Clebsch representation explained

\boldsymbol{v}(\boldsymbol{x}) is:

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

where the scalar fields

\varphi(\boldsymbol{x}) , \psi(\boldsymbol{x}) and \chi(\boldsymbol{x}) are known as Clebsch potentials or Monge potentials, 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. 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{v} has (locally) to be bounded, continuous and sufficiently smooth. For global applicability \boldsymbol{v} 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{\nabla}\chi is in general not solenoidal, the Clebsch representation does not in general satisfy the Helmholtz decomposition.

Vorticity

The vorticity

\boldsymbol{\omega}(\boldsymbol{x}) is equal to

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

\boldsymbol{\nabla} x (\psi \boldsymbol{A}) = \psi(\boldsymbol{\nabla} x \boldsymbol{A}) + \boldsymbol{\nabla}\psi x \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.

References

• • • • • • • Encyclopedia: Encyclopedia of Mathematical Physics . 2 . 593–600 . 2006 . Hamiltonian fluid mechanics . P.J. . Morrison . Philip J. Morrison . http://web2.ph.utexas.edu/~morrison/06EMP_morrison.pdf . Elsevier . 10.1016/B0-12-512666-2/00246-7 . Hamiltonian Fluid Dynamics . 9780125126663 .