Is it possible, for example, for a structure to be linearly stable but non-linearly unstable i.e. the structure will resist small perturbations but will evolve away from equilibrium when the perturbation is larger. In the opposite case, linear instability but non-linear stability, small perturbations will cause evolution to start but the non-linear terms will damp and stop this evolution. Of course, the linear and non-linear terms may act in a consistent manner.
One important consequence of non-linear stability is that perturbations in one direction may be stabilised while those in the other direction may grow. Thus, a structure may be able to cool but not to heat, or may be able to rise but not to fall.
Another new direction is that of combining the effects of dynamics and second-order terms.
Nonlinear Thermal Stability in Two Dimensions with MH Ibanez
Non-linear Thermal Stability in Optically Thin Plasmas with MH Ibanez and E Sira.
Prominences are regions of cool, dense material embedded in the hotter corona. They are known to be associated with the magnetic field. Recent observations have shown them to possess fine structure rather than being 'blocks' of cool material. Prominences can last for up to six months and are often seen to erupt into space. Unanswered questions on prominences include the following.
It is believed that prominences exist in dips in the magnetic field i.e. there is an upwards magnetic tension force counteracting the downwards gravitational force. Two common configurations are the 'normal-polarity proninence' where the overlying magnetic field points in the same direction as the field in the prominence and the 'inverse-polarity prominence' where the opposite is true. Much work has been done on putting this on a firm mathematical basis. One line of work has been to incorporate the fibril structure. This has been achieved by dividing the prominence up into hot and cool regions in the horizontal direction and has been generalised by incorporating variations in the vertical direction.
Forming a cool region at the centre of a hot region involves some process to cause heat to travel from the cooling region to the warmer surroundings. It is thought that the process of 'thermal instability' is responsible for this.
If a prominence has existed in the same situation for weeks or even months, it may seem strange for it to begin erupting at speed. However, it is believed that the eruption is caused by either the equilibrium or the stability of the equilibrium being lost. This phenomenon has much in common with the eruption of coronal mass ejections.
These are often observed above erupting prominences. In fact a configuration normally consists of three components, an erupting prominence, a void or cavity above it, and a shell or bubble above this, all erupting outwards. Normally eruption speeds start off low but then increase before gradually tending towards a constant velocity.
The question was addressed of how the configuration can remain in roughtly the same position for weeks or months and then change so rapidly. It is believed thatfor the 'life-time' of a prominence, it is in a stable equilibrium. As conditions change, the configuration will evolve towards a neighbouring stable equilibrium. This can happen at speeds close to quasi-static.
However, it is possible that the configuration can evolve towards a situation where there is no neighbouring equilibrium. In this case, a small perturbation will cause the system to move away from the equilibrium and the speed of this move will increase. Later, as the configuration (coronal mass ejection or CME) moves further away from the Sun and forces decrease, the speed levels off. This has been put on a mathematical basis.
Please contact me if you are interested in this work. For prospective post-graduate students, funding may be available.
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