[8] Figure 1a shows sea-surface height (SSH) contours showing the Loop Current and a newly-shed warm ring. Figure 1b shows the three-dimensional surface of inertial energy ((*u*^{2} + *v*^{2})/2 = 0.03 m^{2} s^{−2}) from the model, 6 days after Katrina (Sep/03/12:00) [cf. *Wang and Oey*, 2008]. Energetic inertial chimneys (amplitude ≈0.24 m/s) penetrate to 1000-m depth to the right of the storm in the Loop Current and warm-core ring. As seen in animation (not shown) and ray-tracing (below), the chimneys are advected anticyclonically around the rims of the Loop Current and warm ring. Under the Loop Current, penetration is deeper on the eastern side (>1200 m compared to 800 m in the west). Regions outside the Loop Current and warm ring are void of these strong inertial chimneys. The region of low SSH or cyclonic vorticity between the Loop Current and the warm ring will be shown to be where super-inertial waves are produced by the storm.

[9] Katrina winds [∣*u*_{a}∣ > 60 m/s] produced a strong near-inertial response at the mooring. This consists of clockwise-rotating currents that propagate downward (Figure 1c–g; hurricane Rita is included for comparison). In the case of Katrina, the response penetrates and amplifies to depths of about *z* = −640 m with amplitudes exceeding 0.3 m/s around Sep/05. The amplitude attenuates at *z* = −760 m (not shown) to approximately 0.15 m/s, and quite abruptly drops to <≈ 0.1 m/s at *z* = −1005 m (Figure 1h). The response to Rita is less both in terms of amplitudes and depths of penetration.

#### 4.1. Empirical Mode Decomposition Analysis

[10] We use Empirical Mode Decomposition to extract various Intrinsic Mode Functions then compute their Hilbert spectra [*Huang et al.*, 1998]. Unlike FFT, Huang et al.'s method can accommodate rapid frequency variations with little spurious harmonics. The time series can be non-stationary as well as non-linear. The method is efficient; for our time series, it yields only nine intrinsic modes each of which (after the Hilbert transform) gives frequency and amplitude as a function of time. The first mode is of O(hours) period and of very small amplitude, while the ninth is the ‘residue’ which is (nearly) constant (in time) and also has very small amplitude (rms ∼ 10^{−4} m/s).

[11] The second and third modes have near-inertial periods. Their Hilbert spectra for the 6-month period Jun–Nov/2005 near the surface (*z* ≈> −250 m; not shown) indicate strong near-inertial variability both in amplitude and frequency, and a tendency for sub- (super-)inertial waves to be produced when the mooring is inside (outside) the Loop Current where vorticity *ς* < 0 (*ς* > 0). Figure 2a shows spectra at *z* = −640 m, focusing on the response to Katrina. After the storm, energetic super-inertial (*ω*/*f* > 1) signals arrive at the mooring before sub-inertial (Figure 2a). The distribution of energy is skewed to *ω*/*f* < 1 in the upper 1000 m (Figure 2b) because during the 6-month observation the mooring was located predominantly within the negative-vorticity part of the Loop Current. Near-inertial energy is intensified subsurface near 500–700 m depth.

#### 4.2. Ray Analysis

[12] The Loop Current and warm ring play an important role in horizontally advecting and confining near-inertial waves into “chimneys” (Figure 1). Experiments with initially-level isopycnals (i.e., no Loop Current and rings; not shown) produce near-inertial response confined to the upper 200 m. We now use the model flow field and show by way of ray-tracing how the subsurface intensification (Figure 2b) may be explained by stalling, i.e., vanishing of *u* + *C*_{g}, where *C*_{g} = (*C*_{g1}, *C*_{g2}, *C*_{g3}) is the group velocity, along rays at the base of the Loop Current.

[13] Each ray is traced from the mooring at *z* = −600 m, with initial vertical wavelengths 2*π*/*k*_{z} incrementally looped from 70–140 m, horizontal wavelengths 2*π*/*k*_{h} from 35–70 km and wave–angles tan^{−1}(*k*_{2}/*k*_{1}) from –*π* through *π*. Rays are excluded if they do not pass above *z* = −200 m or if they do, no portion of the ray comes within 100 km on either side of Katrina. These limits are reasonable for inertial energy originating from the storm, and result in two (more manageable) groups of rays represented by Rays#1W and 1E respectively in Figure 2c. Ray#1W (1E) represents super- (sub-) inertial waves originating from the west (east) or cyclonic (anticyclonic) side of the Loop Current in the proximity of the storm's track. Other rays that do not pass through the mooring at *z* ≈ −600 m are also similarly traced using the same ranges of wavelengths and wave-angles, as well as the same “exclusion” criterion; examples are rays 2, 3 and 4 in Figure 2c.

[14] Ray#1W shows that the near-inertial wave energy observed at *z* ≈ −600 m (Figure 2b) originates near the surface (*z* = −100 m) approximately 70 km west and 20 km north of the mooring, i.e., near the Katrina's center on Aug/28/10:00GMT between the Loop Current and warm ring, in a region of positive *ς* so that *f*_{eff} > *f*. The ray propagates towards the base of the Loop Current (*z* ≈ −600 m of the mooring); the arrival time, 4∼6 days later, approximately agrees with that observed (Figure 1). The ray ‘stalls’ near *z* = −600 m (crowding of the daily markers ‘*’, for about 7 days), suggesting an accumulation of energy there. This coincides with the observed intensification of energy near this depth (Figure 2b). We explain the cause of this stalling below.

[15] Loop Current frontal cyclones are often seen in high-resolution satellite SST [e.g., http://fermi.jhuapl.edu/avhrr/gm/averages/index.html]. These cyclones originate as small perturbations along the highly sheared current on the western side of the Loop Current in the Yucatan Channel and amplify (in the model) through baroclinic instability over the north Campeche Bank as the Loop Current enters the deep water of the Gulf of Mexico; the LSU mooring is located where frontal cyclones often pass [*Oey*, 2008]. During Katrina, the model suggests that a subsurface cyclonic meander sat astride the mooring. Figure 3a shows this with *ς*/*f* (color) and velocity at *z* = −600 m where a subsurface cyclone with maximum *ς*/*f* ≈ +0.4 and a diameter of about 70–100 km is seen. From the surface, where *ς*/*f* ≈ +0.23, ray#1W first propagates downward (towards the Loop Current) through an environment of weaker and slightly negative *ς* before encountering the cyclone where *f*_{eff} increases under and east of the ray. The ray's intrinsic frequency tends to *f*_{eff} as (*k*_{h}/*k*_{3})^{2} (see equation 1) and the group velocity (see K85's equation 11) become small near the cyclone. There is also upwelling, *u*_{3} ≈ +30 m/day (not shown), which counters the downward *C*_{g3} and helps to maintain the vertical stalling.

[16] The role of strong positive ς and its gradients on stalling may be made more precise by examining how *k*_{h}^{2} and *k*_{3}^{2} behave along the ray near the cyclone. Taking the dot product of *k* with equation (2b), and using (1):

Here, several small terms involving ∇*N*^{2}, *w*, *u*_{zz} and *v*_{zz} are dropped, and *B* = *gρ/ρ*_{o}. Approaching the subsurface cyclone from northwest and surface, we have (k_{1}, *k*_{2}, *k*_{3}) = (>0, <0, >0); the vector (*k*_{1}, *k*_{2}) makes an angle a little less than 45° clockwise from the *x*–axis so that, since ∇ς points eastward towards the cyclone, we have *k*_{h}˙∇ς > 0. Also, ∂ς/∂*z* < 0 and therefore *k*_{h}^{2} tends to zero from (3a). This is confirmed by plots (not shown) which show that *k*_{h} decreases and *k*_{3} increases near the cyclone. From (3b), the last two terms ∂*N*^{2}/∂*z* and (*k*_{h} × ∇*B*)∣_{z}/*f* are both positive (∇*B* points eastward towards the cyclone's center) so they cannot account for the increase in *k*_{3} and ∂ς/∂*z* < 0 is principally responsible for the increase in *k*_{3}.

[17] *Wang* [1991] (based on *Mooers* [1975]) found that anomalously-low-frequency (*ω* < *f*_{eff}) waves from the cold side of a front can be trapped vertically subsurface where isopycnals become flat [see http://www.aos.princeton.edu/WWWPUBLIC/PROFS/PUBLICATION/oeyetal_footnote_on_alf_waves.pdf]. We find that some rays (about 10%) are anomalously-low-frequency. However, vertical trapping alone cannot explain why a ray stalls. Figure 3a shows that the ray at *z* ≈ ∼600 m comes very near the center of the cyclone (defined as the location where *ς*/*f* is a maximum ≈ +0.4), but does not cross it. This behavior is seen in Figure 3b which displays the *ς*/*f* as a surface towards which the ray propagates from above. In addition to being blocked from below, the ray bends northward being blocked also by the *ς*/*f*-ridge formed by the strong cyclonic meander, consistent with the above discussion on equation (3). Since *k*_{h}^{2} and *k*_{3}^{2} are nonnegative, (3) puts a strong constraint on the allowable space to which ray paths may traverse. As seen in Figures 2c, 3a, and 3b, the Ray1W cannot penetrate below the cyclone, nor to the east of the cyclone where the strong positive ς-ridge is present. Thus, near–inertial motions are stalled inside the cyclone for a relatively long period of about 7days before radiating horizontally and rapidly downward away from the mooring (Figure 2c).

[18] A similar “stalling” occurs for ray#1E (Figure 2c). However, after radiating away from the cyclonic ridge, since this ray is sub-inertial, it stalls a second time at *z* ≈ −950 m. Ray#1E is also strongly influenced by the Loop Current. It follows and remains in the near-surface anticyclone of the Loop Current for a relatively long time (5∼6 days) before propagating downward towards the mooring at *z* = −600 m. Though not shown here, other rays (by varying the initial wavenumbers; see above) originating on the western or cold (eastern or warm) side of the Loop Current behave similarly as ray#1W (1E). Similar results are also obtained for rays through *z* = −650 m (instead of −600 m); but rays below approximately *z* = −650 m are very different as they do not originate from the surface. In summary, observed intensification near *z* = −640 m at the LSU mooring may be explained by an accumulation of energy caused by stalling of near-inertial waves by a subsurface cyclone, whose high *f*_{eff}/*f* > 1 prevents energetic near-inertial motions from reaching greater depths. In the vertical, the cyclone acts like a near-inertial “umbrella” with its top at *z* ≈ −600 m to −650 m. This explains why observed near-inertial amplitudes are weaker for *z* < ≈ −1000 m. The arrival at the mooring of energetic super-inertial waves before sub-inertial (Figure 2a) is due to the strong influence of the Loop Current on the latter waves as they are forced to loop around the anticyclone before escaping to deeper levels.

[19] Other rays in Figure 2c illustrate different aspects of near-inertial spreading. Ray#2 begins near the surface between the Loop Current and warm ring. It propagates into the “chimney” in the ring (where *f*_{eff} is reduced; Figure 1). Ray#3 begins at the western side of the Loop Current but within it, and displays a round-the-Loop Current progression as it is being advected anticyclonically to the eastern side, in rough agreement with the numerical simulation (Figure 1). There is no stalling in these two cases. Finally, ray#4 begins inside the Loop Current in a region of strongly negative *ς*/*f* (≈ −0.4 at *z* = −100 m). This ray stalls at *z* ≈ −900 m where *ς*/*f* reaches a local maximum (≈−0.1) and the ray's intrinsic frequency ≈ *f*_{eff} ≈ 0.94 *f*. However, the modeled *ς*/*f* is complicated and *ς*/*f* decreases (more negative, not shown) below *z* ≈ −900 m. The combination of this and a strong downwelling velocity field, *w* ≈ −50 m/day, allows ray#4 to penetrate deeper.