© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The problem of nonreactive transport in continuous medium is relevant in technical applications such as risk analysis, field monitoring and pollution control. This study aims to determine how solute dispersion depends on initial conditions and what are its ergodic properties. We consider a diffusion process in a fluid with velocity $\mathbf{V}(\mathbf{x})$$\mathbf{V}(\mathbf{x})$V(x)\mathbf{V}(\mathbf{x})$\mathbf{V}(\mathbf{x})$ modeled as a random field statistically homogeneous with finite correlation range, continuously differentiable, and incompressible. We also assume the necessary measurability properties which ensure permutations of averages and integrals. For a given realization $\mathbf{V}(x)$$\mathbf{V}(x)$V(x)\mathbf{V}(x)$\mathbf{V}(x)$ and a constant diffusion coefficient $D$$D$DD$D$, the trajectories of the local advection-diffusion process starting at $t=0$$t=0$t=0t=0$t=0$ from the deterministic initial position ${\mathbf{X}}_0$${\mathbf{X}}_0$X_(0)\mathbf{X}_{0}${\mathbf{X}}_0$ are the solutions of the integral Itô equation

where $l=1,2,3$$l=1,2,3$l=1,2,3l=1,2,3$l=1,2,3$ and $W_l(t)$$W_l(t)$W_(l)(t)W_{l}(t)$W_l(t)$ is a Wiener process of mean zero and variance $2Dt$$2Dt$2Dt2 D t$2Dt$.

The second central moment $s_{ll}$$s_{ll}$s_(ll)s_{l l}$s_{ll}$ of the actual concentration is given by the dispersion $s_{ll}={⟨{[X_l-{⟨X_l⟩}_{D{X}_0}]}^2⟩}_{D{X}_0}$$s_{ll}={⟨{\left[X_l-{⟨X_l⟩}_{D{X}_0}\right]}^2⟩}_{D{X}_0}$s_(ll)=(:[X_(l)-(:X_(l):)_(DX_(0))]^(2):)_(DX_(0))s_{l l}=\left\langle\left[X_{l}-\left\langle X_{l}\right\rangle_{D X_{0}}\right]^{2}\right\rangle_{D X_{0}}$s_{ll}={⟨{[X_l-{⟨X_l⟩}_{D{X}_0}]}^2⟩}_{D{X}_0}$, where the subscripts $D$$D$DD$D$ and $X_0$$X_0$X_(0)X_{0}$X_0$ indicate averages over diffusion and initial distribution of the solute molecules. With ${\tilde{X}}_l=X_l-X_{0l}$${\tilde{X}}_l=X_l-X_{0l}$widetilde(X)_(l)=X_(l)-X_(0l)\widetilde{X}_{l}=X_{l}-X_{0 l}${\tilde{X}}_l=X_l-X_{0l}$,

where $S_{ll}(0)={⟨{[X_{0l}-{⟨X_{0l}⟩}_{X_0}]}^2⟩}_{X_0}$$S_{ll}(0)={⟨{\left[X_{0l}-{⟨X_{0l}⟩}_{X_0}\right]}^2⟩}_{X_0}$S_(ll)(0)=(:[X_(0l)-(:X_(0l):)_(X_(0))]^(2):)_(X_(0))S_{l l}(0)=\left\langle\left[X_{0 l}-\left\langle X_{0 l}\right\rangle_{X_{0}}\right]^{2}\right\rangle_{X_{0}}$S_{ll}(0)={⟨{[X_{0l}-{⟨X_{0l}⟩}_{X_0}]}^2⟩}_{X_0}$ is the second moment of the initial concentration and $m_{ll}=2{⟨[X_{0l}-{⟨X_{0l}⟩}_{X_0}]{\tilde{X}}_l⟩}_{D{X}_0}$$m_{ll}=2{⟨\left[X_{0l}-{⟨X_{0l}⟩}_{X_0}\right]{\tilde{X}}_l⟩}_{D{X}_0}$m_(ll)=2(:[X_(0l)-(:X_(0l):)_(X_(0))] widetilde(X)_(l):)_(DX_(0))m_{l l}=2\left\langle\left[X_{0 l}-\left\langle X_{0 l}\right\rangle_{X_{0}}\right] \widetilde{X}_{l}\right\rangle_{D X_{0}}$m_{ll}=2{⟨[X_{0l}-{⟨X_{0l}⟩}_{X_0}]{\tilde{X}}_l⟩}_{D{X}_0}$ is a "memory term" which describes possible persistent influences of initial conditions. To emphasize the role of one-particle statistics and ensemble averaged center of mass, we add and subtract ${⟨{⟨{\tilde{X}}_l⟩}_{DV}^2⟩}_{X_0}$${⟨{⟨{\tilde{X}}_l⟩}_{DV}^2⟩}_{X_0}$(:(: widetilde(X)_(l):)_(DV)^(2):)_(X_(0))\left\langle\left\langle\widetilde{X}_{l}\right\rangle_{D V}^{2}\right\rangle_{X_{0}}${⟨{⟨{\tilde{X}}_l⟩}_{DV}^2⟩}_{X_0}$ and ${⟨{\tilde{X}}_l⟩}_{D{X}_{0}V}^2$${⟨{\tilde{X}}_l⟩}_{D{X}_{0}V}^2$(: widetilde(X)_(l):)_(DX_(0)V)^(2)\left\langle\widetilde{X}_{l}\right\rangle_{D X_{0} V}^{2}${⟨{\tilde{X}}_l⟩}_{D{X}_{0}V}^2$ in (2) and perform the average over the ensemble of velocity realizations $S_{ll}={⟨s_{ll}⟩}_V$$S_{ll}={⟨s_{ll}⟩}_V$S_(ll)=(:s_(ll):)_(V)S_{l l}=\left\langle s_{l l}\right\rangle_{V}$S_{ll}={⟨s_{ll}⟩}_V$ to obtain

The terms of (3) are as follows: the average over initial positions of the one-particle dispersion $X_{ll}={⟨{[{\tilde{X}}_l-{⟨{\tilde{X}}_l⟩}_{DV}]}^2⟩}_{DV}$$X_{ll}={⟨{\left[{\tilde{X}}_l-{⟨{\tilde{X}}_l⟩}_{DV}\right]}^2⟩}_{DV}$X_(ll)=(:[ widetilde(X)_(l)-(: widetilde(X)_(l):)_(DV)]^(2):)_(DV)X_{l l}=\left\langle\left[\widetilde{X}_{l}-\left\langle\widetilde{X}_{l}\right\rangle_{D V}\right]^{2}\right\rangle_{D V}$X_{ll}={⟨{[{\tilde{X}}_l-{⟨{\tilde{X}}_l⟩}_{DV}]}^2⟩}_{DV}$, the variance of the center of mass $R_{ll}={⟨{[{⟨{\tilde{X}}_l⟩}_{D{X}_0}-{⟨{\tilde{X}}_l⟩}_{D{X}_{0}V}]}^2⟩}_V$$R_{ll}={⟨{\left[{⟨{\tilde{X}}_l⟩}_{D{X}_0}-{⟨{\tilde{X}}_l⟩}_{D{X}_{0}V}\right]}^2⟩}_V$R_(ll)=(:[(: widetilde(X)_(l):)_(DX_(0))-(: widetilde(X)_(l):)_(DX_(0)V)]^(2):)_(V)R_{l l}=\left\langle\left[\left\langle\widetilde{X}_{l}\right\rangle_{D X_{0}}-\left\langle\widetilde{X}_{l}\right\rangle_{D X_{0} V}\right]^{2}\right\rangle_{V}$R_{ll}={⟨{[{⟨{\tilde{X}}_l⟩}_{D{X}_0}-{⟨{\tilde{X}}_l⟩}_{D{X}_{0}V}]}^2⟩}_V$, the average over the ensemble of velocity realizations of the memory term $M_{ll}={⟨m_{ll}⟩}_V$$M_{ll}={⟨m_{ll}⟩}_V$M_(ll)=(:m_(ll):)_(V)M_{l l}=\left\langle m_{l l}\right\rangle_{V}$M_{ll}={⟨m_{ll}⟩}_V$, and the variance with respect to the initial position of the one-particle center of mass $Q_{ll}={⟨{[{⟨{\tilde{X}}_l⟩}_{DV}-{⟨{\tilde{X}}_l⟩}_{DV{X}_0}]}^2⟩}_{X_0}$$Q_{ll}={⟨{\left[{⟨{\tilde{X}}_l⟩}_{DV}-{⟨{\tilde{X}}_l⟩}_{DV{X}_0}\right]}^2⟩}_{X_0}$Q_(ll)=(:[(: widetilde(X)_(l):)_(DV)-(: widetilde(X)_(l):)_(DVX_(0))]^(2):)_(X_(0))Q_{l l}=\left\langle\left[\left\langle\widetilde{X}_{l}\right\rangle_{D V}-\left\langle\widetilde{X}_{l}\right\rangle_{D V X_{0}}\right]^{2}\right\rangle_{X_{0}}$Q_{ll}={⟨{[{⟨{\tilde{X}}_l⟩}_{DV}-{⟨{\tilde{X}}_l⟩}_{DV{X}_0}]}^2⟩}_{X_0}$ [4].

Using (1), the one-particle dispersion takes the explicit form

where $u_l(\mathbf{X}(t))=V_l(\mathbf{X}(t))-{⟨V_l(\mathbf{X}(t))⟩}_{DV}$$u_l(\mathbf{X}(t))=V_l(\mathbf{X}(t))-{⟨V_l(\mathbf{X}(t))⟩}_{DV}$u_(l)(X(t))=V_(l)(X(t))-(:V_(l)(X(t)):)_(DV)u_{l}(\mathbf{X}(t))=V_{l}(\mathbf{X}(t))-\left\langle V_{l}(\mathbf{X}(t))\right\rangle_{D V}$u_l(\mathbf{X}(t))=V_l(\mathbf{X}(t))-{⟨V_l(\mathbf{X}(t))⟩}_{DV}$ is the one-particle velocity fluctuation. For the assumed properties of the random velocity field, the velocity sampled on trajectories $\mathbf{V}(\mathbf{X}(t))$$\mathbf{V}(\mathbf{X}(t))$V(X(t))\mathbf{V}(\mathbf{X}(t))$\mathbf{V}(\mathbf{X}(t))$ has the same one-point probability distribution as $\mathbf{V}(\mathbf{x})$$\mathbf{V}(\mathbf{x})$V(x)\mathbf{V}(\mathbf{x})$\mathbf{V}(\mathbf{x})$ [6]. This implies ${⟨u_l\left(\mathbf{X}\left(t^{\prime }\right)\right)⟩}_V=0$${⟨u_l\left(\mathbf{X}\left(t^{\prime }\right)\right)⟩}_V=0$(:u_(l)(X(t^('))):)_(V)=0\left\langle u_{l}\left(\mathbf{X}\left(t^{\prime}\right)\right)\right\rangle_{V}=0${⟨u_l\left(\mathbf{X}\left(t^{\prime }\right)\right)⟩}_V=0$ and the independence of ${\mathbf{X}}_0$${\mathbf{X}}_0$X_(0)\mathbf{X}_{0}${\mathbf{X}}_0$ and the stationarity of the auto-correlation, i.e. ${⟨u_l\left(\mathbf{X}\left(t^{\prime }\right)\right)u_l\left(\mathbf{X}\left(t^{\prime \prime }\right)\right)⟩}_V=u_{ll}(t^{\prime }-t^{\prime \prime })$${⟨u_l\left(\mathbf{X}\left(t^{\prime }\right)\right)u_l\left(\mathbf{X}\left(t^{\prime \prime }\right)\right)⟩}_V=u_{ll}\left(t^{\prime }-t^{\prime \prime }\right)$(:u_(l)(X(t^(')))u_(l)(X(t^(''))):)_(V)=u_(ll)(t^(')-t^(''))\left\langle u_{l}\left(\mathbf{X}\left(t^{\prime}\right)\right) u_{l}\left(\mathbf{X}\left(t^{\prime \prime}\right)\right)\right\rangle_{V} =u_{l l}\left(t^{\prime}-t^{\prime \prime}\right)${⟨u_l\left(\mathbf{X}\left(t^{\prime }\right)\right)u_l\left(\mathbf{X}\left(t^{\prime \prime }\right)\right)⟩}_V=u_{ll}(t^{\prime }-t^{\prime \prime })$. It follows that the last term of (4) vanishes and the one-particle dispersion is a "memory-free" quantity, independent of initial conditions, $X_{ll}(t)=2Dt+{\int }_0^t{\int }_0^t{⟨u_{ll}(t^{\prime }-t^{\prime \prime })⟩}_{D}d{t}^{\prime }d{t}^{\prime \prime }$$X_{ll}(t)=2Dt+{\int }_0^t {\int }_0^t {⟨u_{ll}\left(t^{\prime }-t^{\prime \prime }\right)⟩}_{D}d{t}^{\prime }d{t}^{\prime \prime }$X_(ll)(t)=2Dt+int_(0)^(t)int_(0)^(t)(:u_(ll)(t^(')-t^('')):)_(D)dt^(')dt^('')X_{l l}(t)=2 D t+\int_{0}^{t} \int_{0}^{t}\left\langle u_{l l}\left(t^{\prime}-t^{\prime \prime}\right)\right\rangle_{D} d t^{\prime} d t^{\prime \prime}$X_{ll}(t)=2Dt+{\int }_0^t{\int }_0^t{⟨u_{ll}(t^{\prime }-t^{\prime \prime })⟩}_{D}d{t}^{\prime }d{t}^{\prime \prime }$. For the same reason, $M_{ll}$$M_{ll}$M_(ll)M_{l l}$M_{ll}$ and $Q_{ll}$$Q_{ll}$Q_(ll)Q_{l l}$Q_{ll}$ vanish and from (3) one obtains $S_{ll}=S_{ll}(0)+X_{ll}-R_{ll}$$S_{ll}=S_{ll}(0)+X_{ll}-R_{ll}$S_(ll)=S_(ll)(0)+X_(ll)-R_(ll)S_{l l}=S_{l l}(0)+X_{l l}-R_{l l}$S_{ll}=S_{ll}(0)+X_{ll}-R_{ll}$.

Numerical investigations [3] and perturbation approaches [1] indicate that for Gaussian fields with small fluctuations and point-like support of the initial concentration $X_{ll}$$X_{ll}$X_(ll)X_{l l}$X_{ll}$ has a diffusive behavior $\sim t$$\sim t$∼t\sim t$\sim t$ in the limit of large times and that the single realization dispersion $s_{ll}$$s_{ll}$s_(ll)s_{l l}$s_{ll}$ tends in the mean square limit to $X_{ll}$$X_{ll}$X_(ll)X_{l l}$X_{ll}$. Much less is known about the behavior of dispersion at finite times and for large sources. However, homogeneous fields with finite correlation range are ergodic and for large support of initial concentration distribution the space average with respect to $X_0$$X_0$X_(0)X_{0}$X_0$ of the velocity $V_l(\mathbf{X}(t))$$V_l(\mathbf{X}(t))$V_(l)(X(t))V_{l}(\mathbf{X}(t))$V_l(\mathbf{X}(t))$ tends to its ensemble average (with respect to $V$$V$VV$V$ ). Using (1) and ergodicity of the velocity field, one can argue that for increasing source dimensions $R_{ll}$$R_{ll}$R_(ll)R_{l l}$R_{ll}$ approaches to zero. Therefore, the average dispersion can be approximated by $S_{ll}\approx S_{ll}(0)+X_{ll}$$S_{ll}\approx S_{ll}(0)+X_{ll}$S_(ll)~~S_(ll)(0)+X_(ll)S_{l l} \approx S_{l l}(0)+X_{l l}$S_{ll}\approx S_{ll}(0)+X_{ll}$. A similar ergodicity argument suggests that in the case of Gaussian velocity fields, when the auto-correlation is also ergodic, the second term in (2) approximates the one-particle variance $X_{ll}$$X_{ll}$X_(ll)X_{l l}$X_{ll}$ and the actual dispersion can be estimated by $s_{ll}\approx S_{ll}(0)+X_{ll}+m_{ll}$$s_{ll}\approx S_{ll}(0)+X_{ll}+m_{ll}$s_(ll)~~S_(ll)(0)+X_(ll)+m_(ll)s_{l l} \approx S_{l l}(0)+X_{l l}+m_{l l}$s_{ll}\approx S_{ll}(0)+X_{ll}+m_{ll}$. Thus, the standard deviation of the single realization memory terms, $SD\left(m_{ll}\right)\approx SD\left(s_{ll}\right)$$SD\left(m_{ll}\right)\approx SD\left(s_{ll}\right)$SD(m_(ll))~~SD(s_(ll))S D\left(m_{l l}\right) \approx S D\left(s_{l l}\right)$SD\left(m_{ll}\right)\approx SD\left(s_{ll}\right)$, quantifies the "ergodicity in a large sense" [ 3,4 ] of the actual dispersion $s_{ll}-S_{ll}(0)$$s_{ll}-S_{ll}(0)$s_(ll)-S_(ll)(0)s_{l l}-S_{l l}(0)$s_{ll}-S_{ll}(0)$ with respect to the memory-free stochastic model $X_{ll}$$X_{ll}$X_(ll)X_{l l}$X_{ll}$. Of course, this is true only for large enough extensions of the source. For small sources, the "ergodicity range" ${⟨{(s_{ll}-S_{ll}(0)-X_{ll})}^2⟩}_V^{1/2}$${⟨{\left(s_{ll}-S_{ll}(0)-X_{ll}\right)}^2⟩}_V^{1/2}$(:(s_(ll)-S_(ll)(0)-X_(ll))^(2):)_(V)^(1//2)\left\langle\left(s_{l l}-S_{l l}(0)-X_{l l}\right)^{2}\right\rangle_{V}^{1 / 2}${⟨{(s_{ll}-S_{ll}(0)-X_{ll})}^2⟩}_V^{1/2}$ has to be computed without the above approximation for $s_{ll}[3,4]$$s_{ll}[3,4]$s_(ll)[3,4]s_{l l}[3,4]$s_{ll}[3,4]$.

For illustration we present some results based on two-dimensional "global random walk" simulations [2,3,5]. An incompressible Gaussian velocity field was approximated with satisfactory precision by 640 periodic modes $[1,2]$$[1,2]$[1,2][1,2]$[1,2]$. The Péclet number was fixed to $Pe=U\lambda /D=100$$Pe=U\lambda /D=100$Pe=U lambda//D=100P e=U \lambda / D=100$Pe=U\lambda /D=100$, where $U$$U$UU$U$ is the ensemble mean velocity and $\lambda$$\lambda$lambda\lambda$\lambda$ is a measure of the correlation range. Ensemble averages were performed over 5632 velocity realizations [2]. The initial condition consisted of uniform concentration distributions in transverse slabs ( $\lambda ,L\lambda$$\lambda ,L\lambda$lambda,L lambda\lambda, L \lambda$\lambda ,L\lambda$ ). The terms $Q_{ll}$$Q_{ll}$Q_(ll)Q_{l l}$Q_{ll}$ were found to be negligible small as compared with $2Dt$$2Dt$2Dt2 D t$2Dt$. The transverse mean memory terms $M_{22}$$M_{22}$M_(22)M_{22}$M_{22}$ (derived from (3) for $Q_{22}\approx 0$$Q_{22}\approx 0$Q_(22)~~0Q_{22} \approx 0$Q_{22}\approx 0$ and for one-particle dispersion $X_{22}$$X_{22}$X_(22)X_{22}$X_{22}$ estimated from point source simulations) remain relatively small up to $L=200$$L=200$L=200L=200$L=200$ (Fig. 1). But the corresponding standard deviations, $SD\left(m_{22}\right)\approx SD\left(s_{22}\right)$$SD\left(m_{22}\right)\approx SD\left(s_{22}\right)$SD(m_(22))~~SD(s_(22))S D\left(m_{22}\right) \approx S D\left(s_{22}\right)$SD\left(m_{22}\right)\approx SD\left(s_{22}\right)$, are two orders of magnitude larger and show an important increase with the source dimension (Fig. 2). The longitudinal memory terms $m_{11}$$m_{11}$m_(11)m_{11}$m_{11}$ are negligible and $s_{11}$$s_{11}$s_(11)s_{11}$s_{11}$ behave ergodically with respect to $X_{11}$$X_{11}$X_(11)X_{11}$X_{11}$ at early times, provided that $L\ge 50$$L\ge 50$L >= 50L \geq 50$L\ge 50$.

Acknowledgements This work was supported in part by Deutsche Forschungsgemeinschaft grant SU 415/1-2, Romanian Ministry of Education and Research grant 2-CEx06-11-96, NATO Collaborative Linkage Grant ESP.NR.CLG 981426, and RFBR Grant 06-01-00498.

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