AN ABSTRACT OF THE THESIS OF
Hongbo Qi for the degree of Doctor of Philosophy in Oceanography presented on May 4, 1995.
Title: Propagation of Near-inertial Internal Waves in the Upper Ocean
Redacted for Privacy
Abstract approved:
Roland A. de
Sz4e
Current meter data from two sites were analyzed for near-inertial motions generated by three major storms during the Ocean Storms experiment in the northeast Pacific Ocean. The most striking feature of the inertial wave response to storms was the almost instantaneous generation of waves in the mixed layer, followed by a gradual propagation into the thermocline that lasted many days after the initiation of the storm. The phase of nearinertial currents propagated upward below the mixed layer, confirming the downward energy radiation. The average downward energy flux during the storm periods was between 0.5 and 2.8 mWm2. The estimated vertical and horizontal wavelengths ranged from 150 m to 1500 m and from 140 km to 410 km, respectively. The propagation directions of near-inertial waves estimated from the phase relation between density and velocity mainly lay between northeast and south, indicating sources west of moorillgs. The di-

rections tended to rotate clockwise with increasing depth, consistent with the expected effect of the earth's curvature. The estimated horizontal wavelength and propagation direction were consistent with the horizontal phase difference between inertial currents at the two sites.
A ray-tracing model was developed to determine the ray paths of nearinertial wave groups observed at the two sites. The near-inertial waves were locally generated within a area 150 by 200 km east of storm tracks and west of the moorings. The horizontal wavelengths calculated by the model and estimated from the data both suggested that the observed near-inertial responses consist of a wide spectrum of near-inertial internal waves. The ranges they represented did not agree well. The partial initial surface inertial current fields calculated at the two sites demonstrated similar features during the three storm events and had similar patterns to the drifter measurements made in October. Ray paths showed the propagation asymmetry in north-south and vertical directions, indicating equatorward energy propagation because of the existence of turning latitude. The near-inertial responses described by the ray paths were overall consistent with the observations, but disagreed in certain details. Wave dynamics was inadequate to describe the near-inertial responses at 60 m before day 280 in October.

Propagation of Near-inertial Internal Waves in the Upper
by
Hongbo Qi
A THESIS submitted to
Oregon State University in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
Completed May 4, 1995
Commencement June 1995

Doctor of Philosophy thesis of Hongbo Qi presented on May 4, 1995
APPROVED:
Redacted for Privacy
Maj or Professor, represeting Oceanog4hy
Redacted for Privacy
Dean of College of Oceanic and ftmJspheric Sciences
Redacted for Privacy
\
I understand that my thesis will become part of the permanent collection of
Oregon State University libraries. My signature below authorizes release of my thesis to any reader upon request.
Redacted for Privacy
Qi, Author

ACKNOWLEDGEMENTS
First and foremost I thank my major advisor, Dr. Roland de Szoeke, for his excellent guidance, solid support, and sincere encouragement and friendship throughout my graduate study. Roland has been not only unselfishly sharing his professional wisdom with me but also kindly providing advice in my personal life. This thesis would not have been accomplished without his tremendous patience and investment of time over the years.
It has been a truly rewarding experience for me to work with his strict professionalism.
I am very grateful to the other members of my committee. Dr. Clayton
Paulson provided crucial support during the final stage of the thesis work. His data, valuable advice and insight are indispensable to this thesis. I greatly appreciate his unwavering support and ullderstanding. Dr. Murray Levine also provided much needed support at one time and has always been readily available source of advice and constructive comments. Attending Dr. Bob
Higdon's lectures was such an enjoyment and pleasure that I miss it even today.
Dr. George Somero's willingness to serve as committee member is much appreciated.
My thanks are extended to Dr. Charles Eriksen, who generously made the NP mooring data available to me and contributed to the first paper, and to Drs. Eric D'Asaro and Eric Kunze, who both made thoughtful comments on individual manuscripts. I am also grateful to many other faculty members who taught me various courses. Steve Gard, Eric Beals, Tom Leach and Dr.
Priscilla Newberger really deserve to be specially thanked for their timely

and unfailing help in dealing with the cyberspace, especially when the bits did bite.
Many fellow students and colleagues are always inspirational, friendly and supportive. It has been very delightful and educating to communicate, interact, and be friend with Ayal Anis, Fred Bahr, Lynn Berkery, Bo Huang,
Hongyan Li, Shusheng Luan, Alberto Mestas-Nünez, Chaojiao Sun, Robin
Tokmakian, Yunda Yao, Ed Zaron, Vassilis Zervakis, Shanji Zhang and so on. Friends and colleagues I have not specifically mentioned, thank you all.
I treasure those exciting and entertaining memories shared with all the friends from my beautiful hometown Qingdao. Guys, you are a truly talented group. Your companionship, party creativity, gifted vocal and dancing performance, and bravery in the "Goji" field were colorful enough to keep me from being frequently nostalgic. My life in Corvallis could not have been as comfortable and smooth as it has been without the great courtesy and hospitality of Jo Shaw and her family. My wife and I were certainly blessed and honored for being able to share those enjoyable moments in every holiday atmosphere.
I feel homely to have been integrated into the small community of Corvallis, a city I love very much despite its allergic grass and flowers. A special thanks is due to Dixon recreation center for providing superb exercise facilities, which are solely responsible for one of my non-academic successes reshaping my body with additional 20 plus lb. pure muscle, and a chance to know several new friends.
The greatest and most important support did come from my wife, Pei, my parents, and my brother and sister. They expected and helped me to

succeed with their perpetual love, intensive care, hearty encouragement, and unbelievable patience.
This study was supported by the Office of Naval Research.

CONTRIBUTION OF AUTHORS
Drs. Roland de Szoeke and Clayton Paulson were involved in the development, analysis, and writing of Chapter 2, and assisted in the interpretation of results in Chapter 3.
Dr. Clayton Paulson also provided the Cl mooring data.
Dr. Charles Eriksen provided the NP mooring data and made comments on Chapter 2.

Table of Contents
General Introduction
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
The Structure of Near-inertial Waves During Ocean Storms
Introduction
.............................
6
4
Observations
............................
Storm-generated near-inertial currents
Vertical wave propagation
............
10
....................
23
Horizontal wave propagation
Summary and discussion
..................
33
.....................
44
8
References
..............................
48
3
3.1
3.2
Propagation of Near-inertial Waves During Ocean Storms
Introduction
.............................
54
52
A ray-tracing model
........................
55
3.2.1
3.2.2
Basic equations and dispersion relation
Ray-tracing equations
..................
58
56
1

4
3.3
3.4
3.5
3.6
3.2.3
Applicability of the model
Sensitivity tests
...........................
63
62
Sensitivity to mixed layer depth and stratification 3.3.1
64
3.3.2
Sensitivity to direction of propagation and horizontal wavelength
.....................
68
3.3.3
Sensitivity to stretched vertical wavelength
.
.
.
Application to Ocean Storms
..................
77
74
3.4.1
Model inputs
........................
78
3.4.2
3.4.3
Generation locations and initial horizontal wavenumbers
...........................
79
Surface inertial current fields
.............
92
3.4.4
Comparison of the modeled and estimated horizontal wavelengths
....................
98
3.4.5
Ray paths of near-inertial waves
Summary and discussion
...........
103
.....................
114
References
..............................
118
General Conclusions 122
125 Bibliography
Appendix
Derivation of the Single Equation for v
131
............
132

List of Figures
Figure Page
2.1.
2.2.
Chart showing location of moorings Cl, NP, CO and W deployed during Ocean Storms
..................
9
Profiles of buoyancy frequency N(z) calculated from the vertically smoothed PCM profile data at mooring NP.
.
.
11
2.3. The temperature and density anomaly profiles as determined from the thermistor chain and PCM data, which were low-pass filtered (40 hour half-power) and decimated to daily values
........................
12
2.4. Wind stress and the amplitudes of complex-demodulated inertial currents at (a) Cl and (b) NP during the whole period of the Ocean Storms experiment
............
14
2.5.
Complex-demodulated inertial currents (left) and the estimated inertial shifts (right, %) at Cl and NP during the October storm
.........................
16
2.6.
Complex-demodulated inertial currents (left) and the estimated inertial shifts (right, %) at Cl and NP during the January storm
.........................
18

Figure Page
2.7.
Complex-demodulated inertial currents (left) and the estimated inertial shifts (right, %) at Cl and NP during the March storm
...........................
20
2.8.
2.9.
Conceptual picture of inertial wave generation by a largescale storm
..............................
22
Correlations and phases for various depth pairs calculated from the zero-lagged complex inertial amplitude correlation coefficient
........................
26
2.10. Estimated vertical wavelengths (a) and vertical group velocities (b) in the stretched coordinate. The 95% confidence limit is obtained from the bootstrap method.
.
.
.
28
2.11. Inertial energy density and vertical energy flux calculated over the individual storm periods
............
31
2.12. Averaged propagation directions of near-inertial waves and the correlations between potential density anomaly and rotary currents during each storm period. The 95% confidence limit is obtained from the bootstrap method.
40
2.13. Horizontal correlations and phases between inertial cur-
43
3.1. The relation between propagation time and mixed layer depth with different stratifications:
cph) and measured buoyancy frequency profiles
.......
66
3.2. Ray paths of near-inertial waves originating from different mixed layer depths in three-dimensional space and their projections on xy-, xz-, and yz-planes
.........
67
3.3.
Contours of propagation time calculated with different directions of propagation and horizontal wavelengths.
.
.
69

Page Figure
3.4. Ray paths of near-inertial waves propagating toward different directions in three-dimensional space and their projections on xy-, xz-, and yz-planes
.............
71
3.5. Ray paths of near-inertial waves with different horizontal wavelengths in three-dimensional space and their projections on xy-, xz-, and yz-planes
.................
73
3.6. The variation of propagation time with stretched vertical wavelength for given horizontal wavelength = 300 km.
The tested directions of propagation are
90°, _450, 450, and 90°
................................
75
3.7. Ray paths of near-inertial waves with different stretched vertical wavelengths in three-dimensional space (dashed lines) and their projections on xy-, xz-, and yz-planes
(solid lines)
..............................
76
3.8. The estimates of direction of propagation and inertial shift as model inputs, and the initial directions of propagation calculated from the model during the three storm periods at moorings Cl and NP
.................
80
3.9.
Generation locations and initial horizontal wavenumber vectors of near-inertial wave groups calculated by the model at moorings Cl and NP during the three storm periods
.................................
85
3.10. Contours of amplitudes of surface inertial currents calculated from the near-inertial energy observed over every one-day period at different depths at moorings Cl and
NP during the three storm events
................
94
3.11. Horizontal wavenumber spectra of surface inertial currents calculated at moorings Cl and NP during the three storm periods .............................
97

Figure Page
3.12. Comparison of horizontal wavelengths calculated from the model and estimated from the data during the three storm periods at moorings Cl and NP
.............
99
3.13. Ray paths of near-inertial waves in three-dimensional space and their projections on xy-, xz-, and yz-planes during the three storm periods at moorings Cl and NP.
.
105
3.14. Ray paths of near-inertial waves calculated at 0000 UT every day during the three storm periods at different depths at moorings Cl arid NP
.................
109

List of Tables
Table Page
2.1.
2.2.
Comparison of near-inertial wave parameters estimated from the Ocean Storms experiment with previous observations and predictions from Price's hurricane-response model
..................................
34
Horizontal wavelengths estimated from the horizontal phase differences between near-inertial currents at the same depth at two moorings during the October storm. .
47

Near-inertial oscillations are commonly observed in the upper ocean, dominating the internal wave spectrum.
In the ocean surface layer, they are primarily driven by atmospheric forcing, often during the passage of storms. Newly-generated near-inertial motions can persist for many days in the mixed layer before decaying. The decay of mixed-layer inertial energy is believed to be principally due to the propagation of near-inertial internal waves deeper into the ocean (Pollard, 1970, 1980; Kroll, 1975; Gill, 1984;
D'Asaro, 1985). Near-inertial internal wave plays an important role in the energy balance of the upper ocean because it represents a major contribution to the kinetic energy (D'Asaro, 1985) and provides an energy source for maintaining higher frequency internal waves through nonlinear interaction
(McComas and Muller, 1981; Munk, 1981; Muller et al., 1986). Propagation

of near-inertial internal waves is of particular interest because it redistributes the near-inertial energy in the ocean. Vertical propagation below the mixed layer transfers the near-inertial energy generated at the surface into the ocean interior. Observations of near-inertial motions below the mixed layer usually reveal upward phase propagation and therefore downward energy radiation
(Leaman and Sanford, 1975; Kundu, 1976; D'Asaro and Perkins, 1984). The efficiency of vertical transfer of energy through the thermocline is dependent on the horizontal length scale imposed by wind forcing o the mixed-layer inertial oscillations and the 3-effect, i.e., the latitudinal variation of the Conohs parameter (Gill, 1984; D'Asaro, 1989). Horizontal propagation makes it possible to exchange energy within the near-inertial internal wave band
(Munk, 1981) and to geographically homogenize the internal-wave energy distribution. To study the propagation of near-inertial internal waves in the thermocline is the theme of this dissertation.
Our understanding of the generation and propagation of near-inertial motions in the upper ocean has been greatly advanced only in the last a quarter of century as a result of numerous observations and modelings (Pollard,
1970; Pollard and Millard, 1970; Kroll, 1975; Price, 1983; Rubenstein, 1983;
Gill, 1984; Kundu and Thomson, 1985; D'Asaro, 1989). One of the latest observational efforts, the Ocean Storms experiment, was carried out in the northeast Pacific Ocean from August 1987 to June 1988 with the aim of improving understanding of upper ocean response to atmospheric forcing. This thesis presents an analysis of data from two moorings in this experiment and a modeling study of near-inertial wave propagation in the upper ocean.
In chapter 2, titled "The Structure of Near-inertial Waves During Ocean

3
Storms", the characteristics of near-inertial oscillations generated by three major storms which occurred in October, January and March during this experiment are described. Wave parameters and vertical energy flux are estimated from the vertical wave propagation by using some results from linear internal wave theory. A relation between density and velocity that gives the horizontal directionality of internal waves is derived. The propagation directions of near-inertial waves during the individual storm events are estimated using this relation.
This paper was co-authored with Drs. Roland A. de
Szoeke, Clayton A. Paulson, and Charles C. Eriksen and has been accepted for publication in the Journal of Physical Oceanography (Qi et al., 1995).
Hence it is copyright by the American Meteorological Society.
In chapter 3, titled "Propagation of Near-inertial Waves During Ocean
Storms", the near-inertial responses at depths below the mixed layer are viewed as the successive arrivals of wave groups propagating along ray paths from distant generation locations in the mixed layer. A ray-tracing model based on linear internal wave dynamics and the WKB approximation is developed to determine the ray paths. This paper was co-authored with with
Drs. Roland A. de Szoeke and Clayton A. Paulson and will be submitted for publication to a suitable journal.

Hongbo Qi, Roland A. de Szoeke and Clayton A. Paulson
College of Oceanic and Atmospheric Sciences, Oregon State University
Charles C. Eriksen
School of Oceanography, University of Washington
Submitted to Journal of Physical Oceanography,
American Meteorological Society, 1995, in press.

5
Current meter data from two sites were analyzed for near-inertial motions generated by storms during the ten-month period of the Ocean Storms experiment in the northeast Pacific Ocean. The most striking feature of the inertial wave response to storms was the almost instantaneous generation of waves in the mixed layer, followed by a gradual propagation into the thermoclirie that often lasted many days after the initiation of the storm. The propagation of near-inertial waves generated by three storms in October, January and
March were studied by using group propagation theory based on the WKB approximation. It was found that wave frequencies were slightly superinertial, with inertial shifts 1 3% in October and March and around 1% in
January. The phase of near-inertial currents propagated upward below the mixed layer, confirming the downward radiation of energy by these waves.
The average downward energy flux during the storm periods was between 0.5
and 2.8 mW m2. The vertical wavelengths indicated by the vertical phase differences ranged from 150 m to 1500 m. The vertical group velocity was estimated from the arrival times of the groups at successive depths. Using this in the dispersion relation, horizontal wavelengths ranging from 140 km to 410 km were obtained. A relation between density and velocity that gives the horizontal directionality of internal waves was derived. During the storm periods examined, the propagation directions of near-inertial waves mainly lay between northeast and south, indicating sources west of moorings. The directions tended to rotate clockwise with increasing depth, consistent with the expected effect of the earth's curvature. The estimated horizontal wave-

length and propagation direction were consistent with the horizontal phase difference between inertial currents at the two sites.
2.1
Introduction
Storms generate near-inertial oscillations in the upper ocean.
Newlygenerated near-inertial motions can persist for many days in the mixed layer before decaying. This decay is thought to be principally due to propagation of near-inertial internal waves deeper into the ocean (Pollard, 1980; Gill,
1984). This phenomenon has been observed and studied by many investigators (Webster, 1968; Pollard and Millard, 1970; Geisler, 1970; Kroll, 1975;
Muller et al., 1978; Price, 1983; Gill, 1984; Kundu and Thomson, 1985;
D'Asaro, 1985, 1989). The Ocean Storms experiment was conducted in the northeast Pacific Ocean from August 1987 to June 1988 with the aim of improving understanding of upper ocean response to atmospheric forcing.
This paper presents the results of analysis of data from two moorings in this experiment. The data are described in Section 2.2.
The Ocean Storms data provide an interesting opportunity to study the generation and propagation of wind-forced near-inertial motions. A most striking feature in the data is the nearly instantaneous generation by storms of near-inertial waves in the mixed layer, followed by the gradual advance of near-inertial wave packets into the thermocline, a process lasting many days after the commencement of the storm. The near-inertial peak is a prominent feature of the internal wave spectrum. Vertical propagation of near-inertial

7 waves below the mixed layer is of particular interest because it transfers the near-inertial energy generated at the surface into the ocean interior (Kroll,
1975; Leaman and Sanford, 1975; Price, 1983; Gill, 1984).
Presumably, the wind imposes a horizontal length scale on mixed-layer inertial currents, though this may be modified by variation of the Coriolis parameter during subsequent propagation (D'Asaro, 1989). The rate of vertical transfer of energy through the thermocline varies inversely with this length scale.
An accurate estimate of the horizontal scale of near-inertial currents is thus esseiltial for understanding the vertical energy budget.
We concentrated our analysis on three energetic storms which occurred in
October, January and March during the experiment, and studied the propagation of near-inertial waves from the standpoint of group propagation theory based on the WKB approximation. The characteristics of near-inertial motions during the individual storm events are described in Section 2.3. By examining the relations between the near-inertial currents at various depths, the phase-propagation and wavenumber in the vertical direction are estimated. The vertical group velocity is estimated from the arrival times of the groups at successive depths. Applying this to the dispersion relation, the horizontal wavenumber can be obtained. The average downward energy flux during the storm periods is calculated. These tasks are described in Section 2.4. A relation between density and velocity that gives the horizontal directionality of internal waves is derived in Section 2.5. The propagation directions of near-inertial waves during the individual storm events are estimated using this relation. The horizontal wavenumber is independently estimated from the directional information and the horizontal phase differ-

ro ences at two moorings.
2.2
Observations
The Ocean Storms experiment was conducted in the northeast Pacific
Ocean from August 1987 to June 1988. It consisted of current, temperature, salinity, and surface flux (momentum and heat) measurements. Its main objective was to study the three-dimensional response of the upper ocean to severe storms. Vector measuring current meters (VMCM) were deployed at seven depths on the subsurface mooring at site Cl (47°25.4'N, 139°17.8'W,
Fig. 2.1). They recorded velocity and temperature every 15 minutes at 20m intervals from 60 rn to 160 m and at 195 rn (Levine et al., 1990). Five
Seacat temperature-conductivity recorders on mooring Cl sampled every 6 minutes until 1200 on 25 January 1988 at 70 rn and 89 m and until 1200 on 4 March 1988 at 109, 128 and 150 m, and every 12 minutes thereafter.
The conductivity cell at 150 m failed early in the deployment. The upper two Seacat instruments also contained a pressure sensor, sampling every one hour until 1200 on 25 January 1988 and every two hours after that, to monitor the mooring motion. A profiling current meter (PCM) was installed on mooring NP (47°34.7'N, 139°23.3'W). This instrument profiled every 4 hours, averaging current, temperature and salinity into 5-rn depth bins from
195 m to 35 m. The data were further filtered with a 20-m-wide centered triangular window. The VMCM instrument at 120 m on Cl failed after 11
March 1988. PCM data at 35 m are usually unavailable because currents

141°W 140°W 139°W ci
4100
138°W
48°N 48°N
Co
43p0
60°N
50°N
4ON
160°W 14OW 120°W
139°W
47°N
138°W
Figure 2.1. Chart showing location of moorings Cl, NP, CO and W deployed during Ocean Storms.

10 drew the mooring too deep for the profiling range to reach this depth bin
(Eriksen et al., 1982). The useful ranges are from 50 m to 195 m in October, from 40 m to 190 m from November to May, and from 40 m to 195 m in the remaining months. The two moorings (Cl and NP) were 18.5 km apart in water of depth about 4200 m.
The changing buoyancy frequency profile N(z; t) was calculated from the
PCM data set, averaged over ten-day blocks (Fig. 2.2). Upper ocean density structure at the Ocean Storms site was characterized by a double pycnodine consisting of an upper seasonal thermocline and a deeper permanent halocline. The mixed layer depths were determined from temperature measured with a thermistor chain with 11 thermistors between 9 m and 105 m on mooring W (47°28.O'N, 139°59.2'W) before November 1 (when the mixed layer was shallow, Fig. 2.3); and from the profiles furnished by the PCM after November 1. A time series of wind speed and direction was obtained from an anemometer on mooring CO (47°28.9'N, 139°15.4'W), and was used to calculate a wind-stress time series (Fig. 2.4).
2.3
Some features of near-inertial currents observed during Ocean Storms are described in this section. We first isolate the signals of near-inertial oscillations from the raw time series and then attempt to interpret their spatial and temporal distributions. Near-inertial currents during Ocean Storms manifest all the important features and properties previously documented. The near-

11
[SI
50
-c 100
0
Q)
150
200
0 2
'1
4 6
Buoyancy Frequency (cph)
8
October
January
March
10
Figure 2.2. Profiles of buoyancy frequency
N(z) calculated from the vertically smoothed PCM profile data at mooring
NP. N(z) is averaged over the second tell-days of October and March and the third ten-days of January.

12
6
T (°c)
10 14
Temperature and density profiles during Ocean Storms at
25262728
I shift = 0.6°C
50
-E aa)
C
100
150
200
0
25 26 27 28
50
100
0 a)
C
150
200 time interval = 1 day
Figure 2.3. The temperature and density anomaly profiles as determined from the thermistor chain and PCM data, which were low-pass filtered (40 hour half-power) and decimated to daily values. The thick dashed line indicates the bottom of the mixed layer as determined from the maximum in dT/dz and dut/dz.

13 inertial responses at moorings Cl and NP are compared.
We used the method of complex demodulation (Perkins, 1970; Pollard and Millard, 1970) to separate the near-inertial velocities from the total
v are the eastward and northward velocity components, we calculated the complex amplitude
D(T) given by
D(T)
=
2T
T-T
W(r
(2.1) where W(t) is a tapered data window of length 2T. The magnitude and argument of D give the amplitude and phase of the inertial current at time
'r. The complex demodulation was performed at frequency o = 0.0625 cph rather than the local inertial frequency f (0.06153 cph at Cl and 0.06169
cph at NP) and on a window length of six demodulation periods (2T = 96 hours). This makes 2T an even multiple of the different sampling rates in the two data sets. For W(t) we used a triangular window, which is very effective in suppressing semidiurnal and diurnal tides. The window filter has a half-power bandwidth of 0.01329 cph, so that its main lobe encompasses the inertial frequency band.
The amplitudes of the complex-demodulated inertial currents at corresponding depths at the two moorings for the whole experimental period are shown in Fig. 2.4. Many storms struck the Ocean Storms instrumental array during the ten-month period. Strong storm-generated inertial responses due to events on 4 October, 16 November, 4 December, 13 January, 4 March and 2 April are evident, though a storm on 14 September apparently did not generate inertial currents. Storms in October, January and March gen-

E z
U)
U)
1)
-U
C
220 260 300 340 380 days
420 460 500 540
14
80m
U)
E
(.)
20 a-
E
1 00m
1 20m
1 40m
1 60m
195m
-D
0
E
U,
E
0
60m
80m
1 00m
1 20m
1 40m
1 60m
1987 1988
Figure 2.4. Wind stress and the amplitudes of complex-demodulated inertial currents at (a) Cl and (b) NP during the whole period of the Ocean Storms experiment. The cyclone symbol in the wind stress plot denotes storms on 4
October, 13 January and 4 March.

15 erated the most vigorous inertial oscillations. We concentrate on the time periods containing these three storms in the analysis presented in the following sections. The starting times of complex demodulation were set at
1200 on 4 October, 13 January and 4 March respectively, for both moorings. The resulting phase of the complex-demodulated inertial currents was back-rotated by (cr f)T, so that the phase of a purely inertial oscillation would not change with time. The phase of
D(T) drifts slowly in time during the main burst of amplitude. From the rate of change of this drift, we can estimate the central frequency w of the near-inertial currents. This usually differs by a few percent from the local f.
We call the proportional difference, ii = (w f)/f, the inertial shift. If the low-frequency vorticity of mesoscale variability, which would provide background vorticity to bias f and Dopplershift the frequency, were neglected, a positive inertial shift is indicative of the shallow vertical propagation angle of near-inertial internal waves.
The complex-demodulated inertial currents and the estimated inertial shifts during October are shown in Fig. 2.5. The pre-storm mixed-layer base was about 42 m to 45 m (Fig. 2.3), so the top of the PCM mooring was
thermocline. The storm struck on year day 277 and generated a prominent burst of inertial current amplitude. The VMCM at 60 m responded about one day after the storm. There was even further delay of response at greater depths. At 120 m, for example, the near-inertial response at Cl began on day
285, some 8 days after the passage of the storm. It took about 10 days for

inertial symbol
The speeds
-c
120
40
160
40
200
120
160
200
276 280
-
284 288
Days
292
40(cm/s)
296
Cl
300 276 304 280 284 288
Days
292 296
Cl
300 304
/
4 8 shifts
2.5. greater inertial denotes current
(right, than
12 the
OCT
16 cm/s
%)
8 at and
Cl at vectors
20 24
NP and commencement
28 of are inertial
NP
Complex-demodulated inertial plotted shifts the during storm
4 8 currents the at on
10
4 m between 0%
(left)
12
OCT
16
October and and
20
October. interval.
5% the storm.
24
The are
28 shaded. estimated
Inertial cyclone
....
NP
16

17 the inertial oscillations to propagate from 60 m to 195 m. Inertial currents at 60 m reached their maximum speed (about 0.3 m/s) after about a week and persisted at least another two weeks before decaying to the pre-storm level. In the early response, the amplitudes decreased with depth. After day
298, a core of maximum inertial energy was formed between 80 m and 120 m. The penetration of inertial oscillations became weaker at depths greater than 140 m. The overall pattern of the response was quite similar at the two moorings. The separation of the two moorings (18.5 km) was smaller than the frequently reported horizontal coherence scale of near-inertial currents, which ranges from tens to hundreds of kilometers (Webster, 1968; Pollard and Millard, 1970).
The inertial currents were dominantly superinertial in the envelope of high amplitudes associated with the storm. The inertial shift during the October storm was 1 3% and tended to increase with depth. This suggests that the near-inertial oscillations are propagating internal waves and that the waves found at greater depths have a more northerly origin (Kroll, 1975). Inertial currents rotated clockwise with increasing depth, indicating upward phase propagation and downward energy flux (Leaman and Sanford, 1975).
b. January storm
By January the mixed layer had cooled and deepened. The mixed-layer base before the storm on day 378 was between 100 m and 110 m. The response of the mixed-layer inertial currents to the storm on day 378 was nearly instantaneous and vertically uniform with only a few hours delay at the mixed layer base (Fig. 2.6). Another storm on day 389 enhanced the near-

120
180
180
-c
120
40
Days Days
376 380 384 388 392 396 400 404 376 380 384 388 392
I
396 400 404
1
1!II.UIIIMUI
40
160
200
1it1ii
S
.1.
kL1
-
160
200
12 16 20
JAN
24 28 4
FEB
8 12 16 20
JAN
24 28 4
FEB
8
Figure 2.6. Complex-demodulated inertial currents (left) and the estimated inertial shifts (right, %) at Cl and NP during the January storm. Inertial speeds greater than 8 cm/s and inertial shifts between 0% and 5% are shaded.
The inertial current vectors at NP are plotted at 10 m interval. The cyclone symbol denotes the commencement of the storm on 13 January.
LI

19 inertial currents in the mixed layer. After that, several less intense storms continued to strike the region and the inertial oscillations in the mixed layer persisted over 35 days. The amplitudes and phases were remarkably uniform throughout the mixed layer during this period.
During the first several days, inertial oscillations were largely confined to the mixed layer. The response at 120 m began gradually on day 385, and even later at greater depths. Correspondingly, the inertial shifts became positive, with values around 1%. The response pattern at the two moorings was quite similar, except that at Cl a local speed minimum was found between 120
140 m on days 380 384. This inertial-wave hiatus is clearly seen in the raw current time series (Fig. 2.8). Below the mixed layer, clockwise rotation of inertial currents with depth was seen, suggesting downward energy flux.
In March the mixed-layer base deepened to 120 m, which was about its maximum seasonal penetration. The strongest wind speed during the whole experimental period occurred
Ofl day 430, though a less intense storm had moved through the array one day earlier. The near-inertial response of the mixed layer resembled that in January, though the amplitudes were significantly smaller (Fig. 2.7).
Inertial response in the mixed layer may have been affected by the combination of wind stress and mixed layer depth
(Pollard and Millard, 1970), or by the phase of pre-existing inertial currents
(D'Asaro, 1985). Mixed layer depth seemed not to be the factor in this case because the pre-storm mixed layer depth was nearly the same in January and
March (Fig. 2.3). Stronger inertial currents existed before the March storm.

20
40
428 432
Days
436 440 444 448 452
40(cm/s) Cl
Days
456 428 432 436 440 444 448 452 456 ci
-c
120
160
200
40
120
40(crn/s) NP NP
160
200
4 8 12 16
MAR
20 24 28 4 6 12 16
MAR
20 24 28
Figure 2.7. Complex-demodulated inertial currents (left) and the estimated inertial shifts (right, %) at Cl and NP during the March storm.
Inertial speeds greater than 8 cm/s and inertial shifts between 0% and 5% are shaded.
The inertial current vectors at NP are plotted at 10 m interval. The cyclone symbol denotes the commencement of the storm on 4 March.

21
If the phase of the inertial currents generated by the March storm interfered with that of pre-existing inertial currents, the resulting currents in the mixed layer would be decreased. A portion of energy input by the wind stress would be lost to enhanced turbulent mixing. During the first few days after the onset of the storm, there were no near-inertial currents below the mixed layer base at Cl. The downward penetration of inertial currents became apparent after day 438. There were large positive inertial shifts associated with this downward penetration, while the inertial shifts were 1 3% in the envelope of high inertial amplitudes. Negative inertial shifts occurred in the mixed layer between days 438 and 446 at both moorings. Such subinertial frequencies are frequently observed and may be due to the interference of waves of different modes (Kundu and Thomson, 1985), or to the location of the moorings on the warm side of a mesoscale eddy which Doppler-shifted the inertial wave band (Kunze, 1985). The persistence of mixed-layer inertial currents was much shorter than in January. As in October and January, the responses at the two moorings were quite similar. Upward phase propagation, suggesting downward energy flux, was seen.
A conceptual picture of the generation of inertial waves by a storm is shown in Fig. 2.8. The inertial wave front that advances into the thermocline crosses each successive depth at a later time. Behind the front is an envelope of near-inertial waves. The response at each depth is made up of wave groups that arrive along almost-horizontal, slanting, ray-paths (perhaps multiple paths) from distant generation locations at the surface or in the mixed layer.

22
CM
CM
CM
CM
CM
CM o surface 0(100 km)
X to ti t3t4 mixed layer x4
60m t
80m lOOm
120m i 50'
-50 ..E-
1 60m
JAN FEB
Figure 2.8. Conceptual picture of inertial wave generation by a large-scale storm. The storm strikes simultaneously at the surface over a large area at time t0.
The mixed-layer inertial currents respond immediately. An inertial-wave front advances into the thermocline, crossing each current meter at moorings Cl, NP at a different time. The envelope behind the front is made up of groups of inertial waves arriving from distant locations along almost-horizontal, slanting ray-paths. The u-component current time series at Cl during the January storm is shown on the right.

23
2.4
In this section we use some results from linear internal wave theory to infer certaill properties of groups of near-inertial waves. Wave parameters, such as vertical and horizontal wavenumbers and vertical group velocity and energy flux, can be estimated from the Ocean Storms data sets by using these properties.
We take the view that storms generate groups of near-inertial internal waves made up of components of the form exp{ i
[kx + ly + Jm(z)dz wt]}, (2.2) where k, 1, rn(z), are wavenumber components and frequency.
Vertical wavenumber rn varies in the sense of the WKB approximation as the buoyancy frequency
N(z) varies according to the dispersion relation rn2(z) = K
N2(z)
(2.3) where K
= k2
+ 12. The WKB approximation is formally valid for vertical wavenumbers large compared to the scale of variation of the medium. This condition will turn out not to be very well satisfied. Even so, we will persist with the estimation. We will return to this point in the discussion. Vertical group velocity is given by c9z = ow
=
_I2
_
N2(z) hwm3(zY
(2.4)
It is convenient to introduce a stretched vertical coordinate
(2.5)
1

24 to eliminate the effect of vertical inhomogeneity (Leaman and Sanford, 1975);
Nr is a reference buoyancy frequency (taken to be 3 cph). The advantage of this is that the stretched vertical wavenumber
Nr
= N(Z)m(z)
(2.6) is constant, as is the stretched vertical group velocity
C9
N(z)
Nr
N2
= _K1!_.
Wit3
(2.7)
Consider the lagged complex inertial-amplitude correlation coefficient
F12(r) = yi2(r)e12(T)
D(t)D2(t+r)
[D(t)Di(t)]2 [D(t + T)D2(t
r)]
2
1
(2.8) based on inertial amplitudes computed from (2.1) at depths z1, z2.
If one assumes a single wave dominates, the phase of the zero-lagged correlation coefficient c12(0) gives the vertical wavenumber m = z2
12 z1
w/m. By varying the lag
T to give maximum y12(r), the time delay of the near-inertial wave group as it propagates vertically can be estimated, and hence its group velocity
U'gz
Z2Z1
.
Tmax
(2.9)
(2.10)
Given
Cgz,
(2.4) may then be used to estimate the horizontal wavenumber
Rh.
This formula is better conditioned for w f than (2.3).

25
We calculated the complex correlation coefficient F12(0) for various depth pairs. The time averaging was performed over 20 days, starting at 5 days after the onset of the storm so that the phase difference vs. depth was fully developed (Gill, 1984). Fig. 2.9 shows the correlations and phases of all depth pairs at mooring Cl, as well as their 95% confidence intervals obtained by
Monte Carlo simulation based on the bootstrap method (Efron et al., 1983).
For comparison, the corresponding depth pairs at NP are also shown, except that 190 m instead of 195 m was used for January and March because of data gaps. A negative phase means that inertial currents at z2 are turned clockwise by that amount relative to z1. In almost all cases involving pairs below the mixed layer, q12(0) was negative. Hence phase almost always propagated upward during the three storm events at both moorings. The only exception occurred at Cl in the 140-160 m pair in October. Pairs within the mixed layer exhibited negligibly small phase differences and correlations close to unity; the mixed layer response was nearly simultaneous. Internal consistency of the vertical turning below the mixed layer, i.e.,
= q22
+q52, was satisfied only for small separations, indicating that the single wave assumption is locally valid.
In October the clockwise turning from 60 m to 195 m was 107° at Cl and 112° at NP, but the amount of turning from 60 m to 120 m at Cl (83°) was nearly twice that at NP (43°). For pairs above 120 m, the turning at Cl was greater; below 120 m, the relation is reversed. A secondary peak in the buoyancy frequency occurred at 120 m, and this feature of the permanent py-

80
120
160
200
80
120
160
200
80
120
-
-c 160
0
200'
80
C
8 120 ci)
(I)
1 60
0.4
Cl: OCT
Correlation
0.6
0.8
,_fr-1
1.0 150
Cl: JAN
NP: OCT
i)i ii-,-i
I-:
:
I_-4
Phase (°)
100 50
I.
-''
/
I
.-.-
-4 y4H
I-4
J.-
-.-
Hi-(
-
200
80
120
NP: JAN
H4-4
0
160
200'
80'
120'
NP- MAR First Depth i°Om
160
200
0.4
0.6
0.8
Correlation
1.0 150 100
Phase
50
(°)
0
Figure 2.9. Correlations and phases for various depth pairs calculated from the zero-lagged complex inertial amplitude correlation coefficient. The upper
(first) depth of a pair is distinguished by different lines. The lower (second) depth of a pair is indicated by the ordinates. A negative phase means the second depth leads in phase. The 95% confidence limit is obtained from the bootstrap method.
26

27 dnocline appears to divide the water column and alter the propagation properties of inertial waves. In January most of the vertical turning was greater at Cl than at NP, except for pairs between depths below 140 m. However, the difference between the two moorings was less than 10°. In March inertial currents at Cl consistently showed greater turning. The greatest difference
(about 30°) resulted from the largest separation (60 195 m). For the same separation, the vertical turning in January was greater than in March.
In January mixed layer and thermocline currents were highly correlated
(> 0.8), but in March the correlations were less than 0.6. This is perhaps owing to the difference in the amplitude of the vertical group velocity (Pollard, 1980). A slowly moving wave group is more likely to be affected by the other variabilities in the upper ocean and exhibit lower correlations.
Least-square fits to the local estimates of the stretched vertical wavelength were obtained and are shown in Fig. 2.lOa with their 95% confidence intervals. At NP, only the depths corresponding to those at Cl were used in the estimation. The N(z)-scaled vertical wavelength was obtained from (2.6).
Corresponding to the maximum and minimum values of N(z) in each month, the ranges of vertical wavelengths in the thermocline are compared with previous observations (Pollard, 1980; Brooks, 1983) and with predictions from
Price's (1983) hurricane response model in Table 2.1. Their results are consistent with the lower and upper ends of our estimates, respectively. Pollard's estimate suggests higher internal modes, though Brooks' observation and
Price's model prediction indicate that low-mode structure is significant.

OCT Cl
OCT NP
JAN Cl
JAN NP
MAR Cl
MAR NP
400
Stretched Vertical Wavelength (m)
600 800 1000
(a)
I
I
FH
I
I
I
OCT Cl
OCT NP
JAN Cl
JAN NP
MAR Cl
MAR NP
(b) x x x x x
0.01
0.02
0.03
Stretched Vertical Group Velocity (cm/s)
Figure 2.10. Estimated vertical wavelengths (a) and vertical group velocities
(b) in the stretched coordinate. The 95% confidence limit is obtained from the bootstrap method.

29
We assume that the near-inertial wave groups originate from the mixed layer base, so only the depths below the mixed layer were used for January and March. At NP, only the depths corresponding to those at Cl were used.
The time averaging at z1 in (2.8) is always performed over the period of a prominent burst of inertial amplitudes in order to trace the maximum inertial energy. Fig. 2.lOb shows the least-square fits to the local estimates of the stretched vertical group velocity C. The magnitude of Cg was successively smaller in October, January and March. The unstretched vertical group velocity in the permanent pycnocline showed the same property because the
N(z) values remained nearly unchanged with seasons. The N(z)-scaled vertical group velocities are listed in Table 2.1.
C92 was greater at Cl than at
NP in October, but smaller in January and March.
Horizontal wavelength, Ah = 2'lr/Kh, is recorded in Table 2.1. In October near-inertial waves had comparable horizontal wavelength at the two moorings. The shortest horizontal wavelengths were found in January at both moorings. The horizontal wavelengths of near-inertial waves in March differed markedly. Since their 95% confidence intervals correspond to the vertical wavelength estimations, the uncertainty in the estimates was also quite large.

30
The energy input from strong atmospheric forcing to the surface mixed layer is an energy source of the deep-ocean internal wave field. The energy transfer from mixed layer to thermocline is accomplished by the downward propagation of near-inertial internal waves. The estimation of the vertical energy flux is thus important for understanding the energy budget of the deeper ocean internal waves. The horizontal kinetic energy of near-inertial waves can be estimated by computing the rotary current spectrum in the near-inertial frequency band f+Sw
1
Eh = po /
2 Jf-6w
S(w)dw, (2.11) where S(w) is the clockwise rotary current spectrum; we use 6w = 0.08f for the bandwidth. The rotary spectrum was computed on a 20-day time series, starting at the group arrival time determined by (2.8), so that the maximum near-inertial energy was followed. A boxcar window was applied to reduce the smearing effect. Total energy differs from horizontal kinetic energy
Eh by only vEh
(Fofonoff, 1969), where ii is the inertial shift, a few percent. The vertical energy flux in the thermocline was then estimated as
F = EhC9.
The spline-fitted profiles of energy density are shown in Fig. 2.11.
In the thermocline the inertial currents were most energetic in January because the energy transfer is more efficient with larger horizontal wavenumber. The subsurface energy peak in October appeared at Cl, but was smeared at NP due to the 20-day averaging. The energy level was higher at Cl than at NP in October and January, and mostly higher at Cl in March except at the anomalous depths between 140 m and 160 m.

31
50
0 20 40
Energy Density
(J/m3)
60 0 20
Cl
40 60
NP
'? 100
*.
ci a 1)
150
200
50
OCT
JAN
MAR
1
100
-c
-4-,
0 ci) a
150
200
0 2 4 6 0 2
Vertical Energy Flux (mW/rn2)
4 6
Figure 2.11. Inertial energy density and vertical energy flux calculated over the individual storm periods.

32
The profiles of the vertical energy flux in the thermocline are also shown in Fig. 2.11. They vary with depth by a factor of 3 5 depending on month and location, indicating that the WKB scaling assumption was not entirely fulfilled. The vertical energy fluxes were averaged over depths 60-190 m at
Cl and 50-190 m at NP in October, 100-190 m in January, and 115-190 m in March, respectively. They are recorded in Table 2.1 and showed temporal and spatial variability. The estimates of vertical energy flux for the
October, January and March storms at Cl are 2.66, 2.79 and 0.51
mWm2, respectively; the values at NP are 1.30, 1.68 and 0.47 mWm2. The typical rate for sources and sinks of internal waves is about 1 mW m2 (Briscoe,
1983). The annual average energy flux into the mixed layer by mid-latitude storms has been calculated as 1.44
mWm2
(D'Asaro, 1985).
If one assumes that dissipation takes place within the depth of the pycnocline (about
200 m), these vertical energy fluxes imply dissipation rates of 2 x 10-6 to
1 x i0 W m3. Such values compare reasonably with other observations during Ocean Storms (Crawford and Gargett, 1988) and to previous upper ocean observations (Osborn, 1980).
If the loss of inertial energy from the mixed layer is balanced by energy gain in the thermocline, a decay time scale
Td can be estimated from the ratio of the total energy above a certain depth zd, chosen as at 60 m in October,
105 m in January and 115 m in March, to the energy flux at zd
=
çsf
C Eh(z)dz
JZd
F(zd)
(2.12)
We assumed uniform energy level in the mixed layer and used whatever current meters were available in the mixed layer. In the October storm, when the

33 shallowest current meter was below the mixed layer, the energy density was extrapolated up to the mixed layer base. Horizontal energy flux divergence is neglected in this account of near-inertial energetics. The decay time-scales are listed in Table 2.1. They were comparable to the observed durations of inertial oscillations in the October and January storms. However, they were much longer than the inertial-oscillation duration in the March storm. This suggests that wave dispersion alone was insufficient to deplete the upper layer near-inertial energy and that some small-scale dissipative mechanism such as mixing played an important role in reducing the energy.
2.5
In this section we show how the horizontal directionality of internal waves can be obtained from the phase relation between density and rotary current at a single location. Conventionally, if a horizontal array of current meters is available, directionality of internal waves can be determined from horizontal phase relations between currents in the array (Schott and Willebrand, 1973;
Muller et al., 1978). By combining horizontal direction with the horizontal phase relation between near-inertial waves at two sites, an independent estimate of horizontal wavelength can be calculated and compared to the estimates obtained from the method in Section 2.4.
Neglecting the effects of mean shear, friction and horizontal inhomogeneity, propagating internal waves are governed by the following linearized equa-

Table 2.1. Comparison of near-inertial wave parameters estimated from the Ocean Storms experiment with previous observations and predictions from Price's hurricane-response model.
Cl
October
NP
Ocean Storms
January
Cl NP Cl
March
NP
Pollard
(1980)
"Site D"
Wavelengths
Vertical (m)
Horizontal (km)
Vertical group velocity (mday1)
Vertical energy flux (mWm2)
Decay time scale (days)
230-1300 240-1400 150-780 150-810 190-1100 260-1500 100-240
250 290 140 140 260 410 700-1700
7-39 6-34 5-25 5-27 4-20 4-22 0.03-3
2.66
1.30
2.79
1.68
0.51
0.47
1.4
16 19 23 29 54 70 5 za (m) 60
* moorings N, C and S in the Gulf of Mexico
60 105 105 115 115
Brooks
(1983)
N, C, S
1000
370
60
0.5-1.5
5
Price
(1983)
Model
1000
480
100
6

35 tions of motion:
(
+ if) (u + iv) =
(
Ou Ov
+ +
Ow
+ p,
(2.13)
(2.14)
(2.15)
Op
,ON2w = 0.
(2.16)
In these equations p is pressure perturbation divided by the average density
, p is the density perturbation, g is the acceleration due to gravity, u, v and w are velocity components corresponding to Cartesian coordinates x, y and z measuring distance eastward, northward and upward, respectively, and t is time. The hydrostatic and Boussinesq approximations have been made. By assuming plane-wave solutions of the form
P
=
Po i(kx++fmdzt)
Po
J the relation between density and rotary current amplitudes is
(2.17) gp0
(wf)mp Khe2),
(2.18) where
Kh is the magnitude of horizontal wavenumber vector, and 0 = arctan
(i/k) is the direction of horizontal phase propagation. Hence the phase difference between rotary current and density gives 0, by means of
(2.19)

36
The complex correlation between potential density anomaly and rotary current at the inertial frequency is oo* (t ) D (t)
_________ 1 _________ 1'
[ao*(t)ae(t)]2 [D*(t)D(t)]2
(2.20) where o8(t) is the complex demodulated potential density anomaly, similar to
D(t)
(Eq.
(2.1)).
The magnitude of F ( 1) measures the overall correlation.
If the near-inertial band were a sum of waves of the form
(2.17) over all horizontal directions, the expected value of F would be zero.
If the nearinertial band were dominated by a wave from one direction, the magnitude of F should be large, 0(1), and its phase would give an estimate of the left side of
(2.19), and hence the wave direction.
The above reasoning presumes that observations of current and density were made at fixed depths. Because of mooring motion, current-density time series from instruments fixed on the mooring line were made at a variable depth. By considering the correlation between pressure and density at fixed instruments on the mooring it is possible to correct the preceding analysis for the effect of mooring motion. The PCM time series, because they are constructed by binning and averaging the data as the instrument passes through a pressure range, do not need to be so corrected.
Assume that the near-inertial internal wave field currents and potential density anomaly consist of a clockwise-rotating component at frequency w and vertical wavenumber m:
(u + iv)(z, t) = Uoe mzwt)
(2.21)

37 where
U0
=
UO flcTe,
(2.22) ao(z, t) = a ei(m_wt) + (z) + e,
U0 =
9cj j is the mean potential density anomaly profile. The dependence on horizontal coordinates has been suppressed. The contribution by the mean current shear is negligibly small. On an oscillating mooring, current and density are observed at oscillating depths z(t) = + z0e i(mwt+cbz)
+ (2.23)
(the term m has been added to the phase q for convenience). The terms u, n, n are uncorrelated noises in the observations. Hence the time series observed at an instrument on the mooring is approximately
(i + i)(t) =
UO e mrwt+u)
[1 +
O(mz0)] + nu,
(2.24)
= ao0 ei(m t+r0) [1
+
O(mz0)]
+ ___zoei(n1r_wt+ + nç0,
(2.25) az where n'9 = n + n a/az.
The pressure data at 70 m and 89 m on mooring Cl indicate that the rms and maximum values of mooring oscillation are about 1 m and 10 m, respectively. Since the characteristic vertical scale of near-inertial waves is at least 0(100 m), the O(rnz0) terms are 's-' 0.01 to
0.1 and can be neglected.
The complex correlation between potential density anomaly and rotary current is
=
1 ________ 1
* aej 1
LIU + zv121 e1(o) [i
+
[1
+ 2'ycos( q5) + (]
[1+0 u02
f,2\]1
2)1
(2.26)

The relation, z0 =
-y has been used, with the gain y at the inertial frequency given by
1
Szz(f)]
(f) = C(f) [
,
(2.27) oo(f) where C0 is the coherence between potential density anomaly and mooring oscillation, and and ST9 are the autospectra of mooring oscillation and potential density anomaly, respectively. If the mooring oscillation and density are in phase, i.e., q
= no modification to the observed phase differeilce between rotary current and density is needed. Likewise, if there is no coherence between density and mooring oscillation, 'y = 0, no modification is required. However, if the mooring oscillation and density are coherent and not in phase, the true phase difference between rotary current and potential density anomaly is obtained from the apparent phase difference (u
'1cre)os, which is estimated from
(2.20), by means of
-
= arctan
[ tan
(qfu
9)obs e)obs +
1] where is given by sin(
1+cos(9)
(2.28)
(2.29)
Since there were no salinity measurements at the seven VMCM depths at mooring Cl, we infer the density time series at these depths from the
Seacat data and from the a0-T relation. The time series of potential density anomaly calculated from the Seacat data were linearly interpolated every 15 minutes to make the sampling rate consistent with the current measurements.

39
The permanent halocline lay in the depths 90 150 m. Within the halocline, the time series of aB at 100, 120 and 140 m were obtained by using a second degree polynomial to vertically interpolate and extrapolate o from depths
70, 89, 109 and 128 m. The use of second degree polynomial is justified by experimenting with the PCM data for the smallest phase difference between the measured and the inferred o. Outside the halocline, the time series of a at 60, 80, 160 and 195 m were derived from the temperature data measured by VMCMs by using the a0-T relation fitted by fifth-degree polynomials during the 24-day period at the same depths at NP.
We computed the inertial correlation and phase over record lengths of
20 days encompassing the response from each storm. At mooring Cl, the directions of propagation were estimated with mooring motion correction.
The differences of the direction of propagation estimated with and without mooring motion correction are in no case larger than 26° and mostly less than
10°.
This is because the magnitude of dimensionless parameter
'y
which figures in the correction, is never larger than 0.46, and usually nearer
0.1 or smaller.
Fig. 2.12 shows the correlations between potential density anomaly and velocity, and the estimated directions of propagation, during the periods of storm, as well as their 95% confidence intervals obtained by
Monte Carlo simulation using the bootstrap method.
At NP, the estimates are shown at 10-rn depth intervals starting 10 m below the mixed layer. Certain qualitative information on the sources of near-inertial waves may be obtained from the direction of propagation esti-

0.0
50
100
150
200
50
Cl: OCT
NP:OCT
100'
150
200'
50'
Cl: JAN
100
150
I
-c 200'
NP: JAN
Correlation
0:4
I
I
I
I
II
I
I
0:8
I..
I-H
I
:
1-1-4
1--I--I
1H
100
150'
200
50'
Cl: MAR
100'
150'
200
50
100
150'
NP: MAR
I
I-$-4
I
1-4-4
H-I
I-I--I
1--I-I
I-H
1-I--I
0.0
I I
0.4
Correlation
I
0.8
S
Direction of Propagation
45°
E 45°
H-I
I.
I-H
H
II
I
I
I:
I
I
I
H-I
I-I
H-I
HI
H-I
S
45°
E 45°
Direction of Propagation
Figure 2.12. Averaged propagation directions of near-inertial waves and the correlations between potential density anomaly and rotary currents during each storm period. The 95% confidence limit is obtained from the bootstrap method.

41 mated at one location. During all three events near-inertial waves mainly propagated in directions between northeast and south, indicating sources west of the moorings. Waves propagating toward the northeast will eventually be reflected at their turning latitudes where the north-south component wavenumber 1 becomes zero. Waves propagating toward the southeast, however, may come directly from sources in the northwest, or they may have been reflected at their turning latitudes.
The horizontal direction of propagation tends to rotate clockwise with increasing depth. This phenomenon is believed to be due to the /3-effect proposed by D'Asaro (1989). The latitudinal variation of f produces a temporal drift of north-south wavenumber, dl/dt
= -/3.
North-south wavenumber therefore decreases with time, while east-west wavenumber k remains constant, causing a southerly rotation with the age of the wave.
Inconsistency with the beta-dispersion occurred at about 80 100 m in
October and below 150 m at NP in January and March. In October, modeling efforts (D'Asaro, 1995a) showed that eddy activity played an important role in drawing mixed layer energy rapidly downward to form the energy maximum at about 100 m. The phase structure of near-inertial waves might be modulated by the eddy activity over the 20-day period. In January and
March at NP, the correlations at depths below 150 m were low and the uncertainty in the estimates was relatively large.
Combining the directional information about the near-inertial waves with the horizontal phase differences observed between the two moorings furnishes

an independent estimate of horizontal wavelength. The horizontal phase difference between the two moorings is given by the direct application of (2.8) to the complex-demodulated inertial currents at the same depth at Cl and
NP (represented by subscripts 1 and 2 respectively). Back rotation of the phase of the demodulated inertial currents is not necessary in order to compare the phase relation at the same frequency. The phases and correlations at corresponding depths are shown in Fig. 2.13. The inertial currents were highly correlated because the mooring separation is well within the estimated horizontal wavelengths. The inertial currents at NP led in phase at all depths during all three storm events. At least three moorings are needed to estimate horizontal direction and wavelength from current-current phase differences.
Failing this, a horizontal wavelength may be estimated from the horizontal phase difference
Iti(/-xcosO + LysinO), (2.30)
Ly)
(-6.9 km, 17.3 km) is the separation vector between the two moorings, and 0 = (O + 02)12 is the mean direction of propagation.
The estimation was carried out only at 60, 80, 100 and 195 m in October because the near-inertial waves at these depths propagated nearly in the same direction at the two moorings. The estimated horizontal wavelengths are shown in Table 2.2. They are comparable in order of magnitude to the estimates in Table 2.1.

43
0.85
I!I
120
.1
OCT
Correlation
0.90
0.95
1:1
H
H-I
H-I-F
III
1:1
1.00 40
Phase (°)
30 20 10
0
1:1
F-I--I
H
1:1
I
80
120 a
0
160
JAN
1.1
I
I
H H-F
1:1
II
III
HI
III I
MAR
120 iI.
200
0.85
H
H
H
F-H
F-H
0.90
0.95
Correlation
1.00 40
30 20 10
Phase
(°)
0
Figure 2.13. Horizontal correlations and phases between inertial currents at the same depth at moorings Cl and NP. A negative phase means the inertial currents at NP lead in phase. The 95% confidence limit is obtained from the bootstrap method.

44
2.6
the northeast Pacific were analyzed for near-inertial motions generated by storms. The data were taken during a ten-month period from August 1987 to June 1988. We concentrated on three major energetic storms which occurred in October, January and March, for which we studied the propagation of near-inertial internal waves into the thermocline.
The most striking feature of the inertial-wave response to the storms was the almost instantaneous generation of waves in the mixed layer, followed by a gradual propagation into the thermocline that lasted many days after the initiation of the storm. Our conceptual picture of what happened is summarized in Fig. 2.8. A storm strikes simultaneously over a wide area sending off groups of near-inertial waves along rays in many directions. These rays propagate at a very slight angle below the horizontal. In Fig. 2.8 we show just those rays that reach a given depth on a mooring, such as Cl, at a given time. Examination of the relations between the near-inertial currents at various depths revealed the following properties of the near-inertial wave groups that traveled along these rays.
1. Wave frequencies were slightly superinertial, with inertial shifts 1 3% in October and March and around 1% in January.
2. Inertial responses in the mixed layer were nearly in phase. The phase of near-inertial currents propagated upward below the mixed layer, confirming the downward radiation of energy by these waves. The average

45 downward energy flux during the storm periods was between 0.5 and
2.8
mWm2.
3. The phase-propagation speed and wavelength in the vertical direction were estimated. For the latter we obtained values in the range 150 m
1500 m.
4. The vertical group velocity was estimated from the arrival times of the groups at successive depths. From the dispersion relation, horizontal wavenumber can then be obtained. We obtained horizontal wavelengths in the range 140 km 410 km.
5. A near-inertial response in density, due to isopycnal displacement, was observed. The phase difference between near-inertial current and density indicates the direction of propagation. We found the dominant directions of propagation were between northeast and south, indicating sources west of the moorings. The dominant directions tended to rotate clockwise with increasing depth.
6. The horizontal wavelength from 4) and the propagation direction from
5) were consistent with the phase difference between inertial currents at the two moorings Cl, NP.
The estimation of these wave parameters is based on linear internal wave propagation theory and the use of the WKB approximation, and overlooks dissipative processes. However, mixing may affect the generation and propagation of near-inertial waves, especially in the upper ocean. D'Asaro's calculations (1995a, D'Asaro et al., 1995) indicate that linear wave dynamics are

insufficient to account for the evolution of near-inertial motions observed during Ocean Storms. The WKB scaling assumptions may not be fully satisfied in the upper ocean. Generally, when the vertical variation of the buoyancy frequency is rapid, wave reflection, which the WKB approximation does not countenance, may occur. The validity of the WKB approximation where wave scales are comparable to or larger than the scales of inhomogeneity is questionable. Kunze (1985) encountered a similar situation. Still, he showed that the errors due to the application of the WKB approximation beyond its formal limits of validity were small.
The spectral densities of the internal wave band are remarkably uniform in space and time, although the likely sources are very non-uniform. This suggests that low-frequency internal waves propagate horizontally to homogenize the internal-wave energy distribution. D'Asaro (1991) pointed out that only low-mode, low-frequency internal waves can propagate substantial distances (> 1000 km) in the ocean. Near-inertial internal waves are notably anisotropic in the upper ocean, and are a strong candidate for accomplishing this redistribution.
Acknowledgments.
The authors are grateful to Murray Levine for helpful discussions, and to Eric D'Asaro for comments on the manuscript. The wind data from mooring CO was provided by R. Davis, W. Large, and P.
Niiler.
This work was supported by the Office of Naval Research under
Grant N00014-90-J-1133 (H. Qi and R. A. de Szoeke) and N00014-90-J-1037
(C. A. Paulson).

47
Table 2.2. Horizontal wavelengths estimated from the horizontal phase differences between near-inertial currents at the same depth at two moorings during the October storm.
Depth (m)
01
60
80
100
6°
02
8°
'2
22°
& Ah
(km)
80
21° 26°
150
14° 1°
16°
100
195
60° 45° 35°
180

2.7
References
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Brooks, D. A., 1983: The wake of Hurricane Allen in the Western Gulf of
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Crawford, W. R., and A. E. Gargett, 1988: Turbulence and Water Property Observations During the Ocean Storms Program in the Northeast
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Canadian Data Report of
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D'Asaro, E. A., 1985: The energy flux from the wind to near-inertial motions in the surface mixed layer. J. Phys. Oceanogr., 15, 1043-1059.
D'Asaro, E. A., 1989: The decay of wind-forced mixed layer inertial oscillations. J. Geophys. Res., 94, 2045-2056.
D'Asaro, E. A., 1991: A strategy for investigating and modeling internal wave sources and sinks, in Dynamics of oceanic internal gravity waves,
Proceedings, 'Aha Huliko'a Hawaiian Winter Workshop, edited by P.
Muller and D. Henderson, pp. 451-465, University of Hawaii at Manoa.
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Phys. Oceanogr., in press.
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de Young, B. and C. L. Tang, 1990: Storm-forced baroclinic near-inertial currents on the Grand Bank. J. Phys. Oceanogr., 20, 1725-1741.

49
Efron, B., arid G. Gong, 1983: A leisurely look at the bootstrap, the jackknife and cross-validation. Amer. Statist., 37, 36-48.
Eriksen, C. C., 1991: Observations of near-inertial internal waves and mixing in the seasonal thermoclirie, in Dynamics of oceanic internal gravity waves, Proceedings, 'Aha Huliko 'a Hawaiian Winter Workshop, edited by P. Muller and D. Henderson, pp. 71-88, University of Hawaii at
Manoa.
Eriksen, C. C., J. M. Dahien and J. T. Shillingford, Jr., 1982: An upper ocean moored current and density profiler applied to winter conditions near Bermuda. J. Geophys. Res., 87, 7879-7902.
Fofonoff, N. P., 1969: Spectral characteristics of internal waves in the ocean.
Deep-Sea Res., 16, suppl., 58-71.
Fu, L. L., 1981: Observations and models of inertial waves in the deep ocean.
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Geisler, J. E., 1970: Linear theory of the response of a two-layer ocean to a moving hurricane. Geophys. Fluid Dyn., 1, 249-272.
Gill, A. E., 1982: Atmosphere-Ocean Dynamics. Academic Press, 662 pp.
Gill, A. E., 1984: On the behavior of internal waves in the wakes of storms.
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Greatbatch, R. J., 1984: On the response of the ocean to a moving storm:
Parameters and scales. J. Phys. Oceanogr., 14, 59-78.
Jenkins, G. M., and D. G. Watts, 1968: Spectral Analysis and Its Applications. Holden-Day, 525 pp.
Johnson, C. L., and T. B. Sanford, 1980: Anomalous behavior of internal gravity waves near Bermuda. J. Phys. Oceanogr., 10, 2021-2034.
Kroll, J., 1975: The propagation of wind-generated inertial oscillations from the surface into the deep ocean. J. Mar. Res., 33, 15-51.

50
Kundu, P. K., 1976: An analysis of inertial oscillations observed near Oregon coast. J. Phys. Oceanogr., 6, 879-893.
Kundu, P. K., and R. E. Thomson, 1985: Inertial oscillations due to a moving front. J. Phys. Oceanogr., 15, 1076-1084.
Kunze, E., 1985: Near-inertial wave propagation in geostrophic shear. J.
Phys. Oceanogr., 15, 544-565.
Leaman, K. D., and T. B. Sanford, 1975: Vertical energy propagation of inertial waves: A vector spectral analysis of velocity profiles. J. Geophys.
Res., 80, 1975-1978.
Levine, M. D., C. A. Paulson, S. R. Card, J. Simpkins and V. Zervakis,
1990: Observations from the Cl mooring during Ocean Storms in the
N.E. Pacific Ocean, August 1987 June 1988. Data Report 151, Ref.
90-3, College of Oceanography, Oregon State University, 156 pp.
Levine, M. D., and V. Zervakis, 1995: Near-inertial wave propagation into the pycnocline during Ocean Storms: Observations and model comparison.
J. Phys. Oceanogr., in press.
Matear, R. J., 1993: Circulation within the Ocean Storms area located in the Northeast Pacific Ocean determined by inverse methods. J. Phys.
Occanogr., 23, 648-658.
Muller, P., D. J. Olbers and J. Willebrand, 1978: The IWEX spectrum. J.
Geophys. Res., 83, 479-500.
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264-291, MIT Press, Cambridge, Mass.
Osborn, T. R.,1980: Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr., 10, 83-89.
Paduan, J. D., R. A. de Szoeke, and R. A. Weller, 1989: Inertial oscillations in the upper ocean during the Mixed Layer Dynamics Experiment
(MILDEX). J. Geophys. Res., 94, 4835-4842.

51
Perkins, H., 1970: Inertial oscillations in the Mediterranean. Ph. D. thesis,
MIT-Woods Hole, 155 pp.
Pollard, R. T., 1980: Properties of near-surface inertial oscillations. J. Phys.
Oceanogv., 10, 385-398.
Pollard, R. T., and R. C. Millard, 1970: Comparison between observed and simulated wind-generated inertial oscillations. Deep-Sea Res., 17, 813-
821.
Price, J. F., 1983: Internal wave wake of a moving storm. Part I: Scales, energy budget and observations. J. Phys. Oceanogr., 13, 949-965.
Schott, F., and J. Willebrand, 1973: On the determination of internal wave directional spectra from moored instruments.
134.
Wang, D. P., 1991: Generation and propagation of inertial waves in the subtropical front. J. Mar. Res., 49, 619-633.
Webster, F., 1968: Observations of inertial period motions in the deep sea.
Rev. Geophys., 6, 473-490.
Weller, R. A., and R. E. Davis, 1980: A vector measuring current meter.
Deep-Sea Res., 27, 565-582.
Zervakis, V., and M. D. Levine, 1995: Near-inertial energy propagation from the mixed layer: Theoretical considerations.
J. Phys.
Oceanogr., in press.

Storms
Hongbo Qi, Roland A. de Szoeke and Clayton A. Paulson
College of Oceanic and Atmospheric Sciences, Oregon State University
52
To be submitted to Journal of Physical Oceanography,
American Meteorological Society, 1995.

53
A ray-tracing model based on linear internal wave dynamics and the WKB approximation was developed to study the propagation of near-inertial internal waves generated by three major storms during the Ocean Storms experiment at two sites in the northeast Pacific Ocean. The near-inertial response at depths below the mixed layer was viewed as being made up of wave groups that arrived along ray paths from distant generation locations in the mixed layer. Using direction of propagation and wave frequency estimated from the data as model inputs, the ray paths describing the near-inertial responses at various times and depths as well as their generation locations were determined. The near-inertial waves observed at the two sites were locally generated and their generation locations were within an area 150 by 200 km east of storm tracks and west of the moorings. The horizontal wavelengths calculated by the model and estimated from the data both suggested that the observed near-inertial responses consist of a wide spectrum of near-inertial internal waves. The ranges they represented did not agree well. The partial initial surface inertial current fields calculated at the two sites showed similar patterns during the three storm events and were similar to the drifter measurements in October. Ray paths demonstrated the propagation asymmetry in north-south and vertical directions. The near-inertial energy propagation is eventually equatorward because of the existence of turning latitude. The near-inertial responses described by a group of ray paths either reaching or passing by the moorings were overall consistent with the observations, but disagreed in certain details. Wave dynamics was inadequate to describe the

near-inertial responses at 60 m before day 280 in October.
3.1
Introduction
Near-inertial oscillations commonly observed in the ocean surface layer are primarily generated by wind forcing, often during the passage of storms.
It is generally believed that propagation of near-inertial internal waves into the thermocline and deeper ocean is principally responsible for the subsequent decay of mixed-layer inertial energy (Pollard, 1970, 1980; Kroll, 1975;
Gill, 1984; D'Asaro, 1985). This in turn provides an important energy source for the ocean interior. The decay process has a time-scale of many days, depending
011 the horizontal scale of the wind forcing and the i effect, i.e., the variation of the Coriolis parameter with latitude (D'Asaro, 1989).
Strong near-inertial motions were detected at depths extending to 1500 m during the Ocean Storms experiment, which was conducted in the northeast Pacific Ocean from August 1987 to June 1988 to study the interaction of the atmosphere with the ocean during severe storms. Qi et al. (1995) investigated the propagation of near-inertial internal waves generated by three major storms at moorings Cl and NP in this experiment and described the near-inertial responses at depths below the mixed layer as being made up of wave groups that arrived along ray paths from distant generation locations in the mixed layer. This paper pursues this concept further by developing a ray-tracing model based on linear internal wave dynamics and the WKB approximation to determine these ray paths. The organization of this paper

55 is as follows. The ray-tracing model is established in Section 3.2. We stress the importance of the 3 effect in the model because near-inertial waves are very close to their turning latitude, poleward of which they cannot propagate.
Three basic invariants that prescribe the spatial structures of ray paths are derived. The ray-tracing equations can be analytically integrated.
In Section 3.3 we use the propagation time as an indicator to test the model sensitivity to mixed layer depth, stratification and various wave parameters constituting the three basic invariants. We apply the ray path equations to the propagation of near-inertial internal waves generated by the three major storms at moorings Cl and NP in Section 3.4. Using direction of propagation and wave frequency estimated from the data as inputs, ray paths describing the near-inertial responses at various times and depths are determined. Section 3.5 presents a summary and discussion.
3.2
A ray-tracing model
In this section we develop a ray-tracing model to study the propagation of near-inertial internal waves. The local dispersion relation based on the WKB approximation is derived from the basic equations of linear internal wave dynamics on a mid-latitude ,i3-plane. From the propagation and refraction equations governing the global changes of the wave parameters, we derive three basic invariants prescribing the propagation properties of near-inertial internal waves. The ray paths of wave groups can be analytically determined by specifying the initial conditions of the rays. The applicability of the ray

56 path equations is described.
3.2.1
Neglecting the effects of mean shear, friction and horizontal inhomogeneity, propagating internal waves are governed by the following linearized equations of motion: fv =
(3.1) av
+ fu = (3.2) o=_ap
P z p g, au av ow
(3.3)
(3.4) op
1öN2w = 0.
(3.5)
In these equations p is pressure perturbation divided by the average density
, p is density perturbation, g is the acceleration due to gravity, u, v and w are velocity components corresponding to Cartesian coordinates x, y and z measuring distance eastward, northward and upward, respectively, f is the
Coriolis parameter, N is buoyancy frequency, and t is time. The hydrostatic and Boussinesq approximations have been made.
Combining Eqs. (3.1) (3.5) (see Appendix), a single equation for v may be obtained. This is
/02
' 3/i 0)]
Ov v + z Ox
+
(3.6)

57
92/öx2
+ 92/öy2, and 9
= df/dy is the variation of the Coriolis parameter with latitude.
We take the view that groups of near-inertial internal waves are made up of components of the form exp[i
(kx + ly + mz
(3.7) where k, 1, rn are the x, y, z components of wavenumber vector K and w is the wave frequency, and that wave solutions are locally valid in the sense of the WKB approximation, i.e., the medium and wavenumber vector vary gradually on a scale of wavelength and wave period so that
/3
10K'
1 and
N Oz
10K'
1 where i = 1, 2 and 3, representing x, y, and z coordinates, respectively, ,\ is wavelength, and T is wave period.
Substituting (3.7) into (3.6) and neglecting all the terms of small order except the
/3 term, the local dispersion relation for linear near-inertial internal waves is
N2 /3kN2 w2 = f2 + (k2 + l2)+ m2 w m2
(3.8)
Since near-inertial waves are near their turning latitude, the latitudinal vanation of f becomes critical in affecting the propagation of near-inertial waves.
The /3 term is therefore retained to reflect such dynamics. By defiling
(3.9) = f2 + (k2 + l2),

two roots of (3.8) can be approximated by
=
+{+'r w0 m2]
(3.10)
The third root
73k
2 k2+l2+m2
(3.11) represents Rossby waves, which are not the interest of this study. The positive root in (3.10) describes clockwise-rotating upward-propagating near-inertial internal waves in the northern hemisphere according to (3.7). The dispersion relation can be further simplified by binomially expanding (3.10) and neglecting terms higher than 0(732) j3k N2
(3.12)
This is the dispersion relation to be used in formulating the ray-tracing equations.
I.J
LII.]
3.2.2
The propagation and refraction of wave groups are governed by a system of six ordinary differential equations (Lighthill, 1978) dx
dt5k =
Cgx =
(Ak +
\ N2
2) aw dy
5w dz
-
=
N2
= Ali, m
= Cgz = [A (k2 + 12) +
N2
(3.13)
(3.14)
(3.15)
- Sw dtSx = 0, (3.16)

59
= =Af@,
(3.17) where A = dt
1/w0
= _ =
(i + 12)
(3.18)
{
+
]
2m Oz
/3kN2/wm2, has the unit of time, and
Cgx, C99,
Cgz are the x, y, z components of group velocity C9. The dispersion relation
(3.12) has been used to obtain the right-hand-side expressions of the equations.
The appearance of the buoyancy frequency gradient in
(3.18) reflects the concept of the WKB approximation, i.e., waves sense the spatial variability of the medium only through propagation and refraction, not locally in the dispersion relation.
Three basic invariants are embedded in Eqs.
(3.13) (3.18).
Eq.
(3.16) indicates the conservation of east-west wavenumber k k
= constant along a ray.
(3.19)
The ratio of
(3.15) to
(3.18) yields d / N2
dtk m2)
(3.20) or its equivalent
N2
- = constant along a ray, m2
(3.21) indicating that stretching or shrinking of the wavenumber vector occurs along the gradient of buoyancy frequency. Using /3
= df/dy, the ratio of (3.14) to
(3.17) can be arranged as d (f2
+ l2 m2) = 0.
dt \
(3.22)

This leads to the invariant f2
+
N2
= constant along a ray, ni2
(3.23) which implies a planetary wave guide delimited by north and south turning latitudes where the north-south wavenumber 1 becomes zero and waves are reflected toward the equator. Waves will be amplified as they approach their turning latitudes. Wave frequencies w0, w as well as parameter A consist of the three basic invariants and are therefore conserved along a ray.
Wave energy propagates at the group velocities determined by (3.13)
(3.15). For near-inertial waves, the group velocities are inversely proportional to f, suggesting that waves generated at lower latitude propagate faster. For waves generated at the same latitude, however, larger vertical wavelengths make the energy transfer more efficient both horizontally and vertically. A well-known property indicated by (3.15) is that the rate of vertical transfer of energy varies inversely with the horizontal wave scale (Gill, 1984). In the northern hemisphere, wave groups decelerate while propagating northward, but accelerate while propagating southward. The /3 term in the dispersion relation introduces slight asymmetry of propagation in the east-west direction.
The initial conditions of the ray paths are specified as at t= 0:
X X0, YYo, f=f0 =f(y°), and z = z0.
(3.24)

61
It is straightforward to integrate Eq. (3.13) along a ray since the right hand side of the equation is composed of the invariants only. We relate the variation of wavenumber 1 to the Coriolis parameter f by defining the invariant
(3.23) as
(3.25) f'2 f2 + in2 so that
Tfl' 1 = sgn(l)(f ,2 f
2'\
)
, where sgn(l) defines a wave's meridional direction of propagation:
(3.26) sgn(l) =
1
+1, 1> 0 northward rays,
1,
1 < 0 southward rays.
Eq. (3.14) thus can be integrated in the form of the Coriolis parameter f = f'0 sin +
(3.27)
arcsin(f0/f'0) is determined from the initial condition. Northsouth wavenumber 1 varies sinusoidally with time
1 = cos
[ao
+ sgn(l)A/3±t}.
(3.28)
A wave propagating northward will travel (f'0 f)/i3 distance meridionally to reach its turning latitude in time (ir/2 ao)rn/A3N.
A more familiar expression of the temporal variation of wavenumber 1 is obtained by expanding
(3.28) into power series and neglecting terms higher than
0(E2)
1 = sgn(10) lo
9 t, (3.29) where
10 is the initial north-south wavenumber.
North-south wavenumber 1 therefore decreases with time, while east-west wavenumber k remains

constant, causing a southerly rotation with the age of the wave. The zcomponent propagation equation (3.15) can be transformed into an expression made up of only the invariants by introducing a stretched vertical coordinate
dz (3.30) and substituting (3.28) for wavenumber 1; N,. is a reference buoyancy frequency (taken to be 3 cph). Therefore, the three components of the ray path integrated by following a wave group are
N,.((
N2
2w) yo) = f'0 sin
[ao
+
fo,
sgn(l)
{f'sin2
[co +
[
A
+
\ m2 2
+ wm2jm
N t, where is the stretched vertical coordinate corresponding to z0.
(3.31)
(3.32)
''
1
(3.33)
62
3.2.3
Storms generate near-inertial oscillations in the mixed layer. The subsequent downward propagation of energy into the thermocline consists of a full spectrum of near-inertial internal waves, which may be represented by an infinite number of rays emanating from the forced region. Each ray path described by equations (3.31) (3.33) represents a single wave group of constant frequency. The variation of wave frequency at a fixed point thus implies

63 the arrivals of different wave packets. The recorded near-inertial oscillations are the superposition of wave packets described by a bundle of neighboring rays detected by the instruments on a mooring.
At a given point (x, y, ), the ray path is a unique function of the initial conditions
(x0, Yo,
) and the three basic invariants. The invariants can be constructed from wave parameters such as the horizontal direction of propagation and the horizontal and vertical wavenumber. Unlike conventional ray-tracing problems, the initial conditions appear here in the ray path equations as unknowns. However, can be observationally determined since the advance of near-inertial wave packets into the thermocline is usually thought to originate from the mixed layer base (Kroll, 1975; Gill, 1984). Hence, any one of the four variables among the horizontal and vertical wavenumbers
(Kh = (k2 + l2) and m), the direction of propagation (0 = arctan(l/k)) and the propagation time (t), together with the source location of a ray path
(x0,
Yo), can be uniquely determined from the ray path equations (3.31) (3.33) provided that the other three variables are known.
3.3
Sensitivity tests
The property of a ray path is determined by several quantities including mixed layer depth, stratification and various wave parameters constituting the three basic invariarits. To test the model sensitivity to these quantities is the task of this section. Horizontal wavelength and direction of propagation combine to give x and y component wavenumbers k and 1.
For given

64 horizontal wavelength and stratification, vertical wavelength varies monotonically with inertial shift, which indicates the vertical propagation angle of near-inertial internal waves. We vary each of these quantities within its reasonable range while keeping the others unchanged and construct the three basic invariarits at a fixed observation point intercepting the ray path. The ray paths equations (3.31) (3.33) are then solved for the wave's generation location
(x0, yo) and propagation time t. The latter is used as an indicator of model sensitivity.
The center of the mid-latitude 9-plane (0, 0) is set at mooring Cl (47°25.4'
N, 139°17.8'W, Qi et al., 1995), where the local inertial frequency f is 0.06153
cph. The fixed observation point (x, y, z) is chosen to be at (0, 0, 140) m. In tests of model sensitivity to direction of propagation and horizontal and vertical wavelength, we use the buoyancy frequency profile measured in October 1987 during the Ocean Storms experiment (Qi et al., 1995) to calculate the stretched vertical coordinates and assume a constant mixed layer depth of 40 m.
3.3.1
The model sensitivity to mixed layer depth and stratification is tested for two cases: case 1, mixed layer depth varies from 10 rn to 120 m at 10-rn interval, using a constant buoyancy frequency profile (N = 5 cph) below the mixed layer; case 2, mixed layer depth varies from 40 m to 120 m at 10-m interval, using the appropriate buoyancy frequency profiles measured during

65 various periods of the Ocean Storms experiment. We use the horizontal wavelength
.\h
= 300 km and the inertial shift v = 0.75%. These correspond to a stretched vertical wavelength ) = 755 m. The directions of propagation at the observation point are chosen to be 0 = +45°, representing near-inertial wave groups propagating northward and southward from the west.
The relation between propagation time and mixed layer depth is intuitively simple: it takes longer time for waves originating from shallower mixed layer to reach the same observation depth (Fig. 3.1). Originating from the same mixed layer depth, however, southward rays take longer time to arrive at the site due to their generation and propagation at higher latitudes
(therefore smaller wavenumber 1). The propagation times calculated in case
2 are greater than those in case 1 because the averaged value of buoyancy frequency between the mixed layer depth and z = 140 m is greater than
5 cph which results in greater values for the stretched vertical coordinates.
The spatial structures of the ray paths are shown in Fig. 3.2. For mixed layer depths shallower than 60 m in case 1 and 80 m in case 2, southward rays initially propagate northward and are reflected southward at the turning latitude 47.658°. These rays take more than twice as much time as the northward rays to reach the observation point. The horizontal structure of the ray paths is solely determined by the three basic invariants. The horizontal projections of ray paths originating from different mixed layer depths overlap each other because of the same invariants. Though the vertical structure of the ray paths is influenced by the buoyancy frequency profile, wave groups originating from different mixed layer depths in case 2 propagate along distinct ray paths in the upper 120 m where there is seasonal variability. The generation

1.IsI
- -
9=-45°, consnt N
9=45°, constant N
9=-45°, measured N
9= 45o, measured N 400
300
E
0
0
I.,
200
100
0 h=3001(m
X=755m v = 0.75%
10 20 30 40 50 60 70 80 90 100 110 120 130
Mixed Layer Depth (m)
Figure 3.1. The relation between propagation time and mixed layer depth with different stratifications: constant (N 5 cph) and measured buoyancy frequency profiles. 0 = +45° represent northward and southward directions of propagation, respectively. Wave parameters are horizontal wavelength
= 300 km, inertial shift i-'
0.75%, and stretched vertical wavelength
= 755 m.

-2
-
-
-
__
-.
00
-50 F
I
1
-50
:-'°:
_-
100
- -- I t
-
Figure 3.2. Ray paths originating from mixed layer depths (a) 10 m to 120 at 10-m intervals (constant N) and (b) 40 m to 120 m at 10-rn intervals
(measured N) in three-dimensional space (dashed lines) and their projections on xy-, xz-, and yz-planes (solid lines). The observation point is at (0, 0,
140). The generation locations are marked by dots (southward rays) and asterisks (northward rays). Wave parameters are:
A,,
= 300 km, m, v = 0.75%, and 0 = +45°.
Ac
= 755
67

locations of southward rays are further west than those of northward rays originating from the same mixed layer depth since a wave's traveling distance in the east-west direction varies linearly with propagation time.
3.3.2
The model sensitivity to directionality and horizontal wavelength is tested by varying the direction of propagation in the range 180° to 180° and the horizontal wavelength in the range 50 km to 2000 km for given (a) stretched vertical wavelength ) = 755 m and (b) inertial shift ii = 1.5%. Inertial shift in case (a) and stretched vertical wavelength in case (b) thus vary with the horizontal wavelength.
The propagation times calculated for different directions of propagation and horizontal wavelengths are shown in Fig. 3.3. A common feature in both
(a) and (b) is the approximate symmetry about 0 = ±90°. An exact east-west symmetry of propagation exists to the lowest order (neglecting the 3 term in Eq. 3.13). For 0 = ±90°, the east-west wavenumber k = 0 and a wave's propagation is confined only in north-south direction. For any given horizontal wavelength, propagation time increases from 0 = 180° to its maximum
0 = 180° mean that waves are observed at their turning latitudes. In the case of fixed stretched vertical wavelength, propagation time is less sensitive to directionality for horizontal wavelengths shorter than 200 km, which correspond to larger inertial shifts. As the horizontal wavelength increases,

1800
0
(a)
I
100
10
I I I I
I I I I I I
200 300 400 500 600 700
800 900 1000 1100 1200
Propagation Time (hours)
Inertial Shift (%)
1
10
-120°
-180°
102
Horizontal Wavelength (km)
Stretched Vertical Wavelength (m) io3
180°
120°
0
60°
0°
(b)
-60°
-120°
- 180° v=1.5%
102 io3
Horizontal Wavelength (km)
Figure 3.3. Contours of propagation time calculated with different directions of propagation and horizontal wavelengths, for given (a) stretched vertical wavelength ) = 755 m and (b) inertial shift ii = 1.5%.

70 the horizontal propagation becomes more dominant and the sensitivity to directionality is more evident.
For any given direction of propagation, the variation of propagation time with horizontal wavelength behaves differently in case (a) and (b). In case (a), the vertical propagation angle of near-inertial waves changes. Propagation time increases with increasing horizontal wavelength for waves propagating northward (0° < 0 < 180°). For waves propagating southward (-180° < 0 <
0°), however, propagation time increases with
)h to its maximum value at a critical horizontal wavelength and then decreases with increasing
\h. .Ah
itself varies with directionality, for instance, it has minimum value
)'h =
271 km for 0
450
There exists a very sensitive range in horizontal wavelength for waves propagating southward. The dramatic increase of propagation time in this range indicates the transition from waves propagating directly southward to those being reflected at their turning latitudes. The group velocity of near-inertial waves becomes smaller near the turning latitude as the north-south wavenumber 1 approaches zero. In this particular case, the range is from 200 km to 300 km.
It increases with increasing stretched vertical wavelength, but is independent of the observation depth. In case (b), the vertical propagation angle of nearinertial waves is fixed. Propagation time generally decreases with increasing horizontal wavelength because larger horizontal wavelength corresponds to larger vertical wavelength and hence greater group velocity.
Ray paths of near-inertial waves propagating toward directions from 180° to 150° at 30° intervals are shown in Fig. 3.4. The horizontal wavelength and the stretched vertical wavelength are chosen to be
Ah
= 300 km and

71
-50
E
-c
0-
-i00
-i50
_700
0
-50
E
-i00 a-
-iSO
200
-
-
Figure 3.4. Ray paths of near-inertial waves propagating toward different directions in three-dimensional space (dashed lines) and their projections on xy-, xz-, and yz-planes (solid lines). The observation point is at (0, 0, 140).
The dots of decreasing size indicate the generation locations of ray paths propagating toward _1800 to 150° at 30° intervals. Wave parameters are: horizontal wavelength
)'h
= 300 km, stretched vertical wavelength = 755 m, and inertial shift ii = 0.75%.

72
= 755 m, respectively. These correspond to an inertial shift v = 0.75%.
For the given wave parameters, all the five southward propagating waves initially propagate northward and are reflected at their turning latitudes. The propagation times are longer for southward rays. Waves propagating toward
0 = 90° have the highest turning latitude and have crossed the site twice.
The aforementioned propagation symmetry in east-west direction is evident.
Fig. 3.5 shows the ray paths of near-inertial waves with horizontal wavelengths from 100 km to 1000 km at 100 km intervals. The stretched vertical wavelength is chosen to be ) = 755 m. The assumed directions of propagation are +45°. Among the southward rays (0 = 45°), wave groups with
= 100 km and 200 km propagate directly southward from their generation locations. The other wave groups with larger horizontal wavelength initially propagate northward and are reflected southward at their turning latitudes.
The wave reflections occur at locations closer to the observation depth for larger horizontal wavelength. For both northward and southward rays, the generation locations are distributed anticlockwise from north to south as the horizontal wavelength increases. Wave frequencies of these ray paths decrease accordingly. It seems that for a given vertical propagation distance, near-inertial waves need to locally maintain appropriate vertical propagation angle, which is determined by the local inertial shift. The generation locations for waves propagating northward are in a more confined region. The initial directions of propagation tend to be more meridionally oriented as the horizontal wavelength becomes large. This enables waves to have limited east-west wavenumber k and more dominant north-south wavenumber
1 so that the wavelength can become very large as they propagate to higher

73
-50
0
.
-r
Figure 3.5. Ray paths of near-inertial waves with horizontal wavelengths from 100 km to 1000 km at 100 km intervals in three-dimensional space
(dashed lines) and their projections on xy-, xz-, and yz-planes (solid lines).
The observation point is at (0, 0, 140). The dots of decreasing size indicate the generation locations of ray paths with increasing horizontal wavelengths.
The stretched vertical wavelength ) = 755 m. The directions of propagation are (a) 0 = 45° and (b) 0 = 45°.

74 latitudes.
3.3.3
We examine the variation of propagation time with the stretched vertical wavelength ranging from 100 m to 2000 m for given horizontal wavelength
= 300 km. The corresponding inertial shifts are in the range 0.01% to 5%. Because of the propagation symmetry in the east-west direction, the directions of propagation are selected to be _900, _450, 45°, and 90°, respectively.
Propagation time generally decreases with increasing stretched vertical wavelength because group velocity is proportional to vertical wavelength for a given horizontal wavelength (Fig. 3.6). For waves propagating toward 90°, the transition from initial northward propagation to direct southward propagation causes the rapid decrease in propagation time, in this case, at about
= 1050 m. Propagation time takes maximum and minimum values for o =
90° and 90°, respectively. At very large vertical wavelengths, propagation time becomes less sensitive to horizontal directionality due to more dominant vertical propagation.
Fig. 3.7 shows the ray paths of near-inertial waves with stretched vertical wavelengths from 100 m to 1000 m at 100 m intervals and at 1500 m, propagating toward +45°. The pattern of the ray paths is rather similar to that of varying horizontal wavelength examined above, but in opposite sense.
For southward rays, wave groups with larger stretched vertical wavelength
= 1000 m and 1500 m are lot reflected. The turning latitudes are closer

75
10_i
Inertial Shift (%)
Iø1
4000
MIII1 j' 3000
2500
2000
C
1500
1000
5'
'S
'S
"S
'S
'S
'S
5'
'S
5'
'5
5'
"S
5'
'S
5'
'5'
"N
'N
"N
'N
'5"
5555,
'S
5555,
555'S
'5
55
55
55
'5
500
0_I-
102
Xh3001
Stretched Vertical Wavelength (m)
1
O=-90°
AO
9=45°
Figure 3.6. The variation of propagation time with stretched vertical wavelength for given horizontal wavelength A,, = 300 km. The tested directions of propagation are 90°, 45°, 45°, and 90°.

r
E
-c
0-
-iOO
-150
200
0
-
-50
E
-150 cia)
0
J200
76
200
-- -
-
Figure 3.7. Ray paths of near-inertial waves with different stretched vertical wavelengths in three-dimensional space (dashed lines) and their projections on xy-, xz-, and yz-planes (solid lines).
The observation point is at (0,
0, 140). The source locations are marked by dots of decreasing size for stretched vertical wavelengths from 100 m to 1000 km at 100 m intervals and 1500 m. The horizontal wavelength
'\h propagation are (a) 0 = 45° and (b) 0 = 45°.
= 300 km. The directions of

77 to the observation depth for smaller stretched vertical wavelength. The generation locations are distributed anticlockwise from north to south as the stretched vertical wavelength decreases. To maintain the constant horizontal wavelength
)h
= 300 km at the observation depth, the initial horizontal wavelength is smaller than 300 km for wave groups originating from south of observation site and greater than 300 km for wave groups originating from north of observation site.
3.4
In this section we use the ray-tracing model established in Section 3.2 to investigate the propagation of near-inertial internal waves generated by three major storms, which occurred in October, January and March during the
Ocean Storms experiment (Qi et al., 1995), at two moorings (Cl and NP).
We view the near-inertial responses at each depth below the mixed layer as successive arrivals of individual wave groups propagating along distinct ray paths which originate from the mixed layer base in the forced region and start at the onset of peak wind stress. The periods of main burst of amplitude at each depth are divided into 4-hour intervals at which the three basic invariants are constructed as functions of horizontal wavenumber by using direction of propagation and wave frequency estimated from the data.
The ray path equations (3.31) (3.33) then can be solved uniquely for the horizontal wavenumber
Kh and generation location
(x0, yo).
Consequently, the ray paths of near-inertial waves are completely determined.

78
The mid-latitude /3-plane is centered at mooring Cl (47°25.4'N, 139° 17.8'
W). Mooring NP (47°34.7'N, 139°23.3'W) is located at (-6.92, 17.27) km.
The commencements of wave propagation were set at 900 on 4 October 1987,
1100 on 13 January 1988 and 1200 on 5 March 1988 respectively, as determined from the wind stress data (Qi et al.,
1995). We assume constant mixed layer depths throughout the forced region and use the shallowest mixed layer depths estimated from the data in the calculations, which were 35 rn in October, 95 m in January and 105 m in March, respectively. The buoyancy frequency profiles averaged over twenty days after the onset of each storm are used in the calculations.
At both moorings, ray paths are calculated at 190 or 195 m and at 20-rn intervals from 60 m to 160 m in October, at 20-rn intervals from 120 m to
160 rn in January, and at 140 m and 160 m in March. We present the model inputs and results at both moorings during each storm event in the following.
3.4.1
Model inputs
The direction of propagation of wave groups is estimated every 4-hours over a one-day period from the phase difference between complex-demodulated inertial currents and potential density anomaly by using the method described in Qi et al.
(1995); the central wave frequency is estimated from the temporal drifting rate of the phase of complex-demodulated inertial currents.
Using the estimated direction of propagation and wave frequency in the geometric and dispersion relations, we express the three basic invariants as functions of horizontal wavenumber. The ray path equations (3.31) (3.33)

79 thus have three unknowns,
Kh, x0 and Yo, and can be solved uniquely.
Our ray-tracing model is valid only for propagating near-inertial internal waves, which are of positive inertial shift in the absence of mean shear.
Hence we solve the ray path equations only when the estimated wave frequency is greater than the local inertial frequency. The estimated directions of propagation and inertial shifts as model inputs are shown in Fig. 3.8. The inertial shifts during the three storm events were mostly less than 5% and varied considerably with time at each depth, implying the successive arrivals of different wave groups. Associated with the earliest inertial responses, the inertial shifts tended to decrease with time, a phenomenon due to the dispersive process of near-inertial internal waves, during which shorter waves (with higher wave frequency) propagate faster. The directions of propagation at both moorings during all three storm events were mainly between northeast and south and varied with time at each depth, more smoothly in October and March than in January in the envelope of high amplitudes.
3.4.2
The generation locations and initial horizontal wavenumber vectors of near-inertial wave groups calculated by the model at both moorings during the three storm events are shown in Fig. 3.9, respectively. The Ocean Storms site and the dated principal tracks of cyclone centers at sea level (Lindsay,
1988) are also shown. These generation locations indicated that the observed near-inertial waves were locally generated within an area 150 by 200 km,

(a)
280 290
Days
300 310
0
0.
o
120
60
-120
-180
140m ;:
6
4
2
0 rI)
,
I
)
160m
LU LD 3U
I
10 1)
OCT
D
NOV
10
Figure 3.8. The estimates of direction of propagation (solid lines) and inertial shifts (dashed lines) as model inputs, and the initial directions of propagation (heavy lines) calculated from the model in (a) October, (b) January and (c) March at Cl, and in (d) October, (e) January and (f) March at NP.
The arrows indicate the arrival times and input values of the ray paths to be plotted in three-dimensional space.

(b)
380
120m
C
120 o
0
-120
-180
140m
390
Days
400 410
2
0
6
4
CD
C o
120
60
0
-120
-180
195m
10 15 20
JAN
25
301
5 10
FEB
15 20 c)
Days
6
4
2
CD a
C-
MAR
Figure 3.8. (continued) roll
J]

(d)
280 290
Days
300 310
0
120
0
-180
-120
Figure
3.8. (coiltinued)
5 10
OCT
15 20 25 30
I
5
NOV
10
4
6 a
0
82

0
120 a o
-120
-180
(e)
380 390
Days
400 410
6
4
2 riD] nJ
0
0 o
120
60
0
-120
-180
(f)
430
JAN
440
Days
MAR
Figure 3.8. (continued)
FEB
450
6
4-
2
0'

east of storm tracks and predominantly west of moorings Cl and NP. The generation locations of near-inertial waves observed in January and March were in a smaller region than those in October because of deeper mixed layer.
The patterns of the generation locations of near-inertial waves were consistent with the movement of the storm systems and their relative positions to the moorings. The single strong storm recorded on October 4 was moving toward northeast.
Its closest position was about 600 km northwest of mooring Cl. Two major independent storm systems were present in the northeast Pacific Ocean around January 13. However, the inertial oscillations on January 13 at the Ocean Storms site were most likely due to the northern storm, whose center was about 900 km northwest of mooring Cl. Modeling study (Levine and Zervakis, 1995) indicated that use of the horizontal scale imposed by the translation of this storm and the bearing of storm track produced reasonable model results as compared to the observations. On the other hand, the center of the southern storm was about 2600 km southwest of mooring Cl on January 13 and translated nearly zonally, almost twice as fast as did the northern storm.
It should therefore impose a much larger horizontal scale on the generated inertial oscillations. Approaching January
14, the southern storm curved northeastward and moved within about 1100 km of the Ocean Storms site to complicate the inertial oscillations in the surface mixed layer. The strongest storm during the experimental period occurred at 1200 on March 5 and centered at only about 250 km north of mooring Cl. Considering the typical scale of mid-latitude storms as about
0(1000) km, the storm systems crossed over the moorings during the three storm events. Storms presumably generated stronger inertial currents at the

142°W
150
100
+ 60m
80m o lOOm
120m
0 140m
160m
* 195m
50
>
E
0
141°W
Longitude
140°W
4x1O(m)
). Ix1O(m1)
139°W
October
L
'G.
'I
7 p
48°N
60°N
/
47°N
I
50°N lilt
40°N
I.
I 60°W 140°W 120°W
-100
X (km)
-50 0 50
Figure 3.9. Generation locations and initial horizontal wavenumber vectors of near-inertial wave groups calculated by the model at moorings Cl and NP during the three storm periods. The orientation of the arrows indicates the initial direction of propagation. The length of the arrows represents the magnitude of horizontal wavenumber. All the thin arrows represent larger horizontal wavenumber than the thick ones. The mooring locations are marked.
The Ocean Storms site and the dated principal track of cyclone centers at sea level (Lindsay, 1988) are shown in the inset.
CD r

142°W
150
'
120m o 140m
160m
* 195m
100
Longitude
141°W
1 x io
(m')
+ 1x10(m)
140°W
I
139°W
I
January
48°N
50
.\
0
*4/
60°N
50°N
40°N
160°W 140°W 120°W
-100
X (km)
-50
II
0 50
47°N
Figure 3.9. (continued)

150
142°W o 140m
160m
* 195m
100
14!°W
Longitude
4x10(m')
- 1x10(m)
140°W 139°W
March
48°N
50
E
0
4
60°N
50°N
40°N
160°W 140°W 120°W
-100 x (km)
-50
I
0 50
47°N
4I1
Figure 3.9. (continued)

142°W
150
100
>
+
0
* o
60m
80m lOOm
120m
140m
160m
195m
50
Longitude
141°W
4x10(m') -
1x10(m')
140°W
E
*\'\
I
NP
:4
0
60°N
50°N
40°N
/
160°W 140°W 120°W
-100
X (km)
-50
4
/ 4
411
I
0
139°W
October
50
48°N
47°N
Figure 3.9. (continued)

150
142°W
120 m o 140m
160m
* 190m
100
141°W
Longitude
140°W
1 x i0
(m')
- 1x10(m1)
139°W
January
48°N
50
E
0
0
60°N
50°N
40°N
160°W 140°W 120°W
-100
X (km)
-50
II
0
P
50
47°N
CD
Figure 3.9. (continued)

150
142°W o 140 m
160m
* 190m
100
141°W
I
Longitude
-> 4 x 10
(m')
+ 1x10(m)
140°W 139°W
I
March
48°N
50
E
0
60°N
50°N
40°N
160°W 140°W 120°W
-100
X (km)
-50 0 50
47°N
0-
CD
Figure 3.9. (continued)

91
Ocean Storms site than on the other side of the storm tracks because the wild stress turns clockwise with time on the right side of the track and is therefore well-coupled to mixed layer inertial currents (Price, 1983).
The initial horizontal wavenumber field demonstrates a wide spectrum of near-inertial internal waves. The general tendency is that wave groups observed at shallower depths were generated at locations closer to the mooring and had larger horizontal wavenumber because shorter waves propagate faster. Wave groups had consistent southerly rotation from their initial directions of propagation due to the decrease of north-south wavenumber 1 with time caused by the latitudinal variation of the Coriolis parameter. The amount of turning at each depth tended to vary inversely with the inertial shift (Fig. 3.8). For near-inertial waves with the same vertical wavelength, the vertical and horizontal group speeds decrease as the wave frequency approaches the inertial frequency, prolonging the temporal drift of north-south wavenumber 1 before reaching a specific depth. Both northward and southward initial propagations were found during the three storm events.
For near-inertial waves propagating southward at the moorings, those with lower frequency (usually inertial shifts < 1%) mostly initially propagated northward and were reflected at their turning latitudes; those with higher frequency (usually inertial shifts > 1%) propagated southward directly from their generation locations.

92
3.4.3
Changes of amplitudes of near-inertial waves along ray paths can be derived from the conservation of wave action (Olbers,
1981;
Lighthill,
1978)
=0,
Dt wJ ()]
[
(3.34) where
E is energy density. If waves of constant frequency are considered so that the wave pattern at a fixed point in space is stationary, then Eq.
(3.34) takes the form v
{c9
()]
= 0,
(3.35) indicating that the wave action flux varies along ray tubes inversely with their cross-sectional area, i.e.,
C9
(3.36) where 5a is an infinitesimal cross section with unit vector n. Consider a ray tube intercepted by two horizontal planes at depth z and mixed layer base z0, the small element
8a0 in the plane z = z0 at
(x0, yo) is mapped into an element Sa with area
D(x, y)
D(x0, yo)
(3.37)
Thus by applying
(3.36), the conservation of vertical wave action flux is
[c
O(x,y)
()] zD(xo,yo)
{
C9
/E\1
( \Wi
)] zo
,
(3.38) indicating that the wave action varies along rays if there are changes in the group velocity or if the ray tubes are squeezed. Hence the surface inertial energy can be computed from the energy of near-inertial waves observed at

different depths. The Jacobian is calculated from the ray path equations
(3.31) (3.33) and is unity.
The contours of amplitudes of surface inertial currents calculated from the near-inertial energy observed over every one-day period at different depths are shown in Fig. 3.10.
They only represent partial initial surface inertial energy. The surface inertial current fields calculated at moorings Cl and NP manifested similar patterns during the three storm periods, but with smaller amplitudes at NP. A local maximum of inertial speed is found west of the two moorings in October. The amplitudes of surface inertial currents increased toward the northwest corner during all three storm events. The features of surface inertial current field in October are similar to those demonstrated by the drifter measurements (D'Asaro et al., 1995).
The horizontal wavenumber spectra of surface inertial currents can be constructed by averaging the calculated energy density within different initial horizontal wavenumber bands and are shown in Fig. 3.11. The confidence intervals determined from one standard deviation and the number of data points averaged within a wavenumber band are also shown. Surface inertial currents consisted of a wide spectrum of near-inertial internal waves during the three storm events, with horizontal wavelength ranging from 2 km to 900 km. The averaged slopes of the spectra are between 1.3 and 0.7.

C)
150
100
50
142°W
0)
I
20 30
Inertial Speed (cmls)
141°W
Longitude
140°W
I
40
I
50
139°W
October
I-
48°N
0.
-50
-100
150
100
50
-200
142°W
-ISO -100 -50
141°W
Longitude
140°W
0 50
139°W
October
I-
48°N
94
-50 47°N
-100
-200 -150 -100
X (km)
-50 0 50
Figure 3.10. Contours of amplitudes of surface illertial currents calculated from the near-inertial energy observed over every one-day period at different depths at moorings Cl and NP during the three storm events.

150
100
50
142°W
0
P
0
I
20
I
30
Inertial Speed (cm/s)
141°W
Longitude
140°W
I
40
I
50
139°W
January
48°N
C
47°N
-50
-100
-200
150
100
50
142°W
-150
141°W
-100
X (km)
Longitude
-50
140°W
0 50
139°W
January
I-
48°N
C-
47°N
-50
-100
50
-200
-150 -100
X (km)
-50 0
Figure 3.10. (continued)
95

150
100
50
0
142°W
10
I I
20 30
Inertial Speed (cm/s)
Longitude
141°W 140°W
I
40
I
50
139°W
March
48°N
-50
-100
150
100
50
-200
142°W
-150
141°W
-100
X (km)
Longitude
-50
140°W
0
139°W
March
50
47°N
48°N
-50
-100
-200 -150 -100
X (km)
-50
Figure 3.10. (continued)
0 50
47°N

n
1
U
10'
0 io3
1
Horizontal Wavenumber
Kh
(cpkm) io
10_2
10_i
2
Cl
October io
10_i -
10-
I
7
Cl
January
I
NP
October
NP
January
97
-iO3
.
0
U in2
101
0
March March
10 if13 io' io
10_2 10_i
Horizontal Wavenumber
Kh
(cpkm)
Figure 3.11. Horizontal wavenumber spectra of surface inertial currents calculated at moorings Cl and NP during the three storm periods. The confidence intervals are determined by one standard deviation and not plotted for negative values. The number of data points averaged in the wavenumber band is shown on the top.

3.4.4
Sillce the vertical wavelength can be estimated from the vertical phase difference between inertial currents at different depths (Qi et al., 1995), an independent estimate of horizontal wavelength is obtained by using the estimated wave frequency in the dispersion relation for linear near-inertial internal waves, in which the inertial shift is proportional to the ratio of vertical and horizontal wavelength squared. The horizontal wavelength so estimated is then compared to that calculated from the model. Downward energy propagation (upward phase propagation) corresponds to a clockwise-rotation of inertial currents with depth (Leaman and Sanford, 1975), i.e., a positive vertical wavenumber. Comparisons of horizontal wavelengths are thus made whenever the estimated vertical wavenumber is positive, and are shown in
Fig. 3.12.
At each depth during the three storm events, both the modeled horizontal wavelength hmOI and the estimated horizontal wavelength
Ahest exhibited considerable variation with time, suggesting that the responses consist of a wide spectrum of near-inertial internal waves. The time-average at each depth ranged from 50 km to 600 km for hmodel and from 150 km to 700 km for
'\hest
The estimated and modeled horizontal wavelengths did not agree with each other very well, except at 100 m at Cl and at 120 m at NP in
October, at 160 m and 195 m at Cl and at 190 m at NP in January, and at
195 m at Cl in March. At these depths at least half of the modeled horizontal wavelengths were within 50% of the estimated horizontal wavelengths.

(a)
280 290
Days
300 310
iO3
10
N
10
100m iO3
0
N
0
1-
102
101
1
0
I
D 10 1)
OCT
Z0 ZD iU
I
NOV
10
Figure 3.12. Comparison of horizontal wavelengths calculated from the model (solid lines) and estimated from the data (dashed lines) in (a) October, (b) January and (c) March at Cl, and in (d) October, (e) January and (f) March at NP.

100
(b)
380
120m
.
io3
102
N
0
101
140 m
160m
390
Days
400
I
410
10
195rn
20
JAN
25 30
1
5 10
FEB
15 k I
20
(c)
430 440
Days
450
102
101
0
N
0 iO3
102
1101
10
102
101
0
N
0
.
I
D
Figure 3.12. (continued)
II.)
1)
MAR
LU LD j

101
(d)
280
80mJ../
290
Days
300 lOOm
V 10
5)
10
MOm
310
-
-iO3
102
C a.
C
101
195m
1
5 10 15
OCT
20 25 30
1
5
NOV
10
Figure 3.12. (continued)

(e)
380
120m
55 io3
10
N
0
10
160m
390
Days
400
410 iO3
102
51
('5
0
N
0
0
10'
190m
III
19 20
JAN
'25
301
5
JJ
10
FEB
15 20
(1)
430 440
Days
450
140 m
.
io
10
1101
160 m
\}
-
1
190m
'cm '16702530
MAR
102
101
('5
0
N
0
0
Figure 3.12. (continued)
102

103
The estimated and modeled horizontal wavelengths differed markedly at 140 m and 160 m in March at both moorings, with he3t greater than
Ahmodel mostly by a factor of 5. The horizontal translation of a storm imposes a horizontal length scale on mixed-layer inertial currents (Kundu and Thompson, 1985). Levine and Zervakis (1995) estimated that the imposed initial horizontal wavelengths of mixed-layer inertial currents were about 900 km in
October, 550 km in January and 860 km in March, respectively. The subsequent horizontal wavelengths should be smaller than these initial horizontal wavelengths unless waves went through the wavelength-enlarging process, i.e., the north-south wavenumber 1 decreases as waves propagate northward.
A close examination in Fig. 3.8 and in Fig. 3.12 indicated that waves with the modeled horizontal wavelengths greater than the above imposed horizontal wavelengths at both moorings all initially propagated northward and lay to the north of their generation locations, either before or after the reflection at their turning latitudes.
3.4.5
Whenever the ray path equations can be solved at certain times and depths, we determine the ray paths between the mixed layer base and 195 m
(190 m at NP in January and March).
At each depth, we choose one ray path, whose modeled horizontal wavelength closely matched the estimated horizontal wavelength, to demonstrate

104 the three-dimensional spatial structure between the generation locations and the moorings in Fig. 3.13. The different arrival times of these ray paths are marked by arrows in Fig. 3.8, which also indicate the input values of the estimated directions of propagation and inertial shifts. The time interval between ticks on a ray path is two days.
The ray paths shown all originated from west of moorings and mostly mitially propagated northward, except the ray paths at 80 m at Cl in October and at 160 m at NP in March, which propagated directly southward. Starting at different latitudes, wave groups were of different group velocity and hence contributed to different stage of near-inertial responses. Some wave groups may be generated at locations very close to their turning latitudes, as indicated by the ray paths calculated at 100 m and 120 m at Cl and at
80 m at NP in October. Wave frequencies of the ray paths tended to increase with depth, with only a few exceptions.
Generally, waves with higher frequency are of higher turning latitude; as is the case in October at NP, where ray paths at greater depth came from further north. However, since the turning latitude is actually determined by f2
+ 12N2/m2 and the direction of propagation may change the weight of north-south wavenumber 1 in total wavenumber
(k2 + l2), so the turning latitude of certain waves may be even higher than that of waves with slightly higher frequency. This scenario can be seen at 140 m at Cl in October. The reflection of wave groups occurred often at upper depths, for instance, above 70 m in October, above 120 m in January and above 130 m in March, because the wave frequencies were very close to inertial. Asymmetry between north and south propagation was evident. The shrinking arid stretching of y-distance between ticks on a ray

105
0
(a)
It
-50
I-
-iSO
200
-
-20
0
T
I
I
I
0
I
-I
-iSO
0-
200 t t
1
Figure 3.13. Ray paths of near-inertial waves in three-dimensional space
(dashed lines) and their projections on xy-, xz-, and yz-planes (solid lines) in
(a) October, (b)January and (c)March at Cl, and in (d)October, (e)January and (f)March at NP. The time interval between ticks on a ray path is two days. The dots of decreasing size indicate the generation locations of ray paths derived from increasing depths.

0
(c)
-
-150 -2O0'
1
-
-
--
II
-'
(d)
-50
-50 i
-200
0
106
-200
-
Figtj
313
(Continued)

-150
_100
I
(e)
50'
0
-50
1
-ioor
E
1
-200
- -
--
(1)
0
-100k
-150
200çjLt7/_/150°
I
-200
-
-L
Fig 3.j3
(Continued)
107

path indicates that wave groups decelerated while propagating northward and accelerated while propagating southward. The near-inertial energy is eventually carried to the equator. In the vertical, wave groups propagated faster in weaker stratification. The equal x-distance between ticks on a ray path reflects the linear relation between x and propagation time t.
The successive arrival of numerous rays at each depth constitutes the persistent near-inertial responses. For clarity, we show only those ray paths calculated at 0000 UT every day during the period of strong inertial oscillation as well as the amplitudes of complex-demodulated inertial currents at each depth in Fig. 3.14. The estimated earliest arrival times of wave groups at each depth (Qi et al., 1995) are marked by ticks in the figures. At a given time and depth, a mooring may be surrounded by an infinite number of ray paths. Rays calculated at a specific time and depth would cross the mooring string exactly at that time and depth and then propagate away from the mooring. Some neighboring rays passing by the mooring may also contribute to the near-inertial responses at the mooring if the horizontal distance between the mooring and the position of rays is smaller than the horizontal wavelength of wave packets. It is the superposition of these ray paths (solid lines in the figures) that makes up the observed near-inertial oscillations.
As an expected feature of the model, the calculated ray paths represent the observed near-inertial responses at a specific depth by overlapping the high amplitudes of near-inertial currents. The majority of the ray paths were responsible for the near-inertial oscillations at all depths. Nevertheless, we

109
-50
E
-
-100
-150
-200
-50
1)
-100
-150
-200
-50
-
-100
-150
-200
-50
-
-100
-150
0
(a)
280 290
Days
300 310
I
280 290
Days
300 310
10
OCT
'20
30
NOV
10
-200
10
OCT
20 30 10
NOV
Figure 3.14. Ray paths calculated at 0000 UT every day during the inertial responses at different depths in (a)October, (b)January and (c)March
plex-demodulated inertial amplitudes are darkly shaded, with the scale shown at the upper-left corner. The ticks indicate the estimated earliest arrival times of wave groups. A solid (dashed) ray path represents wave group with horizontal wavelength larger (smaller) than its horizontal distance from the mooring.

-50
-100
0
(c)
430
-150
-200
-50
V
-100
-150
Days
440
410 380 390
Days
400 410
-50
E
-100
-150
0
(b)
380
-200
-50
-
V
-100
-150
-200
10
20
JAN
390
Days
400
30 10
FEB
450
20 10 30 20
JAN
10
FEB
I iii
Days
-
:ti
-
I-i
MAR
Y.
-
20
-200
1
10
MAR
20 ii)
Figure 3.14. (continued)
110

øiJ
50
-100
-150
-200
-100
150
-200
-50
-100
-150
-100
-150
-200
-50
0
(d)
280
-50
290
Days
300 310
10
OCT
20 30
NOV
10
1
280 290
Days
300 310
10
OCT
20 30
NOV
10
Figure 3.14. (continued)
111

-50
E
-100
-150
0
(e)
380
-200
-50
()
-100
-150
-200
10 20
JAN
390
Days
400
30
410
10
FEB
20 10
380
20
JAN
390
Days
400
30
410
10
FEB
20
430
Days
440 450
0
-50
-
5)
-100
-150
-200
-50
-
5)
-100
-150
-200
1
430
Days
440
10
MAR
20
450
30
Figure 3.14. (continued)
10
MAR
20
112

113 compare these ray paths to the observations in a variety of ways. Ray paths describing the late near-inertial responses at upper depths (above 120 m in
October and above 140 m in January and March) often did not contribute to the observed near-inertial responses at greater depths; they either arrived at those greater depths long after the observed near-inertial oscillations or propagated too far away from the moorings. Likewise, some ray paths did not generate near-inertial oscillations at upper depths at the moorings, as shown by the dashed lines
111 the figures, until they reached the positions where the moorings were within the range of horizontal wavelengths.
In October, certain ray paths calculated at 100 m, 120 m, 160 m and 195 m at Cl and at 140 m and 160 m at NP nearly matched most of the observed earliest arrival times of wave groups below 60 m, suggesting that it may be adequate to simplify the observed near-inertial oscillation as the evolution of a single wave packet, but such assumption evidently lacks support in a detailed comparison. There were no ray paths that could adequately describe the near-inertial response at 60 m before day 280. Although several ray paths did pass that period, they all indicated spurious near-inertial responses below
60 m well before the observed arrival times of wave packets. Other modeling studies (Crawford and Large, 1995; D'Asaro, 1995a; Levine and Zervakis,
1995) demonstrated that the inertial currents generated by the October storm extended 20 30 m below the mixed layer due to turbulent mixing, and linear internal wave theory can not replicate the timing of the observed energy propagation from the mixed layer during that period. Ray paths at Cl and
NP exhibited rather similar pattern. There were fewer ray paths available in January because of the extended period of negative inertial shift at all

114 depths. Still, the ray paths could make up most of the observed near-inertial oscillations. Ray paths at 120 m at NP propagated much slower than those at Cl. Ray paths calculated below 160 m propagated too fast in the upper depths and hence indicated the near-inertial responses much earlier than the observation. It is inadequate to use the model of single wave evolution to describe the near-inertial oscillation in January.
Ray paths calculated ill March were overall consistent with the observation and similar at the two moorings. As in January, ray paths derived from the early responses at greater depths tended to initially propagate too fast and thus caused spurious near-inertial oscillations earlier than the observed arrivals of wave groups. The observation could not be simulated by the ray path of a single wave group.
3.5
A ray-tracing model based on linear internal wave dynamics and the WKB approximation was developed to study the propagation of near-inertial internal waves generated by storms. We applied this model to three major storm events, which occurred in October, January and March during the Ocean
Storms experiment, at moorings Cl and NP in the northeast Pacific Ocean.
We took the view that near-inertial responses at depths below the mixed layer were made up of successive arrivals of individual wave groups propagating along distinct ray paths originating from the mixed layer base in the forced region. These ray paths were determined by using the directions of

115 propagation and wave frequencies estimated from the data as model inputs and revealed the following features.
1. The near-inertial waves observed at moorings Cl and NP were locally generated and their generation locations were within an area 150 by
200 km east of storm tracks and west of the moorings.
2. The horizontal wavelengths calculated by the model and estimated from the data both suggested that the observed near-inertial responses consist of a wide spectrum of near-inertial internal waves. However, the ranges they represented did not agree well.
3. The partial initial surface inertial current fields calculated at moorings
Cl and NP showed similar patterns during the three storm events and were similar to the drifter measurements in October.
4. Ray paths demonstrated the propagation asymmetry in north-south and vertical directions. Because of the existence of turning latitude, the near-inertial energy propagation is eventually equatorward.
5. The near-inertial responses described by a group of ray paths either reaching or passing by the moorings were overall consistent with the observations, but disagreed in certain details.
6. There were no ray paths that could adequately describe the nearinertial responses at 60 m before day 280 in October, implying non-wave dynamics involved. The near-inertial responses at all depths might be

116 approximately attributed to the evolutioll of a single wave packet in
October, but not in January and March.
A major limitation of the model is its inability to determine the ray path when inertial shift is negative, which may result from the Doppler-shifting by mean flow or the straining of mesoscale eddy (Kunze, 1985) and the interference of different wave modes (Kundu and Thomson, 1985). At the Ocean
Storms site, the surface mean flows observed in October were prominently northeastward (D'Asaro et al.,
1995).
Considering the calculated generation locations being west of moorings, the Doppler shifts were always positive when near-inertial waves propagated northward and were either positive or negative when near-inertial waves propagated southward, depending on the direction of propagation.
For positive Doppler shifts, waves' intrinsic frequency is nearer inertial. Waves may encounter turning points as they penetrate the mean flows and therefore be reflected. Otherwise, waves' propagation may be refracted by the mean flows. The background mesoscale eddy field in October was dominantly anticyclonic (D'Asaro et al.,
1995). At moorings Cl and NP, the vorticity was weak, about O.02f, and slightly modulated the propagation of near-inertial waves according to D'Asaro (1995a).
However, their modeling study showed that near-inertial energy was trapped in region of positive vorticity, which is opposite to the results revealed by
Kunze's theory (1985). What is more important in affecting the ray path structure is the horizontal flow field below the mixed layer, which was unfortunately not densely surveyed during the Ocean Storms experiment. The inverse analysis by Matear (1993) demonstrated that the mean flow at 200

117 m rotated southward relative to the surface flow and more small eddy features developed. We expect these would have a measurable impact on the ray paths. The direction of propagation and wave frequency estimated from the phase information were subject to phase interference of different modes.
Moreover, when there were multiple forcings as in January and March, phase information presumably related to earlier wave groups might be contaminated by that of later wave groups because vertical phase propagation was much faster than vertical group propagation.

References
Anderson, D. T., aild A. E. Gill, 1979: Beta dispersion of internal waves
Geophys. Res., 84, 1836-1842.
Bell, T., 1978: Radiation damping of inertial oscillations in the upper ocean.
J. Fluid Mech., 88, 289-308.
Crawford, G. B., and W. G. Large, 1995: A numerical investigation of resonant inertial response of the ocean to wind forcing. J. Phys. Oceanogr., in press.
D'Asaro, E. A., 1985: The energy flux from the wind to near-inertial motions in the surface mixed layer. J. Phys. Oceanogr., 15, 1043-1059.
D'Asaro, E. A., 1989: The decay of wind-forced mixed layer inertial oscillations. J. Geophys. Res., 94, 2045-2056.
D'Asaro, E. A., 1995a: Upper ocean inertial currents forced by a strong storm, Part III: Modeling. J. Phys. Oceanogr., in press.
D'Asaro, E. A., 1995b: Upper ocean inertial currents forced by a strong storm, Part IV: Interaction of inertial currents and mesoscale eddies. J.
Phys. Oceanogr., in press.
D'Asaro, E. A., C. C. Eriksen, M. D. Levine, P. P. Niiler, C. A. Paulson, and P. Van Meurs, 1995: Upper ocean inertial currents forced by a strong storm, Part I: Data and comparisons with linear theory. J. Phys.
Oceanoyr., in press.
de Young, B. and C. L. Tang, 1990: Storm-forced baroclinic near-inertial currents on the Grand Bank. J. Phys. Oceanogr., 20, 1725-1741.
Fu, L. L., 1981: Observations and models of inertial waves in the deep ocean.
Rev. Geophys. Space Phys., 19, 141-170.
Gill, A. E., 1982: Atmosphere-Ocean Dynamics. Academic Press, 662 pp.
Gill, A. E., 1984: On the behavior of internal waves in the wakes of storms.
J. Phys. Oceanogr., 14, 1129-1151.

119
Greatbatch, R. J., 1984: On the response of the ocean to a moving storm:
Parameters and scales. J. Phys. Oceanogr., 14, 59-78.
Hebert, D., and J. N. Moum 1994: Decay of a near-inertial waves. J. Phys.
Oceanogr., 24, 2334-2351.
Kroll, J., 1975: The propagation of wind-generated inertial oscillations from the surface into the deep ocean. J. Mar. Res., 33, 15-51.
Kundu, P. K., and R. E. Thomson, 1985: Inertial oscillations due to a moving front. J. Phys. Oceanogr., 15, 1076-1084.
Kunze, E., 1985: Near-inertial wave propagation in geostrophic shear. J.
Phys. Oceanogr., 15, 544-565.
Large, W. G., and G. B. Crawford, 1995: Observations and simulations of the upper ocean response to wind events during the Ocean Storms experiment. J. Phys. Oceanogr., in press.
Large, W. C., J. Morzel and C. B. Crawford, 1995: Accounting for surface wave distortion of the wind profile in low-level Ocean Storms wind measurements. J. Phys. Oceanogr., in press.
Leaman, K. D., and T. B. Sanford, 1975: Vertical energy propagation of inertial waves: A vector spectral analysis of velocity profiles. J. Geophys.
Res., 80, 1975-1978.
Levine, M. D., C. A. Paulson, S. R. Gard, J. Simpkins and V. Zervakis,
1990: Observations from the Cl mooring during Ocean Storms in the
N.E. Pacific Ocean, August 1987 June 1988. Data Report 151, Ref.
90-3, College of Oceanography, Oregon State University, 156 pp.
Levine, M. D., and V. Zervakis, 1995: Near-inertial wave propagation into the pycnocline during Ocean Storms: Observations and model comparison.
J. Phys. Oceanogr., in press.
Lighthill, J., 1978: Waves in Fluids. Cambridge University Press, 504 pp.

120
Lindsay, R. W., 1988: Surface meteorology during Ocean Storms field program. Technical Report, APL-UW TR 8823, Applied Physics Laboratory, University of Washington.
Matear, R. J., 1993: Circulation within the Ocean Storms area located in the Northeast Pacific Ocean determined by inverse methods. J. Phys.
Oceanogr., 23, 648-658.
Munk, W. H., 1981: Internal waves and small-scale processes, in Evolution of Physical Oceanography, edited by B. A. Warren and C. Wunsch, pp.
264-291, MIT Press, Cambridge, Mass.
Olbers, D. J., 1981: The propagation of internal waves in a geostrophic current. J. Phys. Oceanogr., 11, 1224-1233.
Paduan, J. D., and P. P. Niiler, 1993: Structure of velocity and temperature in the northeast Pacific as measured with Lagrangian drifters in fall
1987. J. Phys. Oceanogr., 23, 585-600.
Pollard, R. T., 1970: On the generation by winds of inertial waves in the ocean. Deep-Sea Res., 17, 795-812.
Pollard, R. T., 1980: Properties of near-surface inertial oscillations. J. Phys.
Oceanogr., 10, 385-398.
Pollard, R. I., and R. C. Millard, 1970: Comparison between observed and simulated wind-generated inertial oscillations. Deep-Sea Res., 17, 813-
821.
Poulain, P. -M., 1990: Near-inertial and diurnal motions in the trajectories of mixed layer drifters. J. Mar. Res., 48, 793-823.
Price, J. F., 1983: Internal wave wake of a moving storm. Part I: Scales, energy budget and observations. J. Phys. Oceanogr., 13, 949-965.
Qi, H., R. A. de Szoeke, C. A. Paulson and C. C. Eriksen, 1995: The structure of near-inertial waves during Ocean Storms.
J. Phys.
Oceanogr., in press.

121
Rubenstein, D. M., 1983: Vertical dispersion of inertial waves in the upper ocean. J. Geophys. Res., 88, 4368-4380.
Wang, D. P., 1991: Generation and propagation of inertial waves in the subtropical front. J. Mar. Res., 49, 619-633.
Zervakis, V., and M. D. Levine, 1995: Near-inertial energy propagation from the mixed layer: Theoretical considerations.
J. Phys.
Oceanogr., in press.

122
We investigated the propagation of near-inertial internal waves generated by storms in the upper ocean through data analysis and modeling. The most striking feature of the inertial-wave response to the storms was the nearly instantaneous generation of waves in the mixed layer, followed by the gradual advance of near-inertial wave packets into the thermocline, a process lasting many days after the commencement of the storm. We took the view that near-inertial responses at each depth below the mixed layer were made up of successive arrivals of individual wave groups propagating along distinct ray paths originating from distant generation locations in the mixed layer.
In Chapter 2, the following properties of the near-inertial wave groups that traveled along these rays were found:
Wave frequencies were slightly superinertial, with inertial shifts 1 3% in October and March and around 1% in January.
Inertial responses in the mixed layer were nearly in phase. The phase of near-inertial currents propagated upward below the mixed layer, con-

123 firming the downward radiation of energy by these waves. The average downward energy flux during the storm periods was between 0.5 and
2.8
mWm2.
The phase-propagation speed and wavelength in the vertical direction were estimated. For the latter we obtained values in the range 150 m
1500 m.
The vertical group velocity was estimated from the arrival times of the groups at successive depths. From the dispersion relation, horizontal wavenumber can then be obtained. We obtained horizontal wavelengths in the range 140 km 410 km.
A near-inertial response in density, due to isopycnal displacement, was observed. The phase difference between near-inertial current and density indicates the direction of propagation. We found the dominant directions of propagation were between northeast and south, indicating sources west of the moorings. The dominant directions tended to rotate clockwise with increasing depth.
The horizontal wavelengths estimated from the horizontal direction of propagation and horizontal phase difference between inertial currents at the two sites were consistent with those estimated from the dispersion relation.
In Chapter 3, we developed a ray-tracing model to determine the ray paths of near-inertial wave groups observed at different depths. The model results revealed the following features.

124
The near-inertial waves observed at moorings Cl and NP were locally generated and their generation locations were within an area 150 by
200 km east of storm tracks and west of the moorings.
The horizontal wavelengths calculated by the model and estimated from the data both suggested that the observed near-inertial responses consist of a wide spectrum of near-inertial internal waves. However, the ranges they represented did not agree well.
The partial initial surface inertial current fields calculated at moorings
Cl and NP showed similar patterns during the three storm events and were similar to the drifter measurements in October.
Ray paths demollstrated the propagation asymmetry in north-south and vertical directions. Because of the existence of turning latitude, the near-inertial energy propagation is eventually equatorward.
The near-inertial responses described by a group of ray paths either reaching or passing by the moorings were overall consistent with the observations, but disagreed in certain details.
There were no ray paths that could adequately describe the nearinertial responses at 60 m before day 280 in October, implying non-wave dynamics involved. The near-inertial responses at all depths might be approximately attributed to the evolution of a single wave packet in
October, but not in January and March.

125
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128
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132
To derive the single equation for v, we first combine Eqs. (3.1) and (3.2) by eliminating u
v =
/
OyOt) p.
(Al)
Then we differentiate (3.3) with respect to t and use (3.5) to replace the density perturbation term. The resulting equation is further differentiated with respect to z so that the vertical velocity gradient ow/az can be replaced by the continuity equation (3.4), in which u is substituted by (3.2),
3
DY)
1
32 32
8 \ 1
.
(A2)
Combining equations (Al) and (A2) by O(A1)/Ox
_(D2
3215/i a\1
DyOt a
Ox)
(A3)
Define the following differentiating operators:
=
02
3t2

133
3
£2
£3
=
£4 =
02
5
52
32
a
Eqs. (Al) and (A3) then become
£1v =
£2p,
£3V = £4p.
(A4)
(A5)
Since the order of operators
£2 and
£4 is interchangeable, thus pressure p can be eliminated from (A4) and (A5)
£4(L1 v) = £2(JL3v), (A6) i.e.,
32
011
(A7) z
32/0x2 +
02/0y2, and /3
= df/dy is the variation of the Coriolis parameter with latitude. (A7) has been integrated once with respect to time.