Numerical simulations of wire array Z-pinches under variations of global magnetic fields

Transcription

1 Pontificia Universidad Católica de Chile Facultad de Física Numerical simulations of wire array Z-pinches under variations of global magnetic fields by LUIS ALBERTO DONOSO TAPIA Thesis presented to the Faculty of Physics of the Pontifical Catholic University of Chile as a requirement to opt for the Master s in Physics Degree Advisor : Prof. Felipe Veloso Comission : Prof. Edgardo Dörner Prof. Mario Favre December, 2017 Santiago Chile c 2017, Luis Alberto Donoso Tapia

2 1. c 2017, Luis Alberto Donoso Tapia Total or partial reproduction is allowed for academic purposes, by any mean or procedure, but must include a bibliographical citation which credits the work and its author.

3 Acknowledgements I want to start by thanking my family for their support during this prolonged work period. Without their support and encouragement, the accomplishment of this goal would have been impossible. I thank my parents María Soledad and Luis, my brother Damián, my aunt Mireya and my loyal dog Mingo, who has been with me for more than half of my life, and also my friends. I want to thank my advisor Dr. Felipe Veloso. His support and patience were a critical part of achieving this goal. His unceasing willingness to answer questions and his constant concern about my situation, are gratefully acknowledged I also extend my gratitude to Prof. Jerónimo Maze and Prof. Rafael Benguria, who both showed such good will and understanding of the difficulties I had to overcome during this period, and gave me the opportunities I needed to continue working for my Master s Degree. I give special thanks to Prof. Christopher Feuillade, who accepted me as his doctoral student and understood how I needed to divide my time between my doctoral and master s studies during this period. Also, of fundamental importance for this work was the help of Prof. Jeremy Chittenden from Imperial College, London, who kindly allowed me and the rest of i

4 ii the plasma physics group to use his GORGON code. This work would have not been impossible without the permission to use this code and the continued support he has given throughout the work. I want to thank the rest of the plasma physics group, in particular Professors Mario Favre and Edmund Wyndham, and also my fellow students Vicente Valenzuela, Gonzalo Muñoz and Milenko Vescovi. Also, I need to acknowledge some important people within the physics department who supported me during this period, including Juan Acuña, Cynthia Castillo, Giliana Pérez, and many others.

5 Abstract During the last five years a lot of work has been done with the computational code GORGON [1,2] to simulate the dynamics of transient plasmas. The main work has been focused on studying plasmas in different configurations of wire arrays. These configurations include conical arrays [3,4,5], asymmetrical cylindrical arrays [6]. This thesis focuses on two primary areas. First, how does spatial resolution affect the reliability of observed simulation results? This was studied by simulating conical wire array Z-pinches with varying resolutions. Second, the dynamics of Z- pinch plasmas generated by asymmetrical wire arrays are examined. This causes the magnetic field topology to become asymmetrical, thus causing the plasma to move away from the geometrical center [6]. Regarding the conical arrays studied in this work, a low electronic density region was initially observed, which was located between two high density regions, similar to what has been observed in stellar phenomena like the Herbig-Haro objects [7]. However, subsequent simulations, with resolution increased by a factor of 2,5, have disproved this supposition. The second main aspect of this work shows how undermassed aluminium wire configurations can be used for accurate control of plasma dynamics, by modifying iii

6 iv the governing magnetic field topology of the system. This leads to an acceptable method for controlling the direction and velocity of the plasma precursor within a time period similar to the associated pulsed power generator rise time [6,8,9].

7 Contents 1 Introduction What is plasma? Z-pinches Wire arrays Cylindrical wire arrays The GORGON Code Resistive magnetohydrodynamics From physics to coding Effects of spatial resolution over the simulations Introduction Code setup Simulation Results µm simulations Moving to 100 µm resolution Results of the 100 µm resolution simulations Lessons learned from the resolution change Jet structure Electronic density v

8 CONTENTS vi 3 Asymmetrical arrays with different currents Introduction Electrical currents Undermassed, evenly massed and overmassed arrays wire array Ablation times Magnetic field topology Plasma densities wire array Ablation Times Precursor motion Dimensionless parameters wire array comparison wire array comparison Importance of these dimensionless magnitudes Conclusions 69

9 List of Figures 1.1 Basic scheme of a Z-pinch. The black arrows represent the electrical current. The green circle is the associated magnetic field. The red arrows represent the plasma pressure gradient pointing outwards Examples of MHD instabilities observed on a Z-pinch Plasma. On the left a m = 0 sausage instability is shown. On the right an m = 1 kink instability is shown. The m numbers represent the zeroes of the associated Bessel functions [24,36] Scheme of a cylindrical wire array. The pale blue arrow represents the global magnetic field. The yellow arrow indicates the current, and the red arrow indicates the Lorentz force [27] Core-corona schematic. The central black circle represents the cold and solid core of the remaining wire. The pale blue circle indicates the hot and conductive plasma that emanates from the wire when a current flows through it Summary of input and output data used by GORGON GORGON flowchart describing how the code works Image of the HH34 object taken by the Hubble Space Telescope [21].. 14 vii

10 LIST OF FIGURES viii 2.2 Experimental arrangement for attempting to reproduce the Herbig- Haro objects by using GORGON. On the left there is an end-on view. On the right there is a side-on view In red the experimental current trace measured from the Llampüdkeñ generator can be visible, which was used for the combined array. The green curve represents the current used by the individual arrays, which is exactly the half of the one depicted by the red curve Cutaway of the simulations setup. Here it can be seen how the central wires were put in the simulation. Note: The image on the left is only schematic. The internal and external wires are not coplanar in the combined simulation Combined view of electronic density visualizations. The gap and plasma jet are highlighted. The top row corresponds to the simulation of the nested arrays (both inner and outer arrays combined). The middle row corresponds to the inner array only. The bottom row corresponds to the outer array only Electronic density plots along the jet axis. In the right is possible to see a close up to the region to clearly show the lowering of the plasma density Electronic density profiles for the internal (left) and external (right) arrays Electronic density measurements obtained from 100 µm resolution simulation of the nested conical array Electronic density measurements obtained from the 100 µm simulation of the nested conical array. The arrows represent the position of the gap in the 250 µm simulation at 500 ns (green), 600 ns (blue) and 700 ns (purple). There was no gap visible at 400 ns

11 LIST OF FIGURES ix 2.10 Comparison between electronic density profiles for simulations made at resolutions of 250 and 100 µm along the plasma jet, for 400, 500, 600 and 700 ns. The red arrows indicate the position of the gap for the 250 µm curves. There is no gap visible at 400 ns Electronic density visualization. On the right it is possible to see a smooth stream of plasma coming from the wire. In the left, on the higher resolution case, it is possible to see a modulated stream, which exhibits alternating regions of higher and lower density regions of the plasma stream coming from the wire. The green arrows indicate higher density regions. The yellow arrows show the direction of the plasma stream. The diagonal gray lines represent the partially ablated wire End-on schematic of an asymmetrical cylindrical array. The two thick wires (25 µm) are represented by the red dots, while the thin wires (10 µm) are represented by the white dots End-on view of a GORGON simulation of an asymmetrical wire array consisting of 2 thin (top and right) and 2 thick wires (bottom and left) with diameters of 10 µm and 25 µm, respectively [4]. The green arrow at 320 ns shows the general direction of the precursor which moves from the geometrical center towards the area between the two thick, remaining wires Both current traces used for these comparisons, which correspond to Llampüdkeñ and MAGPIE generators End-on schematic of an 8 wire asymmetrical cylindrical array

12 LIST OF FIGURES x 3.5 The top two rows show end-on and side-on views of the asymmetrical 8 tungsten wire experiment with the Llampüdkeñ generator current. The bottom two rows show end-on and side-on views of the symmetrical 8 tungsten wire experiment with this same current. The end-on views are complemented with magnetic contour lines Detail of the end-on, 500 ns view of Figure 3.5 for the Llampüdkeñ case. Here it is possible to see a horned 10 T magnetic contour line which surrounds the plasma precursor, also extending to the dense streams coming from the thick wires towards the center, without actually reaching the thick wires from where they come. The blue arrow highlights the location of this contour line [4] End-on interferometry of the experiment. The red arrow points towards the higher plasma density region [4] Side-on shadowgraph of the experiment at times 200 and 437 ns. The red arrows denote the plasma flares coming from the wires [35] End-on schematic of an asymmetrical cylindrical array. In broad terms the idea is to make the two red wires thicker than the others Electronic density visualization for the same 4 array simulation made with MAGPIE and Llampüdkeñ currents Side-on electronic density visualization from the 4 aluminium wires MAGPIE case for times 30, 60 and 100 ns. The upper part shows an end-on view, and the bottom part shows the X-Z cross-section End-on (top) and midplane side-on (bottom) electronic density comparison of both generators at their peak currents (t d = I d = 1) Midplane side-on electronic density view of the MAGPIE case at t d = 0,

13 LIST OF FIGURES xi 3.14 Close ups of the thick wires while ablating for both cases. The yellow arrows indicate less dense areas where the wire breakage has occurred. This image corresponds to t d = End-on (top) and midplane side-on (bottom) electronic density comparison of both generators at time t d = 0, End-on (top) and midplane side-on (bottom) electronic density comparison of both generators at time t d = 0, End-on (top) and midplane side-on (bottom) electronic density comparison of both generators at time t d = 0, Close up of the nail and ring plasma structures which surround the wires in the Llampüdkeñ case. The yellow dots were added to the bottom row to help seeing the wire position. Is important to mention that these plasma nails and rings are present at different times

14 List of Tables 2.1 Experimental parameters used in GORGON in attempt to reproduce a Herbig-Haro Object Experimental parameters inserted in GORGON for the simulation described Experimental parameters inserted in GORGON for the simulation described xii

15 Chapter 1 Introduction 1.1 What is plasma? Plasma corresponds to the fourth state of matter, the others being solid, liquid and gas. It is estimated that plasma corresponds to 99 percent of visible matter. Plasmas can occur in both natural forms, and be artificially generated [10]. Natural plasmas occur, for example in stars and auroras [10]. While artificial plasmas are created in a laboratory environment. For example, they may be created using high power lasers [11]. In this case, they are generated when a laser beam is focused on a dense medium surrounded by an ambient gas. The energy delivered by the laser by its photons makes the electrons on the medium to be excited. After this initial excitation the ambient gas will start to absorb laser radiation, and after reaching a critical value of available electrons, the radiation absorption by the gas will greatly increase, which transform all this ambient gas into plasma. Another way to generate plasmas is by generating variable electric fields at radiofrequencies around the MHz region [12]. These fields excite the electrons and 1

16 CHAPTER 1. INTRODUCTION 2 make them oscillate, with similar frequencies and cause a continuous process of separation and recombination between electrons and atoms. This thesis will be focused on Z-pinch plasmas generated by the use of high current electrical discharges produced by pulsed power generators [4,9,13]. They are generated when thin metallic wires, with diameters on the order of micrometers, are put in cylindrical [14], conical [3,4,5], x-pinch [15], radial [16], and some other geometrical configurations [17,18,19]. A high current pulse on the order of hundreds to thousands of ka flows through the wires causing them to ablate into the plasma state. This work is focused on conical and cylindrical wire configurations. Both of them produce Z-pinch plasmas, which will be described in detail later in this work. 1.2 Z-pinches A Z-pinch is one way to confine a plasma [21]. This particular plasma configuration is characterized by generating its own azimuthal magnetic field, which confines it via the current which passes through it. At the same time, this same current is responsible for ionizing the initially solid material (wire) and increasing its temperature by ohmic heating [23]. Figure 1.1 schematically depicts the magnetic field and the current. When these two elements are present the, Lorentz force appears, which has a direction perpendicular to both the current density and the magnetic field and is responsible for keeping it confined within a roughly cylindrical shape. When the configuration is azimuthally symmetrical the Lorentz force usually points to the center of the system. The Lorentz force, which points towards the center of the Z-pinch, is mathematically represented as force per unit volume by J B. Here, J is the current

17 CHAPTER 1. INTRODUCTION 3 Figure 1.1: Basic scheme of a Z-pinch. The black arrows represent the electrical current. The green circle is the associated magnetic field. The red arrows represent the plasma pressure gradient pointing outwards. density, and B is the magnetic field. At the same time, there are forces pointing outwards, opposing the Lorentz force. They correspond to the plasma pressure gradient. The Z-pinch will remain in equilibrium and stable when the Lorentz force and the plasma pressure gradient are balanced. An interesting effect that appears in a Z-pinch precursor or surrounding the wires used to generate plasma, is the occurrence of magnetohydrodynamic instabilities [25]. It is important to note that these effects can also appear in other kinds of Z-pinches such as gas puffs or plasma focus.these can be seen as plasma which appears around the wires, whose shapes may be represented by Bessel s equations solutions. These differences in the plasma geometry along the wire cause variations on the plasma implosion, which is directed towards the center of an array. The implosion of a wire array Z-pinch occurs when a wire eventually runs out of solid material, it all having being transformed into plasma, which is then pushed by the

18 CHAPTER 1. INTRODUCTION 4 Figure 1.2: Examples of MHD instabilities observed on a Z-pinch Plasma. On the left a m = 0 sausage instability is shown. On the right an m = 1 kink instability is shown. The m numbers represent the zeroes of the associated Bessel functions [24,36]. Lorentz force towards the center of the array [23]. 1.3 Wire arrays Cylindrical wire arrays The purpose of this kind of distribution is that, when parallel metallic wires are placed between a pair of electrodes connected to a pulsed power generator [9,26], a large current may be passed through them, usually of hundreds or thousands of kiloamperes, which can reach their peak values in a few hundred nanoseconds. The current may be evenly distributed along the wires, by using wires of similar materials and diameters. Nevertheless, this thesis describes a configuration of different wire

19 CHAPTER 1. INTRODUCTION 5 Figure 1.3: Scheme of a cylindrical wire array. The pale blue arrow represents the global magnetic field. The yellow arrow indicates the current, and the red arrow indicates the Lorentz force [27]. diameters within a cylindrical wire array, producing an uneven current distribution along the configuration. The physics behind this type of array involves a lot of factors, for example, the ablation of the wires, the plasma generated, the Lorentz force which pushes the plasma inwards, and finally the implosion with its associated x-ray emission [21]. The plasma column formed at the center of the system is called the precursor, and it s being fed by plasma coming from the ablating wires and pushed inwards by the Lorentz force (see Figure 1.3).

20 CHAPTER 1. INTRODUCTION 6 One important aspect of flowing high currents through metallic wires corresponds to the ablation they undergo. When a big current flows through a thin wire in a short lapse of time, the following phenomena take place: First the wire will heat up due to the Joule effect. After this a process known as ablation occurs. Here the plasma gradually undergoes a state transition from solid to plasma, progressing from its outer layer inwards. Although plasma emerges from the outer layers, more plasma is injected from the inner layers, showing that the ablation process, while fast, is not instantaneous [21]. Figure 1.4: Core-corona schematic. The central black circle represents the cold and solid core of the remaining wire. The pale blue circle indicates the hot and conductive plasma that emanates from the wire when a current flows through it. The ablation is driven mainly by ohmic heating, which corresponds to the relation between a current flowing through a wire, and the resistance caused by the heating process it undergoes. After plasma surrounds the generating wire, an effect known as the core-corona appears (see Figure 1.4). When looking at the wire from above, a solid and cold core can be seen, which is the remnant of the wire. The plasma is seen to surround this core, corresponding to the corona. Considering that the plasma has a much smaller

21 CHAPTER 1. INTRODUCTION 7 electrical resistance (or higher conductivity) than the solid remnant of the wire, the current will progressively flow within the plasma instead of within the solid remnant. 1.4 The GORGON Code Before describing the code itself, it is important to describe the hardware used for the simulations that will be shown on this work. The computer used was a 64- core Altus-2804i cluster, equipped with 128 GB of RAM. The OS utilized was Linux Mint bit [28]. This cluster was a significant improvement over the previously used computer, which was an AMD processor equipped gamer computer. The upgrade to the Altus cluster reduced the simulation times by as much as 75%, while also permitting simulations of greater resolution. In simple terms, a 600 x 600 x 200 cell 3-dimensional simulation could be completed in a period of 2 or 3 weeks, and produced approximately 1 terabyte of data. This type of simulation allowed to study the dynamics of plasmas produced by asymmetrical wire arrays, which are the core of this work. These studies would have been impossible without hardware of this power. The GORGON code was developed by Prof. Jeremy Chittenden of Imperial College London [25]. The basic principle on which this code is based is resistive magnetohydrodynamics (MHD-r). This means, that the plasma is simulated as a fluid, in contrast to other codes which use PIC (particle in cell) [29] or hybrid principles [30].

22 CHAPTER 1. INTRODUCTION Resistive magnetohydrodynamics Unlike ideal MHD, resistive magnetohydrodynamics adds an additional term to the total balance of forces present in the plasmas modelled. This is summarized in the following equation: E + v B = ηj + 1 J ne B 1 ne P + m e J ne 2 t (1.1) In this equation, the terms on the left side represent the electric field and the Lorentz force, expressed as velocity times magnetic field. On the right side, there are resistive, Hall effect, pressure gradientm and inertial terms respectively. The details are as follows: 1. Electron-ion resistance. The η term represents electrical conductivity, and J is the current density. It is important to note that this is the additional term added by resistive magnetohydrodynamics. If this term were zero, the physics would revert to ideal MHD. 2. Hall effect. Electrons and ions are able to separate and move independently from each other. n represents the plasma density, and e is the fundamental charge. 3. P represents a pressure gradient. This is present because of the existence of gradients in the electronic density. 4. Inertial term equivalent to a F = m a, Newton s second law. m e is the electron mass, and t is the time. The term J can be written as ρ v if the electrical t t charges remain constant during time. Here an acceleration term is clearly visible. ρ represents the mass density and v represents the charge velocity.

23 CHAPTER 1. INTRODUCTION 9 The reason why GORGON takes a resistive MHD approach for studying this kind of plasmas is due to the temperatures and densities that are attained in the simulated experiments. It is important to bear in mind that every simulation gives a plasma density of approximately particles per cubic centimeter, with temperatures reaching dozens of ev. Any attempt to follow this amount of particles (as particle-in-cell simulations operate) would extremely overwhelm any available computing capability. Other ways to study plasmas, such as PIC, use lower temperatures and densities. Other equations built into the code are indicated below: 1. Continuity equation 2. Gauss Law for magentism ρ t + (ρ u) = 0 (1.2) B = 0 (1.3) 3. Faraday s induction law 4. Overall energy balance B t = ( u B) + η 2 B (1.4) p t + u p + γp u = (γ 1)η j 2 (1.5) Some important terms to highlight in this set of equations the are pressure, macroscopic drift velocity, adiabatic parameter, conductivity, and current density, denoted by p, u, γ, η and j respectively.

24 CHAPTER 1. INTRODUCTION 10 Even though GORGON is not the only code used to simulate Z-pinch plasmas used worldwide, it has demonstrated to be one of the most complex and complete code for simulating these experiments. Other codes also used for this purpose are PERSEUS from Cornell University [31] or AEGIS developed at the University of Texas at Austin [38]. 1.6 From physics to coding The code is written in FORTRAN. It both calculates and presents the data using a 3 dimensional cartesian grid. The user can decide both the amounts and sizes of the individual cells. They can be either cubic, or straight prisms. In every case, the opposite sides of the cells are parallel, and contiguous sides always meet at 90 degree angles. The code solves heat and magnetic diffusion equations using two independent temperatures. One of those is for ions and the other for electrons. Another important effect corresponds to local thermodynamic equilibrium (LTE) [26]. Newer versions of the code are being tested omitting this effect, however all the simulations shown in this thesis use the LTE version of GORGON. Problems can arise in the case of to extremely low numbers. This was prevented in GORGON by establishing a minimum simulated mass density of 10 4 kg/m [26], which is present in the background. This approximately corresponds to an electronic density of m 3. The code has built-in initial values for many different magnitudes. For example the demagnetizing H field will always be initialized at a value of A/m, and the vector potential A will start at values of V s/m. These values can be modified

25 CHAPTER 1. INTRODUCTION 11 by the user, but these low values model the absence of an electrical current at the beginning of each simulation without needing to set them exactly at zero, due to the computing problems that might arise. All the units used by the code are in the MKS system, with the exception of temperature, which is displayed by the code in ev. Figure 1.5: Summary of input and output data used by GORGON. The wires are initialized as cold and dense gas columns. This means that the phase transition simulated is from gas to plasma (non-ionized gas to ionized gas). The phase transitions from solid to plasma, which occur in a laboratory test, are not modelled [26]. No important limitations are caused by this approximation. The time that the transition from solid to plasma takes, is of about 10 ns. Considering that these studies are focused in times of hundreds of nanoseconds, this approximation is not relevant.

26 CHAPTER 1. INTRODUCTION 12 Figure 1.6: GORGON flowchart describing how the code works.

27 Chapter 2 Effects of spatial resolution on the simulations 2.1 Introduction An attempt to reproduce the behavior of astrophysical plasma ejection in laboratory environments was made by using conical wire array Z-pinches. These astrophysical plasma ejections are called Herbig-Haro objects and have been well observed by the Hubble Space Telescope. Also they have been widely described in the literature [7,32,33]. One of the important results obtained during this time convened the effect of spatial resolution of the simulations on the results. This is extremely important, because the results strongly depends on the reliability of the simulations. During the first simulations a low electronic density region could be observed in a simulated conical wire array arrangement. This low density region lays between two high density regions, similar to what has been observed in the Herbig-Haro 13

28 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS 14 Figure 2.1: Image of the HH34 object taken by the Hubble Space Telescope [21]. objects [7] which can be seen in Figure 2.1. HH objects are characterized by their episodic emissions of plasma through the interstellar gas. Instead of observing a single and uniform jet coming off the source, is possible to see several episodic bursts of material coming from it (see Figure 2.1). In these objects the source is a newborn star, which expels gas in perpendicular directions with respect to their accretion disks [7]. We proposed that a nested conical wire array system could reproduce these objects in the laboratory. Before trying the actual experiment, simulations with GORGON were conducted, which are shown On the following pages. As far as the author is concerned, no experimental group has experimented with nested conical wire arrays, therefore there are no experimental results to directly compare. The simulation setup consists in two concentrical conical wire arrays, with different inclination angles (see Figures 2.2 and 2.4). This arrangement may produce similar phenomena as seen for the HH objects. The fundamental idea behind this arrangement is that the plasmas generated by each of the arrays will reach the center of the system at different times. It was thought that a direct consequence of this difference in the arrival times would be

29 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS 15 Figure 2.2: Experimental arrangement for attempting to reproduce the Herbig-Haro objects by using GORGON. On the left there is an end-on view. On the right there is a side-on view. the appearance of two clear high density regions, with a low density gap between them. It could be argued that this approach was only forcing the appearance of an HH-like object. However, this is is just an initial step to rest whether future use of this method might produce satisfactory results, as part of a bigger research consisting of the study of plasma emissions.

30 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS Code setup Wire Material Tungsten Wire Diameter 25 µm Lower Diameter of Array 2 and 8 mm Higher Diameter of Array 32 mm Height of Array 14 mm Current Peak 350 ka Current Peak Time 350 ns Cell sizes 250 and 100 µm Table 2.1: Experimental parameters used in GORGON in attempt to reproduce a Herbig-Haro Object. It is important to note that the current peak and, its corresponding time was obtained experimentally from the Llampüdkeñ generator [9], in order to obtain the most realistic parameters possible. This approach presumes that this arrangement will be attempted experimentally at the laboratory in some future time. Similarly important is that, in total, three types of simulations were made. One for the inner array, another for the outer array, and the final simulation with both arrays combined. The simulations with the individual arrays were made with only half of the total current. This means that the total current used for this case was 175 ka in 350 ns. This assumes that the current is divided in equal parts between the two arrays. This is not entirely accurate in the actual experiment, because the amount of current that flow through each wire is directly mandated by their inductances. Longer wires should have smaller currents flowing through them. The length of the inner array wires is of 19,8 mm, while the length of the outer array wires is of 16,1

31 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS 17 Figure 2.3: In red the experimental current trace measured from the Llampüdkeñ generator can be visible, which was used for the combined array. The green curve represents the current used by the individual arrays, which is exactly the half of the one depicted by the red curve. mm. Considering this, the outer array should conduct slightly more current than the inner array. Besides these physical parameters, it was necessary to set up the computational parameters. These purely computational parameters refer to the cell resolution and the cell size. It was decided that the optimal resolution for this experiment was 250 µm, trying to balance the physical accuracy and the total simulation time. The size of the simulation box was of 168 x 168 x 192 cells. When multiplying this amount by the cell size we obtain a simulation space of 42 x 42 x 48 milimeters. Each simulation took 5 hours on the 64-core Altus-2804i cluster used by the plasma group.

32 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS Simulation Results µm simulations Figure 2.4: Cutaway of the simulations setup. Here it can be seen how the central wires were put in the simulation. Note: The image on the left is only schematic. The internal and external wires are not coplanar in the combined simulation. As said in the previous section, three simulations were attempted. In Figure 2.4 it can be seen how the wires were organized. The basal angles are also depicted. The region of the plasma denoted gap in Figure 2.5 corresponds to a significantly lower plasma density region than in the immediate surroundings. This seems to succeed in giving a rough approximation of an episodic emission similar to those observed in HH objects. The idea was to achieve a substantial variation in the electronic density along the length of the jet. This can be clearly seen on the close-up showed in the electronic density figure above. Specifically, values dropped from approximately m 3 in the surrounding areas to m 3 in the gap itself. The length of the gap increases from approximately 2,5 mm to 5 mm during the 200 ns time period shown in Figure 2.6. This figure shows electronic density curves that help to better understand this experiment. To construct this figure, a square section of 2 by 2 cells was taken in the center of the XY plane. Given the size of these cells, this square section represents an area of 500 µm by 500 µm located at the center of the nested array, so that each individual

33 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS 19 Figure 2.5: Combined view of electronic density visualizations. The gap and plasma jet are highlighted. The top row corresponds to the simulation of the nested arrays (both inner and outer arrays combined). The middle row corresponds to the inner array only. The bottom row corresponds to the outer array only. coordinate in the z axis is actually describing an average of the electronic density of these 4 cells taken at each z step. This section tries to follow the jet created by this conical array, with the z coordinate corresponding to the position along the jet s

34 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS 20 Figure 2.6: Electronic density plots along the jet axis. In the right is possible to see a close up to the region to clearly show the lowering of the plasma density. length. There are two reasons for picking those 2 x 2 squares. First, it is important to arrange that the number of cells in the horizontal directions is even, in order to maintain a correct geometrical center. The second reason, which might play against the first one, is that it was necessary to make a very precise study of the electronic density profile. Therefore, it was necessary to make the width of the sample region as thin as possible, which is why 2 cells were selected, instead of a larger number. It was previously stated that more simulations were performed using individual arrays of this nested system. In these cases, profiles of electronic density along the plasma jet were taken, which can be seen in Figure 2.7. The dashed line located at z = 14 mm in Figure 2.7 corresponds to the upper electrode position. Below this value plasma keeps being injected from the wires

35 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS 21 Figure 2.7: Electronic density profiles for the internal (left) and external (right) arrays. which are ablating. Above it the jet is without any external plasma pressure pushing inwards. From the individual density profiles, one immediate effect that can be noticed is that the average density is nearly one order of magnitude greater for the internal array than for the external array. Also, there is a spike in the density at z = 5 mm in the internal array at 500 ns, reaching about 2, m 3. This effect is not seen in the external array. This spike seems to match the arrival of the main plasma stream at the center. However, its causes are not convincingly determined. The vertical velocity (v z ) reached by the high density plasma is lower than m/s for the individual array cases. This value is about the same order of magnitude as the gap velocities seen in the combined plot. This means that the sum of the plasma masses coming from the combined arrays does not generate much extra momentum to increase the velocity of the upward moving plasma, which also means that the plasma coming from both arrays have a weak interaction between them. The average perpendicular velocity of the streams with respect to their producing

36 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS 22 wires is of about 1, m/s. When taking into account the vertical component of this velocity for each of the arrays, the internal array should have a vertical velocity of 1, m/s, while the external array should have a vertical velocity of 7, m/s. It is important to discuss the comparison between the gap density and its immediate surroundings. Regarding this, it is important to note that the difference is less than an order of magnitude, specifically, it drops from m 3 in the surrounding areas to m 3, but is still enough to be explained by the difference in the arrival times of the convergent streams coming from both arrays. This time difference is about 200 ns. This is also sufficiently large to be seen in the linear plots displayed above Moving to 100 µm resolution In order to confirm the factual nature of the previously described phenomena an attempt was made to reproduce the exact simulation at a much higher resolution. Specifically, the resolution was increased from 250 µm to 100 µm. This causes the total amount of cells to increase by a factor or 2, 5 3 = 15, Also increasing the total amount of produced data and the calculation times by roughly the same factor. This leads to an increase in the total calculation time, and for size of the output files of about one order of magnitude. It is important to take this into account, because this is a three-dimensional simulation, and the cubic factors (because these are 3D simulations) will become relevant when adjusting the resolution, and if not handled with sufficient care, the volume of the simulations will exceed the available hardware capability.

37 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS Results of the 100 µm resolution simulations Figure 2.8: Electronic density measurements obtained from 100 µm resolution simulation of the nested conical array.

38 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS 24 It should be noted that the visualization range and color map of the previous figure are exactly the same as seen in Figure 2.5. Considering this similarity, a total absence of the afore-mentioned gap is clearly visible. This is indicated by the absence of a deep and long depression in electronic density, as seen in Figure 2.9. Figure 2.9: Electronic density measurements obtained from the 100 µm simulation of the nested conical array. The arrows represent the position of the gap in the 250 µm simulation at 500 ns (green), 600 ns (blue) and 700 ns (purple). There was no gap visible at 400 ns. To determine the presence or absence of the gap in these high resolution simulations, the electronic density profile along the axis is plotted for the times depicted in Figure 2.9. Is important to mention that in this figure the 2 x 2 cell box previously described has a different size due to the different cell sizes used. Specifically in Figure 2.9 the 2 x 2 box used to integrate has a size of 200 by 200 µm because of

39 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS 25 the use of 100 µm cells, while this same box in Figure 2.6 has a size of 200 by 200 µm. From Figure 2.9, it is seen that it is not possible to distinguish a particular sector as a gap, as would be seen in the low resolution simulations. This means that this effect might be purely computational, rather than physical. In order to understand the full effects caused by the increase in resolution, combined plots, which include electronic densities for both resolutions, are shown. The first notable feature of these plots is how the electronic density shows almost a double peak value for the higher resolution simulation at t = 400 ns. Specifically, this value is seen at 3 mm in the z direction, which coincides with the lowest converging point of the different plasma streams coming from the wires. Still considering the t = 400 ns plot, after the peak value noted at z = 3 mm, the density progressively decays as z increases. This is consistent, since the convergence times of the plasmas are different, depending on the height. It is important also to notice that the electronic density for the higher resolution simulations does not decay monotonously. This means that regions of negative and positive slope are seen along the axis. This phenomena might be due to the modulation of the plasma emission coming from the wires, which will be explained below. The modulation effect refers to the periodic rises and falls of the electronic density. This has been studied previously [34], showing nearly constant values for the wavelength λ, which only varies between materials, and not affected by experimental parameters such as wire diameter or current generated by the associated pulsed power generator.

40 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS 26 Figure 2.10: Comparison between electronic density profiles for simulations made at resolutions of 250 and 100 µm along the plasma jet, for 400, 500, 600 and 700 ns. The red arrows indicate the position of the gap for the 250 µm curves. There is no gap visible at 400 ns.

41 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS 27 Figure 2.11: Electronic density visualization. On the right it is possible to see a smooth stream of plasma coming from the wire. In the left, on the higher resolution case, it is possible to see a modulated stream, which exhibits alternating regions of higher and lower density regions of the plasma stream coming from the wire. The green arrows indicate higher density regions. The yellow arrows show the direction of the plasma stream. The diagonal gray lines represent the partially ablated wire. Its cause can be attributed to the development of m = 0 or Rayleigh-Taylor MHD instabilities along the wires themselves, as well as to the effects of local versus global magnetic fields [36]. To reinforce this concept, it is necessary to take a look into the higher resolution simulations, where the decay is much smoother as the z values increase. The cell size at 250 µm, is too large for the modulation to be perceived. Studies [34] have shown values of λ = 250 µm for tungsten wires, and λ = 500 µm for aluminium wires.

42 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS 28 When examining results for greater values of it, it may be seen that the peak values increase significantly, indicating that more plasma is coming from the wires and converging in the center. For example, in the higher resolution simulations the peak value increases from 1, m 3 to 2, m 3. It is important to recognize that the peak value remains at the same location (z = 3 mm) which reinforces the concept that this place is the first and lowest point at which plasma starts to converge. For the lower resolution simulations a similar effect can be seen. The peak value of the electronic density grows from 0, m 3 to 1, m 3. This more than doubles the value seen previously at 400 ns. A possible explanation for this rapid growth of the electronic density peak value could be related to the lack, or severe diminishing, of the modulation effect. This means that the average value of the electronic density surrounding a wire is greatly increased. It is also important to note that, at 500 ns for the lower resolution simulatiosn there are two very visible dips in the electronic density profile. The first of these is located at 7 mm, and the second is located at 17 mm. This means that, because of the lower resolution used, there is a lack of modulation along the wire, which causes a more or less uniform jet (which can be noticed by the smoothness of the 400 ns curve). And this causes the late appearance of the dips compared to the higher resolution simulations, where the modulation effect is more important. When passing from 500 to 600 ns, perhaps the most interesting effect is noted. In the higher resolution simulation, the peak values of the plasma density begin to decay, while in the lower resolution simulation the peak value of the plasma continues

43 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS 29 to grow. The maximum electronic density visible at 700 ns for the 250 µm simulation is 3, m 3. It increases from the 2, m 3 seen at 600 ns. By contrast, in the higher simulation results, the maximum electronic density decays from 2, m 3 to 1, m 3. A possible explanation for this discrepancy could be found in the lack of modulation explained before for this resolution. A consequence of this is a much smoother plasma emission from the wire, thus artificially increasing the ablation time of the wire. It is important to note that it was not possible to get an accurate value of the time for this type of simulation, due to the long computational time needed. The values of λ = 250 and λ = 500 µm for tungsten and aluminium wires respectively, imply that one 250 µm cell (as used in some of the simulations shown) may contain one full period for the tungsten wires, while one period for aluminium wires would extend to just two cells. This clearly prevents any modulation effect being seen at such low resolutions.

44 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS Lessons learned from the resolution change Jet structure The effects caused by the resolution change, which were explained in the previous pages, can be summarized mostly in two big aspects. First of all, the change in the resolution affects the structure of the jet created at the center of the array. The lower resolution (250 µm) simulation shows a smoother jet from the electronic density point of view, with the gap visible and moving upwards as time passes. In the higher resolution (100 µm) by contrast, the jet shows more changes in the electronic density. Also the gap seen in the lower resolution simulation is not visible anymore. This means that the difference in the arrival times of the streams coming from the two nested arrays is much less important when the resolution is higher. This mentioned time difference might be important for the lower resolution case because the modulation effects explained before are almost non relevant, with the emission coming from the wires being much more smooth. This implies that in the formation of the jet there are two relevant moments, each one being the arrival of the streams from both arrays Electronic density As was mentioned before, the electronic density along the jet for the higher resolution simulation (100 µm) is much more variable than in the lower resolution case (250 µm). This can suggest that the modulation phenomenon takes a significant importance when having a resolution which allows it to appear, causing the jet to exhibit ups and downs on its density (see Figure 2.10) with wavelengths of the same

45 CHAPTER 2. EFFECTS OF SPATIAL RESOLUTION ON THE SIMULATIONS 31 order of magnitude as the one exhibited by tungsten wires (λ W = 250 µm) [34]. Also, as previously mentioned, another factor that could have played a role in the differences between the curves. This corresponds to the difference in sizes of the 2 by 2 cells integration box used. The size of this box in the 100 µm case is 200 by 200 µm, while for the 250 µm case the size is 500 by 500 µm. Still observing Figure 2.10 is possible to see that in 400 and 500 ns the jet has an peak density of almost double in the higher resolution case. This can be explained by more dense material coming from the wires at this stage in this higher resolution. This also indicates that the presence of the modulation effect increases the overall electronic density observed. According to [4] the minimum recommended cell size to study conical arrays is 100 µm. This is because lower resolutions don t allow to properly visualize the unstable feature of the jet. As was possible to see in Figure 2.10, the jet s electronic density profile is much more smooth when using a resolution of 250 µm.

46 Chapter 3 Asymmetrical arrays with different currents 3.1 Introduction In [6] the effect of asymmetries induced on cylindrical wire arrays were studied. All of the experiments were conducted in the Llampüdkeñ pulsed power generator [11], which has a current trace of 350 ka at 350 ns. The author of this thesis participated in detailed studies of the plasma dynamics of asymmetrical wire arrays in the Llampüdkeñ generator [6]. The purpose of this section is extend this work by comparing the dynamics observed on these previous studies [6] with those seen on experiments conducted with the MAGPIE pulsed power generator [26], which generates a current peak of 1,4 MA in 240 ns. The differences between their peak currents and risetimes produce effects which will be studied in this chapter. These comparisons are made by using of numerical simulations obtained via GORGON [25]. 32

47 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 33 The main effects produced by varying the current output affects some plasma features such as the electronic density, ablation times, plasma modulation, among others. Figure 3.1: End-on schematic of an asymmetrical cylindrical array. The two thick wires (25 µm) are represented by the red dots, while the thin wires (10 µm) are represented by the white dots. As a consequence of the previously mentioned aspects, the main idea here is to study the dynamics of the generated precursor of converging plasma, which is strongly affected by the asymmetry imposed on the topology of the surrounding magnetic field. This asymmetry is induced in the following way. A certain amount of wires (two in these cases) are made thicker than the others. In this chapter, cases with 4 or 8 total wires will be considered. In both experiments, two adjacent wires are thicker than the others [6]. Is important to consider that inductance factors will determine how the current is distributed along the wires [6], still. Eventually the thinner wires, because of having less mass than the others, will ablate first. This eventually creates a subarray, made of the remaining thicker wires, through which the rest of the current

48 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 34 keeps flowing. This will have an asymmetric impact on the B = 0 which determines the direction of motion of the plasma. In a perfectly symmetrical case, the B = 0 is expected to remain in the geometrical center of the array in an ideal situation. However, in this asymmetrical case, the zone of zero magnetic field will move towards the center of the sub-array formed by the two thicker wires. This affects the plasma distribution by moving it towards the remaining wires. Figure 3.2: End-on view of a GORGON simulation of an asymmetrical wire array consisting of 2 thin (top and right) and 2 thick wires (bottom and left) with diameters of 10 µm and 25 µm, respectively [4]. The green arrow at 320 ns shows the general direction of the precursor which moves from the geometrical center towards the area between the two thick, remaining wires. It is important to note that, from a numerical point of view, it is more accurate to talk about a minimum magnetic field spot instead of a B = 0 spot. This is because, in the simulations, a numerical value of exactly zero is never observed.

49 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 35 According to Figure 3.2 at the initial time of 140 ns the configuration is mostly symmetric, this can be seen from the 3, 5 T magnetic contour line, which has an almost symmetric gear shape. This shape is also seen in the 5 T line in the 200 ns image. However asymmetries are much more noticeable from 260 ns onwards. Specifically, at 260 ns, the thinner wires located at the top and right of the schematic are almost ablated, which causes the remaining two thicker wires to form a type subarray, which attracts the B = 0 towards their location, shifting the converging center of the plasma towards their center. Finally, at the 320 ns time stage, high density regions of plasma form a thick line between the two remaining thick wires, with mass densities of approximately 1 kg/m 3 or more. The inner contour is clearly displaced to the bottom left of the image, towards the center of the thick wires, which clearly shows that the B = 0 is not centered within the array. These figures illustrate how this minimum magnetic field zone can be thought of as the main governor of the precursor motion. Overall, this is a brief introduction to illustrate the main phenomena expected from an asymmetrical wire array. In sections 3.2, 3.3 and 3.4, two separate comparisons between simulations with different current traces are shown Electrical currents Regarding the electrical currents used for these simulations, it is important to repeat that two different pulsed power generators were simulated, which are Llampüdkeñ [11] and MAGPIE [26]. Their current outputs are plotted in Figure 3.3.

50 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 36 Figure 3.3: Both current traces used for these comparisons, which correspond to Llampüdkeñ and MAGPIE generators. It is important to observe that in Figure 3.3, the Llampüdkeñ curve seems much less smooth than the MAGPIE curve. This occurs because it is possible to insert the current into GORGON in two ways. The first way, is to insert experimental current traces into the code (which are obtained directly from the equipment that measures the current on a pulsed power generator). In this case the current has electrical noise, as can be seen on the Llampüdkeñ curve in the afore-mentioned figure. The second way, is to insert the current as a function of time into GORGON. In MAGPIE, the function depicted in Equation 3.1 was used for inserting the electrical current into the code [25, 26]. The use of this function for the MAGPIE case, explains why its curve looks smoother than the Llampüdkeñ curve in Figure 3.3: I(t) = A sin 2 (ωt) ; A = 1, ampere ; ω = π 2t p, (3.1)

51 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 37 In Equation 3.1 t p = 240 ns corresponds to the current peak time. For the Llampüdkeñ current, the generator is structured to follow a similar current shape [9]. Of course, in an experimental environment, electrical noise is present as was explained before. However, it can still be roughly thought of as following a sin 2 shape. The Llampüdkeñ generator has had some overhauls during its existence [6,9], which have modified its current output. The current trace obtained from the generator for use with these simulations corresponds roughly to a 350 ka peak obtained at a time of 350 ns. An important parameter for comparing generatorsis the risetime, which is A/s for Llampüdkeñ and 5, A/s for MAGPIE, resulting a difference of less than an order of magnitude between them. This is still large enough to indicate using the two for different purposes [4,6]. These risetimes were calculated from their current peaks and their time locations Undermassed, evenly massed and overmassed arrays There are three ways on how the load (the wires and their weight) placed on a pulsed power generator is related to the current. Based on this the arrays can be defined as undermassed, evenly massed or overmassed. An array is called undermassed if the wires completely ablate before the current peak. On the other hand, an array is called evenly massed if the wires finish their ablation process precisely at the current peak. Finally, an array is overmassed when the wires ablate after the current peak or if they don t ablate at all.

52 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 38 The rate of mass per unit length removed (transformed from solid to plasma) from the ablated wires can be written from the momentum balance [34]: V dm dt = µ 0I 2 4πR 0 (3.2) Where V is the assumed constant speed on which the stream moves from the dm wire towards the center. represents the rate at which the wire is losing mass. dt µ 0 is the vacuum permeability, I is the current and R 0 is the array radius [34]. By rearranging terms is possible to solve this differential equation for m, which gives: m(t ) = µ 0 4πR 0 V T 0 I 2 (t) dt (3.3) However for this thesis the currents of both generators can be approximated to an A sin 2 (ωt) function, so the integral can be easily solved. m(t ) = µ 0A 2 4πR 0 V T 0 sin 4 (ωt) dt (3.4) Equation 3.4 represents the total amount of mass lost by each wire while ablating for a certain time T. It works when the current is modelled with a squared sine function. By substracting this expression, evaluated on the time where the peak current is produced, to the initial mass placed at the generator, is possible to anticipate if an array is overmassed, evenly massed or undermassed. Specifically: m i m(t p ) > 0 (3.5) Where m i is the initial mass and t p represents the peak time. Expression 3.5 indicates that the array is overmassed, because after the peak time there is still some solid wire left, whereas: m i m(t p ) < 0 (3.6)

53 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 39 Indicates that at the peak time each wire has lost more mass than what they initially had, which, of course, is impossible, and indicates that the total mass was lost before reaching the peak time. This is the case of an undermassed array, and finally: m i m(t p ) 0 (3.7) Represents an evenly massed array, which indicates that the wire finishes ablating approximately when the current peak is reached. Equation 3.4 allows to anticipate whether an array will be under, evenly or overmassed before the actual experiment is conducted.

54 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS wire array Wire Material Tungsten Wire Diameter 2 x 25 µm (red wires) and 6 x 10 µm (white wires) Array diameter 8 mm Array height 7,08 mm Cell size 30 µm Table 3.1: Experimental parameters inserted in GORGON for the simulation described. The first case studied in detail corresponds to an 8 tungsten wire array. The specifications are listed in Table 3.1. It is important to mention that two cases will be shown. First, using the Llampüdkeñ current, and, second, with the MAGPIE current Ablation times Figure 3.4: End-on schematic of an 8 wire asymmetrical cylindrical array. When taking into account the current Llampüdkeñ generator, the thin wires ablate at times between 450 and 460 ns. The thicker wires ablate outside the time

55 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 41 range of the simulation, which ends at 730 ns. This delay in the ablation times must necessarily be compared with the Llampüdkeñ current peak time, which is 350 ns. The mass per unit length of this system is about 2, kg/m. Figure 3.5: The top two rows show end-on and side-on views of the asymmetrical 8 tungsten wire experiment with the Llampüdkeñ generator current. The bottom two rows show end-on and side-on views of the symmetrical 8 tungsten wire experiment with this same current. The end-on views are complemented with magnetic contour lines. By applying equation 3.4 to this system, about 88% of the total mass of the wires has been ablated at the time when the current peak is reached. This is taking into account an experimental variation of up to 5% of the current peak and up to

56 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 42 5% on the peak time. Considering this, this system when used with the Llampüdkeñ generator can be considered as overmassed, because when reaching the peak time there is very little mass remaining (less than 12%). The overmassed feature of this load when placed in this generator makes is a possibly useful tool to control plasma dynamics. This can be seen from the magnetic contour lines on the top row of Figure 3.5. On the other hand, the bottom two rows of Figure 3.5 show a completely symmetrical system, consisting of an equivalent load made of 8 wires with a diameter of 17 µm. The mass per unit length of this system is about 3, kg/m. It is necessary to recall that 2 wires of 25 µm and 6 wires of 10 µm were used in the asymmetrical simulation. Regarding ablation times, the symmetrical system will ablate outside the 550 ns time range shown. To be precise, the collapse occurs at approximately 650 ns. This shows that this system is also overmassed when used with the Llampüdkeñ generator, which peaks at 350 ns. To understand the differences between ablation times, the results obtained with the MAGPIE generator will be shown in Section 3.4. Is important to remember that MAGPIE has roughly 4 times more peak current than Llampüdkeñ and peaks 100 ns earlier, which greatly affects the ablation times and magnetic field topology. The complete breakage of the thin wires occurs at roughly 230 ns, which is approximately half the time of the breakage seen for the Llampüdkeñ case. The thick wires collapse at approximately 450 ns.

57 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 43 It is necessary to again apply equation 3.4 to this system but now with the MAGPIE current. It has an 1,4 MA current peak reached at 240 ns [26]. This calculation is again made in order to see how the mass of the load relates to the current output. According to this calculation, all the mass of the wires is ablated at approximately 220 ns, making this system evenly massed. Visualizations of the MAGPIE case can be appreciated on section 3.4 of this work Magnetic field topology It can be seen in Figure 3.5, that the magnetic contour lines are almost symmetrical from an azimuthal point of view, and this feature is not expected to be significantly altered until the ablation of the thin wires has begun (see Figure 3.5). This is seen specifically when studying the 500 ns time step of the experiment, where contour lines of higher magnitude than the rest of the system, around 12 T, start to appear around the thick wires, which are located at the bottom left and left sides of the scheme. At this same time, a horned shaped 10 T contour line appears (see Figure 3.6). Its geometry surrounds the plasma precursor, and also extends to the dense streams coming from the thick wires towards the center, without reaching the thick wires from which they come from. The apparition of this line can indicate the end of the azimuthally symmetric phase of this scheme. It was confirmed from these simulations, visualized in Figures 3.5 and 3.6, that by following the lowest magnetic field region, it is possible to approximately obtain the position of the precursor, which is displaced towards the thicker, surviving wires. This displacement is caused by the ablation of the thinner wires, which changes the topology of the magnetic field, from surrounding the system as a whole, to sur-

58 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 44 Figure 3.6: Detail of the end-on, 500 ns view of Figure 3.5 for the Llampüdkeñ case. Here it is possible to see a horned 10 T magnetic contour line which surrounds the plasma precursor, also extending to the dense streams coming from the thick wires towards the center, without actually reaching the thick wires from where they come. The blue arrow highlights the location of this contour line [4]. round just the surviving wires. This will also displace the region of lowest magnetic field from the geometrical center towards these wires. It is theoretically possible to have a B = 0 region at r = 0, if a cylindrical coordinate system is used, with its origin located at the geometrical center of the array. Due to the Lorentz force, the plasma converges to the geometrical center. Also contributing to this fact is that this lowest magnetic field region is located at the geometrical center. Here, magnetic pressure is the mechanism responsible for containing the plasma in this lower magnetic field sector, which by definition corresponds to magnetic confinement. As seen in the simulations, the lowest magnetic field region is not precisely equal to zero. Its values have ranged between 1 and 5 T. A possible explanation

59 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 45 for this could be the cell size, which is 30 µm. This cell size, while small for computational purposes, may be large for physical purposes. This also opens the question of how wide this B = 0 region would be in the real experiment. This may be very difficult to measure. It may be of infinitesimal size. When simulating the MAGPIE generator, it can be understood that a generator with a higher current output and an earlier peak time, is more useful when trying to alter the magnetic field topology. From 60 to 220 ns it can be seen how the system remains almost symmetrical. This can be seen both in the magnetic field contours, which keep showing an uniform gear shape, especially in the 17 T line at 220 ns, and in the side-on visualization, which shows how the precursor remains at the geometrical center of the system. The visualization depicted at Figure 3.6 shows that is possible to control de displacement of the B = 0 placing thicker wires in the direction where the plasma is desired to move. Also is possible to control the time at which the movement wants to be achieved, it can be done either by adjusting the thickness of the thin wires, or by working in a generator with a different current peak and rise time. The MAGPIE case is especially helpful for when seeing the motion of the precursor, which is not placed exactly at the magnetic field minimum. Instead, it is located roughly 2 mm behind it. This delay in the displacement shows how the precursor tends to follow the B = 0. It appears that the plasma requires some critical time to accommodate to the changing magnetic field topology Plasma densities By checking these simulations, and also comparing them with data from the real experiment, it may be said that the precursor has a small displacement of about

60 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 46 Figure 3.7: End-on interferometry of the experiment. The red arrow points towards the higher plasma density region [4]. 1 mm [4]. This is taken from the higher electronic density region shown in Figure 3.7, which is assumed to be the precursor. The diagnostic seen in this Figure is end-on interferometry. This displacement is small considering that 500 ns have passed since the beginning of the current pulse (150 ns after the current peak). This means that at this time, this system still has a high degree of symmetry, which is reinforced by observing Figure 3.7, where the 8 T and 10 T contour lines are highly symmetrical. Also from this simulation is possible to measure the drift velocity of the precursor as m/s. Recalling Figure 3.7, the the geometrical center is marked with a blue X, with the higher density region located about 1 mm off-center in the direction of the thick wires. The magnified region visible on the right of the Figure shows the displacement from the geometrical center in greater detail.

61 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 47 Further information can be obtained from the experiment, from the simulation, and also from the literature [4,36]. Specifically, end-on interferometry (Figure 3.7), along with the side-on shadowgraphy from the experiment (Figure 3.8), can be used get a better and more precise perspective of the experiment. From the 200 ns side-on shadowgraphy seen in Figure 3.8, several interesting features can be seen. As mentioned in [6], and as the left yellow shadow indicates, the plasma precursor is slightly displaced from the center, pointing towards the thick wires located on the left of the image. It is important to note that the precursor looks hollow from this diagnostic. However, this does not mean it has lower plasma density at the center. The shadowgraph diagnostic works with refraction index gradients. This means that the gradients are low inside the precursor. The yellow boxes located at the top parts of Figure 3.8 show the plasma flares coming from the thinner wires. This perspective does not define the precise directions of these flares. However, it is possible to say that they point approximately towards the geometrical center of the array. Also at this time the displacement of the precursor is not significant, so it is not expected that the flares could be pointing in other directions. It is important to note the slight upward pointing direction of the flares. This is especially noticeable in the bottom, magnified images of Figure 3.10, where the flares are indicated with the red arrows. In Figure 3.8, a side-on shadowgraph at a later time of 437 ns is shown. The main feature noted at these experiments is that the ablation phase comes at a much later stage. The time obtained from the experiment is close to the ablation time predicted by the simulation.

62 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS Figure 3.8: Side-on shadowgraph of the experiment at times 200 and 437 ns. The red arrows denote the plasma flares coming from the wires [35]. On the right side of Figure 3.8 (437 ns) it is possible to note, indicated by the red arrows, that the plasma flares clearly have a component which points towards the thick wires instead of the geometrical center. This clearly suggests that a significant change in the magnetic field topology is occurring at this stage. Also at this time the flares are not pointing upwards as seen at the 200 ns stage. 48

63 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 49 On the other hand, it is important to note the top yellow arrow at the 437 ns stage, which shows the precursor at a more developed stage. It shows a very clear m = 1 MHD instability. This instability has been reported to occur in cases where there is finite pressure pressing the precursor [36], and the very existence of this instability indicates that there is current flowing through the plasma that creates it. In this case, the plasma pressure coming from the wires plays a significant role. But even more importantly, the presence of this MHD instability indicates that a significant amount of current is flowing through the precursor. In single wire experiments [36], it is possible to see the development of an m = 0 instability. However, this kind of instability is not formed here, probably because they tend to be present in environments with a much lower ambient pressure [36].

64 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS wire array Figure 3.9: End-on schematic of an asymmetrical cylindrical array. In broad terms the idea is to make the two red wires thicker than the others Wire Material Aluminium Wire Diameter 2 x 25 µm (red wires) and 2 x 10 µm (white wires) Array diameter 8 mm Array height 7,08 mm Cell size 30 µm Table 3.2: Experimental parameters inserted in GORGON for the simulation described.

65 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS Ablation Times Figure 3.10: Electronic density visualization for the same 4 array simulation made with MAGPIE and Llampüdkeñ currents.

66 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 52 The first, and most notable effect, is the time each array takes to implode, as is summarized in the previous figure. For the MAGPIE case, at about 70 ns, the thinner 10 µm wires (located at the top and right positions) are gone and have been completely transformed into plasma. For the thicker 25 µm wires, the ablation process is completed at approximately 100 ns. It is important to notice that all times are referenced to the beginning of the current pulse. On the other hand, for the Llampüdkeñ case, at about 270 ns the thinner 10 µm wires complete the ablation process. For the thicker 25 µm wires, the ablation process is finished at 340 ns. This first result leaves the initial impression that, for the MAGPIE case, the array is extremely undermassed, since all the wires are gone by 100 ns, and the current peak only occurs at 240 ns. On the other hand, for the Llampüdkeñ case, the system is overmassed. The main explanation for the time difference between the ablation of both arrays corresponds to the difference between the risetimes, which is almost 5 larger in MAGPIE than in Llampüdkeñ. Again, to be more strict with this concept, is necessary to again apply equation 3.4 to study the relationship between the load placed on the generators. This system has a mass per unit length of 3, kg/m. This load is almost one order of magnitude lighter than the tungsten case. It is important to consider that tungsten is approximately 7 times lighter than aluminium. In the Llampüdkeñ case, at the peak time, about 20% of the mass still remains this makes this system a bit overmassed for this generator. On the other hand, for the MAGPIE case, all the wires are ablated at approximately 120 ns. This makes this system extremely undermassed. All these calculations were made with equation

67 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS The ablation times are an important factor when these simulations are intended to be performed as experiments in the laboratory. Attempts to reproduce the MAG- PIE case as an experiment can be difficult, due to the limitations in hardware and available diagnostics when taking into account the critical times of the system caused by its severe undermassing. Considering that the Llampüdkeñ case is almost evenly massed when observing the simulation, this will be a better choice if experiments on the laboratory are to be performed Precursor motion These asymmetrical arrays, as explained above, will cause the plasma to move strongly towards the thick wires, which require more time than the thinner wires to ablate. This occurs because of a change in the topology of the magnetic field after the complete ablation of the thinner wires, building a sub-array made of the remaining thick wires, which creates a global magnetic field that surrounds these arrays. This change in the topology will cause the B = 0 (or the minimum magnetic field zone) to progressively shift its position from the center of the complete array to the point between the remaining two thick wires. The topology of the global magnetic field will be affected by the input current, the amount of wires, and the wire material, which causes the explained movement of the plasma. In the case of the 4 aluminium wires with diameters of 2 10 µm and 2 25 µm, with an array diameter of 8 mm, and with the current from the MAGPIE

68 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 54 generator (1,4 MA, 240 ns) [26], it is seen that the system is extremely undermassed, considering the amount of current generated by MAGPIE. All of wires are gone in less than 110 ns, so that the dynamics of this system are extremely fast compared to any other case discussed here. Figure 3.11: Side-on electronic density visualization from the 4 aluminium wires MAGPIE case for times 30, 60 and 100 ns. The upper part shows an end-on view, and the bottom part shows the X-Z cross-section. Plasma densities in the precursor are extremely high, lying between and m 3. Electronic densities surrounding the wires can reach values above m 3. This is similar to the case where aluminium wires were used as well, and where the electronic densities are much higher expected or what has been shown in previous experiments [4,36].

69 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 55 From Figure 3.11, it is important to notice how, for a period of 30 ns, the system appears to be perfectly symmetric. This is due to the lack of a precursor at this time, as is be more visible in the end-on visualizations (Figure 3.11). As the pressure increases, and time advances, the instability starts to take a more characteristically m = 1 shape. It is particularly visible in Figure 3.11 at 60 ns. Also at this time is possible to see how the thin wire (located at y = 13 mm) starts its breakage process from the lower parts. However, the system still keeps most of its symmetry when looking at the precursor. This strongly suggests that the times are still too short for the magnetic field to lose its azimuthal symmetry. At 100 ns the thin wires are completely gone, and the precursor is rapidly displaced to a more central position between the two surviving thick wires, which are also starting to break up at this time. The estimated precursor velocity for this case is 5 x 10 5 m/s, which is much faster than for any of the previous cases. To notice the change in magnetic field topology, it is necessary to take a look at the end-on views with magnetic field contours. The symmetry of the contour lines is almost perfect for 30 ns. However, it can be seen how it starts to break down just before the complete thin array breakage, which occurs at 60 ns. The 29 T contour line is helpful in seeing this effect. It tries to circumscribe both of the thicker (25 µm) wires, located at left and bottom of the figure, but still does not have sufficient magnitude to achieve this. No similar contour can be seen trying encompass join the thin wires. However, when seeing this end-on figure, and its corresponding 60 ns sideon figure (see Figure 3.11), the asymmetry in the magnetic field topology is not sufficient to cause an important displacement of the precursor from the geometrical

70 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 56 center. When observing the diameters of the 29 T contours which surround every wire at this time stage, it is possible to note that all of them have similar diameters of approximately 1 mm. This indicates that similar amounts of current are flowing through each wire, with a calculated value of 145 ka. The position of the precursor can be located more accurately in the side-on view. In this case the electronic density of the precursor is lower than the electronic density of the plasma that surrounds the wires. The average electronic density of the precursor at 60 ns is about m 3. By contrast, the electronic density of the plasma that surrounds the wire is approximately 2 orders of magnitude greater (see Figure 3.11). This much lower density of the precursor indicates two things. First, at this stage, not enough plasma is being fed to it from the wires. Second, a small amount of current (compared to the current flowing through the wires) is flowing through it. This is further reinforced by the observation of an irregular, but centered, 19 T contour line with a rough diameter of 1 mm, which is lower than the 29 T contour that surrounds the wire at this time. For this case, at 100 ns, the thin wires have ablated completely. This means that this configuration is extremely undermassed. In other words, the wires are completely turned into plasma well before the current peak. As a consequence, this case is not very useful for trying to direct a plasma precursor in a particular direction and with a particular velocity in an experiment, because the critical times (ablation times) are very short, creating difficulties for an experimental setup with different plasma diagnostics.

71 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS Dimensionless parameters When comparing two different generators a more precise analysis can be conducted if the following dimensionless parameters are introduced: t d t t p (3.8) And: I d I(t) I p (3.9) Which respectively represent dimensionless time and dimensionless current. The idea behind these concepts is to take the time and current with respect to the current time peak and the current peak themselves. Is possible to assign the following numerical values to these parameters: t p = 240 ns ; I p = 1, 4 MA, (3.10) For MAGPIE [26] and: t p = 350 ns ; I p = 350 ka, (3.11) For Llampüdkeñ [9]. It s important to indicate that when comparing both generators at the same t d, MAGPIE is delivering 4 times more current than Llampüdkeñ wire array comparison The first important comparison to make is how the experiments made by both generators look at their peak current. The same 8-wire array shown before will be used now.

72 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 58 Figure 3.12: End-on (top) and midplane side-on (bottom) electronic density comparison of both generators at their peak currents (t d = I d = 1). By looking at Figure 3.12 is possible to note a few interesting facts. At first is possible to observe that the plasma being emitted by the thick wires for the MAGPIE case is slightly more dense than for Llampüdkeñ, however both remain at the same order of magnitude of approximately m 3. However, the precursor area as a whole shows maximum sustained electronic density values of approximately m 3. Also at the end-on view of the MAGPIE case is possible to observe how a stream of plasma of an electronic density of approximately m 3 is leaving the array to the 2 o clock direction. This can be explained by a destabilization of an MHD m = 1 instability formed at the plasma precursor at an earlier time, which was not able to sustain itself after the breakage of the thinner wires which changed

73 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 59 Figure 3.13: Midplane side-on electronic density view of the MAGPIE case at t d = 0, 625. the magnetic field topology of the system. Some plasma of this instability was facing this direction and then departed the array when the magnetic field couldn t hold it in place. This instability can be appreciated at Figure 3.13, where the roughly alternating regions of plasma going left and right from the precursor suggest the presence of an m = 1 kink instability. MHD instabilities requires a high amount of current flowing through the plasma that generates it. It seems that Llampüdkeñ s current is not enough to create this kind of instability at the precursor. The most important and noticeable aspect is to observe how at its peak time MAGPIE was able to completely ablate the thinner wires, this, of course, can be explained by the total absolute current delivered by MAGPIE is approximately 4 times higher as of Llampüdkeñ. Still at the ablation aspect, is interesting to observe how the core of the wires for the MAGPIE simulation seems to be more solid at this stage than for the Llampüdkeñ case, which seems to contradict the fact that the MAGPIE wires ablated first.

74 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 60 Figure 3.14: Close ups of the thick wires while ablating for both cases. The yellow arrows indicate less dense areas where the wire breakage has occurred. This image corresponds to t d = 1. By looking at Figure 3.14 is possible to confirm the Llampüdkeñ wires are in a more advanced stage of ablation than for the MAGPIE case. This suggests that the absolute time passed since the current pulse started governs the breakage of the wires more than the absolute current flowing through them. Is important to remember that at this stage for the Llampüdkeñ case it s 110 ns later than for the MAGPIE case. In Figures 3.14 and the bottom of Figure 3.12 the side-on visualizations were made at exactly the same plane, so visualization problems are completely ruled out. This visualization cuts exactly at the center of the wire, because of this is possible to confirm the breakage in the wires on the Llampüdkeñ case. Now is interesting to find the time at which the thin wires in the MAGPIE case completely broke apart and compare that with the Llampüdkeñ case at the same stage. At Figure 3.15 is possible to observe how at time t d = 0, 83 the thin wires have finished their ablation and have fully turned into plasma on the MAGPIE case,

75 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 61 Figure 3.15: End-on (top) and midplane side-on (bottom) electronic density comparison of both generators at time t d = 0, 83. however, for the Llampüdkeñ case the ablation is still ongoing for all the wires. As was said before, the ablation times for the thin wires for this case is t d = 1, 3. Beyond the peak time. This means this system is undermassed for MAGPIE and overmassed for Llampüdkeñ. Still at this stage, both generators are delivering currents of I d = 0, 93, almost reaching their peaks. However, the absolute current value for MAGPIE is much higher, this explains why the wide plasma streams coming from the wires are much denser for this case, specifically around one order of magnitude. This difference in the absolute current is also responsible for observing an m = 1 instability only in MAGPIE and not in Llampüdkeñ. This can be observed at both Figures 3.13 and

76 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS wire array comparison Figure 3.16: End-on (top) and midplane side-on (bottom) electronic density comparison of both generators at time t d = 0, 25. When observing Figure 3.17 is possible to observe how at about 100 ns the thick wires are finishing their ablation for the MAGPIE case. This time corresponds to t d = 0, 42. As a consequence, this load is extremely undermassed when being put at this generator. On this section, the same 4-wire array shown before will be compared. In Figure 3.16 an electronic density visualization is used to compare both cases at t d = 0, 25. The relevance of this time is because here the MAGPIE generator

77 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 63 finishes ablating the thin (10 µm) wires. For this case, when looking at the side-on visualization, it is possible to see a possible m = 1 instability. This can be seen by observing some regions of plasma going left, and others going to the right. Also on the end-on view is possible to see a region of dense plasma going to the top left direction. Considering that all these end-on views are planes and not full views, this indicates that plasma coming from exactly the middle height of the precursor goes on this direction. Figure 3.17: End-on (top) and midplane side-on (bottom) electronic density comparison of both generators at time t d = 0, 42. At this same time for the Llampüdkeñ case, the thin wires don t ablate yet. As was explained before, in this case they will ablate at 230 ns, which corresponds to t d = 0, 66. An interesting feature to observe at this time for this case is a nail

78 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 64 of plasma which partially surround every wire, the electronic density of this nail is of approximately m 3. This is around 2 or 3 orders of magnitude lower than the wire they surround. This nail will evolve into a ring which can be observed in Figure Still on the Llampüdkeñ case the precursor has not formed yet. Is possible to see plasma at the center of the system with an electronic density of approximately m 3. However it is still too wide (approximately 1,5 mm) and not concentrated enough to be a precursor. The MAGPIE generator was able to create a far denser precursor at this stage, with electronic densities varying between and m 3, and as it was said before, it shows an MHD instability which strongly indicates that current is flowing on it. Still on MAGPIE, the bottom part of the thin wires has completely ablated, from 0 to 2 mm when measuring from bottom to top, however the rest still has some solid remains on it. Figure 3.17 shows the same comparison but at t d = 0, 42. At this stage most of the thick wires are ablated on the MAGPIE case. Also the plasma that remains from the thin wires is trying to move towards the center of the two thick wires. This occurs because of the sudden magnetic field topology change, and the minimum magnetic zone, as was explained before, moves to in between both surviving wires. On the Llampüdkeñ case the precursor has been formed, and it exhibits an average electronic density of approximately m 3. Its thickness is also around 2 mm. It s mostly symmetrical, considering that the magnetic field topology still

79 CHAPTER 3. ASYMMETRICAL ARRAYS WITH DIFFERENT CURRENTS 65 resembles one of a completely symmetric wire array. No wire has ablated at this stage. As was said before, the thin wires will ablate at t d = 0, 66 while the thick will do at t d = 0, 97. Considering that the thick wires for this case ablate so closely to the current peak time, this system can be considered as evenly massed. On the other hand, for the MAGPIE case, it s extremely undermassed as was explained before. Figure 3.18: Close up of the nail and ring plasma structures which surround the wires in the Llampüdkeñ case. The yellow dots were added to the bottom row to help seeing the wire position. Is important to mention that these plasma nails and rings are present at different times. Going back to the nails and rings, on Figure 3.18 they can be seen more clearly, on the top row the wires are the dense red points, while on the bottom row, the wires are represented by the yellow dots. Is important to indicate that the nails enclose themselves to become rings. From this figure it can be seen that they are bigger when they surround thick (25 µm) wires than when they surround the thin (10 µm) ones. Specifically, the rings have a diameter of approximately 0,4 mm when surrounding