Propagation of NIWs

We observe the largest near-inertial event in temperature oscillations between days 370 and 390 (January of 2003). We propose that these oscillations (as well as near inertial to diurnal band energy in general) are due to impinging internal waves that are forced by the surface winds to the north of the ridge, and propagate southward intersecting the Kaena Ridge. To support this hypothesis we first calculate the ray path of an idealized, equator-ward propagating, inertial wave following Garrett (2001) (his equation 16).

$$\frac{dy}{dz}\simeq-\frac{N}{(2\omega\beta y)^{1/2}}$$ (1)

where y is the meridional, equatorward distance of propagation of an inertial wave generated below the surface mixed layer at a latitude where its frequency is equal to the local Coriolis frequency ( $\omega = f$ ). $z$ is the depth, $\beta=\partial f / \partial y$ is the meridional rate of change of the Coriolis frequency $f$ , and the buoyancy frequency is approximated here by $N=N_0e^{-z/b}$ . The values of $N_0$ and $b$ can be obtained from a best fit to the climatological vertical profile of $N$ at station ALOHA between the surface and $3000 m$ depth. This leads to values of $N_0=7.1 \times 10^{-3} s^{-1}$ and $b=1.25 km$ , with an RMS difference between the approximate and measured $N$ profiles of $9\times 10^{-4} s^{-1}$ .

Equation 4.3.1 can be integrated between the surface $z=0$ where $y = 0$ and a depth Z:

$$Y=\frac{1}{2}\left(\frac{9N_0^2b^2}{\omega\beta}\right)^{1/3}(1-e^{\frac{-Z}{b}})^{2/3}$$ (2)

Setting $Z = 2420 m$ and the latitude to $22^o N$ , gives $Y = 380 km$ . An inertial wave ray, if propagating equator-ward, would originate $380 km$ north of the mooring site, near $25^o N$ , if there were no previous bottom bounces. An inertial wave generated at this latitude has a period of $28.2 hours$ (or $0.85 cpd$ ).

To estimate the propagation time, we use equation 25 from Garrett (2001) :

$$\frac{dy}{dt}=\left(\frac{N_0b}{j\pi}\right)^2\frac{\beta t}{\omega}$$ (3)

where $j$ is the vertical mode number. Equation 4.3.3 can also be integrated between 0 and $Y$ to give a travel time $T$ :

$$T=\left(\frac{2\omega Y}{\beta}\right)^{1/2}\frac{\pi j}{N_0b}$$ (4)

For our choice of parameters, this leads to a propagation time $T$ of $5.8 j days$ , where $j$ is the wave vertical mode number.

We now try to relate the arrival of these waves deep along the ridge flank to the remote surface wind forcing. We consider the surface winds around the dates of interest in the expected area of generation, north of the mooring at $25^\circ N$ (Figure 4.4). We examine the wind analysis product from the operational Global Data Assimilation Scheme (GDAS), with winds converted to $10 m$ height assuming neutral stability. The period covers the entire length of the mooring deployment, and the sampling is every 3 hours. We band-pass filter (periods between 24 and 32 hours) and separate the wind into clockwise and anticlockwise rotating components. The lagged correlation between the time series of the near-inertial clockwise component of the wind at $25^\circ N$ , $158^\circ W$ and the near-inertial kinetic energy remains statistically insignificant (0.38), but shows a small peak at a lag of $10.5 days$ . In comparison, the expected lag is $\sim 6 days$ for a mode 1 wave, or $\sim 12 days$ for a mode 2 wave (equation 4.3.4).

The relationship between wind stress to the north and near-inertial kinetic energy at Kaena Ridge is encouraging but inconclusive. A more elaborate slab layer model forced by the wind (D'Asaro, 1995; Alford, 2003), or a multilevel numerical model (Nagasawa et al., 2000) would be useful for investigating the timing between the generation of NIWs at the surface, and the observed near-inertial variability deep along the Kaena Ridge slope. Nonetheless, we believe that incident NIWs are still a plausible explanation for the intermittent energy in the near-inertial band, and consistent with the shadowing effect proposed for the southern ridge flank.