Let us first find out under which conditions velocity profile \eqref{eq:keplerian} becomes unstable. Following \cite{rayleigh1917dynamics}, consider the fluid element rotating at a radius $r_0$ from the axis of rotation, like in Figure \ref{fig:disk_scheme}b. Its angular momentum is $L_0=m r_0^2 \Omega$, and corresponding centrifugal force is $L_0^2/ m r_0^3$. Imagine this element was displaced to another radius $r>r_0$. The element still possesses the same amount of angular momentum $L_0$, but the acting centrifugal force at the new radius changes: $L_0^2/m r^3$. In order for the element to return to its initial orbit, this force has to be smaller than the equilibrium centrifugal force $L^2/m r^3$ balanced by the pressure gradient. In other words, $L_0^2$ should be smaller than $L^2$:
\begin{equation}
L^2-L_0^2>0.
\end{equation}

Expanding $L$ in Taylor series around the point $r_0$, we get the stability condition in more general form:
\begin{equation}\label{eq:ang_mom_stab}
L \frac{d L}{d r}>0. …show more content…
Then

\begin{equation}
\frac{d L}{d r} \sim \frac{d (r^2 \Omega)}{d r}>0.
\end{equation}

That is, the flow is stable to the small perturbations, if the angular momentum increases with