2.1 LSM in spherical polar coordinates

From here on, it is more convenient to work in terms of spherical polar coordinates. The transformation of (2.1) is given by Ito’s formula

(2.5) $$\begin{eqnarray}\text{d}f(\tilde{\boldsymbol{x}})=\left(\tilde{a}_{i}\frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\tilde{x}_{i}}+\frac{1}{2}(\tilde{\boldsymbol{b}}\tilde{\boldsymbol{b}}^{\text{T}})_{ij}\frac{\unicode[STIX]{x2202}^{2}f}{\unicode[STIX]{x2202}\tilde{x}_{i}\unicode[STIX]{x2202}\tilde{x}_{j}}\right)\text{d}t+\tilde{b}_{ij}\frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\tilde{x}_{i}}\,\text{d}W_{j}\quad i,j=1,\ldots ,6\end{eqnarray}$$

for any function $f(\tilde{\boldsymbol{x}})$ where $\tilde{\boldsymbol{x}}=(x_{1},x_{2},x_{3},u_{1},u_{2},u_{3})$ , $\tilde{\boldsymbol{a}}=(u_{1},u_{2},u_{3},a_{1},a_{2},a_{3})$ and $\tilde{\boldsymbol{b}}=\sqrt{2C_{0}\unicode[STIX]{x1D700}}\,\text{diag}(0,0,0,1,1,1)$ . Specifying the basis vectors as

(2.6a-c ) $$\begin{eqnarray}\boldsymbol{e}_{r}=\frac{\boldsymbol{x}}{r},\quad \boldsymbol{e}_{\unicode[STIX]{x1D703}}=\cos \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D719}\boldsymbol{i}+\cos \unicode[STIX]{x1D703}\sin \unicode[STIX]{x1D719}\boldsymbol{j}-\sin \unicode[STIX]{x1D703}\boldsymbol{k},\quad \boldsymbol{e}_{\unicode[STIX]{x1D719}}=-\!\sin \unicode[STIX]{x1D719}\boldsymbol{i}+\cos \unicode[STIX]{x1D719}\boldsymbol{j},\end{eqnarray}$$

where $\unicode[STIX]{x1D703}=\arccos (x_{3}/r)$ and $\unicode[STIX]{x1D719}=\arctan (x_{2}/x_{1})$ are the polar and azimuthal angles respectively, we define the components of $\boldsymbol{u}$ in spherical polar coordinates as

(2.7a-c ) $$\begin{eqnarray}u_{\Vert }=\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{e}_{r},\quad u_{\bot }=\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}},\quad u_{\vdash }=\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D719}},\end{eqnarray}$$

where $u_{\Vert }$ is the longitudinal component and $u_{\bot }$ and $u_{\vdash }$ are the transverse components. Using (2.5), we can obtain a stochastic differential equation (SDE) for any function of $\tilde{\boldsymbol{x}}$ . The SDEs for $u_{\Vert }$ , $u_{\bot }$ and $u_{\vdash }$ are given by

(2.8) $$\begin{eqnarray}\displaystyle & \text{d}u_{\Vert }=a_{\Vert }\text{d}t+\sqrt{2C_{0}\unicode[STIX]{x1D700}}\,\text{d}W_{\Vert }, & \displaystyle\end{eqnarray}$$

(2.9) $$\begin{eqnarray}\displaystyle & \text{d}u_{\bot }=a_{\bot }\,\text{d}t+\sqrt{2C_{0}\unicode[STIX]{x1D700}}\,\text{d}W_{\bot }, & \displaystyle\end{eqnarray}$$

and

(2.10) $$\begin{eqnarray}\text{d}u_{\vdash }=a_{\vdash }\,\text{d}t+\sqrt{2C_{0}\unicode[STIX]{x1D700}}\,\text{d}W_{\vdash },\end{eqnarray}$$

where

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle a_{\Vert }=\boldsymbol{a}\boldsymbol{\cdot }\boldsymbol{e}_{r}+\boldsymbol{u}\boldsymbol{\cdot }\dot{\boldsymbol{e}}_{r}=\frac{\boldsymbol{a}\boldsymbol{\cdot }\boldsymbol{x}}{r}+\frac{u_{\bot }^{2}+u_{\vdash }^{2}}{r}, & \displaystyle\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle a_{\bot }=\boldsymbol{a}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}}+\boldsymbol{u}\boldsymbol{\cdot }\dot{\boldsymbol{e}}_{\unicode[STIX]{x1D703}}=\boldsymbol{a}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}}+\frac{u_{\vdash }^{2}\cos \unicode[STIX]{x1D703}}{r\sin \unicode[STIX]{x1D703}}-\frac{u_{\Vert }u_{\bot }}{r}, & \displaystyle\end{eqnarray}$$

and

(2.13) $$\begin{eqnarray}a_{\vdash }=\boldsymbol{a}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D719}}+\boldsymbol{u}\boldsymbol{\cdot }\dot{\boldsymbol{e}}_{\unicode[STIX]{x1D719}}=\boldsymbol{a}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D719}}-\frac{u_{\Vert }u_{\vdash }}{r}-\frac{u_{\bot }u_{\vdash }\cos \unicode[STIX]{x1D703}}{r\sin \unicode[STIX]{x1D703}},\end{eqnarray}$$

with $\text{d}W_{\Vert }=\text{d}\boldsymbol{W}\boldsymbol{\cdot }\boldsymbol{e}_{r}$ , $\text{d}W_{\bot }=\text{d}\boldsymbol{W}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}}$ and $\text{d}W_{\vdash }=\text{d}\boldsymbol{W}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D719}}$ . We see that $a_{\Vert }$ , $a_{\bot }$ and $a_{\vdash }$ are equal to the component of $\boldsymbol{a}$ in the relevant direction plus a ‘metric’ term arising from the curved coordinate system. The SDE for the absolute perpendicular velocity, $u_{p}=\sqrt{u_{\bot }^{2}+u_{\vdash }^{2}}$ , is given by

(2.14) $$\begin{eqnarray}\text{d}u_{p}=a_{p}\,\text{d}t+\sqrt{2C_{0}\unicode[STIX]{x1D700}}\,\text{d}W_{p},\end{eqnarray}$$

where

(2.15) $$\begin{eqnarray}a_{p}=\boldsymbol{a}\boldsymbol{\cdot }\frac{\boldsymbol{u}-u_{\Vert }\boldsymbol{x}/r}{u_{p}}+\frac{C_{0}\unicode[STIX]{x1D700}}{u_{p}}-\frac{u_{\Vert }u_{p}}{r}\end{eqnarray}$$

and $\text{d}W_{p}=\text{d}\boldsymbol{W}\boldsymbol{\cdot }(\boldsymbol{u}-u_{\Vert }\boldsymbol{x}/r)/u_{p}$ . The SDEs for $r$ , $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D719}$ are simply $\text{d}r=u_{\Vert }\,\text{d}t$ , $\text{d}\unicode[STIX]{x1D703}=(u_{\bot }/r)\,\text{d}t$ and $\text{d}\unicode[STIX]{x1D719}=(u_{\vdash }/r\sin \unicode[STIX]{x1D703})\,\text{d}t$ . Because of our assumption of isotropy, $p_{E}$ depends only on $u_{\Vert }$ , $u_{p}$ and $r$ and, to ensure $\boldsymbol{a}$ is consistent with our assumption of isotropy, $a_{\Vert }$ and $a_{p}$ must be chosen to depend only on $u_{\Vert }$ , $u_{p}$ and $r$ . Thus, in isotropic turbulence, equations (2.8), (2.14) and $\text{d}r=u_{\Vert }\,\text{d}t$ are sufficient for determining the relative dispersion statistics. In effect, the 3-D model has been reduced to a Q2D model.

The Fokker–Planck equation corresponding to (2.8)–(2.10) and the equations for $\text{d}r$ , $\text{d}\unicode[STIX]{x1D703}$ and $\text{d}\unicode[STIX]{x1D719}$ is

(2.16) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\hat{p}}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}u_{\Vert }\hat{p}}{\unicode[STIX]{x2202}r}+\frac{\unicode[STIX]{x2202}(u_{\bot }/r)\hat{p}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}+\frac{\unicode[STIX]{x2202}(u_{\vdash }/r\sin \unicode[STIX]{x1D703})\hat{p}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}+\frac{\unicode[STIX]{x2202}a_{\Vert }\hat{p}}{\unicode[STIX]{x2202}u_{\Vert }}+\frac{\unicode[STIX]{x2202}a_{\bot }\hat{p}}{\unicode[STIX]{x2202}u_{\bot }}+\frac{\unicode[STIX]{x2202}a_{\vdash }\hat{p}}{\unicode[STIX]{x2202}u_{\vdash }}=C_{0}\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FB}_{\boldsymbol{u}}^{2}\hat{p}.\quad\end{eqnarray}$$

Here $\hat{p}$ is the density function of the distribution of all pairs in $(r,\unicode[STIX]{x1D703},\unicode[STIX]{x1D719},u_{\Vert },u_{\bot },u_{\vdash })$ -space and equals $r^{2}\sin \unicode[STIX]{x1D703}\,p(\boldsymbol{u},\boldsymbol{x},t)$ , with $r^{2}\sin \unicode[STIX]{x1D703}$ being the Jacobian of the transformation between the two coordinate systems, while $\unicode[STIX]{x1D6FB}_{\boldsymbol{u}}^{2}$ denotes $\unicode[STIX]{x2202}^{2}/\unicode[STIX]{x2202}u_{\Vert }^{2}+\unicode[STIX]{x2202}^{2}/\unicode[STIX]{x2202}u_{\bot }^{2}+\unicode[STIX]{x2202}^{2}/\unicode[STIX]{x2202}u_{\vdash }^{2}$ . This can also be derived by transforming (2.2) to the new coordinate system (Risken Reference Risken1989, pp. 88–91). Note that, if we regard the $(x_{1},x_{2},x_{3},u_{1},u_{2},u_{3})$ -coordinate system as orthogonal, then the $(r,\unicode[STIX]{x1D703},\unicode[STIX]{x1D719},u_{\Vert },u_{\bot },u_{\vdash })$ -coordinate system is non-orthogonal. This is because if we consider a vector along a trajectory with either $\unicode[STIX]{x1D703}$ or $\unicode[STIX]{x1D719}$ changing and with the remaining coordinates in the $(r,\unicode[STIX]{x1D703},\unicode[STIX]{x1D719},u_{\Vert },u_{\bot },u_{\vdash })$ -coordinate system held constant (i.e. what one might call a vector in the $\unicode[STIX]{x1D703}$ or $\unicode[STIX]{x1D719}$ direction), then this vector has components in some of the $(u_{1},u_{2},u_{3})$ directions as well as in the $(x_{1},x_{2},x_{3})$ directions (i.e. $(u_{1},u_{2},u_{3})$ changes along the trajectory). Hence we cannot use results for orthogonal coordinate systems to make the transformation. As with (2.2), the model should be such that (2.16) is satisfied by $\hat{p}_{E}=r^{2}\sin \unicode[STIX]{x1D703}p_{E}$ . Because of isotropy, $p_{E}$ has no dependence on $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D719}$ (for fixed $r$ , $u_{\Vert }$ , $u_{\bot }$ and $u_{\vdash }$ ) and so, for stationary turbulence, we obtain

(2.17) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}u_{\Vert }\hat{p}_{E}}{\unicode[STIX]{x2202}r}+\frac{u_{\bot }\hat{p}_{E}\cos \unicode[STIX]{x1D703}}{r\sin \unicode[STIX]{x1D703}}+\frac{\unicode[STIX]{x2202}a_{\Vert }\hat{p}_{E}}{\unicode[STIX]{x2202}u_{\Vert }}+\frac{\unicode[STIX]{x2202}a_{\bot }\hat{p}_{E}}{\unicode[STIX]{x2202}u_{\bot }}+\frac{\unicode[STIX]{x2202}a_{\vdash }\hat{p}_{E}}{\unicode[STIX]{x2202}u_{\vdash }}=C_{0}\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FB}_{\boldsymbol{u}}^{2}\hat{p}_{E}.\end{eqnarray}$$

It is convenient, in our new coordinate system, to follow the split $\boldsymbol{a}=\boldsymbol{a}^{(1)}+\boldsymbol{a}^{(2)}$ introduced above and to split $a_{\Vert }$ , $a_{\bot }$ and $a_{\vdash }$ into (i) terms proportional to $C_{0}$ which balance the right-hand side of (2.17), (ii) metric terms arising from $\dot{\boldsymbol{e}}_{r}$ , $\dot{\boldsymbol{e}}_{\unicode[STIX]{x1D703}}$ and $\dot{\boldsymbol{e}}_{\unicode[STIX]{x1D719}}$ and (iii) terms related to the conditional mean accelerations (the metric terms have no equivalent in the Cartesian coordinate system used above), giving

(2.18) $$\begin{eqnarray}\displaystyle & \displaystyle a_{\Vert }=C_{0}\unicode[STIX]{x1D700}\frac{\unicode[STIX]{x2202}\ln p_{E}}{\unicode[STIX]{x2202}u_{\Vert }}+\frac{u_{\bot }^{2}+u_{\vdash }^{2}}{r}+a_{\Vert }^{\prime }, & \displaystyle\end{eqnarray}$$

(2.19) $$\begin{eqnarray}\displaystyle & \displaystyle a_{\bot }=C_{0}\unicode[STIX]{x1D700}\frac{\unicode[STIX]{x2202}\ln p_{E}}{\unicode[STIX]{x2202}u_{\bot }}+\frac{u_{\vdash }^{2}\cos \unicode[STIX]{x1D703}}{r\sin \unicode[STIX]{x1D703}}-\frac{u_{\Vert }u_{\bot }}{r}+a_{\bot }^{\prime } & \displaystyle\end{eqnarray}$$

and

(2.20) $$\begin{eqnarray}a_{\vdash }=C_{0}\unicode[STIX]{x1D700}\frac{\unicode[STIX]{x2202}\ln p_{E}}{\unicode[STIX]{x2202}u_{\vdash }}-\frac{u_{\Vert }u_{\vdash }}{r}-\frac{u_{\bot }u_{\vdash }\cos \unicode[STIX]{x1D703}}{r\sin \unicode[STIX]{x1D703}}+a_{\vdash }^{\prime }.\end{eqnarray}$$

Conceptually, in Cartesian coordinates, $\boldsymbol{a}$ is both the conditional mean acceleration vector in the immediate future and the deterministic term in the SDE, with $\boldsymbol{a}^{(2)}$ corresponding to the conditional mean acceleration at the current time and $\boldsymbol{a}^{(1)}$ being the difference between that and the conditional mean acceleration in the immediate future. In the non-Cartesian coordinates, the conditional mean acceleration vector in the immediate future and the deterministic term in the SDE are no longer equal, with the metric terms reflecting this difference, and with the $C_{0}$ terms and $(a_{\Vert }^{\prime },a_{\bot }^{\prime },a_{\vdash }^{\prime })$ being the vectors $\boldsymbol{a}^{(1)}$ and $\boldsymbol{a}^{(2)}$ re-expressed in terms of $\boldsymbol{e}_{r}$ , $\boldsymbol{e}_{\unicode[STIX]{x1D703}}$ and $\boldsymbol{e}_{\unicode[STIX]{x1D719}}$ .

Because of isotropy, $p_{E}$ and $a_{\Vert }^{\prime }$ must depend only on $u_{\Vert }$ , $u_{p}$ and $r$ (as noted above). Isotropy also implies that $(a_{\bot }^{\prime },a_{\vdash }^{\prime })=(u_{\bot },u_{\vdash }){\mathcal{A}}^{\prime }$ for some ${\mathcal{A}}^{\prime }$ (since $(u_{\bot },u_{\vdash })$ defines the only possible direction for $(a_{\bot }^{\prime },a_{\vdash }^{\prime })$ ) with ${\mathcal{A}}^{\prime }$ depending only on $u_{\Vert }$ , $u_{p}$ and $r$ . We note that ${\mathcal{A}}^{\prime }=a_{p}^{\prime }/u_{p}$ where $a_{p}^{\prime }$ is the non- $C_{0}$ and non-metric part of $a_{p}$ . It follows that $a_{\Vert }^{\prime }$ and ${\mathcal{A}}^{\prime }$ should satisfy

(2.21) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}u_{\Vert }r^{2}u_{p}p_{E}}{\unicode[STIX]{x2202}r}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}u_{\Vert }}\left(a_{\Vert }^{\prime }+\frac{u_{p}^{2}}{r}\right)r^{2}u_{p}p_{E}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}u_{p}}u_{p}\left({\mathcal{A}}^{\prime }-\frac{u_{\Vert }}{r}\right)r^{2}u_{p}p_{E}=0\end{eqnarray}$$

(here $r^{2}u_{p}p_{E}$ is the density function of the well-mixed distribution in $(r,u_{\Vert },u_{p})$ -space).

Our approach to designing the model is to choose a form for $p_{E}$ and then select $a_{\Vert }^{\prime }$ and ${\mathcal{A}}^{\prime }$ to be consistent with (2.21). We can choose either ${\mathcal{A}}^{\prime }$ or $a_{\Vert }^{\prime }$ and then calculate the other quantity by integrating (2.21). We assume that $p_{E}$ is well behaved in the sense of tending to zero sufficiently rapidly as $u_{\Vert }$ or $u_{p}\rightarrow \infty$ and remaining bounded as $u_{p}\rightarrow 0$ . We also assume that the chosen value of $a_{\Vert }^{\prime }$ (or of ${\mathcal{A}}^{\prime }$ ) gives zero fluxes at infinite velocity (and, for ${\mathcal{A}}^{\prime }$ , across the boundary at $u_{p}=0$ ). However this also needs to be true for the calculated quantities, that is, if $a_{\Vert }^{\prime }$ is calculated, we require that $a_{\Vert }^{\prime }p_{E}$ tends to zero at both $u_{\Vert }=-\infty$ and $u_{\Vert }=\infty$ while, if ${\mathcal{A}}^{\prime }$ is calculated, we require that $u_{p}^{2}{\mathcal{A}}^{\prime }p_{E}$ tends to zero at both $u_{p}=0$ and $u_{p}=\infty$ . One of these limits can be satisfied by adjusting the constant of integration, but satisfying both limits imposes a restriction on the initial choice of ${\mathcal{A}}^{\prime }$ (or $a_{\Vert }^{\prime }$ ). ${\mathcal{A}}^{\prime }$ needs to satisfy

(2.22) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\int _{-\infty }^{\infty }u_{\Vert }r^{2}u_{p}p_{E}\,\text{d}u_{\Vert }+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}u_{p}}\int _{-\infty }^{\infty }u_{p}\left({\mathcal{A}}^{\prime }-\frac{u_{\Vert }}{r}\right)r^{2}u_{p}p_{E}\,\text{d}u_{\Vert }=0,\end{eqnarray}$$

while $a_{\Vert }^{\prime }$ needs to satisfy

(2.23) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\int _{0}^{\infty }u_{\Vert }r^{2}u_{p}p_{E}\,\text{d}u_{p}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}u_{\Vert }}\int _{0}^{\infty }\left(a_{\Vert }^{\prime }+\frac{u_{p}^{2}}{r}\right)r^{2}u_{p}p_{E}\,\text{d}u_{p}=0.\end{eqnarray}$$

We note that the second of these constraints amounts to saying that the model must satisfy the well-mixed condition for a Q1D model when averaged over $u_{p}$ .

2.2 The Eulerian p.d.f. $p_{E}$ and the model functions $a_{\Vert }^{\prime }$ and ${\mathcal{A}}^{\prime }$

There is considerable freedom in designing the model, both in the choice of $p_{E}$ and the choice of $a_{\Vert }^{\prime }$ and ${\mathcal{A}}^{\prime }$ . Although these choices could be based on experimental data, as investigated by Sawford & Yeung (Reference Sawford and Yeung2010), our main aim here is to investigate the usefulness of relatively simple idealised model formulations. We discuss a range of idealisations, including cases where $p_{E}$ is separable, either in terms of $u_{\Vert }$ and $u_{p}$ or in terms of $u_{\Vert }$ , $u_{\bot }$ and $u_{\vdash }$ ; cases with Gaussian velocity distributions, either just for $u_{\bot }$ and $u_{\vdash }$ or for $u_{\Vert }$ , $u_{\bot }$ and $u_{\vdash }$ ; cases where $a_{\bot }^{\prime }$ and $a_{\vdash }^{\prime }$ are quadratic functions of velocity; and cases where the model reduces to a Q1D model. We then describe a particular set of choices to be used for the model simulations presented below.

A natural simplification is to assume that $p_{E}$ is separable in $u_{\Vert }$ and $u_{p}$ and can be written as

(2.24) $$\begin{eqnarray}p_{E}(u_{\Vert },u_{\bot },u_{\vdash })=\frac{p_{u_{\Vert }}(u_{\Vert })p_{u_{p}}(u_{p})}{2\unicode[STIX]{x03C0}u_{p}}.\end{eqnarray}$$

A consequence of this assumption is that any dependence of the transverse and longitudinal velocity components on each other that may exist in reality cannot be taken into account. This cannot be an exact assumption because it implies that the fixed-point moment $\langle u_{\Vert }u_{p}^{2}\rangle$ is zero, while, in the inertial subrange, Kolmogorov’s four-fifths law $\langle u_{\Vert }^{3}\rangle =-(4/5)\unicode[STIX]{x1D700}r$ combined with incompressibility implies otherwise (Pagnini Reference Pagnini2008, p. 365). In addition, evidence from DNS indicates that the p.d.f. of $u_{p}$ conditional on $u_{\Vert }$ depends strongly on $u_{\Vert }$ (Sawford & Yeung Reference Sawford and Yeung2010). These results imply that $p_{E}(\boldsymbol{u})$ is not separable. However, we assume (2.24) in order to make the model tractable; it will be of interest to see how well the model performs despite this limitation. A consequence of this assumption is that because $\langle \boldsymbol{u}\boldsymbol{\cdot }\text{D}\boldsymbol{u}/\text{D}t\rangle =3(\langle u_{\Vert }^{3}\rangle +\langle u_{\Vert }u_{p}^{2}\rangle )/(2r)$ in the inertial subrange of turbulence (for both reality and LSMs of the type considered here), our model cannot satisfy both the four-fifths law and the result that in the inertial subrange $\langle \boldsymbol{u}\boldsymbol{\cdot }\text{D}\boldsymbol{u}/\text{D}t\rangle =-2\unicode[STIX]{x1D700}$ (Mann, Ott & Andersen Reference Mann, Ott and Andersen1999).

A further possible assumption is that the p.d.f. of $(u_{\bot },u_{\vdash })$ is separable with $u_{\bot }$ and $u_{\vdash }$ being independent:

(2.25) $$\begin{eqnarray}p_{u_{p}}(u_{p})=2\unicode[STIX]{x03C0}u_{p}p_{u_{\bot }}(u_{\bot })p_{u_{\vdash }}(u_{\vdash }).\end{eqnarray}$$

Because of isotropy and the circular symmetry of $u_{\bot }$ and $u_{\vdash }$ , a consequence of the assumption that $u_{\bot }$ and $u_{\vdash }$ are independent is that $p_{u_{\bot }}$ and $p_{u_{\vdash }}$ are Gaussian (e.g. Papoulis Reference Papoulis1991, p. 134). It is, of course, equivalent to assume directly that $p_{u_{\bot }}$ and $p_{u_{\vdash }}$ are Gaussian. We denote the common variance of $u_{\bot }$ and $u_{\vdash }$ by $\unicode[STIX]{x1D70E}_{\bot }^{2}$ .

Even once $p_{E}$ is specified, there is still a lot of freedom in choosing $a_{\Vert }^{\prime }$ and ${\mathcal{A}}^{\prime }$ . We consider choices where the transverse terms $a_{\bot }^{\prime }$ and $a_{\vdash }^{\prime }$ are quadratic in the velocity components. This implies ${\mathcal{A}}^{\prime }$ is equal to $u_{\Vert }$ times a function of $r$ i.e.

(2.26) $$\begin{eqnarray}(a_{\bot }^{\prime },a_{\vdash }^{\prime })=(u_{\bot },u_{\vdash }){\mathcal{A}}^{\prime }=(u_{\bot },u_{\vdash })u_{\Vert }\unicode[STIX]{x1D719}(r)\end{eqnarray}$$

(although, as for the Gaussian case, this still does not imply a unique model, with $\unicode[STIX]{x1D719}$ still to be determined). This choice is often made in models where $p_{E}$ is assumed to be Gaussian (Thomson Reference Thomson1987; Borgas & Sawford Reference Borgas and Sawford1994). It includes a broad class of Q1D models (see discussion below), and is, when combined with the $(u_{\Vert },u_{p})$ -separability assumption, a convenient way of ensuring that (2.22) is satisfied. When (2.26) is combined with the $(u_{\Vert },u_{p})$ -separability assumption, equation (2.21) implies

(2.27) $$\begin{eqnarray}a_{\Vert }^{\prime }=-\frac{u_{p}^{2}}{r}-\frac{1}{p_{u_{\Vert }}}\left(\frac{\unicode[STIX]{x2202}\ln \tilde{p}_{u_{p}}}{\unicode[STIX]{x2202}r}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}+2\unicode[STIX]{x1D719}(r)+u_{p}\left(\unicode[STIX]{x1D719}(r)-\frac{1}{r}\right)\frac{\unicode[STIX]{x2202}\ln \tilde{p}_{u_{p}}}{\unicode[STIX]{x2202}u_{p}}\right)\int _{-\infty }^{u_{\Vert }}u_{\Vert }^{\prime }p_{u_{\Vert }}(u_{\Vert }^{\prime })\,\text{d}u_{\Vert }^{\prime },\end{eqnarray}$$

where $\tilde{p}_{u_{p}}=p_{u_{p}}/2\unicode[STIX]{x03C0}u_{p}$ and we have used the condition that the flux $a_{\Vert }^{\prime }p_{u_{\Vert }}$ must tend to zero as $u_{\Vert }\rightarrow -\infty$ . Note that, if we had chosen $a_{\bot }^{\prime }$ or $a_{\vdash }^{\prime }$ to contain a term which is linear in the velocity components, then ${\mathcal{A}}^{\prime }$ would have a term which is a function of $r$ only, which would give rise to a term in $a_{\Vert }^{\prime }$ that increases rapidly as $u_{\Vert }\rightarrow \infty$ with the flux $a_{\Vert }^{\prime }p_{u_{\Vert }}$ not tending to zero in this limit. Also, because $(a_{\bot }^{\prime },a_{\vdash }^{\prime })=(u_{\bot },u_{\vdash }){\mathcal{A}}^{\prime }$ , it is impossible to include a constant term in $a_{\bot }^{\prime }$ or $a_{\vdash }^{\prime }$ . We remark that, with the choice (2.26) and with $a_{\Vert }^{\prime }p_{u_{\Vert }}\rightarrow 0$ as $u_{\Vert }\rightarrow -\infty$ , the flux also tends to zero as $u_{\Vert }\rightarrow \infty$ . If we had chosen to assume $a_{\Vert }^{\prime }p_{u_{\Vert }}\rightarrow 0$ as $u_{\Vert }\rightarrow \infty$ , then the integral on the right-hand side of (2.27) would be replaced by $-\!\int _{u_{\Vert }}^{\infty }$ . This is equivalent because $p_{u_{\Vert }}$ has mean zero.

In the case of Gaussian transverse velocity components, or equivalently with the assumption of $(u_{\Vert },u_{\bot },u_{\vdash })$ -separability, equation (2.27) becomes

(2.28) $$\begin{eqnarray}a_{\Vert }^{\prime }=-\frac{u_{p}^{2}}{r}-\frac{1}{p_{u_{\Vert }}}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}+\frac{u_{p}^{2}}{\unicode[STIX]{x1D70E}_{\bot }^{2}r}+\left(\unicode[STIX]{x1D719}(r)-\frac{1}{2\unicode[STIX]{x1D70E}_{\bot }^{2}}\frac{\text{d}\unicode[STIX]{x1D70E}_{\bot }^{2}}{\text{d}r}\right)\left(2-\frac{u_{p}^{2}}{\unicode[STIX]{x1D70E}_{\bot }^{2}}\right)\right)\int _{-\infty }^{u_{\Vert }}u_{\Vert }^{\prime }p_{u_{\Vert }}(u_{\Vert }^{\prime })\,\text{d}u_{\Vert }^{\prime }.\end{eqnarray}$$

If we further assume that $p_{u_{\Vert }}$ is Gaussian with variance $\unicode[STIX]{x1D70E}_{\Vert }^{2}$ , equation (2.28) becomes

(2.29) $$\begin{eqnarray}a_{\Vert }^{\prime }=-\frac{u_{p}^{2}}{r}\left(\frac{\unicode[STIX]{x1D70E}_{\bot }^{2}-\unicode[STIX]{x1D70E}_{\Vert }^{2}}{\unicode[STIX]{x1D70E}_{\bot }^{2}}\right)+\frac{\text{d}\unicode[STIX]{x1D70E}_{\Vert }}{\text{d}r}\frac{(\unicode[STIX]{x1D70E}_{\Vert }^{2}+u_{\Vert }^{2})}{\unicode[STIX]{x1D70E}_{\Vert }}+\left(\unicode[STIX]{x1D719}(r)-\frac{1}{2\unicode[STIX]{x1D70E}_{\bot }^{2}}\frac{\text{d}\unicode[STIX]{x1D70E}_{\bot }^{2}}{\text{d}r}\right)\unicode[STIX]{x1D70E}_{\Vert }^{2}\left(2-\frac{u_{p}^{2}}{\unicode[STIX]{x1D70E}_{\bot }^{2}}\right).\end{eqnarray}$$

In this case, the model of Thomson (Reference Thomson1990) corresponds to

(2.30) $$\begin{eqnarray}\unicode[STIX]{x1D719}(r)=-\frac{\unicode[STIX]{x1D70E}_{\bot }^{2}-\unicode[STIX]{x1D70E}_{\Vert }^{2}}{2\unicode[STIX]{x1D70E}_{\Vert }^{2}r}+\frac{1}{2\unicode[STIX]{x1D70E}_{\bot }^{2}}\frac{\text{d}\unicode[STIX]{x1D70E}_{\bot }^{2}}{\text{d}r}=-\frac{1}{2\unicode[STIX]{x1D70E}_{\Vert }}\frac{\text{d}\unicode[STIX]{x1D70E}_{\Vert }}{\text{d}r}+\frac{1}{2\unicode[STIX]{x1D70E}_{\bot }^{2}}\frac{\text{d}\unicode[STIX]{x1D70E}_{\bot }^{2}}{\text{d}r},\end{eqnarray}$$

where the last equality is a consequence of incompressibility. This form of $\unicode[STIX]{x1D719}(r)$ also ensures that $a_{\bot }^{\prime }$ and $a_{\vdash }^{\prime }$ agree with Thomson (Reference Thomson1990) regardless of whether the model is Gaussian or not. We note that the form of $\unicode[STIX]{x1D719}(r)$ that corresponds to the model of Borgas (Sawford & Guest Reference Sawford and Guest1988; Borgas & Sawford Reference Borgas and Sawford1994, equation (4.2a)) is given by

(2.31) $$\begin{eqnarray}\unicode[STIX]{x1D719}(r)=-\frac{\unicode[STIX]{x1D70E}_{\bot }^{2}-\unicode[STIX]{x1D70E}_{\Vert }^{2}}{\unicode[STIX]{x1D70E}_{\Vert }^{2}r}=-\frac{1}{\unicode[STIX]{x1D70E}_{\Vert }}\frac{\text{d}\unicode[STIX]{x1D70E}_{\Vert }}{\text{d}r}.\end{eqnarray}$$

For the $(u_{\Vert },u_{\bot },u_{\vdash })$ -separable case, and also for the $(u_{\Vert },u_{p})$ -separable case with the restriction that the shape of $p_{u_{p}}$ is independent of $r$ , we note that, with the quadratic assumption (2.26) and the choice

(2.32) $$\begin{eqnarray}\unicode[STIX]{x1D719}(r)=\frac{1}{2\unicode[STIX]{x1D70E}_{\bot }^{2}}\frac{\text{d}\unicode[STIX]{x1D70E}_{\bot }^{2}}{\text{d}r}+\frac{1}{r},\end{eqnarray}$$

we have

(2.33) $$\begin{eqnarray}a_{\Vert }^{\prime }=-\frac{u_{p}^{2}}{r}-\frac{1}{p_{u_{\Vert }}}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}+\frac{2}{r}\right)\int _{-\infty }^{u_{\Vert }}u_{\Vert }^{\prime }p_{u_{\Vert }}(u_{\Vert }^{\prime })\,\text{d}u_{\Vert }^{\prime }\end{eqnarray}$$

(from (2.28) or (2.27)). In the case where $p_{u_{\Vert }}$ is Gaussian, we then have

(2.34) $$\begin{eqnarray}a_{\Vert }^{\prime }=-\frac{u_{p}^{2}}{r}+\frac{\text{d}\unicode[STIX]{x1D70E}_{\Vert }}{\text{d}r}\frac{(\unicode[STIX]{x1D70E}_{\Vert }^{2}+u_{\Vert }^{2})}{\unicode[STIX]{x1D70E}_{\Vert }}+\frac{2\unicode[STIX]{x1D70E}_{\Vert }^{2}}{r}\end{eqnarray}$$

(from (2.29)). It follows that, for these cases, $a_{\Vert }$ does not depend on $u_{p}$ . In effect, the separation process has become one-dimensional. Indeed, in this case the model reduces to the Q1D model of Kurbanmuradov (Reference Kurbanmuradov1997, equations (3.1), (4.7) and (4.8)). It can be shown that, provided $p_{E}$ is separable in $u_{\Vert }$ and $u_{p}$ , then, for Q1D models, the quadratic assumption (2.26) holds if and only if the shape of $p_{u_{p}}(u_{p})$ is independent of $r$ , and in such cases (2.32) is satisfied. The Q1D model is known to overestimate the growth rate of an ensemble of particle pairs in the inertial subrange of homogeneous isotropic turbulence (Kurbanmuradov Reference Kurbanmuradov1997; Sawford & Yeung Reference Sawford and Yeung2010; Devenish & Thomson Reference Devenish and Thomson2011) and, along with the limitations of Q1D models discussed in § 1, suggests (2.32) is not an appropriate choice for formulating quantitatively realistic LSMs.

We note that, for models based on (2.25) and (2.26) in the classical self-similar inertial subrange, we have $\unicode[STIX]{x1D719}(r)=\hat{\unicode[STIX]{x1D719}}/r$ for some constant $\hat{\unicode[STIX]{x1D719}}$ . Together with $p_{u_{\Vert }}$ and $C_{0}$ , the value of $\hat{\unicode[STIX]{x1D719}}$ characterises the model and is equal to $-1/3$ , $1/6$ and $4/3$ for the Borgas, Thomson and Q1D models respectively.

2.3 Model formulation

We now describe the model configuration chosen for our numerical simulations. We assume $(u_{\Vert },u_{\bot },u_{\vdash })$ -separability of $p_{E}$ with the quadratic assumption (2.26) and we choose $\unicode[STIX]{x1D719}(r)$ to take the form given in (2.30) so that the non-Gaussian model reduces to that of Thomson (Reference Thomson1990) in the case that $p_{u_{\Vert }}$ is Gaussian. This avoids the Q1D assumption and allows us to include the important skewness in the longitudinal velocity distribution, while providing a modelling framework without too many remaining degrees of freedom ( $p_{u_{\Vert }}$ remains to be specified). The $(u_{\Vert },u_{p})$ -separability of $p_{E}$ is probably the weakest assumption here as this is known not to be exact.

The approach we follow in constructing an analytical form for $p_{u_{\Vert }}$ is that developed for modelling single-particle dispersion in the convective atmospheric boundary layer. For the convective atmospheric boundary layer the p.d.f. is positively skewed, i.e. there is a relatively low probability of strong updraughts versus a high probability of weak downdraughts, while for our application $p_{u_{\Vert }}$ is negatively skewed. Following Baerentsen & Berkowicz (Reference Baerentsen and Berkowicz1984), Luhar & Britter (Reference Luhar and Britter1989) and Hudson & Thomson (Reference Hudson and Thomson1994) we superimpose two Gaussian distributions

(2.35) $$\begin{eqnarray}p_{u_{\Vert }}=\mathop{\sum }_{i=1}^{2}\frac{A_{i}}{\sqrt{2\unicode[STIX]{x03C0}}\unicode[STIX]{x1D70E}_{u_{\Vert }i}}\exp \left(-\frac{(u_{\Vert }-\overline{u}_{\Vert i})^{2}}{2\unicode[STIX]{x1D70E}_{u_{\Vert }i}^{2}}\right),\end{eqnarray}$$

where $A_{i}$ , $\unicode[STIX]{x1D70E}_{u_{\Vert }i}$ and $\overline{u}_{\Vert i}$ are as yet unspecified functions of $r$ . In order to determine these unknowns and to construct a p.d.f. that gives the correct first three moments, $p_{u_{\Vert }}$ must satisfy

(2.36) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{-\infty }^{\infty }\,\text{d}u_{\Vert }\,p_{u_{\Vert }}(u_{\Vert })=1, & \displaystyle\end{eqnarray}$$

(2.37) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{-\infty }^{\infty }\,\text{d}u_{\Vert }u_{\Vert }\,p_{u_{\Vert }}(u_{\Vert })=0, & \displaystyle\end{eqnarray}$$

(2.38) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{-\infty }^{\infty }\,\text{d}u_{\Vert }u_{\Vert }^{2}\,p_{u_{\Vert }}(u_{\Vert })=\unicode[STIX]{x1D70E}_{\Vert }^{2} & \displaystyle\end{eqnarray}$$

and

(2.39) $$\begin{eqnarray}\int _{-\infty }^{\infty }\,\text{d}u_{\Vert }\,u_{\Vert }^{3}p_{u_{\Vert }}(u_{\Vert })=m_{3},\end{eqnarray}$$

where $m_{3}$ is the third moment of $u_{\Vert }$ . This method is simply a way of fixing the first three moments and is not expected to be accurate for higher-order moments.

With $p_{u_{\Vert }}(u_{\Vert })$ given by (2.35), (2.28) becomes

(2.40) $$\begin{eqnarray}\displaystyle a_{\Vert }^{\prime }p_{u_{\Vert }} & = & \displaystyle -\frac{u_{p}^{2}}{r}p_{u_{\Vert }}\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{i=1}^{2}\left\{\left[\frac{u_{p}^{2}}{\unicode[STIX]{x1D70E}_{\bot }^{2}r}+\frac{1}{A_{i}}\frac{\text{d}A_{i}}{\text{d}r}+\left(\frac{u_{\Vert }^{2}}{\unicode[STIX]{x1D70E}_{u_{\Vert }i}^{2}}+1\right)\frac{1}{\unicode[STIX]{x1D70E}_{u_{\Vert }i}}\frac{\text{d}\unicode[STIX]{x1D70E}_{u_{\Vert }i}}{\text{d}r}+\frac{u_{\Vert }}{\unicode[STIX]{x1D70E}_{u_{\Vert }i}}\frac{\text{d}\,\overline{u}_{\Vert i}/\unicode[STIX]{x1D70E}_{u_{\Vert }i}}{\text{d}r}\right.\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\left(\unicode[STIX]{x1D719}(r)-\frac{1}{2\unicode[STIX]{x1D70E}_{\bot }^{2}}\frac{\text{d}\unicode[STIX]{x1D70E}_{\bot }^{2}}{\text{d}r}\right)\left(2-\frac{u_{p}^{2}}{\unicode[STIX]{x1D70E}_{\bot }^{2}}\right)\right]\unicode[STIX]{x1D70E}_{u_{\Vert }i}A_{i}{\mathcal{G}}\left(\frac{u_{\Vert }-\overline{u}_{\Vert i}}{\unicode[STIX]{x1D70E}_{u_{\Vert }i}}\right)\nonumber\\ \displaystyle & & \displaystyle -\,\left[\frac{u_{p}^{2}}{r\unicode[STIX]{x1D70E}_{\bot }^{2}}+\frac{1}{A_{i}}\frac{\text{d}A_{i}}{\text{d}r}+\frac{1}{\overline{u}_{\Vert i}}\frac{\text{d}\overline{u}_{\Vert i}}{\text{d}r}+\left(\unicode[STIX]{x1D719}(r)-\frac{1}{2\unicode[STIX]{x1D70E}_{\bot }^{2}}\frac{\text{d}\unicode[STIX]{x1D70E}_{\bot }^{2}}{\text{d}r}\right)\left(2-\frac{u_{p}^{2}}{\unicode[STIX]{x1D70E}_{\bot }^{2}}\right)\right]\nonumber\\ \displaystyle & & \displaystyle \times \left.\overline{u}_{\Vert i}A_{i}\unicode[STIX]{x1D6F7}\left(\frac{u_{\Vert }-\overline{u}_{\Vert i}}{\unicode[STIX]{x1D70E}_{u_{\Vert }i}}\right)\!\vphantom{\left(\frac{u_{\Vert }^{2}}{\unicode[STIX]{x1D70E}_{u_{\Vert }i}^{2}}+1\right)}\right\},\end{eqnarray}$$

where ${\mathcal{G}}$ is the p.d.f. of the normal distribution with unit variance and $\unicode[STIX]{x1D6F7}(x)=(1+\text{erf}(x/\sqrt{2}))/2$ is the corresponding cumulative distribution function. Note that the term of the form $1+\text{erf}$ on the right-hand side of (2.40) would become $-1+\text{erf}$ if the integral on the right-hand side of (2.28) were replaced by $-\!\int _{u_{\Vert }}^{\infty }$ . This is equivalent because of (2.37). The term can also of course be written simply as $\text{erf}$ . The use of these different expressions in combination with asymptotic expansions of $1+\text{erf}$ and $-1+\text{erf}$ can be useful to get good numerical behaviour as $u_{\Vert }\rightarrow \pm \infty$ . As a check we note this reduces to the Gaussian result (2.29) when we make $p_{u_{\Vert }}$ Gaussian by putting $\overline{u}_{\Vert i}=0$ and $\unicode[STIX]{x1D70E}_{u_{\Vert }1}=\unicode[STIX]{x1D70E}_{u_{\Vert }2}$ .

As is appropriate for the inertial subrange of 3-D (incompressible) turbulence we take $\unicode[STIX]{x1D70E}_{\Vert }^{2}=C(\unicode[STIX]{x1D700}r)^{2/3}$ and $\unicode[STIX]{x1D70E}_{\bot }^{2}=(4/3)\unicode[STIX]{x1D70E}_{\Vert }^{2}$ (e.g. Pope Reference Pope2000, p. 193) with $C$ , the Kolmogorov constant, equal to 2 (e.g. Ishihara, Gotoh & Kaneda Reference Ishihara, Gotoh and Kaneda2009). The third-order moment follows Kolmogorov’s four-fifths law $m_{3}=-(4/5)\unicode[STIX]{x1D700}r$ (e.g. Pope Reference Pope2000, p. 204). It is also assumed, following Baerentsen & Berkowicz (Reference Baerentsen and Berkowicz1984) and Luhar & Britter (Reference Luhar and Britter1989), that $\overline{u}_{\Vert 1}=\unicode[STIX]{x1D70E}_{u_{\Vert }1}$ and $\overline{u}_{\Vert 2}=-\unicode[STIX]{x1D70E}_{u_{\Vert }2}$ . Analytical forms of $A_{i}$ and $\unicode[STIX]{x1D70E}_{u_{\Vert }i}$ can then be found by substituting (2.35) into (2.36)–(2.39) and solving for $A_{i}$ and $\unicode[STIX]{x1D70E}_{u_{\Vert }i}$ . This gives

(2.41a-d ) $$\begin{eqnarray}A_{1}=\frac{\unicode[STIX]{x1D70E}_{u_{\Vert }2}}{\unicode[STIX]{x1D70E}_{u_{\Vert }1}+\unicode[STIX]{x1D70E}_{u_{\Vert }2}},\quad A_{2}=\frac{\unicode[STIX]{x1D70E}_{u_{\Vert }1}}{\unicode[STIX]{x1D70E}_{u_{\Vert }1}+\unicode[STIX]{x1D70E}_{u_{\Vert }2}},\quad \unicode[STIX]{x1D70E}_{u_{\Vert }1}=\unicode[STIX]{x1D70E}_{u_{\Vert }2}+\frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FD}},\quad \unicode[STIX]{x1D70E}_{u_{\Vert }2}=\frac{1}{2}\left(\sqrt{\frac{\unicode[STIX]{x1D6FE}^{2}}{\unicode[STIX]{x1D6FD}^{2}}+4\unicode[STIX]{x1D6FD}}-\frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FD}}\right),\end{eqnarray}$$

where

(2.42a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}={\textstyle \frac{1}{2}}C(\unicode[STIX]{x1D700}r)^{2/3},\quad \unicode[STIX]{x1D6FE}=-{\textstyle \frac{1}{5}}\unicode[STIX]{x1D700}r.\end{eqnarray}$$

5.1 Analytical results for large $C_{0}$

Following the empirical studies of Richardson (Reference Richardson1926), Obukhov (Reference Obukhov1941) proposed that the relative dispersion of a pair of particles be governed by an eddy diffusivity, $K$ , of the form $K(r)=k_{0}\unicode[STIX]{x1D700}^{1/3}r^{4/3}$ . We expect this to be valid for large $C_{0}$ . When the initial separation is small, i.e. it can effectively be taken as zero, then it can be shown (e.g. Monin & Yaglom Reference Monin and Yaglom1975, p. 574) that the p.d.f. of the separation distance, $r$ , is given by

(5.1) $$\begin{eqnarray}p_{r}(r)=\frac{32}{315\sqrt{\unicode[STIX]{x03C0}}}\left(\frac{1287}{8}\right)^{3/2}\frac{r^{2}}{\unicode[STIX]{x1D70E}_{r}^{3}}\exp \left[\left(-\frac{1287r^{2}}{8\unicode[STIX]{x1D70E}_{r}^{2}}\right)^{1/3}\right],\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{r}^{2}\equiv \langle r^{2}\rangle =(1144/81)k_{0}^{3}\unicode[STIX]{x1D700}t^{3}$ . The values of the skewness and kurtosis are ${\mathcal{S}}k_{r}=1.70$ and ${\mathcal{K}}u_{r}=7.81$ respectively. The values of the normalised third and fourth-order moments are $\widetilde{\langle r^{3}\rangle }=1.70$ and $\widetilde{\langle r^{4}\rangle }=3.77$ respectively. Note that the values of ${\mathcal{S}}k_{r}$ and $\widetilde{\langle r^{3}\rangle }$ are not identical but differ by less than 0.003.

When the initial separation is finite, the p.d.f. takes the form

(5.2) $$\begin{eqnarray}p_{r}(r,t)=\frac{3r^{2}}{2\unicode[STIX]{x03C0}k_{0}\unicode[STIX]{x1D700}^{1/3}t(rr_{0})^{7/6}}\exp \left[-\frac{9(r^{2/3}+r_{0}^{2/3})}{4k_{0}\unicode[STIX]{x1D700}^{1/3}t}\right]I_{7/2}\left(\frac{9(rr_{0})^{1/3}}{2k_{0}\unicode[STIX]{x1D700}^{1/3}t}\right),\end{eqnarray}$$

where $\text{I}_{\unicode[STIX]{x1D708}}(x)$ is the modified Bessel function of order $\unicode[STIX]{x1D708}$ . This problem takes the same mathematical form as the 2-D steady diffusion problem for an elevated point source with a horizontal mean wind and vertical eddy diffusivity which are both power law functions of height (e.g. Thomson & Devenish Reference Thomson and Devenish2003; Monin & Yaglom Reference Monin and Yaglom1971, p. 661). Although this expression is derived using a scalar $K$ , it can be shown that the diffusion equation for a tensor $\unicode[STIX]{x1D612}_{ij}$ reduces to the diffusion equation with scalar $K$ in the case of spherically symmetric solutions, provided that the scalar $K$ is interpreted as the longitudinal diffusivity. The second moment of the p.d.f. given by (5.2) takes the form

(5.3) $$\begin{eqnarray}\langle r^{2}\rangle =\frac{\unicode[STIX]{x1D6E4}(15/2)}{\unicode[STIX]{x1D6E4}(9/2)}\left(\frac{4}{9}k_{0}\right)^{3}\unicode[STIX]{x1D700}t^{3}\exp \left[-\frac{9r_{0}^{2/3}}{4k_{0}\unicode[STIX]{x1D700}^{1/3}t}\right]_{1}F_{1}\left(\frac{15}{2};\frac{9}{2};\frac{9r_{0}^{2/3}}{4k_{0}\unicode[STIX]{x1D700}^{1/3}t}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D6E4}(x)$ is the gamma function and $_{1}F_{1}(a;b;x)$ is the confluent hypergeometric function (or Kummer) function. At small times, it can be shown that the second moment becomes

(5.4) $$\begin{eqnarray}\langle r^{2}\rangle -r_{0}^{2}={\textstyle \frac{26}{3}}k_{0}\unicode[STIX]{x1D700}^{1/3}r_{0}^{4/3}t={\textstyle \frac{26}{3}}K(r_{0})t.\end{eqnarray}$$

If $\unicode[STIX]{x2202}\unicode[STIX]{x1D612}_{ij}(\boldsymbol{r})/\unicode[STIX]{x2202}r_{j}=0$ (which is needed to ensure $\langle \boldsymbol{r}\rangle =\boldsymbol{r}_{0}$ ) then $(26/3)K$ can be interpreted as $2\unicode[STIX]{x1D612}_{ii}$ .

A quantitative expression for the diffusivity can be derived from the LSM in the limit of large $C_{0}$ . Following (Thomson Reference Thomson1987, p. 541), we have that

(5.5a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D612}_{ii}=-\int _{-\infty }^{\infty }u_{i}p_{u_{i}}g_{i}\,\text{d}u_{i},\quad \unicode[STIX]{x1D612}_{i\neq j}=0,\end{eqnarray}$$

where $g_{i}$ is a solution of

(5.6) $$\begin{eqnarray}C_{0}\unicode[STIX]{x1D700}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}u_{i}}p_{u_{i}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}u_{i}}g_{i}=u_{i}p_{u_{i}}.\end{eqnarray}$$

Here $p_{E}$ is assumed separable as in (2.24) and (2.25), there is no implied summation over $i$ , and $i\in \{\Vert ,\bot ,\vdash \}$ . Integration by parts of (5.5a ) leads to

(5.7) $$\begin{eqnarray}\unicode[STIX]{x1D612}_{ii}=\frac{1}{C_{0}\unicode[STIX]{x1D700}}\int _{-\infty }^{\infty }\frac{q_{i}^{2}}{p_{u_{i}}}\,\text{d}u_{i},\end{eqnarray}$$

where $q_{i}=\int _{-\infty }^{u_{i}}u_{i}^{\prime }p_{u_{i}}(u_{i}^{\prime })\,\text{d}u_{i}^{\prime }$ follows from (5.6). In the longitudinal direction with Gaussian $p_{u_{\Vert }}$ we get $K_{LL}=(C^{2}/C_{0})\unicode[STIX]{x1D700}^{1/3}r^{4/3}$ . For the non-Gaussian form of $p_{u_{\Vert }}$ given by (2.35) it can be shown that

(5.8)

As with (2.40) above, $\unicode[STIX]{x1D6F7}(x)$ can be written as $\text{erf}(x/\sqrt{2})/2$ as a result of cancellation. Numerical integration of the integral on the right-hand side gives $K_{LL}=(4.20/C_{0})\unicode[STIX]{x1D700}^{1/3}r^{4/3}$ with $C=2$ , a 5 % increase over the Gaussian case.

5.2 Analytical results for $C_{0}=0$

We now consider the opposite extreme, namely the ballistic limit $C_{0}=0$ . We emphasise that we do not expect this limit to be physically realistic and, indeed, we will identify some unphysical aspects below. However understanding this limit is useful for understanding how the behaviour of the model varies with $C_{0}$ . When $C_{0}=0$ and $p_{u_{\Vert }}(u_{\Vert })$ is Gaussian, the model given by (3.1) and (3.2) becomes

(5.9a,b ) $$\begin{eqnarray}\frac{\text{d}u_{\Vert }}{\text{d}t}=\frac{u_{\Vert }^{2}}{3r}+\frac{7u_{p}^{2}}{8r},\quad \frac{\text{d}u_{p}}{\text{d}t}=-\frac{5}{6r}u_{\Vert }u_{p}.\end{eqnarray}$$

Since $u_{\Vert }=\text{d}r/\text{d}t$ it follows from (5.9b ) that $u_{p}\propto r^{-5/6}$ . On substituting this into (5.9a ), and noting that $\text{d}u_{\Vert }/\text{d}t=u_{\Vert }\text{d}u_{\Vert }/\text{d}r$ , it is straightforward to show that

(5.10) $$\begin{eqnarray}u_{\Vert }^{2}=\left(u_{\Vert 0}^{2}+\frac{3}{4}u_{p0}^{2}\right)\left(\frac{r}{r_{0}}\right)^{2/3}-\frac{3}{4}u_{p0}^{2}\left(\frac{r}{r_{0}}\right)^{-5/3},\end{eqnarray}$$

where a subscript ‘0’ indicates the initial value. This is like motion in a potential: $r$ may decrease initially but will eventually increase with $u_{\Vert }$ returning to its initial value but with opposite sign when $r$ returns to $r_{0}$ . Once $r\gg r_{0}$ , the second term on the right-hand side can be neglected and it can be shown that

(5.11) $$\begin{eqnarray}r=\frac{1}{r_{0}^{1/2}}\left(\frac{2}{3}\left(u_{\Vert 0}^{2}+\frac{3}{4}u_{p0}^{2}\right)^{1/2}(t+c)\right)^{3/2},\end{eqnarray}$$

where $c$ is a constant of integration reflecting the particle behaviour before $r\gg r_{0}$ . At large times $c$ will be unimportant and we have

(5.12) $$\begin{eqnarray}r\approx \left(\frac{u_{\Vert 0}^{2}+{\textstyle \frac{3}{4}}u_{p0}^{2}}{r_{0}^{2/3}}\right)^{3/4}\left(\frac{2}{3}t\right)^{3/2},\end{eqnarray}$$

although some particles will take much longer than others to reach this regime. The moments and p.d.f. of $r$ at large times can be derived from (5.12) using the fact that $(u_{\Vert 0}^{2}+(3/4)u_{p0}^{2})/r_{0}^{2/3}$ is the sum of the squares of three independent Gaussian random variables with mean zero and variance $C\unicode[STIX]{x1D700}^{2/3}$ . We then get

(5.13) $$\begin{eqnarray}\langle r^{n}\rangle =\left(\frac{8}{9}C\unicode[STIX]{x1D700}^{2/3}t^{2}\right)^{3n/4}\frac{2}{\sqrt{\unicode[STIX]{x03C0}}}\unicode[STIX]{x1D6E4}(3/2+3n/4)\end{eqnarray}$$

and

(5.14) $$\begin{eqnarray}p_{r}(r)=\frac{9\unicode[STIX]{x03C0}r}{(2\unicode[STIX]{x03C0}C\unicode[STIX]{x1D700}^{2/3}t^{2})^{3/2}}\exp \left(-\frac{9r^{4/3}/4}{2C\unicode[STIX]{x1D700}^{2/3}t^{2}}\right),\end{eqnarray}$$

although we only expect $p_{r}$ to be valid for $r\gg r_{0}$ . We note that (5.14) implies that $p_{r}(r)/r^{2}\rightarrow \infty$ as $r\rightarrow 0$ which seems somewhat unsatisfactory. The values of ${\mathcal{S}}k_{r}$ and ${\mathcal{K}}u_{r}$ are respectively 1.05 and 4.44; the values of $\widetilde{\langle r^{3}\rangle }$ and $\widetilde{\langle r^{4}\rangle }$ are respectively 1.48 and 2.58.

It is possible to repeat parts of the above analysis for other models which differ only in the value of $\hat{\unicode[STIX]{x1D719}}$ in order to investigate their behaviour as $C_{0}\rightarrow 0$ . For the Borgas model (2.31) with Gaussian $p_{u_{\Vert }}$ , each trajectory remains bounded for $C_{0}=0$ , suggesting that $g\rightarrow 0$ as $C_{0}\rightarrow 0$ , consistent with Borgas & Sawford (Reference Borgas and Sawford1994, figure 9). In contrast, for the Q1D model (2.32) with Gaussian $p_{u_{\Vert }}$ , the trajectories have a logarithmic correction factor with (5.12) replaced by

(5.15) $$\begin{eqnarray}r\approx \left(\frac{u_{\Vert 0}^{2}}{r_{0}^{2/3}}+\frac{14}{3}C\unicode[STIX]{x1D700}^{2/3}\log \frac{r}{r_{0}}\right)^{3/4}\left(\frac{2}{3}t\right)^{3/2},\end{eqnarray}$$

suggesting $g\rightarrow \infty$ as $C_{0}\rightarrow 0$ , consistent with Kurbanmuradov (Reference Kurbanmuradov1997, figure 1). These differing behaviours as $C_{0}\rightarrow 0$ exemplify contrasting intuitive views of the role of $C_{0}$ in relative dispersion, namely the view that larger velocity correlation times tend to increase the diffusivity and hence the separation rate (a fairly standard diffusivity scaling argument), and the view that velocities have to change in order for particles to separate rapidly (as used for example in the argument of Novikov which is given by Monin & Yaglom (Reference Monin and Yaglom1975, p. 547)).

Although completing a similar analysis for our non-Gaussian model seems intractable, it is possible to obtain a result analogous to (5.10). This can be expressed as conservation of

(5.16) $$\begin{eqnarray}\tilde{p}_{u_{p}}^{\ast }(u_{p}^{\ast })\left|\int _{-\infty }^{u_{\Vert }^{\ast }}u_{\Vert }^{\ast ^{\prime }}p_{u_{\Vert }}^{\ast }(u_{\Vert }^{\ast ^{\prime }})\,\text{d}u_{\Vert }^{\ast ^{\prime }}\right|\end{eqnarray}$$

along trajectories, with $^{\ast }$ indicating scaled quantities (i.e. $u_{\Vert }^{\ast }=u_{\Vert }/\unicode[STIX]{x1D70E}_{\Vert }$ , $u_{p}^{\ast }=u_{p}/\unicode[STIX]{x1D70E}_{\bot }$ , $p_{u_{\Vert }}^{\ast }(u_{\Vert }^{\ast })=\unicode[STIX]{x1D70E}_{\Vert }p_{u_{\Vert }}(u_{\Vert })$ , $\tilde{p}_{u_{p}}^{\ast }(u_{p}^{\ast })=\unicode[STIX]{x1D70E}_{\bot }^{2}\tilde{p}_{u_{p}}(u_{p})=\unicode[STIX]{x1D70E}_{\bot }^{2}p_{u_{p}}(u_{p})/2\unicode[STIX]{x03C0}u_{p}=\exp (-u_{p}^{\ast 2}/2)/2\unicode[STIX]{x03C0}$ ) and $u_{p}^{\ast }=u_{p0}^{\ast }(r/r_{0})^{-7/6}$ . As $r$ increases, $u_{p}^{\ast }$ decreases and $u_{\Vert }^{\ast }$ needs to increase to maintain conservation. Equation (5.10) can be derived from this result for the special case of Gaussian $p_{u_{\Vert }}$ by taking the log of the conserved quantity. Again similar results are obtainable for the Borgas and Q1D models, with similar conclusions to those for the Gaussian case above.

It is interesting to note that the relative velocity of separating particles at separation $r$ tends to be larger than a typical velocity selected from the Eulerian distribution of $u_{\Vert }$ at that $r$ . This is clearest if we consider Q1D models: here we have deterministic paths in $(r,u_{\Vert })$ -space (because we are considering $C_{0}=0$ ) and a sketch of the possible trajectories, consistent with the well-mixed condition, shows that the pairs with small initial separation must end up in the extreme positive tail of the distribution and have $u_{\Vert }\gg \unicode[STIX]{x1D70E}_{\Vert }$ . This is less clear for our 3-D model where the separating velocities are not much larger than $\unicode[STIX]{x1D70E}_{\Vert }$ . However because of the effect of $u_{p0}$ in (5.10) and (5.16), the distribution of velocities will still be skewed towards the positive tail of $p_{u_{\Vert }}$ . The effect is reversed in the Borgas model, with $u_{\Vert }$ becoming zero and then negative, associated with the boundedness of the trajectories.

5.3 Variation of Richardson’s constant with $C_{0}$

Figure 6 shows the values of $g$ calculated from the non-Gaussian forwards and backwards models and from the Gaussian model for different values of $C_{0}$ . Also shown in figure 6 is the diffusion-equation result $g=1144k_{0}^{3}/81$ (Monin & Yaglom Reference Monin and Yaglom1975, p. 574) where $k_{0}=4.20/C_{0}$ as derived above. It can be seen that, for all three LSMs, $g$ decreases with increasing $C_{0}$ with $g$ reaching its diffusive value for sufficiently large $C_{0}$ (see also Kurbanmuradov Reference Kurbanmuradov1997). The value of $g$ from the non-Gaussian backwards model is consistently larger than that for the non-Gaussian forwards model though they both approach the same asymptote for large $C_{0}$ . For $C_{0}=0$ we believe this is explained by the tendency, identified above, for the particle velocities to be skewed towards the positive tail of $p_{u_{\Vert }}$ , with this tail being longer for the backwards model. This process remains active for non-zero $C_{0}$ unless $C_{0}$ is very large, when the short velocity memory time scale causes the forwards and backwards values to become equal in agreement with the analysis of § 5.1.

Figure 6.

The variation of $g$ for different values of $C_{0}$ for the non-Gaussian forwards model (red $+$ ), the non-Gaussian backwards model (blue $\ast$ ) and the Gaussian model (green $\times$ ). The solid black line is $g=1144k_{0}^{3}/81$ where $k_{0}$ takes the value $4.20/C_{0}$ appropriate to our non-Gaussian model. The dashed black line is $g=2C_{0}$ .

If one assumes that the two-particle acceleration covariance is negligible in the inertial subrange, then it can be shown that $g=2C_{0}$ (Monin & Yaglom Reference Monin and Yaglom1975, p. 547). This is unlikely to be the case for real turbulence. Nevertheless it serves as an upper bound on the value of $g$ (Thomson Reference Thomson1990, appendix B). The mean-square distance between a particle and the position it would have if it followed a straight line trajectory is $C_{0}\unicode[STIX]{x1D700}t^{3}$ while the mean-square distance between a particle and the centroid of the set of particles released close to the particle is $(g/2)\unicode[STIX]{x1D700}t^{3}$ . The former is equal to the latter plus the mean-square displacement of the centroid from the straight line trajectory and so $g\leqslant 2C_{0}$ with equality being implausible.

Figure 6 shows that, at $C_{0}=0$ , our model has a value of $g$ which is greater than zero. This exceeds the limit $2C_{0}$ and so implies that the model is behaving unphysically. For the Gaussian case, the value of $g$ from the simulations agrees well with the theoretical value $(8C/9)^{3/2}4/\unicode[STIX]{x03C0}^{1/2}$ ( ${\approx}5.35$ for $C=2$ ) derived from (5.13). We note that, as may be expected from the discussion in § 5.2, $g$ calculated from the Borgas model lies within the bound (see Borgas & Sawford Reference Borgas and Sawford1994, figure 9) but $g$ calculated from the Q1D model does not (see Kurbanmuradov Reference Kurbanmuradov1997, figure 1). However it is not clear that it is advantageous for a model to respect the condition $g\leqslant 2C_{0}$ for all $C_{0}$ when small values of $C_{0}$ are themselves unphysical.

5.4 Variation of moments of $r$ with $C_{0}$

Figure 7 shows the evolution of $\langle r^{2}\rangle$ calculated from the non-Gaussian forwards model for different values of $C_{0}$ : once the particles are no longer in the ballistic regime, the linear behaviour predicted by (5.4) can be observed for sufficiently large $C_{0}$ . Similar results can be obtained from the non-Gaussian backwards and Gaussian models.

Figure 7.

The evolution of $\langle r^{2}-r_{0}^{2}\rangle$ for different values of $C_{0}$ (top to bottom): $C_{0}=0$ (black dotted); $C_{0}=1$ (green dot-dashed); $C_{0}=6$ (red solid); $C_{0}=20$ (cyan long-dashed); $C_{0}=50$ (magenta short-dashed); $C_{0}=100$ (blue solid). The straight lines are proportional to $t$ , $t^{2}$ and $t^{3}$ as shown in the figure.

Figure 8.

The variation with $C_{0}$ of the skewness, kurtosis and normalised third- and fourth-order moments. In each panel the non-Gaussian forwards model is indicated by a red solid line ( $+$ ), the backwards model by a dashed blue line (*) and the Gaussian model by a dotted green line ( $\times$ ). Simulations with $C_{0}>100$ were performed with a larger time step i.e. $C_{1}=C_{2}=C_{3}=C_{4}=C_{5}=C_{6}=10^{-1}$ in (3.3). The horizontal lines are the appropriate values of the skewness, kurtosis and normalised third- and fourth-order moments derived from (5.1):  ${\mathcal{S}}k_{r}=1.70$ , $\widetilde{\langle r^{3}\rangle }=1.70$ , ${\mathcal{K}}u_{r}=7.81$ and $\widetilde{\langle r^{4}\rangle }=3.77$ . The equivalent values from a distribution with shape $r^{2}\exp (-r^{2})$ are ${\mathcal{S}}k_{r}=0.49$ , $\widetilde{\langle r^{3}\rangle }=1.23$ , ${\mathcal{K}}u_{r}=3.11$ and $\widetilde{\langle r^{4}\rangle }=1.67$ . The crosses on the vertical axes represent the appropriate value of the statistics for $C_{0}=0$ as derived from (5.13).

Figure 8 shows the variation with $C_{0}$ of ${\mathcal{S}}k_{r}$ , ${\mathcal{K}}u_{r}$ , $\widetilde{\langle r^{3}\rangle }$ and $\widetilde{\langle r^{4}\rangle }$ calculated from the non-Gaussian forwards and backwards models along with the Gaussian model. It can be seen that for $C_{0}=0$ the Gaussian model agrees well with the values derived from (5.13). It can also be seen that all three models tend towards the diffusive limit as $C_{0}$ increases: the Gaussian model tends to the appropriate values faster than the non-Gaussian forwards model while the non-Gaussian backwards model overshoots before it approaches the large- $C_{0}$ asymptote. The slower convergence of the non-Gaussian model in the limit of large $C_{0}$ was also noted by Sawford et al. (Reference Sawford, Yeung and Borgas2005) for Q1D models. For $C_{0}>100$ we employed a larger time step to reduce the computational cost. It is most likely that the slight discrepancy between the analytical and numerical values for the Gaussian model for $C_{0}>100$ is due to the larger time step; tests for $C_{0}=200$ with a smaller time step confirm this. For very large $C_{0}$ there is some indication that the non-Gaussian forwards model may oscillate as it approaches the asymptote (but computational costs prevent us from exploring this behaviour for still larger values of $C_{0}$ ). It is not clear why the variation with $C_{0}$ is non-monotonic for the non-Gaussian model. We note that ${\mathcal{S}}k_{r}$ and ${\mathcal{K}}u_{r}$ , for both the forwards and backwards models, and $\widetilde{\langle r^{3}\rangle }$ and $\widetilde{\langle r^{4}\rangle }$ , for the forwards model only, reach minima close to values of $C_{0}$ typically found in homogeneous isotropic turbulence. Moreover, these values are closer to the values expected for a distribution with shape $r^{2}\exp (-r^{2})$ , which describes the p.d.f. of $r$ once the particles have decorrelated, than either the equivalent values for the Gaussian model or the values derived from (5.1).

5.5 Variation of the p.d.f. of $r$ with $C_{0}$

The p.d.f. of the pair separation is shown in figure 9 for the non-Gaussian and Gaussian models. For realistic values of $C_{0}$ , the p.d.f. of all three models is less peaked than the diffusive p.d.f. given by (5.1), with the Gaussian model closer to (5.1) than the non-Gaussian models, which is consistent with figure 8. Only once $C_{0}$ becomes sufficiently large do the models agree well with (5.1): figure 9 clearly shows that as $C_{0}$ increases the Gaussian model tends more rapidly to the form given by (5.1) than the non-Gaussian forwards model which is consistent with the results in figure 8. It is also evident that the non-Gaussian backwards model approaches (5.1) more rapidly than the forwards model.

Figure 9.

The p.d.f., $p_{r/\unicode[STIX]{x1D70E}_{r}}(r/\unicode[STIX]{x1D70E}_{r})$ , for (a) the non-Gaussian forwards model, (b) the non-Gaussian backwards model and (c) the Gaussian model for different values of $C_{0}$ at $t=350r_{0}^{2/3}/\unicode[STIX]{x1D700}^{1/3}$ : $C_{0}=0$ (orange ▴); $C_{0}=1$ (red $+$ );  $C_{0}=6$ (green $\times$ );  $C_{0}=10$ (blue *); $C_{0}=50$ (magenta ▫); $C_{0}=100$ (cyan ▪). Equation (5.1) is indicated by a black line.

The form of $p_{r}$ in (5.14) is confirmed in figure 10, including the somewhat unsatisfactory blow up of $p_{r}(r)/r^{2}$ as $r\rightarrow 0$ (at least for $r\gg r_{0}$ ). Similar behaviour of $p_{r}$ is observed in the non-Gaussian model for $C_{0}=0$ . The singularity is however absent for $C_{0}>0$ .

Figure 10.

The p.d.f., $p_{r}(r)$ , for $C_{0}=0$ at $t=350r_{0}^{2/3}/\unicode[STIX]{x1D700}^{1/3}$ where $r^{\ast }=\unicode[STIX]{x1D700}^{1/2}t^{3/2}$ : the non-Gaussian forwards model (red $+$ ), the non-Gaussian backwards model (blue *) and the Gaussian model (green $\times$ ). The solid black line is (5.14).