Transverse mixing of solutes in steady state transport is of utmost importance for assessing mixing-controlled reactions of compounds that are continuously introduced into the subsurface. Classical spatial moments analysis fails to describe mixing because the tortuous streamlines in heterogeneous formations cause plume meandering, squeezing, and stretching, which affect transverse spatial moments even if there is no mass transfer perpendicular to the direction of flow. For transverse solute mixing, however, the decisive process is the exchange of solute mass between adjacent stream tubes. We therefore reformulate the advection-dispersion equation in streamline coordinates (i.e., in terms of the potential and the stream function values) and analyze how flux-related second central moments of plumes increase with dropping hydraulic potential. We compare the ensemble behavior of these second central moments in random two-dimensional heterogeneous flow fields with the moments in an equivalent homogeneous system, thus defining an equivalent effective transverse dispersion coefficient. Unlike transverse macrodispersion coefficients derived by traditional moment analysis, our mixing-relevant, flux-related coefficient does not increase with travel distance. We present closed-form solutions for the mean enhancement of transverse mixing by heterogeneity in two-dimensional isotropic media for linear laws of local-scale transverse dispersion. The mixing enhancement factor increases with the log conductivity variance but remains fairly low. We also evaluate the variance of our cumulative measure of transverse mixing, showing that heterogeneity causes substantial uncertainty of mixing. The analytical expressions are compared to numerical Monte Carlo simulations for various values of log conductivity variance, indicating good agreement with the analytical results at low variability. In the numerical simulations, we also consider nonlinear models of local-scale transverse dispersion.
 In the past 3 decades, the influence of spatial variability in hydraulic properties on solute transport has intensively been analyzed by stochastic theory [e.g., Dagan, 1989; Gelhar, 1993; Rubin, 2003]. The main focus was on longitudinal dispersion phenomena, describing longitudinal spreading of solute plumes and mixing in the flow direction. To a lesser extent, dispersion theories were also applied to transverse solute transfer. Traditional ensemble dispersion quantifies how quickly the second central spatial moments of the ensemble-averaged concentration increase [e.g., Gelhar and Axness, 1983; Dagan, 1984; Neuman et al., 1987; Dagan, 1988]. The classical transverse ensemble dispersion coefficient starts at the value of the local transverse dispersion coefficient, increases with travel time or distance, exhibits a peak when a few integral scales have been sampled by the plume, decreases again, and approaches an asymptotic value, which is slightly higher than the local transverse dispersion coefficient [Dagan, 1994; Attinger et al., 2004]. The intermediate maximum, however, expresses the uncertainty of determining the transverse first moment of individual plumes and thus does not quantify mass exchange between neighboring stream tubes. In effective dispersion, also denoted relative dispersion [Kitanidis, 1988; Dagan, 1990; Andricevic and Cvetkovic, 1998; Attinger et al., 1999; Dentz et al., 2000a, 2000b; Cirpka and Attinger, 2003; Dentz and Carrera, 2003], the order of taking second central moments and ensemble averages is exchanged, such that the uncertainty in estimating the first moment has been factored out. For longitudinal mixing, Cirpka  claimed that effective dispersion for point-like injection is a good predictor of solute mixing. In the transverse direction, however, values of effective dispersion coefficients are very small and most likely prone to higher-order effects [Dagan, 1994; Attinger et al., 2004]. Also, individual plumes can get wider and narrower by spatial variability of advection alone, so that traditional spatial moments analysis can be misleading with respect to actual mixing across streamlines [Rahman et al., 2005; Rolle et al., 2009].
 Contrasts in hydraulic conductivity lead to convergence and divergence of streamlines, which locally focuses and defocuses the flow field [Cirpka and Kitanidis, 2000; Werth et al., 2006; Chiogna et al., 2011]. This phenomenon changes locally the characteristic length and time scales for transverse mass transfer by diffusion and hydromechanical (local-scale) dispersion whenever the solute plume travels through focused and defocused regions of the flow field. Furthermore, most of the mass transfer takes place along the fringe of the plume, where high-concentration gradients can be found [see Rubin, 1991; Cirpka et al., 2008; Bauer et al., 2009; Prommer et al., 2009]. Flow focusing (or flow channeling) enhances mass transfer by diffusion and by hydromechanical dispersion to a large degree. When flow is unfocused, mass transfer by these two processes is slowed down since streamlines are farther apart. The general idea by Werth et al.  is that on the average, these two effects do not cancel out but lead to an overall enhancement of transverse mass transfer by hydromechanical dispersion and diffusion. Depending on the degree of contrast between different permeability zones, the mass transfer induced by flow focusing can yield a substantial increase in transverse dilution and mixing.
 Classical linear stochastic theory does not account for effects of flow focusing and defocusing on transverse mixing because (1) the stochastic-analytical equations have been developed for uniform local dispersion tensors at best and (2) integrations of velocity fluctuations are performed along mean trajectories. This implies that squeezing and stretching of solute plumes and their effect on transverse dispersive mass exchange are neglected in these theories. Higher-order extensions to dispersion theory are restricted to the behavior of the ensemble-averaged concentration in normal coordinates [e.g., Hsu et al., 1996; Neuman, 1993] and thus appear insufficient to express the effects of flow focusing on mean mixing and its uncertainty.
 The key objective of the present contribution is to derive a stochastic framework of transverse mixing. The effects of plume meandering, squeezing, and stretching in two-dimensional heterogeneous domains will be accounted for by switching from Cartesian spatial coordinates to streamline coordinates before going into stochastic analysis. As a key target, we analyze the probability of a solute particle to jump into a neighboring stream tube. This is expressed in terms of the second central moment in the stream function value. Because mixing is taking place at the fringes of solute plumes, we analyze the squared width of the plume fringe in streamline coordinates, which is identical to the second central moment for a point-like injection. We propose to formulate dispersion in terms of streamline coordinates in individual realizations. We take expected values only in a succeeding step, thus providing an effective mixing-related dispersion coefficient for transport in streamline coordinates.
Figure 1 schematically illustrates the concept as a series of steps: (1) A theoretically infinitely large ensemble of log conductivity fields with identical statistical properties is generated. (2) Flow and (3) steady state transport are simulated on each of these fields. (4) The concentration fields are spatially transformed into streamline coordinates, and (5) second central transverse moments are evaluated (in streamline coordinates) in each realization. Figure 1 shows the injection of the solute over half the volume flux at the inlet; the moments, however, are computed for an equivalent point-like injection at the transition point (for an explanation, see section 5.1). (6) After the steps illustrated in Figure 1, ensemble averaging is performed. (7) Finally, we define an equivalent transverse dispersion coefficient by matching the second central transverse moments in streamline coordinates obtained in heterogeneous domains with those obtained in a homogeneous domain with identical effective hydraulic conductivity.
 This paper is organized as follows: Section 2 reviews the governing equations in different systems of coordinates and derives the equivalent transverse dispersion coefficient and the uncertainty of cumulative mixing in principal terms. In sections 3 and 4, stochastic properties of the log conductivity field are propagated to flow and transport in order to obtain closed-form analytical expressions of mixing and its corresponding uncertainty. In section 5, the steps visualized in Figure 1 are explicitly followed with finite ensembles of conductivity fields in numerical simulations, and the results are compared to the analytical expressions derived in sections 3 and 4. Finally, we present our conclusions in section 6. Details of the analytical derivations are included in Appendix A.
2.1. Governing Equations in Different Systems of Coordinates
 We consider steady state two-dimensional groundwater flow in an infinite domain with uniform aquifer thickness and spatially variable, random isotropic hydraulic conductivity K [L/T]:
subject to the mean hydraulic gradient J:
in which ϕ [L] is the hydraulic head, J (dimensionless) is the absolute value of the mean hydraulic gradient oriented into the longitudinal direction, and angle brackets denote the ensemble expectation. At each point within the domain, the specific discharge vector q [L/T] is defined by Darcy's law:
 Solute transport is described by the advection-dispersion equation, which reads, in standard Cartesian coordinates x = (x1, x2), as
subject to initial and boundary conditions. In equation (4), c is concentration [M/L3], θ is the porosity (considered uniform) (dimensionless), and D is the local dispersion tensor [L2/T] discussed in more detail below.
 We may locally rotate the system of coordinates into the local longitudinal coordinate xℓ oriented in the direction of q and the transverse coordinate xt perpendicular to it. Then, equation (4) becomes
in which q is the absolute value of q [L/T], Dℓ is the longitudinal dispersion coefficient, and Dt the transverse one [L2/T].
 In the following, we will focus on quantifying transverse dispersion and its impact upon mixing and dilution under different parameterizations of Dt. Table 1 lists different models for Dt. In model I, pore diffusion is considered the only transverse mixing mechanism. The pore diffusion coefficient Dp [L2/T] is the molecular diffusion coefficient Dm [L2/T] reduced by the tortuosity of the medium (here assumed to be θ−1). Model II is the standard linear model of Scheidegger  with a constant transverse dispersivity αt [L]. In model III, we take into account that the transverse dispersivity αt is typically assumed to scale linearly with grain diameter [Bear, 1979, chapter 4]. As a link to hydraulic conductivity, we assume that K approximately scales with a characteristic grain diameter squared [Hazen, 1892]. This results in a transverse dispersion coefficient Dt with a hydrodynamic contribution that scales with . Model IV, introduced by Chiogna et al. , accounts for incomplete mixing in pore junctions [Klenk and Grathwohl, 2002; Olsson and Grathwohl, 2007], resulting in a nonlinear dependence of Dt on velocity and in compound-specific behavior over a wide range of velocities.
Models I, II, and III are commonly used in the literature [e.g., Scheidegger, 1961; Bear, 1979], while model IV was recently proposed by Chiogna et al. . Dm, molecular diffusion coefficient; Dp, pore diffusion coefficient; αt, transverse dispersivity; d, grain diameter; Pe, grain Péclet number; q, longitudinal specific discharge; θ, porosity.
 Since the orientation of q is spatially variable in a heterogeneous domain, it is convenient to perform a conformal mapping into streamline coordinates, namely, the hydraulic head ϕ and the stream function value ψ [L2/T] [see Batchelor, 2000; Bear, 1972]:
Finally, the advection-dispersion equation in equation (5) can be expressed in streamline coordinates as
 The advantage of the streamline coordinate system is that it is naturally defined by the flow field [e.g., Aris, 1989]. The orientation of xℓ and xt, used in equation (5), varies within the domain, which makes it difficult to detect which point in space is upstream of another point in the domain. In contrast, the streamline coordinates ψ and ϕ are naturally oriented in the flow direction: by construction, two points with identical ψ values are on the same streamline. Similar to the Cartesian coordinate system, streamline coordinates employ an orthogonal basis. However, they account for the tortuous patterns of flow because of the spatial heterogeneity of K. That is, meandering, squeezing, and stretching of plumes due to spatial variability of the flow field are automatically corrected for. For example, if we assess the transverse growth rate of the plume width in streamline coordinates, we automatically exclude the artifacts of (1) the uncertain centroid position, (2) irregular plume outline due to spreading, and (3) possible oscillations of plume growth due to focusing and defocusing from all later steps. Instead, we can immediately measure the additional flow lines captured by transverse mass exchange, which is the quantity that truly matters.
 In the following, we consider steady state transport (continuous release conditions) for a point-like injection at the potential line ϕ0. We formulate transport for a point-like injection because it is equivalent to studying mass transfer at the plume fringe. We also neglect longitudinal dispersion. This is a justifiable assumption for continuously released contaminant sources because the longitudinal gradient of the resulting concentration plumes is practically negligible when compared to the transverse one. Thus, we arrive at
in which A [MT/L] is a scaling factor to obtain the unit of a concentration and δ denotes the Dirac delta function. The second term in equation (11) may be expressed as
in which the first expression may be addressed as a drift term. For a point injection, the plume remains narrow over large distances [Dagan, 1991; Dentz et al., 2000a]. Also, in heterogeneous media with sufficiently small heterogeneity, neither K nor q nor Dt show strong transverse gradients. This implies that the drift term can be neglected and that θKDt can be considered approximately constant over the width of plume, where ∂2c/∂ψ2 is large. Under such conditions, equation (11) becomes
with negligible variation of θKDt in the ψ direction over the width of the plume.
 In summary, the main set of assumptions used to arrive at equation (14) are (1) steady state flow is uniform on the average, (2) steady state transport has point-like injection, (3) longitudinal dispersion is neglected, (4) heterogeneity is small, and (5) the plume (or its fringe) is considered narrow. These five assumptions are necessary in order for the later theoretical results to hold.
2.2. Flux-Related Moments
 In order to quantify the distribution of the solute flux over the volume flux, we compute the lateral concentration moments in terms of the stream function ψ:
in which mkψ(ϕ) [M L2k−3/Tk] is the kth flux-related moment and mkcψ(ϕ) [M L2k−3/Tk] is the corresponding central moment. The normalized first moment is given by μψ(ϕ) [L2/T], while wψ2(ϕ) [L4/T2] is the normalized second central moment, or squared characteristic plume width expressed in units of squared cumulative flux.
 The insight to separate uncertainty in the plume centroid from the average spread of the plume has significantly improved our understanding of dispersion processes in heterogeneous porous media (see Rubin [2003, chapter 10] for an extensive review). Our approach is that in order to quantify actual mixing in the transverse direction, yet another distinction is necessary: We have to follow the direction of flow and take transverse moments in terms of cumulative flux in each individual realization, kept specific to the longitudinal flow coordinate, before averaging over the ensemble. In this way, we can separate effects of tortuous flow paths that may influence spatial moments from the transfer of solute mass across streamlines. This is conveniently achieved by using the potential ϕ as the longitudinal coordinate and the stream function value ψ as the transverse one.
 With the above definitions given in equations (16)–(19) and equation (14) and assuming a narrow plume (or narrow plume fringe) in a mildly heterogeneous domain, the squared plume width wψ2(ϕ) follows:
in which ϕ0 [L] is the potential at the release point of the solute and Δϕ [L] is the potential drop from the inlet to an observation line.
2.3. Stochastic Formulation
 So far, we have considered the behavior of a point-related plume in an individual hydraulic conductivity field. We now want to cast this into a stochastic framework. We consider a locally isotropic log conductivity field Y with uniform mean hydraulic gradient J (uniform average flow) and effective hydraulic conductivity Keff [L/T] [e.g., Rubin 2003]:
 If we want to replace the ensemble of heterogeneous conductivity, flow, and concentration fields with a single equivalent homogeneous field each, the corresponding hydraulic conductivity value would be Keff. Now, we require that the ensemble average 〈wψ2〉 of the effective plume width squared behaves like in the individual realizations according to equation (20). However, we replace the random variable K with the effective value Keff in equation (20) and the potentially random variable Dt with an equivalent homogeneous transverse dispersion coefficient Dteq:
Rearranging equation (24), we obtain a general expression of the equivalent transverse dispersion coefficient Dteq, which is valid under the physical assumptions listed in section 2.1 and the statistical assumptions given in the beginning of this section:
This result is independent of the dispersion model chosen (see Table 1). Another observation is that equation (26) refers only to one-point statistics. It can be evaluated readily from analytical or numerical flow statistics. Furthermore, it does not depend on any assumption related to the multi-Gaussianity of the K field.
Equation (26) is the first key result of our analysis. By performing the transformation of coordinates from standard Cartesian coordinates x to streamline coordinates (ϕ, ψ) prior to ensemble averaging, we have eliminated the scale dependence of dispersion. This scale dependence was one of the most prominent outcomes and nuisances of traditional moment analysis [Dagan, 1989; Gelhar, 1993; Rubin, 2003]. While the key step in computing the traditional transverse ensemble dispersion coefficient is to integrate the transverse velocity covariance function along particle trajectories, no such integrals are needed in our definition of the mixing-related equivalent transverse dispersion coefficient Dteq. We only need one-point statistics, namely, the expected value of KDt. This implies, for stationary conductivity and velocity fields, that Dteq is uniform throughout the domain.
 We expect that KDt has a high uncertainty because K is a random variable and Dt is as well, unless one restricts the analysis to pure pore diffusion (model I in Table 1). In fact, various previous studies have shown that transverse mixing is mainly taking place in a few hot spots of mixing, associated with high-permeability zones [e.g., Werth et al., 2006; Cirpka and Kitanidis, 2000; Cirpka, 2002; Rolle et al., 2009]. With Dteq expressing the rate at which the effective plume width squared changes along the flow direction, the cumulative quantity wψ2 should exhibit a decreasing uncertainty with travel distance [Dagan, 1990; Fiori, 1998; Eberhard, 2004; de Barros and Rubin, 2011]. This can be quantified by its variance, denoted by [L4/T2], which can be computed by
In the following (equation (29)), we normalize expression (27) to the coefficient of variation = (dimensionless). The coefficient of variation is a relative measure for the uncertainty of the squared effective plume width wψ2.
Equation (27) involves integrations in the potential ϕ. This is inconvenient for the derivation of analytical expressions because correlations of log conductivity and dependent variables are given in spatial distances. We therefore approximate these integrals by replacing dϕ with Jdx1, which is an approximation permitted only for small heterogeneity (like the replacement of the individual particle trajectory by the mean trajectory in the Lagrangian analysis of macrodispersion). Then, performing the integration along the mean trajectory of the plume centroid yields
3. Analytical Approximation of the Equivalent Transverse Dispersion Coefficient
 In this section, we will derive analytical expressions that illustrate the impact of flow heterogeneity in enhancing pore diffusion and hydromechanical dispersion. We take equation (26) as the starting point (valid for the assumptions given in section 2.1). Analytical results are obtained for dispersion models I (pure pore diffusion) and II (linear pore-scale dispersion with spatially constant dispersivities) presented in Table 1.
 All the following closed-form solutions are further based on the assumption that the log conductivity Y = ln(K) is multi-Gaussian, second-order stationary, and statistically isotropic. Also, we assume that the mean hydraulic gradient is uniform. Under these conditions, the following relations hold:
in which Kg [L/T] is the spatially uniform geometric mean of hydraulic conductivity, Y′ (dimensionless) is the perturbation of log conductivity, h [L] is a spatial separation vector, CYY(h) (dimensionless) is the autocovariance function of log conductivity, and σY2 (dimensionless) is the variance of log conductivity. Results are obtained for an isotropic exponential covariance function CYY(h).
3.1. Pure Pore Diffusion (Model I)
 If Dt in equation (26) is represented by molecular pore diffusion (Dt = Dp; see model I in Table 1), we arrive at the following identities:
 Because we assume a multi-Gaussian distributed, isotropic, second-order stationary 2-D Y field and a uniform mean hydraulic gradient, Keff and 〈K〉 are given by
 The factor (σY2) in equation (39) represents the enhancement of diffusion-related transverse dilution and mixing due to contrasts of hydraulic conductivity. As we will see in the comparison to numerical simulations, the linear approximation of the enhancement factor, 1 + σY2, describes the enhancement better than the exact expression exp(σY2). This finding was observed for several Péclet numbers. We attribute this effect to the simplifications made to arrive at equation (14), which included a restriction to small values of σY2.
3.2. Linear Local-Scale Dispersion With Constant Transverse Dispersivity (Model II)
 If Dt in equation (26) is represented by hydrodynamic dispersion, following Dt = Dp + αtq/θ with constant transverse dispersivity αt, we arrive at the standard linear local-scale dispersion model commonly used in the literature [Scheidegger, 1954; Bear, 1979]. This is model II in Table 1. In this case, 〈KDt〉 becomes
in which the first term on the right-hand side has been discussed in section 3.1 and the second term can be expressed via the cross covariance between conductivity and velocity fluctuations, K′ and q′, about their respective mean values:
in which q1′ expresses specific discharge fluctuations in the mean flow direction. For the conditions mentioned previously (multi-Gaussian, isotropic, second-order stationarity, 2-D Y field, and uniform mean hydraulic gradient), the term 〈Y′q1′〉 can be approximated to first order in σY2 as [Gelhar, 1993]
 Note that we have used the geometric mean for the hydraulic conductivity (denoted by Kg) in lieu of Keff, which is valid here because we assumed the log conductivity to be a second-order stationary random space variable with isotropic covariance function [Gelhar, 1993; Rubin, 2003].
4. Uncertainty of Cumulative Mixing
 The variance of the width of the plume's fringe is expressed in equation (29) of section 2.3. In the following, we will provide closed-form results for the variance of the plume width for models I and II in Table 1. The variance of wψ2 is computed by first-order perturbative methods. That is, we analyze the first-order deviation of wψ2 from the zero-order mean, square it, and take expected values. To obtain the coefficient of variation we have to normalize the standard deviation by the zero-order mean approximation of wψ2. Details of the derivation are provided in Appendix A. The closed-form solutions are given for an isotropic exponential covariance function CYY with correlation length λ.
 For the case where Dt = Dp, we arrive at the following result for the coefficient of variation based on equation (A9):
whereas for Dt = Dp + αtq/θ, the closed-formed expression for the coefficient of variation is given by (see Appendix A for details)
with the hydromechanic dispersion contribution to transverse dispersion in a homogeneous system with identical effective velocity, hydraulic gradient, and transverse dispersion coefficient,
 In order to verify our concepts as well as the analytical expressions derived, we perform extensive numerical Monte Carlo simulations of flow and transport. The ensemble of simulations includes 10,000 realizations for each model of local-scale transverse dispersion and each value of the log hydraulic conductivity variance σY2. We test all four local-scale dispersion models listed in Table 1 and vary σY2 from 0.25 to 2.5 in steps of 0.25. This leads to a total set of 10 × 4 × 10,000 = 400,000 realizations. In order to minimize nonstationarity effects of fixed-head or no-flow boundary conditions on the flow fields, we perform all flow simulations in periodic domains applying periodic boundary conditions with a mean hydraulic gradient in the longitudinal direction x1.
Table 2 summarizes the parameters used in the numerical simulations. The selected hydraulic parameters are representative of typical sedimentary groundwater systems. The range of log hydraulic conductivity variance (i.e., from 0.25 to 2.5) is close to the one reported by Freeze  on the basis of data collected at different field sites: σY2 ≈ 0.2–2.0. Many extensively investigated aquifers are characterized by values of σY2 in the selected range. For instance, the Borden [e.g., Sudicky, 1986] and Cape Cod [e.g., Garabedian et al., 1991] aquifers exhibit σY2 values of 0.29 and 0.24, respectively. Studies at the Columbus field site [Rehfeldt et al., 1992] and at the Biscayne aquifer [Genereux and Guardiario, 2001] report values of σY2 close to the upper limit of the considered interval (2.7 and 2.53, respectively). Concerning the transport parameters, the most sensitive one is transverse dispersion, which is significantly smaller than its longitudinal counterpart. We selected local transverse dispersion parameters (e.g., transverse dispersivity for model II) typically observed in bench-scale flow-through experiments [e.g., Olsson and Grathwohl, 2007; Chiogna et al., 2010] leading to upscaled transverse dispersion values similar to the ones observed for conservative and reactive solute transport at high-resolution monitored sites [e.g., Garabedian et al., 1991; Thierrin and Kitanidis, 1994; Fiori and Dagan, 1999; Prommer et al., 2009]. Because our analytical results are restricted to isotropic two-dimensional heterogeneity, whereas the heterogeneity at the mentioned field sites was anisotropic, we had to make a choice between typical horizontal or vertical values. We have chosen the vertical one because transverse mixing is most important in that direction.
 Because the columns of the streamline-oriented grids are bounded by potential lines and all rows are bounded by streamlines, we can compute transverse moments of the steady state concentration in ψ for fixed values of ϕ by simple columnwise summation.
 As the inflow boundary condition, we assume a concentration in the inflow of unity in the upper half of the incoming flux and a concentration of zero in the lower one. All other boundaries of the streamline-oriented grids are closed for dispersive solute flux. This resembles a Heaviside function in the transverse direction, with the discontinuity at ψ = 0.5Q. Here Q [L2/T] is the total volumetric flux through the domain per unit thickness. In the following, we denote the concentration distribution resulting from the Heaviside-type boundary condition as cH (ϕ, ψ). Because advective-dispersive transport is linear in c, the concentration of the step-like injection is related to that of a point-like injection at ψ = 0.5Q by integration:
 This identity permits us to evaluate the flux-related moments for a point-like injection mkψ(ϕ), defined in equation (16), from the truncated moments for a step-like injection, here denoted Tkψ(ϕ) [e.g., Jose and Cirpka, 2004]:
With these identities and equations (17)–(19), we can compute wψ2 (ϕ) for a point-like injection from the numerical results with a transverse Heaviside injection. Finally, Dteq is computed by equation (25), in which the expected value is replaced by the ensemble mean. It may be worth noting that performing the numerical simulations on streamline-oriented grids makes integrating over ψ and taking derivatives with respect to ϕ even more simple than spatial integrals and derivatives.
5.2.1. Mean Mixing Enhancement Factor
Figures 2–5 show plots of the equivalent transverse dispersion coefficient Dteq for the various models of local transverse dispersion listed in Table 1. These plots compare results from the analytical approximations presented in section 3 and the numerical simulations described in section 5.1. In all cases, the Dteq values are scaled by the corresponding Dt values from a homogeneous system with K = Kg (with all transport parameters being identical to the heterogeneous cases). This yields a dimensionless enhancement factor for transverse mass exchange due to heterogeneity, relative to the mass exchange rates in homogeneous systems used as a reference. These reference values are denoted Dthom. In the analysis of Dthom, we discriminate between pore diffusion, which is uniform, and mechanical dispersion, denoted by Dmechhom. Equation (48) shows the definition of Dmechhom for model II, i.e., linear Scheidegger dispersion. In the simulations of models III and IV, the definition of Dmechhom needs to be modified according to the respective local transverse dispersion laws listed in Table 1.
Figure 2 shows the simulated transverse mixing enhancement factor Dteq/Dthom as a function of dimensionless distance for the four models of local transverse dispersion listed in Table 1. Results shown here are obtained numerically. The example shown in Figure 2 is for σY2 = 2.0. The plot confirms that the equivalent transverse dispersion coefficient Dteq exhibits no distinct trend with distance, regardless of the model for local transverse dispersion. This scale independence, predicted by equation (26), was confirmed for all values of σY2 and Péclet numbers tested (not shown here), with σY2 ranging from 0.25 to 2.5.
Figure 3 depicts the mixing enhancement factor Dteq/Dthom for the pure pore diffusion model of local transverse dispersion (model I in Table 1) as a function of σY2. The crosses indicate the results from the numerical simulations, and the solid line represents the second-order analytical expression according to equation (39). The agreement between the first-order analytical and numerical results is excellent. For values of σY2 larger than 1.5, the numerical and analytical results start to deviate. It is worth noting that the first-order approximation of equation (39) is a better predictor than the seemingly exact expression Dteq = exp(σY2/2)Dp. We attribute this to counterbalancing simplifications leading to equation (14). In addition, numerical errors in approximating high-frequency components of velocity fluctuations on the computational grids may contribute to an erroneous reduction of mixing enhancement.
 The mixing enhancement factor Dteq/Dthom for the linear Scheidegger model of local transverse dispersion (model II in Table 1) as a function of σY2 is shown in Figure 4. Again, crosses indicate numerical results. The solid line represents equation (45), in which Dteq is approximated to second order in σY as (1 + σY2)Dp + (1 + σY2)Dmechhom. Quite obviously, this relationship is not a good predictor of equivalent transverse dispersion for values of the log conductivity variance σY2 ≫ . We explain this by (1) the simplification leading to equation (26) being more restrictive in the case of velocity-dependent local transverse dispersion than in the velocity-independent one and (2) the linearization of 〈Kq〉 used to arrive at equation (45) being invalid for σY2 ≫ . Since the pore diffusion approximation behaved well (as clearly shown in Figure 3) and because pore diffusion and hydromechanic dispersion are additive in the evaluation of the local transverse dispersion coefficient, we fitted a fourth-order correction to the mechanical dispersion contribution of equivalent transverse dispersion. The resulting fitted expression is given by
which is plotted as a dash-dotted line in Figure 4. The factor of 0.7 appearing in equation (52) is obtained by fitting and most likely depends on the transverse Péclet number.
 For the nonlinear local-scale transverse dispersion models III and IV, closed-form expressions of the equivalent transverse dispersion coefficient Dteq are more difficult to derive than for the linear models I and II. In Figure 5, we show the numerical dependence of Dteq for these two models of local-scale dispersion on σY2. As can be seen, the relationship is even less linear than in the case of model II. Keeping again the mixing enhancement of the pore diffusion part, we have fitted the following fourth-order-in-σY relationships (shown as lines) to the numerical results (illustrated by markers):
The coefficients in equations (53) and (54) are fitted and hold only for the specific settings of the test case. Extended higher-order analysis will be necessary to obtain general upscaling rules for nonlinear local-scale dispersion laws. It is noteworthy that the compound-specific model IV leads to a slightly weaker enhancement of transverse mixing by heterogeneity than model III. This finding is consistent with a local-scale transverse dispersion model that scales less than linearly with velocity [e.g., Olsson and Grathwohl, 2007; Chiogna et al., 2010].
5.2.2. Uncertainty of Cumulative Mixing
Figures 6–8 show the coefficient of variation of the squared plume width wψ2 in streamline coordinates as a function of dimensionless head loss for the various models of local-scale transverse dispersion and all values of σY2 tested. As explained above, is the relative uncertainty of the cumulative mixing the plume has undergone from the injection point to the cross section of observation. Because the analytical expressions of given in equations (46) and (47) scale with the standard deviation σY of log conductivity, the results of obtained from both analytical approximations and numerical simulations are normalized by σY. The dimensionless head loss Δϕ/(J × λ) is equivalent to the mean travel distance expressed in multiples of correlation lengths λ.
Figure 6 shows as a function of distance for the pure pore diffusion model (model I in Table 1). The numerical results are plotted as a solid line, whereas the analytical approximation, given by equation (46), is shown by crosses. The numerical results for the various values of σY2 tested practically collapse to a single line when is normalized by σY. Also, the agreement between the numerical and analytical expressions is striking, even though equation (46) is based on first-order perturbations of K.
 The implication of equation (46), confirmed by the results shown in Figure 6, is that the uncertainty of transverse mixing is fairly high, even in the case of uniform local-scale transverse dispersion, and it decreases only slowly with travel distance. For example, in the case of a small to moderate log conductivity variance of σY2 = 1, the relative standard deviation of our mixing measure starts at a value of unity and drops to a value of 20% only after passing 50 correlation lengths (which may depend on the transverse Péclet number). In the numerical tests, we have also analyzed the statistical distribution of wψ2 at any given distance and variance (results not shown here). A lognormal distribution appears to be a good approximation, which is consistent with wψ2 being a strictly nonnegative quantity and K being also lognormally distributed.
Figure 7 shows as a function of distance for the linear Scheidegger model (model II in Table 1). As before, the numerical results are shown as lines, and the analytical approximation of equation (47) is given by crosses. Unlike the pure pore diffusion case, the numerical results for the various values of σY2 do not fully (but still almost) collapse to a single line when is normalized by σY. We have indicated the results for the different levels of variability by color coding, in which the result for the smallest value of σY2 is indicated by a blue line and that of the highest is indicated by a red line (see legend of Figure 7). The higher σY2 values cause somewhat higher relative uncertainty of cumulative mixing, even after scaling with σY. The differences are most pronounced at short travel distances of up to about three correlation lengths. In this range, the analytical expression of equation (47) systematically underpredicts if σY2 exceeds a value of 0.5. These higher-order effects vanish at larger travel distances, and equation (47) becomes a good approximation of even for the highest values of σY2 tested.
 The relative uncertainty of mixing is higher for the linear Scheidegger model (model II) than for the pure pore diffusion model (model I). If we could neglect Dp in model II altogether, equation (47) would predict a value of of ≈1.54σY at zero distance. Also, for the linear Scheidegger model II decreases more slowly when compared to the case of the pore diffusion model (compare Figure 7 with Figure 6). At a distance of 50 correlation lengths, we would still have a value of ≈ 0.38σY. The higher uncertainty can be explained by the uncertain local transverse dispersion coefficient, correlating with hydraulic conductivity.
Figure 8 shows as a function of distance for nonlinear models III and IV of local-scale transverse dispersion. For these models, we have not derived analytical expressions of such that the crosses shown in Figures 6 and 7 are not present in Figure 8. We use the same color coding to express the differences caused by different levels of variability. The results of models III and IV are plotted as solid and dashed-dotted lines, respectively, but they are practically indistinguishable. Because models III and IV are based on a higher variability of local-scale dispersion than model II, which is again correlated to the variability of K, the uncertainty of mixing is yet higher. Also, nonlinear effects lead to a stronger deviation of the lines for the different levels of variability than in the case of linear Scheidegger dispersion with uniform transverse dispersivity.
6. Discussion and Conclusions
 We have presented analytical and numerical results of transverse dispersive mixing in two-dimensional steady state plumes in heterogeneous formations. By reformulating the advection-dispersion equation in streamline coordinates and taking second central moments in this framework, we relate the degree of mixing to the number of stream tubes across which the solute flux is distributed. We separate the effects of streamline meandering as well as squeezing and stretching of stream tubes from the actual mass exchange between stream tubes. Only the latter contributes to real mixing in the transverse direction.
 We have derived a stochastic-analytical framework for effective transverse dispersion in the streamline coordinate framework. Our equivalent macroscopic transverse dispersion coefficient Dteq relates the ensemble mean rate of change of second central moments in stream function values with the dropping hydraulic head to the behavior in an equivalent homogeneous medium with identical effective hydraulic conductivity. In contrast to traditional spatial moment analysis, the equivalent transverse dispersion coefficient expressing actual mixing does not change with travel distance. Instead, it is based on one-point statistics only, as expressed in equation (26): Dteq = 〈KDt〉/Keff. This key finding is independent of the chosen model for local-scale transverse dispersion. Equation (26) would also hold for nonstationary flow fields if the effective conductivity was chosen adequately, but in such a case, Dteq would vary in space. Our mixing-relevant, flux-related transverse dispersion coefficient Dteq behaves differently from the well-known expressions of macroscopic dispersion coefficients based on traditional spatial moment analysis [e.g., Dagan, 1989; Gelhar, 1993; Dentz et al., 2000a; Rubin, 2003]. The latter are computed by integrating the velocity covariance function along the particle trajectory and depend on the travel distance. By choosing a different system of coordinates in our approach, our mixing-related transverse dispersion coefficients do not exhibit any scale dependence at all. Our numerical studies confirm this behavior.
 We have presented closed-form expressions of Dteq for the cases of pure pore diffusion and linear Scheidegger dispersion with uniform transverse dispersivities as local-scale transverse dispersion models. Our closed-form expressions, developed by first-order approximation, scale linearly with σY2. The maximum enhancement factor of mean transverse mixing was about 20 and refers to model III (Scheidegger parameterization of hydromechanic dispersion with αt ∝ ) with high variance of log(K) and a geometric mean of the local transverse Péclet number of ≈6150. In synthetic cases in which local-scale transverse dispersion was dominated by pore diffusion (not shown here), the enhancement due to heterogeneity was smaller.
 Our results confirm the well-known result that pore diffusion remains an important contribution to transverse mixing even in heterogeneous cases, at least in the velocity range considered here. The pore diffusion contribution to local-scale dispersion does not vanish in the enhancement due to heterogeneity. This implies that the effective macroscopic transverse dispersion coefficient should not be replaced by αt,effv, as is commonly done in classical macrodispersion [e.g., Gelhar and Axness, 1983]. Also, local-scale dispersion is not simply additive to macrodispersion. Hence, we have to consider upscaling of local-scale, mixing-relevant transverse dispersion including the compound-specific pore diffusion component.
 We have also presented closed-form expressions of the uncertainty of cumulative mixing for pure pore diffusion and the linear Scheidegger model with uniform transverse dispersivity. The comparison between numerical and analytical results showed an even better agreement than for the mean mixing coefficient. A key finding is that the uncertainty of cumulative mixing is high and slowly decreases with travel distance. This implies that coefficients of variation larger than unity have to be expected in many natural situations. In summary, heterogeneity increases slightly the mean transverse mixing but causes substantial uncertainty. Such uncertainty can lead also to a high probability that mixing of a plume in a heterogeneous aquifer is smaller than in the homogeneous case. The uncertainty analysis presented in this paper goes beyond previous analytical results of effective mixing [e.g., Dentz et al., 2000a; Cirpka and Attinger, 2003], which were restricted to the expected value of a dispersion coefficient. We are aware of only one study that models scale-dependent dispersion tensors as a random function (thus addressing its uncertainty) [de Barros and Rubin, 2011].
 Our findings open a series of new opportunities in stochastic analysis. In the construction of statistical distributions of concentration, we can now account for the uncertainty of transverse mixing. This has not yet been considered in previous approaches that relied on the transverse effective dispersion coefficient [Schwede et al., 2008; Cirpka et al., 2008], on two-particle covariance matrices of transverse displacements [Fiorotto and Caroni, 2002; Caroni and Fiorotto, 2005], or on a generic constant dispersion tensor that parameterizes mixing [Dentz and Tartakovsky, 2010]. With quantifiable uncertainty of transverse mixing, we will also be able, in the future, to predict the uncertainty of reactive plume lengths [Cirpka and Valocchi, 2007, 2009; Cirpka et al., 2006; Liedl et al., 2005; Ham et al., 2004]. The latter, however, will also require accounting for the uncertainty of the effective source width [de Barros and Nowak, 2010]. Both the improvement of analytical concentration probability density functions and improved probability density functions for the length of reactive plumes are currently being investigated by the authors of this study.
 The analysis presented in this study is for two-dimensional flow and transport only. The extension to three spatial dimensions is nontrivial because the concept of stream functions is restricted only to specific cases of three-dimensional flow fields. In particular, we believe that important effects of twisting streamlines, which lead to substantial deformation of plumes [Jankovic et al., 2009], will require a different framework.
Appendix A:: Derivation of Analytical Expressions for the Uncertainty of Cumulative Mixing
A1. General Procedure
 The starting point of the analysis is the approximation for given by equation (29), choosing x0 = 0 without losing generality:
which is a first-order perturbative approximation. In expression (A2), we have made use of stationarity, that is, 〈Dt(x1*)〉 = 〈Dt(x1**)〉 = 〈Dt〉 and 〈K(x1*)〉 = 〈K(x1**)〉 = 〈K〉. Now, we expand ξ(x1*)ξ(x1**),
 In the pure pore diffusion case, the local-scale transverse dispersion coefficient is uniform, so that Dt′ = 0 and 〈Dt〉 = Dp. Under these conditions, substituting equation (A8) into equation (A1) yields
The double integral for the exponential isotropic covariance function can be evaluated by making use of Cauchy's algorithm:
which is valid for generic f and stationary functions [see Dagan, 1989]. In the specific application appearing in equation (A9), we have
 Finally, Γ3 resembles the expression derived for the pure pore diffusion model:
 While the double integral of CYY has already been solved by equation (A11), we still need expressions for Cq1Y and Cq1q1. We start with the cross covariance between specific discharge and log conductivity along the longitudinal direction, applying again Cauchy's algorithm, equation (A10):
in which γ ≈ 0.577216 is the Euler-Mascheroni constant and E1(x) = ∫x∞ exp(−t)/tdt is the exponential integral function.
 As for the integration over the velocity autocovariance, we will make use of the solution provided by Rubin  (not reproduced here; valid for the 2-D isotropic exponential covariance function of log conductivity):
 Recalling the approximation of in equation (A15), we arrive at the final result:
and the coefficient of variation of wψ2 becomes
 We want to thank Aldo Fiori and two anonymous reviewers for constructive comments that helped to improve the paper. F.P.J.B. and W.N. acknowledge the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart. G.C. and M.R. acknowledge the support of the DFG Research Group FOR 525 “Analysis and modeling of diffusion/dispersion-limited reactions in porous media” (grants GR 971/18-1 and GR 971/18-3).