Chapter 7  Some Mathematics: The Equations of Motion
In this chapter we consider the response of a fluid to internal and external forces. This leads to a derivation of some of the basic equations describing
ocean dynamics. In the next chapter, we will consider the influence of viscosity, and in chapter 12 we will consider the consequences of vorticity.
Fluid mechanics used in oceanography is based on Newtonian mechanics modified
by our evolving understanding of turbulence. Conservation of mass, momentum,
angular momentum, and energy lead to particular equations having names that
hide their origins (Table 7.1).
Table 7.1 Conservation Laws Leading
to Basic Equations of Fluid Motion
Conservation of Mass: 
Leads to Continuity Equation. 
Conservation
of Energy: 
Conservation
of heat leads Heat Budgets. 
Conservation of mechanical
energy leads to Wave Equation 
Conservation of Momentum: 
Leads to Momentum (NavierStokes)
Equation. 
Conservation of Angular
Momentum: 
Leads to Conservation
of Vorticity. 
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7.1 Dominant Forces for Ocean Dynamics
Only a few forces are important in physical oceanography: gravity, buoyancy
due to difference in density of sea water, and wind stress (Table 7.2). Remember
that forces are vectors. They have magnitude and direction.
 Gravity is the dominant force. The weight
of the water in the ocean produces pressure.
Changes in gravity, due to the motion of sun and moon relative to Earth
produces tides, tidal currents, and tidal mixing in the interior of the
ocean.
Buoyancy is the upward or downward force
due to gravity acting on a parcel of water that is more or less dense than
other water at its level. For example, cold air blowing over the sea cools
surface waters causing them to be more dense than the water beneath. Gravity
acting on the difference in density results in a force that causes the water
to sink.
Horizontal pressure gradients are due to the varying weight of water in different
regions of the ocean.
 Friction is the force acting on a body
as it moves past another body while in contact with that body. The bodies
can be parcels of water or air.
Wind stress is the friction due to wind blowing
across the sea surface. It transfers horizontal momentum to the sea, creating
currents. Wind blowing over waves on the sea surface leads to an uneven distribution
of pressure over waves. The pressure distribution transfers energy to the
waves, causing them to grow into bigger waves.
 Pseudoforces are apparent forces that arise from motion in curvilinear
or rotating coordinate systems. Thus, writing the equations for inertial
motion in a rotating coordinate system leads to additional force terms called
pseudo forces. For example, Newton's first law states that there is
no change in motion of a body unless a resultant force acts on it. Yet a
body moving at constant velocity seems to change direction when viewed from
a rotating coordinate system. The change in direction is attributed to a
pseudoforce, the Coriolis force.
Coriolis Force is the dominant pseudoforce influencing currents moving
in a coordinate system fixed to the Earth.
Table 7.2 Forces in Geophysical Fluid Dynamics
Dominant Forces 
Gravity 
Gives
rise to pressure gradients, buoyancy, and tides. 
Corilois 
Results from motion in
a rotating coordinate system. 
Friction 
Is
due to relative motion between two fluid parcels.
Wind stress is an important frictional force.

Other Forces

Atmospheric Pressure 
Results in inverted barometer
effect. 
Seismic 
Results in tsunamis
driven by earthquakes. 
Note: that the last two forces
are much less important than the first three.
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