Methods of representation of a vector field

A three-dimensional vector requires that three quantities be specified in order to uniquely determine the vector. Consequently, in order to represent a field of a vector in general three sets of equiscalar surfaces are required.

The velocity field in the ocean, for the most part, can be considered roughly as a two dimensional velocity field. That is, the vertical velocities in the ocean are so small compared with the horizontal velocities that within a small region of the ocean (small range of latitude) the currents could be considered all in one plane. For such a two dimensional velocity field, two sets of equiscalar surfaces are required to specify the vectors at all geographical positions and all depths. For a given level, the currents at that level can be described by the intersections of the two sets of equiscalar surfaces with the level surface in question. We will refer to these intersections as the isolines in a given plane.

There are a number of different combinations of isolines which can be used to describe a given vector. In the case of the two dimensional velocity field some combinations of isolines which could be used are

1. Isolines of u and isolines of v,
2. Isolines of V and ioslines of , and
3. Isolines of V and streamlines.

Here u and v are the components of the two dimensional velocity , V is the magnitude of the velocity vector, and is the direction measured from some fixed reference. The isolines of V are referred to as isovels as noted earlier. The most common system employed is (3) or a variation of this system. Usually, instead of showing isovels, it is common to find current charts with an array of arrows indicative of the streamlines, with the length of the arrorws denoting the magnitude of V according to some stated scale.

Examples of different types of flow depicted by streamlines are given by Sverdrup et al. (1942), pp. 419, 420, 429.

It should be noted in passing that it is not necessary that the field of velocity be two dimensional in order to actually represent the direction of flow in a given plane by streamlines. In the general case, the streamlines can be considered as representing the projection of flow orientation on the plane of the figure.

Consider the moving vortex system described by the velocity field

where a, b, c are positive constants. Construct: (a) a pattern of streamlines in the x-y plane at the instant ; (b) the pattern of isovels (lines of equal velocity magnitude at the instant ); and the trajectory for a particle which starts at the point , at the initial instant . Hint: It may be found more convenient to deal with the velocity components in polar coordinates.