Clebsch representation

In physics and mathematics, the Clebsch representation of an arbitrary three-dimensional vector field $\mathit{v}(\mathit{x})$ is: $$\mathit{v}=\mathrm{\nabla }\phi +\psi \mathrm{\nabla }\chi ,$$

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

Vorticity

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

$$\mathit{\omega }=\mathrm{\nabla }\times \mathit{v}=\mathrm{\nabla }\times (\mathrm{\nabla }\phi +\psi \mathrm{\nabla }\chi )=\mathrm{\nabla }\psi \times \mathrm{\nabla }\chi ,$$

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

References

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