Chapter 7 - Some Mathematics:
The Equations of Motion

Chapter 7 Contents

7.6 Momentum Equation

Newton's Second Law relates the change of the momentum of a fluid mass due to an applied force. The change is:

(7.8)

where F is force, m is mass, and v is velocity; and where we have emphasized the need to use the total derivative because we are calculating the force on a particle. We can assume that the mass is constant, and (7.8) can be written:

(7.9)

where fm is force per unit mass.

Four forces are important: pressure gradients, Coriolis force, gravity, and friction. Without deriving the form of these forces (the derivations are given in the next section), we can write (7.9) in the following form.

(7.10)

Acceleration equals the negative pressure gradient minus the Coriolis force plus gravity plus other forces. Here g is acceleration of gravity, Fr is friction, and Ω is the Rotation Rate of Earth, 2π radians per sidereal day or

(7.11)

Momentum Equation in Cartesian Coordinates:
Expanding the derivative in (7.10) and writing the components in a Cartesian coordinate system gives the Momentum Equation:

(7.12a)
(7.12b)
(7.12c)

where Fi are the components of any frictional force per unit mass, and φ is latitude. In addition, we have assumed that w << v, so the 2 Ω w cos φ has been dropped from equation in (7.12a).

Equation (7.12) appears under various names. Leonhard Euler (1707-1783) first wrote out the general form for fluid flow with external forces, and the equation is sometimes called the Euler equation or the acceleration equation. Louis Marie Henri Navier (1785-1836) added the frictional terms, and so the equation is sometimes called the Navier-Stokes equation.

The term 2 Ω u cos φ in (7.12c) is small compared with g, and it can be ignored in ocean dynamics. It cannot be ignored, however, for gravity surveys made with gravimeters on moving ships.

Figure 7.3 Sketch of flow used for deriving the pressure term in the momentum equation.

Derivation of Pressure Term
Consider the forces acting on the sides of a small cube of fluid (Figure 7.3). The net force δFx in the x direction is

δFx = p δyδz - (p + δp) δyδz
δFx = -δp δy δz

But

and therefore

Dividing by the mass of the fluid δm in the box, the acceleration of the fluid in the x direction is:

 
(7.13)

The pressure forces and the acceleration due to the pressure forces in the y and z directions are derived in the same way.

The Coriolis Term in the Momentum Equation
The Coriolis term exists because we describe currents in a reference frame fixed on Earth. The derivation of the Coriolis terms is not simple. Henry Stommel, the noted oceanographer at the Woods Hole Oceanographic Institution even wrote a book on the subject with Dennis Moore (Stommel & Moore, 1989).

Usually, we just state that the force per unit mass, the acceleration of a parcel of fluid in a rotating system, can be written:

(7.14)

where R is the vector distance from the center of Earth, Ω is the angular velocity vector of Earth, and v is the velocity of the fluid parcel in coordinates fixed to Earth. The termv is the Coriolis force, and the term Ω Ω R) is the centrifugal acceleration. The latter term is included in gravity (Figure 7.4).

The Gravity Term in the Momentum Equation The gravitational attraction of two masses M1 and m is:

where R is the distance between the masses, and G is the gravitational constant. The vector force Fg is along the line connecting the two masses.

The force per unit mass due to gravity is:

(7.15)

where ME is the mass of Earth. Adding the centrifugal acceleration to (7.15) gives gravity g (Figure 7.4):

g = gf - Ω x (Ω x R)
(7.16)

Figure 7.4 Downward acceleration g of a body at rest on Earth's surface is the sum of gravitational acceleration between the body and Earth's mass gf and the centrifugal acceleration due to Earth's rotation. Ω x (Ω x R). The surface of an ocean at rest must be perpendicular to g, and such a surface is close to an ellipsoid of rotation. Earth's ellipticity is greatly exaggerated here.

Note that gravity does not point toward Earth's center of mass. The centrifugal acceleration causes a plumb bob to point at a small angle to the line directed to Earth's center of mass. As a result, Earth's surface including the ocean's surface is not spherical but it is a prolate ellipsoid. A rotating fluid planet has an equatorial bulge.

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