VERTICAL
DIFFUSIVITY COEFFICIENT
A THERMOCLINE
IN
H. E. Sweersl
Department
of Energy,
Mines and Resources, Marine
615 Booth Street, Ottawa
Sciences Branch,
ABSTRACT
Two equations are presented giving the vertical eddy diffusivity
coefficient
K, in the
thermocline
of a well-stratified
lake as a function of its lake mean rate of downward
displacement and intensity.
Both equations are derived by transforming
the heat transport
equation onto a quasi-Lagrangian
coordinate system, the origin of which moves down with
the lake mean thermocline
depth. Application
of the equations to Lake Ontario yields
summer mean values of K, in the thermocline
of 0.02 to 0.07 cm”/sec. During an exceptionally
quiet period in July and August 1907, K, approached
the value of molecular
diffusivity.
The eddy diffusivity
coefficient can bc
determined by direct methods, such as dye
releases or shear measurements, or by indirect methods making use of changes in the
temporal and spatial distribution
patterns
of naturally occurring parameters. In layers with strong vertical density gradients,
such as thermoclines,
special problems
arise
that complicate determination of the
eddy diffusivity
coefficient by direct measurement; indirect methods, however, may
still be applicable.
Using thermal data from Lake Ontario, I
have developed a quasi-Lagrangian
equation to estimate a mean value for the vertical coefficient of eddy diffusivity
in the
thermocline of a stratified lake. In contrast to the more traditional Eulerian formulations, the quasi-Lagrangian
diffusivity
equation is written with reference to a
moving coordinate system. The origin of
this system is fixed with respect to the
thermocline rather than the surface, and
thus moves up and down with internal
waves, seiches, and seasonal changes in
thermocline depth. The quasi-Lagrangian
l I am indebted to Dr. D. B. Rao who, through
many stimulating
discussions, greatly contributed
to the work described in this paper. I am aIso
grateful to the Marine Sciences Branch for their
kind permission
to let me complete
the work
started during my earlier assignment with the Inland Waters Branch.
equation can take one of two different
forms, depending on the number of simplifying assumptions that can be made. In a
large, deep lake KZ can be expressed as a
function of the lake mean rate of downward movement and the intensity of the
thermocline.
In shallower lakes an additional heat-flux term is introduced.
Derivations
of the Eulcrian and both
quasi-Lagrangian
equations for the diffusivity coefficient are presented below. Lake
Ontario
data have been interpreted using
all three equations; the quasi-Lagrangian
equations, within the narrow limits of their
applicability,
yield more meaningful results
than the Eulerian method.
METHOD
Local temperature changes in a lake can,
in the absence of sources and sinks, and
assuming specific heat to be constant, be
described by the heat transport equation:
where X, y, and x are the two horizontal
and the vertical axes of a rectangular coordinate system with its origin at the sur273
27’4
H.
E. SWEERS
face; u, v, and w the velocity components
and K,, KY, and K, the diffusivity
coefficients in these directions respectively; 2’
the temperature and t time. The vertical
axis is measured positive downward. Integration over the area A of the lake, assuming advective and turbulent
transports
through the boundaries to be zero, gives:
where the bar denotes horizontal means
over the area of the lake and KZ the average
value of the vertical eddy diffusivity
coefficient defined by
In the following,
the caret over K, is
omitted.
The frequently used heat transport equation ( I-Iutchinson 1957; Khlopov 1958; and
others) can easily be derived by integrating
equation (2) over depth from a level x to
the bottom ~a, assuming the second term
o’n the left-hand side to bc negligibly smalI:
%
c3T(x)
a%
-=
at S
x
--dz:=-Kay,
at
(3)
where H, is the lake mean heat content
below a unit area at depth x. For individual stations a similar heat transpo,rt equation, giving K, for any depth x as a function
of local temperature structure and vertical
heat transport, can be derived directly
from equation ( 1) if it is assumed that
lateral advective and diffusive transports
arc negligible.
In the derivation of equation (3) the
lake mean net advective heat transport
term WT has been assumed to be negligible. This is not always the case. Changes
in the distribution of thermocline depth can
give rise to large, reversible net upward
or downward transports of heat, without
any change in total heat content below the
thermocline, For example, assume a model
lake with a horizontal thermocline sepa-
rating two uniform layers of water with
temperatures T1 and Tz. Furthermore, assume that the mean thermocline depth remains constant at depth Zn but that the
thermocline tilts at time t > to, reaching a
minimum depth of x1 at one side and a
maximum depth of x2 at the opposite side
of the lake. The total heat content H,
below a level XI< x < ~2 then increases
because a fraction of the colder water formerly below x is now replaced by warmer
water. If the thermocline levels out again,
the net heat transport is reversed and the
original distribution restored, but the heat
content below the thermocline
remains
constant throughout this process. Actually,
the lake mean heat flux at any level is
difficult to estimate, and a return to equation (2) to evaluate K, therefore is not
always possible.
A second objection, which becomes especially significant
when there are strong
vertical gradients, can be made to equation
( 3). The diffusivity
coefficient normally
is a function of location as well as of depth,
since it is inversely related to the stability
S. In freshwater, S is maximal in the thermocline layer. Local variations in thermocline depth thus may give rise to large
fluctuations in K, along a horizontal surface, and, as a result, equation (3) can give
rather erratic results when applied to levels
around the mean thermocline depth. Other
factors affecting K,, such as internal circulation patterns and bottom topography, are
probably less important for large, regularly
formed lakes, at least at levels around the
mean thermocline depth.
To meet these objectives, the diffusion
equation can be rewritten using a coordinate system fixed with respect to a
characteristic structural clement in the temperature distribution
rather than to the
surface. In particular, toI estimate K, in
the thermocline layer, the thermocline itself
can be used as a reference level. The vertical cocfficicnt
of eddy diffusivity
at a
distance x’ below the thermocline then can
be calculated from the rate of downward
dispIaccment of the thermocline and the
temperature profile,
This will be illus-
VERTICAL
DIFFUSIVITY
COEFFICIENT
trated for a simplified lake model, but the
results are equally valid for any lake with
a well-developed thermocline.
Assume a model lake with a horizontal
thermocline at a depth x,( f ), where xe is
a function of time, and a rectangular coordinate system (x’, y’, x’) with its origin
on the thermocline. The depth x’ and vertical velocity W’ of a point in the new
system then are given by:
x’ = x - x0(t),
IN
275
A TIIERMOCLINE
TEMPERATURE
(4)
w’=w-we,,
(5)
where zwOis the rate of downward movement of the thermocline.
Equation
(2)
then can be transformed to the quasi-Lagrangian coordinate system:
w,
(6)
where T( x’) and K', are the temperature
and diffusivity
coefficient at a depth x’
respectively. K', for x = 0 is numerically
equal to Kz, since the thermocline is horizontal.
Integration
over depth from a depth
x’ = 0 to the bottom ~‘b, assuming thermocline depth to be defined by the isotherm
T( 0)) gives:
d
at,
O’b
S
T(d)
dx’- T(&)
At the bottoam w’ ( x8) = 0 and w’ ( xb) thus
is equal to -w,; in the thermocline T( 0) is
constant by definition and w’( 0) = w (0) E=-We.
Furthermore it can be assumed
that diffusive and advective heat transports
at the bottom are negligible. Equation (7)
then can be reduced to:
&&,
+ w,?-(O) = -K'*(O) [v]
a':0
>(8)
where Hz!, is the mean heat content at the
time t in a column of water extending over
the interval ( 0, x’~ ) ,
FIG. 1. Graphic illustration
of the downward
movement
of the lake mean temperature
curve
according to the assumptions given on p. 275.
An even simpler expression for K’, can
be derived from equation (6) if two additional assumptions are made (see Fig. 1) :
the vertical temperature gradient near the
bottom is zero, and the temperature profile
remains parallel to its original position as
it moves downward. A direct consequence
of these assumptions is that bo’th the bottom temperature T( ~‘b) and the heat content B = J:,y { T( xl) - T( x’b) } dx’ remain
constant with time. Integration
between
x’ = 0 and the bottom of equation (6) now
gives :
we { T(0) - T(&) } = -K',(O)
aq
xf
)
c 1
~
&zf
* (9)
X'=O
The two assumptions made to derive
equation (9) will generally be satisfied
only in large, deep lakes. Their validity for
Lake Ontario is analyzed below. Equation (8) can be applied to any lake with
a well-developed
thermocline.
Actually, the thermocline will seldom be
horizontal but will vary in depth with location. Derivation
of the equations then
gives rise to higher order terms, which can
be neglected in a first order approximation
of K’,, since horizontal gradients of xB usu-
276
10
H.
E. SWEXRS
-
v)
e
Y '5
-
I-
_
Y
z
20 -
25 -
25 i-
II
JII
FIG. 2.
1
I
I
I
I
I
I
I
Mean depth of the isotherms
I
I
I
TO
LAKE
1
”
11
1
’
for 6 cruises in the summers of 1966 (left)
ally remain very small ( < MP2 m/m), The
lake mean vertical temperature profile in
this case should be determined by averaging the depth, x( T), of the isotherms rather
than by determining the mean temperature,
T(X), at each level (Sweers 1969a). It can
be shown that the vertical gradient of x ( T )
for x’= 0 equals the average of the gradients of the individual
temperature-depth
curves, provided that the thermocline does
not reach the surface anywhere. The mean
gradient of T( x ) can be much smaller,
particularly when the depth of the thermocline varies widely over the lake. Another
important property of x( T) is that it is a
direct measure of the heat content, IIT,
below any isothermal surface T.
APPLICATION
II
ONTARIO
During the 1966 and 1967 field seasons,
Lake Ontario was sampled at 2-week intervals at about 50 stations. The stations are
distributed evenly over the lake, and their
planned positions remain fixed throughout
11
11
1
’
4
and 1967 (right).
each field season. The temperature data
discussed in this paper are bathythermograph readings, calibrated with a reversing
thermolmeter.
A lake mean temperature profile is determincd for each cruise by averaging the
depth of isothermal surfaces over the area
of the lake as described above. The depth
X( Ts) of an isothermal surface Ti is taken
equal to zero wherever it reaches the lake
surface.
The level of the 1OC isotherm is used to
define the thermocline [depth of the 1OC
isotherm is very close to the depth of maximum vertical gradient for the data studied
( Sweers 1969b) 1, and all calculations are
performed for this level. The quasi-Lagrangian method is applicable only to a
well-developed
thermocline, so only data
for the cruises between early July and 20
September are used. During both summers
the 1OC isotherm occasionally reaches the
surface, but never at more than 7% of the
stations.
The conditions imposed on the model
VERTICAL
DIFFUSIVITY
COEFFICIENT
IN
A THERMOCLINE
277
of the simpler, but less accurate, equation
(9) leads to similar results.
It is immediately obvious from the table
that the Eulerian appro.ach yields values
of K, that are on average much higher and
more variable than those given by the
quasi-Lagrangian
method. In both years
the summer mean of K, is about twice as
high as that of K’,, and three to four times
as variable. The difference between the
numerical values of bolth K, and K’, for the
two summers is mainly related to climatological conditions and will be discussed
later.
To evaluate the importance of the vertical advective transport term wT in equation (3), the heat content below the mean
thermocline depth, H,, has been calculated
for each cruise, and compared with the
heat content below the 1OC isothermal surface, HP, and with the standard deviation
of thermocline depth, SD,. The latter is a
convenient measure of the variability
in
thermocline depth, increasing with increasing internal wave and seiche activity and
approaching zero for a strictly horizontal
thermocline.
An increase in SD, will generally be caused by an upward displacement of colder water in some areas and a
downward displacement of warmer water
in others. The net result of this will be a
reversible downward transport of heat; reversible because the original thermo,cline
distribution
can, at least theoretically,
be
restored. As a consequence of this, HT
will be smaller than H, unless SD, = 0, when
( HZ - IIr) reaches a minimum of zero.
Table 1 illustrates this strong correlation
RESULTS
between SD, and (H, - HT) as well as the
Equations (3), (S), and (9) have all fact that HT < Hz in all cases, The term
been applied to evaluate the vertical coef- wT in equation (3) therefore cannot be
neglected unless SD, remains fairly constant
ficient of eddy diffusivity
in the thermoover the period considered. Large positive
cline of Lake Ontario,
The results are
or negative changes in SD, between consecsummarized in Table 1 and displayed
utive cruises may result in exceptionally
graphically in Fig. 3. The heat contents,
large or negative values of K,. The negaH, are all given in calories per square tive values of K, in early August and midccntimctcr, assuming H to be zero for a September 1967 both arc coupled to fairly
column of water of 4C. In this section K, large decreases in the variability
of therand K’, obtained by equations (3) and (8)
mocline depth. The total heat content of
respectively, are discussed in detail. Use the lake increased by 6 to 236 and by 14
lake to derive equation (9) are satisfied
reasonably well in Lake Ontario.
Temperatures well below the thennob
Cline remained fairly constant throughout
the summers of 1966 and 1967. At the 50-m
level, the lake mean temperatures were in
the ranges of 4.264.42C and 4.05-4.18C,
respectively, and at the 75-m level, 4.004.06C and 3.89-3.99C. Fluctuations are apparently random and sho,w no systematic
increase except during the last cruise in
each summer, which in three out of the
four cxamplcs given determines the upper
limit of the range. At the 100-m level the
range is reduced to O.O4C, or excluding the
last cruise, 0.02C in both years.
Temperature profiles below the thermocline also remain fairly constant. Lake
mean temperature profiles for all cruises
(Fig. 2) have been obtained by averaging
the depth of isotherms, x( T). For the sake
of clarity, the vertical axis denotes true
depth, not depth below the thermocline.
Below the IOC isotherm the shape of the
profile remains remarkably constant, and
the second assumption thus is good in
a first order approximation.
An analysis
in more detail, however, indicates that
changes in heat content below the therrnoCline may contribute up to1 30% to the lefthand side of equation (S), and both equations therefore have been applied to the
data. The differences between the eddy
diffusivity
coefficients derived from equations (8) and (9)) although not negligible,
are much smaller than between those derived from equation ( 3) and either ( 8)
or (9).
278
I-1. E. SWEERS
1.
Vertical
SD, (10)
HP
TABLE
Median
cruise date
d10)
eddy diffusivities
in 1966 and 1967
Ha-H,
K#
eq. (3)
K’,
eq. (W
Kf,
wt. (10)
0.29
0.106
0.150
0.09
0.071
0.109
0.23
0.079
0.079
0.00
0.012
0.02,2
0.06
0.096
0.128
1966
6 Jul
11.1
4.2
41.8
49.1
7.3
20 Jul
13.8
6.4
37.1
56.7
19.6
3 Aug
16.2
5.3
32.0
48.61
16.6
17 Aug
18.0
7.6
32.1
58.8
26.7
31 Aug
18.9
7.4
29.6
53.4
23.8
14 Sep
21.4
5.7
25.9
40.0
14.1
16.6
6.1
33.1
51.1
0.13
0.073
0.098
5.6
7.0
0.12
0.037
0.050
0.30
0.012,
0.016
Summer
Mean
mean
SD
1967
12 Jul
14.0
4.6
34.6
40.1
5.5
27 Jul
14.3
8.7
34.2
64.3
30.1
7 Aug
14.0
7.1
34.4
53.7
19.3
23 Aug
15.2
7.0
32.1
54.2
22.1
7 Sep
16.2
7.6
30.2
54.8
24z.6
18 Sep
18.2
5.8
31.1
35.6
4.5
15.3
6.8
32.8
50.4
1.9
10.6
-0.14
Mean
SD
mean
to 258 Cal/cm2 respectively over these same
periods, whereas the heat content below
the mean thermocline depth decreased by
10 to 54 and by 20 to 36 cal/cm2.
The relationships between H,, SD~, and
K, and between x ( lo), HF, and K’, are
shown graphically in Fig. 3. The independence of K, and K’, and the smaller variability of the latter are well illustrated by
the graphs. The small decrease in mean
thermocline
depth during early August
1967 is due to limitations in the accuracy
of determining
an “instantaneous”
lake
mean value of x( T) from a small number
(about SO) of nonsynoptic data.
Another important feature shown in the
table is the large difference between the
summer mean values of K’, for 1966 and
-0.008
0.11
0.037
0.054
0.08
0.022
0.032
0.053
0.049
0.06
0.023
0.049
0.17
0.039
0.020
-0.06
Summer
-0.007
1967. This difference is probably significant and related to weather conditions
during the two field seasons. The summer
of 1967 was exceptionally
quiet. Mean
windspeeds at To,ronto International
Airport were 281 cm/set vs. a 30-yr mean of
375 cm/set, a record low since meteorological observations started in this location
in 1938. Over the 4-week period between
12 July and 7 August, mean winds were
even lower (264 cm/set) and strong winds
Hourly mean winds
almost nonexistent.
exceeding 540 and 800 cm/set occurred
during 5.3 and 0.2% of the time, vs. 22.8
and 4.1% respectively in a normal year.
The low values of K’, in July and early
August 1967 therefore can be considered as
real. Unfortunately,
the data are not ade-
VERTICAL
DIFFUSIVITY
COEFFICIENT
IN
279
A THFJtMOCLINE
--01
1967
-02
J1,lv
6
I
20
I
3
I
AUglFit
17
31
I
Septembl
14
1
27
I
FIG. 3. Changes between consecutive
pairs of cruises of the lake mean
thermocline
depth H, and below the 1OC isotherm HT, and of the mean x(
SD~ of the thermocline
depth. Eddy diffusivity
values K, and K’, are means
twecn each pair of cruises, and are calculated using equations (3) and (8)
dates are indicated
along the bottom.
quate to determine the exact magnitude
of K’, during this period. On a lake as
large as this, it is impossible to obtain a
really synoptic picture of the thermal structure over its entire volume, so that a more
accurate K’, during a quiet period cannot
be calculated by this method. It is obvious, however, that vertical diffusivity
is
extremely small and approaches the value
of molecular conductivity (0.12 x 10d2 cm2/
set).
DISCWSSION
The vertical coefficient of eddy diffusivity is a mathematical tool used to describe
eddy mixing processes taking place in a
lake. Its magnitude not only depends on
the spatial scale taken into account, but
varies also with time and location. The
September
August
July
12
I
7
I
23
I
7
t
18
I
heat contents below mean
10) and standard deviation
for the e-week periods berespectively.
Median cruise
major factors causing downward movement
of the thermocline are wind-induced turbulence and vertical convection due to cooling (Tully and Giovando 1963). Both are
essentially intermittent,
and the thermoclinc thus may be stationary for prolonged
periods, with short spurts of relatively
rapid descent. During a stationary period,
vertical diffusion probably is determined
mainly by molecular thermal conductivity
(0.0012 cm2/sec, p, 472 in Hutchinson
1957). During storms or during night or
autumn cooling, on the other hand, the
thermocline
may be pushed downward
rapidly, and the diffusivity
coefficient for
such periods may become fairly large.
Spatial irregularities
in the magnitude of
K, also may bc due to such factors as
circulation patterns within the epilimnion,
280
I-1. E. SWEEZS
variations in de-pth of the thermocline, and
geographical factors.
The long-term averages of the vertical
coefficient of eddy diffusivity
in the thermocline layer derived in this paper can
only be used to give an analytical description of the downward movement of the
thermocline in general terms, They cannot
be used to describe the actual physical
processes causing this movement, nor even
the changes in thermocline
depth over
short time intervals or in small fractions of
the lake,
The low values for Lake Ontario of the
coefficient in the thermocline are not entirely unexpected. Csanady ( 1964), in a
series of dye experiments in Lakes Huron
and Erie, concluded that the thermocline
acts as a diffusion floor. Horizontal and
vertical eddy diffusivities in the epilimnion
are several orders of magnitude larger than
vertical diffusivity in the thermocline. Typical values emerging from dye experiments
are 500 to 1,000 cm2/sec and 7 cm2/sec for
horizontal and vertical diffusivity
in the
upper layers respectively ( Csanady 1964,
1966), as compared with 0.001 to 0.15
cm2/sec for K, in the thermocline.
Numerical values of K, approaching molecular conductivity have been reported by
Hutchinson ( 1957). His examples are observations in Sodon Lake (Michigan), Linsley Pond (Connecticut), and Lake Mendota
(Wisconsin),
which give values of 0.007,
0.0033, and 0.025 cm2/sec respectively.
These have been calculated assuming an
exponential decrease in temperature below
the level of maximum gradient and a constant value of K, over the depth interval
used for the calculations. Neither of these
assumptions has to be made for the present
calculations. It must also be noted that all
of these lakes are several orders of magnitude smaller than Lake Ontario.
The
largest, Lake Mendota, has an area of 39
km2 and a mean depth of 12 m, compared
with 1,600 km2 and 80 m for Lake Ontario.
Application of equation (3) to observations in the Black Sea (Khlopov
1958) also
yields values of K, smaller than 1 cm2/sec.
The results. however. show the same large
variability as my calculations for Lake Ontario using equation (3) and therefo’re are
of doubtful value.
CONCLUSIONS
The quasi-Lagrangian
diffusivity
equation appears to be a useful tool to estimate
a mean value for the vertical coefficient of
eddy diffusivity
in the thermocline of a
strongly stratified lake. Better and more
accurate methods may be available, but
special sampling programs would have to
be designed to collect the necessary data.
A major advantage of this method is that
it can give some insight into, diffusion
processes in a large lake as a byproduct
of heat or chemical budget studies.
Vertical diffusivity
of heat in a well-developcd thermocline
may, under quiet
weather conditions and during the heating
season, reach values approaching the value
of molecular co,nductivity.
Seasonal avcrages are much higher (0.02 to 0.07 cm2/
set) but are nevertheless well belo’w values
in the epilimnion
and confirm that the
thermocline effectively acts as a diffusion
floor throughout most of the summer.
REFEI3ENCES
CSANADY, G. T. 1964. Turbulence
and diffusion
in the Great Lakes.
Mich.
Univ.,
Great
Lakes Res. Div. Publ. 11, p. 326-339.
-.
1966. Dispersal
of foreign
matter by
the currents and eddies of the Great Lakes.
Mich. Univ. Great Lakes Res. Div. Publ. 15,
p. 283-294.
HUTCHINSON, G. E. 1957. A treatise on limnology, v. 1. Wiley.
KI-mopov, V. V. 1958. The variation of the diffusion coefficient
according to Black Sea observations.
Bull. Acad. Sci. USSR, Gcophys.
Ser. 2: 129-133.
SWEF;RS, II. E. 1969a. Two methods of describvertical temperature
distriing the “average”
bution of a lake.
J. Fisheries Res. Board
Can. 25: 1911-1922.
1969b.
Structure, dynamics and chem-.
Investigations
based
istry of Lake Ontario:
on monitor cruises in 1966 and 1967. Dep.
Energy,
Mines Rcsonr. Can., Mar. Sci. Br.
Rep. 10.
TULLY,
J. P., AND L. F. GIOVANDO.
1963. Seasonal temperature
structure
in the eastern
subarctic Pacific Ocean, p. 10-36.
In M. J.
Dunbar [ea.], Marine distributions.
Roy. Sot.
Can. Spec. Publ. 5.