$\varepsilon >0$, the Laplace-Helmholtz equation $$\nabla^4 \psi - \varepsilon^2 \nabla^2 \psi=0$$ has a solution of the form $$\psi = e^{\mu \tau}\left[r
-\biggl(1+
\frac{2
   K_1(\varepsilon )}{\varepsilon  K_0(\varepsilon
   )}
\biggr) 
\frac{1}{r}
+\frac{2 K_1(r \varepsilon )}{\varepsilon 
 K_0(\varepsilon )}
 \right] \sin (\theta ) \label{tdsf}$$ that satisfies $$\begin{split}
\psi = 0 = \frac{\partial \psi}{\partial r} \ {\rm if }\  r= 1\\
\psi \sim  e^{\mu \tau} r \sin(\theta) \  {\rm as}\  r \rightarrow \infty
\end{split}$$ In other words, no matter how small the value of $\mu$, the paradox does not exist.

To summarize: If we have some time-dependence in the simplified form of exponential growth in vorticity the linearization is precisely the Laplace-Helmholtz problem rather than the biharmonic problem. Addition of this type of extra variability, no matter how small, shows that the paradox evaporates, i.e. the existence of the paradox is unstable. Villas Boas [@mario], who has considered the problem from a three-dimensional perspective, also demonstrates that the paradox evaporates when extra [*spatial*]{} variability is incorporated.

In subsequent analysis we will use the Laplace-Helmholtz model as a approximation for the time-[*independent*]{} 2D case, but where the modification is regarded as a simple way as approximating the combined effect of the non-linearities via a linear term. 