Turning depths: Evanescent to propagating wave kinetic energy density

Tidal flow over oceanic topography generates internal waves when the natural frequency $N$ of the water is greater than the tidal frequency $\omega$. When $N<\omega$, evanescent waves are generated. Although the amplitude and kinetic energy of evanescent waves decay rapidly, if the wave reaches a turning depth, where $N=\omega$, and moves into a region where $N>\omega$, the evanescent wave becomes an internal wave. This work expands upon previous research of varying stratifications by investigating the kinetic energy density in internal waves generated by evanescent waves passing through a turning depth. An analytical model is presented and compared to synthetic schlieren experiments of two Gaussian-shaped topographies. The model and experiments both indicate that the kinetic energy density of internal waves increases with decreasing topographic slope, when the distance between the topography and the turning depth decreases and when the average Froude number in the evanescent region is close to one. The model is used to estimate the normalized kinetic energy density of internal waves generated from an oceanic feature located within an evanescent region.

Figure 1

(a) Propagating internal wave and (b) vertically decaying evanescent wave. (c) The turning depth indicates the boundary between the evanescent and propagating regions, with the evanescent wave becoming an internal wave as it passes through the turning depth.

Figure 2

Measured density (red points) and the exponential curve fit (black line).

Figure 3

Experimental tank and visualization system schematic: (a) Front view of the tank with internal wave regions and turning depth as labeled. (b) Side view of the setup with camera and mask for synthetic schlieren imaging.

Figure 4

Fourier amplitudes of $\mathrm{\Delta }{N}^2$ in (a) the evanescent region and (b) the propagating region, shown in contours increasing by 0.0025 for each line. Both figures use the same scaling. The highest values for the contour lines are (a) 0.01 ${\mathrm{s}}^{-2}$ and (b) 0.0125 ${\mathrm{s}}^{-2}$.

Figure 5

Kinetic energy as a function of height, shown for the scenarios of including or excluding $dq/dz$ and $dm/dz$ when calculating the horizontal velocity. The turning depth is shown by the dashed horizontal line and the dash-dotted line indicates the height of a 10% increase in $N$ from the turning depth.

Figure 6

Normalized kinetic energy as a function of height for 4 cases. The solid lines are experimentally calculated ${\text{KE}}^{*}$ while the dotted represent model results. Data are from (a) and (b) cases 1 and 8, which used the medium topography, and (c) and (d) cases 18 and 20, which used the steep topography. The turning depth location $z_{td}$ is marked with a dashed line. The black $\times$ markers indicate the distance over which kinetic energy is averaged in the propagating region.

Figure 7

Contours of $\mathrm{\Delta }\widetilde{N^2}$ for case 15 as a function of ${\omega }^{*}$ and $k^{*}$ in the (a) evanescent and (b) propagating regions. The $\mathrm{\Delta }\widetilde{N^2}$ values have been scaled by a factor of ${10}^3$.

Figure 8

Average normalized kinetic energy in the propagating region as a function of $H/D$ for both the medium and steep topographies with experimental and model values. Red circles represent the medium topography, with closed circles representing experimental data and open the model. Steep topography data are represented with black triangles, again with closed triangles representing experimental data and open the model. The inset contains five steep topography model points with normalized kinetic energy values less than ${10}^{-5}$.

Figure 9

Plot of $\overline{{\text{KE}}_2^{*}}$ as a function of $\overline{{\text{Fr}}_1}$. The symbols and lines follow the same legend as shown in Fig. 8.

Figure 10

Plot of $\overline{{\text{KE}}_2^{*}}$ as a function of $H/D$ and $\overline{{\text{Fr}}_1}$ for the analytical model. The solid red line indicates the medium topography, while the dashed black line is the steep topography. Markers for $H/D$ are as shown.

Figure 11

Plot of $\overline{{\text{KE}}_2^{*}}$ as a function of $W/H$, $a/{\rho }_0$, and $bH$, showing the effects of (a) topographic shape and (b) and (c) an exponential density profile. In (a) the square and circle indicate $W/H=0.45$ and 1.8, or the steep and medium topographies, respectively.

Figure 12

Data from the GEBCO worldwide bathymetry map indicated by the shaded portion, with a Gaussian curve fit through the topographical feature analyzed in this work.

Figure 13

(a) The WOCE data are used to calculate $N^2$, indicated by the red dashed line, while the black line is the curve fit of the data used for the model analysis. (b) Normalized kinetic energy calculated from the model as a function of depth.