In fluid dynamics and plasma physics, the Clebsch representation provides a means to overcome the difficulties to describe an inviscid flow with non-zero vorticity – in the Eulerian reference frame – using Lagrangian mechanics and Hamiltonian mechanics.^[5][6][7] At the critical point of such functionals the result is the Euler equations, a set of equations describing the fluid flow. Note that the mentioned difficulties do not arise when describing the flow through a variational principle in the Lagrangian reference frame. In case of surface gravity waves, the Clebsch representation leads to a rotational-flow form of Luke's variational principle.^[8]

For the Clebsch representation to be possible, the vector field ${\displaystyle {\boldsymbol {v}}}$  has (locally) to be bounded, continuous and sufficiently smooth. For global applicability ${\displaystyle {\boldsymbol {v}}}$  has to decay fast enough towards infinity.^[9] The Clebsch decomposition is not unique, and (two) additional constraints are necessary to uniquely define the Clebsch potentials.^[1] Since ${\displaystyle \psi {\boldsymbol {\nabla }}\chi }$  is in general not solenoidal, the Clebsch representation does not in general satisfy the Helmholtz decomposition.^[10]

The vorticity ${\displaystyle {\boldsymbol {\omega }}({\boldsymbol {x}})}$  is equal to^[2]

with the last step due to the vector calculus identity ${\displaystyle {\boldsymbol {\nabla }}\times (\psi {\boldsymbol {A}})=\psi ({\boldsymbol {\nabla }}\times {\boldsymbol {A}})+{\boldsymbol {\nabla }}\psi \times {\boldsymbol {A}}.}$  So the vorticity ${\displaystyle {\boldsymbol {\omega }}}$  is perpendicular to both ${\displaystyle {\boldsymbol {\nabla }}\psi }$  and ${\displaystyle {\boldsymbol {\nabla }}\chi ,}$  while further the vorticity does not depend on ${\displaystyle \varphi .}$