Chapter 7 - Some Mathematics:
The Equations of Motion

Chapter 7 Contents

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 (Navier-Stokes) 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.

  1. 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.

  2. 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.

  3. Pseudo-forces 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 pseudo-force, the Coriolis force.

    Coriolis Force is the dominant pseudo-force 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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