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.

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