\varphi^0\, \mathbf{e}_z )$. The condition for the vanishing of nonlinear terms then becomes $$\mathbf{\nabla} \left( \mathbf{\nabla}^2 \, \varphi^0 \right) \times
\mathbf{\nabla} \varphi^0 = 0, \quad \textrm{with} \quad \varphi^0 = \varphi^0 \left(x, y\right).$$ This condition is then satisfied by those fields for which the laplacian is constant along the streamlines of the field. As $\omega = - \mathbf{\nabla}^2 \, \varphi$ this is equivalent to the statement that the *vorticity is constant along the streamlines*. This condition is in general not verified, unless very symmetric functions are chosen, e.g. in cartesian geometry by any 1D function like $\varphi^0 = f(x)$, and in polar coordinates by any radial function $\varphi^0 = g(r)$.

Generally speaking even in such peculiar configurations non-linear interactions will arise due to the onset of instabilities. We defer discussion of these extreme examples to a subsequent paper, the random photospheric fields  (\[eq:bc\]) discussed here always giving rise to non-vanishing forces .

Inserting the linear evolution fields  (\[eq:blin\]) in the expression for the integrated energy flux (\[eq:tsz\]), we find $$\label{eq:slin}
S = c_{\mathcal A} \int\! \mathrm{d} a\, | \mathbf{u}^L - \mathbf{u}^0 |^2 
\cdot \frac{t}{\tau_{\mathcal A}},$$ i.e. the Poynting flux $S$ grows linearly in time until such a time that non-linear interactions 