Clebsch representation

$${\displaystyle \mathit{v}=\mathbf{\nabla }\phi +\psi \mathbf{\nabla }\chi ,}$$

where the scalar fields ${\displaystyle \phi (\mathit{x})}$${\displaystyle ,\psi (\mathit{x})}$ and ${\displaystyle \chi (\mathit{x})}$ are known as Clebsch potentials or Monge potentials, named after Alfred Clebsch (1833–1872) and Gaspard Monge (1746–1818), and ${\displaystyle \mathbf{\nabla }}$ 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. 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.

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