In general, a given thermal or kinematic quantity in the ocean depends not only upon position but upon time as well. For example, the fields of temperature and salinity vary with time. The only stationary field which will be rpesumed is that of the earth's gravity. The magnetic field of the earth is known to vary with time.

Consider any scalar quantity *S* which is a continuous
function of the four independent variables *x*, *y*, *z* and
*t*, with *t* being time, for which the space and time
derivatives exists.

The total rate of change of *S* with time is in general
given by the expression

where the differential displacements *dx*, *dy*, *dz* are specified
for the elapsed time *dt*. If we let

be the components of fluid velocity in the *x*, *y*, *z* directions
at the point *x*, *y*, *z*, then the expression (22) takes on
an important enough meaning to attach a special symbol to
the operation therein indicated. We shall define
the operator *D*/*Dt* such that

This will be referred to hereafter as the
**material time rate of change** of S.
The operator can also be written in the form

where

is the fluid velocity vector. The
term represents the
local time rate of change of the quantity *S* at a fixed
point.
The term is a
scalar representing the advectional or field changes in the fluid
associated with the motion of the fluid. The latter quantity
can vanish in one of three ways:

If one of the above conditions is true then there is no
advection of the property *S* by virtue of the fluid velocity and

However, in general the material rate of change differs from the
local rate of change of *S*.

We can attach a physical significance to the term
*DS* / *Dt* if we select a particular small particle of fluid.
Let the symbol *A* identify this particular parcel.
Then it can be shown that

where denote the value of *S* of the parcel of fluid
at any time *t*, and the subscript on the right ahnd term of (28)
denotes the value of the material derivative when each term
of the expression is evaluated at the point occupied by
the parcel *A* at the instant *t*. More specifically,
if represent the position of the
parcel *A* at time *t*, then

where it is understood that are functions
of time which describe the path of the particular
parcel *A*. The term implies that each
term of the expression

is to be evaluated at the position
at the instant *t*, after carrying
out the partial differentiations indicated.

It is clear that physically represents the time
rate of change of *S* in the fluid contained in the small
parcel *A* as the latter is carried along with the fluid
velocity. For this reason we might refer to
or as the individual time rate of change
of *S* for the parcel *A*. Consequently, we can intepret
*DS*/*Dt* in general as the individual time rate of change
of that parcel which happens to occupy the
position *x*, *y*, *z* at the instant *t*.

If we are interested in the time rate of change of a given
property for a particular parcel of fluid, it is uually
somewhat easier to compute *DS*/*Dt* and then evaluate
this at the position of the parcel, rather than evaluate
and take its time derivative. This is
particularly true if the field of is specified.
This leads into the two methods by which the field of motion
can be described.

We can specify either the field of or we can
specify the paths (trajectories) of the particles of
fluid. The former is referred to as the
**Eulerian description** of motion while the latter is
endowed with the title of **Lagrangian description**
of motion. (It is believed however that Euler is
actually responsible for both approaches.)

Specifically, in the Eulerian system we specify implying that

are known.

In the Lagrangian system we specify

for each particle of fluid.

The connection between these two approaches leads to the differential equation for the trajectories, namely:

The above equations determine the path of the particle *A* if
the position is specified at some initial instant in its path
history.

We will first illustrate the connection between the Eulerian and Lagrangian descriptions of motion with a simple one-dimensional problem.

Suppose

where *a* and *b* are constants. Then the equation for the
trajectory of *A* is

This is a linear equation of the type discussed by Sokolnikoff and Sokolnikoff (1941) (Sec. 85). The solution of (31) can be shown to be

where

it being required to specify the initial position
for the particle *A*. By substituting the solution for
in Eq. (31) we find that the velocity of
particle *A* at any time *t* is

which approaches the constant value *b*/*a* with time, if
.

The Eulerian and Lagrangian modes of depicting the fluid motion for this example are compared graphically on the diagram of Fig. 1.03-1.

The diagram shows isolines of *u* (dashed lines) and also the
path histories of several different particles which start at
different points equally spaced along the *x* axis at the
initial instant. For simplicity, the values of *a* and *b*
were taken as unity in the illustration. It is evident
from this picture that, because of the crowding together of
the particles as time progresses, the fluid density must
be increasing with time in the neighborhood of a given
particle as we follow along with the motion of that
particle. Obviously this type of motion, if it existed at
all, could only occur in a compressible fluid.

Now suppose the salinity of the fluid varies with time and distance according to

where , and *m* are constants and *S* is now
the salinity of the fluid. It will be presumed that this
relationship holds in the presence of the foregoing velocity
distribution. The time rate of change of the salinity
following along with parcel *A* can be computed in two
ways.

From Eqs. (32) and (35) we see that the salinity of parcel
*A* is

Therefore

but in view of Eq. (34) this can be written

This same result can be obtained without even knowing the form
of the trajectory. The material derivative *S* for this
one-dimensional case is simply

Inserting *S* in terms of *t* and *x* from (35) yields

When this is evaluted for the parcel *A* the result on the
right is identical with that of Eq. (38). Thus we have at least
one specific check on the validity of the general
statement Eq. (28).

We might next consider a slightly more complicated form of velocity field and investigate the particle trajectories which it generates.

Suppose

where , *k* and *C* are constants. This might represent
the velocity field associated with a longitudinal acoustic
wave in a tube. In this case *C* is the sound velocity
in the fluid, is the wavelength, *kC* is the
frequency, and is the amplitude of the oscillatory
particle velocity. The form of the field of *u* indicates that
energy is being propagated down the tube at a velocity *C*.
The maximum particle velocity in an acoustical wave
is extremely small compared with *C* even for very high
acoustical energy levels (loud noises).

In general, the trajectory of a particle *A* is governed by

However, if as in the case of an ordinary acoustic wave, it can be shown that the variation of with time has very little effect on the right hand side of the equation, and can be taken therefore as the mean value, which happens to be the value at in this example.

Thus

for , and hence

where is the frequency of the sound. It we taken then Eq. (43) reduces to

Thus the fluid particle *A* moves back and forth with a frequency
and with an amplitude
equal to about the mean
position . It will be noted that the condition
is equivalent to
, i.e., the amplitude of vibration
of the particles must be much less than , where
is the wavelength of the acoustic wave.

If we have a situation of a velocity field represented by
Eq. (41) but is not small compared with *C*, then the
particles do not simply vibrate about a fixed position
but rather also have a net forward motion. In order to
solve Eq. (42) in this case, it is convenient to make the
substitution

Then Eq. (42) takes the form

which can be solved for *t* as

The integral can be found in standard tables. Upon evaluation of the integral and subsequent eliminatin of through Eq. (44), the desired solution is

whcih holds as long as is not equal to or greater than *C*.
The new terms introduced in Eq. (47) are

where is the value of at , as before. When is very small then Eq. (47) reduces to the special case given by Eq. (43), however, the algebra is somewhat tedious. It is more readily shown from Eq. (46) that when is very small then the integral can be approximated by

which leads to

which can be verifed as being essentially equal to Eq. (43).

In the limiting case where then Eq. (47) is no longer applicable and we must return to Eq. (46). The solution is readily found to be of the form

In this case the particle which starts at the origin, i.e.
, simply travels with the constant
speed , since the second term in Eq. (49)
vanishes in this case. Particle starting at other positions
at do not travel with speed *C* at first but as time
increases the second term in Eq. (49) approaches zero and
eventually all of the particles that started between
and lump together and travel with speed *C*.
This is best illustrated graphically in the diagram
(see Fig. 1.03-2). The dashed lines in Fig. 1.03-2 are
isolines of equal particle velocity, while the full lines are
trajectories of different particles which start out at positions
equally spaced along the *x* axis at . The effect
is somewhat analogous to that of Case A, except that only
those particles contained initially in an interval
lump together, there being formed many such compressional
regions at intervals of one wavelength.

For the trajectory given by Eq. (47) has an oscillatory
component as well as a net motion. A graph of the trajectory
for a particle starting at when
is given in Fig. 1.03.3 (curve 2). The curve (1) corresponds
to the case of very much smaller than *C*, and curve (3)
corresponds to the extreme case of , each curve
applying to that particle which starts at the origin.

The *net* displacement of a particle which starts
at the origin for an elapsed time of one period is plotted
in Fig. 1.03-4 as a function of . It is apparent
that the magnitude of net displacement is a small fraction of
in one period as long as *u* is less than
2/3 *C* and is negligible if *u* is less than about
*C*/10.

As a third illustration of the interrelationship between the Eulerian and the Lagrangian representations of fluid motion, we will consider the following two-dimensional case:

In this case there are two equations which determine the
trajectory of a particle *A*; these are

Suppose it is desired to find the trajectory of that particle which starts at the origin at , i.e.

Equations (51 a,b) are to be satisfied simultaneously. Consequently, if we differentiate (51a) and eliminate using (51b) we obtain the second order equation

the general solution of which is

From Eq. (51a) we find therefore that

But Eqs. (52a,b) require that

Consequently, the final solutions are

for that particle which starts at the origin at .
These are the parametric equations of a cycloid with a cusp at
the origin and other cusps spaced along the *y* axis at intervals
of (see Fig. 1.03-5, curve 1). Values of
*t*/*T* where are plotted along the
trajectory to indicate the time taken for the particle to
traverse the distance from the origin to the point in question
along the indicated path in the *x*-*y* plane.

In order to compare the trajectory with the field of velocity
with which it is associated, it is convenient to construct
the instantaneous streamlines at a given time.
The **streamlines** are defined as those lines which are
everywhere tangent to the velocity vectors at a given
instant. Thus the streamlines indicate merely the direction
of the flow at a given time, i.e. they give the synoptic flow
pattern. Like trajectories there is no number associated
with the lines, and the spacing is quite arbitrary.
Additional isolines are required in general to give the
magnitude of the velocities (such a set of lines will
be referred to as **isovels**).

In general, the streamlines and the trajectories differ. Only in those cases where the field of velocity is steady is the pattern of streamlines and trajectories identical. In the case at hand the velocity field is changing with time and we must therefore expect to find a difference between the streamlines and trajectories.

The evaluation of the shape of the streamlines at a given instant ismost readily accomplished by setting time dependent terms in the equations for the velocity field equal to a constant and then evaluate the trajectories. That is, we find those trajectories which would exist if the velocity field were frozen, which is the special case where the streamlines are identical with the trajectories.

Thus if we let and represent the positions along a particular streamline at a given time then

where (a constant) is the instant for which the streamline pattern is to be evaluated. These equations lead to the homogeneous second order equation

for which the solutions are of the form

and

These equations can be written in the more convenient form

(where *r* and are constants) which are the
parametric equations of a circle of radius *r*, centered at
the point , ; i.e.

where the radius *r* is arbitrary.

Thus the streamlines for Case C at any instant are concentric
circles, the center of which is on the *y* axis at a position
equal to . This streamline pattern
progresses along the *y* axis consequently at a speed
. The streamlines at are shown
in Fig. 1.03-5.

The magnitude of the velocity *V* at any instant is given by

or

If we let *x*, *y* be the positions along a particular streamline
at instant then we see from Eq. (62) that

or

for this particular example. Thus the velocity on a particular streamline is constant and its magnitude varies from one streamline to the next directly proportional to the radius of the streamline. These results represent a very special type of motion corresponding to a traveling deformationless vortex. The closest thing to this in nature is the horizontal circulation in th eye of a hurricane.

The streamlines and isovels could also be obtained for this
example by use of polar coordinates. This is convenient when a
radially symmetical pattern of this sort is being dealt with.
In order to convert *u* and *v* to the polar velocity
components we must refer to Figure 1.03-6. Consider a point
*x*, *y* also described by *r* and . The components
of along the radius vector and

perpendicular to it (in the sense of positive ) are and , respectively. It will be noted from the diagram that

and also that

Now in this example

Inserting these last expression in Eqs. (66) and simplifying gives

However, if we refer the velocities to a relative coordinate
system whose origin moves with speed along the
*y* axis, then the relative polar coordinates *r* and
are given by

and the polar velocity components and relative to the moving coordinates are

Thus has the same form as , the velocity along the streamline at any instant.

Returning now to Eqs. (54) and (55) we may investigate the shape of other trajectories. One case of interest is the particle which starts at the point , at . For this trajectory we find from Eq. (54) and from Eq. (55). Thus, the parametric equations of the trajectory are

Thus the particle which starts at and
at simply moves along a straight line
parallel to the *y* axis
and travels at constant speed . This is to be expected
since at the initial position of this particle, the fluid velocity
is given by

and since the fluid velocity pattern is traveling at the rate , the particle always remains at the same relative position on the streamlines and maintains the velocity . The trajectory is shown in Fig. 1.03-5 (curve 2).

As a third example trajectory, consider that a particle which starts at the position , at . In this case we find from Eqs. (54) and (55) that

Thus

These are the parametric equations of a degenerate troichoid which
has loops at intervals of along the
*y* axis. This trajectory corresponds to curve (3) of
Fig. 1.03-5.

One more example (not shown on figure) is that of a particle starting at , at . The equations of this trajectory are

The latter trajectory has a minimum value of *x* of
and a maximum value of
. The curve is a troichoid with rounded
peaks rather than cusps (at the minimum *x* values). This
curve would lie intermediate between curves (1) and (2).

Those particles which start at a position between
and would have a path similar to that
just described except that the range of variation of *x* would be
smaller and approach zero for
as seen from curve (2). Those paths which lie very close to curve
(2) would have very nearly a sinusoidal shape.

In general, it can be stated that the trjectories of all particles which are subject to the moving vortex under consideration are troichoids of some type, with curves 1, 2 and 3 being special cases of this general class of curves. It may be remarked in passing that the field of velocity discussed does not lead to any compressional effects whatever. That is, the separation distance between particles remains constant. This will be proven for this example in a later section (Art. 1.07). For the time being, it is sufficient to note that although the trajectories converge and cross each other, the particles do not actually pile up because in no case to two particles reach the crossing point simultaneously.

Mon Dec 1 08:50:29 CST 1997