2D theory and simulations of Neoclassical Tearing Modes

Transcription

1 2D theory and simulations of Neoclassical Tearing Modes Jacco Heres FOM-institute DIFFER - Dutch Institute for Fundamental Energy Research; Institute for Theoretical Physics, Utrecht University Thesis Supervisors: prof. dr. René van Roij ITF-UU, prof. dr. Wim Goedheer DIFFER, UU Daily supervisors: dr. Egbert Westerhof DIFFER, dr. Jane Pratt DIFFER July, Generated: July 25, 2013

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3 Abstract Fast escape of charged particles due to the presence of large magnetic islands limits the performance of modern toamas, toroidally-shaped fusion-energy devices. Magnetic islands are closed magnetic fieldline topologies that form in a fusion plasma because of magnetic reconnection. When magnetic islands grow in size, they are commonly called neoclassical tearing modes NTMs by plasma physicists. The theoretical wor that describes the growth-rate of NTMs is called the generalized Rutherford equation GRE. The GRE assumes that in the poloidal coordinate the perturbation of the helical flux, responsible for the generation of the magnetic island, can be described by a single Fourier mode. In this wor, we developed a two-dimensional MHD simulation to test this assumption of the GRE. The goal is to produce a clear understanding of the plasma effects internal to the magnetic island that are neglected in the GRE. To develop a simulation relevant to toama plasma conditions that also successfully produces magnetic islands, we must formulate new boundary conditions. The formulation of these boundary conditions is a significant step forward for this type of simulation, and was not theoretically clear from the beginning of this wor. We describe in detail the options and consequences of these boundary conditions on the plasma, and present the results of a benchmaring study performed with the code. We discuss possible future extensions to the code to investigate the localized character of a non-inductive current, which could be added to the simulation to suppress growing magnetic islands. 3

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5 Contents 1 Introduction Motivation This thesis The Rutherford equation The reduced MHD approximation The role of resistivity The ideal outer region Coupling the outer region with the resistive layer The resistive layer Nonlinear behaviour Derivation of the Rutherford equation The generalised Rutherford equation The 2D reduced MHD model Perturbing the system Rescaling to dimensionless units The Fourier expansion Symmetry of magnetic islands in the helical direction Sine and cosine series The vorticity and the solution for the stream field Linear stability analysis Viscosity Boundary conditions Central boundary condition for ψ Initial state Parameter normalisation and range Normalisation Physical ranges for the parameters Numerical ranges Sensitivity of the 2D reduced MHD model to the boundary conditions Relevance of the boundary conditions Coupling vorticity and stream function Possible boundary conditions for the vorticity Possible boundary conditions for the stream function Results of the boundary condition tests Results of a benchmar study Conclusions and outloo Conclusions Outloo Physical Numerical improvements A Appendix: Sine and cosine series 47 5

6 6 CONTENTS B Appendix: The numerical schemes 49 B.1 The boundaries B.2 Initial field B.3 Programme structure References 55

7 Chapter 1 Introduction The increasing energy demands of our society, the depletion of fossil fuels and their effects on climate maes the search for new sources of energy an immediate need. Producing electrical energy through thermonuclear fusion is one possible future energy resource. To mae thermonuclear fusion an efficient and reliable energy source, one of the most important hurdles is the understanding of plasma instabilities and development of techniques to suppress them. Instabilities can cause violent disruptions that can damage the fusion device. They also lower the energy levels that can be produced in a plasma. A neoclassical tearing mode NTM is one such instability. It leads to faster losses of particles and energy from the machine. NTMs can also cause a plasma disruption, when the plasma violently explodes out of the fusion machine. NTMs grow around surfaces of constant magnetic flux called flux-surfaces where the angle of the helical field lines is such that the magnetic field lines close onto themselves after a number of toroidal cycles rather than never returning to their previous position and eventually filling the entire surface. These type of flux-surfaces are called rational surfaces, because the ratio between the number of toroidal going around the whole torus and poloidal windings going around a circular cross section perpendicular to the toroidal direction of the magnetic field is a rational number on these surfaces. See Figure 1.1 for an example of the shape of magnetic field lines in a toama. The magnetic field is twisted less on the surfaces at the outside of the toama then on the inside, thereby creating a magnetic shear at each point and giving each flux-surface a unique ratio of the toroidal and poloidal magnetic field. When there is a small perturbation at some point on a rational surface, a particle that moves mainly along the magnetic field line feels the same perturbation each time as it completes a full cycle over the surface. The perturbation grows larger and larger, and the magnetic surface breas up into a chain of magnetic islands, which are the characteristic of NTMs. The name tearing modes comes from the tearing and reconnection of the magnetic field lines that produce the magnetic islands. Figure 1.1: The nested flux-surfaces that are formed by the helical magnetic field lines in a toama, with some field lines shown in red. 7

8 8 CHAPTER 1. INTRODUCTION After small magnetic islands have arisen from small perturbations, the island initially grows exponentially in size within the a thin layer around the rational surface, where resistive effects dominate the physics. The width of this resistive layer is determined by the distance from the rational surface where large gradients in the magnetic field lead to large currents. Within this layer a linear approximation of the perturbation is possible, and this phase is therefore called the linear phase. When the island width exceeds the width of this resistive layer, the mode enters the Rutherford phase, where a nonlinear theory describes the behaviour of the mode. The importance of this neoclassical transport effect is the reason that these ind of tearing modes are called neoclassical. Various other effects, including the bootstrap current[10], ion polarisation current [20] and the rotation of the island can lead either to a saturation of the magnetic island or to a plasma disruption. Even if the disruptive character of magnetic islands is controlled, growing or saturated islands mae the fusion energy production less efficient. In order to produce energy by fusion, particles must be confined in the hot, dense core of the plasma. Particle and energy transport are fast along the magnetic field lines, which typically do not point in the radial direction. In the radial direction transport is due to diffusion and turbulence, and therefore comparatively slow. When a magnetic island is present, some of the magnetic field lines are partially directed in the radial direction, thus transporting particles and heat from the center to the edge quicly. The magnetic field line configuration that causes this is shown in Figure 1.2. Figure 1.2: The effect of the magnetic islands on the confinement, seen in a poloidal cross-section of a toama. In a the toama runs without islands and the surfaces of constant magnetic flux consist of nested tori circles in this 2D cross-section. The pressure, density and temperature are high in the center and decrease towards the edge, the derivatives of these quantities are always negative except at the center. The graph to the right in a shows a setch of the profile of normalised pressure, mass density or temperature along the radial cut in the poloidal plane of a toama with minor radius a. In b two symmetrically positioned magnetic islands shaded green have formed. As particle and energy transport is fast along the magnetic field lines that border on the island, the pressure and temperature profiles are flattened across the island. The central temperature and pressure therefore are lowered. The larger the island grows, the larger the area where the pressure/density/temperature profile are flattened and the more severe the loss of central temperature/pressure.

9 1.1. MOTIVATION Motivation In past decade a great deal of effort and progress has been made in the physics and control of NTMs [10], [3], [18], [9], [4], [12]. This has resulted in efficient ways to suppress NTMs in theory and experiment, for example sawteeth control [19], frequently interrupted regime-ntms [13] and externally applied static helical field [27], radio frequency current drive rf CD to increase classical tearing stability [23], rf CD to replace the missing bootstrap current and lower hybrid current drive LHCD [21]. In ITER, electron cyclotron current drive ECCD is planned to be the main method of suppression of NTMs, LHCD might probably be an alternative [10]. In a recent wor Comisso and Lazzaro [5] develop a 2D numerical model in a slab geometry with periodic boundary conditions in both directions. This model is used to simulate tearing modes and their suppression localized current drive lie ECCD. They draw the wide-reaching conclusions about the effect of the current drive on length scales below the magnetic island width. They observe that: The main characteristics of the response are the appearance of undetectable current sheets and further nonlinear filamentation.. The suggest that different diagnostics and strategies have to be considered to suppress neoclassical tearing modes. However because of their choice of boundary conditions and the parameters they use, the applicability of Comisso and Lazzaro s results to realistic toama scenarios is in question. 1.2 This thesis In this thesis we use parameters more relevant for toamas. We design a model that is fully comparable to the generalised Rutherford equation GRE, the zero-dimensional theory that is the commonly used to describe NTM growth. In this GRE, the growth rate of the island with is proportional to a constant called the stability index. This stability index is determined by the solution of the perturbation of the magnetic flux function in a region far away from the island. We use these boundary conditions in our model in contrast to the periodic boundary conditions of Comisso and Lazzaro. Finally we use a more realistic value of the resistivity. The model results in a simulation tool that can be used to investigate where the assumptions leading to the GRE brea down. It is especially interesting to loo at the small-scale effects, as the GRE assumes that the perturbation can be approximated by a single Fourier harmonic. This could be for instance important when ECCD is applied, as its length scales could be small. Also it is possible to add currents that are not a function of the magnetic flux, but applied independently. We choose to investigate this with a two-dimensional code in order to limit the number of possible effects that could influence our results. In this way it is clear what the consequences of including small-scale effects are, and the results can be compared easily with theory. We start from theory and progressing to models that are numerically solved and can be extended to more realistic situations. We will start with the derivation of the GRE from the 2D reduced magnetohydrodynamics RMHD, which is an approximation to the full magnetohydrodynamics equations. To derive the GRE the relevant fields of RMHD describing the velocity and magnetic field are averaged over the volume of the island. To be able to calculate the necessary integrals, it is assumed that the mode can be approximated by only the lowest order Fourier harmonic in the periodic poloidal direction that is resonant with the magnetic field at the flux-surface where the island is growing. It is reasonable to use only one Fourier harmonic when one is only interested in how the island is growing, and where only spatial scales at the order of the island size play a role. But this assumption could remove interesting and relevant physical effects of localized processes inside the island. These processes are interesting, for example, if ECCD is applied to the interior of the island to replace the loss of current. Therefore, we include higher harmonics in our model, corresponding to smaller spatial scales. We derive a set of coupled nonlinear ordinary differential equations that we solve numerically. The main motivation for setting up this model is to create a simulation tool that implies the same boundary condition on the perturbation of the magnetic flux function as in the GRE. However, in this theory the term governing the velocity stream function drops out of the calculation during an averaging procedure over the magnetic flux-surfaces. For our numerical model we initially examine boundary conditions used in hydrodynamics, but these produce unphysical results. In this thesis we derive the boundary conditions for this problem from a linear analysis. We discuss how these boundary conditions agree with theory and how they produce results realistic to toama plasmas. In Chapter 2 of this thesis the theory of neoclassical tearing modes is treated, and the generalised Rutherford equation is derived. The argument leading to the boundary conditions is discussed in detail, because this thesis focuses on these parts of the theory. In Chapter 3 the 2D model used to study the NTMs is presented. The 2D RMHD equations are wored out and the right parameter range is

10 10 CHAPTER 1. INTRODUCTION determined. Chapter 4 contains a detailed discussion and analysis of boundary conditions. The results of a benchmar study are presented that shows that the simulations are behaving well and describe small tearing modes and NTMs. In Chapter 5 presents conclusions on the boundary conditions and model derived in this wor. Further improvements to our model are suggested. An outline of future wor hat will be performed with this simulation tool is presented and what further improvements can be made to the model.

11 Chapter 2 The Rutherford equation In this section we describe the theory that is the starting point of the numerical analysis. We start with the full magnetohydrodynamic MHD equations and mae some general assumption to simplify them to the reduced MHD model. We introduce the concept of the resistive layer which is the only place where the magnetic reconnection that is needed for magnetic islands to grow, can tae place. This again simplifies the equations in the different region of the toama, but also introduces the stability index that is used to fit the solutions in the different regions together. We investigate the linear solutions in the different regions and its meaning for the growth of a magnetic island. In the final stage we loo at a relatively large island, where a nonlinear treatment is needed. With this nonlinear approach we can finally derive an equation for the growth rate of a macroscopic magnetic island. This calculation was done first by P.H. Rutherford [18] and therefore the final equation caries his name. After deriving the Rutherford equation we consider the influences of neoclassical transport and additional currents. These lead to the generalized Rutherford equation GRE which describes the evolution of the neoclassical tearing modes NTMs. A second generalisation that can be made by truncating the Fourier series after a higher order; this more accurate generalisation is explored in the next chapter. 2.1 The reduced MHD approximation The equations describing the macroscopic behaviour of an electromagnetic fluid lie a plasma are nown as the MHD equations. They consist of the continuity equation and the Navier-Stoes equation from fluid physics and an equation of state coupled with Maxwell s laws and Ohm s law. They are for instance derived in Freidberg [7]. We are using the resistive Ohm s law, neglecting the Hall effect, electron inertia and electron diamagnetism. In this case the MHD equations can be written as, dρ + ρ v = 0 Continuity, 2.1 dt ρ dv = J B p Momentum or Navier-Stoes, 2.2 dt d p dt ρ γ = 0 Internal energy, 2.3 B t = E Faraday s law, 2.4 J = B µ 0 Ampère s law, 2.5 B = 0 Absence of magnetic monopoles, 2.6 E = v B + ηj Resistive Ohm s law, 2.7 In these equations ρ is the mass density, v the velocity of the ions, J the electric current, p the total electron and ion pressure, B the magnetic field, E the electric field, µ 0 the magnetic permeability through the vacuum, γ the ratio of specific heats, and η the resistivity. In Ampère s law 2.5, the time derivative of the electric field is neglected because it enters the equation with a 1/c 2 prefactor, which maes it very small for plasma phenomena where the velocity is non-relativistic v c, lie those considered in this 11

12 12 CHAPTER 2. THE RUTHERFORD EQUATION wor. In the fluid equations 2.1, 2.2 and 2.3, we have used the convective derivative, d dt t + v. 2.8 This derivative is also nown as a substantial derivative or Lagrangian derivative and describes the movement of an element of fluid in a flow. Finally, we tae the plasma flow as incompressible, i.e. v = 0. This limit is true when the typical plasma velocities which are part of the dynamics an thus only directing in the poloidal plane are much smaller then the velocity of compressional waves Alfvén velocity. It follows from the continuity equation 2.1 that the density is constant, and unity when we normalise the system to a non-dimensional one. Figure 2.1: The configuration of the cylindrical approximation to a toama grey that can be used to reduce the MHD equations. The torus left is cut open into a cylinder right with periodic boundary conditions in the z-direction. The main magnetic field is in the toroidal φ-direction red arrow, in the RMHD this is the z-direction and almost constant. The equilibrium magnetic field lies on concentric tori, that are also bend into cylinders green. The radial direction r is defined to be 0 on the axis of the cylinder, while θ is the angle in the poloidal plane. In the original MHD equations there are seven independent variables, namely three velocity components, two magnetic field components one is determined by B = 0, the pressure and the density. Solving for all these variables simultaneously can be complicated. Therefore there are many simplifications that are approximately equivalent to the MHD equations for a specific physical problem. For the configuration that is relevant for us we mae use of the reduced MHD approximation RMHD, see Figure 2.1. This approximation is generally used in astrophysical plasmas, but here we explain how it can be formulated for a toama. Apart from the incompressible fluid approximation, there are 3 approximations: approximate the toama by a cylinder, neglect all dynamics in the toroidal direction and assume the toroidal magnetic field to be constant. In the following we will wor out these approximations. For the first approximation, we use the large aspect-ratio approximation, i.e. the major radius of the torus is much larger than the minor radius, R/a 1. We therefore approximate the system as cylindrical, which is much simpler than a toroidal geometry. The axial or z-direction in the cylinder corresponds to the torodial direction in a toama. In the following we will still call this direction the toroidal direction and we will sometimes write it as z = Rφ. Second, in a toama the magnetic field in the toroidal direction is much stronger than in the poloidal direction. The magnetic energy contained in this field is much stronger than the thermal and inetic plasma energies, so ρv 2 p Bz/2µ 2 0. Therefore changes on small scales or equivalently large gradients can almost only occur in directions perpendicular to the main toroidal magnetic field. Changes along B z correspond to a bending of the magnetic field lines, which in this approximation are assumed to be smooth due to the strength of the magnetic field. Hence derivatives in the z-direction are assumed to be much smaller than those in the radial or θ-direction, z f f, where means the gradient perpendicular to the z-direction. We can conclude that in RMHD all relevant dynamics tae place in the poloidal plane. Third, the toroidal magnetic field B z is assumed to be approximately constant with respect to variations in space and time. Applying this approximation, we neglect B z and t B z. Because B = 0, we can write the magnetic field perpendicular to z in purely terms of the z component of the magnetic potential A, 1 A z B = B r, B θ, B z = A = r θ, A z r, B z = ê z A z + B z ê z = ê z ψ + B z ê z, 2.9

13 2.2. THE ROLE OF RESISTIVITY 13 where we have defined the poloidal flux function ψ = A z. This expression is divergence free if B z = 0. We have thus made 3 assumptions R/a 1, ρv 2 p B 2 z/2µ 0 and B z constant, leading to 3 approximations the cylindrical approximation, z f f for any function f, and neglecting the derivatives of B z respectively. With these approximations at hand, we can derive the reduced MHD equations as an less complicated theoretical framewor to study NTMs. We fill in Ohm s law 2.7 into Faraday s law 2.4, write B = A and integrate the curl out. When integrating, we have to insert the electric potential ϕ, whose gradient is the right integration constant. We get A t = v B ηj ϕ We loo at the perpendicular components first. Because B z = A ê z is considered to be independent of time, we can use the gauge freedom to set t A = 0. Besides this, from Ampère s law 2.4 we can see that j only contains derivatives of B z or z-derivatives of B r and B θ and it can therefore also be neglected because of the assumptions of neglecting dynamics in the toroidal direction and B z being constant. We are left with v B v B z ê z = ϕ To obtain this relation, we have used the fact that according to the momentum equation 2.2, t v z 0, so v z is small compared to v. With help of B z B we now that v z B v B z. Hence for the velocity perpendicular to the main magnetic field we get v = êz ϕ B z The expression in equation 2.12 agrees with the condition for incompressibility, namely v = v = 0. We insert the expression for the velocity of eq into the equation for ψ/ t minus the z-part of eq and get ψ t = v B + ηj z + ϕ z = B ϕ + ηj z B z We compute the current contribution with Ampère s law 2.5, j z = 2 ψ/µ 0. Using the definition of the derivative parallel to the magnetic field, = B/B, the equation for the flux function reduces to ψ t = ϕ + ηj z The parallel gradient of the electric potential can be written ϕ = ê z ψ ϕ/b z. Using the vector identity A B C = A C B we rewrite 2.14 in terms of the convective derivative see eq. 2.8 of the magnetic flux function ψ dψ dt = ηj z Equation 2.15 tells us that the magnetic flux function ψ can change with the flow of the plasma or decay with resistivity. The equation for the electric field φ is obtained from the z-component of the curl of the momentum equation 2.2. dv dt = d êz ϕ ê z = J z B ê z 2.16 dt B z t + v 2 ϕ =B j z In equation 2.17 we have rescaled the electric field by the constant toroidal magnetic field ϕ/b z ϕ. Note that we can interchange the curl and the convective derivative only because of the symmetry betweenv = ê z ϕ and B = ê z ψ. The equations 2.15 and 2.17 for the flux function ψ and the normalised electric potential or stream function ϕ are the reduced MHD equations, and the equations that we will solve numerically in this wor.

14 14 CHAPTER 2. THE RUTHERFORD EQUATION Figure 2.2: The process of magnetic reconnection. Shown are the magnetic field lines, the red ones originally pointing to the right, the blue ones to the left. The arrows point out the velocities or stream field, which pushes these fields together. In the diffusion region within the green ellipse magnetic field lines can reconnect, in our case due to the resistivity. The two magnetic field lines that are reconnected going from the left to the right are shown in darer shade then the others. Reproduced from [24] 2.2 The role of resistivity We now turn our attention to the conditions necessary for simulating tearing modes with reduced MHD equations. This nowledge is usefull because it gives information on which terms of the equations are important in different region of the toama plasma, and thus helps to solve the equations and understand the solutions. For magnetic islands to be bred, the topology of the magnetic flux-surfaces has to be changed by magnetic reconnection of field lines see Figure 2.2. This is a complicated process that is still not fully understood, more about this process can be read in [2, chapter 6] and [1, chapter 10]. However, the only thing we need to now is what the conditions are for the magnetic field to change. If we insert Ohm s law 2.7 and Ampère s law 2.5 into Faraday s law 2.4 we get an induction equation B t = v B η µ 0 B, = v B + η µ 0 2 B Hence there are two terms determining the evolution of the magnetic field, the first term in eq describes the advection of the magnetic field with the plasma flow and the second describes the role of resistivity. In the case of a constant velocity equilibrium situation the first term reduces to v B. In the limit of zero resistivity this combines to db/dt = 0 so the magnetic field moves with the flow. This is called a frozen in magnetic field, the magnetic field lines are frozen into the fluid flow. Such a field cannot change its topology to form island structures. Hence resistivity must be non-negligible when magnetic islands start to grow. The relative magnitude of the two terms at the RHS of eq can be deduced by noting that the advection term of eq is of order v A B/L while the resistivity term is of order ηb/l 2 µ 0. Here v A is the Alfvén speed B/ µ 0 ρ, a characteristic speed of the plasma, ρ is the density of the plasma, and L a characteristic length-scale of the plasma. In a toama in equilibrium the minor radius a is a reasonable choice for the characteristic length scale. The dimensionless number that results from the ratio of these two terms is called the magnetic Reynolds number or the Lundquist number is introduced, which is given by S = µ 0v A L η When the Lundquist number is much larger than unity, the coupling between the magnetic field and plasma flow is strong, and the frozen-in approximation is reasonable. When the Lundquist number is small the coupling is wea, and the magnetic field lines can reconnect easily. The advection term is much more important than the resistive term for most of the plasma in a realistic toama S 1. However small resistive layers exist where reconnection can occur. This does not mean that the resistivity is higher in these resistive layers. It is the Laplacian of the magnetic field 2 B, which is large in this narrow region. This maes the resistive term in eq non-negligible. Hence we can split up the plasma in a toama in two regions: the ideal =non-resitive outer region in which the resistivity is negligible and the resistive layer, where a magnetic island can form. In this way we can use the ideal MHD equations, which are easier to solve, in the outer regions, and a simplified geometry in the resistive

15 2.3. THE IDEAL OUTER REGION 15 layer. In the next paragraphs we will focus on the solutions of the reduced MHD equations in these two regions separately, and discuss the method of fitting these solutions together. Figure 2.3: The possibility of magnetic reconnection with and without resistivity. Without resistivity up there is no diffusive region where field lines can reconnect. The field lines are frozen in and will always preserve their topology. With resistivity down reconnection can occur, and in this situation a chain of islands can form. The reconnection occurs at the point where two islands meet called the X-point as the gradient of the magnetic field is very high near that point. 2.3 The ideal outer region On the large scale of the whole toama resistivity plays a less important role, but the toroidal geometry should be taen into account. The magnetic field is frozen in the plasma flow due to the high Lundquist number. The reduced MHD equation of ψ 2.15 is approximately dψ/dt = 0, so the plasma is characterized by constant magnetic flux-surfaces moving with the flow. As we are looing at a mode that is evolving at a relatively slow resistive timescale τ R = a 2 µ 0 /η, and all relevant timescales in the ideal outer region are much smaller then that, we can neglect the inertial terms. If there is an deviation from a equilibrium solution in the outer region, the fast plasma processes which move at the Alfvén timescale τ A τ r would immediately move the system to an equilibrium state while the relatively slow resistive tearing mode does not change. Therefore eq reduces to 0 = B j. The perturbation has to be resonant with the magnetic field lines at the rational surface, which close onto themselves after going round m times in the poloidal θ direction and n times in the toroidal φ direction. Therefore the flux psi must be 2π-periodic in the helical angle ξ = mθ nφ. We thus assume a cylindrical geometry with one resonant surface at r = r s and a perturbation of the form ψ = ψ r expmθ nφ, > 0 an integer. We insert ψ = ψ + psi, where the bar stands for an equilibrium value, into B j = 0. One can show [2, p75. ff] that up to first order in this perturbation the equation for the perturbed magnetic flux function outside the magnetic island is d 2 ψ dr r dψ dr m 2 + r d j/dr ψ = B θ r1 nqr/m In this equation, jr is the equilibrium current in the z-direction, and qr = B φ r/b θ r is the safety factor. Usually this equation must be solved numerically for a given current profile jr and a safety factor profile qr. One can however see without solving that the derivative of the solution has a discontinuity at r = r s where qr = m/n, and that the value of d j/dr is important in determining the size of this discontinuity. Far away from the rational surface this last term is be neglected because d j/dr becomes negligible. We are left with 1 d r dr r dψ 2 m dr = ψ, 2.21 r

16 16 CHAPTER 2. THE RUTHERFORD EQUATION which can be solved by ψ r = c 1 r m + c 2 r m Taing the limits of r to 0 and and requiring that the flux must vanish in both limits, we see that the only valid solution is { ψ r = c 1 r m for r < r s ψ r = c 2 r m 2.23 for r > r s. And by requiring continuity across the resistive layerthat we treat as if it is infinitesimal small at the resonant surface we now that the constants c 1 and c 2 have to satisfy c 1 r m s = c 2 r m s A typical overall plot of the perturbation of the magnetic flux function ψ is shown in Figure 2.4. Figure 2.4: The plot of a typical perturbed magnetic flux function ψ in cylindrical geometry. There is a jump of the derivative around the resistive layer which has to be matched by the solution of the perturbed flux function inside or around the resistive layer zoomed region with the blue border. Usually the largest part of the perturbed flux is concentrated in the region within the rational surface 0 < r < r s. For large r, dj/dr is small and the perturbation falls of lie r m. Note that inside the resistive layer we have subtracted a constant gradient from the flux function such that the gradient is zero at the rational surface. This can be done because the flux will be expressed in a helical plane defined in eq Coupling the outer region with the resistive layer Before we loo at how the magnetic flux behaves inside the resistive layer, we have to determine the coupling between the inner and outer solution. We want the perturbed flux function ψ and its first derivatives to be continuous. For ψ itself this is done in eq. 2.24, but as indicated in Figure 2.4, there is a discontinuity in the first derivative that should be matched by the change in this derivative across the resistive layer. Again, from outside the resistive layer, the layer can be treated as an infinitesimally small layer around the resonant surface. On the other hand, inside the resistive layer the boundaries are asymptotically at infinity. We examine the difference in logarithmic derivatives of the outer solution = lim ɛ 0 d log ψr dr rs+ɛ d log ψr dr rs ɛ ψ r s + ɛ ψ r s ɛ = lim ɛ 0 ψr s Here ɛ is a small number that is used to tae the limit of the radial coordinate towards r s from both sides of the resistive layer. The quantity is nown as the stability index and it determines whether the island will be growing or not. When we fill in the appropriate forms in eqs and 2.24, into

17 2.5. THE RESISTIVE LAYER 17 the definition of, we get a restriction for each of the possible components of the perturbation of the magnetic flux function ψ. We reach at [ mc1 r s + ɛ m 1 mc 1 r s ɛ m 1 ] = lim ɛ 0 ψr s = lim ɛ 0 [ m r s 1 + ɛ r s m 1 m r s 1 ɛrs m 1 ] = 2m r s The solution of ψ within the layer has to match the change in the fraction ψ /ψ over its whole domain. We thus at the same fraction, but tae the limit of the radial coordinate to infinity. This limit has to match the same value of the stability index, that therefore functions as a boundary condition for the solution in the resistive layer, ψ ɛ ψ ɛ = lim ɛ ψr s One can wonder why only the change in ψ /ψ is matched with the solution in the resistive layer and not ψ /ψ or ψ itself. It will turn out that the growth of the perturbation in the resistive layer only depends on this change in the fraction ψ /ψ. The stability index is a global property of the plasma and depends only on the boundary conditions and the equilibrium plasma values. In the model that we develop in this thesis the boundary condition for the magnetic flux function is based on this argument of asymptotic fitting. It can be calculated see [8] that the change of the magnetic energy in the presence of an island is δw = r ψ2 s /4. We can therefore loo at as a measure of the energy that is available to let the magnetic island grow. 2.5 The resistive layer Figure 2.5: Unfolding the layer around the magnetic island to a slab geometry with Cartesian coordinates. To study the 2D reduced MHD equations within the more complicated resistive layer, it is usefull to move to a coordinate system where the equilibrium magnetic field is exactly perpendicular to the described plane at the resonant surface. This will simplify the computations. Because we are only looing at a thin layer around the rational surface, we use a slab geometry, gain by unfolding the cylindrical layer, see Figure 2.5, with ordinary Cartesian coordinates. As the radial component of the magnetic field is always zero in equilibrium, we choose x = r r s as our first coordinate. The second coordinate can be derived from the toroidal and poloidal angle. The magnetic field lines at the resonant surface close onto themselves after n toroidal and m poloidal windings, so the field lines are pointing in the direction of nr s ê θ + mrê φ, where R is the maior radius of the toama. Thus the second coordinate has to be in the direction mrê θ nr s ê φ. We define the helical coordinate ξ ξ = ê θ nr s mr êφ The resulting plane with a radial and helical direction is shown in Figure 2.6. The equilibrium flux function in this plane is defined in the same way as ψ h = ψ n χ/m where χ is the toroidal flux function

18 18 CHAPTER 2. THE RUTHERFORD EQUATION Figure 2.6: The plane in which the Rutherford equation will be formulated, spanned by the radial and helical direction green is shown onto the rational surface grey with a magnetic field line red on it. and the bar indicates an equilibrium value. Because in the radial and helical direction the magnetic field is zero at the resonant surface x = 0, and we are free to add a constant of ψ due to the gauge freedom, we approximate this helical equilibrium magnetic flux function in this helical plane near the rational surface with a second order Taylor expansion ψ h = ψ n m χ x=0 + x x =0 + x 0 + x2 ψ 2 x x = x2 2 ψ 2 x 2 2 ψ q x=0 x 2 ψ 1 q x=0 x q s x x=0 q s ψ n m χ x=0 + x2 2 2 x 2 ψ n m χ x=0 + Ox 3 1 q 2.29 q s x=0 = x2 2 q s q s ψ x. x=0 Here we have used the fact that the safety factor q = χ/ ψ and that at the rational surface the safety factor is m/n. We write the equilibrium magnetic flux in this way to calculate a realistic range more easily. For the equilibrium stream function ϕ this approximation means that we can set ϕ h = 0, because the right hand side of eq is zero due to the third derivative of ψ. We define ψ = q s ψ q s x x=0 and write the total helical magnetic flux so including a small perturbation from the equilibrium as x2 ψ h x, ξ = ψ 2 + ψx, ξ 2.30 x2 = ψ 2 + ψ xe iξ, 2.31 we can write the perturbation as a Fourier series as the helical plane crosses the same magnetic field line when going from ξ to ξ + 2π and therefore the perturbation as all other physical quantities is periodic in ξ over 2π. We simplify this expression using the assumption that the dominant contribution is due =0

19 2.5. THE RESISTIVE LAYER 19 to the lowest resonant Fourier harmonic. This is reasonable because it can be shown that the lower the stability index, the more the perturbation is damped. This is especially true for higher, because of the condition derived in eq However, this assumption is also the main assumption we wish to evaluate in this thesis, as it is possible to introduce small scale structures such as ECCD-currents. The total helical magnetic flux function can thus be written ψ h x, ξ = ψ x2 2 + ψ 1x cosξ We use this expression to show the 2D topology of a magnetic island see Figure 2.7. In most toamas, ψ is negative. The sign of the perturbation is not important, a substitution ψ 1 ψ 1 leads to an island configuration that is identical except that the O-point and X-point are switched. We tae the perturbation positive to have the O-point at ξ = 0. As the X-point occurs then always at ξ = ±π, the values of the helical flux function at the O and X-point is ψ h 0, 0 = ψ 1 and ψ 1 respectively. The separatrix is situated at x S = 2 ψ 1 / ψ, because ψ x 2 S /2 = 2ψ 1 The island width is thus w = 4 ψ 1 / ψ. Figure 2.7: The contours of the total helical magnetic flux function ψ h from eq for 2 islands π ξ 3π. In this figure the point with the maximal flux is called the O-point. The boundary between the field lines that have been reconnected and the field lines that still have the same topology as the equilibrium field lines is called the separatrix. The point where the edges of the separatrix cross is called the X-point. The width of the separatrix at the O-point is maximal and used as the width of the magnetic island. For higher mode numbers m and n, more islands exist next to each other, forming a chain of islands. Because of the definition of the helical angle, only one island is situated between π ξ π, regardless of m and n. To simplify eq. 2.32, we assume that the dominant first Fourier harmonic ψ 1 x is almost constant over x. This is true for islands that are much smaller then the minor radius of a toama. This implies ψ 1 / x 0, but we do not restrict 2 ψ 1 / x 2. We insert this helical magnetic flux function ψ h and the perturbation of the stream function ϕ 1 into the 2D reduced MHD equations 2.15 and For small islands and therefore perturbation functions that are small with respect to the equilibrium only at the linear terms are significant, and so the evolutions of the magnetic flux and stream function reduce to ψ 1 t i x r ψ ϕ 1 = η 2 ψ 1, s µ t 2 ϕ 1 = i x r ψ 2 ψ 1. s 2.34 These equations can be solved straightforward see [22, section 6.8]. We construct the solution in detail in section in Chapter 3 of this thesis. The solutions for ψ 1 and ϕ 1 are two exponentially growing

20 20 CHAPTER 2. THE RUTHERFORD EQUATION functions. The growth factor γ is the same in both exponentials and is given by [ 3/5 η mbθ q ] 2/5 γ = 0.55 µ 0 µ0 ρ rq 1 4/5 r=r s m = 0.55η 3/5 ψ 2/5 r 1 4/5, 2.35 s where we have written γ in normalised units ρ = µ 0 = 1 and in terms of the equilibrium flux function in the second line. From this linear analysis we also can calculate a characteristic length δ, called the resistive layer width, which emerges as a normalisation factor of the radial coordinate x to dimensionless units. The resistive layer width is defined ρ 1/5 2/5 2/5 δ = ηrq µ 0 B θ mq = /5 ηrs r=r s m ψ In realistic fusion plasmas the resistive layer width is smaller then the ion gyroradius, less then 1 mm. Theoretically the solutions of the fields inside the resistive layer cannot be used explicitly, when δ is so small. However, as long as the island width stays smaller then this resistive layer width, the linear approximation remains valid and we can treat the solutions as exponentially growing. 2.6 Nonlinear behaviour The resistive layer width is below the minimal observation threshold in most toamas. As the linear analysis is meritorious for islands smaller then the restive layer width, this cannot be the relevant description for macro-scale size islands, which can be up to 20% of the minor toama radius. For instance in JET [9] islands of 15 cm 15% of the toama minor radius are observed. In analysing larger island we need to include the nonlinear terms in 2D reduced MHD equations 2.15 and To tacle these more complicated equations we have to switch from the radial coordinate to the flux label Ω. The flux label is defined as ψ h Ω = sign ψ = 8 x2 cosmξ 2.37 ψ 1 w2 where w is the island with. We see that the flux label is defined so that Ω = 1 at the O-point and Ω = 1 at the X-point. Together with the sign of x this label can substitute the radial coordinate. With the coordinates Ω, ξ it is possible to tae the flux-surface average, an integral over each flux-surface, which will simplify eq to such an amount that further analysis is possible. For a general function f it can be written as f. We specify different integrals for inside and outside of the separatrix or Ω 1 and Ω > 1 respectively. This naturally separates topologically different flux-surfaces, because the flux-surfaces inside the separatrix are closed, ellipse-lie surfaces while outside there are of two separate mirrored branches at x > 0 and x < 0 enclosing a whole surface around the core of the toama. The integral outside the separatrix Ω > 1 is defined as f out = 1 N dξ fσ, Ω, ξ, π Ω + cosmξ where N is a normalisation constant defined by the same integral but with 1 inserted instead of f, and σ = signx. The term under the square root comes from differentiating x = w Ω + cosmξ/2 2. Inside the separatrix we have to care about the two branches on both sides of the resonant surface that form one flux-surface. The integral is defined f in = 1 N ξ0 ξ 0 dξ 2π f1, Ω, ξ + f 1, Ω, ξ 2, 2.39 Ω + cosmξ where ξ 0 = arccos Ω/m, such that xσ, Ω, ξ 0 = w Ω + cosmξ 0 /2 2 = 0. An important property of the flux-surface averaging operator is that it annihilates the parallel gradient of any scalar function f, so f = 0. This can be proven by Stoes Theorem. The poloidal and radial magnetic field is the

21 2.7. DERIVATION OF THE RUTHERFORD EQUATION 21 tangent vector of the flux-surface, so the integral can be written as a line integral along the contour of the surface Σ enclosed by the flux-surface. We get f = B f = 1 f dl N Σ = 1 f = 0, 2.40 N because the curl of a gradient is always zero. 2.7 Derivation of the Rutherford equation Σ The stability index as the boundary condition and the flux-surface averaging operator are the conceptual elements we need to derive the Rutherford equation. The island with has become much larger then the resistive layer width, so we describe the island within in a region around the resistive layer, but still small compared to the whole toama. We can approximate eq by B j z = 0, with the same argument leading to eq Therefore j z ψ h is a function of the total flux. From Ampère s law 2.5 we can the compute 2 ψx, ξ = µ 0 j z ψ h x, ξ. Because the distances in the radial direction are much smaller then in the poloidal direction we can approximate 2 ψ with 2 ψ/ x2. To extract the m-th Fourier component, we multiply by cosmξ and integrate over ξ on both sides. Besides that, the solution of ψ must satisfy the boundary condition 2.27, and therefore we integrate over x to get a first derivative out of 2 ψ/ x2. These steps yield the integral equation dx 2 ψ dξ x 2 cosmξ = Using ψ = ψ 1 cosmξ the LHS becomes dx dξ 2 ψ 1 x 2 cos2 mξ =π =π dx dξµ 0 j z cosmξ [ ψ1 x dx 2 ψ 1 x ] = π 1ψ In eq we assume that ψ 1 is constant over x. To couple the RHS of eq with the time derivative of the perturbed flux function, we use the flux-averaged version of the first 2D reduced MHD equation 2.14, ψ = ϕ + η j z = η j z = η j z t Here the parallel derivative vanishes because of the property of the flux-surface average derived in eq. 2.40, and the flux-surface average of η j z is the same as the original η j z, because j z is a flux function. This simplifies our equation so that we can substitute it into eq Together with eq this gives π 1ψ 1 = µ 0 η = µ 0 η =µ 0 ψ 1 t dx dω 1 = µ 0 η g wπ g 1 = 1 dω 2 1 dξ ψ cosmξ 2.45 t ψ dξ dx cosmξ 2.46 t dω dω w 4 2 cosmξ cosmξ dξ 2.47 Ω + cosmξ ψ 1 t In eq we have introduced g 1 as a numerical factor depending on the geometry of the island. This is defined 2 ξ cosmξ 2π Ω+cosmξ ξ 2π Ω+cosmξ

22 22 CHAPTER 2. THE RUTHERFORD EQUATION From the definition of the island width, we can write 1 dψ 1 = 2 dw ψ 1 dt w dt We finally write eq in the typical form of the Rutherford equation, r s 1 = g 1 µ 0 rs 2 dw r s η dt = g τ r dw 1 r s dt Eq is expressed in a dimensionless form, using the resistive timescale τ r = µ 0 r 2 s /η. In the Rutherford regime 1 is a constant and the island width grows or shrins linearly. The exponential growth in the linear phase is followed by an algebraic growth phase which has a constant absolute growth rate instead of a constant relative rate. For large islands, usually decreases linearly with the island width, until w = 0 and the island has reached saturation. 2.8 The generalised Rutherford equation Different non-inductive currents can influence the growth of a magnetic island. In the derivation of the Rutherford equation, these extra currents could be added into the Faraday-Ohm s law 2.15 and then taen into account all along the way. We include the perturbed bootstrap current and ECCD-current, the flux averaged version of eq becomes ψ t The Rutherford equation 2.51 is modified too and becomes = η j z ψ + j bs ψ + j ECCD ψ g 1 τ r r s dw dt = r s 1 + r s j bs + j ECCD The stability index due to an arbitrary non-inductive current j is given by j = 16µ 0R πw 2 ψ dx dξ j cosmξ 2.54 Eq is a generalised formulation of the Rutherford equation GRE. An important example of a non-inductive current for NTMs is the bootstrap current. This current rises from the diffusion of particles that are trapped in banana-orbits on the low field outer side of the toama. Because these particles collide with particles that are not trapped, these passing particles carry a net current see e.g. [7, subsection ]. In [7, eq ] the bootstrap current is given as R T n j bs q r B φ r In eq T is the electron temperature and n the electron density. Inside the magnetic island however, the density is flattened and the bootstrap current is negligible. Therefore it must be subtracted from the inductive current. One can calculate [25] that β θ L q w 2 j bs = g 2 ɛ w L p w 2 + wχ In this expression, g 2 = 32/3π is a geometrical factor, ɛ = a/r is the aspect ratio, L q = q/ q/ r is the q scale length, L p = p/ p/ r is the pressure scale length, and w χ is the critical island width determined by the ratio of the transport coefficients of the parallel an perpendicular transport χ and χ. For large enough islands, the bootstrap current has therefore a considerably stabilising effect on a magnetic island, which in this case is called a neoclassical tearing mode NTM due to the importance of the neoclassical transport effect of the bootstrap current. Another non-inductive current that can be generated is the Electron Cyclotron Current Drive. Simplified, this current can compensate for the bootstrap current gap and thereby maing the mode stable.

23 Chapter 3 The 2D reduced MHD model We create a 2D model in a narrow layer around the rational surface where a magnetic island can grow. This layer lies within a larger equilibrium plasma that drives the growth of an NTM via the boundary conditions. We use a slab geometry that is derived from toroidal geometry by unfolding the layer around the island into a rectangular shape, as shown in Figure 3.1. This slab is described by two coordinates. First by the radial coordinate x = r r s, x [ L, L] with w sat L r s, where r s is the radius of the rational surface and w sat is the width of the saturated island. Second by the helical angle ξ [ π, π], defined by ξ = θ nφ/m, which would be an angle in a curved geometry lie a toama, but in our slab this is a linear quantity. The length scale corresponding to the helical angle ξ is y = r s ξ/m so that y has a range of 2πr s /m which is the poloidal length between two O-points of a tearing mode with mode number m. The helical angle is defined in such a way that the direction of the magnetic field at x = r s is perpendicular to both x and ξ see 2.28, this direction is defined as the z-direction. Figure 3.1: In a the 3D slab geometry around two magnetic islands is shown. The 2D model represents dynamics in a plane perpendicular to the magnetic axis of the island, such a planar cut is shown in red. The slab geometry is obtained by unfolding the circular geometry of a toama. Surfaces of constant magnetic flux often called simply flux surfaces are shown in shaded blue. The separatrix, the surface between the inner area where the magnetic flux-surfaces are really disrupted and the outer area where the flux-surfaces are only deformed, is also plotted. In b the contour plot of the magnetic flux function ψ in a 2D slab geometry is shown. The relevant 2D reduced MHD equations are gain from eqs and 2.17 and given by, t + v ψ =ηj δj BS δj CD E, 3.1 t + v 2 ϕ = B j. 3.2 Here ψ is the helical magnetic flux function, which together with the constant toroidal magnetic field B z can be used to reconstruct the full 3D magnetic field B = ê z ψ + B z ê z. In these equations 23

24 24 CHAPTER 3. THE 2D REDUCED MHD MODEL the velocity is defined by v = ê z ϕ, η is the resistivity in terms of the vacuum permeability µ 0, and ϕ is the normalised electric potential or stream function. The current j = j z can be written as j = 2 ψ p = 2 ψ + n 2 χ/m, where ψ p is the poloidal magnetic flux function and χ the toroidal magnetic flux function. The constant electric field E in eq. 3.1 is introduced to compensate for the resistive decay of the current j in equilibrium. In eq. 3.1 we inserted a perturbed bootstrap term δj BS and ECCD current term δj CD. The effects of these terms are described in section 2.8. They are included for theoretical completeness, and will not be treated numerically in this wor. The implementation of these terms are discussed in the outloo Perturbing the system The relevant plasma dynamics are described by two fields, the magnetic flux function ψ and the normalised electric potential ϕ. ϕ is also called a stream function because the plasma flows along contours of constant ϕ. We expand these two fields into an equilibrium part, denoted by a bar, and a perturbed part, denoted by a tilde: ψ = ψ + ψ and ϕ = ϕ + ϕ. The perturbed part depends only on x and ξ. To model the equilibrium helical flux function ψ = ψ x 2 /2, we use the leading, i.e. second order, term from a Taylor expansion. This approximation yields a simple form for the equilibrium magnetic field B = B ê z ψ = B z ê z + x ψ ê ξ. 3.3 The equilibrium current is j = ψ + n 2 χ/m. There is no perturbation in the toroidal magnetic flux function χ, so the perturbation of the current is simply j = 2 ψ. The equilibrium electric field is defined as Ē = η ψ + n 2 χ/m. There is no equilibrium electric field aside from Ē, which we introduce to eep the current constant in equilibrium. Thus the electrical potential is constrained so that the equilibrium field ϕ is zero, and ϕ = ϕ. The equilibrium velocity v is zero, because ϕ = 0. For these choices, eqs. 3.1 and 3.2 are satisfied in equilibrium. On the LHS both the time derivative and gradient along the velocity vanish. On the RHS the current and electric field compensate each other in eq. 3.1 and the gradient of j is zero. Throughout the literature it is conventional to denote the equilibrium values by the subscript zero. Because we want to use the subscript for the number of the Fourier harmonic, we adopt the bar notation for equilibrium values in this wor in order to eep the notation clear For numerical convenience we express eqs. 3.1 and 3.2 purely in terms of the magnetic flux function ψ and electric potential ϕ. Inserting the perturbed flux function and electric potential into eq. 3.1 and in eq. 3.2, we produce evolution euqations for the perturbed quantities ψ t + ê z ϕ ψ + ψ = η 2 ψ, 3.4 t 2 ϕ + ê z ϕ 2 ϕ = B j + B j + j = B z ê z + x ψ ê ξ 2 ψ + êz ψ ψ + 2 ψ. 3.5 Eqs. 3.4 and 3.5 can be written explicitly in terms of the radial coordinate x and helical angle ξ using = x, y, z = x, m/r s ξ, z and simplify to t ψ = m ϕ r s ξ t 2 ϕ = m ϕ r s x ψ + ψ m ϕ ψ x r s x ξ + η 2 ψ, 3.6 ξ x ϕ 2 ϕ + mrs x x ξ ψ ξ + ψ x ξ ψ 2 ψ. 3.7 ξ x Here we have also used the fact that = x, y, z = x, m/r s ξ, z, ψ = 0, ψ = x ψ ê x. The perturbed magnetic flux function, ψ, is independent of z, so 2 ψ êz ; the same holds for the perturbed electric potential ϕ, so ϕ ê z. We can mae the spatial coordinates dimensionless by replacing x by Lx, where L is the length from the resistive layer to the edge of the simulation volume

25 3.2. RESCALING TO DIMENSIONLESS UNITS 25 and the new x runs from 1 to 1. The equations become t ψ = m ϕ L 2 x r s L ξ ψ + ψ m x r s L t 2 ϕ = m ϕ r s L ξ x ϕ x ξ + m L 2 x r s L ψ ξ + ψ x ξ ψ ξ ϕ ψ x ξ + η 1 L 2 2 ϕ x 2 + m2 x 3.2 Rescaling to dimensionless units r 2 s 2 ϕ ξ ψ L 2 x 2 + m2 rs ψ L 2 x 2 + m2 rs 2 2 ψ ξ 2, ψ ξ Using dimensionless units simplifies the computational results, and maes it easier to compare the size of different terms. We scale the time t n using the resistive timescale τ R = rs 2 /η. Similarly dimensionless forms of the magnetic flux function ψ n and stream function ϕ n are produced using L 2 /τ A, where τ A = 1/ ψ is a modified form of the Alfvén time-scale. We introduce the ratio of the two length scales f = ml/r s and the ratio of time scales g = τr τ A. In dimensionless form eqs. 3.8 and 3.9 are t ψ = xfg ϕ ξ + fg ϕ t 2 ϕ = fg ξ x ϕ + fg x ξ + ψ x ψ ϕ x 2 ϕ x ξ ξ ϕ ψ x ξ x 2 + f 2 2 ϕ ξ 2 x ξ ψ ξ + m2 f 2 2 ψ 2 2 ψ + f x2 ξ 2 2 ψ 2 2 ψ + f x2 ξ 2, Because in toamas f and g tae values of about 0.1 and 10 8 respectively see section 3.9 that the time evolution of helical magnetic flux ψ is dominated by the linear term xfg ξ ϕ, while the evolution of the stream function ϕ is determined largely by fgx ξ 2 ψ. 3.3 The Fourier expansion Because the model is periodic in the ξ-direction, the perturbed terms can be expanded in a Fourier series in the direction of the helical angle ξ. They can be written as ψx, ξ, t = ϕx, ξ, t = = = ψ x, te iξ, 3.12 ϕ x, te iξ The periodicity in the ξ-direction shows that it is more efficient to wor in Fourier space in this direction. A standard way to compute the nonlinear terms in equations 3.10 and 3.11 would be to use a Fast Fourier Transform FFT to real space, multiply the fields, and use a FFT bac into Fourier space. This method is called pseudo spectral [6], [16]. However, Yu et al. [26] show that a small number of harmonics in their case three is sufficient to resolve tearing modes. We therefore compute the nonlinear terms via ordinary convolution products in Fourier space. This is more efficient for a small number of harmonics. We can insert the Fourier series 3.12 and 3.13 into eqs and 3.11 to arrive at equations for the Fourier coefficients ψ x, t and ϕ x, t, t ψ = δ,1 i 1 fgxϕ 1 + m2 f 2 ψ 1 f 2 1ψ fg δ,1+ 2 i1 ϕ 1 ψ 2 i 2 ψ 2 ϕ 1, 1 1, 2 = ifgx ψ ϕ + m2 f 2 ψ f 2 2 ψ + ifg 1 1 ϕ 1 ψ 1 1 ψ 1 ϕ 1, 3.14

26 26 CHAPTER 3. THE 2D REDUCED MHD MODEL and t ϕ 2 ϕ = 1 δ 1 xg if 1 ψ 1 if 1 3 ψ 1 + g 1, 2 δ 1+2 [ if1 ϕ 1 ϕ 2 f 2 2 ϕ 2 ϕ 1 if 2 ϕ 2 if 2 3 ϕ 2 ] + [ ψ 1 if 2 ψ 2 if 2 3 ψ 2 if 1 ψ 1 ψ 2 f 2 2 ψ 2 ] = ifgxψ f 2 ψ + ifg [ 1 ϕ 1 ϕ 1 f 1 2 ϕ 1 1 ϕ 1 ϕ 1 f 1 2 ϕ 1 ] 1 + [ f 1 ψ 1 ψ 1 f 1 2 ψ 1 1 ψ 1 ψ 1 f 1 2 ψ 1 ] In these equations δ l is the Kronecer-delta, that appears when we equate Fourier components. We have suppressed the explicit dependency of the Fourier components on x and t, and written / t as t, and f/ x as f Symmetry of magnetic islands in the helical direction Magnetic islands that are symmetric in ξ are mathematically simple and a good starting point to investigate NTMs. We can show that symmetry of ψ around the O-point and anti-symmetry of ϕ is preserved during time by our model. When ψ is symmetric in ξ, t ψ t ψ vanishes. Using this fact in eq gives 0 = ifgxϕ + ϕ + ifg 1 [ 1 ϕ 1 ψ 1 ψ 1 1 ψ 1 ϕ ψ 1 ϕ 1 ], 0 = ixϕ + ϕ + i 1 [ 1 ψ 1 ϕ 1 + ϕ 1 1 ψ 1 ϕ 1 + ϕ 1 ], 3.16 where we have used that ψ and ψ are symmetric in, and a transformation 1 1 is performed. If the RHS is zero for all, x and t, then ϕ must be antisymmetric in. As long as ϕ remains antisymmetric, ψ also remains symmetric, because higher-order time derivatives of ψ always have to be symmetric in. Time derivatives of 2 ϕ + ϕ reveal that as long as ψ is symmetric, an originally purely antisymmetric ϕ always remains antisymmetric. The initial conditions we chose will have these properties and therefore all our magnetic flux functions are symmetric in ξ Sine and cosine series Because both ψ and ϕ are real, we can write them as also a cosine and sine series, ψx, ξ, t = ϕx, ξ, t = ψ x, t cosξ, =0 ϕ x, t sinξ =0 The equations for these series are wored out in appendix A. These series tae advantage of the real numbers, because there only about half as many terms involved. On the downside, the time evolution equations 3.14 and 3.15 get more complicated and therefore we do not choose to adopt these series and stic with the complex Fourier series. 3.4 The vorticity and the solution for the stream field Simply rewriting the time evolution equation 3.2 for ϕ has resulted in a time evolution equation for 2 ϕ, instead of ϕ itself. To be able to calculate the value for ϕ after each time step, we introduce the plasma vorticity ω 2 ϕ. Computing the differential equation for the cosine terms of ω and ϕ produces 2 ϕ x 2 f2 ϕ = L 2 ω, 3.18

27 3.5. LINEAR STABILITY ANALYSIS 27 which is a 1D screened Poisson equation. The homogeneous part can be simply solved by ϕ,hom x = a e fx + b e fx Boundary conditions are used to determine the constant coefficients a and b in section 3.7. A solution for the inhomogeneous equation is ϕ,part x = x 0 sinh[fx x 1 ] dx 1 ω x f Adding these two solutions gives the solution of the 1D screened Poisson equation. 3.5 Linear stability analysis We apply a linear stability analysis to our equations 3.14 and 3.15 to calculate the linear solutions. Linearising the equations 3.14 and 3.15 gives t ψ = ifgxϕ + m2 f 2 ψ f 2 ψ, 3.21 t 2 ϕ = ifgxψ f 2 ψ Substituting t by the growth factor γ, which we want to determine, transforming ϕ iϕ so that ϕ becomes real and substituting 2 ψ from eq into 3.24, we find γψ =fgxϕ + m2 f 2 2 ψ, 3.23 γ 2 ϕ x 2 = fgx 2 ψ = f 4 2 g 2 m 2 x 2 ϕ f 3 gγ m 2 xψ, 3.24 We have substituted 3.23 into 3.24 eliminate the second derivative of ψ. We simplify equation /4 further by defining δ = f 4 g 2 2 and substituting m 2 γ δ x z and 3.25 gm/γ 1/4 ϕ /ψ y We arrive at = 2 y z 2 =z2 y z = z1 zyz We can derive the growth rate γ by relating the stability index to an integral of ψ 2 ψ x 2 = γd L η = γd L η ψ dx = γl2 η 1 f ψ 2 d4 ηγl 2 f ψ 1 x γ ηl f ψ d 3 ϕ dx ψ ϕ Lx ψ d d Lx 1 zyz dz We used transformations 3.25 and 3.26 to simplify the integral. We numerically compute the solution to 3.27 and produce γ = /5 η 3/5 f ψ 2/5 L 6/ This is the same growth rate as in [22] eq and [2] eq d

28 28 CHAPTER 3. THE 2D REDUCED MHD MODEL 3.6 Viscosity The flow field ϕ grows without any dissipation unless viscosity is introduced to the system, this is necessary at least for numerical purposes. The equation for ϕ 3.2 with viscosity ν is t + v 2 ϕ = B j + ν 4 ϕ Including viscosity in eq gives t ω =ifgxψ f 2 ψ + τ R τ ν ϕ 2f 2 ϕ f 4 ϕ + ifg 1 [ 1 ϕ 1 ϕ 1 f 1 2 ϕ 1 1 ϕ 1 ϕ 1 f 1 2 ϕ 1 ] + [ 1 ψ 1 ψ 1 f 1 2 ψ 1 1 ψ 1 ψ 1 f 1 2 ψ 1 ], 3.31 where τ v = L 2 /ν is the timescale of viscous dissipation. Realistic systems exhibit viscous dissipation, but in the theory of tearing modes the viscosity is not taen into account. We assume therefore that the dissipation is mainly due to the resistivity, so ν η. 3.7 Boundary conditions The differential equations 3.6 and 3.7 for the magnetic flux ψ and electric potential ϕ are first order in time and third order in both our spacial coordinates x and ξ. We require solutions to seven boundary conditions for each function, fourteen in total. The temporal boundary conditions are given by the values of ψ and ϕ at t = 0. We do not introduce thermal fluctuations in our model, hence we always have to start with a very small magnetic island that can grow by using the free energy available in the plasma. The radial boundary conditions for ψ are determined by the coupling of the solutions around the resistive layer to the outer region. For higher harmonics we derived the value of in eq In our model this equation becomes ψ 1, t ψ 1, t = 2mL = 2f for ψ 0, t r s The variable is the tearing mode stability index. The boundary conditions on the first resonant harmonic determines the growth of the island. Because we want to study an island that starts small and grows due to the free energy available in the plasma, we constrain the first resonant harmonic of the tearing mode stability parameter to be positive, 1 = 2C, where C is a positive constant and a factor of 2 is introduced for notational convenience. For = 0 the boundary conditions are the same as for 2, so 0 = 0. Boundary conditions derived from the stability index require complex formulation and implementation. The matching condition of eq depends on the values of ψ or ψ at three different places. Since we are also only interested in the simple case of islands symmetric in the radial direction, we assume ψ 1, t = ψ 1, t for all. If we then also divide by ψ ±1, t instead of ψ 0, t the boundary conditions simplify to ψ ±1, t ψ ±1, t = f for 2, ψ 1±1, t ψ 1 ±1, t = C > 0, ψ 0±1, t = The periodic boundary conditions in the ξ direction determine that the values of ψ and ϕ and all their derivatives have to be the same for ξ = +π and ξ = π. Since the equations are of third order in ξ, this gives three spatial boundary conditions on both ψ and ϕ. Those boundary conditions are automatically implemented by only allowing Fourier harmonics. We have to loo carefully at the vorticity again. Because we need the value of the vorticity at the boundaries to compute the stream function, we need to now how the vorticity behaves near x = ±1. However, the value of ω at the boundary is independent of the choice of a and b that determine the stream functions at that point. We therefore have to specify a boundary condition for the vorticity too, which we can do by looing at the linear theory again. From eq and neglecting the term of order f 3 we see that the vorticity is linear in the radial coordinate x. We thus can extrapolate the vorticity in

29 3.8. INITIAL STATE 29 a linear way from the points inside the grid were we are able to calculate the change of vorticity directly. We thus imply the boundary condition ω x = D, 3.34 x=±1 where the constant D is determined by the radial derivative inside the grid, D = lim x 1 ω / x. The argumentation for the boundary condition of the stream function is discussed in 4.4, we show only the solution here. The constants of the Poisson integral are set such that ϕ x = a sinhfx + x 0 sinh[fx x 1 ] dx 1 ω x f The constant a is defined as 1 0 dx sinh fx x1 f cosh fx x 1 ω x 1 a = f coshf 2 1 sinhf Summarizing, we have found one temporal, two radial and three angular boundary conditions for both ψ and ϕ. We are thus left with one degree of freedom for each function. This is exactly how it should be, as only ψ and ϕ are physical quantities, hence ψ and ϕ have a non-physical degree of freedom which is not determined by physical boundary conditions, given by an arbitrary constant. This constant is determined by choosing ψ and ϕ at t = 0 and does not change during time Central boundary condition for ψ An alternative to the approach above is to divide the two derivatives at the edges by the central value of ψ, so eq will change to ψ ±1, t = f for 2, ψ 0, t ψ 1±1, t ψ 1 0, t = ± C 2, C > 0, ψ 0±1, t = Initial state Because thermal fluctuations are not included in our model, if ψ and ϕ are initially zero, they will remain zero. We instead set all fields to zero, except for the first Fourier harmonic of the magnetic flux function ψ 1. This ψ 1 must initially be symmetric around zero. The choice of a simple exponential, e.g. ψ 1 x = ψ 1,x=0 exp 1 x, which has the advantage that not only ψ 1±x/ψ 1 ±x = ± 1/2 for x = 1 but for all x 0, 1]. However, this also means that there is a discontinuity in the first derivative at x = 0. This is not physically realistic, so an exponential cannot be used Instead we adopt a series form ψ 1 x = ψ 1,x=1 [1 + n i=1 1 x 2i 1 ], ni where ψ 1,x=1 is a constant equal to ψ 1 1. This expression does not have a discontinuity at x = 0 but also does not have a constant ratio ψ 1x/ψ 1 x. For n this function converges to a infinite-well function with the bottom at ψ 1,x=0 and infinitely high borders at x = ±1. The ratio between the derivative and the value at x = 0 changes in a limited way when we truncate the series at n = 1. We apply this initial condition as [ ψ 1 x = ψ 1,x= x 2 1 ] In Figure 3.2 the two possibilities are shown. We set the initial value of ϕ to zero if the initial island

30 30 CHAPTER 3. THE 2D REDUCED MHD MODEL Figure 3.2: Two contourplots for the two options of initial island configuration. In A the configuration of eq is shown and in B the configuration of eq For both configurations we have chosen ψ = 1, 1 = 0.5 and ψ 1,x=0 = We choose a negative value for ψ 1,x=0 because otherwise the islands O-point would be at ξ = ±π. In A the separatrix is at ψ = and in B at ψ = width w is much smaller then the resistive layer. For larger initial islands we use the linearised reduced MHD equation for ϕ, eq. 3.22, to calculate ω 1 at t = 0, fx ψ ω 1 = γl 2 ψ 1 f 2 ψ 1 = 1.82 f ψ xψ η 3 5 L 2 1,x=1 5 2 f x To derive this eq we have used 3.40 and We calculate ϕ 1 at t = 0 from ω 1 via the screened Poisson integral The numerical schemes without the normalisation of t, ψ and ϕ are shown in appendix B. 3.9 Parameter normalisation and range We want to apply our results from our non-dimensional model to experimental data from realistic toamas. Some translation between non-dimensional variables and physical variables is required Normalisation In deriving the Rutherford equation in Chapter 2 we have made a few normalisations. The normalisation of ϕ is ϕ/b z ϕ/b φ ϕ. The normalisation of ψ has been done more subtly using dimensions where the density ρ and the vacuum permeability µ 0 are unity. Reinserting them into the incompressible 2-D MHD equations gives dψ ψ dt = η 2 µ 0 and ρ d 2 ϕ dt ] = Bψ [ 2, 3.42 ψµ0 where B is linear in ψ. The parameters µ 0 and ρ vanish in the equation on the right for ψ/ ρµ 0 ψ, but then in the equation on the left we have to renormalise η as well, η/µ 0 η. By introducing the ratio f and rescaling x Lx we also rescaled f and /L Physical ranges for the parameters The parameters that we use are the toroidal magnetic field B φ, the mass density ρ, the Lundquist number S and toama minor and major radius a and R. The radial length scale L has to be an order of magnitude smaller than the radius of the rational surface, which is of the same order as the minor radius a. On the other hand the radial length scale must be much larger than the width of the resistive layer, which is given by see [2] eq 4.91 δ η 2/5 1 1/5 ψ 2/5 = η SI /µ 0 2/5 1,SI 1/5 ψ SI/ ρµ 0 2/5 3.43

31 3.9. PARAMETER NORMALISATION AND RANGE 31 Table 3.1: The normalisation, physical orders and approximate numerical ranges for the parameters of the model. Parameter Normalisation Physical values Range L - δ L a 0.1m dx - δ m f - L/a 0.1 dt or h - τ H s η η SI /µ 0 v A a/s m 2 /s ν - η m 2 /s 1 L 1,SI 2f 0.2 ψ ψ SI / ρµ 0 B φ /R ρµ s 1 in our model this becomes δ = η 2/5 1/L 1/5 f ψ 2/5 due to the normalisation of the length scales. The helical length scale is of the same order as the toama minor radius, so f L/a. The timesteps have to be smaller then the timescale of the relevant dynamics. In case of tearing instabilities this is the hybrid timescale τ H τ 2/5 A τ 3/5 R, where τ A = a/v A = a ρµ 0 /B φ is the Alfvén timescale and τ R = µ 0 a 2 /η SI = a 2 /η is the resistive diffusion time. In a toama τ A τ H τ R, explaining the term hybrid, between resistive and Alfvén timescales. The physical range of the resistivity is determined by the Lundquist number S from equation 2.18, η = η SI /µ 0 = v A a/s we choose the value of the numerical viscosity to be of the same order. The value of is determined by 3.32 for 1, for = 1 we choose of the same order, only positive, 1 2Lm r s = 2f. Finally the physical value of ψ can be calculated via eq as ψ SI = q s q s ψ r rs B θ a B φ R Numerical ranges With help of typical toama values a 1m, R 5m, B φ 3T, n m 3, m 0 2.5µ g a mixture of deuterium and tritium and S 10 8, we can calculate the actual ranges of the parameters. The results are listed in Table 3.1.

32 32 CHAPTER 3. THE 2D REDUCED MHD MODEL

33 Chapter 4 Sensitivity of the 2D reduced MHD model to the boundary conditions 4.1 Relevance of the boundary conditions A main motivation for this research is to apply the boundary conditions from the Rutherford theory to a 2D numerical simulation. The recent article of Comisso and Lazarro [5] raises issues about the small-scale of the ECCD current, but their choice of boundary conditions and parameter range are not consistent with a realistic toama configuration. In their article they numerically solve a reduced extended MHDmodel, which includes four time-evolution equations: the poloidal magnetic flux function ψ, the parallel ion flow vorticity ω = 2 ϕ, the parallel ion flow velocity v i and the electron pressure in the parallel and perpendicular direction. The Comisso and Lazarro model is normalised and solved in a periodic slab geometry, π x π and L y y L y, 4.1 where L y > L x. They choose to have periodic boundary conditions in both directions, and therefore have to pic a periodic equilibrium flux function, ψx, y = cosx. They insert a perturbation of the form ψ x expγt + iy. Their choice of resistivity leads to a Lundquist number of S = 10 3, which is an order of five away from realistic toama resistivities. They simulate up to a time of 2τ R, which in their case is approximately equivalent to 31τ H or 2000τ A, so the difference in their timescales are much smaller then usual for NTMs. The problem solved by Comisso and Lazzaro is that of a periodic current layer and differs from the problem of the evolution of a tearing mode in a toama on three essential points. First, in the periodic current layer, the second derivative of the flux function has the same sign as the current at the resonant radius, while in a toama with normal shear the two have opposite signs. While in a toama a noninductive co-driven current driven in the same direction of the overall plasma current is predicted to be stabilizing, it is predicted to be destabilizing in the case of a current layer. Second, in the case of the periodic current layer, it is the current layer on the resonant surface itself that is unstable, while the stability of the mode in a toama is determined by the global equilibrium configuration and current density profile well outside the resonant surface. Third, the islands studied by Comisso and Lazzaro reach widths of the order of the equilibrium scale length, while the islands in a toama generally remain small compared to the minor radius. The Rutherford equation only applies as long as the island is small compared to the equilibrium scale length. The choice of the periodic boundary conditions in the radial direction maes the model distinct from a real toama NTM. The comparison with Rutherford theory cannot be made, as the underlying assumptions of this theory do not match the periodic boundary conditions. Our motivation is to simulate island growth without the periodic boundary conditions in the radial direction. Instead, we stay as close as possible to the Rutherford theory by developing and using boundary conditions that readily allow comparison with theory and toama experiments. In this way we obtain a model that is relevant to toamas and also allows for close examination of the Rutherford theory. One target is to loo for the point where the underlying assumptions of the Rutherford equation brea down. Because it is impossible to simulate an infinitely large grid, we choose to implement the boundary conditions at a finite radial distance from the rational surface. When the island is small compared to the whole simulation volume, 33

34 34CHAPTER 4. SENSITIVITY OF THE 2D REDUCED MHD MODEL TO THE BOUNDARY CONDITIONS this is reasonable. Figure 4.1: The position of the four vortices with respect of the island. Contours of the stream function are shown in colour and the contours of the total magnetic flux are shown in blac. Yellow cells are moving clocwise, red cells counterclocwise as v = ê z ϕ. In this particular case the four cells lie exactly inside the simulation volume, and ϕx = ±1 = 0. If different boundary conditions for ϕ are adopted, the vortices would partially lie outside the grid. For example for x ϕx = ±1 = 0, only the side closest to the rational surface lies within the grid. The boundary conditions for the perturbed magnetic flux function ψ are well defined by our goal to match Rutherford theory. Boundary conditions for the stream function and vorticity are not that common in literature. In [14] the conditions ω ±L = 0 and x ϕ ±L = v bac are used, where v bac is the phase velocity of the island in the frame where the bacground electric field vanishes. In [15] there is asumed that for from the resonant surface all scalar fields depend on the magnetic flux. Practical, the same boundary condition for the vorticity is used, but the stream function must satisfy ξ ϕ x ϕ = ξψ x ψ. 4.2 Both of these papers do not implement the Rutherford boundary conditions to the magnetic flux function. As only the perturbation of the magnetic flux function is important to measure the size of the island, and the stream function vanishes from the reduced MHD equation 2.15 after the flux-surface averaging, there is little mentioned about the stream function in Rutherford theory. Only in the linear stability analysis of section 3.5 and in [22, section 6.9] the solution of the stream function is mentioned briefly to determine the growth rate and the restive layer width. From theory we expect that there are 4 vortices around each magnetic island, preserving the shape of the island and providing magnetic reconnection in case of a growing island see Figure 4.1. Heuristically, one can see from Figure 4.1 that the four vortices eep the form of the island stable. While the magnetic field lines tend to straighten themselves due to

35 4.2. COUPLING VORTICITY AND STREAM FUNCTION 35 a lower magnetic energy, the plasma stream pushes the field line in the direction of the rational surface near the X-point, while near the O-point the flow is moving away from the rational surface and thus bending the field lines outwards. The physical and numerical examination of this wor reveals that the solutions are extremely sensitive to the precise boundary conditions of the stream function. The shape and number of vortices appeared to depend greatly on the chosen boundary conditions, which in turn influenced the shape and growth of the magnetic island. In this chapter, we explore the effects of several different boundary conditions for the stream field and the vorticity. We evaluate problems that result from simulations with these boundary conditions. We discuss the physical meaning of the boundary conditions that are finally employed 4.2 Coupling vorticity and stream function We have to deal with the change of the vorticity in the reduced MHD-equations, instead of computing the change in the stream function itself. This not only changes the numerical schemes, but also requires two different boundary conditions for the stream function and vorticity. The choice of the two integration constants from the Poisson equation 3.18 do not affect on the vorticity. Therefore we discuss the boundary conditions for the vorticity and stream function separately, although these boundary conditions must be complementary. 4.3 Possible boundary conditions for the vorticity The boundary conditions for the vorticity are applied in the same way as the boundary conditions for the magnetic flux function ψ, by extrapolating the values for the vorticity inside the grid towards the edges. This boundary condition does not hold only at the edge, but also in the whole area outside the grid the ideal outer region. There are several ways in which this extrapolation could be done, and here we review possible boundary conditions that we have considered: 1. Dirichlet boundary condition. The vorticity is set to zero outside the simulation volume. This choice produces no vortices moving inwards. The resistive layer decoupled as much as possible from the outer region. This choice is also made in [14] and [15], because the Dirichlet boundary condition approximates 2 xϕ = 0, the condition for a freely propagating island. It would also be needed if the stream function has to be an even function around the boundaries. 2. Neumann boundary conditions. Here the radial derivative of the vorticity is set to zero at the boundary. This wors better with a non-slip boundary conditions or other Neumann fixed velocity conditions on ϕ. A Neumann boundary condition is necessary if the solution of ϕ is required to be odd on the boundaries. 3. Extrapolate with the help of the boundary conditions on ψ and ϕ. It is possible to rewrite the numerical schemes at the edges of the grid in such a way that we can compute the vorticity directly from the values of the fields inside the simulation volume. Using this approach we extrapolate ψ and ϕ outside the simulation volume. Although this seems elegant and could be fully consistent with the other boundary conditions, it is quite cumbersome, especially with elaborate boundary conditions on ϕ. More important, however, is that we imply the boundary conditions on ϕ and ψ two times instead of one. Also, because each numerical extrapolation will introduce some discontinuity in the derivatives, this method suffers strongly from large disturbances near the edges. For instance, if ϕ is set to zero on the edge and outside the simulation region Dirichlet boundary condition for the stream function, the discrete second radial derivative of ϕ at the boundary point x 1 is 2 ϕ x 2 x=x1 = ϕ2 2ϕ1 + ϕ0 δx 2 = ϕ2 δx Here we have written ϕ i for the value of ϕ at x i, and δx for the radial distance between the grid points. As δx 1 this derivative can be considerably larger then the same derivative calculated further inside the simulation grid. Of course one could hope that the solution of the stream function near the boundary is sufficiently smooth, such that ϕ 2 δx2 meanϕ. As depicted in 4.5 this is not the case.

36 36CHAPTER 4. SENSITIVITY OF THE 2D REDUCED MHD MODEL TO THE BOUNDARY CONDITIONS 4. Linearly extrapolation. In the linear stability equation 3.22 the time derivative of the vorticity is approximately linear in x, because the small perturbations in the ξ-direction f 1 can be neglected. This produces t 2 ϕ = t ω = ifgxψ. 4.4 Because ψ is approximately constant in the radial direction proportional to, we linearly extrapolate ω outside the simulation volume from the vorticity values inside. This reproduces a Neumann boundary condition where the value of the radial derivative is not determined from outside but from the inside. Apart from the choice of the boundary conditions, we must choose where to implement them. It would be simplest and the most efficient to apply both boundary conditions for ψ and ω exactly at the edge, as far as possible from the rational surface for a given grid size. In this way there would be as many points inside the simulation volume as possible. However, this results in two possible discontinuities at the same point. Especially the computation of the vorticity suffers from small deviations in the first derivative of ψ, as these sometimes mean that large deviations in higher order derivatives of ψ arise. 4.4 Possible boundary conditions for the stream function We review the possible boundary conditions that we have considered for the stream field: 1. Dirichlet boundary condition. The stream function could be set to a fixed constant at the boundary, the most obvious choice being zero. The motivation behind such a choice is the same as for the Dirichlet boundary conditions for the vorticity: it would prevent influences from outside the simulation volume to effect the simulation. If no flow streams into or out from the simulated region, the only coupling with the outer region would be via the ψ-field, where the boundary condition is determined by the Rutherford theory. The solution of φ can be calculated from the general solution 3.35 and becomes ϕ = x 0 sinh fx x 1 dx 1 ω x 1 sinhfx f sinhf 1 0 sinh f1 x 1 dx 1 ω x f Because setting the stream field to zero everywhere outside the grid would create a discontinuity in all radial derivatives, one could choose to initialize ϕ to be antisymmetric around zero. This choice coincides with setting ω to zero at the boundary. This could improve the vorticity values that are calculated near the boundary, especially if the vorticity is extrapolated with the help of the value of the fields outside the grid. However, this choice is arbitrary and is not based on physical arguments. We assume there is another vortex just outside the simulation volume, while this is generally not the case. 2. Neumann boundary conditions. Mitello et al. [14], state that velocity in the ξ-direction at the edge should be matched with the phase velocity of the island with respect to the bacground. In the simplest case this would mean x ϕ x=±1 = 0. The solution is smoother by requiring the solution of ϕ is required to be even on the edge. 3. Zero homogeneous solution. In an attempt to let the stream function evolve as freely as possible, without possible unwanted effects from the boundary, we could set the homogeneous solution of the Poisson equation to zero, leaving only the non-homogeneous solution. This concept for treating the boundary does not determine a complete set of boundary conditions, because values of the vorticity near the boundaries need to be determined before the stream function can be calculated. Also the physical argumentation for this choice is not very clear. 4. Zero total solution at infinity. Instead of looing at the value of the stream function at the edge we could choose to push the implementation of the boundary further away. A natural choice could be to require the stream function to vanish at infinity. This requires that the vorticity also vanishes at infinity. Therefore this boundary condition only wors in combination with ω x = 0 for x 1. For x we can approximate sinhfx x 1 as sinhfx x 1, because x x 1 and x 1/f. The condition therefore reduces to [ lim ϕ = lim sinhf x a + x x 1 0 ] ω x 1 dx 1 = f

37 4.4. POSSIBLE BOUNDARY CONDITIONS FOR THE STREAM FUNCTION 37 This criterium can thus be met by setting a = 1 0 dx 1 ω x 1 f in the Poisson integral Linear analysis Our final choice for the boundary conditions for the stream function is to couple them to the boundary conditions of ψ. We calculate radial boundary conditions for the stream functions ϕ using the linearised equation 3.21 in applying this equation, we neglect the resistive terms because it is applied outside the resistive layer. This results in a time derivative of the helical flux function and its spacial derivative t ψ = ifgxϕ t ψ = ifgxϕ ifgϕ. 4.7 The time derivative of the ratio ψ /ψ, which should be zero at the boundary, yields the further condition: t ψ ψ = 1 ψ ψ t ψ ψ ψ 2 t = ifgxϕ ifgϕ ψ ψ ifgxϕ ψ 2 = Hence xϕ ψ = ϕ xψ ψ 4.9 ϕ ±1, t ϕ ±1, t = ψ ±1, t ψ ±1, t 1 x±1, t = ± An additional condition results from our focus on islands that are symmetric around x = 0. Symmetric islands are produced by a stream function ϕ that is antisymmetric. A symmetric islands and an antisymmetric stream function stay symmetric and antisymmetric respectively. This follows from an identical argument to that made in the helical direction, see eq Because an asymmetric function has a symmetric first derivative, the derivatives at x = ± x are identical and hence the ratio ϕ /ϕ x=1 = ϕ /ϕ x= 1. Asymmetry can be reached by setting a = b in the homogeneous Poisson solution 3.19, which becomes a sinhfx. We have redefined 2a as a for simplicity reasons. The remaining boundary condition at x = 1 is therefore ϕ 1, t ϕ 1, t = fa coshfx + a sinhfx + x x 0 dx 1 cosh fx x 1 ω x 1 0 dx 1 sinh fx x1ω x 1 f x=1 The remaining constant of the homogeneous solution a has to satisfy a = 1 0 dx sinh f1 x1 f cosh f1 x 1 ω x 1 f coshf 2 1 sinhf = Note that this constant is depending all the values of ω for 0 x 1. The constant can be inserted in the Poisson integral of the added homogeneous and non-homogeneous solution ϕ x = a sinhfx + x 0 sinh[fx x 1 ] dx 1 ω x f In Figure 4.2 a typical form of the solution for such a stream field is shown.

38 38CHAPTER 4. SENSITIVITY OF THE 2D REDUCED MHD MODEL TO THE BOUNDARY CONDITIONS Figure 4.2: Left Typical radial profile of stream field ϕ for the boundary conditions derived by linear analysis. Here the vorticity is chosen to be x, the boundary is implemented at x = 1 and 1 = Right Contour plot of the same stream function color and contours of the total helical magnetic flux function blac lines. The center of each vortex lies just within the simulation volume as long as 1/2 1 < 0. Figure 4.3: Typical radial profile of stream field ϕ for the boundary conditions derived by linear analysis, extended outside the simulation volume. The borders of the simulation volume are indicated by the vertical lines at x = ±1. Outside the simulation volume the solution could be extended via the differential equation The general solution for this equation is ϕ,out = C 1 x e 1 2 x x The integration constant C 1 is determined by requiring continuity at the boundary, C 1 = ϕ,in 1 exp 2. For a small 1, the solution falls off lie 1/x outside the simulation volume, giving the radial profile shown in Figure 4.3

39 4.5. RESULTS OF THE BOUNDARY CONDITION TESTS Results of the boundary condition tests Figure 4.4: Strongly-polarized structure in the time derivative of the vorticity at the edge of the simulation grid. The contour plots of the time derivative of the vorticity are shown at t = 0.0 s and t = s. There is a Neumann boundary condition for the stream field, and the boundary condition for the vorticity is set equivalent to x ϕ = 0 outside the simulation volume Two main problems occur commonly for most choices of boundary condition. First, the vorticity structures grows large near to the edge see Figure 4.4. Sometimes these structures were seen in the stream function as well. These structures disturb the bul of the simulation so that the relevant physical processes are strongly influenced. For example, in Figure 4.4 on the RHS the second Fourier harmonic of a substantial amount compared to the first harmonic. This maes the second Fourier harmonic of the stream function, larger than physically could be expected. Also the first Fourier harmonic of the stream function was not growing as expected from theory exponential in the linear phase, quadratic in the Rutherford phase. This effect is lowered when the boundary conditions on the vorticity are implemented not at the edge but more inwards. The effect mainly occurs when the vorticity was extrapolated with the help of ψ and ϕ, but also sometimes and less severe with Dirichlet and Neumann boundary conditions. Second, after the first four vortices had grow for some time, they shrin and new vortices entered the simulation see Figure 4.5. Some of these cells are larger and stronger, some are smaller and weaer, but they all shrin after they fully enter the simulation regime and a chain of vortices forms surrounding the resonant surface. The movement of these cells towards x = 0 appears wave-lie, although not completely regular. These could be Alfvén waves, created because of a mismatch between the profile of the magnetic flux function and the stream function. The timescale on which these waves move matches the Alfvén timescale 10 6 s, however a more thorough investigation of these waves was not relevant to this wor. Sometimes the incoming streams are not all of the same order of magnitude. In these case, small disturbances enter the simulation volume, while on longer timescales the flow reverses see Figure 4.6. These direction reversed vortices sometimes reverted the sign of the perturbation of ψ, creating shapes lie in Figure 4.7. The wave-behaviour is persistent, and occurs at almost all combinations for the boundary conditions of the vorticity and stream function. Even when we strongly manipulate the stream function by extrapolating from regions near the rational surface or from averaging over parts of the radial direction, the flow reverses from time to time and incoming wave-lie perturbations grow. The only exception for this wave behaviour are simulations where the two boundary conditions from linear analysis are used. Other issues that occurred only in some of the simulations were: very rapid growth of the stream function near the edge, disturbing the magnetic field lines; growth of higher order Fourier harmonics in the vorticity, stream function and or magnetic flux for example the vorticity in Figure 4.4.

40 40CHAPTER 4. SENSITIVITY OF THE 2D REDUCED MHD MODEL TO THE BOUNDARY CONDITIONS Figure 4.5: Vortices shrining and moving in from the edge. The four contour plots of the stream field at t = s, t = s, t = s and finally t = s clearly show the incoming Alfvén waves. In this case the boundary conditions for the stream function are ϕx = ±1 = 0, but it also occurred for the the Neumann boundary condition, the zero homogeneous solution and the zero total solution at infinity. After an initial phase of growth, the four vortices, the size of the island and the strength of the flow all decrease. New islands form at the boundary. In the final stage, we see growth of the = 2 Fourier harmonic of the stream function near the rational surface. The simulations where we use the boundary conditions derived by linear analysis for the vorticity and the stream function do not suffer from the domination of waves in the stream field. These boundary conditions successfully reproduce realistic behavior of a tearing mode, both in the linear and Rutherford phase. From now on our model will use the boundary conditions described in eqs. 4.4 for the vorticity and 4.12 for the stream function. In the next section we use those boundary conditions to examine tearing-mode growth in detail and to benchmar tearing-mode growth rates from our simulations against theoretical scaling law and compare with 3D simulations.

41 4.5. RESULTS OF THE BOUNDARY CONDITION TESTS 41 Figure 4.6: Plot of the velocity stream ϕ function along a radial cut in the plasma for a simulation with Dirichlet boundary conditions for the vorticity and stream function. On the left t = s vortices move in from the left and right simultaneously. A short time later, the flow in the simulation volume has reversed. After a similar time elapses right, t = s, the flow reverses again into the original direction. Figure 4.7: Reversed island near the edge. left A contour plot of the total magnetic flux, on the right a plot of the first harmonic of the perturbed flux function over the radial coordinate x. Although the magnetic field lines remain relatively unchanged around the rational surface, the field lines near the edge are shifted ξ = π, due to the large vortices with opposite flowing direction near the edge. In the radial plot right one can see that the shape of the perturbed flux has no relation with which boundary conditions are used. There are 2 points near the edge of on this scale that have almost the same value. The value of ψ at the outermost grid point edge is extrapolated from the value at the next point via the boundary condition on ψ 1, but this does not match the shape of the perturbed flux function near the edge.