The expansion or spreading out of a vector field; also, a precise measure thereof. In mathematical discussion, divergence is taken to include convergence, that is, negative divergence. The mean divergence of a field F within a volume is equal to the net penetration of the vectors F through the surface bounding the volume (see divergence theorem). The divergence is invariant with respect to coordinate transformations and may be written

where ∇ is the del operator. In Cartesian coordinates, if F has components Fx, Fy, Fz, the divergence is

Expansions in other coordinate systems may be found in any text on vector analysis. In hydrodynamics, if the vector field is unspecified, the divergence usually refers to the divergence of the velocity field (see also mass divergence). In meteorology, because of the predominance of horizontal motions, the divergence usually refers to the two-dimensional horizontal divergence of the velocity field

where u and v are the x and y components of the velocity, respectively. This divergence is denoted by any of the following symbols:

where the last two quantities involve derivatives in the isobaric surface. The order of magnitude of the horizontal divergence in meteorological motions is of considerable dynamic importance: The geostrophic wind has divergence of the order of 10−6s−1; the wind field associated with migratory cyclonic systems, 10−5s−1; motions of smaller scale (such as gravity waves, frontal waves, and cumulus convection) have characteristic divergence one or two orders of magnitude greater. See balance equation, deformation; compare diffluence, curl, vorticity.