JOURNAL
OF GEOPHYSICAL
RESEARCH, VOL. 86, NO. C7, PAGES 6426-6428, JULY 20, 1981
The Influenceof OpticalWater Type on the Heating Rate
of a ConstantDepth Mixed Layer
J. RONALD V. ZANEVELD, JAMES C. KITCHEN, AND HASONG PAK
Schoolof Oceanography,
OregonState University,Corvallis,Oregon97331
A simpleheatbudgetmodelfor a radiation-dominated
mixedlayerof constantdepthis presented.In
this model the influenceof the verticalirradiance(heat flux) profileis examinedby meansof the Jerlov
[1976]opticalwatertypeclassification.
It is shownthat the verticalirradianceprofileis importantin determiningthe mixedlayerheatingrate.The heatingrate variesgreatlyasa functionof watertype, mixed
layer depth,and diffusivitybeneaththe mixedlayer,rangingfrom 0.098øC/dayfor oceanicwater type I
with a mixedlayerdepthof 20 m anddiffusivitybeneaththemixedlayerof 1.0cm2 s-• to 0.316øC/day
for coastaltype9 with a mixedlayerdepthof 10m and zerodiffusivitybeneaththe mixedlayer,a variation of more than a factor of 3.
INTRODUCTION
Heat budget considerationsare of importancein determining the dynamicsof the upper ocean,particularly the surface
mixed layer [Niiler and Kraus, 1977].In this paper we address
one specificaspectof the heat budget,the influenceof the vertical distributionof solar heat flux (irradiance)on the vertical
temperaturedistribution in the upper ocean.
This work does not take into consideration fluctuations in
mixed layer thickness.When the wind stressis small, mixed
layer thicknessis nearly constant[Denman,1973].When the
wind stressis large,all solarenergycan be consideredas a surface input [Niiler, 1975].The valuesderivedhere for heating
ratesare thereforeupper limits exceptfor time periodsshorter
than a day.
Heat budgetmodelsusually employ a schemein which the
vertical profile of the absorption of heat is given by an exponential[Denmanand Miyake, 1973]or in which the net heat
flux is consideredto be a surfaceinput [Bowden,1977;Niiler,
1975].The vertical distributionof solarheat flux (irradiance)
may intuitively seemunimportant sincemost solar energy is
absorbedin the top few metersand 99% of all solar energy is
absorbedin the top 75 m. We will show, however, that the
vertical
distribution
of irradiance
can influence
the mixed
layer heatingrate significantly.
In an earlier paper [Zaneveldand $pinrad, 1980]we showed
that an arctangentmodel of total solarirradiancepenetration
best describesthe greater than exponential decreaseof irradiance near the surface and the exponential decrease at
greaterdepths.The model is given by the equation
E(z) = E(0)e-X,•[1- K: tan-' (Kaz)]
(1)
where E(z) is the downwardvectorirradiance(radiant flux on
the upper face of an infinitesimallysmallelementof a surface,
divided by the area of that element) and KI, K•, and K3 are
parametersdeterminedby the radiation absorptionand scattering characteristics
of the water mass.The depth (z) is defined positivedownward.
Jerlov [1976]describeda classificationschemefor the total
solar irradiance penetration profiles in the ocean, and presenteda chart displayingthe regional distributionof optical
water types.By usingJerlov'sclassification
schemeand (1) it
0148-0228/81/001C-0121 $01.00
MODEL
Ignoring all horizontalterms,and assumingno vertical advection,the equationcontrollingthe temperaturedistribution
in a water column
is
=
+
6426
(2)
where T is temperature,z is depth (positivedownward),p is
the densityof water,c• is the specificheat of water,E is the
irradiance,and A is the eddy diffusivityfor temperature.We
do not take into accountheat lossesfrom a layer due to backscatteringor heat gains by attenuationof upwelling irradiance. These effectscounteracteach other to somedegree.
Themaximum
ratioof upwelling
'irradiance
to downwelling
irradiance is 10% for blue light in the clearestoceanicwater
[Jerlov,1976].The typical ratio is much less(on the order of
3%), and the effectof ignoringupwellingirradianceis accordingly small.
Equation (2) is a parameterizationof-the heat conservation
equation(it hasbeen assumedthat the turbulentheat flux can
be modeledby A i)T/i)z) and can be approximatedby the following differenceequation:
T,+,.,
= T,.,+ • (E,.,_•
- E,.,+•)
+ • [A,.,+•(T,.,+,
- T,.,)
+ A,.,_•(T,.,_.- T,.,)]
Copyright¸ 1981by the AmericanGeophysical
Union.
Paper number 1C0121.
is possibleto investigatesystematicallythe influenceof irradianceprofileson mixed layer heating. Table 1 showsthe coefficientsK•, K2, and K3 for the variouswater types.
We have carriedout the calculationsof mixedlayer heating
for two typical surfacemixed layer thicknesses
(10 and 20 m)
and for various diffusivitiesbeneath the mixed layer in the
thermocline(0, 0.1, and 1.0 cm2/s).
For demonstrationpurposes,we have used the incident
heat flux and heat lossvaluesdeterminedby Bowden[1977]
off northwestAfrica. Bowdenalsoshoweda simpleheat budget model with which we comparedour results.More importantly, his measurements
were made in the northwestAfrican
upwelling region in which there is a juxtapositionof turbid
and clear waters.In that case,a large horizontalgradientin
heatingcan have an important effecton the dynamics.
(3)
ZANEVELD
ET AL..' INFLUENCE
TABLE 1. Parametersof the ArctangentModel of Solar Irradiance
Penetration
(Equation
(1)) for VariousOpticalWaterType•
WaterType
K•(10-2 cm-•)
K2
K3(10-2 cm-•)
I (clearest
oceanic)
0.0440
0.3963
4.4547
IA
lB
II
III
0.0490
0.0574
0.0670
0.1250
0.3981
0.4103
0.4158
0.4234
4.4236
4.0725
3.9865
3.7062
I
0.1360
0.4500
3.3772
3
5
7
0.2231
0.3541
0.5028
0.4495
0.4626
0.4789
3.7049
3.6806
3.7150
9 (mostturbid
coastal)
0.5913
0.5427
3.6026
OF OPTICAL
WATER
TYPE
6427
oceanicwater types,as much as 50% of the incident radiation
penetratesbelow shallow pycnoclines.In situationssuch as
coastalupwelling where turbid coastalwater types are juxtaposedwith clear oceanictypes,differentialheating ratesof as
much as 0.2øC/day acrossa front might be produced especially if the surfacemixed layersof the two water typeswere
of different thicknesses.This could have a large effect on the
dynamicsof the region.If, for example,a front is formed by
the boundary of oceanic water type III with a mixed layer
depth of 20 m and coastalwater type 5 with a mixed layer
depth of 10 m, the differential heating rate acrossthe front
would be 0.19øC/day. Beam transmissionmeasurementoff'
the Oregon coast [Kitchen et al., 1978; Zaneveld and Pak,
1979]indicatethat sucha front is not at all unlikely. Unfortu-
nately,irradiancemeasurements
werenot availabletO accu-
rately determinewater types.
The thermal gradientsproducedin the modelsfrom penewhere ? is the time increment and H is the depth increment tration of irradiance were at most 1øC per 10 m, which are in
for the model. The boundaryconditionsare that the heat flux the sameorder of the observedgradientspresentedby Bowden
acrossthe air-seainterfaceis equal to the sum (B) of back ra- [1977]. Bowden'sassumptionthat all solar energyis a surface
diation, conductionto the atmosphereand evaporation. The input is thus adequatefor very turbid water. However, if a
productof the density,p, and the specificheat, Cp,is so close large temperaturegradient can be maintainedby other mechto 1.00 that theseterms were droppedin the numerical calcu- anisms(e.g., vertical shearin horizontal advection),diffusion
lation. This differenceequation is known to be stableif A ?/ may become extremely important. This situation could be
1t: _<« [Richtmyer
andMorton, 1967].
modeled as follows:
dT
PARAMETERS
Qs
-•---pcpD
[1- (1- K2
tan-!(K3D))
The values of incident heat flux, Q•, and heat loss,B, are
taken from an example (leg 1) given by Bowden[1977]: Qs -559 cal cm-2 day-! and B -- 242 cal cm-2 day. The incident
flux is applied sinusoidallyover a 14-h daylight period, and
the heat lossesare applied equally over the entire 24-h day.
The relativedecreaseof the irradiancewith depth is computed
by (1). The parametersK!, K2, K3 are givenin Table 1.
The vertical eddy diffusivity is generally reported to be in
B
GA
exp
(-K•D)]
- p-•D
+D
where G is the thermal gradient (generally negative) externally maintained in the thermocline.If G were of the order
of løC/m, then one could ignore the diffusivityterm in the
caseof a 20-m mixed layer of a coastalwater type only if A <<
0.4 cm•/s. For the same20 m mixedlayerwith the 0.08øC/m
the range0.1 - 100 cm• s-• [Ichiyeet al, 1972;O'Brienand
temperaturegradientgeneratedby the model after 5 days of
Wroblewski,1973]. To producea surfacemixed layer, diffusiheating,we can ignorethe diffusivityterm if A << 4.6 cm2/s.
vitiesof the order 100cm• s-! werefoundto be necessary.
DifThe modelparameter0.1 cm• s-! marginallysatisfies
theserefusivitiesof 0.1 to 1 cm• s-! beneaththe surfacelayer proquirementsas verified by the resultsin Table 2 for the 20 m
ducedgradientsoften observedin the thermocline.In the case
mixed layer.
of zero diffusivitybelow the mixed layer and large diffusivities
in the mixed layer (corresponding to the Bowden [1977]
CONCLUSIONS
model) our equation reducesto
It is shownthat the profile of solarirradiance(heat flux) is
B
(4) important in determiningthe heating rate of mixed layers in
dt pcpD
the absenceof strongwind-inducedmixing. The heating rate
of the mixed layer increaseswith decreasingthicknessof the
which givesdaily temperatureincreasedfor the mixed layer
mixed layer and increasingturbidity. At the beginningof a
directly. D is the depth of the mixed layer. A few examples
seasonal thermocline formation, the warm surface water will
were calculatedwith zero diffusivitybeneaththe mixed layer
be present in a thin layer. For example, off Peru a shallow
to verify that (4) givesthe sameresultsas (3).
(<10 m) mixed layer was observed(K. H. Brink, personal
communication,1980)in which the temperatureand thickness
DISCUSSION
fluctuateddiurnally. The high temperaturein the thin surface
The averagedaily heatingrates(after 5 daysof heatingas mixed layer is conduciveto rapid growth of phytoplankton,
predictedby the numericalmodel,equation(3)) for eightwa- which in turn increasesturbidity and hence the heating rate.
ter typesundertwo mixed layer thicknesses
(10 and 20 m) and This physical/biologicalfeedback loop would contribute to
diffusivitiesbeneaththe mixed layer of 0, 0.1, and 1 cm• s-! the further progressof a seasonalthermocline.A phytoplankare givenin Table 2. As waspreviouslystated,a diffusivityof ton bloom can rapidly increasethe concentrationof sus100cm•/s wasusedfor the mixedlayer.In mostof the coastal pended matter. Each doubling in concentrationwould apwater types,very little radiant energypassesthrough even a proximately double the parameter K, in Table 1. It is thus
10-m mixed layer. The mixed layer losessignificantamounts seen that a bloom which doubles phytoplankton in a day
of heat by diffusionthroughthe thermocline,however.In the could transforma water massfrom oceanictype II to coastal
aT_- {l-I1-
exv
(-K•D)}
pcpD
6428
ZANEVELD ET AL.: INFLUENCE OF OPTICAL WATER TYPE
TABLE 2. HeatingRate of the SurfaceMixed Layerin øC/day for VariousOpticalWater Types,Mixed Layer Depths,and Diffusivities
Beneaththe Mixed Layer
Ditfusivity
Mixed
Beneath
Mixed
Layer,
Layer
Depth, m
cm2 s- 1
I
II
III
I
2
5
7
9
10
10
lO
20
20
20
0
0.1
1.0
0
0.1
1.0
0.178
o. 163
o. 132
0.114
0.110
0.098
0.214
o. 196
o. 156
0.132
0.126
0.111
0.261
0.238
o. 184
0.150
0.142
0.122
0.272
0.246
o. 190
0.152
0.144
0.124
0.298
0.268
0.200
0.157
0.148
0.126
0.312
0.278
0.204
0.158
0.150
0.126
0.316
0.280
0.206
0.158
0.150
0.126
0.316
0.280
0.206
0.158
0.150
0.126
Diffusivityin mixedlayer-- 100cm2/s.
type 5 in a matter of days, increasingthe heating rate by Ichiye, T., N.J. Bassin,and J. E. Harris, Diffusivityof suspended
matter in the Caribbean Sea,J. Geophys.Res., 77, 6576-6588, 1972.
0.1øC/day. In such a situation as well as in frontal zones
Jerlov,N. G., Marine Optics,Elsevier,New York, 1976.
where shallow,warm, and turbid layersflow over clearer and
Kitchen, J. C., J. R. V. Zaneveld, and H. Pak, The verticalstructureof
colder water masses,it is thus conceivablethat the biological
sizedistributionsof suspendedparticlesoff Oregonduringthe upprocesses
can affect the dynamicsvia heatingof the surface
welling season,DeepSea Res.,25, 453-468, 1978.
Niiler, P. P., Deepeningof the wind-mixed layer, J. Mar. Res., 33,
layer.
405-422, 1975.
In lessextremecasesit is still clear that the irradiance proNiiler, P. P., and E. B. Kraus, One-dimensionalmodelsof the upper
files shouldbe measuredif useful modelsof the upper ocean
ocean,Modellingand Predictionof the UpperLayersof the Ocean,
are to be constructed.If, for modeling purposes,solar irraeditedby E. Kraus,Pergamon,New York, 1977.
dianceprofilesare requiredin the absenceof measurements, O'Brien, J. J., and J. S. Wroblewski, A simulation of the mesoscale
distributionof the lower marine trophiclevelsoff westFlorida, Inv.
one can use existingmaps of oceanicwater types such as in
Pesq.,37, 193-244, 1973.
Jerlov[1976],l•utsleovskaya
andKhalemskiy[1977],and Spin- Richtmyer,
R. D., and K. W. Morton, DifferenceMethodsfor Initial-
rad et al. [1979] and obtain the profile by means of (1) and
Table
1.
Acknowledgments.This researchwas supportedby the Office of
Naval Researchthrough contractN00014-79-C-0004 and by the Departmentof EnergycontractDE-AM06-76RL02227.
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(ReceivedJuly 3, 1980;
revisedJanuary 13, 1981;
acceptedJanuary 13, 1981.)