3.1. Evolution of the wave amplitude in a flow with constant shear

Figure 2(a) shows the zonal velocity perturbation evolution of an inertial wavepacket with central frequency ${\it\omega}_{c}=0.10N_{0}$ and spectral width ${\it\sigma}_{{\it\omega}}=0.15{\it\omega}_{c}$ that propagates in a background flow with $\mathit{Ri}_{0}=100$ and $\tilde{f}=f/{\it\omega}_{c}=0.05$ . The reference height is taken as $z_{i}=0$ ; this is the height at which the wavepacket is launched initially. The height of the central critical level of an internal gravity wavepacket is given by $z_{cl}={\it\omega}_{c}/(kd_{z}U)$ . In what follows the vertical coordinate is non-dimensionalized as $\tilde{z}=z/z_{cl}$ , so that the critical level height of the internal gravity wavepacket is $\tilde{z}_{cl}=1$ . The lowest central inertial critical level will be at a height slightly lower than $z_{cl}$ , $({\it\omega}_{c}-f)/(kd_{z}U)=0.95z_{cl}$ . The critical region, where each (non-negligible) spectral component (2.12) has its own critical level, goes from $z=({\it\omega}_{c}-{\it\sigma}_{{\it\omega}}-f)/(kd_{z}U)=0.8z_{cl}$ up to $z=({\it\omega}_{c}+{\it\sigma}_{{\it\omega}}-f)/(k\text{d}_{z}U)=1.1z_{cl}$ .

Figure 2.

(a) Zonal velocity perturbation evolution of an inertial gravity wavepacket. Vertical straight lines represent ${\it\omega}_{c}t=0$ (dashed), ${\it\omega}_{c}t=21.6$ (solid) and ${\it\omega}_{c}t=144$ (dotted). (b) Vertical profiles of the zonal perturbation at ${\it\omega}_{c}t=0$ (dashed), ${\it\omega}_{c}t=21.6$ (solid) and ${\it\omega}_{c}t=144$ (dotted). The wavepacket width is ${\it\sigma}=0.1{\it\omega}_{c}$ ; the background flow is characterized by $\mathit{Ri}_{0}=100$ .

The packet propagates upwards with large vertical group velocity at short times; then the vertical group velocity goes to zero as the packet gets closer to the central critical level. As the wavepacket propagates towards the central critical level, the central vertical wavenumber of the packet increases because of Doppler shifting. The vertical width of the wavepacket is broad at the initial time, then reaches a minimum at about ${\it\omega}_{c}t=100.8$ and finally goes to an asymptotic constant value for long times ( ${\it\omega}_{c}t\gg 100$ ). The amplitude of the wavepacket increases at short times, it achieves a maximum and then at long times it decreases asymptotically towards a constant (see also the evolution of the wavepacket envelope amplitude in figure 3 a). The evolution of the wavepacket envelope amplitude is discussed below. Figure 2(b) shows the vertical profile of the zonal velocity perturbation at the initial time, at the time of maximum amplitude ( ${\it\omega}_{c}t=21.6$ ) and when the packet is close to the asymptotic behaviour ( ${\it\omega}_{c}t=144$ ). The amplitude, width and vertical wavenumber dependences already mentioned are clearly identified in these profiles. Note that, as the packet gets closer to the critical level, the envelope is composed by several vertical wavelengths, which is the assumption of both ray tracing and Gaussian beam approximations.

Figure 3.

Amplitude evolution of an inertial wavepacket. (a) Wavepacket widths of ${\it\sigma}=0.1{\it\omega}_{c}$ and ${\it\sigma}=0.17{\it\omega}_{c}$ and $\mathit{Ri}_{0}=100$ . (b) Background flow characterized by $\mathit{Ri}_{0}=100$ and $\mathit{Ri}_{0}=400$ and a wavepacket width of ${\it\sigma}=0.1{\it\omega}_{c}$ . For each case, three curves are shown: Gaussian beam approximation for inertial waves (solid), Gaussian beam approximation for internal waves (dashed) and the numerical integration for inertial waves (triangles). The asymptotic values for inertial waves are shown with dotted straight lines.

The analytical calculations for the Gaussian beam approximation are conducted for a background flow with constant vertical shear, $\text{d}u_{0}/\text{d}z={\it\Gamma}$ . The evolution of the maximum amplitude of the wave envelope under the Gaussian beam approximation in a flow with constant shear using (2.22) is

(3.1) $$\begin{eqnarray}a(t)={\hat{u}}_{{\it\omega}}\frac{(1+{\it\gamma}_{c}\mathit{Ri}_{0}^{-1/2}t)^{1/2}[{\it\gamma}_{c}^{2}+f^{2}(1+{\it\gamma}_{c}\mathit{Ri}_{0}^{-1/2}t)^{2}]^{1/2}}{{\it\omega}_{c}\left\{1+{\it\sigma}_{{\it\omega}}^{4}\,\displaystyle \frac{\mathit{Ri}_{0}}{{\it\gamma}_{c}^{6}}[{\it\omega}_{c}-({\it\gamma}_{c}^{2}+f^{2}(1+{\it\gamma}_{c}\mathit{Ri}_{0}^{-1/2}t)^{2})^{1/2}(1+{\it\gamma}_{c}\mathit{Ri}_{0}^{-1/2}t)^{2}]^{2}\right\}^{1/4}},\end{eqnarray}$$

where ${\it\gamma}_{c}^{2}={\it\omega}_{c}^{2}-f^{2}$ . Equation (3.1) collapses for $f=0$ to the amplitude evolution found for internal gravity waves in Pulido & Rodas (Reference Pulido and Rodas2008).

Figure 3(a) shows the evolution of the maximum envelope amplitude of a transient wavepacket under the Gaussian beam approximation (continuous line) that propagates in a background flow with $\mathit{Ri}_{0}=100$ and $\tilde{f}=f/{\it\omega}_{c}=0.05$ . The evolution of two waves with different spectral widths, ${\it\sigma}=0.1{\it\omega}_{c}$ and ${\it\sigma}=0.17{\it\omega}_{c}$ , is shown. In both cases, the amplitude increases for short times up to when it achieves a maximum and then it goes asymptotically to a constant value. The amplitude of the wave field depends on two factors (see (2.22)). One factor is the two terms that depend explicitly on the frequency in (2.22); these terms represent the changes in the amplitude because of refraction. The other factor is the last term in (2.22), which could be interpreted as the inverse of the square root of the vertical width of the packet, and so it depends on dispersion effects. The refraction terms are in the numerator in (3.1), while the dispersion effects are represented in the denominator of (3.1). At short times, the vertical width of the packet reaches a minimum. The wavepacket compacts vertically because the highest components of the wavepacket find a stronger background flow so that their vertical group velocity diminishes with respect to the components that are in the lower part of the envelope. At the same time, the intrinsic frequency decreases as the wavepacket propagates upwards in short times. These two effects produce the maximum amplitude at short times (see figure 3 a). Then, the components of the wavepacket are dispersed and so the amplitude starts to decrease. The asymptotic tendency for long times of the refraction terms (numerator of (3.1)) and the denominator of (3.1) are equal, so that the amplitude decreases asymptotically to a constant value.

From figure 3(a), the narrower the frequency spectrum, the larger the wave amplitude maximum. This is coherent with a monochromatic inertio-gravity wave that has an unbounded amplitude at the critical level. The amplitude during the decay period is also larger for a narrower frequency spectrum. Figure 3(a) also compares the behaviour of the inertio-gravity wave with the corresponding internal wave (dashed lines) for each case. There are practically no differences in the amplitude during the growth period. However, inertial effects in gravity waves during the decay period produce larger amplitudes than without them. Indeed, the amplitude for inertio-gravity waves tends to a constant value for long times, while the amplitude of internal waves tends to zero. The asymptotic behaviour for the amplitude of inertio-gravity waves is

(3.2) $$\begin{eqnarray}a_{t\rightarrow \infty }={\hat{u}}_{{\it\omega}}(1-\tilde{f}^{2})^{3/4}\tilde{f}^{1/2}{\it\beta}^{-1},\end{eqnarray}$$

where ${\it\beta}=\mathit{Ri}_{0}^{1/4}{\it\sigma}_{{\it\omega}}{\it\omega}_{c}^{-1}$ . This non-dimensional parameter ${\it\beta}$ is directly proportional to the frequency width of the wavepacket, so that it represents a measure of how transient is the disturbance, i.e. highly transient waves are characterized by a large  ${\it\beta}$ . This constant asymptotic amplitude (3.2) for long times is also obtained with spatial ray tracing. On the other hand, the amplitude of an internal gravity wave that propagates towards the critical level decays as $t^{-1/2}$ . Therefore, during the decay stage of transient waves in a rotating frame, the inertial effects must always be considered to represent properly the wave field for any initial condition.

If the transient inertio-gravity wavepacket propagates in a background flow without shear, the evolution of the amplitude is given by (3.1) for $\mathit{Ri}_{0}\rightarrow \infty$ , which results in

(3.3) $$\begin{eqnarray}a(t)=\frac{{\hat{u}}_{{\it\omega}}}{\left[\displaystyle 1+\frac{{\it\sigma}_{{\it\omega}}^{4}}{{\it\gamma}_{c}^{4}}\,\frac{(2{\it\omega}_{c}^{2}+f^{2})^{2}}{{\it\omega}_{c}^{2}}\,t^{2}\right]^{1/4}},\end{eqnarray}$$

so that the dependence of the amplitude with the changes in the intrinsic frequency disappears (refraction effects) and the term that remains is related to the width of the packet. The amplitude of the wavepacket diminishes with time because of the dispersion of the components of the wavepacket. At long times, the amplitude of the wavepacket decays as $t^{-1/2}$ for waves in background flows without shear, while in a background flow with shear the amplitude of an inertio-gravity wave tends to a constant value because the effects of refraction in wave amplitude balance dispersion effects in wave amplitude, as mentioned above.

The Gaussian beam approximation is evaluated against the numerical experiment in figure 3(a). The maximum amplitude of the wave envelope in the numerical experiment is determined from the numerical complex wave field for each time frame as the maximum of the absolute value. Figure 3(a) shows a close agreement between the numerical solution and the Gaussian beam approximation, only slight differences being found for long times. These differences, a small oscillation of the numerical amplitude around the Gaussian beam solution, are produced by the numerical algorithm (Pulido & Rodas Reference Pulido and Rodas2008). As already mentioned, the numerical modelling of transient gravity waves propagating towards the critical level is particularly difficult since each spectral component finds its own critical level where the amplitude of the spectral component is unbounded there, so that the numerical model must be able to capture the strong destructive interference that occurs in those critical regions. On the other hand, the Gaussian beam approximation is expected to be convergent to the exact solution, like the spatial ray path approximation, for very long times. At short times, in particular when the maximum of the amplitude occurs, the Gaussian beam approximation is very close to the numerical solution.

Figure 3(b) shows the evolution of a transient inertio-gravity wave with a spectral width of ${\it\sigma}=0.15{\it\omega}_{c}$ that propagates upon background flows with different $\mathit{Ri}_{0}$ . The asymptotic behaviour depends on the Richardson number for inertio-gravity waves. On the other hand, the decay of internal gravity waves (dashed lines in Figure 3 b) is independent of the Richardson number.

3.2. Convective instability

One of the main mechanisms of gravity wave breaking is the convective instability (e.g. Hodges Reference Hodges1967; Lindzen Reference Lindzen1981) produced by the differential advection induced by gravity waves of heavier air over lighter air. The convective instability threshold occurs when the buoyancy frequency of the flow becomes zero. Under linear theory, the background flow and the perturbation contribute to the buoyancy frequency,

(3.4) $$\begin{eqnarray}N^{2}\approx N_{0}^{2}+g\partial _{z}\!\left(\frac{{\it\theta}_{1}}{{\it\theta}_{0}}\right)\!,\end{eqnarray}$$

where ${\it\theta}_{0}$ is the background potential temperature and ${\it\theta}_{1}$ its perturbation. Using the polarization relations, the minimum of the buoyancy frequency is given by

(3.5) $$\begin{eqnarray}\text{min}(N^{2})\approx N_{0}^{2}\left(1-\frac{ka}{{\it\Omega}}\right),\end{eqnarray}$$

so that when the wave amplitude reaches

(3.6) $$\begin{eqnarray}\frac{ka}{{\it\Omega}}=1\end{eqnarray}$$

the overturning condition is met.

The asymptotic behaviour for long times of the minimum buoyancy frequency can be obtained analytically,

(3.7) $$\begin{eqnarray}\text{min}(N^{2})_{t\rightarrow \infty }\rightarrow N_{0}^{2}\left[1-{\tilde{u}}\frac{(1-\tilde{f}^{2})^{3/4}}{\tilde{f}^{1/2}{\it\beta}}\right]\!,\end{eqnarray}$$

where ${\tilde{u}}={\hat{u}}_{{\it\omega}}(k/{\it\omega}_{c})$ is the non-dimensionalized initial wave amplitude. The asymptotic constant value, (3.7), depends on the inverse of the quartic root mean Richardson number, i.e.  $\mathit{Ri}_{0}^{-1/4}$ , and the inverse of the frequency width ${\it\sigma}_{{\it\omega}}$ .

From the asymptotic buoyancy frequency values (3.7), we deduce that inertial waves with an initial amplitude smaller than

(3.8) $$\begin{eqnarray}{\tilde{u}}<\frac{\tilde{f}^{1/2}}{(1-\tilde{f}^{2})^{3/4}}{\it\beta}\end{eqnarray}$$

never reach the convective instability. The equality is satisfied for inertial waves that attain the convective instability condition at $t\rightarrow \infty$ at the inertio-critical level.

The evolution of the minimum of the buoyancy frequency obtained with the Gaussian beam approximation for internal and inertial waves is shown in figure 4(a). Two cases are shown for ${\it\sigma}=0.1{\it\omega}_{c}$ and ${\it\sigma}=0.17{\it\omega}_{c}$ (the same wave fields as those shown in figure 3 a). The inertial wave with broader frequency spectrum never reaches the convective instability, contrary to internal waves, which always reach the convective instability. Indeed, the initial amplitude of this broad frequency wave spectrum satisfies (3.8). The quasi-monochromatic inertial waves, for small ${\it\beta}$ , do reach the convective instability.

Figure 4.

Evolution of (a) the minimum normalized buoyancy frequency and (b) the minimum Richardson number at each time. The Gaussian beam approximation for inertial waves is shown as a solid line and for internal waves as a dashed line. The wavepacket widths are ${\it\sigma}=0.1{\it\omega}_{c}$ and ${\it\sigma}=0.17{\it\omega}_{c}$ and the background flow is characterized by $\mathit{Ri}_{0}=100$ and $\tilde{f}=0.05$ .

3.3. Dynamical instability

Inertio-gravity waves may reach saturation through dynamical instability induced by the strong velocity shear of the wave field. The dynamical instability threshold may occur at a lower height than the convective instability (e.g. Dunkerton Reference Dunkerton1997) so that the breaking mechanism may differ from the characteristic wave overturning of internal gravity waves. In the last case, the convective instability occurs at regions of minimum shear (Hodges Reference Hodges1967). Howard (Reference Howard1961) and Miles (Reference Miles1961) showed that the necessary condition for the dynamical instability in an inviscid shear flow is that the Richardson number must be smaller than $\frac{1}{4}$ . In this work, we take the $\frac{1}{4}$ threshold for the instability assuming it is also a relevant reference value for gravity waves in a rotating frame. In the turbulence onset by an inertio-gravity wavepacket propagating upwards, Achatz (Reference Achatz2007) have identified non-modal perturbations with rapid growth even when the amplitude of the wave is not large enough to produce $\mathit{Ri}$ numbers smaller than $\frac{1}{4}$ . Thus, the threshold we use may be considered a (slightly) conservative value.

The Richardson number of the flow is defined as

(3.9) $$\begin{eqnarray}\mathit{Ri}=\frac{g{\it\theta}^{-1}\partial _{z}{\it\theta}}{(\partial _{z}u)^{2}+(\partial _{z}v)^{2}},\end{eqnarray}$$

where ${\it\theta}$ is the potential temperature of the flow, and $u$ and $v$ are the $x$ and $y$ components of the velocity of the flow.

We assume the flow is given by a background flow plus a perturbation, ${\it\theta}={\it\theta}_{0}+{\it\theta}_{1}$ and $(u,v)=(u_{0},0)+(u_{1},v_{1})$ . The perturbation is given by the solution of the Gaussian beam approximation for an inertio-gravity wave that propagates in a background flow. Then the Richardson number, (3.9), is

(3.10) $$\begin{eqnarray}\mathit{Ri}=\frac{\displaystyle \left[1+\left(\frac{k}{{\it\Omega}_{c}}a\right)\cos {\it\psi}_{c}\right]\left(1-\frac{f^{2}}{{\it\Omega}_{c}^{2}}\right)}{\displaystyle \left[\mathit{Ri}_{0}^{-1/2}\left(1-\frac{f^{2}}{{\it\Omega}_{c}^{2}}\right)^{1/2}+\left(\frac{k}{{\it\Omega}_{c}}a\right)\sin {\it\psi}_{c}\right]^{2}+\left[\frac{f}{{\it\Omega}_{c}}\left(\frac{k}{{\it\Omega}_{c}}a\right)\cos {\it\psi}_{c}\right]^{2}}.\end{eqnarray}$$

The phase ${\it\psi}_{c}$ that is taken in (3.10) is the one that gives the minimum Richardson number for each time. This determines the first time that the Richardson number threshold is reached in the wave field. In what follows, we always evaluate the Richardson number at the phase that gives the minimum value.

Figure 4(b) shows the evolution of the Richardson number for two wavepackets of different spectral widths, for the same cases as in figure 4(a). The inertio-gravity wave with ${\it\sigma}_{{\it\omega}}=0.1{\it\omega}_{c}$ reaches the dynamical instability at a similar time as the corresponding internal wave (the same wavepacket but in a medium without rotation) also attains the instability, which is expected to be a convective instability in this case (Hodges Reference Hodges1967). The inertio-gravity wave with ${\it\sigma}_{{\it\omega}}=0.17{\it\omega}_{c}$ only attains the dynamical instability. In the case of an internal gravity wave with this spectral width, it destabilizes the flow convectively at a later time than the inertial wave destabilizes the flow dynamically. These results show that Coriolis effects favour the development of the dynamical instability against the convective instability, particularly when the instability occurs at long times and close to the critical levels. This is in agreement with previous studies (Fritts & Rastogi Reference Fritts and Rastogi1985).

3.4. Height of the wave breaking

The height of the wave breaking is a critical parameter for gravity wave parametrizations (e.g. Alexander & Dunkerton Reference Alexander and Dunkerton1999). We thus examine the dependence of the breaking height on the wave and background flow parameters using the Gaussian beam approximation. The height of the threshold for convective instability $z_{s}$ is non-dimensionalized as $\tilde{z}_{s}=z_{s}/z_{cl}$ , where $z_{cl}$ is the height of the central critical level of an internal gravity wave (so that the results can be compared directly between inertial and internal waves).

Using the convective instability condition, (3.6), with the wave amplitude given by (3.1), and then expressing it as a function of $z$ by means of (2.20), the threshold for convective instability $\tilde{z}_{s}$ is constrained by

(3.11) $$\begin{eqnarray}{\it\beta}^{4}=\frac{{\tilde{u}}^{4}(1-\tilde{f}^{2})^{3}-[(1-\tilde{z}_{s})^{2}-\tilde{f}^{2}]^{3}}{\left\{\left[\displaystyle \frac{(1-\tilde{z}_{s})^{2}-\tilde{f}^{2}}{1-\tilde{f}^{2}}\right]^{3/2}-(1-\tilde{z}_{s})\right\}^{2}}.\end{eqnarray}$$

Equation (3.11) gives the convective threshold height as a function of the transient wave parameters ${\it\beta}$ , $\tilde{f}$ and ${\tilde{u}}$ . For $\tilde{f}\rightarrow 0$ , the convective threshold height given for internal waves in Pulido & Rodas (Reference Pulido and Rodas2008) is recovered from (3.11).

The height at which the transient wavepacket reaches the threshold of dynamical instability, $\tilde{z}_{d}=z_{d}/z_{cl}$ , is defined as the height obtained from (2.20) evaluated at the (minimum) time at which $\mathit{Ri}$ , from (3.10), reaches the value 0.25. This threshold is taken as a mean reference value, but note that Achatz & Schmitz (Reference Achatz and Schmitz2006) and Fritts et al. (Reference Fritts, Vadas, Wan and Werne2006) show that turbulence can be produced by gravity waves at larger Richardson number values. It is not possible to obtain an expression for $\tilde{z}_{d}$ analytically, so that a numerical algorithm is employed to obtain $\tilde{z}_{d}$ as a function of the wave parameters.

Figure 5(a) shows the height where the dynamical (continuous line) and convective (large dots) instabilities for inertial waves are attained as a function of ${\it\beta}$ (a measure of the transient effects in the wave). The height of the convective instability threshold for internal waves is also shown (dotted line, small dots). For inertial waves, the case with $\tilde{f}=0.1$ is taken. The height of the lowest inertial critical level of the central component of the wavepacket for this case is $0.9z_{cl}$ . The shaded contours represent the time difference between the time that the wave requires to reach the dynamical instability threshold and the time for the convective threshold. The time required to develop the dynamical instability depends on the growth rate. Therefore, the time required for the wave to reach the dynamical instability threshold plus the time required for the instability development must be shorter than the time required to reach the convective threshold if the dynamical instability mechanism is dominant. A long time difference between the dynamical instability threshold and the convective threshold shows that the dynamical instability mechanism is more likely to be dominant for the same growth rate.

Figure 5.

(a) Height of the dynamical instability threshold $\tilde{z}_{d}$ (solid) and of the convective threshold $\tilde{z}_{s}$ (larger dots) as a function of ${\it\beta}$ for different initial amplitudes ${\tilde{u}}=0.05,\ 0.15,\ 0.25$ . The height of the convective instability threshold for internal waves is shown with a dotted line. The height at which the maximum amplitude of the wavepacket is achieved is shown with a dashed-dotted line. The shaded contours represent the time difference between the dynamical instability threshold and the convective threshold. (b) Vertical wavenumber as a function of height for inertial waves (solid) and for internal waves (dotted). (c) Plot of $f/{\it\Omega}$ as a function of height.

Figure 5 gives a general picture of the importance of inertial effects for wave breaking. Waves that reach the instability threshold far from the critical level are not affected by the inertial effects. Specifically, far from the inertial critical level, the inertial waves for small ${\it\beta}$ attain the convective instability practically at the same height as the internal waves. We identify the transition roughly at $\tilde{z}_{s}<0.7$ . An extensive analysis of the wave breaking mechanisms for internal waves in the same context is given by Pulido & Rodas (Reference Pulido and Rodas2008). The present results show that the analysis for internal waves is also valid for inertial waves with small ${\it\beta}$ that break far from the critical level. In particular, given the agreement found between the internal and inertial waves, these waves for $\tilde{z}_{s}<0.7$ are expected to break through convective overturning for small ${\it\beta}$ . Note that $\mathit{Ri}$ also becomes zero when $N^{2}$ attains the convective overturning, but in this case the height of the threshold is close to the wave phase of minimum shear for internal waves (Hodges Reference Hodges1967).

Figure 5(b) shows that the central vertical wavenumber is relatively the same for internal and inertial waves at the height of the breaking for $\tilde{z}_{s}<0.7$ . Both the central vertical wavenumber and the central intrinsic frequency only depend on height, so that waves that attain the threshold at a given height for any combination of the parameters ${\it\beta}$ and ${\tilde{u}}$ break with the same central vertical wavenumber and central intrinsic frequency. From figure 5(c), the central intrinsic frequency at the breaking height, ${\it\Omega}_{b}$ , for $\tilde{z}_{s}<0.7$ is $f/{\it\Omega}_{b}<0.35$ , in agreement with the results found by Dunkerton (Reference Dunkerton1997). For any practical application, e.g. gravity wave parametrizations, waves in the atmosphere that break with $f/{\it\Omega}_{b}<0.35$ and small ${\it\beta}$ are well represented by the non-rotating assumption.

On the other hand, the breaking threshold for inertial waves that reach the instability close to the critical level $z/z_{cl}>0.7$ , which corresponds to $f/{\it\Omega}_{b}>0.35$ (see figure 5 c), occurs at lower heights than for internal waves. For these cases, the dynamical instability threshold for inertial waves is attained at lower heights than the convective instability condition. Therefore, there are two effects that act jointly to give a lower breaking height for highly transient inertial waves with respect to internal waves: the appearance of the inertio-critical level at a height lower than the classical (non-inertial) critical level; and that the dynamical instability condition is met at a lower height than the convective instability condition. The height and time differences between the dynamical and convective breaking thresholds increase for larger  ${\it\beta}$ .

Waves with large ${\it\beta}$ that reach the instability threshold for $z/z_{cl}>0.86$ in figure 5(a) satisfy the condition (3.8), so that they do not attain the convective instability and they break through dynamical instability. This is shown in figure 5(a) with the dark shading contour. Transient effects are important for inertial waves that reach the instability criterion at a higher altitude than the maximum amplitude height (dashed-dotted line in figure 5 a). In this case the amplitude of the wavepacket is expected to be small and the time difference between the time required to meet the dynamical instability and the convective instability time increases for larger ${\it\beta}$ . For instance, for $\tilde{z}_{s}=0.8$ the time difference is less than 1 h for small ${\it\beta}$ while it is 4 h for large ${\it\beta}$ , indicating that these highly transient waves are likely to break through dynamical instability instead of convective instability. Therefore, while the (upper) threshold for the dynamical instability is $\tilde{z}_{s}>0.86$ for medium-range ${\it\beta}$ , the (upper) threshold is expected to be somewhat smaller, say $\tilde{z}_{s}>0.75$ , for highly transient waves, large ${\it\beta}$ , considering the height and time differences between the dynamical and convective instabilities.

For waves that propagate close to the inertio-critical level, dissipation by viscosity and thermal conductivity becomes a dominant factor. A readily performed calculation, assuming a constant kinematic viscosity of ${\it\nu}=10^{-3}\ \text{m}^{2}\ \text{s}^{-1}$ , which is a representative value in the atmosphere at 33 km height, gives that the waves have lost 10 % of their amplitude at a height of $z/z_{cl}=0.88$ . For altitudes higher than $z/z_{cl}=0.88$ , the dissipative effects should be considered.

Figure 6(a) shows the breaking regions as a function of ${\it\beta}$ and ${\tilde{u}}$ for $\tilde{f}=0.1$ for which the convective or dynamical instability criteria are satisfied. Three regions are identified in the figure.

1. (i) For waves with a small ${\tilde{u}}/{\it\beta}$ ratio, as specified in (3.8) and shown in figure 6(a) with dark grey shading, the convective instability is never met. Thus, an inertio-gravity wavepacket that propagates towards its critical level in this region is expected to produce a dynamical instability with characteristics similar to the instabilities found by Lelong & Dunkerton (Reference Lelong and Dunkerton1998a ). The limit of this region is a straight line with a slope of $\tilde{f}^{1/2}/(1-\tilde{f}^{\,2})^{3/4}$ , which coincides with $f/{\it\Omega}_{b}=0.7$ except for small ${\it\beta}$ . This limit for $\tilde{f}=0.1$ corresponds to $\tilde{z}_{s}=0.86$ , except for small  ${\it\beta}$ .

2. (ii) There is an intermediate region, which is composed of inertio-gravity wavepackets with intrinsic frequency in the range $0.35<f/{\it\Omega}_{b}<0.70$ . The instability to be initiated by the wave in this region is a hybrid between convective and dynamical instabilities (Dunkerton Reference Dunkerton1997). The dominant behaviour will depend on the growth rate of the dynamical instability, so that the time needed by the dynamical instability to be developed occurs before the convective instability criterion is reached.

3. (iii) In the large-amplitude extreme, for transient inertial waves with ${\tilde{u}}$ and ${\it\beta}$ located in the region above the $f/{\it\Omega}_{b}=0.35$ curve (the light shaded region in figure 6 a), these waves have similar propagation characteristics to internal waves, so that we expect the breaking of these waves to be dominated by convective instability.

Figure 6.

Instability regions for transient inertio-gravity waves as a function of ${\tilde{u}}$ and ${\it\beta}$ for (a)  $\tilde{f}=0.1$ and (b)  $\tilde{f}=0.05$ . Wavepackets that reach instability for $f/{\it\Omega}_{b}=0.2$ , 0.35, 0.7 and 0.9 are represented with solid lines. The shaded dark grey region (region I) represents wavepackets that only reach dynamical instability. The light grey region (region III) represents wavepackets for which the inertial effects, even close to the convective instability threshold, may be considered negligible. The intermediate region (region II) represents wavepackets that do not have a dominant criterion, so that a hybrid convective–dynamical instability may be expected in this regime.

If $\tilde{f}$ is diminished (wavepackets with higher central frequency), as shown in figure 6(b), the slope of regime I limit is diminished, see (3.8), but the breaking frequency does not change, $f/{\it\Omega}_{b}=0.70$ . The hybrid regime, region II, is narrower. The convective instability is dominant (region III) in this small $\tilde{f}$ case for smaller wave amplitudes ${\tilde{u}}$ . Figure 6(a,b) also shows that the dynamical instability is more likely to be developed for transient waves with broader spectral width. In other words, wavepackets with larger ${\it\beta}$ and the same initial amplitude ${\tilde{u}}$ attain the dynamical instability with larger $f/{\it\Omega}_{b}$ so that the inertial effects are more important.

3.5. Propagation in a realistic environment

The performance of the Gaussian beam approximation is evaluated in a realistic environment. We examine the propagation of a Gaussian wavepacket in a background wind that is a realistic smoothed zonal wind profile characteristic of high latitudes in winter in which the subpolar and the stratospheric nocturnal jet are both present. The zonal background wind profile is shown in figure 7(a). This is a particularly difficult case since the zonal wind is non-monotonic. The background buoyancy frequency and Coriolis parameters are typical values, $N_{0}=0.02\ \text{s}^{-1}$ and $f=10^{-4}\ \text{s}^{-1}$ . The background Richardson number as a function of height is shown in figure 7(b). A wavepacket with initial parameters ${\it\sigma}_{{\it\omega}}=0.1{\it\omega}$ , ${\it\omega}=N_{0}/10$ and ${\it\omega}/k=55\ \text{m}\ \text{s}^{-1}$ is launched in this environment. The evolution of the normalized amplitude is shown in figure 8(a). The amplitude of the wavepacket achieves two local maxima. Both peaks in amplitude are achieved when the wavepacket is in the height ranges where the background zonal wind increases with height. These two maxima are produced due to refraction and dispersion effects. In altitudes where the background zonal wind increases with height, refraction effects produce an increase of the amplitude. At the same time, the highest components of the wavepacket are in a stronger background wind so that their group velocity diminishes with respect to the components that are in the lower part of the envelope. Therefore, the dispersion effects are weaker in regions where the background zonal wind increases. The amplitude given by the Gaussian beam approximation shows a close agreement to the numerical integration (figure 8 a). Indeed, the oscillation that appears in the numerical integration at long times is an artifact, so the Gaussian beam approximation probably gives a better field than the numerical experiment (see Pulido & Rodas (Reference Pulido and Rodas2008) for a discussion of this numerical error). The evolutions of the Richardson number and buoyancy frequency are shown in figure 8(b). The case has only dynamical instability, while convective instability is never reached. A close agreement between the Gaussian beam approximation and the numerical solution is found particularly for the buoyancy frequency evolution.

Figure 7.

(a) Background zonal wind as a function of height and (b) background Richardson number.

Figure 8.

(a) Amplitude evolution of an inertial wavepacket with ${\it\sigma}=0.1{\it\omega}$ that propagates in the background zonal wind shown in figure 7 for the Gaussian beam solution (solid line) and the numerical experiment (triangles). (b) Minimum Richardson number (same curve convention as amplitude) and the minimum normalized buoyancy frequency for the Gaussian beam solution (dashed line) and the numerical experiments (diamonds).