3.1 Instability growth rates

In figure 2, we present the growth rate $\unicode[STIX]{x1D70E}$ (2.18) over the $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane for nine different pairs of base flow parameters $(A,\unicode[STIX]{x1D6F7})$ . As noted in § 2, the growth rates are periodic in $\unicode[STIX]{x1D719}_{0}$ over $[0^{\circ },180^{\circ }]$ and symmetric about $\unicode[STIX]{x1D703}_{0}=0^{\circ }$ . In figure 2, the internal wave base flow progresses from steep ( $\unicode[STIX]{x1D6F7}=15^{\circ }$ ) in the leftmost column to shallow ( $\unicode[STIX]{x1D6F7}=75^{\circ }$ ) in the rightmost column, and from small-amplitude ( $A=0.1$ ) in the top row to large-amplitude ( $A=10$ ) in the bottom row.

For $(A,\unicode[STIX]{x1D6F7})=(0.1,15^{\circ })$ , i.e. small-amplitude, steep internal waves (figure 2 a), we observe, in the limit of 2-D perturbations ( $\unicode[STIX]{x1D703}_{0}=0^{\circ }$ ), two thin instability peaks centred at $\unicode[STIX]{x1D719}_{0}=61^{\circ }$ and $\unicode[STIX]{x1D719}_{0}=119^{\circ }$ . These instability peaks extend towards regions of 3-D perturbations ( $\unicode[STIX]{x1D703}_{0}>0^{\circ }$ ) to form two instability ridges on the $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane; the two instability ridges, labelled I and II in figure 2(a), seem to have similar maximum growth rates, both of which occur on the $\unicode[STIX]{x1D703}_{0}=0^{\circ }$ axis. Significantly, the growth rates on the $\unicode[STIX]{x1D703}_{0}>0^{\circ }$ portions of the ridges are comparable to those on the $\unicode[STIX]{x1D703}_{0}=0^{\circ }$ axis, suggesting that a continuous range of perturbation wave vectors on these ridges are of similar growth rates. It is also noteworthy that the variation of $\unicode[STIX]{x1D70E}$ with $\unicode[STIX]{x1D703}_{0}$ on ridges I and II is different, with ridge II retaining larger values of $\unicode[STIX]{x1D70E}$ up to larger $\unicode[STIX]{x1D703}_{0}$ . As $\unicode[STIX]{x1D703}_{0}$ approaches a threshold value of $29^{\circ }$ , the two instability ridges approach each other; above the threshold $\unicode[STIX]{x1D703}_{0}$ , ridges I and II vanish. As $\unicode[STIX]{x1D703}_{0}$ increases towards even larger values, a third, notably weaker, thin instability ridge appears (labelled III in figure 2 a). Ridge III seems to extend all the way to $\unicode[STIX]{x1D703}_{0}=0^{\circ }$ , the axis of 2-D perturbations. Interestingly, ridge III becomes aligned with $\unicode[STIX]{x1D719}_{0}=\unicode[STIX]{x1D6F7}$ (vertical dotted lines in figure 2), indicating the emergence of shear-aligned instabilities as the ridge approaches the $\unicode[STIX]{x1D703}_{0}=0^{\circ }$ axis. The existence of distinct instability bands in an otherwise stable landscape suggests the occurrence of select resonant wave–wave interactions in the small $s$ limit, which we discuss further in §§ 3.2–3.4.

Figure 2.

Growth rate, $\unicode[STIX]{x1D70E}$ , presented as a function of $\unicode[STIX]{x1D719}_{0}$ and $\unicode[STIX]{x1D703}_{0}$ for $A=0.1$ , $1$ , $10$ (top, middle and bottom rows, respectively), and $\unicode[STIX]{x1D6F7}=15^{\circ }$ , $45^{\circ }$ , $75^{\circ }$ (left, middle and right columns, respectively). Stable regions of the $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane where $\unicode[STIX]{x1D70E}<10^{-10}$ have been coloured white. The dotted vertical lines in each panel denote $\unicode[STIX]{x1D719}_{0}=\unicode[STIX]{x1D6F7}$ .

At a moderate internal wave angle, i.e. $\unicode[STIX]{x1D6F7}=45^{\circ }$ , as shown in figure 2(b), we similarly observe only three thin instability ridges over a $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane that is mostly stable. The range of 3-D wave vectors to which ridges I and II extend is smaller when compared with the case of $\unicode[STIX]{x1D6F7}=15^{\circ }$ in figure 2(a). Ridge I appears to be noticeably stronger than ridge II in figure 2(b). Moreover, a third instability ridge is also identified in figure 2(b), although it is nearly indiscernible given its relatively weak growth rates. For a shallow propagating internal wave, as shown in figure 2(c) for $\unicode[STIX]{x1D6F7}=75^{\circ }$ , only three ridges are faintly visible due to the growth rates being an order of magnitude weaker than their $\unicode[STIX]{x1D6F7}=15^{\circ }$ and $45^{\circ }$ counterparts, and for the range of $\unicode[STIX]{x1D703}_{0}$ to which they extend being notably limited.

For the moderate-amplitude, steep-propagation case of $(A,\unicode[STIX]{x1D6F7})=(1,15^{\circ })$ (figure 2 d), the growth rate distribution is qualitatively similar to that of figure 2(a), but with wider regions of the $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane that are unstable. The difference between regions I and II is more dramatic than what is observed at small-amplitude (figure 2 a). Moreover, region III (which was a ridge in figure 2 a) contains growth rate magnitudes comparable to that of regions I and II. The larger instability regions on the $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane imply that there is now a wider, continuous range of perturbation wave vectors whose growth rates are of comparable magnitude. At an even larger value of $A=10$ (figure 2 g), nearly the entire $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane is unstable, with a fourth region – narrowly separated from region III – that encompasses purely transverse ( $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ ) perturbations (which was not the case for weaker amplitude internal waves in figure 2 a,d). As in figure 2(a), the outer boundary of region III in both figure 2(d,g) aligns with $\unicode[STIX]{x1D719}_{0}=\unicode[STIX]{x1D6F7}$ (shear-aligned perturbations) as $\unicode[STIX]{x1D703}_{0}\rightarrow 0^{\circ }$ .

At a moderate-amplitude and intermediate internal wave angle, i.e. $(A,\unicode[STIX]{x1D6F7})=(1,45^{\circ })$ (figure 2 e), the instability islands cover a large part of the $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane, but are nestled among a still sizeable region of stability. When compared with figure 2(b,e) has an additional region of instability (region IV) close to the $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ axis, apart from the fact that the original instability regions from $A=0.1$ have become wider and stronger for $A=1$ . Similar to what is seen in figure 2(d,g), as $A$ increases to $10$ for $\unicode[STIX]{x1D6F7}=45^{\circ }$ (see figure 2 h), the entire $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane becomes effectively unstable. Strikingly, there are at least two more separate instability regions for $A=10$ (figure 2 h) when compared with $A=1$ (figure 2 e). The largest growth rate for $(A,\unicode[STIX]{x1D6F7})=(10,45^{\circ })$ occurs for purely transverse perturbations ( $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ ). There is, however, a ridge-like feature spanning nearly all of $\unicode[STIX]{x1D703}_{0}$ along $\unicode[STIX]{x1D719}_{0}\approx 45^{\circ }$ , on which the growth rate is quite close to the maximum growth rate (which occurs on $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ ). Interestingly, the perturbation wave vectors corresponding to this near-vertical ridge are shear-aligned, i.e. $\unicode[STIX]{x1D719}_{0}\approx \unicode[STIX]{x1D6F7}$ , in the $(x,z)$ -plane of the base flow.

In the shallow internal wave regime of $\unicode[STIX]{x1D6F7}=75^{\circ }$ , the instability landscape of moderate amplitude internal waves of $A=1$ are characterised by an array of several relatively thin unstable ridges, as shown in figure 2(f). We recall that only three instability ridges are discernible at $A=0.1$ in figure 2(c). As the internal wave amplitude increases to $A=10$ (figure 2 i), the instability landscape changes significantly. The unstable ridges in figure 2(f) all seem to widen such that the entire $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane, except for a small patch around $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})=(75^{\circ },0^{\circ })$ , is unstable. Remarkably, there is now a vertical ridge (spanning the entire $\unicode[STIX]{x1D703}_{0}$ range of $0^{\circ }\leqslant \unicode[STIX]{x1D703}_{0}\leqslant 90^{\circ }$ ) at $\unicode[STIX]{x1D719}_{0}=75^{\circ }$ along which the growth rate is large and nearly uniform. The overall maximum growth rate in figure 2(i) still occurs on $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ although the shear-aligned wave vectors on the vertical ridge are not far behind. Furthermore, at $A=10$ , the maximum growth rates at $\unicode[STIX]{x1D6F7}=75^{\circ }$ are now comparable in magnitude to the growth rates at $\unicode[STIX]{x1D6F7}=45^{\circ }$ and $15^{\circ }$ , suggesting that the dependence of maximum growth rate on $\unicode[STIX]{x1D6F7}$ is weaker at larger amplitudes.

Some of the distinct instability ridges seen across $\unicode[STIX]{x1D6F7}$ for $A=1$ in figure 2(d–f) appear to be amplified extensions of the thin instability ridges observed at $A=0.1$ in figure 2(a–c). For example, the first three instability regions (I, II and III) in figure 2(d,e) clearly have thinner, weaker counterparts in figure 2(a,b). Similarly, the broad and barely separated regions of instability in figure 2(g–i) at $A=10$ seem to be extensions of the $A=1$ unstable regions in figure 2(d–f), but not necessarily with one-to-one correspondence between scenarios of the same $\unicode[STIX]{x1D6F7}$ . For example, the fourth instability region in figure 2(g) does not have a counterpart in figure 2(d). Additionally, the number of seemingly separate instability regions in figure 2(h) is larger than the number in figure 2(e). The unstable regions I–III at $\unicode[STIX]{x1D6F7}=15^{\circ }$ , $45^{\circ }$ and $75^{\circ }$ for $A=0.1$ (in figure 2), suggest that the instabilities occurring at large $A$ may be linked to instabilities at small $A$ , in line with Klostermeyer (Reference Klostermeyer1991), who proposed a connection between small $A$ resonances and all instability modes at all internal wave angles. Sonmor & Klaassen (Reference Sonmor and Klaassen1997) reaffirmed the conjecture of Klostermeyer (Reference Klostermeyer1991) by concluding that there exist no instabilities in finite-amplitude plane internal waves that cannot be traced back to resonant instabilities at small  $A$ . The connection between small $A$ resonances and finite $A$ instabilities is addressed more rigorously in §§ 3.4 and 3.5.

3.2 Two-dimensional PSI

In the limit of 2-D perturbations, i.e. $\unicode[STIX]{x1D703}_{0}=0^{\circ }$ and small $A$ , the local stability approach shows the existence of two distinct values of $\unicode[STIX]{x1D719}_{0}$ at which thin instability peaks are present (figure 2 a–c). In this section, we verify that these peaks arise from a resonant wave triad, and proceed to establish correspondence with existing results on 2-D PSI. Furthermore, the local stability approach facilitates an investigation of how 2-D PSI continuously extends towards large internal wave amplitudes as well as 3-D perturbations (we discuss the latter in § 3.3).

Resonant triad interaction (RTI) represents a mechanism by which a pair of subharmonic or superharmonic internal waves extract energy from a primary internal wave through the formation of a resonant triad (Mied Reference Mied1976). Such a resonant triad is formed when the spatial and temporal resonant conditions are satisfied:

(3.1) $$\begin{eqnarray}\displaystyle \boldsymbol{k}_{1}+\boldsymbol{k}_{2}=n\boldsymbol{k}, & & \displaystyle\end{eqnarray}$$

and

(3.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{1}+\unicode[STIX]{x1D714}_{2}=n\unicode[STIX]{x1D714}, & & \displaystyle\end{eqnarray}$$

where the internal wave $(\boldsymbol{k},\unicode[STIX]{x1D714})$ resonantly interacts with the internal waves $(\boldsymbol{k}_{1},\unicode[STIX]{x1D714}_{1})$ and $(\boldsymbol{k}_{2},\unicode[STIX]{x1D714}_{2})$ , and $n$ is any positive integer (Hasselmann Reference Hasselmann1967; Drazin Reference Drazin1977). In the context of RTI, $(\boldsymbol{k},\unicode[STIX]{x1D714})$ denote the primary wave properties presented in (2.7), the subscripts $1$ and $2$ denote subharmonic daughter waves and $n+1$ is the order of the resonance. In this section and the next (§ 3.3), we will focus on second-order wave interactions, which take place for $n=1$ ; higher order resonances ( $n>1$ ) are discussed in § 3.4.

To illustrate the main features of 2-D RTI, i.e. resonant triads formed in the plane of the primary wave, we begin by considering an arbitrary pair of daughter waves, $(\boldsymbol{k}_{1},\unicode[STIX]{x1D714}_{1})$ and $(\boldsymbol{k}_{2},\unicode[STIX]{x1D714}_{2})$ , for a given primary wave $(\boldsymbol{k},\unicode[STIX]{x1D714})$ . The conditions (3.1) and (3.2) with $n=1$ , together with the dispersion relation (2.8) for all three waves, can be solved to obtain a relation between $m_{1}$ and $k_{1}$ , where $\boldsymbol{k}_{1}=(k_{1},0,m_{1})$ (see Bourget et al. Reference Bourget, Dauxois, Joubaud and Odier2013, equation (3.29)). A plot of $m_{1}$ (normalised by the primary vertical wavenumber $m$ ) as a function of $k_{1}$ (normalised by the primary horizontal wavenumber $k$ ) is shown in figure 3(a). Any point on this resonance curve describes the tip of $\boldsymbol{k}_{1}$ , which when subtracted from the primary wave vector $\boldsymbol{k}$ – located at $(1,1)$ in figure 3(a) – will yield the $\boldsymbol{k}_{2}$ that completes the resonant triad (Phillips Reference Phillips1977). An example resonant triad is presented in figure 3(a).

Figure 3.

(a) The $\boldsymbol{k}_{1}$ resonance curve (solid black line) for a primary wave $\boldsymbol{k}=(k,0,m)$ (depicted by the arrow starting at the origin and ending at point $(1,1)$ ) obtained from the solution to the set of equations comprising the resonant conditions (3.1) and (3.2) (for $n=1$ ) and the dispersion relation (2.8). Superposed onto the resonance curve is a representative PSI resonant triad: $\boldsymbol{k}$ (——), $\boldsymbol{k}_{1}$ (— ⋅ —), and $\boldsymbol{k}_{2}$ ( $\cdots$ ). (b) The instability growth rates (solid line), as derived by Koudella & Staquet (Reference Koudella and Staquet2006), corresponding to the resonance curve in panel (a) for $(A,\unicode[STIX]{x1D6F7})=(0.1,45^{\circ })$ . The asymptotic values of the growth rate in the PSI limit for branch-A and branch-B (denoted by ○ and ▵, respectively), as derived by Sonmor & Klaassen (Reference Sonmor and Klaassen1997), are consistent with Koudella & Staquet (Reference Koudella and Staquet2006).

Of particular relevance to our study is the limit where the magnitudes of $\boldsymbol{k}_{1}$ and $\boldsymbol{k}_{2}$ are significantly larger than $\boldsymbol{k}$ , which translates to short-wavelength perturbations (i.e. daughter waves) relative to the base flow (i.e. primary wave). In this limit ( $|k_{1}|\rightarrow \infty$ and $\boldsymbol{k}_{1}$ lying on the resonance curve in figure 3 a), $\boldsymbol{k}_{1}$ and $\boldsymbol{k}_{2}$ become increasingly anti-parallel, and both $\unicode[STIX]{x1D714}_{1}$ and $\unicode[STIX]{x1D714}_{2}$ approach $\unicode[STIX]{x1D714}/2$ . The class of resonant triads characterised by long, anti-parallel perturbation wave vectors with frequencies that are half the primary wave frequency is referred to as PSI (McComas & Bretherton Reference Mccomas and Bretherton1977; Dauxois et al. Reference Dauxois, Joubaud, Odier and Venaille2018).

Since the local stability approach and PSI both concern small-scale perturbations, we proceed to investigate the extent to which the local stability approach recovers 2-D PSI characteristics, i.e. the daughter wave vectors and associated growth rates. We focus on the case presented in figure 2(b), i.e. $(A,\unicode[STIX]{x1D6F7})=(0.1,45^{\circ })$ , which corresponds to an isopycnal steepness of $s\approx 0.07$ . For 2-D perturbations ( $\unicode[STIX]{x1D703}_{0}=0^{\circ }$ ), the local stability approach identifies instabilities at $\unicode[STIX]{x1D719}_{0,1}\approx 69^{\circ }$ and $\unicode[STIX]{x1D719}_{0,3}\approx 111^{\circ }$ , and (by virtue of the periodicity of $\unicode[STIX]{x1D70E}$ with $\unicode[STIX]{x1D719}_{0}$ ) at $\unicode[STIX]{x1D719}_{0,2}=\unicode[STIX]{x1D719}_{0,1}+180^{\circ }$ and $\unicode[STIX]{x1D719}_{0,4}=\unicode[STIX]{x1D719}_{0,3}+180^{\circ }$ . The wave vectors $\unicode[STIX]{x1D73F}_{0,1}=\unicode[STIX]{x1D73F}_{0}(\unicode[STIX]{x1D719}_{0,1},\unicode[STIX]{x1D703}_{0}=0^{\circ })$ and $\unicode[STIX]{x1D73F}_{0,2}=\unicode[STIX]{x1D73F}_{0}(\unicode[STIX]{x1D719}_{0,2},\unicode[STIX]{x1D703}_{0}=0^{\circ })$ , together with the base flow wave vector $\boldsymbol{k}$ , satisfy the spatial triadic resonance condition (3.1) as

(3.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D73F}_{0,1}+\unicode[STIX]{x1D73F}_{0,2}=0, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D73F}_{0,1}$ and $\unicode[STIX]{x1D73F}_{0,2}$ are anti-parallel to each other, and represent wave vectors of much larger magnitude than the base flow wave vector due to the short-wavelength approximation. Thus, the wave vectors $\unicode[STIX]{x1D73F}_{0,1}$ and $\unicode[STIX]{x1D73F}_{0,2}$ are consistent with the daughter wave vectors in the PSI limit shown in figure 3(a).

If $\unicode[STIX]{x1D73F}_{0,1}$ and $\unicode[STIX]{x1D73F}_{0,2}$ are to represent internal wave vectors, then the corresponding frequencies based on the dispersion relation should be $\unicode[STIX]{x1D714}_{1}/N=\cos \unicode[STIX]{x1D719}_{0,1}\approx 0.36$ and $\unicode[STIX]{x1D714}_{2}/N=-\text{cos}\,\unicode[STIX]{x1D719}_{0,2}\approx 0.36$ , both of which equal approximately half of the base flow wave frequency $\unicode[STIX]{x1D714}/N=\cos \unicode[STIX]{x1D6F7}\approx 0.71$ . Given that the local stability framework allows the perturbation wave vectors to evolve in time, we verify that the PSI triadic resonance conditions are satisfied over the entire time period $0\leqslant t\cos \unicode[STIX]{x1D6F7}<2\unicode[STIX]{x03C0}$ by computing $\unicode[STIX]{x1D73F}_{0,1}(t)$ and $\unicode[STIX]{x1D73F}_{0,2}(t)$ via (2.17) and confirming them to be effectively constant in time, with ${\lesssim}5\,\%$ variation (the variation decreases as $A$ decreases). We have thus established that the internal waves at the unstable wave vectors $\unicode[STIX]{x1D73F}_{0,1}$ and $\unicode[STIX]{x1D73F}_{0,2}$ from the local stability approach indeed satisfy the classical PSI triadic resonance conditions with the base flow wave vector. A similar calculation reveals that the internal waves at the unstable wave vectors $\unicode[STIX]{x1D73F}_{0,3}$ and $\unicode[STIX]{x1D73F}_{0,4}$ are also in PSI triadic resonance with $\boldsymbol{k}$ . In summary, each of the two pairs of unstable 2-D perturbation wave vectors from the local stability approach represent a daughter wave pair that is in PSI resonance with $\boldsymbol{k}$ . We proceed to investigate the growth rate at these unstable wave vectors.

For small isopycnal steepness ( $s\ll 1$ ) of the primary wave, the growth rate corresponding to any resonant triad in the plane of the primary wave is (Koudella & Staquet Reference Koudella and Staquet2006, § 4.3.1)

(3.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}_{\text{KS}}=A\cos ^{2}\unicode[STIX]{x1D6F7}\sqrt{S_{1}S_{2}}, & & \displaystyle\end{eqnarray}$$

where

(3.5) $$\begin{eqnarray}\displaystyle S_{p}=\frac{m_{r}k_{q}-m_{q}k_{r}}{4k_{3}^{2}|\boldsymbol{k}_{p}|^{2}}[(|\boldsymbol{k}_{q}|^{2}-|\boldsymbol{k}_{r}|^{2})+|\boldsymbol{k}_{p}|(|\boldsymbol{k}_{q}|-|\boldsymbol{k}_{r}|)]; & & \displaystyle\end{eqnarray}$$

the subscripts $p,q,r=1,2,3$ form a circular permutation, with $1$ and $2$ denoting the daughter waves and $3$ denoting the primary wave.

Figure 3(b) shows $\unicode[STIX]{x1D70E}_{\text{KS}}$ as a function of $k_{1}/k$ , from which we see that the growth rate approaches two different asymptotic values at the PSI limits $k_{1}\rightarrow \pm \infty$ . The larger asymptote (corresponding to the maximum PSI growth rate) occurs as $k_{1}\rightarrow \infty$ , and is referred to as branch-A PSI. The smaller asymptote occurs in the limit $k_{1}\rightarrow -\infty$ , and is referred to as branch-B PSI. The growth rates in these two limits are (Sonmor & Klaassen Reference Sonmor and Klaassen1997, § 3):

(3.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}_{\text{SK}}=\lim _{k_{1}\rightarrow \pm \infty }\unicode[STIX]{x1D70E}_{\text{KS}}=\frac{A}{16}\cos \unicode[STIX]{x1D6F7}[\pm 2\sin ^{3}|\unicode[STIX]{x1D6F7}|+(1+2\cos ^{2}\unicode[STIX]{x1D6F7})\sqrt{4-\cos ^{2}\unicode[STIX]{x1D6F7}}], & & \displaystyle\end{eqnarray}$$

where the upper and lower signs correspond to branch-A and -B, respectively (in figure 3(b) we confirm that $\unicode[STIX]{x1D70E}_{\text{KS}}$ approaches $\unicode[STIX]{x1D70E}_{\text{SK}}$ at the PSI limits). We denote the 2-D PSI growth rates in (3.6) associated with branch-A and branch-B as $\unicode[STIX]{x1D70E}_{A}$ and $\unicode[STIX]{x1D70E}_{B}$ , respectively.

We find, for $(A,\unicode[STIX]{x1D6F7})=(0.1,45^{\circ })$ , the local stability growth rates to be $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D719}_{0,1},\unicode[STIX]{x1D703}_{0}=0^{\circ })=\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D719}_{0,2},\unicode[STIX]{x1D703}_{0}=0^{\circ })=0.01970$ , which is within $0.002\,\%$ of $\unicode[STIX]{x1D70E}_{A}=0.01966$ . Similarly, $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D719}_{0,3},\unicode[STIX]{x1D703}_{0}=0^{\circ })=\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D719}_{0,4},\unicode[STIX]{x1D703}_{0}=0^{\circ })=0.01349$ , which is within $0.006\,\%$ of $\unicode[STIX]{x1D70E}_{B}=0.01341$ . Summarising our investigations at other $\unicode[STIX]{x1D6F7}$ , we conclude that the local stability approach, for small $A$ internal waves and 2-D perturbations, recovers sharp instability peaks at $\unicode[STIX]{x1D719}_{0,1}=\cos ^{-1}(\cos \unicode[STIX]{x1D6F7}/2)$ and $\unicode[STIX]{x1D719}_{0,2}=\unicode[STIX]{x1D719}_{0,1}+180^{\circ }$ , the growth rates at which match with that of branch-A PSI. The local stability approach also recovers sharp instability peaks at $\unicode[STIX]{x1D719}_{0,3}=\cos ^{-1}(-\text{cos}\,\unicode[STIX]{x1D6F7}/2)$ and $\unicode[STIX]{x1D719}_{0,4}=\unicode[STIX]{x1D719}_{0,3}+180^{\circ }$ , the growth rates at which match with that of branch-B PSI. We now perform a more comprehensive comparison of 2-D PSI growth rates derived by Sonmor & Klaassen (Reference Sonmor and Klaassen1997) ( $\unicode[STIX]{x1D70E}_{A}$ and $\unicode[STIX]{x1D70E}_{B}$ ) and our local stability approach ( $\unicode[STIX]{x1D70E}_{A}^{L}$ and $\unicode[STIX]{x1D70E}_{B}^{L}$ ).

Figure 4.

2-D PSI growth rates (a–c) and corresponding wave vector orientations (d–f) presented as a function of $A$ for $\unicode[STIX]{x1D6F7}=15^{\circ }$ , $45^{\circ }$ and $75^{\circ }$ (left, middle and right columns, respectively). The solid and dashed lines in panels (a–c), respectively, denote the branch-A and branch-B PSI growth rates derived by Sonmor & Klaassen (Reference Sonmor and Klaassen1997) in (3.6); the solid and dashed lines in panels (d–f) denote the corresponding perturbation wave vector orientations given by $\cos ^{-1}(\pm \cos \unicode[STIX]{x1D6F7}/2)$ . Our local stability results are denoted by ○ and ▵ for branch-A and branch-B PSI, respectively. The vertical lines associated with our results in panels (d–f) denote the width of the PSI peaks, with the colour intensity indicating the growth rates normalised by the corresponding growth rates plotted in panels (a–c). The value $s=0.1$ is indicated in each panel by a dotted vertical line, indicating the upper limit of the small $s$ regime.

For a given $(A,\unicode[STIX]{x1D6F7})$ , we calculate the maximum growth rate $\unicode[STIX]{x1D70E}_{A}^{L}$ within the instability peak associated with $\unicode[STIX]{x1D719}_{0,1}=\cos ^{-1}(\cos \unicode[STIX]{x1D6F7}/2)$ for 2-D perturbations ( $\unicode[STIX]{x1D703}_{0}=0^{\circ }$ ); the corresponding location of locally maximum growth rate is denoted $\unicode[STIX]{x1D719}_{0,1}^{\ast }$ , which equals $\unicode[STIX]{x1D719}_{0,1}$ only for sufficiently small $A$ since the peaks around $\unicode[STIX]{x1D719}_{0,1}$ (and $\unicode[STIX]{x1D719}_{0,3}$ ) expand into finite-width instability regions as $A$ increases. Similarly, the maximum growth rate within the instability peak around $\unicode[STIX]{x1D719}_{0,3}=\cos ^{-1}(-\text{cos}\,\unicode[STIX]{x1D6F7}/2)$ is denoted as $\unicode[STIX]{x1D70E}_{B}^{L}$ , and the corresponding location of locally maximum growth rate is $\unicode[STIX]{x1D719}_{0,3}^{\ast }$ . In the top row of figure 4, we plot $\unicode[STIX]{x1D70E}_{A}^{L}$ and $\unicode[STIX]{x1D70E}_{B}^{L}$ as a function of $A$ for three different internal wave orientations ( $\unicode[STIX]{x1D6F7}=15^{\circ }$ , $45^{\circ }$ , and $75^{\circ }$ in figure 4 a–c, respectively). For all three $\unicode[STIX]{x1D6F7}$ , there is excellent quantitative agreement between $\unicode[STIX]{x1D70E}_{A,B}^{L}$ (denoted by ○ and ▵ for branch-A and -B, respectively) and the $\unicode[STIX]{x1D70E}_{A,B}$ growth rates of Sonmor & Klaassen (Reference Sonmor and Klaassen1997) (denoted by solid and dashed lines for branch-A and -B, respectively) for $s<0.1$ ( $s=0.1$ is indicated by the vertical dotted lines in all panels of figure 4). The bottom row of figure 4 shows the variation of $\unicode[STIX]{x1D719}_{0,1}^{\ast }$ and $\unicode[STIX]{x1D719}_{0,3}^{\ast }$ with $A$ , which are in excellent agreement with $\unicode[STIX]{x1D719}_{0,1}$ and $\unicode[STIX]{x1D719}_{0,3}$ , respectively, for $s<0.1$ . Also shown in the bottom row of figure 4 (using vertical bars) are the widths, along $\unicode[STIX]{x1D719}_{0}$ of the corresponding instability regions; the widths remain close to zero for $s<0.1$ at all $\unicode[STIX]{x1D6F7}$ .

As $A$ , and thus $s$ , increases, there is a noticeable deviation of the local stability growth rates from the PSI growth rates of Sonmor & Klaassen (Reference Sonmor and Klaassen1997) (top row of figure 4). Around $s\approx 0.5$ , PSI theory breaks down regardless of the specific $A$ and $\unicode[STIX]{x1D6F7}$ values. For each $\unicode[STIX]{x1D6F7}$ , in the large $A$ limit, $\unicode[STIX]{x1D70E}_{A}^{L}$ and $\unicode[STIX]{x1D70E}_{B}^{L}$ converge and saturate at ${\approx}1$ , which, in dimensional time, corresponds to a growth rate of ${\approx}N$ . In summary, branch-A PSI dominates branch-B PSI at small $A$ , but the growth rates of both branches converge to a similar value at large $A$ . We note that the intriguing convergence of the local stability PSI growth rates in the large $A$ limit revealed by the local stability approach is beyond the scope of classical PSI theories. We further note that as $A$ increases, the unstable perturbation wave vectors initialised at $\unicode[STIX]{x1D73F}_{0}(\unicode[STIX]{x1D719}_{0,1-4}^{\ast },\unicode[STIX]{x1D703}_{0}=0^{\circ })$ begin to substantially evolve along the base flow fluid particle trajectories and can no longer be classified as internal waves.

For $s>0.1$ , $\unicode[STIX]{x1D719}_{0,1}^{\ast }$ and $\unicode[STIX]{x1D719}_{0,3}^{\ast }$ deviate noticeably from the branch-A and -B PSI theoretical estimates $\unicode[STIX]{x1D719}_{0,1}$ and $\unicode[STIX]{x1D719}_{0,3}$ , respectively. As $\unicode[STIX]{x1D6F7}$ increases, however, the deviations of $\unicode[STIX]{x1D719}_{0,1}^{\ast }$ from $\unicode[STIX]{x1D719}_{0,1}$ decrease, while the deviations of $\unicode[STIX]{x1D719}_{0,3}^{\ast }$ from $\unicode[STIX]{x1D719}_{0,3}$ increase (for $\unicode[STIX]{x1D6F7}=75^{\circ }$ , $\unicode[STIX]{x1D719}_{0,1}^{\ast }\approx \unicode[STIX]{x1D719}_{0,1}$ even at large $A$ ). We note that the branch-A ( $\unicode[STIX]{x1D719}_{0,1}^{\ast }$ ) and branch-B ( $\unicode[STIX]{x1D719}_{0,3}^{\ast }$ ) instabilities become anti-parallel to each other at large $A$ (i.e. separated by $180^{\circ }$ ), consistent with $\unicode[STIX]{x1D70E}_{A}^{L}$ and $\unicode[STIX]{x1D70E}_{B}^{L}$ converging at large $A$ . As wave steepness increases (or as $A$ increases for fixed $\unicode[STIX]{x1D6F7}$ ), the branch-A peak gradually widens, although, for large $\unicode[STIX]{x1D6F7}$ , the width of the peak is quite small even at large $A$ . The width of the branch-B peak increases with $A$ as well, with the increase being larger than that of branch-A; for $\unicode[STIX]{x1D6F7}=75^{\circ }$ (figure 4 f), the width of the branch-B peaks becomes very large, spanning $90^{\circ }\lesssim \unicode[STIX]{x1D719}_{0}\lesssim 250^{\circ }$ at large $A$ . The wider PSI peaks at large $A$ indicate a larger range of instability, as we have seen in figure 2.

3.3 Three-dimensional PSI

The instability ridges for $(A,\unicode[STIX]{x1D6F7})=(0.1,45^{\circ })$ (figure 2 b), at $\unicode[STIX]{x1D703}_{0}=0^{\circ }$ , comprise branch-A ( $\unicode[STIX]{x1D719}_{0,1}\approx 69^{\circ }$ and $\unicode[STIX]{x1D719}_{0,2}=\unicode[STIX]{x1D719}_{0,1}+180^{\circ }$ ) and branch-B ( $\unicode[STIX]{x1D719}_{0,3}\approx 111^{\circ }$ and $\unicode[STIX]{x1D719}_{0,4}=\unicode[STIX]{x1D719}_{0,3}+180^{\circ }$ ) PSI peaks. The extension of these peaks towards 3-D perturbation wave vectors, i.e. $\unicode[STIX]{x1D703}_{0}>0^{\circ }$ , suggests the existence of 3-D PSI. To confirm this, we first compute $\unicode[STIX]{x1D70E}$ over the $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane for $A=0.01$ and various values of $\unicode[STIX]{x1D6F7}$ in the small $s$ regime, and then trace the instability ridges that emerge from the $\unicode[STIX]{x1D703}_{0}=0^{\circ }$ axis (figure 5 a–c). The 3-D perturbation wave vectors corresponding to these traces satisfy the spatial resonant condition,

(3.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D73F}(\unicode[STIX]{x1D719}_{0},\pm \unicode[STIX]{x1D703}_{0})+\unicode[STIX]{x1D73F}(\unicode[STIX]{x1D719}_{0}+180^{\circ },\mp \unicode[STIX]{x1D703}_{0})=0, & & \displaystyle\end{eqnarray}$$

along the curves in figure 5(a–c) (we recall from § 2 that $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})=\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D719}_{0}+180^{\circ },\unicode[STIX]{x1D703}_{0})$ and $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D719}_{0},\pm \unicode[STIX]{x1D703}_{0})=\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D719}_{0},\mp \unicode[STIX]{x1D703}_{0})$ ). The equal and opposite wave vector components in $y$ for the two daughter waves in (3.7) are a consequence of the primary wave vector $\boldsymbol{k}$ being entirely in the $(x,z)$ -plane.

Requiring the two daughter waves in (3.7) to satisfy the 3-D internal wave dispersion relation,

(3.8) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D714}}{N}=\frac{|\boldsymbol{k}_{xy}|}{|\boldsymbol{k}|}, & & \displaystyle\end{eqnarray}$$

(where $\boldsymbol{k}_{xy}=(k,\ell ,0)$ is the wave vector projected onto the $xy$ -plane), and the temporal resonance condition in (3.2) for $n=1$ yields the relation

(3.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{0}=\cos ^{-1}\left(\pm \sqrt{\frac{1}{4}\frac{\cos ^{2}\unicode[STIX]{x1D6F7}}{\cos ^{2}\unicode[STIX]{x1D703}_{0}}-\tan ^{2}\unicode[STIX]{x1D703}_{0}}\right)\quad \text{for }|\unicode[STIX]{x1D703}_{0}|\leqslant \sin ^{-1}\left(\frac{|\text{cos}\,\unicode[STIX]{x1D6F7}|}{2}\right), & & \displaystyle\end{eqnarray}$$

with the daughter waves at frequency $\unicode[STIX]{x1D714}/2$ . In figure 5(a–c), we superpose points corresponding to (3.9) onto the instability traces; the excellent agreement confirms that the temporal resonant condition is satisfied by the 3-D perturbation wave vectors that comprise each trace. Furthermore, we verify that the perturbation wave vectors on these instability traces are effectively invariant in time, with ${\lesssim}1.5\,\%$ variation over $0\leqslant t<2\unicode[STIX]{x03C0}/\cos \unicode[STIX]{x1D6F7}$ . In summary, the perturbation wave vectors $\unicode[STIX]{x1D73F}(\unicode[STIX]{x1D719}_{0},\pm \unicode[STIX]{x1D703}_{0})$ and $\unicode[STIX]{x1D73F}(\unicode[STIX]{x1D719}_{0}+180^{\circ },\mp \unicode[STIX]{x1D703}_{0})$ are in 3-D PSI triadic resonance with the primary wave vector $\boldsymbol{k}$ , where $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ is any point on the curves shown in figure 5(a–c).

Figure 5.

Traces of 3-D PSI ridges in the $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane (a–c), and corresponding growth rates (d–f) for a base flow of $A=10^{-2}$ and $\unicode[STIX]{x1D6F7}=15^{\circ }$ , $45^{\circ }$ , $75^{\circ }$ (left, middle and right columns, respectively); the corresponding isopycnal steepnesses are, respectively, $s\approx 0.003$ , $0.007$ and $0.01$ . The trace segments associated with branch-A and branch-B PSI are denoted by the solid and dash-dotted lines, respectively. Superposed onto panels (a–c) are discrete points of (3.9) (denoted by the circle markers), and the threshold values of $\unicode[STIX]{x1D703}_{0}$ (denoted by the asterisk markers).

Having confirmed that the instability ridges that exist at small $s$ are indeed a 3-D manifestation of PSI, we proceed to elucidate the growth rate variation along the curves in figure 5(a–c). Each ridge has a point at which the growth rate approaches zero, which we use to separate the trace into two segments, branches-A and -B, based on which 2-D PSI branch the segment is on at $\unicode[STIX]{x1D703}_{0}=0^{\circ }$ (ridges I and II in figure 2 a, respectively). The growth rates corresponding to each trace are presented in figure 5(d–f). For 2-D perturbation wave vectors ( $\unicode[STIX]{x1D703}_{0}=0^{\circ }$ ), the maximum growth rate corresponds to branch-A regardless of $\unicode[STIX]{x1D6F7}$ . For $\unicode[STIX]{x1D6F7}=15^{\circ }$ in figure 5(d), however, branch-B is dominant for practically the entire range of $\unicode[STIX]{x1D703}_{0}>0^{\circ }$ . Moreover, for both $\unicode[STIX]{x1D6F7}=15^{\circ }$ and $75^{\circ }$ (figure 5 d,f, respectively), the maximum branch-B growth rate is comparable to the maximum branch-A growth rate. Looking at 2-D PSI alone for $\unicode[STIX]{x1D6F7}=75^{\circ }$ , one would conclude that branch-A clearly dominates branch-B; in 3-D, however, the maximum branch-B growth rate is almost as strong as branch-A is in two dimensions.

As $\unicode[STIX]{x1D6F7}$ increases, we observe five general features of 3-D PSI: (i) the range, in $\unicode[STIX]{x1D719}_{0}$ , of PSI decreases; (ii) the range, in $\unicode[STIX]{x1D703}_{0}$ , of branch-A (solid line in figure 5) shortens; (iii) the branch-B growth rate curve evolves from one that is monotonically decreasing to one that increases and achieves a maximum growth rate in the vicinity of the threshold $\unicode[STIX]{x1D703}_{0}$ (asterisk marker in figure 5 a–c); (iv) the overall magnitudes of $\unicode[STIX]{x1D70E}$ decrease; and (v) the threshold value of $\unicode[STIX]{x1D703}_{0}$ decreases substantially, thereby restricting the three-dimensionality of the resonant wave interaction between the 3-D perturbation wave vectors and the internal wave base flow.

3.4 Higher-order resonances

To understand the origin of various large $A$ instabilities, we plot, in figure 6, the growth rates in the $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane for $\unicode[STIX]{x1D6F7}=15^{\circ }$ $45^{\circ }$ , and $75^{\circ }$ for three base flow amplitudes that reasonably span the transition from small to large $A$ : $A=0.3$ , $0.5$ and $0.7$ . Moreover, we superpose on figure 6 the contours along which the corresponding perturbation wave vectors are in second- or higher-order triadic resonance with the primary wave, i.e. equation (3.9) modified to include any order of resonance ( $n\geqslant 1$ ):

(3.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{0}=\cos ^{-1}\left(\pm \sqrt{\frac{n^{2}}{4}\frac{\cos ^{2}\unicode[STIX]{x1D6F7}}{\cos ^{2}\unicode[STIX]{x1D703}_{0}}-\tan ^{2}\unicode[STIX]{x1D703}_{0}}\right)\quad \text{for }|\unicode[STIX]{x1D703}_{0}|\leqslant \sin ^{-1}\left(\frac{n}{2}|\text{cos}\,\unicode[STIX]{x1D6F7}|\right), & & \displaystyle\end{eqnarray}$$

where the daughter waves are now at frequency $n\unicode[STIX]{x1D714}/2$ . The second-order ( $n=1$ ) resonance contour is closest to the $\unicode[STIX]{x1D703}_{0}=0^{\circ }$ -axis; as $n$ increases, the higher-order resonance curves expand outwards and approach the $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ axis. As $\unicode[STIX]{x1D6F7}$ increases, the frequency $\unicode[STIX]{x1D714}/N$ decreases [see (2.8)], which, in turn, allows for a greater number of higher-harmonics (i.e. $n$ ) to fit within the internal wave passband $0<n\unicode[STIX]{x1D714}/2<N$ . From (3.10), the highest possible order of resonance is specified by $n=2$ for $\unicode[STIX]{x1D6F7}=15^{\circ }$ and $45^{\circ }$ , and $n=7$ for $\unicode[STIX]{x1D6F7}=75^{\circ }$ . Interestingly, for quasi-2-D perturbations ( $\unicode[STIX]{x1D703}_{0}\approx 0^{\circ }$ ), (3.10) simplifies to $\unicode[STIX]{x1D719}_{0}\approx \unicode[STIX]{x1D6F7}$ (i.e. perturbations that are shear aligned) for $n=2$ (third-order resonance). This explains the orientation of ridge III near the $\unicode[STIX]{x1D703}_{0}=0^{\circ }$ axis in figure 2(a,b) given that ridge III corresponds to third-order resonance, as we will show below.

Figure 6.

Growth rate, $\unicode[STIX]{x1D70E}$ , in the $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane for $\unicode[STIX]{x1D6F7}=15^{\circ }$ , $45^{\circ }$ and $75^{\circ }$ (left, middle and right columns, respectively) for $A=0.3$ , $0.5$ and $0.7$ (top, middle and bottom rows, respectively). Superposed onto each panel are the higher-order resonance curves (dashed lines) given by (3.10); the innermost curve corresponds to $n=1$ , with $n$ incrementally increasing outwards. Stable regions of the $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane where $\unicode[STIX]{x1D70E}<10^{-10}$ have been coloured white.

For $(A,\unicode[STIX]{x1D6F7})=(0.3,15^{\circ })$ (figure 6 a), the instability ridges are of finite width. The $n=1$ resonance contour overlaps with its corresponding instability ridges I and II, as was reported in § 3.3. In contrast, the $n=2$ resonance contour does not necessarily capture the most unstable locations within the corresponding instability region. Instead, the $n=2$ contour accurately captures the upper boundary of instability region III. As $A$ decreases, however, the instability region III of $\unicode[STIX]{x1D6F7}=15^{\circ }$ becomes progressively thinner (and weaker) and overlaps with the $n=2$ resonance contour. For $\unicode[STIX]{x1D6F7}=45^{\circ }$ and $75^{\circ }$ (figure 6 b,c, respectively), the theoretical resonance curves nicely capture all of the instability ridges, although the ridges are very thin and weak for $\unicode[STIX]{x1D6F7}=75^{\circ }$ .

Slightly increasing $A$ to $0.5$ (figure 6 d–f) yields results similar to $A=0.3$ . At $\unicode[STIX]{x1D6F7}=15^{\circ }$ , a deviation from theoretical resonance happens in the form of a widening of the instability ridges, although the signature of the higher-order resonance curves are still clearly present. The upper boundary of instability region III is still nicely captured by the $n=2$ contour. Deviation of the instability ridges from higher-order resonance curves is notable at $\unicode[STIX]{x1D6F7}=75^{\circ }$ , although there is overlap near $\unicode[STIX]{x1D719}_{0}=90^{\circ }$ . The deviations taking place for $\unicode[STIX]{x1D6F7}=75^{\circ }$ are unsurprising given that $s$ increases with $\unicode[STIX]{x1D6F7}$ , and we expect deviations to occur beyond the PSI regime for large $s$ . Interestingly, a new instability region appears in the vicinity of the $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ axis for $\unicode[STIX]{x1D6F7}=45^{\circ }$ (figure 6 e), and it seems to have no corresponding theoretical resonance contour in its neighbourhood. We investigate this new instability region in § 3.5.

The results for $A=0.7$ depict a departure from alignment between the instability ridges and the higher-order resonance curves. Either the instability ridges widen substantially to cover a larger band of the $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane (as for $\unicode[STIX]{x1D6F7}=15^{\circ }$ and $45^{\circ }$ in figure 6 g,h, respectively), or they remain relatively thin but completely fall out of alignment and into a non-trivial pattern (as for $\unicode[STIX]{x1D6F7}=75^{\circ }$ in figure 6 i). What is most striking, however, is the connection observed between the instability ridges and the higher-order resonance curves for small $A$ and the persistence of that connection as $A$ increases. It appears that many features of the instability landscape, especially those corresponding to the maximum growth rates, originate from higher-order resonances in the small $A$ regime, a result that is consistent with the conclusions of Klostermeyer (Reference Klostermeyer1991) and Sonmor & Klaassen (Reference Sonmor and Klaassen1997).

3.5 Transverse instabilities

In figure 6(e,h), we note the existence of an instability region near $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ that is seemingly independent of the higher-order resonance curves. Equation (3.10) informs us that the higher-order resonance curves approach the $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ axis as $\unicode[STIX]{x1D6F7}\rightarrow 90^{\circ }$ . The appearance of the instability region at and around $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ for $\unicode[STIX]{x1D6F7}=45^{\circ }$ (figure 6 e,h), and its absence for $\unicode[STIX]{x1D6F7}=15^{\circ }$ (figure 6 d,g) and $75^{\circ }$ (figure 6 f,i), suggests that the transverse $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ instability is unrelated to higher-order resonances. We note that the perturbation wave vector giving rise to the purely transverse instability is normal to the plane of the base flow, and thus invariant in time [by virtue of (2.12)].

Figure 7.

(a) Growth rate as a function of $(A,\unicode[STIX]{x1D6F7})$ for $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ ( $\unicode[STIX]{x1D719}_{0}$ is arbitrarily set to $0^{\circ }$ ). Stable regions of the $(A,\unicode[STIX]{x1D6F7})$ -plane where $\unicode[STIX]{x1D70E}<10^{-10}$ has been coloured white. The origin of the largest instability tongue at $A=0$ , which occurs at $\unicode[STIX]{x1D6F7}_{1}\approx 48^{\circ }$ , is marked by the left-most vertical dotted line. Additional tongues occur at half-integer multiples of $\sec \unicode[STIX]{x1D6F7}_{1}$ , as shown in panel (b). The threshold for gravitational stability, (3.14), is denoted by the dashed line, and the modified gravitational stability threshold, (3.15), is denoted by the dash-dotted line.

To better understand this transverse instability, we plot the growth rate for $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ (and $\unicode[STIX]{x1D719}_{0}$ arbitrarily set to $0^{\circ }$ ) as a function of $(A,\unicode[STIX]{x1D6F7})$ in figure 7. Interestingly, there are distinct instability tongues spanning the $(A,\unicode[STIX]{x1D6F7})$ -plane. These tongues seem to originate at specific points along the $\unicode[STIX]{x1D6F7}$ -axis at $A=0$ , where the growth rates are infinitesimal. Substituting purely transverse perturbations ( $\unicode[STIX]{x1D73F}_{0}=\hat{\boldsymbol{e}}_{y}$ ) into equations (2.13) and (2.14) yields a set of coupled ordinary differential equations for $a_{x}$ and $a_{z}$ :

(3.11) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{d}^{2}a_{x}}{\text{d}t^{2}}+A\cos ^{2}\unicode[STIX]{x1D6F7}\sin \unicode[STIX]{x1D6F7}\sin (\unicode[STIX]{x1D717}_{0}-\cos \unicode[STIX]{x1D6F7}t)a_{x}\nonumber\\ \displaystyle & & \displaystyle \quad =-A\sin ^{2}\unicode[STIX]{x1D6F7}\cos \unicode[STIX]{x1D6F7}\sin (\unicode[STIX]{x1D717}_{0}-\cos \unicode[STIX]{x1D6F7}t)a_{z}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,A\sin \unicode[STIX]{x1D6F7}\cos (\unicode[STIX]{x1D717}_{0}-\cos \unicode[STIX]{x1D6F7}t)\left(\cos \unicode[STIX]{x1D6F7}\frac{\text{d}a_{x}}{\text{d}t}+\sin \unicode[STIX]{x1D6F7}\frac{\text{d}a_{z}}{\text{d}t}\right),\end{eqnarray}$$

(3.12) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{d}^{2}a_{z}}{\text{d}t^{2}}+[1-A\sin \unicode[STIX]{x1D6F7}(1+\cos ^{2}\unicode[STIX]{x1D6F7})\sin (\unicode[STIX]{x1D717}_{0}-\cos \unicode[STIX]{x1D6F7}t)]a_{z}\nonumber\\ \displaystyle & & \displaystyle \quad =A(\cos ^{3}\unicode[STIX]{x1D6F7}+\cos \unicode[STIX]{x1D6F7})\sin (\unicode[STIX]{x1D717}_{0}-\cos \unicode[STIX]{x1D6F7}t)a_{x}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,A\cos \unicode[STIX]{x1D6F7}\cos (\unicode[STIX]{x1D717}_{0}-\cos \unicode[STIX]{x1D6F7}t)\left(\cos \unicode[STIX]{x1D6F7}\frac{\text{d}a_{x}}{\text{d}t}+\sin \unicode[STIX]{x1D6F7}\frac{\text{d}a_{z}}{\text{d}t}\right).\end{eqnarray}$$

After neglecting the forcing and damping terms on the right-hand side of (3.12), we obtain a Mathieu-like equation for $a_{z}$ which describes a parametrically forced system that has instability tongues in the $(A,\unicode[STIX]{x1D6F7})$ -plane corresponding to regions of parametric resonance. These tongues originate at $A=0$ from the following points (Bender & Orszag Reference Bender and Orszag1999, chap. 11.4):

(3.13) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}=\cos ^{-1}\left(\frac{2}{n+2}\right),\quad n=1,2,3,\ldots \,. & & \displaystyle\end{eqnarray}$$

The points given by (3.13), which are denoted by vertical dotted lines in figure 7, accurately predict the points along the $\unicode[STIX]{x1D6F7}$ -axis from which the tongues emanate. As $A$ increases, the points widen into instability regions with finite and increasing growth rates. We do observe, however, a detachment of the tongues (especially for large $\unicode[STIX]{x1D6F7}$ ) from the $A=0$ axis, which we attribute to the damping and forcing terms in the right-hand side of (3.11) and (3.12) (Nayfeh & Mook Reference Nayfeh and Mook1995, chap. 1.5).

The first tongue originating at $\unicode[STIX]{x1D6F7}_{1}\approx 48^{\circ }$ spans the largest region of the $(A,\unicode[STIX]{x1D6F7})$ -plane, and the bulk of the tongue appears to take form at relatively smaller values of $A$ . It is this tongue that we observe in figure 6(e) at $(A,\unicode[STIX]{x1D6F7})=(0.5,45^{\circ })$ ; the other tongues at larger $\unicode[STIX]{x1D6F7}$ have not taken form by $A=0.5$ , which explains the absence of a transverse instability in figure 6(f) at $\unicode[STIX]{x1D6F7}=75^{\circ }$ . We note that it is the peculiar shape of the instability tongues (the inclined upper boundary in particular) that accounts for $(A,\unicode[STIX]{x1D6F7})=(1,45^{\circ })$ having no purely transverse instabilities (figure 2 e), in contrast to smaller $A=0.5$ and $0.7$ for $\unicode[STIX]{x1D6F7}=45^{\circ }$ (figure 6 e,h) having purely transverse instabilities. Another consequence of the non-trivial formation of transverse instabilities is region IV in figure 2(e). Although it has detached from the $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ axis, region IV is an extension of the transverse instability seen at smaller $A$ in figure 6(h). (For $\unicode[STIX]{x1D6F7}=45^{\circ }$ , we observe the detachment of region IV from the $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ axis at some $A$ in $0.95<A<0.96$ .)

The density of tongues grows as $\unicode[STIX]{x1D6F7}\rightarrow 90^{\circ }$ and remapping the growth rates to the $(A,\sec \unicode[STIX]{x1D6F7})$ -plane better reveals the formation of instability tongues (figure 7 b). We superpose on each plot in figure 7 the threshold amplitude below which statically unstable layers do not occur at any time during the internal wave period:

(3.14a,b ) $$\begin{eqnarray}\displaystyle \max _{0\leqslant t<2\unicode[STIX]{x03C0}/\cos \unicode[STIX]{x1D6F7}}\left(\frac{\text{d}\bar{\bar{\unicode[STIX]{x1D70C}}}}{\text{d}z}+\frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}z}\right)<0,\quad \text{or}\quad A<\frac{1}{\sin \unicode[STIX]{x1D6F7}}, & & \displaystyle\end{eqnarray}$$

where $\bar{\bar{\unicode[STIX]{x1D70C}}}(z)$ is the background density stratification and $\bar{\unicode[STIX]{x1D70C}}(\boldsymbol{x},t)$ is the density field of the internal wave base flow. The minimum $A$ at which the instability tongues form increases with $\unicode[STIX]{x1D6F7}$ and appears to approach the gravitational stability curve (dashed line in figure 7) at large $\unicode[STIX]{x1D6F7}$ . The region above the gravitational stability curve is largely unstable, although thin bands of stability separate the unstable tongues.

Without parametric pumping (i.e. the time dependency in the coefficient of $a_{z}$ ), equation (3.12), with the right-hand side set to zero, would have neutrally stable oscillatory solutions if

(3.15) $$\begin{eqnarray}\displaystyle A<\frac{1}{\sin \unicode[STIX]{x1D6F7}(1+\cos ^{2}\unicode[STIX]{x1D6F7})}, & & \displaystyle\end{eqnarray}$$

which appears to be a slightly modified version of the static gravitational stability criterion (3.14). The inclusion of parametric pumping, however, allows for unstable solutions (i.e. parametric resonance) – even if the coefficient of $a_{z}$ is positive over the entirety of the forcing period – provided that the pumping occurs at half-integer multiples of the natural period of the system (3.13). In an effort to separate the influence of parametric pumping on wave stability, we identify, in (3.15), a heuristic measure of stability by requiring the coefficient of $a_{z}$ in (3.12) to be positive at all times. The criterion in (3.15), which is plotted as a dash-dotted curve in figure 7, adequately separates the bulk of the instability tongues from the stems connecting them to $A=0$ . The region above (3.15) in figure 7 is overwhelmingly unstable due to both parametric and gravitational-like instability given that the coefficient of $a_{z}$ in the left-hand side of (3.12) is negative for a substantial portion of the wave period; the thin instability stems below (3.15) exist entirely due to parametric resonance.

Although the transverse instability regions and other instability ridges seen in figure 6 both originate at small $A$ , the former arises from parametric resonance and the latter from RTI. The instabilities associated with transverse perturbations, therefore, offer another mechanism, separate from RTI, by which large $A$ instabilities can form. Our findings remain in line with the position of Klostermeyer (Reference Klostermeyer1991) and Sonmor & Klaassen (Reference Sonmor and Klaassen1997), who conclude that all large $A$ instabilities are linked to small $A$ resonances.

3.6 Dominant instability modes

To identify the dominant instability mode for a given $(A,\unicode[STIX]{x1D6F7})$ , we calculate $\unicode[STIX]{x1D70E}$ over the entire $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane and then select the maximum growth rate, which we denote $\unicode[STIX]{x1D70E}^{\ast }$ . The corresponding $\unicode[STIX]{x1D719}_{0}$ and $\unicode[STIX]{x1D703}_{0}$ of $\unicode[STIX]{x1D70E}^{\ast }$ , denoted $\unicode[STIX]{x1D719}_{0}^{\ast }$ and $\unicode[STIX]{x1D703}_{0}^{\ast }$ , respectively, reflect the initial wave vector orientation of the most unstable perturbation. Figures 8(a)–8(c), respectively, plot $\unicode[STIX]{x1D70E}^{\ast }$ , $\unicode[STIX]{x1D703}_{0}^{\ast }$ and $\unicode[STIX]{x1D719}_{0}^{\ast }$ over a range of base flow parameters $(A,\unicode[STIX]{x1D6F7})$ . Moreover, we derive from figure 8 a regime diagram for the various dominant instability modes, which we present in figure 9. We superpose on each plot in figures 8 and 9 the boundary below which the gravitational stability criterion given by (3.14) is satisfied (dashed line), and the boundary below which the classical shear stability criterion is satisfied (dotted line):

(3.16a,b ) $$\begin{eqnarray}\displaystyle \min _{0\leqslant t<2\unicode[STIX]{x03C0}/\cos \unicode[STIX]{x1D6F7}}Ri>\frac{1}{4}\quad \text{or}\quad A<\frac{2}{\sin ^{2}\unicode[STIX]{x1D6F7}}, & & \displaystyle\end{eqnarray}$$

where $Ri=N^{2}/(\text{d}\bar{u}/\text{d}z)^{2}$ is the instantaneous Richardson number associated with the base flow without accounting for the unsteady density gradients associated with the internal wave. We note that (3.14) represents the stability criterion for a steady, one-dimensional (1-D), vertical density gradient with no shear flow; similarly, (3.16) is strictly valid only for 1-D shear and density profiles along the direction of gravity. Despite their limitations, (3.14) and (3.16) do provide points of reference for evaluating the relevance of classical gravitational and shear instability criteria to the emergence of various instabilities in the local stability approach. Interestingly, if we include the unsteady density gradients associated with the internal wave base flow in the definition of the Richardson number, i.e. $Ri=(N^{2}-(g/\unicode[STIX]{x1D70C}_{0})\,\text{d}\bar{\unicode[STIX]{x1D70C}}/\text{d}z)/(\text{d}\bar{u}/\text{d}z)^{2}$ , then the boundary given by $\min _{0\leqslant t<2\unicode[STIX]{x03C0}/\cos \unicode[STIX]{x1D6F7}}Ri>1/4$ coincides with the gravitational stability boundary in (3.14).

Figure 8.

(a) Maximum growth rate, $\unicode[STIX]{x1D70E}^{\ast }$ , as a function of $A$ and $\unicode[STIX]{x1D6F7}$ , and the corresponding perturbation wave vector orientations (b) $\unicode[STIX]{x1D703}_{0}^{\ast }$ and (c) $\unicode[STIX]{x1D719}_{0}^{\ast }$ . The dashed (– –) and dotted ( $\cdots$ ) lines denote threshold boundaries of gravitational and shear stability, respectively.

Figure 9.

Regime diagram indicating the dependence of the dominant instability mode on the base flow orientation, $\unicode[STIX]{x1D6F7}$ , and amplitude, $A$ . The thresholds for gravitational and shear stability are denoted by the dashed (– –) and dotted ( $\cdots$ ) lines, respectively.

In general, $\unicode[STIX]{x1D70E}^{\ast }$ monotonically increases with $A$ for a given $\unicode[STIX]{x1D6F7}$ . For a given $A$ , however, $\unicode[STIX]{x1D70E}^{\ast }$ decreases with $\unicode[STIX]{x1D6F7}$ for $A\ll 10$ because the dominant instability in that regime is PSI (as discussed below and shown in figure 9), for which we know that $\unicode[STIX]{x1D70E}_{A}$ decreases with $\unicode[STIX]{x1D6F7}$ (as shown in figure 4 a–c). For $A\gtrsim 10$ , $\unicode[STIX]{x1D70E}^{\ast }$ decreases with $\unicode[STIX]{x1D6F7}$ below some $\unicode[STIX]{x1D6F7}$ value, past which $\unicode[STIX]{x1D70E}^{\ast }$ abruptly begins to increase. This dual-behaviour of $\unicode[STIX]{x1D70E}^{\ast }$ for $A\gtrsim 10$ is linked to a transition of the dominant instability mode, as discussed later in this section.

The corresponding $\unicode[STIX]{x1D703}_{0}^{\ast }$ values, shown in figure 8(b), clearly indicate a number of distinct regimes of the dominant instability mode. Up to $A\sim 1$ , the dominant instability is 2-D by virtue of the perturbation wave vectors having $\unicode[STIX]{x1D703}_{0}=0^{\circ }$ . As we have shown in § 3.2, the dominant instability mode for small $A$ and 2-D perturbations is 2-D PSI. As further evidence, we note that the corresponding $\unicode[STIX]{x1D719}_{0}^{\ast }$ values in this regime (shown in figure 8 c) are independent of $A$ and approach $90^{\circ }$ as $\unicode[STIX]{x1D6F7}$ increases, both of which are in agreement with the behaviour of branch-A PSI in figure 4(d–f). The boundary for classical shear stability (dotted line in figures 8 and 9) appears to nicely delineate this limit of the PSI regime for sufficiently large $\unicode[STIX]{x1D6F7}$ .

Beyond the upper boundary of the 2-D PSI regime, there is another distinct regime where $\unicode[STIX]{x1D703}_{0}^{\ast }$ is close to $90^{\circ }$ (figure 8 b), which corresponds to a dominant mode that is transverse to the base flow. This 3-D instability dominates for large $A$ , with the threshold in $A$ decreasing as $\unicode[STIX]{x1D6F7}\rightarrow 90^{\circ }$ . We note that there are only certain points in this regime where $\unicode[STIX]{x1D703}_{0}^{\ast }$ is precisely $90^{\circ }$ ; the mean and standard deviation of $\unicode[STIX]{x1D703}_{0}^{\ast }$ are approximately $81^{\circ }$ and $6.5^{\circ }$ , respectively. In figure 7, we see that the instability tongues for purely transverse perturbations are separated by thin stability bands along the $\unicode[STIX]{x1D6F7}$ axis at large $A$ . The transverse dominant instability mode may, therefore, be slightly offset from $\unicode[STIX]{x1D703}_{0}^{\ast }=90^{\circ }$ , as we see in figure 8(b), due to the presence of these thin stability bands. Despite transverse perturbations having the maximum growth rate in the $(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D703}_{0})$ -plane at large $A$ , we recall that there are instabilities spanning nearly the entire range of $\unicode[STIX]{x1D703}_{0}$ at $\unicode[STIX]{x1D719}_{0}^{\ast }\approx \unicode[STIX]{x1D6F7}$ with growth rates close to $\unicode[STIX]{x1D70E}^{\ast }$ (see, for example, figure 2 i).

Sandwiched between the distinct 3-D transverse and 2-D PSI regimes are two intermediary regimes: one where the dominant perturbation wave vector is effectively 2-D and shear-aligned, and another regime where the perturbation wave vector is 3-D over a range of oblique angles. We identify the shear-aligned regime as one where $\unicode[STIX]{x1D719}_{0}^{\ast }$ is within $15^{\circ }$ of $\unicode[STIX]{x1D6F7}$ and is thus aligned with the direction of shear in the base flow. Moreover, the corresponding $\unicode[STIX]{x1D703}_{0}^{\ast }$ values are all less than $15^{\circ }$ , which translates to perturbation wave vectors that are effectively two-dimensional. From figures 8(c) and 9, we can see that the shear-aligned regime hugs the lower boundary of the 3-D transverse regime and, for small $A$ and large $\unicode[STIX]{x1D6F7}$ , overlaps with the 2-D PSI regime. The overlap of the shear-aligned and 2-D PSI regimes is to be expected since the $\unicode[STIX]{x1D719}_{0}$ values for branch-A PSI approach $90^{\circ }$ as $\unicode[STIX]{x1D6F7}\rightarrow 90^{\circ }$ (as can be seen in figure 4 d–f). The lower boundary of the non-PSI shear-aligned regime is nicely captured by the shear stability curve, as shown in figure 9.

The classical shear stability curve nicely separates the shear-aligned regime from the remaining regime, which is located below the shear stability curve and above the 2-D PSI regime. This regime corresponds to perturbation wave vectors with oblique 3-D $\unicode[STIX]{x1D703}_{0}^{\ast }$ ( $40^{\circ }\lesssim \unicode[STIX]{x1D703}_{0}^{\ast }\lesssim 50^{\circ }$ ) and a range of $\unicode[STIX]{x1D719}_{0}^{\ast }$ that is strongly dependent on $A$ and $\unicode[STIX]{x1D6F7}$ . Interestingly, this 3-D oblique regime is situated between two regimes that are predominantly two-dimensional.

Figure 9 presents a summary diagram of the four dominant instability regimes (listed in increasing $A$ ): 2-D PSI, 3-D oblique, quasi-2-D shear-aligned, and 3-D transverse (with the quasi-2-D shear-aligned and 2-D PSI regimes overlapping for $\unicode[STIX]{x1D6F7}>70^{\circ }$ ). The shear stability curve separates the 2-D PSI and 3-D oblique regimes from the quasi-2-D shear-aligned and 3-D transverse regimes. Given how our calculations are restricted to small-scale instabilities, it is only our 2-D branch-A PSI regime, at small $A$ , that is in agreement with the dominant instability regime diagram of Sonmor & Klaassen (Reference Sonmor and Klaassen1997, figure 22); both of our studies also recover a shear-aligned regime, but the shear-aligned regime of Sonmor & Klaassen (Reference Sonmor and Klaassen1997) overlaps with our 3-D transverse regime in the $(A,\unicode[STIX]{x1D6F7})$ -plane. A more quantitative comparison of the other regimes in figure 22 of Sonmor & Klaassen (Reference Sonmor and Klaassen1997) are precluded because they are derived from finite-scale instabilities, which are beyond the scope of the local stability approach.