Internal wave transmission through a thermohaline staircase

With the aim to predict energy transport by internal waves incident upon observed density staircases in the ocean, the theory for internal wave tunneling through a single finite-size mixed region is extended to predict the transmission of internal waves through a piecewise-constant density profile with an arbitrary number of steps. An analytic solution is found if all steps have equal vertical extent. From this solution, bounds are established for the relative incident horizontal wave number and frequency of waves that have negligible transmission, a result that is independent of the number of steps. If the step sizes vary randomly about a mean vertical extent, the transmission can be computed numerically for several random realizations. Provided the horizontal wavelength is much longer than the total vertical extent of the staircase, the standard deviation of the computed transmission coefficients is small and the mean agrees well with the analytic prediction for equally spaced steps. The results are applied to thermohaline staircases observed in the ocean in order to put lower bounds on the wavelengths of long inertia gravity waves that significantly transmit to great depth.

Figure 1

Schematics showing the density profiles (thick black lines) of a staircase with $J$ equal steps of depth $L$ (left) and a staircase with $J$ irregular steps having mean depth $L$ (right). In both plots the diagonal light dashed line indicates the mean stratification in the absence of a staircase. The extent of the thick gray line segments on the right indicate the change in the depth of the density jump from the case of equal-depth steps.

Figure 2

Transmission through a single mixed region in cases with (a) no background rotation $f=0$ and (b) $f=0.1N_0$. The color scale for both plots is indicated at the top left in (a).

Figure 3

Same as in Fig. 2 but showing the transmission through a staircase composed of (a) $J=2$, (b) $J=5$, and (c) $J=10$ evenly spaced steps. In all cases $f=0.1N_0$. The dashed lines show the predicted upper bound given by (25) on the region containing the sequence of transmission spikes.

Figure 4

Transmission as a function of $kL$ for (a) $\omega /N_0=0.105$ ($\omega /f=1.05$), (b) $\omega /N_0=0.12$ ($\omega /f=1.2$), and (c) $\omega /N_0=0.2$ ($\omega /f=2.0$). Plots show transmission through $J=5$ steps (thin solid line), 10 steps (thin red dashed line), and 20 steps (thin blue dotted line). The thick short-dashed lines in each plot gives the approximation (27) to $T$ for $kL\ll 1$ and $\omega \gtrsim f$ for the case $J=5$ and the thick long-dashed line in (c) gives the approximation (26) to $T$ for $kL\ll 1$ and $\omega \gg f$ for the case $J=5$. Note that the vertical scale in (c) is larger than in (a) and (b). In theory all peaks do reach unity, but the finite vertical resolution of the plots ($kL$ is computed in steps of $0.0001$) means that the peak at unity is not always captured.

Figure 5

(a) Same as in Fig. 3, but showing transmission through $J=10$ unevenly spaced steps computed from one random realization of $\left\{{\sigma }_j\right\}$ with $\epsilon =0.1$. The average transmission (solid lines) computed in 100 random realizations is plotted as function of $kL$ for (b) $\omega /N_0=0.105$ ($\omega /f=1.05$), (c) $\omega /N_0=0.12$ ($\omega /f=1.2$), and (d) $\omega /N_0=0.2$ ($\omega /f=2.0$). The thin-dashed lines in the bottom three plots show the mean plus and minus the standard deviation. The thick-dashed lines show the approximate analytic prediction (27).