Explanation for the next steps of lamb-chaplygin dipole

Question

Can someone explain me the steps please (I mean steps 2 and 3)? I didn't understand it.

1) Any flow field represented by a stream function $\psi$(x,r), defined as:

• v = - $\frac{\partial \psi}{\partial x}$, u = $\frac{\partial \psi}{\partial y}$

2) With (u,v) the velocity components in the Cartesian coordinate system (x,y), is a solution of the two-dimensional Euler equations for incompressible fluid provided that it satisfies the equation:

• $\frac{\partial ^2 \psi}{\partial x^2} + \frac{\partial ^2 \psi}{\partial y^2} = f(\psi)$

3) where $f(\psi)$ is an arbitrary function of $\psi$.

any help appreciated!

Answer 1

When introducing the stream function, the steps that you usually take are as follows.

1. Replace $u$ and $v$ by the streamfunction.
2. Derive the horizontal momentum equation (for $u$) with respect to $y$ and the other with respect to $x$.
3. Eliminate the pressure term, to end op with a single equation in $\psi$.