Flows induced by Coriolis-influenced vertically propagating two-dimensional internal gravity wave packets

Theory is developed to predict the flows induced by small-amplitude two-dimensional vertically propagating internal wave packets under the influence of rotation. While the long-wave response originally predicted by Bretherton [J. Fluid Mech. 36, 785 (1969)] dominates if the influence of background rotation is negligible, the induced waves are found to be evanescent if the Coriolis parameter is sufficiently large. Explicitly, evanescent induced flows dominate if the Coriolis parameter is greater than the forcing frequency, which is set by the ratio of the vertical group velocity of the wave packet divided by the vertical scale of modulation of the wave packet. The predicted structure of the induced flow is confirmed by numerical simulations. For an upward and rightward propagating Gaussian wave packet that dominantly induces a long-wave response, the flow is rightward across the upper flank and leftward across the lower flank of the wave packet. In contrast, the induced flow for a dominantly evanescent response is negative over the middle of the wave packet. This qualitative change in structure is anticipated to influence the modulational stability of moderately large-amplitude wave packets.

Figure 1

Horizontal velocity associated with the wave-induced flow for a Gaussian wave packet with $k{\sigma }_x=k{\sigma }_z=20$, containing waves with $m/k=-1$: (a) induced flow computed with $f=0$, (b) induced flow computed with $f/N=0.01$ and its decomposition into the flow associated with (c) long waves and (d) evanescent disturbances. In all cases the velocity is normalized by $kN{A}_0^2$. The white circle centred at the origin (appearing as an ellipse in these stretched coordinates) is drawn with radius $20/k$, indicating the spatial extent of the wave packet. Note the reduced scale in (d) that serves to show better the relatively small evanescent disturbances.

Figure 2

As in Fig. 1 but for an internal wave packet containing waves with $m/k=-3$.

Figure 3

As in Fig. 1 but for wave packet containing waves with $m/k=-5$. Note the reduced scale in (c).

Figure 4

First four coefficients $C_n\left({\tilde{\mu }}_0\right)$ used in the approximation to $u^{\left(2\right)}(X=0,Z)$ for $|Z/{\sigma }_z|\ll 1$, as defined by (26) for a Gaussian wave packet.

Figure 5

Predicted horizontal flow (black curves) at $X=0$ for (a) $m/k=-1$, (b) $-3$, and (c) $-5$. Asymptotic approximations for $\left|Z\right|/{\sigma }_z\ll 1$ (25) and $\left|Z\right|/{\sigma }_z\gg 1$ (27) are plotted as short- and long-dashed lines, respectively.

Figure 6

Contours of $k/{\mu }_0$ ($=k{\sigma }_z/{\tilde{\mu }}_0^{★}$ at transition) indicating whether the induced flow is dominated by long waves or evanescent disturbances. For given $k{\sigma }_z$, parameter values to the left (right) of this contour are dominated by long waves (evanescent disturbances). A solid line is drawn for the contour with $k{\sigma }_z/{\tilde{\mu }}_0^{★}=20/\left(2.4\right)$, which is used for the examples presented here. The open circles indicate the parameters corresponding to the theoretical results shown in Figs. 1–3. The crosses indicate the parameters corresponding to numerical simulation results presented in Sec. 3. The red triangle corresponds to observations of mountain waves [22], and the red circle corresponds to observations of convectively generated waves [23].

Figure 7

Results of numerical simulation with $f=0.1N$ and initialized only with a Gaussian wave packet (without the predicted Eulerian induced flow initially superimposed) having $m=-k$, $A_0=0.01/k$ and ${\sigma }_x={\sigma }_z=20/k$, showing the wave-packet-filtered horizontal flow (a) at $Nt=10$, and (b) after evolving to time $Nt=200$. In both plots, the color scale is the same, as indicated in (a), and the white circles of radii $20/k$ indicate the location and extent of the wave packets.

Figure 8

Results from simulations with (a) $f=0.1N$, (b) $f=0.05N$, and (c) $f=0.02N$ showing (left) the wave-packet-filtered horizontal flow at $Nt=0$ and (right) the same field at $Nt=200$ with the predicted Eulerian induced flow initially superimposed. The fields on the right are plotted with the same scale as that indicated in the corresponding plot to the left, and the white circles of radii $20/k$ indicate the location and extent of the wave packets.

Figure 9

Vertical profiles of the horizontal flow across the center of the wave packet as predicted by theory (black line) and extracted from simulations at $Nt=200$ (blue lines) as shown in the right panels of Fig. 8.

Figure 10

As in Fig. 9, comparing vertical profiles of horizontal velocity from simulations (solid lines) with theory (dotted lines) with $m=-k$, (a) $f=0.1N$, (b) $f=0.05N$, and (c) $f=0.02N$, and with wave-packet horizontal extent $k{\sigma }_x$ as indicated by the colored lines toward the upper left of each plot. The cases with $k{\sigma }_x=20$ are reproduced from Fig. 9 with the black dotted and solid curves corresponding, respectively, to the black and blue curves shown in Fig. 9.