Clebsch variables

Question

I should find if it is true that I can formalize the velocity in BE-condensate with a clebsch decomposition. We know that $$u=\nabla \phi.$$ with $\phi$ the phase of the madlung tranform $$\phi=\sqrt\rho \exp^{i\phi}.$$ So we know that $\phi$ is a multivalued-function. Do you think it is possible to write $$v^*=df-\phi^* \theta. $$ The formal definition of clebsch decompsition is here https://drive.google.com/file/d/1skVTCT1K1gGcfUhTjbFRyxZl6e2f6OdQ/view