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 vχ is irrotational, that is, H × vχ = 0; vψ is solenoidal, that is, H · vψ = 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.