Clebsch representation

In physics and mathematics, the Clebsch representation of an arbitrary three-dimensional vector field formula_1 is:
where the scalar fields formula_3formula_4 and formula_5 are known as Clebsch potentials or Monge potentials, named after Alfred Clebsch (1833–1872) and Gaspard Monge (1746–1818), and formula_6 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 formula_7 has (locally) to be bounded, continuous and sufficiently smooth. For global applicability formula_7 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 formula_9 is in general not solenoidal, the Clebsch representation does not in general satisfy the Helmholtz decomposition.
Vorticity.
The vorticity formula_10 is equal to
with the last step due to the vector calculus identity formula_12 So the vorticity formula_13 is perpendicular to both formula_14 and formula_15 while further the vorticity does not depend on formula_16 