Clebsch representation

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

$\boldsymbol = \boldsymbol \varphi + \psi\, \boldsymbol \chi,$

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

Vorticity

The vorticity $\boldsymbol(\boldsymbol)$ is equal to

$\boldsymbol = \boldsymbol\times\boldsymbol = \boldsymbol\times\left( \boldsymbol \varphi + \psi\, \boldsymbol \chi\right) = \boldsymbol\psi \times \boldsymbol\chi,$

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

Notes

References

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Vector calculus
Fluid dynamics
Plasma physics