JOURNAL OF GEOPHYSICALRESEARCH,VOL. 85, NO. C4, PAGES 1910-1916,
APRIL 20, 1980
TheBatchelor
Spectrum
andDissipation
in theUpperOcean
THOMAS M. DILLON AND DOUGLAS R. CALDWELL
School
of Oceanography,
Oregon
StateUniversity,
Corvallis,
Oregon
97331
Observations
of vertical
temperature
microstructure
at ocean
station
P duringthemixedlayerexperiment(Mile)indicate
thattheshape
ofthehigh-frequency
temperature
gradient
spectrum
depends
onthe
relativestrengths
of turbulence
andstratification.
ForlowCoxnumber((dT/dz)2)/(dT/dz)• thelinear
rangeof theBatchelor
spectrum
isnotwellapproximated
byobserved
spectra,
whileforhighCoxnumbera remarkably
closecorrespondence
to theBatchelor
spectrum
isfound.Dissipation
ratescalculated
by the temperaturegradientspectrumcutoffwave numbermethodshowa dramaticcontrastin turbu-
lencebetween
lowandhighwindspeed
periods
separated
byonly3 hours,
showing
thattheresponse
of
themixedlayerandtransition
zoneto windforcingis rapid.Someindicationis foundthatthe thermoclinemay alsorespondrapidlyto surfaceforcing.
INTRODUCTION
and foundthat the temperaturegradientspectrumwasalmost
Numerous
comparisons
ofthetheoretical
universal
temper-alwaysresolvedto an easilyidentifiablecutoff.With this data
atureand temperaturegradientformspredictedby Batchelor set a test can be made of the universalityof the Batchelor
[1959]have beenmade with spectrameasuredin natural wa- spectrum in natural stratified waters.
In addition, the particularmeteorologicalconditionsenters[e.g.,Grantet aL, 1968;Gregg,1976,1977;Nasmyth,1970;
ElliotandOakey,1976;MarmorinoandCaldwell,1978].While counteredduring the microstructurecastson Julian date 244
suchmeasurements
oftencomparefavorablywith theviscous- at stationP enableus to usethe Batchelorscalingto address
pertinentto mixedlayerdynamics:
(1) What is the
convective
Batchelorspectrumby showinga linearrange,it questions
of thesurfacewatersto surfaceforcing?
has seldombeenpossibleto resolvethe entiretemperature timescaleof response
distributedvertically,and doesthe disgradientspectrum
for a varietyof reasons,
includinglimited (2) How is dissipation
change?
(3) Canthevertior unknownsensorfrequencyresponse
characteristics
and too tributionvaryassurfaceconditions
muchnoise.Whenthespectrum
hasbeenresolved
beyondthe cal distribution of dissipationbe associatedwith features of
(4) Doesthe upperseasonalthermoclineremicrostructure
peak,a detailedcomparison
with the Batchelor the stratification?
spond
significantly
to surfaceforcing?
spectrum
hasnot provedfavorablein certaincases[Nasmyth,
1970;Elliot and Oakey,1976],promptingat leastoneinvestiTHEORETICAL BACKGROUND
gator[Elliotand Oakey,1976]to proposea differentmodelfor
The Batchelorspectrumof temperaturegradientfluctuathehigh-frequency
spectrum.
Sincetheapplicability
of a Batchelor spectrumallows one to make an estimateof the kinetic
tionsin one dimensionmay be written [Gibsonand Schwarz,
energydissipation
rate from a temperature
spectrumalone,it
1963]
is importantthat the natureof the viscous-convective
spectrum be well understood
and that the universalityof the Bat-
S(k) -- (q/2)•/2X•s-•D-•f(a)
wherek is thewavenumber(herein unitsof radiansper unit
A companionpaperdescribes
a testof the hypothesis
that length),q is a universalconstant,Xois the temperaturevarithe cutoffwave numberin the spectrumis identifiablewith ance dissipationrate satisfying
the Batchelorwavenumber[Caldwellet al., 1980].The result
chelor spectrumin stratified fluids be tested.
is favorable when turbulent fluctuations are weak in com-
parison to the mean stratification, but because it assumes
strong stratification, the test is not relevant to intense turbu-
Xo6D
foøø
S(k)
dk-6D
((-•-z
]l
(2)
diffusivity,
T' is a temperature
fluctuation,
lence.The presentpapershowsthat the Batchelorspectral D is themolecular
form is followed more closelywhen the fluctuationsare z is depth,ks is theBatchelor
wavenumber(Ep-•D-•)•/4,• is
strong. It is thus plausible that the cutoff wave number is re- thekinetic
energy
dissipation
rate,v isthekinematic
viscosity,
lated to the Batchelor wave number in both casesand so can
anda is a nondimensional
wavenumberkkB-•(2q)
•/'•.The
be usedto estimatedissipation.The definitivetestof the rela-
nondimensional
shapeof thespectrum
is givenby
tionbetweenthecutoffwavenumberandthedissipation
will
be direct simultaneousmeasuresof the smallestscalesof both
(3)
velocityandtemperature.
Thisdifficulttaskhasnot yet been
accomplished,
and so the presentpaperand its companion
mustbe assessed
as indirectevidenceonly,showingthat the If an inertial subrangeexists,the transitionfrom inertial to
relationsbetweenthe cutoffwave number,Batchelorscale, viscous-convective
behavior
occurs
atthewavenumber
k, ---
and dissipationrate are plausibleand consistent.
C,Pr-•/2kB,
andforwavenumbers
in theequilibrium
range
is givenby
During the Mile experimentat oceanstationP (50øN, smallerthank, thespectrum
145øW),August-September
1977,we performeda numberof
castswith a verticalmicrostructure
profilingdevicein the surfacelayerand seasonal
thermocline
duringa moderatestorm
Copyright¸ 1980by the AmericanGeophysical
Union.
Paper number 9C 1648.
0148-0227/80/009C-1648501.00
S(k) --•]•Xt•-l/3ki/3
Here both C. and fi are universalconstants,
and Pr is the
Prandtl number v/D.
1910
(4)
DILLON
AND CALDWELL:
DISSIPATION
1.0
IN THE UPPER OCEAN
1911
[1959] suggeststhat even when the Reynolds number is so
small that an inertial subrange does not exist, the Batchelor
form may be found.
'
The effect of stratificationon the Batchelor spectrum,however, has not been explored. In natural waters (Prandtl number of the order of 10), temperature fluctuations are usually
causedby turbulent stirring in the presenceof a mean gradi-
_\
0.1
S(k)ke D
q,/'•'•Xe
ent. The
level of turbulence
relative
to the stratification
is
O.Ol
likely to be important in determiningthe scaleat which local
isotropy is approachedand in determining whether or not a
Batchelor spectrumis found.
o OOl
0.0001
I
0.001
i
0.01
i
O. I
i
A measure
of the turbulence
relative
to the stratification
1.0
may be constructedin the form of a 'stratificationReynolds
O•(cycles/cm)
number'Rs= Ulv-', where U is estimatedas•/31•/3 [Tennekes
Fig. 1. Nondimensionaluniversalspectrumwith a 'fine structure'
and Lumley, 1972] and the length scale is estimated as the
rangefor a < aR, the Richardsonwave number;an inertial subrange
Richardson
length I = e•/2N-3/2,where N is the local buoyfor aR< a < a,; anda viscous-convective
Batchelor
spectrum
for a >
a,. The positions
of theKolmogorov
wavenumberasandBatchelor ancyfrequency.Thus Rsreducesto (ev-')N -2, a squaredratio
wavenumberaB are determinedby choiceof the universalconstantq
of time scales of the turbulence
and stratification.
If E is esti-
(2(3)•/2assumed
here),
whiletheshape
ofthespectrum
fora > a, is matedby the Cox numberCx = ((dT'/dz)2)/(dT/dz) 2 and N
independentof universalconstants.The inertial subrangeis rarely observed in vertical microstructure.
[Caldwellet al., 1980],e is proportionalto DN2Cx, so that Rs
is proportionalto Pr-'Cx. We might thereforeexpectthat the
Nondimensionalizingthe spectrumby the Batchelor scale
kB and X0yields the equilibrium range spectrum(Figure 1)
approach to the Batchelor spectrum in a stratified natural
fluid would depend on the Cox number.
We imagine that for small Cox number (turbulence weak
compared to the stratification),small-scaletemperature features which are possibly remnants of the mean stratification
may exist at wave numbers large enough to coincide with
wave numbers in the viscous-convectivesubrange and alter
the low wave number portion of the Batchelor spectrum. A
wherea, = C,Pr-•/2(2q)'/2
anda• marksthewavenumberat similar situationwas hypothesizedby Gregg[1977],who, upon
whichthe free structureand internalwageeffectsbecomeap- examining spectrafrom the main oceanic thermocline, which
preciable.Continuityat k, establishes
the relationbetweenq, somewhatresembledthe Batchelor spectrumbut failed to reveal a distinct linear range, supposedthat the low wave numfi, and C, as [WilliamsandPaulson,1977]
ber viscous-convectivespectrum was 'contaminated' by the
fi = qC,2/•
(6) fine structure-internal wave spectrum.
In the remainder of this work, following a brief experimenGibson[1968]advanceda theoreticalargumentthat 3'/2 < q
tal description,we report that for large Cox number in the
< 2(3)•/:;Grantetal. [1968]estimated
q -- 3.9ñ 1.5andC, -0.024 +_0.008 (standarderror estimates).Williamsand Paulson oceanicsurfacelayer the Batchelorspectrumcloselyapproxi[1977] found from inertial subrangemeasurementsin air that matesobservedspectra.For smallerCox number the observed
fi -• 0.5 and noted a small but definite dependenceof • upon spectradepart significantlyfrom the Batchelor form, showing
Reynolds number. The atmosphericmeasurementsof • are a broader, lesspeaked shape.We then presentdepth profiles
surelymore accuratethan one could reasonablyexpectfrom of dissipationcalculatedby the cutoff wave number technique
marine measurements,and there is no reasonto expect iner- and compareprofiles taken during low and high winds.
tial subrangeparameterssuchasfi to dependon Prandtl num-
S(k){(q/2)
'/2
Xk-•}-'=f(a
) a>_a,
ber.If q is 2(3)•/2and• is 0.5, C, becomes
0.055,twiceas
EXPERIMENTAL
DESCRIPTION
large as the value found by Grant et al. [1968] and subThe vertical microstructure profiles to be discussedhere
stantiallylarger than the values0.03 and 0.04 found by Gibson were collected at ocean station P during August-September
et aL, [1970]but only haftthe value of 0.1 estimatedby Gibson 1977 as part of the Mile experiment. The microstructureproand Schwarz [1963] from laboratory measurements.There is filer usedwas a small, winged, nearly freely falling devicesimilar to that describedby Caldwell et al. [1975]. The fall rate is
thusa largeuncertainty
regarding
the valueof C,.
If dissipation rates are to be estimated from the cutoff fre- adjustable;for theseruns it was set at 10 cm s-•. The probe
quencyof the Batchelorspectrum,q is the only relevant uni- carried two thermistors, a Neil Brown conductivity sensor,
versal constant.A percentageerror in q is reflectedas a sys- and a pressuresensor.The data were transmittedto the ship
tematic error twice as large in E.This error may not be as large through a small-diameter (1.6 mm) cable containing four
as that resultingfrom direct high-frequencyvelocity determi- pairs of expendablebathythermograph(XBT) wire sheathed
nations of E in water and will not affect relative measures of e, in a Kevlar strengthmember and coated with syntacticfoam;
so that depth variability can be assessedwithout error from the foam density is adjusted so that the entire cable has a
this source.
slight tendency to sink.
Although the Batchelorspectrumwas proposedfor homogOnethermistor
Wasredundantly
amplified,
andthetwosigeneous, locally isotropic turbulence in an unstratified fluid nals were sent to the surfacethrough different wires, where it
with large Prandtl number, it is clear, in contrast to inertial was again amplified, differentiated,filtered with a 12-pole
subrangeturbulence,that the spectralform is not an asymp- Butterworthfilter (3-dB frequencysetat 30 Hz), and then digtotic limit of high Reynolds number turbulence. Batchelor itized and recorded at a rate of 90 s-•. Since most of the noise
1912
DILLON
AND CALDWELL:'
DISSIPATION
IN THE UPPER
OCEAN
the observedspectrumis well resolved,Xomay be found by integratingthe spectrum.Without simultaneoushigh-frequency
velocity measurements,e cannot be measureddirectly. An estimate of the dissipationrate can be made from the spectrum
Spectrum of heated
laminar jet
if the transitionalwavenumberk, can be determined.This
E 1.0
techniqueis of limited applicability, however,becausespectra
exhibiting unambiguous transitions from the inertial to the
viscous-convectivesubrange are found in only a few of the
ß
ß
casts. We believe that this is so because the low wave numbers
ß
of the inertial subrangespectrumcan be seen only in thick
ß
sectionsof the water column. The turbulence properties in
such a thick sectionmay changewith depth, causingthe obß Uncorrected Spectrum
served spectrum to be nonstationary. These low wave numbers may also be contaminated by the fine structure-internal
o Corrected Spectrum,
i,af •' ,bf4.cf 6
wave spectrum [Gregg, 1977].
I
i
I
I
If the entire spectrumhas been resolved,a high wave number normalization can be used.Supposethat the fluid is turbuf(Hz)
lent and the signal is well above noise. Then the gradient
Fig. 2. Frequencyresponseof the P-85 thermistoras determined
by passing
thethermistor
througha heatedlaminarjet at 10cm s-l; spectrumwill exhibit a peak of some sort followed by a very
rapid roll-off. First, we calculateXoby integration;then we dea -- 2.6352x 10-2, b -- -2.1897 x 10-5, and c -- 3.2556x 10-?.
ß
_
fine a cutoff wave number kc as the wave number at which the
contentof the temperaturesignalsoriginatedin the first stage spectrum falls to some fraction, say, one tenth, of its peak
amplifier, these noise sourceswere incoherenton the two value. If the Batchelor spectrumdoes obtain, ks will be some
channels.A calculation of the in-phasepower [Caldwell et al., fraction (which depends on the value of q assumed) of kc.
1969] allowed a significant reduction in noise level to be Wave numbersare then nondimensionalized
by ks-'(2q)'/2,
made. A secondthermistor was differentiatedinside the probe
and the spectrum$(k) is nondimensionalized
by (q/2)'/2
rather than at the surface but otherwise preprocessedin a
xoks-'D-'. Note that whilethe particularvalueof ks (and
manner similar to the first. The conductivity was sent to the also % = ks4v-lD-l) dependson q, the shapeof the nonsurfaceon the remaining XBT pair, and the pressuresignal dimensionalizedspectrum is independent of q. A number of
was transmitted in frequency-modulatedform in the range suchnondimensionalizedspectramay then be ensemble-aver500-1300 Hz.
aged and compared to the Batchelor spectrum. This comThe frequencyresponseof the thermistors(Thermometrics parisonwill be independentof the particular value usedfor q.
P-85) was determinedby a test similar to the Fabula test [FaA slightly improved technique is to find the cutoff wave
bula, 1968],using for the plume a laminar jet emitted from a number kc (and hence ks and es) by performing a nonlinear
0.30 x 12.7mm nozzle.The velocityof thejet was2-3 cm s-' weighted least squares fit to the Batchelor spectrum. The
at the heightwherethe probe,movingat 10 cm s-', passed weightingused correspondsto the degreesof freedom of the
through.A detailedmap of the jet revealeda nearly Gaussian band-averagedspectralestimates[Otnesand Enochson,1972];
temperaturedistributionwith a spectralpower 3 dB in wave- hence the high-frequencyroll-off portion of the spectrumis
length of 1.9 mm. The frequency response, which sub- much more heavily weighted (by a factor of 25-50) than the
stantiallyagreeswith a similar testperformedby M. C. Gregg lower-frequency, prepeak linear region of the spectrum. In
(personalcommunication,1979)on anotherP-85, could not
be well describedby either single or double pole response
functions;for computationalpurposes,a power seriesin even
powersof frequencyup to the sixthpowerwasusedinstead.It
describesthe responseto 30 Hz with 3% standarderror (Figure 2).
I
The data were collected on Julian date 244, 1977, between
the hours 0500 and 1200 UT in the midst of a moderate storm
(maximumsustained
wind speedsof 16 ms-'). This wasthe
secondstorm encounteredduring the Mile experiment. The
low-pressurecenterof the stormpasseddirectlyover stationP,
and a dramatic change in wind speed (Figure 3) was noted,
along with a reversalof wind direction,at 0200 hours.Casts
during the earlier part of the samplingperiod, when winds
were low, showmarkedly lessactivity in the surfacelayer and
thermocline than later castsin higher winds [Dillon and Caldwell, 1977], showinga rapid responseof the upper waters to
increasingwind stress.We sampled,in a shortperiodof time,
water with greatlydifferentlevelsof turbulencerelativeto the
mean stratification.
COMPuTATIoNAL TECHNIQUES
A nondimensional form of the high-frequency gradient
spectrummay be determinedif both Xoand e are known. If
1600
2400
0800
1600
TIME (GMT)
,1ULIAN DATES 245,244
Fig. 3. Surfacestressand wind directionduring the secondsignificant storm encounteredduring the Mile experiment. Wind speed
droppedto near zero, and wind directionchangedas the low-pressure
storm center passedover station P. Times of the five microstructure
dropsduring light winds and the 10 dropsduring stormwinds are indicated as vertical bars on the lower portion of the figure.
DILLON
•
1.0
AND CALDWELL:
DISSIPATION
IN THE UPPER OCEAN
i
1.0
1913
•
,
o.,-
0.1
S(k)kBD
'•,
0.01
0.01
o.ool
0.001
...-'
,,,
-
-
cx>2.500
Cx<
500
150
samples
samples
i
o.oool
O.Ol
o.oool
i
O.Ol
i
i
o. i
i.o
/
1.0
o. i
k
Fig. 4. Ensemble-averagednondimensional spectrum for Cox
numberslessthan 500. The lower wave number portion of the spectrum departssignificantlyfrom the Batchelorspectrum(solid curve).
Error bars are the standard error of the mean based on the standard
deviation. The dashedenvelope showsthe averagepositive and negative deviationfrom the mean;that is, half of the spectralestimateslie
within the envelope. Thus although the mean of the spectral values
has a small uncertainty (the bars), the estimatesin an individual spectrum may lie quite far from the mean.
Fig. 6. Ensemble-averagednondimensionalspectrum for large
Cox numbers.The low wave number portion of the spectrumis in remarkably close agreement with the Batchelor spectrum. Thus althoughthe mean of the spectralvalueshas a small uncertainty(the
bars), the estimatesin an individual spectrummay lie quite far from
the mean.
sembleaveraginggroupsof spectrawe will know that we have
found a Batchelor form if the low wave number region of the
ensembleaverageagreeswith the linear region of the Batcheaddition, since the spectrumrolls off so rapidly, the fitting lor spectrum. The high wave number roll-off region is conprocedurewill be very sensitiveto the slopeof the spectrumin strained by the fit to follow the Batchelor spectrumas well as
the cutoff range of wave numbersand insensitiveto the spec- it may, but the low wavenumberregion is not.
tral estimatesat low, prepeak wave numbers.
To minimize vertical intermittency, the data were broken
After performingthe high-frequencynormalizationand en- into 512 point segments(--60 cm). (Even for these thin samples, however, there is no guaranteethat the turbulenceis homogeneousthroughoutthe segment.)A Gaussianwindow was
applied to each segmentto reducespectralleakage,and a fast
Fourier transform was applied. After band averaging, the
spectral estimateswere corrected for thermistor attenuation.
Only fully resolvedspectraare included in this paper; a speco.i
trum was operationallydefined as 'resolved'if it fell to 10%of
1.01 I
its peakv.alue and wasstill abovenoiselevelsbeforethe 3-dB
S(k)kBD
frequency(30 Hz) of the analoguefilter was reached.Approximately 90% of the spectrain the surfacelayer met this criterion.
0.01
SPECTRAL
SHAPES
Nondimensional spectra from all surface layer runs on
Julian day 244 were groupedtogetherin rangesof Cox number for comparisonto the Batchelor form (the surface layer
was defined as the region of temperaturewithin 0.25o C of the
surfacevalue). Spectrawith low Cox numberswere markedly
89 samples
different from those with a high Cox number. The low Cx
spectra (Cx < 500) characteristicallyexhibit a broad maxi0.0001
I
O.Ol
o. i
mum before roll-off and do not agree well with the Batchelor
k
form in the linear range (Figure 4). Spectrawith higher values
of Cx progressivelyapproach the Batchelor spectrum at low
Fig. 5. Ensemble-averagednondimensionalspectrum for inter- wave number (Figure 5), and for Cx > 2500 the low-fremediate Cox numbers.The low wave number portion of the spectrum
quency fit to the Batchelor form is quite remarkable (Figure
o.ool
-
500
<cx<2500
is closer to the Batchelor spectrum than the low Cox number spectrum but still is significantlydifferent. Thus althoughthe mean of the
spectralvalueshas a small uncertainty(the bars), the estimatesin an
individual spectrummay lie quite far from the mean.
6).
The same trend of the averaged spectra to approach the
Batchelor spectrum is found if they are grouped by Rs =
1914
DILLON
0 2õ
i
AND CALDWELL:
i
i
DISSIPATION
IN THE UPPER OCEAN
i
COX
ß Cx2 2500
0.::'0 -
I00
ß
0 Cx< I00
10-4
i
150.15
I01
102
I
I
I
NUMBER
I0$
I
104
io5
ios
I
•e (cm2/s$)
i0-$
i
10-2
i0-•
i
i
-
S(k)kBD
0.10 -
0.05
• 30
W
.......
-
0.1
1.0
2
k
4O
Fig. 7. Variance-preserving plot of ensemble-averagea nondimensional spectrafor large (solid circles) and small (open circles)
Cox numbers compared to the Batchelor spectrum (solid curve).
Slightly elevatedvaluesof the large Cox number spectrumnear the
peak may indicateviolation of Taylor's hypothesis.
es•,-'N-2. Similarbehavioris exhibitedwhen the spectraare
,
I
I
I
I
I
12
TEMPERATURE
DAY
244,IIii
WIND SPEED
(øC)
/5.5 m/$
Fig. 9. Profilesof es, Cox number,and temperature
duringstrong
winds. No systematictrend is seen in es. Note the vertical intermittency in es in the thermocline below 32 m.
classified and averaged in terms of cutoff wave number or
buoyancy frequency; that is, low kc and high N spectraare lent fields in which the temperature gradient spectrum debroad and flat, while high kc and low N spectraresemblethe pendsonly on Xoand es and thosein which it dependson adBatchelor spectrum, although the approach to the Batchelor ditional parameters (Cx, for example). In the former case,
form is less pronounced for these classificationsthan for fluctuations in the viscous-convectivesubrange will not declassificationby Cx or Rs.
pend on the history of the fluid but simply reflectthe stateof
We have at present no means of knowing whether the the local dissipation.In the latter casethe mean stratification,
broad spectraseenat low Cox number are indicative of small- which certainly dependson the history of the fluid, seemsto
scale temperature structuresattributable to the 'mean' strati- be important (in a way not yet clearly defined). Until we have
fication or whether they have a fundamental meaning for tur- evidence to show otherwise, we assume that although the
bulencein a stratifiedfluid. Although one might claim that all stratificationmay be important in the viscous-convectivesubtemperature fluctuations are in a sense due to a turbulent range at low wave numbers in the prepeak region, the wave
field, a clear distinction can be made between stratified turbunumber of the spectral roll-off dependsonly on the local dissipationrate; that is, the highest-frequencyturbulenceis independentof stratification.The effectof stratificationis simply
COX NUMBER
to add a mean 'background' spectrum at low viscous-coni0ø iOt
i0z 103 104 105
% (cmZ/s
$) ,
,
,
•
t
,
vectivewave numbers,as was suggestedby Gregg[1977]. This
point of view has some empirical backing, since the dissipationrate calculatedfrom the cutoffwave number is highly
correlated with the dissipation rate as calculated from N, D,
and Cx [Caldwell et aL, 1980].
20
Stratified regions may exhibit 'fossil turbulence' [Turner,
1973], that is, temperature microstructurepersisting after velocity gradientshave disappeared.Schedvin[1979] statesthat
ß 25
i
I
i
when(es/v)'/2 _<N, the microstructure
may be fossil.If es can
be approximatedas NeDCx [Caldwellet aL, 1980],this condition for fossilizationis equivalent to Cx < Pr, a condition
rarely seenin the actively mixing surfacelayer. For the data
35
examinedhere, (es/v)'/: waslessthan N only 6% of the time,
and these cases were confined to the thermocline.
40
I
6
I
I
8
I
TEMPERATURE
DAY 244, 0542
I
I0
I
I
12
(øC)
WIND SPEED 5, 5m/$
Fig. 8. Profilesof es, Cox number, and temperatureduring light
winds. Note the intermittency in es and Cox number and a trend toward increasinges as the seasonalthermoclineis approached.The
dashedline indicatesa region where the spectrumwas unresolved.Estimatesof Cox number in the surfacelayer may be particularly uncertain becauseof the small temperature gradient.
The effect of
fossilization or the approach to fossilizationon the shape of
the viscous-convectivespectrumis as yet unknown.
On a variance-preservingplot (Figure 7), the low wave
number difference appears less pronounced, and differences
near the peak are accentuated.The high Cox number spectrum is significantlyhigher than the Batchelor spectrumnear
the peak; we attribute this to a violation of Taylor's hypothesis caused by a larger turbulent intensity for the high Cox
number spectra[Lumley, 1965]. The first-order correctionfor
this effect depends on the turbulent intensity and is particu-
DILLON
((cm2/s$)
<;6 •d5 •d4 •d3 •6z 0
I i i I i I
,
AND CALDWELL:
N (cycles/hr)
•0
20
i
i
I
[
DISSIPATION
I
IO
_>
ß
ß
IN THE UPPER
OCEAN
1915
In modeling the surface mixed layer it is usually assumed
that both the energy available for mixing and the dissipation
are proportional to the cube of the wind speed [Niiler and
Kraus, t 977]. The 'high' winds have about 3 times the speedof
the 'low' winds, so the energy available for mixing is expected
to increase by a factor of about 30. The calculated increase in
dissipation is of approximately this magnitude. A more detailed comparison of the variation of dissipation with wind
speed is not warranted from these data becausethe surface
wave zone, in which dissipationis expectedto be largest, is
not adequately sampled. For this reason a depth-integrated
dissipationcannot be calculatedand related to changingwind
stress.
ca 3o
ß Low
Winds
o High Winds
Fig. 10. Average of • and buoyancyfrequencyin low (solid
circle) and high (open circle)wind speed.During low winds,• increaseswith depth until the therm•line is reachedand showstwo regionsof hi• dissipationassociated
with the top of the transitionzone
near 20 m and the top of the thermoclineat 33 m. During high winds,
• is nearly •nstant with depth and is much larger than during low
winds. The volume averageof • shouldnot be attempted•1ow 30
larly simple for the Batchelorspectrum,but sincewe do not
measure the turbulent intensity, we can make no correction
The method usedto calculatethe mean profileswas to average all resolveddissipationsduring low or high winds; those
caseswhere the spectrumwas unresolvabledue to any cause
were not included in the average.The spectrawere almost always resolvedabovethe seasonalthermocline,so lack of resolution had little effect in the surface layer and the transition
zone. However, the thermocline spectra were often unresolvable, usually becauseno turbulence was found sufficiently
intense to produce a microstructurepeak significantly above
the fine structurebackground.Thus the profile averagesbelow
35 m are biasedtoward larger values,so that dissipationrates
in the thermocline region may not be representativeof the actual volume averagethermocline dissipation.The average of
the resolved spectrain the thermocline is, however, significantly greaterin high winds than in low winds, indicating that
when turbulenceis found in the upper seasonalthermocline, it
is moreintensein higl{windsthan in low. This indication
must, however, be treated rather more cautiously than confor this effect.
clusionsregarding the mixed layer and transition zones.
DISSIPATION
RATES
One is then led to inquire how energy is transmitted
through the zonesof large stratificationtypical of the seasonal
Dissipationratesin the surfacelayer and upper thermocline
were estimatedfrom the spectraby usingq -- 2(3)I/2. In the thermocline. While only speculationsmay be offered, two
early, quiet casesthe dissipationrate profile was intermittent mechanisms are possible: (t) intense shearing within the
in the vertical and showed a definite increase with depth as thermoclineand (2) transmissionthrough the stratifibationby
internal waves. Of these,the latter seemsmore plausible, bethe thermoclinewas approached(Figure 8). This may reflect
causeclassicalmean shearingimplies an exchangeof momenincreased shear production in the mixed layer-thermocline
tum by actual masstransport,which would causea more contransition region. In the region of maximum temperature gratinuous distribution of turbulence, while transmission by
dient the dissipationdecreasesand is even more intermittent
internal waves generated at the top of the thermocline reand smaller below.
As the wind speedincreased,the dissipationrate became quiresno massexchange.If energyleakageby internal waves
through the transition zone to an already saturated internal
larger and more uniform in the vertical (Figure 9). Again, a
wavefieldwereto o•cur,internalwaveinstabilities
wouldresharp falloff occurredwhere the temperature gradient was
leaseenergy in the thermoclineregion in a necessarilypatchy
largest,but in this zone and below, the dissipationrate was
and highly intermittent distribution. It should be noted that
significantlylarger than it was earlier.
the two mechanismsare not mutually exclusive,for shearing
Much of the scatterdue to intermittency was removed from
of the mean flow can intensify internal wave instabilities.
the dissipationrate profilesby averagingall profilesfrom simA significantdifferencebetweenthe storm studiedhere and
ilar wind conditions(Figure t0); the depth for each cast was
the first, stronger storm during the Mile experiment is that
normalized to the t0 ø C isotherm to minimize the effect of induring this secondstorm, no large-scalecatastrophicmixing
ternal waves. The low-wind profile showsa definite trend towas observed.The temperature inversionsobservedhere were
wardincreasing
dissipation
withdeptha•ovethethermocline,
with smallpeaksof dissipation
bothat the top andbottomof all of moderatesize (---1-m vertical scaleor less),while in the
first Mile storm on Julian dates 235-236, larger-scaleevents
the transition zone; this may be indicative of two distinct
may
have been the dominant mechanism of heat and mass
shear zones. In the seasonal thermocline and below, the distransport [Dillon and Caldwell, 1978].
sipationrate is severelyattenuated.In contrast,the later high-
wind profile averageis nearly constantwith depth. No in-
SOURCES OF ERROR
tensification is seen in the transition layer, and stratification
The sourcesof error in computing the dissipation rate Ea
above the seasonalthermocline has no observableeffect upon
the profile. In addition, dissipationsare larger during high from the cutoff wave number are both random and systemwinds than during low winds. We concludethat the responses atic. The random error associatedwith determining the cutoff
frequency from the spectral estimatesis 5-10%, negligible
of the turbulence to surfaceforcing is very rapid.
1916
DILLON
AND CALDWELL:
DISSIPATION
when comparedwith either systematicerrorsor the variability
found in the upper waters. Another random error, associated
with violation of Taylor's hypothesisdue to possiblylarge turbulent intensity [Lumley, 1965],may causea 40% overestimate
of EBfor intenseturbulence.Surfacewave orbital velocity may
also introduce an uncertainty in the estimate of EB,for while
the microstructureprofiler respondswell to vertical velocities
at surface
wavefrequencies
[DillonandCaldwell,
1980],
itsresponseto a fluctuating horizontal velocity field is unknown.
An imperfect responsewould causefluctuationsin the angle
of attack of the sensor and deviations
from the constant value
assumedfor application of the Taylor hypothesis(T. R. Osborn, personal communication, 1979). Systematicerror in all
estimatesof e.• resultsfrom uncertaintyin the universalconstant q. While this does not affect comparisonsof the dissipation (Figure 9), it may be the major sourceof error in
computing volume averages of the total mixed layer dissipation.
Another sourceof error pertinentto volume averageslies in'
the fact that an estimate of Es in intermittent turbulence is
biased toward the larger values.This may be seenby considering the extreme hypothetical caseof a 60-cm segmentwhich
contains
a turbulent
core with a vertical
scale of a few cen-
timeters surrounded by nonturbulent water. In this case the
turbulent core would completely determine the cutoff wave
number, but an Escomputed from this should not be taken as
being typical of the entire 60-cm segment.An attempt to delineate such regions of turbulent and nonturbulent fluid
within a segmentwould be a painstaking and highly subjective task, possiblysubject to interpretational errors as severe as imputing the single value of Es to the entire segment.
Such circumstancesmay occur in the seasonalthermocline, so
that caution should be used in volume averaging in highly
stratified and intermittent regions.
Since most of theseuncertaintiesare impossibleto assessat
this time, we chooseto place no error estimateon EBand simply caution the reader that large uncertaintiesexist. It should
be noted, however,that the conclusionsof this studyare based
upon vertical variations in Es of an order of magnitude or
more and so are independent of the above uncertainties.
CONCLUSIONS
IN THE UPPER OCEAN
stantiallyagreedwith our own. This work wassupportedby the Ofrice of Naval Research, contract N00014-76-C-0067.
REFERENCES
Batchelor, G. K., Small-scalevariation of convectedquantities like
temperature in turbulent fluid, J. Fluid Mech., 5, 113-133, 1959.
Caldwell, D. R., F. E. Snodgrass,and M. H. Wimbush, Sensorsin the
deep sea, Phys. Today, 22(7), 34-42, 1969.
Caldwell, D. R., S. D. Wilcox, and M. Matsler, A relatively simple
freely-falling probe for small-scaletemperature gradients, Limnol.
Oceanogr.,20, 1035-1047, 1975.
Caldwell, D. R., T. M. Dillon, J. M. Brubaker,P. A. Newberger,and
C. A. Paulson,The scalingof vertical temperaturegradient spectra,
J. Geophys.Res., 85, this issue, 1980.
Dillon, T. M., and D. R. Caldwell, Temperature microstructureprofiles at ocean station P: Preliminary resultsfrom the Mile experiment, Ref 77-22, 215 pp., Sch. of Oceanogr., Oreg. State Univ.,
Corvallis, 1977.
Dillon, T. M., and D. R. Caldwell, Catastrophiceventsin a surface
mixed layer, Nature, 276, 601-602, 1978.
Dillon, T. M., and D. R. Caldwell, High-frequencyinternal waves at
ocean station P, J. Geophys.Res., 85, in press, 1980.
Elliott, J. A., and N. S. Oakey, Spectrumof small-scaleoceanictemperature gradients,J. Fish. Res. Bd. Can., 33, 2296-2306, 1976.
Fabula, A. G., The dynamic responseof towed thermistors,J. Fluid
Mech., 34, 449-464, 1968.
Gibson,C. H., Finestructureof scalarfieldsmixed by turbulence,1, 2,
Phys. Fluids, 11, 2305-2327, 1968.
Gibson,C. H., and W. H. Schwarz,The universalequilibriumspectra
of turbulent velocity and scalar fields, J. Fluid Mech., 16, 365-384,
1963.
Gibson,C. H., R. R. Lyon, and I. Hirschsohn,Reactionproductfluctuations in a sphere wake, •4L4•4 J., 8, 1859-1863, 1970.
Grant, H. L., B. A. Hughes,N.M. Vogel, and A. Moilliet, The spectrum of temperature fluctuations in turbulent flow, J. Fluid Mech.,
34, 423-442, 1968.
Gregg, M. C., Finestructureand microstructureobservationsduring
the passageof a mild storm,J. Phys. Oceanogr.,6, 528-555, 1976.
Gregg,M. C., Variationsin the intensityof small-scalemixing in the
main thermocline,J. Phys. Oceanogr.,7, 436-454, 1977.
Lumley, J. L., Interpretation of time spectrameasuredin high-intensityshearflows,Phys.Fluids,8(6), 1056, 1965.
Marmorino, G. O., and D. R. Caldwell, Temperaturefinestructure
and microstructureobservationsin a coastalupwellingregionduring a period of variable winds (Oregon,summer 1974), Deep Sea
Res., 25, 1073-1106, 1978.
Nasmyth, P. W., Oceanic turbulence,Ph.D. thesis,69 pp., Inst. of
Oceanogr.,Univ. of B.C., Vancouver, 1970.
Niiler, P. P., and E. B. Kraus, One dimensionalmodelsof the upper
ocean, in Modelling and Prediction of the Upper Layers of the
Ocean, edited by E. B. Kraus, Pergamon, New York, 1977.
Otnes,R. K., and L. Enochson,Digital Time Series•4nalysis,467 pp.,
John Wiley, New York, 1972.
Schedvin,J. C., Microscalemeasurements
of temperaturein the upper
ocean from a towed body, Ph.D. thesis, Univ. of Calif. at San
Diego, La Jolla, 1979.
1. The observedviscous-convective
subrangespectrumof
temperature gradients approachesthe Batchelor spectrum as
the Cox number increases.For low Cox number the spectrum
is broader and flatter than the Batchelor spectrum.
2. Kinetic energy dissipationrate profilesas inferred from
H., andJ. L. Lumley,,4 FirstCourse
in Turbulence,
300pp.,
the spectralcutoff wave number are highly intermittent, and Tennekes,
MIT Press,Cambridge,Mass., 1972.
values grow larger with increasingwind. The responseto sur- Turner,J. S.,Buoyancy
Effectsin Fluids,367pp.,Cambridge
Univerface forcing occurs on time scales comparable to the time
sity Press, New York, 1973.
Williams, R. M., and C. A. Paulson,Microscaletemperatureand vescalesof the forcing.
locity spectrain the atmosphericboundarylayer, J. Fluid Mech., 82,
•4cknowledgments.We thank P. A. Newberger, M. Matsler, and A.
Robinson for their aid in collecting these data; S. Wilcox for constructingthe electronicelementsof the probe; and S. Blood for computational aid. M. C. Gregg generouslyprovidedus with preliminary
resultsof a thermistorfrequencyresponsetest of the P-85 which sub-
547-567, 1977.
(Received June 20, 1979;
revised November 5, 1979;
acceptedNovember 15, 1979.)