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.