Figure 1

(Color online) Lagrangian approximation (symbols) of Eq. (1) with $b=1$ and operator (9) vs the semianalytical solution (lines) developed by Schumer et al. [44]. The time scale index $\gamma$ varies from 0 to 1, while the space scale index $\alpha$ is limited to 2. The classical second-order advection-diffusion equation $\partial p∕\partial t=-v\partial c∕\partial x+D{\partial }^{2}C∕\partial x^2$ is also shown (dashed line) for comparison. (a) The spatial distribution of $p(x,t)$, with parameters $v=1$, $D=0.1$, $t=30$, $\gamma =0.1–0.9$, $\beta =0.1$, and the instantaneous source at $x=0$. (c) The temporal evolution of $p(x,t)$, with $v=1$, $D=0.1$, $\gamma =0.2–0.95$, $\beta =0.2$, the initial source at $x=0$, and the control plane at distance 5. (b) The semilog plot of (a), and (d) the log-log plot of (c).Reuse & Permissions

Figure 2

(Color online) Lagrangian approximations (symbols) of Eq. (1) with $b=1$ and operator (9) vs the implicit Eulerian solutions after time-subordination operation (lines), for four different cases. Some parameters are the same in all cases, including: $v\left(x\right)=2+0.04x$, $D\left(x\right)=8+0.15x$, $\gamma =0.5$, an instantaneous source at $x=50$, and the control plane at distance 80. (b) The log-log plot of (a).Reuse & Permissions

Figure 3

Operational time vs clock time. Different lines represent a different sample path in time.Reuse & Permissions

Figure 4

The 2D density described by Eq. (11) with $b=1$ at time $t=1$ with $v=0$, $D_1=D_2=1$, ${\alpha }_1=1.3$, and ${\alpha }_2=1.7$. The mixing measure is assigned along two discrete directions: $M_1=M(30°)=0.4$, $M_2=M(-35°)=0.6$. (a) Semianalytical solution with $\beta =0$. (b) Lagrangian solution with $\gamma =0.98$ and $\beta =0.5$. (c) Lagrangian solution with $\gamma =0.3$ and $\beta =0.2$. The circle (exaggerated for display) in (a) denotes the particle source location (0,0) at time 0.Reuse & Permissions

Figure 5

(Color online) The measured Bromide plumes at days 503 (a) and 126 (e) vs the simulated plumes [(c)–(e)] using the FDE (11). The parameters (either predicted or fitted) are the space scale index ${\alpha }_x=1.1$, ${\alpha }_y=1.5$, time index $\gamma =0.35$, coefficient $\beta =0.08$ ${\text{days}}^{-0.65}$, $v_x=0.30\mathrm{m}∕\text{day}$, $v_y=0$, and $D_y=0.30{\mathrm{m}}^{1.5}∕\text{day}$. (a) The measured plume at day 503. (b) The best-fit three-zone mixing measure with seven discrete directions and weights. (c) The simulated plume at day 503 using constant $D_x$ $(D_x=0.30{\mathrm{m}}^{1.1}∕\text{day})$. (d) The simulated plume at day 503 using space-dependent $D_x$ $(D_x=0.30+0.0017x{\mathrm{m}}^{1.1}∕\text{day})$. (e) The measured plume at day 126 vs the simulated plumes using constant or linear $D_x$. In (e), the simulated plume using the constant $D_x$ is similar to that with linear $D_x$, due to the truncation of plumes. The diamonds in (a) and (e) denote the locations of samplers at the MADE site.Reuse & Permissions