Clebsch representation

In physics and mathematics, the Clebsch representation of an arbitrary three-dimensional vector field [math]\displaystyle{ \boldsymbol{v}(\boldsymbol{x}) }[/math] is: [math]\displaystyle{ \boldsymbol{v} = \boldsymbol{\nabla} \varphi + \psi\, \boldsymbol{\nabla} \chi, }[/math]

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

Vorticity

The vorticity [math]\displaystyle{ \boldsymbol{\omega}(\boldsymbol{x}) }[/math] is equal to

[math]\displaystyle{ \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, }[/math]

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

References

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• Clebsch, A. (1859), "Ueber die Integration der hydrodynamischen Gleichungen", Journal für die Reine und Angewandte Mathematik 1859 (56): 1–10, doi:10.1515/crll.1859.56.1, https://zenodo.org/record/1448884 
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• Wesseling, P. (2001), Principles of computational fluid dynamics, Springer, ISBN 978-3-540-67853-3 
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