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Numerical simulation of the tokamak scrape-off layer (SOL) is an essential tool for the prediction of the conditions to be expected in future fusion reactors such as the ITER project, now under construction in Southern France. One particularly important issue regards the estimation of the expected transient power loads on plasma-facing components (PFC) due to magnetohydrodynamic plasma relaxations, known as Edge Localised Modes (ELMs). These loads are a major cause of concern for ITER owing to the very severe restrictions on PFC lifetime (especially the divertor targets) that they will impose if their amplitude is not maintained below a given size. Even though SOL plasma modelling has reached a comparatively high level of sophistication (the ITER divertor is being designed in part with complex edge plasma codes), the majority of simulations are performed for steady state conditions, necessarily excluding the description of transient events. This thesis explores the utility and validity of the fluid plasma, Monte-Carlo neutrals approach to SOL and divertor modelling in the presence of time dependent ELM phenomena. It aims to test the most complex tool of this type currently available, the fluid (B2.5)-neutral Monte-Carlo (EIRENE) code package SOLPS5, against a variety of ELM sizes in two very different tokamaks, TCV and JET. Although the SOLPS package has been the modelling tool of choice for ITER design, it has not yet been systematically used for the study of ELM transients. A key element throughout is rigorous benchmarking – seeking the best possible agreement between both experiment and simulation and between different codes for the same experiment, using as many different measurements as possible to constrain the model. Such benchmarking attempts are still, unfortunately, comparatively rare on today's machines. Fully time-dependent simulations (2-D plasma, 3-D neutrals) have been performed of four H-mode plasmas, two each on TCV and JET, covering Type III and Type I ELMs over a range of pedestal collisionality and energy expelled per ELM from ΔWELM ∼ 0.005 → 0.7 MJ. The high end of this limit corresponds to the current maximum ΔWELM which is thought to be tolerable on ITER for acceptable divertor target lifetime. The two tokamaks differ radically in size, input power and divertor geometry, but share carbon as the main PFC material. The SOLPS5 simulations have thus been performed with all carbon charge states included but do not feature activated poloidal drift terms. The approach is first to seek the closest match to experimental upstream, pedestal/SOL and downstream target profiles during the inter-ELM phase. This is achieved through the specification of radially varying perpendicular particle and heat diffusivities and/or convective radial velocity in order to account for the different transport levels in the edge and SOL regions. Poloidal variation of these transport coefficients is also applied to distinguish between main chamber SOL and divertor regions. This is important in a device like TCV with rather unconventional divertor geometry. Similar reasoning applies even more to the ELM itself, which is known to burst into the SOL in the outboard, unfavourable curvature region and is thus extremely poloidally localized. This has also been accounted for especially in the attempts to simulate the TCV ELM events. The complexity of the ELM instability continues to prevent a complete theoretical description of the evolution of transport during the event. In SOLPS5, the simplest and currently only method by which the ELM can be simulated is to increase the anomalous transport coefficients used to simulate the pre-ELM state during a brief interval corresponding to the ELM duration, such that the total energy expelled during this time is compatible with that measured experimentally. In the case of TCV Type III ELM, where reasonable upstream and downstream data are available and for which the largest number of sensitivity studies have been performed in this thesis with SOLPS, agreement is good in the pre-ELM phase and reasonable, but less satisfactory during the ELM. This ELM is a largely convective event in terms of pedestal temperature collapse and is, by virtue of its low ΔWELM, the "least kinetic" of the four events studied. Nevertheless, comparison of the SOLPS5 simulation results at the divertor target with those from dedicated Particle-in-Cell kinetic transport code calculations for the same ELM, demonstrate that kinetic effects are important and must be properly accounted for (by appropriate adjustment of kinetic coefficients in the fluid simulations). This presumably becomes even more important as the ELM size increases, but can only be tested to the extent that the appropriate experimental data is available. As a consequence, the tentative conclusion from the work presented here is that the use of SOLPS in a predictive sense for ITER would at best provide indicative results. In addition to the code-experiment benchmark, a code-code comparison has also been performed, checking SOLPS5 against published and well known time dependent Type I ELM simulations obtained with the dedicated JET code suite EDGE2D-Nimbus. A benchmark of this complexity has not previously been attempted and has been reassuringly somewhat successful, albeit with some unresolved discrepancies. A key feature of ELM boundary physics occupying much current research are the energy deposition asymmetries observed at the targets, which favour the inner target during the ELM for forward toroidal field direction and which appear to reverse when the field direction is inverted. These trends are opposite to the behaviour seen in inter-ELM phases, behaviour which is conventionally understood to result from toroidal geometry and the contribution of poloidal drift physics. Added complexity comes from magnetic geometry, a prominent feature of the TCV-JET comparisons described in this thesis, the results of which seem to influence the simulation results (which do not include drift effects). A recent development has been the suggestion that the ELM, in convecting plasma from pedestal to SOL regions, carries with it memory of the high toroidal rotation velocity known to characterise the H-mode pedestal on all devices. This hypothesis has been tested here in a preliminary manner, and for the first time in this kind of simulation, by imposing a toroidal velocity inside the magnetic separatrix in the simulations and studying the radial transport of this toroidal momentum into the SOL. Applied in the first instance to the TCV Type III ELM, the indications are that transfer of this rotation into the SOL can drive target asymmetries in the direction seen experimentally, though there are significant negative consequences for the resulting target profiles in other parameters for which a potential resolution would require protracted further study which has not been possible here.