The generation of strong magnetic fields during the formation of the first stars

Cosmological hydrodynamical simulations of primordial star formation suggest that the gas within the first star-forming halos is turbulent. This has strong implications on the subsequent evolution, in particular on the generation of magnetic fields. Using high-resolution numerical simulations, we show that in the presence of turbulence, weak seed magnetic fields are exponentially amplified by the small-scale dynamo during the formation of the first stars. We conclude that strong magnetic fields are generated during the birth of the first stars in the universe, potentially modifying the mass distribution of these stars and influencing the subsequent cosmic evolution. We find that the presence of the small-scale turbulent dynamo can only be identified in numerical simulations in which the turbulent motions in the central core are resolved with at least 32 grid cells.Comment: Accepted for publication in ApJ Letter

Eddy current losses in transformer windings

Målet med denne masteroppgaven er å undersøke en semi-analytisk tapberegningsmetode, spesielt når man tar hensyn til virvelstrømstap i transformatorviklinger. Dette gjøres gjennom sammenligninger mellom simulerte og målte resultater, og gjennom sammenligninger av målte og nominell data fra produsenten. Alt dette utføres på en trefase 230/400 V testtransformator. En stor del av denne masteroppgaven er å utvide arbeidet som er gjort i fordypningsprojektet med samme navn, ved å ta i betraktning sirkulerende strømmer og utføre målinger på transformatoren. De simulerte resultatene ble funnet ved bruk av COMSOL multiphysics. Dette ble gjort ved å modellere en 3D-modell av transformatoren for å finne de sirkulerende strømningene og ved å modifisere 2D-modellene som ble konstruert i fordypningsprosjektet. De simulerte sirkulerende strømmene ble brukt til å modifisere trefase-tapene som ble beregnet i fordypningsprosjektet. Dataene som ble samlet inn fra disse simuleringene ble behandlet ved hjelp av Microsoft Excel. De nominelle dataene som ble brukt til sammenligning ble anskaffet fra produsenten av transformatoren, Noratel. De forskjellige type målinger som ble utførst varierte. En kortslutningstest ble utført og sammenlignet med de nominelle dataene. Motstandsmålinger av transformatoren ble utført for å beregne gjennomsnittstemperaturen på viklingene etter en varmekjøringstest og motstanden til viklingene ved romtemperatur. Måling av sirkulerende strømmer på primærsiden av transformatoren ble utført ved bruk av Rogowski-spoler. Dette ble gjort for å sammenligne disse strømmene med det simulerte resultatet. Termoelementer ble brukt til å måle temperaturøkningen i forskjellige vindinger, og ble brukt til å beregne kobbertapet i de respektive vindingene. Dette ble gjort på 3 forskjellige strømnivåer. For å behandle disse dataene ble MATLAB og Microsoft Excel brukt. Kortslutningstesten viste at kobbertapene som ble oppgitt i de nominelle dataene ble utført ved nominell temperatur i viklingene. Ved nominell temperatur var forskjellen mellom det målte tapet og det nominelle tapet 0,7 %. Trefase-tapene som ble funnet ved romtemperatur i fordypningsprosjektet ved 841,3 W økte til 931,7 W når man tok i betraktning sirkulerende strømmer, en økning på 10,7 %. Det korrigerte tapet er fremdeles ikke på 1,11 kW, som er kobbertapene som er registrert ved romtemperatur for transformatoren. Når man sammenligner de simulerte sirkulerende strømmene i primærviklingen og de målte sirkulerende strømmene, er det tydelig at forskjellen er ganske liten, både i amplitude og fasevinkel. Fra motstandsmålingen ble det klart at primærviklingen har størst motstand. Det var en viss forskjell mellom den målte og beregnede verdien av motstanden ved romtemperatur, 7.648 \% i sekundærviklingen og 8.215 \% i primærviklingen. Denne forskjellen antas å skyldes unøyaktige antagelser om geometri fra transformatoren og unøyaktig data om resistivitet og areal av lederen. For motstanden etter varmekjøringstest ble det klart at den beregnede gjennomsnittstemperaturen i primærviklingen stemte godt overens med den målte temperaturen. For temperaturmålingene påviste lineariteten i de første 10 minuttene av målingene at antakelsen om tilnærmet adiabatisk oppvarming var gyldig. Det ble også tydelig at den aktive effekten var proposjonal med temperaturøkningen. Når man brukte disse temperaturøkningene for å beregne effekttapet i svingene, var det noen forskjeller mellom det målte og simulerte resultatet avhengig av parallellen i viklingen, uavhengig av strøm. Årsaken til dette ble ikke funnet, og videre målinger vil være fordelaktig. Sammenligningen mellom den magnetodynamiske og semi-analytiske tapberegningsmetoden når man vurderer sirkulerende strømmer, viste små forskjeller mellom metodene, og det antas derfor at de kan brukes om hverandre ved 50 Hz. For sammenligningene mellom tapsberegningsmetodene med og uten sirkulerende strømmer, ble forskjeller tydelige avhengig av parallellene i viklingene og kan forklares ved hjelp av lekkfeltet og viklingens geometri.The aim of this master thesis is to investigate a semi-analytical loss calculation method, especially when taking into account eddy current losses in transformer windings. This is done through comparisons between simulated and measured results, and through comparison between measurements and rated data from the producer. All of this is performed on a three phase 230/400 V test transformer. A large part of this thesis is to expand the work done in the specialization project of the same name, by taking into consideration circulating currents and performing measurements. The simulated results were found using COMSOL multiphysics. This was done by modelling a 3D model of the transformer to find the circulating currents and by modifying the 2D models created for the specialization project. The circulating currents were used to modify the three phase losses calculated in the specialization project. The data gathered from these simulations were processed using Microsoft Excel. The rated data used for comparison was acquired from the producer of the transformer, Noratel. The types of measured result that were gathered varied. A short circuit heat run test was performed, being compared with the rated data. The resistance of the windings were measured both at room temperature, and after the heat run test. This was done to calculate the average temperature of the windings after a heat run test and the resistance of the windings at room temperature. Measurement of the circulating currents on the primary side of the transformer was performed using Rogowski coils. This was done to compare these currents with the simulated result. Thermocouples were used to measure the temperature increase in different turns, and was used to calculate the copper loss in the respective turns. This was done at 3 different current levels. To process this data, MATLAB and Microsoft Excel were used. The short circuit test showed that the copper losses given in the rated data was performed at operating temperature of the windings. At the operating temperature, the difference between the measured and the rated loss was at 0.7 %. The three phase losses found at room temperature in the specialization project at 841.3 W increased to 931.7 W when taking into consideration circulating currents, causing the losses to increase with 10.7 %. It is still not at 1.11 kW, which is the copper losses registered at room temperature for the transformer. When comparing the simulated circulating currents in the primary winding and the measured circulating currents, it becomes clear that the difference is rather small, both in amplitude and phase angle. From the resistance measurement, it became clear that the primary winding has the largest resistance. There were some differences between the measured and calculated value of the resistance at room temperature, 7.648 \% in the secondary winding and 8.215 \% in the primary winding. This difference is assumed to be due to inaccurate geometry assumptions of the transformer and inaccurate data about the area and resistivity of the conductor. For the resistance after the heat run test, it became clear that the calculated average temperature of the primary winding matched well with the measured temperature. For the temperature measurements, the linearity in the first 10 minutes of measurements showed that the assumption of approximately adiabatic heating was valid. It was noted that the higher the current, the bigger the temperature increase would be. When using these temperature increases to calculate the power loss in the turns, there were some differences between the measured and simulated result depending on the winding parallel, regardless of current being applied. The cause of this was not found and further research would be beneficial. The comparison between the magnetodynamic and the semi-analytical loss calculation method when considering circulating currents showed minute differences between the methods, and it is therefore assumed that they can be used interchangeably at 50 Hz. For the comparisons between the loss calculation methods with and without circulating currents, differences became clear depending on position of the winding parallel and can be explained using the leakage field and the geometry of the winding

Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

We design stable and high-order accurate finite volume schemes for the ideal MHD equations in multi-dimensions. We obtain excellent numerical stability due to some new elements in the algorithm. The schemes are based on three- and five-wave approximate Riemann solvers of the HLL-type, with the novelty that we allow a varying normal magnetic field. This is achieved by considering the semi-conservative Godunov-Powell form of the MHD equations. We show that it is important to discretize the Godunov-Powell source term in the right way, and that the HLL-type solvers naturally provide a stable upwind discretization. Second-order versions of the ENO- and WENO-type reconstructions are proposed, together with precise modifications necessary to preserve positive pressure and density. Extending the discrete source term to second order while maintaining stability requires non-standard techniques, which we present. The first- and second-order schemes are tested on a suite of numerical experiments demonstrating impressive numerical resolution as well as stability, even on very fine meshe

Statistical analysis of the mass-to-flux ratio in turbulent cores: effects of magnetic field reversals and dynamo amplification

We study the mass-to-flux ratio (M/\Phi) of clumps and cores in simulations of supersonic, magnetohydrodynamical turbulence for different initial magnetic field strengths. We investigate whether the (M/\Phi)-ratio of core and envelope, R = (M/\Phi)_{core}/(M/\Phi)_{envelope} can be used to distinguish between theories of ambipolar diffusion and turbulence-regulated star formation. We analyse R for different Lines-of-Sight (LoS) in various sub-cubes of our simulation box. We find that, 1) the average and median values of |R| for different times and initial magnetic field strengths are typically greater, but close to unity, 2) the average and median values of |R| saturate at average values of |R| ~ 1 for smaller magnetic fields, 3) values of |R| < 1 for small magnetic fields in the envelope are caused by field reversals when turbulence twists the field lines such that field components in different directions average out. Finally, we propose two mechanisms for generating values |R| ~< 1 for the weak and strong magnetic field limit in the context of a turbulent model. First, in the weak field limit, the small-scale turbulent dynamo leads to a significantly increased flux in the core and we find |R| ~< 1. Second, in the strong field limit, field reversals in the envelope also lead to values |R| ~< 1. These reversals are less likely to occur in the core region where the velocity field is more coherent and the internal velocity dispersion is typically subsonic.Comment: 12 pages, 8 figures, accepted for publication in MNRA

Magnetic fields during the early stages of massive star formation - I. Accretion and disk evolution

We present simulations of collapsing 100 M_\sun mass cores in the context of massive star formation. The effect of variable initial rotational and magnetic energies on the formation of massive stars is studied in detail. We focus on accretion rates and on the question under which conditions massive Keplerian disks can form in the very early evolutionary stage of massive protostars. For this purpose, we perform 12 simulations with different initial conditions extending over a wide range in parameter space. The equations of magnetohydrodynamics (MHD) are solved under the assumption of ideal MHD. We find that the formation of Keplerian disks in the very early stages is suppressed for a mass-to-flux ratio normalised to the critical value \mu below 10, in agreement with a series of low-mass star formation simulations. This is caused by very efficient magnetic braking resulting in a nearly instantaneous removal of angular momentum from the disk. For weak magnetic fields, corresponding to \mu > 10, large-scale, centrifugally supported disks build up with radii exceeding 100 AU. A stability analysis reveals that the disks are supported against gravitationally induced perturbations by the magnetic field and tend to form single stars rather than multiple objects. We find protostellar accretion rates of the order of a few 10^-4 M_\sun yr^-1 which, considering the large range covered by the initial conditions, vary only by a factor of ~ 3 between the different simulations. We attribute this fact to two competing effects of magnetic fields. On the one hand, magnetic braking enhances accretion by removing angular momentum from the disk thus lowering the centrifugal support against gravity. On the other hand, the combined effect of magnetic pressure and magnetic tension counteracts gravity by exerting an outward directed force on the gas in the disk thus reducing the accretion onto the protostars.Comment: 22 pages, 17 figures, accepted for publication in MNRAS, updated to final versio

Vertical structure of a supernova-driven turbulent magnetized ISM

Stellar feedback drives the circulation of matter from the disk to the halo of galaxies. We perform three-dimensional magnetohydrodynamic simulations of a vertical column of the interstellar medium with initial conditions typical of the solar circle in which supernovae drive turbulence and determine the vertical stratification of the medium. The simulations were run using a stable, positivity-preserving scheme for ideal MHD implemented in the FLASH code. We find that the majority (\approx 90 %) of the mass is contained in thermally-stable temperature regimes of cold molecular and atomic gas at T < 200 K or warm atomic and ionized gas at 5000 K < T < 10^{4.2} K, with strong peaks in probability distribution functions of temperature in both the cold and warm regimes. The 200 - 10^{4.2} K gas fills 50-60 % of the volume near the plane, with hotter gas associated with supernova remnants (30-40 %) and cold clouds (< 10 %) embedded within. At |z| ~ 1-2 kpc, transition-temperature (10^5 K) gas accounts for most of the mass and volume, while hot gas dominates at |z| > 3 kpc. The magnetic field in our models has no significant impact on the scale heights of gas in each temperature regime; the magnetic tension force is approximately equal to and opposite the magnetic pressure, so the addition of the field does not significantly affect the vertical support of the gas. The addition of a magnetic field does reduce the fraction of gas in the cold (< 200 K) regime with a corresponding increase in the fraction of warm (~ 10^4 K) gas. However, our models lack rotational shear and thus have no large-scale dynamo, which reduces the role of the field in the models compared to reality. The supernovae drive oscillations in the vertical distribution of halo gas, with the period of the oscillations ranging from ~ 30 Myr in the T < 200 K gas to ~ 100 Myr in the 10^6 K gas, in line with predictions by Walters & Cox.Comment: Accepted for publication in ApJ. Replacement corrects an error in the observed CNM pressure distribution in Figure 15 and associated discussio

A method for reconstructing the variance of a 3D physical field from 2D observations: Application to turbulence in the ISM

We introduce and test an expression for calculating the variance of a physical field in three dimensions using only information contained in the two-dimensional projection of the field. The method is general but assumes statistical isotropy. To test the method we apply it to numerical simulations of hydrodynamic and magnetohydrodynamic turbulence in molecular clouds, and demonstrate that it can recover the 3D normalised density variance with ~10% accuracy if the assumption of isotropy is valid. We show that the assumption of isotropy breaks down at low sonic Mach number if the turbulence is sub-Alfvenic. Theoretical predictions suggest that the 3D density variance should increase proportionally to the square of the Mach number of the turbulence. Application of our method will allow this prediction to be tested observationally and therefore constrain a large body of analytic models of star formation that rely on it.Comment: 8 pages, 9 figures, accepted for publication in MNRA

A robust numerical scheme for highly compressible magnetohydrodynamics: Nonlinear stability, implementation and tests

The ideal MHD equations are a central model in astrophysics, and their solution relies upon stable numerical schemes. We present an implementation of a new method, which possesses excellent stability properties. Numerical tests demonstrate that the theoretical stability properties are valid in practice with negligible compromises to accuracy. The result is a highly robust scheme with state-of-the-art efficiency. The scheme's robustness is due to entropy stability, positivity and properly discretised Powell terms. The implementation takes the form of a modification of the MHD module in the FLASH code, an adaptive mesh refinement code. We compare the new scheme with the standard FLASH implementation for MHD. Results show comparable accuracy to standard FLASH with the Roe solver, but highly improved efficiency and stability, particularly for high Mach number flows and low plasma beta. The tests include 1D shock tubes, 2D instabilities and highly supersonic, 3D turbulence. We consider turbulent flows with RMS sonic Mach numbers up to 10, typical of gas flows in the interstellar medium. We investigate both strong initial magnetic fields and magnetic field amplification by the turbulent dynamo from extremely high plasma beta. The energy spectra show a reasonable decrease in dissipation with grid refinement, and at a resolution of 512^3 grid cells we identify a narrow inertial range with the expected power-law scaling. The turbulent dynamo exhibits exponential growth of magnetic pressure, with the growth rate twice as high from solenoidal forcing than from compressive forcing. Two versions of the new scheme are presented, using relaxation-based 3-wave and 5-wave approximate Riemann solvers, respectively. The 5-wave solver is more accurate in some cases, and its computational cost is close to the 3-wave solver.Comment: 26 pages, 17 figures, Journal of Computational Physics, publishe