N PS ARCHIVE
1964
MCDONNELL, J.
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APPLICATION OF SIMILARITY THEORY
TO FORECASTING THE MIXEDLAYER
DEPTH OF THE OCEAN
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john r. McDonnell
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LIBRARY
U.S. NAVAL POSTGRADUATE SCHOOL
MONTEREY, CALIFORNIA
APPLICATION OF SIMILARITY THEORY TO
FORECASTING THE MIXED LAYER DEPTH OF THE OCEAN
jf <jf a/ % <$g
John R. McDonnell
APPLICATION OF SIMILARITY THEORY TO
FORECASTING THE MIXED LAYEF 3EPTH OF THE OCEAN
by
John R. McDonnell
Lieutenant, United States Navy
Submitted in partial fulfillment of
the requirements for the degree of
MASTER OF SCIENCE
United States Naval Postgraduate School
Monterey, California
19 6 4
ffp L/5RAJ5Y
VS NAVAL POSTGRADUATE SCHOOL
MONTEREY. CALIFORNIA
APPLICATION OF SIMILARITY THEORY TO
FORECASTING THE MIXEDLAYER DEPTH OF THE OCEAN
by
John R. McDonnell
This work Is accepted as fulfilling
the thesis requirements for the degree of
MASTER OF SCIENCE
from the
United States Naval Postgraduate School
ABSTRACT
The thermal structure of the ocean, especially the uppermost mixed
layer, greatly affects sonar ranges. In this paper, similarity theory
is applied to the problem of forecasting the depth of the mixed layer
during the warm season, assuming the controlling processes are secular,
nonadvective , and nondivergent. The resulting forecast method consists
mainly of two equations. Parameters used are wind, coriolis effect, the
coefficient of thermal expansion and a measure of the excess heat within
the mixed layer. The constants in the equations were determined using
data from OWS Papa (50N, 145W). The forecast method treats both seasonal
and transitional thermoc lines . The method was tested with data from OWS
Pap* and OWS November (30N, 140W). The tests apparently indicate wide
applicability of this forecast method and thus tend to corroborate the
proposal by Kitaigorodsky that the mixed layer depth is a function of a
universal coefficient.
ii
TABLE OF CONTENTS
Section Title Page
1. Introduction 1
2. Description of Similarity Theory 5
3. Variables and Processes that Affect the MixedLayer
Depth During the Warm Season 7
4. Selection of Parameters to Represent Controlling
Processes 8
5. Practical Application of Similarity Theory 13
6. Application and Test of Equations 22
7. Conclusions 29
Bibliography 30
Appendix
I. Basic Procedures for Application of JT Theorem 32
II. Relationship between Variance of the MLD and the Strength
of the The rmoc line 36
v
III. Methods Used for Determining Values of Parameters 38
IV. Forecasting Method Proposed by Kitaigorodsky 43
iii
LIST OF ILLUSTRATIONS
Figure Page
1. Typical Thermal Structure of the Upper Layer of the
Ocean During the Warm Season at OWS Papa 3
2. Two Possible Thermal Structures with Same Excess Heat
Present in Warm Layer 11
3. P versus N Based on Monthly Cliraato logical Data for the
Seasonal Thermocline at OWS Papa 15
4. P versus N for OWS Papa Data (Transitional Thermocline)
with Least Squares Best Fit Line 17
5. P versus N for OWS Papa Data (Seasonal Thermocline)
with Least Squares Best Fit Line 20
6. P versus N for OWS November Data Compared to OWS Papa
P versus N Best FitLine (Seasonal Thermocline) 26
7. Variance of MLD about the Mean versus Temperature
Change Through the Uppermost Thermocline 37
8. Representation of Area T Used in Determining Q™ 39
9. Representation of Areag Used in Determining Q s 41
10. Values of P versus N Computed Using Equations
Developed by Kitaigorodsky 45
iv
LIST OF TABLKS
Table Page
1. Data Used to Determine Values of P and N for Mixed
Layer Depths Associated with Transitional Thermoclines 18
2. Data Used to Determine Values of P and N for Mixed
Layer Depths Associated with Seasonal Thermoclines 21
3. Forecast Verifications 23
4. Statistical Results of Forecasts 24
5. Data Used to Determine Values of P and N for Mixed 
Layer Depths Associated with Seasonal Thermoclines
at OWS November 27
6. Values for the Coefficient of Thermal Expansion
Based on a Salinity of 32,5°/oo 42
LIST OF SYMBOLS AND ABBREVIATIONS
/3 coefficient of thermal expansion
MLD mixed layer depth
Q Excess heat present in upper layer of water
Q F Net heat flow to and from upper layer of water
Q T Excess heat present in upper layer of water associated with a
transitional thermocline
Qq Excess heat present in upper layer of water associated with a
seasonal thermocline
TS Temperature at the surface of the ocean
W wind speed or a representative wind speed
f coriolis parameter
4
A. modified coriolis parameter (f x 10 )
to angular velocity of earth
latitude
vi
1. Introduction.
The detection of enemy submarines by sonar is one of the major un
solved problems of this decade. Despite recent technological advances
in many fields of science, the nuclear submarine is still practically
invulnerable. Even though sonar technology, in particular, has advanced
rapidly, we are still unable to position our ASW forces or their sonars
for optimum performance. This is largely because sonar ranges are either
enhanced or reduced by refraction of sound energy in the surface layers
and we are not able to forecast accurately the thermal structure which
greatly determines refraction.
Many authors have devised systems for forecasting the thermal struc
ture of the upper layer of the ocean based on either dynamical analysis
or empirical relationships. The equations resulting from dynamical
analysis are either too complicated, if all the physical processes are
considered; or are impractical, if many simplifying assumptions are made.
Empirical relations have been determined for certain locations and for
limited time periods but do not appear to be valid universally.
Another possible method for developing a forecasting system uses
similarity theory. Kitaigorodsky CO investigated the application of
similarity theory to the ocean thermalstructure forecasting problem.
Although his results have certain drawbacks, as noted in appendix IV,
still he has shown the applicability of the method. Consequently, the
method used by Kitaigorodsky is also applied by this author, with some
modifications of parameters, in an effort to develop a more practical
result. The form of P, a dimensionless coefficient inherent in the
application of similarity theory, is determined by use of data from
Ocean Weather Ship Papa (50N 145W).
Figure 1 depicts a typical ocean thermal structure in the wartn sea
son at OWS Papa. Characteristically , there is a quasiisothermal layer
that extends from the surface to the upper boundary of a negative tem
perature gradient.
The depth to which the quasi isothermal layer extends is usually
referred to as the mixed layer depth. In this paper the mixed layer
depth (MLD) is defined as that depth at which the temperature of the
water first becomes 1C less than that of the water at the surface.
A transitional layer of negative temperature gradient between layers
of relatively small temperature variation is referred to as a thermocline
Four main types of thermocline can be classified, primarily according to
degree of permanence: diurnal, transitional, seasonal, and permanent.
Diurnal thermocline s it results from a net heat gain during
the day, a small thermocline (AT < 1C, see fig. 1) being formed
close to the surface by late afternoon. With a net heat loss at
night, the thermocline will be destroyed by morning.
Transitional thermocline: a moderately large thermocline
(AT > 1C) is formed when diurnal heat input exceeds losses. After
a few days or weeks it joins with the seasonal thermocline as the
added heat diffuses downward.
Seasonal thermocline: the transition zone which lies between
the surface waters warmed during the summer and the colder water
below. At OWS Papa the seasonal thermocline depth is about 20
meters in the late summer, and about 50 meters in the spring and
fall. It is not present in the winter.
TEMPERATURE CO
UJ
2
a.
UJ
o
Diurnal
Thermocline
Transitional
Thermocline
 Seasonal
Thermocline
80 u 
Figure 1
Typical Thermal Structure of the Upper Layer of
the Ocean During the Warm Season at OWS Papa
l .<.. .
Permanent thermocline; in many localities a deep therraocline
that is relatively persistent in depth exists throughout the year.
Below a permanent thermocline the temperature gradient (usually
negative) is very small all the way to the bottom.
The entire thermal structure of the ocean, though of interest, is
not immediately as important to sound propagation with present sonars as
the depth of the uppermost mixed layer of the ocean. As far as this
mixed layer is concerned, there are two basic seasons at OWS Papa. These
are the warm season, when mixing of the upper layer is mainly due to the
wind, ind the cool season, when mixing is mainly due to thermoh aline con
vection. Thus the physical processes during the two seasons are quite
different and are usually treated separately.
As in the paper by Kitaigorodsky, similarity theory will be applied
to develop a method for forecasting the mixed layer depths associated
with the transitional and seasonal thermoc lines during the warm season
at OWS Papa. Henceforth, diurnal therraociines will be ignored and the
terms "MLD" and "depth of the thermocline" will be synonymous and will
refer only to mixedlayer depths associated with either transitional or
seasonal therraociines. Only those changes in the MLD which are secular,
nonadvective, and nondivergent will be considered.
2. Description of Similarity Theory.
There are three basic steps in the application of similarity
theory,
1) Determine the physical processes that control the physical
phenomenon of interest.
2) Select parameters that accurately represent the controlling
physical processes.
3) Apply the ff theorem Q2j to the chosen parameters.
The TT theorem is a method for determining a dimensionaiiycorrect
relationship for a given set of parameters. A short description given
by Binder Z^3 follows.
Let A^A2A3....A n be n physical quantities which are involved in
some physical phenomenon. Examples of these physical quantities
are velocity, viscosity, and density. Let m be the number of all
the primary or fundamental units (such as length, mass, and time)
involved in this group of physical quantities. The physical equa
tion, or the functional relation between these quantities, can
be written as
j (A, A,. A 3 A m ) =
The IT theorem states that the foregoing relation can be written as
where each If is an independent dimensionless product of some of
the A's.
Thus, if there are n physical variables in a particular problem, and m
fundamental units, then the physical relationship can be expressed in a
form involving (nm) dimensionless ratios. An elaborate and formal
proof of the theorem is given by Buckingham _2j. A simplified stepby
step procedure and an example are given in appendix I.
As can be seen in appendix I, to equate these (nm) ratios, the TT
theorem introduces a dimensionless coefficient (P) which serves to make
5
the dimensionless ratios numerically equivalent. A practical example
for the necessity of P is given in the second paragraph of section 5.
The form of P is normally determined from actual test data. P can be a
constant or it can be a function of some or all of the parameters used
to determine the ratios. Buckingham [_2J suggests several ways to deter
mine the form of P.
One method is to let P be a function of all of the parameters of
the process with the exception of the one for which the relationship is
being developed. For ease of reference, this combination of parameters
is called N. The 1f theorem is entered with the chosen parameters and
an expression for the dimensionless ratio N is obtained. Pairs of
values of p and N, determined using observed data, are plotted together.
Then the form of P(N) can be determined from this plot.
Once the dimensionless ratios and the form of p have been deter
mined, then the results can be used to forecast the mixed layer depth.
3. Variables and Processes that Affect the MixedLayer Depth During
the Warm Season.
There is general agreement among many researchers that during the
warming season the mixed layer depth varies with:
1) wind (Geary C4] , Tabata [5], Mazeika [S~] f Munk and Anderson [7]);
2) the heat flow between upper and lower layers (Mazeika [6J >
Rossby and Montgomery QsQ , Kitaigorodsky CQ)!
3) coriolis effect (Rossby and Montgomery CC » Kitaigorodsky Ckl)*
4) divergence in the upper layers of the ocean (Mazeika [b] , Tabata
and Giovando V]);
5) internal wave action (Mazeika M , Tully [lOJ , Tabata and Gio
vando K]);
6) advection (Mazeika C€]» Tully [jQ] , Tabata and Giovando C?] ).
Only the first three of these processes will be considered in developing
a forecast method. The remainder of the influences contribute to the
scatter of the results.
4, Selection of Parameters to Represent Controlling Processes.
An aid in determining representative parameters is to write the
relevant dynamical equations describing the processes and to examine
them term by term.
The equations of motion in the xy plane, assuming an unaccelerated
flow in an unbounded and horizontally homogeneous ocean, reduce to:
O « f \ri /o d s
(1)
Q = —jM. +
Oil
/° a ^
*#
where u and v are the components of velocity, ~TL V ar *d "fl are the com
ponents of stress on the horizontal plane, and + is the coriolis param
eter. At the interface, the stress is given by:
fV^W 1 (2)
where ~\ is the stress on the water surface, W is the mean wind speed and
o is a coefficient that is a function of wind speed and the height at
which the wind is measured. This indicates that the water motions, in
cluding the eddies that carry both heat and momentum, depend upon the
surface wind speed. Thus a measure of wind speed that is representative
of the stress effects should be introduced as a parameter in the applica
tion of the IT theorem. However, the depth to which wind effects reach
depends also on the coriolis parameter as shown theoretically by
Ekman \JL1J. Thus, the coriolis parameter must also be considered in
application of IT theory.
The maintenance of a thermal structure involves heat flow. A common
equation for temperature changes due to heat flow is:
8
where J° is the density of water, c is the specific heat of water, T is
the temperature of the water, and K is the eddyconductivity coefficient
for heat. This indicates that the local rate of change of temperature
is a function of K as well as of the differences in temperature between
layers. But K depends upon the field of motion and the stability. Param
eters to represent motion have already been chosen in the previous para
graph. Thus, to complete the representation of the variables in the heat
flow process, the only additional parameters that need be introduced are
those representative of stability.
Thus it appears that, for the conditions and assumptions given in
section 3, the IT theorem should be entered with parameters that are
representative of:
1) wind stress;
2) coriolis effect;
3) stability.
WIND STRESS. Here it is necessary to find a value of wind speed
which is representative of the influence of wind on the MLD over a period
of time. The wind enters into the mixedlayerdepth problem in the form
of stress, and the stress coefficient ( o) is an increasing function of
wind speed. Thus equations (I) and (2) show that higher winds should be
weighted more heavily in estimating the influence of wind stress upon
the MLD.
One therefore should avoid a linearly averaged wind, The author
has devised an empirical method which qualitatively and objectively
weights the stronger winds in order to obtain a suitably representative
wind parameter. The parameter is called the "representative maximum
wind" and is defined as:
the average of the five highest winds of the eight usually
reported during a chosen 24hour period. The chosen period is
that one having the highest winds in the interval from 72 hours
to 12 hours prior to observation time.
The stepbystep procedure for determining the "representative maximum
wind" is given in appendix III.
CORIOLIS EFFECT. The standard equation for the coriolis parameter
times 10 will be used to represent coriolis effects:
_n_ =x f x io + = a lj sin <p x io
t
The constant, 1x10 , was introduced arbitrarily for convenience.
STABILITY. The density gradient, and thus the stability, can be
represented by a combination of the coefficient of thermal expansion
( /3 ) and the temperature difference between layers; the following two
parameters have been chosen to represent stability:
1) the coefficient of thermal expansion (/3 ) ;
2) the excess heat (Q) in the mixed layer over that in lower layers.
Values of /3 used in this paper were interpolated from values given
by Sverdrup \\2\ assuming a constant salinity of 32.5 % . These values
are tabulated in appendix III. Constant salinity was assumed at OWS
Papa during the warm season as in Tabata £fQ.
The stepbystep methods used to determine Q are given in appendix
III. One difficulty with the use of Q as a measure of stability is
illustrated in figure 2. This shows that thermal structures of obviously
10
T— >
1
P
T^
Q, =Q
Figure 2
Two Possible Thermal Structures with
Same Excess Heat Present in Warm Layer
11
different .stability can prevail with the same Q. However, the combina
tion of Q, A , and the MLD give a good approximation of the stability
of the mixed layer. (The MLD enters the forecast model in the form of
P(N) since P(N) is initially determined from raw data that includes
MLDs . )
Thus, besides the MLD itself, four parameters have been chosen for
entry into the n theorem:
1) the representative maximum wind (W);
2) the coriolis parameter (f);
3) the coefficient of thermal expansion (/3);
4) the excess heat in the upper layer (Q).
12
5. Practical Application of Similarity Theory.
In section £ , f our parameters were chosen as representative of the
processes effective in forming and maintaining a mixec layer at OWS Papa
in the warm season. When the 1T theorem is applied to these four param
eters the results are:
Q/3rt (3)
Both of these equations are derived in the example in appendix I.
Because the relationship between ml I) and the other parameters is
complex, the coefficient P(N) is not in general a constant and must be
found as a function of some of the parameters, as in appendix I. As an
example of the ambiguity which must be eliminated by use of P, consider
again the parameter Q; as shown in fig. 2, two quite different thermal
structures may have the same excess heat Q. Thus a value of Q alone
does not uniquely specify the thermal structure. Rather the structure
is a result of the interaction of all the controlling parameters. That
is, a wide range of possible mixedlayer depths exists for each value of
a particular controlling parameter, whereas only one MLD exists for each
combination of parameters. To specify the form of P(N), which is a
function of all the prevailing parameters, several pairs of values of
P and N are determined from experimental data and plotted together.
Then the form of P(N) is given by this plot.
If , in the application of similarity theory, parameters truly repre
sentative of the controlling processes are chosen, then the plot of P
13
versus N should have little scatter. If the parameters used are not
truly representative, then large scatter results.
To test equations (3) and (4) they were used to obtain paired values
of P and N; these values were based on mean monthly data at OWS Papa
tabulated by Tabata and Giovando [_9J. Values of the parameters, W, ft ,
_TL , and Q s (where the subscript S refers to seasonal thermoclines) were
determined in accordance with Appendix III. The paired values of P and
N are plotted in figure 3. The points have very little scatter; and, in
fact, a straight line gives a very good leastsquares fit. However, the
use of figure 3 is limited because it represents a climatological MLD,
whereas the MLD for a particular day can vary considerably from the clima
tological MLD. Figure 3 is important in that the small scatter of the
paired values apparently indicates that the proper parameters have been
used in equations (3) and (4).
In order to make equations (3) and (4) useful in short range predic
tion of the MLD, P(N) was determined using daily data from OWS Papa. The
data were from June to October during the years 1958 through 1962. In
order to filter out internal wave effects, data were used only from those
days for which six or more bathythermographs were available; the MLD
used is a mean of the six (or more) bathythermographs. This requirement
limited the available data such that only 22 pairs of P and N are avail
able for transitional MLDs and 29 pairs for seasonal MLDs. Although the
number of paired values appears small, they represent over 200 BTs.
Pairs of values of P and N were first obtained for MLDs associated
with transitional thermoclines with values of the parameters W, /3 ,
Si, and Q T determined as stated in Appendix III. Note that for a
14
16
14
12
10
x 8
•', »■
Number by each point indicates
month of year point represents,
NX 10
Figure 3
8
P versus N Based on Monthly Climatological Data
for the Seasonal The rmoc line at, OWS Papa
15
transitional thermocline the general parameter Q in equations (3) and
(4) is represented by the more precise parameter Q^.. Figure 4 is a plot
of pairs of P and N resulting from data listed in table 1. The least 
squares bestfit straight line has the equation:
P = .a b"xio~^ + z.^i N
Substituting this into equation (3) gives:
MLD.^XI0 + (^+^ (£)
(5)
Thus equation (5) can be used to forecast a transitional MLD without
determining a value of P.
It is interesting to note the similarity between equation (5) and
the equation derived by Rossby and Montgomery [_8J :
MLD— V**+K a $> (6>
W z wind speed measured at the sea surface.
= latitude
Based on 39 observations, they arrived at a value for K 2 of 2.38 per
second. It was concluded that K^ was negligible. The MLD data for these
39 observations were obtained by interpolating between readings of tem
perature and salinity which are 1025 meters apart. The average MLD for
the 39 observations was 32 meters, and they considered a fivemeter aver
age error as a plausible assumption. Since the data used by Rossby and
Montgomery appeared to be of poor quality compared to the data available
at OWS Papa, it is suggested that equation (5) is more applicable than
(6) for forecasting the transitional thermocline. Tests in section 6
16
•'.*'
10
i
/ 1
V
»
)•
;
9
* /
8
/ • '
7
1
6
■
o
x S
0.
. , •
•/
• /
■
1
4
•
\ 1
• /
/•

3
• /
/ •
.
2
1 •
/ **
/ •
/ ■'•'.•''
1
.
n
I t
»
L
I
1
>
. 1
NX1CT
Figure 4
P versus N for OWS Papa Data (Transitional Thermocline)
with LeastSquares BestFit Line
17
TABLE 1
DATA USED TO DETERMINE VALUES OF P AND N
FOR MIXEDLAYER DEPTHS ASSOCIATED WITH TRANSITIONAL THERMOCLINES
W Qt MLD TS
DATE
(KNOTS)
*■
(Kg cal/cnO
(METERS)
(°C)
PxlO 4
NxlO 4
61758
27.4
7.21
26.4
10.3
2.0
0.96
61858
27.4
6.39
23.4
10.3
1.6
0.85
62058
27.8
8.06
28.5
10.4
2.4
1.10
71758
20.0
3.78
20.4
13.2
1.8
0.80
81658
25.6
18.29
34.1
12.5
8.6
2.95
71062
20.0
9.00
27.2
10.7
4.1
1.3
71262
21.0
7.25
30.0
10.8
5.0
1.71
8066l"
19.6
13.50
23.9
13.5
8.0
3.00
90661
26.6
21.40
34,8
13.3
9,7
3.40
91461
30.6
24.00
39,7
13.4
9.4
3.33
62362
12.3,
2.98
13.9
10.5
2,3
0.92
62762
24.0
5.86
27.3
9.5
2.1
0.86
63062
24.0
4.01
27.4
10.0
1.5
6.10
70262
20.0
6.32
32.4
10.1
4.1
1.15
70662
25.0
9.63
30.3
10.4
3.7
1.40
70762
25.0
10.11
32.4
10.3
4.2
1.47
71462
20.
5.38
27.6
11.0
3.1
1.04
71662
20.0
5.87
28.0
11.2
3.5
1.13
72062
17.2
4.86
24.3
11.6
3.4
1.12
72662
13.2
1.10
14.1
13.0
0.8
0.35
72862
17.0
1.83
19.8
13.0
1.2
0.46
73062
22.8
5.34
23.0
12.7
2.1
0.96
18
Indicate this to be true and also show that the terms of equation (5)
are all of the same order of magnitude,
Although equation (5) apparently is applicable for MLDs associated
with the transitional thermoelinej, tests indicated that (5) was no
longer applicable when the transitional MLD joins the seasonal MLD,. In
this case, a slightly different approach is necessary, apparently due to
the increased stability associated with a seasonal type thermocline. To
take this increased stability into consideration Q s (defined in A>pendix
III) is used in equations (3) and (4) instead of Q T .
The general procedure for finding values of P and N for the sea
sonal MLD was the same as for values associated with the transitional
MLD with the exception that Q<, was used instead of Q_,, Pairs of values
of P and N for seasonal MLDs, based on data listed in table 2, are
plotted in figure 5. The leastsquares bestfit line for these points
was:
p = 6.1 X lo~ 4 h3. 8=1 N
Substituting into equation (3) gives:
M LD=i.lxio"^ Q/3 ][jij + 3  8 Hit:) (7)
Equation (7) is applicable to MLDs associated with the seasonal therao
c 1 ine .
An objective method has been devised to determine whether a trans i
4
tional or seasonal condition exists When N < 3o5 x 10 , the therm©
4
cline is defined as transitional . When N > 3.5 x 10 „ a seasonal
thermocline is said to exist. This is not a rigid rule. Obviously, even
when N < 3.5 x 10 s if the forecast MLD using equation (5) is deeper
than a prevailing seasonal thermae 1 ine , Chen equation (7) should be used
rather than (5).
L9
I °
;'l
0)
00
(O
in
o
X
z
W
3
60
<*)
c
O
o
B
h e
C *J
O «H
W U.
W I
CJ u
«8 3
a. e"
«8 C/S
a, i
u
w Vi
3? «
O a>
hS:
O £
VM *=»
ss &
(0
3
(A
b
4)
>
■:■
CM
o
o
o
CM
OlXd
20
TABLE 2
DATA USED TO DETERMINE VALUES OF P AND N FOR MIXEDLAYER DEPTHS
ASSOCIATED WITH SEASONAL THERMOCLINES
W
QS 2
(Kg cal/cnO
MLD
TS
DATE
(KNOTS)
(METERS)
c°c)
PxlO 4
Nx 10'
62762
24.0
28.3
27.3
9.5
10.3
4,
91958
50.0
44.1
49.8
LI. 6
7.6
3.5 
61858
27.4
22.5
23.4
10.3
5.6
3.0
91161
27.2
43,2
40.5
13.3
21c 8
6.7
82761
21 = 4
40.6
32.2
14.5
27.7
8.5
82361
21.0
41.4
31.3
13,8
27.7
8.6
80661
19.6
35.7
23.9
13.5
21.0
7.9
72062
17.2
28.5
24.3
11,6
20.2
6.6
90661
26.6
39.7
34.8
13,3
18,0
6.3
91461
27.2
40,6
39.7
13.4
20.1
6,3
73062
23.0
31.8
22.8
12.;
12.3
5.7
90961
27.2
40.7
37.6
13.4
19.1
6.3
71462
20.0
31.0
27.6
11.0
18.0
6.0
71662
20.0
27.5
28.0
LI. 2
16.2
5.3
91461
30.6
39.5
39.7
13.4
15.5
5.5
81261
28.4
32.5
30.3
13.2
11.3
4.9
71262
21.0
29.6
30.0
10.7
16.6
5.3
71062
20.0
27.5
27.2
10.8
15.4
5.2
70262
20.0
26.2
32.4
10.1
16.9
4,8
80861
28.0
29.6
26.8
13.3
9.3
4.5
81461
32.0
37.4
34.7
12.8
11.4
4.8
63062
24.0
29.1
27.4
10.0
11.0
4.4
70662
25.0
30.5
30.3
10.4
11.7
4.5
70762
25.0
27.6
32.4
10,3
11.4
4.0 '
61758
27.4
29.5
26.4
10.4
8.2
3.9
61958
27.4
29.8
25,5
10.4
8.0
4.0 '
62058
27.8
29.3
28.5
10.3
8.6
3.8
81658
25.6
39.8
34.1
12.5
18.6
6.4
91858
50.
43.7
49.4
11.5
7.4
3.4
21
6. Application and Test of Equations.
Equations (5) and (7) can be used to forecast MLD over any length
of time for which the parameters can be accurately predicted. Thirteen
tests of the equations were made using independent data for forecast
periods of from one to four days. Five of the tests were based on data
from OWS Papa (50N, 145W) and eight on data from OWS November (30N, 140W).
Three of the tests were for transitional thermoclines and ten were for
seasonal thermoclines. The tests were made using a mean MLD computed
from four to six BTs from each day as a representative MLD for that day.
Results are shown in table 3.
The verification of a forecast is made difficult by the fact that
the MLD varies randomly. One objective way to measure verification suc
cess is to determine if the forecast value lies within the normal range
of variability of the MLD as measured by the standard deviations of the
individual values. In table 3 the 13 values listed under "verified MLD"
actually represent 63 BTs. The standard deviation with respect to the
daily mean for these 63 bathythermographs was 3.47 meters. The percent
age of forecasts that fell within one and two standard deviations of the
individual MLDs is given in table 4. The rootmean square and the alge
braic mean of the forecast differences are also given in table 4„
The algebraic mean of the forecast was computed to determine if the
forecasts were biased. Forecasts using the equation proposed by Rtssby
and Montgomery were biased positively whereas those based on equations
(5) and (7) were less biased and negative. This suggests that the nega
tive term neglected by Rossby and Montgomery (see section 5) should be
included as in equations (5) and (7).
22
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23
TABLE 4
STATISTICAL RESULTS OF FORECASTS
FORECASTS BY
EQUATIONS
(5) AND (7)
FORECASTS BY
PERSISTENCE
FORECASTS BY
ROSSBY AND
MONTGOMERY
EQUATION
X FORECASTS
WITHIN 1 STANDARD
DEVIATION
69
46
15
% FORECASTS
WITHIN 2 STANDARD
DEVIATIONS
92
61.5
46
ROOT MEAN SQUARE
OF DIFFERENCES
(METERS)
4.3
6.3
11. 1
ALGEBRAIC MEAN
OF DIFFERENCES
(METERS)
2.9 1.06
1
8.1
24
To test how the function P(N) at OWS November compares with that
found at OWS Papa, eight paired values of P and N at OWS November were
computed. The BTs upon which the verifications were based in table 3
were also used to obtain these paired values in the same manner as for
OWS Papa (section 5). Figure 6 is a plot of paired values of P and N
for the data listed in table 5; the straight line represents P(N) for
the constants derived from data at OWS Papa. The function P(N) appears
to be very nearly the same at both OWS Papa and OWS November.
Two examples of the application of equations (5) and (7) to fore
casting follow.
Example 1. Forecast to be made for 23 August 1957 based on BTs
taken on 19 August at OWS Papa.
1. From the BTs taken on 19 August 1957, Q~ was determined to be
2
1.74 Kg cal/cm by methods described in Appendix III.
2. Using the August climatological data computed by Tabata and
Giovando [V] , the average net heat transfer downward across the air
sea boundary at OWS Papa is approximately +0.1 Kg cal/cnr per day.
2
Using this information, 0.4 Kg cal/cm were added to Q„ during the
fourday interval up to the verification day giving?
Q T = 1.74 + 0.4 = 2.14 Kg cal/cm 2
(Where enough current meteorological data are available, the methods
for computing the net heat transfer proposed by Laevastu [.13J could
be applied instead of climatology.)
3. Values of /3 , Sl , and W were determined by the methods described
in Appendix III.
4. Equation (4) was entered and N was found to be less than
25
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26
TABLE 5
DATA USED TO DETERMINE VALUES OF P AND N FOR MIXED LAYER DEPTHS
ASSOCIATED WITH SEASONAL THERMOCLINES AT OWS NOVEMBER
DATE MLD W Qs P x 10 4 N x 10 4
(METERS) (KTS) (Kg cal/cra Z )
61657 27.9 13.2 32.9 26.8 7,5
70757 43.0 17.1 31.6 13.0 4.7
71057 37.5 19.3 22.7 11.5 4.3
72157 44.5 23.5 25.2 24.9 6.4
72757 49.0 17.8 28.3 28.5 9.7
82558 34.7 15.1 39.6 34.2 10.5
73159 43.5 18.5 33.0 23.1 6.9
70159 32.0 18.0 18.2 9.9 3.9
27
3.5 x 10" . Consequently, equation (5) was used, giving a forecast
MLD of 16.1 meters.
5. The seasonal MLD was at 32 meters on 19 August 1957, According
to the above forecast, the wind was not strong enough to cause the
transitional thermocline to merge with the seasonal thermocline.
Thus persistence is used to forecast the depth of this seasonal
thermocline.
6. The forecast thermal structure for 23 August 1957 consists of a
mixed layer down to a transitional thermocline at 16.1 meters with
a seasonal thermocline below it at 32 meters. The verification
showed a transitional thermocline at 18.3 meters and a seasonal
thermocline at 30.5 meters.
Example 2. For the forecast MLD of 1 August 1959 at OWS Papa, steps
1, 2 and 3 as done in example 1 resulted in N >3.5 x 10" . Conse
quently, the procedure for a seasonal thermocline was folio' ed.
That is, Qc was determined and equation (7) was used to forecast
the MLD. The forecast MLD was 35 meters and 39 meters verified.
28
7. Conclusions.
The agreement between forecast and observed MLDs tends to verify
the findings of Kitaigorodsky that similarity theory is useful in this
forecasting problem. Equations (5) and (7) were successful in forecast
ing both large and small changes in the MLD, in the positive as well as
in the negative sense. The concept of a universal function P(N), as pro
posed by Kitaigorodsky, is strengthened.
Equations (5) and (7) appear to be useful in forecasting the mixed
layer depth to a reasonable degree of accuracy. Some of the deviations
can be attributed to the effects of divergence, advection and internal
waves which were not evaluated in this study.
Future rest arch could well be applied to determining more paired
values of P and N for other ocean locations. This would not only help
to fix the constants of equations (5) and (7) better, but would help to
demonstrate further the universality of P(N), if it exists.
29
BIBLIOGRAPHY
1. Kitaigorodsky, S, A. On the computation of the thickness of the
windmixing layer in the ocean. Academy of Sciences, USSR, Geo
physics Series, no. 3, March 1960,
2. Buckingham, E. On physically similar systems; illustrations of the
use of dimensional equations. Physical Review, vol. 4, 1914, pp
345376,
3. Binder, R. C. Fluid mechanics, Prentice Hall Inc. 1943.
4. Geary, J, E. The effect of wind upon the mixed layer depth. M.S.
thesis, U. S. Naval Postgraduate School, Monterey, Calif., 1961.
5. Tabata, S. Temporal changes of salinity, temperature, and dis
solved oxygen content of the water at Station "P" in the northeast
Pacific Ocean, and some of their determining factors. Journal
Fisheries Research Board of Canada, vol. 18, no. 6, 1961.
6. U. S. Navy Jydrographic Office. Prediction of the thermocline depth
by Mazeika, P. A. TR 104, June 1960.
7. Munk, W. H. and E. R. Anderson. Notes on a theory of the thermo
cline. Journal of Marine Research, vol. 7, no. 3, pp 276295, 1948,
8. Rossby, C. G. and R. B. Montgomery. The layer of frictional influ
ence in wind and icean currents. Massachusetts Institute of Tech
nology and Wood*; Hole Oceanographic Institution, Papers in Physical
Oceanography and Meteorology, vol. Ill, no. 3.
9. Pacific Oceanographic Group, Nanaimo, B. C. The seasonal thermo
cline at OWS Papa during 1956 through 1959 by Tabata, S. and L. F.
Giovando. Fisheries Research Board of Canada, no. 157, April 23,
1963.
10. Pacific Oceanographic Group, Nanaimo, B. C. Oceanographic domains
and assessment of structure in the North Pacific Ocean by Tully,
J. P. Fisheries Research Board of Canada, File N613(4), 16 Sept.
1963.
11. Ekman, V. W. On the influence of the earth's rotation on ocean
currents. Ark. f . Mat, Astr. och Fysik. K. Sv. Vet, Ak. , Stock
holm, 190506, vol. 2, no. 11, 1905.
12. Sverdrup, H. V. The oceans. Prentice Hall Inc., 1942.
13. Laevastu, T. Factors affecting the temperature of the surface layer
of the sea. Societas Scientiarum Fennica Commentationes Physico —
Mathematicae XXVI, 1960.
30
14. Monin, A, S. and A. Mo Obukhov. Basic laws of turbulent mixing in
a ground layer of the atmosphere. Transactions of the Geophysical
Institute, Academy of Sciences, USSR, no. 2 (151), 1954.
15. . Oceanic observations of the Pacific: 1955. The Norpac
Data prepared by the Norpac Committee, University of California
Press and University of Tokyo Press, Berkeley and Tokyo, 1960.
16. . Manuscript Report Series, Oceanographic data records,
Ocean Weather Station "P", Fisheries Research Board of Canada,
19561962.
17. Budyko, M. I. The heat balance of the earth's surface. Translation
by Office of Technical Services, U. S. Department of Commerce,
Washington 25, D. C, 1958.
31
APPENDIX I
BASIC PROCEDURES FOR APPLICATION OF TT THEOREM
Let n represent the number of parameters (A, A9....A ) chosen to be
representative of the processes involved in a particular physical phe
nomenon. Let m represent the number of fundamental dimensions (length,
time, mass etc.) present in all of the parameters. Then follow these
steps.
1. Determine the number of " TT dimension less ratios" required to
incorporate all the chosen parameters (given by nm).
2. Combine m + 1 parameters into an equation of the following form:
where (A, A^^.A ) must together include all of the fundamental
dimensions.
3. Substitute the fundamental dimensions into equation (8) and
equate like powers on each of the fundamental dimensions to zero,
a* a* .A 4., = L M°T t
A. A 3  rvw+l
4. Solve the resulting m equations with m unknowns to obtain x. ,
yi and z .
5. Substitute the values of x.,y ,z. into the equation for TT\ to
obtain the first dimensionless ratio.
6. Repeat steps 2 through 5 after changing A to another parameter
m+1
not previously used for any A and solve for TT 2 . Repeat as neces
sary until Tf. , TT , ....TT ratios are obtained.
7 1' 2' nm
7. The ratios represented by TT,, TT, ....TT are all dimensionless
12* nm
32
and can be equated one to the other to obtain a functional relation
ship that is dimensionally correct. The ratios are then made numer
ically equivalent by use of a dimensionless coefficient P.
An example illustrates the application of the foregoing step by step
procedure.
Parameters Dimensions
Q (total heat present) LT
W (wind speed) L/t
XL (coriolis effect) l/t
/3 (coefficient of thermal expansion) 1/T
H (mixedlayer depth or ML.D) L
n = 5 m = 3
STEP 1 nm = 2 " If ratios" will be required
STEP 2 11^ = Q w SL. ft and %= Q. *wV H
To find the powers (x,y,z) which make the above products dimension
less:
STEP 3
For Til : (Lrf' (L/t/'(l/tf (1/T) = L_° T° t°
STEP 4 (L) rh*"fr\ ~ °
(T) ^.1=0
i
(t)
W " % = °
*, =1
33
= 1
STEP 5
\}\ = q' w~'jn' /3
or
Q. TL/3
ir t 
STEP 6 For If: (i_ T f ( L /t)^ ( l/t f ( L) = L°T°t°
(L)
^2. + ^ + 1 = O
(T) ^ = o
Ct)
'^1 fr
^ = o
fc
= 1
T, = q°w"'TL H
ir  ^ H
STEP 7 Since both Tf^ and TT 2 are dimension less , either can be
inverted in order to give a relation that agrees with what is observed
physically. In this case it is known that the MLD is an increasing func
tion of the wind, and u^ and TT2 ma y De combined into?
mp^
where P is a dimension less coefficient whose form must be determined
experimentally. Substituting for ff an< * "9 an< * so * vin 8 f° r tne mixed
layer depth gives;
" L D = P( Q ^)
34
In his paper on TT theory, Buckingham [2] gave several methods to
determine the form of p. Of these, the same method used by Kifcai*,
sky £l] will be given as an example here. Since P may be considered a
function of all the original parameters » they can be used to determine P
with experimental data. If P does not turn out to be constant, it can
be made a function of certain of the parameters and the form of that
function can be found from the data. Thus P is made a function of N,
which, in this example h is made to depend upon all the parameters except
MLD, the quantity to be predicted. Use of tT theory, as before, yields
a dimensionless ratio of these parameters °
1M W
Entering the equations for both P and N with the same data and plot
ting the corresponding pairs gives P(N) in graphical form.
35
APPENDIX II
RELATIONSHIP BETWEEN VARIANCE OF THE MLB
AND THE STRENGTH OF' THE THERMOCLINE
Mazeika jjQ used 356 groups of BT data t© determine that the MLD
at OWS Echo had fluctuations with a mean amplitude of 17,85 feet and
standard deviation ©f 10.55 feet about the mean. He attributed these
fluctuations to internal waves.
On the basis of 100 BT observations from OWS Papa for the months of
September and October of the years 1958 through I960* the author deter
mined a system for estimating the amount of fluctuation of the MLD that
might be expected at verification time. The temperature difference be
tween the sea surface and the temperature of the near isothermal water
just below the thermocline was used as a measure ©f the strength of the
thermocline. The variance of each observation was computed from a five
day running mean. The average variance and the average AT for each two
meter increment of misted layer depth was then computed and plotted as
pairs in figure 7. Figure 7 can be used to estimate the variance that
can be expected at verification time.
In order to smooth out the fluctuations of the MLD when an accurate
verification Is desired 9 it is apparently best to take a series of BT
observations over a period of at least 12 hours and then use the average
of all the BTs to verify the forecast.
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(gSJd}dtN) ODUeueA
37
APPENDIX 111
METHODS USED FOR DETERMINING VALUES OF PARAMETERS
WIND (W)
1. Out of the 72hour period preceding verification time, but not
including the 12 hours immediately preceding verification time,
determine the 24hour period during which the highest average winds
prevailed or are forecasted to occur.
2. Fro® the eight wind reports given for that 24 hour period,
choose the five highest speeds . The average of these five is the
"representative maximum wind."
EXCESS HEAT IN UPPERMOST LAYER (Q)
When finding values of P and N 9 the computation of Q was based on
the average BT for each particular 12hour period. When using equations
(5) and (7) to forecast an MLD, the most recent BTs available should be
used to compute Q. In either case the following steps are applicable in
determining the excess heat in the uppermost layer.
1. Assume a transitional warm layer exists. For this situation
compute Q™ as follows;
a) Determine "AREA^'s, depicted in figure 8. The dashed line
is a vertical drawn from the point of maximum curvature of the
BT trace (roughly the bottom of the thermocline) to the surface.
b) Or & s then given by
i
Q T ^CpAREA X 10" (Kgc^l/c^)
For OWS Papa in summer />c s 0.975 for salinity 32.5 °, oo ,
assumed to be constant (pc n Ira JL®JL_ and AREA T in M°C).
38
Figure 8
reseiitati©© of'Area*
d in Deterraiaing 'Q_,
4
2. If N > 3.5 x 10 using Q~> then a seasonal type thertaocline is
assumed to exist. The excess heat for a seasonal therrnocline (Qe)
is found by the following objective method.
a) Determine "AREAe'\ depicted in figure 9(d). The dotted
lines, figure 9(a) 8 is a vertical drawn from the intersection of
the BT trace asad 200 meters. HH is determined by equalizing
areas A and B as in figure 9(b). Then
AREAg r C  B
b) , .
Q s =/=>c p AREA X 10 (Kgc^l/cm^j
40
T >
T >
Q 2
200
/ Trace
«
200 
(a)
<b)
T >
T >
(c) ,' (d)
: Figure 9
Representation ©f Area g Used In Determining Q«
41
TABLE 6
VALUES FOR THE COEFFICIENT OF THERMAL
EXPANSION BASED ON A SALINITY OF 32.5°/oo
Temperature (°C)
( <10 6 (1/°C)
>
14.0
205
14.0

13.6
200
13.5

13.1
195
13.0

12.6
190
12.5

12.1
186
12.0

11.6
182
11.5

11.1
178
11.0

10.6
174
10.5

10.1
168
10.0

9.6
163
9.5

9.1
158
9.0

8.6
152
8.5

8.1
147
8.0

7.6
141
7.5

7.1
135.5
7.0

6.0
124.5
<6
.0
113
42
APPENDIX IV
FORECASTING METHOD PROPOSED BY KtTAIGORODSKY
Kitaigorodsky [ij , using similarity theory as described by Monin and
Qbukhov [141 , developed an equation for forecasting the MLD. In develop
ing the equation Kitaigorodsky assumed that thermal convection was neg
ligible and that the vertical gradient of salinity was zero. This limits
the use of the equation to the warm season when a stable layer exists.
The equation does incorporate parameters that represent all the processes
and variables that were considered in section 3 to cause or affect the
MLD at OWS Papa.
Equation developed by Kitaigorodsky using similarity theory;
MLD=P
where i T a s a parameter proportional to the tangentia
stress of the wind on the sea surface
Qp  rate of heat flow from the atmosphere (LT/t),
2
g s acceleration of gravity (L/t );
/3 s coefficient of thermal expansion of sea water (l/T)
P  a dimensionless coefficient that is a function of
Na both N and P are determined empirically;
(Mi is dimensionless)
i\_ s coriolis parameter (1/t).
The value of P in equation (8) is given by a plot of 14 correspond
ing values of P and N, obtained by Kitaigorodsky from data from the
43
NORPAG ATLAS [l5] . The large areal extent which the data represents
makes the results widely applicable throughout the ocean during the warn
season.
The present author, using data from OWS Papa jj.6] has determined 11
more pairs of values of P and N. Each of these 11 pairs represents from
four to six BTs averaged together in an attempt to smooth out internal
wave effects. The paired values of P and N given by both Kitaigorodsky
and the present author are plotted on figure 10.
In calculating P and N, it appears on the basis of the reference [l 7]
that Kitaigorodsky used an average (climatological) heat flow to find Qp.
The present author also used an average heat flow, the average heat flow
since the time that the upper layer of the ocean was last isothermal.
This average heat flow was arrived at in three steps.
1. Calculate the heat added in the surface layer since warming
began. The method described for Q_ in A>pendix III was used for
this calculation.
2. Determine the length of time elapsed since warming began.
3. Divide value determined in 1 above by the value determined in
2 above.
Despite the different locations of the NORPAC and OWS Papa data and
despite the differences in Qp, both sets of the points P versus N appear
to fit the same curve (in figure 10). Still both methods for obtaining
Q„ seem weak. In section 4, Qp is replaced by another parameter to
represent stability in order to eliminate that weakness.
Another problem in applying the method proposed by Kitaigorodsky
dp
lies in the steepness of the slope, ^, for the small values of N.
44
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 10
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Va
s of P versus H
ted Using Equations Developed by Kltaigorodsky
45 , : ■■,•■''
To avoid these problems, the author derived by similarity methods
an equation with more readilyavailable parameters and a more stable
P(N).
46
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