expressions for $q^\smalL(\x,t)$ and the other fields. After introducing the increase of trajectory separation in a time $t$: $\delta\z(t|\x,\y)=\z(t|\x,0)-\z(t|\y,0)-(\x-\y)$ Eqn. (2.7) becomes, in the new variables: $$\partial_tq^\smalL(\x,t)+\int\d^2y\gamma(\x-\y+\delta\z(t|\x,\y))q^\smalL(\y,t)=
\int\d^2y h(\x-\y+\delta\z(t|\x,\y))\xi(\y,t)
\eqno(2.11)$$ which must be coupled with the equation for $\delta\z$; inverting Eqn. (2.2): $$\partial_t\delta\z(t|\x,\y)=\frac{1}{2\pi}\int\d^2r[{\bf G}(\x,\r)-{\bf G}(\y,\r)]q^\smalL(\r,t)
\eqno(2.12)$$ with $${\bf G}(\x,\r)=
\frac{(\x-\r+\delta\z(t|\x,\r))_\perp}
{|\x-\r+\delta\z(t|\x,\r)|^2}
\eqno(2.13)$$ We see then that the natural expansion parameter of the theory is: $$\frac{\delta z(t|\x,\y)}{|\x-\y|}
\sim\frac{|\u(\x,0)-\u(\y,0)|}{|\x-\y|\gamma_{|\x-\y|^{-1}}}
\sim\rho^{-1}
\eqno(2.14)$$ i.e. the relative amount of particle separation increase in an eddy lifetime. The zero order of the theory, which is Gaussian and is described by Eqn. (2.6) after substituting $\u\to\u^\smalL$, corresponds to neglecting trajectory separation in an eddy lifetime, while keeping the uniform large scale sweep, implicit in the Lagrangian field $q^\smalL$.

Although the results which follow in the present paper are all obtained to the lowest order in the $\rho$ expansion [@note1], associated with neglecting all non-Gaussian effects in $\u$, a diagrammatic expansion of Eqns. (2.12-13) in terms of the fields $q^\smalL$, $\delta\z$ and their conjugate could be obtained by means of the Martin-Siggia-Rose formalism [@martin73]. This expansion would only be valid locally around $t=0$, since, at long times, trajectory separation becomes dominant. $[$To be consistent, this perturbation expansion should 