ANALYSIS AND INTERPRETATION OF
TIDAL CURRENTS IN THE COASTAL BOUNDARY LAYER
by
PAUL WESLEY MAY
B. S., Southern Missionary College (1972)
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
and the
WOODS HOLE OCEANOGRAPHIC INSTITUTION
May, 1979
Signature of Author
Joint Program in Oceanography, Massachus6ts Institute of Technology
- Woods Hole Oceanographic Institution, and Department of Earth and
Planetary Sciences, and Department of Meteorology, Massachusetts
Institute of Technology, May, 1979.
Certified by .........
.
.
..
Thesis Supervisor
Accepted by ..........................................................
Chairman, Joint Oceano r
mittee in Earth Sciences,
Massachusetts In
5
hd4
1
- Woods Hole Oceanographic
Institution.
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ANALYSIS AND INTERPRETATION OF
TIDAL CURRENTS IN THE COASTAL BOUNDARY LAYER
by
PAUL WESLEY MAY
Submitted to the Massachusetts Institute of Technology - Woods
Hole Oceanographic Institution Joint Program in Oceanography on
May 1, 1979, in partial fulfillment of the requirements for the
degree of Doctor of Science
ABSTRACT
Concern with the impact of human activies on the coastal region of the
world's oceans has elicited interest in the so-called "coastal boundary
layer"-that band of water adjacent to the coast where ocean currents adjust
to the presence of a boundary. Within this zone, roughly 10 km wide, several
physical processes appear to be important. One of these, the tides, is of
particular interest because their deterministic nature allows unusually
thorough analysis from short time series, and because they tend to obscure the
other processes.
The Coastal Boundary Layer Transect (COBOLT) experiment was
conducted within 12 km of the south shore of Long Island, New York to
elucidate the characteristics of the coastal boundary layer in the Middle
Atlantic Bight. Analysis of data from this experiment shows that 35% of the
kinetic energy of currents averaged over the 30 m depth are due to the
semidiurnal and diurnal tides.
The tidal ellipses, show considerable vertical structure. Near-surface
tidal ellipses rotate in the clockwise direction for semidiurnal and diurnal
tides, while near-bottom ellipses rotate in the counterclockwise direction for
the semidiurnal tide. The angle between the major axis of the ellipse and the
local coastline decreases downward for semidiurnal and increases downward
for diurnal tides. The major axis of the tidal ellipse formed from the depth
averaged semidiurnal currents is not parallel to the local shoreline
2
but is oriented at an angle of -15 degrees. This orientation "tilt" is a
consequence of the onshore flux of energy which is computed to be about 800
watts/m.
A constant eddy viscosity model with a slippery bottom boundary
condition reproduces the main features observed in the vertical structure of
both semidiurnal and diurnal tidal ellipses. Another model employing long,
rotational, gravity waves (Sverdrup waves) and an absorbing coastline explains
the ellipse orientations and onshore energy flux as a consequence of energy
dissipation in shallow water. Finally, an analytical model with realistic
topography suggests that tidal dissipation may occur very close (2-3 km) to the
shore.
Internal tidal oscillations primarily occur at diurnal frequencies in the
COBOLT data. Analysis suggests that this energy may be Doppler-shifted to
higher frequencies by the mean currents of the coastal region. These motions
are trapped to the shore and are almost exclusively first baroclinic mode
internal waves.
Thesis Supervisor: Gabriel T. Csanady
Title: Senior Scientist
Woods Hole Oceanographic Institution
ACKNOWLEDGEMENTS
This thesis
contains contributions
few of which can be mentioned in
would
like
to thank. my
thesis
support, encouragement, and
member
of
my
this short space.
advisor,
patience.
committee,
thesis
kindly
only a
from many individuals,
Dr.
Dr.
G.
R.
In particular, I
T.
C.
provided
Csanady,
for his
Beardsley,
another
the
from
data
the
CMICE experiment as well as a number of helpful discussions.
Tom Hopkins and Jim Lofstrand,
Dr.
Laboratory,
COBOLT
spar
braved
were very helpful
buoy data.
inclement
In
weather
in
both from Brookhaven National
obtaining a useable version
addition, Bert Pade
and
seasickness
to
and Neal
perform
of the
Pettigrew
the
daily
hydrographic surveys.
The
the
data processing was
Woods
Pennington,
Hole
in
aided
Oceanographic
particular,
was
greatly by
Institution
consulted
on
the "Data
Buoy
Dollies"
Group.
numerous
of
Nancy
occasions
and
cheerfully corrected many of my programming errors.
I would also like
to thank Doris Haight and May Reese
for their
help in typing the various versions of this thesis.
This
Brookhaven
359133-S.
work
was
National
supported
by
Laboratory
the
Department
subcontract
of
numbers
Energy
325373-S
under
and
TABLE OF CONTENTS
ABSTRACT .
.
.
.
. .
.
ACKNOWLEDGEMENTS . . .
TABLE OF CONTENTS. . .
LIST OF FIGURES. . .
LIST OF TABLES .
I.
II.
.
.
. .
The Coastal Boundary Layer and COBOLT Experiment
A.
Introduction . . . . .
B.
The COBOLT experiment.
C.
The experiment site. .
D.
Coastal measurements .
E.
The COBOLT instrumenta tion .
F.
COBOLT experiments and data. . .
G.
Data processing. . . . . . .
.
. .a
Nearshore Tidal Current Observations
A.
Introduction .
B.
Tidal analysis . .
C.
The tidal ellipse.
. .
.
.
.
.
.
.
.
4.
.
.
.
.
.
.
Tidal observations in the Middle Atlantic Bight.
Analysis of the COBOLT tidal signal. .
t.
.a..
5
F.
Results of the semidiurnal analysis. . . . . . . .
G.
Band structure of the semidiurnal admittances. . .
H.
Results of the diurnal analysis. . . . . . . . . .
I. Consequences and conclusions . . . . . . . . ...
III.
Tidal Dynamics and Theory
A.
Tidal theory . . . . . . . . a.
.. .
. .......
B.
Vertical structure of tidal currents . .
89
C.
Effects of friction on tidal propagation . . . . .
99
D.
The Sverdrup-Poincare
E.
The Sverdrup wave--no reflection
.
.
.
.
101
.
.
.
.
107
F.
The Poincare wave--perfect reflection. .
107
G.
The combination Sverdrup-Poincare wave .
113
H.
Comparison of the model to observations.
123
wave model
I. The effects of local topography. . . . .
J.
IV.
Summary. . . . . . . . . . . . . . . . . . . . . .
128
144
Observations of Coastal Internal Tides
Introduction . . . . . . . . . . . . . . . . . . .
147
B. Dynamical theory of the internal tides . . . . . .
148
C.
Solutions for constant Brunt-Vaisala frequency . .
150
D.
Solutions for an arbitrarily stratified fluid. . .
152
E.
The mean fields of the COBOLT experiment .....
156
A.
6
G.
Internal tidal oscillations. . . . . .
. .
. . .
166
H.
Modal structure of the internal tides. . . .
. . .
178
I.
Comparison to theory .
.
. . .
. . .
190
J.
Energy and flux of the internal tide .
. . .
. . .
190
K.
Conclusions . . . . . . . .
a.. ....
REFERENCES . . . . . . . . . . .
. .
. . . . .
. . . . . .
. . . . . . .
BIOGRAPHICAL NOTE. . . . . . . . . .
.
. . . . . .
.
192
. . . .
. .
.
. .
. . .
194
LIST OF FIGURES
FIGURE
1-1
PAGE
Cross section and classification of the Middle Atlantic
Bight continental shelf
14
1-2
The South shore of Long Island
17
1-3
Configuration of a typical COBOLT spar buoy
22
1-4
Locations of the four spar buoys of the May, 1977
experiment
1-5
Depth profile off Tiana beach, location of spar buoys,
instrument packages, and hydrographic survey stations
1-6
25
26
Calendar for the May, 1977 experiment showing buoy
duration and hydrographic surveys
27
1-7
Configuration of MESA-CMICE mooring #5
30
2-1
Definition sketch of the tidal ellipse
42
2-2
Cophase contours of the Middle Atlantic Bight from
to Swanson, 1976
2-3
Tidal ellipses in the New York Bight from Patchen, Long,
and Parker, 1976 (MESA)
2-4
2-5
45
49
Variance preserving plot of the kinetic energy of
depth-averaged currents at buoy 2
55
Semidiurnal tidal ellipses in the COBOLT experiment
59
2-6
Semidiurnal tidal ellipses in the CMICE experiment
60
2-7
Vertical profile of semidiurnal ellipticity
61
2-8
Vertical profile of semidiurnal orientation
63
2-9
Vertical profile of semidiurnal kinetic energy
64
2-10
Semidiurnal ellipses for depth-averaged currents
66
2-11
Band structure of semidiurnal admittances
70
2-12
Vertical profile of diurnal ellipticity
73
2-13
Vertical profile of diurnal orientation
74
2-14
Vertical profile of diurnal kinetic energy
76
2-15
Diurnal ellipses for depth-averaged currents
77
3-1
Theoretical vertical profiles of semidiurnal ellipticity
93
3-2
Theoretical vertical profiles of semidiurnal orientation
94
3-3
Theoretical vertical profiles of diurnal ellipticity
96
3-4
Theoretical vertical profiles of diurnal orientation
97
3-5
Geometry of the step shelf tidal model
3-6
Ellipticity versus offshore distance for perfect
reflection (r'=1) and several positive incidence angles
3-7
110
Orientation angle versus offshore distance for perfect
reflection (r'=1) and several incidence angles
3-9
109
Ellipticity versus offshore distance for perfect
reflection (r'=1) and several negative incidence angles
3-8
102
112
Ellipticity angle versus offshore distance for imperfect
reflection (positive incidence angle)
115
3-10
Ellipticity versus offshore distance for imperfect
reflection (negative incidence angle)
3-11
Orientation angle versus offshore distance for imperfect
reflection
3-12
139
Kinetic energy versus offshore distance for depth
dependent model
3-21
138
Ellipticity versus offshore distance for depth dependent
model
3-20
131
Orientation angle versus offshore distance for depth
dependent model
3-19
130
Comparison of modelled depth dependence with actual depth
profile off Tiana Beach
3-18
124
Plot of ln(H -H) versus offshore distance at COBOLT
site, plus linear curve fit
3-17
122
Ellipticity versus Orientation angle at y=O with COBOLT
observations plotted for three semidiurnal frequencies)
3-16
121
Ellipticity versus Orientation angle at y=O; varying
incident wave frequency
3-15
120
Ellipticity versus Orientation angle at y=O; reflection
coefficient varied while holding incidence angle constant
3-14
117
Ellipticity versus orientation angle at y=O; incidence
angle varied while holding reflection coefficient constant
3-13
116
141
Dissipation versus offshore distance for depth dependent
model
143
3-22
Free surface elevation versus offshore distance for depth
144
dependent model
4-1
Twenty-two day average sigma-t cross section at the
COBOLT site
4-2
161
Twenty-two day average Brunt-Vaisala frequency profiles
at buoys 2-4
164
4-3
Salinity time series from instruments 31 and 34
167
4-4
Sigma-t time series at all levels, buoy 3
168
4-5
Sigma-t energy density spectrum from insts.
170
21-23 & 31-33
4-6
Temperature energy density spectrum from
insts. 21-23 & 31-33
171
4-7
Onshore velocity energy density spectrum, insts 32-32
172
4-8
Alongshore velocity energy density spectrum,
4-9
Kinetic energy density at 15 m for Sept.,
4-10
First three vertical velocity modes at buoy 3
179
4-11
First three horizontal velocity modes at buoy 3
181
4-12
Fitted first baroclinic mode at buoy 2:
insts 32-33
1975 experiment
Fitted first baroclinic mode at buoy 2:
velocities
177
onshore
184
velocities
4-13
173
alongshore
185
LIST OF TABLES
TABLE
PAGE
1-1
Summary of data returns from Tiana Beach site
31
2-1
The wavelength of the semidiurnal tide
47
2-2
The tidal components used to construct the reference
time series
52
2-3
The semidiurnal admittances and coherences
56
2-4
The band structure of the semidiurnal admittances
69
2-5
The diurnal admittances and coherences
71
4-1
Numerical values of T, S, N, and sigma-t: buoy 3
161
4-2
Longshore mean currents at all instruments
165
4-3
Isopycnal displacements for buoys 2 & 3 for all levels:
12 hr.,
4-4
18 hr., and 24 hr. periods
175
Energy distribution among modes 0-2; buoys 2-4; 12 hr.,
18 hr., and 24 hr. periods
183
CHAPTER I
THE COASTAL BOUNDARY LAYER AND THE COBOLT EXPERIMENT
A.
Introduction
The coastal regions of the world's oceans have been the subject of
increased interest among physical
oceanographers
This narrow band of shallow water surrounding
been regarded as too insignificant
deep
ocean,
theories.
and
as
too
The economic
fisheries
and energy
scientific
interest
important study in its
have
also
spurred
(see
by
reviews
conform
the
activities,
dynamics
own right.
interest
shallow water dynamics
to
and
are not
Niiler
of
to
simple
however, have
the
led
has long
the
offshore
promoted
continental
to
dynamical
of
seas
Improved measurement
have
decade.
the great volume of the
and environmental considerations
related
in
the continents
to affect
complicated
in the last
new
as
an
capabilities
realization
that
as complicated as originally supposed
(1975)
and
Winant
A
(1978)).
complete
understanding of the interaction of these regions with the rest of the
ocean may yet prove the shelf's importance
to the deep ocean
if
only
as a boundary condition.
The breadth of the continental
areas
within
Fleming,
shelf
the
1942),
break
or
one hundred
meter
shelf
is
by definition
isobath
(Sverdrup,
limited
Johnson,
to
and
though shelf studies often pass beyond the continental
continental
slope
in
order
to
include
important
conditions
in
the
transition of
shallow to deep ocean flow.
east coast of the United States,
Middle
Atlantic
Bight,
distance of 100 km.
the
a region known as the
specifically in
shelf
extends
Off the
typically
to
an
offshore
A representative cross section of this particular
region is shown in figure 1.
The eastern continental
areas
depicted
in
figure
boundary
dissimilarities
layer
of
the
a
1:
known as the shelf break;
coastal
shelf is
often subdivided
region
of
sharp
further into the
topographic
inner and outer self regions;
(CBL)
inner
close
and
to
outer
the
shore.
shelf,
and
and,
a narrow
The
the
change,
dynamical
shelf
break,
often noted as the basis of this classification scheme, are summarized
in Beardsley, Boicourt, and Hansen (1976).
The
This
layer.
wide,
region
that
interest
here
is
the
boundary
coastal
applied to a band of water on the order of 10 km
term is
which is
of
is
small
compared
to the width of
the continental
shelf,
but large compared to the several hundred meter width of the surf zone
or
littoral
layer is
zone.
From a physical
the region where offshore
standpoint,
the coastal
boundary
to the presence of
currents adjust
the coast.
Early
features
work
which
on
are
Great
the
peculiar
observational
particular,
(Csanady,
Lakes
to
evidence
the
and
coastal
primary
transient
mechanism
meteorological
by
which
forcing.
the
layer.
modelling
led
In
to
1977 for more details)---
nearshore
With
revealed
has
boundary
theoretical
the concept of a coastal "jet" (see Csanady,
the
1972)
regard
waters
to
the
respond
to
relatively
20
40
60
80
100
\ A!00
150
200
250
Figure 1-1
Cross section and classification of the Middle Atlantic Bight continental shelf
uncomplicated
dynamics
of
large
lakes,
this
model
has
substantially
increased the understanding of coastal boundary layer processes.
While
application
boundary layers
difficult
are
presence
the
And,
than
of
what
the
strong
current
equivalent
tidal
are important in
theory
observations
do
Great
currents
So,
jet
observational
observations
with the rest of the shelf.
scales
coastal
straightforward,
since suitable
rare.
interpret
is
of
exist
oceanic
confirmation
in
the
are
Lakes
more
appears
scale
that
coastal
is
more
coastal region
difficult
observations
and large
it
to
due
flows
to
to
the
associated
two additional
the oceanic coastal boundary layer:
time
the mean
circulation, and tidal frequency motions.
As part of the Coastal
thesis
is
directed
toward
Boundary Layer
developing
Experiment
an
understanding
frequency motions of the coastal boundary layer.
by presenting a description of the
followed by a conceptual model
features
or
of the barotropic
baroclinic
tides
is
that reproduces
addressed
tide.
of
the
This goal is
tidal currents
or surface
(COBOLT),
tidal
pursued
of the coastal
many of
this
zone
the observed
The question of internal
a
detailed
description
and
Transect
(COBOLT)
experiment
was
with
comparison to existing models.
B.
The COBOLT experiment
The
designed
COastal
BOundary
specifically
to
Drawing from experience
of
newly
developed
Layer
study
the
complexity
of
the
coastal
zone.
gained on the Great Lakes and taking advantage
instrumentation,
it
was
planned
to
provide
a
16
detailed
spatial
boundary
layer,
with
the
the
large
motivation
for
stations
little
was
currents
picture
induced
circulation
experiment
offshore,
known
project
temporal
scale
the
power
The
and
represents
the
the
with
joint
by
the
efforts
and
the
shelf.
proposals
to
realization
coastal
the
of
coastal
interaction
continental
the
about
wind-driven
tides,
provided
together
the
by
of
was
observationally
of
locate
that
boundary
Woods
The
very
layer.
Hole
Ocean-
ographic Institution and Brookhaven National Laboratory (BNL).
C.
The experiment site
The
southern coast of Long Island was chosen for the site of the
COBOLT experiment because of
coastline.
This region is
The approximate
30'W.
The
figure 2.
shown in
km west of Montauk Point,
and 60
Apex,
coordinates
site
enjoys
the
of
easy
idealized
Tiana Beach,
and
is
the
terminus
from
straight
the shore
of Long Island.
400
is
720
and
of
waters
about 6 km east of
distance
reasonable
45'N
protected
the
Inlet which
also within
are
experiment
access
Shinnecock Bay through Shinnecock
Tiana Beach,
an
135 km east of New York City and the New York Bight
is
location point,
its similarity to
of
Brookhaven
National Laboratory.
Geographically,
the
coast
of
Long
northern boundary of the Middle Atlantic Bight.
virtually continuous barrier
for
entrances
water
to
topography
protected
is
formed
sand bar,
bays
from
in
forms
Island
The coast itself
with only
its
loose,
part
the
is
a
four or five breaks
km extent.
The
large-grained
sands
195
of
shallow
and
is
AN
0
50
uN 0
410
0
NG
TIANA
BEACH
OCEAN
ATLANTIC
40*30'
Figure 1-2 The south shore of Long Island
ffimffikwwwwmiav
18
as
1973)
et al.,
(Swift
features
smooth with minor "swale"
remarkably
the only irregularities.
topographic
While
uncomplicated
are
smooth
there
are
other
dynamics,
and
Swanson,
show
1976)
close-to-resonant
response
vicinity of Montauk Point
to point.
aberrations
strong
gives
and
rise
the Sound
Also,
(Ketchum and Corwin, 1964).
phases
tidal
of the data.
is
(Redfield, 1958
a major
propagation
tidal
in
to the
source
of
the
in
rapidly
change
A
Sound.
currents
large
that
COBOLT
the
likely to have some
is
entrance
to very
relatively
of
observations
to 50 km away from the
up
characteristics
point
Tidal
COBOLT measurements.
on
to
lead
features
for example,
The presence of Long Island Sound,
effects
and
the interpretation
complicate
which may
site
experiment
features
from
fresh water
Since the runoff from Long Island itself
of any fresh-
is relatively minor, the Sound is probably the origin
ening that occurs at the COBOLT site.
In
addition,
measurements.
Inlet
the proximity of Shinnecock
is
Though it
narrow (about
may influence
200 m wide)
and less
the
than 5
m deep at most points, visual surveys indicate that the plume of tidal
discharge
as
2-3 km out to sea and is
reaches
Thus,
6 km.
it
conceivable
is
visible as
that moorings
which
far down-shore
are
close
to
shore may show the effects of being near to the inlet.
D.
Coastal measurements
One
of
measurement
in
mounting
major
hurdles
encountered
program is
that of
choosing adequate
the
a
near-shore
instrumentation.
It
19
is
well
known
profoundly
carefully
that
affected
by
conceived
Instruments which
vane),
current
such as
meters
high
sampling
mounted
frequency
and
near
surface
are
waves
even
when
gravity
the
averaging
schemes
are
employed.
sample speed and direction (via Savonius
the VACM or Aanderaa current meters,
rotor
and
are particularly
susceptible to rectification of wave-induced orbital velocities, even
when mean velocities
Taut rope moorings
are
Strong currents,
cause
sizable
layer
comparable
magnitude
(McCullough,
also contribute to measurement
ways.
surface
of
errors
such as those encountered in
vertical
excursions
fluctuations
to contaminate measurements
of
the
in several
the coastal zone,
instrumentation.
can be transmitted
at deeper
1977).
down
Also,
the flexible
instruments.
Finally,
rope
the lack
of torsional rigidity may introduce directional errors.
The presence of a nearby coast
In
own.
the
addition to the increased
nearness
polarized in
of
the
coast
adds measurement problems
error
causes
low
current of
orientation
velocity--an
frequency
currents
to
be
the alongshore direction and increases the probability of
alongshore
in
its
possibility of human interference,
measuring important- onshore velocities
strong
of
can
amount which is
50
incorrectly.
cm/sec,
cause
a
1
as
little
cm/sec
comparable
For example,
as
error
in
one degree
in
the
a
of
onshore
to the true mean value of the
onshore currents.
To
mooring
the
list
of
design must
salinity be measured.
difficulties
be
added
the
to
be
demand
overcome
that
Unlike the deep ocean,
in
both
instrument
and
temperature
and
where tight temperature-
20
salinity
properties
make
possible and eliminate
shallow
coastal
a
functional
(somewhat)
waters
have
relationship
between
the need for salinity
-no such
links.
Density
the
two
time series,
variations
are
controlled by salinity at certain times of the year and by temperature
at
other
times,
and
both
signals
are
usually
large.
In
order
to
separate dynamic effects, time series of both parameters are essential.
Despite
the
difficulties,
carried out in the coastal
several
zone
of
conventional measurement techniques.
useful
experiments
the Middle Atlantic
have
been
Bight
using
Two of the most notable of these
are the EG&G Little Egg Inlet experiment (EG&G, 1975) and the New York
Bight MESA project (Charnell and Hansen,
successes,
eliminate
a concerted
the potential
ation and moorings,
in
effort
earlier
was
sources of
1974).
made in
These
view of these
the COBOLT experiment
error in conventional
and to add measurement
studies.
Even in
capabilities
requirements
to
instrument-
not available
necessitated
a
radical
departure from common deep water mooring design and instrumentation.
E.
The COBOLT instrumentation
The
Spar",
mooring
platform
the
COBOLT
instruments,
was developed for coastal work off La Jolla,
constructed of sections
Brush,
for
1972).
The
the
California.
of 2 1/2" diameter PVC pipe (Lowe,
moorings
utilize
specially
without sacrificing
designed
pipe.
Since
it
is
too much of -the inherent
torsionally
rigid
(torsional
It
Inman,
joints to allow the spar to articulate freely at the several
points,
"Shelton
is
and
universal
junction
rigidity of the
variations
are
estimated
by
the
manufacturer
requires only one compass
current meters
buoyancy
mounted
element
typically 100 in
to
be
less
than
10),
the
mooring
to determine the orientation of the
on it
employed,
in
the
rigid steel cages.
mooring
a 50 cm/sec current.
also
four
With the large
tilts
very
little;
Thus much of the vertical and
rotational movement of conventional moorings is eliminated.
Instrument packages consist of two temperature probes--one "local"
and
one
"remote"--and
Marsh-McBirney,
Inc.
induction-type
Model
711
conductivity
electromagnetic
sensor,
and
current meter.
a
The
current meters have two orthogonal sets of electrodes mounted on a 2
cm diameter vertically oriented cylinder.
The principles of operation
of the electromagnetic current meter are discussed in Cushing (1976).
A typical
mooring configuration,
pictured
four of the instrument packages described above,
orthogonal
tilt
transmitter.
sensors,
an
in
situ
data
in
figure
3,
employs
plus one compass,
processor,
and
two
a radio
Sensor outputs are low-pass filtered in real time with a
five second time constant (the stated response time for the sensors is
typically
processor.
one
second)
Averaged
and
continuously
values
of
the
integrated
measured
in
the
data
parameters
are
then
transmitted, on command, to a shore station at Tiana Beach.
can therefore adjust
the sampling rate or detect faulty instruments
while the experiment is
in progress.
typically to one month periods,
transmitter.
al. (1976).
Operators
Experiment duration is
limited,
by the large power consumption of the
Further technical details are available in
Dimmler,
et.
Radio Telemetry
link to BLN
Floatation collar
3.8m-
ECM
7.8 m-
ECM
Data processor
16.0 m
ECM
Compass and inclinometer
Shelton spar
28.4m~
-*
Fu *-
Configrat
6
io
.
ECM
.-
Battery
container
---.-
*
.
o
a
t
C
-
30.8m
spar
buoy
Figure I1--3* Configuration of a typical COBOLT spar buoy
In spite of the care taken in its design, the COBOLT moorings have
not been perfected yet.
An experiment somewhat related to COBOLT, the
Current Meter Inter-Comparison- Experiment (CMICE),
was conceived as an
opportunity to test the merits of the spar system against coventional
moorings and instruments.
In this experiment,
Beardsley,
six moorings were deployed off Tiana Beach
et.
al.
(1978),
described in
in a line parallel to the shoreline and 6 km from the beach.
detail by
Four of
the moorings were conventional taut rope moorings instrumented with a
variety of
current
type), while
meters
(mostly of
the
Savonius
the remaining two moorings were
rotor
and
vane
the Shelton spars.
A
comparison of the measurements of these instruments suggest that there
are some deficiencies in
the COBOLT moorings and intrumentation.
The
sources of possible error in the COBOLT velocity measurements are:
1.
Errors
due
to
mis-orientation
of
the
single
compass
or
misalignment of current meters with respect to the compass.
2.
Errors due to a shift in
the zero point of either or both of
the current meter axes.
3.
Errors due to asymmetric gain adjustment of the two current
axes or non-cosine response of the sensors.
F.
COBOLT experiments and data
After some pilot studies, the full COBOLT array of four spar buoys
was
first deployed in May, 1977.
buoys
and
their
relationship
to
The location of each of the four
surrounding
features
is
shown
in
24
figure 4.
The buoys were placed approximately 3 km,
km away from the shore,
and stand
in
20 m,
28 m,
6 km,
30 m,
9 km and 12
and 32 m of
water respectively.
The
instrument
configuration,
daily hydrographic
casts
ically in
figure 5.
numbers:
the first
the instrument
Instruments
mounted,
starting at the top,
place
instruments
below
the
(described
corresponding
is
in
at
surface;
bottom
profile,
subsequently)
are
and
is
standard
identified by a sequence of two
to the number of the buoy on which
mounted.
is
depths:
intermediate
the
to the order,
An attempt was made
shallowest
instruments
at
7.4
at
to this rule with one instrument
to
3.8 meters
meters
meters; and the deepest at 2.4 meters above the bottom.
exception
of
schemat-
shown
and the second corresponding
which it
location
and
16.0
Buoy 1 is the
at 12.3 meters
instead
of
16.0 meters.
The
spars
were
launched
on
recovery from all four buoys was
non-uniform power
significantly.
until May 24;
April
initiated
drain, endurance
Buoys
29,
1 and 3 were
of
1977,
and
on April
the different
operational until
and buoy 4 until May 17.
regular
30.
data
Because
of
moorings
varied
May 29;
buoy 2
The operation period of the
experiment is summarized in figure 6.
The
quality
temperature(2),
of
instrument
records
(containing
temperature(l),
salinity, X velocity and Y velocity).is
good,
with the
exclusion of buoy 1 which suffered numerous irrecoverable data gaps.
These gaps were uniformly spread throughout the data and amounted to a
total
of
140
hours
out
of
a
total
duration
of
about
700 hours
or
I
I
I
1
0
1
2
3km
3.
4,
30'
Figure 1-4
Locations of the four spar buoys of the May, 1977 experiment
MAY 1977
hydro.
20
.c
S
4-
30
40
5
10
km
Figure 1-5 Depth profile off Tiana beach, location of spar buoys, instrument packages,
and hydrographic survey stations
MRY
12 3
APR. 30
START
BUOYS
H 77
7
HH
H
2
3
4
a
H
H
H
6 hr gap
buoys 3, 4
H 16
H
H
H
H
H
H
H
H
H
END
BUOY 4
22H
23
H 2 4H
2
S
END
BUOY
25
H
H
2
H
E ND
BUOY
3
H Hydrographic survey conducted
Figure 1-6 Calendar for the May, 1977 experiment showing buoy duration
and hydrographic surveys
one-fifth of the total time.
Eree
of
isons,
long
but
gaps
and
consequently
the rest of
tidal analysis.
One stretch of
the record
ten days was
can be used
was
abandoned
for
as
Data from the other three moorings,
only occasional, short
data gaps
during periods
relatively
limited
compar-
unacceptable
buoys 2-4,
for
showed
of high speed
flow.
measurements,
daily
These gaps never exceeded 6 hours in length.
In
conjunction
hydrographic
surveys
ments were made
along a line
stations,
with
of
continuous
the area were
from a small
vessel
coincident with the
about 1 km,
the coastal
the
was
chosen
boundary layer
the spar buoys.
Although
buoy
conducted.
at
spar
These STD measure-
ten semipermanent
transect.
to give more
The spacing of the
detailed resolution
of
than was provided by the 3 km spacing of
they were performed
only in
and although they are aliased by tidal fluctuations,
surveys are a valuable
locations
source of information in
fair weather,
the hydrographic
interpreting
the spar
data.
In view of
the
questions
quality of the spar system,
of the tidal
analysis
that
and in
to follow,
be included in the discussion:
have
arisen
an effort
results
concerning
the
data
to assure the generality
from two other moorings will
a "reference" mooring
from the CMICE
experiment, and the COBOLT pilot mooring.
The mooring chosen from the CMICE experiment was deployed by the
MESA
New York Bight
project
and has
been used extensively
in
their
field program.
The
current meters;
three mounted on a subsurface taut wire mooring, and a
instrumentation
consisted
of
four Aanderaa RCM-4
fourth mounted
biases.
The
instrument
at
experiment
was
beneath
mooring
11
a
surface
is
shown
meters
below
conducted
at
spar
buoy
to
schematically
the
surface
the COBOLT
reduce
in
did
site
figure
not
convention
used here,
it
is
One
function.
The
1976
with
at approximately the
same location as buoy 2 of the May COBOLT experiment.
the
7.
in February,
this particular mooring positioned 6 km offshore
designated as #5 in the CMICE experiment
wave-induced
The mooring was
and since this conforms
retained in Table
I and
to
in further
references.
The
COBOLT
pilot
mooring,
launched
in
September,
single mooring placed 11 km offshore at roughly
buoy
4
of
the
May,
packages at 7.8 m,
details
1977
16.0 m,
concerning
this
experiment.
It
the same
had
and
the
was
instrument
32 m of water.
others
a
location as
working
and 27.0 m and was in
mooring
1975,
employed
in
The
this
analysis are summarized in Table 1.
Although it
seems a bit capricious
taken during different
year,
the
there
same
seasons and
are elements
throughout
stratification
are
istically related
the
the
COBOLT
separated
Aanderaa current meters
time by more
than
year.
Even
the
if
tidal
forcing
meteorological
signal
at all
should
times.
be
Including these
allow comparison between certain
spar
buoys
and
the
relatively
of the CMICE experiment,
and
determin-
aspects
of
well-understood
and will also assure
that measurements are somewhat representative of different seasons
conditions.
a
signal which are expected to remain
to well-known forces
experiment
in
of the
different,
additional moorings will
to compare current observations
and
Spar buoy
W~AJ
Auxiliary
float
9.0 M-
51
3.0m
I1# buoyancy
10.0 m-(
955t#: buoyancy
1 1.0 m-
Aanderoo 52
3/16" Wire rope
15.7 m~n
Aanderoa 53
25.0 in-
Aande raa 54
A coustic release
Dan forth
anchor
Figure 1-7
Clump anchor
Configuration of MESA-CMICE mooring #5
TABLE 1-1
SUMMARY OF DATA
RETURNS FROM THE TIANA BEACH SITE
Water
Dist.
Depth of working
Depth
offshore
Current meters
No.
Duration
Sept., 1975
0
640 hr
32.6 m
Feb., 1976
5
697
26.5
6
3.0, 15.7, 25.0
May, 1977
1
240
20.3
3
3.8, 7.8, 12.3, 17.9
2
577
27.7
6
3.8,
7.8,
16.0,
25.3
3
700
30.8
9
3.8,
7.8,
16.0,
28.4
4
385
32.3
12
Date of Exp.
11 km
4.2 m,
16.5 m,
29.7 m
3.8, 7.8, 16.0, 29.9
G.
Data Processing
The data sampling scheme is unique to the spar system and presents
minor problems
and
at
of its own.
intervals
that
Buoys were
ranged
from
interrogated at separate
five
minutes
to
several
times
hours.
Since ordinary time series analysis demands that sampling intervals be
uniform
time,
and
the
one-hour
that
measurements
COBOLT
data
sampling
were
over
a
one-hour
closest
whole
comparison
adjusted
interval
interval in the data).
for
(one
to
hour
a
time
period
hour.
The
and
then
taken
common
was
This was achieved
be
by
base
the
most
scheme
common
with
a
common
averaging all data
interpolating
interpolation
a
time
far
by first
at
was
values
a
to
third
the
order
polynomial that used four data points (two on either side of a gap) to
determine the value of the function on the hour.
advantage
of
interpolation,
consistent
eliminating
and
curves.
of
For
the
filling
sharp
bends
gaps
periodic
in
one-half of
a period.
Using
this
as
introduced
strong
functions,
nomial interpolation gives a good visual
This method has the
fit
for
by
tidal
linear
flows
with
the
poly-
example,
for record gaps of up to
a guideline, COBOLT
data gaps
were filled only if they were less than or equal to 6 hrs in duration;
that is, half a semi-diurnal tidal period.
The
X and Y component
were converted
velocities
output
from
to east and north components
the current meters
using
the headings
the single on-board compass.
Then the coordinate system was
by
local
220
to
uncertainties
conform
usually
to
the
associated
coastline
with
this
at
Tiana
maneuver
are
from
rotated
Beach.
quite
The
small
here
The
to
due to
result
the uniformity of the coastline
is
and
features.
system with the X axis aligned
alongshore
the Y axis
the north-
a coordinate
the east-northeast
and topographic
pointing onshore to
northwest.
The
special
salinity time
attention.
conductivities)
ments
series
and by as much
measurements.
salinities
Mean
differed
from the May, 1977 experiment
by
as much
(computed
as
3
o/oo
These aberrant
suggest
that
these
salinity measurements
aberrations
constant calibration offset,
two different procedures.
to eliminate
all density
were
the
the first,
inversions,
instru-
from nearby
resulted in
STD
large,
Since there was nothing
other
than
the
result
an effort was made to correct
In
measured
from adjacent
as 2 o/oo from values obtained
persisent inversions in the computed density.
to
from
required
of
them using
salinities were offset enough
while in
the
second,
salinities
were made to conform to nearby daily hydrographic survey salinities
a least-squares sense.
credence
to
the
notion
a
in
These adjustments agree quite closely and give
that
errors
were
due
only
offsets and not to instrument drift or malfunction.
to
calibration
CHAPTER II
NEARSHORE TIDAL CURRENT OBSERVATIONS
A.
Introduction
An examination of the current records from any coastal experiment
in
at
the Middle Atlantic Bight shows that they are dominated (vis.ually
least)
by
oscillations
properties
large
tidal
do
of
not
the
amplitude
oscillations.
transport
water
of the tidal
the records--particularly
consequence,
column
if
an understanding
Even
mass,
such
momentum,
(except
signal
though
or
short
other
in non-linear
often obscures
period
passive
cases),
the
other aspects
the observation period is
short.
of
As a
of some of the slower and less obvious
processes of the coastal region may be improved by an understanding
of the tides.
Certain
controlled
aspects
of
coastal
or influenced by
dynamics
may
the surface tides.
also
be
Internal
directly
waves,
for
example, are known to be generated by tidal currents interacting with
the
topographic
There is
currents
coastal
found
in
coastal
control the high background level
regions--acting,
in
effect,
like
to the question of tidal
presumed to occur on the continental
(Munk,
areas
(Rattray,
also evidence (Bowden and Fairbairn, 1956)
closely related
is
features
1968).
Little
is
that
1960).
the
tidal
of turbulence observed
in
a stirring
is
dissipation,
rod.
This
much of which
shelves of the world's oceans
known about the mechanisms
by which this
is
accomplished
or the
regions
in
which
it
occurs.
A study of coastal
tides may serve to illuminate the subject.
Because of the deterministic nature and relatively high frequency
of tidal currents,
information can be extracted from relatively short
duration experiments.
1977 is
The
thirty
days
of
data
gathered
during May
suitable for some forms of tidal analysis and will be used in
the hope of elucidating
region, comparing
other systems,
the
some of the local
performance
and as a
first
of
dynamics
the
step in
of the nearshore
COBOLT mooring
system
obtaining detided records
to
for
analysis of low frequency phenomena.
B.
Tidal Analysis
Tidal
method
which
analysis
introduced
forcing
is
traditionally
carried
by Lord Kelvin in 1867.
occurs,
are
obtained
from
out using
The
the harmonic
frequencies,
expansions
of
., at
the
tidal
potential (Doodson and Warburg, 1941) and used in the expression
F(t)
which
is
adjusting
=
a. cos (W.t + #.)
1
then
fitted
the
constants
to
the
a.
1
1
data
in
and *.
,
(1)
a
least-squares
This
method
sense
requires
by
long
records, typically greater than a year, in order to resolve some of
the closely spaced constituents, and to provide statistical stability
since
weak
Also,
the similarities
tidal
"lines"
in
are
often
obscured
by
background
noise.
responses to given forcing are concealed
in
the multitude of different amplitudes
and phases.
So it
is
not well
suited to the analysis of short records.
In
harmonic
analysis,
statistical stability is
at the expense of resolution.
different
in
frequencies
(Bendat
spectra
but
and
improves
Piersol,
the
In
differences
adjacent
example,
fifteen
days
the
since
components
these
Averaging
spectral
Thus,
minimum
lunar
differ
estimates
resolving capabilities
analyzing
to
resolve
different
the
spectral
estimates
short
frequencies.
the
principal
spectral
ability
time
series
the averaging procedure obscures
is
resolution of
the
reliability of
1971).
since
between
or averaging across frequency bands
reduces
problem is critical
days.
That is, averaging the spectra of many
pieces or realizations,
individual
usually maintained
In
tidal
record
by
over
one
n
cycle
frequency
that
solar
in
spectral
analysis,
length
and principal
estimation
or harmonic- techniques
long term observations.
of
the
allows
fifteen
bands
tidal
for
constituents
days.
limits
to signals which differ by n cycles in
reliable
this
the
fifteen
constituents
by
depends on the availability of fairly
If this
criterion is not met the
so-called
"admittance approach" offers a viable alternative.
The
approach,
one
method
is
used
x(t),
the "output",
analyze
the
COBOLT
data,
described by Munk and Cartwright
(1966).
hypothesizes
series,
to
a
linear,
which is
the most
causal
termed
general
relationship
the "input",
linear
can be defined by the convolution integral,
admittance
Basically, if
between
and y(t),
relationship
the
which
between
two
is
time
termed
the
two
y(t)
where h(t)
is
f x(t') h(t-t')
=
known as the
dt'
(2)
impulse response
function.
Defining the
Fourier transform by capital letters, i.e.,
=
F()
f f(t) e
dt ,
(3)
and taking the transform of equation (2) gives
=
Y()
H()
X(W) ,
(4)
where H(o) is the transfer function or admittance.
Since
one
rarely
works
with direct
transforms,
but
rather
with
spectra, the following definitions are useful:
AUTO-SPECTRUM
S (W)
=
X(o) X*(o)
(5)
CROSS-SPECTRUM
(where
(4),
it
*
indicates
S
a
xy
(o)
complex
=
X*(W)
Y(W)
conjugate)
;
from which,
using
equation
follows that
S
(W)
=
H(w) S (w).
(6)
38
If x(t) is a periodic function, say
x(t)
a exp iwt
=
(7)
,
equation (2) assumes a particularly simple form
y(t)
This form
since it
the
is
especially
=
H(W) x(t) .
useful
in
(8)
generating
the output
is more easily computed than equation (2).
conceptual
basis of
the
admittance;
it
function,
It also reveals
is a measure
of
the
spectral linkage between the input and the output functions.
The primary advantage of the admittance analysis
to reduce noise to well-defined levels without
tion.
This
Smoothness"
amplitudes
bands.
is accomplished
by
(Munk and Cartwright,
and
phases
This is
invoking
the
are relatively
that
are
being
forced
at
sacrificing
so-called
smooth
over
resolu-
"Credo
of
broad
frequency
that the response of most
physical systems does not change too abruptly if
altered.
the ability
1966) which states that admittance
based on the observation
forcing or input is
is
the frequency of the
Exceptions to this argument are systems
close-to-resonant
successful use of the admittance approach
frequencies.
The
does not depend on a high
degree of resolution because the admittance function varies so slowly
with frequency that any structure in
records or low resolution analysis.
it
can be discerned with short
Because the input is
generally a
39
well-known
function
for
resolution
analysis
of
equation (8)
which
long
output
time
time
series
series
are
can
available,
be
high
obtained
once the form of the admittance function is
from
known.
Instead of resolution and stability, the questions to be answered
in
the
admittance
function.
output
The
that
Smoothness";
time
ideal
the
it
periods
approach
center
input
function
admittances
is
to
available
allow
the
on the
is
proper
related
necessarily
(or
choice
so
conform
of an input
closely
to
can be constructed)
the
to
the
"Credo
of
for long enough
desired
resolution;
and,
another
important
advantage.
it
is
free
of
noise.
The
analysis
admittances
offers
are formed from ratios,
they tend to divide out some of
the numerical effects of the finite Fourier transform.
of interest in
Because
This is again
the processing of short time series where
information
from narrow frequency bands is spread out into relatively broad bands
by
the
effective
transform
manner,
alters
filtering
both
the
of
the
input
transform
and
output
process.
functions
Because
in
the
a similar
these effects are minimized with the use of the admittance.
Finally,
the
analysis
provides
a
measure
output is coherent or phase-locked to the input.
of
how
much
of
the
This measure is the
squared coherence, defined as (Bendat and Piersol, 1971)
Y2
S (m)
2
W(SS
x
2
xy
)
y
(9)
)S
That part of the signal which has random variations
phase,
such
as
weather
fluctuations
or
in
amplitude and
intermittent
baroclinic
40
effects,
is
summarily classified
admittances have,
as a result,
of the coherence.
real
and
as noise.
Ensemble
well-defined
averages
of the
errors expressed in
terms
A particularly simple form for the variance of the
imaginary
parts
of
the
(which
admittance
are
distributed
normally) is given by Munk and Cartwright as
JH(W) ~ 1 - y(
2
V
where N is
2N
2
the number of statistical
degrees of freedom and Y is the
true coherence.
Traditionally
function
when
the
equilibrium
analyzing
short
tide
duration
is
tide
chosen
gauge
as
or
the
input
current meter
data (see Filloux, 1971 and, Regal and Wunsch, 1973). The equilibrium
tide,
however,
assumption that
ocean.
the
of
the
from
the
entirely
presence
it
astronomical
forcing comes
the
the earth is
coastal waters
forcing by
at
computed
tidal
covered by
of
is
land
masses
well-known
under
bodies
plays
and
1961)
only a minor
instead through interaction with the
continental
shelf
outer
continuity
topography.
boundaries.
Here
to drive more energetic
in
flows
that
role.
the
on the
discussion of this subject is
the
following chapter.)
For this
tides
water depth and act
ical
(Further
The main
pceanic tidal
through
forcing alone.
In
direct
deep ocean
shelf than could be achieved
in
the
an infinitely deep
(Defant,
currents are constricted by the rapid decrease
through
potential
It consequently does not account for variations that occur as
result
shallow,
is
continental
the action of direct astronomcontained
reason a series of coastal
tide
height
function
analysis
for
procedure
suggested
a
presumably
is
observations
of
by
coastal
tidal
Cartwright,
series
station is
used as the input function in
So,
and
input
a
following
Zetler
computed from the tidal harmonics
(1969),
a
of a nearby tide
analyzing the COBOLT data.
The tidal ellipse
The presentation of the data is
the use of the tidal ellipse.
are periodic
v, which
iv
appropriate
fields.
Munk,
reference
C.
more
much
can be formed.
complex
(-)
Given the orthogonal velocities
A+
and
and
A~,
decomposed
rotating
in
(+).
counterclockwise
into two
opposite
u and
vector u +
with some frequency w, the complex
This vector may be
vectors
clockwise
conveniently accomplished through
constant,
directions:
Algebraically
this
is
expressed as
u + i v
+
A
=
iWt
e
-
+ A
-iWt
add to,
alternately
These rotating vectors
e
(11)
or
subtract
from one
another producing the characteristic shape of the tidal ellipse.
phases
of
the
vectors
which
determine
direction
the
The
is
ellipse
oriented.
The
parameters
ellipticity
and the
succinctly
which
orientation.
describe
They
are
the
ellipse
illustrated
in
are
the
figure
and defined (respectively) by:
-
LA= i-
A
A+ + A-
12)
I
ELLIPTICITY
E= m/M
ORIENTATION
TIDAL
Figure 2-1
ELLIPSE
Definition sketch of the tidal ellipse
argA
+ arg A
2
(13)
In geometric terms, the ellipticity is the ratio of the minor axis of
the
ellipse
to its
axis.
major
It
is:
positive
if
the complex
current vector u + iv rotates in a positive sense (counterclockwise);
negative if the vector rotates in a negative sense (clockwise); equal
to one if the vector traces a perfect circle; and equal to zero if
the ellipse degenerates into a line.
The orientation measures the angle between the major axis of the
ellipse and the positive x axis.
(The x axis will point alongshore
and the y axis onshore throughout this work.)
definition, to fall between
These
easier
quantities
are
to visualize, but
dynamics of the tides.
constant
phase)
and
It
is
by
constrained,
±900.
introduced, not only to make the results
as
tools
diagnostic
for
determining
the
While the free surface co-phase (lines
(lines
co-amplitude
of
constant
of
amplitude)
contours are valuable in this respect, the velocity field is quite
sensitive
sensitivity
introduces
dynamics.
to
other
dynamic
is a consequence
ellipse
(e.g.,
of
the
characteristics
frictional)
rotation
that
are
of
effects.
the
peculiar
This
earth which
to
certain
Thus, it is advantageous to employ information from both
surface and velocity fields in attempting any interpretations.
D.
Tidal observations in the Middle Atlantic Bight
Under a common classification
from
the
amplitudes
constituents,
characterized
the
as
of
four
tides
of
predominantly
scheme which uses a ratio
prominent
the
semidiurnal
Middle
semidiurnal.
and
Atlantic
This
formed
diurnal
Bight
ratio
are
(Defant,
1961),
Ky1 +0O1
M + S '
2
2
(14)
ranges from 0.19 at Sandy Hook, New Jersey to 0.33 at Montauk Point,
New York,
and averages about 0.25 for the Middle Atlantic Bight in
general.
The
M2
constituent
is
the
largest;
the
ratio
M2 :S2
typically being about 5:1 (Shureman, 1958).
The Atlantic Ocean
semidiurnal
tide
arrives
everywhere
at
the
edge of the continental shelf at roughly the same instant (Dietrich,
1944)
and
progresses
with
Jersey-Delaware shore.
Sound
affects
cophase
To the north,
propagation
contours
the
characteristics
to-resonant response (Swanson, 1976).
paralleling
the
New
presence of Long Island
markedly with
its
near-
Cophase lines (see figure 2,
taken from Swanson's work) bunch up around Montauk Point and distort
normal tidal patterns many kilometers away from the Sound itself.
a consequence,
the tide propagates to the east (towards
As
the entrance
to the Sound) along eastern Long Island and the west along central
and western Long Island
(in conformity to the rest of the shelf).
The contours also show that the propagation pattern divides somewhere
near the COBOLT region (station 20 on Swanson's map).
Thus this area
marks the transition between the tidal regime of the Bight and that
Figure 2-2
Cophase contours in the Middle Atlantic Bight (from Swanson)
TraWW-- M
- tloction
of the Sound, and complicated interactions between the regions may be
expected.
A crude
estimate of the
semidiurnal
tidal wavelength,
which will
be valuable in the ensuing discussion, may be made by using the phase
lags
from
the
NOAA
Tide
Tables
with
the
kinematic
relationship
between wave speed and wavelength,
=
Wavelength
Phase Speed
x
Period.
These figures suggest that this wavelength is
The
but
diurnal
seem
to
perpendicular
tides
to the
from
is
difficult
to
north
to
south
as
with
the
semidiurnal
cophase
contours
isobaths and coastline rather than parallel
them (Dietrich, 1944).
it
about 1500 km (Table 1).
are not so well documented
progress
(15)
to
In view of the lack of published information,
characterize
them
propagation patterns differ noticeably
except
in
noting
from those of the
that
their
semidiurnal
constituents.
Tidal
current
measurements
appropriate analysis,
current measurements
in
one of
lightships
the
on
from about fifty
located
dangerous
shoals
nearshore
tidal
currents.
however.
First
of
and
shelf,
are generally sparse.
earliest studies
were
the
at
of
the
consequently
all,
Some
tidal
accompanied
Haight
light ships
tidal
large
Most
observations
are
compiled
usually
of
harbors
very complicated
general
ellipses
to
the
on the East Coast
currents.
entrance
are
(1942)
by
these
or
on
examples
of
may
very
be
made,
elongated
TABLE 2-1
SEMIDIURNAL WAVELENGTH COMPUTATION
Guage
Location
Distance
from
High & Low
Water Interval
Phase
Speed
Wavelength
Sandy Hook
Shinnecock
Inlet
138 km
Fire Island
Jones Inlet
39
0.83 hr
1.13 hr
144 km/hr
1791 km
0.63
0.48
114
1413
0.32
0.45
104
1295
(the
ellipticity
more
circular
almost
is
at
much
less
offshore
exclusively
in
the
than one)
at nearshore
And,
points.
clockwise
velocity
direction;
locations
vectors
at
and
rotate
94% of Haight's
observation points, according to Emery and Uchupi, 1972.
Form measurements
on
the
to 50% of the total variance
the
combined
(1976),
using
ellipses
are
direction,
effects
the
of
while
data
and
set,
circular
diurnal
Flagg (1977)found
at individual
diurnal
same
virtually
outer shelf,
up
current meters was due
to
semidiurnal
notes
and
ellipses
that
that
oriented
are
tides.
semidiurnal
in
very
Traschen
the
tidal
cross-isobath
elliptical
and
are
oriented along isobaths.
Nearshore
current measurements,
and Parker
(1976)
effects
a nearby shoreline,
of
not influenced
If
there
is
diminished
condition
(low
a
to
the
solid
the
ellipticity)
Apex,
of harbors
show
if
or bays
onshore
tidal
boundary
condition
tidal
ellipses
to elongate
Figure
semidiurnal
3,
tidal
taken
along
at
the
axes
show
a
noticeable
orientation.
Typically
anywhere
5 0-10 -- and,
from
this
it
deviation
orientation
does
not
Patchen,
local topographic or shoreline features.
from
to
shore.
must
shore.
from their
"tilt"
seem
that
are
be
This
into very eccentric
addition to the elongation of the tidal ellipses,
major
pronounced
velocities
from
ellipses
the
Long,
the measurements
the
forms.
the
those of Patchen,
particularly
boundary,
satisfy
causes
as
the New York Bight
by the presence
Parker, shows
In
in
such
it
a
is
Long,
and
experiment.
noted that
shore-parallel
angle
is
correspond
small-to
any
8 meters above bottom
73*30'
74%0
S
New%York
Rockaway Pont
Sandy Hook
50
0
cm/sec
New Jersey
1 520.000
N
IStatute Miles
0
5
0
f
Nautecalm
74%0'
0
73*MY
3 meters or less above bottom
74*0D'
I
uperO
Say
New York
Rockaway Pont
Sandy Hook
S6
50
0
cm/sec
New Jersey
I 520.000
0
5
10 K
5
N0
74*00'
Rotation is predominantly clockwise at 8 m (26 ft) above
the bottom, counterclockwise at 3 m (10 Ift)or
less above the bottom.
The centers of ellipses are the station locations.
m0
73*M'
Arrows indicate direction of progress of the maximum M2
flood current velocity.
Current velocity vectors rotate 3600 in 12 to 42 hours.
Figure 2-3
Tidal ellipses in the New York Bight
Source: From Patchen et &11976
Metcatr Poject ion
Other
tidal
generalizations
ellipses
ellipses
near
ellipticities
circular
in
direction.
(8
can be
m) are
regarding
made
from
the
bottom
(3
than
those
near
shape)
and
By contrast,
almost
this
m
rotate,
the
away)
the
structure
experiment.
It
usually
surface
generally,
tidal ellipses
always more
vertical
in
the
further
eccentric
appears
exhibit
(here
of
the
that
different
they
are
more
counterclockwise
away from the bottom
and rotate
in
the clockwise
direction.
Measurements
coastal
near Little Egg
Inlet,
series available for comparison,
presence of the inlet.
exception)
J.
(EG&G,
1975),
another
are highly influenced by the
This, as was the case with Haight's analysis,
makes generalizations difficult.
predominantly
N.
clockwise
The experiment does show, however,
rotation
and emphasizes
the
point
of
tidal
that
ellipses
large
(with
amounts
of
one
variance
are due to the tides; 33% for year-long records in this case.
There
coastal
appear
tidal
increased
to
currents
interest
usually
focus
mention
of
be
on
tidal
in
the
few
in
this
lower
phenomena.
other
the
relevant
Middle
Atlantic
frequency
on
signal
other
of
Bight
Measurements
region.
Work
studies
despite
which
and neglect
shelves
nearshore
do
the
exist
altogether
(e.g.
Petrie,
1975), while serving as a useful comparison, will not be pertinent to
the
Middle
Atlantic
Bight
because
forcing and topographic features.
of
different
deep
ocean
tidal
E.
Analysis of the COBOLT tidal signal
the
step in
first
The
gauge
Sandy
at
the
assumed to be in
moorings are
operated over
one to have
closest
the
also
necessary to obtain stable values
the
has,
It
prediction.
in
tidal regime of the open shelf and
to that of Long Island Sound.
to be too closely related
not
the COBOLT
if
a reasonable choice
This is
west of the COBOLT site.
to the
140 km
approximately
New Jersey,
Hook,
the input
The station chosen was the
time series for the admittance procedure.
tide
the
involved
COBOLT data
the
from which to generate
tide station
of a reference
choice
of
analysis
It
is
of time
the long period
of tidal amplitudes
and phases
been operational
for more than
fact,
for
a
hundred years.
used to construct the reference time series
constants
The tidal
were taken from Shureman (1958) and represent the results of harmonic
The components employed are listed in
analysis of ten years of data.
table
2 along with appropriate
phase
relative
to
the
of
transit
Non-astronomical
meridian).
amplitudes,
periods,
tides,
over the
the mean moon
such as
due
those
and components with amplitudes
and radiational effects,
(the
epochs
and
to
Greenwich
non-linear
that are less
than 2% of the M2 amplitude were ignored.
The
input
procedures
as
overlapping
Fast
by
Fourier
1/15
generated
function
were
data
followed
360 hours
pieces
Transform
cycles per
with
routine
day.
These
was
then
to give
meter
current
the
(15
subjected
days)
long
spectral
estimates
fall
were
to
same
the
data;
i.e.,
used with
estimates
a
separated
approximately
on
the
TABLE 2-2
TIDAL COMPONENTS OF
REFERENCE TIME SERIES
COMPONENT
PERIOD
11.96723
AMPLITUDE
2.9 cm
PHASE
243 deg
12.00000
13.0
246
12.19162
3.4
203
12.42060
70.0
218
12.65835
15.9
201
23.93447
9.0
102
24.06589
3.2
105
25.81934
4.3
98
S2
and
N2 '
M.
the
spectrum,
12.00
hr;
diurnal,
with
and
adjacent
estimates
approximately
with
estimates
on
at
-at
the
25.71
12.86
0
hr
portion
semidiurnal
the
for
bands
frequency
and
hr,
K -P
24.00
and
of
12.42
hr,
and
bands
for
the
hr.
The
Fourier
coefficients
were hanned to reduce leakage of energy from the strong
tidal
into
lines
spectral
the
estimates
weaker
between
ones,
the
and
then
reference
used
series
to
and
form
the
cross-
individual
The admittances for each 360 hr piece were then
velocity components.
calculated according to equation (6).
Three
Besides
types
the
different
of
averaging
standard
pieces
and
practice
across
among the COBOLT moorings
effects
the
are
dynamic
effects
frictional
error,
and
these
show
baroclinic
averaging
used
in
this
and
real
enterprise
vertical
systematic
topography
variations
instruments
presence
of
on
different
lie in
moorings
the same
background
horizontal
"noise"
are
and
expected
due
on
admitto
the
due
to
sources
of
presumably subject
be
(see
small;
acceptable
plane.
cross-isobath vari ations at these locations.
since
other
of the region
seems
taken
lengths
variations
the
should
over
were
variations
Unlike
data.
done to reduce the
for buoys 2-4 where the depth changes only 4 m in 6 km.
instruments
COBOLT
averages
This was
horizontal
should be
that horizontal
the
cross-spectra
bands,
influences.
Examining the
on
errors and short record
real
topography
variations
to prediction.
1) suggests
of
to
of
themselves.
Caution must be
expected
utilized
frequency
of individual instrument
results.
tances
were
especially
So averaging
provided
Other errors
to
chapter
swamp
the
and the
any
real
As
mentioned
mooring
5
before,
from
represent
two
the
parallel
CMICE
different
analysis
experiment.
seasons--winter
and
provide a check for baroclinic effects.
will
furnish
comparison
between
the CMICE
performed
these
Since
will
a
also
was
experiments
spring--comparing
Also,
on
the
them
experiments
instrumentation
and
the
data
is
newer COBOLT instrumentation.
F.
The results of the semidiurnal analysis
The
presence
exhibited
by
of
tidal
spectral
frequency
analysis
averaging was
independent
or
variance
preserving"
combined,
plot.
the
In
time
this
account for about
"barotropic"
velocities--a
the
to
series.
some extent, the
The
area
in
this
35% of
the total
fairly
typical
beneath
it makes
and
energy
each
to the
"variance
diurnal
tides
observed in
proportion
4.
true depth
so-called
semidiurnal
The
figure
shown. in
the contribution
the
COBOLT
time
at buoy 2 is
series
case,
in
velocity
velocities.
proportional to
of
the
to isolate,
barotropic
frequency band is
total
done
of
currents
spectrum of depth-averaged
This
motions
in
the
coastal
waters.
The
table
admittances
3
for
averaged over
12.86 hr,
hours at
and
each
for
of
the
the
semidiurnal
instruments
components
deployed.
are
These
entered
have
in
been
three frequency bands covering periods from 12.00 hr to
over T/360
the buoy in
pieces,
question.
where
is
Also included
intervals computed from equation (10)
oF the true coherqnce,
T
the
record
length
in
are the 95% confidence
and using the unbiased
estimate
.8
Ld
U)
.6
.4
(9
a:
LU
LU
C)
FLU
.2
0
.05 .10
FREQUENCY (C/HR)
,C1
.50
Figure 2-4 Variance preserving plot of kinetic energy of depthaveraged currents at buoy 2
TABLE 2-3
ADMITTANCE AMPLITUDES AND PHASES
FOR THE SEMIDIURNAL TIDE
INST.
21N
21E
22N
22E
23N
23E
24N
24E
AMPL.
0.058
.162
.037
.153
.053
.148
.053
.095
ERROR
0.019
.017
.015
.011
.009
.010
.013
.007
PHASE
-65
58
-79
58
-110
47
-170
34
32N
32E
33N
33E
34N
34E
0.052
.154
.052
.138
.048
.106
0.010
.010
.010
.012
.005
.007
-85
57
-92
50
41N
41E
42N
42E
43N
43E
44N
44E
0.028
.153
.039
.150
.070
.122
.052
.104
51N
51E
53N
53E
54N
54E
0.047
.147
.054
.184
.036
.118
ERROR
18
6
21
4
10
4
13
4
COH1.
0.83
.98
.78
.99
.94
.99
.90
.99
36
11
4
11
5
6
4
0.92
.99
.92
.98
.97
.99
0.029
.027
.029
.027
.014
.033
.009
.018
-44
61
-15
65
-99
54
-152
30
46
10
37
10
11
15
10
10
0.55
.97
.68
.97
.96
.93
.97
.97
0.009
.016
.016
.020
.009
.015
-88
53
-85
50
-150
40
11
6
16
6
13
5
0.91
.97
.85
.97
.88
.98
-153
Quantities listed under ERROR are the 95% confidence
limits of the admitance amplitude and phase.
COH is the true coherence of tidal currents with
the reference series
^2
Y
where N is,
=
NY
Y2__
N - 1
(16)
as before the number of degrees of freedom,
and Y is
the
coherence estimate formed from equation (9).
While the number of degrees of freedom are fairly low (generally
less than 12)
due to the short
record lengths, the coherences are
high, especially for the alongshore components.
statistical
nents,
errors
are kept to manageable
As a consequence,
Onshore
levels.
compo-
though much noisier, exhibit a certain stability (with the
exception of instruments 41 and 42)
that suggests that these numbers
are also trustworthy.
Though longer records would do much to clear up doubts about the
apparent discrepancies,
certainly
reliable.
the admittances exhibit some trends which are
Most
noticeable
is the disparity between
onshore and alongshore admittance amplitudes.
of the adjustment of velocities
This is
the
a consequence
to the presence of the shore and may
partly cause the lower coherences evident in the onshore admittances
since signal to noise ratios are presumably decreased also.
Vertical trends are also evident.
decrease
towards
drastically for
the
onshore
bottom,
while
components.
Admittance amplitudes usually
phases
This
decrease
also--quite
tendency is
always most
evident in the current meter that is nearest to the bottom and may
indicate the presence of a frictional boundary layer.
Systematic differences between the May 1977 data and the February
1976
data
experiments.
are
small
Slightly
despite
larger
the
differences
admittances
for
between
the
CMICE
the
two
data
and
slight discrepencies
in phase might be noted,
but all variations are
well outside the resolving capabilities of the analysis
and conse-
quently cannot be argued with much certainty.
The semidiurnal tidal ellipse for each of the available instruments is
moorings
shown in its appropriate location in figure 5 for the COBOLT
and
in
figure
6
for
the
CMICE
mooring.
The
striking
features of nearshore tidal flow are immediately apparent from these
The ellipses are all very eccentric but are not oriented
diagrams.
negative
inclination
near the bottom.
for
they have a small but persistent
Instead,
parallel to the shoreline.
(-100 to -150)
Moreover,
which becomes more
the sense of rotation, which is
noticeable
clockwise
all shallow and intermediate instruments, reverses to counter-
clockwise for all bottom instruments.
Some of the ellipses have noticeably different
characteristics
from ellipses at the same level on other moorings, or from adjacent
Ellipses at two instruments already
ellipses on the same mooring.
alluded
to, 41
and 42,
have
slightly
different
orientations
than
other instruments, while the ellipse at instrument 21 appears to have
a slightly different ellipticity.
consequence
of
problems
outlined
These discrepencies are probably a
in
chapter
1 (i.e.,
instrument
related errors).
Horizontal averaging provides a mean vertical profile of ellipse
and tidal characteristics.
The vertical profile of averaged ellip-
ticity in figure 7, for example, shows plainly the characteristics
described above.
Ellipticities through most of the water column are
COBOLT
21
10 cm/sec
41
22
32
42
23
33
43
24
34
44
Figure 2-5
Semidiurnal tidal ellipses inthe COBOLT experiment
-4--
51
CMICE
10
cm/sec
53
54
Figure 2-6
Semidiurnal tidal ellipses in the CMICE experiment
61
ELLIPTI CITY
0
-0.2
-0.4
0
0.2
95%
10
E
20 -
30
Figure 2-7
Vertical profile of semidiurnal ellipticity
constant
change
and
negative
abruptly,
indicating
however,
clockwise
somewhere
surface.
Ellipses near
the bottom,
positive
ellipticities
indicating
rotation.
Unfortunately,
this
rotation.
between
in
to those above,
change
feature
values
16 m and 25 m below the
contrast
a
These
is
to
not
have
counterclockwise
well
resolved
and
little can be said about the structure in the region where it changes
most
rapidly.
Also
agreement between
though
the
from
this
diagram
the COBOLT and CMICE
ellipticity
is
subject
it
is
data
to
apparent
sets
large
is
that
very
errors
good
because
the
even
it
is
formed from a small difference of two large numbers (equation (12)).
The
averaged
orientation
angle
with
respect
to
alongshore
the
direction (figure 8)
has an almost linear trend with depth instead of
changing
It is more homogeneous
suddenly.
than for
the COBOLT
experiment,
uncertainties
involved
distribution.
More
these
though
profiles
importantly,
for the CMICE experiment
in view of
could
be
orientation
the
the
statistical
part
of
angle
is
the
same
negative
and significantly different from zero at all levels.
Figure
energy.
9
This
shows
the
energy
vertical
was
structure
formed from
of
the
averaged
the admittances
kinetic
by multiplying
them by the 70 cm amplitude of the principal semidiurnal constituent,
M2, at
Also
Sandy Hook.
the
is
included
energy
from the harmonic
analysis of a four day period when the stratification,
the
internal
tidal
oscillations,
was
admittance approach and averaging in
baroclinic
signal
is
evident
by
strongest.
and presumably
The success
of the
eliminating much of the unwanted
the
vertical
uniformity
of
the
63
ORIENTATION
-30
-20
(0)
-10
0
95%
10-
E
-
CL
20
& COBOLT
o
CMICE
May 77
Feb 76
30
Figure 2-8
Vertical profile of semidiurnal orientation
ENERGY (Cm2/sec 2)
60
40
20
80
x
95%
10
E
1-
20
o
COBOLT
o CMICE
May 77
Feb 76
X 4 day harmonic
mean
30
Figure 2-9
Vertical profile of semidiurnal kinetic energy
65
energy.
The CMICE mooring does show some variability in the vertical
structure,
but this is most likely due to the fact that the surface
instrument was not attached directly to the mooring but to a tethered
spar.
As a result, the top meter shows much less of the surface wave
contamination common in conventional moorings and instrumentation.
In view of the homogeneity of
well-covered water column,
it
tidal energy and the reasonably
seems natural to form the "barotropic"
ellipse by integrating the data vertically.
Accordingly, the real
and imaginary parts of the admittances were summed using the ordinary
trapezoid rule to approximate
integration.
The velocity equations
(using again the 70 cm tidal
amplitude to
convert
admittances
to
velocities) are:
COBOLT
(17)
U
=
9.0 cos (Wt-
v
=
2.8 cos (Wt+ 1020)
510)
and
CMICE
(18)
u
=
10.2 cos (Wt - 500)
v
=
2.9 cos (Wt+ 990).
The depth integrated ellipses and their respective parameters are
compared
in
figure
10.
Although
it
is difficult
uncertainties of the integration procedure,
agree very well,
to
define
the
the two ellipses seem to
further supporting the assertion that the barotropic
66
SEMIDIURNAL
T
COBOLT
10 cm/sec
E
-0.13 *.02
db= - 16
± 40
CMICE
10 cmn/sec
E=
>=
Figure 2-10
-0.15
E.03
-13**5*
Semidiurnal ellipses for depth-averaged currents
primary features of the mid-depth
i.e.,
depth averaged ellipses;
local
shoreline
sense.
These
regime
and
agree well
and
seem
features
that
the
tidal ellipses are preserved in
the
with
also shows
figure
they are significantly inclined to the
contours,
depth
mooring
spar
COBOLT
the
that
The
results.
given accurate
has
system
and
resolved
have been
currents
tidal
be
to
the
they rotate
and
a
of
part
in
nearshore
tidal
of Patchen,
Long,
the
MESA measurements
a clockwise
and Parker, 1976.
The band structure of semidiurnal admittances
G.
The
to
of the
structure
be
explored.
This
admittances
is
structure
Certain conditions may alter this
place,
the
height
tidal
at
to such
factors
as
distance
vary
Secondly,
considerably;
17:70:13
about
energetic
cm.
the
the
to
while
be
fairly
amplitudes
implies
more
2,
to
the
constituents
example
for
uncertainties
which do not enjoy
due
are passed along in
N2 :2 :
ratio
closely
or proximity
the different
of
first
fine differences
Island Sound
from Long
yet
smooth.
In the
undoubtedly
probably contains
amplitude
This
constituents,
Hook,
These differences
the Hudson River estuary.
admittances.
expected
has
bands
statement slightly.
Sandy
related to that at Shinnecock,
frequency
across
in
the high signal
the
is
less
to noise
ratio of the M 2 tide.
The
admittances
were
obtained
the
admittance
by
and
depth
and
then
phases
for
the
three
integrating
the
real
averaging
the
values
most
and
energetic
imag.nary
for
buoys
bands
parts
2-4.
of
The
results,
including
ellipse
parameters,
are
entered
in
table
4
and
displayed in figure 11.
Li
Both
increased
and
v
admittance
frequency.
The
amplitudes
phases
of
both
decrease as the frequency increases.
viewed
with
errors.
As
of
slight
far as
frequency
decreases
skepticism
the ellipse
changes
are
tend
These results,
parameters
the
are
visible
dramatically with increased
decrease
components
considering
most
to
in
also
the
frequency.
tend
however,
to
must be
magnitude
concerned
with
of
the
the effects
ellipticity
which
The orientation,
by
contrast, is totally unaffected.
H.
The results of the diurnal analysis
Analysis of the semdiurnal band is much simplified by reason of
its
great
about
energy
200:1.
content.
Furthermore,
variance observed.
of
the
The
total
By
M2
it
contains
contrast,
variance
and
signal:noise
almost
ratio
30%
is
of
in
the
fact
total
the diurnal band has only about
its
principal
resolved
component,
5%
K
-
P1, has a signal:noise ratio of only 4:1 in the COBOLT experiment.
As a consequence,
more
uncertainty
the diurnal
associated with
over the four frequency bands
table 5.
It
is
the semidiurnal
fact,
coherences
confidence)
for
admittances are liable
apparent
them.
the coherences
analysis, especially
in
are not significantly
most
of
These admittances,
from 22.50 hr to 27.69 hr,
that
the
to have much
are shown in
are much lower than for
the onshore
components.
different from zero
admittances
averaged
of
buoy
4
In
(with 95%
and
for
3
69
TABLE 2-4
BAND STRUCTURE OF
SEMIDIURNAL ADMITTANCES
ADMITTANCE
PERIOD
12.86 hr
12.42
12.00
COMP.
AMPLITUDE
PHASE
ELLIPT.
U
0.137
.007
51
3
V
0.051
.007
-96
7
U
0.132
.005
51
2
V
0.041
.003
-102
5
U
0.131
.009
V
0.045
.007
48
-113
4
9
-0.19
ORIENT.
-18
-. 13
-16
-. 10
-18
0
ADMITTANCE
AMPLITUDE
I
I
PHASE
I
I
I
I
.161-
-A60*
.14
~-440*
U
.121-
200
10-
.08
.06
-80*
.04i
-100*
V
021-
-120*
I
I
I
N2 M2 S
2
Figure 2-11
I
II
I
N2 M2 S
2
I
Band structure of semidiurnal admittances
TABLE 2-5
ADMITTANCE AMPLITUDES AND PHASES
FOR THE DIURNAL TIDE
INST.
21N
21E
22N
22E
23N
23E
24N
24E
AMPL.
0.16
.31
.09
.35
.11
.37
.06
.23
32N
32E
33N
33E
34N
34E
0.19
.39
.07
.34
.10
.24
41N
41E
42N
42E
43N
43E
44N
44E
0.30
.48
.20
.38
.20
.43
.17
.33
51N
51E
53N
53E
54N
54E
0.17
.48
.29
.44
.15
.17
ERROR
0.09
.10
.06
.13
,
.35
.05
PHASE
-75
42
-74
51
-122
26
59
.10
55
0.10
.07
-68
44
-53
34
24
60
.07
.03
.09
ERROR
28
18
35
20
43
37
23
27
11
12
16
21
-144
.08
.10
.09
.19
.26
0.08
.19
.12
.26
.07
.10
1
122
25
-2
43
26
76
-25
47
-106
43
134
19
10
15
24
24
38
25
21
23
31
25
31
Quantities listed under ERROR are the 95% confidence
limits of the admitance amplitude and phase.
COP is the true coherence of tidal currents with
the reference series
Starred coherences indicate that these quantities are not
significantly different from zero at the 95% confidence
level
COH.
0.67
.83
.58*
.80
.10*
.46*
.54*
.50*
0.65
.92
.10*
.90
.84
.75
0.26*
.97
.00*
.93
.84
.84
.32*
.66*
0.69
.75
.72
.59
.69
.59
components
number
on
of
buoys
2
degrees
and
of
The
3.
freedom,
CMICE
shows
mooring,
with
significant,
a
greater
albeit
low,
coherence at all levels.
Because
of
the
large
considered separately
structure
of
but
band-averaged
diurnal
only as
the admittances
resolution)
semidiurnal
but
uncertainties,
useful
and
case.
is
components
lumped diurnal
consequently
information
depth-averaged
is
This averaging, it appears,
admittances
since
it
is
the
is
be
The
(and
it
the
from
the
in
the
for
the
with
available
admittances,
not
admittances.
lost
still
will
just
as
essential
only way to achieve
significant
coherences.
The
figure
surface
vertical
12.
of
the
Like the semidiurnal
and
furthermore,
at all
structure
increases
with
diurnal
ellipticity
depth.
This makes the
diurnal
the semidiurnal ellipses at all
levels.
mooring
but
The
never become positive and in
depths.
follow
abruptly
the COBOLT ellipticities
go
offscale
at
ellipticity
the
it
is
is
shown
negative
COBOLT
at
in
the
ellipticities,
fact remain less than -0.2
ellipses more circular
Ellipticities
at
the CHICE
at surface and middle
bottom.
This
is
than
depths
undoubtedly
a
spurious result.
COBOLT
orientation
angles
(figure
13)
progress
almost
from large negative values at the surface to a positive
bottom.
roughly
By
contrast,
the
the opposite slope.
in noise or
CICE mooring
is
linearly
angle at the
non-monotonic
and has
These variations appear to be submerged
influenced by non-barotropic effects.
indicate the consequences of the low coherence.
Large
error bars
ELLIPTICITY
0
-0.2
-0.4
0
1
1
1
0.2
1
0.4
1
%
-95
1-9
*
COBOLT
May 77
o
CMICE
Feb 76
I0E
a20
30
Figure 2-12
Vertical profile of diurnal ellipticity
74
ORIENTATION
-30
-20
-10
0
(0)
10
20
30
95%
10
0
COBOLT May77
0
CMICE Feb76.
E
LL
20
30
Figure 2-13
Vertical profile of diurnal orientation
The
vertical
distribution
constant
values
except
decreases
very
sharply
near
at
of
the
energy
bottom.
the bottom
(figure
The
14)
CMICE
instrument,
shows
mooring
again
almost
energy
calling
the
result into question.
In
the
spite of major differences
depth-integrated
similar.
Compared
to
ellipses
the
in
ellipse parameter distributions,
(figure
15)
semidiurnal
appear
ellipse,
to
be
the diurnal
quite
ellipse
rotates in the same direction but is oriented at less of an angle to
the shoreline and is slightly-more circular.
The velocity equations,
with phases relative to Sandy Hook high tide, are:
COBOLT
(19)
u
=
3.5 cos (Wt-
y
=
1.3 cos (wt + 800)
400)
and
(20)
CMICE
T.
=
3.7 cos (wt -
v
=
1.1 cos (Wt + 900)
420)
Consequences and conclusions
It
should be apparent
coastal
by now that
in
the COBOLT data.
the ellipticity (including
non-zero
the main features
of previous
tidal current observations in the Middle Atlantic
also evident
are
U
orientation
angle.
Bight
are
The most prominent characteristics
sense of rotation)
The
vertical
and the
structure
small,
of
but
these
0
ENERGY
4
2
0
I-*--*---1----**-*---------1*--------------1----*
(cm2 /seC
6
I
I
2
)
10
8
I
I
I
95%
10 -
E
LU
20
30
I
I
Figure 2-14
I
I
I
0
COBOLT
May 77
o
CMICE
Feb 76
I
I
I
I
Vertical profile of diurnal kinetic energy
-
DIURNAL
COBOLT
- 5 cm/sec
(Z:E
E=
<
=
-0.27 *.08
-
5*
+
6*
CMICE
5 cm/sec
-0.20*
d>= -Il* *
E=
Figure 2-15
.l1
80
Diurnal ellipses for depth-averaged currents
parameters
is
another
feature
that
agrees
qualitatively
with
other
experiments.
A
curious,
presence
of
and
small
possibly
scale,
loose-grain sediments.
that,
on average,
with
the
is
features
it
difficult
further
Swift,
topographic
Duane,
220 ±160
of
conceivable
these
is
wave-like
observation,
and McKinney
that
and may account
to argue
evidence
for
in
the Middle
tidal
currents
the
inclination
the
formed
(1973)
Atlantic
are
have noted
Bight.
It
responsible
for
And,
although
that this
swale
topography
angle
tidal
ellipses,
of
by
form an acute angle
for their persistence.
persuasively
concerns
features
the "crests" of these features
shoreline
certainly
related
is
the
possibility is rather intriguing.
These features seem,
or
surface
Bight.
The
tides
of
depth
structure
the energy profile)
nature
of
then,
the
the
to be characteristic
nearshore
region
of the ellipse
tide
itself,
to
relatively
unstratified
insignificant
have
levels.
conditions
reduced
Also,
of
Middle
parameters
Finally,
depth
integration
the
the
February
support the conclusion that the barotropic
baroclinic "noise"
the
Atlantic
(particularly
suggests that the time and buoy averaging,
velocities
resolved.
in
of the barotropic
baroclinic
comparison
is
good
or the
tidal
with
the
enough
to
tidal components have been
has
certainly
reduced
remained and has exposed the true barotropic
what
tidal
currents.
This
tant
is
not
or do not
to say that
influence
the internal
these
tidal currents are unimpor-
calculations.
This
energy
primarily
affects
the
barotropic
admittances
in manageable
introducing low coherences into the measurements.
particularly true
diurnal
for
tide--both
baroclinic
the
of
tides.
ways;
Further
show
evidence
discussion
of
of
this
by
This appears to be
on-shore velocity components
which
namely
and for
interference
the
from
matter, however,
is
deferred until chapter 4.
The vertical
structure
of
the ellipse parameters
point to
the importance of friction.
occur
the
in
parameters
are
smooth
seems
also
to
The depth variations
that
do
(at
least
for
tides)
and are accentuated near the bottom where
should
be
strongest.
This
matter
will
also
be
the
semidiurnal
frictional effects
explored
in
more
detail (chapter 3).
In
terms of importance to continental
layer dynamics, the most interesting
shelf and coastal boundary
and significant observation
is
that the depth-averaged tidal ellipse has a marked inclination to the
shoreline.
of
both
within
Such an inclination is
energy
and
momentum.
12 km of shore,
vanishingly small
these
in order
indicative
Since
fluxes
to conform
all
of shoreward
the
COBOLT
are normally
transport
moorings
considered
are
to be
to zero flux boundary condi-
This appears not to be the case, however.
tions at the shore.
The energy equation for long waves is obtained from the Laplace
tidal equations (see
chapter 3)
by forming the vector dot product bv
with the momentum equation, and adding it
the continuity equation.
2t
(V
to the product of g
and
This gives the energy conservation equation,
V + gC ) + g V
hv
0,
(21)
or
aE+
where E is
=
-
+
0
,
(22)
the kinetic plus potential energy of the water column, and
F is the energy flux.
Averaging over a wave period,
indicated by
brackets, < >, gives the average tidal energy flux,
F
=
gh <cv>
.
(23)
Before forming this product for the COBOLT data, the reference
tide amplitude and phase must be shifted in some manner from Sandy
Hook to Shinnecock Inlet.
This shift to local tide is
the
tenuous
phase
and
measurements in
hour
in
is rather
the Shinnecock area.
estimating
directions
due
the
phase
for the energy flux.
can
to
the
lack
primarily in
of
Errors of as little
result
in
tide
as half an
radically
The calculation,
gauge
however,
different
is
very
revealing even if some errors are present.
Analysis
by
Swanson
(1976)
suggests that
the tide offshore of
Shinnecock Inlet precedes that at Sandy Hook by about 1.0 ± 0.1
(error inferred from Swanson),
about
the
50
cm.
semidiurnal
available
This
amounts
tide.
to
this adjustment,
and that the M2 tidal amplitude is
a phase correction
Unfortunately,
for the diurnal tide.
hr
no
such
of 300± 30 for
information
is
Despite the obvious shortcomings of
these figures are used in obtaining a rough estimate
of the tidal energy flux for the semidiurnal tide.
Noting the 30
phase
correction in the free surface equation,
(17) can be used with equation (23) to give the energy flux,
=
Flux onshore
800
150 watts/m
(24)
=
Flux alongshore
The
energy
alongshore
flux has
component
1000
300 watts/m
a significant
to
the
east
onshore
(towards
component
Long
Island
and
an
Sound).
Furthermore, the magnitude of the energy flux is quite large compared
with shelf-wide estimates such as those of Miller (1966).
Miller,
using a frictional dissipation equation due to Taylor (1919),
CdU
=
E
(with Cd = 0.002 and typical
3
,
(25)
tidal current speeds),
found that the
energy flux on the eastern coast of the United States averaged less
that
250
scale.
watts/m
Though
and
was
comparisons
argument and the direct
COBOLT
relatively
calculations
between
local
seem to
unimportant
this
on
a world-wide
shelf-wide
dissipation
flux calculation are difficult, the
indicate
that Miller's
values
are
an
underestimate.
A more unusual fact is
distance from the shore.
Taylor's is
supposed,
that the flux is
so high at such a short
If a bottom friction mechanism similar to
this rate requires tidal current amplitudes of
30 cm/sec shoreward of the COBOLT moorings--about
than those observed.
three times higher
Another candidate
for this dissipation,
Shinnecock Bay, has only
the narrow (200 m) and shallow (5 m) inlet to admit energy.
Even a
gross overestimate of energy entering the bay, made by assuming that
all
the energy of the incoming tide
is dissipated,
results
in
an
energy flux of only 10 watts/m at the COBOLT site.
It
is possible that the flux can be accounted for by considering
the divergence
of alongshore energy flux.
This supposition requires
that the alongshore flux increase towards the entrance to Long Island
Sound
Sound.
by
roughly
While
divergence
of
100
there
watts/m
is no
for
direct
this magnitude,
the
every
kilometer
evidence
that
dissipation
closer
might
rates
to
the
dispute
a
in the Sound
would have to be 5-10 times greater than are expected in order to
accomodate the divergence.
the alongshore flux is
As suggested,
of Long Island Sound.
The large tidal
probably due to the presence
currents of the
imply relatively large dissipation rates.
Sound also
This flux does not mean
that tidal currents at the COBOLT site are dominated by Long Island
Sound tidal flow.
Atlantic Bight,
Co-oscillating tides,
generate
such as those of the Middle
substantial velocities but transport little
energy because of their standing wave characteristics.
The uncertainties of
evaluation
the
flux
calculations
are
absent
of Reynold's stress terms due to tidal velocities.
in
the
These
depend only on the time averaged product of the velocity equations,
(17) and (18).
For
the
results in
tide,
semidiurnal
difference
phase
of
1500
about
a momentum flux of
<uv> =
This is
a
onshore
transport
-13
2
cm /sec
2
(26)
.
of westward momentum
comparison with other forces
if
its
divergence
and
is
is
very
small
uniform across
in
the
10 km coastal region.
The momentum flux does not show any divergence
across
within
buoys
2-4
(to
6 km
of
shore),
however,
and
may yet
prove of significance very close to shore.
The diurnal momentum flux is
the
semidiurnal
negligible,
and
Mann
flux
it makes
(1978)
and
an
who
has
an order
the
interesting
found
large
components on the Scotian shelf.
ments,
the Scotian shelf
of magnitude
same
sign.
In
semidiurnal
Although
comparison with
momentum
fluxes
contrast
smaller
totally
Smith,
in
than
Petrie,
both
tidal
to the COBOLT measure-
and diurnal
fluxes had
opposite
signs.
To
analysis
summarize:
of the COBOLT
performance
conventional
experiment
boundary
with
of
the
a
condition
COBOLT
continental shelf.
From
be
a
physical
questions
applied
exceptions,
considerable
instrumentation
important
to
well-defined
data promotes
techniques.
raises
few
at
in
the
confidence
comparison
standpoint,
in
with
the
concerning
the
the
boundary
inner
tidal
the
more
COBOLT
proper
of
flux
the
CHAPTER III
TIDAL DYNAMICS AND THEORY
A.
Tidal dynamics
Theoretical
ment
of
the
Although
interest in
tidal
the
the tides dates back to Newton's develop-
potential
potential
which
did
appeared
explain
the
in
Principia
origin
of
in
1686.
tide-producing
forces,
the real beginnings of the dynamic theory of the tides can be
traced
to
Laplace;
appeared in
done
that
is
particular
So
1799.
it
in
to
the subject
difficult
to
is
his
old,
Mechanique
Celeste
and
work has
find a problem
enough
that has
which
been
not been ap-
proached in some manner before (see e.g., Ferrell's (1874) discussion
of non-linear bottom stress
tidal
theory is
(1960),
(1970)
while
and
summarized in
more
recent
Hendershott
and tidal friction).
Lamb (1932),
reviews,
(1973),
such
emphasize
Much of the early
Proudman (1953),
as
Munk
the
and
areas
or Defant
Hendershott
that
are
of
concern to modern investigators.
A dynamic
theory of
the tides begins by considering
equations of motion for a fluid in
the Eulerian
a rotating frame of reference.
their most general form they are (Krauss, 1973):
(
+ (V - V)v + 2Q x v)
-Vp-
VC} + V (1)
-;
+V- v
=0
,
where the symbols are defined as follows:
In
85
p
is the density of the fluid,
v = (u,v,w)
is the Eulerian velocity,
is the rotation vector of the observer's
frame of reference,
is the fluid pressure,
p
is the gravitational potential,
is the stress tensor,
and the other quantities have well known meanings.
As
traditionally
are
and
earth.
these
stand,
they
The
equations,
usual
equations
for
simplified
lead
fluid is
not
the
to
have been critically examined by Miles
used here with one exception--the
to
application
which
approximations,
to
difficult
too
much
are
tides
Laplace
(1974)
solve
on
the
tidal
and will be
considered
friction-
less.
Basically,
important
the
approximations
and
idealizations
employed, and the modifications required of (1) are:
1.
a homogeneous, incompressible fluid:
-t
+
v
'
p=
0;
(2)
2.
small disturbances relative to uniform rotation:
(V
3.
V)v
=
0;
(3)
a uniform gravitational field (which implies
the neglect
of tidal self-attraction);
4.
a rigid ocean bottom; and
5.
a shallow, or hydrostatic, ocean:
p
g .
known
as
(4)
9z
This
tion"
last
simplification,
(Eckart,
1960),
involves
not
the "traditional
only
the
neglect
approximaof
accelerations but also the neglect of the vertical Coriolis
to the horizontal velocity.
vertical
force due
The omission of the latter term (and the
approximation
itself) has come under some attack (Phillips,
1966)
no
instance
use
would
be
the
nature
of
specific
has
come
to
light
where
its
but
misleading.
In
tidal
addition
to
dynamics
on
simplifications.
6.
the
simplifications
the
continental
allows
certain
other
a plane earth coordinate system:
f k;
(5)
retention of stresses on horizontal planes only:
V - T
where
8.
shelf
above,
They are:
2 Q
7.
listed
T=
=3T/z
(6)
X i + Yj; and,
the omission of direct tidal forcing:
V+ =
0 .
(7)
The
plane
modelling
after
earth,
coordinate
a
of
the
f-plane
approximation
the dynamics of small scale oceanic
noting
as
or
that
the
Coriolis
function of the
Taylor
earth,
latitude, G.
expansion
of
f
commonly
phenomena.
parameter
system on -a spherical
is
of
f =
a
a
21Q[ sinG,
given
It
local
The ratio of the
around
used
is
for
made
Cartesian
varies
first
slowly
two terms
latitude
forms
the
criterion for applicability of equation (5),
ta
t an 6where AG
is
a
latitude
<<1,
increment and RAG is
problem (R = 6000 km being the radius
dimension
of 100
(8)
km, for
example,
the length
of the earth).
gives
scale of the
A typical shelf
(8) a value of
about
2 x
10-2 for mid-latitudes.
According
equation
(1),
are
"stresses"
known
stresses (except possibly in
the
frictional
familiar
Reynold's
forces
in
the oceanic case,
In
of stress gradients.
consequence
turbulent
to
to
dominate
a
fluid
to
viscous
very thin layers near boundaries).
decomposition
(Bowden,
these
1962)
a
due to
forces
due
forces
are
With
turbulent
forces are (in the x direction)
(V
the bracket
where
velocity
and
. T) - i
a
+
=
indicates
perturbations.
vertical
length
right-hand side of (9),
a
time
average
Introducing
scale,
is
H,
(9)
,
a
the
and the
horizontal
ratio
of
indicate
primes
length
the
scale,
terms
on
L,
the
88
H
< UVI>
(10)
< u'w'> L
The validity of assumption
For tidal waves
104 ;while
on the
.
7 requires
11 =
shelf,
measurements
direct
and <u'w'> = 1 cm 2 /sec 2 .
this ratio be very small.
that
100m and L = 1000
that <u'v'>
show
km,
so
cm2 /sec 2
1O
=
I/L =
Thus, this criterion is met.
Finally, tidal phenomena on the shelf are generally assumed to be
independent
1960).
of
the
direct
of
the
tidal
potential
(Defant,
They are instead generated by the inertia of the deep ocean,
acting through continuity,
tides
forcing
are
then
termed
propagating waves.
at the edge of the continental
"co-oscillating"
and
are
treated
shelf.
as
The
freely
The condition for the validity of this approxima-
tion is that
-
gVC
where
V <<
,
(11)
VC
n is known as the equilibrium tide and is
given by Lamb (1932)
as
3 M
where M/E is
earth,
and
the ratio of
R/D
the
is
distance to the moon.
1 a
k
_
-
R 3
(E)
2ED
=
I
the mass
ratio
1
3
R (cos 3
of
-
(12)
3),
of the moon
the
radius
of
to
the
the
mass
earth
of
the
to
the
The gradient is typically
3M
R3
-8
sin 20
=
5
x
10
(13)
89
of this
A comparison
latitudes.
for middle
quantity with the order
of magnitude of the surface gradient made from estimates of the tidal
(VC
wavelength
10-
)
the
that
indicates
ratio
is
(11)
small--
about 0.05.
With
the
using
the free
relation to replace the pressure with
equation (1)
and
above,
described
changes
surface
to the x and y momentum equations,
reduces
hydrostatic
the
function,
C,
and continu-
ity:
Bu
at
-
-g
fv
+ fu
=
3x
-
au
av
3x
ay
p az
1y
p
3y
at
B.
+
(14)
3z
aw
+
z
The vertical structure of tidal currents
To examine the effects of
tidal
friction on the vertical
currents, the stress must
be related to
manner to close the set of equations (14).
=
where K
is
Kv
structure of
the currents
in some
The form chosen,
(15)
,
a constant eddy viscosity, models
a well-mixed water column, away from boundaries.
turbulent
effects
in
It
be
is
treated
(14).
apparent that the free surface,
as
a
forcing
This results
prescribed
in
a "local"
on
the
horizontal
of z,
can
velocities
in
calculation of current structure for
tidal height variations.
by i and adding the two,
(15))
function
being independent
Multiplying
leads (including
the
second
equation
the stress parameterization
to the single, complex, second-order equation,
+ i
fq -
K
~2
=
P ,
(16)
where q = u + i v and
P = g(
)
+ i
i ((w+f)A+eit-
(w-f)A e
is the arbitrary forcing function expressed in
)
terms that will relate
it easily to the tidal ellipse (see chapter 2).
The
solution
to
this
equation can be
expressed
as the
sum of
a
clockwise and a counterclockwise rotating solution,
q(z)
=
qj(z) e Wt + q 2 (z) e-it
(1)
where
q(z) = C exp((1+i)z/d ) + C2 exp(-(1+i)z/d)
+ A+
for all W, and
(18)
q2 (z) = C 3 exp((1±i)z/d 2 ) + C exp(-(1±i)z/d 2 ) + A
(using the plus sign for W < f and the minus sign for o > f)
integration constants Cn, and defining the parameters
d =
K/(w+f)
d
/-fl
with the
This problem has been treated before by Sverdrup
recently by Butman
lies
in
took
to be non-slippery.
does
not model
account
the
for
The difference
(1975).
choice
of
the
a
between
(1926)
and more
the two
analyses
Sverdrup
bottom boundary
condition,
which
From a physical
standpoint,
this condition
turbulent boundary layer
correctly.
the existence of the so-called "wall
In
layer",
order
to
Butman used
instead a slippery boundary condition,
K
where
r
is
=
(19)
at z = 0,
r q
an adjustable drag coefficient.
Both investigations used
the same no-stress surface boundary condition
3q/az
=
0
at z = H.
(20)
With these boundary conditions, equation (18) becomes
r d
q (z) = A+ eiot (1 ~ K1
cosh(1+i)(z-H)
v'1
(21)
q2 (z) = A~ e-i i(
-
r d
K 2 sinh(1±i)( -),
vQ2
2
where
Q
for n = 1,
= (1±i) sinh(1±i)H/d
+ (rd /K ) cosh(1±i)H/d
2 and where the minus sign is
used for W > f and the plus
sign for o < f.
The analogy
is
to
seen clearly in
the ellipse
equations
equation (equation
(17)
and (21).
(11)
Indeed,
of chapter 2)
for
the case
of
no
bottom
equation
stress,
and
(17)
shows
reduces
that
the
exactly
forcing
to
the
parameters,
complex
A+
ellipse
and A~,
can
be
identified with the frictionless, barotropic tidal current ellipse.
Because
the
description,
forcing
the actual
parameters
numerical
orientation are also arbitrary.
are
arbitrary
values
of
independent
parameters
vertical
structure
frictional
forcing
of
the ellipse
constants,
"local"
and ellipse
However, it has been shown by Butman
tidal ellipticity
the
the
ellipticity
that the vertical structure of
of
in
A+
parameters
and
A-.
depends
a fact which was
r and Kv;
and orientation is
Thus
solely
the
on
the
used by Butman to
make estimates of these constants.
The vertical profiles of ellipticity and orientation are shown in
figures 1 and 2 for values of the dimensionless quantities,
Y
=
r/fH
and
(22)
1
that
gave
good
visual
fits
/
2K
to the
observed
semidiurnal
profiles
of
the same ellipse parameters (presented in chapter 2).
These
of chapter
figures were made to correspond
roughly to figures
2 by adjusting A to match the observed thickness
tional
influence
to the
total
( A
depth
of
is
actually the
the water
ratio
column)
7 and 8
of fric-
of the Ekman layer
depth
and then varying Y to match
ELLIPTICITY
-0.2
-0.4
0.2
0
0.2-
0.4-
~
z/H
0.6-
0.8-
0. 2
y =0.5
A =0.2
1.0
Figure 3-1
Theoretical vertical profiles of semidiurial ellipticity
ORIENTATION
-100
-23*
0*
0.2-
0.4-
_
z/H
=0.2
=O.5
0.6
0.8
A=0.2
1.0
Figure 3-2 Theoretical vertical profiles of semidiurnal orientation
95
the
range
of values
assumed
by
the
ellipticity
or orientation.
was found that large values of Y, which correspond (in
approaches
result
infinity) to Sverdrup's
The
actual
the limit as Y
no-slip bottom boundary condition,
in excessively large ranges
ellipticity.
It
observed
for
the
range
ellipse
of
this
orientation
parameter
and
supports
the use of the slippery boundary condition (19).
The model reproduces the main features of the observations
well
considering
constant
values
near
its
the
(currents
simplicity.
surface
The
and then changes
can even
rotate
in the middle of the water
Thus,
it
rather
for
near
is
to higher
the bottom.
slope towards negative angles as
column, but diminishes
shows the effects
example,
sharply
counterclockwise)
The orientation has an almost linear
bottom.
ellipticity,
fairly
sharply near the
of friction a little
further away
from the boundary than the ellipticity does.
Since
the
values
of Y and A were
vertical structure of semidiurnal
chosen
to
fit
the
observed
ellipticities and orientations,
and
since the actual values of these numbers were determined by judicious
choice
of
surprising
the
that
forcing
the
parameters,
agreement
A+
between
good.
An independent test may be made,
values
of y and A to
the
diurnal
tidal
and
A~,
theory
however,
ellipse
it
may not
and
be
observation
too
is
by applying the satme
vertical
structure.
Figures 3 and 4 are these predicted vertical profiles.
The
except
layer is
diurnal
that
it
ellipticity
has
deeper for
profile
is
much
a slightly more gradual
frequencies
that
like
the
slope since
are closer
semidiurnal,
the boundary
to inertial
(see
the
ELLIPTICITY
-0.4
-0.2
0
A =0. 2
0.2-
0.4-
z/H
0.6-
0.8 -=0.2
1.0
Figure 3-3
Theoretical vertical profiles of diurnal ellipticity
ORIENTATION
- 20C
0
-10*0
10*
200
A
0.2
0.4-
~
z/H
0.6-
0.8y=0.2
1.0
Figure 3-4
Theoretical vertical profiles of diurnal orientation
for
expressions
tation,
on the
d
following
n
other hand,
f--it
increase.s
slope
is
12-13
in
with
chapter
is
is
rather
found
2).
The
(18)).
The
totally different
depth
exactly what
equation
in
than
the
model
diurnal
orien-
at frequencies
decreases.
below
This
opposite
COBOLT observations
(figures
also
shows
a
greater
range
of
orientation angles--again a feature that is found in the data.
Given
the
viscosity
model
observations,
provide
independent
it
and
is
acceptable
the
agreement
COBOLT
COBOLT
the best
site,
to suppose
values
the
and
agreement
semidiurnal
reasonable
of
Using the definitions of equation
the
between
the
range
and
that
friction
(22),
(A =
constant
diurnal
parameters,
K
depth of
for y and A that
0.20± .05
eddy
frequency
this model will
the nominal
of values
with observations
the
and y =
also
and
r.
30 m at
produced
0.4± .1)
gives
K
v
= 10-25 cm 2/sec
(23)
r
These
(see
values
e.g.,
estimate
agree
Butman,
made
by
= 0.09-0.014 cm/sec
well with
1975).
Scott
low frequency currents.
other
The
estimates
value
and Csanady
for
(1976)
r
at
made in
also
shallow water
agrees
the COBOLT
with
an
site using
C.
The effects of friction on tidal propagation
Besides
affecting
friction
currents,
istics of the
the
also
local
vertical
affects
tides (this
the
"global"
structure
global
surface
subject
has
investigators
important in
In
space
have
time,
conditions.
ignored
friction
It
resistance
more apparent.
also
field
order
in
the
of
attention
the
character-
of equation (16)).
basic
in
While
past,
the
conditions
on
as
the
shortens
the
expected
nature
the
water
by
most
that
are
of
attenuating waves
the
wavelength
column has
less
the
It
largely ignored fact,
is this
effects
shallow water,
to explore
simplest
terms,
of bottom
that is
the global
or
initial
1955).
inertia
In
and hence,
friction
are much
the enhancement
vertically
of the
tidal
at
fv
=
-g
+ fu
=
-g
3x
ay
+5
+
integrated
versions
pH
wh e re
H is
the depth of the fluid;
B is
the bottom stress; and
(24)
PH
- + -(Hu) + -c(Hv)
at
3x
9y
of
elevation
equations (14) will be used:
at
in
investigated here.
structure
the
boundary
(Proudman,
to change,
frictional effects in
In
some
acts
depending
shallow water, where
less
considerable
propagation
tidal
the coastal region; namely rotation and depth variations.
general,
or
received
the
tide was impressed on the
previous solution as the arbitrary forcing
this
of
0
of
100
u and v are the vertically averaged velocities defined by
H
v(x,y,t) = 1/H
v(x,y,z,t) dz
0
In
direct analogy
to equation (19),
bottom friction will be taken
as proportional to the depth-averaged velocity, i.e.,
B
Although
this
introduce
a
form
has
Physically,
should
used
be
some
dissipative
dynamics.
in
-r
v
.
(25)
shortcomings
mechanism,
velocities
But
(25).
albeit
above
for
(Rooth,
a
the
tidal
crude
1972),
it
one,
into
frictional
oscillations,
does
the
wall
layer
the
depth-
integrated velocities are a good approximation.
The equations can be solved quite simply if
be
constant.
The
elimination
of
all
the depth is
variables
except
taken to
the
surface
elevation leads to the equation,
((
at
+ r)2 + f 2 )_-_
H
at
gH(
3t
+ r
H
=2C
0.
(26)
Rearranging this into frictional and non-frictional expressions
a((a
+ f2)
at t 2
)gv)
22
gH2)
gives
+
(27)
+
r (2
II
Thus
the
equation
2
at2
+ r
G
H at
separates
gHV 2C)
neatly
For small bottom friction or large depth,
(upper
part)
governs
the
dynamics
of
0.
into
two
well-known
forms.
the familiar wave equation
tidal
waves,
whil-
for
large
101
friction
or small
.equation
(lower
solutions.
depth,
the dynamics
part) which is
fact for
In
are
governed by the telegraph
known to have wave-like
the large
friction-shallow
and diffusive
water
case,
(27)
reduces approximately to the parabolic partial differential equation,
2 2
gi_
r
Dt
The
is
scaling factor
This
r/wil.
0.2-0.4)
but
that
factor
rapidly
is
r/wH
This point,
marks
the
reaches
which is
0
indicates
small
becomes
decreasing water depth.
structure,
=
at
which dynamics are
the
COBOLT mooring
important
Using r = 0.1,
a
(28)
value
of
nearshore
appropriate
site
because
(about
of
the
as inferred from the vertical
unity
in
about
about 1 km from the shore at
outer boundary of frictional
dominance
10 m of water.
the COBOLT site,
in
depth dependent
dynamics.
D.
The Sverdrup-Poincare wave model
Analytical
solutions
of
(24)
are
extremely
complicated
if
both
rotation and variable depth are retained in
the model.
To simplify,
one must
two or
to model
either make a
choice
between
the
try
the
frictional effects of shallow water in some other manner.
The obvious choice for an alternate model is a step discontinuity
to simulate shoaling water.
coordinate
system
envisioned,
dynamics
the
that
deeper
dominate,
and
The
will
of
the
be
the
geometry of
used
two
shallower
is
sides
a
solution dominates\ (t'ough this region is
this
situation and the
shown
in
is
region
region
a
figure
where
not examined in
5.
As
where
wave
the
diffusive
detail).
102
Figure 3-5
Geometry of the step shelf tidal model
103
In
the deep region,
the familiar
this
wave refraction
problem is
posed in
problem with
the
incident, reflected, and transmitted waves.
the same manner
solutions
formed
determine
of
the
(29)
H'v'
the relationships between the various components.
first
condition,
Kelvin
direction
(alongshore),
must
they will
decay
shallow
in
the
from
The boundary conditions,
at y = 0 ,
Hv
as
waves,
be
which
excluded
region
travel
from
but
not
only
Because
in
consideration
in
the
deep,
the
x
since
and
a
mismatch at the boundary will inevitably result.
The governing equation in the deep region is (from (24))
2
(
+
f2)C
-
gH V2C
0
,
(30)
at
with the velocity fields defined by
2
(a2
2
+ f2)u =_g(
+ f a)
y
tx
at22
(31)
22_2
(D
at
2
2
+ f2)v =-g(
at3y
ax
Assuming a solution consisting of an incident and a reflected wave,
c
where
k
=
a (exp i(kx+ty-wt) + R' exp i(kx-ty-wt) ,
and k are
equation (30),
real
wave
numbers,
gives the dispersion
and
substituting
(32)
it
relation for the elementary
into
long
104
gravity
wave
in
a
reference--the
rotating
Sverdrup
(Sverdrup,
wave
1926):
2 _
The
friction
solution
is
in
assumed
across the step.
It
g
=
shallow
to
(k
water
absorb
must,
(33)
+ 22)
is
relatively
all
the
however,
have
energy
'
where 2' can be
=
T'
complex
exp i(kx+9'y-wt)
to allow for
that
the same
the x direction as the deep water solution.
unimportant
is
since
transmitted
functional form in
It is taken to be
,
(34)
frictional
attenuation
in
the
offshore direction.
Applying
the boundary
conditions,
with
the aid of
equation
(31)
prescribes the relationships:
R'
+
1
=T'
(35)
gi
22
(Wl(R'-1)+ifk(R'+1))
=
igH' T'
((r/H-i22
(fk-(r/H--io))
which can be solved for the complex amplitude of the reflected wave,
R'
H
-1 IQ
--
I'Q'
j~
HQ* + H'Q'(
where
Q
Q'
Eqiati on (36)
oz -
=
ifk
wZ' - ifk
can be reformulated
into the more conveni rnt form,
(36)
105
(Q/Q*) r'
R'=
where
r'
will
be called
(37)
,
the "reflection
coefficient"
and
is
defined
by
In
the present
(38)
1 - (H'Q'/HQ)
1 + (H'Q'/HQ*)
r
context H'/H is
presumed to be small in
keep the absorption of energy to small
values.
In
this
order to
case r'
can
be approximated to order (H'/H)2 by
r' r = 1 -1 H' Real (Q) Q, .(39)
In
r'
general
is
complex,
but
for
small
H'/H
it
can,
for
all
practical purposes, be considered real since the imaginary part of r'
is
very small in
real
since both
For the case of no rotation,
this instance.
Q
and
Q'
become real.
Also, r' is less
r'
is
than unity
for H' < H, indicating an absorbing boundary.
Replacing
R'
in
equation (32)
C
(r'Q e '+
with equation
(37)
gives
the wave
solution,
Q* e
) ei(kx-tt)
(40)
and the velocity distributions,
V
ag Q12 1
, e-ity
ity)
i(kx-ot)
o -_f ~ Q*
(41)
u
=
ag
2
o-
1 (r'SQ e
2 Qe
f Q*
- S*Q* e
) ei(kx-ot)
106
where
S
These solutions
whose
elementary
ift.
-
consist of incident
and reflected
are
characteristics
well-known
invoked as an explanation of tidal phenomena.
Sverdrup waves
but
are
seldom
The ellipse character-
istics are easily examined from equations (40) and (41).
This boundary condition bears a close relation to one proposed by
Proudman (1941)
continental
in
shelf
an attempt to model the dissipative effects of the
as
a
boundary
to
the
deep
ocean.
The
Proudman
boundary condition,
V
c
at y = 0 ,
takes the place of the boundary conditions
some of the numerical models
It
1977).
was
solutions
California with
included
and has been used in
deep ocean
tides
(Hendershott,
also used by Tendershott and Speranza (1971)
strongly localized
These
of the
(29)
(42)
coastal
dissipation at the end of a long channel.
were
applied
some
success.
implicitly
in
to model
to
the Adriatic
Finally,
the
(1978)
model
Redfield's
Sea
and
absorbing
the
Gulf
boundary
of Long Island
of
is
Sound
and several other basins.
To show the analogy,
the
solution (32)
is
used in
with (42) as a boundary condition to give
-
Q
+ a(2_ f2)
2 2
gQ
gQ* - a(o
f
0*3'
)
where
22
1 + a(wo ~f )/gQ
1
-
_ ?)
,
l
equation (31)
107
is
a clear analogy to equations (37)
r
II +
+ 2
2
=
and (38).
-
) Real
_(~-f
g
Thus a must
be
complex
to
agree
If a/g is
small,
(
(45)
.
jQ[
completely
with
equation
(38)
and
also must be negative to assure that r' < 1.
E.
The Sverdrup wave--no reflection
The plane Sverdrup wave is
0,
which
is
the case
of
investigated by setting r'
no reflection
or
perfect
= 0 and k
absorption.
=
For
this wave, the ellipticity is defined by
C = -i (v/u) = -(f/o).
For middle latitudes,
of about -2/3--i.e.,
minor
axis:major
(46)
the Sverdrup tidal ellipse has an ellipticity
a clockwise
axis
ratio of
rotation
2:3.
of the
current
The orientation
vector and
angle
of
Sverdrup tidal ellipse coincides with the direction of propagation
the
so
in this instance is identically zero.
F.
The Poincare wave solution--perfect reflection
Examining the opposite extreme,
r'
=
1,
shows
the characteristic
the case of perfect reflection or
tidal
ellipses
of
two superimposed
Sverdrup waves whose onshore velocities exactly cancel one another at
the shore.
1971).
result
wave
This combination is
Often
in
invoked
as
an eigenvalue
numbers
(Defant,
a
known as the Poincare wave (Platzman,
solution
in
a channel,
where
problem and a set of discrete
1960),
this
solution
is
boundaries
cross-channel
also valid
for
long,
108
straight
coasts
and Wimbush,
or
1970)
continental
topographies
(Munk,
Snodgrass,
where only one boundary plays an important
The resulting
the dynamics.
shelf
free
surface
role in
and velocity distributions
are:
(o, cos ky + fk sin ky) ei(kx-wtl
=
-i2ag
2
v =
u =
2 2 22
2 ((W -f) f
( 2
2ag
22 2
i(kx-wt)
(k +P, )) sin ky e
)Wf sin 9y) ei(kx-ot)
cos Ly + (k2 +,
2)k
(47)
o -f )
As
in
the non-rotating,
standing wave problem,
surface nodes is apparent from the
first of the
Otherwise,
the
wave
complicated
than the non-rotating
rotating
standing
case.
the possibility of
equations
characteristics
Turning
again
to
in
(47).
are
more
the
tidal
ellipse to elucidate the signature of the Poincare wave,
v
_
u
where
Besides
highly
dependent
2
?
+ f /g)
sin ky
kk cos ky + wf/gH sin Zy
the dispersion
sion.
(
relation has
been used
to simplify the expres-
depending on the distance offshore,
on
the
wave numbers,
or
48)
'
the ellipticity
is
equivalently
the
angle
By
the
angle of
of
incidence
of
the incoming
incidence
as
the angle between the wave vector and the shoreline (see
Sverdrup
wave.
defining
figure 5),
I
the wave
properties
=
tan~(- ),
(49)
can be plotted as a function of offshore distnace
1.0
0.8
r'=
0.6
0.4
I= 30*
I= 60*
0.2
00
CL
-J
0.2
0.4
0.6
0.'8
ky3
-0.2
-0.4
-0.6
-0.8
-1.0
Figure 3-6 Ellipticity versus offshore distance for perfect reflection
and several positive incidence angles
1.0
0.8
0.6
0.4
0.2
0
-0.2
-0.6
-1.0
Figure 3-7 Ellipticity versus offshore distance for perfect reflection
and several negative incidence angles
111
The
negative values
(the
is
ellipse
distance offshore
velocity
becomes
becomes
circular.
but
equal
the
to
this
From
the
on,
point
and
velocities
the onshore
the
and
velocity
alongshore
short
or negative if .k > 0
away from shore,
some distance
At
a
positive
becomes
to the right)
zero
identically
the ellipticity is
polarized)
alongshore
than
greater
= 0)
k < 0 (shore
(shore to the left).
and
positive
both
assuming
greatly,
(y
linearly
if
Note
on the angle of incidence and the distance
depending
At the shore
offshore.
for
scaled by the wavelength.
these plots is
varies
ellipticity
6 and 7
figures
and positive incidence angles.
the direction of propagation)
that the offshore distance in
in
done
on the right-hand when looking
representative negative (shore
several
in
is
This
angles.
incidence
various
for
ellipse
velocities
onshore
again
tend
ellipticities
are
toward smaller values.
this
at
Also
changes
ellipse
tion between
is
tation
the
point,
abruptly
u and v is
constrained
to
always
be
for
the
same values
to
00
from
from shore and all incidence
angle
orientation
900
or
900
incidence
the
rela-
the orien-
all
for
of
phase
the
Figure 8 shows
angles.
axis
major
equation (47))
(see
00
the
Since
900.
either
of the
of
distances
the orientation
angle (both
positive and
negative) as the previous figures.
It
clear that
is
the COBOLT data differs substantially
Sverdrup and Poincare wave models.
ellipse
does
ellipse
is
rotate
in . the
much too circular
While
clockwise
the Sverdrup
direction,
the
from both
wave
shape
current
of
to agree with COBOLT observations.
the
The
I=60*
80*0
I=30
60
40
r=
20
00
0.4
0:2
1
-20
0:6
0:8
ky
-40
-60
-
- 80*
0
I -60*
Figure 3-8 Orientation angle versus offshore distance for perfect reflection
and several incidence angles
113
Poincare
current
observations,
ellipse,
is
orientation
unlike
any
oriented exactly alongshore
tendencies
close
to
the
wave progressing with the shoreline
wave model
ellipses degenerate
rotate counterclockwise
So as
of the
they stand,
ellipses
in
the
and exhibits
shore.
Also,
for
on its right-hand
COBOLT
no onshore
an
incident
side,
Poincare
into lines very near to the shore and
further
away,
again unlike
the observations.
the Sverdrup and Poincare waves are not capable of
reproducing the results of the COBOLT experiment individually.
G.
The combination Sverdrup-Poincare wave
A combination of these waves offers a third alternative,
the case of 0
pure Poincare and Sverdrup waves as references,
1 is
less
difficult
to
plus
incident Sverdrup wave
wave;
or
a
equivalently,
-
consists
amplitude
reflected
plus
smaller
wave
a
r'
of
an
Sverdrup
amplitude
The solutions for this case are:
incident Sverdrup wave.
= -
a smaller
Poincare
solution
This
interpret.
and with
a((r'+1)(oiZ cos Zy + fk sin Zy
Q-
-
i(r'-1)(fk cos iy + wk sin ky)) e
(50)
u =
-a
ag
Q*(w"-f
-
v =
2
_
2
((r'-1)((o -f2)k
2
2
cos Ly + (k +z. )Wf sin ky)
-
)
i(kx-)t)
2 2
2 2
i(r'-1)((k +9 )of cos Ly + (W -f )kZ sin ky)) e ~
a g
(U2
2
2 + f2
2
2
Q*(2'-f 2)
-((r'-1)
cos Ly
-
.
.
i(kx-wt)
i(r'+1) s tn Zy) e.
114
Tn
natural
cient will
situations
generally be
it
is
close
likely
that
the reflection
to unity because
the
to reflect much of the energy impinging on it.
true
very
all
possible
near
to
shore.
Nevertheless,
reflection
coefficients
it
shore
This is
is
the
two
Sverdrup
diagrams
are
waves
of
representative
Sverdrup waves
of equal
be interpreted
as
waves.
of
amplitude
all
to
ellipse
the
range
the range of possibilities
Sverdrup
waves
ellipses
that can occur under
and may provide
between
parameter
interactions
the
Poincare
distance
the
of relection
that exist
two
of
two
can then
interacting
coefficients
for
some insight into
a wide
as
consider
interactions
The offshore
difference
entire
(just
possible
amplitude).
the phase
Considering
establish
unequal
certain
particularly
diagrams will then be representative of all the possible
of
is
valuable
since
coeffi-
two
will
interacting
the forms of tidal
range of reflection
conditions
and distances from shore.
Plots
of the
for different
the
the
sharp
transitions
the
parameters
real values of
combination
between
tidal
two waves
the Poincare wave
exhibit
becomes
smoother
less
the ellipse parameters
reduce
Sverdrup wave
orientation
with an
figures
9 through 11
the reflection coefficient.
noted in
waves
are shown in
to
of
ellipse pairameters,
profiles
important.
Tnstead
In
as
interference
the r'
= 0
limit
those of the uniformly srooth pure
angle
equal
to
the complement
of
the angle of incidence and an ellipticity equal to -f/w.
The ellipticity,
characteristics,
tion
coefficient
as
a result
increasi-ngly
drops.
of
favors
Thus,
the
transition
negative
clockwise
to Sverdrup wave
values
rotation
as
of
th.
the
reflectidal
1.0
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1.0
Figure 3-9
Ellipticity versus offshore distance for imperfect reflection
1.0
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1.0
Figure 3-10
Ellipticity versus offshore distance for imperfect reflection
800
60
40
z
0
20
O0
z
-20
-40
-60
-80*
Figure 3-11
Orientation angle versus offshore distance
for imperfect reflection
118
ellipses
graph
becomes
over
observing
more
which
probable
the
(as
ellipticity
perfectly
the
extent
negative).
The
probability of
is
(c =
circular
measured by
1)
is
quite
small
seen
sharp
that
orientation angles
jumps
With these
from
00
general
to
900
trends
(figure
as
(11))
were
no longer
seen
established,
in
the
the
Also, it
exhibit
Poincare
extrapolations
the
since
ellipticity tends to range somewhere between zero and -f/w.
is
of
can
the
wave.
easily be
made for cases not shown.
Considering that
to
the tidal wavelength is
shelf dimensions,
parameters
9-11
may
that,
allows
be
most
interesting
found
for ky
<< 1.
the
Poincare
wave
unlike
both
the
non-zero
generally large
orientation
It
aspects
is
model,
angles
and
of
apparent
the
compared
the
ellipse
from
figures
combination
non-zero
wave
ellipticities
near the shore (near y = 0)--features which reproduce the results of
the COBOLT experiment.
In
particular,
interest
to
plotting
the
angles
as
the
a
angle
constant.
axis,
conditions
COBOLT
possible
function
coefficient.
the
the
of
experiment,
of
incidence
the
are
combinations
The various
Since
at y =
the
angle
curves
while
diagram
in
of
of
which
ellipticity
incidence
angles
is
limited
to
negative
the most
summarized
and
by
orientation
reflection
12 are formed
by varying
the
symmetric
and
of
the
reflection
about
only negative orientation angles are shown.
incidence
are
conveniently
figure
holding
is
0,
the
00 orientation
Also,
values,
coefficient
the range of
-80o < I < -100
119
with the
indicated by dots),
are
increments
of incidence
angle
(100
exception of the asymptotes which are indicated by dashed lines.
As
alternative
an
r',
constant
a
constant
figure
r'
is
examining
can
plot
similar
incidence
13
for
while
in
ellipse
ellipse
characteristics
by
the
made
the
from 0.6-1.0
varied
of
be
varying
holding
reflection
angle
coefficient.
for
of
In
dots indicating reflection
with
By considering both of these figures,
coefficient increments of 0.1.
the variations
the
characteristics
that occur
due
to
changes
reflection coefficient or angle of incidence should be evident.
From
figure
that while
angles
the
from
12
-900
regardless
ficient.
For higher
angles
since
negative
the
ellipticities
angles
to
+900,
of
the angle
in
result
is
plot
can
about
cover
00)
the
ellipticities
of
incidence
small
is
always
furthermore,
angles
small,
and
is
quite evident
angles
are
coefficients
that
small
alongshore
wave
numbers)
figure
difficult
in
incidence
range
negative
coef-
the bulk
(positive
negative
or
ellip-
and small
any
real
situation
where
It is also evident from
angles
(waves
with
larger
orien-
associated
with
small
negative
orientations
also
show
variations
are
of
13 and would make determin-
close to unity.
negative
13
in
apparent
or reflection
orientation
symmetric)
it
complete
are
coefficients,
reflection
incidence
of
reflection
figure
reflection
This convergence of lines to small orientations
ticities.
ation
its
orientation
nearshore
of incidence
(and
tation angles.
The
ellipticities
and
depend on the frequency of the incoring wave.
that
Figure 14 shows curves
120
ORIENTATION
-80
-60
I
I
I=-8*
-40
I
I
-20
00
I
1~;
-0.2
I
P
T
I=- 10
-0.4
-0.6
0.8
Figure 3-12
E
L
L
Ellipticity versus orientation angle at y = 0;
incidence angle varied while holding reflection
coefficient constant
C
I
T
Y
121
ORIENTATION
-80
-60
-40
-20
O0
-0.2
E
L
L
P
T
C
- 0 .4 c
T
Y
-- 0.6
-
Figure 3-13
Ellipticity versus orientation angle at y = 0;
reflection coefficient varied while holding
incidence angle constant
-0.8
122
ORIENTATION
-10
-20
-30
0
E
L
L
P
T
--0.1 1
/I
/
/1
T
Y
I=-80*
r'=0.9
-- 0.2
Figure 3-14
Ellipticity versus orientation angle at y = 0;
varying the incident wave frequency
123
angle
-700 < I < -20'.
range,
previous
figures
is
from
clear
different
waves
as
ellipse
that
14
o/f = 1.5 was used in
Although
representative
figure
= 0.9 and for the incidence
for r'
frequencies
different
for several
the mid-latitude
of
characteristics.
particular,
In
higher
it
slightly
frequency
the
(i.e.,
ellipticities
less negative nearshore
tend to have
tide,
have
frequencies
tidal
other
M2
the
ellipticity approaches zero) than lower frequency waves.
it
Again,
angles
incidence
be
should
on
(shore
around
the
this
operation
is
00
left
angle
orientation
that
of
wave
a
positive
looking
if
in
the
can be constructed simply by reflecting
direction of propagation)
plots
the
for
diagrams
similar
that
noted
all
The
axis.
angles
orientation
affect
main
become
the
of
positive when
the coast is on the left of an incoming wave.
H.
Comparison of the model to observations
comparison
A
ellipticities
and
on
components
observations
COBOLT
by plotting
accomplished
is
Poincare wave model
tations
the
of
12.
figure
of
three
the
15
Figure
the observed
semidiurnal
resolved
shows
an
Sverdrup-
the
with
orientidal
corner
enlarged
of
figure 12 with these observed values in place.
The positions of the observations
that
of
incidence
an
incident
angle)
fact
predicted
hand
side
oE
by
the
wave
since
eminating
observed
on the plot are consistent with
from
the
orientation
deep
angles
ocean
negative;
are
the modelling of waves with a coast on
wave
(looking
in
the
direction
of
(negative
a
the right-
propagation).
124
ORIENTATION
-20
-30
-10
O0
-- 0.2
E
L
L
P
T
-- 0.4
-- 0.6
Figure 3-15
Ellipticity versus orientation angle at
y = 0 with COBOLT observations plotted
for three semidiurnal frequencies
I
T
Y
125
Furthermore,
incidence
data are
the
angles
(since
also consistent with predictions
observed
large reflection coefficients
to
zero).
Finally,
suggested
tidal
by
the
component,
the
(since
data
is
the
closest
angles
are
small)
observed ellipticities
exhibit
theory--i.e.,
S2,
orientation
for large
the
to
are close
dependence
highest
zero
and
on
frequency
frequency
semidiurnal
while
ellipticity,
the
lowest frequency component, N 2 , is furthest away.
The
notion
dently, to
of
some
small
degree,
It
chart (figure 2-2).
incidence
by
angle
with
exists,
the
south
examining
shows
dicular to cophase contours,
shore
due to the curvature
angle
can
be
Swanson's
confirmed
semidiurnal
that the wave vector (which
or wave crests)
of Long
is
cophase
perpen-
does indeed form a small
Island.
of cophase
indepen-
lines,
While
some
this angle
ambiguity
appears to
be approximately -20O.
Physical
cient
is
considerations
also
a reasonable
suggest
that
result.
By
a
large
requiring
onshore energy fluxes match observed energy fluxes,
compute
(chapter
a
reflection
2)
and
the
coefficient.
From
Sverdrup-Poincare
reflection
the
free
that
it
energy
surface
coeffi-
theoretical
is
po>sible to
flux
and
equation
onshore
velocity distributions (equation (50)), the onshore flux is
F
=
gH Real
<Cv*>
-
(51)
2 2
_ag H _1
2
2
- r'
2
2
Substituting the following numerical values into (51):
126
and
equating
it
to
a
=
50 cm
H
=
30 m
o
=
2 /24
k
=
2 /1500 km,
the
energy
hr
flux estimates
of
chapter
2
(COBOLT
measurements and Miller's shelf-wide estimate) gives
=
r'
a value
that
supports
0.98 ± .01 ,
the large
(52)
reflection
coefficient
deduced
from
the tidal ellipse parameters.
Since
the
flection
ellipse
coefficient
determine
the
parameters
and
values
angle
of
In
could be
used to find the values
principle,
the three
because of
the
on
an
this
obtained
angle
reflection
is
from Swanson's
known
it
coefficient
orientation.
Doing
is
independent
of r'
this
for
re-
is
impossible
at
a single
to
fre-
frequencies
and T, but this procedure
complexity of
work offers
a
it
unknowns,
semidiurnal
the
equations
is
involved
The estimate of incidence
an
a straight-forward
from
two
observation
and the uncertainties of the observations.
angle
the
incidence,
of both with
quency.
very difficult
depend
alternative
since
once
task to determine
single
observation
the M2
tidal
of
frequency
ellipticity
with
the
and
I
= -200
gives a reflection coefficient of
=
r'
0.95
,
(53)
which is close to the estimate derived in (52).
Other
observations
Sverdrup-Poincare
in
wave model
the
too.
Middle
Atlantic
The model
Bight
suggests
support
that
the
clocktwise
127
current
should
ellipses
rotating
tidal
stances.
Emery and Uchupi
(1972)
long
gravity
phases in
the
waves
(without
appropriate
of the
for
most
to
implies
modelling
tidal
ellipses
left-hand
and
for
side (New
a
wave
Jersey)
that
the Sverdrup
to a rotating system,
tidal
phenomena
propagating
looking in
for a wave propagating with
(Long
Island).
Both
ellipse orientation in
angle
for
the
of
with
in
cases
a
show
to
the
nearshore measurements--a
shore-to-the-left
case
and
wave,
is
the
the
region.
(figure 2-3)
shoreline
the direction
a shoreline
these
fit
elevations
Finally, the results of Patchen, Long, and Parker (1976)
show
circum-
this was true at 94% of
rotation)
long gravity wave
solution
in
Redfield's (1958) empirical
the Middle Atlantic Bight
extention
norm
the
found that
the sites surveyed by Haight (1942).
of
be
to
the
of propagation,
the right-hand
predicted
side
sense
of
positive orientation
and
a
negative
angle
model
is
clearly
an oversimpli-
region,
even
for
the
shore-to-the-right case.
The
two-wave,
fication of
does
Sverdrup-Poincare
the complex tidal
explain many of the observed
does not
account for geometries
coasts.
Corners,
the coast
from
regime
which scatter
not
features.
other
considered.
two-wave model
the
wave
propigation
order of magriitude
The model,
than infinitely
additional
and the
Small
incoming wave energy
And,
the
though
it
for example,
long,
straight
such as that formed by the coast of Long Island and
of New Jersey,
them are
of
(towards
predicts
scale
(Mysak,
an energy
the west
at
the
reflections
are
also
ignored.
the direction of
flux in
COBOLT
result
irregularities,
coastal
1978),
that
site)
Larger than that which is observed.
that
is
an
128
Kelvin waves have also been ignored,
a serious
flaw.
The
generally progress
fluxes
that
are
Kelvin wave,
attenuated
in
semidiurnal
tides
this
is
not likely to
of the Middle Atlantic
Bight
the onshore direction and have the small energy
characteristic
of standing waves.
a possible solution (though
offshore
though
and have
a large
not in
By contrast,
this model),
alongshore
energy
the
would be
flux.
These
features are not seen in the observations.
Although
necessary
to
more
complicated
account
for
models
these
are
additional
possible
features,
and
the
may
be
comparison
with observations is good enough to suggest that:
1.
The
classical
Sverdrup wave
tion for the semidiurnal
it
accounts
for
many
is
the
fundamental
mode of propaga-
tides of the Middle Atlantic Bight since
of
the
observed
features
of
the
tidal
currents.
2.
The coastal
region absorbs a small
of the Sverdrup wave,
probably
water
10
shallower
than
amount
through
This
m.
of the incident energy
frictional
absorbed
dissipation
energy,
small fraction of the incident wave energy, is
larger
although
in
a
than some
previous studies have suggested.
I.
The effects of local topography
It
is
of considerable
Sverdrup-Poincare
that is
interest
evident
interest
wave model
in
of
to
consider
one particular
the
complicating
the COBOLT region--topography.
to consider depth variations
effects
to determine
on
the
feature
Not only is it
of
how far away from
129
the step model,
shore can be correctly considered a small distance in
but
some
to shed
also
In
dissipation may occur.
the
on
light
other words,
it
the
of where
question
is
inferred
hoped that this model
how good is the vertical wall assumption,
will answer the questions:
and where does the large amount of energy observed to be propagating
onshore dissipate?
The
long wave equations, with no bottom
for a variety of bottom profiles.
particularly well-suited
tical expression.
bottom
The COBOLT region, however, has a
profile
to
=
H (1 -
straight-line
of
fit
16.
In
the
distance
values
of H
linear
fit
of
area.
y is
a simple
analy-
These
the
this
outermost
depth
parameters
(54)
the best
were
constants
ln(H -H)
fit
to the actual
chosen
from
versus y shown in
a
figure
only extended to 14 km--slightly greater than
shown, H0 =
for
approximate
the
give
on the graph of
this plot,
by
b e~s ) ,
s, and b are chosen to
profile
approximate
This function is
H(y)
where H0 ,
friction, can be solved
in
35 m
COBOLT
mooring.
appears
to give
b =
and
gives
equation
sulting computed profile with
the
0.9
(54).
actual
Of
the
the best
s =
different
fit.
The
0.2 km~
as
the
of
the
re-
A comparison
depth profile is
shown
in
figure 17.
With
omitted
the
in
depth
equations
variations
(24),
retained
elimination
equation for the free surface,
of
and
the
bottom
the velocities
stresses
gives
the
130
100
50
E
10
0
X
0.
0
2
4
6
8
10
12
OFFSHORE (km)
Figure 3-16
Plot of in(Ho- H) vers us offshore distance at
the COBOLT site, plus linear curve fit
10-
20
ac
-C
4.-
bc
'I,
30-
H(y)=H0 (I-e-sy
IJ
10
0
5
10
km
Figure 3-17
Comparison of modelled depth dependence with
actual depth profile off Tiana Beach
15
132
Taking
gHV 2
+ f2 )%
(
the
depth
to
-
be
VH -
a
only
-
(V
function
of
= 0.
fk x V)
the
offshore
(55)
variable
allows a solution,
F(y) exp i(kx-wt) ,
C(x,y,t)
which,
when
substituted
into
(55),
results
in
(56)
the
second
order,
ordinary differential equation,
1 dH dF
d2F
+---+
(
W2f 2
g
H dy dy
dy2
fk
dH
2
+-- - - -k)F
w H dy
.
=0
(57)
A change of the independent variable to
z = b exp (-sy)
(58)
and substitution of the depth profile (54),
transforms
2
d2F
dF
2
z (l-z) 2 + z(1-2z) - + (a2 + Sz)F
dz2
(57)
0
,
to
(59)
dz
where
2
22
_
2
1
(W -f
2
-
k
f
k
k ( f
S
S
Equation (59)
is
k2
g H
o
)
one of the many variations
equation (Morse and Feshbach, 1953).
z
= 0,
1,
dom.tin of
+o
(or
at sy = +o,
ln b,
intere.-t provided b < 1.
of the hypergeometric
It has regular singularities
-o);
all
of which
lic, outside
at
the
133
by
solutions,
Stegun
much
more
illuminating
to
equation
solve
the
(59)
Abramowitz and
in
tabulated
to obscure
as
so general
but are
(1964)
are
functions,
hypergeometric
Its
(1970).
Wimbush
and
Snodgrass,
Munk,
by
and
(1967)
Ball
context before
an oceanographic
in
This equation has been studied
results.
directly
It
using
is
the
method of Frobenius.
Substituting the infinite series,
zp
=
F(z)
c
(60)
zn,
and equating lowest order terms gives
into equation (57)
the indicial
equation,
p2
This
two independent
higher
(61)
which
are
associated
of the second order problem.
solutions
determines
terms
order
0.
roots
distinct
two
has
equation
=
2
the
constants
the
Equating the
for n > 0,
cn,
with
by
the
equation
c
n
= (p+n)(p+n-1)
2
(p+n) 2
to eliminate p,
Or, with the use of (61)
c
n
=c
(62)
6)1
n-i
(n(n-l)
For large sy (i.e., z
-
n(n
-+
-
a22 )
12a)
.
i (2n-1)
n-1
0) the solution becomes
(63)
134
F(y)
lim
(64)
b e Iasy
sy +0
a2 > 0.
provided
This
solution which is
flat
the
steep
solution
appropriate,
portion of
the
be pursued by examining
and a reflected
apparent
from
topography)
or
that
plane
Sverdrup
for large y (that
large
s
this and
(that
is,
over
for a
step model
the
just as before.
sa = k is
wave
is,
(using the plus sign in
wave
(minus sign)
the
for
between
an incident
and (59)
(63)
simply
apparently,
The similarities
slope).
(64))
is
equation
It
an appropriate
can
is
also
definition
for the wave number in the offshore direction.
Both of the previous studies have examined the case for b = 1 and
emphasized
consisting
modes
merely
Wimbush
studies
previous
topographic,
of
These
are
the
and
Kelvin,
shore-trapped
edge
point
examined
that
they
shapes
of
out
the
exist.
Snodgrass,
Neither
of
the
only
their
specific
depth
solutions;
the
Ball
waves.
ignores the trigonometric solutions while Munk,
completely
and
where a2 < 0.
solutions
spectra.
In
form of
the final
profile,
to
interest of examining
the
consist
of an
incident
the
of this
effects
solutions
the trigonmetric
and reflected
wave.
is
rearranged
From equations
(57)
and (60):
=
where
hofore,
(A e
c
=
ity
1.
c
+
b
The
by demanding
n
-sny
en
B
+ B e
absorbing
that
-ity X-
c
boundary
the reflected
b
n
- sny)
en)
condition
e
i(kx-ot)
,
is
wave amplitude
(65)
applied
be
less
as
than
135
the incident wave amplitude
for r'
vanish
=
at the coast
for r'
of the reflected wave.).
v
1 (This
and that the onshore velocity
condition
determines
the phase
The onshore velocity is
-sny-(A1
y
=
c bn(Q +iswn)e-sny
2_ 2 (Ae
V
< 1,
(66)
where
Q
is
c bn(Q*-isn)e-sny) ei(kx-Wt)
n
(Be ity
-
defined
in
equation
(36).
This
determines
the
coefficients,
=
A
c* bn(Q*.-iswn)
n
r'
r'B*
(67)
c
=
B
bn(Q +iswn)
Here it has been noted from equation (63) that c
=
c
The final form of the solution closely parallels
c.
that of the step
model (compare to equation (40) and (41)):
v
=
-
u
ei(kx-wt)
+ r'F(y) eY)
(F*(y) e
g
W -f
(r'e'y (Q F(y) + isw G(y))
e
(Q*
(68)
F*(y)-
isw G*(y))) e i(kx-wt)
(S* F(y) -
2(r'e'k
=
-
sf G(y))
+
W -f
+
e
zy (S F*(y) + sf G*(y))) ei(kx-ot)
where
e-sny
F(y)
=
B*
c
bn
G(y)
=
B*
c
bn n e-sny
136
and
S is
defined
calculated
as
in
numerically,
equation
(41).
but several
These
features
functions are
easily
can be noted before
the
solution is presented.
One
of
topography
is
the
is
to be answered
an adequate
accomplished
terms in
questions
by
is
whether
or not
the step
representation of the COBOLT recion.
examining
the
ratio
c /c
since
the
This
successive
the series are a measure of the extent to which the solution
deviates from a plane wave.
This ratio,
c
1C
+2.
-
+ a
) + ia
(69)
1 + i2a
c
0
is
small
if
both a and
3 are
small.
These
parameters
are
indeed
small for the semidiurnal frequency waves since
c k/s
a a P/s
are
the
ratios
lengths.
result
to
within
the topographic
length
scale
tidal
the
does not
imply
characteristics
region
characteristic
of
that
very near
topographic
topographic
the topography
scale
is
to
change.
the
Off
tidal
the
the COBOLT moorings
Far
the
from
the coast,
for tidal models.
however,
will be
wave-
shore,
unimpor-
particularly
Tiana
Beach,
1/s = 5 km so variations
current meter records between
wall
to
The ratio (69) is estimated to be no greater than 0.03.
This
tant
of
(70)
shore
are
in
the
the
a possibility.
can be considered
a vertical
137
For perfect reflection
simplify consiAerably
ence
of
a
vertical
since
= I) the expressions
all
three are
number
and
its
term,
exp
i(kx-ot),
dependendent
quantities
(r'
and
v
is
complex
boundary model,
the
So,
900.
Again,
necessary
to
an
energy
provide
just
velocities,
phase and ellipse orientation angles
or
means
as
C
in
and
and u
the
v,
the
are
x
real
flat
bottom,
900
out-of-
are
are restricted to be either 00
absorbing boundary
the
differ-
Disregarding
that
u
equation (68)
the suri or
conjugate.
this
imaginary.
either
in
orientation
condition
(r'
needed
to
"tilt"
< 1)
fit
is
the
observations.
The
absorbing
shoaling water,
boundary
however,
condition
shows
that
angle
this
as
a
effect
much
more
effective
in
since the incoming wave crests are refracted
by the topography to parallel
orientation
is
the shoreline.
function
is
of
Moreover,
offshore
a plot of the
distance
only evident very near
(figure
to the
18)
shore--too
close to be detected by the COBOLT moorings.
A plot
absorbing
of
the
computed
boundary
topography
ellipticity
condition
is
also
(figure
needed
19)
in
the
to bring the ellipticity to the magnitude
observations.
Again,
the greatest
variations
shows
occur
that
an
presence
of
and sign of the
closer
to
shore
than could have been detected with the operational COBOLT moorings.
These computations
observed
also
in
support
crease
suggest
that
a region very close
the
inter-mooring
the statistical
the
strongest
(within 2-3 km)
averaging
confidence
in
the
used
variations
may be
to the coast.
in
chapter
tidal current
2
They
to
in-
measurements
138
00
-10
-20
-30
-40*
5
10
y (km)
Figure 3-18
Orientation angle versus offshore distance
for the depth dependent model
139
0.4
0.2
0
-.
02
=
45*
-0.4
SI
I
I
I
I
i
i
1
10
5
0
i
y (km)
Figure 3-19
Ellipticity versus offshore distance for
the depth dependent model
140
the
since
depth-varying
model
suggests
little
variation
in
ellipse
parameters beyond 5 km from shore.
The question
and whether it
cients
sy
=
the
inferred
great enough
can be answered
previously,
at
is
of where
in
to account
the reflection
(59).
this equation has a singularity at z = 0,
In
kept
for
part by examining equation
b,
which
lies
very
close
exactly at the shoreline for b = 1.
are
tidal dissipation might occur
finite
constants,
at
A and B,
this
energy be reflected
case
of
perfect
(the
reflection
the
i.e.,
by
by
Sommerfeld radiation
(r'
=
1) leads
to
a
the
that
on the other hand,
become
singularity,
infinite
at
the
surface
integration
all
incoming
condition).
standing
fairly uniform energy over the entire nearshore region.
purely progressive wave,
b < 1 and
for
and free
choosing
demanding
As noted
or equivalently
shore
The velocities
singularity
correctly;
wave
to
coeffi-
This
wave
and
Allowing
permits the velocities
which
in
turn
allows
a
a
to
large
energy level near to the singularity.
Examining
for several
the kinetic
different
energy as a function of offshore distance
values
of
r' confirms
this.
Figure
20
is
a
plot of
1
2
2
2 H (u + v2)
KINETIC ENERGY
computed
decreases
energies
for
the
depth-dependent
nearshore
for
r'
for
< 1 show
the
the
model.
case
The kinetic
of
presence
(71)
perfect
of
the
energy actually
reflection,
singularity
while
by rising
141
1.5
KINETIC
ENERG
1.0
1.0
I= -45*
0.5 - 0.9
0.8
I
A
I
I
I.
-I
I-
10
5
0
I
y (km)
Figure 3-20
Kinetic energy versus offshore distance for
the depth dependent model
142
slightly
very
close
to
the
shore.
The
depth
in
factor
(71)
diminishes this effect somewhat.
It
is
reasonable
to assume that dissipation rates are great where
the energy content of
is
not included
in
the wave
the model.
field
is
great,
even
The dissipation rate
though
in
friction
this instance
can be estimated using Taylor's (1919) dissipation equation,
DISSIPATION
Figure
21 shows
varying model,
uniform
rate,
the
Cd
U3 = Cd (u2 + v2)3/2
computed values
again for
offshore
as
=
rate of
at
about
cannot be detected
1.
function
values
The
r' <
1).
km
by
the COBOLT moorings.
times greater than the offshore
from
the
depth-
of r' and with a
dissipation
the shoreline only
This increase starts to be
5
distance
for
nearshore
a marked increase near
for imperfectly reflected waves
apparent
this
several different
dissipation
expected, shows
of
(72)
the
shore
and,
once
Furthermore,
rate and is
it
again,
is
many
certainly able to account
for the additional dissipation needed to explain the observed onshore
energy flux.
In
surface
contrast
to
(figure
22)
the
kinetic
does
not
shore for any of the realistilc
It
energy
and
show a rise
in
dissipation,
amplitude
values of the reflection
exhibits instead the linear trend characteristic
tides (Petrie, 1975).
with
coastal
the
near
free
to
the
coefficient.
o-f co-oscillating
The relatively flat wave amplitude also agrees
observations
and
implies
that
high
dissipation
could not be detected by observations taken from tidal
stations.
rates
143
0
5
10
y (km)
Figure 3-21
Dissipation versus offshore distance for
the depth dependent model
144
1.6
1.4
r
51.
O,0.9, 0.8
1.2
I= -45*
I
I
I
0
I
t
I
I
5
I
I~
I
I
10
y (km)
Figure 3-22
Free surface elevation versus offshore
distance for the depth dependent model
I
-
145
Retaining
appears
the
to
a real istic
confirm
COBOLT tidal
moorings
are
vertical-wall
topography
some notions
records;
minimal
coastline,
be possible in
shallow water
that
can
and that
the Laplace
that have
namely,
and
in
been advanced
variations
be
tidal
modelled
the inferred
equations
to
across
explain
the
three
by
effectively
tidal dissipation
a
may
inside the coverage region of the COBOLT
experiment.
J.
Summary
Three idealized analytical models have been examined ii
to
illuminate
coastal
the effects
region.
structure
friction
They were:
model;
vertical
of
a
a
reflected
coastline;
and,
on
tidal measurements
constant
Sverdrup
an effort
eddy
viscosity,
wave model
with
an
a reflected Sverdrup wave model
in
the
vertical
absorbing
vith simple
topography.
From the
first
of these models,
viscosity parameterization
description
of
the
of
vertical
ellipticity and orientation,
are
applied.
(1926)
must
layer
bottom
appears
that a constant
frictional
stresses
structure
of
provided
boundary
the
be modified
to
near the bottom.
the
law,
bottom
results
the observations.
which
for
the
presence
to
vertical
the
boundary
bottom velocity
profiles
which
agree
diurnal
and
in
no-slip
of
eddy
an adequate
proper boundary
condition,
A more appropriate
stress
in
\
account
allows
semidiurnal
investigation was the physically inadequate
relates
drag
The
it
conditions
Sverdrup's
condition,
a turbulent
wall
condition
which
through
a
quite
well
linear
with
146
a boundary which perfectly reflects
In the second model,
Sverdrup
tidal
was
waves
ellipse
not
capable
characteristics.
of
Specifically,
the correct sense of current vector
orientation
nearshore
or
the
effects
in
condition.
which
shallow water,
The
resulting
nearsbore
the model
rotation,
presence
dissipative mechanism,
frictional
boundary
angle,
explaining
barotropic
did not show
the correct major axis
of
onshore
is
due
was
incident
energy
to the
modelled
flux.
A
enhancement
by an
of
absorbing
Sverdrup-Poincare waves
success-
fully reproduced the features mentioned above.
Finally,
was
realistic
topography
found that the vertical wall geometry is
reality,
to
by including
the
if
it
is
stipulated that wavelengths
topographic scale and
the
the
in
a
third model,
it
a good approximation
are large in
to
comparison
region of interest is outside
this scale.
For the COBOLT region these conditions are met since the
topographic
length
suggests
scale
is
short;
that dissipation rates
about
5 km.
are large
within
and could account for the apparent
coastal region.
The
the
analysis
scale
also
distance
absorption of tidal energy in
the
CHAPTER IV
OBSERVATIONS OF COASTAL INTERNAL TIDES
A.
Introduction
With
the
ability
to measure
deep
ocean
realization that tidal frequency motions
the surface or barotropic tide.
tidal
currents
are not
has
come
the
exclusively due
to
Internal tides, or internal waves at
or near-tidal frequencies,
are
present
in almost
all
oceanic
current records and may even dominate velocity measurements at some
locations and frequencies (Gould and McKee, 1973).
By
obvious extension, much of the observational
literature
The
of
extent
internal waves
of
this
internal
waves
specific
to
by
the
material
Briscoe
internal
generation process,
is
applicable
is
evident
(1975).
tides,
In
much
to
in
the
a
internal
general
addition,
of which
and theoretical
the
is
is summarized in an excellent
tides.
review
of
literature
devoted
to
the
review by Wunsch
(1975).
Observations
e.g.,
Petrie,
on
1975,
the
continental
Halpern,
1971,
shelf and in shallow seas
Lee,
1971,
or Apel,
et. al.,
(see
1975)
show that internal tides are common--probably because nearby regions,
particularly the
are
generated
despite
slope,
(Prisenberg, Wilmot,
Sandstrom, 1962).
rare,
continental
are
areas
where
and Rattray,
internal
1974,
tides
and Cox
and
Current measurements near the coast, however, are
theoretical
interest
in
this so called "corner
region"
148
Measurements
(Wunsch, 1969).
that have been made (Winant,
primarily concerned with freely propagating waves.
motions,
such
as
those
of
the mid-latitude
observed near topographic features
are
Trapped internal
diurnal
such as
1974)
seamounts
tide,
have
(Hendry,
been
1975),
however, and presumably may be important near the coast.
The presence or absence of an internal tidal signal in the COBOLT
data
It
is
is
of relevance
well-known
source
that velocities
of confusion in
(Regal and Wunsch,
addition
mittent
both
to the barotropic
efforts
1973 and Magaard and McKee,
of internal
amplitudes
structure
vertical
of
to internal
and
the
internal
structure
of
tidal current
tides
phases
tides
are
a major
to interpret deep-ocean tidal currents
of energy to barotropic
nature
due
tidal analysis of chapter 2.
results
in these
tides
in
large
affect
tidal
Besides
estimates,
the
the
such
of
vertical
analysis
parameters
the
the inter-
uncertainties
Also,
estimates.
may
barotropic
1973).
of
the
as
those
a
brief
associated with the tidal ellipse.
B.
Dynamic theory of the internal tides
The
analysis
of
the
COBOLT
data
is
strengthened
The development
exposition of the theory relevant to internal tides.
of the theory here follows
the
subject
such
as
closely
Eckart
approximations
(1960),
involved
(1966).
The
equations
are similar to those made
(see chapter 3).
those given in
Phillips
in
standard
(1966),
formulating
for the Laplace
The primary differences
are
by
texts on
or
the
Krauss
model
tidal equations
that the fluid
is
no
149
longer considered to be homogeneous and the effects of a mean current
will be included in the dynamics.
Consider then a stratified ocean with a uniform mean current U,
in geostrophic
equilibrium, and
in hydrostatic
equilibrium.
case letters)
waves,
Small
from this mean state,
governed by
are
a mean density distribution p 0(z),
deviations
lower
such as those caused by internal
Boussinesq
incompressible,
linearized,
the
(indicated by
equations:
fk x v
5t+
Dt
p
=
az
Dt
V -v
a
where p is
density
is
density
to
equal
VP-
!
0
p
k
0
0
(1)
0,
perturbation
the
sum
such
of p and
that p << p0
p0 , k
is
and
a
the
unit
total
vector
pointing in the +z direction, and
_
D
Dt
a
at
+
U
(2)
ax
The equations which relate the velocity and pressure fields are:
D2
2
+*
D(
2+ f ) V =
Dt
1 (DV
h
- ( - Vh
0
(3)
and
2 + N2) W
2- +Nw
Dt2
where Vh is
the horizontal
1 D
Dt 3zp
0
gradient
'
operator D /D x i + a / y
the Brunt-Vaisala frequency is defined as
j
and
150
N 2 (z)
Combining
=
(1) into
-
a
g (p/az)/
single
p0
(4)
equation
in
the
vertical
velocity
gives
2
2
(
f2)
+ (
Dt2
az
Dt2
+ N2) V2 w
h
0,
=
(5)
with associated boundary conditions
w(z=0)
=
=
w(z=-H)
0
This "rigid lid" approximation removes
.
(6)
the surface wave
solutions
of
equation (5).
C.
Solutions for constant Brunt-Vaisala frequency
For
constant
2
N (z)
2
N
a
0
solution
to
equation
(5)
is
the
plane wave,
w(x,y,z,t)
where the wave vector
=
K
W0 exp i(kx+ky+mz-cot)
ki +
2 j + mk, and
measured by a stationary observer.
,
w is
(7)
the wave frequency
The boundary conditions demand a
discrete set of vertical wavenumbers m
mH
=
n
while substuting the solution (7)
sion relation,
(8)
i
into equation (5)
gives the disper-
151
((G- Uk)2_
denote this case)
sec
N
N2- Q- Uk)2)(k2
2
from a to
-1
the dispersion relation can be further
and
N
2
10
=
0
-4
sec
where
regions,
coastal
and
frequencies
tidal
2
2 =
the absence of a mean current (removing the tilda
In
for
2
-2
2
10-8
=
that W
assuming
by
,
simplified
2
<<
Then
0
(W2 _
2 (k2 + 2
2 =
2
The similiarity between this
surface gravity waves
(10)
0
dispersion relation and that for long
chapter 3)
(see
is
apparent
if
an "equivalent
depth"
(N /m ) 2 = (N0H
g h
mT) 2
(11)
is defined, reducing (10) to
W2
these equations
Because
+2 ).
solutions
(12)
are similar to the Laplace tidal equa-
any solutions implicit in
tions,
as
= g h (k
2
the tidal equations can also appear
In particular, there
of the internal wave equations.
are free and trapped modes that are equivalent
to the Sverdrup
Kelvin wave solutions of the Laplace tidal equations.
is clear
numbers
case
that
k
for w2
and P,--both
both k2
and
2
2
two
possibilities
demanding
are positive
that
k 2+
2
From (12)
and
it
exist
for
the wave-
> 0.
In
the
first
and freely propagating internal
152
waves
result.
In
the
second
case k2 < 0
1
and
<1k 21
imply
exponential decay or growth for the solution (7).
2
For w2 <
Ik21 --implying
freely in
coast.
only
one
waves
of
possibility
this
frequency
the y-direction but are trapped
In
addition,
the
exists--
wavelengths
range
1221
£2 < 0
and
do
propagate
not
to a boundary such as
of
both
free
and
the
trapped
internal motions are considerably smaller than the equivalent surface
waves since
g h
n
Taking
.
N
2
0
=
10
that first mode
about
2
=
-4
(N H/nl) <<g
o
sec
(n =
-2
and
H
.
=
(13)
30
1) internal waves
10 km--a factor of
100 less
m,
for
example,
should have
suggests
wavelengths
than the equivalent
of
surface wave-
lengths.
D.
Solutions for an arbitrarily stratified fluid
For an arbitrarily stratified fluid equation (5) has a solution
w(x,y,z,t) = W(z) exp i(kx+ky-wt)
,
(14)
if (with w2<< N2 and notation of equation (16) retained)
d2W
N2(z) K2
+
2
2
dz 2
- f2
(15)
0,
W
where K = ki + tj is
now the horizontal wave vector (1K
= K) and the
boundary
of
a
conditions
equation
(6)
remain.
This
is
classical
>
153
eigenvalue problem whose solutions are a countable number of eigenfunctions,
Fn(z),
eigenvalues
and
each
with
an
associated
eigenfunctions,
for
eigenvalue
arbitrary
Kn.
N2(z),
These
can
be
computed numerically by one of any number of integration techniques.
One such method,
Krol
(1974).
a matrix diagonalization technique,
The
rigid
lid boundary
conditions
is
also
described by
assure
that
eigenfunctions are orthogonal,
K2
Kn
H
=(--nP)
N2(Z) W2( z) dz
2
2
f
2
J )dW
H
n
0
dz
dz
0
H
N2(z) Wm (z) Wn (z) dz
f
=
(16)
0
0
dW
dW
H
m
n
dz-
0
=
for
m
n.
0
Other perturbation fields are related to the eigenfunctions by
n
(Z)
wk + 2i f dWn
dz
=
n
v (z)
n
n p
1
dW
n
- i fk
wK 2
dz
n
.
o-
.w
2
nn
f
2
n
2
(17)
dW
n
dz
N2
pn (z)
= i p
0
g W
W(z)
n
154
Important
relative
relationships
phases
between
developed from (17).
defining
the
the
various
amplitude
measured
ratios
quantities
and
can
be
For trapped waves, where Z2 < 0,
u
n
_
v
o
k + fX
+ fk
2
p
pKN
on
2 W
n
'
dz
where the decay parameter has been redefined as:
able equations for free waves
(18)
ndWn
k + fA1
un
X = -i2..
are found by taking
Compar-
X= 0 since
the
direction of propagation is arbitrary.
The phase
propagation
relation
(u) and that
types of waves,
0
0
or
180
o
between
while
depending
the
velocity
perpendicular to
in
the
direction
it (v) is
900 for both
the phase relation between u and
on
the
sign
of
dW /dz.
n
of
These
p is
either
phases
are
an important consideration in determining the propagation direction
of the internal wave.
The energetics of an internal wave field are examined by multiplying the momentum equation by v, the continuity equation by -p,
the density conservation
equation by pg/N ,
and adding
and
the results
to obtain the energy conservation equation,
Po D
2
2
2
2 Dt (u + v + w
Making
vertical
the
assumption
kinetic
2 2
+ ( pg/0 ) IN ) +V-pv
that w2<< N2
energy,
w2.
allows
Integrating
one
(19)
0.
to
(19)
disregard
over
the
the
water
155
column
and
for free and trapped waves
slightly different expressions
results in
are substituted into (19):
when the relations of equation (17)
2
2
(<u > + <v >)
PO
=
KINETIC ENERGY
-
2
4
W2 K 2 f
dz
dz
dz
+ (
f
+fk)
2 4
4
FREE
2
2
_1 (ok +fX)
dz
2
dW
2
1
2
dW
dz
(_n)
dz
=
4o2
ENERGY FLUX
=
0 2
of
that there
(k i +
1
wK2
po
internal
is
) >/N
waves.)
N
ratio
(20)
FREE
TRAPPED
W
FREE
dz
n
2
wk + fX
KN2
2K
The
=
=
kj)
no vertical
dz
&
<py>dz
/
=
=
(Note
<(g P /
N 2 W2
nd
1
TRAPPED
2
p2
POTENTIAL ENERGY
brackets < >)
by
(indicated
average
time
a
performing
2
Wn
energy flux in
of
TRAPPED
dz
potential
the modal description
to
horizontal
kinetic
energy, another important diagnostic quantity, is
P.E.
K.E.
_
w2 _
2
+
2
FREE
F(
(21)
156
(W _
)(k2 _
)
2
2
W
(cok + fX) (coX + fk)
TRAPPED.
(21)
cont.
Finally, the total energy for both wave types is
E = K.E. + P.E. =
1
2(w 2_f
TN WR dz
2
)
FREE
(22)
22
_I (wk+fX) + (wX+fk)
4w2 (
+
N2 W2 dz
TRAPPED.
K2 (W2_ 2)
This quantity, the total energy per unit surface area, is that energy
carried along by a wave packet.
p v
=
Consequently
E C,
(23)
4
where C is the group velocity,
C
C
=
3k i +W/UZ
A
A
(24)
.j+Dw/@m k
E. The mean fields of the COBOLT experiment
The average sigma-t cross section formed from the twenty-two days
of profiling
figure 1.
The most noticeable feature is
a fairly distinct pycnoBecause this
cline about 12 m deep which broadens toward the shore.
feature exists in
the individual daily cross sections,
able to assume that this reduction of stratification is
physical process
(such as the enhanced mixing) and is
artifact of the averaging procedure.
shown in
is
that coincided with the buoy measurements
it
is
reason-
due to some
not solely an
MAY 1977 AVERAGE
0
24.8
8_25.2
16
24.6
25.0
25.4
2 5.6
25.8
5. 24
26.0
32
40
Ut
0
5
10
km
Figure 4-1
Twenty-two day average sigma-t cross section at the COBOLT site
15
158
salinity, and sigma-t at the
The numerical values of temperature,
location
listed
of spar buoy 3 are
a profile
representative
as
in
day
twenty-two
the
possible
To determine
1.
table
as
average
profiles of the two transect stations located on either side of a
given mooring (averages
were also performed for buoys 2 and 4) were
Also included in the
combined to make the estimates in the table.
table are the standard deviations of the averaged temperatures and
salinities.
These profiles suggest that density during the month of May was
primarily controlled by the salinity.
This can be checked quantita-
tively by noting that the partial derivatives of density with respect
to temperature and salinity (obtained directly from the equation of
state) are quite different, i.e.,
= 7.5 x 10- 4
ap
as
=
DT
the
10 -4
3
(25)
3
10
c.3
./0
cm o/oo
At buoy 3 a temperature contrast across
of about 30 results in a density change of 4.2 x
thermocline
halocline
-1.4 x
and S = 32 o/oo.
gm/cm3.
3
cm 0/00
ap
at T = 8
gm
The
giving
a
salinity
change
was
density
change
of
about
7.5
twice as great as that due to temperature.
x
1
o/oo
10~4
across
the
gm/cm 3--almost
The late-spring measure-
ment period and the large amounts of fresh water discharge from the
Connecticut River (Ketchum and Corwin, 1964) are probably responsible
for this result.
159
TABLE 4-1
NUMERICAL VALUES OF T, S, SIGMA T, AND N2
AT BUOY 3
DEPTH
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Standard
deviation:
TEMP.
SALIN.
( 0C)
(0/ 0)
10.71
10.68
10.60
10.50
10.39
10.26
10.08
9.86
9.63
9.40
9.17
8.94
8.70
8.46
8.25
8.07
7.92
7.80
7.67
7.56
7.47
7.40
7.32
7.23
7.15
7.08
7.02
6.97
6.91
6.86
6.84
31.96
31.96
31.97
31.98
31.99
32.01
32.05
32.09
32.15
32.21
32.28
32.36
32.44
32.52
32.59
32.65
32.70
32.75
32.78
32.81
32.85
32.88
32.90
32.92
32.94
32.95
32.96
32.97
32.99
33.01
33.02
1.5
0.1
SIGMA T
24.46
24.46
24.48
24.51
24.54
24.58
24.63
24.70
24.78
24.87
24.96
25.06
25.16
25.26
25.34
25.41
25.48
25.53
25.58
25.62
25.65
25.69
25.72
25.75
25.77
25.79
25.81
25.83
25.85
25.87
25.87
N2
PERIOD
(SEC-2)
(MIN)
0.73x10~4
1.78
2.27
2.85
3.99
5.49
6.86
7.86
8.57
9.14
9.60
9.75
9.32
8.37
7.23
6.17
5.27
4.54
3.97
3.58
3.34
3.11
2.76
2.30
1.87
1.67
1.76
2.01
1.89
0.81
12
8
7
6
5
4
4
4
4
3
3
3
3
4
4
4
5
5
5
6
6
6
6
7
8
8
8
7
8
12
160
Also included in table 1 (and displayed in figure 2 for all three
mooring
The
locations) are
equivalent
the values of
periods
less
.quite short--generally
are
frequency.
the Brunt-Vaisala
ten
than
minutes--implying that resolution of high frequency internal waves is
not possible because of the one hour buoy averaging period.
high
frequency
internal
waves
will
alias
not
the
However,
low
frequency
signals either.
The COBOLT instruments also recorded significant mean currents.
Onshore mean currents are generally quite small (less than 2 cm/sec)
The alongshore
and can be ignored as an influence on internal tides.
mean currents,
intervals,
in
presented
table
two
2 for
averaging
different
are substantially stronger than onshore currents and more
comparable to
the phase speed
of internal
flowed to the west during the experiment,
These
waves.
and,
currents
given the distance to
Montauk Point and the entrance to Long Island Sound (60 km),
suggest
an upper layer advective time scale of 7-14 days.
Although the measurement period over which these averages were
made was quite variable,
been included.
Figure 3,
showing the salinity time series of buoy 3
at 3.8m and 25.Om (instruments 31
changes measured during the
crossed
the
site
causing
the
Higher salinities
approximately
ten days,
from
about
32.5
and 34),
experiment.
homogeneous.
sharply
that the major time scales have
appears
it
water
to
the abrupt
On May 10 a large
column
20,
when
31.7
o/oo.
storm
to become practically
then persisted at the
until May
o/oo
illustrates
the
surface
salinity
This
for
changed
freshening
is
161
N
(sec-2 )
10
15x10
10
E
15
0
C
20
25
30
Figure 4-2
Twenty-two day average Brunt-Vaisala frequency
profiles at bouys 2, 3 and 4
162
TABLE 4-2
MEAN ALONGSHORE CURRENTS
FOR BUOYS 2, 3, & 4;
MAY 1977
CURRENTS AT BUOY NO.
2
3
4
APR. 30-MAY 15
LEVEL
-
-4.5 cm/sec
2
-6.2
3
-2.8
-6.6
-7.4
4
-0.6
-2.4
-2.6
-10.5
APR. 30-MAY
LEVEL
-11.9 cm/sec
1
-
1
-3.2 cm/sec
2
-5.6
-8.2
3
-2.9
-5.0
4
-1.2
-2.3
*Negative values are to the west
-10.2
33.5
33.5
33.0
33.0
32. 5
32.5
9..
z 32.0
32.0 z
cc
an
31.5
31.5
31.0
31.0
Figure 4-3
Salinity time series from instruments 31 and 34
164
due
probably
from Long Island
an influx of water
to
nearest and most logical source of fresh water).
Sound
(the
Furthermore, the
ten day time estimate obtained from the salinity series agrees with
the advective time scale suggested by the mean currents and supports
the notion that a thirty day averaging period is reasonably representative.
F.
Internal tidal oscillations
COBOLT
The
internal tides.
experiment
is well-suited
for
observations
of
the
The three kilometer spacing of moorings was intended
to provide a coherent array for internal (albeit very low frequency)
While the orientation of the transect line (i.e.,
motions.
to
dicular
shore)
limits
sensitivity
directional
to
perpen-
the offshore
direction, the propagation of free waves is probably biased in this
direction by the local topography.
The sigma-t time series of all four instruments on buoy 3 (figure
4) shows
plainly
the
variations
of
interest
here:
the
internal
tides.
The large regular oscillations at instruments 32 and 33 are
typical
of measurements
tions
at instruments
near
the pycnocline while
smaller oscilla-
32 and 33 indicate reduced stratification and
adjustment to top and bottom boundaries.
To identify
the major periodicities of pycnocline oscillations,
energy spectra of sigma-t time series were computed.
the averaged
sigma-t
Figure 5 shows
energy density of instruments 21-23 and 31-33
for the twenty-five day period over which both buoys were operational
26.5
26.5
26.0
- 26.0
2.4
m
25.5
- 25.5
25.0
25.0
a- 24.5
24.5 a-
(f)
0
24.0
24.0
23.5
-
I
01
77
-
I r-
06
11
r1
AA
gureu
-
oVaL61t:bVt
I i
cI
16
-
6
I
21
Ld I
levelsbuo
26
3UY%
23. 5
166
o-t
100
D I SD
A
10-1
10-2
10-3
195%
0.01
0.10
1.00
FREQUENCY (cycles/hr)
Figure 4-5
Sigma-t energy density spectrum from insts. 21-23 & 31-33
167
(April 30-May 24),
with the values of diurnal
inertial (18.35 hr),
for reference.
and semidiurnal (12.42
hr)
(D = 24.00 hr),
local
included
frequencies
These instruments were selected as representative of
upper layer and pycnocline fluctuations.
The internal
oscillations
have two resolved maxima at which energy rises above the background
level:
a broad peak centered around the
inertial frequency and a
sharp peak centered on the principal semidiurnal frequency.
the
"inertial"
frequencies,
it
peak
is
so much
contains
almost
broader
three
than
times
accounts for most of the regular oscillations
figure
4.
By
contrast,
the
temperature
that
as
at
much
Because
semidiurnal
energy
and
that catch the eye in
spectrum
from
the
same
instrument packages (figure 6) does not show any energy significantly
above the continuum.
Further insight into the nature of the internal oscillations can
be obtained by examining the kinetic energy spectrum.
A comparison
of the energy density of onshore velocities (figure 7) with that of
alongshore velocities (figure 8) shows a marked disparity.
energy
is much
less
than
alongshore
semidiurnal frequency ranges,
In fact,
alongshore currents
energy
for both
Onshore
diurnal
and
but not for frequencies near inertial.
are almost ten times as
energetic as
onshore currents for the semidiurnal band and almost three times as
energetic for the diurnal.
Currents in
the inertial
range,
on the
other hand, have comparable energies in both directions.
A convenient way
to characterize
the
sigma-t variations is
to
convert the energy in a particular frequency band to an equivalent
isopycnal displacement using the differential
168
T
101
D I SD
100
10-
10-2
195
0.01
0.10
1.00
FREQUENCY (cycles/hr)
Figure 4-6 Temperature energy density spectrum from insts. 21-23 & 31-33
169.
PERID
9
HRS.
10
+4100
10 ii
(II
ED
H-
CrN
+3
10
CD
+2
10
cn
LLJ
Z
11
10
cz
CD
+0
10
Figure 4-7
Onshore velocity
energy density
spectrum from insts.
32 and 33
1 1-1
-I
II
FUN0.01
0.1
FREQUENCT,.CYCLES/HRS.
170
PERIDD,
+4100
10 11
1i
HRS.
10
1
(NJ
CL
LL
(iT
N
(NJ
+3
10
(I
10
LJ
c>
Lu
+1
10
LJ
Li
--
+[)
10%-1
Figure 4-8
Alongshore velocity
energy density
spectrum from
insts. 32 and 33
1 0-1I
'.0 1
O. 01
0.1
FREQUENCY, CCLES/HRS.
171
Aa
These quantities
= Da/Dz Az
=
N2 AC .
are averaged .for buoys
(26)
2 and 3 at each instrument
level and presented for the three periods of interest in table 3 with
the isopycnal displacement expected for a long surface wave of 100 cm
amplitude.
This last item is
velocity decreases
computed by assuming that the vertical
linearly to zero at the bottom and
that it
is
equal to the time derivative of the displacement, i.e.,
w(z) = aw (H-z)/H
and
(27)
;/t
where a is
= w(z),
the amplitude of the surface wave,
?
is the isopycnal
displacement, and H is the depth of the water.
Since the semidiurnal
surface tide has an amplitude of about 1 m in
the COBOLT region and
the diurnal surface tide an amplitude of about 10 cm,
these numbers
give a fair indication of the isopycnal displacements due to surface
tides alone.
A comparison of the measured and computed displacements shows to
what degree the surface tide can account for the sigma-t variations
at each frequency.
In the semidiurnal band displacements due to the
surface tide and those computed from density variations are virtually
identical (except
into
for instruments at level 4 where energy peaks fade
the continuum) implying
that
the narrow energy peak at
this
172
TABLE 4-3
ISOPYCNAL DISPLACEMENTS
FOR BUOYS 2 & 3;
ALL LEVELS
INST
N2
12 HR.
18 HR.
24 HR.
(cM)
(CM)
(CM)
3.9x10- 2
70
2.3x10-2
73
2.4x10- 2
45
2.8
79
4.9
11
4.4
4.8
50
2.4
100
4.8
52
2.5
24
2.4
96
2.3
67
1.6
42
1.0
31
2.3
170
4.0
130
3.1
91
2.1
32
6.9
62
4.3
78
5.4
78
5.4
33
7.2
60
4.3
86
6.2
60
4.3
34
2.0
180
3.5
65
1.3
110
2.2
21
3.3x10 4
22
6.2
23
102
DISPLACEMENTS FROM SURFACE WAVE
WITH 1 METER AMPLITUDE
DEPTH
AC~
1
4 m
87 cm
2
8
73
3
16
53
LEVI L
173
is almost
frequency
tide.
The
greater
exclusively
band
diurnal
can be
than
a
consequence
however,
displacements,
expected
from
is
there
reasonable
to
no
assume
that
horizontal
barotropic
5-10
are
times
tide.
And finally,
it
displacement
around
18
hours,
this
energy
is
baroclinic
all
of
and
is
in
the large horizontal scale (barotropic) and
In this sense
nature.
small
surface
the
surface wave contributions
therefore must be in part due to the baroclinic
since
of
(baroclinic)
scale
fluctuations
are
sorted
by
frequencies in the May experiment.
Unlike the May
This sorting of dynamics does not always occur.
1977 experiment, where kinetic energy is found in a broad band at and
above
diurnal
density for
spread
frequencies,
the September
a comparable
1975
spectrum
(figure
experiment
fairly symmetrically about 24 hours,
inertial frequency.
of kinetic
9)
shows
energy
energy
and not as high as the
The semidiurnal energy peak is again very sharp,
as in May, 1977, and is centered at 12.42 hours.
G. Modal structure of the internal tides
It
flows,
is
possible to discriminate between barotropic and baroclinic
and between the
examining
the
modal
different modes
structure
of
density distribution (see table 1).
three
vertical
velocity
density distribution at
(1974).
These
modes
the
(figure
10)
the
computed
currents
Consider,
eigenfunctions,
buoy 3 by
of baroclinic motions,
for example,
Wn(z),
computed
from
by
the
the first
from
the
procedure outlined by Krol
generally have
large
amplitudes
lwm'
j
174
PERIOD,
1
U I
10
100
I
HRS.
I
I
I
C\i
C
IL
162
Z
cc
LL)
w
CD
HU
L-
1 1
Figure 4-9
10
Kinetic energy
density at 15 m
for September,
1975 experiment
F-I
0.01
0.1
FREQUENCY, CTCLES/HRS.
175
VERTICAL
MODES
Figure 4-10 First three vertical velocity modes at buoy 3
176
where N2 is
large,
toward
top
the
i.e.
and
in
the pycnocline,
bottom,
where
velocity adjusts to the boundaries.
N2
and smaller amplitudes
weakens
and
the
vertical
These are the features that are
evident in the sigma-t variations of the COBOLT experiment.
The
dW /dz,
n
horizontal
velocity
modes,
are shown in figure 11.
which
are
proportional
to
Unlike vertical velocities, which
must be extrapolated from other fields, the horizontal velocities are
directly measured quantities
and can be used in a straightforward
manner in interpreting the distribution of energy among the modes.
The COBOLT data were analyzed by fitting the calculated eigenfunctions to the observed velocities in a least squares sense.
a
continuous
ments
u. at
eigenfunction
each
U (z) =
n
M different
of
dW /dz
n
points
Given
and velocity measurein
the
vertical,
it
is
possible to determine a coefficient an, for each mode, such that
( u.
i
is a
minimum.
The
-
aU (z.))2 h.
n n
i
(8
i
(28)
factors,
weighting
h.,
trapezoid integration rule to favor instruments
vertical range.
are
that
chosen
by
the
cover a large
Minimizing (28) with respect to an determines
the
values of the coefficients,
Nh.u.U (z.)
ai
n
i(29)
i n
h.U2 (z
i=1
i
n
1
Because the eigenfunctions have arbitrary amplitudes,
a more useful
177
HORIZONTAL
Figure 4-11
MODES
First three horizontal modes at buoy 3
178
quantity is the ratio of the fitted eigenfunction variance to the
observed variance,
M
Pn
2 2
h.a U (z.)
i n n2
i=1
(30)
11
which is expressed as the percentage of observed variance (or energy)
that can be accounted for by a fit of the nth eigenfunction.
The results of the eigenfunction analysis and mode fitting for
three different frequency bands are summarized in table 4. With only
four instruments measuring velocity (three
than four modes is not possible.
(n = 0) and
barotropic
first
at buoy 3) a fit
In practice it
two
baroclinic
accounted for virtually all the variance,
of more
was found that the
modes
(n =
1 & 2)
so fitting of higher order
modes was not necessary.
Table
4
confirms
analysis by showing:
largely barotropic;
clinic; and,
also shows
the
results
of
the
isopycnal
displacement
that the vertical variance in the 12 hr band is
the variance in
the 18 hr band is
largely baro-
that the variance in the 24 hr band is mixed.
that onshore velocities
at
The table
12 and 24 hr have a higher
percentage of variance in the baroclinic modes
than do alongshore
velocities.
The mode
fitting of
assertion that energy in
18 hour
variance
this band is
is consistent with
the
internal tidal energy that has
been Doppler-shifted away from the diurnal band,
since 80-95% of the
variance at all three COBOLT moorings can be attributed to the first
179TABLE 4-4
PERCENTAGE OF VARIANCE IN
BAROTROPIC AND FIRST TWO BAROCLINIC
MODES
% AT
12 HR.
B
U
M
0
0
D
Y
E
NO.
EA.
2
0
55
95
16
2
2
0
0
0
81
96
1
3
1
6
1
3
4
0
% AT
18 HR.
WAVE
% AT
24 HR.
LENGTH
(KM.)
12
HR.
24
HR.
33
12
17
0
1
5
7
4
90
64
92
82
4
28
15
22
0
1
6
2
3
6
9
2
4
3
11
71
51
NO.
EA.
T.
NO.
EA.
1
0
1
5
87
63
3
97
99
98
90
9
0
0
0
0
2
2
0
1
1
89
95
2
2
T.
90
95
86
T.
44
94
33
1
6
81
81
81
55
9
24
16
23
1
0
0
0
0
0
1
0
0
7
9
NO.
= Onshore component
EA.
= Alongshore Component
TOT. = Total variance
180
baroclinic mode.
mode,
The 18 hr velocities at buoy 2 and the fitted first
shown together
in figures
12
and
13,
visually
confirm
the
baroclinic nature of this freqency band.
The
modal
eigenvalue,
analysis
also
supplies
Kn = (27r /wavelength).
the
magnitude
These wavelengths
(see
are approximately 15 km for the first internal mode in
of
the
table 4)
the COBOLT
region, and around 5 km for the second internal mode.
For trapped waves, more information is needed to determine the
wavelength--i.e.,
there must
be some method of choosing the decay
scale or e-folding distance.
Traditionally the Kelvin wave problem
is
modelled
boundary
in
an ocean
condition.
with
Equation
a
vertical
(17)
wall
shows
and
that
no-normal-flow
this
condition
(assuming y is the onshore direction),
v
&w-
ifk
-i (wX + fk)
(31)
,
demands that
A =-fk/w,
(32)
so that the eigenvalue is
K2
n
W2_f2
o -f
As
a result,
Kelvin wave
the wavelength of
k2
n
W2
(33)
a first mode semidiurnal
internal
is almost half that of a free wave. In comparison, a
first mode diurnal internal Kelvin wave has a wavelength of around 20
km.
Furthermore,
it
is
necessary,
in
order
to
have
exponential
offshore decay, for these wave to propagate alongshore to the west
(the negative x direction)--the same direction as the mean current.
181
-4
4 cm/sec
-2
Figure 4-12
Fitted first baroclinic mode at buoy 2:
onshore velocities
-
-ffi~
182
-4
-2
Figure 4-13
0
2
Fitted first baroclinic mode at buoy 2:
alongshore velocities
4 cm/sec
183
H.
Comparison to theory
This
subject
last
fact
to Doppler
Kelvin wave
makes
the
shifting.
internal
The
Kelvin
frequency
wave
particularly
of a diurnal
internal
superimposed on a mean current of 10 cm/sec would be
measured by a stationary observer as
w
=
+ Uk = 1/24 hr + 10 cm/sec x 1/20km
= 1/17 hr
This
estimate, using
(34)
.
realistic
values
for
all
of
the
parameters,
results in a Doppler-shifted frequency that is remarkably close to
the
sigma-t energy maximum observed in the COBOLT data.
In fact,
smaller amplitude mean currents would bring the frequency estimate
more into line with the observed sigma-t energy peak at 19-20 hour
periods.
This evidence again favors the hypothesis that energy peaks
at near
inertial frequencies in the May 1977 data are a result of
Doppler-shifted internal tides of diurnal period.
Vertical coherence at 18 hour periods is high among the COBOLT
instruments
confidence
(always
level)
significantly
so
computed
accurately.
equation
(18).
u-velocities
phases
These
An average
different
between
phases
of
(alongshore east)
all
from
measured
agree
with
COBOLT
zero
quantities
the
50, indicating a clockwise rotation of ellipses.
the
99%
can
be
predictions
of
instruments
lag v-velocities
at
shows
that
(onshore) by 910 +
Also,
lower layer
184
(level 3) alongshore
variations
by
velocities lead
1700 + 10 --again
in
thermocline (level 3) sigma-t
close
agreement
to
predicted
values.
Horizontal
coherence
periods and phases
between
buoys
is also
generally small (less
high
at
18 hour
than 200) indicating that
the wave crests are parallel to the mooring transect (perpendicular
Phase differences
to the shore).
that do exist can be explained by
considering the different mean alongshore velocities at each of the
moorings.
The
comparison
of
observations
boundary Kelvin wave model
condition used
demands
that
Furthermore,
to choose
fails in
onshore
the
the
the
idealized
certain respects.
vertical
The boundary
offshore decay scale (equation (33))
velocities
magnitude
with
of
be
the
identically
decay
scale,
zero
everywhere.
using appropriate
values of W, f, and k, suggests that the e-folding distance should be
less
than 3 km.
velocities
Observations, by contrast,
are comparable
to
alongshore
indicate that onshore
velocities
(u/v ~ 0.9)
and
that their magnitudes do not show any measureable decrease offshore,
even out to 12 kilometers (buoy 4).
It is apparent from the discussion that led to equation (34) that
the absence of onshore currents and the choice of a decay scale are
both a consequence of the no-normal-flow boundary condition.
of the relatively gentle bottom slope (see chapter 3),
In view
this boundary
condition and the traditional Kelvin wave model are probably inappropriate for the COBOLT region.
The pycnocline intersects
the bottom
185
several
kilometers
from
the
and
shore
suggests
that
the boundary
condition may be modelled more correctly by demanding that
v dz
=0.
(35)
already met by the baroclinic modes and cannot be
This condition is
Unlike the Kelvin wave,
used to determine a decay scale.
the integral condition does
allow onshore velocities
however,
(see equation
(18)) for decay scales other than that obtained in equation (32).
Although the traditional, vertical boundary, internal Kelvin wave
model
fails
for
to account
possible that similar,
observations.
Also,
some of the
observed
trapped-wave dynamics are responsible
because
of
the
strength
and
it is
features,
for the
persistence
of
coastal mean currents, it is reasonable to assume that the broadening
of
the
kinetic
energy
of
around
surface
diurnal
and
frequencies
Doppler-shifted
is
due
to
internal
the
tidal
combined
presence
motions.
Wunsch (1975) suggests that a broadening of energy peaks is
one of the noticeable features of the internal tide,
and that it
can
be used to distinguish the respective contributions of barotropic and
baroclinic tides to current meter records.
peak
in
both
fall
and
spring
measurements
The presence of a broad
suggests
a
persistent
generation mechanism such as the barotropic tide since other possible
generation processes
(e.g., wind stress) are intermittent and quite
different for the two seasons.
186
Despite
these arguments, however, it is difficult to establish
unequivocally the reality of the Doppler-shifting mechanism without
examining other
effects
which may be
direct forcing, mean shear,
topography,
important.
etc.,
Factors
such
as
may be responsible for
the unusual results of the May, 1977 experiment.
Only more inclusive
models and further examination of the data will resolve this question.
I. Energy and flux of the internal tide
The energy content of the internal diurnal
the aid of equation (24).
tide is
examined with
From table 4 (assuming that the Doppler
mechanism is operating) it is apparent that not all of the diurnal
shifted to the 18 hour band,
internal tidal energy is
since 24 hour
period velocities still show a substantial amount of variance in the
first
baroclinic
mode
(about
30%).
It
is estimated
that
this
contribution is about one-fourth of the contribution from the 18 hour
band and is
ignored in the following calculations.
Depth integrated
values of potential and kinetic energies for the 18 hour band are
averaged for buoys 2-4 to give
This is
the
K.E.
=
33 ±5
P.E.
=
5 ± 1
Joules/m 2
Joules/m
2
(36)
.
roughly half of the energy content that can be computed for
surface
diurnal
tide
and less
surface semidiurnal tide.
The ratio of energies is
than one-tenth
of that
of the
187
=
P.E./K.E.
0.15 ± 0.05
,
(37)
again in contrast to the Kelvin wave model.
decay scale in equation (21)
Using the Kelvin wave
suggests equipartition of energy (i.e.,
P.E./K.E. = 1) while a larger decay scale, more in line with the
observations, gives an energy ratio of less than one, as observed.
It is also possible to perform a crude energy flux calculation.
Unlike
ties
the surface wave flux calculation,
(free
surface
computation,
the
knowledge
of
equation
(23)
elevation
and
flux calculation
the
dynamics.
can be
For
used
if
velocity of the wave is not
velocities)
for internal
a
it is assumed
quanti-
were
used
waves
requires
shore-trapped
that
for
some
internal
wave,
the
group
the
i.e. C = 20 km/24 hr = 23
In this case
Energy Flux = 9 watts/m
alongshore to the west.
While this figure is
(38)
small with respect to
computed surface semidiurnal flux rates (chapter
to
the
too different than the phase velocity
(they are identical for the Kelvin wave);
cm/sec.
where measureable
deep water
internal
tide
2)
it
is
comparable
fluxes measured by Wunsch and Hendry
(1972).
If the internal wave
west,
the
Topographic
source
of
features
this
is assumed to progress
energy
is probably
at the entrance
alongshore to the
Long
Island
Sound.
to the Sound itself are quite
188
pronounced
and
undoubtedly
provide
the
correct
length
scale
for
The alongshore length scales of topographic features to
generation.
the south of Long Island,
by contrast,
are not well-matched
of the internal tide, but are generally much longer.
to those
Thus a topo-
graphic generation process such as that proposed by Baines (1973)
is
more likely to occur at the entrance to the Sound than locally along
Furthermore,
the South Shore.
wide with respect
to
the entrance to Long Island Sound is
the internal
tide wavelength and undoubtedly
will prevent any transmission across from the southern coast
England (see Buchwald, 1971).
of New
It is also possible, though the matter
is open to speculation, that the semi-permanent density front known
to exist where the fresh waters of the Sound come into contact with
the
saline waters
of
the Mid-Atlantic Bight,
play
a role
in the
generation process.
It
is not
likely that
the
internal
tide
evident
periods results from generation in Long Island Sound.
at
12 hour
Internal waves
at semidiurnal frequencies are free waves and are therefore able to
radiate away from the .generation region in
likely source
would be
onshore generation
from offshore
regions.
Because
barotropic currents, analysis of
difficult.
nearshore
(e.
all directions.
g.,
A more
the shelf break)
or
dominated
by
the records
are
the internal oscillations
is very
Even so, there are indications in the COBOLT data that
density
flucuations
generation in the coastal zone.
lead
those
further out;
evidence
of
Until longer records are available,
it is not feasible to resolve this question fully.
PAGES (S) MISSING FROM ORIGINAL
190
J. Conclusions
effects
The
the
of
internal
tides
chapter 2 should be clear at this point.
little
baroclinic
analysis.
energy present
Onshore velocities,
on
the
Diurnal velocities,
of
Semidiurnal velocities show
to interfere with
the
barotropic
where the baroclinic effects were the
strongest, were indeed subject to the most variations
2).
analysis
tidal
during the May 1977 experiment,
(see chapter
experienced
a bit more interference from internal tides but not nearly so much as
might have occurred in
the absence of
a mean current.
In either
case, the fact that baroclinic variance is primarily in the first
mode promotes the success of the vertical integration as a way of
reducing the effects of the baroclinic tides on the results of the
barotropic analysis.
REFERENCES
Apel, J. R., H. M. Byrne, J. R. Proni, and R. L. Charnell. 1975.
Observations of oceanic internal and surface waves from the
Earth Resources Technology Satellite. Journal of Geophysical
Research. 80:865-881. .
Abramowitz, M. and I. Stegun. 1964. Handbook of Mathematical
Functions. National Bureau of Standards, Applied Mathematics
Series 55.
1046pp.
Ball, F. K. 1967.
Edge waves in
Research 14:79-88.
a ocean of finite depth.
Deep-Sea
Baines, P. 1973. The generation of tides by flat bump topography.
Deep-Sea Research. 20:179-205.
Beardsley, R. C., W. Boicourt, L. C. Huff, and J. Scott.
1977. CMICE
76: A current meter intercomparison experiment conducted off
Long Island in Feb.-Mar. 1976.
Woods Hole Oceanog. Inst.
Technical Report WHOI 77-62:
123 p.
Beardsley, R. C., W. Boicourt, and D. Hansen. 1976. Physical
oceanography of the Middle Atlantic Bight.
in Middle Atlantic
Continental Shelf and the New York Bight. vol 2. M. Grant
Gross, ed.
American Society of Limnology and Oceanography.
p.
20-34.
Bendat, J. S. and A. G. Piersol. 1971. Random Data:
Measurement. Wiley-Interscience, N. Y.: 407 p.
Bowden, K. F. 1962. Turbulence. The Sea. Vol. 1.
Hill. Interscience Pub., N. Y.: 802-825.
Analysis and
Ed. by M. N.
Bowden, K. F. and L. A. Fairbairn. 1956. Measurements of turbulent
fluctuations and Reynolds stresses in a tidal current.
Proceedings of the Royal Society of London. A 237:422-438.
Briscoe, M. 1975. Internal waves in the ocean.
Geophysics and Space Physics. 13:591-645.
Reviews of
Buchwald, V. 1971. The diffraction of tides by a narrow channel.
Journal of Fluid Mechanics. 46:501-511.
Butman, B. 1975. On the Dynamics of Shallow Water Currents in Mass.
Bay and on the N. England Continental Shelf. Doctoral
Dissertation.
Massachusetts Institute of Technology and Woods
Hole Oceanographic.
192
Cartwright, D., W. Munk, and B. Zetler.
measurements. EOS. 50:472-477.
1969.
Pelagic tidal
Charnell, R. L. and D. Hansen. 1974. Summary and Analysis of
Physical Oceanography Dat'a Collected in the New York Bight During
1969-1970. NOAA/MESA Report No. 74-3. Boulder, CO.
Cox, C. and H. Sandstrom. 1962. Coupling of internal and surface
waves in water of variable depth. Journal of the Oceanographic
Society of Japan. 30th Anniversary Volume. 449-513.
Csanady, G. 1972. The coastal boundary layer in Lake Ontario.
Journal of Physical Oceanography. 2:41-53, 168-176.
Csanady, G. 1977. The coastal jet conceptual model in the dynamics
of shallow seas. in The Sea. Vol. 6. E. D. Goldberg, Ed. John
Wiley and Sons, N.Y. 117-144.
Cushing, V. 1976. Electromagnetic water current meter.
Oceans 76 Conf. Proc 25C. 1-7.
Defant, A.
1961.
N.Y. 560 p.
Physical Oceanography.
Vol.
2,
MTS and IEEE
Pergamon Press,
Dietrich, G. 1944. Die Schwingungssystems der half - und eintagingen
tiden in den ozeanen.
Veroeffentl. Inst. Meerskd. Univ. Berlin.
A41:7-68.
Dimmler, D. G., D. Huszagh, S. Rankowitz, and J. Scott. 1976.
controllable real-time collection system for coastal
oceanography. Ocean 76 Conf. Proc. 14E-1:1-13.
A
Doodson, A. T. and H. D. Warburg. 1941. Admiralty Manual of the
Tides. His Majesty's Stationery Office, London:270 pp.
Dronkers, J. J. 1964. Tidal computation in
waters. North-Holland Pub., Amsterdam.
rivers and coastal
173 pp.
1960.
Hydrodynamics of Oceans and Atmospheres.
Eckart, C.
Press, N.Y. 290 pp.
Pergamon
EG&G, Environmental Consultants. 1975. Summary of Oceanographic
Observations in New Jersey Coastal Waters near 39 28'N Latitude
and 74 15'W Longitude during the period May, 1973 through May,
1974. A Report to the Public Service Electric and Gas Co.,
Newark, N.J.
Emery, K. 0. and E. Uchupi. 1972. Western North Atlantic.
Assoc. of Petroleum Geologists, Tulsa, Okla. 532 pp.
Amer.
193
Ferrel, Wm.
1874.
Tidal Researches.
U. S. Coast Survey Rept.
Govt. Printing Office, Washington, D.C. 268 pp.
U.
S.
Filloux, J. H. 1971. Deep-sea tide obs. from the northeastern
Pacific. Deep-Sea Research. 18:275-284.
Flagg, C. N.
1977.
The Kinematics and the Dynamics of the New
England Continental Shelf and Shelf/Slope Front. Doctoral
Dissertation, Massachusetts Institute of Technology-Woods Hole
Oceanographic Institution. 207 pp.
Gould, W. and W. McKee. Vertical structure of semidiurnal tidal
currents in the Bay of Biscay. Nature. 244:88-91.
Haight, F. J. 1942. Coastal Currents along the Atlantic Coast of the
United States. U. S. Coast and Geodetic Survey Spec. Publ. No.
230. 73 pp.
Halpern, D. 1971. Semidiurnal internal tides in Massachusetts Bay.
Journal of Geophysical Research. 76:6573-6584.
Hendershott, M. C.
1973.
Ocean tides.
EOS 54:76-86.
Hendershott, M. C.
1977.
Numerical models of ocean tides.
Sea.
Vol. 6.
E. D. Goldberg, Ed.
John Wiley and Sons,
47-95.
The
N.Y.
Hendershott, M. C. and W. Munk. 1970. Tides. Annual Review of Fluid
Mechanics.
Vol. 2.
M. Van Dyke, Ed.
Annual Reviews, Inc.
Palo
Alto, CA. 205-224.
Hendershott, M. C. and A. Speranza.
1971.
Co-oscillating tides in
long, narrow bays; the Taylor problem revisited. Deep-Sea
Research. 18:959-980.
Hendry, R. M. 1975. The Generation, Energetics and Propagation of
Internal Tides in the Western North Atlantic. Doctoral
Dissertation, Massachusetts Institute of Technology-Woods Hole
Oceanographic Institution. 345 pp.
The persistence of "winter" water
1964.
Ketchum, B. and N. Corwin.
on the continental shelf south of Long Island, New York.
Limnology and Oceanography. 9(4):467-475.
Krauss, W. 1966. Methoden und Ergebnisse der Theoretischen
Ozeanographie: Interne Wellen. Gebruder Bor'ntraeger, Berlin.
248 pp.
194
Krauss, W. 1973. Methods and Results of Theoretical Oceanography:
Dynamics of the Homo. and Quasihomo. Ocean. Gebruder Bor'ntraeger,
Berlin. 302 pp.
Krol, H. 1974. Some numerical considerations: Eigenvalue problems
in the theories of underwater sound and internal waves. NATO
SACLANT ASW Memorandum SM-40.
Lamb, H.
Lee,
1932.
Hydrodynamics.
Dover Pub., N.Y.
0. 1961.
Observations on internal waves in
Limnology and Oceanography. 6:312-321.
738 pp.
shallow water.
Lowe, R. L., D. L. Inman, and B. M. Brush. 1972. Simultaneous data
system for instrumenting the shelf. 13th Conf. on Coastal Eng.,
Vancouver, B.C.
Magaard, L., and W. McKee. 1973.
Semidiurnal tidal currents at Site
D. Deep-Sea Research. 20:997-1009.
McCullough, J. 1977. Problems in measuring currents near the ocean
surface. MTS & IEEE Ocean 77 Conf. Proceedings 468:1-7.
Miles, J. 1974.
On Laplace's tidal equations.
Mechanics. 66:241-260.
Journal of Fluid
Miller, G. 1966.
The flux of tidal energy out of the deep ocean.
Journal of Geophysical Research. 71(10):2485-2489.
Mooers, C. N. K. 1975. Several effects of a baroclinic current on
the cross-stream propagation of inertial-internal waves.
Geophysical Fluid Dynamics. 6:245-275.
Morse, P. and H. Feshback.
McGraw-Hill, New York.
1953.
Methods of Theoretical Physics.
Munk, W. 1968. Once again--tidal friction.
Astron. Soc. 9:352-375.
Quart. Jour. Roy.
Munk, W. and D. Cartwright.
L966.
Tidal spectroscopy and
prediction. Proc. Roy. Soc. London. A259:533-581.
Munk, W., F. Snodgrass, and M. Wimbush. 1970. Tides off-shore:
Transition from California coastal to deep-sea waters.
Geophysical Fluid Dynamics. 1:161-235.
Mysak, L. 1978.
Wave propagation in random media, with oceanic
applications. Reviews of Geophysics and Space Physics.
16(2):233-261.
195
Niiler, P. 0.
1975.
A report on the continental shelf circulation
and coastal upwelling.
Reviews of Geophysics and Space Physics.
13(3):609-659.
Patchen, R., E. Long, and B. Parker. 1976. Analysis of Current Meter
Observations in the New York Bight Apex, August, 1973-June, 1974.
NOAA Technical Report 368-MESA 5.
U. S. Department of Commerce.
24 pp.
Peregrine, D. H. 1976. Interaction of water waves and currents. in
Advances in Applied Mechanics, vol.16. C. S. Yih, Ed. Academic
Press, N. Y. p9-117.
Petrie, B. 1975. M2 surface and internal tides on the Scotian
shelf and slope. Journal of Marine Research. 33:303-323.
Phillips, N. A. 1966. The equations of motion for a shallow rotating
atmosphere and the "traditional" approximation.
Journal of
Atmospheric Science. 23:626-628.
Phillips, 0. 1966.
Press. 261p.
Dynamics of the Upper Ocean.
Cambridge Univ.
Platzmann, G. 1971. Ocean tides and related waves. Lectures on
Applied Mathematics, v. 14. Ed. by W. Reid. American
Mathematical Society, Providence, R.I.
Prinsenberg, S. J., W. Wilmot, and M. Rattray, Jr.
1974.
Generation
and dissipation of coastal internal tides. Deep-Sea Research.
21:263-281.
Proudman, J. 1941. The effect of coastal friction on the tides.
Geophys. Suppl. Mon. Nat. Roy. Astrono. Soc. 5(l):23-26.
Proudman, J.
Rattray, M.
Tellus.
1953.
Dynamical Oceanography.
John Wiley, N.Y.
409 pp.
1960. On the coastal generation of internal tides.
12:54-62.
Redfield, A. 1950.
The analysis of tidal phenomena in narrow
embayments. Papers in Physical Oceanography and Meteorology.
11(4):1-36.
Redfield, A. 1958. The influence of the continental shelf on the
tides of the Atlantic coast of the United States. Journal of
Marine Research. 17:432-448.
Redfield, A.
1978.
The tide in
Research. 36:255-294.
coastal waters.
Journal of Marine
196
Regal, R. and C. Wunsch. 1973. M 2 tidal currents in the western
North Atlantic. Deep-Sea Research. 20(5):493-502.
Rooth, C. 1972. A linearized bottom friction for large scale oceanic
motions.
Journal of Physical Oceanography.
2(2):509-510.
Scott, J. and G. Csanady. 1976. Nearshore currents off Long Island.
Journal of Geophysical Research. 81(30):5401-5409.
Scott, J., T. Hopkins, R. Pillsbury, and E. Davis.
1978.
Velocity
and Temperature off Shinnecock, Long Island in October-November,
1976.
Oceanographic Services Division, Brookhaven National
Laboratory (unpublished manuscript).
Shureman, P. 1958. Manual of Harmonic Analysis and Prediction of
Tides. U. S. Coast and Geodetsic Survey, Special Publication No.
98. U. S. Government Printing Office, Washington, D.C. 317 pp.
Smith, P., B. Petrie, and C. Mann. 1978. Circulation, variability
and dynamics of the Scotian Shelf and Slope. Jour. Fish. Res.
Board Can. 35:1067-1083.
Sverdrup, H.
1926.
Dynamics of tides on the North Siberian shelf.
Geophys. Publ. 4(5):1-75.
Sverdrup,- H., M. Johnson, and R. Fleming.
1942.
Prentice-Hall, Inc., Englewood Cliffs, N.J.
The Oceans.
1087 pp.
Swanson, R.
1976.
Tides.
MESA New York Bight atlas Monograph 4.
New York Sea Grant Inst., Albany, N.Y. 34 pp.
Swift, D., D. Duane, and T. McKinney. 1973. Ridge and swale
topography of the Middle Atlantic Bight, North America: Secular
response to the Holocene hydraulic regime. Marine Geology.
15:227-247.
Taylor, G. I. 1919. Tidal friction in the Irish Sea.
Roy. Soc. A220:1-93.
Phil. Trans.
Traschen, J. 1976.
Tides of the New England continental shelf.
Woods Hole Oceanographic Institution (unpublished manuscript).
Whitham, G. 1960. A note on group velocity.
Mechanics. 9:347-352.
Journal of Fluid
Winant, C. 1974. Internal surges in coastal waters.
Geophysical Research. 79:4523-4526.
Winant, C. 1978. Coastal current observations.
of Geophysics and Space Physics.
Journal of
To appear in Reviews
197
Wunsch, C.
1969.
Progressive waves on slopes.
Mechanics. 35:131-144.
Journal of Fluid
Wunsch, C., and R. Hendry. 1972. Array measurements of the bottom
boundary layer and the internal wave field on the continental
slope. Geophysical Fluid Dynamics. 4:101-145.
Wunsch, C. 1975. Internal tides in the ocean.
and Space Physics. 13:167-182.
Reviews of Geophysics
BIOGRAPHICAL NOTE
Paul May was born on June
spent
most
of
his
younger
years
14,
in
1950
in Chicago,
the
suburb
of
Illinois.
Hinsdale.
Paul
graduated from high school in 1968 with high scholastic honors.
attending
Southern Missionary
four years,
class
of
in
in
Collegedale,
1972.
As a graduate
Sciences
coming
to
at
student,
he enrolled
the University
Woods
Hole
of
in
for
of his
the Department
Chicago
Oceanographic
After
Tennessee
he graduated as a physics major and as president
Geophysical
before
College
He
for
one
year
Institution
and
Massachusetts Institute of Technology.
Paul is interested in a variety of outdoor activities, including
back-packing,
scuba-diving,
photographer.
He
Falmouth,
months!).
is
Massachusetts
married
and
to
skiing.
the
and will have
former
a
child
He
is
Sandra
in
also
an
Watkins
the near
avid
of
E.
future
(5