to an absorption that increases rapidly with $J$ and $|{\it\nu}|$ : the asymmetry that favours perturbations with ${\it\nu}>0$ over those with ${\it\nu}<0$ between the inertial levels is of little significance compared to this absorption.

The WKB solution represents the exact solution accurately even for moderately large $J$ . For $J=5$ , for instance, the two solutions almost coincide everywhere except in regions close to the inertial levels (see figure 3). The WKB approximation deteriorates around the lower inertial level for $|{\it\xi}+1|\lesssim 0.05$ (see figure 3 b), corresponding to an $O({\it\mu}^{-1})$ distance to the inertial level, as expected. An analogous breakdown of the WKB approximation occurs for ${\it\xi}\approx 1$ , but in this case the solutions are very small.

Figure 3.

Vertical velocity $W$ for the exact solution (2.4a )–(2.4c ) ( $\text{Re}\,W$ , black solid; $\text{Im}\,W$ , black dashed) and the WKB approximation (3.4a )–(3.4c ) ( $\text{Re}\,W_{WKB}$ grey solid and $\text{Im}\,W_{WKB}$ grey dashed) for $J=5$ , ${\it\nu}=0.1$ : (a) global view; (b) zoom on the lowest inertial level.

Figure 3 also illustrates how the solution behaves as an upward-propagating wave – with real and imaginary parts in quadrature – outside the inertial levels (see figure 3 a), and as an evanescent perturbation – with real and imaginary parts in phase – between the inertial levels (see figure 3 b). This behaviour can be interpreted as a form of tunnelling, analogous to, but different from, the classical tunnelling between turning points as described, for instance, by Bender & Orszag (Reference Bender and Orszag1978). The continuity of the arguments of the last exponential factors in (3.4) shows that the amplitude decrease of the transmitted signal by the factor $\text{e}^{-{\it\mu}{\rm\pi}}$ can be immediately related to the decay of the solution $W_{WKB}^{(II)}$ between the inertial level as $\exp (-{\it\mu}({\rm\pi}/2+\sin ^{-1}{\it\xi}))$ since this decreases by the same factor $\text{e}^{-{\it\mu}{\rm\pi}}$ . In the tunnelling interpretation, the valve effect seems to play no role, because the amplitude jump $\text{e}^{{\it\nu}{\rm\pi}/2}$ through the lower inertial level is exactly balanced by the inverse jump $\text{e}^{-{\it\nu}{\rm\pi}/2}$ at the upper inertial level.

An obvious difference with classical tunnelling is the absence of a fully conserved flux. In classical tunnelling, wave-action flux conservation leads to the relation $|T|^{2}+|R^{2}|=1$ that constrains the transmission and refection coefficients. Here, the analogous conservation is that of the EP flux, but because this can jump at the inertial level (Eliassen & Palm Reference Eliassen and Palm1961; Grimshaw Reference Grimshaw1975), there is no associated constraint on $T$ and $R$ , making the absence of reflection $R=0$ compatible with partial (indeed very weak) transmission $|T|<1$ . Regarding the EP flux, we note that the WKB solution between the inertial levels (3.4b ) suggests a zero flux, since it is evanescent (see for instance Gill Reference Gill1982). The flux is non-zero, however, and can be captured by extending the WKB analysis to take into account the Stokes phenomenon that arises at ${\it\xi}=1$ . The relevant computation, carried out in LPV12, reveals the presence of an exponentially small solution increasing exponentially with ${\it\xi}$ to be added to (3.4b ). Taking this into account gives a non-zero flux, deduced from equation (2.31) in LPV12 to satisfy

(3.5) $$\begin{eqnarray}\frac{F^{z(II)}}{|F^{z(I)}|}\sim \text{e}^{-2{\rm\pi}{\it\mu}}(\text{e}^{2{\rm\pi}{\it\nu}}+1),\end{eqnarray}$$

consistent with (2.12) for ${\it\mu}\gg 1$ .

4. Valve effect and inertial instability

To describe further the valve effect, we return to the exact solutions and first present vertical profiles of $W$ when $J=5$ and for ${\it\nu}=\pm 1$ in figure 4. The global views in figure 4(a,c) show that the solutions are almost identical. Even though a closer examination of the solution at the lowest inertial levels (figure 4 b,d) shows that the solution with ${\it\nu}>0$ is amplified whereas that with ${\it\nu}<0$ is attenuated, this difference matters little because of the very rapid decay with ${\it\xi}$ of both solutions between the inertial levels. Also in figure 4, we see that the EP flux is very small between the inertial levels, consistent with expression (2.12) for the ratio $F^{z(II)}/|F^{z(I)}|$ of the EP flux between the inertial level to the incident EP flux. For $J\gg 1$ , this ratio is approximated by (3.5) which shows, as discussed, that the valve effect plays only a minor role in modulating it.

Figure 4.

Vertical velocity $W$ for the exact solution (2.4a )–(2.4c ) ( $|W|$ thick grey solid, $\text{Re}\,W$ grey solid, and $\text{Im}\,W$ grey dashed) and EP flux $F^{z}$ from (2.11) (black solid) for $J=5$ and ${\it\nu}=\pm 1$ . The EP flux is normalized by its incident value: (a,c) global views for ${\it\nu}=1$ and ${\it\nu}=-1$ , respectively; (b,d) corresponding zooms on the lowest inertial level.

For $J<1$ , the situation is different, and the valve effect can lead to an intensification of the perturbation between the inertial levels. This is illustrated in figure 5 which shows profiles of $W$ and $F^{z}$ for $J=0.5$ and ${\it\nu}=\pm 2$ . When ${\it\nu}=2$ in figure 5(a,b), the incident wave is clearly amplified as it passes through the lower inertial level (figure 5 b), and the solution between the inertial levels does not display the almost exponential decay with altitude that characterizes the solution when $J\gg 1$ . The behaviour for ${\it\nu}=-2$ illustrated in figure 5(c,d) is then completely different since the incident wave is almost entirely absorbed at the lowest inertial level. This pronounced difference between positive and negative ${\it\nu}$ is clearly apparent in the EP flux ratios in (2.12a ) as figure 2 confirms. More importantly, (2.12) tells us that the disturbances with ${\it\nu}>\sqrt{(J-1/4)/(1-J)}$ are amplified between the inertial levels when $J<1$ (in this case the ratio is larger than 1). More than this, for $|{\it\nu}|\gg 1$ ,

(4.1) $$\begin{eqnarray}\frac{F^{z(II)}}{|F^{z(I)}|}\sim \exp \left(2{\rm\pi}({\it\nu}-\sqrt{J}|{\it\nu}|)\right),\end{eqnarray}$$

so the amplification is arbitrarily large for ${\it\nu}\rightarrow +\infty$ when $J<1$ .

Figure 5.

Same as figure 4 but for $J=0.5$ , ${\it\nu}=\pm 2$ . (a,c) global views for ${\it\nu}=2$ and ${\it\nu}=-2$ , respectively; (b,d) corresponding zooms between the inertial levels. In (a,b), the EP flux is normalized by its amplitude between the inertial layers. Note the change of scale in (d).

It is interesting to relate this result to inertial instability, which also occurs when $J<1$ (Stone Reference Stone1966), a condition that implies that the background flow potential vorticity is negative (Bennetts & Hoskins Reference Bennetts and Hoskins1979). Inertial instability, or symmetric instability, arises for perturbations with $k=0$ , that is, ${\it\nu}=\infty$ , a case that we have not treated explicitly. It is examined in Stone (Reference Stone1966) and can be recovered easily by replacing the non-dimensional variable ${\it\xi}$ defined in (2.3) and clearly inappropriate for $k=0$ by the original $z$ . This reduces the equation for $W$ to

(4.2) $$\begin{eqnarray}(f^{2}-{\it\omega}^{2})W_{zz}+2\text{i}{\it\Lambda}\,f\,lW_{z}-J{\it\Lambda}^{2}l^{2}W=0\end{eqnarray}$$

which admits the simple solution $W\propto \text{e}^{\text{i}mz}$ for

(4.3) $$\begin{eqnarray}{\it\omega}^{2}=f^{2}(J{\it\delta}^{2}+2{\it\delta}+1),\end{eqnarray}$$

where ${\it\delta}=({\it\Lambda}l)/(\,fm)$ . A purely imaginary frequency ${\it\omega}$ , corresponding to symmetric instability, occurs for $J<1$ , with maximum growth rate $\text{Im}\,{\it\omega}=f\sqrt{1/J-1}$ for ${\it\delta}=-1/J$ .

Although our analysis is not adapted to treat this case, because when $k\rightarrow 0$ the inertial levels go to infinity and it becomes impossible to impose radiation conditions beyond them, the condition $J<1$ for IGW amplification between the inertial levels establishes a clear connection between the valve effect and the inertial instability criteria. This connection with inertial instability is not the only one. For instance, the fact that waves with large ${\it\nu}$ are amplified between the inertial levels suggests that the shear layer response to a random incident IGW field can become very large between the inertial levels and preferentially symmetric, as are inertial instabilities. Also, the response in this transverse plane will be tilted in the direction of the isentropes (see figure 2(d,h) in LPV12) as is also the case for inertial instability since ${\it\delta}<0$ for them (see (4.3)). Note also that a stabilization of the flow by the disturbance is expected to occur in the nonlinear regime. Indeed, the large increase of the EP flux at the lower inertial level accelerates the mean flow there, whereas the decrease at the upper inertial level decelerates it there. As a result, the positive shear tends to be reduced: as is often the case, the development of the instability weakens its cause.