2.1. General formulation

The present analysis assumes an unbounded, incompressible, uniformly stratified Boussinesq fluid with constant buoyancy frequency $N_{*}$ and includes the effect of background rotation under the $f$-plane approximation. Using non-dimensional variables with $1/N_{*}$ as the time scale and $L_{*}$ as the length scale (to be specified later) the governing equations for the velocity field $\boldsymbol {u}$, the reduced density $\rho$ and pressure $p$ are

(2.1a)\begin{gather} \boldsymbol{\nabla}\boldsymbol{\cdot}{\boldsymbol{u}}=0, \end{gather}

(2.1b)\begin{gather}\rho_{t}+\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla} \rho = \boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{j}, \end{gather}

(2.1c)\begin{gather}\boldsymbol{u}_{t}+\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{u} + \boldsymbol{f}\times \boldsymbol{u}={-}\boldsymbol{\nabla}{p}-\rho\boldsymbol{j}+\nu\nabla^2{\boldsymbol{u}}. \end{gather}

Here, $\boldsymbol {j}$ is a vertical unit vector pointing upwards, $\boldsymbol {f} = f\boldsymbol {j}$, where $f$ is the local Coriolis parameter normalized with $N_{*}$, and $\nu =\nu _{*}/N_{*}L_{*}^{2}$ is the inverse Reynolds number, where $\nu _{*}$ denotes the kinematic viscosity.

In the inviscid limit ($\nu =0$), (2.1) admits time-harmonic plane wave solutions in the form of beams (Tabaei & Akylas Reference Tabaei and Akylas2003) that feature general profile in the cross-beam ($\eta$) direction, and are uniform in the along-beam ($\xi$) and transverse horizontal ($z$) directions (figure 1a). Specifically, the beam velocity components $(u,v,w)$ in the coordinate system $(\xi , \eta , z)$, along with $\rho$ and $p$, are given by

(2.2a)\begin{gather} (u,v,w) = (u_{0}, 0, w_{0}) \equiv \left(1, 0, \frac{\text{i} f\cos\theta}{\omega}\right)U(\eta)\text{e}^{-\text{i}\omega t} + \text{c.c.}, \end{gather}

(2.2b)\begin{gather}\rho = \rho_{0}(\eta,t) \equiv{-}\frac{\text{i}\sin\theta}{\omega} U(\eta)\text{e}^{-\text{i}\omega t} + \text{c.c.}, \end{gather}

(2.2c)\begin{gather}p = p_{0}(\eta,t) \equiv \text{i}(1-f^{2}) \frac{\sin\theta\cos\theta}{\omega} \int^{\eta}U(\eta')\,{\rm d}{\eta'}\text{e}^{-\text{i}\omega t} + \text{c.c.}. \end{gather}

Here, the complex amplitude $U(\eta )$ specifies the beam profile and $\theta$ is the beam inclination angle to the horizontal (figure 1a), which is tied to the beam frequency $\omega$ via the (non-dimensional) dispersion relation

(2.3)\begin{equation} \omega^{2} = f^{2} + (1-f^{2})\sin^{2}\theta. \end{equation}

Furthermore, ($\xi$, $\eta$) are related to the horizontal and vertical coordinates $(x,y)$ by $\xi = x\cos \theta -y\sin \theta$ and $\eta = x\sin \theta +y\cos \theta$. Generally, finite viscosity ($\nu \neq 0$) will introduce along-beam variations in (2.2) (Mowbray & Rarity Reference Mowbray and Rarity1967; Thomas & Stevenson Reference Thomas and Stevenson1972). These viscous effects are ignored here.

Figure 1.

(a) Schematic of uniform beam with frequency $\omega$ and inclination $\theta$ to the horizontal, in keeping with (2.3). The coordinate system $(\xi ,\eta ,z)$ is defined by the along-beam, cross-beam and transverse directions, respectively. Dotted lines indicate the finite width of the beam and the arrows within show the along-beam velocity $u_{0}$. The perturbation wavevector $\boldsymbol {k}$ is inclined by the angle $\chi$ to $\eta$ and by $\phi =\theta -\chi$ to the vertical $y$. (b) Beam profile (3.1) for $\epsilon =0.1$. Solid, dashed and dotted lines correspond to the real part, imaginary part and $|U|$, respectively.

The focus of the present analysis is on the stability of the primary wave (2.2) to infinitesimal perturbations. As (2.2) does not depend on $\xi$ or $z$, we superimpose normal modes in the form

(2.4)\begin{equation} u = u_{0} + \hat{u}(\eta,t) \exp({\text{i} (\mu \xi + m z)}) , \end{equation}

where the real parameters $\mu$ and $m$ specify the along-beam and transverse wavenumbers of the perturbation, with similar expressions for $v,w,\rho$ and $p$. After inserting into (2.1) and linearizing with respect to the perturbations, we obtain the following equations (dropping the hats):

(2.5a)\begin{gather} 0 = \text{i} \mu u + v_{\eta} + \text{i} m w, \end{gather}

(2.5b)\begin{gather}\rho_{t} ={-} \text{i} (\mu u_{0} +m w_{0}) \rho - \rho_{0\eta}v - u\sin\theta + v\cos\theta, \end{gather}

(2.5c)\begin{gather}u_{t} ={-} \text{i} (\mu u_{0} +m w_{0}) u - u_{0\eta}v -\text{i} \mu p +\rho \sin\theta -wf\cos\theta+\nu \mathscr{L} u, \end{gather}

(2.5d)\begin{gather}v_{t} ={-} \text{i} (\mu u_{0} +m w_{0}) v -p_{\eta} -\rho \cos\theta -w f \sin\theta +\nu \mathscr{L}v, \end{gather}

(2.5e)\begin{gather}w_{t} ={-} \text{i} (\mu u_{0} +m w_{0}) w - w_{0\eta}v - \text{i} m p +uf\cos\theta + vf\sin\theta+\nu \mathscr{L}w, \end{gather}

where $\mathscr {L}\equiv -\mu ^{2} + \partial ^{2}_{\eta }-m^{2}$ takes into account the full effects of viscosity on the perturbations. As the basic state (2.2) is periodic in $t$ (period $T_{0}=2{\rm \pi} /\omega$), (2.5) is a linear equation system with periodic coefficients, which can be solved by Floquet theory to determine stability for a given beam profile $U(\eta )$ and parameters $\mu$ and $m$.

2.2. Sinusoidal waves

For a sinusoidal plane wave, the primary wave profile in (2.2) is taken in the form $U=\epsilon \exp (\text {i}\eta )$, where the amplitude parameter $\epsilon =U_{*}/(N_{*}L_{*})$, $U_{*}$ being half the along-$\xi$ velocity amplitude, and the wavelength has been normalized to $2{\rm \pi}$ (which fixes the length scale $L_{*}$). In this instance, it is straightforward to solve (2.5) via Floquet theory by expanding the perturbations in Fourier series in $\eta -\omega t$, and thus recover the thoroughly studied instability of a sinusoidal plane wave (e.g. Sonmor & Klaassen Reference Sonmor and Klaassen1997; Yau et al. Reference Yau, Klaassen and Sonmor2004). Out of this body of work, of special interest here is the nearly inviscid ($\nu \ll 1$) small-amplitude limit ($\epsilon \ll 1$). In this regime, the most unstable perturbation is two-dimensional (i.e. $m=0$) and involves only two subharmonic frequency components that are freely propagating sinusoidal internal waves and form a resonant triad with the primary wave. Specifically, these subharmonic waves, with frequencies $(\omega _{1},\omega _{2})$ (where $|\omega _{1,2}|<\omega$) and wavevectors $(\boldsymbol {k}_{1},\boldsymbol {k}_{2})$, each satisfy the dispersion relation (2.3), and together satisfy the triad resonance conditions

(2.6a)\begin{gather} \omega_{1} + \omega_{2} = \omega, \end{gather}

(2.6b)\begin{gather}\boldsymbol{k}_{1}+\boldsymbol{k}_{2} = \boldsymbol{k}_{0}, \end{gather}

where $\boldsymbol {k}_{0} = \hat {\boldsymbol {e}}_{\eta }$ is the primary wavevector. This class of instabilities, which encompasses all such pairs of $(\omega _{1},\boldsymbol {k}_{1})$ and $(\omega _{2},\boldsymbol {k}_{2})$, is thus known as TRI.

For inviscid flow conditions, in the event that the beam inclination angle $\theta \gtrsim 43^{\circ }$ for $f=0$, or when $\omega \approx 2f$ for any $\theta$, the most unstable pairs of subharmonic waves feature $\omega _{1}\approx \omega _{2}\approx \omega /2$ and $|\boldsymbol {k}_{1}|\approx |\boldsymbol {k}_{2}|\gg 1$ (e.g. Yeh & Liu Reference Yeh and Liu1981; Sonmor & Klaassen Reference Sonmor and Klaassen1997). This particular form of TRI is widely known as PSI. However, it is important to note that other flow conditions may favour other forms of TRI over PSI. For instance, the most unstable triads for $\theta \lesssim 43^{\circ }$ and $f=0$ involve subharmonic waves with $|\boldsymbol {k}_{1}| < 1 < |\boldsymbol {k}_{2}|$, which is sometimes referred to as the ‘branch-C’ instability (Sonmor & Klaassen Reference Sonmor and Klaassen1997).

2.3. Locally confined beams

Moving away from the sinusoidal wave, KA14 and KA17 examined the possibility of PSI for locally confined beams in the small-amplitude nearly inviscid limit. As discussed in § 1, these approximate models of PSI are based upon the assumption of triad resonance, which strictly holds for a small-amplitude sinusoidal primary wave. Here, to assess the validity of KA14 and KA17, we perform a formal stability analysis of a locally confined beam profile $U(\eta )$ based on the full stability problem (2.5). In general, this task has to be carried out numerically.

The stability problem (2.5) is more challenging to handle numerically for a general profile $U(\eta )$ than for the sinusoidal wave profile: in addition to Fourier expansion in $t$, it is necessary to use a separate discretization in $\eta$. To tackle this complication, we follow the approach taken by Onuki & Tanaka (Reference Onuki and Tanaka2019) in their stability analysis of finite-amplitude internal wave beams under oceanic flow conditions. This approach was used earlier for inertial wave beams (Jouve & Ogilvie Reference Jouve and Ogilvie2014) and also in a very recent study of finite-amplitude thin internal wave beams under laboratory flow conditions (Fan & Akylas Reference Fan and Akylas2020).

As our interest centres on the relevance of PSI to small-amplitude locally confined beams, we focus on two-dimensional perturbations ($m=0$). It should be noted that three-dimensional ($m\neq 0$) instabilities are possible and may be important at large amplitudes (Onuki & Tanaka Reference Onuki and Tanaka2019). Furthermore, other three-dimensional instability mechanisms such as streaming, which involves the generation of a resonant mean flow due to finite transverse variations of a beam (Kataoka & Akylas Reference Kataoka and Akylas2015; Fan, Kataoka & Akylas Reference Fan, Kataoka and Akylas2018; Fan & Akylas Reference Fan and Akylas2020; Jamin et al. Reference Jamin, Kataoka, Dauxois and Akylas2021) and thus falls beyond the scope of linear stability analysis, may be important in other contexts.

Following Onuki & Tanaka (Reference Onuki and Tanaka2019), we solve (2.5) using the ‘monodromy’ matrix, which can be easily computed by integrating (2.5) over one period of oscillation of the primary wave, namely $T_{0} = 2{\rm \pi} /\omega$. The Floquet exponents $\lambda =\lambda _{r}+\text {i}\lambda _{i}$, where $\lambda _{r}>0$ implies instability, as well as the Floquet modes, are then obtained via the eigenvalues and eigenvectors of the monodromy matrix (see Fan (Reference Fan2020) and Fan & Akylas (Reference Fan and Akylas2020) for details). Finally, we repeat this procedure for various $\mu$ in order to find the instability modes with the highest growth rate. In implementing this procedure, (2.5) was typically discretized using 1024 Fourier modes in the computational domain $\eta \in [-10, 10]$ and integrated using a pseudo-spectral method with fourth-order Runge–Kutta time stepping and a typical $\Delta t=0.02$. The ensuing eigenvalue problem was then solved using standard MATLAB algorithms.

3.2. Floquet stability versus near-inertial PSI

The Floquet stability results for a small-amplitude beam ($\epsilon = 0.01$) with $\omega =2f$ discussed above suggest that the instability dynamics assumed by KA17 for near-inertial PSI is valid only as long as the fine-scale perturbations have moderately large $|\boldsymbol {k}|$ (see figure 2b,e): when $\mu$ and thereby $|\boldsymbol {k}|$ is increased while keeping $\epsilon$ fixed (e.g. figure 2c,d,f), the main hypothesis of PSI breaks down since the instability dynamics is no longer dominated by the subharmonic frequencies $\pm \omega /2$. These findings are consistent with the asymptotic scaling $|\boldsymbol {k}|=O(\epsilon ^{-1/2})$ of KA17, which is clearly violated as $|\boldsymbol {k}|\rightarrow \infty$ for fixed $\epsilon$. As a result of this restriction on $|\boldsymbol {k}|$, PSI does not capture the dominant inviscid instability (largest growth rate) which arises as $|\boldsymbol {k}|\rightarrow \infty$ (figure 2a). More importantly, however, quantitative comparison between the growth rates predicted by KA17 and our Floquet analysis revealed noticeable discrepancies even when PSI is valid (e.g. for $\mu =2.4$ in figure 2b,e).

To understand and resolve these issues, we now turn to an asymptotic stability analysis of finite-width beams in the joint limit $\epsilon \ll 1$ and $|\boldsymbol {k}|\gg 1$, based on the full stability problem (2.5).

4.1. Preliminaries

Since PSI is a two-dimensional instability, the following analysis assumes from the outset perturbations that only vary in the plane of the basic state (i.e. $m=0$ in (2.4)). In this setting, rather than $u$ and $v$, it is more convenient to introduce a stream function $\psi (\xi ,\eta ,t)$, in terms of which $u=\psi _{\eta }$ and $v=-\psi _{\xi }$. Thus, the incompressibility equation (2.1a) is automatically satisfied, and $\psi$, $w$ and $\rho$ are governed by the vorticity equation

(4.1)\begin{equation} \nabla^2\psi_{t} - \cos\theta\rho_{\xi} - \sin\theta\rho_{\eta} - f\sin\theta w_{\xi} + f\cos\theta w_{\eta} + J(\nabla^2\psi, \psi) - \nu \nabla^{4}\psi = 0, \end{equation}

along with the transverse momentum and continuity equations

(4.2a)\begin{gather} w_{t} + f\sin\theta\psi_{\xi} - f\cos\theta\psi_{\eta} + J(w, \psi) - \nu \nabla^{2}w = 0, \end{gather}

(4.2b)\begin{gather}\rho_{t} + \cos\theta\psi_{\xi} + \sin\theta\psi_{\eta} + J(\rho, \psi) = 0, \end{gather}

where the Jacobian $J(a,b) = a_{\xi }b_{\eta }-a_{\eta }b_{\xi }$. Furthermore, in terms of the stream function, the basic state (2.2) that describes a uniform internal wave beam is replaced by

(4.3a)\begin{gather} \psi_{0} = \epsilon \,Q(\eta)\text{e}^{-\text{i}\omega t} + \text{c.c.}, \end{gather}

(4.3b)\begin{gather}w_{0} = \epsilon\frac{\text{i} f \cos\theta}{\omega}Q_{\eta}\text{e}^{-\text{i}\omega t} + \text{c.c.}, \end{gather}

(4.3c)\begin{gather}\rho_{0} ={-}\epsilon\frac{\text{i} \sin\theta}{\omega}Q_{\eta}\text{e}^{-\text{i}\omega t} + \text{c.c.}, \end{gather}

where $Q(\eta )$ and the beam profile $U(\eta )$ in (2.2) are related by $\epsilon Q_{\eta } = U$, and $\epsilon$ is the amplitude parameter used earlier in (3.1). For convenience, here $\epsilon$ has been made explicit in (4.3) as the focus of this section centres around small-amplitude beams.

As in (2.4), we superimpose infinitesimal perturbations to this basic state in the form of normal modes

(4.4)\begin{equation} \psi = \psi_{0} + \hat{\psi}(\eta,t)\text{e}^{{\text{i}\mu\xi}}, \end{equation}

with similar expressions for $w$ and $\rho$, where $\mu$ is a real along-beam wavenumber parameter. Furthermore, in keeping with Floquet theory, $\hat {\psi }$ is expanded as a Fourier series in $t$ with period $2{\rm \pi} /\omega$, multiplied with an exponential term that contains the Floquet exponent $\lambda$,

(4.5)\begin{equation} \hat{\psi} = \text{e}^{\lambda t}\sum_{n={-}\infty}^{\infty}\hat{\psi}^{(n)}(\eta)\text{e}^{\text{i} n \omega t}, \end{equation}

with analogous expressions for $\hat {w}$ and $\hat {\rho }$. Inserting then (4.4) and (4.5) into (4.1) and (4.2) and linearizing leads to a set of infinite homogeneous equations for the perturbation amplitudes $\hat {\psi }^{(n)}, \hat {w}^{(n)}$ and $\hat {\rho }^{(n)}$, where $n=0,{\pm }1,{\pm }2,\ldots$. This eigenvalue problem for $\lambda$, which determines stability, is of course equivalent to the one obtained earlier from (2.5) and can be tackled numerically by a similar procedure as in § 2.3.

Rather than proceeding numerically, however, here we seek an asymptotic approximation to this Floquet eigenvalue problem in the limit of a small-amplitude beam ($\epsilon \ll 1$) subject to fine-scale perturbations ($|\boldsymbol {k}|\gg 1$) in a nearly inviscid fluid ($\nu \ll 1$), the conditions under which PSI comes into play according to the numerical results in § 3.1. Specifically, as seen in figure 2(e), the spatial mode shapes $\hat {\psi }^{(n)}(\eta )$ in (4.5) take the form of wavepackets with a common carrier wavenumber $\gamma$ in the cross-beam ($\eta$) direction. Therefore, we write

(4.6)\begin{equation} \left(\hat{\psi}^{(n)},\hat{w}^{(n)},\hat{\rho}^{(n)}\right) = \left(A^{(n)}(\eta), W^{(n)}(\eta), R^{(n)}(\eta)\right) \text{e}^{\text{i} \gamma \eta}. \end{equation}

Furthermore, we express the along- and cross-beam wavenumbers as

(4.7a,b)\begin{equation} \mu ={-}k\sin\chi, \quad \gamma =k \cos\chi, \end{equation}

where

(4.8a,b)\begin{equation} k=\sqrt{\mu^{2}+\gamma^{2}}, \quad \chi=\tan^{{-}1}(-\mu/\gamma) \end{equation}

are the magnitude and inclination angle to the $\eta$ direction, respectively, of the carrier wavevector $\boldsymbol {k} = \mu \hat {\boldsymbol {e}}_{\xi }+\gamma \hat {\boldsymbol {e}}_{\eta }$ (see figure 1a). Finally, to bring out the perturbation components with frequency $\omega /2$, without loss of generality, we take

(4.9)\begin{equation} \lambda \rightarrow \lambda-\text{i}\frac{\omega}{2}. \end{equation}

As a result of the substitutions (4.6)–(4.9), the Floquet mode in (4.4) takes the form

(4.10)\begin{equation} \hat{\psi}(\eta,t)\text{e}^{\text{i}\mu\xi} = \text{e}^{\lambda t}\text{e}^{-\text{i}\omega t/2} \exp({\text{i} k(\eta\cos\chi-\xi\sin\chi )})\sum_{n={-}\infty}^{\infty}A^{(n)}(\eta) \text{e}^{\text{i} n \omega t}, \end{equation}

with similar expressions for $\hat {w}$ and $\hat {\rho }$. Furthermore, upon substituting (4.4) and (4.10) into (4.1) and (4.2), linearizing and collecting equal harmonics, it follows that the perturbation amplitudes $\tilde {A}^{(n)} = kA^{(n)}, W^{(n)}$ and $R^{(n)}$ ($n=0,\pm 1,\pm 2,\dots$) satisfy the infinite equation system

(4.11a)\begin{gather} \left(\lambda+\text{i}(n-1/2)\omega\right)\mathbb{L}\tilde{A}^{(n)} + \text{i}\left(\sin\phi R^{(n)} - f\cos\phi W^{(n)}\right) + \frac{1}{k}\left(\sin\theta R^{(n)} _{\eta} - f\cos\theta W^{(n)} _{\eta}\right)\nonumber\\ - \text{i}\sin\chi\left\{\epsilon k\left(Q_{\eta}\mathbb{L}\tilde{A}^{(n+1)} + Q^{*}_{\eta}\mathbb{L}\tilde{A}^{(n-1)}\right) + \frac{\epsilon}{k}\left(Q_{\eta\eta\eta}\tilde{A}^{(n+1)} + Q^{*}_{\eta\eta\eta}\tilde{A}^{(n-1)}\right)\right\}\nonumber\\ +\nu k^{2}\mathbb{L}^{2}\tilde{A}^{(n)} = 0, \end{gather}

(4.11b)\begin{gather} \left(\lambda+\text{i}(n-1/2)\omega\right)W^{(n)} - \text{i} f\cos\phi \tilde{A}^{(n)} - \frac{f\cos\theta}{k} \tilde{A}^{(n)} _{\eta} +\nu k^{2}\mathbb{L}W^{(n)} \nonumber\\ - \text{i}\sin\chi\left\{\epsilon k\left(Q_{\eta}W^{(n+1)} + Q^{*}_{\eta}W^{(n-1)}\right) - \frac{\text{i} \epsilon f\cos\theta}{\omega}\left(Q_{\eta\eta}\tilde{A}^{(n+1)} - Q^{*}_{\eta\eta}\tilde{A}^{(n-1)}\right)\right\} = 0, \end{gather}

(4.11c)\begin{gather} \left(\lambda+\text{i}(n-1/2)\omega\right)R^{(n)} + \text{i}\sin\phi \,\tilde{A}^{(n)} + \frac{ \sin\theta}{k} \tilde{A}^{(n)} _{\eta} \nonumber\\ - \text{i}\sin\chi\left\{\epsilon k\left(Q_{\eta}R^{(n+1)} + Q^{*}_{\eta}R^{(n-1)}\right) + \frac{\text{i} \epsilon\sin\theta}{\omega}\left(Q_{\eta\eta}\tilde{A}^{(n+1)} - Q^{*}_{\eta\eta}\tilde{A}^{(n-1)}\right)\right\} = 0, \end{gather}

where

(4.12)\begin{equation} \mathbb{L} = 1-\frac{2\text{i}\cos\chi}{k}\frac{\textrm{d}}{\textrm{d}\eta} - \frac{1}{k^{2}}\frac{\textrm{d}^2}{\textrm{d}\eta^2}, \end{equation}

and $\phi =\theta -\chi$ is the inclination of $\boldsymbol {k}$ to the vertical (see figure 1a).

4.2. PSI regime

The eigenvalue problem (4.11), while equivalent to the one discussed in § 2.3, is a convenient starting point for examining the limit $\epsilon \ll 1$ and $k\gg 1$ appropriate for PSI. Specifically, for $k\gg 1$, the amplitudes $(\tilde {A}^{(n)},W^{(n)},R^{(n)})$ of the various frequency components that comprise the Floquet mode (4.10) correspond to the envelopes of the wavepackets found numerically in figure 2(e,f). According to (4.11), these amplitudes are coupled via the terms in the curly brackets which are $O(\epsilon k)$ to leading order. Therefore, when $\epsilon k = O(1)$, all frequency components may participate equally in the instability dynamics of a small-amplitude ($\epsilon \ll 1$) beam. This accounts for the broadening of the frequency spectrum of the instability at fixed $\epsilon$ as $\mu$ (and therefore $k$) is increased (figure 2c,d).

By contrast, when

(4.13)\begin{equation} \epsilon k\ll 1 \end{equation}

in (4.11), the amplitudes of the frequency components of the Floquet mode are weakly coupled. In this instance, it is possible to asymptotically truncate the Fourier series in (4.10) so that a finite number of frequency components dominate the instability dynamics, as is the case in PSI. Thus, we now turn our attention to the joint limit

(4.14a,b)\begin{equation} \epsilon \ll 1, \quad k\gg1, \end{equation}

under the condition (4.13) and, as seen in figure 2(b), we assume that the frequencies $-\omega /2$ and $\omega /2$, which correspond to $n=0$ and $n=1$ in (4.10), dominate the instability. Then, from (4.11), the rest of the harmonics are less important

(4.15)\begin{equation} \left(\tilde{A}^{(n)},W^{(n)},R^{(n)}\right) = \begin{cases} O\left((\epsilon k)^{n-1}\right) \tilde{A}^{(1)}, & (n\geq2), \\ O\left((\epsilon k)^{|n|}\right)\tilde{A}^{(0)}, & (n\leq{-}1). \end{cases} \end{equation}

Furthermore, after eliminating $W^{(0)}$, $W^{(1)}$, $R^{(0)}$ and $R^{(1)}$, and making use of (4.13)–(4.15), the equations for the dominant $n=0$ and $n=1$ components of (4.11) can be approximated as

(4.16a)\begin{align} &\text{i}\varOmega \tilde{A}^{(0)} + \omega\left(\lambda+\nu k^{2}\beta\right)\tilde{A}^{(0)} + \frac{\omega c}{k} \tilde{A}^{(0)}_{\eta} - \text{i}\frac{3\omega^{2}}{4k^{2}}\tilde{A}^{(0)}_{\eta\eta}\nonumber\\ &\qquad + \epsilon k \text{i}\sin\chi \left\{\varOmega\frac{2}{\omega} Q_{\eta}\tilde{A}^{(1)} - Q^{*}_{\eta}\left(\sin\phi\,R^{({-}1)}-f\cos\phi W^{({-}1)} + \frac{\omega}{2}\tilde{A}^{({-}1)}\right)\right\}\nonumber\\ &\qquad -\epsilon^{2}k^{2}\text{i}\sin^{2}\chi\left(1+\frac{4\varOmega}{\omega^{2}}\right) |{Q_{\eta}}|^{2}\tilde{A}^{(0)} + \epsilon\frac{\sin\chi}{\omega}\left\{3\delta Q_{\eta\eta}\tilde{A}^{(1)} + 2\omega c Q_{\eta}\tilde{A}^{(1)}_{\eta} \right\}\nonumber\\ &\quad= O\left(\frac{\epsilon}{k},\epsilon^{2}\right), \end{align}

(4.16b)\begin{align} &-\text{i}\varOmega \tilde{A}^{(1)}+ \omega\left(\lambda+\nu k^{2}\beta\right)\tilde{A}^{(1)} - \frac{\omega c}{k} \tilde{A}^{(1)}_{\eta} + \text{i}\frac{3\omega^{2}}{4k^{2}}\tilde{A}^{(1)}_{\eta\eta}\nonumber\\ &\qquad + \epsilon k \text{i}\sin\chi \left\{\varOmega\frac{2}{\omega} Q^{*}_{\eta}\tilde{A}^{(0)} + Q_{\eta}\left(\sin\phi R^{(2)}-f\cos\phi W^{(2)} - \frac{\omega}{2}\tilde{A}^{(2)}\right)\right\}\nonumber\\ &\qquad +\epsilon^{2}k^{2}\text{i}\sin^{2}\chi\left(1+\frac{4\varOmega}{\omega^{2}}\right)|{Q_{\eta}}|^{2}\tilde{A}^{(1)} + \epsilon\frac{\sin\chi}{\omega}\left\{3\delta Q^{*}_{\eta\eta}\tilde{A}^{(0)} + 2\omega c Q^{*}_{\eta}\tilde{A}^{(0)}_{\eta} \right\}\nonumber\\ &\quad= O\left(\frac{\epsilon}{k},\epsilon^{2}k\right), \end{align}

where

(4.17a)\begin{gather} \varOmega = \sin^{2}\phi+f^{2}\cos^{2}\phi - \frac{\omega^{2}}{4}, \end{gather}

(4.17b)\begin{gather}\beta = \frac{1}{2}\left(1+\frac{4f^{2}\cos^{2}\phi}{\omega^{2}}\right), \quad \delta = \sin\phi\sin\theta + f^{2}\cos\phi\cos\theta, \end{gather}

(4.17c)\begin{gather}c = \frac{2}{\omega}\left(\delta - \frac{\omega^{2}}{4}\cos\chi\right). \end{gather}

Here, in deriving (4.16), it is assumed that

(4.18)\begin{equation} \lambda \sim \nu k^{2}\ll 1, \end{equation}

anticipating that in the weak-coupling limit (4.13) instability is weak, and including only the leading-order effects of viscosity.

The $O(1)$ terms in (4.16) drop out as the carrier wavevector $\boldsymbol {k}$ has to adhere to the dispersion relation (2.3) for the dominant frequencies $\pm \omega /2$, so $\varOmega =0$ in (4.17a). Furthermore, from (4.11), the amplitudes of the frequency components $\pm 3\omega /2$, to leading order, are given as

(4.19a)\begin{gather} \tilde{A}^{({-}1)} ={-}\epsilon k\frac{\sin\chi}{\omega}Q_{\eta}\tilde{A}^{(0)}, \quad \tilde{A}^{(2)} = \epsilon k\frac{\sin\chi}{\omega}Q^{*}_{\eta}\tilde{A}^{(1)}, \end{gather}

(4.19b)\begin{gather}\left(R^{({-}1)},R^{(2)}\right)=\frac{2\sin\phi}{\omega} \left(\tilde{A}^{({-}1)},-\tilde{A}^{(2)}\right), \end{gather}

(4.19c)\begin{gather}\left(W^{({-}1)},W^{(2)}\right)=\frac{2f\cos\phi}{\omega} \left(-\tilde{A}^{({-}1)},\tilde{A}^{(2)}\right). \end{gather}

As a result, inserting (4.19) into (4.16), we find that all interaction terms at $O(\epsilon k)$ and $O(\epsilon ^{2}k^{2})$ vanish, leaving the following closed system for $\tilde {A}^{(0)}$ and $\tilde {A}^{(1)}$:

(4.20a)\begin{align} &\left(\lambda+\nu k^{2}\beta\right)\tilde{A}^{(0)} + \frac{c}{k} \tilde{A}^{(0)}_{\eta} - \text{i}\frac{3\omega}{4k^{2}}\tilde{A}^{(0)}_{\eta\eta}\notag\\ &\qquad + \epsilon\frac{\sin\chi}{\omega^{2}}\left\{3\delta Q_{\eta\eta}\tilde{A}^{(1)} + 2\omega c \,Q_{\eta}\tilde{A}^{(1)}_{\eta} \right\}= O\left(\frac{\epsilon}{k},\epsilon^{2}k\right), \end{align}

(4.20b)\begin{align}&\left(\lambda+\nu k^{2}\beta\right)\tilde{A}^{(1)} -\frac{c}{k} \tilde{A}^{(1)}_{\eta} + \text{i}\frac{3\omega}{4k^{2}}\tilde{A}^{(1)}_{\eta\eta}\notag\\ &\qquad+ \epsilon\frac{\sin\chi}{\omega^{2}}\left\{3\delta Q^{*}_{\eta\eta}\tilde{A}^{(0)} + 2\omega c \,Q^{*}_{\eta}\tilde{A}^{(0)}_{\eta} \right\} = O\left(\frac{\epsilon}{k},\epsilon^{2}k\right). \end{align}

These two coupled equations (along with the boundary conditions $\tilde {A}^{(0)},\tilde {A}^{(1)}\rightarrow 0$ as $\eta \rightarrow \pm \infty$) define an eigenvalue problem for $\lambda$ that governs the stability of small-amplitude finite-width beams under the conditions (4.13), (4.14a,b) and (4.18), appropriate to PSI. This reduced system captures the effects of the group velocity and dispersion of the subharmonic perturbation wavepackets, as well as the leading-order effects of the interaction of these wavepackets with the underlying beam, and weak viscous dissipation.

In view of (4.13) and (4.14a,b), the $O(1/k)$ group velocity effect in (4.20) always dominates the $O(\epsilon )$ interaction of the perturbation with the beam, the only term that can cause instability. On these grounds KA14 argued that a small-amplitude finite-width beam cannot suffer PSI in general. Furthermore, from this scaling argument follow two notable special cases: (i) near-inertial PSI (considered in KA17 and revisited in § 5 below), where the group velocity $c$ in (4.17c) happens to be small; and (ii) PSI of a beam with nearly monochromatic profile (considered in KA14 and revisited in § 6 below), where the triad interaction of the subharmonic perturbations with the beam is finely tuned and can lead to instability.

Finally, turning to the full Floquet stability eigenvalue problem (4.11), a small-amplitude ($\epsilon \ll 1$) beam with general locally confined profile could be unstable to fine-scale ($k\gg 1$) perturbations with $\epsilon k =O(1)$. In this regime, while the Floquet mode frequency spectrum broadens, figure 2(a) suggests that the instability growth rate remains $O(\epsilon )$. Thus, as explained in § 7, it becomes possible to set up a separate asymptotic model for $\epsilon \ll 1$, $k\gg 1$ and $\epsilon k = O(1)$, which reveals a new, broadband instability of small-amplitude beams.

5.1. Reduced eigenvalue problem

We now revisit near-inertial PSI in the light of the small-amplitude theory of § 4 and the numerical Floquet analysis of § 3. To allow for slight deviations from the critical frequency $\omega =2f$, we write

(5.1)\begin{equation} \omega = 2f + \sigma, \end{equation}

where $|\sigma |\ll 1$ is a detuning parameter.

Here, it is important to note that when $\omega <2f$ (i.e. $\sigma <0$), perturbations at half this frequency are subinertial ($\omega /2 < f$) so in (4.16) the dispersion relation $\varOmega =0$ cannot be satisfied by a real wavevector $\boldsymbol {k}$. However, it is still possible to allow for slightly subinertial subharmonic perturbations in our asymptotic approach by inserting (5.1) into (4.11) from the onset. This results in two additional terms, $\omega \sigma \tilde {A}^{(0)}/2$ in (4.16a) and $\omega \sigma \tilde {A}^{(1)}/2$ in (4.16b), which incorporate the effect of small detuning. Since $\omega /2=f$ to leading order, with the inclusion of these additional terms it is sufficient now to take, without loss of generality, $\phi =0$ (i.e. $\boldsymbol {k}$ points vertically) so that $\varOmega =0$ and the $O(1)$ terms in (4.16) again drop out. Furthermore, for this choice of $\phi$, the group velocity terms in (4.20) can be neglected in view of (4.17). The remaining terms of (4.20) – which represent the effects of wavepacket dispersion, coupling to the underlying beam, viscous dissipation and departure from the critical frequency – may then be formally balanced by the scalings

(5.2a–d)\begin{equation} \lambda\rightarrow\epsilon\lambda, \quad \nu=\alpha\epsilon^{2}, \quad k=\frac{\kappa}{\epsilon^{1/2}}, \quad \sigma=\epsilon f \hat{\sigma}, \end{equation}

consistent with (4.14a,b) and (4.18). In this ‘distinguished limit’, making also use of (4.17), the eigenvalue problem (4.20) reduces to

(5.3a)\begin{gather} \left(\frac{\hat{\lambda}}{\omega/2}-\text{i}\frac{\hat{\sigma}}{2}\right) \tilde{A}^{(0)} - \text{i}\frac{3}{2\kappa^{2}} \tilde{A}^{(0)}_{\eta\eta} + \frac{3\sin2\theta}{4\omega} Q_{\eta\eta}\tilde{A}^{(1)} = 0, \end{gather}

(5.3b)\begin{gather}\left(\frac{\hat{\lambda}}{\omega/2}+\text{i}\frac{\hat{\sigma}}{2}\right) \tilde{A}^{(1)} + \text{i}\frac{3}{2\kappa^{2}} \tilde{A}^{(1)}_{\eta\eta} + \frac{3\sin2\theta}{4\omega} Q^{*}_{\eta\eta}\tilde{A}^{(0)} = 0, \end{gather}

where

(5.4)\begin{equation} \hat{\lambda} = \lambda + \alpha\kappa^{2}. \end{equation}

For a given beam profile $Q$, the eigenvalues $\hat {\lambda }$ depend on the parameters $\omega , \hat {\sigma }$ and $\kappa$, and instability arises when $\hat {\lambda }>\alpha \kappa ^{2}$. Accordingly, $\hat {\lambda }$ can be interpreted as the inviscid growth rate.

The stability eigenvalue problem (5.3) is similar, but not entirely equivalent, to the one obtained in KA17 for near-inertial PSI. Specifically, the present asymptotic analysis, which is based on the full Floquet stability problem (4.11), reveals the importance of the ${\pm }3\omega /2$ frequency components (4.19) in the PSI dynamics: including these disturbances cancels out the terms proportional to $|Q_{\eta }|^{2}$ in (4.16), which thus do not appear in the final stability equations (5.3). In contrast, KA17 focuses solely on the ${\pm }\omega /2$ frequency components so these ‘nonlinear refraction’ terms erroneously are present in the stability eigenvalue problem of KA17 (cf. their (28)). This resolves the discrepancy noted in § 3.2 between KA17 and the numerical Floquet analysis: the predictions of the eigenvalue problem (5.3) are in excellent agreement with Floquet numerical results for near-inertial PSI, as discussed below. Furthermore, it should be noted that KA17 assumes $\omega \geq 2f$, whereas our analysis permits the detuning parameter to be of either sign, encompassing both superinertial and subinertial perturbations. The present, more general, treatment reduces to that of KA17 by setting $\hat {\sigma }=\sigma _{{KA}}^{2}(1-f^{2})/f^{2}$ and $(\tilde {A}^{(0)}, \tilde {A}^{(1)}) \rightarrow (\tilde {A}^{(0)}, \tilde {A}^{(1)})\exp (\text {i}\kappa \sigma _{{KA}}\eta /\sin \theta )$, where $\sigma _{{KA}}$ is the detuning parameter used in KA17.

5.2. Instability growth rates

We now compare the predictions of (5.3) with those of the numerical Floquet analysis. To this end, $Q$ in (5.3) is chosen as

(5.5)\begin{equation} Q(\eta) = \frac{1}{\sqrt{8{\rm \pi}}}\int_{0}^{\infty}\text{e}^{{-}l^{2}/8}\text{e}^{\text{i} l \eta}{\rm d}{l}, \end{equation}

which corresponds to the same beam profile (3.1) used earlier (and also matches the profile used in KA17). Furthermore, as in § 3, $\omega = 0.1$ and all reported growth rates $\lambda _{r}$ will be in the scaled time $T=\epsilon t$ consistent with (5.2a–d). Numerically, the eigenvalue problem (5.3) was discretized in $\eta$ using eighth-order centred differences and the resulting matrix eigenvalue problem was solved using standard numerical eigenvalue algorithms in MATLAB for the eigenvalue $\hat {\lambda }$. The computational domain was varied between $\eta \in [-5, 5]$ and $[-100,100]$, depending on the choices of $\kappa$ and $\hat {\sigma }$, with a typical grid size of 2000 points.

Figure 3 compares the instability growth rates $\lambda _{r}$ predicted by the numerical Floquet analysis and the near-inertial PSI theory (5.3) under various flow conditions. Specifically, the PSI growth rates are plotted as a function of the perturbation wavenumber $\kappa >0$ for viscosity parameter $\alpha =0$ and $10^{-3}$, and detuning parameter $\hat {\sigma }=-4,-2,0,4,10$ and 20. The corresponding Floquet growth rates are shown for beam amplitude $\epsilon =0.002$, with $\kappa$ computed using (4.8a,b) and (5.2a–d), where $\gamma$ is taken to be the average of the carrier wavenumbers (in $\eta$) of the two subharmonic frequency components. For this $\epsilon$, the largest detuning parameter chosen here, $\hat {\sigma }=20$, corresponds to $\omega /f = 2.04$, which translates to a frequency detuning of approximately 2 %. A discussion of results for varying $\epsilon$ is presented later (see figure 5).

Figure 3.

Comparison of predicted instability growth rates (in terms of the scaled time $T=\epsilon t$) as a function of the scaled perturbation wavenumber $\kappa$ between the near-inertial PSI theory (using beam profile (5.5)) and the Floquet analysis (using beam profile (3.1) with beam amplitude $\epsilon =0.002$) for primary beam frequency $\omega =0.1$. Results are shown for viscous parameter (a,c) $\alpha =0$ corresponding to inviscid flow and (b,d) $\alpha =10^{-3}$. Panels (a,b) show results for detuning parameter $\hat {\sigma }=0$ ($\bullet$, solid line), $\hat {\sigma }=4$ ($\triangle$, dashed line), $\hat {\sigma }=10$ ($\square$, dashed-dotted line) and $\hat {\sigma }=20$ ($\lozenge$, dotted line). Panels (c,d) show results for $\hat {\sigma }=0$ ($\bullet$, solid line), $\hat {\sigma }=-2$ ($\times$, dashed line), $\hat {\sigma }=-4$ ($*$, dashed-dotted line). In all plots, shapes correspond to the Floquet growth rate while lines correspond to the near-inertial PSI growth rate.

Overall, we observe excellent quantitative agreement between the fully numerical and the asymptotic instability growth rates. In the inviscid case ($\alpha =0$) shown in figure 3(a,c), $\lambda _{r}=\hat {\lambda }_{r}$ saturates to an approximately constant value for large $\kappa$ that depends on the detuning $\hat {\sigma }$. As $|\hat {\sigma }|$ increases, i.e. as $\omega /2$ moves away from $f$, this limiting growth rate is diminished, a stabilizing effect that is stronger for $\hat {\sigma }<0$ (i.e. $\omega /2 < f$). In this instance, the computed PSI growth rates have zero imaginary part ($\lambda _{i}=0$) so in view of (4.10), subinertial subharmonic perturbations have frequency exactly equal to $\omega /2$, also in agreement with the numerical Floquet analysis. For finite viscosity, as illustrated in figure 3(b,d) for $\alpha =10^{-3}$, instability is confined to a finite range of $\kappa$. This is to be expected based on the near-inertial PSI theory: as $\kappa$ is increased, the quadratic $\alpha \kappa ^{2}$ will eventually exceed $\hat {\lambda }_{r}$, the inviscid growth rate, which is approximately constant at large $\kappa$. Here, it should be noted that $\alpha =10^{-3}$ with $\epsilon =0.002$ sets the inverse Reynolds number $\nu =O(10^{-9})$ and corresponds to nearly inviscid conditions, as would be found in oceans.

Figure 4 plots the maximum instability growth rate, taken over $\kappa$, as a function of the detuning $-6\leq \hat {\sigma }\leq 20$. Again, there is excellent agreement between the near-inertial PSI theory and the Floquet analysis. As illustrated in figure 4, instability is completely suppressed regardless of viscous dissipation when $\omega /2$ is sufficiently subinertial ($\hat {\sigma }\lesssim -5$). For inviscid flow conditions (figure 4a), the maximum instability growth rate, which is found as $\kappa \rightarrow \infty$ for a given detuning $\hat {\sigma }$, shows very weak dependence on $\hat {\sigma }$ for $\hat {\sigma }\geq 0$. Under viscous flow conditions ($\alpha =10^{-3}$), shown in figure 4(b), the maximum growth rate decreases for large $\hat {\sigma }$, indicating that instability is weakened far enough from near-inertial conditions. Interestingly, though, for finite viscosity, instability is strongest at small but finite $\hat {\sigma }>0$, rather than exactly at the critical $\omega =2f$ ($\hat {\sigma }=0$). This is due to the behaviour of the growth rates near the onset of instability: as illustrated in figure 3(a,b), the growth rates for $\hat {\sigma }=4$ arise at slightly lower $\kappa$ and have larger values in the range $1\lesssim \kappa \lesssim 5$ than the growth rates for $\hat {\sigma }=0$.

Figure 4.

Maximum instability growth rate as function of detuning $-6\leq \hat {\sigma }\leq 20$ for the same primary beam as in figure 3. (a) Inviscid flow conditions ($\alpha =0$). In this case, since the maximum growth rate for a given $\hat {\sigma }$ occurs as $\kappa \rightarrow \infty$ (see figure 3a,c), results are presented for the near-inertial PSI growth rate at $\kappa =12$ (solid line) and $\kappa =100$ (dotted line). Floquet results ($\circ$) are shown at $\kappa \approx 12$. (b) Viscous flow conditions ($\alpha =10^{-3}$). In this case, since the maximum growth rate is achieved at finite $\kappa$ (see § 5.2), plotted growth rates correspond to the maximum over all $\kappa$. Solid line is the near-inertial PSI growth rate, while ($\circ$) is the Floquet growth rate.

Finally, figure 5 compares the Floquet and PSI growth rates under inviscid flow conditions ($\alpha =0$) at three different values of the detuning $\hat {\sigma }=-4, 0$ and 20 for the beam amplitudes $\epsilon =0.002, 0.01, 0.05$ and 0.1. For $\hat {\sigma }=0$, the Floquet growth rates for the larger $\epsilon =0.2$ are also shown. As expected, the best agreement between the PSI theory and the Floquet analysis is found when the assumptions $\epsilon \ll 1$, $\epsilon k\ll 1$ ($\kappa \lesssim \epsilon ^{-1/2}$) and $|\hat {\sigma }|=O(1)$ made in the PSI theory are well satisfied. It should be noted that typical values for oceanic internal waves, with characteristic velocity $U_{*}\approx 0.1\ \textrm {m}\ \textrm {s}^{-1}$, characteristic beam width $L_{*}\approx 500\ \textrm {m}$, and buoyancy frequency $N_{*}\approx 2\times 10^{-3}\ \textrm {s}^{-1}$ (see e.g. Gerkema, Lam & Maas Reference Gerkema, Lam and Maas2004), result in the dimensionless amplitude $\epsilon \approx 0.1$.

Figure 5.

Inviscid instability growth rate as a function of the scaled perturbation wavenumber $\kappa$ predicted by the Floquet analysis for beam amplitude $\epsilon =0.002$ ($\circ$), 0.01 ($\triangle$), 0.05 ($\square$), 0.1 ($\lozenge$). The same primary beam profile and frequency as in figures 3 and 4 are used here. Plots correspond to detuning parameter $\hat {\sigma }=-4$ (a), $\hat {\sigma }=0$ (b) and $\hat {\sigma }=20$ (c). The results for $\hat {\sigma }=0$ in panel (b) include Floquet growth rates for $\epsilon =0.2$ ($*$). Solid lines correspond to the near-inertial PSI growth rate.

5.3. Instability dynamics

The results presented above indicate that the instability growth rates predicted by the numerical Floquet analysis agree well with the near-inertial PSI theory. However, as illustrated in figure 2 and discussed in § 4.2, the main assumption of PSI theory, namely that the frequency components at ${\pm }\omega /2$ dominate, eventually breaks down as $\kappa$ is increased with $\epsilon$ fixed. More precisely, the PSI theory fails when $\kappa = O(\epsilon ^{-1/2})$, in keeping with (4.13) and (5.2a–d), and unstable Floquet modes contain broadband frequency spectra.

To confirm these theoretical predictions, we return to the instability growth rate curve in figure 2(a). When plotted in terms of the scaled wavenumber magnitude $\kappa$ rather than the along-beam wavenumber $\mu$, this curve corresponds to the growth rate curve in figure 5(b) for $\hat {\sigma }=0$ and $\epsilon =0.01$ and is nearly indistinguishable from the predictions of the PSI theory. However, despite this agreement, the time-frequency spectra in figure 2(b–d), which correspond to $\kappa \approx 3,10$ and 20, respectively, indicate that the instability dynamics for $\kappa \gtrsim 10$ does not resemble PSI at all, consistent with the estimate $\kappa \approx \epsilon ^{-1/2}=10$ for the breakdown of the PSI assumption. Furthermore, figure 6 compares the time-frequency spectra of the most unstable inviscid Floquet modes at $\kappa =10$ for beam amplitudes $\epsilon =0.002$ and 0.05 and detuning $\hat {\sigma }=-4,0$ and 20. Since these Floquet modes exactly correspond to the growth rates shown in figure 5 at $\kappa =10$, the Floquet growth rate is in excellent agreement with the PSI theory for both beam amplitudes. However, for the larger amplitude $\epsilon =0.05$ (figure 6a–c), the frequency spectra indicate that the instability dynamics does not correspond to PSI, consistent with $\epsilon k\approx 2$ not being small.

Figure 6.

Time-frequency spectra of the fastest-growing Floquet mode found at scaled perturbation wavenumber $\kappa =10$ for the same primary beam profile and frequency used throughout § 5. Panels (a–c) show modes for beam amplitude $\epsilon =0.05$ while panels (d–f) show modes for $\epsilon =0.002$. Panels (a,d), (b,e) and (c,f) correspond to detuning parameter $\hat {\sigma }=-4, 0$ and 20, respectively. The vertical dotted lines indicate the frequencies ${\pm }f/\omega$.

In summary, when $\epsilon k\ll 1$ the near-inertial PSI theory is in excellent agreement with the numerical Floquet analysis. When $\epsilon k$ is not small, instability is not dominated by the subharmonic frequencies at ${\pm }\omega /2$, consistent with the conclusion reached in § 4.2. Remarkably, however, PSI theory, even though it is not formally valid in this regime, still provides reliable predictions for the instability growth rate. A similar observation was made by Sonmor & Klaassen (Reference Sonmor and Klaassen1997) for the instability of a sinusoidal wave (see their § 4a). The reason for the wider than expected validity of PSI theory is discussed in § 7.

6.1. Reduced eigenvalue problem

Aside from near-inertial conditions, PSI is also possible for small-amplitude locally confined beams with a nearly monochromatic profile, as pointed out in KA14. This study assumed no background rotation ($f=0$), so the beam frequency obeys $\omega =\sin \theta$ according to (2.3), and omitted the contribution of the ${\pm }3\omega /2$ frequency components to PSI dynamics. However, unlike near-inertial PSI, these frequencies do not affect the stability eigenvalue problem derived in KA14.

Here, we compare the predictions of PSI theory with those of Floquet stability analysis for a nearly monochromatic beam under the conditions assumed in KA14. To this end, rather than quoting KA14, it is more economical to derive the PSI eigenvalue problem directly from the small-amplitude limit of the Floquet stability problem discussed in § 4. Specifically, we return to (4.20) and consider a beam with profile

(6.1a,b)\begin{equation} Q(\eta) = q(H)\text{e}^{\text{i}\eta}, \quad H=\hat{\epsilon}\eta. \end{equation}

Here, $H$ is a stretched envelope coordinate and $\hat {\epsilon }\ll 1$ is a small parameter (made precise in (6.4a–d) below) that controls the width of the slowly varying envelope $q(H)$ relative to the characteristic length scale $L_{*}=\varLambda _{*}/2{\rm \pi}$, where $\varLambda _{*}$ is the carrier wavelength of the primary wave beam. It should be noted that under the present conditions, the dispersion relation $\varOmega =0$ in (4.16) is exactly satisfied by choosing $\sin \phi =\omega /2$.

In view of (6.1a,b), the perturbation amplitudes $\tilde {A}^{(0,1)}$ take the form

(6.2a,b)\begin{equation} \tilde{A}^{(0)}(\eta)\rightarrow\tilde{A}^{(0)}(H)\text{e}^{\text{i}({\eta}/{2})}, \quad \tilde{A}^{(1)}(\eta)\rightarrow\tilde{A}^{(1)}(H) \text{e}^{-\text{i}({\eta}/{2})}, \end{equation}

and inserting (6.1a,b) and (6.2a,b) into the stability equations (4.20), we find at leading order

(6.3a)\begin{gather} \left(\lambda+ \frac{1}{2}\nu k^{2} + \text{i}\frac{c}{2k}+\text{i}\frac{3\omega}{16k^{2}}\right)\tilde{A}^{(0)} + \hat{\epsilon}\frac{c}{k} \tilde{A}^{(0)}_{H} - \epsilon\sin\chi\cos^{2}\left(\frac{\chi}{2}\right) q\tilde{A}^{(1)} = 0, \end{gather}

(6.3b)\begin{gather}\left(\lambda+ \frac{1}{2}\nu k^{2} + \text{i}\frac{c}{2k}-\text{i}\frac{3\omega}{16k^{2}}\right)\tilde{A}^{(1)} - \hat{\epsilon}\frac{c}{k} \tilde{A}^{(1)}_{H} - \epsilon\sin\chi\cos^{2}\left(\frac{\chi}{2}\right)q^{*}\tilde{A}^{(0)} = 0. \end{gather}

The various terms in (6.3) may then be formally balanced in a similar fashion to the near-inertial PSI analysis (cf. (5.2a–d)), by the scalings

(6.4a–d)\begin{equation} \lambda\rightarrow\epsilon\lambda, \quad \nu=2\alpha\epsilon^{2}, \quad k = \frac{\kappa}{\epsilon^{1/2}}, \quad \hat{\epsilon}=\frac{\epsilon^{1/2}}{D}. \end{equation}

Here,

(6.5)\begin{equation} D=2{\rm \pi} N \epsilon^{1/2}=O(1) \end{equation}

is a scaled width of the beam envelope in terms of $N$, the number of carrier wavelengths contained within the beam. Inserting (6.4a–d) into (6.3) and making the substitutions $\tilde {A}^{(0,1)}\rightarrow \tilde {A}^{(0,1)}\exp [-3\text {i}\omega H/(16c\hat {\epsilon }k)]$ and $\lambda \rightarrow \lambda -\text {i} c/(2 k)$, which do not affect stability, we find that the PSI of nearly monochromatic beams is governed by

(6.6a)\begin{gather} \hat{\lambda}\tilde{A}^{(0)} + \frac{c}{D\kappa} \tilde{A}^{(0)}_{H} - \sin\chi\cos^{2}\left(\frac{\chi}{2}\right) q\tilde{A}^{(1)} = 0, \end{gather}

(6.6b)\begin{gather}\hat{\lambda}\tilde{A}^{(1)} - \frac{c}{D\kappa} \tilde{A}^{(1)}_{H} - \sin\chi\cos^{2}\left(\frac{\chi}{2}\right)q^{*}\tilde{A}^{(0)} = 0, \end{gather}

where

(6.7)\begin{equation} \hat{\lambda} = \lambda+\alpha\kappa^{2}. \end{equation}

As noted earlier, the final stability eigenvalue problem (6.6) is identical to that found by KA14 (see their (4.10)), as the frequency components at ${\pm }3\omega /2$, omitted in KA14, ultimately do not play a role in the PSI stability analysis of nearly monochromatic beams.

6.2. Comparison with Floquet analysis

We now compare the Floquet stability analysis of a nearly monochromatic beam with the predictions of the PSI theory (6.6). Specifically, we chose the Gaussian envelope profile

(6.8)\begin{equation} q(H) = \tfrac{1}{2}\exp({-}8H^{2}). \end{equation}

Combined with (6.1a,b), (6.4a–d) and (6.5), (6.8) corresponds to the overall beam profile

(6.9)\begin{equation} U(\eta) = \text{i}\frac{\epsilon}{2} \exp \left(-\frac{2\eta^{2}}{{\rm \pi}^{2}N^{2}} \right) \exp(\text{i} \eta) \end{equation}

in (2.2) for the Floquet analysis. Figure 7 plots the beam profile (6.9) for $N=1$ and 7. We recall that, according to (6.5), $N=O(\epsilon ^{-1/2})$ in the PSI theory. Numerically, the eigenvalue problem (6.6) was discretized in the domain $H\in [-5, 5]$ using eighth-order centred differences with a grid size of 500 points, and the resulting matrix eigenvalue problem was solved using standard numerical eigenvalue algorithms in MATLAB for the eigenvalue $\hat {\lambda }$. The discussion below will focus on the beam inclination angle $\theta =45^{\circ }$ (which is tied to the beam frequency via the dispersion relation $\omega =\sin \theta$) and inverse Reynolds number $\nu =10^{-5}$. The effect of varying $\theta$ is discussed in Fan (Reference Fan2020, § 3.6.3).

Figure 7.

Beam profile (6.9) for non-dimensional amplitude $\epsilon =1$ and (a) $N=1$ and (b) $N=7$, where $N$ corresponds to the number of carrier wavelengths contained within the beam width. Solid lines indicate the real part, while dotted lines indicate the imaginary part.

Figure 8(a) plots the maximum instability growth rates (taken over $\kappa$ and given in terms of the slow time $T=\epsilon t$) predicted by the PSI theory and the Floquet analysis as a function of $N$, the number of carrier wavelengths of the beam profile, for various values of $0.025\leq \epsilon \leq 0.4$. As expected, in the limit of $N\rightarrow \infty$, the PSI growth rates converge onto the sinusoidal wave growth rate. Furthermore, the Floquet growth rates show excellent agreement with the PSI theory for all $\epsilon \leq 0.4$ and $N\geq 2$, even though in these parameter ranges neither the assumption of small amplitude (i.e. $\epsilon \ll 1$) nor the assumption of scale separation between the carrier and the envelope (i.e. $N\gg 1$) are necessarily well satisfied.

Figure 8.

(a) Computed instability growth rates (in terms of the scaled time $T=\epsilon t$) according to the Floquet stability analysis as a function of $N$, the number of carrier wavelengths contained in the beam width, for the profile (6.9) with viscosity $\nu =10^{-5}$, beam frequency $\omega =\sin 45^{\circ }$ and beam amplitudes $\epsilon =0.025$ ($\triangledown$), 0.05 ($\circ$), 0.1 ($\lozenge$), 0.2 ($\square$) and 0.4 ($\triangle$). The corresponding asymptotic PSI growth rates, as predicted by (6.6), are overlaid and labelled. The PSI growth rate for a sinusoidal wave (i.e. $N\rightarrow \infty$) is indicated by the horizontal dashed line. (b) Minimum amplitude $\epsilon _{c}$ for instability as a function of $N$ for $\nu =5\times 10^{-4}$ ($\triangleright$, dotted line), $10^{-3}$ ($*$, dashed line) and $2\times 10^{-3}$ ($\triangleleft$, solid line). Shapes correspond to Floquet analysis results while lines correspond to the amplitude threshold (6.10) predicted by the PSI theory.

Furthermore, based on PSI theory, KA14 proposed a lower bound on $N$, the number of carrier wavelengths contained in the beam envelope, for PSI to arise (see their (5.14)). By rearranging their expression for this lower bound, which depends on various parameters including the inverse Reynolds number $\nu$ and beam amplitude $\epsilon$, we deduce a minimum amplitude threshold for instability, at fixed $N$,

(6.10)\begin{equation} \epsilon_{c} = \left(\frac{1}{2\mathbb{C}_{c}}\right)^{1/3}\frac{c^{2/3}}{\varGamma}\frac{\nu^{1/3}}{(2{\rm \pi} N)^{2/3}}. \end{equation}

Here, $\mathbb {C}_{c} \approx 3.35 \times 10^{-3}$ is a coefficient set by the choice of envelope profile (6.8), $c$ is defined in (4.17c) and $\varGamma =\sin \chi \cos ^{2}(\chi /2)$. Figure 8(b) compares (6.10) with the minimum amplitude threshold found numerically using the Floquet analysis for various $2\leq N \leq 10$ and for three different viscosities $\nu = 5\times 10^{-4}, 10^{-3}$ and $2\times 10^{-3}$. We observe excellent agreement between (6.10) and the numerical results.

Figure 9 plots the time-frequency spectra of the Floquet modes corresponding to the growth rates shown in figure 8(a) for $\epsilon =0.05$ and 0.2, and for $N=2$, 10 and 20. Similar to the results found in § 5.3 for near-inertial conditions, the frequency spectra widen as the beam amplitude $\epsilon$ is increased, corresponding to the emergence of multiple instability frequencies other than ${\pm }\omega /2$. When this happens, again we find that PSI theory, although it is no longer strictly valid, still provides accurate predictions of the instability growth rates.

Figure 9.

Time-frequency spectra of the fastest-growing Floquet mode for various configurations shown in figure 8(a). Panels (a–c) correspond to beam amplitude $\epsilon =0.2$ and 0.05, respectively. Panels (a,d), (b,e) and (c,f) correspond to $N=2$, 10 and 20, respectively. Vertical dotted lines in each plot correspond to the frequencies ${\pm }\omega /2$. The wavenumber $\kappa$ for each mode is displayed.

Finally, it is worth noting that, according to (6.10), $\epsilon _{c}\rightarrow 0$ in the inviscid limit ($\nu =0$) regardless of $N$. This hints that small-amplitude ($\epsilon \ll 1$) locally confined beams with general (not necessarily nearly monochromatic) profile are possibly susceptible to instability. This point is discussed below.

7.1. Revised Floquet analysis

Returning to the Floquet analysis in § 4, we adopt the frame riding with the primary wave

(7.1)\begin{equation} \xi' = \xi - \int^{t} \psi_{0\eta}{\rm d}{t'}, \end{equation}

where $\psi _{0}=O(\epsilon )$ is given by (4.3a). This key step (suggested by an anonymous referee) makes it possible to factor out the (along-beam) advection of the Floquet mode due to the underlying wave and thereby eliminate the $O(\epsilon k)$ terms in (4.11). Specifically, in view of (7.1), the Floquet mode in (4.4) and (4.10) is replaced by

(7.2)\begin{equation} \hat{\psi}'(\eta,t)\text{e}^{\text{i}\mu\xi'} = \text{e}^{\lambda t} \text{e}^{-\text{i}\omega t /2} \exp({\text{i} k (\eta \cos\chi - \xi'\sin\chi)})\sum_{n={-}\infty}^{\infty}A'^{(n)}(\eta) \text{e}^{\text{i} n \omega t}, \end{equation}

with similar expressions for $\hat {w}'$ and $\hat {\rho }'$. Substituting these revised modes into the linearized form of (4.1) and (4.2) governing the perturbations, and collecting equal harmonics leads to an infinite equation system, analogous to (4.11), for the amplitudes $\tilde {A}'^{(n)}=kA'^{(n)}$, $W'^{(n)}$ and $R'^{(n)}$ ($n=0,\pm 1,\pm 2, \dots$). Unlike (4.11), however, the coupling terms due to the interaction with the primary wave are now $O(\epsilon )$ instead of $O(\epsilon k)$. Thus, for $\epsilon \ll 1$, $k\gg 1$ and $\epsilon k =O(1)$, it is possible to asymptotically truncate the Fourier series in (7.2) and obtain a reduced stability eigenvalue problem for the dominant perturbation amplitudes.

In this new eigenvalue problem, the $O(\epsilon )$ instability growth rate derives from the $O(\epsilon )$ coupling of the perturbation with the primary wave, which now may be balanced with the $O(1/k)$ group velocity effect and the $O(\nu k^{2})$ viscous dissipation. This ‘distinguished limit’ is achieved by the scalings

(7.3a–c)\begin{equation} \lambda \rightarrow \epsilon \lambda, \quad \nu = \alpha' \epsilon^{3}, \quad k = \frac{\kappa'}{\epsilon}. \end{equation}

Thus, compared with near-inertial PSI (cf. (5.2a–d)), here the growth rate is of the same order, but the perturbation length scale is shorter and the Reynolds number much higher. Furthermore, although the Fourier series in (7.2) may be truncated, the Floquet mode as a whole features broadband frequency spectrum due to the factor $\exp (-\text {i} k \xi ' \sin \chi )$, which in view of (7.1) comprises all harmonics when $\epsilon k = O(1)$.

7.2. Reduced eigenvalue problem

We now sketch the derivation of the asymptotic stability eigenvalue problem appropriate to the limit $\epsilon \ll 1$, $k\gg 1$ and $\epsilon k =O(1)$. Upon implementing (7.3a–c), the equation system governing the Fourier amplitudes $\tilde {A}'^{(n)}$, $W'^{(n)}$ and $R'^{(n)}$, correct to $O(\epsilon )$, can be reduced to the following system for $\tilde {A}'^{(n)}$ ($n=0,{\pm }1,{\pm }2,\ldots$):

(7.4)\begin{align} &\text{i}(\varOmega-n(n-1)\omega^{2})\tilde{A}'^{(n)}\notag\\ &\quad- \epsilon\left(\lambda\frac{\varOmega+\left(n^{2}-n+\frac{1}{2}\right)\omega^{2}}{\left(n-\frac{1}{2}\right)\omega} - \alpha'\kappa'^{2}\frac{f^{2}\cos^{2}\phi + \left(n-\frac{1}{2}\right)^{2}\omega^{2}}{\left(n-\frac{1}{2}\right)\omega} \right)\tilde{A}'^{(n)}\nonumber\\ &\quad +2\frac{\epsilon}{\kappa'}\left(\delta - \omega^{2}\left(n-1/2\right)^{2}\cos\chi\right) \frac{\textrm{d}\tilde{A}'^{(n)}}{\textrm{d}\eta}\notag\\ &\quad -\epsilon \left(n-1/2\right)\omega\sin2\chi\left(Q_{\eta\eta}\tilde{A}'^{(n+1)} + Q^{*}_{\eta\eta}\tilde{A}'^{(n-1)}\right)\nonumber\\ &\quad +\epsilon\frac{n-\frac{1}{2}}{\omega}\sin\chi\left(\sin(\phi-\chi)\sin\phi+f^{2}\cos(\phi-\chi)\cos\phi\right) \notag\\ &\quad \times\left(\frac{Q_{\eta\eta}\tilde{A}'^{(n+1)}}{n+\frac{1}{2}} - \frac{Q^{*}_{\eta\eta}\tilde{A}'^{(n-1)}}{n-\frac{3}{2}}\right) = O(\epsilon^{2}), \end{align}

where $\varOmega$ and $\delta$ are given in (4.17a) and (4.17b). Then, setting $\varOmega =0$ makes the $O(1)$ terms in (7.4) for $n=0,1$ drop out, and the dominant Fourier amplitudes $\tilde {A}'^{(0)}$ and $\tilde {A}'^{(1)}$ satisfy the closed system

(7.5a)\begin{gather} \hat{\lambda}\tilde{A}'^{(0)} + \frac{c}{\kappa'}\frac{\textrm{d}\tilde{A}'^{(0)}}{\textrm{d}\eta} + \frac{\sin\chi}{\omega^{2}}\left(\delta+\frac{\omega^{2}}{2}\cos\chi\right)Q_{\eta\eta}\tilde{A}'^{(1)} = 0, \end{gather}

(7.5b)\begin{gather}\hat{\lambda}\tilde{A}'^{(1)} - \frac{c}{\kappa'}\frac{\textrm{d}\tilde{A}'^{(1)}}{\textrm{d}\eta} + \frac{\sin\chi}{\omega^{2}}\left(\delta+\frac{\omega^{2}}{2}\cos\chi\right)Q^{*}_{\eta\eta}\tilde{A}'^{(0)} = 0, \end{gather}

where

(7.6)\begin{equation} \hat{\lambda} = \lambda + \alpha'\kappa'^{2}\beta, \end{equation}

with $\beta$ and $c$ given in (4.17b) and (4.17c). Equations (7.5) (along with the boundary conditions $\tilde {A}'^{(0)},\tilde {A}'^{(1)}\rightarrow 0$ as $\eta \rightarrow \pm \infty$) define an eigenvalue problem for $\hat {\lambda } = \hat {\lambda }_{r} + \text {i}\hat {\lambda }_{i}$. Here, in view of (7.6), $\hat {\lambda }_{r}$ may be interpreted as the inviscid instability growth rate whereas $\lambda _{r} = \hat {\lambda }_{r} - \alpha '\kappa '^{2}\beta$ accounts for the effect of viscous dissipation.

As remarked earlier, the instability described by (7.5) is distinct from PSI because the Floquet mode (7.2) features a broadband frequency spectrum and the scaling $\nu \propto \epsilon ^{3}$ appropriate here requires larger Reynolds number than PSI, where $\nu \propto \epsilon ^{2}$ according to (5.2a–d) and (6.4a–d). Therefore, as discussed in § 7.3, this broadband, essentially inviscid instability may affect beams that are not generally susceptible to PSI. Furthermore, the same type of instability replaces PSI under inviscid flow conditions in the limit where the wavenumber of subharmonic perturbations is large enough so that $\kappa '=\epsilon k =O(1)$. For instance, at inertial conditions $\omega =2f$, where according to (4.17) $\phi =0$, $\delta = f^{2}\cos \theta$, $\beta =1$ and $c=0$, (7.5) reduces to

(7.7a)\begin{gather} 2\hat{\lambda}\tilde{A}'^{(0)} + \tfrac{3}{4}\sin2\theta Q_{\eta\eta}\tilde{A}'^{(1)} = 0, \end{gather}

(7.7b)\begin{gather}2\hat{\lambda}\tilde{A}'^{(1)} + \tfrac{3}{4}\sin2\theta Q^{*}_{\eta\eta}\tilde{A}'^{(0)} = 0, \end{gather}

which agrees with the limiting form, when $\kappa =\epsilon ^{1/2}k=\kappa '/\epsilon ^{1/2} \gg 1$, of the near-inertial PSI stability problem (5.3) at the critical frequency $\omega =2f$ ($\hat {\sigma }=0$). This explains why (5.3), which was derived under the assumption $\epsilon k \ll 1$, provides reliable predictions of instability growth rates even when $\epsilon k = O(1)$ and the Floquet mode is no longer dominated by the subharmonics at $\pm \omega /2$ (see figures 3, 5 and 6).