JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 92, NO. C5, PAGES 5035-5044, MAY 15, 1987
The Upper Ocean Internal Wave Field'
Influence of the Surface and Mixed Layer
MURRAY
D. LEVINE
College of Oceanography,OregonState University,Corvallis
A model is developed to calculate the upper ocean internal wave spectrum as modified by the surface
boundary and mixed layer. The Garrett-Munk spectrumis assumedto describethe deep ocean wave
field. The main effectof the surfaceand mixed layer is to align the vertical structureof the wavesforming
vertically standingwaveslocally; this contrastswith the assumptionof random alignment in the GarrettMunk model.Model spectraand coherences
are calculatedfor idealizedbuoyancyfrequencyprofilesand
compared with the Garrett-Munk model. Measurementsin the upper 200 m from the Mixed Layer
Dynamics Experiment in the northeast Pacific in 1983 are also compared with the model results. The
most striking successof the model is predictingthe observedhigh coherenceand 180ø phasedifference
acrossthe mixed layer of horizontal velocity.
1.
INTRODUCTION
A kinematic descriptionof oceanicinternal wavesis usually
based on the hypothesisthat the wave field is composedof a
random sea of linear waves.For deep ocean internal wavesthe
most widely known descriptionis the one given by Garrett and
Munk [1972, 1975] and summarized by Munk [1981] (hereinafter referred to as GM). The GM formulation assumesthat
the ocean has no top or bottom boundary and that the buoyancy frequency N(z) varies slowly with depth compared with
the fluctuations of the waves. This permits the vertical variation of wave amplitude and vertical wavelength to be scaled
by N(z) (WKB approximation). Other models, such as the
Internal Wave Experiment (IWEX) model [Mfi'ller et al.,
1978], differ in detail but usually retain the basic tenets of the
GM model. The GM model provides a standard with which
to compare other models. In the upper ocean, say from the
surface to 200 m, the GM model is expected to have limited
application even discountingthe possibleeffectsof strong surface forcing or dissipation.This is becauseN(z) is no longer a
slowly varying function and because the surface acts as a
boundary which can reflect waves. Despite these limitations,
the GM spectrum has been compared with upper ocean
measurements made by acoustic doppler profilers [Pinkel,
1984], moored current meters, and thermistor chains [Kiise
and Siedler, 1980; Levine et al., 1983a, b-I, towed thermistor
chains [Bell, 1976], and vertical conductivity, temperature,
and depth profilers [Pinkel, 1975, 1981]. These comparisons
indicate that while some aspects of the GM spectrum are
consistent with the data, many are not. Pinkel [1981] has
discussed and modeled
some of the effects of the surface on the
internal wave spectrum; this model is mostly concerned with
the effects of a frequency dependent waveguide thickness but
does not explicitly considerthe effectsof wave reflection at the
surface.
The purpose of this paper is to develop a model that extends the GM formulation of the deep ocean internal wave
field into the upper ocean by explicitly including the surface
boundary and realistic upper ocean stratification. The deep
ocean is still assumed to be composed of a collection of
random waves with the amplitude distribution in frequency
wave number space specified by GM. The modification of
each upward-propagating wave is followed as the wave enters
the upper ocean stratification and reflects off the surface; a
standing wave in the vertical is formed locally. The statistical
measures of the upper ocean wave field can be predicted by
summing the contribution from all the waves. This procedure
is analogous to that used to extend the GM model into the
region near a turning point [Desaubies,1973].
The spectral quantities calculated from the model can then
be compared with upper ocean observations.The model provides a useful framework for interpreting internal wave
measurementsin the upper ocean in the same way as the GM
spectrum does for the deep ocean. This comparison is a first
step in studying the signature of other upper ocean processes,
such as wind stressand solar heating, on internal wave observations.
The unique set of observations available from the Mixed
Layer Dynamics Experiment (MILDEX) provides a good opportunity to compare model predictions with data. MILDEX
took place in the northeast Pacific Ocean about 700 km west
of Santa Barbara, California, during October-November 1983.
Measurements were made in the upper 200 m of temperature
and velocity from beneath a drifting buoy and temperature
from a towed thermistor chain. From these data, frequency
and horizontal wave number spectraand coherenceswere calculated and compared with the model.
The upper ocean spectral model is developed in section 2.
Spectra and vertical coherencesfor idealized N profiles are
then calculated from the model (section 3). In section 4, data
from MILDEX are compared with the results of the model
using the actual N profile. A summary and conclusion are
presentedin section 5.
2.
MODEL
The first step in forming a statistical model of the upper
ocean
Copyright 1987 by the American GeophysicalUnion.
internal
wave
field
is to determine
the structure
of a
single wave component as it encounters the upper ocean.
Since the model is linear, all of the components can then be
added together using the amplitide distribution that will yield
Paper number 7C0227.
0148-0227/87/007C-0227505.00
5035
5036
LEVINE:
UPPEROCEANINTERNAL
WAVEFIELD
LogN
•
•
Note that the assumptionthat fl is slowly varyinghas been
used in deriving (5). For a specifiedstratificationthe wave
function½(z)can be solvednumericallyfor a givenincident
wave with amplitudeI, wave numbera, and frequencyco.
There is only one solutionof ½; this is not an eigenvalue
problem that resultsin an infinite number of modal solutions.
-d
This boundaryvalue problemwas solvedusingan efficient
finitedifference
methoddueto LindzenandKuo [ 1969].
Spectralquantitiescan be calculatedby summingthe contributionfrom all the wavecomponents.
It is assumedthat the
collectionof wavesis horizontallyisotropic,and there is no
correlationamongthe components.
The frequency
crossspec-
traforhorizontal
velocity
Suandvertical
displacement
Scbetweendepthsz• andz2 canthenbe written[e.g.,Willebrandet
Fig. 1. Samplesolutionof ½ and {pfor a typicalN profile.Below al., 1977]
-d, a constantN regionis usedto determinethemaximaof ½ and {p
needed
formatching
to thedeepoceanGM spectra
(equation
(8)).
Su(co,
Z1, Z2) •-
((_02
._[_
f2)(CO2
__f2)1/2
2N(-d)
nmax
ß • en(CO)•Pn(CO,
Z•)Ckn(co,
%)
deep oceanstatisticsconsistentwith the GM spectrum.
Assumethat the horizontal velocity and vertical displacement of a single,horizontally propagating plane wave can be
written
z,, z,)=
as
u(x, z, t) = •2•p(z)exp [i(ax - cot)]
(la)
•(x, z, t) = •½(z)exp[i(ax - cot)]
(lb)
(6a)
n=l
((_/)2
__f2)1/2....
N(-d) nllen((D)½n((-O'
Z1)½n((D'
Z2) (6b)
wheref is theinertialfrequency
anden(co)
is thespectral
distribution function.The wavefunctionsspecifythe verticalstructure of a plane wave at frequencyco and horizontal wave
numberan,
where•p= a-•(O½/Oz),a is the horizontalwavenumber,cois
the frequency,and the real part of the right-hand side is implied.
The wave function ½(z)is determinedby solving
½"
=0
(2)
where
•2(z)
=a2N2(z)
_co2
092
_f2
(3)
The surfaceis definedto be at z = 0, and z = -d is the depth
below
which
the
internal
waves
statistics
are
assumed
-d
the wave function
can be written
The indexn is oftenreferredto asa "mode"number.However,
in thisapplication
the verticalmodesare not determined
by
solvingthe boundaryvalueproblemoverthe entiredepthof
theocean.Thespectral
distribution
function
en(co)
specifies
the
magnitude
of eachwavecomponent
at coandanandis determinedby matching
thespectrum
to thedeepoceanGM spectrum
at z = -d.
In the GM model,a is expressed
as a functionof n and co
(dispersion
relation)usinga modifiedWKB approximation:
as
½(z) = I exp [-ilS(z)z] + R exp [ilS(z)z]
at z = -d
nr•(co2_f2)•/2
an((D)
-- --
(4)
where/5(z) is a slowly varying function of z, and I and R are
the complex amplitudesof an incident (upward propagating
energy) and reflected wave (downward propagating energy),
respectively.Recall that for an internal wave the phasepropagates upward (downward) while the energy propagatesdownward (upward). The z dependenceis the sameas usedby GM
and is a simplifiedform of the WKB approximation.
To solve (2) for an arbitrary N(z) in the region above z --d, boundary conditions at z = 0 and -d are needed. With
little error the surfaceis assumedto be rigid, •p(0)= 0. Specifying the boundary at z = -d is more subtle sincethe phase
of the reflection is not known before solving the problem.
Sinceno energyis lost upon reflection,III- IRI, but arg I •
arg R in general.Combining (4) and the derivative of (4), R
can be eliminated, and the lower boundary condition can be
expressedas a function of the incident wave amplitude only,
½' - ifl½ = - 2iflI exp [-ilS(z)z]
to,z)
Equation(6) represents
the spectraof a collection
of tlma
x
planeinternalwaves;
fortheresults
presented
here,t/ma
x-- 20.
to
behave as those specified by GM (Figure 1). Hence below
z -
z)=
(5)
b
NO
(7)
whereb is thedepthscaleof N(z) (1300m), N ois thebuoyancy
scale(5.24 x 10-3 s-• or 3 cph),and it is assumedthat co<<
N. We choosethe versionof the GM formulationthat partitionsthe energyinto a discretesetof wavenumbers("modes")
[Munk, 1981] rather than distributingit over a continuum
[Garrett and Munk, 1972' 1975]. Note that a is not a function
of z' the horizontalwavenumberdoesnot changeas a wave
propagatesthroughstratificationthat only variesin the vertical.
The mostpracticalmethodto determine%(co),
suchthat the
spectrum matchesthe G M spectrum at z-- -d, is to extend
the numericalsolutionof ½ndeeperthan d into a regionof
constantN = N(-d) (Figures1). In this constantN region,½
has a sinusoidalshapewith verticalwave numberfl determined from (3).
The reason for solving ½ in this "artificial," constant N
regionis to determinethe amplitude(maximumof ½n(z))of
LEVINE.' UPPER OCEAN INTERNAL WAVE FIELD
this wave component relative to the amplitude of the incident
wave I. By equating the upper ocean spectrum (6) to the GM
spectrum I-Munk, 1981] the spectral distribution function is
5037
Buoyancy Frequency (cph)
0
5
,
I0
I
200• ' ,
,
0
5
I
I0
I
i
i
5
I0
5
I
I0
I
5
I0
15
I,
I
I
I
I
found
2 Hn
Eb2Nof
5O
En(f-D)smax [½n2]
r• 12(_D
(8)
where the parameter E is the nondimensional energy (6.3
E
'"'
I00
x 10-5) andtheweighting
function
H, =/•- • (j,2 + n2)- •.
The parameterj, setsthe wave numberbandwidthof the
spectral quantities and has a value of 3 in the GM model. The
normalization
constant H is chosen such that
Cose'
In practice,I is set to an arbitrary constant,and (2) is solved
numericallyfor ½n(Z)using(5) as the bottom boundarycondition. The matching condition at z = -d (equation (8)) then
Io
lb
,
2(]
I
2b
5
Fig. 2. Buoyancy frequency profiles used in calculating model
spectra.Case 1 is constantN, case2 is a stepin N, and case3 is the N
profile from MILDEX.
determines%(co)relative to this arbitrary constantsuchthat
the absolute value of the spectrum (equation (6)) equals the
GM
The geometricfactor in the denominator of the integrand accounts
level.
for
the
contribution
to
the
one-dimensional
wave
To comparewith data, two typesof statisticalquantitiesare
investigated'moored and towed. This terminology follows the
convention adopted by GM and refers to observations that
are made from instruments which are fixed in space and instrumentswhich profile horizontally. The moored spectrumof
number • from all wave numbers greater than • in a horizontally isotropic wave field. This integral was solvednumerically
by first transforming the integrand to remove the singularity
in the denominator using the method of Iri, Moriguti, and
Takasawa [Atkinson, 1978]. The towed vertical coherenceof
verticaldisplacement,
MSs,is a frequencyautospectrum
at a
verticaldisplacement,
TVCe is the coherence
in horizontal
fixed location in spaceand is given by (6b)
MS•(ro,z) = S•(ro,z, z)
(9)
wave number at a fixed vertical separationand is given by
TVC•(o•,Z1,Z2)=
The moored vertical coherence of vertical displacement,
[TS•(o•,
z•)TS•(o•,
g2)]1/2
MVC e is thecoherence
in frequency
for a fixedverticalsepa-
.n•"
S•((O,
Z2)
d(o(13)
....fN(z)
(•En
2Zl,
"•-•E'-•-I/2
ration and is given by
S•(O),
Z1,Z2)
The correspondingmoored and towed spectral quantities for
MFG,(o);
z1,
z2)
=[S•(o),
z1,
z1)S•(o)
,z2,
z2)]
1/2(10)horizontal velocity have the exact same forms with the subThe towedspectrum
of verticaldisplacement,
TSe is thehori-
script • replacedby u.
zontal wave number spectrumat a fixed time and is given by
2
z)da)
• n•'•
n [•v{•)
OranS•(a),
(•En
2 --z,
•E2)
1/2
TS•(•, z) .....
TABLE
Buoyancy Frequency
la
lb
2a
constant, N -- N o
constant,N = 3.2 N O
step profile:
N = N Obelow 50 m
N = 0 upper 50 m
step profile:
N = 3.2N0 below 50 m
N = 0 upper 50 m
2b
3a
MILDEX
3b
MILDEX
3c
MILDEX
Spectral Composition
GM
GM
GM
GM
GM
GM
RESULTS--IDEALIZED
N PROFILES
with idealized N profiles (Table 1): constant (case 1) and step
profiles (case 2) (Figure 2). These caseswere chosento isolate
the effectson the spectrumdue to the presenceof a surface
boundary alone (case1) and due to a mixed layer (case2). The
step profiles consistof a 50-m-deep mixed layer with N = 0
above a region of constantN. To examinethe dependenceon
the absolutevalue of N, two valueswere used: N = N O(cases
(12)
1. Summary of Model Cases
Case
MODEL
To illustrate the behavior of the model, two cases were run
(11)
where
(Dn
2=f2
+(o•
-•
bNø•
?2
3.
but no contribution
fromn= 1, ex=0
GM with frequency
dependent waveguide,
bN o--• h(o) N
la and 2a) and N = 3.2 N O(caseslb and 2b). The value off
was chosento be the sameas at MILDEX (8.0 x 10-5 s-•).
In all the spectral calculations the model spectrum was
matchedto the GM spectrumat a depth of d = 200 m.
ConstantN profile. A sampleof the wave functions½ and
•p that determine the vertical structure of the spectra are
shown in Figure 3. SinceN is constant,the wave functionsare
purely sinusoidal.At higher N (case lb) the vertical wave
number is higher for a given an becauseof the dispersion
relation (3).
The spectral shapesof the model in frequency(MS) and
wave number (TS) nearly follow the GM prescriptionat all
depths and thereforeare not shown. The spectrallevelsas a
function of depth are shownin Figure 4. Accordingto the GM
5038
LEVINE:UPPEROCEANINTERNALWAVE FIELD
C].
1
0
2
i
Mode Number
.3
i
4
i
5
6
7
8
9
10
9
10
i
•"'150
200
•.
1
2
3
Mode
Number
4
5
6
7
8
0
this simplemodel.The negativecoherences
are a resultof the
deterministicalignmentof the wave functionscausedby the
reflection at the surface.Specifically,the reflectionforcesthe
maxima of 4) and the zerosof •J to occurat the surface.In the
GM model there are no negative coherencesbecausethe
alignmentof the wavefunctionsis random; all verticalcoherencesgo to zero monotonicallyas the separationincreases.
Other deviations from the GM model occur near the surface. In the GM model at constant N the contours would be
straight,parallellinesat a 45ø angleindicatingthat the coherenceis a functionof verticalseparationonly and not of absolute depth. The deviation from straight, parallel contours
occursprimarilyoverthe adjustmentregion.The presence
of a
reflectingsurfacehas a relatively small effect on the near-
50
100
150
200
surface coherences.
C.
1
2
.3
1
2
.3
Mode Number
4
5
6
7
8
9
10
9
10
0
50
100
150
200
d.
0
i
'•100
i
"
ModeNumber
4
i
5
i
6
i
7
8
'
: ,," ,," ('
Step profile. For the step profile the assumptionof a
slowlyvaryingN(z), as usedin the GM model,appearsto be
violated. Also, the GM spectrumis not defined within the
mixed layer where N = 0. This model providesa method for
extendingthe GM spectruminto the mixed layer, wherevertically propagatinginternal wavescannotexist.
In the stepprofilethe wavefunctions,and hencethe spectra,
differ from the constantN caseprimarily in the mixed layer.
There is a turning point (co= N(z)) at the base of the mixed
layer for waves of all frequencywhere the wave functions
changebehavior from oscillatory(co< N(z)) to exponential
•150 / [ : ', ",
200
MSu
e.
1
O"
2
i
.3
4
i i
5
i
i
6
7
8
9
10
i
i
i
'
'
50 i ,iJ i..i..:..'...'...'....'
•' 100
ß
ø200
i/ ! ",
TSu
101
10 o
Mode Number
10 2
0
101
10 2
,
: : : :'.:'•l
,,,;-• ; ;::::
. :ß::: ß.:,::::
lI
5O
,, ",, ) ,',,
/
II
/
!
I
100
Fig. 3. Wave functions of vertical displacementqJ(solid lines) and
horizontal velocity 4) (dashed lines) as a function of depth for (a) case
la, (b) case lb, (c) case2a, (d) case2b, and (e) case3.
model, the spectral level should be constant with depth in a
constant N ocean.However, in the model there is somedepth
dependenceof the spectrain an "adjustment region" near the
150
2
2OO
la
:1:
b.
: ::::
o
surface.
The verticaldisplacement
spectra,
MS• and TSo goto
zero at the surface as a result of the upper boundary condition; the wavefunctionsqJ,are zero at the surface.At higher
N the vertical wave numbersare higher, and correspondingly,
the adjustment region is smaller. The horizontal velocity spectra, MS, and TS,, are also affectedby the surfaceboundary
with an adjustment region that is approximately the same as
for verticaldisplacement.
At the surfacethe wave functions4),
are at a maximum for all n, resulting in a spectrallevel that is
2-3 times greater near the surfacethan at depth.
Examples of the moored vertical coherences(MVC, and
MVC<) at 1.4 cph and towedverticalcoherences
(TVC, and
TVC•)at 10-4 cpmarecontoured
in Figures
5 and6. The
feature of the vertical coherencesthat differs most strikingly
from the GM model is the presenceof "negative"coherences.
The sign of the coherenceis used as shorthand notation to
expressthe phase of the coherence:positive is in-phase (0ø),
negativeis out-of-phase(180ø).No other phasesare possiblein
50
,.,lOO
,
10o
.: : :2.
.............
01
J
,
102
........
i.!,
oj .[
??..
101
,
....
,
:0
2
MS(.
TS½
Fig. 4. Modelspectra
(a) MS,,,(b) TS,, (c) MSe and(d) TS• asa
function of depth for case la (light dashed), case lb (bold dashed),
case 2a (light solid), and case 2b (bold solid). Absolute units of the
spectra
arearbitrary.Mooredspectra
(MS,,andMSs)areevaluated
at
1.4cph;towedspectra
(TS.andTS•)areevaluated
at 10-'•cpm.
LEVINE' UPPER OCEAN INTERNAL WAVE FIELD
0
50
o
100
I
150
I
\
0
I
50
100
I
I
\
150
[ .......
0
200
100
150
150
0
50
100
150
200
o
50
•
0
50
lOO
100
150
0
lOO
150
15o
•
lOO
150
150
15o
200
2000 5b 160 1•0
200
5b l C•01•0 200
Depth (m)
Contour plot showing the model vertical coherencesbe-
tween any two depthsfor casela (N = No). The 1 contour is on the
diagonal; the 0 contour is bold. Contour interval is 0.2; negative
for the GM
50
100
Depth (m)
values are shaded. The 0.5-contour
o
\
50 1
5b l C•01•0 200
Depth (m)
100
0
50
so
Fig. 5.
0
===========================
100
sb
150
"::i:::::::i:!?::
::::!:i:!:::
50
0
100
I ......
==============================================
\
200
50
5039
model is shown as a
dashedline. The mooredcoherences
(MVC) are evaluatedat 1.4 cph;
the towedcoherences
(TVC) are evaluatedat 10-4 cpm.
Depth (m)
Fig. 7. Contour plot showing the model vertical coherencesbetween any two depthsfor case2a (Step of No). The 1 contour is on
the diagonal; the 0 contour is bold. Contour interval is 0.2; negative
values are shaded. The 0.5 contour
for the GM
model is shown as a
dashedline. The moored coherences(MVC) are evaluatedat 1.4 cph;
the towedcoherences
(TVC) areevaluatedat 10-4 cpm.
(• > N(z)). The wavefunctions½nare nearlyconstant,and ½n mixed layer whetherthe adjustmentregionis larger or smaller
are nearly linear in the upper 50 m (Figure 3). At large vertical than the mixed layer depth. The velocityspectra(MS, and
wavelengths(low modes)the wave functionsare similar to the TS,) are nearlyuniformin the mixedlayer becausethe ½nare
constantN cases;the effectof the relativelysmall, mixed layer nearly uniform.However,the spectrallevel in the mixed layer
is slight.For waveswith a verticalwavelengthcomparableor is slightly higher than the deep spectrumat low N (case2a)
smallerthan the mixedlayer there are zero crossings
in •pand and becomeslower at high N (case2b).
maxima in •bnear 50 m depth (Figure 3).
The main differencesin the coherencesof the step profile
Below 50 m the spectraare similar to the constantN spec- compared with the constant N case also occur in the mixed
tra (Figure4). In the mixedlayer,verticaldisplacementspectra layer (Figures 7 and 8). At a fixed vertical separation the
(MScand TSc)decrease
to zero at the surfaceas in the con- coherencewithin the mixed layer is higher than it is for the
stant N case. The difference is that the decrease occurs in the
50
100
150
0
50
100
150
0
same separation in the constant N case. The differences are
200
o
o
50
100
lOO
MVCu
150
TVCu
0
•
15o
150
150
150-
200
200
5b 160 1•0 200
Depth (m)
tweenany two depthsfor caselb (N = 3.2 No). The 1 contouris on
the diagonal; the 0 contour is bold. Contour interval is 0.2; negative
values are shaded. The 0.5 contour
for the GM
model
is shown as a
100
150
200
100
MVCu '
150
0
x:
100-
lOO
MVC•
0
sb
Dep(h (m)
Fig. 6. Contour plot showing the model vertical coherencesbe-
50
5O
100
5b 160 1•0 0
0
o
100
0
150
lOO
15o
50
200
100
50
o
50
50
o
o
Dep(h (m)
Fig. 8. Contour plot showing the model vertical coherencesbetweenany two depthsfor case2b (Stepof 3.2 No). The 1 contouris on
the diagonal;the 0 contouris bold. Contour interval is 0.2; negative
values are shaded. The 0.5 contour
for the GM
model is shown as a
dashed line. The moored coherences(MVC) are evaluated at 1.4 cph;
dashed line. The moored coherences(MVC) are evaluated at 1.4 cph;
the towedcoherences
(TVC) areevaluatedat 10-4 cpm.
the towedcoherences
(TVC) areevaluatedat 10-4 cpm.
5040
LEVINE' UPPER OCEAN INTERNAL WAVE FIELD
Buoyancy Frequency (cph)
10 -1
ß
10 ¸
I
i I tllltl
101
!
i i Iiiiii
lOOO--
102
i
I I , ,Ill
--
2000-
__
--
__
--
--
__
5000Fig. 9.
from mode
1 is eliminated.
This is
accomplishedby setting H•- 0 and changingthe normalization constantH' H,for n > 1 is unchanged(equation(8)).
Case 3b demonstratesthe effect of increasingthe horizontal
wave number that has the most energy. As will be seen,the
motivationfor eliminatingmode 1 is to improvedramatically
the agreement between model and observed coherences. It
may seem ad hoc to eliminate mode 1' however, the observaof mode 1 in the
GM model is not conclusive.The resultsfrom IWEX [Miiller
et al., 1978] showthat mode 1 may be relativelylessenergetic
than the GM prescription. Other observations have also
shownthat themodelfit is improvedusinga highervalueofj,
[e.g.,Levineet al., 1986]'theeffectof higherj, is to reducethe
----
4000-
case 3b the contribution
tional evidence that indicates the dominance
--
3000-
Two other caseswere also run with the MILDEX N profile
but usingmodifiedspectraldistributionfunctions(Table 1). In
• ',..............
I I IIIIII
Buoyancyfrequencyprofile for MILDEX.
magnitude of mode 1 relative to the other modes. While totally eliminating mode 1 is extreme, this case serves as an
indication of the effect on the spectrum for a decreasein the
relative magnitude of mode 1.
In case 3c the GM form of the dispersionrelation (7) is
modified to model more realistically a variable N profile. In
the GM model the vertical scale of the wave is a constant, b. A
more accurate parameterization allows for a variation in the
due to the absenceof zero crossingsin the wave functions in
the mixed layer. Below the mixed layer the coherencestructure is nearly the same as the correspondingconstant N case.
4.
MILDEX:
MODEL
AND OBSERVATIONS
The statistical model developed in section 2 is compared
with observations made during the Mixed Layer Dynamics
Experiment (MILDEX). Three casesof the model are run with
the N profile observed during MILDEX. The casesdiffer in
the details of the spectraldistribution function (Table 1). One
verticalscaleof the waveguideas a functionof frequency,h(co).
For MILDEX an estimate of h(co)is taken to be the depth
range below the mixed layer where co< N(z). To be consistent,
a modificationof the stratificationscaleN Oin (7) is alsomade.
The scaleN Ois replacedby the averagebuoyancyfrequencyN
over the waveguideh(co).At low frequency,h(co)approaches
the depth of the oceanand N is a minimum.As frequency
increases,h(co)becomessmaller while N increases'the net
effectfor MILDEX is a decreaseof the produceh(co)N as
frequencyincreases.The consequence
of replacingNob by
of the cases(3a) usesthe samee,(co)as in section2; the other Nh(co)in (7) is to modify the horizontal wave number as a
two cases(3b and 3c) are small modificationsof the weighting function of n and co.At high frequency,• has a larger value
used in the GM model. Observed spectra and coherencesare
then compared with the model results.
Model. The only data neededto calculate the model spec0
tra for the MILDEX site is the buoyancyfrequencyprofile. A
I
composite N profile of the entire water column was created
I
from several sources(Figure 9). The stratification in the upper
I
•e !
ß Ii
I
75 m is taken from a typical single profile observed by the
;
;'
I
_
_./.
(..
Rapid Sampling Vertical Profiler [Newberger et al., 1984]. A
I
'" --.
ß
single realization is more representativethan an average profile where the sharp thermocline is smeared out due to the
averaging of many profiles each of which has been displaced
100
vertically by internal waves.The profile was completedusing
observations to 300 m made aboard Research Platform Flip
(R. Pinkel, personal communication, 1986) and profiles to 800
m made aboard the R/V Acania (T. Stanton, personal com150
munication, 1986). Deeper values of N were estimated from
historical data averagedin 1ø x 1ø squaresby Levitus [1982].
An enlargedview of the N profile in the upper 200 m is shown
in Figure 2.
I
I
200 ,
10-1
10 ¸
10 •
103
First, model spectra and coherenceswere calculated (case
_
_
'
3a) usingthe sameGM parameters
E, j,, b, and N Oand the
samespectraldistributionfunction,•,(co),as in section2. For
this case the model spectra are qualitatively similar to the
results from the idealized step profile (case2). In case 3a, N(z)
decreases below 50 m, while in case 2, N is constant below the
mixed layer.
b
Velocity Spectrum, MSu (cm2 s-2 cph-1)
Fig. 10. Moored velocityspectra,1/2 (MS,, + MSv),as a function
of depth for frequencybands (a) 1.5-3.0 cph, (b) 0.375-0.75 cph, and
(c) 0.0937-0.1875 cph. Data are shown as solid circles; model results
are plotted as solid lines (case3a) and dashedlines (case3c). The 95%
confidencelimits are indicated on the deepestvalues.
LEVINE' UPPER OCEAN INTERNAL WAVE FIELD
I
I
I
I
I
I
I
$041
prising as there are many other sourcesof variability in the
mixed layer that are clearly not due to free, linear internal
waves. While the model spectral level does not agree perfectly
with the observations, it is certainly better than the GM
model, which is not defined in the mixed layer where N goes
to zero. The observed spectral level in the mixed layer decreasesrelative to 70 m as frequency increases.Model case 3c
with the frequency dependent waveguide also exhibits this
same general behavior.
I
The depthdependence
of the TS• asobserved
by the towed
thermistor chain in the wave number band from 10 -4 to 10-3
cpm is shown with the model results in Figure 11. As with
MS,, the model cases3a, 3b, and 3c are similar. The data
follow the model remarkably well indicating a slight maximum near 50 m. The absolute values are best fit by case 3c
although there is lessthan a factor of 1.5 separatingthe cases.
The GM model has no maximum but increasesmonotonically
with depth following 1IN scaling.
Although data are not available to estimate all possible
autospectra,it is interestingto compare the depth dependence
100
I
I
I
I
I
I
I
of the modeledMS<,MS,, TS<,and TS, (Figure12).As the
I
10 4
103
Displacement
Spectrum,
TS½
(m2 cpm-1)
Fig. 11. Towed spectrumof verticaldisplacementas a functionof
depthfor the wavenumberband 10-4-10-3 cpm.Data are shownas
solid circles;model resultsare plotted as a solid line (case3a) and a
dashedline (case3b). The 95% confidencelimits are indicatedon the
deepestvalue.
than in case 3a at the same coand n. As a result, the spectral
distance below the mixed layer increases, the spectra more
nearly follow the N scalingexpectedfrom the WKB approximation, that is, velocity spectrascaleas N(z) and vertical displacementspectrascaleas 1IN(z). In the mixed layer, where N
scalingis obviouslynot valid, the u spectraare nearly constant
with depth, and the • spectradecreaseuniformly to zero at the
surface,similar to the idealizedstep profile (case2b).
A comparison of the frequencyand wave number depen-
denceof the observedand modeledautospectraMS,, MSs,
and TScare examinedin Figures13, 14, and 15. Frequency
quantities for case 3c are a much stronger function of fre- spectraMS, from 15 and 141 m are plotted with the model
quency,especiallyat high frequency.
In all of these calculations no quantities have been calcu-.
lated at frequenciesabove N(--d)= 3.5 cph. This is because
Spectral Density (units arbitrary)
there is no energy above local N in the GM model; above N,
10 o
1 01
1 02
freely propagatinginternal wavescannot exist.It is possibleto
extend the analysis by using a different method at high frequency to set the spectrallevel, but this would complicatethe
straightforwardobjectiveof this paper.
Observationscomparedwith model. The model is compared
with
MILDEX
observations
from a drifter
and a towed
ther-
mistor chain. The drifter consisted of a free-floating toroid
buoy that was instrumentedbelow with a string of 15 VectorMeasuring Current Meters from 3 to 141 m and 4 Aanderaa
thermistor chains from 70 to 800 m [Levine et al. 1984]. The
deploymentwas near 33ø51' N, 126ø42' W; recoveryoccurred
19 days later about 80 km northeast.The thermistor chain
was towed in the MILDEX area from the R/V Wecoma a
total of nearly 2000 km [Baumannet al., 1985]. Twenty-eight
temperaturesensorsspannedthe surfaceto 95 m depth.
First, the autospectraof the data and model are compared.
,50
,--
,'/
//
100
TS• are shownwiththemodelresultsin Figures10 and 11.
150
The observedand modeled MS, are plotted together as a
function of depth for several frequency bands (Figure 10).
Below the mixed layer there is not much to distinguishamong
200. . .......
The depth dependenceof the observedautospectraMS, and
,
MS
--
/,,,,,
l••'•••/-i
!/ : '.'.'.."..'
the model cases 3a, 3b, and 3c. The absolute level and the
Fig. 12. Model spectraas a function of depth for case3a. Abso-
depth dependenceof the data reasonablyfollowsthe model;
luteunitsof thespectraarearbitrary.Mooredspectra(MS,,andMSc)
however,the agreementis worseat low frequency.Within the areevaluated
at 1.4cph;towedspectra
(TS, andTS•)areevaluated
at
mixed layer the spectrallevelsvary more in the vertical than 10-4 cpm. Buoyancyfrequencyscaling(WKB approximation)is
indicated by the internal wave model. This is not too sur- shown as dashed lines.
5042
LEVINE: UPPER OCEAN INTERNAL WAVE FIELD
.--.105
•-•103
i
, i i l,,,l
i
i i ! Illll
I
I I ,
'
'
'
'
,
,
' '''''1
' '''''1
, ,i,,
I
!
o 10 4.
e• 102
141m
I
E
E 10 3
:• 102
•_1 0 o
•10•
o
10-1
• 10-2
•100
o
o
>10-1
•5 10-3
10-1
10-2
10 0
101
: '.'.',::'.:'. :'.'.'.'.'.'.
'. :: :','.'.
10-2
Frequency(cph)
10-1
10 0
101
Frequency (cph)
Fig. 13. Mooredvelocityspectra,« (MS, + MSv),asa functionof
Fig. 14. Mooredverticaldisplacement
spectrum,
MSs,at 85 m.
frequency at 15 (no offset) and 141 m (offset by factor of 100). Data
are shown as solid circles. Model results are plotted as solid lines
(case 3a). The portion of model case 3c that differs significantlyfrom
Data are shown as solid circles;model results(case3a) are plotted as
case 3a is shown
indicated.
180ø phase differencewith a higher coherencein general than
the smaller separationsof 55 m (15-70 m) and 71 m (70-141
m). Case 3a reproducesthe observed phase structure; how-
as a dashed
line. The
95%
confidence
limits
are
spectra in Figure 13. Again, only case 3a is plotted as the
model results for the other cases are nearly the same. The
greatestdifferenceamong the casesoccursin case 3c at high
frequencywhere there is a sharp decreasein the spectrumin
the mixed layer. The model spectralshaperepresentsthe data
reasonablywell; however,there are differencesin the levels.At
15 m in the mixed layer the model is a bit higher than the
observations;the G M spectrum is not defined where N = 0.
At 141 m the model
is lower
than the data. This difference
a solid line. The 95% confidence limits are indicated.
ever, the coherence between 15 and 141 m is much smaller
than observed.When mode 1 is eliminated (case3b), the coherenceincreasesdramatically. This is becausethe most energetic"modes"now have a singlezero crossingbetween15 and
141 m (Figure 3). The agreementof the model with the other
is
not due to a difference between the model and GM spectrum
sincethey are virtually identical at this depth.
A goodestimateof the verticaldisplacement
spectrumMSc
is available only at 85 m from the moored thermistor chain
(Figure 14). As with the MS,, the differencesamong the cases
and the GM model are slight; however, the agreement with
the observationsis good.
The wavenumberdependence
of the observedTScat 50 m
is compared with the model in Figure 15. The agreement between the model and data is good, especiallyin the midband
of wave numbers. The model level is higher than the GM
spectrum at low wave numbers and fits the data better. The
model resultsbecomelessvalid at high wave numberssinceno
contributionsfrom frequenciesbeyond 3 cph were used.
In contrast to the autospectra the model coherencesvary
substantially among cases3a, 3b, and 3c. A detailed compari-
son of the model MI/C, with three observedcoherencesis
presentedin Figure 16. In the GM model the MI/C, is not a
function of the absolute depth but only of the vertical separation; the larger the separation,the smaller the coherence,with
the phase always zero. Clearly, the data to not follow this
behavior. The largest separation of 126 m (15-141 m) has a
10-5
10-4
10-3
10-2
Wove
number(cpm)
Fig. 15. Towedspectrum
of verticaldisplacement,
TSc,at 50 m.
Data are shown as solid circles' model results(case 3a) are plotted as
a solid line. The G M model is plotted as a dashed line.
LEVINE' UPPER OCEAN INTERNAL WAVE FIELD
Frequency
1 0 -2
1.00
e
o
1 0 -1
1 0 -2
ß
0.50-
o
0.25
10-1
o.
I
15-70m
ß ;
r::
Frequency
1 O0
1 0-2
1 0-•
ß
-•
'
b
70-141m
.....
'
-'i
-
-
0.00
1 O0
I
15-141m
_
.....
', ',', ' .... I
-
........
ee
ß
ß
.
•
90--
•_
0
'
10
.
.
m
.
'
ß ß
"
ß
'
--
-
c.
ßß
ß
ß
ß'
..... ..ee
ß
'
ß
eeee
ß
-
ß ß
.............
-90 -
L
ß
ß
"
ß
ß
.
"
........
ß
ß
180--
I
Iø
ß
e,e
O
O
Frequency
10 0
........ I ........ I ........ I
0.75-
••-
5043
"'
ß
ß
ß
ß
ß
'
ß
ß
ß
ß
ß
•- ...
ß
ßß
ß
"
ß
'
.
- 180
, ,-......• ........ • ........
10 -2
10-•
100
10•
10-•
100
101
10-•
100
1
Fig. 16. Mooredverticalcoherence
oœvelocity,MVC,, between(a) 15 and70 m, (b) 70 and 141m, and(c) 15 and 141
m. Data arc plottedas solid½ir½1½s.
Model coherences
œor½as½
3a (hold dashed),½as½
3b (hold solid),and ½as½
3½(light
solid)arcalsoshown.Data abovethelightdashedlinearcs•gn•cant]ynonzeroat the90% confidence
coherencesis not as good; however, it does show significantly wave is formed.The underlyingpremisehere is that there is a
lower coherence at a smaller separation. Case 3c with the strong tendencyin the wave field toward random phases
variable waveguide scaling is similar to case 3a at low fre- among the modes but that the phase relationshipsbecome
quency;hencethe agreementwith the observationsis not very more deterministicin the vicinity of a reflectingsurface.
The bestway to apply this model to a data setis to solveit
good. However, at high frequencyin the 1.5- to 3-cph band
the model coherencehas a peak with a 180ø phase difference with the appropriateN profile.However,a qualitativeindicabetween 15 and 141 m which agrees well with the data, al- tion of the effecton the spectracan be obtainedby examining
though no better than case 3b. Case 3c also predicts a peak the results from two idealized N profiles (Table 1): constant
with a 180ø phasedifferencebetween 15 and 70 m; while this (case1) and a stepprofilewith N = 0 in the upper50 m mixed
does not quite agree with the observations,a similar feature layer (case2) (Figure 2). The effectof the surfaceboundary
alone can be examined in the constant N ocean (case 1); varidoesoccur at a slightly lower-frequencyband.
5.
SUMMARY AND CONCLUSIONS
The main purposeof this paper is to developa formalismto
calculate the effect of the surface boundary and the mixed
layer on the spectralquantitiesof the internal wave field. It is
assumedthat the deep ocean wave field follows the GarrettMunk description.The alteration of the wave field in the
upper oceanis assumedto be due to the verticalvariation of
N(z) and the presenceof a surfaceboundary.The structureof
each wave component is tracked as it propagatesinto the
upper oceanfrom the GM oceanbelow. All the wave components are then added together to produceestimatesof spectral
quantitiesin the upper ocean.
The results of the model are useful in interpreting upper
ocean measurementsby indicating the modificationsin spectra and coherencesthat can be attributed directly to the kinematic distortion of the deep ocean internal wave field. In
order to studyother aspectsof the upper oceanwave field it is
first necessaryto understandthe passiveeffect of the surface
and the upper oceanstratification.This is especiallyimportant
when comparingobservationsmade in regionswith radically
different N profiles.
The spectralfeaturesof the upper oceanmodel that differ
from the GM spectrumare due to the nonrandomalignment
in depth of the verticalwave functions.In the GM model the
phaserelationshipamong the vertical"modes"is assumedto
be random. This may be a resonableassumptionin the deep
ocean.However, near a reflectingboundary, suchas the ocean
surface or the base of the mixed layer, a vertically standing
ations of the spectralquantitiesin the vertical are causedby
the particularalignmentof all wave componentsand not fluctuationsin N(z). Specificfeaturesthat differ from GM are as
follows:(1) the deviationsfrom the GM spectrumand coherencesoccur near the surfacein an adjustmentregion,which is
smaller at higher N, (2) at the surface,vertical displacement
spectra
(MScand TSc)go to zero,andvelocityspectra
(MS,,
and TS,) are a maximum(Figure4), (3) "negative"coherences
are possible,and (4) verticalcoherences
of • (MVCc and
TVCc)decrease
towardthesurface
in theadjustment
region
while thoseof u (MVC, and TVC•) increase;very near the
surfaceall coherencesincrease(Figures 5 and 6).
The addition of a mixed layer (case2), whosedepth is small
relative to the horizontal wavelengthof the wave,causesthe 4•
wave functionsto be nearly uniform and the • wave functions
to be nearly linear in the mixed layer (Figure 3). Below the
mixed layer the verticalwavelengthof each componentis a
functiononly of N (equation(7)), and hencethe verticalwavelengthsin casesl a and lb are the sameas in cases2a and 2b,
respectively.
At lower wavenumbers(lowermodes)the mode
shapesfor the step profile are nearly the same as for the
constantN case(Figure 3). However,at higherwave number
(highermodes)the phaseof the verticalstructurefor the step
profile is significantlyshiftedrelative to the constantN case;
the depth of the shallowestzero crossinghas been shifted
deeperin cases2a and 2b comparedwith casesl a and lb for
the same mode. The major differencesin the spectralquantities betweenthe stepprofile cases(cases2a and 2b) and the
constantN cases(casesla and lb) occur in the mixed layer
5044
LEVINE: UPPER OCEAN INTERNAL WAVE FIELD
and can be summarized as follows: (1) in the mixed layer the
spectraof velocity(MS u and TSu)are nearly uniform, and the
spectral level is relatively higher for a lower value of deep N
at MILDEX. There is also evidenceof a frequencydependent
verticalwaveguideat high frequency(case3c).
(Figure4), (2) Spectraof verticaldisplacement
(MS• and TS•)
decreasein the mixed layer to zero at the surface(Figure 4),
and (3) All vertical coherencesare higher than the corresponding constant N casewhen at least one of the two depthsare in
the mixed layer (Figures7 and 8).
To apply the model to a realistic situation, the spectral
quantities were calculatedfor the N profile observedduring
the Mixed Layer DynamicsExperiment(MILDEX) (Figures2
and 9). The farther below the mixed layer, the more closelythe
Acknowledgments.
I thank JamesRichmanfor orchestratingthe
Drifter experimentand providingme with the currentmeterdata, and
Roland deSzoeke, Lloyd Regier, and Russ Davis for their role in
planning,preparing,and deployingthe Drifter. I alsothank Clayton
Paulsonfor allowing me to use the towed thermistorchain data, and
Robert Pinkel and Tim Stantonfor supplyingthe N profiledata. The
effortsof Eric Bealsin carryingout the numericalanalysisand Steve
Gard in preparing the final figures are greatly appreciated.Discussionswith Phyllis Stabeno and Teresa Chereskin were most useful.
The support by the Office of Naval Researchunder contract N0001484-C-0218is gratefullyacknowledged.
modeledautospectra
(TSc,MSc,TS•, MSs)followthe N scal-
REFERENCES
ing of the WKB approximation (Figure 12). The deviation
from N scaling is greatestjust below the mixed layer and, of
course,in the mixed layer itself where N scalingis not possible. It may seemsomewhatsurprisingthat the WKB approximation works as well as it doessincethere is significantvertical variation in N compared with the wave functions, especially at the energetic,low modes. This result implies that
significant deviations from N scaling of observedspectra are
Atkinson,K. E., An Introductionto NumericalAnalysis,p. 269, John
Wiley, New York, 1978.
Baumann, R. J., L. M. deWitt, C. A. Paulson, J. V. Paduan, and J. D.
Wagner, Towed thermistor chain observationsduring MILDEX,
Ref 85-12, 135 pp., Coll. of Oceanogr.,Oreg. StateUniv., Corvallis,
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Bell, T. H., The structureof internal wave spectraas determinedfrom
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due to real differencesin the GM model and not just kin- Desaubies,Y. J. F., Internal wavesnear the turning point, Geophys.
Fluid Mech., 5, 143-154, 1973.
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Garrett, C. J. R., and W. H. Munk, Space-time scalesof internal
Some features of the depth dependenceof the spectral
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modelcan be foundin the observed
spectra,TS• and MSs.
Estimates
of TS• at severaldepthsfollowsthe modelremark-
Garrett, C. J. R., and W. H. Munk, Space-time scalesof internal
waves:A progressreport, J. Geophys.Res.,80, 291-297, 1975.
ably well (Figure 11); the slight maximum at 50 m does not K/•se, R. H., and G. Siedler,Internal wave kinematicsin the upper
tropical Atlantic, Deep Sea Res.,26, suppl.,161-189, 1980.
occur in the N-scaled GM model. The comparison of the
Levine, M.D., R. A. deSzoeke,and P. P. Niiler, Internal waves in the
observedand modeled MS• is perhaps not as convincing;
upper oceanduring MILE, J. Phys.Oceanogr.,13, 240-257, 1983a.
however, some features of the model do occur in the data
Levine, M.D., C. A. Paulson, M. G. Briscoe, R. A. Weller, and H.
Peters, Internal waves in JASIN, Philos. Trans. R. Soc.London,Ser.
(Figure 10). For example,the observedspectrumin the mixed
A, 308, 389-405, 1983b.
layer tends to be uniform with depth. Some of the measured
Levine, M.D., S. R. Gard, and J. Simpkins,Thermistor chain observariability may be due to the variation of the structure of the
vationsduring MILDEX, Ref 84-9, 159 pp., Coil. of Oceanogr.,
mixed layer over the duration of the experiment; the increase
Oreg. State Univ., Corvallis, Oreg., 1984.
near the surfaceis probably due to direct atmosphericforcing. Levine,M.D., J. D. Irish, T. E. Ewart, and S. A. Reynolds,Simultaneousspatial and temporal measurementsof the internal wave field
At higher frequencythe data show lower spectral levels in the
during MATE, J. Geophys.Res.,91, 9709-9719, 1986.
mixed layer relative to 70 m. This behavior is mimicked by
Levitus, S., ClimatologicalAtlas of the World Ocean,NOAA Prof.
model case3c with the frequencydependentwaveguide.
Thefrequency
dependence
of themodeled
MSuandMS• are
Pap. 13, U.S. Departmentof Commerce,Washington,D.C., 1982.
Lindzen, R. S., and H.-L. Kuo, A reliable method for the numerical
compared with observationsin Figures 13 and 14. The model
integrationof a largeclassof ordinaryand partial differentialequa-
providesa reasonablefit to the observedMS• at 15 m, cer-
tions, Mon. Weather Rev., 97, 732-734, 1969.
tainly better than the G M spectrumwhich is undefined in the
mixed layer.
The greatest successof the model is in explaining the high
vertical coherencewith a 180ø phase between 15 and 141 m
(Figure 16). In the GM model the coherencedecreasesmonotonically as a function of vertical separation with zero phase
difference.Case 3a predicts the 180ø difference,but the coherenceis much too small. While a systematicinvestigationof the
effect of modifying the GM spectrumis beyond the scopeof
this paper, somesimple alterationsto the GM spectrumwere
made in cases3b and 3c. When mode 1 is eliminated (case3b),
the agreement between data and model is much improved;
this case also follows the data by predicting lower coherence
between
15 and 70 m and between 70 and 141 m than between
Milllet, P., D. J. Olbers, and J. Willebrand, The IWEX spetrum,J.
Geophys.Res., 83, 479-500, 1978.
Munk, W. H., Internal waves and small-scaleprocesses,in Evolution
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edited by B. A. Warren and C. Wunsch,
pp. 264-291, MIT Press,Cambridge,Mass., 1981.
Newberger,P. A., H. H. Dannelongue,and D. R. Caldwell,The rapid
samplingverticalprofiler,MILDEX, October-November1983,Ref.
84-4, 123 pp., Coil. of Oceanogr., Oreg. State Univ., Corvallis,
Oreg., 1984.
Pinkel, R., Upper ocean internal wave observationsfrom Flip, J.
Geophys.Res., 80, 3892-3910, 1975.
Pinkel, R., Observations of the near-surface internal wavefield, J.
Phys. Oceangr.,11, 1248-1257, 1981.
Pinkel, R., Doppler sonar observations of internal waves: The
wavenumber-frequency
spectrum,J. Phys. Oceanogr.,14, 12491270, 1984.
Willebrand,J., P. Miiller, and D. J. Olbers,Inverseanalysisof the
trimooredinternal wave experiment(IWEX), Number20a, Ber. aus
15 and 141 m. At high frequency the frequency dependent
dem Inst. fiir Meereskunde an der Christ.-Albrechts-Univ.,Kiel,
waveguide(case3c) provides a reasonablefit to the data. This
FederalRepublicof Germany, 1977.
is consistent with the improved agreement in case 3c of the
M. D. Levine, Collegeof Oceanography,Oregon State University,
velocity spectral level MS• in the mixed layer (Figure 10).
Corvallis, OR 97331.
Overall, case 3b with no contribution from mode 1 fit the data
better than 3a. This suggeststhat the lowest horizontal wave
(ReceivedAugust 18, 1986;
acceptedJanuary 14, 1987.)
number accordingto the GM scalingis not the most energetic