Modeling the depth dependence of Cs-137 concentration in Lake Onuma

Lake Onuma on Mt. Akagi (Gunma Prefecture, Japan) is a closed lake with an average water residence time of 2.3 years. The activity concentration of radioactive cesium in the lake was high shortly after the Fukushima accident. According to Suzuki et al. [1] and Watanabe [2], after a filtration process, Cs-137 are separated into two groups: particulate form and dissolved form. These two forms appears to have very different concentration profiles with each other,  when the Cs-137 concentration plotted against the sampled water depths. In the present study, we are going to model those behavior of particulate/dissolved forms with an emphasis on the depth dependency.

We consider a creation-annihilation process of plankton for the model of the particulate form, since diatom shells are found to be a major constituent of the particulate Cs-137 [2]. We set  ∂P/∂t = f(x,t)  and  f(x,t) = χ(x) cos(ωt) (0 ≤ x ≤ L(water column height), t > 0),  where P=P(x,t) is the activity concentration of the particulate form. The term f(x,t) is the rate of the net production of the plankton at a specific location x at a specific time t. Seasonal cycle is also taken into account by the cosine function (we neglect the phase shift here). The function χ(x), depends solely on water depth x, is responsible for dynamics or inhomogeneity of lake water, such as circulation, stratification or a thermocline. We assume that such a water structure relates to the production rate of plankton through the function χ(x). Thus, we may obtain the concentration of particulate Cs-137. For the dissolved concentration S(x,t), we use the classical diffusion equation with the diffusivity K being dependent on both space and time (i.e. K(x,t)), namely ∂S/∂t =  ∇•(K(x,t) ∇S). Here S=S(x,t) is the activity concentration of the dissolved form. The total activity concentration C(x,t) is the sum of P(x,t) and S(x,t). Using the pair of the equations, we can reproduce the followings. (1) depth profiles of each of the soluble- and particulate activity concentration and (2) depth profiles of the total Cs-137 concentration.

 [1] Suzuki, K. et al., Sci. Tot. Env. (2018)

 [2] Watanabe, S. et al.,  Proc. 20^th Workshop on Environmental Radioactivity (2019)