JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. C9, PAGES 19,715-19,729,SEPTEMBER 15, 2001
Observationsand modeling of coastalinternal wavesdriven
by a diurnal sea breeze
J. A. Lerczak
Department
of PhysicalOceanography,
WoodsHoleOceanographic
Institution,
WoodsHole,
Massachussets,USA
M. C. Hendershott
and C. D. Winant
ScrippsInstitutionof Oceanography,
Universityof California,SanDiego,La Jolla,California,USA
Abstract. DuringtheInternalWaveson the ContinentalMargin (IWAVES) field experimentsof 1996 and 1997 off of MissionBeach,California(32.75øN), we observedenergetic,
diurnal-bandmotionsacrossthe entirestudysitein waterdepthsrangingfrom 15 to 500 rn
andspanning
a cross-shore
distance
of 15 km. The spectral
peakof thecurrentswasat the
diurnalfrequency
(av• = 1 cpd)andwassufficiently
wellresolved
to beclearlyseparated
from the slightlyhigherlocalinertialfrequency(f = 1.08 cpd). Thesemotionswere
surfaceenhanced
andclockwisecircularlypolarizedandhadan upwardphasepropagation
speed
of• 68md-l, suggesting
thatthemotions
weredriven
predominantly
bythediurnal
seabreeze. However,the downwardenergy(upwardphase)propagationseemsirreconcilablewith the subinertialdiurnalperiod,and moreover,the intermittentdiurnalcurrent
eventswerenot obviouslyassociated
with diurnalseabreezeevents.We rationalizethese
features
usinga flat-bottomed
linearmodalsuminternalwavemodelthatincludesadvection
andrefraction
dueto subtidalalongshore
flow,V(z, t). Fluctuations
in V at theobserving
sitecanchangethe "effective"local Coriolisparameterf + Vz by as muchas 50%, thus
makingthediurnalmotionsat differenttimeseffectivelyeithersubinertial
or superinertial.
Themodelis integrated
numerically
for 200 daysat a latitudeof 32.75øNunderdifferent
windandsubtidalflow conditions:
purelydiurnalwindsandno V, purelydiurnalwinds
anda time-independent
V, narrow-band
diurnalwindsandno V, andnarrow-band
diurnal
windsand subtidal,time-dependent
V. Model diurnalcurrentsforcedby narrow-band
diurnalwindsandsubtidalV showcomplexoffshorestructurewith realisticintermittency
andspectralbroadening.This studysuggests
thatcontinental
marginsin the vicinityof
the30ø latitude(whereav• = f) areregionsthatcouldpotentially
produceenergetic,
sea
breeze-driven
baroclinicmotionsandthatthesemotionscouldbe regulatedby the vorticity
of the local subtidal currents.
1. Introduction
Duringthe 1996 and 1997 InternalWaveson the ContinentalMargin(IWAVES) fieldstudiesoff of MissionBeach,
California, diurnal-band(definedhere as between0.727 and
1.33 cyclesper day) currentswere energetic,with ampli-
tudesaslargeas25 cm s-•. Theirstructure
wassimilar
to that of near-inertialmotionsobservedin the open ocean
[Leaman, 1976; Kunzeand Sanford, 1984; D'Asaro, 1984]
and on continentalshelves[Kundu, 1976; Denbo and Allen,
1984]. Currents were enhanced at the surface and were
clockwisepolarized(v led u by 90ø). Lines of constant
phasepropagatedtowardthe surface,suggestinga downwardenergyflux and a surfacesourcefor the motions.For
Copyright2001 by theAmericanGeophysicalUnion.
Papernumber2001JC000811.
0148-0227/01/2001JC000811 $09.00
thisreason,we believethatthesemotionswerenotforcedby
the diurnalsurfacetide but by the local diurnalseabreeze.
In contrast to near-inertial motions observed elsewhere,
currentvarianceobservedduring IWAVES was not peaked
at or slightlyabovef (1.08 cpd,latitude- 32.75ø) but was
peakedat the slightlysubinertialdiurnalfrequency(av• = 1
cpd). In the open oceanand in many coastalregionswith
strongwinds the wind forcing tendsto be broad-band,and
the nearlyresonantinertial frequencyis excitedmore effectively than other frequencies.At the IWAVES study site,
however,windswere generallyweak but had a sharppeak
at av•. Becausethe wind forcingwaspredominantlyat this
singlefrequency,the responseof the oceanwas peakedat
thatfrequency.
The strongresponseof the coastaloceanto the seabreeze
duringIWAVES remainssurprisingbecauseav• is slightly
subinertialand only a weak evanescentresponsewithin the
mixed layer would be expected.During IWAVES, however,
19,715
19,716
LERCZAK
ETAL.'
INTERNAL
WAVES DRIVEN
BY A DIURNAL
SEA BREEZE
diurnalcurrentspenetratedconsiderably
below the mixed
layer and consistentlyhad an upward phasepropagation,
which is unexpectedfor evanescent,subinertialmotions.
Moreover, diurnal motions were much more intermittent
than would be expectedfor the regularforcingof the sea
breeze. Consequently,
the diurnalspectralpeakof the currentswas muchbroaderthan the sharpspectralpeak of the
wind.
We proposethatlow-frequency
background
currentsplay
a critical role in settinghow effectivethe sea breezeis at
pumpingenergyinto the ocean. A mesoscale
eddy field
can significantlychangethe dynamicsof near-inertialmotions.In particular,therelativevorticityof theeddyfieldcan
changethe effectiveCoriolisparameter"felt" by the near- Figure 1. Internal Waves on the Continental Margin
inertial motions, and this can have several results: the hor- (IWAVES) study site. Circles mark the locationsof the
izontal spatialscaleof the near inertial motionscan be set mooringsof the summerarrays(Table 1). Opencirclesmark
by the horizontalscaleof the eddy field [Balmforthet al., mooringsdeployedin the summerof 1996; solid circles
1998]; the rate of dispersionof near-inertialenergycanbe mark summer1997 moorings;shadedcirclesmarkmoorings
deployedin the summersof bothyears.Depthsare givenin
enhanced[Balmforthet al., 1998; van Meurs, 1998]; trapmeters. The 1996 and 1997 fall mooringdeploymentsare
ping andamplificationof near-inertialmotionscanoccurin not shown(see Table 1).
regionsof negativerelativevorticity[Kunze,1985;D'Asaro,
1995]; andspectralbroadeningof the near-inertialpeakcan
occurdue to temporalchangesin the backgroundvorticity 2. Field Studies
field [D'Asaro, 1995].
During1996and1997,arraysof moorings
weredeployed
Low-frequencycurrentsobservedduringIWAVES were
predominantlyorientedin the alongshore
directionandhad off of MissionBeach,California(Figure1), twiceeachyear
From approximately
the end
amplitudes
ashighas50 cms-1. Thevorticity
of these
cur- with differentconfigurations.
of Juneto the end of Augusteachyear,the arrayspanned
rentsVx changedslowlyovertime andrangedfrom +0.5 f.
range
This rangewas sufficient,at times, to changethe effective depthsfrom 15 to 500 m, coveringa cross-shore
Coriolis parameterfrom being subinertialto superinertial of • 15 km. Duringthe late summerandearlyfall, moordepthsrangand can explainmuchof the intermittencyobservedin the ingsweretightlyspacedin shallow,nearshore
diurnal currents.
ingfrom 15 to 30 m, withtheexception
of the 100m moor-
We begin,in section2, by givinga brief descriptionof the ing deployedin the fall of 1996(Table1). This studyfoarraysdeployedin
IWAVES field studiesand the data used in theseanalyses. cuseson datafrom thebroadshelf/slope
In section3 we describethe spatialand temporalstructure the summer and fall of 1996 and the summer of 1997.
of the diurnal-bandcurrentsobservedduringIWAVES. In
Eachmooringwasinstrumented
with at leastoneacoustic
section4 we describethealongshore
low-frequency
currents dopplercurrentprofiler(ADCP) to measure
thethreecomand their vorticity. In section5 we summarizethe nature ponents
of velocityas a functionof depth. On the shelf,
of the windsin the vicinity of the IWAVES studysite. We between67 and 80% of the water column was coveredby
concludethat over the cross-shore
rangeof IWAVES (0-15 the ADCPs (Table 1). On the slopethe full watercolumn
km from the coast)the diurnalwinds were nearlylinearly couldnot be sampledbecauseof ADCP rangelimitations.
polarizedand orientedin the cross-shoredirectionwith a In 1996thedeeperportionof thewatercolumnwassampled
phasethat was roughlyconstantwith cross-shore
distance. at the350 m mooring.In 1997two ADCPs(a deepupward
Then, in section6, we describea simplelinearmodelwith lookingoneanda shallowdownward
lookingone)weredewhich we attemptto explain someaspectsof the diurnal ployedonthe350 and500 m mooringlines.Together
they
currentsobservedduringIWAVES. In the model, a coastal coveredapproximately
theupper50% of thewatercolumn
oceanwith a flat bottomis drivenby a cross-shore,
diurnal at eachmooring.Verticalresolutionrangedfrom4 to 16 m
wind actingover the mixed layer and decayingaway from on theslopeand1 to 4 m on theshelf. Samplingintervals
the coast.A barotropicjet flowsin the alongshore
direction. rangedfrom 1 to 4 min. While not reportedon here,sevThe field variablesandtheforcingaredecomposed
intover- eral temperatureloggerswere alsodeployedon eachof the
tical modes,and we assumethereis no alongshore
depen- mooringlines.
Wind speedanddirectionweremonitoredfromtheScripps
dence.First, we describethe casewith purelydiurnaltime
dependence.Next, we describea seriesof time-dependent Institutionof Oceanography(SIO) pier (• 12 km north of
simulations
in whichthediurnal-band
windsarecomparable the studysite,Figure 1) duringthe experiments.Wind data
to thoseobservedduringIWAVES, and the alongshore
jet were also obtained from the SIO Coastal Data Informavariesin amplitudeslowlyover time. A comparison
of the tion Programas well as from National Data Buoy Center
observed and model diurnal-band currents follows in section
(NDBC) and SIO Marine Observatorybuoysin the vicin7.
ity of the IWAVES studysite (Table 2). In addition,con-
LERCZAK ETAL.' INTERNAL WAVES DRIVEN BY A DIURNAL SEA BREEZE
19,717
Table 1. Parameters
of ADCPs DeployedDuringIWAVES a
Mooring
Depth,m
Alongshore
Direction,deg
ADCP
DepthRange,m
Vertical
Resolution,m
Sample
Time, min
Pings/
Sample
ADCP
Frequency,
kHz
Summer 1996
350
100
70
30 c
30 n
30 s
15
337
354
348
351
351
351
360
151-327
20-88
6-62
4-24
4-24
4-24
2-13
100
30
27.5
25
22.5
20
17.5
15
354
351
353
355
356
358
359
360
14-90
4-26
6-22
6-20
5-17
4-14
4-12
2-13
500 u
5001
350 u
3501
120
30
15
359
347
351
360
17-85
143-279
13-89
116-196
24-108
3-27
2-13
30
30
25
25
20
20
15
15
351
351
355
355
358
358
360
360
4-26
3-27
4-22
4-22
2-17
2-17
1-13
2-13
s
n
s
n
s
n
s
n
337
16
4
4
2
2
2
1
Fall 1996
4
2
2
2
2
2
2
1
Summer 1997
4
8
4
4
4
2
1
Fall 1997
2
2
2
2
1
1
1
1
4
1.5
1
1
1
1
1
8
30
24
30
30
30
30
150
300
300
300
300
300
1200
2
1
1
1
1
1
1
1
32
30
30
30
30
30
30
30
150
300
300
300
300
300
300
1200
1
4
1
2
2
1
1
25
13
25
45
45
40
40
300
150
300
300
300
300
1200
2
1
1
1
2
2
2
1
64
107
111
111
260
260
185
30
150
300
300
300
300
300
300
1200
aThelettersc, n, ands designate
mooringsdeployedat, northof, andsouthof the centralonshore/offshore
mooringline.
In 1996northandsouthmooringsat the30 m isobathwere• 1.2 and0.66 kin, respectively,
from thecentermooring.In 1997
the n and s mooringswere separatedby 0.5 km. AcousticDopplerCurrentProfilers(ADCPs) were at the bottom,looking
upward,exceptthat the lettersu and 1 indicatewhere upper (downwardlooking) and lower (upwardlooking) ADCPs,
respectively,
weredeployedon the500 and350 m mooringlinesin the summerof 1997.
tinuousconductivity-temperature-depth
(CTD) yo-yoswere structure
wasobserved
in the spectrafromtheotherdeployconducted
at the mooringlocationson numerousoccasions ments.To mimimizethe spreading
of thediurnalfrequency
for periodsrangingfrom 4 to 24 hours.
intoneighboring
bins,spectrawerecalculated
usinga time
blockthatwasa multipleof 24 hours.Thefull recordlength
3. Descriptionof Diurnal Currents
of eachADCP time seriesto within 24 hourswas used,and a
We definethe diurnalbandasbeingbetween0.727 and
1.33cpd(1/33 and1/18 cph). Thisfrequency
bandincludes
bothrrr,• (1 cpd)andf (1.08cpd).A 5 daytimeseries
of thediurnal-band
currents
at the70 and100m moorings
duringthesummer
of 1996IWAVESdeployment
(Figure2)
Table 2. Wind Stations a
Station
Offshore
Years
Distance,km
Analyzed
1992-1993, 1999
showsmanyof the salientfeaturesof the diurnal-bandcur-
rents.Currents
weregreatest
atthesurface
anddecayed
with
NDBC 46047
220
increasing
depth.Alongshore
currents
v led cross-shore
cur-
San Clemente
110
1998-1999
rentsu by90ø;thatis,thecurrents
wereclockwise
polarized.
Therewasa well-defined
upwardphasepropagation
of the
currents.
Thethickdashed
linesin Figures2a and2bhavean
NDBC 46048
60
1992-1993
Pt La Jolla
8
1998-1999
SIO pier
0
1996-1999
upward
slope
of 50rnd-•. Thisupward
phase
propagation aData from buoys NDBC 46047 and NDBC 46048
isconsistent
witha downward
energyfluxandsuggests
that wereobtainedfrom the NationalData BuoyCenter. San
the source of these motions was at the surface.
3.1. Diurnal-BandRotary Spectra
Rotary spectraof the currentsfrom the summerof 1997
IWAVESdeployment
areplottedinFigure3. Thesamebasic
Clemente and Point La Jolla data were from SIO Marine
Observatory
buoys.The SIO pier dataof 1996and 1997
were from IWAVES observations. The data from 1998 and
1999 were from the SIO Coastal Data Information Program.
19,718
LERCZAK
ETAL.:
INTERNAL
WAVES DRIVEN
BY A D1URNAL
SEA BREEZE
3.2. Variance VersusDepth and Cross-ShoreDistance
Thevariance
ofthediurnal
currents,
{u2)+ Iv2),typically
decreasedwith depth (Figure 4). The only exceptionwas
the shallowest(15 m) mooring of the fall of 1996 (Figure
4b), wherethe variancenearthe bottomwas as high as that
nearthe surface.In 1996 the variancedroppedto < 0.25 its
maximumvalueat a depthof --•50 m. In 1997 the variance
decreasedmore slowly with depth. At the 120 m mooring,
for example,
(u2) + (v2) was< 0.25 its maximum
value
•-] lO
,:.)o
.......0
•:•
1oo
at a depthof ,-• 80 m. The variancesat the 120 and 350 m
mooringsin the summerof 1997 (Figure 4c) trackedeach
othercloselywith depth.At the 500 m mooringtherewas a
subsurface
maximumat a depthof ,-• 37 m.
For H < 100 m, there was a clear increase in variance
-10
with distancefrom the coast.In the summerof 1996 (Figure
4a),forexample,
(u2) + (v2) nearthesurface
approximately
doubledfrom the 30 to 70 m mooring(Az = 4.0 km) and
from the 70 to the 100 m mooring(Ax = 3.4 km).
100L ................
233
234
•
235
.............
..)----........
236
.,.---..........
237
-10
238
Days (1996)
Clockwise
Counterclockwise
120 rn
Figure2. Five daytime seriesof durnal-band
(0.727-1.33
cpd)currents
versus
depthat the70 and100m moorings
of
the summerof 1996 IWAVES deployment.(a) Cross-shore
currentsu and(b) alongshore
currentsv at the70 m mooring.
(c) u and (d) v at the 100 m mooring. The thick vertical
linesin Figures2a and2b marka time whenthe surfaceu
at the70 m mooringwasmaximumonshoreandthesurface
v waschangingfrom beingnorthwardto southward;
thatis,
the currentswere clockwisepolarized. The samewas true
for the currentsat the 100 m mooringas indicatedby the
bold verticallinesin Figures2c and2d. The gray scaleis in
ß
I I
I•' [
ß
800
400
I
o
0.6
0.8
1
1.2
o
0.6
1.4
0.8
1
1.2
1.4
350 rn
1200
cms-1. Zerocurrentis indicated
by thesolidcontours.
The
'd)lI
dashed
lineshaveanupward
slopeof 50 rnd-1 .
ß
800
"
•o
I
•
_
19m
'
,
400
Hanningwindowwasappliedto thetime seriesbeforecalculating the spectra.To maximizethe spectralresolution,the
spectrawere not ensembleaveragedin time. Rotary spectra
are plottedfor five differentADCP bins rangingfrom near
the surface to near the bottom of the water column of each
mooring. The range of the diurnal band is markedby the
verticaldashedlinesin eachpanel.
For all moorings,clockwiseenergy(CW) dominatedover
counterclockwiseenergy(CCW). Variancewithin the diurnal bandtypicallydecreased
with increasingdepth.Variance
was typically peakedat or near the the diurnal frequency.
This resultwas significantas the verticallyaveraged,clockwiserotaryspectraof all mooringsandall deployments(not
just thoseplottedin Figure 3) had peaksat the diurnal frequencyand not at the inertial frequency. This was different from the spectralshapeoftenobservedfor near-inertial
motions,for which the peak in energyis at or a few percenthigherthanthe inertialfrequency[D'Asaro et al., 1995;
Baines,1986]. While energyduringIWAVES wastypically
maximumat thediurnalfrequency,thepeakswerebroadand
spannedmuch of the diurnalband. In Figure 3 this is most
evidentin the spectrafrom the 500 m mooring.
-7 - •
,
i -
0
0.6
0.8
1
1.2
0.6
1.4
--"•
0.8
1
! 27m_
[ 51m
i 158rr
i 194rr
'
1.2
1.4
500 rn
ß
1600[ ,'•
/L_.LI
e" •/ r/•
' '
I,;
•_•..-••
'•ø/
_/
'
800
I'
ol
0.6
I
1
.
.
.
1600
II
AO
800
39 m
'
0.8
1
1.2
Cyclesper day
1.4
59 m:
. 203•r
0
0.6
ß
0.8
1 ' 112
I
2.75
1.4
Cyclesper day
Figure 3. Rotary spectraof diurnal-bandcurrentsfrom the
(a-b) 120, (c-d) 350, and (e-f) 500 m mooringsof the summerof 1997IWAVES deploymentThe dashedverticallines
mark the frequencyrangeof the diurnalbanddefinedfor this
study(0.727-1.33 cpd). Spectraare plottedfor five ADCP
bins for eachmooring. Depthsof the bins are indicatedon
the right side of the right panels. The spectralresolution
Aa of thespectrais indicatedat thetoprightcornerof each
panel. The diurnaland inertialfrequenciesare indicatedby
thick solidlines at the top of eachpanel.
LERCZAK ETAL.' INTERNAL WAVES DRIVEN BY A DIURNAL SEA BREEZE
19,719
latedthe phaseslope•buzversusdepthfor the cross-shore,
diurnal-bandcurrentsof the IWAVES moorings. The rela-
tivephases
ACu,iandsquared
coherence
p2between
neigh-
•150--
--350m
[ --30m
] [/• •120m
150
250
0
II
--100m
150[
/(
25ot F•I1996 • 25o[fi
Su•er
1997
d0
;0
boring ADCP bins were averagedover the diurnal band,
and phaseslopeswere estimatedby centereddifferences.
Fowardandbackwarddifferenceswereusedto calculate•b•z
at the lowestanduppermostADCP bins,respectively.Resultsare summarizedin Figure5.
In the upper100 m of the watercolumn,•b• wasalways
greaterthanzero(upwardphasepropagation)
anddecreased
with increasingdepth. In the upper50 m, wherethe diurnal currentswere mostenergetic,the average•b• of the six
•
0
20
c•/s •
40
•
0
20
c•/s •
40
•
c•/s •
Figure4. Average
diurnal-band
variance
({u2) + {v2})ver-
of Figure5 was5.3øm-• (verticaldashed
linein
susdepthat selectedmooringsof theIWAVESdeployments moorings
Figures5a-5c) corresponding
to an upward, diurnal phase
of the (a) summerand (b) fall of 1996 and (c) summerof
1997. The diurnal band is definedas the frequenciesbetween0.727 and 1.33 cpd.
speed
of t58rnd-•. Apparently,
thephaseslopein theupper 50 m washigherduringthe 1996 deployments(Figures
5a and5b, average
•b•,•was6.0øm-1) thanthesummer
of
1997deployment
(Figure5c,average
•b•,•was4.6ørn-1).
3.3. Polarization and Ellipticity
The rotary spectra(Figure 3) and 5 day time series(Figure2) indicatethatdiurnalCW energydominatesoverCCW
energyin the diurnalband. To make this assessment
more
quantitative,
we calculatedCW/CCW averagedoverthe
upper50 m of the watercolumn(thedepthrangeoverwhich
diurnalbandenergytendedto be high):
CW
CWi
CCW
= •1k CCWi
'
i--1
(1)
Below a depthof 100 m (Figure5c), •b•,zwas somewhat
lower thanit washigherin the watercolumn.At the 500 m
mooringin the summerof 1997, •b•,•was negative(downwardphasepropagation)
overthe depthrange150-210m.
3.5. Intermittency of the Diurnal Currents
Diurnalcurrenteventswereverysimilarat adjacentmooringsbut were intermittentin time. This is apparentin runningestimatesof verticallyaveraged,diurnal-band,horizontal kineticenergy(KE, Figures6-8), calculatedaccordingto
wherethe sumis overtheADCP binswithin theupper50 m
Table 3. Clockwiseto Counterclockwise
EnergyRatio and
Ellipticty
of
Diurnal-Band
Currents
a
andcounterclockwise
variancesat depthbin i, respectively,
of the water column and CWi and CCWi are the clockwise
summed over the diurnal band. For H
over all ADCP
< 30 m the sum is
bins. Results are summarized
in Table 3. For
all IWAVESdeployments,
CW/CCW decreased
fromthe
offshore
moorings
to thecoast.ForH > 30 m, CW/CCW
was often > 10, and was between 1 and 2.5 for H < 30 m.
The squareof the ratio of the minor to major diurnalcur-
rentellipseaxes(e-2) andthe majoraxisorientation
(0)
were averagedover the upper50 m of the water columnat
eachmooringin a similarmannerto CW/CCW (Table3):
1 N
e--20--•
i=1
ei20i'
(2)
Like CW/CCW, e-2 decreased
fromthe offshoremoor-
Standard
Depth,
m
CW
Deployment c cw
e-
2
0, deg Deviation N
0, deg
15
15
15
15
30
30
30
summer 1996
fall 1996
summer 1997
fall 1997
summer 1996
fall 1996
summer 1997
2.5
2.0
1.2
2.2
2.3
2.2
1.6
0.34
0.52
0.29
0.39
0.52
0.61
0.60
30
70
fall 1997
summer 1996
2.0
15.9
0.61
0.90
94
104
99
94
94
120
99
74
.........
100
summer 1996
15.7
0.86
.........
100
fall 1996
9.1
0.74
.........
120
350
summer 1997
summer 1997
6.1
7.6
0.73
0.77
.........
.........
500
summer 1997
12.8
0.69
.........
4.8
15
4.2
7.6
8.0
16
21
10
11
11
11
10
11
12
14
12
ings to the coast. When the currentswere clockwise,cir-
cularlypolarized
(CW/CCW >> 1, H > 30 m), e-2 _> 1.
aCW/CCW is the ratio of clockwise to counterclockwise
Whilethecurrents
at H _<30 m wereclockwisepolarized energy;
e-2 isthesquare
oftheratioof theminortomajor
(CW/CCW > 1), theyweremoreellipticalthanthecur- current
ellipseaxes;and0 is theorientation
of themajor
rentsfartheroffshore.For thenearshore
moorings
(H _<30 ellipseaxis(0) of the diurnal-bandcurrentsof the IWAVES
m), stableestimates
of 0 couldbeobtained,andthemajorel- deployments.
All valueswereaveraged
overtheADCPbins
lipseaxiswasorientedin thealongshore
direction(0 m 90ø) withintheupper50 m of thewatercolumn.Ellipseorientafor all IWAVESdeployments.
tionis only shownfor H _<30 m. At thosemooringlocations,e-2 wassmall,and0 wasstableovertheADCP bins.
3.4. VerticalPhaseSlope,Oq•,/Oz
The standard deviation of 0 is over the number of ADCP
Linesof constant
phaseof thediurnalcurrents
propagatedbins(N) at eachmooring.Whenthemajoraxisis oriented
upwardin the water column(Figure 2). We have calcu- in the alongshoredirection,0 - 90ø.
19,720
LERCZAK ETAL.' INTERNAL WAVES DRIVEN BY A DIURNAL SEA BREEZE
Summer1996
Fall 1996
50
d•/dz
'•
Summer1997
50
t(dCg'/m)'
• II
At the 70 m mooringin the summerof 1996, for example,
thevertically
averaged
alongshore
variance
was103cm2s-2,
50
while the verticallyaveragedcross-shore
variancewas only
5.5 cm2s-2. At times,V waseffectively
barotropic,
while
at other times, it was sheared in the vertical, with currents
'.
/ ......
-5
0
5
•
.o
-5
i)
0
5
10
/ . •t.
-5
0
flowingin oppositedirectionsat differentdepths.
We haveestimatedthe vorticityof the low-frequencycurrents by finite differencingthe vertically averagedV be'
• •.•
• ' 10 tweenneighboringmooringpairs:
50•
o•2
...• 50
o[----7
150 •
250
[
0
l•m
'
150 •
0'5 '
e.)
0
0.5
0
,
(4)
whereV/are the verticallyaveragedV and Az is the crossshoreseparation
of the mooringpair. Verticalaveragingwas
donetwo differentways' over the upper40 m of the water
columnand over the upper 100 m of the water column(the
entire water column on the shelf). We averagedthesetwo
waysto determinehow significantlythe verticalshearof V
l•m
d.) 250
V2 - V•
Vx =
0.5
affected the estimate of Vx.
In the modelwe presentsubsequently,
the sizeof f + V•
Figure 5. (a-c) Diurnal-bandverticalphaseslope•b•zat serelative
to
f,
which
we
define
as
F
(=
1
+
V•
/ f), isa dynamlectedIWAVES moorings.The verticaldashedlinesindicate
the•buz(5.3øm-1) averaged
overtheupper50m of thewater columnandoverthe six mooringsplotted.(d-f) Average
diurnal-band
coherencies
squared
(/92)fromtheneighboring
ADCP pairsusedto calculate½uz.
N
KE-
i=1
+ (Q)'
(3)
where the sum is over ADCP bins in the upper 100 m of
the watercolumn.The temporalaveraging,indicatedby the
angledbrackets,wasover2 day time blocks,andan estimate
of KE wasmadeevery6 hours.
An increasein KE after day 205 in the summerof 1996
was apparentat the 30, 70 and 100 m moorings(Figure6a).
Modulationsof KE with a timescaleof ,-• 5 - 10 days at
the 100 and 70 m mooringstrackedeach other closely. In
the fall of 1996,KE wasmuchhigherat the 100 m mooring
than at the 30 m mooring(Figure 7a), with three energetic
pulsescenteredat days264, 280, and 302.
A decreasein diurnalenergytowardthecoastwasalsoapparentin the KE time series.In the summerof 1997 (Figure
8a), for example,currentswere most energeticat the 500
m mooring. The two pulsesof enhancedKE centeredon
days183 and208 wereapparentfor thethreemooringsplotted. The pulsesappearedto propagateoffshore.The maximum of the first pulse occurredat day 182.1, 182.6, and
183.15for the 120, 350, and500 m moorings,respectively
(indicatedby the dots abovethe peaks). The correspond-
1.0
.....
0.5I- •
180
,
,
200
•
•'
220
V
•
,
:
240
days(1996)
ingoffshore
speeds
were4.8 and9.7 cms-1 forthe120/350
kineticenergyvertim and 350/500 m mooringpairs,respectively.Similar off- Figure 6. (a) Runningdiurnal-band
cally averagedover all ADCP bins from the summerof
shorespeedswere estimatedfor the secondpulse(4.9 and 1996 IWAVES deployment.Runningestimateswere aver10cms-1 forthe120/350m and350/500mmooring
pairs, agedovertime blocks2 daysin length. An estimatewas
respectively).
madeevery6 hours.(b) Low-frequency,
alongshore
currents
V averagedoverthe upper40 m of the watercolumn. (c)
F = 1 + V•/f withV fromFigure6b. (d) V averaged
over
the entirewatercolumn.(e) F = 1 + V•/f with V from
Low-frequency
currents
(or< 0.727cpd)werepredomi- Figure6d. The thick horizontallinesin Figures6c and6e
2If2(=
nantlyoriented
in thealongshore
direction
duringIWAVES. markthevalueof err>
•
4. Low-Frequency
CurrentsandVorticity
LERCZAK ETAL.' INTERNAL WAVES DRIVEN BY A DIURNAL SEA BREEZE
i
!
30a.)
i
19,721
100
m
• •0
10
0
•' .mLb)
'-- 100m
• 0•.•
ß
i
/'x
'
,
•
i
i
i
[ bII)
c• 20 [-
•':
•
o[... 7"•' ..... %
!
0.5
'
'
'
x
5d0m
I
'
350m
.......... •'
' }
,. ^
,'.-.•
'/•/N•
/NA \
'
•/I ',M.\...... • d..W/... :,..•%•.'....•..
i
.-20
• 1.5c.)
' 'I
•
I
I
'
i
I
I
'
I
I
•Wf--120/350
ii
350 / 5OO
•' 1.5
i
260
i
i
280
i
I
i
300
•-- 1.0
days(1996)
Figure 7. SameasFigure 6, but for the fall of 1996 deploy-
•
0.5
ie:)Jun:.•,.*.•.•lY,•,•.,•••
/.iugust
'
ø2'I
2
I
ment.
1+ Vx/f
.
I
I
.
I
180
m
I
200
220
......
-
350 / 5OO
240
days(1997)
2 if2 (= 0.85),an Figure 8. Same as Figure 6, but for the summerof 1997
icallyimportant
quantity.
WhenF > ao•
deployment.At the 500 and350 m moorings,diurnalKE
evanescent
response
to the diurnalwindsis expectedin the wasaveraged
overADCP binsin theupper100m of thewa2 /f2, diurnalinternal tercolumn,andV andI' of Figures8d and8e,respectively,
coastal
ocean,
whereas
whenF < ao•
wavescanbegenerated
by thewinds.We plotF versustime, werealsoaveraged
overtheupper100m. Thedotsin Figure
estimatedfor theIWAVES deployments,
in Figures6-8. For 8a indicatethetimesof maximumKE at thethreemoorings
thesedeployments,
F did notappearto be verysensitive
to for thetwo energeticpulsesreferredto in thetext.
whetherV wasaveragedovertheupper40 m or theupper
100 m.
using21 yearsof measurements
from shipsandfrom land
In the summerof 1996 (Figures6c and6e), F, measured
stationsat the San Diego Airport andthe northwesttip of
betweenthe 70 and 100 m moorings,rangedbetween0.8
SanClementeIsland. We extendedthe diurnalanalysisof
and1.5for thefirst30 daysof thedeployment.
At day208,
Dorman [1982] by studyingsummerwind measurements
2•/f2 andstayed
belowformostof the
F dropped
belowat,
remainderof thedeployment.Thiswasparticularlyevident
in Figure6e. Thiswasthesameperiodwhendiurnalenergy
wasmostenergetic(Figure 6a).
NDBC 46047 (220 km from coast)
SIO pier
b.)
a.)
In thesummer
of 1997' F dipped
below0'2
Di /f2 between
Maj. diurnal
days180-190and203-212.Thiswasparticularly
evidentin
axis
Figure8c. Thesewereroughlythesametimeswhenthetwo
pulsesof enhanced
diurnalenergyoccurred(Figure8a).
However,
coincidence
of F dropping
belowcr2o/f2and
enhanceddiurnalenergydid not alwaysoccur. For exam-
ple,inthefallof 1996(Figure
7),F dropped
belowat,•
twice, arounddays 269 and 292. In contrastto the summer deployments,
diurnal-band
energyat the 100 m moor-
IMindiurnal
axis
I
,,
1
2
0
•lll___l
_,A
1
2
cycles Der day
cycles Der day
ingwasrelativelylowduringthesetwotimeperiods(Figure
7a). In fact, diurnal-band
KE washighestduringperiods Figure9. Spectra
of windvelocityat the(a) SIOpierand
whenF > 1, e.g., days260-268,274-285, and298-305.
(b) NDBC buoy46047 in the summerof 1999. Windswere
rotatedintomajor(thickline) andminor(thinline)axesof
5. Diurnal Winds off of Southern California
thediurnal-band
variance.Themajoraxeswereorientedat
Dotman [1982] characterized the winds between San
angles
of 1015
ø and7t5
ø relativeto truenorthin Figures9a
and9b, respectively.The diurnalband,as definedfor this
DiegoandSanClemente
Island(,-•110km fromthecoast) study,is indicatedby theverticaldashedlines.
19,722
LERCZAK ETAL.' INTERNAL WAVESDRIVEN BY A DIURNAL SEA BREEZE
obtainedfrom the SIO pier (Figure 1) andvariousmeteoroa.)
logicalbuoys(Table2) rangingfrom a few kilometersfrom
the coastto • 220 km offshore. Details of the analysisare
givenby Lerczak[2000].
Representativewind spectraare shownin Figure 9 (lowi
frequencyvarianceis plotted in additionto diurnal-band
variance).Over the summermonths,spectrawere averaged
overthree25 day longensembles.The time serieswerenot • 24
b.) ,
windowedbeforecalculatingthe spectra.
The diurnalpeaksin the wind spectraweremuchsharper • 20
than the corresponding
peaksobservedin the currentspec•
16
tra (Figure 3). Diurnal variability was predominantlyorientedin the major axisdirection;the ratio of majorto minor
12
axisvarianceat the diurnalfrequencywas60 at the SIO pier
and 8.2 at NDBC 46047. The diurnal-bandRMS amplitude,
1.0
' o 1992
however,
wassignificantly
lessatNDBC46047(0.43m s-1)
compared
totheSIOpier(0.88m s-•).
A runningharmonicanalysisof the diurnal(24 hourperiod) windsdemonstrates
thestabilityof amplitudeandphase
over time, especiallyat the coast. Resultsfor the SIO pier
and NDBC
46047
c.)
0
i
d.)
10a). The phase,however,wasremarkablystableovertime
(Figure 10b). Maximum onshorewinds occurred2.7 hours
after local noon. At NDBC 46047 the amplitudeof the di-
....
onshore winds at NDBC
180
i
i
....
o
i
i
i
......... * ......
'
22O
'
110
'
6O
m
8
,
0
Distancefrom shore(km)
46047 occurred 9 hours
after local noon (6.2 hoursafter maximum onshorewinds at
the SIO pier).
Diurnal-bandwind statisticsfrom otheryearsandat other
wind stationsare summarizedin Figure 11. The diurnalbandRMS amplitudeof the majorellipseaxiswindsranged
i
ß 1998
1999
o
windsvariedfrom0.8 to 1.7m s-• overthesummer
(Figure
maximum
i
+ 1993
0.5
in Figure 10. At the SIO pier the amplitudeof the diurnal
was somewhatmore variable than at the pier. On average,
mm
o
ß 1996
[] 1997
in the summer of 1999 are summarized
urnalwindsrangedfrom0.25to 1.25m s-1, andthephase
0
*
Figure 11. Summarystatistics
of diurnal-band
windsat various stationsoff the southernCalifornia coast(Table 2). (a)
RMS amplitude. (b) Relativephase(in hours,relativeto
0000:00 LT) calculatedfrom a harmonicanalysisof the diurnal(24 hourperiod)windsusingindependent
timeblocks,
from0.4 to 1.1ms-1 (Figure1la). In 1999(indicated
by 8 daysin length.Eachindependent
estimateof the relative
stars),winds decayedfrom the coastto 220 km offshore. phaseis plotted.(c) Squareof theratioof theminorto major
diurnal-band
ellipseaxese-2. (d) Orientation
of themajor
•,2.0•.)
.•,
•
•
ß
o
i
•,
' i -- 'SIOpier
'
'
ellipseaxis,relativeto the orientationof the coast(90ø is
perpendicular
to thecoast).
i
.
i
i
i
.
i
i
i
•4
However,a trendin the wind amplitudewasnot obviousfor
mostyearsstudied.
The phaseclearlyincreased
from thecoastto 220 km offshore(Figure 1lb). On average,the time of maximumonshore winds was 6.6 hours later 220 km offshore than at the
coast.
0 , June
, ::
180
,
Jul• , :: ,
200
220
Aug•st,
240
Time (days)
Figure 10. Runningestimatesof harmonicconstants
of diurnal(24 hourperiod)windsat the SIO pierandNDBC buoy
46047 in the summerof 1999. (a) Harmonicamplitudeand
(b) phaselag wereestimatedfor themajorellipseaxiswinds
usingtime blocksfour daysin length.Estimatesweremade
every 6 hours. The phaselag is expressedin hoursand is
relative to 0000:00
LT.
At all stations,diurnalwindswere highly elliptical(Figure 1l c). The squareof the ratio of minor to major ellipse
axese-2 wasusually< 0.25. At theSIO pier(0 km from
thecoast)theaverage
e-: overthefoursummers
sampled
was 0.07. The orientationof the major axis was scattered
around90ø (Figure 1ld); thatis, windswere approximately
normalto the coast. From this analysiswe concludethat,
overtheextentof theIWAVES array,diurnalwindswerecoherent,hadnearlya constantphase,andwerenearlylinearly
polarizedin the cross-shore
direction.
LERCZAK ETAL.: INTERNAL
WAVES DRIVEN
BY A DIURNAL
6. Modeling a Sea Breeze-Driven Coastal
SEA BREEZE
w- E w,•(x,t)ck,•(z).
Ocean
19,723
(10)
n=0
We explain someof the featuresof the diurnal currents
observedduringIWAVES usinga variationof themodeldescribedby Gill and Clarke [1974] and used with modificationsby Zervakisand Levine [1995] and Balmforthand
Young[1999]. The geometryis shownin Figure 12. A
straightcoastline(z = 0) runsparallelto the y axis, and a
continental
shelfwith constantdepthH extendsto the west.
Separability
requires
thattheverticalstructure
functions
(•b,•
and4>,•)arerelatedby
N2
½• =•
2½•=•,
(11)
wherec• is a sep•ationconstant
withunitsof wavespeed.
Theserelationships
leadto theeigenvalue
equation,
A slowlyvarying,barotropicgeostrophic
current,V(z, t),
N2
flowsparallelto the coast.All field variablesare assumedto
½• + •½• - 0.
(12)
be independent
of y. We consideronly the cross-shore
wind
stresswith a predominantly
diurnalfrequency.
boundmyconditions(½• = •
=
The modelequationsare linear,hydrostatic,and Boussi- Rigid-lid, flat-bottomed
0 at z = 0,-H) me assumed,and the normalizationof •
nesqwith rotationandwind forcing:
is definedby
ut - fv = -7r• + X• ,
(5)
vt+f (l+ -f-)u- O,
•m
(6)
dz -- 6nm,
(13)
where •nm is the Kroneckerdelta and the asteriskindicates
N2w = -7rzt,
(7) complexconjugation. The coefficientsc•n that couplethe
u• + w• = 0.
(8)
wind to the nth internalwavemodeare thereforegivenby
o
The variable•r is the pressureperturbationdividedby the
meandensity,(u,v,w) are velocitycomponents
in the (x,y,z)
directions,and f is the local Coriolisparameter.The ocean
(14)
t) -H
is forcedby a prescribed
windstress,
X (x, z, t), actingover The resultingequationsfor un, vn, andPn are
the mixed layer. We assumethe wind blowsonly in the x
Unt - f Vn = -Pn• + C•nX ,
direction.
(15)
Like Gill and Clarke [ 1974], we assumeall field variables
(includingthe wind forcing)are separablein the horizontal
and verticalcoordinatesand decomposethem into normal
v.t + fru.
= 0,
(16)
modes:
Pnt+ Cn2 Unx= 0,
U• V• 7r•Xz =
E{Un(X,t),
Vn(X,t),
pn(x,t),
C•nX(x,t)}
•Pn(Z),(9)
n=O
wherer = 1 + Vx/ f. From(15)-(17)a singleequation
for
u,• can be obtained:
- (&t + f2r)u =
/•/•Along shore
(17)
(18)
6.1. Stratification
The modelbuoyancy
frequency
profile(Figure13a) is
similarto thatobserved
in the upper100 m off of Mission
BeachduringIWAVES:
Diurnalwindstress
1[
Z--Zm)]
[l+eZ•-•]
. (19)
N(z)-No• l+tanh(
This profilehas a mixed layer of thicknesszm, risesto a
maximumvalueat thebaseof themixedlayer,anddecays
exponentially
below.To matchconditions
at MissionBeach,
Figure12. Schematic
of themodel.A straightcoastline
runs
we
take
No
=
5.12x10
-3
s
-1,
zm
=
-10
m, Zo= -40.7m,
parallelto the y axis,anda continental
shelfwith constant
N (period= 3.1
depthH extendsto thewest.A slowlyvaryinggeostrophic6 = 15.6m, ande = 2 m. Themaximum
periodat
current,V(z, t), flowsparallelto thecoast.Thisflowis as- min)occursat a depthof 12.8m. Thebuoyancy
sumedto be independent
of y.
theseaflooris 20 min (Figure13a).
19,724
LERCZAK ETAL.' INTERNAL WAVES DRIVEN BY A DIURNAL SEA BREEZE
Seabreezecouplingconstants
Buoyancyfrequency
b.)
a.,,....L•
CD isa dragcoefficient
(1.1x10-3),andU istheamplitude
of theseabreezeat thecoast(1.5m s-1). Usingthesevalues,r•: = 0.03dyncm-2. Wecalculated
thepurelydiurnal
solution
for a seabreezedecayscale/•-1 of 50 km at latitudeswhere the diurnal frequencyis superinertial(29.5ø)
and subinertial(32.75ø) (Figure 14). The subinertiallatitude is the samelatitudeas the IWAVES experiments.The
ocean'sresponsenear the coastis not very sensitiveto/•.
0.1
The first 100 vertical modes were solved for in the calcula0
100
0
20
40
tion. Whenthe diurnalfrequencyis superinertial,
a beamof
internalwavesradiatesaway from the coastalongan interFigure 13. (a) Buoyancyfrequencyprofileusedin the nal wave characteristic(dashedline) startingat the baseof
model.(b) Couplingconstants
c•,•of themodelwindto the the mixed layer (Figures14a and 14b). Maximum currents
0
0.01
0.02
N(7•
0.03
mode nnmher
(rad/•
n
nth internal wave normal mode.
in the beam are 14 cm s-1. At a distance of 5 km from the
coast(Figure14d), thereis a sharpphaseshift in the crossshorecurrentsu at the baseof the mixed layer. The currents
6.2. Coupling to the Wind
directlyrespondingto the wind in the mixed layer have a
How thewindcouplesto thedifferentinternalwavemodes constantphasewith depth. At a distanceof 40 km from the
dependson the underlyingstratification,whichdefinesthe coast,currentshave a higher amplitudethan closerto the
vertical structure of the internal wave modes and the struc-
ture of the wind-drivenEkman layer. As in Gill and Clarke
[1974] andBalmforthand Young[1999], we assumethe sea
breezeforcingactspredominantly
in thesurfacemixedlayer,
and we giveit the followingform:
II
Xz- • l+tanh(
) X(x,t)
(20)
100
q-•
100
400
•.•
200
30
20
1o
o
Cross-shoredist (kin)
Cross-.shore
dist(kin)
o
I r•(x,t)
x = --
Po
40
0
501
5kin
moorin
Zm
The resultantcouplingconstants
a,• decayrapidlywith mode
number(Figure 13b). For analyticalconvenience,
we considera wind thatdecaysexponentiallyawayfrom the coast,
0
o
2
2
Time (days)
Time (days)
o
6.3. Exactly Periodic Solutions
Considerthe long time responseof the oceanto a sea
breezewith a singlefrequencya whenthe backgroundflow
is steady.Assumethat all transientshavepropagatedaway
from the vicinity of the coast,and that the oceanoscillates
withthesame
frequency
asthewind(u,• • e-i•t). Equation
lOO
•
4•
•
2•
40
0
30
20
10
0
Cross-shoredist,(kin)
Cross-shore
disl.(kin)
••0
(18) becomes
0
u•
+
rr2 - f2F
Cn2
u•-
•ct•X.
Cn2
(22)
is
1
-i,•,•x
(eu• - e
)
½n
2PoZmn2n
-Jr-•t
2'
o
2
Tin• (days)
Figure14. Modelsolutions
of u2 for a purelydiurnaldependent
seabreeze-driven
coastalocean(summation
of
When thereis no backgroundflow (F = 1), we can solve
for Un analytically. At the coast,Un goesto zero, and a
radiationconditionis imposedat x -• -o•; thatis, eitherthe
solutiondecaysawayfrom thecoast,or wavesareallowedto
propagateawayfrom butnottowardsthe coast.The solution
ia 1 fox
u• - a•----
2
Time (days)
100vertical
modes
of (23)).Theseabreeze
frequency
is 1
cpd,itsoffshore
decay
scale/•-1is50km,anditsamplitude
atthecoast
is 1.5ms-1. Theocean's
response
wascalculatedat(a-d)a latitude
of 29.5øand(e-h)32.75ø(thelatitude
of theIWAVESfieldstudy).Figures14aand14eshowu2
versus
cross-shore
distance
anddepth.Figures14band14f
arethe sameasFigures14aand 14ebut for the first45 km
from the coast. The thick vertical lines indicatethe loca-
(23)
tionsof synthetic
moorings.
Figures
14c,14d,14g,and14h
show
u2 versus
timeanddepth
atthetwosynthetic
moor-
ings.Thegrayscaleis0 - 50 cm2s-2 forFigures
14a-14d
wheren,•_
- f: )/c2•.
and0 - 1 cm2s-2 forFigures
14e-14h.Thedashed
line
The diurnalwindstressat thecoast,r•:, wasestimated
to in Figure14aindicates
thediurnal
characteristic
emanating
bePairCDU
2, wherePairisthedensity
of air(1.2kgm-2), from the coast.
LERCZAK
ETAL.'
INTERNAL
WAVES DRIVEN
coast.Themixedlayercurrents
havea constant
phasewith
BY A DIURNAL
SEA BREEZE
19,725
We solved(22) numericallyfor the first50 modesusing
methodof Gaussian
elimination
described
depth,
whilethephases
in thebeambelowthemixedlayer thegeneralized
andKuo[1969]witha zerocross-shore
flow
propagate
upward.
Thewidthofthebeamextends
fromthe by Lindzen
boundary
condition
at the coastanda radiationboundary
baseof themixedlayerto 40 m belowthesurface.
At the subinertiallatitude(Figures14e-14h),currentsare condition 1000 km offshore. The latitude was set to 32.75 ø,
muchweaker;
themaximum
amplitude
of u is 1 cms-•.
the sameasthe location of IWAVES. The varianceof the cur-
distanceanddepthis plotWithinthemixedlayer,currents
decayoffshore
withthede- rentsasa functionof cross-shore
15c.Outside
oftherange
where
F < 0'2/f2,
cayscaleof theseabreeze.Thereis a 180ø phase
shiftin tedinFigure
is low. Thetwolocations
whereF = 0'2if2
thecurrentsat thebaseof themixedlayer.The currentsjust thevariance
dottedlines)actascaustics
forinternalwavereflecbelowthe mixedlayeroccurbecauseof the coastalbound- (vertical
complicated
internal
arycondition.
Theiroffshore
decayscaleis thedeformationtion.WhereF < o'2/f2,anenergetic,
occurs,
withbeams
beingreflected
off thebotradiusof thefirstinternalwavemode(c•/f, 6.1 km in this wavepattern
case).
tom and at the baseof the mixed layer.
6.4. ExactlyDiurnal Response
with r :/: 0
6.5. Time-dependentWind and V
In theabove,we assumed
thatthereis notime dependence
Theparameter
F - 1 istheratioof therelativevorticityof
currents
andthatthewindblowspurely
thebackground
current
totheplanetary
vorticityf. Thisrel- to thebackground
high modeswere
ativevorticitychanges
theeffectiveCoriolisconstant
felt by at the diurnalfrequency.Consequently,
thediurnalmotions[Kunze,1985;D'Asaro, 1995;Balmforth present
in thesolutions,
resulting
in complicated
spatial
pat-
andYoung,
1999].WhenF > a2/ f2, (22)iselliptic
in(2;,z). ternsof theinternalwavefield (Figure15c). However,high
background
Theocean's
response
will beevanescent
asin thesubinertial modestravelslowly,andin a time-dependent
nothavetimetoevolveandmakea
case
ofFigures
14e-14h.
However,
when
F < a2if2, (22)is fieldtheywouldprobably
hyperbolic
in (2;,z),andinternalwaveswill freelypropagate. coherentcontributionto the solution.For example,consider
To demonstratethis, we solved (22) with a Gaussianjet thetime it wouldtakea particularmodeto propagateacross
theregion
whereP < o'2/f2 inFigure15b(At -- A2;/c•).
flowingin thealongshore
direction(Figure15a):
For the first mode, At is • 0.5 days. For the 50th mode,
V- Voe
-«('"•-•')•'
.
(24) At is • 30 days,longerthanthetimescaleof variabilityof
V duringtheIWAVESexperiments
(Figures
6 to 8). There-
In all subsequent
analyses,
Vo - 40 cm8-1 , 2;0 - -25 fore, for realisticvariabilityin V we needto studythe timekm, and A - 10 km. The resultingF is shownin Figure dependentproblem.
15b. Over the cross-shore
range5.4-22 km from the coast,
We solved(15)-(17) numericallywith a time-dependent
F
and realisticdiurnal-bandwind variability at the same latitudeasIWAVES. The equationswere finite differencedover
a 400 km domain in the cross-shore direction.
a.)
,
I
"•20
1
6O
40
20
The model winds were constructed
+
I
•
ß
!
60
40
20
An Adams-
Bashforthtime steppingschemewas usedto integratethe
equations
forwardin time, andthe time stepwasmadeshort
enoughso that the Courant-Friedrichs-Lewy
stability condition was met for all vertical modessolvedfor. A sponge
layerwasaddedto the offshoreendof the domain.
i
0
to have the statisti-
cal variability of the diurnal-bandwinds observedduring
IWAVES (Figure9a). The windswere peakedat the diurnal
frequencywith a uniform,low backgroundlevel acrossthe
diurnalband. In orderto avoidgeneratinglargetransientsat
the startof a modelrun, we rampedthe windsup from zero
amplitude
to a finalstationary
amplitude
of 1.5rns-• over
the first 20 days (Figure 16a). As before, the winds were
givenan offshoredecayscaleof 50 km.
The background
jet V hadthe shapeshownin Figure 15a,
but its amplitudeVo was modulatedover time. Three simulation caseswere run. In caseI the jet amplitudewas set
to zero for the entiresimulation.In caseII (Figure 16e), Vo
Figure 15. (a) Cross-shore
profileof the alongshore
jet used was setto zero for the first 50 daysandwas thenrampedup
in themodel. Thejet is Gaussianin shape,centeredat 25 km
overthe next 10 daysto a maximumnorthwardamplitudeof
offshore,with a width of 20 km. The maximumamplitude
withthisamplitude
forthenext
is 40 cms-•. (b) Theresultant
I' fromthejet. (c) (u)2 + 40 cms- •. Thejet remained
70
days
and
was
then
ramped
down
to
zero
over
a periodof
(v2) fromthepurelydiurnalmodelsolution
of (22) witha
diurnalwind forcing and the F of Figure 15b. The model 10 days. It remained at zero for the rest of the simulation.
was calculated for a latitude of 32.75 ø.
Like caseII, Vo of case III was set to zero for the first 50 and
19,726
LERCZAK ETAL.' INTERNAL WAVES DRIVEN BY A DIURNAL SEA BREEZE
Simulation
Case I
Simulation
•-2
Case II
Simulation
-2
0
50
1 O0
15o
200
Case III
-2
o
50
lOO
150
200
0
50
100
150
200
1.4
ß
1.2
[-
1.0
0.8
0.6
.4
•..)
.................
1.4
il
................
'Of.-.
•- / t 1'0
.2
1.2
1.8
0.8
•.6
0
50
lOO
15o
200
...........
•'•12
'•'
-I-
•
8
0.6
50
100
150
200
' ßß
0
50
100
150
200
150
200
...................
g:)..... 2'kmfrom
coast
2
12
8
8
15km
............ 250 km
i
i
i
i
4
0
o
0
50
100
150
200
50
Days
100
150
200
'--
0
50
Days
100
Days
Figure 16. Time seriesfrom threetime-dependentsimulationcases.(a-c) Cross-shorewind at the coast.
The windsdecayedoffshorewith a decayscaleof 50 km. (d-f) F at 15 km from the coast(Figure 15b).
(g-i)Vertically
averaged
u2 + v2 atdifferent
cross-shore
locations.
last50 days(Figure16f). From days50 to 150,however,Vo
oscillated
about+40 cms-1 witha periodof 20 days.All
simulationswere run for 200 daysandthe first25 modes.
We considertheresponseof theoceanat threedifferentlocationsfromthecoast:2 km (inshoreof theregionwhereI'
at 15 km. The varianceat 250 km roseat aboutday 100
andremainedcomparatively
highuntilaboutday 180,again
apparentlydueto theradiationof internalwavesawayfrom
the coast.
Vertically averaged,clockwisepolarizedspectraat the
canbelessthano'2/f2),15km(thelocation
withthelowest threecross-shore
locationsare plottedin Figure 17. Counvalueof I' when thejet flows northward),and 250 km (beyondthe influenceof the wind andthejet). The vertically
averaged
variance
(u2 + v2) versus
timeis plottedfor the
2 km fromcoast
50 '1
'a.) '" ' '1
I
I
threeoffshorelocationsandthreesimulationcasesin Figures16g-16i.
When I' = 1 (caseI), the variancewas low at all three locations(Figure 16g) but was highest15 km from the coast.
Therethe varianceincreased
slowlyoverthe first20 days
(theperiodoverwhichthe windswererampedup). Afterward,the variancewasmodulatedovertimebut apparently
remainedat a stationarylevel. The modulationin the variance was apparentlydue to the modulationof the diurnal
15 km from coast
50 '1
b.)' ' ' ' '1'
I
I
-II '
I
I
I
I
I
I
I
I
I
I
I
I
0 .II ....
0.6
1
50 "a.)' '"
, ,II ,
1.4
- ',
0 .II . . /•A
0.6
1
,II
1.4
5o[ ',',,)' j ,' ' 'l'
0 .II .
0.6
II
1
1.4
_
4 -II f.) ' '-
06
,I
1
50 I g) '"
1.4
' '1
0 .I .
0.6
1
50 I h.) '"
1.4
' -I
I
0
0.6
I
I
.I . . • A-^^.I
1
I.
14
0
06
I
I
.I .
I
I
1
1.4
0
0.6
_
II
I
'
I
I
0 .I . . )k•,..
_
I
I
wind.
Over the first 45 days,caseII (Figure 16h) was identical
to caseI. Oncethejet wasturnedon at day50, however,the
250 km fromcoast
4 II C.
) '
.I .
I
.
.
!
1
1.4
4 iI i.) ' ' - -I'
I
0
0 6
I
I
I
I
I
I
,
.I .
_
J
1
_
1.4
variance
rapidlyincreased
at the 15km location.A slight
cpd
cpd
cpd
risein thevariance
atthe2 kmlocation
wasalsoapparent.
Figure17. Vertically
averaged,
clockwise
rotaryspectra
of
Afterday145,whenV wasrampeddown,thevariance
at 15 the currentsfrom the modelsimulations.Counterclockwise
kmgradually
decayed.
Thevariance
at2 kmroseabruptly
at variances
were2-3 ordersof magnitude
lowerthanclockday155,apparently
theresultof internal
waveenergy
being wisevariances.
Thealongshore
jet modulation
for(a-c)sim-
released
fromthetrapping
region.A slightrisein variance
at ulation
case
I, (d-f)case
II, and(g-i)case
III isshown
inFigthe250kmlocation
occurred
atday170andwasapparentlyures16d-16f. Spectraareshownfor threedifferentlocations
Thevertical
dashed
linesmarktherange
of
dueto internalwaveenergyradiating
awayfromthecoast. fromthecoast.
In caseIII (Figure 16i) the rise in varianceat 15 km was
not asdramaticasin caseII. Pulsesin varianceoccurred,-• 5
days after maximum northwardV. Peaksin the varianceat
thediurnal
bandof thisstudy(0.727-1.33
cpd).Thethick
solidlinesat thetopof eachpanelmarkthediurnal(left
line)andinertial(rightline)frequencies.
Notethatthescale
of the spectraof the currents250 km from the coastis dif2 km occurred
roughlyat thetimeof theminimain variance ferentthanfor the othertwo cross-shorelocations.
LERCZAK ETAL.: INTERNAL
2 km from coast
15 km from coast
WAVES DRIVEN BY A DIURNAL
.j .
lOO
o
i
2
cm2/ s2
3
o
lO
20
30
o
cm2/ s2
0.2
0.4
19,727
ent). In caseI, KE at 15 km from the coastwas low and was
250 km from coast
o
SEA BREEZE
0.6
confinedto the mixed layer and slightlybelow. In caseII,
KE wasconsiderablygreaterthanin caseI. A maximumoccurredbelowthe mixed layer at a depthof • 16 m. Most of
the KE wasconfinedto the upper40 m of the watercolumn,
but someenergywas presentdown to the seafloor.Energy
was presentbelow the mixed layer in caseIII but did not
penetrateasdeeplyas in caseII. A smallamountof KE was
presentat the seafloor.
7. Discussion
cm2/ s2
Figure 18. Kinetic energy versus depth from timedependentsimulationsat (a) 2 km, (b) 15 km, and (c) 250
km from the coast. For all threecases,the kineticenergy
wasaveragedoverthe periodbetweendays50 and 150 of
the simulations(thetime periodwhenthejet wasturnedon).
Note that the kinetic energyscalesof Figures 18a-18care
different.
terclockwise
variance(not shownin Figure 17) was2-3 or-
dersof magnitude
lowerthanclockwise
variance.
Clockwise
spectral
levelswerelow at 2 km fromthecoast.In caseI,
therewasa peakat thediurnalfrequency
anda smallerone
attheinertialfrequency.
In caseII, therewasa broadinertial
peakanda somewhat
smaller
andnarrower
diurnalpeak.In
caseIII, variancewasmoreevenlydistributed
fromthediurnalfrequency
to theupperboundof thediurnalband.
At 15 km in caseI, therewere two distinctpeaksat the
Despiteits simplicity,the model that we havepresented
demonstrates
that backgroundcurrentscan be importantin
regulatingthe degreeto which the seabreezecanpumpenergy into the coastalocean. The model predictsthat at a
latitudeof 32.75ø, little diurnal energywould be pumped
intotheoceanwithoutthepresenceof a negativebackground
vorticity.Even with a constantdiurnalforcingby the wind,
a time varyingbackgroundflow can producean oceanresponsethatis highly intermittent.
The model diurnal-band
currents resembled the currents
observedduringIWAVES. Both were predominantlyclockwise polarized. During IWAVES, currentsnear the coast
(H < 30 m) were elliptical and orientedin the alongshore
direction,whereascurrentsfartheroffshorewere circularly
polarized. In the model the polarizationis dependenton F
and,consequently,
offshore
distance
(Ivl/ll - rfla, (16)).
In orderfor the modelto explainthe polarizationpatternobservedduringIWAVES, F wouldneedto be consistently
> 1
forthevalues
of e-2
diurnalandinertialfrequencies.
Theinertialpeakwaslarger (V• > 0) nearthecoast.Forexample,
thanthediurnalpeakdespitetheconsiderably
stronger
forc- typicallyobservedat the 15-30m moorings(• 0.4, Table3),
ingatthediurnalfrequency
(thediurnalwindpeakwasmuch F wouldneedto be • 1.5. However,this probablydid not
largerthanthebackground
levelat f, Figure9a). Thenear- occurduringIWAVES becauseV couldbe eithernorthward
resonant
response
of theoceanat f, apparently,
caused
the or southward,andsincethesecurrentstypicallydecayedtolargeresponse
despitetheweakforcing.Not surprisingly,ward the coast,V• near the coastcould be either positive
the 15 km spectrum
wasmuchmoreenergetic
for caseII. or negativeand was probablynot consistently> 0. Thus
dependence
of polarizationobservedduring
The largestpeakin CW variancewasat the diurnalfre- the cross-shore
quency,
butaninertialpeakwasonlyslightlysmaller.Vari- IWAVES cannotbe explainedby the model.
Both duringIWAVES and in the model,diurnal-bandKE
ancewasalsoapparentat frequencies
lowerthana•,• and
higherthanf. In caseIII, energywasalsohighestat the was highestnear the surfaceof the water column and dediurnalfrequency.
However,thevariancewasbroadlydis- cayedwith depth. However,the model had maximumKE
tributedbetweenaD• andtheupperboundof the diurnal belowthemixedlayerwhichwasnottypicallyobserveddurband.
ing IWAVES, exceptat the 500 m mooringin the summer
Thescalesof thespectra
for the250 km cross-shore
loca- of 1997 (Figure 4c), where a maximumin KE occurredat a
tionareexpanded
relativeto thespectrafromthelocations depthof 37 m.
Diurnal-band variance increased from the coast to 10 km
closerto shore.Here,beyondthedirectinfluenceof thewind
andthejet, variance
wasat andhigherthantheinertialfre- offshoreduringIWAVES as well as in the model(Figures4
quency.This wasapparently
dueto slightlysuperinertialand 14, respectively).In the modelthis was a consequence
internalwavesradiatingaway from the coast. This near- of the coastalboundarycondition(u = 0 and,consequently,
profile chosen
inertialvariancewashighestin caseIII, the casein which v = 0 becauseof (6)) and the cross-shore
V was modulated with the shortesttimescale.
forF (Figure15b),whichwasalways> rr2/f2 atthecoast
diurnalmotions
Horizontalkineticenergyversusdepthis plottedfor the <11< 5,4 km) anddidnotallowsignificant
threecasesin Figure18. The kineticenergywasaveraged to be driventhere(Figure 15c).
At 15 km from the modelcoast,currentspectrafrom the
over the time periodbetweendays50 and 150 of the simmodel
were much broaderthan the spectmmof the wind,
ulations,the periodwhenthejet was turnedon in casesII
as
was
observedduring IWAVES. The broadeningin the
andIII. In all threecases,KE washighestat 15 km from the
model
apparently
was causedby the slow changesin V,
coast(notethat the KE scalesof Figures18a-18care differ-
19,728
LERCZAKETAL.: INTERNAL WAVESDRIVEN BY A DIURNAL SEA BREEZE
whichallowedforfrequency
mixing.Theslowchanges
in V gradients
in densitymusthavebeenpresent.Hayesand
alsocausedthecross-shore
locationof thecaustics
(location Halpern[1976]showed
thatcross-shore
gradients
in den-
where
F = ae/fe) tochange
overtime,further
complicat-sitycanmodifytheinternal
wavedispersion
relationship.
ing thetemporalandspatialresponse
of theocean.
In thepresence
of analongshore
geostrophic
current,
interIn the summers
of 1996and 1997,diurnal-band
KE ap- nalwaves
canfreelypropagate
withinthefrequency
range
peared
tobeenhanced
whenF < ae/re (Figures
6 and8). (f2r - M4/N2)1/2and
N, where
M2- -gP•/Po.
When
However,in thefall of 1996,KE at the 100m mooringap- M = 0,freeinternal
waves
canexistwhen
F < 0-2if2 (the
pearedto be greatest
whenF > I (Figure7). Thisdiscrep- relevant
relationship
inthemodel
presented
here).Thepresancybetween
themodelandtheIWAVESobservations
may ence
ofacross-shore
gradient
inp,however,
willmodify
this
havebeencaused
by severalfactors.First,themodelmay relationship.
Whether
M eispositive
ornegative,
it willtend
notcontainall therelevantphysicsnecessary
to describe
the to reducethelowerlimit of thefree internalwaveband.Thus
diurnalintermittency
observed
duringIWAVES.We will ad- slowchanges
in cross-shore
density
gradients,
in addition
to
dressthis in more detail below. Second,the finite difference
thevorticity
ofthebackground
currents,
could
havechanged
estimate
of Vxbetween
the100and30 m moorings
maynot theeffective
Coriolis
parameter
at theIWAVESstudysite
havebeenrepresentative
of thetruerelativevorticityat the andcontributedto the intermittentbehaviorof the observed
100 m mooring. The low-frequency
currentsat the 30 m diurnal currents.
mooringwereweakand,for thepurposeof estimating
Vx,
Acknowledgments.
TheIWAVESfieldprogram
andsubseeffectively
zero. Thus,whenV at the 100m mooringwas
efforthasbeenfundedby theOfficeof NavalRenegative,theestimateof V• waspositive,andwhenV at the quentanalysis
search.
Theauthors
wishtothank
LouGoodman
forhissupport
100 m mooringwaspositive,V• wasnegative(Figure7). of
thisproject.
Wealsothank
Charles
Coughran,
PaulHarvey,
and
Whenmooringsfartheroffshorewereusedto estimateV•, Jerry
Wanetick
fortheirhelpinmooring
development
anddeployhowever,
theopposite
couldoccur.Forexample,
in thesum- ment,instrument
maintenance,
datacollection
andmaintenance,
support.
Thanks
alsoto CliveDorman,
whopromerof 1997betweendays205 and213 (Figure8b), V was andcomputer
helpininterpreting
thewinddata,andtoDaveChapman
for
negative
atthe120,350,and500m moorings,
anda positive vided
hiscareful
reviewof themanuscript.
V• wouldhavebeenestimated
if thefinitedifferencing
was
donebetween
oneof thesemoorings
andthe30 m mooring.
However,V wasmostnegativeat the 120m mooringand References
leastnegativeat the 500 m mooring,andthe resultantVx
was negative.
Baines,P. G., Internal tides, internal waves and near-inertialmotions,in BaroclinicProcesses
on ContinentalShelves,Coastal
The idealized,flat-bottomed
modelmostlikelydoesnot
EstuarineSci.,vol. 3, editedby C. N. K. Mooers,pp. 19-31,
containall the physicsthatcontributed
to the complicated AGU, Washington,D.C., 1986.
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and W. R. Young,
a slopingbottomon the propagation
of the diurnalinternal Enhanceddispersionof near-inertialwaves in an idealized
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geostrophic
flow,J. Mar. Res.,56, 1-40, 1998.
Balmforth,
N.J., andW. R. Young,Radiative
damping
of near-
thediurnalenergywasconcentrated
in theupperportionof
inertial
oscillations
inthemixed
layer,
J.Mar.Res.,
57,561-584,
thewatercolumn,theinternalwavesprobablyradiatedout
1999.
of theIWAVESstudysitebeforereflecting
off thebottom. D'Asaro,E. A., Windforcedinternal
waves
in thenorthPacific
and
Sargasso
Sea,J.Phys.Oceanogr.,
14,781-794,1984.
Therefore
thesloping
bottomin thevicinityof thestudysite
may not be relevant.
D'Asaro,
E.A.,Upper-ocean
inertial
currents
forced
bya strong
storm,partIII, Interaction
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andmesoscale
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wasalsoignored.Again,thisis probablynot
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25,2953-2958,1995.
a problemin the regionwherethe diurnalmotionsweredi- D'Asaro,
E. A., C.C.Eriksen,
M.D. Levine,
P.Niiler,C.A. Paulrectlyforcedby thewinds(e.g.,theentireIWAVESstudy son,
andP.VanMeurs,
Upper-ocean
inertial
currents
forced
bya
site) sincemostof the energyis near the surfaceand not
strong
storm,
partI, Dataandcomparisons
withlineartheory,
J.
25, 2909-2936,1995.
subjected
to bottomfriction.However,
thedistant
response Phys.Oceanogr.,
Denbo,
D.
W.,
and
J. S. Allen,Rotary
empirical
orthogonal
of themodel(farawayfromtheseabreeze
forcing)
isprobfunction
analysis
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neartheOregon
coast,
J. Phys.
ablynotrealistic
because
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asthey Oceanogr.,14, 35-46, 1984.
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C. E., Windsbetween
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Is-
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in the
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upwelling,
coastal
curmodel. If therewas alongshore
dependence
in the diur- Gill,A.E.,andA.J.Clarke,
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DeepSeaRes.,21,325-345,1974.
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couldhavebeengenerated.
Thesewaveswouldpropagate coastal
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offtheOregon
coast,
J.Mar.Res.,
3,247-267,
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upthecoastandmayhaveintermittently
entered 1976.
theIWAVESstudysite.Alongshore
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in V could Kundu,P.K., An analysis
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nearOrehavecauseddiurnalenergyto be advected
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wereoftenverticallysheared
dur- Kunze,E. L., and T. B. Sanford,Observations
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ing IWAVES,but the modelassumed
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14,566-581,1984.
was barotropic.SinceV was oftenbaroclinic,cross-shore Leaman,
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onthevertical
polarization
anden-
LERCZAK ETAL.: INTERNAL
WAVES DRIVEN
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BY A DIURNAL
SEA BREEZE
19,729
M. C. Hendershott
andC. D. Winant,Centerfor CoastalStudies,
0209, ScrippsInstitutionof Oceanography,
La Jolla,CA 920930209(mch@coast.ucsd.
edu;cdw@coast.ucsd.
edu)
J. A. Lerczak,Department
of Physical
Oceanography,
MS#21,
WoodsHole Oceanographic
Institution,WoodsHole, MA 02543
integrationof a large classof ordinaryand partial differential (jlerczak@whoi.edu)
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