**7. Conclusions**

The present work was based on an Eulerian approach to determine dispersion of radioactive contaminants in the PBL. To this end the diffusion equation for the cross-wind integrated concentrations was closed by the relation of the turbulent fluxes to the gradient of the mean concentration by means of eddy diffusivity (K-theory). We are completely aware of the fact that K-closure has its intrinsic limits so that one would like to remove these inconsistencies. However, comparisons of predictions by this approach to experimental data have shown that there are scenarios where this lack is not significantly manifest, which we use as a justification together with its computational simplicity to perform our simulations based on this approach.

Since the consistency of the K-approach depends crucially on the determination of the eddy diffusivity considering the turbulence structure of the PBL in its respective stability regimes, we elaborated parametrisations for the eddy diffusivity coefficients based on the micro-meteorological parameters that were extracted from meso-scale WRF simulations, that allowed to take into account the realistic orography of the larger vicinity of a reactor site in consideration. The approach proposed here for the determination of the eddy-diffusivity coefficient is based on the Taylor statistical diffusion theory and on the spectral properties of turbulence. The assumption of continuous turbulence spectrum and variances, allows the parametrisations to be continuous at all elevations, and in stability conditions ranging from a convective to a neutral condition, and from a neutral to a stable condition so that a simulation

of a full diurnal cycle is possible. Simulating micro-meteorology for a short period for the Fukushima Nuclear Power Station Accident may be considered a first step into a direction where the impact of the contamination of radioactive material in the site may be simulated and evaluated for the whole period of the accident until today. Thus the present work may be understood as one tile in a larger program development that simulates radioactive material dispersion using analytical resources, i.e. solutions. In a longer term we intend to build a library that allows to predict radioactive material transport in the planetary boundary layer that extends from the micro- to the meso-scale.

16 Will-be-set-by-IN-TECH

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**7. Conclusions**

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power plant for 3 hours, 48 hours and 93 hours after the beginning of the release.

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**Figure 5.** The logarithmic concentration distribution of radioactive substances released from the nuclear

The present work was based on an Eulerian approach to determine dispersion of radioactive contaminants in the PBL. To this end the diffusion equation for the cross-wind integrated concentrations was closed by the relation of the turbulent fluxes to the gradient of the mean concentration by means of eddy diffusivity (K-theory). We are completely aware of the fact that K-closure has its intrinsic limits so that one would like to remove these inconsistencies. However, comparisons of predictions by this approach to experimental data have shown that there are scenarios where this lack is not significantly manifest, which we use as a justification together with its computational simplicity to perform our simulations based on this approach. Since the consistency of the K-approach depends crucially on the determination of the eddy diffusivity considering the turbulence structure of the PBL in its respective stability regimes, we elaborated parametrisations for the eddy diffusivity coefficients based on the micro-meteorological parameters that were extracted from meso-scale WRF simulations, that allowed to take into account the realistic orography of the larger vicinity of a reactor site in consideration. The approach proposed here for the determination of the eddy-diffusivity coefficient is based on the Taylor statistical diffusion theory and on the spectral properties of turbulence. The assumption of continuous turbulence spectrum and variances, allows the parametrisations to be continuous at all elevations, and in stability conditions ranging from a convective to a neutral condition, and from a neutral to a stable condition so that a simulation

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The quality of the solution may be estimated by the following considerations. Recalling, that the structure of the pollutant concentration is essentially determined by the mean wind velocity **U**¯ and the eddy diffusivity **K**, means that the quotient of norms *�* = ||**K**|| ||**U**¯ || defines a length scale for which the pollutant concentration is almost homogeneous. Thus one may conclude that with decreasing length ( *� <sup>m</sup>* and *m* an increasing integer number) variations in the solution become spurious. Upon interpreting *�*−<sup>1</sup> as a sampling density, one may now employ the Cardinal Theorem of Interpolation Theory ([30]) in order to find the truncation that leaves the analytical solution almost exact, i.e. introduces only functions that vary significantly in length scales beyond the mentioned limit.

The square integrable function *χ* = *<sup>r</sup> c dt dx d* ¯ *<sup>η</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>2</sup> (*<sup>η</sup>* <sup>=</sup> *<sup>y</sup>* or *<sup>z</sup>*) with spectrum {*λi*} which is bounded by *m�*−<sup>1</sup> has an exact solution for a finite expansion. This statement expresses the Cardinal Theorem of Interpolation Theory for our problem. Since the cut-off defines some sort of sampling density, its introduction is an approximation and is related to convergence of the approach and Parseval's theorem may be used to estimate the error. In order to keep the solution error within a prescribed error, the expansion in the region of interest has to contain *<sup>n</sup>* <sup>+</sup> 1 terms, with *<sup>n</sup>* <sup>=</sup> int *mLy*,*<sup>z</sup>* <sup>2</sup>*π�* <sup>+</sup> <sup>1</sup> 2 . For the bounded spectrum and according to the theorem the solution is then exact. In our approximation, if *m* is properly chosen such that the cut-off part of the spectrum is negligible, then the found solution is almost exact.

Further, the Cauchy-Kowalewski theorem ([8]) guarantees that the proposed solution is a valid solution of the discussed problem, since this problem is a special case of the afore mentioned theorem, so that existence and uniqueness are guaranteed. It remains to justify convergence of the decomposition method. In general convergence by the decomposition method is not guaranteed, so that the solution shall be tested by an appropriate criterion. Since standard convergence criteria do not apply in a straight forward manner for the present case, we resort to a method which is based on the reasoning of Lyapunov ([6]). While Lyapunov introduced this conception in order to test the influence of variations of the initial condition on the solution, we use a similar procedure to test the stability of convergence while starting from an approximate (initial) solution **R**<sup>0</sup> (the seed of the recursive scheme). Let <sup>|</sup>*δZn*<sup>|</sup> <sup>=</sup> � <sup>∑</sup><sup>∞</sup> *<sup>i</sup>*=*n*+<sup>1</sup> **R***i*� be the maximum deviation of the correct from the approximate solution Γ*<sup>n</sup>* = ∑*<sup>n</sup> <sup>i</sup>*=<sup>0</sup> **R***i*, where �·� signifies the maximum norm. Then strong convergence occurs if there exists an *<sup>n</sup>*<sup>0</sup> such that the sign of *<sup>λ</sup>* is negative for all *<sup>n</sup>* <sup>≥</sup> *<sup>n</sup>*0. Here, *<sup>λ</sup>* <sup>=</sup> <sup>1</sup> �Γ*n*� log <sup>|</sup>*δZn*<sup>|</sup> |*δZ*0| .

For model validation one faces the drawback, that the majority of measurements are at ground level, so that one could think that a two dimensional description would suffice, however the present analysis clearly shows the influence of the additional dimension. While in the two dimensional approach the tendency of the predicted concentrations is to overestimate the observed values, this is not the case for the results of the three dimensional description, mainly because it does not assume turbulence to be homogeneous. Moreover the solution of the advection diffusion equation discussed here is more general than shown in the present context, so that a wider range of applications is possible. Especially other assumptions for the velocity field and the diffusion matrix are possible. In a future work we will focus on a variety of applications and introduce a rigorous proof of convergence from a mathematical point of view, which we indicated in sketched form only in our conclusions.

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