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Free boundary extension of the SIESTA code and its application to the Wendelstein 7-X

(2018)
Author
Promoter
Raul Sanchez Fernandez, Joachim Geiger, Jose Miguel Reynolds Barredo and (UGent)
Organization
Abstract
The race for obtaining new sources of clean energy is going in many directions. One of them, which is being studied since the 1950’s, is nuclear fusion. Nuclear fusion, which consists on joining two light atoms, occurs naturally at the core of stars. The process is driven by the high pressure at the core due to gravitational effects. The high pressure increases the energy of the atoms in the core to the point of them having enough energy to overcome the electrostatic repulsion when colliding. In this process an impressive amount of energy is liberated which, as with nuclear fission, can be transformed to electrical energy. Clearly the conditions met in the core of stars is not fulfilled on the surface of Earth. Nevertheless there are methods for reproducing conditions on Earth under which fusion is possible, the most promising one being the construction of a magnetic trap containing gas which is then heated up to temperatures high enough for attaining fusion. Due to the high temperatures the gas is converted to plasma which means that it can be contained using electromagnetic fields. Nevertheless it is difficult to contain the highly energetic particles and complex designs of magnetic traps are required. The two main designs being explored nowadays are called the tokamak and the stellarator, both have a toroidal geometry but different operational properties. Plasma dynamics in magnetic confinement devices is a very complex topic to study, mainly because turbulent processes are involved, but also because the interaction between the charged particles, the fields their movements create and the external electromagnetic fields is a system for which the solution to the dynamics equations is hard to obtain. Any deviation from the expected behaviour can result in a considerable alteration of the electromagnetic fields which may cause important deviations of the magnetic configuration from the designed one, preventing fusion to take place and torpedoing the confinement properties of the devices. This is why the study of plasma dynamics is extremely important. In order to understand the dynamics of magnetically confined plasmas it is necessary to have reliable tools which can properly describe confined plasmas, taking into account the possible interactions they may go through at laboratory conditions in fusion devices. The ideal magnetohydrodynamic (MHD) theory is capable of describing the global behaviour of fusion plasmas, therefore it is broadly used for the computation of magnetic equilibria in fusion devices. Other theories, which consider the particle movement of the ions and electrons inside of plasmas, are used whenever a more detailed solution is needed, for example in the study of energy transport properties in fusion plasmas. The equations constituting the ideal MHD theory are also hard to solve, therefore numerical codes which do the work of finding solutions are needed. These codes are of utmost importance to analyse equilibria of different experiments. The well known VMEC code (Variational Moments Equilibrium Code) does the three-dimensional ideal MHD analysis assuming nested magnetic surfaces, i.e. a layered set of torii. VMEC is broadly used in the fusion community in the analysis of the MHD equilibria. It is so broadly used that many codes rely on its equilibria solutions to run further simulations. SIESTA (Scalable Iterative Equilibrium Solver for Toroidal Applications) is one of such codes that rely on VMEC, taking a step further on the equilibrium solution. Using VMEC’s solution, SIESTA computes the ideal MHD equilibrium solution of the problem under study, without the assumption of nested magnetic surfaces. This results in the possible development of magnetic islands and stochastic regions. These type of structures are important in the study of magnetically confined plasmas because they can greatly damage the confinement properties of fusion devices. Although in some cases, as is the case of the Wendelstein 7-X (W7-X) stellarator, island structures can be used as part of the design of the experiment. In the case of the W7-X , a chain of magnetic islands help the correct functioning of the plasma divertor, which are plates taking up high energy fluxes coming out of the confined plasma region. In this special case, a modification on the magnetic island structure could lead to energy deposition on the plasma vessel, which may undermine the operation of the device. The divertor island structure in the W7-X is located just outside the last closed flux surface (LCFS), this being outside of VMEC’s computational domain because outside this limit it is not able to find a solution for a closed (nested) magnetic surface. SIESTA, as was originally conceived, has the limiting aspect that it solves the equilibrium inside of the LCFS found by VMEC without modifying this boundary. This condition implies that the results obtained for equilibria where there are possible instabilities or perturbations close to the LCFS are not well computed since SIESTA leaves the LCFS untouched. In this work a free-plasma-boundary version of SIESTA is developed in order to overcome this original limitation. The approach used consists on extending the analysis domain given by VMEC, in such a way that the vacuum region, or at least the most important part of it, is contained within the analysis volume of SIESTA. This requires the extension of the numerical analysis mesh guaranteeing the continuity of the metric elements on the mesh, a good approximation of the magnetic field solution in all the volume and a pressure solution which couples with the magnetic field. These three requirements have been solved and the extended version of SIESTA has been implemented. The field has been computed using the vector magnetic potential, which ensures the continuity of the magnetic field solution. The new version of SIESTA is applied to the specific case of the Wendelstein 7-X stellarator, at the IPP Greifswald (Germany), making comparisons with previous studies of equilibria showing the development of neoclassical bootstrap currents which cause the divertor island chain to shift its position. The previous studies were carried out with the VMEC-EXTENDER code combination, which is the general tool for ideal MHD equilibrium studies used in IPP. This code combination takes into account the equilibrium solution of VMEC to compute a magnetic field created by plasma currents by use of the virtual casing principle (VCP). The solution is then complemented by the vacuum field calculation derived from a Biot-Savart integrator code. Because of the combination of different solutions coming from different codes, the method is not fully consistent. Nevertheless, it has shown to be correct for the vacuum case and has been tested to be close to the experiment. This lack of consistency is avoided in the new version of SIESTA.
Keywords
Nuclear fusion, magnetohydrodynamics, Wendelstein 7-X, MHD equilibrium

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Citation

Please use this url to cite or link to this publication:

Chicago
Peraza Rodriguez, Hugo. 2018. “Free Boundary Extension of the SIESTA Code and Its Application to the Wendelstein 7-X.”
APA
Peraza Rodriguez, H. (2018). Free boundary extension of the SIESTA code and its application to the Wendelstein 7-X.
Vancouver
1.
Peraza Rodriguez H. Free boundary extension of the SIESTA code and its application to the Wendelstein 7-X. 2018.
MLA
Peraza Rodriguez, Hugo. “Free Boundary Extension of the SIESTA Code and Its Application to the Wendelstein 7-X.” 2018 : n. pag. Print.
@phdthesis{8563350,
  abstract     = {The race for obtaining new sources of clean energy is going in many directions. One of them, which is being studied since the 1950{\textquoteright}s, is nuclear fusion. Nuclear fusion, which consists on joining two light atoms, occurs naturally at the core of stars. The process is driven by the high pressure at the core due to gravitational effects. The high pressure increases the energy of the atoms in the core to the point of them having enough energy to overcome the electrostatic repulsion when colliding. In this process an impressive amount of energy is liberated which, as with nuclear fission, can be transformed to electrical energy. Clearly the conditions met in the core of stars is not fulfilled on the surface of Earth. Nevertheless there are methods for reproducing conditions on Earth under which fusion is possible, the most promising one being the construction of a magnetic trap containing gas which is then heated up to temperatures high enough for attaining fusion. Due to the high temperatures the gas is converted to plasma which means that it can be contained using electromagnetic fields. Nevertheless it is difficult to contain the highly energetic particles and complex designs of magnetic traps are required. The two main designs being explored nowadays are called the tokamak and the stellarator, both have a toroidal geometry but different operational properties.
Plasma dynamics in magnetic confinement devices is a very complex topic to study, mainly because turbulent processes are involved, but also because the interaction between the charged particles, the fields their movements create and the external electromagnetic fields is a system for which the solution to the dynamics equations is hard to obtain. Any deviation from the expected behaviour can result in a considerable alteration of the electromagnetic fields which may cause important deviations of the magnetic configuration from the designed one, preventing fusion to take place and torpedoing the confinement properties of the devices. This is why the study of plasma dynamics is extremely important.
In order to understand the dynamics of magnetically confined plasmas it is necessary to have reliable tools which can properly describe confined plasmas, taking into account the possible interactions they may go through at laboratory conditions in fusion devices. The ideal magnetohydrodynamic (MHD) theory is capable of describing the global behaviour of fusion plasmas, therefore it is broadly used for the computation of magnetic equilibria in fusion devices. Other theories, which consider the particle movement of the ions and electrons inside of plasmas, are used whenever a more detailed solution is needed, for example in the study of energy transport properties in fusion plasmas.
The equations constituting the ideal MHD theory are also hard to solve, therefore numerical codes which do the work of finding solutions are needed. These codes are of utmost importance to analyse equilibria of different experiments. The well known VMEC code (Variational Moments Equilibrium Code) does the three-dimensional ideal MHD analysis assuming nested magnetic surfaces, i.e. a layered set of torii. VMEC is broadly used in the fusion community in the analysis of the MHD equilibria. It is so broadly used that many codes rely on its equilibria solutions to run further simulations. SIESTA (Scalable Iterative Equilibrium Solver for Toroidal Applications) is one of such codes that rely on VMEC, taking a step further on the equilibrium solution. Using VMEC{\textquoteright}s solution, SIESTA computes the ideal MHD equilibrium solution of the problem under study, without the assumption of nested magnetic surfaces. This results in the possible development of magnetic islands and stochastic regions.
These type of structures are important in the study of magnetically confined plasmas because they can greatly damage the confinement properties of fusion devices. Although in some cases, as is the case of the Wendelstein 7-X (W7-X) stellarator, island structures can be used as part of the design of the experiment. In the case of the W7-X , a chain of magnetic islands help the correct functioning of the plasma divertor, which are plates taking up high energy fluxes coming out of the confined plasma region. In this special case, a modification on the magnetic island structure could lead to energy deposition on the plasma vessel, which may undermine the operation of the device. The divertor island structure in the W7-X is located just outside the last closed flux surface (LCFS), this being outside of VMEC{\textquoteright}s computational domain because outside this limit it is not able to find a solution for a closed (nested) magnetic surface.
SIESTA, as was originally conceived, has the limiting aspect that it solves the equilibrium inside of the LCFS found by VMEC without modifying this boundary. This condition implies that the results obtained for equilibria where there are possible instabilities or perturbations close to the LCFS are not well computed since SIESTA leaves the LCFS untouched.
In this work a free-plasma-boundary version of SIESTA is developed in order to overcome this original limitation. The approach used consists on extending the analysis domain given by VMEC, in such a way that the vacuum region, or at least the most important part of it, is contained within the analysis volume of SIESTA. This requires the extension of the numerical analysis mesh guaranteeing the continuity of the metric elements on the mesh, a good approximation of the magnetic field solution in all the volume and a pressure solution which couples with the magnetic field. These three requirements have been solved and the extended version of SIESTA has been implemented. The field has been computed using the vector magnetic potential, which ensures the continuity of the magnetic field solution.
The new version of SIESTA is applied to the specific case of the Wendelstein 7-X stellarator, at the IPP Greifswald (Germany), making comparisons with previous studies of equilibria showing the development of neoclassical bootstrap currents which cause the divertor island chain to shift its position. The previous studies were carried out with the VMEC-EXTENDER code combination, which is the general tool for ideal MHD equilibrium studies used in IPP. This code combination takes into account the equilibrium solution of VMEC to compute a magnetic field created by plasma currents by use of the virtual casing principle (VCP). The solution is then complemented by the vacuum field calculation derived from a Biot-Savart integrator code. Because of the combination of different solutions coming from different codes, the method is not fully consistent. Nevertheless, it has shown to be correct for the vacuum case and has been tested to be close to the experiment. This lack of consistency is avoided in the new version of SIESTA.},
  author       = {Peraza Rodriguez, Hugo},
  keyword      = {Nuclear fusion,magnetohydrodynamics,Wendelstein 7-X,MHD equilibrium},
  language     = {eng},
  pages        = {136},
  school       = {Ghent University},
  title        = {Free boundary extension of the SIESTA code and its application to the Wendelstein 7-X},
  year         = {2018},
}