# Clebsch representation

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

${\boldsymbol {v}}={\boldsymbol {\nabla }}\varphi +\psi \,{\boldsymbol {\nabla }}\chi ,$ where the scalar fields $\varphi ({\boldsymbol {x}})$ $,\psi ({\boldsymbol {x}})$ and $\chi ({\boldsymbol {x}})$ are known as Clebsch potentials or Monge potentials, named after Alfred Clebsch (1833–1872) and Gaspard Monge (1746–1818), and ${\boldsymbol {\nabla }}$ 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. 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 ${\boldsymbol {v}}$  has (locally) to be bounded, continuous and sufficiently smooth. For global applicability ${\boldsymbol {v}}$  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 $\psi {\boldsymbol {\nabla }}\chi$  is in general not solenoidal, the Clebsch representation does not in general satisfy the Helmholtz decomposition.

## Vorticity

The vorticity ${\boldsymbol {\omega }}({\boldsymbol {x}})$  is equal to

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

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