# Helmholtz's theorem

From Glossary of Meteorology

## Helmholtz's theorem

The statement that if

*F*is a vector field satisfying certain quite general mathematical conditions, then*F*is the sum of two vectors, one of which is irrotational (has no vorticity), the other solenoidal (has no divergence).Thus, the horizontal velocity field, for example, may be expressed by where

**v**_{χ}is irrotational, that is,**∇**_{H}×**v**_{χ}= 0;**v**_{ψ}is solenoidal, that is,**∇**_{H}·**v**_{ψ}= 0;**k**is a unit vector directed vertically; and χ and ψ are scalar functions that may be computed from the given wind field.Charney, J. G. 1955. J. Mar. Res.. 14. 477–498.

Salby, M. L. 1996. Fundamentals of Atmospheric Physics. 375–377.