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Clebsch representation

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In physics and mathematics, the Clebsch representation of an arbitrary three-dimensional vector field v ( x ) {\displaystyle {\boldsymbol {v}}({\boldsymbol {x}})} is:

v = ∇ φ + ψ ∇ χ , {\displaystyle {\boldsymbol {v}}={\boldsymbol {\nabla }}\varphi +\psi \,{\boldsymbol {\nabla }}\chi ,}

where the scalar fields φ ( x ) {\displaystyle \varphi ({\boldsymbol {x}})} , ψ ( x ) {\displaystyle ,\psi ({\boldsymbol {x}})} and χ ( x ) {\displaystyle \chi ({\boldsymbol {x}})} are known as Clebsch potentials or Monge potentials, named after Alfred Clebsch (1833–1872) and Gaspard Monge (1746–1818), and ∇ {\displaystyle {\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 v {\displaystyle {\boldsymbol {v}}} has (locally) to be bounded, continuous and sufficiently smooth. For global applicability v {\displaystyle {\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 ψ ∇ χ {\displaystyle \psi {\boldsymbol {\nabla }}\chi } is in general not solenoidal, the Clebsch representation does not in general satisfy the Helmholtz decomposition.

Vorticity

The vorticity ω ( x ) {\displaystyle {\boldsymbol {\omega }}({\boldsymbol {x}})} is equal to

ω = ∇ × v = ∇ × ( ∇ φ + ψ ∇ χ ) = ∇ ψ × ∇ χ , {\displaystyle {\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 ∇ × ( ψ A ) = ψ ( ∇ × A ) + ∇ ψ × A . {\displaystyle {\boldsymbol {\nabla }}\times (\psi {\boldsymbol {A}})=\psi ({\boldsymbol {\nabla }}\times {\boldsymbol {A}})+{\boldsymbol {\nabla }}\psi \times {\boldsymbol {A}}.} So the vorticity ω {\displaystyle {\boldsymbol {\omega }}} is perpendicular to both ∇ ψ {\displaystyle {\boldsymbol {\nabla }}\psi } and ∇ χ , {\displaystyle {\boldsymbol {\nabla }}\chi ,} while further the vorticity does not depend on φ . {\displaystyle \varphi .}

Notes

  1. ^ Lamb (1993, pp. 248–249)
  2. ^ Serrin (1959, pp. 169–171)
  3. Benjamin (1984)
  4. Aris (1962, pp. 70–72)
  5. Clebsch (1859)
  6. Bateman (1929)
  7. Seliger & Whitham (1968)
  8. Luke (1967)
  9. Wesseling (2001, p. 7)
  10. Wu, Ma & Zhou (2007, p. 43)

References

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