## Info

Equation 6-17 (nonrotating) and Eq. 6-21 (rotating) are equivalent statements of the balance of forces. The distinction between them is that the former is expressed in terms r

5Note that the balance Eq. 6-17 cannot hold exactly in our experiment, because radial accelerations must be present associated with the flow of water inward from the diffuser to the drain. But if these acceleration terms are small the balance Eq. 6-17 is a good approximation.

6By doing so, we thus eliminate from Eq. 6-19 the centrifugal term q2r associated with the background rotation. We will follow a similar procedure for the spherical Earth in Section 6.6.3 (see also GFD Lab IV in Section 6.6.4)

of Vq, the latter in terms of vq. Note that Eq. 6-21 has the same form as Eq. 6-17, except (i) n (measured relative to the reference parabola) appears rather than H (measured relative to a flat surface), and (ii) an extra term, -2Qvq, appears on the right hand side of Eq. 6-21; this is called the Coriolis acceleration. It has appeared because we have chosen to express our force balance in terms of relative, rather than absolute, velocities. We shall see that the Coriolis acceleration plays a central role in the dynamics of the atmosphere and ocean.

### Angular momentum

Fluid entering the tank at the outer wall will have angular momentum, because the apparatus is rotating. At r1 the radius of the diffuser in Fig. 6.5, fluid has velocity Qri and hence angular momentum Qr^. As parcels of fluid flow inward, they will conserve this angular momentum (provided that they are not rubbing against the bottom or the side). Thus conservation of angular momentum implies that

where Vq is the azimuthal velocity at radius r in the laboratory (inertial) frame given by Eq. 6-15. Combining Eqs. 6-22 and 6-15 we find

We thus see that the fluid acquires a sense of rotation, which is the same as that of the rotating table but which is greatly magnified at small r. If Q > 0, meaning that the table rotates in an anticlockwise sense, then the fluid acquires an anticlockwise (cyclonic7) swirl. If Q < 0 the table rotates in a clockwise (anticyclonic) direction and the fluid acquires a clockwise (anticyclonic) swirl. This can be clearly seen in the trajectories plotted in Fig. 6.6. Equation 6-23 is, in fact, a rather good prediction for the azimuthal speed of the particles seen in Fig. 6.6.

We will return to this experiment later in Section 7.1.3 where we discuss the balance of terms in Eq. 6-21 and its relationship to atmospheric flows.

### 6.6.2. Transformation into rotating coordinates

In our radial inflow experiment we expressed the balance of forces in both the nonrotating and rotating frames. We have already written down the equations of motion of a fluid in a nonrotating frame, Eq. 6-6. Let us now formally transform it in to a rotating reference frame. The only tricky part is the acceleration term, Du/Dt, which requires manipulations analogous to Eq. 6-18 but in a general framework. We need to figure out how to transform the operator D/Dt (acting on a vector) into a rotating frame. Of course, D/Dt of a scalar is the same in both frames, since this means ''the rate of change of the scalar following a fluid parcel.'' The same fluid parcel is followed from both frames, and so scalar quantities (e.g., temperature or pressure) do not change when viewed from the different frames. However, a vector is not invariant under such a transformation, since the coordinate directions relative to which the vector is expressed are different in the two frames.

A clue is given by noting that the velocity in the absolute (inertial) frame uin and the velocity in the rotating frame urot, are related (see Fig. 6.9) through

where r is the position vector of a parcel in the rotating frame, Q is the rotation vector of the rotating frame of reference, and Q x r is the vector product of Q and r. This is just a generalization (to vectors) of the transformation used in Eq. 6-15 to express the absolute velocity in terms of the relative velocity in our radial inflow experiment. As we shall now go on to show, Eq. 6-24 is a special case of a general ''rule'' for transforming

7The term cyclonic (anticyclonic) means that the swirl is in the same (opposite) sense as the background rotation.

FIGURE 6.9. On the left is the velocity vector of a particle u,n in the inertial frame. On the right is the view from the rotating frame. The particle has velocity urot in the rotating frame. The relation between u,n and urot is u,n = urot + Q x r, where Q x r is the velocity of a particle fixed (not moving) in the rotating frame at position vector r. The relationship between the rate of change of any vector A in the rotating frame and the change of A as seen in the inertial frame is given by:

FIGURE 6.9. On the left is the velocity vector of a particle u,n in the inertial frame. On the right is the view from the rotating frame. The particle has velocity urot in the rotating frame. The relation between u,n and urot is u,n = urot + Q x r, where Q x r is the velocity of a particle fixed (not moving) in the rotating frame at position vector r. The relationship between the rate of change of any vector A in the rotating frame and the change of A as seen in the inertial frame is given by:

the rate of change of vectors between frames, which we now derive.

Consider Fig. 6.9. In the rotating frame, any vector A may be written

where (Ax, Ay, Az) are the components of A expressed instantaneously in terms of the three rotating coordinate directions, for which the unit vectors are (x, y, z). In the rotating frame, these coordinate directions are fixed, and so i DA \ ^DAx ^DAy ^DAz \~Dt)rot = + + 'D.

However, viewed from the inertial frame, the coordinate directions in the rotating frame are not fixed, but are rotating at rate Q, and so

Therefore, operating on Eq. 6-25,