In tokamaks, turbulence is a key contributor to cross field transport. However, it is also responsible for the spontaneous generation of large scale structures such as zonal ows. These are of relevance to fusion plasmas as they can create transport barriers which aid plasma confinement. The interaction between drift waves and zonal ows can be investigated using reduced models such as the Hasegawa- Mima and Hasegawa-Wakatani equations. A four-wave truncated model is developed for the Extended-Hasegawa-Mima (EHM) equation. This produces a set of four ordinary differential equations (ODEs) that are used to investigate the modulational instability (MI), a mechanism by which drift waves can produce a zonal ow. These equations are linearised to produce a dispersion relation for the MI which is used to produce a set of maps of the linear growth rate of the MI. These show how additional modes become unstable as the gyroradius is increased. The truncated model and dispersion relation are then compared to measurements taken from simulations of the full EHM partial differential equation (PDE) which has been seeded with an appropriate initial condition. Good agreement is found when the pump wave has no component in the direction of the density gradient. A similar truncated model is derived for the Extended-Hasegawa-Wakatani (EHW) equations. As the EHW system has separate equations for density and potential this leads to a set of eight ODEs. The linearisation technique used for the EHM system cannot be applied here. Instead, approximations based on the built in EHW instability are made to calculate a linear growth rate for the zonal ow using the ODEs describing it. These analytical predictions are then compared to a full PDE simulation of the system, which is initialised using random noise. It is found that for particular sets of waves the ODEs provide a good prediction of the linear growth rate. A driving term is added to the EHM equation to reproduce the effect of the built in instability of the EHW equations. This causes a drift wave spectrum to grow when full EHW PDE simulations are seeded with random noise. The four-wave ODE model is updated to include this driving. The ODE model again produces good predictions for the growth rate of the zonal flow.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:595754 |
| Date | January 2013 |
| Creators | Gallagher, Stephen J. |
| Publisher | University of Warwick |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://wrap.warwick.ac.uk/59703/ |
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