NPS ARCHIVE
1966
KELLEY, R.
ROBERT D, l<ai£T
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■^ POSTGRADUATE SCHOOL
.SY, CALIF. 93940
feferrLfe! ^T.=:?rriu...^
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VERIFICATION OF MCDONNELL'S
MIXED -LAYER DEPTH FORECASTING MODEL
by
Robert D. Kelley
Lieutenant, United States Navy
B. S., Ohio University, 1958
Submitted in partial fulfillment
for the degree of
MASTER OF SCIENCE IN PHYSICAL OCEANOGRAPHY
from the
UNITED STATES NAVAL POSTGRADUATE SCHOOL
October 1966
K)?s fl^^H\\;e
e/<is
v^eLiX^,^.
ABSTRACT
A model based on Kitaigorodsky's application of similarity theory
and modified by McDonnell to forecast the mixed-layer depth was studied.
The model applies during the warming season and is based on the theory
of similarity. The parameters involved in the model were determined
from bathythermograph data recorded at Ocean Weather Stations November
(latitude SON, longitude 140W) and Bravo (latitude 56 30N, longitude
51W) . Parameters were evaluated daily and grouped by months. Both
seasonal and transitional MLD situations were treated.
From these parameters, the form of the dimensionless function P(N) ,
claimed by Kitaigorodsky to be universal, was determined by least squares
fit to be best approximated by a second order polynomial. Forecasting
equations involving P(N) were developed for each month and tested with
data from the following years for both OWS ships.
There is general agreement between the observed MLD and that found
from the prediction equation based on the last year's P(N) for the same
month and location. Month-to-month and spatial differences in P(N) cast
considerable doubt on its universality, at least as determined by the
parameters as currently defined.
NAv'AL POSTGRADUATE IScftSOC
; :>!TI^REY. CALIF, 93^40
TABLE OF CONTENTS
Section Page
1. Introduction 11
2. Review of McDonnell's model 13
3. Area study selection 15
4. Calculation of parameters 17
5. The form of the function P(N) 36
6. A possible universal function P(N) 52
7. Procedure for forecasting and testing 55
8. Evaluation of results 64
9. Conclusions and acknowledgement 66
10. Bibliography 67
Appendix
I Method used for determining the parameter Q 68
LIST OF TABLES
Table Page
1. Monthly number of BT data cards analyzed and 19
number of paired values determined.
2. Parameters used to determine values of P and N 21
for June 1957 at OWS November.
3. Parameters used to determine values of P and N 22
for July 1957 at OWS November.
4. Parameters used to determine values of P and N 23
for August 1957 at OWS November.
5. Parameters used to determine values of P and N 24
for September 1957 at OWS November.
6. Parameters used to determine values of P and N 26
for October 1957 at OWS November.
7. Parameters used to determine values of P and N 28
for June 1960 at OWS Bravo.
8. Parameters used to determine values of P and N 30
for July 1960 at OWS Bravo.
9. Parameters used to determine values of P and N 32
for August I960 at OWS Bravo.
10. Parameters used to determine values of P and N 34
for September 01-09, 1960 at OWS Bravo.
11. Parameters used to determine values of P and N 34
for September 19 - 30, 1960 at OWS Bravo.
12. Parameters used to determine values of P and N 35
for October 1960 at OWS Bravo.
13. Coefficients for each month used in the forecast- 37
ing equation.
14. Forecast of MLD's for June 1958 at OWS November. 57
15. Forecast of MLD's for July 1958 at OWS November. 57
16. Forecast of MLD's for September 1958 at OWS 58
November.
17. Forecast of MLD's for October 1958 at OWS 59
November.
LIST OF TABLES (Cont'd)
Table Page
18. Forecast of MLD's for June 1961 at OWS Bravo. 60
19. Forecast of MLD's for July 1961 at OWS Bravo. 61
20. Forecast of MLD's for August 1961 at OWS Bravo. 62
21. Forecast of MLD's for September 1961 at CWS Bravo. 63
22. Forecast of MLD's for October 1961 at CWS Bravo. 63
23. Combined statistical analysis of forecasts for 65
seasonal MLD's.
24. Coefficient of thermal expansion (/Ox 10^) of sea water 71
at sea level for different temperatures and salinities.
LIST OF ILLUSTRATIONS
Graph Page
1. Least Squares Best Fit Curve using OWS Papa ^0
Transitional and Seasonal Data.
2. Least Squares Best Fit Curve OWS November 30 ^1
OON 140 OOW June 1957.
3. Least Squares Best Fit Curve OWS November 30 ^2
OON 140 OOW July 1957.
4. Least Squares Best Fit Curve OWS November 30 ^3
OON 140 OOW August 1957.
5. Least Squares Best Fit Curve OWS November 30 ^^
OON 140 OOW September 1957.
6. Least Squares Best Fit Curve OWS November 3C ^5
OON 140 OOW October 1957.
7. Least Squares Best Fit Curve for OWS Bravo 56 ^^
30N 51 OOW June I960.
8. Least Squares Best Fit Curve for OWS Bravo 56 ^7
30N 51 OOW July 1960.
9. Least Squares Best Fit Curve for OWS Bravo 56 ^8
30N 51 OOW August 1960.
10. Least Squares Best Fit Curve for OWS Bravo 01 ^9
thru 09 September I960.
11. Least Squares Best Fit Curve for OWS Bravo 19 50
thru 30 September 1960.
12. Least Squares Best Fit Curve for OWS Bravo 56 51
30N 51 OOW October 1960.
13. Least Squares Best Fit Curve June thru September 54
OWS Bravo 1960 and OWS November 1957.
Figure
1. Representation of the AREA used in calculating 70
the parameter Q.
LIST OF SYMBOLS AND ABBREVIATIONS
ASW anti-submarine warfare
BT bathythermograph
C specific heat of sea water at constant pressure
P
f coriolis parameter
MLD mixed-layer depth
MLD seasonal mixed-layer depth
MID transitional mixed-layer depth
Q excess heat in upper layer associated with seasonal
the rmoc line
Q excess heat in upper layer associated with transitional
the rmoc line
TS temperature at surface of ocean
W representative maximum wind
^ coefficient of thermal expansion
p density of sea water
0 latitude
JTL modified coriolis parameter (f x 10^)
O^ angular velocity of earth
1. Introduction.
Extensive studies have been made on the ensonified bands of water
in the sea in an effort to utilize better their potential for sound pro-
pagation. Sound transmission in the upper layers of the ocean is for the
most part determined by the vertical temperature regime. The need for
more information about this thermal structure to increase the effective-
ness of our ASW equipment and perhaps develop new ideas from this know-
ledge is urgent.
Various methods have been devised for forecasting the ocean thermal
structure. Statistical predictions of the thermocline depth and sub-
surface thermal structure have been the recent trend. The tools of this
statistical approach have been either multiple linear-regression techni-
ques or harmonic analysis of temperature cycles at various depths.
The bulk of applied research, however, is still based on either
dynamical models or on parametric empirical relationships. Inherent in
d3mamical analysis is the problem of mathematical complexity if all pro-
cesses are considered; if simplifying assumptions are made, the reality
of the model becomes questionable. Forecasting techniques based on
empirical relationships are only locally valid with monthly or seasonal
adjustments required.
As pointed out by McDonnell [5] in his paper "Application of Simi-
larity Theory to Forecasting the Mixed-Layer Depth of the Ocean", the
theory of similarity represents an alternative approach in building a
forecasting model. Kitaigorodsky [4] was the first to investigate the
application of similarity theory as proposed by Monin and Obukhov [6]
to predict the thermal structure in the upper layer of the ocean. In
11
the development of this model, Kitaigorodsky assumed that purely thermal
convection due to unstable density stratification was negligible and that
vertical gradients of salinity are equal to zero. This imposed a season-
al limitation on the resulting equations. Generally speaking, a stable
density stratification exists in the upper layer during the warming season
when the thickness of the nearly isothermal layer can be considered main-
ly a function of wind mixing. Heat fluxes across the air-sea interface
during the summer are positive (inward) and tend to build and strengthen
the seasonal thermocline.
With these assumptions, McDonnell applied the method of Kitaigorodsky,
with some modification of parameters to develop a practicable forecasting
model. In McDonnell's conclusion a recommendation was made that future
research be applied in determining the form of the dimensionless func-
tion P(N), inherent in the application of similarity theory, for various
oceanic locations in order to test Kitaigorodsky 's contention that P(N)
is a universal function.
The present author studied two distinct geographical areas using
McDonnell's mixed-layer depth forecasting model in an effort to estab-
lish the form of P(N). In this way, the form of the function P(N) could
be better fixed and the possibility of its universality tested. Further-
more, the practicability of McDonnell's model and parameters could be
tested if realistic mixed-layer depths could be forecast using his
method.
12
2. Review of McDonnell's model.
McDonnell used data recorded at OWS Papa and the theory of similar-
ity to develop a method of forecasting the mixed-layer depths associated
with transitional and seasonal thermoclines during the warming season.
The mixed-layer depth (MLD) was defined as the depth at which water
first became IC colder than the water at the surface. Usually, this
depth could be accepted as the top of the seasonal thermocline. Transi-
tional thermoclines were identified as those having a temperature differ-
ence from the surface of less than IC with a certain degree of permanence
so as not to involve those of diurnal period. McDonnell considered the
term "MLD" and depth of the thermocline synonymous and refers only to
mixed-layer depths associated with either transitional or seasonal thermo-
clines. Only secular, non-advective, and non-divergent processes were
considered as influencing the MLD. Other processes contribute to MLD
behavior which deviates from the model.
The relationships developed by McDonnell are:
MLD - m^, <»
where: (j = total heat present or excess heat in the upper
wind mixed-layer,
^^ = representative maximum wind,
Xi = coriolis parameter times 10^ (2(jd sin0 x 10^)
/^ = coefficient of thermal expansion,
13
P = a dimensionless function of N with the form
of a first degree polynomial.
To specify the form of P(N), equations (1) and (2) were solved for
P(N) and N respectively. Then measured values of the parameters provid-
ed 200 paired values of P(N) and N which were plotted together. The form
of P(N) was found by curve fitting to this plot. Seasonal and transition-
al MLD's were separately treated, a linear function P(N) being determined
for each of these situations.
McDonnell pointed out that, if the parameters chosen truly represent
the controlling processes, then the plot of P versus N would have little
scatter. Large scatter indicates assumptions were inadequate, e.g.,
divergence and advection are certainly important during some intervals.
McDonnell's final equations incorporating the linear relationship
for P(N) were:
MLD = ^'°^)d -.'25>^I0" -^, (3)
' "-^ si Q/Sn?"
where
(4)
for transitional MLD and
where
^4
(5)
P(m) ^ l.iD^^ - 4.Uio
(6)
for the seasonal MLD,
14
3. Area study selection.
Several basic considerations governed the choice of the data used
in this study. The first requirement was dependability, i.e., the
measurements must be of acknowledged accuracy and recorded at a fixed
location with appropriate frequency as nearly continuous as possible
during the periods of interest; the second requirement was immediate
availability, an important matter because of the limited time available
for preparation of the study; the third requirement was that data be
suitable to measure the phenomena the thesis attempts to describe, which
means mainly that the effects of extraneous processes, such as internal
wave activity, convection and advection be minimized or, at least, evalu-
ated; and a fourth consideration was that the data come from geographi-
cally and climatologically dissimilar areas and from different times so
that the possibility of a universal function and its application to fore-
casting could be examined.
The requirements having to do with quality, frequency and continuity
are satisfactorily met by the data from OWS ships; in fact there are
few other sources for suitable data. The particular weather ships from
which data were used were chosen in large part because of their being
on hand in large quantities, thus providing economy of both time and
money.
Specifically, data available for the study represented two dis-
tinct geographical locations, one in the Atlantic (OWS Bravo 56 30N, 51W)
and one in the Pacific (OWS November 30N 140W) . In addition comparison
was available with McDonnell's work at OWS Papa (SON, 145W) .
According to Tully [8], OWS November is contained in the eastern
extremity of the large Subtropic Region in which the mid-ocean flows
15
are zonal and the waters respond to surface processes. Advectlon o£ ,
thermal regimes are minimal since no major current system is present.
The location coincides with the mean position of the permanent Pacific
anticyclone for the summer months, but effects of convergence in deep-
ening the MLD can be estimated from Fofonoff's [1] mass transport cal-
culations.
OWS Bravo, however, located in the eastern sector of the Labrador
Sea does not possess these ideal conditions. Random advective influ-
ences may be present due to meandering of adjacent current patterns.
Additionally, monthly mean patterns of atmospheric circulation show the
presence of a deep low over this location; therefore horizontal diver-
gence can be expected in the upper layers. To some extent, as at OWS
November, this effect can be estimated.
(The West Greenland Current (warm) on the north and Labrador
Current (cold) to the south could provide advective influences.)
16
4. Calculation of parameters.
The start of the warming season is evidenced by the onset of the
seasonal thermocline; it remains in effect until after the autumn e<fiiiaox
when the seasonal thermocline settles to lower depths by convection and
decays. Data to cover this period were selected from the months June
through October.
To determine the parameter MLD, observed values of MLD were plotted
against time for each month, MLD's being read directly from the BT trace.
Plots were made with the time interval three hours, the normal spacing
of BT observations aboard ocean weather stations (OWS) ships. Both
seasonal and transitional MLD's were plotted from the six to eight BT's
available per day. A smooth curve representing the top of the thermo-
cline or actual MLD was then sketched connecting the plotted points. In
this manner an observation time with a missing BT report could be assign-
ed an interpolated MLD.
A mean MLD was computed from the four plotted MLD's during each
twelve-hour interval starting with midnight Greenwhich. If more than
one interpolated MLD was contained in the averaging process, the inter-
val was not accepted. By assessing the MLD in this manner, the ambient
variations due to internal waves hopefully were reduced.
To determine Q, a BT trace was selected from each 12— hour interval
studied that best represented the mean seasonal (and transitional, if
it existed) MLD for that interval. The value of the parameter Q was
determined from this trace representing the total heat in the uppermost
layer. A step-by-step procedure for determining the value of Q is
explained in appendix I with appropriate illustrations. The technique
17
used by the author represents a modification of McDonnell's method.
The parameter W (representative maximum wind) defined by McDonnell
is an average of the five highest winds reported in a 24-hour period
that precedes the 12-hour interval of interest by up to 72 hours.
The values of a , the coefficient of thermal expansion, are list-
ed in table 24 as given by Sverdrup [7]. The value of the parameter^
is selected by entering table 24 with the surface temperature of the
representative BT for the 12-hour interval being studied and the appro-
priate salinity.
Table 1 is a breakdown by OWS ship and month of the nearly 1500 BT's
which provided the data for determining 628 paired values of P and N
subsequently used in evaluating the form of the function P(N). Of the
total paired values, 473 represent seasonal and 155 represent transi-
tional thermoclines.
The following equations were used to obtain the paired values of
P and N from the parameters calculated for each 12-hour interval.
W (2)
Tables 2 through 12 give the values of the parameters and the corres-
ponding paired values of P and N for each observation time. The only
irregulatiry in this process was September 1960 at OWS Bravo where the
available data represented only the first 10 and last 11 days of the
month. During the 10 day segment missing, the surface temperature became
(Normally eight wind reports are available in a 24-hour interval)
18
TAEIE 1
KOKTIHY IXKBKR OF BT DATA C/JtDS j^IiiLYZI-JD
AI^D iajI-iBEii OF PAIHiD VALUES DETKH1-1II;ZD
OVJS i;OVE^ER
jf OF BTs # OF MD ' s r/ OF MD ' s /i^ OF PAIRED VALUES
KOIITH yiliVR AIJALiZED SEASOlI/iL TRAIISIEIIT SE/iSOII/i ITA'ISIEIIT
June 1957 96
July 1957 134-
Aug. 1957 129
Sept. 1957 150
Oct. 1957 196
June I960 176
July I960 177
Aug. I960 1A7
Sept. I960 134-
Oct. I960 123
96
55
35
19
134
0
45
0
112
5
4.B
0
l^J.
U2
52
44
196
0
62
0
O'JS
BRiWO
157
114 ■'
50
17
118
168
51
56
114
43
55
11
129
15
39
8
123
0
38
0
19
less by 3.5C and the MLD increased by over 30 meters, indicating that
other processes than those considered in the model may be involved.
Therefore the data for September were split into two segments and treat-
ed separately.
With this change of season, the heat fluxes across the air-sea inter-
face, although not computed, may well be negligible. During the follow-
ing month, October, (as the cooler continental air masses became more
prominent) instability mixing due to density increases created by
evaporation may influence the depth of this isothermal layer. The in-
fluence of evaporation, not considered in this model, would be indicated
by the scatter in the paired values of P and N.
20
TABLE 2
PARAMETERS USED TO DETERMINE VALUES OF P AND N
FOR JUNE 1957 AT jOWS IJ0VE14BER
DATE
. w
Qs
^tp
MLDg
KI.Dt
TS
X
1.48
yN„
\
.N+
(KlIOTS)
13.0
(Kg c
sal/cm'^)
(METERS)
(°c)
21.1
10^^
.94
10^^
060157
6.2
27.9
060157
12.6
4.8
27.3
22.2
1.22
.77 •
060257
11.0
5.8
27.7
22.2
1.97
1.07
060257
10.8
4.8
26.6
21.7
1.57
.87
060357
10.2
4.9
24.6
22.2
1.71
.98
060457
10.2
5.8
25.0
23.3
2.12
1.18
060557
10.2
7.4
.76
26.3
8.8
23.3
2.83
1.50
.10
.15
060557
9.0
7.7
.68
23.8
10.9
22.8
3.34
1.74
.14
.15
060657
8.6
5.9
1.17
25.5
8.7
22.2
3.01
1.40
.20
.27
060657
10.6
6.3
1.05
24.8
10.0
22.8
2.05
1.21
.14
.20
060757
10.6
6.5
.68
26.4
13.1
22.2
2.25
1.25
.13
.14
060757
10.6
7.7
1.53
28.2
11.8
22.8
2.85
1.48
.24
.30
060857
10.6
5.4
1.32
24.0
11.7
22.8
1.71
1.79
.20
.26
060857
10.2
7.3
.65
21.7
8.9
22.8
2.25
1.45
.08
.14
060957
9.2
7.6
1.05
22.0
8.5
21.7
2.83
1.62
.14
.22
060957
8.6
6.7
1.35
22.2
7.6
22.2
2.97
1.57
.20
.31
061057
8.8
8.5
1.71
25.7
12.. 8
22.2
4.16
1.95
.42
.39
061057
8.8
9.5
2.55
27.0
11.8
23.2
5.04
2.24
.59
.60
061157
8.0
8.9
1.78
25.0
11.7
23.3
5.29
2.31
.49
.46
061157
7.2
7.6
2.04
23.7
10.1
22.4
5.14
2.14
.58
.57
061257
6.0
10.6
3.00
26.7
8.5
23.4
11.96
3.67
1.07
1.04
061357
7.0
11.4
2.68
27.0
9.1
23.6
9.54
3.38
.75
.79
061457
7.8
9.3
2.47
28.2
12.1
23.3/
21.8^
6.56
2.48
.74
.65
061457
9.2
9.1
1.77
28.6
13.7
4.41
1.95
.41
.38
061557
11.4
1.68
14.6
22.5
.28
.30
061757
12.6
9.0
23.8
22.2
1.99
1.45
y
061757
12.6
11.4
28. 1^
22.1
2.98
1.83
061857
13.8
11.6
25. 9 \
22.1
2.33
1.71
061957
14.8
11.9
31.1 ^
22.2
2.49
1.63
061957
14.6
11.7
27.8
22.5
2.25
1.63
062057
14.2
8.9
29.2
21.7
1.84
1.24
062057
14.8
12.3
28.2
21.8
2.28
1.63
062157
14.8
11.6
27.5
22.1
2.14
1.59
062157
14.8
10.7
29.8
21.8
2.09
1.42
062257
14.0
10.7
30.3
22.1
2.37
1.51
062257
14.2
12.9
31.3
•
22.1
2.96
2.05
21
TABLE 3
PARAMETERS USED TO DETERMINE VALUES OF P AIJD N
FOR JULY 1957 AT OWS NOVEI^BER
DATE
W
Qt,
MLDc
(knots) (Kg cal/cm2) (METERS)
MLDt
070957
16.3
12.09
071057
14.3
11.0
071057
12.2
9.8
071157
10.8
8,4
071157
9.8
9.5
071257
7.2
13.2
071257
6.0
11.2
071357
6.4
10.9
071357
6.4
12.0
071457
6.0
11.6
071457
6.0
10.1
071557
10.2
13.0
071557
14.4
10.5
071657
18.4
11.9
071657
19.6
11.5
071757
22.6
13.3
071757
22.0
9.7
071857
23.0
13.2
071857
23.0
8.2
071957
23.0
8.5
071957
21.0
11.0
072057
21.0
9.5
072057
21.0
12.0
072157
19.8
9.8
072157
20.0
9.6
072257
20.0
11.0
072257
19.6
10.7
072357
18.4
9.0
072357 •
• 13.0
9.3
072457
12.2
9.6
072457
13.4
10.3
072557
14.2
11.8
072557
14.2
13.2
072657
16.6
12.1
072657
16.8
11.0
072757
16.8
10.7
072757
15.2
11.4
072857
18.8
9.3
072857.
20.6
8.8
072957
20.6
12.3
072957
20.6
8.1
073057
19.8
13.0
073057
19.4
13.8
073157
17.8
12.5
073157
18.0
12.3
34.2
36.0
42.3
40.3
41.2
42.0
47.9
38.5
44.2
37.1
38.3
41.2
39.1
40.3
39.6 .
35.9
35.3
38.0
34.0
35.3
43.3
42.9
44.5
41.2
44.4 \
39.2
43.6
41.0
44.1 \
35.7
43.6
44.4
49.2
50.5
46.4
47.4
47.0
39.8
33.7
44.7
36.7
47.7
49.3
48.6
48.7
TS
(^c)
X
^o^^
X 10^ ^
23.8
2.37
1.55
23.9
2.94
1.60
23.8
4.23
1.67
23.7
4.41
1.61
23.8
6.19
2.02
24.1
16.77
3.94
24.9
23.37
4.01
24.3
16.06
3.67
24.4
20.31
4.03
24.9
18.74
4.15 .
24.0
16.34
3.62
24.3
8.07
2.73
24.3
3.11
1.57
24.2
2.22
1.39
'24.2
1.86
1.27
24.2
1.47
1.27
24.1
1.11
.95
23.6
1.44
1.20
23.6
.80
.74 •
23.8
.87
.77
'
23.8
1.65
1.09
24.0
1.41
.98
23.6
1.84
1.18
23.8/ 1.56
1.03
23.7^
1.62
1.00
23.6
1.63
1.14
23.9
1.84
1.14
23.9
1.66
1.02
23.8
3.68
1.49
23.6
3.50
1.64
23.4
3.79
1.60
24.0
3.95
1.78
24.2
5.05
2.00
24.1
3.47
1.57
24.1
2.84
1.41
23.5
2.73
1.32
23.8
3.53
1.56
21.4
1.50
.97
21.0
1.00
.84
21.6
1.86
1.17
20.7
.96
.75
21.3
2.27
1.29
21.7
2.59
1.39
21.4
2.75
1.38
21.4
2.63
1.33
22
TABLE 4
PARAMETERS USED TO DETERMINE VALUES OF P AND N
FOR AUGUST 1957 AT OWS NOVEliBER
DATE
W
(Kg cal/cm )
MLD MLD^
s t
TS
P
N
4 ^
(KNOTS)
(METERS)
(^C)
X 10'
080157
19..0
11. 1
40.8
21.4
1.80
1 . 14
080157
19.3
13.1
50.9
21.5
:.. . 44
1.30
080257
19.8
13.5
51.8
21.7
2.56
1.34
080257
19.8
12.7
48.5
21.5
2.25
1.26
080357
17.8
12.4
43.4
21.5
2.71
1.37
080357
17.0
i3.9
54.1
21.4
3.74
1.31
081157
10.0
12.2
49.0
22.5
8.83
2.47
081157
10.0
10.0
37.8
22.9
5.59
2.02
081257
9.6
11.5
37.1
23.6
7.03
2.49
081257
9.4
14.8
35.4
'23.6
9.01
3.27
081357
• 9.6
15.1
44.4
23.7
11.06
3.27
0S1357
9.6
15.5
47.2
23.7
12.07
3.35
081457
11.8
22.0
54.8
23.8
13.16
3.88
081457
13.0
15.9
53.2
23.8
7.60
2.55
081557
13.8
22.5
49.3
23.8
8.86
3.40
081557
13.8
17.5
44.7
23.8
6.25
2.64
081657
13.8
18.3
41.6
23.7
6.08
2.76
081657
11.8
13.3
39.8
23.6
5.78
2.34
081757
11.2
13.5
39.2
23.7
6.42
2.51
081757
10.2
17.9
40.1
23.7
10.49
3.65
081857
10.0
18.5
50.0
23.9
14.06
3.84
081857
9.4
17.9
41.5
23.6
12.78
3.97
081957
9.4
18.4
37.9
23.8
12.15
3.55
081957
9.4
18.4
37.9
23.8
12.00
4.03
082057
10.4
14.8
42.9
23.9
8.93
2.96
082057
10.4
19.3 '
46.3
23.9
12.70
3.86
082157
10.4
14.4
44.9
23.9
9.08
2.88
082157
10.4
11.4
39.2
23.9
6.28
2.28
082257
9.4
14.6
42.8
. 23.7
10.74
3.23
082257
9.0
19.3
48.7
23.8
17.64
4.45
082357
10.4
18.1
47.3
23.9
12.03
3.62
08^357
11.4
19.7
47.1
24.0
10.85
3.59
082457
11.8
19.8
53.0
24.0
11.45
3.49
082457
11.8
19.0
49.8
24.0
10.32
3.35
082557
11.8
23.8
55.7
24.0
14.47
4.19
082557
11.8
23.5
53.5
23.9
13.73
4.15
082657
13.2
28.4
59.5
24.1
15.31
4.28
082657
13.6
20.7
54.1
23.9
9.21
3.16
082757
13.4
18.1
42.9
23.8
6.57
2.81
082757
13.9
22.6
47.9
23.8
8.51
3.38
082857
12.6
21.6
39.1
23.6
8.09
3.56
082857
11.4
21.5
42.3
23.9
10.63
3.93
082957
10.4
14.1
38.3
23.8
7.59
2.83
082957
10.0
11.7
30.5
23.6
5.43
2.44
083057
10.2
16.9
42.3
24.0
10.45
3.45
083057
9.8
22.5
46.3
24.2
17.01
4.93
083157
7.8
17.7
46.5
24.1
21.20
4.87
083157
6.0
23.7
49.1
24.1
50.69
8.48
X 10
23
TABLE 5
PARAMETERS USED TO DETEH^iINE VALUES 0^' P AMD M
FOR SEPTEMBER 1957 AT OWS NOVEMBER
DATE
1
W
(KNOTS)
(Kg^cal/cm^)
(MTERS)
TS
\^c
A
090157
6.3
23.6
.46
45.7
8.5
23.3
41.31
7.80
.15
.15
090157
6.5
22.0
.97
36.6
15.2
23.3
28.96
7.04
.53
.31
090257
6.4
22.8
.56
57.0
9.8
23.3
48.23
7.46
.21
.18
090257
6.4
29.4
.77
59.4
12.2
23
64.81
9.56
.35
.25
090357
7.4
29.8
.83
67.1
9.1
22.8
55.50
8.38
.21
.24
090357
7.4
29.2
.78
61.0
10.7
23.3
49.45
8.21
.24
.22
090457
7.8
22.2
1.03
42.7
12.2
23.3
23.69
5.92
.32
.28
090457
7.8
25.7
1.38
48.8
16.8
23.3
31.34
6.85
.59
.36
090557
7.8
18.5
.54
27.4
6.1
23.9
12.67
4.93
.08
.14
090557
5.4
27.3
.93
45.7
9.1
23.3
65.03
10.54
.45
.36
090657
8.2
28.8
1.38
57.9
12.2
23.9
37.70
7.31
.38
.35
090657
9.2
26.5
.92
51.8
9.1
23.9
24.65
7.38
.15
.21
090757
10.5
28.2
1.15
51.8
11.6
23.9
20.15
5.58
.18
.22
090757
10.5
23.8
1.39
45.7 .
12.2
23.9
15.00
4.72
.24
.28
090857
11.2
21.4
.54
48.8
6.1
23.9
12.65
3.98
.04
.10
090857
11.2
24.8
1.44
45.7
18.3
23.9
13.73
4.61
.32
.26
090957
13.2
19.7
1.19
57.9
15.2
23.9
9.95
3.13
.15
.18
090957
14.2
29.4
2.01
59.4
24.4
23.3
13.17
4.30
.36
..29
091057
14.2
26.1
2.18
48.8
21.3
23.3
9.60
3.83
.35
.32
091057
14.2
22.4
1.07
36.6
25.9
23.9
6.18
3.29
.21
.15
091157
14.2
21.3
1.38
51.8
18.3
23.9
17.47
4.52
.40
.29
091157
19.8
24.1
1.68
51.8
17.1
23.3
19.76
5.12
.46
.36
091257
9.8
23.7
2.00
45.7
19.8
23.9
17.15
5.03
.63
.43
091257
9.8
28.6
1.92
48.8
20.7
23.9/
22.09
6.07
.63
.40
091357
9.8
26.8
.85
57.9
9.1
23.9
48.13
7.96
.24
.25
091357
7.0
21.0
1.65
51.8
25.9
23.9
33.75
6.24
1.33
.49
091457
7.0
21.1
1.46
47.2
12.8
23.9
30.89
6.26
.59
.43
091457
7.0
20.6
1.18
45.7
21.3
23.9
29.20
6.12
1.24
.56
091557
9.8
25.8
1.10
56.4
12.8
23.3
23.03
5.47
.22
.24
091657
9.8
20.3
2'. 46
48.8
28.0
23.9
15.68
4.32
1.09
.53
091757
10.0
23.2
1.54
54.9
19.8
23.9
19.37
4.83
.46
.32
091757
10.2
24.6
2.18
61.0
18.3
23.9
21.92
5.01
.59
.45
091857
16.8
23.1
2.29
57.9
25.0
23.3
7.20
2.85
.31
.28
091857
16.8
23.0
1.57'
57.9
30.5
23.9
7.17
2.85
.25
.19
091957
19.-8
23.5
1.56
48.8
27.4
23.9
4.45
2.46
.29
.25
091957
19.8
22.9
2.45
39.6
30,5
23.9
3.52
2.41
.22
.21
092057
18.4
24.4
1.86
57.9
26.8
23.9
6.35
2.76
092157
18.2
25.0
48.8
23.3
5.59
2.85
092157.
17.8
22.7
.85
48.8
21.3
23.3
5.32
2.66
.08
.10
092257
15.0
22.7
2.44
61.0
27.4
23.9
9.36
3.15
.45
.33
092357
9.0
25.7
2.39
64.6
30.5
24.4
32.14
6.14
1.41
.58
092357
9.0
28.7
2.54
64.0
27.4
23.9
34.46
6.64
1.31
.58
24
TABLE 5 (Cont'd)
DATE
W
%
Q
Ih
MLD
s
MLD
t
TS
P
s
N
s
^
\
(
[KNOTS)
(Kg ca
.1/a
(METERS)
(°C)
X 10^
X 104
092457
6.0
2
.45
27,
.4
24.4
2.92
.88
092557
7.0
20.0
1
.36
61.0
29,
.9
23.9
37.85
5.94
1.26
.40
092557
8.0
23.6
1
.87
67.1
33,
.5
24.4
38.80
6.34
1.53
.50
092657
10.6
27.2
2
.17
54.9
24,
.4
24.4
20.84
5.51
.73
.45
092757
16.4
29.3
67.1
25.0
11.46
3.84
092757
17.6
22.4
54.9
25.0
6.23
2.73
092857
17.6
26.2
61.0
25.0
8.09
3.19
092857
17.6
26.2
61.6
25.0
8.18
3.19
092857
17.6
28.0
67.1
24.4
9.51
3.42
093057
11.6
22.6
45.7
24.4
12.03
4.18
093057
11.6
22.6
45.7
24.4
12.03
4.18
TABLE 6
PARAMETERS USED TO DETERMINE VALUES OF P AND N
FOR OCTOBER 1957 AT OWS NOVEMBER
DATE
W
Q Q^
MLD MLD^
TS
P
N
(Kg cal/cm )
s t
s
4 '
(KNOTS)
(METERS)
CC)
X 10
100157
10.6
19.3
30.2
24.6
8.13
3.91
100157
10.0
24.3
36.0
24.7
13.72
5.22
100257
10.6
22.2
35.6
24.4
11.03
4.50
100257
13.6
25.8
35.1
24.4
7.67
4.08
100357
13.8
21.0
36.6
24.5
6.33
3.26
100357
13.9
27.7
44.2
24.7
9.93
4.28
100457
14.6
23.5
42.2
24.3
7.30
3.46
100457
16.2
26.4
48.5
24.8
7.67
3.51
100557
16.0
22.8
39.8
24.3
5.56
3.06
100557
16.2
23.2
50.1
24.4
6.95
3.08
100657
15.3
19.8
34.0
24.4
4.51
2.77
100657
14.2
23.6
39.4
24.3
7.23
3.55
100757
17.4
22.2
40.9
24.2
4.71
2.75
100757
21.0
24.1
44.9
24.2
3.84
2.47
100857
22.0
23.0
42.4
24.1
3.17
2.24
100857
21.8
22.3
40.4
24.3
2.98
2.20
100957
22.0
28.4
46.3
24.4
4.26
2.77
100957
21.8
21.6
37.1
24.5
2.65
2.13
101057
17.6
22.4
43.8
24.3
4.97
2.73
101057
14.8
20.9
41.7
24.3
6.23
3.03
101157
14.8
20.5
38.0
24.2
5.57
2.98
101157
14.8
22.4
42.3
24.1
6.78
3.25
101257
14.2
26.1
44.7
24.2
9.07
3.95
101257
13.8
26.4
45.2
24.2
9.83
4.11
101357
10.0
19.6
46.9
24.3
14.42
4.21
101357
9.2
22.2
43.5
24.2
17.89
5.19
101457
11.2
20.4
40.6
24.1
10.35
3.91
101457
12.0
21.4
41.8
23.9
9.45
3.72
101557
12.0
21.8
44.3
23.3
10.20
3.79
101557
12.0
19.0
39.6
22.8
7.71
3.22
101657
10.8
21.7
45.3
22.8
12.45
4.07
101657
10.8
21.3
43.1
21.8
11.30
3.88
101757
9.2
20.8
42.3
22.8
15.36
4.59
101757
8.6
20.2
39.8
22.4
16.05
4.76
101857
8.6
20.0
44.0
21.7
17.07
4.58
101857
9.4
21.6
44.0
22.2
15.89
4.65
101957
7.0
20.1
41.9
22.2
25.39
5.82
101957
10.6
22.2
43.1
22.2
12.58
4.25
102057
14.2
22.3
39.8
22.5
6.51
3.19
102057
15.2
25.5
42.7
22.3
6.95
3.41
N.
X 10
26
TABLE 6 (Cont'd)
DATE W Q Q^
s tj
(KNOTS) (Kg cal/cm )
102157
16.6
27.2
102157
18.0
24.8
102257
19.8
23.0
102257
19.8
23.2
102357
19.8
25.7
102357
19.4
26.6
102457
16.8
25.3
102457
11.6
24.1
102557
12.6
24.5
102557
12.6
23.5
102657
12.6
21.4
102657
12.6
22.2
102757
10.0
25.2
102757
10.6
20.5
102857
10.6
25.5
102857
10.8
25.4
102957
9.4
24.6
102957
9.0
22.0
103057
7.4
21.5
103057
12.0
23.2
103157
16.2
27.5
103157
17.2
27.2
MLD MLD^
s t
TS
P
8
N
4 "
(METERS)
rc)
X 10
49.0
22.5
7.14
3.32
42.3
22.5
4.79
2.80
43.9
22.2
3.81
2.36
44.3
22.4
^.88
2.37
47.6
21.7
4.43
2.55
49.2
22.5
5.14
2.78
53.7
22.0
6.90
2.96
44.4
22.2
11.66
4.21
57.0
22.1
12.99
3.83
46.3
22.4
10.13
3.79
43.8
21.8
8.47
3.34
44.7
22.0
8.97
3.46
55.0
22.2
20.47
5.12
44.1
22.5
11.89
3.92
50.5
22.1
16.93
4.89
50.4
22.1
16.21
4.76
53.3
21.9
21.30
5.14
52.8
21.9
20.57
4.80
51.9
21.9
29.29
5.71
50.7
22.2
12.07
3.92
61.0
21.9
9.17
3.34
56,0
21.8
7.39
3.11
N
X 10
27
TABLE 7
PARAMETERS USED TO DETERl^JNE V/vLUES OP P AIJD II
JTJUE I960 AT OV/S BRAVO
DATE
■w
Qs
Qt^
MLDg
MT.Dx
TS
X 1(
.43
,^K
P^ .No.
(KKOTS)
22.6
(Kg C)
2.41
al/cm^)
54.9
w
(°c)
5.0
3^'
.15
X 10^*
I*
060160
060160
25.4
2.94
50.6
4.5
.39
.16
060260
25.8
2.95
54.9
5.0
.41
.16
060260
25.8
4.82
.42
73.2
12
.2
4.8
.89
.26
.01
.02
060460
20.0
4.02
48.8
4.8
.82
.28
060460
20.0
4.15
39.6
4.8
.69
.29
060560
20.0
5.65
36.6
5.0
.86
.39
060560
23.2
4.79
1.18
67.1
24.4
5.0
.99
.30
.09
.07
060760
23.2
1.02
19.8
4.4
.06
.06
060760
23.2
2.20
32.6
4.4
.22
.13
060860
16.4
1.52
39.6
3.9
.37
.13
060860
17.2
.76
25.6
4.4
.11
.06
060960
17.2
.80
25.9
4.4
.12
.06
060960
17.2
.75
21.3
4.4
.09
.06
061060
17.2
.52
18.3
4.4
.05
.04
061060
16.2
1.95
29.0
4.4
.36
.17
061160
14.0
1.10
19.8
4.4
.19
.11
061160
15.2
3.62
25.3
5.3
.66
.33
061260
15.4
3.39
22.9
4.4
.55
.30
061260
17.4
3.92
26.8
5.0
.58
.31
061360
17.4
2.94
18.9
5.0
.31
.23
061360
17.4
3.00
25.6
5.0 /
4.8 /
.42
.24
061460
17.4
2.44
24.4'.
.33
.19
061460
21.6
4.42
37.2 \
5.3
.59
.28
061560
21.6
3.40
31.4
>
5.0
.38
.22
061560
21.6
6.40
34.1
5.1
.78
.41
061660
21.6
7.30
25.9
4.7
.68
.46
061660
20.6
4.50
28.3
5.3
.50
.30
061760
16.6
4.74
27.4
4.9
.79
.39
061760
19.2
4.83
24.1
5.3
.53
.35
061860
19.8
5.70
29.3
4.8
.71
.40
061860
19.8
5.65
29.9
5.4
.72
.39
061960
19.8
7.40
35.7
5.0
1.13
.51
061960
19.2
6.70'
, 35.4
5.8
1.07
.48
062060
•19.2
6.91
.76 •
31.1
6,
.1
5.4
.97
.49
.02
.05
062060
17.8
5.31
.61
25.0^
4,
.6
6.1
.77
.48
.02
.05
062160
17.0
1.95
.93
27.4
9,
,1
6.0
1.21
.62
.05
.08
062160
13.2
7.56
.90
32.3
12.
,8
6.3
2.58
.87
.12
.10
062260
11.8
9.50
i. 13
36.6
11.
,8
5.8
4.17
1.11
.23
.13
062260
9.6
8.97
1.83
33.5
16.
,8
5.6
5.45
1.28
.56
.26
062360
10. 2-
1.91
18.
.3,
6.1
.62
.28
062360
10.2
8.67
1.43
34.7
14.
,3
5.6
4.83
1.17
.33
.19
TABLE 7 (Cont'd)
DATE
W
(KNOTS) (Kg cal/cm )
MLD
MLD.
s t
(METERS)
TS
? N
s s
X 104
P
t
X 104
N.
062460
062460
062560
062560
062660
062760
062760
062860
062960
062960
063060
063060
12.0
12.6
14.0
14.0
14.0
19.0
19.0
19.0
14.8
14.8
11.0
10.6
7.35
7.60
8.37
10.89
8.75
8.12
10.25
11.50
7.28
11.10
.98
.97
2.90
2.17
3.28
29.0
35.1
33.5
35.1
31.4
32.0
36.0
33.5
25.9
27.4
12.8
16.8
14.6
11.9
18.3
.90
16.8
5.7
5.6
5.6
5.8
6.1
6.1
6.1
5.6
6.1
6.1
6.1
6.2
2.25
2.27
2.39
3.59
1.40
1.32
1.71
3.24
1.58
4.63
.80
.75
.82
1.18
.70
.65
.74
1.18
.74
1.52
15
,17
.36
22
56
11
11
28
21
35
25
13
TABLE 8
PARAMETERS USED TO DETEM-aiffi V/JAJES OF P AND 13
FOR JTJLY I960 AT OWS BRAVO '
DATE
1
(KNOTS)
(Kg ctd/cm)
U^IETERS)
TS
(°c)
Ps /s
X 10^
X 10^
070160
14.0
11.26
1.78
55.8
27.3
6.1
5.90
1.22
.46
.19
070160
14.0
13.25
1.66
57.4
26.2
6.7
7.14
1.43
.41
.18
070260
14.0
13.09
2.04
61.2
24.6
6.5
7.52
1.42
.47
.22
070260
12.6
13.92
2.74
63.4
34.4
6.7
10.23
1.67
1.09
.33
070360
12.2
13.64
2.54
67.8
38.3
6.3
11.43
1.69
1.20
.32
070360
13.2
10.13
3.53
57.4
31.7
6.7
6.14
1.16
1.18
.40
070460
13.2
13.75
2.80
68.4
31.7
7.2
10.81
1.72
1.02
.35
070460
13.2
15.05
3.05
65.6
32.8
7.2
11.35
1.88
1.15
.38
070560
12.4
14.88
.68
60.1
16.4
7.5
11.65
1.98
.15
.09
070560
12.4
14.77
4.05
71.1
27.3
6.9
12.57
1.80
1.32
.49
070660
14.4
14.72
3.77
73.8
35.5
6.7
9.64
1.55
1.19
.40
070660
18.2
13.24
4.66
79.3
30.1
7.1
6. 35
1.20
.85
.42
070760
18.2
13.22
2.40
71.1
24.6
7.5
5.68
1.20
.36
.22
070760
18.2
15'.72
3.34
65.6
27.3
7.6
6.49
1.48'
.57
.31
070860
17.0
12.10
3.88
60.1
30.1
7.0
4.63
1.17
.74
.38
070860
12.6
10.35
2.80
54.7
27.3
7.2
7.14
1.35
.96
.37
070960
13.8
13.96
2.86
73.8
26.2
7.2
10.84
1.67
.79
.34
070960
13.8
15.74
3.92
76.6
30.6
7.2
12.68
1.88
1.26
.47
071060
13.8
12.47
3.81
65.6
23.0
7.2
8.60
1.49
.92
.45
071060
13.8
11.01
2.14
61.2
12.0
7.2
7.09
1.31
.27
.26
071160
13.8
15.56
1.74
79.3
16.4
7.9
13.50
1.93
.31
.22
071160
14.4
13.71
4.17
65.6
27.3
8.3
7.8 /
9.42
1.70
1.19
.52
071260
14.8
16.58
5.41
82.0
33.9
12.94
1.92
1.07
.63
071260
14.8
3.14
20.8
8.6
.67
.39
071360
14.8
16.44
2.44
76.6
23.5
8.1
12.49
1.99
.57
.29
071360
14.4
15.75
5.58
67.3
26.2
7.7
10.65
1.88
1.47
.66
071460
13.4
14.60
2.97
75.5
21.9
8.3
13.34
1.95
.79
.40
071460
13.4
15.29
5.00
67.8
25.7
8.0
12.03
1.96
1.49
.64
071560
13.4
15.52
5.52
71.1
27.9
8.1
13.35
2.07
1.86
.74
071560
13.0
16.43
5.08
67.3
27.3
8.3
14.22
2.26
1.78
.70
071660
12.0
14.13
4.90
82.0
24.6
8.3
17.48
2.10
1.82
.73
071660
12.0
19.45
6.63
87.5
32.8
8.3 •
25.68
2.90
3.28
.99
071760
14.6
16.45
6.22
65.6
37:2
8.3
11.00
2.01
2.36
.76
071760
22.8
13.14
4.62
60.1
35.5
8.3
3.30
1.03
.69
.36
071860
22.8
12.41
3.66
59.1
32.8
8.1
3.07
.97
.50
.29
071860
22.8
14.66
6.30
71.1
36.1
8.5
4.36
1.15
.95
.49
071960
20.0
17.57
7.52
76.6
44.8
8;3
7.31
1.57
1.83
.67
071960
20.2
15.14
6.32
62.9
37.7
8.2
5.07
1.34
1.27
.56
072060
22.0
5.40
36.1
7.5
.81
.40
072060
22.0
18.88
6.68
68.4
45.9
7.8
5.56
1.47
1.32
.52
072160
17.4
2.37
19.7
7.8
.32
.23
072160
17.4
17.04
8.86
68.4
44.8
7.7
8.02
1.68
2.73
.87
TABLE 8 (Cont'd)
DATE
W
0
MLD
MLD.
(KNOTS) (Kg cal/cm )
072260
072260
072360
072460
072460
072560
072560
072660
072660
072760
072860
072860
072960
073060
073060
073160
073160
15.2
13.2
13.2
12.6
13.6
13.6
13.6
10.0
7.0
9.0
10.2
11.0
10.6
10.6
10.4
10.4
10.0
17.80
14.00
16.30
15.08
14.20
17.86
15.45
18.27
17.85
19.29
22.12
21.31
7.63
6.55
7.81
7.87
.61
7.42
3.40
8.06
6.73
6.76
2.52
6.14
3.90
3.31
s t
(METERS)
71.1
54.7
71.1
73.8
61,2
71.1
71.1
73.8
68.4
75.5
76.6
76.6
38.3
35.5
42.
38.
13.
49,
27.
33.
32.8
30.1
19.1
31.7
21.9
20.8
TS
Cc)
8.1
8.0
8.3
7.8
8.6
8.2
8.2
7.8
8.5
8.1
8.4
8.7
9.2
9.1
9.4
9.0
8.0
? N
s s
X 104
11.90
9.16
14.45
14.61
10.56
14.92
12.91
22.67
30.27
36.59
36.38
2.09
1.81
2.21
2.05
1.93
2.35
2.03
59.70 4.66
3.00
3.50
4.09
3.94
't
X 104
N.
3.83
2.82
3.86
3.54
.17
16.13
2.49
5.71
4.10
4.23
1.00
4.20
1.77
1.55
1.04
.89
1.03
1.03
.10
1.89
.68
1.41
1.13
1.23
.46
1.13
.69
.61
31
TABLE 9
PARAMETERS USED TO DETERl-'xlKE VALUES OF P AND W
KIR AUGUST I960 AT OWS BRAVO
DATE
W
(Kl^OTS)
(Kg cal/cm'^)
MLDg mn
(METERS)
4. TS
^ (°c)
10^ ^
X
10^ ^
080160
8.8
6.80
15.2
8.9
3.00
1.43
080160
11.8
9.06
18.3
9.3
2.78
1.47
080260
14.0
4.67
15.2
8.9
.81
.62
080260
13.6
8.40
21.3
9.2
2.26
1.19
080460
17.0
9.46
22.9
9.2
1.75
1.07
080560
17.0
15.85
30.5
9.3
3.91
1.79
080560
15.0
10.25
24.4
8.9
2.50
1.27
080660
12.0
6.22
18.3
9.2
1.85
1.00.
080660
12.0
13.51
30.5
10.1
6.89
2.23
080760
10.0
14.42
1.45
29.0
9,
.1
9.4
9.77
2.77
.31
.28
080760
12.5
13.72
25.9
9.5
5.31
2.11
080860
15.2
22.96
3.96
44.2
12,
.2
9.7
10.56
2.99
.50
.52
080860
15.2
10.02
24.4
11.1
2.78
1.43
080960
15.2
14.92
29.0
10.4
4.65
2.00
081060
15.2
10.50
25.9
11.1
3.10
1.49-
081060
20.2
13.02
24.4
11.1
2.05
1.39
081160
20.2
8.66
18.3
11.1
1.02
.93
081160
16.0
12.33
21.3-
.11.4
2.70
1.67
081260
14.0
8.14
15.2
11.7
1.70
1.29
081260
12.0
10.82
18.3
11.8
3.70
2.00
081360
12.0
8.15
15.8
11.9
2.41
1.50
081360
12.0
12.30
19.8
11.7
4.55
2.27
081460
081460
15.6
15.6
13.06
11.66
.28
22.9 '
21.3
3.
•.0
11.7 /3.31
11.6/ 2.75
1.85
1.65
.01
.04
081560
15.6
15.20
29.0
11.1
4.77
2.11
081560
15.6
14.64
27.1,
11.6
4.39
2.08
081660
15.2
16.50
30.5 \
11.7
5.86
3.40
081660
13.8
20.37
6.55
38.4 \
18.
.3
11.7
11.05
3.27
1.69
1.05
081760
13.8
9.62
21.3
11.7
2.89
1.54
081760
13.0
13.06
25.0
11.7
5.22
2.22
081860
14.2
12.50
24.4
11.6
4.07
1.95
081960
25.8
20.58
29.0
11.1
2.36
1.73
081960
25.8
19.05
27.4
10.7
2.02
1.56
082060
25.0
10.52
.25
'23.5
9.
,1
11.4
1.04
.91
.01
.02
082060
20.0
12.90
.17
32.6
3.
,1
11.1
2.77
1.40
.01
.02
082160
13.4
21.30
39.6
11.1
12.36
3.44
082160
10.4
12.00
18.3
11.1
5.34
2.50
082260
14.8
17.30
26.8
11.1
5.57
2.53
082360
14.8
15.28
21.3
.
11.7
4.00
2.28
082460
15.2
15.25
25.0
11.7
4.44
2.22
082460
15.2
15.52
25.6
11.7
4.63
2.26
082560
15.2
9.33
18.3
11.7
1.99
1.36
082560
11.8
12.62
.20
20.7
3.
7
11.3
4.94
2.31
.02
.04
32
TABLE 9 (Cont'd)
DATE
W
0
Q^
MLD
MLD^
TS
P
N
P
N
s
to
s
t
s
s
£
t
(KNOTS)
(Kg cal/cm^)
(METERS)
(°C)
X 10^
X 104
082660
11.8
19.60
36.6
11.7
13.86
3.68
082660
11.8
22.62
.90
39.6
9.1
11.4
16.93
4.15
.15
.16
082760
11.8
21.70
37.8
11.7
15.85
4.07
082760
9.8
16.88
.67
30.5
10.7
11.7
14.42
3.81
.20
.15
082860
9.8
19.62
33.5
11.9
18.41
4.43
082860
9.8
18.78
32.0
12.1
17.20
4.33
082960
9.4
19.07
1.00
32.0
11.3
12.2
19.00
4.59
.35
.24
082960
8.8
13.82
25.0
11.2
11.74
3.40
083060
10.6
15.00
1.22
27.4
15.8
10.6
9.41
2.99
.44
.24
083060
16.0
16.90
22.9
10.6
3.89
2.23
083160
19.4
17.15
27.4
10.4
3.10
1.81
083160
19.4
16.00
26.5
10.1
2.80
1.68
33
TABLE 10
PARAl^iETEHS USilD TO DLTi;,RI«Jl^ VALUiiS OP P Mh M
FOR SEPT. 01-09, I960 AT OWS BRAVO
DATE
V
(OOTS)
(Kg cal/aor)
MT-Dg MLD+
(METERS)
TS
(°c)
X
xo^^
X
zo^"^
090160
19.8
15.78
24.4
10.4
2.44
1.63
090160
19.8
14.90
23.8 .
10.3
2.25
1.54.
090260
19.8
18.15
29.0
10.3
3.33
1.87
090360
25.2
14.15
21.3
10.0
1.14
1.11
090360
25.2
11.40
21.9
10.3
.98
.92
090460
25.2
17.00
25.6
10.2
1.70
1.38
090460
25.0
13.62
22.3
10.4
1.21
1.11
090560
21.0
14.22
21.3
10.6
1.77
1.43
090560
21.0
12.22
20.4
10.7
1.45
1.23
090660
21.0
13.31
21.9
10.6
1.70
1.34
090660
13.6
13.62
20.1
10.6
■ 3. SI
2.12
090760
12.6
10.02
17.7
10.6
2.87
1.68
090760
13.0
15.10
21.3
10.6
4.89
2.46
090860
16.6
19.19
2
.49
34.7
11,
.6
10.6
6.22
2.45
.27
.32
090860
19.2
24.26
11,
.38
42.7
21,
.3
10.6
7.23
2.67
1.69
1.25
091060
20.0
11.42
21.3
10.6
1.56
1.21
091060
20.0
15.05
22.3
10.6
2.17
1.60
:erj
TABLE 11
iED TO DETERl^KE VALUE
PARAMLT
S U£
;S OF P AND H
20.0
FOR SEJ
18.55
>T. 19-30
, 1*
960 AT OWS
7.1 /
1, BRAVO
5.16
1.53
091960
55.5
092060
18.0
20.00
54.9
6.7
6.24
1.68
092060
20.0
22.71
5,
.76
62.5
30.
,5
7.3 .
7.11
1.87
.88
.47
092160
27.0
7.
.70
\
30.
,5
7.1
.65
.47
092160
27.0
23.63
7,
.12
82.3 V
36.
.6
7.2
5.34
1.44
.72
.43
092260
27.0
10.23
2.
.71
42.7 \
24.
.4
6.8
1.10
.57
.17
.15
092260
27.0
11.55
39.6
7.1
1.26
.70
092360
22.8
13.84
2,
.00
52.7
21.
,3
7.8
2.92
1.04
.17
.15
092360
15.0
19.31
56.4
7.3
9.69
2.12
092460
19.8
20.22
2.
.64
61.0
18.
,3
7.1
6.30
1.68
.25
.22
092460
25.0
19.20
54.3
6.7
3.07
1.16
092560
25.0
18.80
47.2
6.7
2.61
1.14
092560
25.0
18.70
61.0
'
7.2
3.66
1.23
092660
28.4
10.15
36.6-
7.5
.92
.59
092660
28.4
16.20
54.3
6.7
2.01
.86
092760
30.0
16.10
58.5
6.6
1.93
.81
092760
30.0
20.40
54.3
'
6.1
2.27
1.03
092860
22.8
18.70
63.4
6.3
4.20
1.24
092860
20.0
17.70
64.6
6.1
5.26
1.34
092960
20.0
18.62
59.4
6.7
5.09
1.41
092960
15.8
19.21
57.9
6.1
8.20
1.84
093060
21.2
16.42
62.5
6.7
4.20
1.17
093060
22.8
20.47
57.3
6.5
4.14
1.35
TABLE 12
PARAMETERS USED TO DETEH-miE VALUES OF P .UID K
K)R OCTOBER I960 AT OWS BRAVO
DATE W Qs Qt ^a.D MLD^.
(KNOTS) (Kg cal/cm2) (mItERS)
100260
25.5
19.75
100360
27.0
19.05
100460
25.0
18.62
100460
25,0
18.80
100560
20.0
20.40
100560
16.8
17.78
100660
18.2
15.25
100660
19.6
17.55
100760
19.6
19.95
100760
19.6
15.32
100860
25.0
18.45
100860
25.0
17.72
100960
23.0
11.20
100960
20.0
15.71
101060
17.0
15.31
101160
20.0
10.38
101260
20.0
10.52
101360
20.0
15.80
101460
22.0
15.55
101560
25.0
15.95
101660
25.0
17.95
101660
25.0
16.50
101760
17.0
18.10
101760
17.0
13.26
101860
17.0
16.62
101960
12.0
17.84
102060
20.2
14.71
102160
20.2
17.35
102260
20.2
17.24
102360
18.0
17.65
102460
1.80
16.60
102460
17.0
16.95
102560
13.0
17.30
102660
20.0
18.52
102860
25.0
15.62
102960
22.0
14.40
103060
22.0
14.57
103160
27.0
16.10
56.4
58.8
55.5
55.5
54.3
59.4
51.2
59.4
62.5
47.2
59.4
57.9
42.7
50.9
46.6
39.6
41.1
48.8
47.9
60.0
58.8 '
64.6
62.5
51.8
62.5'
62.5 \
54.3
64.0
58.5
59.4
56.4
56.4
61.0
67.1
59.7
67.1
62.8
64.6
ts
(°c)
'^10^
Ns
6.1
3.15
1.17
6.0
2.83
1.07
5.8
2.76
1.02
6.7
3.07
1.14
5.9
4.63
1.40
6.1
6.89
1.60
6.3
4.34
1.27
6.0
5.08
1.38
6.1
5.97
1.54
5.6
3.14
1.07
5.0
2.93
1.01
5.8
2.74
.97
5.8
1.51
.67
6.1
3.68
1.19
6.0
4.54
1.36
5.7
1.72
.71
5.4
1.81
.72
6.0
3.55
1.20
5.9
2.57
.97
6.1
2.82
.97
5.8
/ 2.82
^3.14
.99
6.0
1.00
5.8
6.54
1.46
6.1
4.37
1.18
5.6
6.00
1.34
5.8
12.93
2.04
6.1
3.60
1.10
6.1
5.01
1.30
4.7
4.13
1.17
6.1
5.96
1.48
5.8
5.38
1.33
5.9
5.53
1.37
5.7
10.43
1.83
6.1
5.72
1.40
6.1
2.75
.95
6.1
3.67
.99
6.1
3.48
1.00
6.1
2.63
.90
X
10^
N.
5. The form of the function P(N) .
A least squares computer program was used to determine the poly-
nomial of degree K which best fits (in the least squares sense) M data
points. The best fit among those polynomials tested (through third order)
was for K = 2 for each of three groups of points representing about one-
fourth of all paired values of P and N. The coefficients of the poly-
nomial were then computed for each month and tabulated in table 13,
P(N) having the form below,
PfNl) ~ a^H -+ cX^M 4' "\o (8)
The corresponding forecasting equation is
^-^
hLD--a.^/«Q-v^,_W^^o^.
(9)
McDonnell's criteria for acceptable data limited the number of his
paired values to only 22 pairs for transitional MLD's and 29 pairs for
seasonal MLD's. These data, as a result, were from various months of
the warming season during the years 1958 through 1962. Because of the
small number of paired values and the grouping of the seasonal and transi-
tional paired values, only a linear regression separately done for the
two categories was justified. These are equations (4) and (6) of
McDonnell; they do not necessarily represent the most likely form of
the function P(N).
The present author used both seasonal and transitional paired values
together to obtain a single form for P(N). This was then incorporated
into McDonnell's basic equation (1) and used to forecast both seasonal
and transitional MLD's. Graph No. 1 represents the form of P(N) using
36
TABLE 13
COEFFICIENTS FOR EACH MONTH USED IN THE FORECASTING EQUATION
OWS November
«o - lo""
.094
-.089
.081
-.030
4.235
^2 "" ^°
h
June
.721
.500
July
1.117
.401
Aug.
.606
.726
Sept.
.582
.928
Oct.
1.142
-2.890
OWS Bravo
June
1.38
1.63
July
1.93
2.59
Aug.
.87
.36
Sept.
(1-
-10)
1.11
-.59
Sept.
(19-30)
1.66
1.13
Oct.
4.74
-4.83
-.03
-.47
.09
.48
-.12
2.95
OWS November (June through September)
.543 1.289 -.228
OWS Bravo (June through September)
.996 1.815 .023
37
a second-order polynomial as the best fit for the paired values deter-
mined by McDonnell at OWS Papa.
Graphs No. 2 through 12 are the curves of the function P(N) as
determined for each month. All paired values are plotted on each scatter
diagram.
The scatter of the paired values is relatively small for most months
indicating that McDonnell's model may well contain the correct combina-
tion of parameters. Usually the paired values of P and N for transition-
al situations were found near the origin with little scatter. During
low wind conditions, the computation of P is very sensitive to small
errors in wind speed which accounts for much of the excess scatter at
large P. Additional scatter probably results from random fluctuations
not removed by the averaging procedures described in section 4.
One can see that the monthly best fit curves have a variety of
slopes apparently indicating the non-universality of P(N). However,
systematic deviations due to contaminating influences (e.g. divergence),
but included in the computation of the paired values, may account for
the variations in slope of each monthly function. By analyzing incre-
mental changes in P and N associated with small increases in Q and MLD,
general conclusions concerning the influence of divergence and advection
on the paired values can be made. This analysis indicates that reduc-
tion of the MLD by divergence or advection tends to diminish the slope
dP/dN and vice versa.
(Graph No. 5 for September 1957 had 10 points which fell outside
the scale. Graph No. 4 for August had one such point.)
38
Divergence of the Ekman transport was computed from the monthly
Ekman transport at grid points in the vicinity of each location during
the year studied. Meridional and zonal components of Ekman transport
calculated by Fofonoff and Ross [1,2], were used for this. At OWS Bravo,
maximum divergence was during August which has the least slope of any
function for that OWS ship. The same correlations were noted at OWS
November except that the divergence was negative.
Systematic deviations in the paired values as a result of advection
could not be evaluated as easily.
(July and August at OWS November were anamolous months in this
respect.)
39
k
■
.'
g
o
\
9
9
•
Q-l
•
1
y
y
^
ca
,;►
y
y
•
^
/^
9
Q
^
y^
+
+
•
I
f
4
^X^
-j;J>
+
. ^
Si
•
^
X^
'
•
•
^^<^
^
T
y
o
y^
•
mz .0
QQl
.0
d^
.0
^
.0
ae>i
.0
^
.0
dee
N
X-3CflLE - l.QeE+08 LMITS/'IHCH.
Y-3CflLE - 5.00E+0Q UhUTS/'IHCK
LEAST SQUARES BEST FIT CURUE USING OWS PflPP
TRflNSITIONflL AND SEflSOhflL DflTfl GRAPH NO 1
40
/
•'
"
/
•
'.
>
1
\.
■*
■ ■
^
/
•
-
.,
/
/
+/
/'
-l^"*-'^
^
f
erne .e
881 .0
88Q .0
880 . .0
881 .0
885 .0
866
N
X-3CflL£ - Le8E+aa UMITS-'INCH.
f-scflLE - 5.aaE-taa umTSz-iticn
LEAST SQUARES BEST FIT CURUE OWS NOUEMBER
30 mW 14:0 00W JUNE 1957 GRAPH NO 2
u\
.0882
.0880
.088»i
.0885
.0086
N
X -SCALE - 1.8aE-f88 LJNIT5/IHCH.
Y -SCALE - 5.88E+e8 UNITS/' JMCH.
LEAST SQUARES BEST FIT CURUE OWS T^OUE.^BER
30 Qm 110 00W^ JULY 195P GRAPH hO 3
42
«
ft
^
s
•
/
)
•
;
/
/
} ■
■^
-
•
•
>
•
'
/
/
+
+
/
+
^^
,_/
--^^
<r
■
18880
.0
881
.0
882 .0
880 .
.0
881
,0
885-
je
806
N
K-SCflLE - l.eaE+68 LMITS/INCa
Y-SCaE - 5.9aE+88 IMTS/JNCa
LEAST SQUARES BEST FIT CURUE ONS NOUEMBER
30 00M l^tQ 00W AUGUST 1957 GRAPH NO li
Zi**
'
i-
?
^
+
»
•
+ /
^
^
/ +
9
?
J
*>
^
>
'
/
/
"7* y
9
y
+
-
/
/
+
•
>
■t
is
-
'
a.
+ / .
1-
9
•
!>
■«
'
-
-
,/
+
»
+
V
z' +
.+
/^
+
'
^^f^
•
:eSe6 .0
aei
.0
962 .e
980 •
.0
mi .0
685
.e
ees
N
x-3CflLE - LsaE+aa ikits/ihch.
Y-3Ca.E - 6.0aE+00 umTS/JhCli
LEAST SQUARES BEST FIT CUR'JZ OWS NOUEMBER
30 00N lii0 00W SEPT. 1957 GRAPH NO 5
?
■■
/
5
/
>
•
%
1
+
/+
+
+ /
»
»
■
+ /
+
«
-
+
+ /
/+
4-
i
4
+ 1
+
-4?-
-
'.eaoe .0
aai .0
082 .0
Q80 . .0
3QH .0
805 .0
806
N
x-3CflLE - i.aaE+aa uNiT3/'mca
LEAST SQUARES BEST FIT CLIRUE OWS MOUEMBER
0 Oeri 1^0 00W, : OCTOBER IPS?" GRAPH NO 6
•^
o
^
•
X
* /
•
•
•
/
■
o
•
•
•
•
#
N
f
9
' •
/ .
«
./
•
r
•
^e0 .01
361 .01
902 .0
900 .01
d&i .0
906 .0
906
N
K-SuflLE - 1.00E+ee UNITS''i'HCH.
V-3CflLE - F.00E+O0 UNnS/INOi
LEAST SQUARES BEST FIT CURUE FOR OWS BRAUO
56 30N;51 00W JUNE 1960 GRAPH MO 7
46
N
x-scae - 1.O0E+00 units/ihch.
LEAST SQUARES BEST FIT CURUE FOR OWS BRflUO
56 30N 51 00W JULY. 1960 GRAPH NO 8
47
•
1
/
•
■
/
/
®
/
9
1 .
N
•
•
7^
O
•
S
/
•
i
/
■
®
^
^^
,
mo .0
del .0
062 .0
660 .0
d6li .0
D65 .0
Dee
N
::'3CflLE ' 1.60E+06 UNHS^JMCH.
Y-?CaE - F.oeE+e8 UMITS^IMCa
LEAST SQUARES BEST FIT CURUE FOR OWS BRflUO
56 30h 51 00W AUGUST I960 GRAPH NO 9
48
^
t
/
o
•
•
•
•
«
o
•
/
N
L9
I/?
•
/
O
r-t
/
9
(
+/^
•
9
/^-
s
+
1
:
doae .0
981 .01
9eQ .Qi
960 .01
^m .0
965 .0
986
N
X-3CaE - 1.68E+ee IWTVIHCrt
H-XfU • 5.88E+00 UHHS/'IHCH.
LEAST SQUARES BEST FIT CURUE FOR OWS BRflUO
81 THRU 09 SEPTEMBER 1960 GRAPH NO 10
49
"
t ■ /
9
/
,
•
o.
% /
9
*
CLo
/
.
•
/ .
O
•
/
'
9
X
/
«
mo .0
dQi .0
QQQ .61
360 .01
961 .0
^ .0
366
N
Y-SCflLE - &68E+88 UNHS/'INCH.
LEAST SQUARES BEST FIT CURUE FOR OklS BRflUO
19 THRU 30 SEPTEMBER 1960 GRAPH NO 1 1
50
/
k
^
9
/
O
/
'
O
,1
/
■
(
1.0
0
/
/
'
■
i
0.
/
/
'
0
•
iSIr
-*r
t
0
jW^
.
•
deee .0
381 .0
302 .01
300 M
30ti .0
»5 .0
306
N
K'XPLt - 1.O0E+00 UNITS/It€li
'f -SCALE - 5.O0E+00 UMHS/INCH.
LEAST SQUARES BEST FIT CURUE FOR OWS BRflUO
56 30N 51 00W OCTOBER 1960 GRAPH ISO 12
51
6. A possible universal function.
The concept of a universal function P(N) as proposed originally
by Kitaigorodsky was investigated by combining all of the 504 paired
values of P and N for the months of June through September for both OWS
ships. By the least squares best fit method, the second order poly-
nomial for P(N) was found to be
P^n) = .411M0'^sJ*-^-':^.25kj - Au^i^wf (10)
with the resulting universal forecasting equation,
A 4 2.
^ ' SI Q/an^ ^ ^
Graph No. 13 represents the function P(N), equation (10), with
upper and lower bounds of one standard deviation of the residues. The
residues are defined as the difference between the computed and original
ordinates and can be interpreted statistically as the standard error of
estimate of P.
Graph No. 13 also indicates the least-squares best-fit polynomial
for each OWS ship during the same months June through September. The
function P(N) from OWS November remains inside the statistical bounds
indicating that the proposed universal function may be appropriate for
that location. OWS Bravo, located in a more dynamic area, has a func-
tion which exceeds the statistical bounds for high values of N. Pro-
cesses not included in the model may explain this deviation.
The function P(N) for each OWS ship is estimated from the data of
only one warming season and may well be unrepresentative. Investiga-
tion of other years may reveal a closer correlation between different
52
locations and times which would strengthen the idea of a universal
function as well as Improve the estimates of the constants Involved,
53
.0062
^e84
N
K-5CPLE - LOCE-fae miTS/IMai
I
LEAST SQUARES BEST FIT CURUE JUNE THRU SEPT
OWS mu 1957 AND OWS BRflUO 1960 GRAPH MO 13
t. Procedure for forecasting and testing.
Equation (9) can be used to forecast MLD's over any length of time
for which the parameters can be accurately predicted. Data such as
were used to determine the coefficients in (9) were available for the
following years at both OWS ships. A continuous day-to-day forecast was
used to test the appropriate monthly coefficients for equation (9). In
essence the forecast was a test of whether the curves P(N) for a given
year and month were useful in predicting MLD's for the same month in
some other year.
All BT's available for the preceding 24-hour period were used to
2
calculate a mean observed MLD. The parameters >3 > Q» *"^d W were com-
puted by the same methods used in determining the paired values P and N.
Using the parameters /^ > Q» ^^^ ^ i^ ^^^ forecasting equation (9), with
the proper coefficients for the month and location under study, a daily
MLD was computed and compared to the 24-hour mean observed MLD. This
process was continued day by day from the available data with the results
listed in tables 14 through 22. A total of 169 forecasts were made, 20
representing MLD and 149 representing MLD ,
Although forecasts for periods greater than 24 hours were not at-
tempted, equation (9) is assumed to possess this utility. In an extend-
ed forecast, a mean value representing the heat flux across the air-sea
(Only a small number of observations was available for June and
July 1958 at OWS November. August data for the same location were
missing.)
2
(For comparison with the computed daily MLD, a 24-hour interval
was necessary to provide additional BT data for averaging out non-
periodic influences.)
55
interface per day could be applied to modify the parameter Q for heat
accretion during the forecast interval. Monthly climatological data
(Kimball [3] )are available for certain oceanic areas that list the aver-
age net heat flux per day. More important, however, is an accurate wind
prediction. Its importance can be seen by analyzing the terms with the
coefficients a_ and a, of equation (9) from table 13, and noting the ex-
pected changes in the parameters Q and W respectively. The average
change in Q as a result of heat flux is at most about ten percent in a
2
single day, based on approximately .4 Kg. cal/cm per day influx at OWS
November, while the change in W may range from 0 to 30 knots during the
same interval. When considering forecast changes in the seasonal MLD,
the term involving the coefficient a„ then becomes negligible.
Therefore, daily increases in Q were not considered essential in
forecasting seasonal MLD's. The fact that wind through mechanical mix-
ing during the warming season is usually the dominant factor in fore-
casting changes of the seasonal MLD is clearly seen - assuming fluctua-
tions created by internal waves have been averaged out.
The possible universal function derived from all paired values for
June through September was not tested by forecasting.
56
TABLE 14
lORECAST OF l-'XD ' s FOR JUKE 1958 AT OWS 1]0VE-:BER
DATE W Qs Qt
(hi;OTS) (Kg cal/cn^) (^c)
062658' 12.2 17.25
062758 16.8 14-. 58
062353 19.6 1^.85
062958 20.8 16.25
063058 20.2 17.30
Forecast seasonal MD's vjithin one standard deviation (3.1 meters) 80^
Forecast seasonal MLD's vdtliin two standard deviations (6.2 meters) 30^
TABLE 15
FORECAST OF MLD's FOR JULY I958 AT OV/S K0VEI4EER
FOKtC
AST
obse;
aVED
F0REC;^T-^
TS
tiDg
i-:LDt
.LilD^
L2.D.
L;iiS) ^
DIFF DIFF
(°c)
(i-J.TLRS)
(MfiT:
(METERS)
20.0
A0.5
4B.8
-8.3
20.0
39.6
39.6
.0
20.0
42.4.
43.5-
-1.1
20.0
.^5.1
45.9
- .8
20.0
47.2
46.6
.6
071058
13.6
9.26
20.0
33.9
32.0
6.9
071158
18.6
6.64
20.0
32.5
30.3
2.3
071258
14.2
9.20
20.0
35.4
38.7
-3.3
071353
12.8
8.20
21.1
32.1
35.4
-3.3
071/.5S
10.6
9.60
21.3
35.1
37.2
-2.1
071558
10.0
10.02
21.7
35.9
/ a.o
-5.1
Forecast seasonal >iLD's •id. thin one standard deviation (3.7 meters) 67%
Forecast seasonal 1-lLD's viithin tv;o standard deviations (7.2 meters) 100^
(Negative values indicate forecast MLD's were too shallovO
57
TAIiLIii 16
FORECAST OF liLD's l'X)E SiiJ^TH-iBEIi 1953 AT OUS UC^yH-lBER
DATE W Q Q.S TS I'iDg
(KI;OTS) (K2 .cal/cm^) C^C)
09015s
09025s
090353
0904.53
09055s
090653
090758
090;>53
090958
09105s
091253
09135s
091/+5S
09155s
09165s
C9175S
091S5S
091953
092053
092153
092258
09235s
092.^^58
C9255S
092653
09275s
092353
09295s
093058
23.8
21.6
16.0
15.8
15.8
12.6
10.0
10.0
9.0
8.0
12.0
12.0
12.0
11.6
11.6
11.0
15.3
16.2
17.2
16.8
U.O
10.8
16.2
16.8
16. A
17.0
U,2
10.6
10.0
12.60
U.6S
15.73
12.75
16.16
15.60
16.10
U.22
14.. 10
14-44
13.82
16.70
16.50
18.70
lo.oO
18.42
19.77
18.71
18.85
20.70
17.09
22.51
22.72
23.50
21.76
22. SI
24,. 19
24.00
.65
.68
23.3
23.3
22. S
23.6
23.3
23.9
23.3
23.9
23.9
23.3
23.2
23.0
23.2
23.1
22.9
23.1
23.1
23.1
23.1
22.7
22.9
23.1
22.5
22.8
22.7
22.7
22.8
22.7
22.7
I'OREGAST
l-iLDu
4917
50.6
45»5
40.3
45.9
41.8
3S.6
36.3
34.8
33.5
37.4
42.1
4^.8
44.9
45.1
43.7
51.8
50.6
52.1
54.6
45.5
50.1
57.4
57.9
58.6
56.6
54.8
52.5
51.5
10.0
9.4
OBSERVED I'TOIiECAST
IXD i-lD^ DIFl'. DII'Tx
(LiETERS) (LITERS)
37.5
39.0
36.6
38.4
43.6
40.5
40.2
38,1
36.6
35.1
39.0
39.6
39.0
39.6
4~i-.^
42.7
4/;. 2
47.9
4o.o
47.0
44.2
48.2
•49.7
51.5
51. S
51.5
53.9
51.5
54.9
9.1
9.1
12.2
11.6
3.0
1.9
2.3
1.3
• 1.6
- 1.8
- J-.o
- 1.6
- 1.6
2.7
2.S
5.3
4.0
1.0
7.6
2.7
5.5
7.6
1.3
1.9
7.7
6.4
6.3
5.1
.9
1.0
• 3.4
.9
.3
Forecast coasonal i-iLD's within one standard deviation ( 5.8 nieters) 72;^
Forecast seasonal l-IED's vathin t^-o standard deviations (11.6 meters) 97.^5
58
TiBLE 17
I'X)RECAST OF MD«s FOR OCTOBER 1958 AT OUS IlOVa^IilSR
DATE W Qs Q^^
(KNOTS) (Kg cal/cm^) (oq)
100158
9.8
20.27
100258
11.6
21.85
100358
9.5
21.90
100^58
8.4.
20.38
100558
7.6
23.90
100658
6.0
20.85
100758
6.0
24..I5
ICOS58
7.0
27.82
100958
lO.A
22.4.0
101058
11.8
26.35
101153
11.0
25.65
101258
6.0
25.4.5
101358
6.0
25.00
101A5S
6.0
28,52
101553
7.2
23.05
101658
9.U
25.00
101758
16.8
27.55
101858
19.6
23.^5
101953
15.2
25.10
102058
15.2
27.60
10215s
15.2
27.12
102258
12.0
27.95
1024.58
16.0
25.93
102558
16.0
26.10
102658
15.2
26.10
102758
15.2
28.60
102853
12.0
26.62
10295s
16.8
28.4.0
103058
27.8
29.60
103158
25.6
28.35
FORECAST
OBSERVED
FORECAST
TS
H.D^ KLD^t
MD, I'T.D.
DIFF,, DIFIJ
(^c)
(i4tei^)
(I'xTEIlS)
22.7
39.1
A8.8
- 9.7
22,6
a.o
4.7.9
- 6.9
22.7
4.3.7
48.8
-4.1
22.9
4J-.3
47.2
- 5.9
22.9
52.7
54.9
- 2,2
22.9
4.7.4.
51.8
- 4.4
23.3
57.2
54.9
2.3
23.2
65.6
60.7
4.1
23.3
4.6.6
54.9
-. S.3
23.2
52.0
57.0
- 5.0
23.1
51.3
54.9
-3.6
22.9
61.1
57.9
3.2
23.1
59.2
56.4
2.3
22.9
70.4.
65.5
4.9
22.3
51.1
53.3
- 2.2
22.6
52.2
57.0
- 4.3
22.7
50.2
57.9
- 7.7
22.8
51.3
60.4
- 9.1
22.7
^5.8
54.9
- 9.1
22.6
51.2
57.9
-6.7
22.9
50.3
57.3
- 7.0
22.5
53.8
61.0
- 7.2
22.2
4.5.7
59.7
-13.4
21.2
4,6.0
57.9
-11.9
22.1
4,6.3
54.9
- 8.6
21.9
51.8
64.0
-12.2
22.0
50.5
62.5
-12.0
21.9
50.3
61.0
-10.7
22.1
58.1
67.1
- 9.0
21.7
54.2
65.5
-11.3
Forecast seasonal l^lLD^s within one standard deviation (3.2 meters) 17^
Forecast seasonal l-ILD's -within tv;o standard deviations (6.4 nieters) 43/j
59
TilBLE 18
BDRECAST OF MLD « s FOR JUKE I96I AT OWS BRAVO
FORE!
:ast
OBSER^IO
FORECAST
DATE
V/
%
Q+..
TS
MLD.
MT.D+
E.BS)
l-XD
IvXD.
DIFFc
DIFF+
:ers)
(KNOTS)
(Kg cal/cni'^)
(°c)
(!■&]
(kStees) ""
iv^
061961
26.0
6.35
6.1
IJ:1
36.6
7.5
062C61
23.0
3.79
6.7
35.2
32.9
2.3
062161
23.0
3.25
6.1
33.9
27o4
6.5
062261
17.2
2.38
6.7
25.3
15o2
10.1
C6236I
16.8
4.58
6.7
29.4
13.3
11.1
062461
16.4
4.08
6.7
23.0
21.3
7.7
062561
18.6
4.10
6.7
30.7
31.7
-1.0
O6266I
23.8
4.26
6.1
37.2
33.2
5.0
062761
23.8
3.56
6.4
35.6
26.2
9.4
062861
19.6
5. 87
.48
6.1
33.4
14.8
31.1
9.1
3.3
5.7
062961
15.8
5.07
.59
6.7
29.1
15.8
•30.5
9.1
-1.4
6.7
O63C6I
10.0
5.85
.63
6.7
23.0
12.3
29.6
9.1
-6.6
3.2
Forecast seasonal l-iLD's vithin one standard deviation ( 6.6 irieters) 58^
Forecast seasonal MLD's within two standard deviations (13.2 irieters) 100$^
60
TABLE 19
FOREC/iST OF MLD's FOR JD'LY 1961 AT OIJS EPJiVO
FORECAST
(OBSERVED)
FORECAST
DATE
W
Qs
Q4._
TS
1-J.D.
ERS)
M,Do
MLD,
DIFF. DIFF,
(KKOTS)
(Kg cal/cm^)
(°c)
{v^t:
(METERS) ''
(I'ijTERS)
070161
27.6
5.34
6.0
35.4
32.9
2.5
07G261
28.9
8.0/,.
6.1
34.5
36.9
17.6
070^61
28.0
12.38
6.1
73.3
61.9
11.4
070561
20.0
10.54
6.1
58.3
63.1
~ 4.8
070661
18.2
8.51
6.2
49.3
43.3
6.0
070761
20.8
9.16
6.1
54.0
48.8
5.2
070C61
20.8
11.32
6.1
61.8
50.0
11.8
070961
IS. 2
9.46
6.7
52*6
47.9
4.7
071061
12.0
16.41
7.2
66.2
54.9
11.3
071161
12.0
14.62
3.05
5.6
54.6
19.0
54.3
12.2
.1 6.8
071261
12.0
18.13
2.71
5.8
62.8
16.6
60.0
18.3
2.3 - 1.7
071361
18.0
18.45
3.10
5.6
73.6
15.9
65.5
32.0
8.1 -16.1
071A61
20.0
5.72
5.6
35.5
33.5
2.0
071561
20.0
5.85
5.0
36.2
34.4
1.8
071661
15.0
6.83
5.6
37.6
31.4
6.2
071761
15.0
5.90
5.3
34.2
30.5
3.7
073061
8.0
8.00
7.2
35.9
26.2
9.7
073161
10.0
8.52
7.2
40.6
27.4
13.2
Forecast seasonal M-D's vathin one standard deviation ( 9.9 meters) 75^
Forecast seasonal i-ILD's within ti-ro standard deviations (19.8 meters) lOOj^J
61
TABLE 20
FORECAST OF l-iD's FOR /OJGUST 1961 AT 0W3 BRAVO
FORECAST
0B3ERVEJ
• FORECAST
DATE
V
Qs
Qt.
TS
IID^
ERS)
^.IQ„
MIX>.
DIFF,, DIE
(IvIIOTS)
(Kg cal/cn'^)
(°c)
i}^^.
(1-IETERS)
0
(INTERS)
080161
18.0
7.10
9.5
16.8
21.3
- 4.5
080361
16.0
6.52
8.3
14.7
22.9
- 8.2
0804.61
u.o
6.83
1<
.23
7.8
13.7
12.5
2^.4.
17,
,2
-10.7 .3
080561
12.0
1,
.84
6.7
9.3
15,
.2
-5.9
080661
12.0
1.
.50
7.8
9.5
13.
,7
-4.2
0SC761
10.0
1.
.50
7.8
7.7
13.
,7
-6.0
0SCS61
13.8
1.
,00
7.8
5.9
10,
,7
-4.8
080961
17.4-
3,U
Z.3
13.3
21.3
- 8.0
081061
20.0
2,
.20
7.8
16.4.
10.
,7
5.7
081261
20.0
8.U
6.7
17.1
25.0
- 7.9
081361
13.6
U.OO
6.7
16.7
32.0
-15.3
0814-61
lA./V
7.95
6.7
14-. 1
27.4
-13.3
081561
U.4,
5.94.
7.2
12.8
24.4
-13.. 6
081661
11.0
8.75
7.2
14-. 2
24.4
-10.2
081761
12.0
8.31
7.8
14-. 5
24.4
- 9.9
081861
19.2
7,65
7.8
17.1
24.4
- 7.3
081961
19.6
8.18
8.9
13.5
24.4
- 6.9
082061
19.0
8.80
7.6
18.2
29.0
-10.8
082161
19.0
11.12
8.3
21.2
30.5
-10.3
082261
15.2
9.22
8.9
17.7
30,5
-12.8
082361
15.8
12.1 /,
9.4.
22.1
32.0
- 9.9
0324.61
19.8
16.20
9.3
29.3
36.0
- 6.7
082561
23.6
15.32
8.3
28.1
37.2
- 9.1
032661
27.0
17.10
9.3
33.1
38.1
- 5.0
082761
2^,. 8
16.12
9.3
30.9
39.0
- 8.1
082861
22.6
16.02
8.3
23.5
35.1
- 6.6
082961
20.5
18.95
9.5
32.9
37.2
- 4.3
083061
15.0
l/,.4-0
8.9
22,.l
34.1
-10.0
083161
1/^.2
14-. 70
8.9
2^.1
29.6
- 5.5
Forecast seasonal KLD's witliin one standard deviation ( 5.6 meters) 17^
Forecast seasonal MLD's vithin two standard deviations (11.2 meters) ^3%
62
TABLE pi
FORECAST OD MLD»s FOR SEPTEMBER 1961 AT OWS BRAVO
'
FORECAST
OBSERVED
FORECAST
DATE
W
Qa Qt-
TS
MLDg KLDt
MUDg KI.D+
(METERS)
Dlb'Fg DIFFx
(KNOTS) (Kg cel/cm^)
(^c)
(MLTKHS)
(METERS)
090161
17.6
13.39
8.9
19.0
27.A
- 8.4
090261
15.8
16.30
7.8
21.3
29.0
- 7.7
090361
20.0
15.a
9.2
27.6
32.0
-9.4.
O9OA6I
20.0
13.53
8.3
18.9
27.^
-8.5
O9056I
18.0
16,30
8.9
23.0
30.5 ■
- 7.5
090661
17.2
18.00
9.A
26.6
32.9
-6.3
09IO6I
19.0
18.20
7.8
23.8
38.1
-U.3
091261
15.0
23.05
8.3
32.5
U.2
-11.7
091361
15.0
25.80
8.3
36.7
^5.1 ■
- 8.4. .
091761
15.0
22.60
8*U
31.8 ^
a.i
-9.3
091861
20«0
23.91
7.8
31.5 '
; Arf.5
-16.0
091961
20.0
2i^.89
7.8
32.9
A5.7
-12.8
092161
21.0
2^.10
7.8
31.8
^7.2
-15.4.
092261
20.2
25.95
7.8
34.^
57.9
-23.5
092361
16.J,
26.30
7.8
35.5
§7.9
-23.4.
09Zi6l
19.A
26.60
8.3
37.1
5^.9
-17.8
O9256I
20.0
2/^.95
8.3
3A.5
5A.9
-20./^
092661
20.0
26.20
8.3
36.A
57,9
-21.5
092761
15.0
26.30
8.3
37.5
67.1
-29.6
092961
22.0
23.90
7.6
31.3
67.7
-36.4
Forecast seasonal lOiD's idthin one standard deviation (11.5 meters) 40^
/
Forecast seasonal MLD*8 vithin tvo standard deviations (23.0 meters) 80^
TABT.F, 22
FORECAST
OF MLD»s FOR OCTOBER I96I AT 0\« BRAVO
100261
28.0
24.85
7.2
97.4
76.2
21.2
100361
28.0
23.65
7.2
91.9
X 76.2
15.7
100461
20.0
18187
6.7
65.9
50.3
15.6
IOO56I
18.0
16.42
5.8
51.3
42.7
8.6
100661
19.0
17.21
6.7
60.2
51.2
9.0
100761
22.0
17.40
5.6
55.0
53.9
1.1
100861
25.0
17.65
5.6
57.8
51.8
6.0
101161
22.0
17.68
5.6
55.7
67.1
-11.4
101261
26.0
19.27
5.6
61.9
64.0
- 2.2
1014iSl
20.0
13.55
5.6
45.3
64.0
-18.7
IOI56I
20.0
.15.42
5.6
49.0
67.1
-18.1
Forecast seasonal MLD*8 within one standard deviation (10. 6 meters) U5%
Forecast seasonal MLD's within two standard deviations (21.2 meters) 100^
63
8. Evaluating the results.
Table 23 is a condensation of the statistical analysis of pre-
dicted MLD In relation to the observed MLD . Deviations of the fore-
s s
cast from the observed MLD are compared with the standard deviation (^ )
of the dally mean of the observed MLD for each month. Statistics were
not obtained for transient MLD situations since too few of these occurred
during any month for a statistical analysis. Persistence forecasts from
day to day were used for comparison.
Except for the month of October , OWS November had a large per-
centage of forecasts (72%) within one CT » which is significant in that
the average CT (5 meters) is small.
For the same months at OWS Bravo only 40 percent of the forecasts
were within one C (9 meters). The Inability of equation (9) to fore-
cast accurately the MLD may be related to factors, such as divergence,
not included in the model. Use of additional paired values P and N for
each month should Improve forecasts based on the resulting function P(N).
Extension of the monthly study into other years should bring about fur-
ther improvement, as random contaminating processes are smoothed out by
Increase in sample size.
(October was omitted to avoid months containing possible convec'
tlve mixing.)
64
TABLE 23 .
COMBINED STATISTICAL ANALYSIS OF FORECASTS FOR SEASONAL MD's
MONTH
YEAR OWS if OF FOSTS
% FCSTS
WITHIN ONEC
% FCSTS
WIXION TWO^
cr
(METIERS)
June
1958 November
5
80
(50)
80
(75)
3.1
July
1958 November
6
67
(60)
100
(80)
3.7
Sept*
1958 November
29
72
(100)
97
(100)
5.8
Oct,
1958 November
30
17
(62)
iV3
(83)
3.2
June
1961 Bravo
12 ,
75
(73)
100
(100)
6.6
Ju3y
1961 Bravo
12
58
(82)
100
(82)
9.9
Aug,
1961 Bravo
2U
17
(91)
83
(100)
5.6
Sept.
1961 Bravo
20
40
(100)
80
(100)
.11.5
Oct.
1961 Bravo
n
k5'
(85)
100
(90)
10.6
Overall
average of forecast
seasonal MLD's \dthin one
<r
45
(82) % .
Overall
average of forecast
seasonal MLD's \dthin two 0"
81
(92) $
\
\
1
(Values in parentheses are statistical analysis of forecasts by persistence.)
65
9. Conclusions and acknowledgement.
As a result of this study concerning the application of a pro-
posed mixed-layer depth forecasting model, the following conclusions
can be made.
(1) Persistence gives the best short term prediction of MLD
in the locations studied. If no recent observations are available,
predictions utilizing a previous year's P(N) and accurate wind fore-
casts are useful.
(2) The dimensionless coefficient P(N), inherent in the ap-
plication of similarity theory, is best approximated by a second-degree
polynomial.
(3) A single function can be used to represent P(N) for both
seasonal and transitional MLD's.
(4) During the warming season, changes in the MLD are mainly
influenced by variations in the wind speed.
(5) The concept of a universal function P(N) proposed by
Kitaigorodsky may be valid, but its determination requires consider-
able refining of existing data to remove contaminating influences.
For his invaluable aid in the preparation of this manuscript, the
author is deeply indebted to Associate Professor J. B. Wickham,
Department of Meteorology and Oceanography, U. S. Naval Postgraduate
School, Monterey.
66
BIBLIOGRAPHY
1. Fofonoff, N. P. Transport calculations for the North Pacific
Ocean 1957- Pacific Oceanographic Group, Fisheries Research
Board of Canada. MS Rept. Series (Oceanog. and Limnol.)
no. 79, August 1960.
2. Fofonoff, N. P. and C. K. Ross. Transport calculations for
the North Atlantic Ocean 1960. Pacific Oceanographic Group,
Fisheries Research Board of Canada. MS Rept. Series (Oceanog.
and Limnol.) no. 121, April 1962.
3. Kimball, H. H. Amount of solar radiation that reaches the '
surface of the earth on land and on the seas and methods by
which it is measured. Monthly We a. Rev. , vol. 56, 1928.
4. Kitaigorsdsky, S. A. On the computation of the thickness of
the wind-mixing layer in the ocean. Academy of Sciences, USSR,
Geophysics Series, no. 3, March 1960.
5. McDonnell, J. R. Application of similarity theory to forecasting
the mixed-layer depth of the ocean. M.S. thesis, U. S. Naval
Postgraduate School, Monterey, Calif., 1964.
6. Monin, A. S. and A. M. Obukhov. Basic laws of turbulent mixing
in a ground layer of the atmosphere. Transactions of the
Geophysical Itratitute .Academy of Sciences, USSR, no. 2(151), 1954.
7. Sverdrup, H. U. , M. W. Johnson, and R. H. Fleming. The Oceans,
their physics, chemistry, and general biology. Prentice Hall Inc.,
1942.
8. Tally, J. P. Oceanographic domains and assessment of temperature
structure in the North Pacific Ocean. Pacific Oceanographic Group,
Journal Fisheries Research Board of Canada, vol. 21, no. 5, 1964.
67
APPENDIX I
METHOD USED FOR DETERMINING THE PARAMETER Q
The parameter Q is defined as
where the factor (AREA) is given by the integral ( Zdt, T. and T^
being the temperatures of the "isothermal" layer (see fig. 1, slide 1)
below and above the thermocline (either seasonal or transitional) , and
Z is the depth from the surface to the temperature curve. Density is
represented by p and C is the specific heat at constant pressure.
In evaluating the factor (AREA) , the most difficult step is the
choice of T. . It is that temperature, where the water becomes isother-
mal or nearly so. The isothermal condition may continue to great depth
or exist in only a thin layer between temperature gradients. Frequently
this layer is difficult to distinguish, in which case reference must be
made to adjacent BT slides to establish at least a nearly isothermal
condition. In any case the subjectivity in calculating Q by this pro-
cedure probably contributes to scatter of the curves P(N).
Once T. and T are determined, (AREA) is found by replacing J Zdt
by an equivalent rectangle with the area 21 (T„ ~ ^O • The depth of z"
is determined by a horizontal line drawn through the thermocline such
For OWS November during the warming season pC^ = .975 (cal/Ccm )
that equal areas will result above and below Z (see fig. 1, slide 4).
= .y/:3 (cai/ccn
P
for an average salinity of 32.5 /oo and can be considered constant.
For OWS Bravo DC " 1.01 (cal/Ccm ) for an average salinity of 34.5 /oo.
68
A constant factor was calculated that included ft C and a change of
dimensions (from British to Metric and from Fahrenheit to Centigrade)
enabling direct computation of Q from the BT slide. This factor was
1/6.05 for OWS November and 1/5.9 for OWS Bravo.
A sample calculation of Q from slide 4 follows:
1. Determine the difference in temperatures between T„ and
T^. (13.8"F)
2. Read the depth of the horizontal line Z. (150 Ft)
3. If this slide were from OWS Bravo data, divide the product
of steps 1 and 2 by 5.9, giving
Q = (13.8) (150) X lO"-^ = 35 (kg cal/cm^)
® 5.9
Calculations of Q are done in the same manner and usually are an order
of magnitude less than Q .
This method outlined represents a modification to McDonnell's
technique. He constructed T^ so as to intersect the BT trace at 200
meters (656 feet) . This method soon became unreasonable in evaluating
Q for two reasons. First, excess heat in the uppermost layer was poorly
represented. Q represented the excess heat in the layer above 200
meters. Secondly, Q could be evaluated realistically only on slides
from deep BT's which are seldom used. The present author's method, al-
though subjective, better represents the excess heat in the mixed-layer
under study.
69
^1
T.
r:~r~17^'-^'^'>'/'''''r'^'"/J "— -'-^H i — ^ — r' i^-i—-l-i j-.—.-'-i -'-(-•;-__)
P-lt-l:^.i:rr:Li";r:..:.!.! i
^li!;:
;_...; — L .r .' « i-;--f-
;:i:;:l:|
P
Slide 1
PT' // .>^l-l-^-ri-----UJ-i-l-'-J-'-H
•I'TTxri I-' !-i-
^3
gig^s4?i^iir^
j-H-i--
Slide 2
i:i
fHmii;-'
L-_i-:l
Slide 3
Slide L ■
0
iO
2U
3C\r
3:o
F\.
iliiii iBm iiiiiii i^BBi
gmmimmk ifeiiiiiS sMiMilMi i^SMM
illliliiiilp ki^illtejiii gtetmng||||| Ifejgarggggjj
jsc — /t:
^ r-T
2U
4U
Slide 5
A
B
C
D
Slide 6
Slide 7
trjfjpnr^ ,
Slide S
-the transitional mixed-layer depth (llD..),
-the intersection of vertical (T-, ) \^th the ET trace
for transitional situations.
-the seasonal laixed-layer depth (l-iD ),
• -the intersection of the vertical (t^ ) \d.th the 3T trace
for seasonal situations.
Figure 1
Representation of the AREA used
in calculating the parsjueter Q'
70
TABLE 24
COEFFICIENT OF THERMAL EXPANSION (y^xlO^) OF SEA VIATER
AT SEA LEVEL FOR DIFFERENT TEMPERATURES AND SALINITIES
SALINITY 0/00
30
31
32
33
34
35
5
1.01
1.04
1,06
1.08
1.11
1.14
6
1.12
1.15
1.17
1.19
1.22
1.24
7
1.23
1,26
1.28
1.30
1.33
1.35
8
1.34
1.37
1.39
1.41
1.44
1.45
9
1.45
1.48
1.50
1.52
1.55
1.56
10
1.57
1.59
1.61
1.63
1.65
1.67
11
1,67
1.69
1.72
1.73
1.75
1.76
1.77
1.80
1.82
1.83
1.84
1.86
§13
1.87
1.89
1.91
1.93
1.94
1.95
^
1 1^
1.97
1.99
2.01
2.02
2.03
2.04
S
t^ .
2,06
2.08
2.09
2.11
2.13
2.14
16
2.15
2.16
2.17
2.19/
2.21
2.23
17
2.23
2.24
2,26
2.28
2.30
2.31
18
2.32
2,33
2.35
2.37
2.39
2.40
19
2.41
2.42
2,44
2.46
2.47
2.48
20
2.50
2.51
2,53
2.55
2.56
2.57
21
2.58
2.59
2.61
2.63
2.64
2.65
22
2.67
2.68
2.69
2.71
2.72
2.73
23
2.75
2.76
2.77
2.79
2.80
2.81
24
2.83
2.84
2.86
2.87
2.88
2.89
25
2.92
2.93
2.94
2,95
2.96
2.97
71
INITIAL DISTRIBUTION LIST
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73
Unclassified
Security Classification
DOCUMENT CONTROL DATA - R&D
(Security ctmmmiUcatian o/ Utta, body ol abstract and indexing annotation muat be enterad whan tha ovatalt raport la claaaltlad)
1 ORIGINATING ACTIV/ITY (Corporate author)
U. S. Naval Postgraduate School
Monterey, California
2a. REPORT SCCURI TY CLASSIFICATION
Unclassified
26 OROUP
3. REPORT TITLE
Verification of McDonnell's Mixed-Layer Depth Forecasting Model
4 DESCRIPTIVE NOTES (Typa ol raport and inclusive datea)
5 AUTHORC5; (Laat nama. Hrat name. Initial)
KELLEY, Robert D.
6. REPORT DATE
October 1966
7a. TOTAL NO. OF PAGES
72
7b. NO. OF REFS
8a. CONTRACT OR GRANT NO.
6. PROJECT NO.
9a. ORIGINATOR'S REPORT NUMBERfSJ
N/A
N/A
9b. OTHER REPORT NOCS^ (Any othar numbara Oxat may ba maalOnad
thia raport)
d.
N/A
MCkji^
10. AVAILABILITY/LIMITATION NOTICES
Thj.o aocument has been a rr* ~
jrelease and sale; its dismbution la unxiaxiic:
11. SUPPLEMENTARY NOTES
None
12. SPONSORING MILITARY ACTIVITY
U. S. Navy
13. ABSTRACT
A model based on Kitaigorodsky 's application of similarity theory and
modified by McDonnell to forecast the mixed-layer depth was studied. The
model applies during the warming season and is based on the theory of
similarity. The parameters involved in the model were determined from bathy-
thermograph data recorded at Ocean Weather Stations November (latitude 30N,
longitude 140W) and Bravo (latitude 56 30N, longitude 51W). Parameters were
evaluated daily and grouped by months. Both seasonal and transitional MLD
situations were treated.
From these parameters, the form of the dimensionless function P(N) ,
claimed by Kitaigorodsky to be universal, was determined by least squares
fit to be best approximated by a second order polynomial. Forecasting
equations involving P(N) were developed for each month and tested with data
from the following years for both OWS ships.
There is general agreement between the observed MLD and that found from
the prediction equation based on the last year's P(N) for the same month
and location. Month-to-month and spatial differences in P(N) cast consider-
able doubt on its universality, at least as determined by the parameters as
currently defined.
DD
FORM
1 JAN 64
1473
75
Unclassified
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14.
KEY WORDS
LINK A
ROLE
LINK B
ROLE
WT
LINK C
Ocean
Upper
Layer
Thermal
Structure
Similarity
Forecasting
INSTRUCTIONS
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DD
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'MM.
?iV4V.''''!'*
iiklii|^'.X;.;V.
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