Chandrasekhar-Kendall Functions Research Articles

Polarization properties of turbulent synchrotron bubbles: an approach based on Chandrasekhar–Kendall functions

Synchrotron emitting bubbles arise when the outflow from a compact relativistic engine, either a Black Hole or a Neutron Star, impacts on the environment. The emission properties of synchrotron radiation are widely used to infer the dynamical properties of these bubbles, and from them the injection conditions of the engine. Radio polarization offers an important tool to investigate the level and spectrum of turbulence, the magnetic field configuration, and possibly the degree of mixing. Here we introduce a formalism based on Chandrasekhar-Kendall functions that allows us to properly take into account the geometry of the bubble, going beyond standard analysis based on periodic cartesian domains. We investigate how different turbulent spectra, magnetic helicity and particle distribution function, impact on global properties that are easily accessible to observations, even at low resolution, and we provide fitting formulae to relate observed quantities to the underlying magnetic field structure.

Chandrasekhar–Kendall functions in astrophysical dynamos

Some of the contributions of Chandrasekhar to the field of magnetohydrodynamics are highlighted. Particular emphasis is placed on the Chandrasekhar–Kendall functions that allow a decomposition of a vector field into right- and left-handed contributions. Magnetic energy spectra of both contributions are shown for a new set of helically forced simulations at resolutions higher than what has been available so far. For a forcing function with positive helicity, these simulations show a forward cascade of the right-handed contributions to the magnetic field and nonlocal inverse transfer for the left-handed contributions. The speed of inverse transfer is shown to decrease with increasing value of the magnetic Reynolds number.

TURBULENT DYNAMOS IN SPHERICAL SHELL SEGMENTS OF VARYING GEOMETRICAL EXTENT

We use three-dimensional direct numerical simulations of the helically forced magnetohydrodynamic equations in spherical shell segments in order to study the effects of changes in the geometrical shape and size of the domain on the growth and saturation of large-scale magnetic fields. We inject kinetic energy along with kinetic helicity in spherical domains via helical forcing using Chandrasekhar-Kendall functions. We take perfect conductor boundary conditions for the magnetic field to ensure that no magnetic helicity escapes the domain boundaries. We find dynamo action giving rise to magnetic fields at scales larger than the characteristic scale of the forcing. The magnetic energy exceeds the kinetic energy over dissipative time scales, similar to that seen earlier in Cartesian simulations in periodic boxes. As we increase the size of the domain in the azimuthal direction we find that the nonlinearly saturated magnetic field organizes itself in long-lived cellular structures with aspect ratios close to unity. These structures tile the domain along the azimuthal direction, thus resulting in very small longitudinally averaged magnetic fields for large domain sizes. The scales of these structures are determined by the smallest scales of the domain, which in our simulations is usually the radial scale. We also find that increasing the meridional extent of the domains produces little qualitative change, except a marginal increase in the large-scale field. We obtain qualitatively similar results in Cartesian domains with similar aspect ratios.

Nonlinear Evolution of Kink‐unstable Magnetic Flux Tubes and Solar δ‐Spot Active Regions

The motivation for the work described in this paper is to understand kink-unstable magnetic flux tubes and their role in the formation of δ-spot active regions. It has been proposed that, during their rise to the photosphere, a certain fraction of convection zone flux tubes become twisted to the point where they are unstable to the current driven kink instability. These kink-unstable flux tubes then evolve toward a new, kinked equilibrium as they continue to rise to the photosphere, appearing as δ-spot groups upon emergence. Because of their kinked nature, these flux tubes could be highly susceptible to flaring, explaining the very active nature of δ-spot groups. We study the kinking flux tube problem with a three-dimensional numerical model containing only the most basic features of a kink-unstable flux tube. We build on our earlier work describing the linear phase of the kink instability, and follow the evolution into the nonlinear regime: (1) We perform numerical simulations of constant-twist, kink-unstable flux tubes in an initially cylindrical equilibrium configuration in three dimensions, in a high-β pressure-confined environment. We consider many different initial configurations, including the Gold-Hoyle flux tube. (2) These numerical calculations confirm the growth-rate predictions of our earlier work, when viscous dissipation is included. They also confirm our velocity profile predictions. (3) The flux tubes evolve toward new helically symmetric equilibrium configurations. (4) The timescale for saturation to the kinked equilibrium configuration is τsat ~ 10/ω0, where ω0 is the linear growth rate calculated as in the earlier paper. (5) The cylindrically symmetric part of the kinked equilibrium is well described by the m = 0 Chandrasekhar-Kendall functions (i.e., the Lundquist field). The m = 1 helically symmetric part, however, is not well described by the m = 1 Chandrasekhar-Kendall functions. (6) The equilibrium kink amplitudes are not large, at less than one-third of the tube radius. (7) The peak kinetic energy of the instability can be predicted from the initial excess perpendicular magnetic energy. (8) The amplitudes of the kinked tubes are large enough to give a δ-spot region tilt angle of up to 30° away from that of an unkinked tube.

Temporal behaviour of pressure in solar coronal loops

The temporal evolution of pressure in solar coronal loops is studied using the ideal theory of magnetohydrodynamic turbulence in cylindrical geometry. The velocity and the magnetic fields are expanded in terms of the Chandrasekhar-Kendall (C-K) functions. The three-mode representation of the velocity and the magnetic fields submits to the investigation of chaos. When the initial values of the velocity and the magnetic field coefficients are very nearly equal, the system shows periodicities. For randomly chosen initial values of these parameters, the evolution of the velocity and the magnetic fields is nonlinear and chaotic. The consequent plasma pressure is determined in the linear and nonlinear regimes. The evidence for the existence of chaos is established by evaluating the invariant correlation dimension of the attractorD 2, a fractal value of which indicates the existence of deterministic chaos.

Pressure structure of solar coronal loops, III

The theory of ideal magnetohydrodynamic turbulence in cylindrical geometry is used to study the steady-state structure of a coronal loop. The pressure profile is derived from MHD equations by representing the velocity and magnetic fields as the superposition of Chandrasekhar-Kendall functions. Such a representation brings out the three-dimensional structure of the pressure in the coronal loop. The radial, azimuthal, and axial variations of the pressure for a constant density loop are discussed in detail. The pressure has an oscillatory behavior for different azimuthal angles at some radial positions. This study predicts more features in pressure than can be compared with the presently available observations.

Nonlinear magnetohydrodynamics by Galerkin-method computation.

A fully spectral numerical code is used to explore the properties of voltage-driven dissipative magnetofluids inside a periodic cylinder with circular cross section. The trial functions are orthonormal eigenfunctions of the curl (Chandrasekhar-Kendall functions). Transitions are observed from axisymmetric resistive equilibria without flow to helically deformed laminar states with flow, and between pairs of helical laminar states with different pairs of poloidal and toroidal m and n numbers. States of minimum energy dissipation rate seem to be preferred. At high values of the pinch ratio, fully developed magnetohydrodynamic turbulence is observed.

Discrete Eigenstates of Plasmas Described by the Chandrasekhar-Kendall Functions

A mathematical background has been studied for the eigenfunction expansion of solenoidal vector fields based on the force-free fields, which are eigenfunctions of the curl derivatives. Eigenfunctions of the self-adjoint curl operator span the orthogonal complement of the irrotational fields. Explicitly for a cylindrical domain, the Chandrasekhar-Kendall functions with zero flux give the total of the eigenfunction of the self-adjoint curl operator. The Chandrasekhar-Kendall functions have magnetic surfaces, which have been numerically calculated. When we consider multiply connected domain and finite-flux solutions of the eigenvalue problem, we obtain a solution for every complex number as the eigenvalue. The corresponding solution generally has a symmetry when we consider a symmetric domain. Non-symmetric solution bifurcates at the eigenvalue of the zero-flux eigenfunction (the spectrum of the self-adjoint curl operator).

Two-dimensional steady-state pressure structure in a coronal loop

By expressing the magnetic field and fluid velocity in terms of two Chandrasekhar-Kendall functions (n = 0, m = 0; n = 1, m = 0) we investigated the steady-state pressure profile inside a solar coronal loop. For constant density loops, we found a two-dimensional (radial and axial) structure of pressure. This work is the modified version of the work of Krishan (1985). At the base of the loop, the pressure is found to increase steeply outwards along the radius, whereas at the apex it decreases slowly. The radial variation of pressure is found to be minimum around L/5, where L is the length of the loop measured from one foot to another one. But Krishan (1985) found that the rate of increase of pressure at the base was nearly equal to the rate of decrease of pressure at the apex, and the pressure was found nearly constant at L/4. For axial variation, we found that along the loop axis the pressure increases from the base up to z = 3L/8 and then decreases up to the apex, whereas at the surface, the pressure decreases from the base up to the apex. Krishan (1985), however, found the axial variation to be linear.

Discrete Eigenstates of Plasmas Described by the Chandrasekhar-Kendall Functions

Discrete Eigenstates of Plasmas Described by the Chandrasekhar-Kendall Functions

The Chandrasekhar-Kendall functions

We find that the Chandrasekhar-Kendall functions (A) do not satisfy the identity (A·▿)A=(▿×A)×A+1.2▿A2 and, therefore, the results, in magnetohydrodynamics and other fields, obtained by two independent authors may differ from one another.

Two-dimensional pressure structure of a coronal loop

The steady-state pressure structure of a solar coronal loop is discussed using the theory of magneto-hydrodynamical turbulence in cylindrical geometry. The steady state is represented by the superposition of two Chandrasekhar-Kendall functions. This representation, in principle can delinetate the three dimensional temperature structure of the coronal loop. In this paper, we have restricted ourselves to a two dimensional modeling since only this structure submits itself to the scrutiny of the available observations. The radial as well as the axial variations of the pressure in a constant density loop are calculated. These variations are found to conform to the observed features of cool core and hot sheath of the loops as well as to the location of the temperature maximum at the apex of the loop. We find that these features are not present uniformly all along either the length of the loop or across the radius as will be shown in the text. We have also discussed the possible oscillatory nature of these pressure variations and the associated time periods have been estimated.