Vlasov-Maxwell equilibria are characterised by the self-consistent descriptions of the steady-states of collisionless plasmas in particle phase-space, and balanced macroscopic forces. We study the theory of Vlasov-Maxwell equilibria in one spatial dimension, as well as its application to current sheet and flux tube models. The ‘inverse problem' is that of determining a Vlasov-Maxwell equilibrium distribution function self-consistent with a given magnetic field. We develop the theory of inversion using expansions in Hermite polynomial functions of the canonical momenta. Sufficient conditions for the convergence of a Hermite expansion are found, given a pressure tensor. For large classes of DFs, we prove that non-negativity of the distribution function is contingent on the magnetisation of the plasma, and make conjectures for all classes. The inverse problem is considered for nonlinear ‘force-free Harris sheets'. By applying the Hermite method, we construct new models that can describe sub-unity values of the plasma beta (βpl) for the first time. Whilst analytical convergence is proven for all βpl, numerical convergence is attained for βpl = 0.85, and then βpl = 0.05 after a ‘re-gauging' process. We consider the properties that a pressure tensor must satisfy to be consistent with ‘asymmetric Harris sheets', and construct new examples. It is possible to analytically solve the inverse problem in some cases, but others must be tackled numerically. We present new exact Vlasov-Maxwell equilibria for asymmetric current sheets, which can be written as a sum of shifted Maxwellian distributions. This is ideal for implementations in particle-in-cell simulations. We study the correspondence between the microscopic and macroscopic descriptions of equilibrium in cylindrical geometry, and then attempt to find Vlasov-Maxwell equilibria for the nonlinear force-free ‘Gold-Hoyle' model. However, it is necessary to include a background field, which can be arbitrarily weak if desired. The equilibrium can be electrically non-neutral, depending on the bulk flows.

Ces travaux abordent la problématique des mesures de sonde de Langmuir dans un environnement RF. Les mesures expérimentales ont montré que des courants DC négatifs (électroniques) étaient collectés sur la structure d'une antenne ICRF sous tension, pendant que des courants DC positifs (ioniques) sont recueillis par une sonde de Langmuir à l'autre bout du tube de flux magnétique ouvert connecté à l'antenne, la sonde étant au potentiel de la machine. Un modèle de tube de flux asymétrique, de type de sonde double, est présenté. Celui-ci modélise un plasma, confiné le long des lignes de champ magnétique, ayant à chaque extrémité une électrode dont l'une est polarisée à un potentiel RF et l'autre à la masse. L'électrode polarisée modélise le potentiel RF résultant de l'intégration, le long d'une ligne champ magnétique, du champ électrique rayonné par les straps d'une antenne ICRF, tandis que l'autre électrode modélise la sonde au potentiel de la machine. Ce modèle permet d'expliquer l'apparition de courants DC en émettant simplement l'hypothèse qu'il faut à la fois une asymétrie de la source RF par rapport à une masse fixe, une conductivité RF transverse non nulle autorisant des courants RF transverses ainsi qu'une caractéristique courant-tension non linéaire due aux gaines pour favoriser des courants négatifs du côté RF et des courants positifs côté sonde. Ce modèle permet également de modéliser les caractéristiques Courant DC - Tension DC d'une sonde en présence de RF et ainsi d'évaluer les propriétés du plasma. Dans ce cas l'électrode modélisant la sonde n'est plus à la masse, mais à un potentiel donné. Des résultats analytiques sont trouvés dans certaines limites / This work investigates the problematic of probe measurements in RF environment. DC currents flowing along magnetic field lines connected to powered ICRF antennas have been observed experimentally. Negative (i.e. net electron) current is collected on the powered ICRF antenna structure, while positive (i.e. net ion) current is collected by magnetically connected Langmuir probes. An asymmetric model based upon a double probe configuration was developed. The ICRF near field effect is mimicked by a ?driven? RF electrode at one extremity of an "active" open magnetic flux tube, where a purely sinusoidal potential is imposed. The other connection point is maintained at ground potential to model a collecting probe. This "active" flux tube can exchange transverse RF currents with surrounding "passive" tubes, whose extremities are grounded. With simple assumptions, an analytical solution is obtained. We can thus explain how DC currents are produced from RF sheaths. This model also makes it possible to model the characteristics DC Current' DC Voltage of a probe in the presence of RF and thus to evaluate some plasma properties. In this case the electrode at ground potential (probe) is polarized at a given potential. Analytical results are found within certain limits

The Sun, by the virtue of its proximity to Earth, serves as an excellent astrophysical laboratory for testing our theoretical ideas. The Sun displays a plethora of visually awe-inspiring phenomena including flares, prominences, sunspots, corona, CMEs and uncountable others. It is now known that it is the magnetic field of the Sun which governs all these and also the geomagnetic storms at the Earth, which owes its presence to the interaction between the geomagnetic field and the all-pervading Solar magnetic field in the interplanetary medium. Since the solar magnetic field affects the interplanetary space around the Earth in a profound manner, it is absolutely essential that we develop a comprehensive understanding of the generation and manifestation of magnetic fields of the Sun. This thesis aims at developing a state-of-the-art dynamo code SURYA1taking into account important results from helioseismology and magnetohydrodynamics. This dynamo code is then used to study various phenomenon associated with solar activity including evolution of solar parity, response to stochastic fluctuations, helicity of active regions and prediction of future solar cycles. Within last few years dynamo theorists seem to have reached a consensus on the basic characteristics of a solar dynamo model. The solar dynamo is now believed to be comprised of three basic processes: (i)The toroidal field is produced by stretching of poloidal field lines primarily inside the tachocline – the region of strong radial shear at the bottom of the convection zone. (ii) The toroidal field so formed rises to the surface due to magnetic buoyancy to form active regions. (iii) Poloidal field is generated at the surface due to decay of tilted active regions – an idea attributed to Babcock (1961) & Leighton (1969). The meridional circulation then carries the poloidal field produced near the surface to the tachocline. The profile of the solar differential rotation has now been mapped by helioseismology and so has been the poleward branch of meridional circulation near the surface. The model I describe in this thesis is a two-dimensional kinematic solar dynamo model in a full sphere. Our dynamo model Surya was developed over the years in stages by Prof. Arnab Rai Choudhuri, Dr. Mausumi Dikpati, Dr. Dibyendu Nandy and myself. We provide all the technical details of our model in Chap. 2 of this thesis. In this model we assume the equatorward branch of the meridional circulation (which hasn’t been observed yet), to penetrate slightly below the tachocline (Nandy & Choudhuri 2002, Science, 296, 1671). Such a meridional circulation plays an important role in suppressing the magnetic flux eruptions at high latitudes. The only non-linearity included in the model is the prescription of magnetic buoyancy. Our model is shown to reproduce various aspects of observational data, including the phase relation between sunspots and the weak, efficient. An important characteristic of our code is that it displays solar-like dipolar parity (anti-symmetric toroidal fields across equator) when certain reasonable conditions are satisfied, the most important condition being the requirement that the poloidal field should diffuse efficiently to get coupled across the equator. When the magnetic coupling between the hemispheres is enhanced by either increasing the diffusion or introducing an α ff distributed throughout the convection zone, we find that the solutions in the two hemispheres evolve together with a single period even when we make the meridional circulation or the α effect different in the two hemispheres. The effect of diffusive coupling in our model is investigated in Chap. 3. After having explored the regular behaviour of the solar cycle using the dynamo code we proceed to study the irregularities of the Solar cycle.We introduce stochastic fluctuations in the poloidal source term at the solar surface keeping the meridional circulation steady for all the numerical experiments. The dynamo displays oscillatory behaviour with variable cycle amplitudes in presence of fluctuations with amplitudes as large as 200%. We also find a statistically significant correlation between the strength of polar fields at the endofone cycle and the sunspot number of the next cycle. In contrast to this there exist a very poor correlation between the sunspot number of a cycle and the polar field formed at its end. This suggests that during the declining phase of the sunspot cycle poloidal field generation from decaying spots takes place via the Babcock-Leighton mechanism which involves randomness and destroys the correlation between sunspot number of a cycle and the polar at its end. In addition to this we also see that the time series of asymmetries in the sunspot activity follows the time series of asymmetries in the polar field strength with a lag of 5 years. We also compare our finding with available observational data. Although systematic measurements of the Sun’s polar magnetic field exist only from mid-1970s, other proxies can be used to infer the polar field at earlier times. The observational data indicate a strong correlation between the polar field at a sunspot minimum and the strength of the next cycle, although the strength of the cycle is not correlated well with the polar field produced at its end. We use these findings about the correlation of polar fields with sunspots to develop an elegant method for predicting future solar cycles. We feed observational data for polar fields during the minima of cycle n into our dynamo model and run the code till the next minima in order to simulate the sunspot number curve for cycle n+1. Our results fit the observed sunspot numbers of cycles 21-23 reasonably well and predict that cycle 24 will be about 30–35% weaker than cycle 23. We fit that the magnetic diffusivity in the model plays an important role in determining the magnetic memory of the Solar dynamo. For low diffusivity, the amplitude of a sunspot cycle appears to be a complex function of the history of the polar field of earlier cycles. Only if the magnetic diffusivity within the convection zone is assumed to be high (of order 1012cms−1), we are able to explain the correlation between the polar fiat a minimum and the next cycle. We give several independent arguments that the diffusivity must be of this order. In a dynamo model with diffusivity like this, the poloidal field generated at the mid-latitudes is advected toward the poles by the meridional circulation and simultaneously diffuses towards the tachocline, where the toroidal field for the next cycle is produced. The above ideas are put forward in Chap. 6. We next come to an important product of the dynamo process namely the magnetic helicity. It has been shown independently by many research groups that the mean value of the normalized current helicity αp= B (Δ×B)/B2in solar active regions is of the order of 10−8m−1, predominantly negative in the northern hemisphere, positive in the southern hemisphere. Choudhuri (2003, Sol. Phys., 215, 31)developed a model for production of the helicity of the required sign in a Babcock-Leighton Dynamo by wrapping of poloidal field lines around a fluxtube rising through the convection zone. In Chap. 7 we calculate helicities of solar active regions based on this idea. Rough estimates based on this idea compare favourably with the observed magnitude of helicity. We use our solar dynamo model to study how helicity varies with latitude and time. At the time of solar maximum, our theoretical model gives negative helicity in the northern hemisphere and positive helicity in the south, in accordance with observed hemispheric trends. However, we fit that during a short interval at the beginning of a cycle, helicities tend to be opposite of the preferred hemispheric trends. After calculating the sign and magnitude of helicity of the sunspots we worry about the distribution of helicity inside a sunspot. In Chap. 8 we model the penetration of a wrapped up background poloidal field into a toroidal magnetic flux tube rising through the solar convective zone. The rise of the straight, cylindrical flux tube is followed by numerically solving the induction equation in a comoving Lagrangian frame, while an external poloidal magnetic field is assumed to be radially advected onto the tube with a speed corresponding to the rise velocity. One prediction of our model is the existence of a ring of reverse current helicity on the periphery of active regions. On the other hand, the amplitude of the resulting twist depends sensitively on the assumed structure (ffvs. concentrated/intermittent) of the active region magnetic field right before its emergence, and on the assumed vertical profile of the poloidal field. Nevertheless, in the model with the most plausible choice of assumptions a mean twist comparable to the observational results. Our results indicate that the contribution of this mechanism to the twist can be quite find under favourable circumstances it can potentially account for most of the current helicity observed in active regions.