Interaction of NIWs with the Sloping Bottom

The near-inertial band vertical displacements at mooring DN have a rms value of $18 m$ . We seek to determine whether the observed vertical displacements are consistent with the horizontal motions of a freely propagating near-inertial internal wave, or if the vertical displacements are the result of the near-inertial horizontal motions directed across the slope, which must satisfy a no normal flow condition at the boundary. The amplitude of the velocity field associated with a freely propagating internal wave can be expressed as

$$u=\pm\frac{A_0\sqrt{m}}{k}\exp(i(kx+ly+\phi(z)-\omega t)=\pm u_0\exp(i(kx+ly+\phi(z)-\omega t)$$ (5)

$$w=\frac{A_0}{\sqrt{m}}\exp(i(kx+ly+\phi(z)-\omega t))=u_0\frac{k}{m}\exp(i(kx+ly+\phi(z)-\omega t)$$ (6)

The associated vertical displacement $\zeta$ can be written as

$$\zeta=\frac{w}{i\omega}=u_0\frac{k}{i m\omega}\exp(i(kx+ly+\phi(z... ...{i\omega}\sqrt{\frac{\omega^2-f^2}{N^2-\omega^2}}\exp(i(kx+ly+\phi(z)-\omega t)$$ (7)

The amplitude of the vertical displacement for an idealized propagating near-inertial wave is therefore related to the horizontal velocity amplitude by

$$\zeta_0=u_0\frac{1}{i\omega}\sqrt{\frac{\omega^2-f^2}{N^2-\omega^2}}$$ (8)

Alternatively, near the boundary, the flow has to satisfy a zero normal flow condition, so that $u$ and $w$ are related by

$$w=-u\frac{dH}{dx}$$ (9)

Where $z=-H$ is the ocean bottom, and the vertical displacement amplitude is then related to the horizontal velocity amplitude by

$$\zeta_0=-u_0\frac{1}{i\omega}\frac{dH}{dx}$$ (10)

In our case, for a near-inertial wave of frequency $0.85 cpd$ , a buoyancy frequency $N = 16 cpd$ , a local Coriolis frequency $f = 0.75 cpd$ , and a topographic slope of $1/10$ , the ratio of vertical displacement amplitude to horizontal velocity amplitude is $\sim 450$ for a free wave, and $\sim 1600$ for a wave encountering the slope. To compare this theoretical value of the ratio of vertical displacement to horizontal velocity to our measurements, we analyze in more detail the period of high near-inertial energy around day 375 (Figure 4.3 and 4.5). The current and temperature data (Figure 4.3b and d) are low-passed filtered between 0.7 and 0.9 cpd (Figure 4.5). We observe near-inertial across-slope velocities of $0.05 ms^{-1}$ amplitude , associated with a vertical displacement amplitude of $\sim 60 m$ , leading to a ratio of displacement to horizontal velocity of $\sim 1200$ , close to the theoretical ratio of $\sim 1600$ obtained above for a wave along the local topographical slope. The phase between temperature and velocity is such that downslope (upslope) flow is associated with increasing (decreasing) temperature. Unfortunately, no current data were available further than $70 mab$ to compare with the temperature data. Above $100 mab$ , the low-pass filtered temperature shows an average upward phase (downward energy) propagation of $760 mday^{-1}$ (Figure 4.6b). The amplitude of the temperature oscillations also decreases by a factor of 2 between $70$ and $200 mab$ , consistent with the observations from Figure 4.1. We conclude that the incoming, downward propagating, near-inertial wave is constrained by the supercritical slope in the lower $100 m$ to produce vertical displacement larger than theoretically predicted.

We only have measurements for a very limited area of a complicated ridge slope system. Considering the locally supercritical slope, and the small horizontal scales (100's of meters in the horizontal, 10's of meters in the vertical) of the topographic features in the mooring area (Figure 3.12) compared to the horizontal scales of a typical near-inertial wave (10's of km in the horizontal, 100's of meters in the vertical), we did not consider here the theoretical treatment of the reflection of the NIW by a theoretical plane boundary as was done by Eriksen (1982) or Müller and Liu (2000).

Figure 4.1: Temperature variance, as a function of depth and time, calculated over 8 day intervals, for all frequencies (top), the semidiurnal (1.9-2.1 cpd, middle top), the diurnal (0.9-1.1 cpd, middle bottom), and the near-inertial (0.7-0.9 cpd, bottom) bands;

Figure 4.2: Velocity variance, calculated over 8 day intervals, for the measured current at $43 mab$ (blue), for the barotropic current from the TPXO model (green), and for the superposition of tidal constituents obtained from the measured current at $43 mab$ (red), for all frequencies (top), for the semidiurnal (1.9-2.1 cpd, middle top), for the diurnal (0.9-1.1 cpd, middle bottom), and the near-inertial (0.7-0.9 cpd, bottom) bands.

Figure 4.3: 3 day time series of potential temperature between 68 and 220 mab (a), between 28 and 68 mab (colored lines in b), dissipation (black line in b), buoyancy frequency calculated between sensor pairs ( $10^{-3} s^{-1}$ , c) overlaid with the top (crosses) and bottom (circles) of detected overturns (see chapter 6). Velocity, rotated so that the across-slope velocity is in the ordinate direction (d).

Figure 4.4: 10 m winds (vector direction and wind speed) obtained from the GDAS product. The circle has a radius of 320 km around the DN mooring location.

Figure 4.5: 5 day segments of temperature (a), vertical displacements (b) and across-slope current (c) at DN, band-pass filtered between 0.7 and 0.9 cpd. The temperatures (a) and displacements (b) are shown for the 6 sensors located between $28$ and $68 mab$ , the current data (c) shown are for bins $43$ ,$47$ and $51 mab$ .

Figure 4.6: 5 day segments of vertical displacements (a), band-pass filtered between 0.7 and 0.9 cpd, and lines of constant phase (b). The displacements are shown for the 11 sensors located between $28$ and $220 mab$ . '.' and 'x' indicate local maxima and minima used to calculate the phase lines. Phase lines are line fits to the five top sensors (blue) and the bottom 6 sensors (black).