Clebsch representation

In physics and mathematics, the Clebsch representation of an arbitrary three-dimensional vector field is:[1][2]

where the scalar fields and are known as Clebsch potentials[3] or Monge potentials,[4] named after Alfred Clebsch (1833–1872) and Gaspard Monge (1746–1818), and is the gradient operator.


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.[5][6][7] 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.[8]

For the Clebsch representation to be possible, the vector field has (locally) to be bounded, continuous and sufficiently smooth. For global applicability has to decay fast enough towards infinity.[9] The Clebsch decomposition is not unique, and (two) additional constraints are necessary to uniquely define the Clebsch potentials.[1] Since is in general not solenoidal, the Clebsch representation does not in general satisfy the Helmholtz decomposition.[10]


The vorticity is equal to[2]

with the last step due to the vector calculus identity So the vorticity is perpendicular to both and while further the vorticity does not depend on



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