Prog. Oceanog. Vol. 10, pp. 147-171.
0079-6611/81 0701-0147 $05.00/0
Pergamon Press Ltd. 1981. Printed in Great Britain.
A multi-parameter extension of temperature/salinity diagram
techniques for the analysis of non-isopycnal mixing
MATTHIAS TOMCZAK,JR.
Division of Oceanography, CSIRO Marine Laboratories, P.O. Box 21, Cronulla, NSW 2230,
A ustralia.
The technique of water mass analysis based on temperature-salinity curves is extended to
more than three water masses (or water masses defined by temperature-salinity curves rather than
points) by including one or more additional parameters and solving the equations of linear mixing
without additional assumptions. The oceanographical significanceof the result is studied by means of
examples. It is shown that the method is particularly useful for investigations of mesoscale mixing as
observed in frontal zones where interleaving and intrusions are commonly observed, and that it can be
used for obtaining bulk estimates of the impact of non-isopycnal, small-scale mixing events on the
mesoscale fields, Possible parameters are discussed and sources of errors reviewed. Examples are used
to show that major nutrients can be used as parameters to detect non-isopycnal mixing although
limitations in the historical data and low data accuracy lead to considerable error bounds at present.
It is argued that introduction of continuous flowanalysis techniques and continuous vertical profiling
in marine chemistry will greatly improve the potential of the method.
Abstract
1.
INTRODUCTION
SINCE HELLAND-HANSEN (1918) introduced the TS-diagram and JACOBSEN (1927) and
DEFANT (1929) subsequently devised methods for the determination of mixing parameters
from TS-curves, the analysis of the relationship between temperature and salinity in the
ocean has been a valuable tool for the study of mixing processes. WOST (1935) extended its use
by developing his "core layer method" and thus applying TS-diagram analysis to problems of
the oceanic circulation. A substantial part of our knowledge of the distribution and
circulation of the oceans' deep waters still relies on the analysis of TS-relationships, and this
situation is unlikely to change for some time to come.
A basic assumption of classical TS-diagram analysis is vertical layering of the water masses
involved in the mixing and circulation process. The best-known application is the "mixing
triangle" for the quantitative determination of the a m o u n t s of three water masses in a water
sample. As an example, observed temperature and salinity values from 11 "Carnegie" stations
(FLEMING, SVERDRUP, ENNIS, SEATON and HENDRIX, 1945), performed in 1929 in the eastern
N o r t h Pacific Ocean are plotted in Fig. 1. The emerging TS-curve can be interpreted as the
mixture of three progressively deeper water masses ST, SA and P B (MAMAYEV (1975), when
proposing this interpretation, identified P B as Pacific Deep and Bottom Water, SA as
Eastern N o r t h Pacific Subarctic Intermediate Water and S T as Eastern N o r t h Pacific
Subtropical Intermediate Water). Thus, although in principle every combination of
temperature and salinity within the mixing triangle is possible, the majority of the
observations can be expected to occur in the vicinity of the lines P B - S A and SA ST, and the
line P B - S T is only obtained as the theoretical limit where the volumes of both P B and S T are
infinitely large c o m p a r e d to the volume of SA, and the time elapsed since the water masses
came into contact with each other is also infinitely large. Ivanov (whose work became k n o w n
148
M. TOMCZAK
outside the USSR through the book by MAMAYEV(1975)), SHTOKMAN (1946) and later
MAMAYEV (1975) developed the theory of TS-curves on the basis of vertical layering and
extended it to an arbitrary number of water masses.
Soon after the development of the basics of TS-analysis it became obvious that in large
ocean areas nearly linear TS-relationships do not result from mixing between two vertically
layered homogeneous water masses. ISELIN(1939) pointed out that in fact TS-diagrams of
stations in the Atlantic Ocean at mid-latitudes have a striking resemblance to the TScorrelation of the surface layer of the North Atlantic Ocean in winter. This followed the
studies by MONTGOMERY(1938) and PARR (1938) who applied the technique of isentropic
analysis developed in meteorology (which in the ocean can be replaced by isopycnal analysis
as a good approximation) to problems of physical oceanography. Their hypothesis that
lateral mixing is much more effective than vertical mixing, contradicts the assumption of
vertical mixing between vertically layered water masses which forms the basis of the mixing
triangle; this was indeed explicitly pointed out by Parr. Both authors were able to explain the
distribution of temperature and salinity below the surface layer to depths of several hundreds
of metres over large ocean areas.
The importance of lateral mixing was thus clearly established. The consequences for
classical TS-diagram analysis were, however, less dramatic than might have been anticipated.
One obvious consequence is that a linear TS-relationship obtained at mid-depth (or, in fact,
any reasonably smooth TS-curve without inflection points) may simply reflect the TSrelationship at the ocean surface along a path across the region of origin; under such
circumstances, no deep mixing is necessary to explain observed temperatures and salinities,
and the whole mass of water characterised by the observed TS-curve should be considered a
single water mass. Consequently SVERDRUP, JOHNSONand FLEMING(1942) defined the central
water masses of the oceans (i.e. the water masses found between the surface layer and about
1000 m) by TS-curves rather than points. In Fig. l, as an example, the line between SA and S T
20
"C
1SOre. '
ST.
15
lO S
A
51
0 33.5
J
2500m
l -34.0
I
345
L ~o
35"0
FIG. 1. TS-diagram of stations 133-140 and 147-149 from cruise VII of"Carnegie", 1928-1929, in the eastern
North Pacific Ocean. Approximate depths of observations are indicated. For details of the three water masses PB,
SA and S T see text.
Temperature/salinity diagram techniques
149
is one representation of Eastern North Pacific Central Water {ENPCW), and all points along
that line represent "pure" ENPCW. Of course, in any vertical section, temperature and
salinity decrease downwards, and TS-pairs obtained in "pure" ENPCW are arranged in a
monotonic pattern along the definition curve. As a consequence, Pacific Deep and Bottom
Water (PB) underlying ENPCWcan only mix vertically with the SA end of the TS-range of
ENPCW. This shows that the basic assumption of the mixing triangle still holds, but the
splitting of E N P C W into a mixture of SA and ST now has to be seen as a convenient formalism
for the calculation of mixing between PB and ENPCW rather than a description of the
hydrographic situation.
With this restriction, TS-diagram analysis has continued to be useful in a variety of
oceanographic situations, many of which involve more than three water masses. Whenever
four or more water masses are involved, the role of the assumption of vertical layering
becomes even more crucial. In the case of mixing of three water masses there exists a unique
solution to any combination of temperature and salinity within the mixing triangle for the
percentage contribution of the three water masses involved, and it is only when the attempt is
made to calculate mixing coefficients (or, more precisely, the quantity A,,/(pu) where A,, is the
coefficient of vertical mixing, p mean density and u the velocity of the centre layer relative to
the upper and lower layers) that the assumption of vertical layering has to be made. In the
case of four or more water masses there exists a whole range of solutions, and the assumption
of vertical layering is necessary to select a particular solution as the oceanographically most
plausible one. As an example, consider the TS-diagram shown in Fig. 2 (ScHMIDX, 1929).
North Atlantic Central Water (NACW) is represented by the heavy line. Mediterranean
Water enters the region with temperature and salinity values indicated by M, and Atlantic
Deep and Bottom Water is represented by AB. Because of its high density, Mediterranean
Water intrudes into the region between the Deep and Bottom Water and North Atlantic
Central Water and gradually mixes vertically with ,4B and the lower end of NACW. This
2O
"C
10
A6
0
351.0
35L.5
~
36'0
I
36'5
~,,,
FIG. 2. TS-diagram of station 1883 from the "Dana" expedition, 1922, in the eastern North Atlantic Ocean.
Approximate depths of observations are indicated. For details of the water masses see text.
150
M. TOMCZAK
vertical layering forms the basis for a TS-diagram mixing analysis which divides the TSrange spanned by AB, M and N A C W into two triangles by artificially representing N A C W
through the upper and lower TS-combinations C W 1 and C W 2. A similar example for the
South Pacific Ocean can be found in MAMAYEV (1975).
Following HELLAND-HANSEN'S(1918) original notation, we shall call a combination of
temperature and salinity such as C W 1 or C W 2, which can be used to formally represent one
end of the TS-curve of a water mass, a water type. The method of breaking down a given area
on the TS-diagram into a set of mixing triangles can be extended to any number of water
types if only three of them can have simultaneous contact at any point in space (MAMAVEV,
1975), In most oceanic situations this condition is met because the corresponding water
masses are layered vertically, and mixing between them is predominantly vertical. However,
use of continuously profiling S T D (salinity/temperature/depth) equipment has supplied
evidence that this is not always the case. Figure 3 shows a TS-diagram obtained with such an
instrument in the north-east of the Central Atlantic Ocean (BROCKMANN,HUGHES and
TOMCZAK~1977). Two water masses were observed at the station, South Atlantic Central
Water ( S A C W ) and North Atlantic Central Water ( N A C W ) , but they were obviously not
vertically layered, in the strict sense of the term, i.e. with one water mass completely overlying
the other. Instead, S A C W o c c u r s as a layer which intrudes into N A C W a t mid-depth, making
it impossible to construct mixing triangles by assigning water types to both water masses.
Such a situation is possible whenever the density ranges of two water masses overlap and it
does occur quite regularly under such circumstances.
The mechanism by which these intrusions develop is easily understood today. Because
both water masses have a certain density range in common, they can coexist laterally over
that range without becoming statically unstable. Temperature and salinity, on the other
hand, differ sharply at any particular density. As a result, a temperature and salinity front
exists between the two water masses. The situation is illustrated in Fig. 4 which shows that
mixing occurs in an inclined frontal zone between the water masses, with no perturbation of
the density field.
18
16
SAC
14
~'~NACW
c,
3510
355
S
~
~
~
2,0
10
6],'S , T
8
: . . :
6
35 0
35 5
360
:/~o
36.5
FIG. 3. TS-diagramof station 3647 from cruise 36 of"Meteor", 1975,in the North-Eastern Central Atlantic Ocean.
SACW: South Atlantic Central Water; NACW: North Atlantic Central Water. The inset shows the distribution of
water masses calculated on the assumption of isopycnal mixing. The data are processed STD data.
Temperature/salinity diagram techniques
Station 1
Station 2
151
Station 3
Water
,~s
Slatlon 1
S
( 3 - , ~ s ~ . ~ c o n sAt
•J
Station
,~:?S'o'
S
T
Water
m
s
IStahon
3
S
O't
....
T
........
S
FIG.4. Two water masseslayeredhorizontally.Outside the frontal zone isothermsand isohalinesare horizontal and
coincide with isopycnals.The composition of the mixture of water in the frontal zone is calculated along isopycnals.
If it is assumed that mixing in the frontal zone is strictly along isopycnals, it is possible to
devise a method which provides an alternative to the method of successive mixing triangles
described in Fig. 2. For any given TS-combination, the percentage of the two water masses
involved is calculated from the distances along the isopycnal to the intersection with the TScurve of the water mass. The procedure, which does not involve artificial representation of
water masses by water types, is illustrated in Fig. 4 and an application demonstrated in Fig. 3.
TOMCZAK and HUGHES (1980) used this method for an analysis of the water masses in the
Canary Current upwelling region, and their results were in good agreement with the
distribution of water masses inferred from current measurements.
It has been observationally established by WILLIAMS (1975) that TS-fronts of the type
sketched in Fig. 4 are areas of active double diffusion. This process, also known as "salt
fingering", is a very efficient way of vertically mixing heat and salt (TURNER,1978). Its
efficiency is greatly enhanced because of interleaving of the water masses at the front as a
result of current shear. Figure 5 illustrates the effect with an example from the set of stations
(BROCKMANN,HUGHESand TOMCZAK, 1977) used by TOMCZAKand HUGHES (1980). At least 10
inversions of salinity can be seen in the upper 400 m, and salt fingering can be expected to be
present on all of them. The obvious question is: to what extent is it justified to apply isopycnal
TS-diagram analysis to frontal zones if vertical transport of properties at different rates can
be expected to be important along the interfaces?
JOYCE (1977) proposed a model of turbulent mixing in the ocean based on STERN'S (1967)
concept of three scales of motion near a thermohaline front. The large-scale field, forced by
air-sea interaction, is linked with the formation of the TS-curves of water masses and, as far
152
M, TOMCZAK
SACW/°~J"/_
:-%00
oi'~o~
14
,
12
10
i
o0
Z65
Auftrieb
1975
R.R.S. D i s c o v e r y
o
s t a t i o n nr 8 6 9 3
'750
635.0
3~5
l
36.0
%
36"5
FIG. 5. TS-diagram of station 8693 from cruise 69 of"Discovery", 1975, in the North-Eastern Central Atlantic
Ocean, with depths of observation indicated. SACW: South Atlantic Central Water. N A C W : North Atlantic
Central Water. The data are processed STD data, corrected for different response times of conductivity and
temperature sensors. The apparent instabilities in a, reduce to neutral stability in a,.,.p.
as the water masses are layered vertically, with the mixing triangle and its combinations. The
medium-scale field, governed by interleaving and intrusions, is controlled by lateral
(isopycnal) mixing and linked with isopycnal TS-diagram analysis. The small-scale mixing is
driven mechanically and by double diffusive mechanisms and is effective across isopycnals;
since its TS-coherence can be quite low, no particular TS-diagram technique has been
developed for it yet, and it may be impossible to devise one at all.
The controlling parameter of the intensity of salt fingering is the density ratio
R = (~AT)/IflAS) where ctp =-Op/?~T, tip = Op/OS, p is density and AT and AS are
temperature and salinity increments along the TS-curve. IN,HAM (1966) noted that TScurves often fit curves of constant density ratio better than straight lines, which leads SCHMITT
(private communication) to suggest that double-diffusive mixing is important for the
medium-scale field as well. A method for estimating the bulk effect of small-scale nonisopycnal mixing on the medium-scale field, which is required to test this hypothesis, does not
exist.
The present study proposes a generalisation of TS-diagram techniques for the analysis of
any number of water masses, which does not involve assumptions such as vertical layering or
isopycnal mixing. When it is applied to a set of observations and its results are compared with
the results of classical methods it is possible to detect isopycnal mixing and deviations from it.
The method may therefore prove useful for a detailed study of the relative importance of
mixing at the different scales and for the formulation of a generalised TS-diagram analysis
which includes the techniques of the mixing triangle and of isopycnal mixing as two limiting
cases.
The essence of the method and of interpretation of results is contained in the following four
sections which, together with the conclusions, probably cover the interests of most readers.
For those readers wishing to apply the method to their own data, a detailed discussion of
possible parameters and of error sources is included.
Temperature/salinity diagr:lm Icchniques
153
2. T H E M E T H O D
The method is a simple extension of the set of equations which forms the basis of the mixing
triangle. Consider a situation where a certain number of water masses which can be
represented by n water types contributes to the mixture of water encountered at an
oceanographic station. Assume further that n-1 characteristic and independent parameters
are known for each of the water types and readings are obtained for the same n-1 parameters
from water samples at the station. Then, assuming identical exchange coefficients for all
properties, the relative contribution of any water type to the waters at the station can be
determined from the linear system of equations
Ax = B
(1)
Where A is the n × n matrix of the parameter values for the n water types, B is a vector of 17
elements which contains the n-1 observations, and x is a vector of n elements which gives the
relative contributions of the water types. The elements of the last row in A and the last
element of B are identically 1, to express the condition that all contributions must add up to
100 ° o of the observed volume of water.
An obvious example is the mixing triangle which is obtained by setting n = 3 and choosing
temperature and salinity as independent parameters. If (T~,Si) are the values of temperature
and salinity of water type i (i = 1, 2, 3) and (Tobs; Sobs) are observed temperature and salinity,
the system of equations (1) becomes
x 1 T 1 q.- x 2 Z 2 -~- x 3 Z 3 = Tobs
x 1 S 1 71- x 2 S 2 + x 3 S 3 = Sob s
(2)
X 1 q - X 2 -[-X 3 = 1
If more than three water types are involved, additional parameters have to be measured. It
should be noted that the method is not completely free of assumptions. The necessity to
assume identical exchange coefficients seems to exclude application to problems involving
double-diffusive processes; this problem will be addressed again in a later section. Note,
however, that with the determination of values for n-1 parameters the system of equations (1)
results in a unique solution irrespective of the type of mixing involved. I now proceed to
explore the possibilities of the method, restricting attention mainly to the case of two water
masses represented by four water types (which is the case of Figs. 3-5) and one example of a
TS-diagram which involves five water types. As will be seen, the first case corresponds to the
situation often encountered at oceanic fronts and can probably be regarded as an example of
the majority of applications for the method. The second case corresponds to an intrusion of
coastal water at the shelf break.
3. A P P L I C A T I O N T O A QUALITATIVE TEST O F ISOPYCNAL M I X I N G
An application which gives a qualitative idea whether an observed water mass stems from
isopycnal mixing or not is the following. Consider the mixing of two water masses, for
example South Atlantic Central Water (SACW) and North Atlantic Central Water (NA CW)
as shown in Fig. 3. If we restrict our analysis to the density range of 26.473 4 a t ~< 27.160, the
TS-curves of both water masses can be approximated by straight lines which can then be
154
M. TOMCZAK
represented by four water types. For the evaluation of an example, let these representations
be 7"1 = 18.65°C, S~ = 36.760%° for the upper end of the N A C W curve and T2 = 11.00°C,
S 2 = 35.470°/°0 for the lower end; likewise, T3 = 15.25°C, S 3 = 35.700°/°0 and T4 = 9.70°C,
S 4 = 35.177%o for SACW. Assuming that mixing is isopycnal we can then determine the
relative contributions of S A C W and N A C W to water of any TS-combination within the
range set by the four water types, and from the position of the intersection of the
corresponding isopycnal with the straight lines which represent S A C W and N A C W , we can
formally attribute contributions to the water types such that the sum of two of them
represents the contribution of one water mass. The method is illustrated in Fig. 6 where water
of T,,bs = I 1.97°C, Sob~, = 35.575%o is seen to contain 40 °/o SACW, represented by 20'!~o of
water type 3 and 80 °/oof water type 4, and 60 o(, N A C W , represented by 22.5 °/o of water type 1
and 77.5 °/o of water type 2 (the difference of 22.5 !?,/,against 20 ?/obeing due to the nonlinearity
of the equation of state).
A qualitative test of the validity of this isopycnal analysis can be made if definition values of
parameters other than temperature and salinity can be obtained for the water types. It can be
shown that phosphate can be used as such a parameter for S A C W and N A C W ; the definition
values are P1 = 0.00, P2 -- 1.19, P3 = 1.50 and 104 = 2.30/tg-at//. (A full discussion of the
derivation of these definition values is given in TOMCZAK, 1981). Given the contribution of the
four water types to any allowable TS-combination as derived from isopycnal TS-diagram
analysis above, we can calculate from the equation for phosphate of system (1) the phosphate
content at any particular TS-combination that corresponds to conditions of isopycnal
mixing and draw the corresponding isophosphates over the allowable TS-range. This is
conveniently done on a grid which transforms the range of temperatures and salinities in the
TS-diagram into a rectangular grid of water mass content against density as shown in Fig. 6.
Observed phosphate values can be plotted on the same grid and the resulting isophosphates
compared with the theoretical distribution. An example is given in the same figure.
It will be noted that this method is to some degree a statistical one, as usually a certain
number of stations is necessary in order to obtain enough observations to adequately cover
the range of densities and water mass distributions. This in itself can be an advantage because
it reduces the inevitable errors involved in the procedure. However, reasonable results can be
achieved with not too many stations. The example shown in Fig. 6 is based on 18 out of 24
stations, with phosphate data estimated to be precise to +0.1 ~tg-at//. The differences
between the theoretical and observed patterns are probably not significant if all possible
other errors are taken into account, and we are led to the conclusion that mixing of S A C W
and N A C W in the area of observation is isopycnal to a first approximation. On the other
hand, 6 of the 24 stations had to be omitted when contouring the phosphate distributions
because they did not fit into the general pattern; phosphate values differed by up to 1.2 #g-at//
from expected values (ToMCZAK, 1981 ). This shows that the method is capable of detecting
deviations from isopycnal mixing and suggests that such deviations exist in certain areas of
the oceans.
Note also that this qualitative test can be applied to situations where the definition curves
for the water masses deviate from straight lines. In that case no formal breakdown into water
types is possible, but as the water masses are uniquely defined in parameter space, parameter
values for pure N A C W and S A C W at the density of the observation are known, and the
calculation of relative water mass contributions and expected phosphate content proceeds as
before. The approximation of definition curves by straight lines for all parameters was made
here in anticipation of the following discussion where it is indeed crucial.
35 0
1o
D.C
gLZ,.,~°
,4-/-w,~
o\o /
SAC
-
i
360
I
'
~NACW
%:,
0 NACW 20
40
60
27.2IW14 i ~
80
~
610
40
,
80
%
26-8
100%
2641
100
,
,
40
I > ,
60
80
O NACW 20
J "I
0 75
20 SACW O
.
60
,
40
FIG. 6. An example for a qualitative test ofisopycnal mixing. Left: definition of water masses S A C W and N A C W
a n d w a t e r t y p e s ( W T ) 1 4.Centre:the•retica•distributi•n•ftestparameterph•sphate.Right:•bservedph•sphate
distribution based on 18 out of 25 stations from cruise 36 of "Meteor", 1975.
1"01
WT 3
J
26-6
100%
,,5
"
ao
O/o 100
"10
\
,
20 SACW0
U.
e~
_=.
,7
,H
156
M. TOMCZAK
4. A P P L I C A T I O N T O A Q U A N T I T A T I V E D E T E R M I N A T I O N O F WATER MASSES
IRRESPECTIVE O F MIXING C O N D I T I O N S
In order to quantify deviations from isopycnal mixing conditions, the system of equations
(1) has to be solved for individual stations, using n-1 parameters for n water types involved.
The result can then be compared with the solution that corresponds to isopycnal mixing and
conclusions drawn on mixing length scales across isopycnals if certain additional
assumptions are made. The procedure is again illustrated best with an example.
We return to Fig. 6 where, as we have seen, a mixture of water containing 60 "/o N A C W and
40 °~~,,S A C W could be characterized by the combination To~S = 11.97°C, So~~ = 35.57_5o/oo,
P,,h~ = 1.41 ~tg-at/I (which, of course, represents only one possible T S P - c o m b i n a t i o n out of
many others that correspond to a 60 %-40,',o mixture). Table 1 lists the formal breakdown
into the four water types and contrasts it with the result obtained for the combination
11.97°C, 35.575°/°° and 1.20 pg-at// which has the same temperature and salinity and in the
TS-diagram is thus represented by the same point. In order to establish the possibilities and
limitations of the method it is important to understand the differences between both sides of
the table.
In the isopycnal case, the water is composed of 60'I o N A C W and 40'10 S A C W . This is
formally represented in the solution by a break-down into percentages of the four water types
which correspond, in a TS-diagram, to the lengths of the TS-characteristics between the TSpoints of the alternate water type and the intersection with the isopycnal of the observation
point icompare Fig. 6 and note that 13.5/0.60 = 22.5 etc.). In the non-isopycnal case, the
percentages of the four water types obtained from the equations constitute the result of the
analysis and not a formal breakdown a posteriori. Since the only way to obtain pure N A C W
is to mix WT~ with WT~, and likewise to mix W T 3 with W T 4 for S A C W , it is clear that the
N A C W content of the water mass is given by [ W T 1 ] + [WT2] - t h e brackets stand for
"percentage of" --and the S A C W content by [WT3] + [WT41. The water mass content is
thus obtained by summation over the content of its water types. Again, a graphical
representation in the TS-diagram is possible, but interpretation is not as straightforward as
in the isopycnal case.
Figure 7 compares the graphical representations of the two examples. In both cases, the
TS-characteristic of N A C W and S A C W is divided between W T 1 and W T 2 or WT3 and W T 4
in accordance with their respective percentages as given in Table l. The division points A
and A 2 (the points of the isopycnal case A) fall on the isopycnal which cuts through the
observation point. The division points B~ and B 2 of the non-isopycnal case B coincide with
(1} FOR THE MIXING OF S A C W *'tNI)
N A C W (EXPRESSED1N % OF VOLUME)
TABLE I. TWO SOLUTIONS OF EQUATIONS
Water
mass
Water
type
Isopycnal [A)
T = 11.97°C
S = 35.575°/°°
P = 1.41#g al/l
NACW
WT
WT z
13.5/
46.5J 60
fWT~
Non-isopycnal (B)
T = 11.97°C
S = 35.575%°
P = 1.20,ug-at//
6.3}
77.7 84
Temperature/salinity diagram techniques
157
°C
18
16
14
12
SACW
D'~ C'
16'~,
10
wT4
I
35.0
I
I
I
I
36-0
I
I
%0
FIG. 7. Graphical representation of two solutions (A) and (B), based on the assumption that the observed water
properties are the result of mixing between pure NACW and pure SACW. (A) is the isopycnal solution (compare
Table 1).
the end points of a line which cuts through the observation point and is divided by this point
as 84:16 (the relative amounts of N A C W and S A C W in the sample), but they do not fall on
the same isopycnal. The conclusion suggested by the figure is that in example B S A C W of
density a t = 26.67 mixed with N A C W of density a t = 27.13 at a ratio 16:84. Note, however,
that this argument is based on the graphical representation of the solution which in contrast
to the simple figures of Table 1 is based on an important assumption: that the water at the
point of observation was formed by direct mixing of pure N A C W and S A C W . Other
graphical representations are possible if this assumption is relaxed.
For example, starting from the solutions of Table 1, a series of mixing events could be
constructed in which only parts of the respective volumes of the four water types are mixed
successively until all available contributions are exhausted. The result would be identical to
mixing B x and B 2 but this time the last mixing event which results in point B would be the
outcome of a much smaller non-isopycnal mixing event. It could even be an isopycnal mixing
event if all previous steps happened to end up on two points on the line A IA2. Likewise, the
solution represented by A 1 and A 2 can be taken as the end product of a similar series of
mixing events, some of them isopycnal, some non-isopycnal, the only condition being that at
the end of all events the contributions from the four water types add up to the relative
volumes of Table 1.
It is seen from these considerations that the solution to equations (1) does not represent the
local mixing event which created the water represented by the observed parameter values, nor
can the graphical representation of Fig. 7 be taken as an illustration of that event. Within the
frontal zone, pure S A C W and pure N A C W may be nowhere present in the vicinity of the
observation point and a local mixing event would not involve S A C W and N A C W in pure
form. Multi-parameter mixing analysis forces a solution which is based on the mixing of pure
S A C W and pure N A C W as they exist outside the frontal zone, i.e. it constitutes a bulk
158
M. TOMCZAK
analysis by tracing the original components in a water body back to the beginning of the
series of mixing events. The individual events may be isopycnal or non-isopycnal; the
analysis does not help to resolve this but it can indicate what the bulk effect of the series of
events is. Provided the effect of non-isopycnal events such as salt fingering on density is
unidirectional, the effects of successive mixing events on density cumulate and the method
provides a bulk measure for the degree of non-isopycnal mixing.
The ability of the method to discriminate between isopycnal and non-isopycnal mixing is
invariant under changes in the definition of reference water types. This is an important
property because in practical applications the definition of the water types is somewhat
arbitrary and results for their relative contributions depend strongly on their choice. In the
example of Fig. 7, W T I - W T4 were chosen in a way to extend the linear representation of
SACW and N A C W over the maximum possible range without introducing important
deviations from the actual curves. Moving the definitions of WT1-WT4 up or down the lines
which represent SACW or N A C W does not change the locations or inclinations of the
solution lines A~A 2 and BIB z (provided, of course, that the phosphate definition values of
WT~-WT, are adjusted in correspondence with the changes in the temperature and salinity
definitions), and [SACW] and [NACW] given by
[WTi] + [WT2] = [ N A C W ]
(3)
[WTs] + [WT4] = [SACW]
remain the same although the individual contributions vary. In principle, W T t - W T4 could
be defined independent of SACW and N A C W anywhere and the solution lines A1A 2 and
BIB2 would still remain unaffected as long as the new definition values of all parameters were
adjusted proportionally ( [SACW] and [NACW] would not have any meaning in that case).
In practice it proves difficult to define water types without reference to physically existing
water masses, and a definition which uses [SACW] and [NACW] as invariants makes
interpretation of results easier.
5. ON THE POSSIBILITY OF A QUANTITATIVE DETERMINATION OF
NON-ISOPYCNAL MIXING
The last section describes how to calculate "true" percentages of n water types for a
particular mixture from a single observation of n-1 parameters. I now apply the technique to
a series of oceanographic stations. This will reduce the ambiguities involved in the graphical
representation of the results.
An oceanographic station is a set of observations, arranged in order of increasing density,
to which the above analysis can be applied individually and the result plotted in a TSdiagram as before. Figure 8 gives an example (ToMczAK,1981 ). It shows the analysis, for the
depth range 150-500 m, of three stations in the Canary Current area. As before, the system of
equations (1) is solved for n -- 4 with temperature, salinity and phosphate as parameters and
the definitions values given earlier. TS-points for 100 m depth are included for later reference
but not analysed. The graphical representation assumes mixing of pure SACW and pure
N A C W as before, and the relative amounts of SACW and N A C W are therefore expressed
inversely by the lengths of the segments of the lines through the observation points.
Endpoints of lines outside the range defined by WT1-WT4 (for example at station 3659,
400m) signify a negative solution for the adjacent water type caused, for example, by
Temperature/salinitydiagram techniques
159
inaccurate measurement; these effects are discussed in the following sections. It is seen
that at station 3658 the major part of the water is of N A C W origin while S A C W is the main
source of water at station 3660/1.
The stations shown in Fig. 8 are situated in the frontal zone between N A C W and S A C W
off north-west Africa. The distance between 3658 and 3659 is approximately 30 km; between
3659 and 3660/1 it is close to 20 km. Clearly, important changes in mixing conditions occur
over these distances in a frontal zone. Station 3660/1 apparently is very close to isopycnal
mixing conditions, apart from an observation at 150 m depth where reliable determination of
N A C W components is difficult because N A C W content amounts only to 7°/Jo. Likewise,
deviations at station 3658 from isopycnal mixing are not very large if results from 150m
depth where S A C W content is 8 % only are disregarded. In contrast, analysis of station 3659
results in a composition of the waters between 200 and 400 m depth which definitely cannot
be explained by isopycnal mixing. We know that deviations from isopycnal conditions in a
single diagram do not necessarily indicate local non-isopycnal mixing. However, by
comparison with adjacent stations we are able to detect changes in mixing conditions and
provide evidence that non-isopycnal mixing at station 3659 is likely to occur locally within an
area of less than 30 km diameter.
The TS-diagram as used in Fig. 8 is a useful way of showing deviations from isopycnal
mixing. Its disadvantages are that reading off the relative amounts of N A C W and S A C W is
tedious and errors introduced by small admixtures of water masses are magnified
graphically. These drawbacks are eliminated by use of the density/percentage graph
introduced in Fig. 6. Figure 9 gives the results for the three stations discussed above in these
coordinates, including observations at 100m depth. The relative amounts of N A C W and
S A C W as determined from the system of equations is shown by circles which are connected
by a heavy line, and the individual results for the water types which make up the solution are
shown as light lines. The result ofisopycnal analysis is indicated for comparison and given by
the broken curve for the water masses; in addition, shading indicates the corresponding
WT~
18
~c
J
16
WT 3
a
: lOOm
a, 150rn
b ~ 200m
c: 3 0 0 m
e WT2
I
35'0
35.0
I
36"0/~35'0
36-0
36~.0
d
400m
e
500m
I
I
%0
FIG.8. Watermass analysisforstations 3658,3659and 3660/1fromcruise36 of"Meteor", 1975.Labelledpoints are
observations, and lines through points are derivedfrom the assumption that water at all levelsis formedby direct
mixing of pure NACW and pure SACW.
160
M. TOMCZAK
O-t ~ ~ , .
3659
3658/
26.6
NACW
26.8
\
_
~
SACW
///
1//
3660/1 O-t
266
SACW
268
27.0
~ / Nt ~~ t t W
0% 20 40 60 80 100 0% 20 40 60 80 100 0% 20 40 60 80 100
FIG. 9. Another representation of the water mass analysis of Fig. 8. Two solutions are shown, both represented by
the relative percentages of (from left to right) W4and W3 (which sum up to S A C W ) and W1 and W2 (which sum up to
NACW). The isopycnal solution is: W, = left shaded area, W3 = area between W4 and broken curve, WI = area
between broken curve and W2, W2 = fight shaded area. The solution of multi-parameter analysis is: W, = area to
the left of the first full curve, W3 = area between the first fall curve and the heavy curve, W1 = area between heavy
curve and second full curve, Wz = area to the right of the second full curve. Observation depths are 100, 150,200, 300,
400 and 500m.
formal decomposition into water types. U n d e r perfect isopycnal conditions, and given errorfree measurements, the light curves should coincide with the boundaries of the shaded zones
and the heavy curve with the broken curve.
The presentation of Fig. 9 shows the actual percentages of the water masses involved more
clearly than the TS-diagram of Fig. 8, and it offers better possibilities for estimating error
bounds. As an example, let the result be accurate to + 15 ~ water mass content. F r o m Fig. 9 it
is then deduced that conditions at stations 3658 and 3660/1 do not differ significantly from
isopycnal conditions but they do at station 3659 because differences between the heavy and
the b r o k e n curves reach up to 32 ~o at that station. A disadvantage is the difficulty in handling
negative but still meaningful solutions. We shall discuss this in detail when turning to the
discussion of error estimates.
In order to arrive at quantitative estimates of the effect of non-isopycnal mixing, combined
use of both representations seems to be the most appropriate way. Once the significance of
changes from isopycnal to non-isopycnal conditions is established from Fig. 9, Fig. 8 gives
estimates in terms of density: F r o m stations 3659 and 3660/1 it is deduced that a range of a, of
+0.1 corresponds to the assumed +159/oo range in water mass content. The density
deviations observed at station 3659 are __+0.2;hence, the m i n i m u m effect that occurred over a
range of 2 0 - 3 0 k m is mixing of water across density surfaces over a range of +0.1 a, or
+ 100 m. Such a calculation, of course, has to be substantiated by further evidence and by
investigation of other possible courses, such as advection normal to the section along which
the stations were taken (ToMCZAK, 198 1). It shows, however, that estimates of the relative
importance of non-isopycnal mixing processes can be obtained with the method.
Does the assumption of identical exchange coefficients for all parameters invalidate the
results ? Small-scale turbulent mixing is driven by the difference in molecular diffusivities of
heat and salt, but multi-parameter mixing analysis is not applicable to that scale. The effect of
large-scale turbulent diffusion is reflected in the shape of TS-diagrams which would be Sshaped curves, with temperature approaching the final value from both ends m o r e rapidly
Temperature/salinitydiagram techniques
161
than salinity, if the coefficient for large-scale turbulent diffusion of heat were detectably larger
than the corresponding one for salt. S-shaped TS-eurves in the ocean are, however, quite rare.
In terms of STERN'S (1967) scales of motion it can probably be said that diffusivities of
temperature and salinity are of equal magnitude for the large-scale processes but differ at the
small-scale end. What the situation is at the intermediate scale is not known, but it seems
reasonable to assume that isopycnal stirring is the dominant process, not only on a large scale
but also in a frontal zone, where it is coupled with sporadic, non-isopycnal processes which
gradually transform waters from one set of characteristics to another. When devising a
method for estimating the bulk effects of the sporadic, small-scale processes on the mediumscale field, it seems justified to extend the situation known to exist at large scales into the
medium scale and to see whether the results are consistent with our ideas of the physics of
oceanic mixing.
6. POSSIBLE PARAMETERS
As the analysis of n water types involves n-1 parameters, determination of temperature and
salinity alone is insufficient for analysis of more than three water types, and additional
parameters have to be found. They must be conservative and independent; conservative
because they must conserve their value under all processes other than mixing, independent
because any parameter which is a linear function of another parameter causes the
determinant of the system of equations (1) to become identically zero. Furthermore, since
every measurement involves errors and the effects of errors in the measurements of several
variables accumulate in the solution, parameters should preferably have a high
range/precision ratio, i.e. the range over which values vary should be large compared to
observational precision.
Temperature and salinity are ideal parameters because they are definitely conservative and
independent and have an extremely high range/precision ratio of 103: Temperature typically
varies over 10°C in mixing diagrams and can be measured to + 0.01 °C while salinity spans a
range of 1°/oo and can be measured to _+0.001°/oo. Both will therefore continue to be used.
Other parameters are conservative to some degree only, and some of them offer very poor
range/precision ratios today, causing error bounds which render the solution practically
meaningless, although this might change in the future with the introduction of new methods.
Therefore, a choice has to be made, before applying non-isopycnal mixing analysis, of the
proper parameters and the appropriate definition of the water types. This section discusses
possible suitable parameters and considers some ways of defining water types properly.
A starting point for the search for reasonably useful parameters is that they should also be
readily available. This may cause us not to choose the "'most ideal" parameters in the sense of
the above-mentioned criteria but it offers the opportunity to start applying the method to
existing data. With these considerations in mind, obvious candidates are the nutrients like
phosphate, nitrate, silicate etc. Clearly, they are not conservative, but just to what extent they
can be regarded as such determines their use for the present method. In the surface layer,
biological activity causes changes in nutrient content of the water to such a degree that this
area has to be excluded from the analysis. Since both temperature and salinity also undergo
changes not related to mixing in the surface layer which leads to a breakdown of classical TSdiagram analysis in that area, the restriction imposed by increased biological uptake and
release of nutrients near the surface is not very severe, although first experience with the new
method in a coastal upwelling area suggests that the effect reaches somewhat deeper [to
162
M. TOMCZAK
about 100-150 metres) than the influence of the atmosphere on temperature and salinity, and
thus the upper limit of applicability is defined by the nutrients. This is confirmed by studies of
the changes of nutrient content in the surface layer in the same area (TREGUERand LE CORRE,
1979; FRIEDERICH and CODISPOTI, 1979).
Below the surface layer, the distribution of nutrients generally reflects the deep oceanic
circulation, indicating quasi-conservative behaviour, i.e. dominance of advective-diffusive
processes over biochemically induced concentration changes. It is difficult to quantify the
individual effects at present, but it is probably reasonable to say that changes due to biochemical activity cannot be neglected on oceanic scales. They do not seem to pose a
problerri, however, on intermediate scales of the order of some hundred kilometres which is
quite sufficient for a study of the major oceanic fronts if the characteristics of the water types
are defined in the vicinity but well outside the frontal zone. Such a procedure, which has been
shown to be successful (ToMcZAK, 1981 ), avoids another major problem which occurs with
nutrients, viz. the search for definition values of the water types at their place of origin. The
origin of most water masses of the oceans is known today, and characteristic temperatures
and salinities for the areas of origin were already given by SVERDRUP,JOHNSON and FLEMING
(1942). Corresponding values for phosphate, silicate or any other important chemical tracer
are much harder to obtain, and it may take some time before they become available. A few
stations performed on both sides of the frontal zone, well outside the area where mixing is
expected to be important, can supply an ad hoc definition of water types on scales where
nutrients are very nearly conservative.
Apart from these considerations which apply to all nutrients, there are some other points
which have to be taken into account when a tracer is chosen for an analysis. Different tracers
may be useful for different purposes. Values for dissolved silicon (or silicate) for example vary
from almost zero to more than 20 /~g-at// in the upper 1000m of the ocean, and can be
determined to a precision of + 0.1 #g-at/l, yielding a range/precision ratio > 2 x 10 z. This
compares very favourably with phosphate which ranges from zero to about 3 ~g-at/1 while
precision in its determination is of the same order of magnitude resulting in a range/precision
ratio of < 102. On the other hand, data on silicate are still sparse in many ocean areas, and
some older data are unreliable. Phosphate may therefore serve as a useful parameter despite
the above shortcomings because more historical data are available and can be included if
statistical treatment is envisaged. Nitrate seems to be a good compromise between the two,
with a historical data base nearly as good as the data base for phosphate and a range from
close to zero to more than 25 #g-at/l, at about the same precision of determination.
Other possible parameters are, of course, artificial tracers. Their presence is only
temporary which makes ad hoc definition of water types compulsory. Most of them can only
be used for studies on rather restricted scale, but tracers such as radiocarbon have been
introduced on a global scale and may be useful in future studies. A definite advantage of
radiocarbon and similar tracers is that they are not influenced by biological processes to any
important extent and therefore are truly conservative if their rate of natural decay (which is a
known quantity) is properly allowed for.
Finally, it should be pointed out that nutrients are independent parameters with regard to
temperature and salinity but not necessarily in relation to each other. Thus, in a situation
which involves more than four water types and, as a consequence, the use of more than one
nutrient parameter, care has to be taken that nutrients are selected which are truly
independent of each other. As an example, a linear relationship exists between nitrate and
phosphate which holds for all oceans (REDFIELD~ KETCHUM and RICHARDS, 1963), and both
Temperature salinity diagram techniques
163
nutrients cannot be used simultaneously. The concentration of silicate, on the other hand,
varies greatly in relation to phosphate and nitrate content in different water masses, and a
combination of silicate and either phosphate or nitrate can be used in the analysis of mixing
of five water types.
7. SOURCES OF ERRORS
It was demonstrated with the example of three stations in Figs. 8 and 9 that the method can
detect deviations from isopycnal mixing. In view of the various approximations and
uncertainties with some of the parameters involved, a reliable quantitative result can only be
obtained after careful consideration of all possible errors and of alternative interpretations.
To find a mathematical expression for the combined effect of all of them is a task beyond our
present capabilities. Instead, the effect of each of them separately will be considered and one
will try to arrive at some reasonable estimates. A comparison will then be made between these
estimates and the results of the earlier example in order to see whether the observed
deviations from isopycnal mixing are significant.
A systematic error of the method is introduced by the non-linearity of the equation of state.
A mixture of two water types of equal density but of different temperature and salinity is
heavier than the original water types and therefore will not stay on the isopycnal surface from
which it derived its properties but sink to the level of its new density. Nothing is known about
the significance of this effect in the presence of simultaneous active salt fingering, but as a
precaution it may be reasonable to accept the vertical distance which results from sinking due
to the gain in density (or the corresponding density increment) as a minimum below which
deviations from isopycnal mixing cannot safely be regarded as significant. It is easy to obtain
an estimate from a comparison of classical isopycnal analysis and the multi-parameter
method.
Let us return to Fig. 6. Between the two lines which define S A C W and N A C W the
isopycnals are approximated for isopycnal analysis by straight lines, and the non-linear
character of the equation of state is expressed only by the fact that these lines are not
parallel--hence, the difference of 22.5 ~o against 2 0 ~ in the example. Approximation by
straight lines is correct because it is along straight lines in the TS-diagram that the water
mixes. The gain in density due to mixing is expressed by the deviation of an isopycnal from a
straight line over the T S range of interest. Compared to the density increase experienced in
vertical mixing of water masses, the gain in density for two water types of identical density
before mixing is actually quite small. In Fig. 6 for example, the maximum gain in trt is less than
0.02 along an "isopycnal" as compared to 0.1 along the definition curves of N A C W or
S A C W . T h e maximum possible error caused by non-linearity of the equation of state is
therefore 0.02 in a t or 10-20 m in the vertical for the present example. This can be converted
to an error estimate for the determination of [ W T 1 ] to EWT4], using typical distributions like
those shown in Fig. 9; the result is a maximum figure of 2-4 ~o. It will be seen soon that other
factors limit the accuracy of the method to + 1 0 - 1 5 % at present, which makes the
contribution to the total error from the non-linearity of the equation of state rather
insignificant. It should also be noted that the error due to non-linearity is biased, giving
preference to the high-density water types W T 2 and WT4; but there is no systematic deviation
of the percentages calculated for the water types in Fig. 9 towards W T 2 and W T 4. A
conclusion is drawn that depth changes of the order of 10 m and virtual differences in water
mass content of a few per cent due to density increase from mixing are likely but insignificant
in view of other sources of errors.
164
M. TOMCZAK
Another source of systematic errors is the deviation of characteristic TS-curves of water
masses from straight lines. In isopycnal mixing analysis any shape of the TS-curves can be
accounted for in full, since the relative distances to an observed TS-combination along the
corresponding isopycnal can be measured for any curve. The multi-parameter method, on
the other hand, relies on linear definition curves to the extent that it is the basis for adding up
percentages of water types---which depend on the somewhat arbitrary definition of water
types - - to an invariable result, the percentage contribution of a water mass. There are several
ways to handle the problem. First, in order to arrive at conclusive results, the same
representation of the water masses by linear TS-approximations should be used for isopycnal
and multi-parameter analysis. Negative solutions should be avoided as far as possible by
choosing TS-approximations which embrace as many of the observation points as possible.
Naturally, the observations which are used to define the TS-curve of a water mass display
some scatter and the proposed method does not use the mean curve as the definition curve
but places all of these observations on one side of the linear TS-approximation. This
introduces a certain bias, but it does not cause additional errors because it affects both
methods to the same extent. If the deviation of the true TS-characteristic from a straight line
is large, the result will be that fewer stations display one of the water masses in its pure form in
the results than in reality. While this casts doubt on the results in terms of absolute
percentages, it again does not affect the comparison between isopycnal and non-isopycnal
mixing.
In some cases, a TS-characteristic may be approximated by a series of straight lines. This
involves the introduction of additional water types (one for each point where two straight
lines meet) and of the corresponding number of parameters. In this case the increase of the
total error is determined by the properties of the additional parameters such as
range/precision ratio etc. which will be dealt with separately.
An important point to note is the case where the TS-relationship of a water mass is linear
but the relationship between T or S and an additional parameter is not. Since isopycnal
analysis ignores all parameters other than T and S, the non-linearity in the definition curve of
any of those can cause considerable bias in the results of the multi-parameter analysis as
compared to isopycnal analysis. It may therefore be necessary in such circumstances to
introduce an additional parameter in order to be able to improve the approximation of the
water mass characteristics by using three rather than two water types (in the TS-diagram the
points for the three water types would then be collinear). Whether this improves the result or
not depends on the situation.
As an example, consider a situation of two water masses, represented by four water types
and analysed with the aid of temperature, salinity and silicate. Let the temperature/silicate
curve deviate from a straight line to such a degree that multi-parameter analysis of a T-S-Sicombination taken from the linear approximation of the definition curve results in 90 °/0
water mass content instead of 100~o (the result of isopycnal analysis). Assuming for the
moment that due to a low range/precision ratio the use of phosphate as a parameter is linked
with an error of + 15 ~ in terms of water mass content, introduction of phosphate as an
additional parameter would not improve the result because the reduction of the error due to a
non-linear T/Si-curve from a maximum of 10 °/o to, say, 2-3 ~o is more than outbalanced by
an additional error of 15 °/o. The situation would be different if the precision of phosphate
measurements would increase by a factor of 3.
To sum up the discussion of non-linear definition curves for the water masses, it seems
advisable to limit the analysis to a density range in which the definition curves of all
Temperature/salinitydiagram techniques
165
parameters, when plotted against temperature, are very close to straight lines. This limits the
number of parameters to a minimum. If necessary, the analysis can be extended to other
density ranges by using different linear approximations for different density ranges.
After this review of error sources inherent to the problem of mixing and TS-analysis in
general I turn to errors introduced by individual parameters. By far the most important
source of error at present is the limitation imposed by the methodology in the determination
of parameter values. Its influence is best expressed by the range/precision ratio. As has been
mentioned already, temperature and salinity have a ratio of 103 which corresponds to a
determination of water mass content of 0.1%. This is much less than all errors already
discussed and shows that for the purpose of the multi-parameter method temperature and
salinity can be regarded as accurately determined and used for general reference purposes, for
example as an independent variable against which others such as nutrients may be plotted. It
should be noted, however, that a ratio of 103 is not obtained in all cases. In the examples of
Fig. 8 the salinity range spanned by SACW and N A C W decreases from 1°/oo at 14-18°C to
0.3°/°° at 9-12°C if measured along an isopycnal, and it is smaller by a factor of 2 if taken
along an isotherm. Thus, the range/precision ratio varies as a function of density and is less
than 103 by a factor of 3-6 in this particular example. Still, the corresponding uncertainty in
water mass determination of 0.3-0.6 % water type content is very small.
The situation is quite different with some nutrients. Again referring to the same example,
the difference between SACW and N A C W in phosphate content is 1.2-1.5#g-at//
(decreasing with increasing density), measured along an isopycnal (for a full discussion of
nutrients in the frontal zone of SACW and NACW, see TOMCZAK,1981). With an accuracy of
phosphate determination of _+0.1 pg-at/l this results in an uncertainty of 6.7-8.3 '~o water
type content or an average of 15 o0 SACW or N A C W content, respectively. This estimate
should be compared with the distributions shown in Fig. 9. It is then immediately evident that
deviations of the S A C W / N A C W composition from isopycnal mixing conditions are
significant at station 3659 only, perhaps with the exceptions of the data points for G -~ 27.0 at
station 3658 and for a, ~ 26.7 at station 3660/1.
These large error bounds have a number of consequences for a successful application of the
method. First, they can result in negative solutions of the equations which can still be
oceanographically significant. To illustrate this point, we consider station 3658 of Fig. 9 in
more detail.
Table 2 gives the results of the analysis for station 3658 for the observed combinations of
temperature, salinity and phosphate and for two other data sets consisting of the same
temperatures and salinities but phosphate values decreased and increased by 0.1 #g at/l,
respectively. It is seen that negative solutions occur which apparently render part of the
analysis meaningless. However, if the percentages are plotted in the cumulative way of Fig. 9
it can be seen that most of the negative solutions result from the low range/precision ratio of
phosphate. Figure 10 is a graphical representation of Table 2. Here, negative solutions are
indicated by points outside the 0-100 ~o range and by points which lead to a crossing of
corresponding lines. Only at the lowest density level is the percentage content of W T 2 so
highly negative that the sum [WT4] + IWT3] + [WT~] > 100~o in all cases, and the right
shaded error bound area falls completely outside the allowable range 0-100 %. This density
level correspondingly is the only one where the value of [SACW] determined from isopycnal
analysis falls outside the error bounds for [SACW] calculated from multi-parameter analysis
(the central shaded area). It should be stressed that this situation is different from cases of
non-isopycnal mixing discussed before where a valid solution existed but did not coincide
166
M. TOMCZAK
TABLE 2. RESULTS FROM "METEOR" STATION 3658 WITH (a) OBSERVED T, S, P O 4 - P VALUES, (b) ALL
PO 4 P VALUES DECREASED BY 0.1 #g-at/l, (c) ALL P O 4 - P VALUES INCREASED BY 0.1 # g at/I. UNITS ARE
PER CENT WATER MASS CONTENT
Density
(a) W T 1
W~
WT 3
WT 4
(b}
WT 1
WT2
WT 3
WT 4
(c)
WT 1
WT 2
WT 3
W T,~
26.519
26.742
26.858
26.978
27.070
27.156
77.9
- 14.3
19.8
16.7
66.2
18.3
4.3
11.2
28.2
35.0
24.2
12.6
9.1
42.8
23.2
24.9
7.3
59.6
10.1
23.0
4.2
46.4
- 1.4
50.9
74.5
0.2
21.8
3.5
62.9
32.8
6.2
1.9
24,8
49.5
26.2
-0.5
5.8
57.3
25.2
11.8
3.9
74.1
12.1
9.9
0.9
60.0
0.6
37.7
81.2
-28.8
17.8
29.8
69.6
3.8
2.3
24.4
31.5
20.5
22.2
25.8
12.5
28.3
21.2
38. I
10.6
45.1
8.2
36.1
7.6
31.9
-3.4
64.0
with the result of isopycnal analysis. In the present case, the result of the analysis indicates
that an oceanographically valid solution does not exist for the set of parameter values
observed at ~r, = 26.519. The reason for this can be either non-isopycnal mixing with water
outside the range of the analysis, i.e. of density less than ~r, = 26.473 (or larger than
a t = 27.160), or a process not accounted for by the present method. The fact that significantly
negative solutions occur quite frequently at depths above 150 m suggests that nutrients are
significantly affected by local uptake and regeneration from the surface down to more than
100m depth and that the method should not be applied to data taken from such depths.
Apart from the one data point at the lowest density, all results from isopycnal analysis fall
just within the error margin of the method which confirms our earlier statement that mixing
at station 3658 is close to isopycnal. It is, however, important to note that a full test for
significant deviations from isopycnal mixing has to include the following:
266
268
270
20
0 I~,,
20
40
60
80
100
120
140
FIG. 10. The same representation ofresults for station 3658 asin Fig. 10but with error bounds due to possible error~
in phosphate determination as given in Table 2 drawn in. Heavy dots indicate results from isopycnal analysis.
Temperature/salinitydiagram techniques
167
(a) The solution of the multi-parameter method must be oceanographically significant, i.e.
negative solutions must not exceed the range estimated for errors.
(b) For isopycnal mixing, the percentages of all water types into which the mixture of water
is formally broken down have to fall within the corresponding error bounds of the multiparameter solution.
Thus, not only must [SACW] = [WT4] + [WT3]fall within the centre shaded area of Fig.
10, but [WT4] must also fall within the left shaded area and [WT4] + [WT2] q- [WT1] into
the right shaded area. (For the example of station 3658 this can be confirmed by comparison
with Fig. 9). In order to see this it is useful to consider again Fig. 8. In this representation,
[SACW] and [NACW] are given by the relative distances between the definition curves for
N A C W and SACW and the observation point, and a deviation from isopycnal mixing is
indicated by the deviation of the line through the observation point and the isopycnal
through the same point. These distances vary as a function of this deviation, provided the
definition curves of the water masses are not parallel to each other. In fact, the deviation of
[SACW] = [WT3] + [WT4] determined by multi-parameter analysis, from [SACW] as
determined by isopycnal analysis is a function of the angle between the line through the
observation point and the corresponding isopycnal and, at the same time, of the angle
between the definition curves of the water masses, and it goes to zero if the latter degenerates
to zero. The only way to resolve the type of mixing in that situation is to compare [WT1],
[WTz] etc. individually since they correspond to the relative distances along the definition
curves which define the slope of the line through the observation point and can only coincide
with the result of formal isopycnal break-down if this line coincides with an isopycnal.
The problem of very sizable errors being introduced by low range/precision ratios can be
reduced by using silicate instead of phosphate. Unfortunately, this increases the influence of
another source of errors: the uncertainty in the definition curves. It is very much a function of
historically available data and may thus vary considerably between different ocean areas.
First experience with the method (ToMCZAK,1981 ) suggests that the uncertainty can easily be
reduced below the level of errors inherent in the method of determination for phosphate but
not for silicate at the present time. Hence, a choice of silicate instead of phosphate reduces the
error due to range/precision ratio from 15 ojjoto 1.5'!~,, say, but increases the error due to
unprecise definition of nutrient values for W T I - W T 4 from 1.5 ",',, to 15 !~Jo-The net gain is
negligible. At the moment it seems advisable to perform the analysis with both phosphate and
silicate and carefully compare the results.
8. O U T L O O K AND FUTURE APPLICATIONS
After this somewhat sobering review of errors involved it might be helpful to discuss some
future applications of the method and with the aid of another example, show that the method
can give new results even with the large errors which affect it at present. It will then become
clear that by applying statistical techniques and using new methods of data collection in
marine chemistry foreseeable in the near future, the accuracy of the method can be increased
by an order of magnitude in a relatively short time.
Figure 11 shows another TS-diagram of a station from the area and expedition which
supplied most of our material for developing the method. The observed TS-combinations
suggest that only SAC W and NA CW are present at the station and that SA CW has by far the
larger share. Isopycnal analysis results in admixtures of NA C W of between 4 ~o at 150 m to
24 O~at 400 m (the observation at 100 m depth is not considered further because it is outside
168
M. TOMCZAK
°C
18
16
//
14
/f
12
2
~T
lo
2
NT~/
I
L
l
I
35"0
I
I
I
36"0
I
%0
FIG. 11. TS-diagram of station 3630 from cruise 36 of"Meteor', 1975, in the north-eastern Central Atlantic Ocean.
the range of the TS-combinations defined by the four water types). When phosphate is
introduced as another independent parameter, however, and multi-parameter analysis is
applied as before, the results are quite different. Table 3 compares the admixture of NACW
for both methods in column (a) and the first of columns (b). Assuming an overall error of
15-20 ~o, it is seen that the results of both methods agree at all observation levels except
300 m, suggesting isopycnal mixing at all levels but one. The discrepancy at the 300 m level
could of course be interpreted as evidence for non-isopycnal mixing; but with results for both
[WT1] and [WT4] strongly negative (although still within error bounds) this mixing would
have to involve water types over a density range of one unit in az, or a depth range of more
than 400 m.
The result at the 300 m level is improved if it is recalled that in the area off Cap Blanc,
North-West Africa, which is the source of the data, admixtures of warm, very saline coastal
TABLE 3. COMPARISON OF DIFFERENT ANALYSIS METHODS FOR "METEOR" STATION 3630. (a) [NACW] FROM
ISOPYCNAL ANALYSIS,(b) [NACW] = [ W T 1 ] + [WT2] AND INDIVIDUAL RESULTS FOR [ W T 1 ] , . . . , [WT4] FROM 4-
PARAMETER ANALYSIS, (C) INDIVIDUAL RESULTS [WT 1].... , [WT s ] FROM 5-PARAMETER ANALYSIS.ALL UNITS ARE PER
CENT WATER TYPE VIZ. WATER MASS CONTENT
Depth I
(a)
(m) I[NACW] [NACW] [WT 1]
(b)
[WT2]
[WTs]
[ W T 4]
(c)
[WT,] [WT2] [WT3] [WT,]
[WT~]
150
200
4
14
-3.5
31.4
6.3
3.5
-9.8
27.9
94.3
60.1
9.2
8.5
- 7.7
- 14.1
5.0
46.5
106.3
75.2
- 6.3
- 10.9
2.7
3.4
300
18
72.3
- 8.6
80.9
42.5
- 14.8
6.4
65.0
29.7
1.8
- 2.9
400
24
22.7
7.8
14.9
20.8
56.6
80.1
73.4
- 11.5
11.7
-53.7
Temperature/salinitydiagram techniques
169
water masses have been observed quite frequently to occur at comparable depth in the
vicinity of the shelf (I~xERS, 1976). The TS-characteristics of these water masses vary over the
seasons, and few data have been collected on their nutrient characteristics. For the purpose of
the example, we introduce an additional water type W T 5, thought to represent this coastal
water and defined by T5 = 24.5°C, S 5 = 39.5%0 and Ps = 0. It can be shown (ToMCZAK,
1981) that this is a reasonable approximation of water found on the very shallow Banc
d'Arguin off the Mauritanian coast. In order to apply multi-parameter analysis for a test of
possible admixtures of coastal water we introduce silicate as an additional parameter and
S i 1 = O, S i 2 = 5, S i 3 = 9, S i 4 = 21, S i 5 = 0 # g - a t / / a s definition values for silicate. The result
is shown in the last columns of Table 3.
At first glance the result seems worse than the earlier one: The number of negative
solutions is considerably larger. It is, however, quite remarkable that the 300 m level is an
exception to this general trend and shows quite an improvement. This can be understood in
the following way: The coastal water has a density much higher than that encountered
offshore at the same depth. Whenever some of this water is advected offshore, it sinks rapidly
down the continental slope and then spreads at the level of its own density (P~TE~S, 1976).
The density of W T s is a t = 26.92, while observed densities at 200, 300 and 400 m are
a t = 26.75, 26.93 and 27.01, respectively. It is thus likely that an admixture of W T 5 should
only be present at the 300 m level, provided mixing is not strongly non-isopycnal. In view of
the fact that the analysis without W T 5 showed reasonable agreement with isopycnal mixing
conditions at all other levels, non-isopycnal mixing is not very likely at that level as well. The
conclusion which can be drawn is that admixtures of W T 5 are not present at levels other than
the 300 m level, and inclusion of a water type which is not present at all results in considerable
deterioration of the results, because the additional but unnecessary parameter introduces
new errors which amplify the present uncertainties of the method. At the 300 m level, on the
other hand, the improvement of the result strongly suggests the presence of a small amount of
WTs even though the uncertainty in the definition values of the coastal water and the
combined effects of the increased number of errors involved still gives a slightly negative result
for [ W T s ] and thus does not enable us to draw quantitative conclusions.
With this assessment of the method's present capabilities, what improvements can be
made in the near future? Knowledge of definition values of water masses for additional
parameters can certainly be improved and this will happen as the number of observations
increases and allows determination of meaningful averages. The same procedure can also be
used in order to reduce the considerable errors inherent in the use of some parameters at
present. The result would be a statistical analysis of frontal zones based on time and space
averages of some sort. Analysis of individual stations will become of increasing interest with
the development of two major new techniques in marine chemistry: automated analysing
techniques and continuous profiling.
Automated analysing techniques (the so-called continuous flow analyser) have been
introduced into oceanography only recently. While they may not directly solve the problem
of comparability of chemical data between different laboratories and cruises, they have
increased repeatability of observations within a single cruise and have a large potential to
improve the range/precision ratio. More recently, the technique has been used in conj unction
with long tubes and a pump in order to obtain continuous vertical profiles of chemical
parameters.
In physical oceanography, introduction of the STD instrument led to the discovery of
important processes on scales not resolved by the ordinary Nansen bottle cast. A similarly
170
M. TOMCZAK
important development can be anticipated for chemical oceanography. As far as multiparameter mixing analysis is concerned, a continuous vertical profile of all parameters,
although not improving the statistical significance of individual readings, increases the
confidence in the results because deviations from isopycnal mixing can then be observed in
their vertical development and extent. Alternatively, readings can be averaged over depth
intervals of 5-10 m, resulting in an increase of both precision and data density over classical
methods. Eventually, continuous flow analysis will be used with undulating towed sensor
packages, and the resulting high data density will without doubt reduce the parameterinduced errors of multi-parameter mixing analysis to a degree that quantitative water mass
analysis should be accurate enough to enable investigation of all those processes enumerated
above as sources of systematic errors. The increased inflow of data will result in improved
knowledge of definition values of water masses for a number of additional parameters,
opening the possibility of performing the analysis with more than the minimum number of
parameters necessary for the solution and to solve the system by inverse methods.
Acknowledgement--I wish to thank R. SCHMITT for drawing my attention to the work of M. C. INGHAM and for
commenting on an earlier draft of the manuscript, and comments received from an anonymous referee which greatly
improved presentation of the physical interpretation of results.
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