For a fluid with non-zero vorticity, the following representation for the velocity in terms of scalar variables is well-known: $$ \vec{v}=\vec{\nabla}\phi +\beta\vec{\nabla}{\gamma} $$ It is called Clebsch representation. I've been reading that the so called Clebsch variables $\gamma$ and $\beta$ have a zero convective derivative: $$ \dot{\gamma}+(\vec{v}\cdot \vec{\nabla})\gamma=0 $$ $$ \dot{\beta}+(\vec{v}\cdot \vec{\nabla})\beta=0 $$

Why do Clebsch variables have a zero convective derivative?

I've tried to prove it by myself but I haven't been able . I've been searching a lot for a proof trough the Internet, but the only things I've found require lagrangian or hamiltonian formalism. Can I prove these equations by only using Euler fluid equations and/or continuity equation?

  • 2
    $\begingroup$ Been reading where? $\endgroup$
    – Qmechanic
    Feb 25 '18 at 21:30
  • 2
    $\begingroup$ Hint. There is gauge freedom in the definition of these potentials. Can you use it to get out of convective constancy? $\endgroup$ Feb 25 '18 at 22:26
  • $\begingroup$ @Qmechanic In link $\endgroup$ Feb 25 '18 at 23:59
  • $\begingroup$ @CosmasZachos Sorry, which Gauge freedom? $\endgroup$ Feb 25 '18 at 23:59

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