The question is as to whether we can do this in a sensible and self-consistent manner. We shall look at this in a variety of ways, starting with a rather [*ad hoc*]{} approach.

Reorganizing the low Reynolds number\
equations
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The production of a global solution to the equations requires a slightly unusual approach. We take the time-independent equation $$\nabla^4 \psi = R_e \frac{\partial(\psi, \delsq \psi)}{\partial(x, y)} \label{oldfour}$$ as our starting point and note some obvious facts. First, if we have a $\psi_0$ satisfying Laplace’s equation, then it satisfies equation Eq. (\[oldfour\]) identically. Second, in the high Reynolds number limit, although this is a singular limit, we note that the Jacobian term should then vanish identically, so that any solution of $$\delsq \psi = f(\psi)$$ satisfies Eq. (\[oldfour\]) when $R_e \rightarrow \infty$. This will be true of course in the linear case when $f(\psi) \propto \psi$ and this last equation is of Helmholtz type. Now let $\lambda$ be a (possibly complex) number of order 1 and $\beta$ a parameter to be determined. In particular, if we consider solutions, $\psi_1$, of a parametrized Helmholtz equation in the form $$\delsq \psi_1 = \lambda R_e^{2\beta} \psi_1$$ then the Jacobian will vanish and we 