Small-scale solar magnetic fields

As we resolve ever smaller structures in the solar atmosphere, it has become clear that magnetism is an important component of those small structures. Small-scale magnetism holds the key to many poorly understood facets of solar magnetism on all scales, such as the existence of a local dynamo, chromospheric heating, and flux emergence, to name a few. Here, we review our knowledge of small-scale photospheric fields, with particular emphasis on quiet-sun field, and discuss the implications of several results obtained recently using new instruments, as well as future prospects in this field of research.Comment: 43 pages, 18 figure

Quantum MDS Codes over Small Fields

We consider quantum MDS (QMDS) codes for quantum systems of dimension $q$ with lengths up to $q^2+2$ and minimum distances up to $q+1$. We show how starting from QMDS codes of length $q^2+1$ based on cyclic and constacyclic codes, new QMDS codes can be obtained by shortening. We provide numerical evidence for our conjecture that almost all admissible lengths, from a lower bound $n_0(q,d)$ on, are achievable by shortening. Some additional codes that fill gaps in the list of achievable lengths are presented as well along with a construction of a family of QMDS codes of length $q^2+2$, where $q=2^m$, that appears to be new.Comment: 6 pages, 3 figure

Jet Collimation by Small-Scale Magnetic Fields

A popular model for jet collimation is associated with the presence of a large-scale and predominantly toroidal magnetic field originating from the central engine (a star, a black hole, or an accretion disk). Besides the problem of how such a large-scale magnetic field is generated, in this model the jet suffers from the fatal long-wave mode kink magnetohydrodynamic instability. In this paper we explore an alternative model: jet collimation by small-scale magnetic fields. These magnetic fields are assumed to be local, chaotic, tangled, but are dominated by toroidal components. Just as in the case of a large-scale toroidal magnetic field, we show that the ``hoop stress'' of the tangled toroidal magnetic fields exerts an inward force which confines and collimates the jet. The magnetic ``hoop stress'' is balanced either by the gas pressure of the jet, or by the centrifugal force if the jet is spinning. Since the length-scale of the magnetic field is small (< the cross-sectional radius of the jet << the length of the jet), in this model the jet does not suffer from the long-wave mode kink instability. Many other problems associated with the large-scale magnetic field are also eliminated or alleviated for small-scale magnetic fields. Though it remains an open question how to generate and maintain the required small-scale magnetic fields in a jet, the scenario of jet collimation by small-scale magnetic fields is favored by the current study on disk dynamo which indicates that small-scale magnetic fields are much easier to generate than large-scale magnetic fields.Comment: 14 pages, no figur

Large jets from small-scale magnetic fields

We consider the conditions under which a rotating magnetic object can produce a magnetically powered outflow in an initially unmagnetized medium stratified under gravity. 3D MHD simulations are presented in which the footpoints of localized, arcade-shaped magnetic fields are put into rotation. It is shown how the effectiveness in producing a collimated magnetically powered outflow depends on the rotation rate, the strength and the geometry of the field. The flows produced by uniformly rotating, non-axisymmetric fields are found to consist mainly of buoyant plumes heated by dissipation of rotational energy. Collimated magnetically powered flows are formed if the field and the rotating surface are arranged such that a toroidal magnetic field is produced. This requires a differential rotation of the arcades' footpoints. Such jets are well-collimated; we follow their propagation through the stratified atmosphere over 100 times the source size. The magnetic field is tightly wound and its propagation is dominated by the development of non-axisymmetric instabilities. We observe a Poynting flux conversion efficiency of over 75% in the longest simulations. Applications to the collapsar model and protostellar jets are discussed.Comment: 9 pages, 12 figures, accepted for publication in A&A, complementary movies at http://www.mpa-garching.mpg.de/~rmo/pap3/index.htm

Curves of genus 3 over small finite fields

We present a table containing the maximal number of rational points on a genus 3 curve over a field of cardinality q, for all q<100. Also, some remarks on Frobenius non-classical quartics over finite fields are given.Comment: 9 page