Download some observations of short-term heat transfer through the surface

SOME OBSERVATIONS
OF SHORT-TERM
HEAT
THROUGH
THE SURFACE LAYERS OF TIIE
TRANSFER
OCEAN’
David H. Shonting
U.S. Naval
Undcrwatcr
Ordnance
Station,
Newport,
Rhode
Island
ABSTRACT
Short-term
diurnal heating and cooling data from the Tongue of the Ocean, Bahamas,
are prescntcd and discussed. The method of acquiring the time series data and the method
of presentation
arc rcvicwcd.
A 7&y
diurnal heating study is discussed in relation to the
vertical diffusion of heat and its relation to meteorological
parameters.
Results of a 24-hr diurnal heating cxpcrimcnt
arc prescntcd, and thermal eddy diffusion
coefficients
arc dcrivcd and compared with those obtainccl by other methocls. An approximatc heat budget calculation
is performed whcrcby cssentinlly all of the energy input and
output arc accounted for, utilizing
the diurnal time scrics tcmpcraturc
data.
INTRODUCTION
The data prescntcd here were acquired
during an oceanographic sampling program
conducted by tho U.S. Naval Underwater
Ordnance Station, the University of Miami,
and the U.S. Naval Oceanographic Office.
The study was in the central part of the
Tongue of the Ocean, Bahamas. The purpose of these studies was to dcterminc the
time and spatial variability of tempcraturc,
salinity, density, and the speed of sound
and to relate the observed variability with
circulation and meteorological environment.
The data present a definitive picture of
the thermal response of the surface layer
of the subtropical ocean to diurnal solar
heating with varying wind conditions. It is
the purpose of this study to relate the measurements to a simple theory of diffusion
of heat in the surface layers and to consider
an energy budget for a specific 24-hr period.
In general, the temperature data were
obtained under meteorological
conditions
most favorable for the observation
of
diurnal heating and for portraying vertical
Consequently, the clearly
heat diffusion.
defined isotherm variations presented arc
probably
not representative
of average
spring subtropical diurnal heating conditions, since calm winds, clear skies, and low
sea state essential for observing this phenomenon arc not usually present. However,
the temperature variations and heat budget
probably present a realistic picture of the
magnitudes of tho various parameters pertinent to the short-term exchange oE energy
at the air-sea interface.
OBSERVATIONS
The first series of data was collected
aboard the USS San Pablo of the U.S.
Naval Oceanographic
Office while
on
anchor station in the Tongue of the Ocean
(24”40’ N lat, 77’35’ W long) during the
period 20-27 May 1962. The data include
water temperatures at the surface and at
40 m, measured with protected reversing
thermometers during Nanscn casts taken
every 3 hr, surface air temperatures obtained with a standard mercurial thermometer, and wind speeds recorded with a
Bendix Friez Acrovane mounted on the
ship’s bridge.
1 The author wishes to thank the U.S. Naval
The second and third series of obscrvaOceanographic
Office For the USC of the USS Sun
tions included ba thy thermograph records
Pablo oceanographic
clata taken in May 1962 and
the officers and crew of the RV II. C. HazJes for
supplemented
by air temperatures
and
their assistance in obtaining the time station clata
wind speeds, obtained aboard the RV II.
of February
and March 1962. The review and
C. Hayes under charter by the Naval Uncriticism offered by Dr. E. B. Kraus of the Woods
derwater Ordnance Station from Marine
IIolc Oceanographic
Tnstitution
is gratefully
apby the
This study was supported
Acoustical
Services of Miami,
Florida.
prcciatccl.
Task Assignment
of Naval
Weapons
Bureau
These data were collected during two
RU222EOO0/219-l/R004-03-01 to the U.S. Naval
periods, 6-8 February 1962, while the H. C.
Underwater
Ordnance
Station for oceanographic
H~ZJZS was held by a three-point
deep
studies.
576
SHORT-TERM
HEAT
TRANSFER
THROUGH
ocean mooring system, and 13-14 March
1962, while the vessel was in a two-point
bow-stern mooring. During both periods,
the sampling was in the central part of the
Tongue of the Ocean about 10 km east of
Andros Island at about 24’24’ N lat, 77’36’
W long, where the water depth is about
1,500 m. The three-point mooring system
held the vessel within a quasi-circular area
of about 100 m radius as determined by
Dccca “Hi Fix” navigation system. The
two-point
mooring used in March constrained the H. C. Hayes so that it exhibited
an east-west swing of about 1.5km range.
The actual speed of the vessel swinging
about on these mooring systems was of the
order of 1 cm set-I. A discussion of the
mooring is given by Shonting, Cook, and
Marcy ( 1963).
The sampling interval for both series of
observations was 15 min. The bathythermographs used had a depth range of 250 m
an d were manufactured
by Wallace and
Tiernan Inc. Each was calibrated bcforc
each set of observations at the Woods Hole
Oceanographic
Institution.
Air tcmperatures and wind speed measurements aboard
the E1. C. fInyes were made with a standard laboratory thermometer and a Bendix Friez portable anemometer, respectively. During the period 13-14 March, air
temperature data were recorded aboard the
RV Gercla of the University of Miami,
which was moored about 4 km cast of the
II. C. Hayes.
To supplement the discussion of diurnal
heating measurements of 13-14 March
1962, incident and reflected solar radiation
data are prescntcd that were obtained
aboard the USS San Pablo 10-12 March
1960 while on a three-point mooring in the
Tongue of the Ocean at 24’35’ N lat, 77’34’
W long (see Magnitsky and French 1960).
The radiation data were recorded with
Epplcy pyrheliometcrs
mounted on 10-m
staffs outbo’ard of the vessel.
PROCESSING
THE
BATHYTIIERMOGRAPH
DATA
Experience indicates that the bathythermograph is not an accurate thermometer.
However, with careful use it does provide
OCEAN
SURFACE
LAYERS
577
a reasonably accurate method of portraying the vertical tempcraturc gradient.
During both series of bathythcrmograph
and
observatioas, that is, in February
March 1962, there was a nearly isothermal
mixed surface layer about 180 m deep.
This depth is normal for the late winter in
these waters. This layer provided a constant reference background from which to
observe diurnal heating. The serial bathythermograms showed occasional sporadic
variations in the temperature profiles, indicating a net translation of the temperaindicated
turc axis. This displacement
abrupt temperature variations in the isothermal layer of as much as 0.3C. However,
simultaneous measurements made with protected thermometers indicated negligible
changes (less than 0.05C) in the layer.
Rather than discarding these displaced profiles, we applied a smoothing procedure to
each series of observations. A depth was
chosen where it was assumed the tcmpcrature variation
was negligible,
and the
bathythermograph
profiles were adjusted
to intersect this point. This smoo,thing remo,ved all abrupt temperature fluctuations,
but it did not erase the realistic surface
layer temperature variations.
The inherent danger in the above procedure is that any horizontal advection of
an isothermal water layer with a slightly
different tempcraturc would be masked.
However, during the periods of measurcment, no surface temperature variation occurrcd that was not attributable to diurnal
heating, and since this method of study is
approximate, no attempt to differentiate
between instrumentation
error and advection error was made.
In the February bathythcrmograph
study,
the profile was adjusted to read 24.3C at
the 30-m level. The entire trace was essentially isothermal from about 20 to 160
m throughout the sampling period.
During
the March bathythcrmograph
series, simultaneous scmicon tinuous measurcmcnts of temperature were obtained at
four depths between 82 and 260 m on a
moored taut wire thermistor array positioncd about 1.8 km from the 11. C. Hayes.
578
DAVID
TEMPERATURE
H. SHONTING
t
.
FIG. 1. Time variation
in the temperature
file obtained with the bathythermograph.
pro-
The thermistor at the 82-m depth averaged
24.117C, with a standard deviation
of
0.0376C for the entire 60 hr of data points
over the 15 hr of bathythermograph
observations. Therefore, the reference point
of 24.1C at 82 m was used to position the
temperature traces for this series.
The
temperature-depth-time
sections
were made from the adjusted temperature
depth profiles by selecting the 0.2C increments from the individual
profiles and
plotting them with time. Fig. 1 shows the
temperature-depth
profiles as they varied
at five different times during the February
1962 series.
RESULTS
The first series of data considered portrays an example of the thermal response
of the surface water to the diurnal variation of solar radiation for a period of 7
days. Fig. 2 shows the diurnal march of
air temperature ( thin solid line), surface
temperature ( heavy solid line), and 40-m
temperature ( dashed line) for the period
20-27 May 1962. Above the temperature
plot is the wind speed record.
The air temperature generally follows
the diurnal variation of solar radiation.
The sampling interval of 3 hr is so,mewhat
longer than desired to observe the rapid
rise and fall of the air temperature, but one
can estimate the average time of maximum
( 1300 hours ) and minimum ( 0430 hours )
temperatures.
The maximum heating for
a 24-hr period occurs from 0500 to 1100 and
on 21 May had a value of SC. The range
of shipboard air temperature for the 7-day
neriod
is about 5.3C. There is annarent a
net rise in the mean axis of the diurnal
fluctuations for the 7 days of about 1C. It
has been suggested by E. B. Kraus (personal communication)
that these air temperature ranges may be biased toward
high values, that is, shipboard measurements may give higher air temperatures
than occur in the undisturbed
air away
from the vessel. The time plots of air
temperature are to be considered mainly
for phase comparison of the time variation
of air temperature with respect to water
temperature.
The surface water temperature (actually
measured about 1 m beneath the surface)
reflects the diurnal heating with the average time of maximum heating occurring at
1400 hours. The surface water temperature
( and possibly the air temperature) tends to
exhibit a double minimum, one at 2300
hours and the second at about 0700 hours.
The average diurnal temperature variation
for the period is about 0.9C. The apparent
net heating of the air over the 7-day interval is reflected in a similar trend in the
surface water temperature. The observed
increase in surface water temperature for
the 7-day period is about 1C.
The 40-m temperatures fluctuate about
l.lC during the 7-day interval. The oscillations themselves exhibit larger amplitude
during the first 4 days than for the last 3
days.
FIG. 2. Seven-clay records of wind speed, air
temperature,
surface temperature,
40-m temperature, and M2 tidal component
sinusoidal
curve
( lower curve).
SHORT-TERM
6-
HEAT
TRANSFER
THROUGH
OCEAN
LOCAL TIME -HOURS
12
14
16
18
20
22
00
02
04
06
08
’ I
1 ( I , I , 1,
I ( I , I,
I,
I,,
SURFACE
IO
12
14
16
18
20
22
, 1 , I,
, , ,
, , 3
24-,,,,,,,,,,,,,,,,,,,,,,,,,,,
FIG. 3.
Time variations
I
of wind
speed, air temperature,
The frequency of the 40-m temperature
oscillations appears to exhibit a semidiurnal
component as opposed to the diurnal temperature variation at the surface. To portray this semidiurnal effect graphically, a
sinusoidal wave of arbitrary amplitude having a period equal to iU2, the principal
lunar tidal component having a period of
about 12.4 hr, is plotted under the 40-m
curve for comparison.
The cloud cover for the entire daylight
periods (with the exception of one observation 50 cloud cover at 0500, 23 May)
was less than %o and averaged less than
?/lo.
The wind speed (Fig. 2) never exceeded
7 m secl and averaged about 3.5 m set-l
over the 165hr period.
The sea state
records for this period never exceeded
Beaufort scale of 2 (that is, waves were
less than 1 m in height).
Fig. 3 includes results of the February
bathythermograph
study, the measured
579
LAYERS
and isotherm
I
I
I
I
depths for 6-B Felxuary
I
,
,
00
3
I_
1962.
wind speed, the air temperature, and the
time-depth-temperature
section drawn from
the consecutive bathythermograph
profiles
as shown in Fig. 1.
The air temperature fluctuated between
27 and 28C during the period from 1100 to
1700, then dropped sharply at about 1730
hours. The lower temperature between 23
and 24.7C prevailed until the beginning of
diurnal heating at 0800 the following day.
The peak occurred at about 1100, attaining
31.Z. A trailing-off effect was noted from
1300 to midnight.
The cooling effect reflected in the air temperature drop at 1700
was observed within the surface layer from
the surface at 1700 hours to about 10 m
by 1800 hours. This cooling continued
through 1900, with the water surface temperature attaining a low of 24.OC. Between
1930 and 2030 hours, the surface showed
indication of heating that attained a maximum at 24.2C. Meanwhile, the 24.2C isotherm reached to about 14 m, and then
580
DAVID
IO T
iii
r”
o
0
04
1 [
06
I [,
08
,,
IO
,,,,(I
H. SIIONTINC
LOCALTIME- HOURS
12
14
16
18
1 , I
20
,
I
22
,
I
00
,
,
02
,
8-
6r
;I0
27
g 26
-
6
-
I2
E 14
w
0 16
I8
2ot1”““““““““““1i
FIG. 4.
Time variations
of wind
speed, air temperature,
ascended, reaching the surface just after
midnight. The diurnal heating commenced
at about 1100 with a drop in the 24.4C
isotherm, reaching down to 19 m by 1800
hours. The surface water was heated above
24.6C from about 1400 to 1800 hours. By
2300 hours, the diurnal disturbance vanished, and the surface layer was left isothermal at about 24.3C.
The winds during this perio,d were moderately strong bcforc the drop in air temperature at 1700 hours. With the cooling
of the air, the winds decreased to about
2-3 m see-1 and remained light until the
period of diurnal heating the following day.
The results of the March bathythcrmograph study are summarized in Fig. 4. The
air temperatures were recorded only at 3-hr
intervals and thus only approximately incli-
and isotherm
depths for 13-14 March
1962.
cate the heating of the air. The air tempera ture, however,
increased markedly
from 24.OC at 0800 to 27.OC by 1100 hours,
dipped to 25.7C at 1400, rose to 27.3C
at 1700 hours, and finally fell to 24.4C by
2000 hours.
The first heating of the water appeared
at 1000 hours, with the 24.3C isotherms
descending and reaching a maximum depth
of about 12 m by 1500 hours. This time
correponds to the time of maximum heating
at the surface in which the temperature exceeded 25.OC. The path of the downward
moving isotherms is somewhat steeper than
that of the upward trending isotherms From
1700 through to midnight.
At midnight, a
new heating occurred during the 24 hr,
as indicated by the 24.2C isotherm at 6 m.
The wind, as indicated in the upper graph
SHORT-TERM
HEAT
TRANSFER
THROUGH
OCEAN
SURFACE
0800
0700
0900
,581
Jr
-a-w--L
00
0600
LAYERS
1000
II00
---.A
1200
1300
Iwoo
1500
-- ,f
1600
1!
If300
1700
LOCAL TME - HOURS
I
_1
-lo
FIG. 5. Two-day
average of incident radiation
measured in the Tongue of the Ocean, lo-11
Xlarch 1960.
in Fig. 4, was relatively
strong before
0300; in fact, the winds were recorded at
10-11 m set-l at midnight before the bathythermograph sampling that began at 0300
hours.
During the sampling from 0300 to 0700
hours, the winds were moderate, that is,
about 9 m set-l at the beginning of the
sampling and decreasing to about 6 m set-l
by 0700 hours. At the inception of the
heating, there was an isothermal layer of
24.1c.
From 0700 to 2400 hours, the winds were
light, not exceeding 6 m set-l and averaging about 4 m set-l.
Fig. 5 ( stepped curve) shows the Z-day
average of the incident radiation measured
during 10-11 March 1960. During this
period, clear skies prevailed. The vertical
bars represent the one-half-hr integrals of
incident flux in langlies (cal cm-2). The
maximum incident radiation occurs about
between 1230 and 1330 and is about 1.3
ly min-l.
The integral of the incident energy during the 24-hr period (dotted curve) attains
a value 560 ly. This corresponds closely to
the value of the daily insolation for 24.5” N
latitude during March indicated by Haurwitz ( 1941). The time variation of the
incident radiation exhibited a clear sinusoidal variation with the limits of detectable flux beginning at 0600 and vanishing
by 1830.
During the period of incident radiation
measurement, simultaneous measurements
FIG. 6. Albedo of the sea surface calculated
from incident and reflected radiation data, lo-11
March 1960.
were made of reflected radiation at three
positions outboard of the vessel. A plot of
the albedo values computed from the records of the incident and reflected radiation
is presented in Fig. 6.
DISCUSSION
It is of interest to note the apparent net
heating of the surface water for the 7-day
period in Fig. 2, which indicates a temperature increase of approximately 0.1X per
day. According to Smith ( 1940)) the average seasonal range of temperature is about
8C in the region of the Bahama Banks. Let
us assume that the seasonal temperature
T( t ) can be described by a sinusoidal relation of the type
T(t)
= G
sin go,
where Ts = the surface temperature range
( 8C ) , n/360 = the ciruclar frequency ( rad
day-l), and t = time in days. Let us further
assume that the 8C temperature variation
attains a minimum in February and maximum in August. Thus, the period of maximum heating would occur in May, the
period of the 7-day measurements, or when
t equals about 90 days. To evaluate the
slope in May, we take the first time derivative of equation ( 1) and upon solving, T =
O.lC/day.
Since this value is comparable
to the observed slope, it suggests that we
are observing actual seasonal heating. It is
also evident that this heating effect is not
readily detectable at 40-m depth.
Pronounced diurnal heating and cooling
are graphically shown by the plots of air
and surface water temperature in Fig. 2. The
582
DAVID
H. SHONTINC
difference between the thermal and turbulent mixing properties of air and water is
shown by the superposition of the air and
surface water temperatures,
The air cxhibits three to four times the diurnal temperature amplitude of the surface water. It
is interesting to note, however, that temperature differences between the air and
water are not controlled mainly by conduction This is exemplified by the fact that,
in general, the water temperature just after
the noonday heating begins to fall sharply,
while the temperature of the air ranges
Erom about 1.8 to 2.OC above the water
temperature. The water temperature starts
to fall sharply, catching up to and crossing
the descending air temperature at about
1900 hours. It is clear that during the
period when Tnila> Twatcr, there can be no
net heat transfer from the water to the air
by conduction.
Other mechanisms must
cool the surface water. Wind effects are
not to bc neglected, and there is some indication that a diurnal afternoon thermally
driven wind or Andros Island sea breeze is
generated from the hours of 1400 to late
evening. This is apparent in Fig. 2 for the
afternoons of 21, 23, 24, and 25 May. This
afternoon breeze would greatly contribute
to the cooling oE the thin heated surface
layer by promoting turbulence that mixes
the newly formed thin layer of warm water
downward
into the essentially
infinite
source of cooler water below, and by promoting surface evaporation.
The surface
water does, however, seem to exert a damping effect upon the cooling rate of the air
during the evening. Thus, the rate of air
temperature decrease rapidly attcnuatcs as
the water temperature is passed. The average range between the minimum air and
surface water temperatures is about O-SC,
while the maxima show an average difference of about 1.8C. This suggests a
definite damping and an apparent increase
in convective or conducting heat transfer
during the nocturnal hours.
It is suggested that the optimum conditions for observing diurnal heat effects in
the surface layers occur when warm, sunny
conditions are supplemented by calm sea
conditions (see Stommel and Woodcock
1951). In view of this, it is noted in Fig,
2 that the sharpest rise of sea temperature
occurred between 0600 and 1200 hours on
21 May just after a period of calm sea and
beEore winds. Unfortunately,
the sampling
interval of 3 hr is too long to enable one to
follow the heating with precision.
The temperature curves for the heating
an d cooling of the air and surface water
are dissimilar (Fig. 2). The complex problem of the thermodynamics
of the heat
transfer at the air-sea interface is not
within the scope of this paper, but it is
worthwhile to mention some features of the
data.
The tempcraturc of the air immcdiatcly
above the water is markedly different from
that of the sea surface itself. For example,
consider the diurnal heating and cooling
in these two media. Most of the heating in
both the air and water is caused by the
This
absorption of the solar radiation.
means that at the time of maximum heating
the energy input to the water and air far
exceeds the energy output by back radiation, turbulent mixing, and other less important mechanisms.
The air tends to
absorb the incoming radiation rather uniformly within its volume, but in the surface
water it is ab,sorbcd exponentially
with
depth. The intensity of this absorI?ion
depends on the transparency and scattering
properties of the water.
The cooling processes within the air and
water are even more dissimilar than the
heating. The major cooling is by back
radiation at infrared frequencies by the air
an d water, both as approximate blackbody
radiators. The air can reradiate its cncrgy
with complex interactions but nearly uniformly within
its volume, whereas the
water reradiates energy at infrared wavclengths solely at its immediate surface.
Thus, in considering the different geomctries of the two media as absorbers and
as radiators and their different heat cnpacities, it is clear that their heating characteristics arc different.
Concerning the tcmperaturc variations in
Fig. 3, the 40-m temperature oscillates in
SHORT-TERM
HEAT
TRANSFER
THROUGH
such a way that it is difficult to envisage a
diurnal heating effect at this depth. There
is, howcvcr, a semidiurnal oscillation that
closely follows tho M2 principal lunar tidal
component.
May is a poor time of the year at which
to observe the vertical extent of diurnal
solar heating, because the surface layer
tends to bc less isothermal than at other
times of the year. This condition is charactcrized by an erratic near-surface tcmperature profile that can mask small tcmpcrature variations caused by surface heating. It is in May that the spring heating
disrupts the thick isothermal layer, extending to 150 m, formed during early
winter and predominant in February and
March. The spring heating brings about a
modcrate temperature gradient within the
original mixed layer. This is caused by
intensified surface heating such that wind
mixing cannot make the surface layer isothermal below 20-30 m. During this spring
shoaling of the seasonal thcrmoclinc, the
layer cxtcnding to 200 m generally becomes
disturbed with small temperature anomalies
that can be transient in time and that are
observed at the 40-m depth, as in Fig. 2.
Defant (1961) d iscusses diurnal tcmpcrature variations observed on the Meteor. The
50-m temperature variations detected wcrc
less than 0.05C. This stems unlikely inasmuch as this is probably well within the
noise level even in an isothermal mixed layer.
Considering
the gross heat penetrations
shown in Figs. 3 and 4, it is unlikely that a
diurnal temperature variation is dctcctablc
dcepcr than 25-30 m, even with very scnsitive tcmperaturc probes.
The degree of heating and cooling in the
vertical are to a great extent determined by
the wind mixing at the surface and the
duration and intensity of the incident radiation rclativc to the amount of blackbody
radiation cmittcd from the water surface.
It is instructive to examine closely the
patterns of the isotherms as a function of
depth and time. In Fig. 3, the spacing of
the samples prcscnts a jagged pattern both
during cooling, bctwccn 1500 and 2400
hours, and during heating, between 1100
OCEAN
SURFACE
LAYERS
583
and 1200 hours. The vertical extent of the
rcsponsc is quite similar in both cases. In
both heating and cooling, the downward
propagation rate of the isotherms is of the
order of 5-10 m hr-r. It is difficult to arrive at definite conclusions from these sections bccausc of the oversimplification
resulting from the smoothing and contouring. However, the results indicate the extent of the typical diurnal heat wave that
may occur under tropical and subtropical
conditions.
The thermal response appears to be
much faster for cooling than for heating,
as shown by comparing the time lag of
water temperature with respect to air during the diurnal heating. The heating pattern in Fig. 4 is particularly
well dcfincd
and shows the time-variable geometry of the
isotherms. It appears that the early morning winds ( 0300-0600 hours ) bring about
enough vertical mixing to remove any
thermal stratification caused by the diurnal
heating of the previous day. This, coupled
with ideal heating conditions, allows clear
observation of internal heating effects.
If the daily variation of incident radiation (Fig. 5) is compared with the isotherm pattern, the heating lag is about 3
hr from the time of the maximum incident
heat flux to the maximum isotherm penctration. The asymmetry of the isotherm
pattern at about 1600 hours is as striking
as the symmetry displayed by the variation
of incident heat flux. The asymmetry is
probably due to the variation in stability
of the surface layer as it is heated and
cooled through the 24-hr period. Since the
thermal stratification
is a time-variable
quantity, it follows that the turbulent structure of this layer will change.
Let us consider criteria of the magnitude
of the variation of stability as evaluated by
the Richardson number R, given by
(2)
whcrc g = accclcration of gravity (cm set-2),
p = dcnsit y ( g cm-3), and ti = mean current speed at depth x ( cm set-l ) ,
When surface waters arc heated from
584
DAVID
II.
any initial isothermal condition,
the instability or vertical turbulence is reduced
greatly. For example, referring to’ Fig. 4,
the variation in the Richardson number at
the 6-m level is estimated to increase by
a factor of lo” between 0900 and 1400 hours,
This was calculated on the basis of the
time variation of the temperature gradient,
assuming the mean vertical velocity sheer
is held constant,
This rapid change in vertical stability
would account fo,r the lag in the sudden
drop in the isotherms between 1000 and
1200 hours, Starting at 1300 hours, the
strong gradient formed by the close spacing
of the 24%24.6C isotherms increases the
vertical stability by decreasing the density
of the surface layer and tends to dampen
vertical mixing. By 1800 hours, there is
virtually no direct solar energy being added
to the water column. At this time, the isotherm distribution
is at the mercy of the
ambient turbulent characteristics controlling the vertical mixing of the water. This
turbulcncc is a function of the wind-imparted momentum at the surface and the
mixing across the relatively warm isotherms
into the colder isothermal water at 24.1C.
Surface cooling due to evaporation and
conduction is also associated with wind
speed. The wind speed was quite constant
during the period 0930-2400 hours, averaging about 4 m set-I. Assuming that the
wind stress on the water varies as the
square of the wind speed, this would give
a variation of the stress by a factor of about
three. It is clear that even with quasi-steady
wind conditions, the wind-imparted
turbulence may have an appreciable time variation, giving rise to complex interactions
with the isotherm distribution.
Apart from variable wind effects, the
newly formed stability of the layer apparcntly causes a much slower return of the
isotherms to the surface.
THERMAL
EDDY
DIFFUSION
The plot of the iso therms as a function
of time and depth allows one to approximate the diFfusion properties after making
certain assumptions concerning the nature
SIIONTING
of the diffusion mechanism. Let us assume
that the process by which heat is transferred vertically
downward
can bc described by the Fourier heat equation for
heat flow in a solid in one climension, that
is,
--dT =Ka”T
(3)
at
ax2 ’
where T is tcmperaturc, t is time, and K is
the thermal eddy diffusivity,
which is assumed to be constant with depth x. For
the case of diurnal heating, let us assume
a surface time-variable boundary condition
T (0,-t) = To cos wt,
(4)
where To is the amplitude of the tcmpcraturc wave at the surface and w is the frequcncy. The sinusoidal time variation has
often been used in describing diurnal heating within the atmosphere (XX, for cxample, Lettau 1951 and Haltiner 1958).
According to Sutton ( 1953), this form is a
good approximation
of the daily heating
for clear skies.
A solution to equation ( 3) agreeing with
equation ( 4) is
T (x,t) = Toe-+
cos [w&]
* (5)
The first factor in expression (5) is the
amplitude of the temperature wave at a
given depth and thus is independent of
the time of maxima and minima. Following
Dutton and Bryson ( 1962), let us define
the function representing the total range of
temperature at a depth (that is, twice the
amplitude) as
T,&(x) = 2Toe-“&
Writing
P=-,
the frequency
211
we obtain
w
(6)
in terms of period,
-
It is clear that the range of temperature at
a given depth increases with the period of
heating. The variation of Tlz versus depth
as calculated using the data in Fig. 4 is
given
in Table 1.
-
SIIORT-TERM
HEAT
Solving for the eddy diffusivity
tion (7), we obtain
‘1ltANSFER
TIIROUGII
in equa-
OCEAN
SURFACE
calculated
TR
K ( cm3 xx+)
(“c)
1.0
1.0
0.8
0.7
0.6
0.5
0.3
0.1
0.0
0
Upon comparing equation (3) with this
more general form, we observe the contribution of the product term of the vertical
gradient of K’, and the temperature gradient. It is indicated that in a region in the
water column whcrc equation (8) shows
a rapidly changing value of K with depth,
585
1. Vales of the thermal eddy diffusivity
from equation (8) for various depths
TABLE
Depth z ( m )
and hence an estimate of the thermal diffusivity at various depths. Table 1 gives
the values of K calculated from equation
(8) for the case of diurnal heating described
in Fig. 4, where P equals 24 hr (8.64 x
10’) see ) .
The range of calculated values of K is
comparable with values given by Defant
( 1961) for eddy diffusivities derived from
various oceanic tcmperaturc measurements
made within the upper 300 m. It is to be
noted that equation (8) is a statement of
the constancy of K if (3) holds true. The
variation of K with depth indicates that
equation (3) does not hold true. IIowevcr,
the extent of the applicability
of a type of
equation such as (3) (with, albeit, a K
varying with depth) is approximately
indicated by the values of the eddy thermal
diffusivity
that are quite represcntativc as
a first-order estimate of values obtained by
other methods.
As a check on how realistic the heat
equation is in describing the vertical heat
transfer, equation ( 1) was evaluated using
finite difference quantities from Fig. 4 (the
region from 13X5-1345 hours and 4-12 m
depth). The calculated value of K is about
155 cm2 set-l.
It is reasonable to assume the rate of
vertical diffusion (that is, the intensity of
turbulence) should be a function of depth.
This is expressed by an equation of the
form
LAYERS
z
G
8
9
10
12
1.4
111
102
89
61
25
10
0
the nonlinear term in the right-hand side
of equation (9) has become important, owing to a large value of ~3K’,/&z.
Equation (9) explains nothing of how or
why K’, varies with depth. The difficulty
in postulating the exact nature of the diffusivity is that little is known of the parameters controlling
diurnal heating in the
surface waters of the ocean. The same
situation exists in the atmosphere. Haltiner
( 1958) describes solutions to the heat diffusion equation (9) with the diffusivity
K’, varying periodically
with time and exponentially
with height.
I& solves the
equation in a finite difference form, utilizing a high-speed digital computer. This
approach should prove fruitful when applied to oceanic diurnal heating data if one
for
can stipulate a proper form for K’,(x)
the ocean.
HEAT
BUDGET
CALCULATION
Let us consider a vertical column of
water of unit cross section and x m in depth
measured from the surface, in which we
wish to measure the heat budget for a 24-hr
period. To perform this calculation, we
must have estimates of the following quantities: the net heat gain or loss within the
water column QN, the incident solar radiation upon the sea surface Qr, the reflected
radiation Q n, the energy transmitted across
the air-sea intcrfacc by conduction and
evaporation QzI, and the cncrgy reradiated
from the sea surface at infrared wavelengths
QItn.Any heat advected in or out
of the region by horizontal
currents is
ncglccted.
586
DAVID
The heat budget equation
QN
=
Qr
-
Qre
-
QT
II.
is written
-
(10)
Qnn.
In applying this equation to the diurnal
heating measurements made 13-14 March
1962 ( Fig. 4)) th e various components of
equation (10) wcrc derived as follows:
The net heat gained z?y the water, QN-The quantity QN must represent the total
energy absorbed in the water column. In
the present case, one can estimate the heat
intake to a depth limited by the thermal
sensitivity of the bathythermograph
(that
is, to the nearest O.lC). The average depth
of the diurnal thermal perturbation is approximately 10 m. For convenience, assume
that the quantity of heat intake Q,V is made
LIP of two quantities,
Qls and Qx, the first
of which is the energy calculated within
the isotherm layers followed by the bathythermograph.
From Fig. 4, it can be seen
that this is the amount of energy in the
layer bounded by the surface and the 24.lC
isotherm.
The second quantity
is that
radiation
that was absorbed below the
mean depth of the 24.1C isotherm. Its
computation will be discussed later.
The amount of heat absorbed in the
region above the 24.1C isotherm can be
calculated by the following relation:
Qra = C, z L,( T,-1 + 0.05” - 24.1”), (11)
n-1
where C, = specific heat of seawater, 0.93
cal g- 1 ‘C-l for 36sG salinity, L,, = the thickness of the layer bctwcen two adjacent isotherms of cross-sectional arca of 1 cm2, and
T
=
+Ln
tPmncwahlrp
nF
the
nth
isotherm.
SI-IONTING
culated as +105 ly day-’ ( the plus sign
indicates heat taken into the layer),
It is of interest to use equation ( 11) to
calculate the heat required to produce the
isotherm distribution as it occurred at 1500
hours, the time of maximum depth penctration. The amount of energy ncedecl is
about 550 ly. Fig. 5 shows that by 1500
hours only about 480 ly were incident on
the sea surface since sunrise. There is thus
an indication of the absolute error in the
temperature or depth sensors (or both) of
the bathythermographs.
An error of 1 m
in the depth of the isotherms can change
the 1500 hours heat content calculation by
85 ly, giving a 15% error. Error in the estimation of thickness of the upper isotherms
becomes critical to the total integral of the
heat content, because the upper layers contain the most absorbed heat.
The incident s&r w&don,
Q1-This
value is found from Fig. 5 by integrating
the hillf-hourly values of incident radiation.
Although these data were collected a year
before the March 1962 measurements, they
were taken at the same latitude and under
clear skies similar to those on 13-14 March
1962. The integrated 2-day average is 562
ly day-l.
The reflected solar radiation, QIt-This
quantity was measured directly, simultancously with the direct solar radiation. As
a check on the reflected energy values, a
plot oE the calculated diurnal variation of
albedo is included in Fig. 6. The time variation of the seawater albcdo with the sun’s
altituclc and amount of reflected cncrgy for
the whole day agree well with values given
by Svcrdrup, Johnson, and Fleming ( 1942).
-n ,..
x ?...
. .
SHORT-TERM
HEAT
TRANSFER
THROUGH
OCEAN
TABLE 2.
7
SURFACE
587
LAYERS
Twenty-four-hour
energy
butlgct
Incident radiation QI
= 560 ly
= 25 ly
Energy reflected QR
Energy reradiated +
Energy conducted +
= 405 ly
QT
Energy of evaporation 1
Energy absorbed Qrs
= 105 ly
Energy left over = QI - ()I< - QT - VI.4 = 25 ly
FIG. 7. The half-hour
integrals
(Curve A) and the 24-hr integral
( Curve B ) .
of QT + Qm
of QT + QM
where
R = Bowen ratio,
A = 1.5 X lo-” cm-Z set”,
H = 585 cal g-l,
u = wind speed, m set-l,
r = relative humidity, and
e,,. = saturation vapor pressure at the
ambient air temperature.
According to relations derived by Brunt
( 1944)) Dorsey ( 1940), and Budyko (1956),
the infrared radiation emitted from the sea
surface is approximated by
Qrrlc = 0.9&TA4
lO-Z&.)
( 0.39 - 5.04 x
( 1 - 0.6n”),
( 13)
where (J = Boltzmann’s
constant = 8.17 X
lo-l1 cal cm-2 min-l , T-4 = absolute temperature, “K, and n = cloud amount in
tenths.
Combining
these equations we have:
QT + QItlr = 0.985 TK4( 0.39 O.O5dre,,) ( 1 - 0.6n2) +
(1 + R)(l-r)5.26
x
lo-“ue,.
(14)
The value of e,, is determined from TK,
and thus the energy transmitted by long
wave radiation, conduction, and evaporation is reduced to a function of only surface
temperature, TK, the wind speed, u, and the
relative humidity,
r. Kraus and Root11
( 1961) computed the value of QT + QRlr
against surface temperature with varying
equation
wind speeds. For evaluating
( 14)) let n = 0, r = 0.8, and R = 0.1,
values representative
for the period 13
hlarch, 0000-2400 hours. The half-hour in-
tegrals of QT + Q RIGare shown in Fig. 7
( Curve A). The value of the 24-hr integral
QTI + Q1<,: (Curve B ), is about 405 ly.
In summary, the 24-hr period energy
budget is given in Table 2. There is a
moderate excess of 25 ly. The value of
excess energy may be the result of our
inadequacies in estimating the magnitudes
of the variables considered in the budget.
In particular, the value of the net energy
absorbed in the water by 2400 hours (that
is, +105 ly ) may be in error by as much as
15-20% owing to error in estimation of
depth of the isotherms. On the other hand,
the imbalance could be attributed to two
other factors, advection of warmer water
into the region or the previously mentioned
energy Q-r that was transmitted through the
surface layers into the region where the
heating was insufficient to detect with the
bathythermograph.
It is apparent from data collected in the
general area during this period (see Shonting et al. 1963) that the horizontal temperature gradients were small, that is, less
than O.OlC km-l and not monatomically
varying in any one direction in the Tongue
of the Ocean. However, not enough is
known of the current pattern or speed to
assess properly the magnitude of the advective effects.
To estimate the energy which, according
to Beer’s law, would reach a depth below
10 m (the average depth of the 24.1C isotherm), Beer’s law is used in the form
Qz = Q,,c (I~,
(15)
where QZ = energy reaching at depth z
(m), Q. = energy penetrating the surface
water, and a = extinction coefficient, m ‘.
Using a value of N of 0.05 m ‘, which is
appropriate for subtropical Sargasso sea-
588
DAVID II. SIIONTING
-.
1960. Teplovoi
halanz zcmmio poverwater similar to that found in the Tongue
blcn
osti
cjidromcteorologichcskoc
izdatel’stvo.
of the Ocean, it is found that about 70%
Leningrad.
255 p.
of the energy should be absorbed within
DEFANT,
A.
1961. Physical oceanography,
v. 1.
the first 10 m. On the basis of 100 ly day-l
Pcrgamon, New York. 730 p.
absorbed above this level, about 45 ly day 1 DORSEY, E. 1940. Propcrtics of ordinary water
substance.
Am. Chcm.
Sot. Monographs.
would bc absorbed below this level. By
Reinhold, New York. 220 p.
varying tither the extinction coefficient or DUTTON, J. A., AND Ii. A. BHYSON. 1962. Heat
flux in Lake Mendota.
Limnol. Oceanog., 7 :
the depth in the Beer’s law calculation, we
could easily arrive at nearly the 25 ly day-l
HALzIi;z*C: ,
. 1958. The diurnal temperature
cxccss energy found in the previous calcuwave’ wi& a cocfficicnt
of diffusivity
which
lations. This value of the excess energy is
varies periodically
with time and exponentially with height.
J. Meteorol., 15: 317-323.
almost negligible in comparison with inHAURWITZ, B. 1941. Dynamic meteorology.
Mcstrumental errors and horizontally advected
Graw, New York. 36s p.
heat .
KRAUS, E. B., ANT) C. RoovRi.
1961. Tcmperaturc and stcacly state vertical heat flux in the
These budget calculations appear to rccocean surface layers.
Tcllus, 13 : 231-238.
ommend the use of the temperature-depthLETTEAU, H. 1951. Theory of surface temperatime section as a working tool with which
turc heat transfer
oscillations
near a level
to observe mixing and turbulent processes
ground surf act.
Trans. Am. Geophys. Union,
32: 189-200.
in the ocean. The results demonstrate that
MAGNITSKY, A. W., AND H. V. FRENCH. 1960.
an accurate estimate of the oceanic distriTongue of the Ocean research experiment.
bution of solar energy is feasible using
Tech. Rept. No. 94. (Unpublished
manuscript.) U.S. Navy Hydrog. Office, Washingand meteorological
data.
oceanographic
However, as a method of studying turbuu s to:AT;:* l~~E~;OrRAPEIIC
9
OFFICE.
1962.
lent eddy mixing in the ocean, it must be
’ * Oceanogiaphic
Station Data, USS San Pablo
considered as an empirical approach. To
Cruise No. 31947 of May ancl June, 1962.
( Unpublished
manuscript. ) 150 p.
analyze heat diffusion on the ocean surface
D. H., G. S. COOK, AND C. F. MOREY.
layers more properly, it is necessary to have SHONTING,
1963. Oceanographic
data report, Tongue of
knowlcdgc
of the dynamic motions ocof the Ocean, NUOS Cruise 3, March 1962.
NUOS
Consec 349.
(Unpublished
manucurring within the regime of diffusion.
script,)
U.S. Naval
Underwater
Ordnance
This aspect has generally been neglected in
Station, Newport,
Rhode Island.
150 p.
thermal studies in the ocean, but it might
sMyIITII c . L . 1940. The Great Bahama Bank.
lead to a better understanding of oceanic
I.’ General hyclrographic
and chemical fcaturcs. J. Marinc Rcs., 3: 147-170.
diffusion problems.
REFEXiENCES
1944. Physical ancl clynamic metclb<UNT, D.
orology.
Camhridgc,
London.
570 p.
RUIIYKO, M. I. 1956. The heat balance of the
Hydromcteorological
Instiearth’s surface.
(Translated
by Office of
tute, Leningrad.
Technical Scrvicc, U.S. Department
of Commercc. )
STOMMEL, II., ANU A. WOOIXOCK.
1951. Diurnal heating of the surface of the Gulf of
Mexico in the spring of 1942. Trans. Am.
Geophys. Union, 32: 563-571.
McSUTTON, 0. G. 1953. Micrometeorology.
Graw, New York. 333 p.
SVERDIXJP, H. U., M. W. JOHNSON, AND R. H.
PrcnticcFLEMING.
1942. The
oceans.
Hall, Englewood
Cliffs, N.J. 1087 p.