OCEANOGRAPHY 510
Handout Notes #1
The Surface Circulation, Definitions, and the Equation of State of Seawater
§1.1 Introduction
The modern study of the circulation of the ocean has advanced from the point of pure description. It is now
necessary to employ more sophisticated techniques, including higher mathematics, solidly based physical reasoning,
and sophisticated computing, to attempt to uncover the secrets of the inner workings of the ocean. Furthermore,
it is no longer practical to study the ocean from the viewpoint of a single scientific discipline, be it physics,
biology, chemistry, or geology; even though our interest here is on the physics of ocean circulation, these other
elements will often make an important contribution to our understanding of the physics at hand. The emphasis
in this course is on the large-scale ocean circulation averaged over a long time. However, there is an increasing
interest in examining changes in time in the global circulation of both the ocean and the atmosphere, and we
shall attempt at least to touch on some of these changes in this course. Time constraints severely limit what can
and will be discussed here; there will be very little discussion of some topics that are usually discussed in a first
course like this one, such as tides, coastal and estuarine circulation, and simple wave theory. While these are
important and essential topics for professional oceanographers to understand, they will not be dealt with here,
but are discussed in considerable detail in other courses.
As individuals, much of our intuitive notion of how the ocean works comes from having examined the ocean
surface in coastal and estuarine oceans—this is the ocean that most of us see most often. However, the processes
occurring in this most familiar ocean are not necessarily representative of the global ocean, and it is prudent to
examine the large-scale circulation of the global ocean in detail in its own right. Consider the histogram of the
depth of the world ocean shown in Figure 1.1. The mean elevation of land that is above sea level is about 840 m,
a result that is not surprising based upon our experiences. The mean depth of the sea, however, is about 3800
m. If we assume that the average parcel of water in the ocean exists at roughly half this depth, say about 2000
m, then to gain some rudimentary understanding of the working of the ocean requires that we examine processes
occurring in the deep sea in some detail. Once we have some basic, general idea of how the deep sea works, we
will find that this knowledge will help in understanding how the more familiar coastal ocean operates as well.
§1.2 The Surface Circulation
The study of physical oceanography really consists of two basic questions: (1) where does the water go? and (2)
why does it go there? To answer the second question requires a preexisting answer to the first, and yet surprisingly
little is known about the answer to this question, except for perhaps the surface circulation of the world ocean. A
cartoon of the surface circulation (Figure 1.2, adapted from Sverdrup et. al., 1942) suggests that there is a basic
pattern of the circulation at the sea surface in most of the major ocean basins; in general, the surface circulation
is not symmetric about the center of ocean basins, but is instead strongly intensified on the western sides. The
strongest, major currents of the world ocean, including the Gulf Stream in the North Atlantic and the Kuroshio in
the North Pacific, occur along the western boundaries of oceans, and there exist somewhat weaker but important
westward-intensified counterparts in the southern hemisphere, including the Brazil-Falkland Current and the East
Australian Current. At eastern boundaries, such as the west coast of North America, somewhat weaker boundary
currents like the California Current exist in nearly all the oceans. Even a newcomer to oceanography might
assume that the surface currents in the world ocean are primarily driven by the wind, and this would indeed be
a good assumption. However, just how this happens is not so clear, since the wind fields over the major ocean
basins [such as the North Atlantic, as in Figure 1.3] tend to be symmetric about the center of the basin, while
as previously noted the surface currents in the ocean are asymmetric. The cause of the asymmetry is a classic
theoretical problem in physical oceanography that was first solved by Stommel (1948) and is still being studied
in considerable detail today. One can easily see that the North Pacific circulation is not qualitatively unlike the
North Atlantic, as shown in Figure 1.4. In general, since there is some similarity in the surface circulations of
all ocean basins, we can summarize the canonical circulation as in Figure 1.5. Between the trade winds and the
westerlies, there is a subtropical gyre in all ocean basins. Between the maximum in the westerlies and the onset
of the polar easterlies there is a subpolar gyre. Both of these gyres are westerward intensified [n.b. the naming
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convention for winds and ocean currents is, unfortunately, opposite; winds are named by the direction from which
they are blowing, while ocean currents are named by the direction to which they are flowing. Thus a west wind is
blowing to the east while a westward current is flowing to the west]. Near the equator, a series of smaller, intense,
counterotating gyres occurs in most places. The one exception to these generalities is the Indian Ocean; while the
flow is westward intensified, the direction of the western boundary current varies with season, depending on the
direction of the wind. During the northeast monsoon, in northern hemisphere winter, the winds are predominantly
from the Indian subcontinent, as a result of low pressure over the Himalayas. During this time the Somali Current
flows to the south. During the southwest monsoon, in northern hemisphere summer, the winds blow primarily off
the Sahara Desert. During this period the Somali Current flows primarily to the north. The transition between
these two regimes usually occurs in mid-April and requires only a few weeks [see Figure 1.6].
To get some insight into how the ocean circulation is driven, we might consider two very simple models of
circulation as shown in the two hypothetical basins in Figure 1.7. In the first basin (a pan of water, for example)
suppose that the water is homogeneous throughout the basin. In the absence of any applied forces, the surface of
this basin would be flat (later in the course we will define in more detail what is meant by “flat”). If some external
force such as a wind were applied to the basin, simple physical arguments would suggest that the surface of this
basin would be deformed, with water piling up on the downwind side of the basin. If the force were suddenly
removed, the surface of this basin would slosh back and forth until all motion eventually ceased through frictional
processes, until it was once again flat. This is a good prototype of the wind-driven circulation in an ocean basin,
except for the fact that in most oceanic problems the basin is not a pan of water but instead extends for thousands
of kilometers, thus requiring factors such as the earth’s rotation and spherical geometry to be taken into account.
Many of the essentials of the wind-driven surface circulation are present in this simple thought problem, however.
In the second basin in Figure 1.7, suppose that there exist two layers of fluid. Initially, before any forces
have been applied, both the surface of this “ocean” and the interface between the layers are flat. Suppose that
something, by some means (differential heating, for example) is done to this basin that causes the interface to
deflect, much as the surface was deflected in the first example. It will be found that (at least for a fluid like
seawater) the surface of the basin probably won’t be deflected as much as the deflection of the interface; the
surface deflection might be imperceptible by ordinary means. On the other hand, the interface deflection might
be large. Once the external force is removed, however, the interface will slosh back and forth as did the surface
in the first problem, and eventually everything will again be flat, once friction has had a chance to work. We
might term this second example a prototype of the thermohaline circulation in an ocean basin, and it is indeed
not a bad starting point for discussing how the ocean operates internally. As with the first example, however,
some important effects such as rotation and spherical geometry are important in the context of the real ocean
and must be added before this model can be useful.
At a later point in this course we will examine the effects of wind on the ocean circulation in some detail, including
the mechanisms by which the wind actually imparts momentum to the ocean as well as large-scale asymmetries
induced by rotation and spherical geometry. In addition to the wind, other factors, related to atmospheric effects,
can influence the surface circulation of the ocean. The surface salinity (S), through its effect on the density of
seawater, can have a major impact on the flow at the sea surface [see Figure 1.8]. In general, the distribution of
surface salinity can be related to the difference between evaporation (E) and precipitation (P) at the sea surface.
The quantity E P, the net evaporation, averaged over an annual cycle, is strongly correlated with the surface
salinity. Generally this quantity is positive in the tropics and subtropics, where the temperature and humidity of
the atmosphere are favorable to high eveporation rates from the ocean. At subpolar latitudes, where evaporation
is relatively lower and precipitation can be quite high, generally E P < 0. This is particularly true for the North
Pacific, where precipitation can exceed evaporation by nearly one meter per year off the coast of North America.
The global hydrological cycle must be considered in any thorough examination of the large-scale ocean circulation.
Averaged over the northern hemisphere, E 112.5 cm/yr, while P 81.2 cm/yr. Thus, river runoff in the northern
hemisphere must be about 31 cm/yr, the difference between E and P.
If we let
= E P, then we might write that
(1)
where the dot denotes a time derivitive, or, alternatively, that
(2)
2
where K is a constant of proportionality. This differential equation can be integrated to yield
(3)
Thus, larger net evaporation rates are expected to yield larger surface salinities; this is simply because evaporation
removes fresh water from the ocean and leaves salt behind, raising the salt concentration (the “salinity”). It is
even possible to increase S so much by evaporation that the resulting water becomes convectively unstable. This
process occurs in the center of subtropical gyres, where
is quite large, and results in the formation of so-called
subtropical underwater. Negative net evaporation rates can have the reverse effect of making the water column
highly stable. As a global annual average, the surface salinity of the world ocean clearly is closely related to
the global distribution of , as shown in Figures 1.9a and b. The production of sea ice at high latitudes has the
reverse effect of precipitation/evaporation. When sea ice is formed, water is removed preferentially to salt (the
salinity of sea ice is usually less than 10 psu, while that of sea water is usually about 35 psu). The water under
the ice is thus enriched with salt, relative to its pre-ice state, resulting in an increased salinity. Melting the ice
reverses this process, creating a cold, very fresh layer at the sea surface as the ice disappears.
Ultimately, almost all of the driving mechanisms of the atmosphere and ocean circulation are related to differential
heating of the earth by the sun (the obvious exceptions are gravitational tides and geothermal heating). Thus, sea
surface temperature over the globe (Figure 1.10) is strongly dependent on latitude. However, there is a marked
east-west asymmetry to the sea surface temperature, related to the asymmetry of the surface ocean circulation. This
is most evident in the North Pacific and North Atlantic Oceans, where the western boundary currents are especially
strong. It should not be surprising that this heating of the ocean surface by the atmosphere might feed back
into the atmosphere itself; where the proper combinations of sea surface temperature and atmospheric conditions
are present, atmospheric convection may be enhanced and the local cloud cover intensified, as is often the case
in the eastern North Pacific off the coast of North America, from the equator to southern Alaska. On a more
global scale, clearly the temperature of the sea surface and the resulting exchange of heat with the atmosphere
is one component of the global balance of heat, a topic of great contemporary interest. The annually averaged
net heating of the ocean surface (Figure 1.11) suggests that there are regions of the ocean, usually in the western
portions of the subtropical gyres and especially in the northern hemisphere, where the ocean may give up very
large amounts of heat to the atmosphere on a net annual basis. Of course, direct heating of the atmosphere by
the ocean in these regions is only one component of the total heat flux depicted in Figure 1.11, but nonetheless it
is clear that the ocean must play a central role in the global balance of heat, especially at lower latitudes. This
topic will be explored in much more detail in a subsequent section.
The surface circulation is clearly important, as it is that part of the ocean that has direct and continuing contact
with the atmosphere. As noted in the Introduction, however, the “average” parcel of water in the global ocean
exists perhaps 2000 m below the sea surface. Yet it is now suspected that even this deep water plays a central role
in the long-term climatic state of the planet. One contemporary idea that will be examined in considerable detail
later in this course is the so-called conveyor-belt notion of global ocean circulation, as depicted in Figure 1.12,
taken from Broecker and Peng (1983). The basic idea of the conveyor belt is that water becomes convectively
unstable at high latitudes in the Atlantic and sinks to near the bottom of the ocean. From the North Atlantic this
water mass (“North Atlantic Deep Water”) proceeds into the South Atlantic, joins the water mass formed in the
Weddell Sea (“Antarctic Bottom Water”) and proceeds to fill the deep basins of the Indian, South Pacific, and
North Pacific Oceans in that order. This deep circulation takes approximately 1000 years from North Atlantic
to North Pacific. Since deep water is formed in only a few areas, by sinking, this deep water mass must slowly
rise at it fills the rest of the ocean in order that mass be conserved. To feed the sinking at the poles, a supply of
surface water is required. Thus, it is hypothesized that that a shallow return flow, from the North Pacific, through
Indonesia, across the equatorial Indian Ocean, around southern Africa, and into the South and ulitimately North
Atlantic, must exist. While there is increasing evidence that the deep portion of this hypothesized conveyor belt
exists, at least in some form, there is virtually no evidence that the shallow return flow depicted in Figure 1.12 is
correct. Since the near-surface ocean circulation is clearly much easier to observe than the deeper circulation, this
result may at first glance seem unbelievable. However, the problem is that there is a vigorous shallow wind-driven
circulation that exists superimposed upon the long-term thermohaline circulation, whatever it may be. Seawater
parcels spend most of their time going around westward intensified wind gyres in the surface ocean; as little as
one or two years might be required for each circuit. The conveyor belt portion of the shallow circulation, on the
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other hand, might require 1000 years or so. Thus, it is very difficult to separate the relatively weak conveyor
belt from the vigorous shallow wind-driven circulation. This is a problem of great contemporary interest, and it
will be discussed in more detail in a later section.
§1.3 Definitions of Some Important Quantities
The three variables that are most closely tied to the dynamical properties of seawater are salinity, temperature,
and pressure. While conceptually the definitions of these quantites might appear to be obvious, it is necessary
to consider the meanings of these quantities in some detail. The ultimate goal of this analysis is to determine
an equation of state for seawater.
(a) Salinity
There are geochemical aspects to any discussion of salinity. Where did the salt in the ocean originate? One
hypothesis is that the salt in the ocean might have somehow originated from a combination of continental runoff
of freshwater, followed by evaporation at sea. While this is possible, the available geochemical evidence does not
seem consistent with this idea, since the earth’s crust and seawater have different chemical abundances. The earth’s
crust is rich in silica and aluminum, rivers are rich in carbonates, and the oceans are rich in halides and sulfides.
Thus the cycle leading to salt in the ocean must be more complicated than simple continental runoff followed
by oceanic evaporation. Another possibility is that seawater condensed from the original proto-atmosphere as
the earth was formed, and the salt presently in seawater originated at the same time. While these are fascinating
discussions, we will leave the merits of various arguments arguments concerning the origin of salt in the ocean
to a course in chemical oceanography.
Seawater is a complex mixture of nearly all the elements; a table listing the major constituents of seawater is
given in Figure 1.13. These concentrations vary considerably from place to place over the world ocean. Despite
the complexity of its composition, however, seawater has the remarkable property that its principal contituents
occur in nearly constant ratios everywhere in the ocean. This suggests that, whatever the sources and sinks for salt
in the ocean, the ocean has become well-mixed in its constituents over geological time scales. This constancy of
proportions means that to determine the “salinity” of seawater requires only a good measurement of one of the
constituents; once one constituent is well-known, then the others may be determined by knowing their ratios with
the measured consituent. This constancy of proportions is not perfect; there are small departures in the ratios from
ocean to ocean and basin to basin. However, our ability to measure salinity has not yet reached the point where these
small departures in the deep sea are of serious importance to the analytical determination of salinity in the ocean.
There is a classical, formal definition of salinity:
“Salinity is the total amount of solid materials in grams contained in one kilogram of seawater when
all the carbonate has been converted to oxide, the bromine and iodine replaced by chlorine, and all
the organic matter completely oxidized.”
There are a number of ways to actually measure salinity. The simplest conceptual method would be to evaporate a
known quantity of seawater to complete dryness and then weigh the residue. This is not a practical way to measure
salinity, since it might take considerable time to evaporate the sample to complete dryness, and also since the
determination of the weight (actually mass) of the residue is a process with considerable inherent error. A more
practical method, which was used until the 1960s, is to titrate seawater with AgNO3 , precipitate the halogens in
the sample, then determine the amount of Cl. By assuming constant proportions, the salinity can be determined
from the ratios of the other components. This method is historically good to about .02
The basic relation in this case used to determine salinity is
(4)
Since the mid-1970s it has been common to use instruments capable of continuous profiling of temperature,
salinity, and pressure rather than to measure these quantites from individual bottle samples alone. In such cases
the salinity is not measured directly, but is determined from measuring the electrical conductivity of seawater
C(S, T, p) and inferring S as a back calculation from a knowledge of C, T, and p. Using modern profiling CTD
instruments, state-of-the-art salinometers, and UNESCO standard water, salinity can be determined to better than
one part in a million. In terms of “parts per thousand,” this is an error of less than .001
However, since salnity
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is only rarely measured directly anymore, the definition of salinity has been changed to reflect the dependence on
conductivity. Since salinity is not actually measured, units such as “parts per thousand” hardly seem appropriate;
instead, the new units of salinity have been deemed “practical salinity units” or psu. The numerical value of psu is
essentially the same as the old value of parts per thousand, but the use of psu reminds us that salinity is now based
on a determination of electrical conductivity rather than a direct measurement of the dissolved solids in a sample.
How well must we be able to measure salinity? Some feel for the answer to this question can be gained from
examining Figures 1.14 (salinity across 43 S in the South Pacific) and 1.15 (salinity across 18 S in the Indian
Ocean). It is clear from these sections that there are vast regions of the deep sea where S appears to vary by
.05 psu or less, and a thorough knowledge of the deep circulation in such regions requires a determination of
S to at least this limit. It was thus quite difficult to learn very much about the deep circulation of the world
ocean by older methods of salinity determination, although these methods were adequate for surveying the upper
ocean, where variations in S are of the order of 1 psu.
(b) Pressure
Pressure is an important variable in almost all oceanographic problems, in chemistry, biology, and geology in
addition to ocean physics. To completely understand the idea of pressure in seawater, we need to consider first
the difference between solids and fluids (ie, liquids and gases).
The classic text by Batchelor (1967) begins on page 1 of Chapter 1 by stating that “The defining property of
fluids, embracing both liquids and gases, lies in the ease in which they may deformed. A piece of solid material
has a definite shape, and that shape changes only when there is a change in the external conditions. A portion of
fluid, on the other hand, does not have a preferred shape, and different elements of a homogeneous fluid may be
rearranged freely without effecting the macroscopic properties of the portion of the fluid. The fact that relative
motion of different elements of a portion of fluid can, and in general does, occur when forces act on the fluid
give rise to the science of fluid dynamics.” Another way of stating this is that when a solid is at rest there are
many possible combinations of forces that may be acting to achieve this equilibrium. With fluids, on the other
hand, in the most general case applied forces will act to deform a marked fuid element. The exception to this
general statement comes with forces acting in a direction purely normal to the surface of the fluid. A fluid at
rest cannot sustain a tangential force, but it can sustain one normal to its surface. Later in this course we will
investigate the role of tangential forces (or shearing stresses) on a fluid such as the ocean. For now, however,
we will restrict ourselves to the topic of normal forces.
An example of fluid at rest under the action of purely normal forces is fluid in a container. The walls of the
container exert a normal force on the fluid inside, and it is possible that static equilibrium can be achieved with
no other forces acting. In more general terms, if we consider only cases of static equilibrium, then we must only
consider the effects of normal forces. We shall define the inward normal force per unit area on a fluid element
as the pressure. With this simple definition in mind, it is straightforward to proceed to examine the balance
of pressure forces acting a fluid element at rest. Consider the sketch shown in Figure 1.16. The faces of this
macroscopic fluid element are numbered, and the coordinate axes are shown. If we initially consider only the
pressure forces acting in the x direction, then we can write that
(5)
where Fpx denotes the component of pressure force in the x direction per unit volume. This can be easily
simplified to
(6)
We may carry out a similar process in the y and z directions, with the net result that
(7)
5
In the limit that the sides of the cube become vanishingly small, we can replace the finite differences here
with partial derivitives, so that
(8)
Thus, we have shown that the sum of the pressure forces acting on this fluid element is just equal to the pressure
gradient. Here, however, we have examined the net pressure force per unit volume. Usually it is more convenient
to examine the net pressure force per unit mass, since mass is ordinarily conserved following a fluid parcel while
volume may not be. We can write the pressure force per unit mass as
(9)
where is the density of seawater and the subscript m denotes the force per unit mass. Since we will be nearly
always using this result on a per unit mass basis, we will henceforth drop the subscript m. Equation (9) states that
the net pressure force on a seawater parcel per unit mass is proportional to minus the pressure gradient; thus, a
pressure gradient in one direction exerts a proportional force in the opposite direction.
We now consider the possibility of other body forces that might balance the pressure force; the most obvious
candidate is gravity. Thus, in a state of no acceleration and no motion, we might have
(10)
where the second force represents gravity. We can evaluate each of the forces to find that
(11)
where k is a vertical unit vector. This vector equation can be decomposed into three scalar partial differential
equations. The horizontal parts yield simply
(12)
suggesting that there is no pressure variation on horizontal surfaces, while the vertical part is
(13)
Equation (13) is known as the hydrostatic relation, or hydrostatic balance. This is generally the balance of forces
along the vertical axis for the large-scale ocean circulation. It is simple to see the physical significance of this
result; assuming that is not a function of depth (a reasonable assumption in this particular case, since only
varies by a few per cent from top to bottom in the ocean), we may integrate both sides of this equation to find that
(14)
where p(0) is the pressure at the sea surface. Since pressure is a scalar quantity, we may arbitrarily define it
with respect to any reference value that we choose; here, we will take p(0) = 0. With this in mind, equation
(14) simply says that the pressure (ie, the force per unit area) at any depth in the ocean (assuming that only
gravity and pressure forces are acting) is given by the weight of the water column above, per unit area. Note
that depths here are measured positive upwards from the sea surface, so that a depth of 1 m is given by z= 1
m. We can check that the right side of equation (14) is indeed the weight per unit area: the units of this term
are [Length/Time2 ][Mass/Length3 ][Length], which are indeed the proper units for weight per unit area. Since the
hydrostatic pressure is the weight of the overlying water column per unit area, we expect that at great depths
there will be great pressures.
Using the integrated hydrostatic realtion [Eq. (14)] it is possible to examine the relationship between depth and
pressure in the ocean. Taking g=9.8 m/sec2 and =1000 kg/m3 , then we have p=9.8 103 kg/(m2 -sec2 ) per meter
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of depth. However, 1 kg-m/sec2 is more commonly known as 1 newton, so we have p=9.8 103 newtons/m3 per
meter of depth. Recall that 1 Pascal (Pa) = 1 newton/m2 , so that p=9.8 103 Pa = 9.8 KPa per meter of depth. A
non-SI unit for pressure that is commonly used in oceanography is the bar, which is equal to 100 KPa. As a result,
p=.098 bar per meter of depth, or .98 decibars per meter of depth, where 1 decibar = 0.1 bar. Thus, the pressure in
decibars at any depth in the ocean is very nearly equal to the depth in meters there. For this reason, oceanographers
commonly report pressure in the ocean in units of decibars, rather than the more standard units of Pascals.
In general, the large-scale, long-term ocean circulation can be considered to be in hydrostatic balance vertically,
as given by equation (13). Under what conditions will processes in the ocean be hydrostatic? It is simple to
estimate the degree to which the ocean might be hydrostatic in order to answer this question. Consider the
full, time-dependent version of Newton’s Laws for vertical accelerations in the ocean [essentially the vertical
part of eq. (10)] given by
(15)
Here z denotes the vertical coordinate. In purely hydrostatic balance, the left side of this equation is identically
equal to zero, and the system is in equilibrium with terms 2 and 3 balancing. In general, however, the forces
of pressure and gravity will not balance perfectly, implying that term 1 will not be identically zero. For an
approximate state of hydrostatic balance, we need the twin conditions that (a) term 2 and term 3 are nearly equal,
and (b) term 1 << term 2 or term 3. Continuing along this line of reasoning, if we estimate z as some scale
distance z and time as some time t, then for condition (b) to hold we need
(16)
or otherwise the total acceleration (term 1) will be comparable to the pressure gradient and the acceleration due
to gravity (terms 2 and 3). Suppose we take z to be 1 meter, a typical change in the height of the sea surface
in the open ocean. Then for hydrostatic balance we require that
(17)
or about .3 seconds. Thus for a surface change of 1 meter, only motions occuring on time scales much slower
than a second or so can be in hydrostatic balance. Note that for t = 1 day and a surface change of 1 meter, the
acceleration due to gravity is about 107 times larger than the total vertical acceleration, so that the hydrostatic
relation is indeed appropriate for motions with time scales of a day or longer.
(c) Temperature
The final state variable that influences the density of seawater is temperature. The formal definition of temperature
as a state variable (see any thermodynamics book) is the familiar “Two systems in thermal equilibrium with
each other have the same temperature; two systems not in thermal equilibrium with each other have different
temperatures”. In most physical cases heat is actually the more interesting quantity, although it is temperature
that is inevitably the quantity that is measured. A process or system wherein no heat is gained or lost is termed
an adiabatic system, and this is the case that we will most often consider in physical oceanography. In the
context of an adiabatic system, one simple way to think about the meaning of temperature is simply as the heat
content per unit volume of a system of particles.
Consider the situation shown in Figure 1.17. A particle of seawater at the sea surface has a temperature of 5 C in
an ocean with a salinity of 35 psu. Suppose that we consider the case where this parcel of seawater is enclosed by
a hypothetical, flexible bag that can expand or contract, but that will allow no heat conduction or diffusion across
its walls. Thus any changes to the seawater inside the bag are adiabatic. Suppose now that we displace this bag
and its contents from the ocean surface to a depth of 4000 m, and then measure the temperature of the seawater
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inside the bag: it would be found that the seawater inside the bag would now be at a temperature of 5.45 C.
This is because seawater is slightly compressible, and as a result the volume of the bag would have decreased
somewhat as it was displaced to higher pressure. Since the bag has adiabatic walls, however, the heat content
of the water inside the bag would be the same as at the sea surface. Thus, the heat per unit volume inside the
bag (the temperature) increased slightly. If we begin instead with a parcel-in-a-bag at the bottom of the ocean
with a temperature of 5 C and displace the particle to the sea surface, the temperature will decrease slightly, to
4.56 C, as the pressure decreases and the bag expands slightly.
It can be seen that, because of this pressure effect on temperature, some care must be taken when comparing the
temperature of two seawater parcels. Are they at the same pressure? If not, how do we compare them? Consider
for example the situation shown in Figure 1.18: two parcels of seawater, having salinities, temperatures, and
pressures (S1 , T1 , p1 ) and (S2 , T2 , p2 ) are brought together and mixed. What will be the character of the resulting
water mass? To resolve the indeterminacy and begin to answer this question, we shall introduce the concepts of
in situ temperature and potential temperature. The in situ temperature is simply the measured temperature of a
marked seawater parcel. In Figure 1.17a, the in situ temperature of the particle displaced to 4000 m is 5.45 C. The
potential temperature is the temperature that a parcel would have if raised adiabatically to the sea surface. In Figure
1.17a, the potential temperature of the parcel is 5 C. In a similar fashion, in Figure 1.17b the in situ temperature is
5 C, and the potential temperature is 4.56 C. It is standard to denote the in situ temperature by T and the potential
temperature by . Note that as a consequence of this particular definition of potential temperature, ≤ T.
By always referencing the temperature of a seawater parcel to a common pressure, in this case the sea surface (ie,
p=0), the confusion in the interpretation of temperature in Figures 1.17 and 1.18 has been removed; the classical
definition of potential temperature, as given above, is based on a reference at the sea surface. In recent years it
has been common to use other pressures for the reference where it may be more convenient. For example, if we
had measured the temperature from two seawater samples, from depths of 1900 m and 2200 m, and we wished to
know the relative temperatures of the two if they were brought together, we might reference the two samples to
2000 m instead of the sea surface, since this pressure is much closer to the local pressures than p=0.
The value of the potential temperature is determined empirically. In principle, we may relate the potential and in
situ temperatures by describing the increase of temperature with increasing p. This can be quantified by writing
(18)
where p0 is the reference pressure and the quantity , called the adiabatic lapse rate, is defined as
(19)
The subscript S implies that the partial derivitive is to be taken with the salinity held constant. The quantity
is determined empirically, as by Bryden (1973) or Fofonoff (1962), and is shown in Figure 1.19. [This figure is
taken from Fofonoff (1962). The correct modern definition is as given by Bryden (1973); however, the differences
in the definitions of in these two papers are quite small, and Figure 1.19 provides reasonable qualitative
estimates.] One may think of as the vertical gradient that would exist in temperature in an ocean that was
well-mixed adiabatically, and where only variations in temperature due to pressure were possible. In general, it is
possible to get a feel for the difference between T and by examining Figure 1.20, also taken from Fofonoff’s
paper [Again, the correct modern definition is that given by Bryden (1973), but the differences between Fofonoff
(1962) and Bryden (1973) are small.]. An example of how pressure effects become important in deep water is
reflected in the T and fields in the Marianas Trench. It is clear that, based on T alone (Figure 1.21a), it might
be inferred that there is a temperature inversion in the vicinity of the Marianas Trench at depths below about
4000 m; this might imply some sort of convective instability, caused by some unknown process. However, it is
clear from an examination of the field near the trench that no such inversion is present: the apparent inversion
in temperature is simply a consequence of adiabatic effects due to the slight compressibility of seawater and is
not related in any fashion to convective instability.
8
§1.4 The Equation of State
Having defined the principal state variables of seawater, we can now examine the actual equation of state. The
equation of state is simply a relation that can be used to determine the density, given the important state variables.
Examples of equations of state often encountered include the ideal gas law, van der Waal’s equation, and the
Clausius-Claperyon relation. These equations often occur in physics, chemistry, and the atmospheric sciences, and
all are quite simple relations that relate density to the principal state variables. In the case of seawater, however,
no such simple relation is possible. This is due to the fact that liquids in general, including seawater, usually
have quite complicated equations of state. Furthermore, seawater is a binary fluid (that is, it has two important
components, temperaure and salinity), and this greatly increases its chemical and physical complexity. As a result,
no simple equation of state is possible for seawater. Instead, the modern equation of state has been determined
empirically through a large number of precise laboratory measurements. The modern equation of state of seawater
used by oceanographers is simply a polynomial fit to the laboratory measurements.
The equation of state for seawater takes the general form
(20)
where is the density of seawater as a function of salinity, temperature, and pressure, which have been previously
defined. The value of varies by only a few per cent over the world ocean. At the sea surface in the tropics,
has a value of about 1.021 g/cm3 or 1021 kg/m3 . In the deepest trench of the world ocean the value of is
about 1.071 g/cm3 or 1071 kg/m3 . Thus the total variation in is not large, and writing most of the numbers is
redundant. With this in mind, we will define a new variable such that
(21)
where the subsripts cgs and mks denote the system of units used in defining . The variable is convenient because
we don’t need to write the redundant digits every time. This definition of and is called the in situ density.
Just as we have previously seen with temperature, there is a strong pressure effect in density simply due to the
slight compressiblity of sea water; as a parcel is moved to deeper and deeper depths (ie, greater and greater
pressures), the density will change simply due to the fact that the mass of the parcel remains constant while the
effective volume decreases. At low pressures (ie, near the sea surface) the in situ density is a useful definition.
However, at higher pressures density is such a strong function of pressure that the deeper waters of the world
ocean appear nearly identical in density, even though they may have quite different values of temperature and
salinity. Thus, at greater pressures it is useful to remove the pressure effects and to examine only the changes of
density due to temperature and salinity, much as was done with temperature. In order to do this, we can define
a new density t (pronounced “sigma-tee”) such that
(22)
In this case the density is evaluated at the sea surface (p=0) in order to remove all pressure effects. However,
this variable is also good only relatively near the sea surface (but deeper than the in situ density), since T is
also a function of pressure, as we have already seen. With this in mind, we might define yet another variable
(pronounced “sigma-theta”) such that
(23)
In this case the in situ temperature has been replaced by the potential temperature in order to further remove
the effects of pressure on density. [Note that the pressure has not yet been completely removed from density,
however, since itself is a function of p.] Based upon these definitions of density, it can be seen that ≥
≥
would have T increasing with depth as shown in
t generally in the ocean. Clearly an ocean having a constant
Figure 1.22a; this implies that an ocean having a constant
would have t decreasing with depth, as shown in
Figure 1.22b. This is the situation depicted in Figure 1.21 in the Marianas Trench. Typical profiles of salinity,
) for the ocean are shown in Figure 1.23.
temperature, and density (ie,
The actual equation of state of seawater, which determines or for various values of salinity, temperature, and
pressure, cannot be determined from an analytical theory. This is because the nature of seawater does not lend
9
itself to such simple analysis. First, there are three state variables involved, which inevitably leads to a complicated
equation of state. Second, seawater is not an equilibrium solution; that is, there is diffusion in any realistic sample
of seawater, and this leads to quite complicated effects in any attempt to model the nature of seawater. Third, and
most important, seawater is not a dilute solution. Thus, we can’t use most of the principles derived in classical
thermodynamics for use in simple solution theory [see Guggenheim (1950) for a good discussion of these issues].
With these limitations in mind, it is essentially impossible to derive an analytical equation of state for seawater.
It is possible to derive an empirical equation of state for seawater, and that is the origin of the equation used
in modern physical oceanography. The process involved in obtaining this equation requires three distinct steps.
First, the equation of state of pure water at a pressure of 1 atmosphere is determined empirically. In this case,
we have = (T) = w alone (the subsrcipt w denotes fresh water). Second, the equation of state as a function
of both temperature and salinity, written as a perturbation on the fresh water result, is determined, so that 2
= w + (S, T, 0). Third, and finally, the full equation of state is determined by using the second result with
a pressure correction, so that
(24)
The quantity K is known as the secant bulk modulus and is a common measure of the ratio of stress to strain in
a given material. Clearly K must also be determined before the equation of state given in (24) can be useful.
The steps to determine K are analogous to those for . First, a value of K for fresh water, Kw , is determined.
Next, the salinity is included as a perturbation, so that K2 = Kw + K(S, T, 0). Finally, the full K3 = K(S, T, p)
is determined as a correction to K2 . The forms of these functions and constants is given in Figure 1.24a for the
modern International Equation of State (UNESCO, 1981). In practice, all of these functions and constants are
determined empirically in highly controlled laboratory environments. Additionally, a plot showing the general
relationship between salinity, temperature, and density can be found in Figure 1.24b.
The equation of state for seawater given in Figure 1.24 is somewhat unwieldly, and it is difficult to extract any
insight about the behavior of seawater simply by staring at the various terms of the equation. On the other hand,
it is of interest to know relative sizes of the salinity, temperature, and pressure effects in density. Beginning
with = (S, T, p), we can differentiate to find that
(25)
Dividing through by the density yields
(26)
where
(27)
Here we have introduced the quantity , the specific volume, which is simply the reciprocal of the density (ie, =
1/ ). The specific volume is used quite often in atmospheric problems, and we will have occasion later to use it
when we discuss the geostrophic relation and the dynamic method. The three coefficients in equation (26) are
themselves functions of salinity, temperature, and pressure. Some typical mid-water values are as follows:
(28)
As an example of how we might use these values, suppose that we inquire into the changes in salinity, temperature,
and pressure necessary to change
at a seawater parcel from a value of 1.027 to 1.028. This is a change of
10
approximately .1%; thus, from equation (26) this requires approximately at 10 C change in temperature, or a 1 psu
change in salinity, or a 250 decibar change in pressure of the seawater parcel. In the upper ocean, where pressure
effects are slight, it is mainly temperature and salinity that influence the density of seawater. At greater depths,
pressure has a very strong effect; that is why we remove it in most calculations of density.
§1.5 The Stability of a Seawater Parcel
As a final topic to consider at this point concerning the density of seawater, we examine the stability of a parcel
of seawater parcel that has been displaced vertically from its initial position. If this were to occur, the force of
gravity pulling down on the parcel would no longer be balanced by the buoyancy force pushing it up; as a result,
the parcel must accelerate. Consider the situation shown in Figure 1.25a: a parcel of seawater initially at position
1 is displaced vertically to position 2, in the presence of a background density gradient. The positions 1 and 2 are
near enough that the background gradient of density can be adequately approximated by a linear density profile,
as shown (ie,
is a constant). Let the vertical displacement from the initial position z0 be denoted by z.
Then Newton’s Laws (on a per unit volume basis) may be written as
(29)
The right side of Equation (29) may be interpreted as the buoyancy force acting on the fluid parcel. If the right
side of (29) is positive, then the particle is lighter (less dense) than its surroundings after being displaced, and
the particle will continue to accelerate upwards (ie, the left side of (29) is also positive). If the right side of
(29) is negative, then the particle is heavier (more dense) than its surroundings after being displaced, and the
particle will experience negative (downwards) acceleration and will hence return to its initial position. Equation
(29) may be rearranged to yield the differential equation
(30)
or
(31)
where
(32)
The differential equation (31) has the canonical form of a simple harmonic oscillator with frequency N. We may
interpret this to mean that, as long as N2 > 0, a particle displaced from its initial position as shown in Figure
1.25a will oscillate at a frequency N, which is a function of the stratification. The frequency N is known as the
Brunt-Vaisala frequency and appears in many problems in geophysical fluid dynamics. In shallow portions of
) takes on relatively large negative values, N may be as large as 2 /(5
the ocean, where stratification (ie,
minutes). In deeper water, where the stratification is smaller, N may be as small as 2 /(10 hours). A typical
profile of N for the North Atlantic, determined from CTD data, is shown in Figure 1.26.
Let us consider the implications of equation (31) in more detail. As noted, if N2 > 0, then (31) has solutions
of the form
(33)
Thus a particle displaced vertically from its initial position will oscillate at frequency N about its initial position if
< 0. As we have seen previously, this sign for the density gradient is indicative of a stable stratification,
wherein light fluid overlays heavier fluid.
If, on the other hand, N2 < 0, then (31) has solutions of the form
(34)
11
where
(35)
In this case it is possible that if a particle is displaced from its initial position its distance from its initial position
will grow exponentially, and the particle will not return to its initial position. This is indicative of an unstable
situation. Actually, this is just a quantitative way of stating what we already knew, that heavier fluid overlying
light fluid is not a stable situation. As equation (34) states, when heavy fluid overlays light fluid, even the slightest
perturbation of a particle in this fluid can lead to instability (ie, overturning of the fluid).
How much energy is required for such a vertical displacement to occur? It may be helpful to recall the simple
problem of a ball of mass m rolling down an inclined plane. If the plane is of height z0 , then the ball, initially
at rest at the top of the inclined plane, has a kinetic energy (ie, energy of motion) equal to zero and a potential
energy of mgz0 . When it has rolled to the bottom of the inclined plane, it will have kinetic energy equal to
mv2 /2, where v is the ball’s velocity at the bottom of the inclined plane, and a potential energy of zero. In
between the top and bottom, the total energy will be some combination of kinetic and potential energy. Since
total energy must be conserved in this problem, then the total energy at the top of the inclined plane must equal
the total energy at the bottom. We thus find that
(36)
or v = (2gz0 ).5 . Thus the velocity of the ball at the bottom of the inclined plane is determined by the height
of the plane.
With this simple model in mind, we consider the energy necessary to exchange two parcels of seawater in a
background density gradient, as shown in Figure 1.25b. We can compute the change in potential energy of
the system, PE, as
(37)
However, from equation (32) this can be rewritten as
(38)
The change in energy per unit mass required to exchange the two particles is given by N2 z2 . This quantity
is known as the available potential energy or APE of the system and is relevant to many problems in physical
oceanography as well as many biological problems. For example, the APE would be the amount of energy required
by an organism to swim against the density gradient from point 2 to point 1. Since the APE depends upon N2 ,
we expect that, for a given vertical displacement, more energy would be required in the thermocline, where the
stratification is strong, than in deeper water, where stratification is relatively weak. This suggests that vertical
mixing might be somewhat easier to accomplish in the deep waters of the world ocean than in the main thermocline.
12
Literature Cited
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Broecker, W.S. and T.H. Peng (1983) Tracers in the Sea. Eldigio Press, Lamont-Doherty Geological Observatory,
New York.
Bryden, H.L. (1973) New polynomials for thermal expansion, adiabatic temperature gradient, and potential
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Fofonoff, N.P. (1962) Physical properties of seawater. In: The Sea, M.N. Hill, ed., 1, 3–30. Wiley-Interscience,
London.
Guggenheim, E.A. (1950) Thermodynamics: An Advanced Treatment for Chemists and Physicists. North Holland
Publishing Co., New York.
Stommel, H.M. (1948) The westward intensification of wind-driven ocean currents. Transactions of the American
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Sverdrup, H.U., M.W. Johnson, and R.H. Fleming (1942) The Oceans, Their Physics, Chemistry, and General
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