Clebsch representation

In physics and mathematics, the Clebsch representation of an arbitrary three-dimensional vector field $\boldsymbol{v}(\boldsymbol{x})$ is:{{harvtxt|Lamb|1993|pages=248–249}}{{harvtxt|Serrin|1959|pages=169–171}}

$\boldsymbol{v} = \boldsymbol{\nabla} \varphi + \psi\, \boldsymbol{\nabla} \chi,$

where the scalar fields $\varphi(\boldsymbol{x})$ $, \psi(\boldsymbol{x})$ and $\chi(\boldsymbol{x})$ are known as Clebsch potentials{{harvtxt|Benjamin|1984}} or Monge potentials,{{harvtxt|Aris|1962|pages=70–72}} 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.{{harvtxt|Clebsch|1859}}{{harvtxt|Bateman|1929}}{{harvtxt|Seliger|Whitham|1968}} 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.{{harvtxt|Luke|1967}}

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.{{harvtxt|Wesseling|2001|page=7}} 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.{{harvtxt|Wu|Ma|Zhou|2007|page=43}}

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 $\varphi.$

Notes

References

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Category:Vector calculus

Category:Fluid dynamics

Category:Plasma theory and modeling