Computer Modelling of Marine Radiotracers

Mathematical and Computational Aspects

Objectives of the course

• To learn the basic principles on building a numerical model for radionuclide dispersion (essential equations and numerical schemes)

• To give practical training on the implementation of such ideas into a real computational code

• To provide bibliography to access to more complex developments

How do we achieve objectives?

• Theoretical lessons• Practical exercises with computers

with the help of documentation provided

Theoretical contents1 Introduction2 Model structure3 Introduction to the transport equation4 Solving hydrodynamics5 Solving hydrodynamics and dispersion6 Modelling non conservative radionuclides7 Lagrangian dispersion models8 Sensitivity analysis

1 I

Practical exercises

• E1: Using a dispersion model (GISPART)• E2: Solving the 1D transport equation• E3: Solving the 2D hydrodynamic

equations• E4: Solving hydrodynamics and dispersion• E5: Developing a lagrangian dispersion

model

TimingMorning Afternoon

Monday L1 and 2 E1Tuesday L3 E2Wednes. L4 E3Thursday L5 and 6 E4Friday L7 and 8 E5

Documentation:

• Notes on modelling marine radiotracers• CD with:

– Notes in pdf format– Selected papers from the bibliography (pdf)– GNUPLOT plotting software– GISPART model, instructions and codes– Fortran codes

Introduction

1) Why modelling radioactivity dispersion in the environment?

2) What is a model and how do we build it?3) Marine dispersion models: form box

models to full 3D dispersion models for non conservative radionuclides.

Why modelling radioactivity dispersion?• To test hypotheses on the environmental behaviour of

radionuclides• Evaluate impact in response to inputs of radionuclides in a

quantitative manner• Models form part of radiological assessments by

predictions of doses: support the decision-making process in response to accidents

• Help to monitor areas affected by permanent releases• Oceanography: radionuclides can be used as tracers of

oceanic circulation and currents• The model should not only reproduce measurements, but

also give insight of the processes responsible of such behaviour

What is a model?

A model consists of a set of differential equations that capture the dispersion processes to be simulated.

These equations are solved numerically on a regular grid, that covers the area of interest, at discrete time steps.

Building a model

• Select environmental processes to be simulated• Select model structure (spatial and temporal resolution etc)• Set differential equations that capture the selected

processes and structure• Construction of the computational code• Calibration• Validation against different observations• Sensitivity analysis

Models for radionuclide dispersion

Box and dynamic models

Box models•Long term dispersion over thousand km and temporal scales of the order of 103 y.•Boxes exchange radionuclides according to effective rates deduced form oceanographic information•Assumptions

– uniform mixing into each box–instantaneous mixing

•Exchanges water-sediment through equilibrium distribution coefficients:

d

sd C

Ck =

Box structure for surface, mid-depthand deep waters of a box model for the North-Atlantic andArtic regions.It simulates the transport of radionuclidesfrom Sellafield and LaHague to the Artic

Box structure of a model for simulating 137Cs dispersion inthe Mediterranean Sea

Box model essential equation

∑ ∑= =

+−−=n

j

n

jiiiiijjji

i QAkAkAkdtdA

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Ai - activity in box ikij - transfer rate from box i to jki - loss of material from box i (decay, transfer to sediment...) without transfer to anotherQi - source term to box i

•Now modellers have started including time-dependent flow fields:•Box models are still used to estimate doses

•Hydrodynamic equations are solved to obtain currents at each point of the domain and at each instant of time

•Dynamic models have a much higher spatial and temporal resolution than box models

•These currents are used to solve the dispersion equation for radionuclides

Dynamic models

Computed tidal currents in the Irish Sea when water level is increasing at Anglesey.

Only one of each 4 vectors are shown.

Dynamic models were first applied to conservative radionuclides (remain dissolved)

Computed and measured levels of dissolved 137Cs released from Sellafield (pCi/L)

Distributions of 125Sb (dissolved) released from Cap de la Hague at different times during the year 1989.

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t=2 h t=4 h t=6 h

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•Next step: include exchanges between water and sediments (suspended matter and bed sediment)

•A kinetic approach is more adequate than using an equilibrium distribution coefficient

Water Suspendedmatter

Bed sediment

Measured and computed Pu concentrations along the English Channel

Other processes have also been included:

•Reduction-oxidation reactions in the case of plutonium

•Different kinetic models for adsorption-release involving several parallel or consecutive reactions

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One-step model Two-step model % of reduced Pu

Computed distribution of 239+240Pu in sediments (kBq/m2)

•Spatial resolution has evolved with time: from coarse box models to high resolution 2D or 3D models.

•Areas covered range from small coastal areas or estuaries to models for simulating dispersion in an ocean or the whole oceanic space (global models).

Computed distribution of 137Cs in surface water by a global model.

Lagrangian models