In physics and mathematics, the '''Clebsch representation''' of an arbitrary three-dimensional vector field <math>\boldsymbol{v}(\boldsymbol{x})</math> is:<ref name=Lamb>{{harvtxt|Lamb|1993|pages=248–249}}</ref><ref name=Serrin>{{harvtxt|Serrin|1959|pages=169–171}}</ref>
<math display="block">\boldsymbol{v} = \boldsymbol{\nabla} \varphi + \psi\, \boldsymbol{\nabla} \chi,</math>
where the scalar fields <math>\varphi(\boldsymbol{x})</math><math>, \psi(\boldsymbol{x})</math> and <math>\chi(\boldsymbol{x})</math> are known as '''Clebsch potentials'''<ref>{{harvtxt|Benjamin|1984}}</ref> or '''Monge potentials''',<ref>{{harvtxt|Aris|1962|pages=70–72}}</ref> named after Alfred Clebsch (1833–1872) and Gaspard Monge (1746–1818), and <math>\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.<ref>{{harvtxt|Clebsch|1859}}</ref><ref>{{harvtxt|Bateman|1929}}</ref><ref>{{harvtxt|Seliger|Whitham|1968}}</ref> 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.<ref>{{harvtxt|Luke|1967}}</ref>
For the Clebsch representation to be possible, the vector field <math>\boldsymbol{v}</math> has (locally) to be bounded, continuous and sufficiently smooth. For global applicability <math>\boldsymbol{v}</math> has to decay fast enough towards infinity.<ref>{{harvtxt|Wesseling|2001|page=7}}</ref> The Clebsch decomposition is not unique, and (two) additional constraints are necessary to uniquely define the Clebsch potentials.<ref name=Lamb/> Since <math>\psi\boldsymbol{\nabla}\chi</math> is in general not solenoidal, the Clebsch representation does not in general satisfy the Helmholtz decomposition.<ref>{{harvtxt|Wu|Ma|Zhou|2007|page=43}}</ref>
==Vorticity==
The vorticity <math>\boldsymbol{\omega}(\boldsymbol{x})</math> is equal to<ref name=Serrin/>
<math display="block"> \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>\boldsymbol{\nabla} \times (\psi \boldsymbol{A})=\psi(\boldsymbol{\nabla}\times\boldsymbol{A})+\boldsymbol{\nabla}\psi\times\boldsymbol{A}.</math> So the vorticity <math>\boldsymbol{\omega}</math> is perpendicular to both <math>\boldsymbol{\nabla}\psi</math> and <math>\boldsymbol{\nabla}\chi,</math> while further the vorticity does not depend on <math>\varphi.</math>
==Notes== {{reflist}}
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Category:Vector calculus Category:Fluid dynamics Category:Plasma theory and modeling