Download OCEANOGRAPHY 502 Handout Notes #3 The Gravity Field of the

OCEANOGRAPHY 502
Handout Notes #3
The Gravity Field of the Earth, the Coriolis Effect, and Geostrophic Balance
§3.1 Introduction
In the seventeenth century the laws governing the motions of celestial bodies were active topics of study at
Eureopean universities. It was Newton who first suggested that the motions of celestial bodies was essentially
governed by the same laws as the motions of particles on the earth, such as apples falling from a tree (it is
suggested by several of Newton’s biographers that the “apple falling from the tree” incident never actually
occurred). The relationships between “gravity” for a stationary body on a stationary (ie, nonrotating) planet, for a
stationary body on a rotating planet, and for a moving body on a rotating planet are clearly important for a study
of physical processes in the ocean, since seawater is a fluid in motion on a rotating planet. The details of these
relationships, and the circumstances where rotation is or isn’t important, is the subject of these notes.
§3.2 The Gravity Field of the Earth—Stationary Particles on a Stationary Planet
Newton’s work led him to formulate what has become known as the Universal Law of Gravitation, which states
that the force between any two particles having masses m1 and m2 separated by a distance r is an attraction acting
along the line joining the particles and has a magnitude given by
(1)
where G is a universal constant having the same value for all pairs of particles. This is Newton’s law of universal
gravitation; the first particle attracts the second along the line connecting them by a force of magnitude F;
likewise, the second attracts the first in the opposite direction by the same amount. The scalar constant G,
called the constant of universal gravitation, has units of [length]2 /[mass x (time)2 ]. Its presently accepted value
is 6.673 10—11 newton-meter2 /kilogram2 .
This can be seen from an examination of Figure 3.1. The force exerted on m2 by m1 , F21 is directed opposite
to the displacement vector, r12 , of m2 from m1 . The force exerted on m1 by m2 , F12 , is directed opposite to
the displacement vector r21 , of m1 from m2 . Thus, F12 = F21 , and the forces form an action-reaction pair.
We can write in vector form that
(2)
where r12 is the magnitude of r12 . This notation explicitly shows that the force points in the opposite direction to
the displacement. Thus the gravitational force is attractive, and the force obeys an inverse-square relationship. Note
that we could have written the denominator in equation (2) as the square of r12 instead of the cube if we replaced
r12 by a unit vector
. In this case the relationship between force and distance would take its more usual form
(3)
Equation (3), essentially Newton’s result, is useful for gaining an understanding about the attractive properties
of point masses. However, for three-dimensional objects such as the earth some care must be taken in applying
equation (3). The approach necessary in calculating the force of attraction between two finite bodies is to compute
the contribution from each point mass comprising the finite body separately, and then to sum (ie, integrate) over all
the point masses. The net attraction is the vector sum of the contribution from all of the point masses separately.
Consider the gravitational attraction by a spherical shell on a point mass P having a mass m, as shown in Figure
3.2a. Suppose the shell has a thickness t that is small compared to its radius. We assume that each small piece of
the shell exerts on P a force which is proportional to the mass of the small piece and inversely proportional to
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the square of the distance between the small piece and P, and directed along the line joining them. Suppose that
a small piece of the shell at point A attracts m with a force F1 . A small piece of the shell, having mass equal
to that at A, that is equally far from m but diametrically opposite A, attracts m with a force F2 . The resultant of
these two forces on m is then F1 + F2 . Notice, however, that the vertical components of these two forces cancel
one another, and that the horizontal components, F1 cos and F2 cos , are equal. By dividing the spherical shell
into pairs of particles such as this, we can see at once that all transverse forces on m cancel in pairs. A small
mass in the upper hemisphere exerts a force having an upward component on m that will cancel the downward
component of force exerted on m by an equal mass located symmetrically in the shell. To find the resultant force
on m arising from the shell, we thus need only to consider the horizontal components.
Let us take as our element of mass of the shell a circular strip labeled dS in Figure 3.2a. Its length is 2 rsin ,
its width is rd , and its thickness is t. Hence, it has a volume
(4)
If the density of the sphere is , then the mass within the strip is
(5)
The force exerted by dM on the particle at P is horizontal and has the value
(6)
The variables x,
, and
can be related geometrically; it can be seen that
(7)
Using the law of cosines, this can be rewritten as
(8)
or
(9)
By differentiating equation (8), we then find that
(10)
or
(11)
We can then use (8), (9), and (11) in (6) to find that
(12)
This is the force exerted by the circular strip dS on the particle at P.
We must now consider every element of mass in the shell and sum over all the circular strips in the entire shell.
This operation is an integration over the shell with respect to the variable x, where x ranges from a minimum value
2
of R r to a maximum value R r. Carrying out the integration we find that
(13)
where
(14)
is the total mass of the shell. This is exactly the same result we would obtain for the force between particles of
mass M and m separated by a distance R. We have thus proved that a spherical shell of uniform density attracts
an external point mass as if all the shell’s mass were concentrated at its center.
A solid sphere can be thought of as composed of a large number of concentric shells. If each spherical shell
has a uniform density (even though different shells may have different densitites), then the result above applies
to a solid sphere as well as just a shell. Thus, a body like the earth, moon, or sun, to the extent that they
are spheres with radially symmetric densities, may be regarded gravitationally as point masses acting on other
masses located outside of them.
We can also compute the force exerted on a point mass inside the shell, as shown in Figure 3.2b. The situation
is similar to the previous one, except that now R is smaller than r. The limits of integration over x are now
r R and R+r. The integral in equation (13) becomes
(15)
Thus the gravitational force on a mass inside the shell is zero.
For a sphere of uniform mass and radius re , it is thus clear that a particle inside the sphere at some radius r such
that 0<r<re , is attracted only by the spherical mass distribution inside r; the mass between r and re exerts no net
attraction on the point mass. If MI represents the mass inside the point mass’s radius, then
(16)
The magnitude of the gravitational force on the particle is then
(17)
Thus, the force of gravity inside a sphere of uniform mass increases as r. From equation (13), once the particle is
outside the sphere (ie, r>re ), the force of attraction decreases as r—2 . This result is summarized in Figure 3.3.
It is often convenient to define a vector force F in terms of the gradient of a scalar potential, , so that F =
. In general any vector force may be defined in this way, as long as it is a conservative force; that is, the
force field is such that if a particle is displaced from its initial position along some arbitrary path in the field
and then is returned to its initial position, no net work is done on the particle. Many forces, such as gravity or
electromagnetic forces, are conservative, and thus may be defined in terms of a potential. Friction, on the other
hand, it not conservative, since energy (ie, work) is always lost during any particle motion, and a particle cannot
return to its initial position without some loss of energy. In the case of gravitational attraction of a particle of
mass m at radius r from earth’s center (where r≥re ), we have then
(18)
where the left side represents the force per unit mass, and
is called the geopotential and is defined by
(19)
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Here me is the mass of the earth.
Since the force of gravity is a function of distance from the center of the earth, it is useful to inquire as to the
variation of gravity from the bottom to the top of the ocean. Consider the situation shown in Figure 3.4. Let a
be the radius of the earth and h be the depth of the ocean. The quantity av is the average density of the earth,
including the solid earth and the ocean, and is the density of the ocean alone. To pose the question of the
variation of gravity over the depth of the ocean, we examine the ratio
(20)
From equation (17), we know that
(21)
The gravitational attraction at the bottom of the ocean, g(a h), is given by
(22)
Here the volume of the ocean has been approximated by multiplying the surface area of the ocean by its
depth; since h<<a this is quite a good approximation. Forming the ratio of these two, and making use of this
approximation, it is found that
(23)
Taking av = 5.5, = 1.0, h = 4 km, and a = 6.3x103 km, we find that = 1.0007. Thus, gravity is larger by .0007
at the bottom of the ocean than at the top, because the density of the ocean is less than the mean density of the earth
as a whole. But in general the difference in gravity between the top and bottom is small enough that it can safely be
neglected in nearly all calculations concerning the physics of ocean circulation, and so we shall henceforth do so.
§3.3 The Gravity Field of the Earth—Stationary Particles on a Rotating Planet
The results from the previous section are correct insofar as the earth is not rotating or the forces of attraction are
much larger than any effects of rotation. In general, however, a stationary particle on the surface of a rotating
earth will feel both the effects of gravitational attraction and rotation. To examine the effects of rotation, we begin
with the simple example of uniform circular motion, as shown in Figure 3.5. Here a particle is travelling in a
circular trajectory around a point C at a constant speed. If the rotation rate about the center is and the particle
is a distance r from the center, then this speed is just r. The particle’s position is P at time t and P′ at time
t+ t. Although the particle’s speed is constant in time, the velocity of the particle clearly cannot be, since the
direction of the velocity vector is changing at each instant. Thus, the particle is accelerating, and from Newton’s
Laws a net force is implied. If the velocity vector is v at time t and v′ at time t+ t, it is not difficult to show
(see any beginning university-level physics textbook) that the vector difference in velocity, v, over the small
interval t points to the center of the circle, C. Thus, this is the direction of the acceleration. This implies that
there is always an acceleration towards the center of the circle in uniform circular motion. This acceleration is
called the centripetal acceleration. In vector form, if
is the rotation vector and r is the position vector of the
rotating particle, then (again see any elementary physics textbook) the vector velocity of the particle v is
.
The centripetal acceleration is
, or
, pointing towards the center of the circle.
The description of the circular motion by an observer in an inertial coordinate system (ie, one that is stationary
with respect to distant, fixed stars) is that the particle is not in equilibrium (ie, it is accelerating, and the forces
upon it are not balanced). If the particle is a ball on a string, as shown in Figure 3.6, the observer in the
intertial frame would say that the force directed inwards is manifested by the tension T in the string that is
necessary to hold the ball in place.
On the other hand, in oceanographic applications it is nearly always advantageous to pose the problem in the
rotating coordinate system (that is, one embedded in the earth, such as east-west, north-south, and up-down). In
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the case of the ball on the string in Figure 3.6, the rotating coordinate frame would be the coordinate system of
an observer moving so as to always face the ball. In such a case the balance of forces as seen in the inertial
frame is the same as before. However, in the rotating frame the observer would always see the ball in the same
position, and hence the observer would describe the ball as stationary and in equilibrium (ie, with the forces
balanced). This observer would say that the inward-pointing tension in the string T was balanced by an outward
pointing centrifugal force Fc ; the concept of the centrifugal force was essentially invented by the observer in order
to balance the tension. In this sense, centrifugal force is sometimes referred to as a “pseudo-force” or a “fictitious
force”, but there is really nothing fictitious about it. To the observer in the rotating frame, it is a perfectly
legitimate description of what is happening. It is also necessary: Newton’s Laws are only defined for inertial
coordinate systems; if we are to use them in noninertial systems such as the rotating earth, then some adjustment
to Newton’s original laws are necessary, and one of these adjustments is the centrifugal force.
On the earth, the centrifugal force is due to the earth’s rotation. If the rotation vector of the earth is given by
, then a stationary particle on the rotating earth is moving with a velocity v given by v =
where r is
the position vector pointing from the rotation axis of the earth to the particle position. The centrifugal force
Fc is given by
or
. Thus the magnitude of the centrifugal force is 2 r, where is the
magnitude of earth’s rotation
and r is the distance from the rotation axis, and the centrifugal force points
outward from the rotation axis of the earth.
We now proceed to consider the combined effects of centrifugal force and gravitational attraction, which is the
situation for a stationary particle on a rotating earth. The “true” gravity will not be the force of attraction (discussed
in §3.2) alone, but will instead be the vector sum of the attractive and centrifugal forces, as shown in Figure 3.7b.
If the radius of the earth is ro , then the distance from the rotation axis at any latitude is ro cos . Thus, the
magnitude of the centrifugal force is 2 ro 2 cos , and the direction points outward from the axis of rotation; we
will call this direction . It is useful, however, to define the centrifugal force in a coordinate system consistent
with earth-based thinking, such as east-west, north-south, and up-down. The centrifugal force does not point in
any of these directions (Figure 3.7b), and must first be resolved into these components before our definition can
be useful. We will let be a unit vector pointing north, and be a unit vector pointing up (ie, away from earth’s
center along a radius). We can write the centrifugal force (per unit mass) as
(24)
The vector
can be resolved into north and up components to yield
(25)
Putting this into the expression for Fc , we find that
(26)
The force of attraction, by definition, is just
(27)
Thus the total force FT acting on the particle is the vector sum of FC and FA , or
(28)
Note that the geopotential
associated with this force (FT =
) is
(29)
The force FT is the true gravity field of the earth. Note that the true gravity field does not necessarily point
towards the center of the earth, as shown in Figure 3.7a. In general true gravity is a function of latitude: at
the equator ( = 0) the gravity vector points to the center of the earth, but the magnitude of gravity is the least
of anywhere on the earth since the centrifugal force and gravity directly oppose each other. At the poles ( =
/2), the gravity vector again points to the center of the earth, and the magnitude of gravity is largest, since
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the centrifugal force is identically zero along the rotation axis. At mid-latitudes, between the equator and poles,
gravity does not point to the center of the earth, and the magnitude is somewhere between the equatorial and
polar values. However, clearly the true gravity vector points nearly to the center of the earth everywhere, since
at the surface of the earth the force of attraction is much larger than the centrifugal force. This can be seen from
a comparison of the components of the two forces in the
direction:
(30)
Taking = 7.3 10—5 sec—1 , ro = 6.3 108 cm, and g = 980 cm sec—2 , we find that
is about 3 10—3 at
most (ie, at the equator). So the centrifugal force isn’t large, but it may become important at distances from
earth’s center that are larger than earth’s radius (consider an orbiting satellite, for example). [Note that we could
carry out a similar analysis to examine the angle of deflection of true gravity from the axis, and we would
find that the deviation is at most about 0.2 , at a latitude of 45 .] Additionally, in the study of tides it is found
that the centrifugal force induced by one body rotating about another plays a crucial role in the generation of
tidal bulges on opposite sides of the central body.
§3.4 Moving Particles on a Rotating Planet—The Coriolis Effect
The most general case of the combined effects of gravity and rotation occurs for moving particles on a rotating
planet. In this case the effects of attraction, centrifugal force, and Newtonian accelerations all play central roles.
Consider the situation depicted in Figure 3.8. A particle initially at rest at point A is thrown towards point B.
Points A and B are at the same longitude but different latitudes. Newton’s laws tell us exactly what will happen
in this case: assuming no friction is present, the particle at A will accelerate under the influence of gravity only
(since that is the only force present), and the path will be a parabola as the particle moves horizontally and up and
down. If the particle is initially located at the point (xo yo ,zo ), has mass m, and the initial velocities are Vx , Vy , and
Vz in the x, y, and z directions, then it is easy to show using Newtons’ Law in an intertial frame that
(31)
and
(32)
If the particle is launched directly south, then the only east-west initial velocity (Vx ) is the velocity due to the
rotation of the earth. If ro is the radius of the earth, then this velocity is just ro cos (where is the latitude),
as we have already seen. Thus, a stationary particle at A is moving slower than a stationary particle at B. An
observer in the rotating coordinate frame (ie, embedded in the earth) might expect a particle launched at A towards
B to eventually reach B, but will be surprised to find that the particle appears to curve to the right when looking
south, as shown in Figure 3.8. This is just because the particle at A can’t keep up with the stationary particle at B
because of the slower initial velocity at A. Thus, the observer at A will state that a force acted on the particle that
turned it to the right, making it land behind point B. The reverse would happen for a particle launched from B to
A, again with an apparent turn to the right. This “fictitious” force is called the Coriolis force, and is necessary to
explain things consistently in a rotating coordinate system. However, the force is not really fictitious, as it makes
perfect sense to the observer in the rotating coordinate frame.
The task here is essentially to repair Newton’s equations, derived for an inertial coordinate system, to that they
work in a rotating coordinate frame. In the inertial frame, we have
(33)
In the rotating frame this cannot completely describe the balance adequately for the reasons just stated. We can
attempt to repair Newton’s Law by simply adding the Coriolis force, so that
(34)
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Alternatively, we may decide to make the repair by adding an extra acceleration term to the right side of (33)
and referring to is as the Coriolis acceleration, so that
(35)
It is clear that the repair can be done either way, and that FCoriolis = maCoriolis . We will examine the form of the
Coriolis terms shortly. For now, however, it is useful to examine the question of the importance of the Coriolis
effect. If two people a few meters apart throw a ball to each other, clearly they do not take such rotational effects
into account. On the other hand, it is known that such effects are of major importance in the study of atmosphere
and ocean dynamics. How can we know when to take the Coriolis effect into account, and when to ignore it?
We can examine this question with Figure 3.9. A particle moves on a rotating earth from point A to point B. The
distance between these two points is L, and the particle moves a a characteristic speed Uo . Thus, the time tAB
for the particle to move from A to B is L/Uo . The characteristic rotation time for the earth te is just —1 , where
is the rotation rate of the earth. The ratio of these two times is then just
(36)
The quantity Ro is called the Rossby number (after C.G. Rossby, an outstanding meteorologist of the first half
of the 20th century). Our intuition tells us that when tAB is short compared to te (large Rossby number, the
case of two people throwing a ball), then the Coriolis effect is unimportant. Alternatively, in the case when
tAB is long compared to te , (small Rossby number, the case of oceanic and atmospheric flows), the we must
take the Coriolis effect into account.
We can examine some examples of these two extremes. Consider the old adage that water flows down a drain
clockwise in the northern hemisphere and counterclockwise in the southern hemisphere, (presumably) due to the
Coriolis effect. Suppose that the water is flowing at about 10 cm/sec (Uo ) and the drain is about 4 cm across
(L). If the earth is rotating at a rate of once per day, we find that Ro is about 2x105 . Thus, it is unlikely that
the Coriolis effect plays any role at all in the flow down the drain; the old adage can’t possibly be true, as other
things that vary from drain to drain will be much more important in determining the nature of the flow than the
Coriolis effect. On the other hand, consider the Gulf Stream. If we take a typical speed in the upper ocean in
the Gulf Stream to be 50 cm/sec (Uo ) and the width of the Stream to be about 100 km (L), then it is found that
Ro is about 7x10—3 . This is quite small and suggests that the rotation of the earth and Coriolis effects will play
a major role in determining the nature of the Gulf Stream.
What exactly is the form of the Coriolis force? This may be derived rigorously from first principles, and such a
derivation is done in many advanced mechanics textbooks. Essentially one just begins with a system of orthogonal
unit vectors and then allows this system to rotate. Any time derivitive taken in an inertial frame can then be
written in terms of these unit vectors and their time derivitives. If second derivitives are taken, then terms of the
form of accelerations arise, and these may be substituted into Newton’s Law. This yields the form of the Coriolis
terms, which essentially relate the inertial frame accelerations to the rotating frame accelerations. If this procedure
is carried out (see the text Mechanics by K.R. Symon, Addison-Wesley, 1971, for a complete blow-by-blow
derivation), it is found that the Coriolis acceleration per unit mass is
(37)
where
is the rotation vector of the earth and u is the particle velocity. Alternatively, the Coriolis force
per unit mass is just aCoriolis .
We can examine the Coriolis effect in detail by expanding equation (37). As shown in Figure 3.10, the rotation
vector can be resolved into a north-south (ie, component) and up-down (ie, component) to yield (for the
Northern Hemisphere)
(38)
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where
is the latitude. Expanding the Coriolis force as the determinant of the cross-product matrix, we have
(39)
where the velocity u has east, north, and up components (u,v,w) and
is a unit vector pointing east.
For the N. Hemisphere:
Initial u
Coriolis Force (= -2
North (0,v,0), v>0
East
South (0,v,0), v<0
West
East (u,0,0), u>0
South, Up
West (u,0,0), u<0
North, Down
Up (0,0,w), w>0
West
Down (0,0,w), w<0
East
u)
It is clear from the table that the effect of the Coriolis force is always to induce a turn to the right in the Northern
Hemisphere; this is consistent with the reasoning applied to explain Figure 3.8. However, it appears that up-down
motion may also be induced by the Coriolis effect. For purely up or down initial motions, there is a west or
east displacement induced by the Coriolis effect. This can be understood by considering the problem of a ball
dropped from a tower. If the stationary tower stands a height h above the surface of the earth, then it is moving
around the rotation axis of the earth at a speed of approximately (ro +h). A point on the ground, however, is
moving only at a speed of ro . Thus, the ball dropped from the tower will outrun the point on the ground and
will appear to land east of the tower. Alternatively, a ball thrown straight up at from the ground at the base of
the tower will be slightly west of the tower when it reaches the height of the top of the tower, since it is moving
slightly slower than the top of the tower and gets left behind.
The initially purely east-west motions noted in the table also have Coriolis deflections, although that this should
occur is perhaps not readily apparent from an examination of Figure 3.8 alone. The cause of this can be seen
in Figure 3.11. The direction of the Coriolis force for purely east-west initial motions is outward and normal
to the rotation axis. However, in general this vector does not lie along any of the coordinate axes (ie, east,
north, or up). Hence, in general the Coriolis force for initially east-west motions will have both north-south
and up-down components. This is strictly a consequence of spherical geometry; if the earth were a cylinder,
there would be no north-south or up-down displacements for initially east-west motions [this is indeed the case
near the equator, where the earth is locally approximately a cylinder, as can be verified by an examination of
equation (39)]. The Coriolis effect exists because of the rotation of the earth, but the particular form that it takes
results from the presence of spherical geometry.
A further examination of equation (39) suggests that some simplification of the full Coriolis terms can be made.
In the eastward component, we note that
(40)
The cotangent of will generally be of the order of unity or less, except near the equator. Since for large-scale
motions the ratio (w/v) is generally very small (ie, probably something like 10—5 or so), this implies that the
first term in the east component of equation (39) can be safely neglected with respect to the second. Precisely
because the vertical velocities are so small (and essentially unmeasurable) compared to the horizontal ones, we
generally won’t examine the Coriolis effects of the vertical component of velocity or acceleration anyway; clearly
the Coriolis accelerations are much smaller the gravitational accelerations, and it is the gravitational accelerations
that are balancing the pressure gradient term in the hydrostatic relation, the vertical balance of forces appropriate
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for describing the large-scale ocean circulation. As a result, we can also safely ignore the vertical component of
equation (39). The Coriolis effect then becomes, to a good degree of approximation,
(41)
where f = 2 sin is known as the Coriolis parameter. Using this simplification, we can write Newton’s laws
for a rotating coordinate system (per unit mass) as
(42)
This suggests that if we have steady motion in the rotating frame (ie, the accelerations are zero), then unless u
= 0 identically we must have some imposed force. Similarly, if the sum of the forces is zero, then either the
motion must be time-dependent or it must be zero. Additionally, it is clear that the work associated with the
Coriolis effect must be zero. This should be clear, since only one event is occurring, even though observers in
different frames of reference may describe it differently. Since only one event happens, then a fixed amount of
energy is associated with that event and observers in the two frames must agree on that quantity of energy. The
energy (or work) associated with the Coriolis effect is
(43)
Thus, there is no work or energy associated with the Coriolis effect.
§3.5 Geostrophic Balance
We can actually specify the forces on the right side of equation (42); we choose them to be the pressure gradient
force and the force of gravity. The equation of motion in the rotating frame then becomes
(44)
If the motion is approximately steady, so that we can neglect the local accelerations with respect to the Coriolis
effect (ie, the small Rossby number case appropriate for the large-scale ocean), this vector partial differential
equation can be written as three scalar equations of the form
(45)
The first two of these equations are collectively known as geostrophic balance (from the Greek “earth-turned”)
and represent the balance of forces in a large fraction of the global ocean at large scales. The third equation
is just our old friend hydrostatic balance. How geostrophic is the ocean? If we take U as a characteristic
velocity scale and T as a characteristic time scale of some mid-ocean, large-scale process, then the acceleration
terms can be modelled roughly as
(46)
The Coriolis term can be simply modelled as fU. The ratio of the accelerations to the Coriolis term is then just
another form of the Rossby number,
(47)
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as long as we are thinking about processes that occur on time scales long with respect to a day (here Te is the
characteristic rotation time of the Earth, approximatly equal to 1/f; recall that f is roughly the same size as the
rotation rate of the earth, except at the equator). Thus, except near the equator, the geostrophic approximation
should work quite well for large-scale phenomena in the ocean.
The geostrophic approximation puts strong constraints on the nature of the velocity field; the most striking result of
the geostrophic relation is that the flow is not down the pressure gradient (as would be the case for flow in a pipe:
push at one end of the pipe and water comes out the other end), but is instead along pressure contours and across
the pressure gradient. This is the same situation as with high and low pressure areas on a weather map, which
most of us are used to seeing; generally the large-scale weather systems in the atmosphere are also approximately
in geostrophic balance. Thus, for the sea level across the Gulf Stream (Figure 3.12) from north to south, we
expect an eastward flow (ie,
) north of the maximum point in sea level and a weaker westward flow
(
) south of the maximum point. Taking h to be 1 meter in figure 3.12, LA to be 200 km, and the
earth to be rotating once per day, the eastward geostrophic velocity in the Gulf Stream is predicted to be about 50
cm/sec in Figure 3.12, which is not a bad order of magnitude estimate for the surface flow in the Stream east of
Cape Hatteras. The geostrophic approximation is a powerful tool for examining the flow in the large-scale ocean.
This use of sea level and the geostrophic approximation to infer the velocity field suggests that, if sea level could
be monitored on a global basis, the geostrophic portion of the ocean circulation could be deduced, at least near the
sea surface [at deeper levels, where the circulation may be associated with the stratification of the ocean, deducing
the circulation may be a more complicated proposition, as we will see later]. Such monitoring of sea level is
now possible from orbiting satellites, using the technique known as satellite altimetry. The method involves the
transmission of microwave pulses towards the Earth from an orbiting satellite and the reception of the reflected
signal back from the Earth’s surface. By knowing the properties of the medium between the satellite and the
planetary surface, the propagation speed as a function of frequency can be determined, and hence the two-way
travel time of the electromagnetic pulse can be used to yield a distance between the satellite and the sea surface.
The position of the satellite can be determined quite precisely by tracking it from ground stations, and hence the
absolute topography of the sea surface can be determined. In practice, the height of the sea surface can be deduced
to within a few centimeters from this method, yielding a map of sea level in the vicinity of the satellite. Such
satellites can map the entire Earth’s surface on roughly a 10 day basis, thus providing a powerful technique for
inferring the surface geostrophic circulation. An example of this technique are the satellite ground tracks shown in
Figure 3.13, taken from the work of Cheney and Marsh (see Journal of Geophysical Research, V. 86, pp. 473–483,
1981) that used the Seasat data [Seasat was the first altimetry mission, which took place in 1978; the mission
ended when the satellite failed prematurely after only 90 days in orbit]. Figure 3.13 shows a portion of the western
North Atlantic near the Gulf Stream, where numerous Gulf Stream rings are clearly present. The altimetric sea
surface height, shown in Figure 3.14 for various orbits, clearly shows an increase in sea level from north to south
near 38 N of about 1 meter, followed by a slow decrease in sea level towards the Equator. This is similar to the
model sea level profile shown in Figure 3.12. The rapid north to south increase near 38 N represents the surface
height signature of the Gulf Stream (check yourself that geostrophic balance yields an eastward flow), and the slow
decrease towards represents the broad return flow of the subtropical gyre of the North Atlantic. The presence of
several cold rings south of the Gulf Stream can clearly be seen from their sea level signature in Figure 3.14.
On a global basis, the technique of satellite altimetry can potentially be a powerful tool for describing the
large-scale ocean circulation. As shown in Figure 3.15, it is not difficult for an orbiting satellite to map the sea
surface completely over a time interval of 10 days or so, thus providing a complete map of the surface geostrophic
circulation for the whole world ocean over this time frame. There is a serious shortcoming to this method, however:
what must really be known is the height of sea level with respect to the geoid, not with respect to the surface of the
Earth, as shown in Figure 3.16. From the definition of the geopotential ( ), we know that the force of gravity over
the Earth is proportional to
. As a result, the equilibrium sea surface must be a contour of constant , since
there would be a net force, and hence an acceleration, on the sea surface if it were not in the shape of a contour
of constant . For there to be a geostrophic ocean circulation, it is necessary for the sea surface to depart at some
point from a constant contour, thus producing pressure gradients. As a result, the geoid must first be removed
from any profile of sea level before any geostrophic computations can be undertaken. Since published values for
the geoid are generally not good to less than a few meters over most of the Earth (and are considerably poorer
than this in many places), it is often difficult if not impossible to determine the sea level with respect to the geoid.
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On the other hand, even without a good model for the geoid it is possible to examine the variability of the ocean
circulation from an orbiting satellite. The idea in this case is simply to note that, while the ocean may change
from one satellite orbit to the next, the geoid must generally remain the same. Thus, orbit-to-orbit changes in
measured sea level reflect true variability of the ocean, even if the absolute circulation is not known. As an
example, in Figure 3.17 sea level measured from a number of satellite passes over the Gulf Stream is shown; the
ground tracks for these orbits were within a few kilometers of each other, over the 90 day Seasat mission. The
sharp increase in sea level from north to south near 40 N can clearly be seen, denoting the position of the Gulf
Stream, and the weak decrease in sea level between 36 N and the Equator delimits the broad return flow of the
subtropical gyre. On shorter spatial scales, however, it is clear that there is variability in the ocean from orbit to
orbit. This variability can be quantified for the whole world ocean, as shown in Figure 3.18: here, contours of
surface variability over the 90 day Seasat mission have been produced, showing that the western boundary regions
in all the oceans, plus the Antarctic, are areas having the most variability in sea level. From this, we deduce
that the surface height variability in these regions is probably the strongest of anywhere in the world ocean. In
recent years the variabiltiy in sea level has been greatly improved by the Geosat and Topex/Poseidon missions,
whose accuracy in measuring sea level is far greater than what could be achieved by Seasat. Nevertheless, the
most basic conclusions concerning spatial variations in sea level over the ocean as seen by Seasat are consistent
with what is seen today by the more accurate altimetry missions.
§3.6 Geostrophic Balance Applied—The Dynamic Method
For all its power as an analysis tool, however, satellite altimetry can really only tell us about the geostrophic
circulation in the vicinity of the sea surface. Since the ocean is generally stratified in density, the deeper circulation
depends not just on the sea level distribution but also on the internal distribution of temperature and salinity.
Consider the sections of temperature and salinity across the Gulf Stream as it leaves Cape Hatteras, shown in
Figure 3.19: there are clearly internal pressure gradients associated with the roughly 500 meter changes in the
depths of isotherms and isohalines across the Gulf Stream. We expect that these internal pressure gradients will
result in internal geostrophic flows; however, it appears that property surfaces vary in depth internally a good
deal more than change in height of the sea surface. With this in mind, how do these internal pressure gradients
compare to the pressure gradient associated with the slope of sea level?
We can construct a simple model to address this question. Consider the two columns of fluid shown in Figure
3.20, having densities and heights as shown. We may further specify that
(48)
and
(49)
The pressure at the bottom of the left column is then
(50)
In a similar manner, we can write the pressure at the bottom of the right column as
(51)
The difference in pressure
p at the bottom of these two columns is then
(52)
Thus, the difference in pressure between these two columns can be caused by either a difference in the densities
internally, or by a difference in the heights of the interface in the two cases. As an example, consider the air-sea
interface. The density difference between the atmosphere and ocean is nearly 1 g/cm3 ; for a “sea level” change of
1 meter, this amounts to a pressure change of about 1 decibar at the bottom of the left column. On the other hand,
the internal density change (as across the Gulf Stream in Figure 3.19, for example) is only about 0.01 g/cm3 ; this
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implies that h, the interface height, could be as large as about 100 m in order to yield the same change in pressure
of about 1 decibar at the bottom of the column 2. We are left with the impression that large vertical excursions of
the depths of isopycnal surfaces in the water column have roughly the same effect on the pressure field, and thus the
field of geostrophic velocity, as as change in the height of sea level of a meter or so over the same horizontal scales.
In order to apply the geostrophic relation to the case of a stratified ocean, some modifications to equation
(45) are necessary. In particular, suppose that the v component of (45) is differentiated with respect to the
vertical coordinate z:
(53)
Here the first differentiated term inside the brackets has been dropped with respect to the second. It is not
difficult to show that their ratio is about
, where is a typical density of the water column and
is the typical departure from this value; and pH and pV are the typical horizontal and vertical changes
in pressure over the water column for the phenomenon of interest. For the Gulf Stream (an extreme case), for
to be .03 g/cm3 , and the horizontal and vertical pressure changes
example, if we take to be about 1 g/cm3 ,
to be 1 dbar and 3000 dbar respectively, then we find that this ratio is about .001—small enough that we can
neglect the first term with respect to the second.
The expression on the right side of (53) can be further simplified by taking advantage of the fact that we are
interested only in cases where the ocean is approximately hydrostatic, so that
(54)
In a similar fashion we can operate on the u part of (45) to find that
(55)
This pair of equations (54 and 55) are collectively known as the thermal wind equations. They specify the change
in the velocity field with depth that is induced by a horizontal change in the density field. Thus, for the Gulf
Stream section shown in Figure 3.19, we have
> 0, so that
> 0 also. The geostrophic velocity
must be increasingly to the east in this section at higher levels of the water column. A stylized version of this
situation can be seen in Figure 3.21. The density at any level is greater on the northern side of the Gulf Stream,
yielding an eastward velocity that increases at higher levels of the water column. However, it is impossible from
this result alone to determine the absolute velocity; as shown in Figure 3.21, there are an infinite number of
possible velocity fields that can exist for a given horizontal change in the density field (3 possibilities are shown
in Figure 3.21). Thus the thermal wind equations can only tell us about the shear field, not the absolute velocity
itself. This is because the integration of equation (54) with respect to z yields the horizontal velocity u, plus an
unknown constant of integration. It is this unknown constant that accounts for the nonunique solution for the
horizontal velocity, as shown in Figure 3.21. To remove this degeneracy, additional information must be supplied
in order to adequately evaluate the constant of integration.
Before further exploring the vertical integration of equation (54), it is useful to consider exactly what can and
cannot be measured in the ocean. The density field, on the right side of (54), can be measured adequately over
the entire water column using modern shipboard instrumentation such as a profiling CTD system. The depth z,
however, is not easily measurable by any standard means. In fact, depth is nearly a useless variable in most
oceanographic contexts. Pressure, which can be measured by a variety of methods, is a much more useful variable
to convey “depth” than is the vertical coordinate z.
With this in mind, we can integrate (54) to find that
(56)
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The left side of this equation represents the geostrophic velocity relative to some level zo , while the right side
is essentially the horizontal gradient of the hydrostatic pressure. As noted above, however, we would prefer to
use pressure as the vertical coordinate rather than depth, since depth itself is not easily measured. With this in
mind, we can change the variable of integration and the limits of integration in (56) by again using the hydrostatic
relation to relate z and p. By making use of the specific volume, , rather than the density, (56) can be rewritten
as (see Chapter 8 in Pond and Pickard for the details)
(57)
where the quantity po is now the reference pressure. Since specific volume is just the reciprocal of density, and
density changes by only a few percent from a value near 1, we expect that the specific volume also changes
by only a few percent from a value near 1. We can write that = o + , where is called the specific volume
anomaly and represents all of the departure of from its near-unity value. The part of the integral on the right
side of (57) due to o will then be a constant and will just be zero when differentiated with respect to x, and
only the part of the specific volume will contribute to the integral. Making these substitutions, we can write
the stratified geostrophic equations for u and v as
(58)
where the quantity
D is known as the dynamic height anomaly and is given by
(59)
Using equations (58) and (59) to determine the relative geostrophic velocity field from the density field is known
historically as the dynamic method. There are literally thousands of pages of physical oceanographic research
in existence that employ these equations for estimating the oceanic flow field, and they must be considered to
be one of the very most important tools available to the physical oceanographer for determining the circulation.
However, equation (58) still has the undesirable property that the velocities are only determined up to some
unknown constant or reference velocity. Some understanding of the nature of this problem can be seen from
an examination of Figure 3.22. In Figure 3.22a, pressure surfaces are always parallel to depth surfaces. Thus,
through the hydrostatic relation the density and pressure are functions of depth only. As a result, when horizontal
derivitives are taken of density, pressure, or dynamic height in equations (56) or (58), the result is zero, since
density and pressure are only functions of z. In Figure 3.22b, the more common situation in the ocean, pressure
is not parallel to depth, and hence through the hydrostatic relation density and pressure may have horizontal
gradients, and thus some geostrophic velocity may exist. In general, however, it is difficult to measure the absolute
inclination of pressure surfaces with respect to depth surfaces (ie, z). This is because in general the deviation of
pressure in some horizontal direction along a constant depth is quite small—as we have seen, a one meter (or
one decibar) change is about the maximum that one would expect to see across the major current systems of the
world ocean. Thus, it is difficult to remove this degeneracy.
This can seen in more detail in Figure 3.23. Here we have three “water masses”, A, B, and C, arranged in the
sequence cold-warm-cold. In 3.23a, we assume that at some deep level zo the horizontal pressure gradient is zero
(that is, pressure is parallel to depth). At this level there can be no flow, since the horizontal pressure gradient
along this surface is zero. Above this level, however, geostrophic flow is possible. At position 1 (assuming that the
left side of the page is north, and the right side south), we have
> 0, and thus the eastward velocity increases
up (ie,
> 0 also). At position 2, the sign of the density gradient is negative, and the eastward velocity must
decrease upwards. Since the velocity must be zero at the level zo , there is only one possiblity for the absolute
velocity, as shown at the bottom of Figure 3.23a. However, it is also possible that the situation depicted in Figure
3.23b occurs; here, the sea surface is level and there exists a pressure gradient at depth. In this case, the geostrophic
flow at the sea surface is zero, but because since the signs of the density gradients are the same as before, the
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shear field is the same as in 3.23a. The result is the velocity field shown at the bottom of Figure 3.23b. Clearly
the fields depicted in 3.23a and 3.23b are quite different, and they are different only because of a difference in the
assumption of where the level surface occured in the water column. The problem, of course, is that in practice we
cannot tell these situations apart, and hence our ability to know the absolute geostrophic flow field is limited.
How can the absolute geostrophic velocity be determined from hydrographic data? The classical answer is to
assume that the flow is zero at some level (the so-called level of no motion) and to use this level as the point to
set the constant of integration to zero, hence yielding the absolute geostrophic flow at all other levels. This may
sometimes be a simple and useful method for estimating the circulation, especially when quantitative accuracy is
not required. If the circulation was actually known a some level (a level of known motion) then a value for the
constant could be determined from that knowledge and the absolute geostrophic velocity field could be estimated
(note that it is only necessary to know the absolute velocity at a single level in order to determine it at all others).
It is possible in principle to measure the absolute velocity at some level by using moored current meters or freely
drifting floats over some local region of the ocean and to obtain the absolute geostrophic velocity field at all depths
in this region using this approach. However, in practice this approach is difficult and expensive over a large region
of the ocean, such as a single ocean basin, and it has rarely been done. A third approach might be to use the
geostrophic approximation in conjunction with some additional information that might be useful for evaluating the
unknown constants, such as constraints on the mass balance or heat balance over an oceanographic region; this
leads directly to the modern idea of inverse models that apply such constraints in conjunction with geostrophic
and hydrostatic balance to produce an estimate of the circulation over hundreds to thousands of kilometers. While
these methods are extremely powerful, they commonly do not yield a unique solution for the flow field, so they
might not always be an improvement over the older method of simply assuming a level of no motion.
Perhaps the most promising method of deducing the absolute geostrophic circulation is to combine the satellite
methods described in §3.5 with the dynamic method discussed in this section. In principle, the use of satellite
altimetry can yield a good estimate of the flow at the sea surface [nb. the geoid must be known to do this!], by
measuring the height of sea level over the globe, as shown in Figure 3.24a. This then can be used to evaluate the
constant of integration in (58) at the sea surface. The hydrographic data (Figure 3.24b) can then in principle be
used to estimate the relative geostrophic circulation below the surface. If these two techniques were combined,
the absolute geostrophic flow at all levels would be known (there is no Figure 3.24c, since this has not yet
been done). This procedure could in principle be carried out globally, wherever hydrographic data exist and
the flow is geostrophic, and for this reason the combination of satellite altimetry with in situ hydrographic data
represents a technique with great potential that will undoubtedly be employed in the coming years to improve
our estimate of the global ocean circulation.
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