[2] Dispersion in random velocity fields plays a central role in a variety of geophysical phenomena. We start from the hypothesis that local scale transport of a passive scalar *c*(**x**, *t*) can be described by the advection-dispersion equation (ADE)

where **u**(**x**, *t*) is the flow velocity and **D** the dispersion tensor, which here is assumed to be constant. Examples of passive scalars *c*(**x**, *t*) include solute concentration and fluid temperature. In many geophysical applications characterized by large Reynolds numbers *Re*, randomness of **u**(**x**, *t*) is a manifestation of turbulence. In other applications, of which flow and transport in porous media are a prime example, the velocity **u**(**x**, *t*) is often treated as random to reflect spatial heterogeneity and related uncertainty about the ambient environment (e.g., uncertainty about the porosity ϕ(**x**) and hydraulic conductivity *K*(**x**) of a heterogeneous porous medium) even though *Re* might be very small and flow is laminar. In the former class of problems, temporal variability of **u**(**x**, *t*) is essential, while in the latter class velocity is often considered at steady state.

[3] Regardless of application, randomness in, or uncertainty about, the flow velocity **u**(**x**, *t*) renders a solution of (1) stochastic, i.e., given in terms of the probability density function (PDF) of *c*(**x**, *t*). As a practical matter, it is common to focus on the one-point concentration PDF, *p*_{c}(*ψ*; **x**, *t*), which specifies the probability of the random concentration *c* at a point **x** and time *t* to have a value in [*ψ*, *ψ* + *dψ*].

[4] The scalar PDF carries information on the mixing state of a system [e.g., *Villermaux and Duplat*, 2006] and as such plays a central role in the quantification of non-linear mixing-driven reaction systems beyond the mean field [e.g., *Pope*, 2000]. Furthermore, the scalar PDF is a key quantity for the assessment of environmental hazards and risks [e.g., *Tartakovsky*, 2007] because it encodes the uncertainty about the mean behavior.

[5] While derivation of PDF equations for purely advective transport (**D** ≡ **0**) is mathematically rigorous and relatively straitforward [e.g., *Indelman and Shvidler*, 1985; *Chen et al.*, 1989], the quantification of the role of diffusion in the evolution of the scalar PDF remains a fundamental question. Phenomenologically, diffusion reduces uncertainty by smearing out scalar gradients. It sharpens the scalar PDF, or in other words it reduces scalar variance. These mechanisms have been quantified mainly using phenomenological approaches, so called mixing models. Such approaches include the interaction by exchange with the mean model (IEM) [*Villermaux and Devillon*, 1972; *Dopazo and O'Brien*, 1974], mapping approaches [*Chen et al.*, 1989] and stochastic mixing models [*Valiño and Dopazo*, 1991; *Fox and Yeung*, 2003; *Fedotov et al.*, 2005], see also *Pope* [2000] for an overview. In general, the relaxation or degradation mechanisms caused by diffusion have been found difficult to account for [*Chen et al.*, 1989]. An alternative approach is to rely on an assumed form of the PDF, most often by postulating that the concentration PDF is a *β*-distribution [*Girimaji*, 1991]. This approach has proved to be successful in modeling transport in random porous media in steady-state random velocity fields **u**(**x**) [*Caroni and Fiorotto*, 2005; *Bellin and Tonina*, 2007; *Cirpka et al.*, 2008].

[6] This dichotomy points to the need to develop an approach that is equally applicable to transient and steady-state velocity fields **u**(**x**, *t*), posing as few restrictions on its statistics as possible. PDF methods [e.g., *Tartakovsky et al.*, 2009; *Meyer et al.*, 2010] provide a general framework for achieving this goal. In the present analysis we compute the PDF of concentration *c*(**x**, *t*) in (1) by imposing the following two constraints on **u**(**x**, *t*). First, we assume the (steady or transient) velocity to be divergence-free, ∇ · **u**(**x**, *t*) = 0. Second, we assume that **u**(**x**, *t*) is a statistically homogeneous (stationary) multivariate Gaussian random field. The latter assumption is somewhat restrictive, since the spatio-temporally fluctuating random velocity field **u**(**x**, *t*) is determined by boundary conditions and, in the case of porous media flow, the medium heterogeneity. Yet, experimental evidence strongly suggests that **u**(**x**, *t*) is indeed (approximately) Gaussian for turbulent flows [e.g., *Bronski and McLaughlin*, 2000]. The same assumption is routinely made in analyses of transport in random porous media [e.g., *Koch and Shaqfeh*, 1992; *Zhang*, 1995; *Jaekel and Vereecken*, 1997; *Fiori and Dagan*, 2000; *Dentz and Tartakovsky*, 2008]. For non-stationary and non-Gaussian velocity fields, the presented approach can be used to obtain a prior distribution for the concentration PDF, which can be updated through a Bayesian approach or ensemble Kalman filtering to account for non-stationarity and non-Gaussianity.

[7] With these assumptions in mind the mean flow velocity 〈**u**〉 and variance *σ*_{u}^{2} are constant,

The two-point correlation function has the form

Here **u**′(**x**, *t*) denotes zero-mean random fluctuations of the velocity field **u**(**x**, *t*) about its ensemble mean , i.e., **u**(**x**, *t*) = + **u**′(**x**, *t*). The angular brackets 〈·〉 denote the ensemble average over the random field **u**(**x**, *t*). The components *C*_{ij} of the correlation matrix **C** decay on the correlation scales *l*_{ij} and *τ*_{ij}. The advection time *τ*_{u} = ℓ_{u}/∣∣ measures the time for advective transport over the correlation scale ℓ_{u} in the direction of the mean velocity. The diffusion scale *τ*_{D} = ℓ_{D}^{2}/*D* denotes the typical diffusion time over a transverse correlation distance ℓ_{D}. The dimensionless Péclet number is defined as *Pe* = *τ*_{D}/*τ*_{u}. It compares the efficiency of diffusive and advective transport.

[8] The single point PDF of the concentration *c*(**x**, *t*) is defined as

where *ψ* is the sampling variable. The concentration *c*(**x**, *t*) is a functional of the flow field **u**(**x**, *t*), whose concrete form is unknown because there is no closed form solution of (1) for general spatially fluctuating velocity fields.

[9] Here we derive a mapping approach to obtain the concentration PDF *p*_{c}(*ψ*; **x**, *t*) that explicitly takes into account the impact of diffusion on the evolution of the concentration PDF. This approach is based on mapping the disorder distribution as quantified by the joint PDF of the (random) centroid and (random) spatial width of *c*(**x**, *t*) onto the scalar PDF. To this end, we establish an expression for *c*(**x**, *t*) that depends explicitly on these (random) observables. To obtain such an expression, we consider the Langevin equation for the particle trajectories **x**(*t*) that is equivalent to the ADE (1),

where ** ξ**(

*t*) is a Gaussian white noise characterized by zero mean and correlation

where *D*_{ij} are the components of the dispersion tensor **D**. The angular brackets 〈·〉_{ξ} denote the noise average over ** ξ**(

*t*). The Langevin equation (5) contains two stochastic processes: the random velocity field

**u**(

**x**,

*t*) and noise

**(**

*ξ**t*), which models diffusion. The random processes

**u**(

**x**,

*t*) and

**(**

*ξ**t*) are statistically independent. The random concentration

*c*(

**x**,

*t*) in a single disorder realization can be written in terms of the particle trajectories by averaging over the noise

**(**

*ξ**t*), i.e.,

*c*(

**x**,

*t*) = 〈

*δ*[

**x**−

**x**(

*t*)]〉

_{ξ}. The average concentration is obtained by averaging over

**u**(

**x**,

*t*).

[10] Let us rewrite the Langevin equation (5) as

where **v**(*t*), the particle velocity averaged over the noise ** ξ**(

*t*), and

**(**

*η**t*), the velocity fluctuations about

**v**(

*t*), are defined as

Here, **m**(*t*) = 〈**x**(*t*)〉_{ξ} is the centroid of the scalar field *c*(**x**, *t*), and *δ***u**(*t*) = **u**[**x**(*t*), *t*] − 〈**u**[**x**(*t*), *t*]〉_{ξ} the Lagrangian velocity fluctuation.

[11] The random processes ** η**(

*t*) and

**u**(

**x**,

*t*) are not independent. The distribution of

**(**

*η**t*) is conditional to the specific realization of

**u**(

**x**,

*t*),

*p*

_{η∣u}(

**h**;

*t*) = 〈

*δ*[

**h**−

**(**

*η**t*)]〉

_{ξ}.

[12] These definitions imply that ** η**(

*t*) has zero conditional mean, 〈

**(**

*η**t*)〉

_{ξ}=

**0**, and its conditional correlation function

*C*

_{ij}

^{η}(

*t*,

*t*′) = 〈

*η*

_{i}(

*t*)

*η*

_{j}(

*t*′)〉

_{ξ}is given by

Note that **C**^{η}(*t*, *t*′) is a random matrix since its definition does not involve averaging over **u**(**x**, *t*).

[13] The approach of modeling the Lagrangian velocity increment in order to infer concentration statistics has been employed by, e.g., *Fiori and Dagan* [2000] and *Meyer et al.* [2010]. *Fiori and Dagan* [2000] derived the velocity statistics based on a perturbation approach and determined the mean concentration and concentration variance. Similar results were obtained by, e.g., *Graham and McLaughlin* [1989], *Kapoor and Gelhar* [1994], and *Kapoor and Kitanidis* [1998].

[14] Both **u**(**x**, *t*) and ** ξ**(

*t*) are Gaussian processes. We assume that the conditional PDF of

**(**

*η**t*) in (8) is Gaussian,

To confirm the validity of this assumption we conducted random walk simulations conditioned on a realization of a steady random velocity field **u**(**x**). (Numerical simulations correspond to isotropic three-dimensional divergence-free steady Gaussian random velocity fields **u**(**x**) with ℓ_{u} = ℓ_{D} = *l*; all theoretical derivations remain valid for anisotropic velocity fields.) The increment statistics of ** η**(

*t*) as well as the correlation matrix

**C**

^{η}(

*t*, 0) were determined numerically from 10

^{6}realizations of

**(**

*ξ**t*). In Figure 1 we compare the results of these simulations for

*η*

_{1}(

*t*) with the conditional PDF (10). One can see that (10) is indeed a good representation of the conditional PDF of

**(**

*η**t*). We therefore conclude that

**(**

*η**t*) can be modeled as a correlated Gaussian noise in a single realization of

**u**(

**x**,

*t*).

[15] Next, we express the random concentration in a single disorder realization as a conditional average over the correlated noise ** η**(

*t*)

The random particle trajectories **x**(*t*) are obtained by integrating (7) over time,

Substituting the resulting **x**(*t*) into (11) and taking the conditional average over ** η**(

*t*) using (10) gives

where the random matrix (*t*) is

Solution (13) implicitly assumes that the flow domain is infinite and the initial condition is a point source at **x** = **0**. Distributed sources can be accounted for by treating *c*(**x**, *t*) in (13) as a Green function and integrating it over the source distribution.

[16] Expression (13) maps [**m**(*t*), (*t*)] onto *c*(**x**, *t*), that is, *c*(**x**, *t*) = *f*[**x**, *t*; **m**(*t*), (*t*)]. Concentration PDF then is given by

where *p*_{m,κ}(** μ**,

**k**;

*t*) is the joint PDF of

**m**(

*t*) and (

*t*).

[17] In the following, we focus on the impact of fluctuations in the centroid **m**(*t*). Thus, we assume that the fluctuations of (*t*) about its ensemble mean value ^{e}(*t*) = 〈(*t*)〉 are small compared to the fluctuations of **m**(*t*), so that (*t*) can be approximated by its mean value ^{e}(*t*), which describes the effective width of *c*(**x**, *t*) [e.g., *Kitanidis*, 1988]. This assumption allows us to simplify (15) as

where *p*_{m}(** μ**;

*t*) = 〈

*δ*[

**−**

*μ***m**(

*t*)]〉 is the centroid PDF.

[18] The random dynamics of centroid **m**(*t*) are described by averaging (5) over ** ξ**(

*t*),

*d*

**m**(

*t*)/

*dt*= 〈

**u**[

**x**(

*t*),

*t*]〉

_{ξ}. Since

**u**(

**x**,

*t*) is Gaussian, we assume that

**m**(

*t*) is Gaussian as well. It is then completely defined by its mean 〈

**m**(

*t*)〉 =

*t*and variance

The centroid PDF then takes the form

To test the assumption that the centroid PDF is Gaussian, we conducted Monte Carlo simulations (MCS) of dispersion in a Gaussian random velocity field **u**(**x**). The results of 10^{4} realizations of **u**(**x**) were used to compute the PDF of **m**(*t*), its mean 〈**m**(*t*)〉 and variance ^{c}(*t*). Figure 2 compares the centroid PDF obtained via MCS with the Gaussian model (18). This comparison indicates that **m**(*t*) may be modeled as a Gaussian stochastic process for the times and disorder strengths under consideration.

[19] To illustrate the presented mapping approach, we consider the PDF of concentration integrated over the directions transverse to the mean flow and normalized by the maximum concentration,

which varies between 0 and 1. Its PDF is given by (; *x*_{1}, *t*) = 〈*δ*[ − (*x*_{1}, *t*)]〉. Using (16) to map the centroid PDF (18) onto the concentration PDF we obtain

with *α*_{11}(*t*) = κ_{11}^{e}(*t*)/κ_{11}^{c}(*t*). Thus, (; *x*_{1}, *t*) is completely parametrized by the centroid variance κ_{11}^{c}(*t*) and the effective spatial width κ_{11}^{e}(*t*) of *c*(**x**, *t*).

[20] We evaluate this expression for transport in a steady three-dimensional divergence-free random velocity field **u**(**x**). For *σ*_{u}^{2} < 1, the effective width κ_{11}^{e}(*t*) and the centroid variance κ_{11}^{c}(*t*) can be obtained through a first-order perturbation expansion in *σ*_{u}^{2} as [e.g., *Dentz et al.*, 2000]

Alternatively, one can obtain these observables from either measurements or MCS.

[21] Figure 3 illustrates the impact of dispersion on the concentration PDF. As *Pe* increases, i.e., as the dispersion coefficient decreases, (; *x*_{1}, *t*) becomes broader. In other words, increasing dispersion decreases concentration uncertainty. In the limit as *Pe* → ∞ (*α*_{11} → 0) the PDF in (20) reduces to a sum of two delta-functions peaked at = 0 and 1. Typically, the impact of dispersion on the concentration PDF is modeled in terms of closures such as the IEM model. Here we derived a mapping approach to obtain the concentration PDF taking explicitly into account mixing due to dispersion and spatial disorder.

[22] In summary we derived an explicit map between the scalar PDF and the distribution of the random velocity field as quantified in terms of the joint PDF of the (random) centroid and (random) spatial variance of the passive scalar. The explicit analytical expressions for the scalar PDF shed new light on the role of dispersion in the evolution of the concentration PDF.