A Note on Capon's Minimum Variance Projection for Multi-Spacecraft Data Analysis

Capon's minimum variance projection for the multi-point measurements is revisited using the method of likelihood function to derive the minimum variance projection and a simplified error estimate analytically. Theoretical construction of the minimum variance projection assumes a Gaussian form of the likelihood function and also regards the data covariance as a proxy of the noise covariance. The minimum variance projection is extended to the problem of two-spacecraft mode decomposition in the Mercury magnetosphere in which the magnetic field is a superposition of the constant field from the current sheet and the dipolar field from the planet. The extension of the Capon estimator (the data-variance projection) can identify the signal amplitudes of the different fields with a sufficient accuracy when the statistical averaging is properly done. The Capon estimator serves as a powerful analysis tool when the spatial resolution is limited to only a few points

On the applicability of Taylor's hypothesis in streaming magnetohydrodynamic turbulence

We examine the range of applicability of Taylor's hypothesis used in observations of magnetic turbulence in the solar wind. We do not refer to turbulence theory. We simply ask whether in a turbulent magnetohydrodynamic flow the observed magnetic frequency spectrum can be interpreted as mapping of the wavenumber turbulence into the stationary spacecraft frame. In addition to the known restrictions on the angle of propagation with respect to the fluctuation spectrum and the question on the wavenumber dependence of the frequency in turbulence which we briefly review, we show that another restriction concerns the inclusion or exclusion of turbulent fluctuations in the velocity field. Taylor's hypothesis in application to magnetic (MHD) turbulence encounters its strongest barriers here. It is applicable to magnetic turbulence only when the turbulent velocity fluctuations can practically be completely neglected against the bulk flow speed. For low flow speeds the transformation becomes rather involved. This account makes even no use of the additional scale dependence of the turbulent frequency, viz. the existence of a "turbulent dispersion relation".Comment: 31 pages, 3 figures, submitted to Earth, Planets & Spac

On the ion-inertial range density power spectra in solar wind turbulence

A model-independent first-principle first-order investigation of the shape of turbulent density-power spectra in the ion-inertial range of the solar wind at 1 AU is presented. De-magnetised ions in the ion-inertial range of quasi-neutral plasmas respond to Kolmogorov (K) or Iroshnikov-Kraichnan (IK) inertial-range velocity turbulence power spectra via the spectrum of the velocity-turbulence-related random-mean-square induction-electric field. Maintenance of electrical quasi-neutrality by the ions causes deformations in the power spectral density of the turbulent density fluctuations. Kolmogorov inertial range spectra in solar wind velocity turbulence and observations of density power spectra suggest that the occasionally observed scale-limited bumps in the density-power spectrum may be traced back to the electric ion response. Magnetic power spectra react passively to the density spectrum by warranting pressure balance. This approach still neglects contribution of Hall currents and is restricted to the ion-inertial range scale. While both density and magnetic turbulence spectra in the affected range of ion-inertial scales deviate from Kolmogorov or Iroshnikov-Kraichnan, the velocity turbulence preserves its inertial range shape in this process to which spectral advection turns out to be secondary but may become observable under special external conditions. One such case observed by WIND is analysed. We discuss various aspects of this effect including the affected wavenumber scale range, dependence on angle between mean flow velocity and wavenumber and, for a radially expanding solar wind flow when assuming adiabatic expansion at fast solar wind speeds and a Parker dependence of the solar wind magnetic field on radius, also the presumable limitations on the radial location of the turbulent source region.Comment: Pages 18, Figures 5, Discussion paper submitte

On isospin and flavour of leptons and quarks

Isospin emerges naturally from the Lorentz transformation of spinors, if they are based on the vector representation of the Lorentz group. The resulting extended Dirac equation for a massive spin-one-half fermion has two new additional degrees of freedom associated with the up and down components of isospin. This doublet is interpreted as describing the electron and neutrino. It is adjoined with the SU(2) symmetry group. The extended Dirac equation appears in six versions which are connected by similarity transformations. It is argued that this trait may explain the occurrence of the three families of the leptons and suggested that flavour arises genuinely from the algebraic properties of the extended Dirac equation. Its solutions are discussed and the physical role of isospin is elucidated. Isospin symmetry can be gauged, which leads to a weak-interaction-type theory and is valid for finite initial mass. Breaking the isospin SU(2) symmetry yields the correct electric charges of the particles by means of the electroweak unification procedures of the standard model

Fundamental Fermion Interactions via Vector Bosons of Unified SU(2)⊗SU(4) Gauge Fields

Employing the fermion unification model based on the intrinsic SU(8) symmetry of a generalized Dirac equation, we discuss the fundamental interactions under the SU(8) = SU(2) ⊗ SU(4) symmetry group. The physics involved can describe all fermions, the leptons (electron and neutrino), and the colored up and down quarks of the first generation in the standard model (SM) by a complex SU(8) octet of Dirac spinor fields. The fermion interactions are found to be mediated by the unified SU(4) and SU(2) vector gauge boson fields, which include the photon, the gluons, and the bosons Z and W as well known from the SM, but also comprise new ones, namely three colored X bosons carrying a fractional hypercharge of ± 4/3 and transmuting leptons into quarks and vice versa. The full covariant derivative of the model is derived and discussed. The Higgs mechanism gives mass to the Z and W bosons, but also permits one to derive the mass of the colored X boson, for which depending on the choice of the values of the coupling constant, the estimates are 35 or 156 GeV, values that are well within reach of the LHC. The scalar Higgs field can also lend masses to the fermions and fix their physical values for given appropriate coupling constants to that field

Correction to: On isospin and flavour of leptons and quarks

Isospin emerges naturally from the Lorentz transformation of spinors, if they are based on the vector representation of the Lorentz group. The resulting extended Dirac equation for a massive spin-one-half fermion has two new additional degrees of freedom associated with the up and down components of isospin. This doublet is interpreted as describing the electron and neutrino. It is adjoined with the SU(2) symmetry group. The extended Dirac equation appears in six versions which are connected by similarity transformations. It is argued that this trait may explain the occurrence of the three families of the leptons and suggested that flavour arises genuinely from the algebraic properties of the extended Dirac equation. Its solutions are discussed and the physical role of isospin is elucidated. Isospin symmetry can be gauged, which leads to a weak-interaction-type theory and is valid for finite initial mass. Breaking the isospin SU(2) symmetry yields the correct electric charges of the particles by means of the electroweak unification procedures of the standard model

Lorentz invariance and the spinor-helicity formalism yield the U(1) and SU(3) fermion symmetry

In this paper we use the so-called spinor-helicity formalism to represent three-vectors in terms of the Pauli matrices and derive a generalized relativistic wave equation for a massive fermion of spin one-half. We thus extend the Dirac equation by making use of the Pauli-Lubański operator that includes isospin explicitly. As a consequence, we get new degrees of freedom related to isospin helicity, in addition to the two standard ones of the Dirac equation that are associated with the kinetic spin-helicity doublet and the particle-antiparticle pair. Formally, isospin helicity has 2(2s+1) degrees of freedom for an arbitrary general isospin s and has the eigenvalues s and -(s+1) , and thus it reveals a kind of hidden symmetry in any isospin field. The resulting four degrees of freedom for isospin 1/2 are interpreted as being associated with two independent subspaces of dimension 1 related to the U(1) and 3 related to SU(3) symmetry, i.e. to the leptons and quarks

Twisted magnetic flux tubes in the solar wind

Magnetic flux tubes in the solar wind can be twisted as they are transported from the solar surface, where the tubes are twisted owing to photospheric motions. It is suggested that the twisted magnetic tubes can be detected as the variation of total (thermal+magnetic) pressure during their passage through observing satellite. We show that the total pressure of several observed twisted tubes resembles the theoretically expected profile. The twist of isolated magnetic tube may explain the observed abrupt changes of magnetic field direction at tube walls. We have also found some evidence that the flux tube walls can be associated with local heating of the plasma and elevated proton and electron temperatures. For the tubes aligned with the Parker spiral, the twist angle can be estimated from the change of magnetic field direction. Stability analysis of twisted tubes shows that the critical twist angle of the tube with a homogeneous twist is 70$^0$, but the angle can further decrease owing to the motion of the tube with regards to the solar wind stream. The tubes with a stronger twist are unstable to the kink instability, therefore they probably can not reach 1 AU.Comment: 4 pages, 4 figures, accepted in ApJ

Low Frequency Waves Upstream and Downstream of the Terrestrial Bow Shock

The thesis makes use of the four point measurements provided by the Cluster spacecraft mission and determines not only wave vectors but also dispersion relations, propagation patterns, and wave number spectra in the upstream and the downstream regions of the terrestrial bow shock. Furthermore, those wave properties are determined in a proper frame of reference, the plasma rest frame, by correcting the Doppler shift. In the upstream region the dispersion relation exhibits two branches which correspond to the right-hand polarized magnetosonic/whistler mode and the ion beam resonant mode, respectively. The upstream waves exhibit phase velocities oriented toward upstream along the magnetic field in the plasma rest frame. In the downstream region the dispersion relations show a transition to the mirror mode whose properties are frequently detected in the statistical study, too. Interestingly, the mirror modes have finite propagation speeds, possibly coupled to the background inhomogeneities known as the drift mirror mode. While the upstream waves propagate nearly parallel to the background magnetic field, the downstream waves propagate nearly perpendicular. On a statistical average there is an organization of wave propagation pattern: outward divergent in the upstream region; toward the magnetosheath flank region aligned with the plasma flow direction in the downstream region; and inward convergent in the magnetosheath flank. The wave number spectra are also directly determined for the magnetic field fluctuations in the shock upstream region and they are interpreted in light of turbulence. The fluctuations exhibit properties of not fully developed turbulence but intermittency, suggesting that there is not enough time for turbulence to become fully developed.Die CLUSTER-Raumsondemission ermöglicht erstmals Messungen an vier Raumpunkten im erdnahen Weltraum, damit man nicht nur Wellenvektoren, sondern auch Dispersionsrelationen und Ausbreitungsmuster der Wellen sowie Eigenschaften der Turbulenz nahe der Bugstoßwelle der Erde bestimmen kann. Außerdem können solche Eigenschaften durch die Berechnung der Doppler-Verschiebung im richtigen Rahmen, Plasma-Ruhesystem, dargestellt werden. Die Dispersionsrelationen im Upstream-Gebiet (vor der Bugstoßwelle) zeigen zwei Kurven, die der rechthändige zirkular polarisierte Mode und der Ionenstrahl-resonante Mode entsprechen. Im Downstream-Gebiet (hinter der Bugstoßwelle) zeigen die Dispersionsrelationen eine Kurve, die der Mirror-Mode entspricht. Die Eigenschaften der Mirror-Mode ist statistisch am häufigsten im Downstream-Gebiet beobachtet. Interessanterweise zeigen die Mirror-Mode eine endliche Ausbreitungsgeschwindigkeit. Die Mirror-Mode mag also mit der räumlichen Inhomogenetät gekoppelt werden. Während die Upstream-Wellen sich parallel zum Magnetfeld nach außen von der Erde ausbreiten, breiten sich die Downstream-Welle senkrecht zum Magnetfeld und nach innen zur Erde aus. Wellenzahlspektren im Upstream-Gebiet werden auch direkt bestimmt, und die Fluktuationen werden vom Blickwinkel der Turbulenz interpretiert. Die Upstream-Wellen stellen die Eigenschaften der intermittenten Turbulenz dar, d.h. das Upstream-Gebiet hat keine Zeit, um voll entwickelte Turbulent zu werden