Now, by making use of the Lagrangian variables R and t (see Eqs. [2.1] and [2.2]), one can integrate this equation at once to obtain w, m0k d Xi
p p0 dXk where u0(R, 0) and p0(R, 0) are the initial values of u(R, t) and p(R, t). As was shown by Helmholtz, this solution simply means that the particles that compose an absolute-vorticity line at one instant will continue to form an absolute-vorticity line at any subsequent instant. The proof lies in the fact that a tangent to such a line is carried by the fluid so that it always remains tangent to an absolute-vorticity line. Let dXi be the components of the vector representing a line element, at the instant t = 0, of an absolute vortex line. As we follow its motion, we have d X
d Xk where the dxi s are the new components, at time t, of this line element. Now, by hypothesis, we can always write dXi = e —, (2.46)
where e is some constant. From Eqs. (2.44)-(2.46), it follows that m0k d xi mi dxi = e--= e —, (2.47)
P0 dXk p thus implying that the vector with components dxi is also tangent to an absolute-vorticity line. This concludes the proof. By virtue of Eqs. (2.46) and (2.47), we also note that the ratio u/p is proportional to the length of a line element along an absolute-vorticity line. This is known as vortex line stretching or shrinking.
In summary, we have shown that the absolute-vorticity lines move with the fluid in the absence of baroclinicity and friction. However, although one can also construct lines of relative vorticity, it is only the absolute-vorticity lines that may remain coincident with material lines. Moreover, when the last two terms in Eq. (2.42) do not identically vanish, viscous friction allows the absolute-vorticity lines to diffuse across the fluid, with the baroclinicity also being able to create new vortices.
2.3.1 The Taylor—Proudman theorem
Let us consider steady motions in a rotating fluid. Then, if both the Rossby number and the Ekman number of the flow are small, and if the baroclinic vector is identically zero, Eq. (2.42) becomes n ■ grad u = 0, (2.48)
since |Ci <<|n| when Ro ^ 1. This condition implies that the velocity relative to the moving axes must be independent of the coordinate parallel to n. If this vector is along the x3 axis, we can thus write d u1 d u2 d u3
In particular, if we consider a system with solid boundaries perpendicular to the rotation axis so that one has u3 = 0 at some specified value of x3, it follows at once that d u d u2
everywhere in thefluid. The flow is then entirely two dimensional in planes perpendicular to the rotation axis.
Motions that satisfy the Taylor-Proudman constraint can be observed in laboratory experiments (e.g., Greenspan 1968, Fig. 1.2, and Tritton 1988, Sec. 16.4). For example, let us consider a case in which the relative motion between the fluid and an obstacle is perpendicular to the rotation axis. Obviously, the fluid is deflected past the obstacle. However, because the flow must be two dimensional, this deflection also occurs above and below the obstacle. Accordingly, one observes the formation of a column of fluid, extending parallel to the rotating axis from the obstacle, around which the fluid is deflected as if it too were solid. Since the neglected terms never exactly vanish, especially at the edge of the column, there is in fact some interchange between the exterior and the interior of the column. Yet, Eq. (2.48) clearly demonstrates the tendency for coupling of the relative motion in the direction parallel to the vector O.
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