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Tidal Energy Chapter | 3 57

D Ω
(− × R). These terms appear when we write the equation of motion in
dt
a rotational coordinate system. Therefore, we refer to them as apparent or
fictitious accelerations or forces. Assuming that the angular velocity of the Earth
is constant results in d Ω = 0. The centripetal acceleration at the equator is about
dt
0.3% of the Earth attraction, and in measurements it is difficult to differentiate
between the Earth attraction and centripetal acceleration. The resultant force of
Earth attraction and centrifugal force is called gravity, which acts normal to the
surface of the Earth.
Now, consider a rotating frame attached to the Earth, in which x and y
axes correspond to the west-east and the south-north directions, respectively.
The z axis is oriented towards the centre of the Earth. An object moving
northwards in the northern hemisphere is considered; using the cross product,
the magnitude of Coriolis acceleration will be, 2Ω sin φv, where φ is the latitude
(or the angle between velocity and rotation vectors), and the direction of the
Coriolis acceleration will be towards the east, or to the right of moving object
(i.e. perpendicular to the plane containing rotation and velocity vectors). It can
be seen that the magnitude of Coriolis is zero for an object moving northwards
at the equator (φ = 0).
Let us assume that an object is moving from the west towards the east
in the northern hemisphere; the magnitude of Coriolis acceleration (using
cross product) will be 2Ωu, as the rotation vector and velocity vector are
perpendicular. For this scenario, the Coriolis acceleration has two components:
one component towards the south, or negative y direction. With a magnitude of
2Ωu sin φ, and the other component away from the centre of the Earth (z axis),
with a magnitude of 2Ωu cos φ. Therefore, the Coriolis effect turns objects
rightward in the northern hemisphere. We can similarly show that Coriolis turns
objects leftward in the southern hemisphere.
As we know, the Earth rotates 2π in 1 day. At the same time, it also rotates
a little (≈1/365 × 360 degrees) around the Sun. A sidereal day (86,164 s) is
the exact time that it takes for the Earth to make one rotation, and is slightly
shorter than a solar day (86,400 s). Therefore, the angular velocity of the Earth is
2π −5
Ω = = 7.29 × 10 per s. Note that the Coriolis parameter or frequency,
86,164
f, is defined as 2Ω sin(φ).
3.7 KELVIN WAVES
The propagation of tidal waves in the oceans and continental shelves is primarily
affected by the Earth’s rotation. The dynamics of long waves on a rotating
system was originally described by Lord Kelvin. In a rotating reference frame
like the Earth, Newton’s second law of motion—which is valid for an inertial
frame of reference with zero acceleration—cannot be directly applied. By
introducing Coriolis as a ‘fictitious force’, we can use Newton’s second law
in the Earth rotating system as discussed in Section 3.6. In the northern
hemisphere, the Coriolis force causes a deflection of the currents towards the
right of the direction of motion (e.g. poleward propagating currents are deflected