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[MAG39-P02] Time evolution and depth dependence of Cs-137 concentration in Lake Onuma compared with the time-fractional diffusion model
Keywords:Fukushima Accident, Cesium, Diffusion
Lake Onuma on Mt. Akagi (Gunma Prefecture) is a semi-enclosed caldera lake with an average water residence time of 2.3 years. In August 2011, some of the authors found that the activity concentration of wakasagi fish (Japanese pond smelt) 640 Bq/kg, exceeding the provisional standard. Since then, the concentration has been continuously monitored regarding both the wakasagi fish and the lake water.
The radioactivity in the lake water exceeded the regulation value immediately after the Fukushima nuclear accident, and continuous monitoring has been conducted since then. This study aims to reproduce the measured results (both temporal changes and depth profiles) of Lake Onuma using the time-fractional diffusion equation, which is an extension of the classical diffusion model and can handle more general diffusion processes. The time-fractional diffusion equation has a non-integer derivative order such as 0.6, and has a second-order derivative for space, just like the ordinary diffusion equation. The time-fractional diffusion equation arises when the continuous-time random walk (CTRW) is expressed as a partial differential equation. Therefore, it has the property of CTRW, which implies "past concentrations affect current concentrations," and can represent fat tails (slower decay of concentrations than the classical diffusion model). Such a property is often found in observables of the concentrations in environmental pollution. Therefore, the time-fractional diffusion equation may be worth trying as a model for pollutants.
In the sampled lake water of Lake Onuma, Cs-137 is separated into “particulate form” and “dissolved form” after being filtered as explained in Suzuki et al.[1] and Watanabe et al.[2]. When these two forms of Cs-137 measurements are plotted against lake water depth, they have very different profiles from each other. However, this study does not deal with each of the particulate/dissolved form, but with the total concentration, which is the sum of both. The analytical solution is obtained by separation of variables for x (water depth) and t (elapsed days since March 15, 2011). The equations depending only on x or t are eigenvalue problems with the separation constant as a common eigenvalue. The obtained analytical solution that depends only on t is shown in the Figures 1 and 2. Here the solution is adjusted only by a constant prefactor to match the measured values. It can be seen that the time series data is well reproduced by the time-fractional diffusion equation (order of 0.68). In reality, the lake condition is expected to vary greatly from season to season and from day to day; this model does not aim to reproduce such detailed variations. Rather, we aim to estimate the long-term trend by representing the averaged behavior.
The depth profile, on the other hand, is compared with the measured data for the part that depends only on the depth x. Three types of boundary conditions were considered: constant concentration (Dirichlet type), constant flux (Neumann), and mixed boundary condition (Robin). The new knowledge obtained in the present study is that the t-dependent part yields the model parameters which affect the x-dependent part through the eigenvalues; we consistently reproduce the x-dependence of the concentration with these parameters.
The time-fractional diffusion equation is related to the ordinary diffusion equation in the following way. In 2002, Bulgakov et al. [3] used the classical diffusion model for the decadal change of Cs-137 concentration in Lake Svyatoe in the Bryansk region of Russia after the Chernobyl accident. Their formula corresponds to the case where the order of derivative of the time-fractional diffusion equation with order = 1/2. Incidentally, CTRW can be used to model the mass transfer in the bottom sediments of a lake[4].
[1] Suzuki, K. et al., Radiocesium dynamics in the aquatic ecosystem of Lake Onuma on Mt. Akagi following the Fukushima Dai-ichi Nuclear Power Plant accident, Sci. Tot. Env., 622–623 (2018) 1153–1164.
[2] S. Watanabe et al., For elucidate of bottom stop phenomenon in Cs-137 concentration in wakasagi hypomesus nipponensis of Lake Onuma on Mt. Akagi. Proc. 20th Workshop on Environmental Radioactivity, (2019) 84-89.
[3] Bulgakov, A.A. et al., J. Env. Radioact., 61 (2002) 41–53.
[4] Roche, K.R., Aubeneau, A.F., Xie, M., Packman, A.I., Anomalous Sediment Mixing by Bioturbation, American Geophysical Union, Fall Meeting 2013, abstract id. H21D-1091.
The radioactivity in the lake water exceeded the regulation value immediately after the Fukushima nuclear accident, and continuous monitoring has been conducted since then. This study aims to reproduce the measured results (both temporal changes and depth profiles) of Lake Onuma using the time-fractional diffusion equation, which is an extension of the classical diffusion model and can handle more general diffusion processes. The time-fractional diffusion equation has a non-integer derivative order such as 0.6, and has a second-order derivative for space, just like the ordinary diffusion equation. The time-fractional diffusion equation arises when the continuous-time random walk (CTRW) is expressed as a partial differential equation. Therefore, it has the property of CTRW, which implies "past concentrations affect current concentrations," and can represent fat tails (slower decay of concentrations than the classical diffusion model). Such a property is often found in observables of the concentrations in environmental pollution. Therefore, the time-fractional diffusion equation may be worth trying as a model for pollutants.
In the sampled lake water of Lake Onuma, Cs-137 is separated into “particulate form” and “dissolved form” after being filtered as explained in Suzuki et al.[1] and Watanabe et al.[2]. When these two forms of Cs-137 measurements are plotted against lake water depth, they have very different profiles from each other. However, this study does not deal with each of the particulate/dissolved form, but with the total concentration, which is the sum of both. The analytical solution is obtained by separation of variables for x (water depth) and t (elapsed days since March 15, 2011). The equations depending only on x or t are eigenvalue problems with the separation constant as a common eigenvalue. The obtained analytical solution that depends only on t is shown in the Figures 1 and 2. Here the solution is adjusted only by a constant prefactor to match the measured values. It can be seen that the time series data is well reproduced by the time-fractional diffusion equation (order of 0.68). In reality, the lake condition is expected to vary greatly from season to season and from day to day; this model does not aim to reproduce such detailed variations. Rather, we aim to estimate the long-term trend by representing the averaged behavior.
The depth profile, on the other hand, is compared with the measured data for the part that depends only on the depth x. Three types of boundary conditions were considered: constant concentration (Dirichlet type), constant flux (Neumann), and mixed boundary condition (Robin). The new knowledge obtained in the present study is that the t-dependent part yields the model parameters which affect the x-dependent part through the eigenvalues; we consistently reproduce the x-dependence of the concentration with these parameters.
The time-fractional diffusion equation is related to the ordinary diffusion equation in the following way. In 2002, Bulgakov et al. [3] used the classical diffusion model for the decadal change of Cs-137 concentration in Lake Svyatoe in the Bryansk region of Russia after the Chernobyl accident. Their formula corresponds to the case where the order of derivative of the time-fractional diffusion equation with order = 1/2. Incidentally, CTRW can be used to model the mass transfer in the bottom sediments of a lake[4].
[1] Suzuki, K. et al., Radiocesium dynamics in the aquatic ecosystem of Lake Onuma on Mt. Akagi following the Fukushima Dai-ichi Nuclear Power Plant accident, Sci. Tot. Env., 622–623 (2018) 1153–1164.
[2] S. Watanabe et al., For elucidate of bottom stop phenomenon in Cs-137 concentration in wakasagi hypomesus nipponensis of Lake Onuma on Mt. Akagi. Proc. 20th Workshop on Environmental Radioactivity, (2019) 84-89.
[3] Bulgakov, A.A. et al., J. Env. Radioact., 61 (2002) 41–53.
[4] Roche, K.R., Aubeneau, A.F., Xie, M., Packman, A.I., Anomalous Sediment Mixing by Bioturbation, American Geophysical Union, Fall Meeting 2013, abstract id. H21D-1091.