Response to forcing: the governing equations and some basic solutions - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 56

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moving at speed u in a circular path on a non-rotating Earth. The final term 2 u is

the additional acceleration which results from the combination of rotation and the
relative velocity. This acceleration acts at right angles to the Earth's axis of rotation
and, therefore, has a component in the horizontal plane given by:

2 u cos ð 90 L Þ¼ 2 u sin L ¼ fu

which acts to the left of the direction of motion. The negative of this acceleration is
the Coriolis force per unit mass which, therefore, acts to the right of the direction of
motion. It is this term which we must include in the equations of motion if we are
dealing with motion in a coordinate system rotating with the Earth.

3.2.2 The acceleration term: Eulerian versus Lagrangian velocities

There is a further complication in dealing with the acceleration term in Equation (3.6) .
Strictly, the second law is applicable to each particle of a fluid and relates the changes
in momentum (mass velocity) of the particle to the forces acting on it. The velocity

here is that of the particle. In the ocean, velocities are usually measured, not for
a particle, but at the particular point where a current measuring device is located;
i.e. we make measurements of the velocities of lots of particles as they pass through a
fixed point. Such velocities at a fixed point are termed Eulerian in contrast to
Lagrangian velocity measurements which track an individual particle. In applying
the second law, we need, therefore, to relate the acceleration of a particle of fluid
in the x direction (written as Du/Dt) to the acceleration in a fixed Eulerian frame
(

t). This is done by noting that the velocity change experienced by a particle will
be the sum of the local rate of change and the changes due to movement through
the fluid where velocity will be changing with position.
Look at the illustration in Fig. 3.2 . In a small time dt the displacement of the
particle in the x direction will be udt, which will result in a change in u of the particle

@ u/ @

Figure 3.2 The circular particle moving
in the x y plane is displaced by dx ¼ udt
and dy ¼ vdt in a small time interval
dt. If the velocity component u varies
with x and y then the total change of u
following a particle (i.e. in a Lagrangian
frame of reference) will be the sum of the
local change @ u/ @ t (i.e. in an Eulerian
frame) and the additional changes due
to the displacements dx(

Change following the particle:

u u u

δ u = δ t δ x δ y

t x y

Derivative at a point:

u
t

δ y = v δ t

@ u/ @ x)and

dy( y). If the motion is in 3D, there

will an additional term due to the
displacement dz

@ u/ @

¼ wdt.

δ x = u δ t