Given a sufficiently smooth scalar field $\phi$, how to get another scalar field $\psi$ so that their gradients are orthogonal. In 2D case, this is equivalent to:$$\nabla \phi \cdot \nabla \psi=\frac{\partial\phi}{\partial x}\frac{\partial\psi}{\partial x}+\frac{\partial\phi}{\partial y}\frac{\partial\psi}{\partial y}=0$$.
Notice the solution of $\psi$ may be up to a constant. This may be similar to the problem of finding a orthogonal curvlinear coordinate. I just want to know when does the solution exist and how to calculate it.