5.3 Dynamic Model of Tides
171
5.3.2 Coriolis Acceleration
Coriolis acceleration is introduced to allow the application of Newton's second
law to the rotating Earth, as originally Newton's law applied only for motion
in an unaccelerating coordinate system. A rigorous mathematical development
of the Coriolis terms is beyond the scope of this book (for details see, for
example, Le Mehaute, 1976). Rather, in this chapter we will describe the
physical consequences of Coriolis acceleration. For simplicity, let us consider a
fluid particle located on the Earth's surface, in the Northern Hemisphere, and
rotating with the Earth. A rotation about the Earth's axis creates a centrifugal
force, which for a particle resting on the Earth is balanced by the component of
gravity perpendicular to the Earth's axis. Suppose now that this particle moves
south, toward the Equator. Therefore, the particle's distance from the Earth's
axis will increase and according to Eq. (5.10), centrifugal force experienced
by the particle will increase. This larger force cannot be totally balanced by
the component of gravity; thus the particle starts to fall behind the rotating
Earth. Since the Earth rotates from west to east, the fluid particle describes a
path westward (Fig. 5.lOa). For example, if corrections for this deflection are
not made, an aircraft departing from Stockholm, Sweden, on a flight to Lagos,
Nigeria, would drift 4,800 km westward (to the right) and be forced to land in
South America rather than in Africa (Pinet, 1992).
Using similar arguments, we find that when the same particle moves northward from the Equator, it's path turns eastward. This means that again the deflection is to the right of the direction of the motion (Fig. 5.10a). In the Southern Hemisphere the deflection of the particle will be to the left (Fig. 5.lOa).
This is expressed in saying that the Coriolis force deflects tidal flows cum sole
(with the Sun); in the Northern Hemisphere to the right, clockwise, and in the
Southern Hemisphere to the left, anticlockwise (Bearman, 1997).
In Fig. 5.lOb, the deflections associated with the eastward or westward movement of the particle in the Northern and Southern Hemispheres are shown.
If for example, the particle moves eastward with respect to the Earth along
a parallel in the Northern Hemisphere, the particle experiences an additional
centrifugal force because of additional tangential velocity. Therefore, in order
to be balanced by the component of gravitational force, a particle tends to move
further away from the Earth's axis, towards the equator (Fig. 5.lOb). In this
way, the centrifugal acceleration, u; / r (as well as centrifugal force), remains
the same and the balance between centrifugal force and the component of the
gravitational force is restored. In the case of particle movement westward, a
fluid particle is deflected towards the Pole (Fig. 5.lOb). All above deviations
are known as Coriolis effects.
To obtain an expression for the Coriolis acceleration, let us consider the simple
case of a water particle, P, at rest with respect to the Earth, which is rotating
with the frequency WE (Fig. 5.11). From Eqs. (5.7) and (5.8) we find that the
acceleration of the particle with respect to the axis of rotation is:
171
5.3.2 Coriolis Acceleration
Coriolis acceleration is introduced to allow the application of Newton's second
law to the rotating Earth, as originally Newton's law applied only for motion
in an unaccelerating coordinate system. A rigorous mathematical development
of the Coriolis terms is beyond the scope of this book (for details see, for
example, Le Mehaute, 1976). Rather, in this chapter we will describe the
physical consequences of Coriolis acceleration. For simplicity, let us consider a
fluid particle located on the Earth's surface, in the Northern Hemisphere, and
rotating with the Earth. A rotation about the Earth's axis creates a centrifugal
force, which for a particle resting on the Earth is balanced by the component of
gravity perpendicular to the Earth's axis. Suppose now that this particle moves
south, toward the Equator. Therefore, the particle's distance from the Earth's
axis will increase and according to Eq. (5.10), centrifugal force experienced
by the particle will increase. This larger force cannot be totally balanced by
the component of gravity; thus the particle starts to fall behind the rotating
Earth. Since the Earth rotates from west to east, the fluid particle describes a
path westward (Fig. 5.lOa). For example, if corrections for this deflection are
not made, an aircraft departing from Stockholm, Sweden, on a flight to Lagos,
Nigeria, would drift 4,800 km westward (to the right) and be forced to land in
South America rather than in Africa (Pinet, 1992).
Using similar arguments, we find that when the same particle moves northward from the Equator, it's path turns eastward. This means that again the deflection is to the right of the direction of the motion (Fig. 5.10a). In the Southern Hemisphere the deflection of the particle will be to the left (Fig. 5.lOa).
This is expressed in saying that the Coriolis force deflects tidal flows cum sole
(with the Sun); in the Northern Hemisphere to the right, clockwise, and in the
Southern Hemisphere to the left, anticlockwise (Bearman, 1997).
In Fig. 5.lOb, the deflections associated with the eastward or westward movement of the particle in the Northern and Southern Hemispheres are shown.
If for example, the particle moves eastward with respect to the Earth along
a parallel in the Northern Hemisphere, the particle experiences an additional
centrifugal force because of additional tangential velocity. Therefore, in order
to be balanced by the component of gravitational force, a particle tends to move
further away from the Earth's axis, towards the equator (Fig. 5.lOb). In this
way, the centrifugal acceleration, u; / r (as well as centrifugal force), remains
the same and the balance between centrifugal force and the component of the
gravitational force is restored. In the case of particle movement westward, a
fluid particle is deflected towards the Pole (Fig. 5.lOb). All above deviations
are known as Coriolis effects.
To obtain an expression for the Coriolis acceleration, let us consider the simple
case of a water particle, P, at rest with respect to the Earth, which is rotating
with the frequency WE (Fig. 5.11). From Eqs. (5.7) and (5.8) we find that the
acceleration of the particle with respect to the axis of rotation is: