943

NAVAL POSTGRADUATE SCHOOL

Monterey, California

THESIS

OBSERVATIONS OF INERTIO- GRAVITY WAVES
WAKE OF HURRICANE FREDERIC

IN

THE

by

Lynn K. Shay

December 1983

Thesis

Advisor: R. L. El

sb(

?rry

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Observations of Iner tio-Gravity Waves in
the Wake of Hurricane Frederic

5. TYPE OF REPORT & PERIOD COVERED

Master ' s Thesis ;
December 19 8 3

6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR^

Lynn Keith Shay

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Naval Postgraduate School
Monterey, CA 93943

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Naval Postgraduate School
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December 1983

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IS. SUPPLEMENTARY NOTES

IS. KEY WOROS (Continue on tereree mid* II neceeamty and Idontlty by block number)

hurricane Frederic, iner tio-gravity waves, inertial waves,
group velocity, phase velocity, normal modes, barotropic,
baroclinic, complex demodulation, least squares, Brunt Vaisala
frequency

20. ABSTRACT < Continue on reveree eide II nacaaaary and Identity by block number)

Inertial waves excited in the mixed layer by hurricane
Frederic, had horizontal scales of approximately 1 to 2 times
the baroclinic Rossby radius of deformation (50 km) of the
first mode near the DeSoto Canyon. Initially, energy propagated
vertically at about 1.25 km/d and horizontally at about 80 km/d.
These waves spun down over e-foldine scales of four inertial
periods as energy propagated vertically at 270 m/d and horizontal-

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ly at 30 km/d. Inert io-gravity waves in the deep thermocline had
horizontal scales of 25 to 50 km and vertical scales approximate-
ly equal to the water depth. The energy of these waves was domin-
ated by the barotropic mode with some contributions from modes 1
and 2. These waves were not admitted to the shelf region because
the bottom slope was greater than the slope of the internal wave
characteristics.

The mean flow followed the isobaths at all levels, but it wa^
in the opposite direction in the bottom layer. The mean flow
initially decreased along the eastern boundary of the canyon as
the storm forcing readjusted the flow. Near-bo t torn temperature
variations of 4*C were associated with the storm surge and advec-
tion in the along-track direction, particularly along the north
rim of the canyon.

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Cfcservaticns of Inert ic-Gravity Waves in the Wake of

Hurricane Frederic

by

B. S. Oceanocrap

Lync K. Shay
hy, Florida Ins'titut

e of Technology, 1976

Submitted in partial fulfillment of the
requirements for the degree of

MASTER OF SCIENCE IN OCEANOGRAPHY

from the

NAVAL POSTGRADUATE SCHOOL
December 1983

D'JI
ABSTRACT .../. c ^

Inertial waves excited in the mixed layer by hurricane
Frederic, had horizontal scales of approximately 1 to 2
times the tarcclinic Rossby radius of deformation (50 km) of
the first mode near the CeSoto Canyon. Initially, energy
propagated vertically at about 1.25 km/d and horizontally at
80 km/d. These waves spun down over e-folding scales of

four inertial periods as energy propagated vertically at 270
m/d and horizontally at 30 km/d. Inertio-gravity waves in
the deep therirccline had horizontal scales of 25 to 50 km
and vertical scales approximately equal to the water depth.
The energy of these waves was dominated by the barotropic
mode with seme contributiens frcm modes 1 and 2. These
waves were not admitted to the shelf region because the bot-
tom slope was greater than the slope of the internal wave
characteristics.

The mean flew followed the isobaths at all levels, tut
it was in the opposite direction in the bottom layer. The
mean flow initially decreased alcng the eastern boundary of
the canyon as the storm forcing readjusted the flow. Near-
bottcm temperature variations of uoc were associated with
the storm surge and advection in the along-track direction,
particularly along the north rim of the canyon.

TABLE OF CONTENTS

I. INTRODUCTION 13

II. HISTORICAL REVIEW 16

III. DATA AND ANALYTICAL METHODS 23

A. CURFENT METER DATA 23

B. STCFM TRACK 24

C. WINE FIELD 27

D. VERTICAL TEMPERATURE GRADIENTS 27

E. TEMPORAL AND SPATIAL SCALES 30

F. CCMEUTATION OF SPECTRA 32

G. COMPLEX DEMODULATION 34

IV. STORM PEFICD 37

A. INERTIAL RESPONSE 37

B. SUBINERTIAL RESECNSE 63

C. SUFEFINERTIAL RESPONSE 73

V. NORMAL MODES 81

A. THECEY 81

B. EATA ANALYSIS 83

VI. CONCIUSICNS 95

APPENDIX A 98

A. CCMELEX DEMODULATION 98

B. POTENTIAL ENERGY 100

C. LEAST SQUARES FIT 101

APPENDIX E -10U

A. DEEP REGION 104

B. SHAIIOW REGION 110

LIST OF REFERENCES 115

INITIAL DISTRIBUTION LIST 118

LIS! OF FIGURES

Figure 1. Track of Hurricane Frederic 14

Figure 2. Satellite Photograph of Frederic at 2001

GMT 12 Sept. 1979 26

Figure 3. Wind Speed and Direction Measured by NOAA

Eucy 42003 28

Figure 4. Vertical Temperature and Brunt-Vaisala

Freguency Profiles 30

Figure 5. Freguency Response for the Band-pass and

Lew-pass Filters 36

Figure 6. Band-Pass Filtered Mixed layer V Velocity

Component at CMA3 36

Figure 7. Mixed Layer Velocity Components at CMA2. . . 38

Figure 8. Sinrilar to Fig. 7, except at CMA3 40

Figure 9. Time Series of HKE in the Mixed layer .... 41

Figure 10. Similar to Fig. 8f except in the deep

ttermocline (251 m) 43

Figure 11. Progressive Vector Diagrams at CMA3 44

Figure 12. Time Series of HKE at CMA3 47

Figure 13. Normalized HKE Spectra from CMA3 49

Figure 14. Mixed Layer Normalized Rotary Spectra

from CMA3 50

Figure 15. Vertical Scales at CMA3 55

Figure 16. Estimates of Hcrizontal Scales in the

Mixed Layer 58

Figure 17. Hcrizontal Scales in the Deep Thermocline . . 60

Figure 18. Low-pass Filtered PV D at CMA2 66

Figure 19. Lew-pass Filtered PVD at CMA3 67

Figure 20. Thermocline Temperature Time Series from

CMA3 at 251 m. 69

Figure 21. Temperature Tine Series from CMA2 at 324

m 71

Figure 22. Potential Energy Spectra from CMA3 72

Figure 23. Cress-Spectrum Between V-Velocity and

Temperature 75

Figure 24. Mid-depth Velocity Components from CMA1. . . 76

Figure 25. Horizontal Velocity Eigenf unctions at

29.33 N and 87.11 9 84

Figure 26. V Velocity Coefficients at CMA3 86

Figure 27. Same as Fig. 26, except at CMA2 87

Figure 28. V Component Time Series at CMA3 93

Figure 29. Surface PVD fcr a) CMA2 and b) CMA3. ... 105

Figure 30. Bottom PVD for a) CMA2 and b) CMA3 106

Figure 31. Temperature Time Series from CMA3 at 457

m 111

Figure 32. Mid-depth PVC at CMA1 112

LIST OF TABLES

TABLE I. Comparison of Inertial-Internal Wave

Parameters 17

TABLE II. A Synopsis of Storm Period Observations . . 25

TABLE III. Rcssby Radii cf Deformation 32

TABLE IV. Normalized Rotary Spectrum Analysis at

Inertial Frequency 51

TABLE V. Vertical and Horizontal Coherencies at

the Inertial Frequency 53

TABLE VI. Comparison of Vertical and Horizontal

Group Velocity 61

TABLE VII. Normalized Rotary Spectrum Analysis at

Sufcinertial Frequency 70

TABLE VIII. Normalized HKE Spectra at Inertial and

Semi-diurnal Frequency 77

TABLE IX. Normalized Rotary Spectra at the Semi-
diurnal Frequency 79

TABLE X. Variance of the Normal Modes at CMA2 and

CMA3 90

TABLE XI. Normalized HKE Spectral Estimates at 95%

Confidence 107

TABLE XII. Normalized Rotary Spectrum Analysis . . . 109

TABLE XIII. Normalized HKE Spectral Estimates at 95%

Confidence 113

ACKNOWLEDGEMENT

The author extends his gratitude to all those who have
contributed tc the evolution of this thesis. First and
foremost, I would like to express my deepest appreciation to
my faculty advisor, Erof. Russell Elsberry. His guidance
and wisdom during a very difficult time of my life were
exceeded only by his thorough knowledge of the subject mat-
ter. Under his leadership, this thesis work has been the
crescendo of my graduate education. The expertise of Dr.
Andrew Willmott is held ir. high regard. His attention to

mathematical details has irade this thesis a much better man-
uscript. I also wish to thank Prof. Christopher Mooers for
his willingness to share his knowledge on the application of
normal mode theory to the analysis of current meter data,
and fcr providing the computer program, IWEG, to solve for
the vertical structure. His criticism and advice have con-
tributad significantly to my development as an oceanogra-
pher. The wisdom and guidance of Dr. William Hart whenever
encouragement was needed has also been greatly appreciated.

The dedicated efforts cf the personnel who maintain the
computer systems at the W. R. Church Computer Center at NPS
as well as at NAVOCEANO are greatly appreciated. Their wcrk
certainly helped expedite the publishing of this thesis.

10

I also wish to acknowledge the unheralded efforts of the
current meter group of NAVOCEANO . In particular, the fine
work of Messrs. Jose Alonso and Steve Clark is appreciated
and have made this thesis work a reality- The advice of
Drs. Eill Wiseman (LSU) , Zack Ha llock (NORDA),and Ben Korgen
(NAVOCEANO) and Mr. Jack Tamul (NAVOCEANO) was extremely
helpful in the analysis cf the data. Drs. Hank Perkins
(NORDA) and Ortwin vcn Zweck (NAVOCEANO) have shared some
stimulating ideas concerning inertial-internal wave motions.
I further extend my gratitude to Mr. Jerry Carroll for his
assistance in gaining access to this very unique data set.
The Physical Oceanography Branch at NAVOCEANO kindly pro-
vided some of the computer software used in the data analy-
sis. The expertise cf Mr. Steve Lauber in the
pre-processing of the data greatly aided the data analysis.
The drafting cf seme of the figures was done with great care
by Ms. Viola Lutrell, Mr. Chuck Murphy, and Mr. Glenn Vocr-
his cf NAVCCEAKC.

I thank the two most important people in my life, my
wife, Sally and daughter, Missy. Their patience and encour-
agement during the evolution of the thesis has helped me a
great deal.

11

Finally, I would like to thank NAVOCEANO for their sup-
port ever the first three quarters cf my graduate studies
Furthermore, the author extends his appreciation to ONR fo:
support during the final stages of the research.

12

I. INTRODUCTION

" the deer alone learnath "

Nietzsche

The oceanic response tc intense, transient atmospheric
events, such as hurricanes, is dominated by robust insrtial
wave excitation in the mixed layer and the subsequent propa-
gation and dispersion of inertio-gravity waves in the ther-
mocline. The study of these forced waves has been largely
through numerical and analytical investigations, because of
the lack of sufficient data to resolve the scales and ener-
getics of the motion. Hence, observations of ocean currents
and temperatures during a hurricane are neccessary for vali-
dating models and theories dealing with forced inertio-grav-
ity t>aves.

A comprehensive data set of ocean observations was col-
lected by the 0. S. Naval Oceanographic Office (NAVOCEANO)
during the passage of hurricane Frederic in the summer of
1979 (Shay and Tamul, 1980). NAVOCEANO deployed three

moored taut-wire current meter arrays in the northern Gulf
of Mexico southeast of Mobile, Alabama. During the period
of deployment, hurricane Frederic passed within 80 to 130 km

13

of the array sit€S (Fig. 1) at 2 100 GMT 12 September. Five
hours later, Frederic made landfall at Dauphine Island, Ala-
bama. During and subseguent to the passage of Frederic, the
National Hurricane Research Division dropped several expend-
able bathythermographs (AXET) from reconnassiance aircraft
in the area of the current meter deployments (Black, 1983).

SC.IE ft

Figure 1.

Track of Hurricane Frederic. The contours are in
fathoms, A depicts positions of current meter
arrays, (NAVOCEANO) , ■ depicts the position of
AXET drops (Black, 1983) . The track of hurricane
Frederic is based on a report by Hebert (1979) .

1U

This thesis addresses the problem of the vertical and
horizontal dispersion of the forced inertial wave energy
from the mixed layer. The forcing readjusts the mean flow
as well as generating anisotropic inertial motion. The cur-
rent meter observations allow determination of the scales
and the energetics of the forced, anisotropic wave motion
through use cf spectrum analysis and complex demodulation.
The modal structure is examined by formulating the Sturm-
Liouville problem and thsn performing a least squares fit
between the eigenf unctions and the demodulated time series
of the current observations.

15

II. HISTORICAL REVIM

A nonlinear model for a stationary hurricane (O'Brien
and Reid, 1967) was developed to simulate the observations
made in the wake of hurricane Hilda (Leipper, 1967). The
results indicated that upwelling regions were restricted to
the area enclosed by the iiaximum wind regime, whereas down-
welling occurred outside this regime as warm water was
advected away from the stcrm center.

Cne of the most proncunced responses of the ocean to
wind forcing is the generation of inertial oscillations in
the surface mixed layer. Pollard (1970) simulated the gen-
eration of these waves using a two-dimensional, wind-driven
model developed by Pcllard and Millard (1970) . The pre-

dicted amplitudes and decay rates cf the forced oscillations
agreed well with the Woods Hole Oceanographic Institution
(WHOI) * site D1 observations. Using the same data set. Pol-
lard (1980) showed that under relatively strong wind condi-
tions, 67 % of the horizontal kinetic energy (HKE) in the
mixed layer was found near the inertial frequency. The sub-
sequent downward propagation of energy was estimated to be

16

of the order cf 10~3 cm/s ( 1 m/d) . This vertical group
velocity was too small to account for the radiation of ir.er-
tial waves and the subsequent loss of energy from the mixed
layer. The corresponding scales of the inerrial wave motion
were about 100 to 240 d in the vertical and hundreds of
kilometers in the horizontal (Table I).

TABLE I
Compariscn of Inert ial- Internal Wave Parameters

I Length

Brooks
(1983)

S cale s

Horizontal (km) 370

Vertical (m) 1000

Group
Velocity

Horizontal (km/d) 23
Vertical (m/d) 60

From Brocks (1983)

Price
(1983)

480

1000

86
160

Pollard
(1980)

700-1700
100-240

0. 8-17.
0.03-3.

Dsing a linear, two-layer model, Geisler (1970) simu-
lated inertic-gravity waves as part of the baroclinic
response in the wake of a translating storm. The critical
factors governing the generation of these waves were:

• the translational speed of the hurricane must be
greater than the internal wave phase speed; and

• the horizontal scales of these waves must be comparable
to the oceanic Rcssby radius of deformation.

17

He further noted that the ocean response to a moving hurri-
cane is strcngly baroclinic.

The ocean thermal response to hurricane forcing, was
investigated numerically by imposing Ekman layer dynamics in
a mixed layer model (Elsberry et al., 1976). The input of
energy frcm the wind stress was forced to generate inertial
oscillations and cause turbulent mixing via entrainment.
Upwelling alsc enhances the turbulent mixing process in the
upper layers of the ocean. The main result was that advec-
tion dominated the thermal response near the storm track as
opposed to the heat loss from the ocean surface to the
storm.

Strong atmospheric fcrcing enhances the turbulent mixing
process in the surface mixed layer and causes it to deepen.
As the mixed layer continues to deepen, the thermocline
begins to erode as water from the thermocline is entrained
into the mixed layer. However, stratification tends to sup-
press turbulent mixing and aids in the creation of small-
scale internal waves either through entrainment or
Kelvin-Helmholtz instabilites (KH) at the base of the mixed
layer (Pollard et al. , 1973 ; Garwood, 1977) .

18

Eelow the mixed layer, ocean current variability is
linked to vertically pre Plating wave groups of the internal
wave field (Kase and Gibers, 1979) . The actual mechanisms
for the vertical transport of energy from the wind forced
mixed layer tc the thermccline for the generation of large
scale inertial-internal waves are not well understood. Per-
haps the most effective way of creating these large scale
inertial-internal waves is through Ekman suction. Krauss
(1972 a,b, 1976) showed that a horizontally varying wind
stress causes a mass transport 9 0 deg. to the right of the
stress in the Ekman surface layer. This surface layer div-
ergence is accompanied by an upward displacement of the iso-
pycnals or upwelling of cccler water from below which tends
to suppress the turbulent mixing process. During the relax-
ation of the wind, these isopycnals are displaced downward ,
which contributes to the creaticn of large scale inertial-
internal waves at the base of the mixed layer.

Curing the passage of hurricane Belle (Mayer et al.,
1981), ocean current and temperature measurements were
acquired on the continental shelf of the Middle Atlantic
Bight. Their analyses indicated that most of the inertial-
internal wave energy was ccntained in the first mode at the

19

deeper sites (water depth of 70 m) and in a heavily damped
second mode at the shallow sites (water depth of 50 m) . The
variability in the modes was not attributed to the spatial
variability of the wind stress. The bottom slope and mean
velocity fields significantly altered the oceanic response
to hurricane Eelle.

A mixed layer model (Garwood, 1977) was embedded into a
multi-layer, primitive equation, ocean circulation model
(GCM) by Adamec et al., (1980). Using this GCM model, Hop-
kins (1982) simulated the baroclinic response to a forcing
pattern similar to hurricane Frederic and compared the
results to the data collected in the wake of Frederic (Shay
and Tamul, 1980). The model predicted the inertial response
in the mixed layer quite well. However, the predicted ocean
current response in the subsurface layers was less energetic
than the observations, and decayed much too fast. These
discrepancies between the model simulations and observations
are due in part to the imposition of the rigid lid conditior
at the surface, which eliminates the barotropic mode. Fur-
thermore, these model simulations did not include topogra-
phy, even though the observations are from a region where
the bottom topography is rugged and the ocean depth is less
than 1 km deep.

20

Recently, ocean measurements were made during the
passage cf hurricane Alien in the western Gulf of Mexico
(Brocks, 1983) . The spatial scales of the inertial wave
were different from the 'site D1 observations, as shown in
Table I. The vertical scales in the ocean are generally

much greater under hurricane forcing than during the passage
of a cold front, while the converse is true for the horizon-
tal scales cf motion. The vertical group velocity, which
depends on tcth the wave and the Brunt-Vaisala freguencies,
for the Allen observations exceeded the group velocity from
the 'site D' observations. This estimate from the Allen
observations is misleading due tc the lack of upper thermo-
cline and mixed layer data. Moreover, the wind speed is
much greater during the passage cf a hurricane than a fron-
tal passage. Therefore, the amount of turbulent mixing in
the mixed layer and inertial wave excitation should be much
more rapid during a hurricane. Other fea-ures measured in

the Allen response included topographical dependence, and
the vertical phase locking during the first few inertial
pericd (IP) fcllcwing the storm.

The baroclinic response of the ocean to a hurricane was
modeled by (Price, 1983) using a multi-level, inviscid

21

model. The scales and energetics of the simulated inertial
response were similar to mixed layer data collected by a
NOAA data tuoy during the passage of hurricane Eloise. The
horizontal and vertical scales, as given in Table I, were
quite large in comparison to the thickness of the thermo-
cline (200 m) . The maxiiium energy of the inertial-internal
wave motion was predicted to occur at a distance of twice
the radius of maximum winds, hereafter referred to as the
maximum wind regime, which was roughly 80 km for Eloise.

Further, the rate of vertical energy propagation was large
in comparison to both the Allen and 'site D1 observations,
and accounted for the depletion of energy from the mixed
layer. Other mechanisms, such as KH instability (Pollard et
al., 1973 ; Garwood, 1977) and turbulence (Bell, 1978), act
to remove energy from the mixed layer. However, these phe-
nomena were neglected as mixing was not explicitly included
in the model formulation.

Greatbatch (1983) modeled the nonlinear response of the
ocean to a moving stcrm. The major results were that the
transition between upwelling and downwelling zones in the
oscillatory wake was rapid, and that nonlinearities account
for the displacement of the maximum response to the right of
the storm track.

22

III. DATA AND ANALYTICAL METHODS

A. CURRENT METEB DATA

Ten Aanderaa RCM-5 current meters were deployed on three
moored taut-wire arrays ir depths ranging from 100 to 470 m
in the northern Gulf of Mexico. These current meters sam-
pled ccean current speed, direction and temperatures at 10
minute intervals. Two current meter arrays (CMA2, and 3)
were deployed on adjacent sides of the DeSoto Canyon where
bathymetric ccntcurs converge to form the head of the canyon
(see Big. 1) . The other icoring was deployed closer to the
coast in a depth of about 100 m of water where the isobaths
are nearly parallel to the coast. A synopsis is given in
Table II cf the storm period observations, which extends
from a few days prior to hurricane passage (12 September) to
the end of the deployment period (mid-October) .

The guality cf the data is generally good; however, the
lengths of the time series are not all the same. For

instance, the Savonius rctors were eventually lost from all
current meters in the mixed layer due to the large current
speeds. These large mixed layer current speeds are not

23

corrected fcr rctor pumping, which is a function of the
mooring design as well as the large direction vane of the
Aanderaa current meters (Pofonoff and Ercan, 1967). in
addition to the rotor problems, temperature records from all
mixed layer current meters were unrecoverable because the
ocean temperature exceeded the threshold temperature of the
thermistors of 21.5 °C . Sea surface temperatures 325 km
south-southeast of the mooring sites, as measured by NCAA
Buoy 42003, reached 28.8<>C (Johnson and Renwick, 1981) .

B. STORM TRACK

The stcrm track is based on the best position data on
hurricane Frederic's movement through the Gulf of Mexico
(Hebert, 1979) . Because of the intensity of Frederic, recon-
naissance aircraft constantly monitored the storm. Frederic
increased to maximum strength, maximum winds or minimum
barometric pressure, 80 to 130 km west of the CMA sites
about 2100 GM1 12 September (see Fig. 1). A visible photo-
graph of Frederic from a GOES satellite at 2001 GMT 12
September clearly delineates a well developed eye of about
40 tc 50 km in diameter (Figure 2) . The translational speed
of the hurricane at this time was about 7 to 7.5 m/s as it
approached the Gulf Coast, which was much larger than the

24

internal wave phase speed. Thus, an inertio-gravity wave

response is expected in the wake of hurricane Frederic
(Geisler, 1970).

TABLE II
A Synopsis of Stcrm Period Observations

Meter
Depth
(1)

Record
Length

Start

Time

(GMT)

End

Time

(GMT)

Variables

CMA1

21
49
64
92*

0
22.21
16. "6
20.79

No hurricane data

0040 5 Sep. 0600 27 Sep. u,v

0040 5 Sep. 1920 21 Sep. u,v,T

0040 5 Sep. 2040 25 Sep. u,v,T

CMA2

19

24.67

0040 5

Sep.

179

22.63

1920 7

Sep.

3 24

24.16

0040 5

Se p.

1650 29 Sep,
0100 30 Sep
1900 29 Sep,

u,v

u,v,T

u,v,T

CMA3

21

24.91

0100 2

Sep

251

36.00

0100 2

Se p

437

35.69

0100 2

Sep

457

51. S4

O1C0 2

Se p

2200
0100
1730
0000

2 6 Sep

8 Oct

7 Oct

4 Oct

u,v
u,v,T
u,v,T
u,vrT

* time clock synchronization problems
u,v = horizontal velocity components
T = temperature
GMT = Greenwich Mean Time

25

Figure 2. Satellite Photograph cf Frederic at 200 1 GMT 12
y Sept. 1979. Visual satellite imagery is courtesy

of NOAA/NESDIS.

26

C. BIND 'FIELD

Wind field data were obtained from shore stations at
Mobile, Alabama and Pensacola, Florida. These records indi-
cated that the surface wind speed did not exceed 40 m/s.
However, ether reports suggested that wind speeds ranged
between 48 to 58 m/s as Frederic made landfall (Hebert,
1979). These discrepancies are attributed zo local boundary
effects and the non-representativeness of wind data col-
lected in the coastal region. Marine winds were also meas-
ured by NOAA Euoy 42003 (Jchnson and Renwick, 198 1). Wind
speeds at this location never exceeded 35 m/s. The eye

clearly passed over the bucy as indicated by the minimum in
the wind field as the direction changed from 40 to 200 °
True (Fig. 3), with a corresponding decrease in pressure to
959 millibars (mb).

D. VERTICAL TEMEERATURE GRADIENTS

The AXBT data collected by the National Hurricane
Research Division, as reported by Black (1983), are used for
the computation of the Br UEt-Vai sala frequency. These data
were collected after the passage of Frederic in the area of
current meter array deployments near the DeSoto Canyon (see
Fig. 1).

27

254

255

256

I I I I I I I I I I I I I I I I I I I I I M I

15 18 21 0 3 6 9 12 IS

256

254

255

JULIAN DAY/TIME (GMT)

Figure 3. Wind Speed and Direction Measured by NOAA Buoy
42003. The upper panel represents the observed
wind speed time series where the abscissa depicts
time in Julian Days starting Dn 1200 GMT 11 Sept.
to 1500 GMT 13 Sept. 1979. Ths lower panel
represents the concurrent wind direction time
series. The hatched area depicts the period when
the data buoy was in the eye of the hurricane
(Johnson and Renwick, 1981).

28

The vertical density profiles are computed at 10 m
intervals using the AXBT data and climatological data from
NAVOCEANO thrcugh the equation of state

P = pQ(l-a(T - Tq)) 9 (1)

where p is the density, T is the observed temperature, T0
and p0 are the reference temperature and density respec-
tively frcn. climatology, and a is the thermal expansion
coefficient, which is taken to be 0.0002/°C . Some error is
expected due tc the neglect of the salinity term. However,
an examination of the clicatological T-S diagram, for the
DeSoto Canyon area indicates that density variations are due
to temperature rather thai salinity effects. The Brunt-Vai-
sala, N2 , frequency is computed using a centered finite
difference frcm the expression

*T2 g A p

p Az / ^- >

o

where g is the acceleraticc due to gravity andAp/Az is the
vertical density gradient. Since the AXBT data only

extended to about 250 m, it was necessary -o extrapolate the
vertical temperature gradient to 470 m. The temperature is

29

assumed to decrease uniformly at a rata of 0.0311 oc/m from
250 to 470 m (Fig. 4) • • The corresponding uniform increase
in the density produces a constant Erunt-Vaisala profile.

T (°C)

7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5

I ' 1 ' 1 ' 1 ' 1 ' 1 ' 1 ' 1 ' 1

N (cph)

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

Figure 4. Vertical Temperature and Brunt- Vaisala Frequency
Profiles. The abscissa depicts the Brunt-vaisala
frequency (solid) and temperature (dashed) from
Black (1983) .

E. TEMPORAL AND SPATIAL SCALES

The lccal variations in the bottom topography at the
DeSotc Canyon (see Fig. 1) are appproximately parallel and

30

normal to the coast at CM£2 and CMA3. For example, north-
south flow at CMA2 is in the cress-shelf direction, but it
is in the alcng-shelf direction at CMA3. Henceforth, a

Cartesian coordinate system is used to represent the hori-
zontal velocity components in the x and y directions, where
motion to the east and the north corresponds to +u and + v
current velocities, respectively.

The fundamental time scale used in this thesis is an
inertial pericc, which is equal to 24.10, 24.25 and 24.47 h
at CMA 1,2 and 3 respectively. In the following calcula-
tions, the Inertial Period (IP) will be set equal to 24.4 h.
The intrinsic length scales are either the radius of maximum
winds, approximately 40 ka; for Frederic, or the Rossby radii
of deformation, which are given in Table III for the baro-
tropic and first three barcclinic modes. These deformation
radii are based en th€ inertio-g ravity wave phase speed com-
puted from the Sturm-Liouville problem, which will be dis-
cussed in a later section. For convenience, the scales in
the x and y directions will be referenced as cross and
along-track scales respectively, because the storm track was
nearly perpendicular to the coastline.

31

TABLE III
Rossby Radii of Deformation

Phase
Mode Speed

No. (m/s)

D

ef ormati
Radii

(Jsi)

950.

56.
30.
18.

on

0 68.0

1 4.0

2 2.2

3 1.3

F. COMPUTATION CF SPECTRA

The energy and cross spectra are computed in the follow-
ing manner: (1) the mean is computed and is extracted from
-he time series; (2) a Tukey data window is applied to the
data; (3) the data are transformed into frequency space via
the Fast Fourier Transform (FFT) ; and (4) the transformed
data are spectrally averaged over bandwidths which vary in
width with frequency, and are closely tied to the computa-
tion of confidence levels. The number of data points trans-
formed by the FFT used in the spectrua analyses does not
have to be proportional to 2n , where n is an integer. The
general FFT is advantageous in the analysis of events
because of their transient nature. However, in the analysis
of the spectra, equal numbers of data points are used to
avoid leakage and smearing of energy from adjacent frequency

32

bands (Otnes and Enochson, 1978) . Finally, the higher fre-
quency energy abcve the Nyquist frequency, 1/(2 A *) where
^ t is the sampling interval, is eliminated to prevent ali-
asing of the spectra.

The HKZ is decomposed into the clockwise (CW) and coun-
terclockwise (CCW) rotating components using the rotary
spectrum methcd outlined in Gonella (1972) and Mooers
(1972). The main results of this analysis are the rotary
spectrum, ellipse stability and orientation, and rotary
coefficient. The orientation of the ellipse indicates the
direction of the horizontal phase velocity, while the sta-
bility indicates the variability in the ellipse orientation.
A high stability index indicates that the wave motion is
anisctropic, or that the horizontal direction of phase pro-
pagation is unidirectional. Conversely, a low index of sta-
bility implies that the direction of phase propagation is
randcm or isotropic. The rotary coefficient indicates the
amount of confidence that can be placed in the rotary spec-
tral estimate and the rotational component that dominates
the rotary spectrum. For instance, a +1.00 implies that the
moticn is polarized in the CM direction. A more rigorous

mathematical treatment is given by Gonella (1972) and Mooers
(1972).

33

G. COMPLEX DEMODULATION '

A useful method for determining inert ial oscillations in
a time series of ocean current measurements is through the
use cf the complex demodulation method (Perkins, 1970). The
mathematical details are given in Appendix A. This method
can te implemented either ty linear filtering or by perform-
ing a least sguares harmcnic analysis on the time series.
The linear filtering method is preferred because data
points are lost whenever harmonic analyses are used to form
the time-varying amplitudes and phases of frequency depen-
dent motions. This is especially useful in the analysis of
events when all data points are needed to resolve the tran-
sient motions. A major disadvantage of the linear filtering
method is that it only applies to discrete frequencies,
whereas the least squares method can treat all frequencies
within the constraints of the time serias.

Initially, the data are band-pass filtered between 18
and 30 h to limit the frequency content in the time series
to near-inertial motions. These filtered data are multi-
plied by the trigonometric arguments of the inertial fre-
quency. At this point, some higher frequency motions have
been introduced into the time series (Appendix A) and these

34

are removed ty low-pass filtering the data at 22 h. The
resultant time series is manipulated to form the instantane-
ous amplitudes and phases cf the inertial motion. The final
set of time-varying amplitudes and phases provide a local
harmonic analysis, or more explicitly, a time series of
spectral estimates for inertial motion. The filter used in
the demodulation was a Lanzcos square taper window with 24 1
weights (Fig . 5) .

The band-pass filtered, mixed layer velocity components
at CMA3 are excellent examples for using complex demodula-
tion (Fig, 6). As the current velocities increase in
response to the storm, the amplitude of the forced inertial
wave increases as well. The inertial wave amplitudes modu-
late the increases and decreases in the mixed layer veloci-
ties. After the maximum velocities occur, roughly 2 IP
following hurricane passage, the strength of the currents
decreases linearly with e-folding scales of about 4 IP.
This type of e-folding behavior is very common in amplitude-
modulated signals.

35

Figure 5. Frequency
Filters.

represented by the
filter response is shewn

Response for the Band-pass and Low-pass
The tand-pass filter response is

by the solid line while the low-pass
by the dotted line.

Time (Julian Days)

Figure 6. Band-Eass Filtered Mixed layer
Component at CMA3.

7 Velocity

36

IV. STORM PERIOD

In the following analyses, the data are divided into two
periods: 1) the storm period observations as previously
described in Table II; and 2) the quiescent period which
encompasses the period frcm the start of the data to a few
days prior tc the storm. This chapter focuses on the iner-
tial, subinertial and superinertial response in terms of
energetics and scales of notion to hurricane Frederic. The
quiescent period observations and results are given in
Appendix B. For convenience, the terms shallow and deep
will refer tc the regicns where CMA1 and CMA2,3 were
deployed.

A. INERTIflL RESPONSE

Hurricane Frederic passed about 80 km west of the DeSoto
Canyon at 2100 GMT 12 September (JD 255) 1979. The current
speeds began to increase at 070 0 GMT or approximately 14 h
prior to the time of closest approach. The currents in the
mixed layer at CMA2 increased to a maximum within an IP fol-
lowing the passage of Frederic (Fig. 7) . The storm caused
the horizontal velocity to oscillate with periods of about
24.3 h and to rotate in a CW direction, which indicates the

37

presence of inertial waves in the mixed layer. These waves
then decreased in amplitude over e-folding scales of 4 IP.
The storm also caused the u-component to flow in the oppo-
site direction from that cf the pre-storm flew, whereas the
v-component increased markedly in amplitude. The near-bct-
tom current speeds (shown later) began to increase within 3
h after the mixed layer response.

c

c 5
o <"

o

120
IX

60

G0-

40-

20

0-

-20-

-40-

-60-

AAA^^'

2+e

/WwV^

256

260 264

Time (Julian Days)

26?

272

Figure 7. Mixed Layer Velocity Components at CMA2. The

ordinate depicts the magnitude of the horizontal
velocity components in cm/s for the east-west
current (upper) and north-south current (lower) .
The hatched area depicts the period of hurricane
passage.

38

In the mixed layer at CMA3 , the storm amplified the
pre-storm currents to a maximum of 135 cm/s (Fig. 8). As at
CMA2, this increase in the currents is a manifestation of
inertial wave excitation. However, the inertial response at
CMA3 was different from CMA2 in that the kinetic energy of
the waves was larger and persisted longer (approximately 2
IP) . The energy of these waves then decreased over e-fold-
ing scales of 4 IP. The increase in near-bottom current

(shown later) occurred within 4.5 h following the mixed
layer response.

These horizontal differences of the inertial response in
the mixed layer between CMA2 and CMA3 are observed in the
amplitude changes of the HKE (Fig. 9). The initial increase
in kinetic energy at the two arrays was similar. The maxi-
mum in kinetic energy occurred first at CMA2 and led the
mixed layer maximum at CMA3 by about 3 h. The differences
in HKE levels were quite pronounced. For example, the mixed
layer HKE level was roughly 60 % of that found in the mixed
layer at CKA3, even though CMA2 was closer to the eye of the
storm. This large difference over a distance of 34 km may
be due in part tc the stronger mean current at CMA2, which
may have reduced the amplitude of the inertial waves. The

39

Time (Julian Days)
Figure 8. Sinilar to Fig. 7, except at CMA3.

direction of the mean flow and the horizontal phase propaga-
tion cf the inertial waves (shown below) roughly coincided,
which suggests that there may have been some interaction
between them. Following these maxima, the energy decayed
with an e-folding scale cf 4 IP at both arrays. Forced
inertial waves in the mixed layer that vary in space and
time are one mechanism to drive inertio-gravity waves in the
subsurface layers (Krauss, 1972a, b).

40

m
U

n
a)

>-i
td

T3
0)
X .

ST3
0)

a>x:

J= Ui

+j <d

•a

cw

•H

4-J+J
O <t)

w«o
0) c
•H <d
n

c/)»o

•H
U)r-|
B U

•H CO

<i>
M

3

en

•H

0s2

Q91X 009T 0921 0001 09<L 009 092 0

2**(oas/uio) XSJ9U3 orpin)!

41

The vertical differences in the inert ial response were
even more dramatic- In the deep thermocline (251 m) at
CMA3, the increase in the currents was sensed within 2 h
following the nixed layer response (Fig. 10). The initial
maximum occurred within an IP (JD 256) after the winds
relax, which was followed by inertial oscillations in the
horizontal currents. In the presence of stratification, the
particle mcticns are elliptical rather than circular as
observed in the mixed layer. The amplitude of inertio-grav-
ity waves decreased slightly following the initial response.
About 10 IF later, the current increased to a secondary max-
imum that nearly exceeded the initial maximum. The u-velo-
city essentially defined an amplitude-modulation envelope
that collapsed and expanded in time and persisted for 21 IP
following the initial maxiirum. The near-bottom currents at
CMA3 (shown later) behaved in a similar manner except that
the secondary maximum occurs 6 IP following the storm.

The vertical changes in the currents at CMA3 are even
more apparent in the progressive vector diagrams (PVD) ,
which are Lagrar.gian representations of the Eulerian meas-
urements. That is, the ccntinucus current measurements are
integrated with respect tc time to approximate the

U2

24:5 250 255 260 265 270

Time (Julian Days)

275

280

Figure 10. Sinilar tc Pig- 8, except in the deep
thermocline (251 m) .

trajectory of the water parcels at that point. The horizon-
tal velocity components in the mixed layer at CMA3 defi-
nitely showed the classical, CW rotating vector associated
with inertial waves (Fig. 11a). Prior to the storm, small
scale, freely propagating inertial waves were present, and
they were subsequently amplified in response to the storm.
These forced waves were superposed on a southward mean flow
that increased in magnitude in response to the forcing. The
horizontal displacement over successive time intervals of 2

43

IP increased following stcrm passage. After the maximum
current velocities near JD 258, the amplitude of these
forced, near-circular inertial oscillations decreased.

Scale: \

Figure 1 1 .

Progressive Vector Diagrams at CMA3. The *

represents a time span of 48 h starting on JD

245 for a) mixed layer (21 m) , and b) the deep
ttermocline (251 m) .

44

In the thermccline (Fig. 1 1 b) , the storm amplified the
freely propagating inertic-gravity waves. However, the
dimensions of the ellipses are distorted because of the
increases in the mean flew between JD 257 and 261. This
pericd was immediately followed by decreases in the mean
flow until JE 265, when the inertio-gravity response was
more pronounced. The pericd of increasing and then decreas-
ing mean flow corresponds to the approximate modulation
envelope and lag period of 10 IP between JD 256 and 266
(Fig. 10). After JD 266, the modulation envelope started to
collapse as the inertio-gravity waves decayed in amplitude
with an e-fclding scale of about 4 IP.

The vertical propagaticn of the energy, which is a con-
sequence cf the generaticn of inertio-gravity waves in the
thermccline, is indicated by the vertical differences in the
HKE between the mixed layer and the thermoclme (Fig. 12).
In the mixed layer, the naximum HKE occurred approximately
one IP following the storm and persisted over the next 2 IP.
Subsequently, a relative maximum was reached in the deep
thermocline atcut 2 h after the surface maximum, and it was
only 5 to 8 % cf the maximum HKE in the mixed layer. Based
only on these initial phase differences between the velocity

45

components at the two levels, the vertical propagation of
energy would fc€ about 3 cm/s (2.6 km/d) . The thermocline
HKE then remained nearly constant over the next 4 to 6 IP.
On JD 262, tfce thermocline HKE gradually began to increase
as the vertically propagating energy from the mixed layer
accumulated. Using this 10 IP lag period, zhe vertical pro-
pagation of energy is about 0.03 cm/s (26 m/d) . The first
estimate of 2.6 km/d is about 1 to 2 orders of magnitude
greater than the Eloise and Allen observations (see Table
I) . However, the second estimate of 26 m/d is much more
comparable to the Allen estimates. The propagation of

energy will be addressed below using the dispersion relation
of inertio-gravity waves. The phase propagation was clearly
upward over the first 10 IP following the storm. Over the
subseguent 10 IP, the propagation was not as clear as phases
in the mixed layer became erratic, presumably due -co the
decreasing energy levels or spectral broadening.

The inertial peak dominated the HKE spectra at both
locations and at all depths. The change in the inertial HKE
between the surface and bottom layers was about an order of
magnitude, while the energies between mid-depth and interme-
diate levels did not change significantly at CMA3 (Fig. 13) .

U6

G

M -H

Q)

>i (1)

(d U

HO) C

J3 0)

T3H U

Q) 0)

X M-l

•H t*M

e— :H

T3T3

<ua'

JS^rH

-M tf) (0

(dU

U^-H

0^4->

<W U

(DO)

mc>

<:-h

aci-i a;

UU£

OV

-MS

idMlf)

d)-M

W^U

t^-M-H

x a.

u> u>

"4-1 J- 13

0-*-1

<d

Wt3 0)

U>GJ_t

•H<d<d

(-1 •

<D^n3H

tOt3 a>^

•ri,CX

WiHO

eu-t-> u>

•H UJ (0*U

EH-'^C-*-'

f^

0)
M

3

•H

OS-iJ 005J 0521 0001 05i 005 052 0

47

Although the inertial peak was prominent, the variability in
spectral shapes was due tc the differences in the background
continuum cf internal waves which varies with depth (Fu,
1980) and bandwidth averaging.

The HKE spectra are decomposed into CW and CCW rotating
components to clarify the rotational characterise ics of the
inertial wave motion. The spectral peak near the inertial
freguency was shifted about 3-6% higher than the inertial
freguency and it dominated the CW spectra calculated from
the mixed layer velocity measurements at CMA3 (Fig. 14) .
The rotary spectrum estimates for the inertial period
motions are given in Table IV. The CW, inertial HKE vari-
ability estimates are generally 1 to 2 orders of magnitude
larger than the CCW rotating motion. In comparison, the
subinertial (periods of 3 IP) and superinertial (periods of
12 h) CW spectral estimates are more than an order of magni-
tude less than the inertial motion (Tables VII and VIII) .

At all levels at the deep region, the positive rotary
coefficients approached unity, which indicates that the
waves were polarized in the CW direction (Table IV) . The
direction of tie semi-major axes of the inertial ellipses at
CMA2 rotated CW with depth except near the bottom. The

48

o^

ptp

CjO-

>> 1

CflU

dp-:

CD T~

OS

f \A

o5

CD-

CD

a'

O-

21M
251M
437M
457M

K.

M,

CI

I

"I — 1IM

-3

10
Figure 13

10 10

eye les/hr

Normalized HKB Spectra from CMA3 at 21 m
(solid) , 251 m (dashed) , U37 m ( dashed-dot ted)
and U57 m (dotted). K, represents the diurnal
tide frequency, f the mertial frequency and Mx
the semi-diurral tide frequency.

H9

<b_

~o_

*b_

o
\

<u
o

c

s

~o_

CC¥

K.

M,

T 1 — I I I I |

1 1 1 — I I I I I

lrf

10"

1 I I I IT

Figure 14.

10"

i i i iM

101

10'
cycles/hr

Mixed Layer Normalized Rotary Spectra from CMA3
for CW motion (solid) and CCw morion (dashed) .
K, represents the diurnal tide frequency, f the
mertial frequency and M, the semi-diurnal tide
frequency.

50

TABLE IV
Normalized Rctary Spectrum Analysis at Inertial Frequency

Normalized
Rotary

Spectra Ellipse

Depth CW CCW Dir. Rotary

(m) (cm/s) 2/cph Stab. QS.2-) Coeff .

a) CMAJ

49 6.7x10*3 1.3x10 + 3 0.81 18 +0.68

64 2.5x10+3 1.4x10+3 0.20 65 +0.62

b) CMA2

19

1.8x10+5

3.5x10 + 3

0.90

60

+ 0.96

179

4.6x10+*

3. 7x1 0+2

0.95

80

+ 0.95

324

2.8x10+*

1.3x1 0+2

0.30

60

+ 0.96

2) CHA3

21

4.0x10+s

2.0x1 0+3

0.87

50

+ 0.98

251

1.4x10+5

6. 1x1 0+2

0.63

90

+ 0.98

437

1.2x10+5

6. 3x10+2

0.50

150

+ 0.98

457

6.2x10+*

5.0x1 0+2

0.77

160

+ 1.00

direction of the near-bottom ellipse ro-ated CCW with depth
between the thermccline (179 m) and near the bottom (324 m) .
The stability cf the ellipses indicated that the inertial
moticn near the bottom at CMA2 was isotropic. In contrast,
inertial oscillations were strongly anisotropic at all of
the ether depths in the deep region. The inertial wave
response at CMA1 was markedly less intense than at CMA2 or
CM A3, but it was polarized in a CW sense. The reasons for
the semi-diurnal tide at CMA1 being much more energetic than
the inertial motion is explained below.

51

The vertical and horizontal coherences are computed to
estimate the spatial scales. The spatial scale L is com-
puted from the cross -spectral phase relationship using

L = ^Ad , O)

where Ad is either the vertical or horizontal distance
between the current meters and A<f> is the phase difference
between similar velocity components (Pollard, 1980). This
analysis is restricted to the CMA2 and 3 due to the absence
of an energetic inertial response at CMA1 (Table V) . Esti-
mates of vertical scales are more accurate than the horizon-
tal scale estimates because the vertical spacing of the
current meters below the nixed layer is not the same at the
two arrays. These scales calculated from the entire storm
time series will be compared belcw with the scales calcu-
lated from the instantaneous phases.

The vertical cross-spectral analyses (Table V) indicate
that the inertial response in the mixed layer led the near-
bottom response. The normalized cross-spectrum variance
between the vertical levels was about a factor of two
greater at CMA3 than at CMS2. The vertical scales computed
from (3) are about 360 and 540 m at CMA2,3 respectively.

52

That is, the inertial motion had scales of the order of the
water depths at both arrays and was coherent at the 95 %
confidence level.

TABLE V

Vertical and Horizontal Coherencies at the Inertial

Frequency

Nor nalized

Cross-Spectra Squared Phase

Arrajj Variables Variance/c ch Coherence (j|£3«)

Vertical

2 u(s) ,u(b) 3. 2x10+*

2 v (s) ,v <b) 2.7x10+*

3 u(s|'u(b) 6.5x10+*
3 v (s) ,v (b) 6. Cx10+*

Horizontal

3,2 u (s) 1.9x10+5

3,2 v (s) 1.6x10 + 5

0.62

-50

0.60 +

-70

0.70

-55

0.80

-50

0.98

-45

1.00

-55

0.98

-20

0.95

-30

3,2 u (b) 2.3x10+*

3,2 v (b) 2.1x10+*

♦ net significant at 95 f confidence

* spectra assumes an average inertial frequency of 0.0412 coh
(s) = surface layer (CMA2 = 19 m ; CMA3 = 21 m)

(b = bottom layer (CMA2 = 324 m ; CMA3 = 457 m)

The instantaneous phase differences of the velocity com-
ponents between the two records can also be used to estimate
the vertical scales (Fig. 15). Initially, the estimated
vertical scales at CMA3 were much greater (typically 800 to
900 m) than the water depth, but after JD 257 the estimates
were consistently of the order of the water depth. The

53

large vertical scales observed initially agree with these
estimated from the Eloise simulations by Price (1983) and
the Allen observations by Erooks (1983). At CMA2r the ver-
tical scales had similar trends and converge no scales of
the crder of the water depth.

The horizontal cress- spectrum estimates (Table V) indi-
cate both velocity components were coherent in the mixed
layer. The corresponding scales, as estimated from (3), are
20 and 35 km in the east-west and north-south directions
using an array separation of 34 km. Initially, the horizon-
tal scales were set by the atmospheric forcing, but these
estimates may be somewhat biased because the entire storm
time series was used in the computation of the spectra. The
cross-spectrum variance between the two near-bottom records
was about cne order of magnitude less than in the mixed
layer. However, these values are significant at the 95 %
confidence level as indicated by the coherence squared. The
response at CKA2 leads the response at CMA3 by about 2 h.
The horizontal scales near the bottom calculated from (3)
are 20 and 35 km in the east-west and north-south direc-
tions, respectively. The differences in the scales are due
to the larger scales set up by the hurricane in the

5U

m

CO

CD
C\i

CO
CO
C\i

C\2
CO

CO

CO

<u

>%

(0

Q

C

o

cd

CO

"^

cv

3

OS S
lO jr*

c\2 r-«

CO
lO
C\J

000T

09^ 009 092

(sja^a^) apses iftSiiaq

■lO
C\2

CO

hifi

c\?

lO

m

C\2

x:
a>+j

-p c
a) .

sue
o *

4-j cum
jOcr

a) f/Jts

3 U (0

ems
o n

14-1 CN

m»4-t

U

+■> co-p

(didc
-c a)

UJ UL.C

u> o

(0 3 s
uoo

to CU <J
G

fH (d >i
rO-P+J
U G-H

•*i (0 U

4J-M O
M 0J»-1
W C ci>

>-H >

m

(P
M

•H
fa

55

along-track (north-south) than in the cross-track (east-
west) directions. These estimates of scales may be somewhat
misleading because of the instrument depth difference (135
m) , although they agree well with mixed layer estimates.

The instantaneous phase differences between the mixed
layer currents indicate that the horizontal length scales
were set bj the storm initially (Fig. 16). The scales

decreased rapidly and approached constant values of 60 and
100 km in the east-west and north-south directions, respec-
tively. These scales in the mixed layer are roughly 1-2
times the baroclinic Rossby radius of deformation for the
first mode (56 km). The estimates of the horizontal scales
of inertic-gravity waves in the thermocline are computed
from the fcottcn: and mid-depth instantaneous phase differ-
ences in the 0-velocity ccmponent at CMA2 and CMA3 , respec-
tively (Fig. 17) . The east-west and north-south scales in
the thermocline immediately after hurricane passage are
roughly the same as in the mixed layer. After 4 IP, the
scales revert to the pre-storm scales of 25 and 50 km in the
east-west and north-south directions, respectively. Seme
caution is advised in accepting these estimates due to the
large depth difference of 73 m. In a later section, it is

56

suggested that the energy was contained primarily in the
barotropic and first two karoclinic modes. Hence, a depxh
difference of 13 m may be acceptable within the deeper ther-
mocline because of the relatively large displacement of the
isotherms, the large vertical scales, and the amount of
energy contained in the barotropic mode.

The estimates of vertical energy propagation speed are
computed only en the basis of the lag times between the sur-
face and subsurface response (Table VI). In the first case,
the response is felt at the bottom of the water column
within 3 to 4.5 h after increases in the mixed layer are
detected. Tie initial response corresponds to a vertical
scale of roughly 900 m (Fig. 15). This lag of 3 to 4.5 h

results in a vertical energy propagation speed of 2.6 km/d
which is roughly 1 to 2 orders of magnitude greater than the
values calculated by Brooks (1983) and Price (1983). The 10
IP lag in the second maximum observed at 251 m represents a
vertical energy propagation speed of 26 m/d when the verti-
cal scale is about 500 m. This vertical propagation of
energy agrees guite well with the estimates derived from the
Allen observations. However, there is also a second maximum
observed near the bottom atout 6 IP following the storm.

57

021

•H

+>cs

0J4J PS
-p tn (0

CO)

EU'CI'H

in 04J^i
cp * cuw o

J3X! S <ti (/}
4-> CLOd)

U >i
d Ul Oi-O

•H 3 >*d

o-PPHid
w o>-h a)

(U+JU 4->

o-p o)sa
w a > a>

.H-P UKN

+'13+' M
£5.h on it)

O C«aS
NO) <DS3 W

M+J * r4

O 4-"0 <c

SCHICU •

OT3 U) C >

axi) a+j m

WO) O 33

a) (ti ctyt o o
-P-Q d)«— m
rtJ Mwl T3

a u 0/ _c o>

•H<Um<N -p.fi
+J >4M*s: M w
W (O-H B O <0

v£>

0>

u

3
•H

(sj9}9iiion>l) 9I^°S m^u^l

58

Consequently, the vertical (CgE ) and horizontal (Cg^ )
group velocities are calculated from the following expres-
sions (Brocks, 1983)

Cgz " " m ~2 tan 9 7 (4)

0

n _ a N2 2a

CgH " k ^2 tan 9 7 <5)

where cr is the observed frequency, m and k are wavenumbers
in the vertical and horizontal cross-track directions, and
tan0 - k/m. The wavenumbers tn and k are equal to 2 n /L2and
2 tt /Irrespectively, where Lz and LH represent the vertical
and horizontal scales as estimated previously. The assigned
values to the variables in (4) and (5) are

a ■ 7. 54 x 10-s rad/s,
N2 = 3. 30 x 1C-s rad2/sz,
m = 6.98 x 1 0"3 rad/m,
m s 1.16 x 10-2 rad/m,
k ■ 1.05 x 10-* rad/m.
The initial vertical group velocity is 1.25 km/d as com-
puted from the dispersion relation (4) with an initial
scale of 900 m (Table VI) . This estimate is about half of

59

09T

v

m

■V 'US w

0)CU
(U.C l3t0
C! t

UU 3 >

o c=r o M
sajtN en a
t-tum i u

•i-t-x! OJ3

cue* a </i
CD O <D
1) d) T3T3

OW+JC

^o-w+j td

c a <D S-H

•HOfilH

<DO-P o

I/1VQ.01 V)

a) c s (0
(0+j u .a

UC 0)

-U-PE-i 0)
H W-H -P
<tf GO U
4->-H O »-H

c ,-i— .a.

0 0)0)3(1)
N^3 > T3
•H4J «-

O G.CCN M

33 O-tJ^iO

M
3
IT
•H

921 001 9-L 09 98

(sj9^9uionx) 9I130S tft^usq

60

TABLE VI
Comparison cf Vertical and Horizontal Group Velocity.

Vertical Brooks * Price *
Scales (1983) (1983)

900 m 500 m

Vertical

Group

Velocity

Lag Time 2.6 0.03

Disoersion 1.3 C.27 .06 .160

(km/d)

Horizontal

Group

Velocity

Dispersion 80. 30. 23. 86.

(km/d)

* From Brooks (1983)

the cne computed based on the initial lag time and is still
an order of magnitude larger than the estimates computed
from the Eloise simulations. After a few IP, the vertical
group velocity is about 0.27 km/d, which is of the order of
the Eloise estimates cf abcut 0. 16 km/d (Price, 1983) . Fur-
thermore, this estimate is about one order of magnitude
larger than the value of 0.02 km/d that Brooks (1983) esti-
mated from the Allen observations. Clearly, the initial
storm forcing causes energy to propagate vertically at a
significantly higher rate than after a few IP following the
storm.

61

The horizontal propagation of energy is a much more
rapid than the vertical propagation (Table VI). The group
velocity estimate for the Frederic observations is initially
80 km/d; and it decreases to about 30 km/d a few IP follow-
ing the storm. This difference is due to the tan2 depen-
dence on m as previously defined. The initial horizontal
group velocity estimate agrees with the Price (1983) esti-
mate whereas the lower value agrees with the Brooks (1983)
estimate. The horizontal phase speeds of the inertio -grav-
ity waves in the wake of Frederic are about 70 cm/s with a
vertical phase speed of rcughly 1 cm/s.

In summary, inertial waves maintained a constant ampli-
tude over 1 to 2 IP following the passage of Frederic.
These oscillations in the mixed layer then relaxed over an
e-folding scale of about 4 IP. In the subsurface layers,
the forced inertio-gravity waves decayed over similar time
scales after the appearance of a secondary maximum. The lag
between the initial impulse of energy and the second subsur-
face maximum varied over depth. For example, this period
was about 10 IP at 251 m, but it was only 6 IP at 457 m.
These lag values define a modulation envelope which evolves
in time. If vertical propagation of energy from the mixed

62

layer alone accounts for these secondary maxima, why do the
maxima occur later in the deep thermocline than near the
bottom ? Cther mechanisms besides dispersion of energy from
the mixed layer must be responsible. For example, nonlinear
resonant interactions among the inertio-gravity waves could
be responsible (McComas and Bretherton , 1977). Analyses of
these interactions are beyced the scope of the present work.
The initial vertical scales of the inertio-gravity waves
in the wake of Frederic are approximately egual to the
Eloise and Allen estimates. Hcwever, after a few IP the
vertical scale decreased to about 500 ra or approximately the
water depth. The horizontal scales estimated from the Fred-
eric ebservatiens differ greatly from the Allen data and
Eloise simulations. The horizontal scales during Allen were
four times the mixed layer horizontal scales observed immed-
iately after the passage of Frederic. This difference
increased to a factcr of about eight approximately 4 IP
after Frederic. These discrepancies may be due to the rug-
ged bcttom topography in the DeS oto Canyon region.

B. SUBINEBTIAI RESPONSE

The forcing by hurricare Frederic also altered the "mean
flow" and the lower frequency motions (periods greater than

63

3 IP) . Th€ variability and response of the mean flow to the
forcing is depicted in progressive vector diagrams (PVD) .
The ocean current data were low-pass filtered with a half
power point at 26 h to isolate the forced response on the
lower freguency variability.

The mean flow in the nixed layer at CMA2 was directed
towards the east prior to the storm (Fig. 18a). During the
passage of hurricane Frederic near JD 256, there was only a
small deflection in the mean flow. The subsequent horizon-
tal displacements indicate that the mean flow was about 26
km/d, or almost twice the mean current during the pre-stcrm
period. The mean flow at 179 m (not shown) at CMA2 also
increased towards the east.

The bcttca PVD at CMA2 indicates that the mean flow at
that level was mere or less constant (Fig. 18b). However,
the direction of this flow was opposite to that observed in
the mixed layer. Consequently, the bottom PVD was nearly a
mirrcr image of the surface PVD, although the bottom flow
was slower than that of tte mixed layer. There was also a
superposed lower frequency motion of about 3 I? near the
bottom, particularly during the period between the initial
and secondary maxima of HKE (first 6 IP in Fig. 9). The

64

lower frequency oscillations may be due to coastally trapped
waves, which were enhanced by the storm, since waves of the
same period were observed during the quiescent period
(Appendix E) . These waves can be altered by both bottom
topography and continuous density stratification (Wang and
Mooers, 1976). A study of the mechanisms by which the mean
flow is altered by these waves and the rugged terrain is
also beyond the scope of this thesis.

At CMA3, the surface mean flow was southward pricr to
the storm (Fig. 19a). At the time of storm passage, near JD
256, the surface mean flew was deflected outward from the
storm center. Between JD 257 to 269, the mean flow acceler-
ated to about 13 km/d and then decelerated to about 6 km/d
towards the south. Prior to the storm, the mean flow near
the tcttom at CMA3 was ncrthward following the local bottom
topography (Fig. 19b) . There was an eastward (up the canyon
slope) deflection late on JD 256. After completing a loop,
the mean flow veered towards the northwest (down the canyon
slope). Between JD 254 to 263, the mean flow near the bot-
tom was considerably faster than either earlier or later
times. Again, the mean flow near the bottom was reversed
relative to that in the mixed layer.

65

Scale: \-

^64

Figure 18

70 km

Lew-pass Filtered PV C at CMA2. The * represents
a time interval of 48 h starting on JD 248 for
a) mixed layer (19 m) and b) bottom Layer (324

m) .

Thermocline temperatures in the wake of Frederic varied
spatially as well temporally. There were sizeable subiner-
tial oscillations as well as near-inertial oscillations in
the temperature signal (Fig. 20) . These temperature varia-
tions were typically 3.0°C at both array sites and were
superposed on mean temperatures that range between 13.0 to

66

J

Scale: \

Figure 19.

Lew-pass Filtered PV E at CMA3. The * represents

a time interval of 48 h starting on JD 245 for

a) mixed layer (21 m) and b) bottom Layer (457
m) .

14.5°C in the thermocline. At CMA3r the temperature at 251
m started to increase about 0600 GMT on JD 255 and continued
to increase over the next 15 h due to the strong northward
current. The temperatures then decreased by 2<>C due to a
strong southward current. Inert ial fluctuations of abcut
3oc persisted ever the subsequent 8 IP . After this time,

67

the variations underwent a slight increase in frequency
which corresponded to about the time of the secondary maxi-
mum in the horizontal velocity components. For the remain-
der of the record, the temperature variations were
superposed on a stationary mean.

Eottom temperature variability at CMA2 is a clear exam-
ple of advective processes that are associated with lower
frequency motions (Fig. 21). On JD 254 (11 September), a
temperature decrease of 1.3°C over a 16 h period marked the
beginning of the forcing period as cool water was advected
from the DeSoto Canyon. The temperature then increased by
3.70 c over the next 25 h as warm water was advected by the
southward velocity component which was downslope in this
part of the canyon. As the winds relaxed, cooler water was
again advected into the area. Around JD 260, the tempera-
ture dropped fcy 4°C ever an 11 hour period. This drastic
change in temperature was due to the advection of cold water
by these subinertial waves of 3 IP period. From the orien-
tation of the ellipses (Table VII) , the direction of the
wave propagation was northward (upslope, away from the
DeSotc Canyon) . These waves set up cross-shelf oscillations
(relative to CMA2) in the north-south direction. After

68

0'9T

m

z:

u

e
o

M
<H

CO
(J)
•H

<D

CO

d)

s

•H

<U
O
(d

M
d)
O*

a
a»

E-"

a>

•H

fcH(N

o

CM

W
M

3

en

•H
Cm

OH OCT OET

69

these large variations, near-inertial oscillations in the
temperature were superposed on a gradual warming trend.

TABLE VII
Normalized Botary Spectrum Analysis at Subinertial Frequency

Rotary

Coef f .

Depth
(m)

Normalized
Rotary
Spectra
CW CCW

Elli
Stab.

ps

e
Dir.
(Dea.)

324
457

4.5x10*3 3.0x10+3
4.7x10+3 1.0x10 + 2

0.65
0.50

10
10

♦ 0.20

♦ 0.98

The potential energy spectra ware computed using the
temperature time series and the vertical temperature gradi-
ents from the AXET data collected by Black (1983) (Appendix
A) . The mcst prcminent peak in the potential energy spectra
at CMA3 is near the inertial frequency (Fig. 22) . The most
energetic peaks at the semi-diurnal and subinertial frequen-
cies are near the bottom. Potential energy variability in
the bottom layers at CMA2 (not shown) is significant at the
95 % confidence levels only for periods of about 3 IP.

From rotary spectrum analyses at periods of 3 IP near
the bottom: (1) the CW and CCW rotating components at CMA2
were nearly equal, with just a slight tendency towards CW
polarization; and (2) the motion at CMA3 was polarized in a

70

O'tl

cm
ro

(d

S3

u

e
o

M-i

W
Q)
•H
M
0)
CO

0)

M

s

B
EH

U

•H

Cm

0"ZI

(5)

an oot 06
ajn^jaduiaj,

71

10
cycles/hr

Figure 22

Potential Energy Spectra from CMA3
isotherm displacements observed at
(solid) , 437 b (dashed)
represents the diurnal
inertia! frequency and
frequency.

and
tide

for

251

457 m (dot

frequency,

the
m
:ed) .

f the

K<

M- the semi-diurnal tide

72

CW sense (Table VII) . Furthermore, the phase propagation

for these lcnger period waves was northward, and anisotropic
as indicated ry the ellipse orientation and stability.

The cross-spectrum between the temperature and north-
south velocity records at the bottom was also dominated by
the peak at about 3 IP (Fig. 23). The corresponding phase
and coherence estimates are 100 deg. and 0.98 respectively,
with the velocity leading the temperature signal. Hence,
these lower frequency oscillations in the cross-shelf direc-
tion at CMA2 were properly oriented for advecting the temp-
erature pattern. These cross-shelf oscillations were
responsible for the variability in the bottom temperature 4
IP after the passage of Frederic.

C. SOPERINERTIAL RESPONSE

As Frederic passed within 130 lem to the west of the
CMA1 , the dominant response was a significant increase in
the westward current (Fig. 24) . This increase was due to

convergence of flow on the right side of the storm. 3y con-
trast, there was a divergence of flow on the left side of
the hurricane, where water was transported off-shelf by the
winds. For example, a negative tide was recorded at Biloxi,
Mississippi which was indicative of the left side of the

73

storm. The north-south current only increased by about 20
cm/s, whereas the east-west current increased to 80 cm/s.
After the hurricane passage, near-inertial waves persisted
over the subsequent 3 IP. The post-storm semi-diurnal tidal
currents were mere energetic than during the quiescent
period and were superposed on a non-stationary trend. At
the intermediate depth at CMA1, the current speed increased
to 80 cm/s during the period of strong forcing, as the
semi-diurnal tidal currents were enhanced.

The presence of internal tides at the semi-diurnal fre-
quency is indicated by the HK E spectra during the storm
period at CMA1 (Table VIII). The spectral estimates in the
semi-diurnal frequency band are almost equal to the inertial
period estimates. Baines (1973) shows that one necessary
condition for the presence of internal tides is a marked
increase in the semi-diurnal tidal currents. The observed
increase in the semi-diurnal tidal currents should be equal
to or greater than the inertial effects. The storm-spectral
estimates at the semi-diurnal frequency exceed those from
the guiescent period by mere than order of magnitude (Appen-
dix B) . a barotropic tide propagating over the shelf break
generates internal tides cf semi-diurnal tidal period

74

o_

cycles/hr

Figure 23. Cross-Spectrum Between V-Velocity and

Temperature from CMA2 at 324 m. Kv represents
the diurnal tide frequency, f the inertia!
frequency and Mt the semi-diurnal tide
frequency.

75

40-|

*J

•

c
a;

20 1

G*-v

2.5;

o-

c M

-20-

d H

■

i °

-40-

D

.

-<30-

-eo-

60 -i
*j
C
0] 40-

o 9

<~%. 20-

o ,-
o - o

>

-20-1

-40

24£

256 264

Time (Julian Days)

Figure 24. Mid-depth Velocity Components from CMA1. The

ordinate depicts magnitude in cm/s of the east-
west currents (upper) and north-south currents
(lower) .

(Prinsenberg and Rattray, 1975 ; Barbee et al., 1975 ; Tcr-
grimson and Hickey, 1979). Hence, an increased barotropic
tide associated with the storm surge propagating over rugged
bottom topography should enhance the internal tides as mani-
fested in the semi-diurnal tidal currents.

76

TABLE VIII

Normalized HKE Spectra at Inertial and Semi-diurnal

Frequency

Normalized

Meter No. of Data Frequency HKE Spectra

Depth Points Resolution f M2

(§) (£2h) (SI^s) 2/cph

3200

0 .0018

2.0x10*3

1.0x10+3

2410

0.0025

3.0x10*3

6.0x10+2

3000

0 .0020

1.4x10+3

3. 2x10+3

CMA1

49
64
92*

* instrumentation problems with time clock
M2 is the semi-diurnal tide frequency band
f is the ineitial/diurnal tide frequency band

The dominance of internal tides on the shelf at CMA1 can
be explained ty considering the bottom slope in the region.
If the bottom slope is greater than the internal wave char-
acteristic, then the rays will not be admitted to the shelf,
but instead will be reflected into the oceans interior (Bar-
bee et al, 1S75 ; LeBlond and Mysak,1978 ; Torgrimson and
Hickey, 1979). The critical slope for the internal wave
characteristic is given by:

** = (°2 - f2)1/2 (6)

dy \2 _ c2} 1 K }

where f is the local Coriclis parameter, which is equal to
7.18 x 10~s rad/s near the DeSotc Canyon. The remainder of

77

the variables have been defined above. The slope of the
in ertio-gravity wave characteristic is about 4 x 10-3,
whereas in the direction of the canyon axis (45 deg.) the
bottom slope is 6 x 10~3 . Therefore, it is clear that the
inertio-gravity waves will be reflected seaward as the slope
of the characteristic is less than the bottom slope in the
DeSctc Canyon region. Inertial oscillations will be locally
generated on the shelf but they will only persist for a few
IP following the storm. The internal tides, however, will
be admitted to the shelf region because the slope of the
semi-diurnal tide characteristic is 2 x 10~2, which is
almost an order of magnitude larger than the bottom slope.

The semi-diurnal tides were also enhanced at CMA2 and 3,
but they were much less energetic than the inertial waves
(Table IX) . The CW polarized, semi-diurnal tidal variabil-
ity was more than an crder of magnitude greater than the CCW
variability estimates, except in the bottom layers at CMA3.
The directions of the semi-major axes of these waves changed
from 120 tc 20 deg. with depth at CMA2, whereas the opposite
occurred at CKA3, where the direction changed from 30 to 50
deg. Hence, the waves were anisotropic and exhibited a
strong north-south component of flow at CMA2 and to the
northeast at CMA3.

78

TABLE IX
Normalized Bctary Spectra at the Semi-diurnal Frequency.

Normalized
Rotary

Spectra Ellipse

Eepth CW CCW Dir. Rotary

(m) (cm/s) 2/cph Stab. (Peg.) Coef f .

a) CMA2

19 1.3x10+3 U.3X10+2 0.80

179 5.0x10+2 9.0X10+1 0.73

324 4.3x10+2 1.2x10 + 2 0.87

120

+ 0.30

60

+ 0.72

20

+ 0.41

k) £MA3

21

1.0x10+*

5.5x1 0+2

0.58

30

+ 0.55

251

6.7x10+2

1.0x1 0+2

0.92

90

+ 0.70

437

3.2x10+2

1. 3x10+2

0.40

40

+ 0.40

457

2.0x10+2

1.8x1 0+2

0.15

50

+ 0.04

In summary, the superinert ial response in the wake of
Frederic was dominated by the enhanced semi-diurnal tidal
current at all three arrays. The semi-diurnal tidal cur-
rents were markedly more energetic than the inertial cur-
rents on the continental shelf at CMA1 , and agrees with
internal tide theory (Baines, 1973) . The storm surge, which
is associated with the passage cf the storm, increases the
barotropic tide. As this increased barotropic tide propa-
gates over rugged bottom terrain in a stratified ocean, the
increase in the semi-diurnal tidal currents are manifesta-
tions of internal tides (Earbee et al, 1975 ; Prinsenberg
and Rattray, 1S75 ; Torgrimson and Hickey , 1979). Internal

79

tides can propagate freely up the continental slope and onto
the shelf, whereas inertio-gravity waves generated off the
slope are reflected towards the interior of the ocean. The
rotational characteristics of the semi-diurnal -ides, as
well as the direction of phase propagation of these waves,
are consistent with these theories.

80

V. NORMAL MODJS

A. THEORY

The Sturm-Licuville picblem is solved using the vertical
profile of Brunt-Vaisala frequencies and linear wave theory.
The vertical eigenvalues cf the horizontal velocity are then
used to ottain a least sguares fit to the Fourier coeffi-
cient time series computed from the demodulation of the cur-
rent meter records. The amount of variability associated
with each mode is determined to assist in understanding the
modal structure of the inertio- gravity wave response gener-
ated fcy the passage of a hurricane.

The problem is formulated by following the work of
Fjeldstad (1958) and making the following assumptions:

f plane;

continuous stratification;

hydrostatic balance with basic state at rest;

incompressible;

inviscid; and

flat tottcm.

81

The governing vertical structure equation for linear, iner-
tio-gravity waves is given by

2

£-| + y2$(z)W = 0 -> (7)

dz

where W(z) gives the vertical structure of the inertio-grav-
ity waver /f=l5''oa-fx , $(z)=N2- 0-2 , and K2 = fc2 + 12, and k

and 1 are the horizontal wave number vectors in the x and y
directions, respectively. Furthermore, equation (7) is a
second order, homogeneous, ordinary differential equation.
The specification of the problem is completed by imposing
the following boundary conditions:

f£ - guW = 0 z=0 7 (8)

dz '

W = 0 z=-d . (9)

The surface boundary condition (8) is a dynamic boundary
condition which states that at the sea surface pressure is
continuous across the interface. Boundary condition (9)

specifies no normal flow through the bottom. The solution
to equations (7-9), which define a well known Sturm-Liou-
ville problem, yields the structure of the vertical velocity
for constant N, and, is given by

Wn(z) . sin(2J£) n (10)

82

where the mode number n = 0,1,2.... The horizontal velocity
eigenfunct ions 0 (z) are given by

Un(z) ~ cos(H^) 7 (11)

which is the vertical derivative of wn (z) .

The Sturm-Licuville problem is solved numerically by the
predictor-corrector fourth-order method modified by Hamming
(Gerald, 1980). This methcd computes a new vector from four
preceding values. A fourth-order Runge-Kutta method is used
for the adjustment of the initial vertical increments and
the computations of starting values.

The resulting amplitudes of the horizontal velocity
eigenfunctions fcr modes 0 through 3 are shown in Fig. 25 .
The number of zero crossings equals the mode number. Most
of the vertical structure lies in the upper thermocline
regicn near the base cf the mixed layer.

B. DATA ANALYSIS

The horizontal eigenfunctions are then fitted in a least
squares sense to the time-varying amplitudes of the horizon-
tal velocities. The mathematical details are given in
Appendix A with the final matrix equations (16-18). The
time evolution cf the north-south velocity coefficients at

83

o
d

-6.0

-4.0

-2.0

Uh(z)

0.0

2.0

4.0

6.0

o

o
o

o

o
o

02

PL,

o

CO

o

d

ID
CO

O

d

o

q

d
to

o

d
o

lO

LEGEND

MODE 0

MODE 1

"MODE'S"

"MODE "3

Figure 25. Horizontal Velocity Eigenf unctions at 29.33 N

and §7.11 w. The abscissa depicts the amplitude
of the horizontal velocity eigenf unction for
mode 0 (solid) , mode 1 (dashed) , mode 2 (dotted)
ana mode 3 (dashed-dctted) .

8U

CM A3 indicates that the storm excited modes 0,1, and 3 with a
damped mode 2 (Fig. 26) . The amplitudes of modes 0 and 1
reached a peak on JD 256 and then decreased. By JD 262, the
barotropic mode horizontal speed decreased to nearly a con-
stant value of about 10 cm/s. After JD 262, modes 0 and 1
oscillated over the remainder of the record with a period
of about 5 IP. This behavior of the modss in Fig. 26 indi-
cates that a modulation occurred between inertial wave modes

0 and 1. More importantly, these modes define a modulation
envelope which is consistent with the amplitude variations
of the currents below the mixed layer.

At CMA2, the time-varying coefficients of the modes are
markedly different in their behavior (Fig. 27). The spatial
variability in the modal structure between CMA2 and CMA3 is
associated with the differences in the mean flow, which is
topographically controlled. Initially, all of the modes

were excited, with a fairly energetic lode 3. The baro-
tropic coefficient decreased to about 2 cm/s while the mode

1 amplitude approached zero near JD 252. After JD 262,
modes 0 and. 1 were in phase and defined the limits of the
coefficients.

35

►J-25

OOiO

OS

CO

10
CM

CD

^~-^

CM

00

>>

CO

Q

e

CO

- — h

,-^~

r-

3

10

1-3

CM

*— '

0

s

• *H

E-

CO

10

CM

05
CM

OC

02 01 0 01-

(s/uio) apniTjduiy

r
02-

oc-

CM

T3
Pi
(0

a^

o

ow

r\i

en
«« a>

Err-!

u o

a

(0 «.

WO •

COO
0) W 0)

•H (t)4->

0«O-P
•H^O
<4-l «0
4-l«- I
d) IT3

O <1)<D

O W

>iS (0

•H »w
COf*")

0)«-tQ)

> CO

W O

>wa

M
•H

86

Q!QiQ
OOiO

O

CD

CN

C\J

cm

cO

Q

-M

(0

a

CO

a.

CO

3

o

"-8

X

s>— ^

0)

0)

CO

cm

i

02

OT 0 01-

(s/uio) 9pn^Tjdiny

\-\o
cm

CN

•H

Cm

18

0)

s
to

CN

CP
M

3

a>

•H

Cm

02-

0C-

o

CM

87

The mccal current time series are re-computed using the
coefficients from the least sguares fit and the eigenfunc-
tions for each cf the current meter depths at CMA3. For
example, the reconstructed time series are compared in Fig.
28 with the actual demodulated current time series at 251 m .
Recall that a secondary maximum occurred at this site about
10 IF following the passage of Frederic (Fig. 10). The
reconstructed amplitude cf the barotropic mode is greater
than the actual v component of the data between JD 255 and
262 (Fig. 28). However, summing modes 0,1 and 2 accounts
for most cf the observed variability, with the barotropic
mode the most energetic. Over the remainder of the record,
JD 262 to 270, most of the observed variability can be
described cnly by modes 0 and 2 . The contribution of the
barotropic mode was significant and consistent with earlier
estimates that the vertical scales were of the order of the
water depth. The strength of the barotropic mode continued
for the entire time series of the modal coefficients at
CM A3.

Estimates of the modal contributions to the observed
near-inertial variability are given in Table X for both the
storm and poststorm periods at CMA2,3. These estimates are
based on the expression

88

X

2
£(V. - V. )

s = 1. - J - , (12)

j J

where V; and VjM are the observed and modal scalar components
of the horizontal velocity. The index m depicts the summa-
tion of the modes, for example, 0, 0+1, and 0+1+2, and inde
j repesents a summation ever the number of observations in
the period. The storm period is redefined to start at the
time of hurricane passage and continue for 7 IP. The post-
storm period starts where the storm period ends and contin-
ues for roughly 8 IP until the end of record.

The mixed layer variability at CMA2 is dominated by the
barotropic mode during the storm period. The contribution
of the barctrcpic mode tc the observed horizontal current
variability decreases with depth, but by including the
baroclinic modes most of the near-inertial variance can be
described by modes 0,1 and 2. The only exception is at 179
m. The first three modes contribute only 54 and 37% to the
observed horizontal current variance in the east-west and
north-south directions, respectively. Furthermore, the
u-velccity component of the barotropic mode exceeds the

39

TABLE X
Variance cf the Normal Modes at CMA2 and CMA3.

Depth Velocity
(m) Component Period

CMA2

19 u s

19 u ps

19 v s

19 v ps

179 u s

179 u ps

179 v s

179 v ps

321 u s

324 u ps

324 v s

324 v ps

CM A3

Modes

0

0 + 1

0+1 + 2

(%)

(%)

(%)

68

78

100

87

91

100

69

78

100

92

93

100

53

50

54

98

98

95

35

31

37

97

98

95

*

50

67

97

98

99

45

72

85

97

99

99

69

97

100

73

86

100

68

98

100

*

50

100

55

81

98

95

96

99

58

84

98

94

95

99

73

68

97

95

94

91

64

78

94

94

94

92

22

82

85

79

74

99

21

89

84

81

66

99

21 u s

21 u ps

21 v s

21 v ps

251 u s

251 u ps

251 v s

251 v ps

437 u s

437 u ps

437 v s

437 v ps

457 u s

457. u ps

457 v s

457 v ps

s: storm period ( number of observations = 1008)

ps: Dost-stcrm period (number of observations = 1152)

*: an overestimation of the variance (explained below)

observations by a considerable amount and causes equation
(12) to be less than zero (see Fig. 27). This behavior
could be attributed to the bottom boundary layer where infi-
nite energies occur when the slope of the internal wave
characteristic approaches the bottom slope (Prinsenberg and
Rattray, 1975) . Most of the observed variance during the

90

post-storm period is attributed to the barotropic mode. The
modes are much more well-behaved during this period, which
is presumably a manifestation of free waves.

The modes of the horizontal velocity are well-behaved
during the stcrm period at CMA3, except near the bottom
where only 20 to 22% of the observed variance can be attrib-
uted to the barctropic mode. This is probably due to bottom
boundary effects. At the other depths, the barotropic mode
explains 55 tc 73% of the variance, with most of the vari-
ance accounted for by the summation of the barotropic and
first two barcclinic modes. During the post-storm period,

the observations of the nixed layer velocity are less than
that of the barctropic mcde in the north-south direction,
which causes the value to be less than zero. The reason for
this large discrepancy is due to the e-folding scale of the
horizontal velocity in the mixed layer, the persistence of
the coefficient cf the barotropic mode (see Fig. 26) , and
the radiation of inertial waves away from the storm track.
Otherwise, 73 tc 95% cf tbe observed horizontal current var-
iance can be attributed to the barotropic mode, and adding
the first two barcclinic mcdes accounts for more than 90% of
the variance.

91

Hopkins (1982) modeled only the baroclinic response to
hurricane passage and he found that the inertio-gravity
waves decayed toe rapidly in the thermocline and bottom lay-
ers to account for the vertical structure of the observed
variability. Thus, a model with only the baroclinic modes
could not explain the secondary maximum observed in these
layers.

Some caution has to applied in the interpretation of the
normal modes for several reasons. First, the formulation of
the Sturm-Liouville problem assumes that the vertical struc-
ture consists of a standing wave for the N2 = constant case.
From basic physics, a standing wave can be decomposed into
two waves of equal amplitudes propagating in opposite direc-
tions, which assumas no phase propagation. Over the first 7
to 1 0 IP, there is a downward propagation of energy, as well
as upward propagation of phase, which violates the standing
wave argument. Secondly, a flat bottom ocean is assumed for
the application of normal mode theory. rha validity of this
assumption crucially depends on the ratio between the slope
of the internal wave characteristic to the bottom slope.
For example, as this ratic approaches unity, the energy den-
sities in the bottom boundary layer approaches infinity. At

92

245

250

255

260

265

270

20

O GO

> £
>

10-

o

20

O ^-v

0 JQ

10

<y £

1 °

>

0

20

>>

-J-i

• iw4

a ^

o ^

10

> s

1 °

>

245

i i i

MODE 0

MODE 0+1

250 255 260

Time (Julian Days)

265

270

Figure 28. V Component Time Series at CMA3 for the observed
time series (solid) and the reconstructed time
series (dotted) for a) mode 0 , b) mode 0+1 , c)
mode 0+1+2 .

93

that point, dissipation becomes important and must ba
included in the model (Prinsenberg and Rattray, 1975). Con-
sequently, the assumption of in viscid dynamics is no longer
valid. Despite all these approximations, the normal mode
analysis fits the observations quite well. Even the reso-
nance or nodulation envelope which occurred at CMA3 about 10
IP following the storm passage, is represented quite well by
modes 0, 1 and 2.

9U

VI . CO NCL 0 SI CNS

The ocean response near the DeSoto Canyon to hurricane
Frederic was dominated by the excitation of inertial waves
in the mixed layer and inertio- gravity waves in the thermo-
cline. These waves are not admitted onto the shelf because
the slope cf the characteristics of internal wave motion is
less than the bottom slope (Barbee et. al, 1975 ; LeBlond
and Mysak, 1S78 ; Torgrimson and Hickey, 1979). The major
features cf the forced inertio-gravity waves observed at
CMA2 and 3 are:

• initially, the vertical propagation of energy was of
the order cf 1. km/d as the forcing was felt throughout
the water column within 3 to 4.5 H following the'pas-
sage cf Frederic;

• the initial horizontal propagation of energy was about
80 km/d with a phase speed of 70 cm/s ;

• the waves were anisotropic at all depths and had an
upward propagation of phase;

• maximum HKE in the mixed layer occurred at the fringe
of the naximum wind regime with a second maximum in the
thermocline 6 to 10 IE after the storm;

• the EKE decayed on an e-folding scale of about U IP
after the maximum in the mixed layer and the second
maximum in the thermccline;

• the vertical propagation after about 6 IP is of the
order of 102 m/d with a corresponding horizontal propa-
gation of energy of 30 km/d;

• the mixed layer horizontal scales were 60 and 100 km in
the cress and along-track directions, respectively,
while the scales in the thermocline were 25 and 50 km
in the cress and along-track directions;

95

• the vertical scales were cf the order of the water
depth which suggests that the barotropic mode was
important;

• most cf the observed variabilty was in the barotropic
mode, with some contributions from modes 1 and 2;
together they defined a modulation envelope in the deep
thermccline at CMA3;

• the modal coefficients varied in space as well as time
due tc the rugged bcttcm topography and mean currents.

The mean flew was topcgrapically controlled and reversed
direction frcm surfaca tc bottom. The forced response in
the mixed layer caused the mean flow to increase at both
CMA2 and CMA3. The mean flow below the mixed layer was also
changed by the storm. Furthermore, this mean flow was

influenced ty subinertial variability (periods of 3 IP) and
was linked to the advecticn of cooler water cross-shelf rel-
ative to CEA2.

The semi-diurnal tidal currents increased in magnitude
and varied spatially in response to the passage of hurricane
Frederic. The near dominance of the semi-diurnal tidal cur-
rents over the inertial motions at CMA1 was consistent with
internal tide theories. The semi-diurnal tides were admit-
ted to the shelf region because the slope of the internal
wave characteristic exceeded the critical bottom slope

96

(Baines , 1973 ; Prinsenberg and Rattrayr 1975 ; Torgrimson
and Hickey, 1979) .

In a brcader context, the oceanic response to hurricane
Frederic was a geostrophic adjustment problem, which had
both transient and steady state components. After the tran-
sient, inertic-gra vity waves propagated away from the storm
track, a gecstrophically balanced ridge and current system
remained under the storm track. That is, The steady state
currents were due to the balance between the horizontal
pressure gradients and the Coriclis force (Geisler, 1970).
Although some adjustments to the mean flow are described,
additional study is reguired to completely understand the
interelat icnships between the forced mean flow, the forced
wave field and the bottom topography.

97

APPENDIX A

A. COMPLEX DEMODULATION

The motivation for complex demodulation is to isolate
the carrier wave of a certain frequency, which must be known
a priori , and to form amplitudes and phases of the modulated
signal. A linear filtering approach is used to form the
instantaneous amplitudes and phases in contrast to the har-
monic analysis method. Essentially, the amplitude and phase
time series represents a lccal harmonic analysis rather than
discrete estimates averaged over continuous periods (Otnes
and Enochscn, 1978) .

A band-limited time series is multiplied by the trigono-
metric arguments of the carrier wave. The resultant time
series is then lew-pass filtered to eliminate some high fre-
quency noise that is generated. The cosine and sine coeffi-
cients are ther combined tc form the amplitude and phase of
the modulated wave. Consider the time series:

x(i) = A<i) cos ( (2 7T f^At) ♦ 0(i)) , (1)

where 4 is the carrier frequency, At is the sampling inter-
val, </> is the phase , A is the amplitude of the wave, and i

98

represents a time index. Both the phase and amplitude vary
as functions cf time. Multiply equation (1) by the sine and
cosine arguments of the carrier frequency fc ,

xs (i)=A (i)iccs ((2irfciAt) + 0 (i) )1 sin (2 tt fc iAt )
xc (i)=A <i)jccs ((2 7T fc iAt) + 0 (i) ) I cos (2 n fc iAt )

(2)
(3)

where

x, (i) = x(i) sin (2 7rfciAt) , and
xc (i) = x(i) cos (27rfciAt).

0

Equation (2) can be rewritten as

xs (i) =A(i)/2Jsin ((U7rfciAt) * 0 (i) )1 - sin 0 (i) . (U)

Similarly, equation (3) can be expanded and simplified as:

xc (i) =A(i)/2|cos ((47rfciAt) ♦ 0 (i) )|+ cos 0 (i) . (5)

Low pass filtering the abcve x$ ,xt series yields:

y$ (i) = -A (i) /2 {sin 0 (i) \ , (6)

yc (i) = A(i) /2 {cos * (i) I ,

where ys (i) and yc(i) are the low-pass filtered xs (i) and \
(i) . These data are filtered by convolving the input series
with a set cf filter weights in the time domain (Bendat and

99

Pierscl, 1S71). Squaring and summing ys (i) and yc (i)

yields

a2 (i) = (y2 (i) + y| (i)) = A2 /w

or,

A(i) = 2 a(i) (7)

where A (i) is the amplitude as a function of time. Simi-
larly, the phase as a function of time is computed as:

4>6<i) = Tan-i <-ys (i) / yft (i) ) . (8)

B. EOTENTIAL ENERGY

The potential energy is computed -co determine the rela-
tionship of the isotherse displacements to internal wave
motion. Given a temperature time series from a moored array
and the corresponding vertical temperature gradient df/dz,
the isotherm displacement ? is computed in the following
manner:

dT/dz

where T1 is the fluctuating part of the observed tempera-
tures. The potential energy per unit volume is defined by:

100

PE = £■£ N2 V2 , (10)

where N* is the Erunt-Vaisala frequency, ?' is the isotherm
displacement fluctuations, and p0 is the reference density.
which is assumed to be unity.

C. IEAS1 SQUflEES FIT

Given a demodulated time series of the amplitude coeffi-
cients and the vertical structure of the horizontal velocity
eigenfunctions, the problem is to compute the time-varying
coefficients associated with each dynamical mode. The

time-varying amplitudes U(z^rt) at depths z,,za, .... z*
may be written as

0

where EN(Zj) are the eigenfunctions for modes 1,2,3..., and
CN(t) are the coefficients to be calculated in the analysis.
Only three tarcclinic medal coefficients are computed
because current meters were deployed at three depths in the
water column. For example, the current at depth z, , is
written as

0, = C, E, (z, ) * C2E2(z,) ♦ C3E3(z, ) ,

101

and there are siiilar expressions at depth z2 and z3. The
square of the error in the least squares fit ai depth z, isr

6,2 = |o, - (C, E, (z, ) ♦ CaEa(z, ) ♦ C3E3(z, ))| 2 (11a)

and similarly at z and z ,

ef = |ua- (C, E, (z2) + C2E2(z2) ♦ CaE3(Zi))l2 r (11b)

e| = |u3- <C,E, (z3) + CaEa(z3) + C2E3(z3))J2 . (11C)

The error is idnimized by solving the following set

def /%cy ♦ de2. /ac,+ae| /dc,= 0 , (12a)

def /dc2 +^2 /^C2 + ae| /dcz= 0 , (12b)

def /dc3 +aef /dC3+def /dC3 = 0 . (12c)

Differentiating equation (11) with respect -co C, and summing
the expressions according to equation (12a) yields

0, E,(z,) ♦ D2E2(zz) ♦ 03E3(z3) =

C, (E, (z,) S, (z,) * E, (za) E, (Z2) ♦ E,(23) E, (z3))

+ C2(E, (Z,) Ex(z,) + E,(z2) Ea(z2) ♦ Et (z3) E2(z3))

♦ C3(E,(zt) E3(z,) +E,(z2) E3(z2) +E,(z3) E5(z3)),

or

V0CEj(ZL) = C,^E, <zL) E,(z() +CaVE,(zi) E^z;)

lM l»l CI

102

+ ^^y, (zi

) E,(z«)

(13)

i«\

Now if equations (12brc) are applied to equations (11) , the
corresponding equations for C2 and C3 become:

3 3 3

Vu-^fz,) =C|VE,(z,) E2(z-,) ♦CiVE2(zi) E2 (Z| )

+ c3tX<z.> E*<zi>

(1U)

3

^OjE^Z;) =C,]TE,(Zj) E^(z;) ♦CaVE2(z,) S3(Z;)

i«l •=!

3

♦ C3yE3(Zi ) E3(z,)

(15)

By inspection, let

a*M=/JWZi > E*<Zi

1=1
3

for m,n =1,2,3 , (16)

(17)

i»l

Therefore, equations (13-15) can be simplified using the

above definitions into a simple matrix aquation of the form

**■ ftMN* CN

(18)

Equation (18) is a simple symmetric matrix equation and it
was sclved using the IMSL routine LEQT2F on the IBM 3033.

103

SEPENDIX E

A. DEEP REGICS

The ocean currents at CMA2,3 during the quiescent period
clearly showed topographical effects associated with the
DeSotc Canyon. The behavicr of the surface currents is best
described by a EVD (Fig. 29). The strong u component at
CMA2 produced a total displacement of 600 km over 39 days
for a 15. 4 km/d mean current. Inertial waves were super-
posed on this scean flow, particularly during the period from
JD 232 to 239. The mean flow increased during this period.
At CMA3, the mean flow decreased in the southerly direction
from 13.3 to 9.5 km/d (Fig. 29b) . The different orientation
in the mean flow across the entire shelf region is presum-
ably due to the influence of bottom topography.

The trajectory of the near-bottom horizontal currents at
CMA2 (Fig. 30a) was in the same direction as in the mixed
layer until JD 232. After the reversal, the mean flow was
towards the west at about 3.3 km/d. The bottom PVD at CMA3
(Fig. 30t) showed that iritially the mean flow was towards
the north at 2.0 km/d over the first 6 IP. The flow then

104

"P*

Scale.:

4»0kw\

Scale :

130 kw\

223 227 231

2+4

Figure 29. Surface PVD fcr a) CMA2 and b) CMA3. The *

represents a time span of 48 h starting a) JD
2C§ for CMA2 (19 m) and b) JD 213 for CMA3 (2 1
m) .

changed direction and accelerated tc about 2.8 km/d in the
same direction as in the surface layer.

Horizontal kinetic energy spectra for this period are
summarized in Table XI fcr both inertial and semi-diurnal
freguency bands. The vertical variation in the inertial HKE
levels at CMA2 indicated a larger amount of energy in the

105

fe24

Sc&\&:

30Knn

2+4

Figure 30.

Bcttcm PVD for a) CMA2 and b) CMA3. The *

represents a time span of 48 h starting a) JD

209 for CMA2 (324 m) and b) JD 213 for CMA3 (457
m) .

thermocline than at the surface or near-bottom. However,
the difference between the surface and bottom energy levels
was about cne order of magnitude. At CMA3 , the inertial HKE
estimates decreased steadily with depth and there was mere
than an crder of magnitude difference between the surface
and bottom layers. The HKE levels were slighxy greater at

106

CMA3 in tfce surface layer and thermocline, but they were
slighty less near the bottom than at CMA2.

TABLE XI
Normalized HKE Spectral Estimates at 95% Confidence

Normalized
HKE Spectra
f M2

(£lZs) i£</C£h

Meter

No. of Data

Depth

Points

Bandwidth

(1)

cph

a) CMA2

19 5780 0.001 1.7x10+3 2.1x10+2

179 4590 0.0013 3.0x10+3 3.8x10+**

324 5780 0.001 3.1x10+2 8.0x10+**

k) CMA3

21
251
437
457

4680

0.0013

8. 0x10+3

2.3x10+3

4680

0.0013

4. 5x10+3

7.0x10+2

4680

0.0013

1.9x10+3

4. 6x10+3

4680

0.0013

2. 0x10+2

1. 7x10+i

* instrumentation problems with time clock
M2 is the semi-diurnal tide frequency band
f is the inertial/diurnal tide frequency band

The semi-diurnal HKE estimates also varied spatially.
The estimates of HKE in the thermocline and near-bottom lay-
ers at CMA2 were not significant; however, the HKE estimates
at CMA3 were significant at all depths. The HKE estimates
at 437 m even exceeded both the surface layer and thermo-
cline levels by about an crder of magnitude. In comparison
with the near-bottom estimates at 457 m, the kinetic energy

107

levels were mere than two orders of magnitude greater.
These pronounced changes in the semi-diurnal energy levels
with depth indicate the presence of internal tide motion.
In most of tie kinetic energy spectra examined, longer
pericd motions were insignificant, because -heir energies
were smeared acrcss adjacent bands except in the bottom lay-
ers. Subinertial variability at periods of about 3 IP was
quite apparent in the bottcm reccrds at both CMA2 and CMA3.
These motions were generally more energetic than the semi-
diurnal tides.

Rotary spectra are analyzed for the surface and bottom
current records and are given in Table XII. Most of the HKE
variability is consistent with polarized, CW rotating iner-
tial motions. The CW energy level exceeds the CCW level by
about an order of magnitude. The ellipse orientations for

these waves rctates CCW (tacks) with depth. These ellipses
are stable except for the near-bottom inertial period motion
at CMA2. The direction of the semi-major axis of the
ellipse varies over time, which indicates that the inertial
waves are isotropic. The semi-diurnal tidal motion is

polarized in the CW direction. Generally, the semi-diurnal
tidal motions are coherent, although the phase differences

108

between the horizontal velocity components vary between the
arrays.

TSBLE XII
Normalized Rotary Spectrum Analysis

Normalized

R o ta r y

Spectra Ellipse

Freg. Deprh CW CCW Stab- Dir. Rotary

Band (m) (cm/s) »2/cph (Dec[.) Coeff .

a) CMA2

I

19

1.6x10+3

1.9x10+2

0.80

103

+0.90

s

19

1.6x10 + 2

3.6x10+ i

0.39

101

+0.87

I

324

5.0x10+2

1.4x10+1

0.05

90

+0.87

s

324

1.0x10 + 0

1.1x10+o

0.60

169

-0. 18

0.40

157

+0.9 9

0.61

55

+0.63

0.51

138

+0.9 6

0.20

93

+ 0.75

b) CMA3

I 21 1.2x10+* 1.7x10+1

S 21 4.0x10 + 2 3.0x10+1

I 457 5.6x10+2 1.6x10+1

S 457 3.0x10+1 3.9x10+0

I=inertial frequency band

S = Semi-diurnal frequency band

Ocean bottom temperatures ranqed between 8.0 and 12.5 °C
at CMA2 and 8.0 to 9.0 oc at CSA3 (Pig. 31) , but there were
different variations and trends. At CMA2, (not shown) the
temperature decreased gradually to about 8.3 °C near JD 234,
and then increased abruptly without any periodicities. Bot-
tom temperatures at CMA3 were much lower and there was
near-inertial motion superposed on this early cooling trend.

109

The minimum temperature occurred on JD 232, which was about
2 IP before th€ minimum observed at CMA2. These minima in
the temperature, in addition to the cooling trends, are man-
ifestations of the advection of cold water from the DeSotc
Car-yen by the mean flew.

B. SHALLOW REGION

In the thermocline (U9 m) , the inertial oscillations
were not nearly as energetic and did not rotate CW as in the
mixed layer (Fig. 32). The non-stationary trends in the
east-west currents were similar to the mixed layer observa-
tions, but there was an initial mean flow towards the deep
ocean. Superposed on the mean flow were lower frequency
oscillations with periods cf the order of about 10 to 14 IP.
The total displacement over the forty days was about 300 km
which corresponded to a mean current of 7.5 km/d.

Inertial oscillations at 64 m were not as energetic as
at 49 m, but there was a similar non-stationary trend in the
time series. This trend was associated with a lower fre-
quency waves having a pericd of 10 to 14 IP. These oscilla-
tions were superposed on a northward mean flow which was
slighly less than 7.0 Jcm/d. Records from the current meter
nearest to the bottom (92 m) , were out of synchronization

110

06 9*8 99 VQ Z2 09

o

<"*

OJ

•

s

r-

Lf">

o

d-

5

V

(TJ

m

*2

X3

/*™v

(J

w

2? >^

s

S «5

o

Q

u

C

w

03

0)

•»■*

•H

__ . »-t

U

co H

V

33

CO

<U

CD

a

£

•H
EH

•«-1

E-"

0)

"*

M

a

«rt

M

0)

CU

s

<u

E-t

o

1

•

f—

CO

a)

M

3

2

CT>

?3

fa

9^

?3

111

50 k

tvv

Figure 32. Kid-depth PVE at CMA1. The * represents a time
span of 48 h starting on JD 207 at 49 m.

due to internal clock problems. For example, large

increases in the velocity occurred on JD 252 as opposed to
256. Hence, the data were not included.

The HKE spectra for the 21, 49, and 64 m depths are
shown in Table XUI for bcth the inertial and semi-diurnal
frequency tands. The inertial/ diurnal tidal kinetic energy

112

in the surface layer was two orders of magnitude larger than
in the subsurface layers. The semi-diurnal tidal motion

decreased from the surface to 49 m and then increased with
depth such that the HKE near the bottom almost equaled that
near the surface.

TABLE XIII

Normalized HKE Spectral Estimates at 95% Confidence

Normalized
Meter No. of Data Frequency HKE Spectra

Depth Points Resolution f M2

(J) 2£b (cm/s)+2/cph

CMAJ

21 5825 0.001 1.4x10+* 1.5x10+2

49 5825 0.001 1.4x10+3 4.3x10+1

64 5825 0.001 5.0x10+2 8.0x10+1

92* 5825 0.001 1.5x10+2 1.0x10+2

* instrumentation problems with time clock
M2 is the seui-diurnal tide frequency band
f is the inertial/diurnal tide frequency band

Temperature data at bcth 21 and 49 m are not available
because of the thermistor problem. The temperatures ranged
from 18.0QC to the thermistor cutoff of 21.5 °C at 49 m.
Temperature variations were about 0.5OC over an IP, although
most of the variability was associated with some non-sta-
tionary trends and lower frequency motions similar to those
observed in the currents. Jus- prior to the storm, tempera-

113

tures decreased to a relative minimum as cold water was
advected fcy tfce mean flow.

In summary, based on the observations during the quies-
cent period, the mean flew strongly depends on the bottom
topography. Inertial band motion is evident at all depths
during the quiescent period. Seme of this motion is due to
freely propagating inertial waves while the remainder is
part of the forced diurnal tides. Internal tides, as well
as inertial oscillations, are part of the freely propagating
internal wave continuum below the surface layer. The effect
of the topography on the mean flow and the wave motions fur-
ther complicates the dynamics of the region and the resul-
tant circulation . Longer period waves (2.5 to 3 day
pericd) set up north-south oscillations that advect cooler
water from the DeSoto Canycn onto the shelf region. Super-
inertial frequency motions are influenced by the diurnal and
semi-diurnal tides and contribute to the variability.

1 14

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