Clebsch representation

In physics and mathematics, the Clebsch representation of an arbitrary three-dimensional vector field ${\displaystyle {\boldsymbol {v}}({\boldsymbol {x}})}$ is:^[1][2]

$${\displaystyle {\boldsymbol {v}}={\boldsymbol {\nabla }}\varphi +\psi \,{\boldsymbol {\nabla }}\chi ,}$$

where the scalar fields ${\displaystyle \varphi ({\boldsymbol {x}})}$${\displaystyle ,\psi ({\boldsymbol {x}})}$ and ${\displaystyle \chi ({\boldsymbol {x}})}$ are known as Clebsch potentials^[3] or Monge potentials,^[4] named after Alfred Clebsch (1833–1872) and Gaspard Monge (1746–1818), and ${\displaystyle {\boldsymbol {\nabla }}}$ is the gradient operator.

Background

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]

Vorticity

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

$${\displaystyle {\boldsymbol {\omega }}={\boldsymbol {\nabla }}\times {\boldsymbol {v}}={\boldsymbol {\nabla }}\times \left({\boldsymbol {\nabla }}\varphi +\psi \,{\boldsymbol {\nabla }}\chi \right)={\boldsymbol {\nabla }}\psi \times {\boldsymbol {\nabla }}\chi ,}$$

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 ${\displaystyle \varphi .}$