# About the convective constancy of Clebsch potentials

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?

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