D

DUro,

since, by definition

Equation 6-27 is a more general statement of Eq. 6-18: we see that there is a one-to-one correspondence between the terms.

6.6.3. The rotating equations of motion

We can now write down our equation of motion in the rotating frame. Substituting from Eq. 6-27 into the inertial-frame equation of motion Eq. 6-6, we have, in the rotating frame,

Du 1

Coriolis Centrifugal accel" accel"

where we have dropped the subscripts "rot" and it is now understood that Du/Dt and u refer to the rotating frame.

Note that Eq. 6-28 is the same as Eq. 6-6, except that u = urot and "apparent" accelerations, introduced by the rotating reference frame, have been placed on the right-hand side of Eq. 6-28 (just as in Eq. 6-21). The apparent accelerations have been given names; the centrifugal acceleration (-Q x Q x r) is directed radially outward (Fig. 6.9), and the Coriolis acceleration (-2Q x u) is directed "to the right'' of the velocity vector if Q is anticlockwise, sketched in Fig. 6.10. We now discuss these apparent accelerations in turn.

Centrifugal acceleration

As noted above, -Q x Q x r is directed radially outwards. If no other forces were acting on a particle, the particle would accelerate outwards. Because centrifugal acceleration can be expressed as the gradient of a potential,

Q2r2

where r is the distance normal to the rotating axis (see Fig. 6.9) it is convenient to combine V ^ ^r2 ) withgz = V [gz), the gradient

FIGURE 6.10. A fluid parcel moving with velocity urot in a rotating frame experiences a Coriolis acceleration, -2Q x urot, directed ''to the right'' of urot if, as here, Q is directed upwards, corresponding to anticlockwise rotation.

FIGURE 6.10. A fluid parcel moving with velocity urot in a rotating frame experiences a Coriolis acceleration, -2Q x urot, directed ''to the right'' of urot if, as here, Q is directed upwards, corresponding to anticlockwise rotation.

of the gravitational potential gz, and write Eq. 6-28 in the succinct form:

where

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