Clebsch representation

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

$$𝒗=\mathrm{\nabla }\phi +\psi \mathrm{\nabla }\chi ,$$

where the scalar fields $\phi (𝒙)$$,\psi (𝒙)$ and $\chi (𝒙)$ 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 $𝒗$ has (locally) to be bounded, continuous and sufficiently smooth. For global applicability $𝒗$ 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 $𝝎(𝒙)$ is equal to

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

with the last step due to the vector calculus identity $\mathrm{\nabla }\times (\psi 𝑨)=\psi (\mathrm{\nabla }\times 𝑨)+\mathrm{\nabla }\psi \times 𝑨.$ So the vorticity $𝝎$ 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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