Generation of nonlinear internal waves by flow over topography: Rotational effects

We use the forced Ostrovsky equation to investigate the generation of internal waves excited by a constant background current flowing over localized topography in the presence of background rotation. As is now well known in the absence of background rotation, the evolution scenarios fall into three cases, namely subcritical, transcritical, and supercritical. Here an analysis of the linearized response divides the waves into steady and unsteady waves. In all three cases, steady waves occur downstream but no steady waves can occur upstream, while unsteady waves can arise upstream only when there is a negative minimum of the group velocity. The regions occupied by the steady and unsteady waves are determined by their respective group velocities. When the background current is increased, the wave number of the steady waves decreases. In addition, the concavity (canyon or sill), the topographic width, and the relative strength of the rotation play an important role in the generation mechanism. Nonlinear effects modulate the wave amplitude and lead to the emergence of coherent wave packets. All these findings are confirmed by numerical simulations.

Figure 1

The schematic diagram of the coordinate system. The Gaussian topography expressed as $b_{M}exp\left[-{\left(x/l\right)}^2\right]$ where, given the pycnocline locates near the surface when the nonlinear coefficient ${\mu }_0<0$ [see Eq. (A13)], in reality, $b_M<0$ indicates undersea bump, for example, a sill or ridge, while $b_M>0$ implies undersea dent, for instance, a canyon or basin; refer to scale (A26).

Figure 2

Numerical simulations of the linearized (a) and full [(b) and (c)] Ostrovsky Eq. (4), while the results are shown in the scale of Eq. (1). These snapshots are captured at time $t=5000$. Here $b_M=-0.03$ for panels (a) and (b), $b_M=0.03$ for panel (c), and the other common parameters used are $(l,\mathrm{\Delta },\alpha )=(12.6,0,0.01)$, which indicates these scenarios locate on the transcritical case since the criteria ${\mathrm{\Delta }}_m=-0.45$ and ${\mathrm{\Delta }}_M=0.81$.

Figure 3

The layout is the same as in Fig. 2. These snapshots are captured at time $t=2250$. Here $b_M=-0.04$ for panels (a) and (b), $b_M=0.04$ for panel (c), and the other common parameters used are $(l,\mathrm{\Delta },\alpha )=(3.2,-0.6,0.01)$, which indicates these scenarios locate on the subcritical case since the criteria ${\mathrm{\Delta }}_m=-0.5$ and ${\mathrm{\Delta }}_M=1.0$. To better illustrate the results, the axes are broken in the $x$ direction.

Figure 4

The parameters used are $(b_M,l,\mathrm{\Delta })=(-0.04,3.2,-0.6)$, which indicates these scenarios locate on the subcritical case. Panel (a) indicates the rotation $\alpha =0.01$, while (b) for $\alpha =0.03$, and both are plotted at time $t=2500$.

Figure 5

The layout is the same as in Fig. 2. These snapshots are captured at time $t=5000$. Here $b_M=-0.04$ for panels (a) and (b), $b_M=0.04$ for panel (c), and the other common parameters used are $(l,\mathrm{\Delta },\alpha )=(3.2,1.1,0.01)$, which indicates these scenarios locate on the supercritical case since the criteria ${\mathrm{\Delta }}_m=-0.5$ and ${\mathrm{\Delta }}_M=1.0$. Note that the axes are broken in the $x$ direction.

Figure 6

The parameters used are $(b_M,\mathrm{\Delta },\alpha )=(-0.04,1.2,0.01)$, which indicates these scenarios locate on the supercritical case. Panels (a1) and (a2) indicate the width of topography $l=3.2$, while $l=70$ for (b1) and (b2), of which (a1) and (b1) are at time $t=25$ and (a2) and (b2) at time $t=5000$.