## The vorticity equation

To visualize a fluid motion, it is often convenient to construct the streamlines of the flow. Since a streamline is an imaginary line that is everywhere tangent to the fluid velocity v(r, t), the family of such lines is given by the integration of dxi dxn dx3

Vi V2 V3

In steady flows, streamlines and particle paths are identical.

In many instances, however, it is also instructive to describe the flow in terms of the absolute vorticity u = curl v, (2.34)

which represents the local and instantaneous rate of rotation of the fluid measured in an inertial frame of reference. By definition, a continuous line that is everywhere tangent to the vector u(r, t) is called an absolute-vorticity line. The family of such lines is defined impact on the meteorological studies of that time, however, so that few advances in our knowledge of the behavior of rotating fluids were made during the nineteenth century. As a matter of fact, it is not until the late 1850s that the American meteorologist William Ferrel (1817-1891) gave the first mathematical formulation of atmospheric motions on a rotating Earth. Moreover, as we shall see in Section 2.6.1, the importance of the deflective force of the Earth's rotation on wind-driven currents in the oceans was not recognized until the turn of the twentieth century. For comparison, Sir Arthur Eddington (1882-1944) in 1925 noticed that large-scale meridional currents in the radiative regions of a star would be deflected east and west by the star's rotation, but it is not until 1941 that Gunnar Randers (1914-1992) made the first detailed analysis of the steady motion exhibiting a balance between the viscous and deflective forces in a rotating star (see Eq. [4.49]).

by the pair of differential equations dxi dx? dx3