Fast-slow wave transitions induced by a random mean flow

Motivated by recent asymptotic results in atmosphere-ocean fluid dynamics, we present an idealized numerical and theoretical study of two-dimensional dispersive waves propagating through a small-amplitude random mean flow. The objective is to delineate clearly the conditions under which the cumulative Doppler shifting and refraction by the mean flow can change the group velocity of the waves not only in direction, but also in magnitude. The latter effect enables a possible transition from fast to slow waves, which behave very differently. Within our model we find the conditions on the dispersion relation and the mean flow amplitude that allow or rule out such fast-slow transitions. For steady mean flows we determine a finite mean flow amplitude threshold below which such transitions can be ruled out indefinitely. For unsteady mean flows a sufficiently rapid rate of change means that this threshold goes to zero, i.e., in this scenario all waves eventually undergo a fast-slow transition regardless of mean flow amplitude, with corresponding implications for the long-term fate of these waves.

Figure 1

Sample rays in two cases. The first column shows the ray path as dots in physical space, superimposed on the stream function of the mean flow; the spacing of the dots becomes denser and resembles a solid line if the ray slows down. Lighter red color refers to later times for orientation. The second column shows the path of the ray in the phase space and the third column shows the logarithm of the intrinsic frequency $\omega$ as a function of time. Top row: $\alpha =1.2$ and ${\varepsilon }_0=0.5$, corresponding to an ergodic trajectory with no shift in frequency and no fast-slow transition. Bottom row: $\alpha =0.5$ and ${\varepsilon }_0=0.5$, corresponding to a trapped trajectory with shift in frequency, indicative of the fast-slow transition.

Figure 2

Evolution of $k$ in the case $\alpha =0.5$ and ${\varepsilon }_0=0.5$. The wave packet is trapped around $t\approx 100$ and afterwards $k$ exhibits secular growth, with fluctuations around a linear trend (oblique blue line).

Figure 3

Top row: no-transition case with $\alpha =1.2$ and ${\varepsilon }_0=0.5$. Bottom row: fast-slow transition case with $\alpha =0.5$ and ${\varepsilon }_0=0.5$. Left: stream function $\psi$ and 1600 rays at $t=200$. Center: normalized PDF computed by counting the rays on a coarse-grained grid. Right: sample correlation coefficient between $\left|\psi \right|$ and the ray PDFs, which is close to zero in the no-transition case but significantly positive in the fast-slow transition case (several graphs for different mean flow samples are plotted).

Figure 4

Comparison between the $n=3$ case (top row) and the $n=5/3$ case (bottom row), showing a representative stream function (left), the phase diagram of $\varepsilon (t=200)$ in the $(\alpha ,{\varepsilon }_0)$ space obtained by computing the trajectories of 16 rays on ten mean flows (center), and the extracted mean boundary with its standard deviation computed over the ten mean flows (right).

Figure 5

Evolution of the phase diagram of $\varepsilon (t=200)$ in the $(\alpha ,{\varepsilon }_0)$ space in the case of an unsteady mean flow. On the far left is the base case of a steady mean flow. On the far right is the case of a fast varying mean flow that leads to a decorrelated white noise regime for which the boundary between shift and no-shift is a straight line at $\alpha =1$. A gradual transition lies in between and separates these two limiting behaviors.