Diffussion of a scalar field in a compressible turbulent medium

The diffusion of scalar fields (temperature, density number of some admixture) in a compressible medium showing an isotropic, homogeneous and stationary turbulence is considered. The derived formulae for turbulent diffusivity χ_T(ξ) hold up to ξ ≈ 1, where ξ = u₀ τ₀/R₀ (u₀, τ₀, and R₀ are characteristic velocity, life-time, and correlation length of turbulent pulsations, respectively. The velocity field of turbulent motions u(r, t) is assumed to be known and the influence of the scalar field onto u(r, t) is neglected. It is shown that the velocity correlators, which change their signs in dependence on the space corrdinates, may give negative values for ξ_T(ξ) when ξ ≠ 0.     

The structure of the turbulent shock wave propagating in the solar atmosphere across the magnetic field

A consistent account of plasma turbulence in magnetohydrodynamics equations describing transport processes across the magnetic field is presented. The structure of the perpendicular shock wave generated in the solar atmosphere, as a result of either local disturbance of the magnetic field or dense plasma cloud motion with a frozen-in magnetic field, has been investigated. The region of parameters in the solar atmosphere at which the electron-ion relative drift velocity u exceeds the electron thermal velocity V _eand generation of radio emission becomes possible, has been determined. The plasma turbulence inside the front has been shown, under conditions of solar corona, not to cause the oscillation structure of shock front to break down. Under chromospheric conditions, the shock profile is aperiodical. Then, the condition u > V_ecan be satisfied and shock waves having an Alfvén Mach number M which exceeds the critical value M _c 3.3 for aperiodical shock waves can exist (Eselevich et al., 1971a). Arguments are given in favour of the fact that perpendicular shock waves are generated in the Sun's atmosphere when dense plasma clouds, with a frozen-in magnetic field, are expanded.     

Rate of change of vorticity covariance of MHD turbulence in a rotating system

In this paper we are concerned with incompressible MHD turbulence in a rotating system and have derived an equation for the rate of change of vorticity covariance of MHD turbulent flow. The result derived shows that the defining scalars (r,t), (r,t), and (r,t) for the rate of change of vorticity covariance solely depend on the defining scalars of the tensorsW _ij, P_ik,j, F_kj,i, T_ik,j, andR _kj,ialready defined in the text.     

Scaling laws in decaying helical hydromagnetic turbulence

We study the evolution of growth and decay laws for the magnetic field coherence length ξ, energy E_M and magnetic helicity H in freely decaying 3D MHD turbulence. We show that with certain assumptions, self‐similarity of the magnetic power spectrum alone implies that ξ ∼ t^1/2. This in turn implies that magnetic helicity decays as H ∼ t^–2s, where s = (ξ_diff/ξ_H)², in terms of ξ_diff, the diffusion length scale, and ξ_H, a length scale defined from the helicity power spectrum. The relative magnetic helicity remains constant, implying that the magnetic energy decays as E_M ∼ t^–1/2–2s. The parameter s is inversely proportional to the magnetic Reynolds number Re_M, which is constant in the self‐similar regime. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The influence of satellite flexibility on orbital motion

The orbital perturbations induced by the librational motion and flexural oscillations are studied for satellites having large flexible appendages. Using a Lagrangian procedure, the equations for coupled motion are derived for a satellite having an arbitrary number of appendages in the nominal orbital plane and two flexible members normal to it. The formulation enables one to study the influence of flexibility on both the orbital and attitude motions. The orbital coordinates are expanded as perturbation series in =(l/a ₀)²,l anda ₀ being a characteristic length of the satellite and unperturbed semi-major axis of the orbit, respectively. The first order perturbation equations are solved in terms of elastic deformations and librational angles using the WKBJ method in conjunction with the variation of parameter technique. Existence of secular perturbations is noted for certain librational flexural motions. Three specific examples, Alouette II, Radio Astronomy Explorer and Tethered Orbiting Interferometer, are considered subsequently and their possible secular drifts estimated.List of Symbols A _ij, B_ij coefficients in the eigenfunction expansion ofv _i andw _i respectively, Equation (10) - C _k, D_k constants, Equation (21) - EI _i flexural rigidity of theith appendage - E(u₀) ²(1+e ₀ cosu ₀)² h ₀ ³ - F(u₀) perturbation function, Equation (17b) - F ,F ,F functions of librational angles and flexural displacements, Equation (11i) - F ,F ,F F ,F ,F with change of independent variable fromt tou ₀ - I _xx, I_yy, I_zz principal moments of inertia of the undeformed satellite - [J ⁱ] inertia dyadic of the deformedith appendage - [J _d] inertia dyadic of the deformed satellite - M mass of the satellite - P _R, P_u functions of librational angles and flexural displacements, Equation (15d) and (15e), respectively - R _c magnitude ofR _c - R _c0, R₁ unperturbed value and first order perturbation ofR _c, respectively - R _c ,R ₀ position vectors of the c.m. of the deformed and undeformed satellite, respectively - T kinetic energy of the satellite - U potential energy of the satellite - U _e, U_g elastic and gravitational potential energy, respectively - X, Y, Z orbital co-ordinate axes, located at the c.m. of the deformed satellite - Y ₁(u₀), Y₂(u₀) functions ofu ₀, Equation (18b) and (18c), respectively - a semi-major axis - a ₀ unperturbed value ofa - e eccentricity - e ₀ unperturbed value ofe - h ₀ unperturbed angular momentum per unit mass of the satellite - i inclination of the orbital plane to the ecliptic - i, j, k unit vectors alongx (or ),y (or ) andz (or ) axes, respectively - l characteristic length of the satellite - l _i length of theith appendage - [l ⁱ] matrix of direction cosines ofx _i, v_i andw _i - l ,l ,l direction cosines ofR _c - m ₀, m_i mass of the main body andith appendage, respectively - p _i ² - q _m, Q_m generalized co-ordinate and force, respectively - r ₁ R ₁/R_c0 - r position vector of an element of the body referred toxyz axes - r _u position vector of an element after deformation, referred to axes - r _c x _c i+y _c j+z _c k, position vector of the c.m. of the deformed body referred toxyz axes - s x _i/l_i - t time - u true anomaly - u ₀, u₁ unperturbed value and the first order perturbation ofu, respectively - u elastic displacement vector - u _c u–r _c - velocity of an element relative to axes - v _i, w_i flexural deformations - x, y, z body co-ordinate axes with origin at the c.m. of the undeformed satellite - x _i distance of an element of theith appendage from the root - _j jth eigenfunction (normalized) of a cantilever - angle between the line of nodes and vernal equinox - , , components of nondimensionalized angular velocity of the satellite - , , pitch (spin), yaw and roll, respectively - _i nominal inclination of theith appendage in the orbital plane - - small parameter, (l/a ₀)² - _j jth eigenvalue of a cantilever - gravitational constant - _jk constant, Equation (11j) - , , body co-ordinate axes with origin at the c.m. of the deformed satellite - ( i + j + k), angular velocity of the satellite     

A search for the age-dependency of Ap star parameters

Some observational data of the sample of the magnetic chemically peculiar stars (MCP stars) are investigated statistically. For the MCP stars of spectral types later than A2 both the frequency distribution and the R ⋅ sin i-values suggest the existence of a linear relation between stellar diameter and rotation period. The MCP stars of spectral types earlier than B9 show an overpopulation of small R ⋅ sin i which may indicate the existence of a second group with smaller radius in this sample. The equatorially symmetric rotator is used as the magnetic model. With respect to its temporal behaviour the effective magnetic field is separated into dipolar and quadrupolar contribution. Both signs of the axisymmetric quadrupole moment appear with equal frequency. The dipole moment which produces the amplitude of the B_eff(t) curve forms for longer periods two groups which are separated by a distinct gap. Both of the groups exhibit magnetic fields which are the stronger the greater the stellar radius is, contrary to what is expected for frozen-in fields. The dominance of magnetic curves without polarity reversal for longer-period stars is in accordance with predictions of the dynamo theory.     

The Energy Lost by Differential Rotation in the Generation of the Solar Toroidal Magnetic Field

The integrals, I_i(t) = _GL u_i j × B _i dv over the volume GL are calculated in a dynamo model of the Babcock–Leighton type studied earlier. Here, GL is the generating layer for the solar toroidal magnetic field, located at the base of the solar convection zone (SCZ); i=r, , , stands for the radial, latitudinal, and azimuthal coordinates respectively; j = (4)^-1 × B, where B is the magnetic field; u_r,u are the components of the meridional motion, and u is the differential rotation. During a ten-year cycle the energy _cycle I(t)dt needs to be supplied to the azimuthal flow in the GL to compensate for the energy losses due to the Lorentz force. The calculations proceed as follows: for every time step, the maximum value of |B| in the GL is computed. If this value exceeds B_cr (a prescribed field) then there is eruption of a flux tube that rises radially, and reaches the surface at a latitude corresponding to the maximum of |B| (the time of rise is neglected). This flux tube generates a bipolar magnetic region, which is replaced by its equivalent axisymmetric configuration, a magnetic ring doublet. The erupted flux can be multiplied by a factor F_t, i.e., by the number of eruptions per time step. The model is marginally stable and the ensemble of eruptions acts as the source for the poloidal field. The arbitrary parameters B_cr and F_t are determined by matching the flux of a typical solar active region, and of the total erupted flux in a cycle, respectively. If E(B) is the energy, in the GL, of the toroidal magnetic field B = B sin cos , B (constant), then the numerical calculations show that the energy that needs to be supplied to the differential rotation during a ten-year cycle is of the order of E(B_cr), which is considerably smaller than the kinetic energy of differential rotation in the GL. Assuming that these results can be extrapolated to larger values of B_cr, magnetic fields 10⁴ G, could be generated in the upper section of the tachocline that lies below the SCZ (designated by UT). The energy required to generate these 10⁴ G fields during a cycle is of the order of the kinetic energy in the UT.     

Plane symmetric inhomogeneous perfect fluid universe with electromagnetic field in Lyra geometry

A new class of plane-symmetric inhomogeneous cosmological models of perfect fluid distribution with electro-magnetic field based on Lyra’s geometry is obtained by considering a time dependent displacement field. The source of the magnetic field is due to an electric current produced along the z-axis. Only F ₁₂ is a non-vanishing component of electromagnetic field tensor. To get the deterministic solutions, the free gravitational field is assumed to be of Petrov type-II non-degenerate. It has been found that the displacement vector β(t) behaves like cosmological term Λ which is consistent with the recent observations of type Ia supernovae. It is also observed that β(t) affects entropy. Some geometric and physical behaviour of the models are also discussed in presence of magnetic field.      

A study of the contrast of sunspots from photometric images

The thermal contrast , and the umbra-penumbraA _u/A _p, were calculated for 63 sunspots of various sizes and morphologies. Contrary to the assumptions of the PSI model, andA _u/A _p were found to be quite variable. The values of ranged from 0.1807 to 0.4266;A _u/A _p ranged from 0.0089 to 0.4899. The values of andA _u/A _p correlated well (r = 0.6018;p<0.005) and the regression for andA _u/A _p was obtained: = (0.220 ± 0.016) + (0.340 ± O.06)A _u/A _p. The values of andA _u/A _p were then compared with complexity ratings, magnetic field strength, time, and . The quantities andA _u/A _p were found to be independent of the complexity, magnetic field strength, and time factors. The correlation between andA _u/A _p lead to the proposed division of into an umbral thermal contrast _u, and a penumbral thermal contrast _p. These values were calculated from the photometric data: _u = 0.57 ± 0.01 and _p = 0.26 ± 0.006.     

Gravitational field of domain wall in Lyra geometry

In this paper, we study the domain wall with time dependent displacement vectors based on Lyra geometry in normal gauge i.e. displacement vector φ _i ^* =[β(t),0,0,0]. The field theoretic energy momentum tensor is considered with zero pressure perpendicular to the wall. We find an exact solutions of Einstein’s equation for a scalar field φ with a potential V(φ) describing the gravitational field of a plane symmetric domain wall. We have seen that the hyper surfaces parallel to the wall (z=constant) are three dimensional de-sitter spaces. It is also shown that the gravitational field experienced by test particle is attractive.     

Vector magnetic fields in sunspots

Observations of a round, unipolar sunspot in the Zeeman triplet Fe i 6302.5 with the High Altitude Observatory Stokes Polarimeter are used to derive the vector magnetic field in the spot. The behavior of the magnitude, inclination, and azimuth of the field vector B across the spot is discussed. A linear relation is found between the continuum intensity I _c and the field magnitude B. Time series obtained in the umbra show significant power in the magnitude of the field at a period of t 180 s but the other components of the field vector do not display this behavior.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

The Correlation between the Magnetic and Velocity Fields on the Full Solar Disk

A possible correlation between the magnetic and velocity fields has been analyzed based on the SOHO/MDI magnetograms and Dopplergrams. It is found that the observed large-scale weak magnetic field (weaker than 50 G (gauss)) is correlated with the velocity statistically. The curves of u⋅b with latitude, where u and b are the velocity and magnetic fields in a rectangular region (±15^○ in longitude, ±45^○ in latitude) on the Sun, show the same patterns in the years 2000, 2004, and 2007. The patterns indicate that u and b are positively correlated near the equator but are anti-correlated at the middle latitudes. For a strong magnetic field between 50 G and 3000 G, the curves of u⋅b with latitude show the same tendencies at the middle latitudes. Near the equator, however, the slope of the curve is positive in 2000 and is negative in 2004 and 2007. In addition, we give an estimation for the amplitude of the cross helicity h _χ (h_c=[`(u·b)]h_{\chi}=\overline{\mathbf{u}\cdot\mathbf{b}}) inferred from the MDI data, which is of the order of 10³ G m s⁻¹ near the center of the solar disk.     

Kinetic theory of MHD instabilities in a nonuniform plasma

Most of the MHD instabilities originating from the nonuniformity of a plasma excite MHD surface wave. When the excited wave has a frequency _s which corresponds to the local shear Alfvén wave resonance (_s = k v _a (x), where v _a is the Alfvén speed and k is the wave number in the direction of the magnetic field), the surface wave resonantly mode converts to the kinetic Alfvén wave, the Alfvén wave having a perpendicular wavelength comparable to the ion gyroradius and being able to propagate across the magnetic field. We discuss various linear and nonlinear effects of this kinetic Alfvén wave on the plasma including particle acceleration and heating. A specific example for the case of a MHD Kelvin-Helmholtz instability is given.     

Relationship between Powerful Flares and Dynamic Evolution of the Magnetic Field at the Solar Surface

Powerful flares are closely related to the evolution of the complex magnetic field configuration at the solar surface. The strength of the magnetic field and speed of its evolution are two vital parameters in the study of the change of magnetic field in the solar atmosphere. We propose a dynamic and quantitative depiction of the changes in complexity of the active region: E=u×B, where u is the velocity of the footpoint motion of the magnetic field lines and B is the magnetic field. E represents the dynamic evolution of the velocity field and the magnetic field, shows the sweeping motions of magnetic footpoints, exhibits the buildup process of current, and relates to the changes in nonpotentiality of the active region in the photosphere. It is actually the induced electric field in the photosphere. It can be deduced observationally from velocities computed by the local correlation tracking (LCT) technique and vector magnetic fields derived from vector magnetograms. The relationship between E and ten X-class flares of four active regions (NOAA 10720, 10486, 9077, and 8100) has been studied. It is found that (1) the initial brightenings of flare kernels are roughly located near the inversion lines where the intensities of E are very high, (2) the daily averages of the mean densities of E and its normal component (E _n) decrease after flares for most cases we studied, whereas those of the tangential component of E (E _t) show no obvious regularities before and after flares, and (3) the daily averages of the mean densities of E _t are always higher than those of E _n, which cannot be naturally deduced by the daily averages of the mean densities of B _n and B _t.     

Incompressible MHD Turbulence

The inertial range of incompressible MHD turbulence is most conveniently described in terms of counter propagating waves. Shear Alfvén waves control the cascade dynamics. Slow waves play a passive role and adopt the spectrum set by the shear Alfvén waves. Cascades composed entirely of shear Alfvén waves do not generate a significant measure of slow waves. MHD turbulence is anisotropic with energy cascading more rapidly along k _⊥ than along k _∥. Anisotropy increases with k _⊥ such that the excited modes are confined inside a cone bounded by k _∥∝ k _perp ^2/3. The opening angle of the cone, θ(k _⊥)∝ k _⊥ ^-1/3, defines the scale dependent anisotropy. MHD turbulence is generically strong in the sense that the waves which comprise it are critically damped. Nevertheless, deep inside the inertial range, turbulent fluctuations are small. Their energy density is less than that of the background field by a factor θ²(k _⊥)≪. MHD cascades are best understood geometrically. Wave packets suffer distortions as they move along magnetic field lines perturbed by counter propagating wave packets. Field lines perturbed by unidirectional waves map planes perpendicular to the local field into each other. Shear Alfvén waves are responsible for the mapping's shear and slow waves for its dilatation. The former exceeds the latter by θ^-1(k _⊥)≫ 1 which accounts for dominance of the shear Alfvén waves in controlling the cascade dynamics. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the Influence of a Large-scale Magnetic Field on Turbulent Motions in an Electrically Conducting Medium

This paper deals with turbulent motions in a homogeneous incompressible electrically conducting medium in the presence of a magnetic field which is on average homogeneous and stationary. Using a model in which the turbulence is produced by a stochastic body force, and supposing a weak interaction between motion and magnetic field, a method is developed for calculating the pair correlation tensor of the velocity field from that occuring in a zero magnetic field. As an example, the pair-correlation tensor for a homogeneous stationary turbulence, which is isotropic and mirror-symmetric for zero magnetic field, is determined. With obvious assumptions on the correlation for zero field, two results are obtained. Firstly, the turbulent velocity is reduced by the magnetic field, the component parallel to the field, however, less than those perpendicular to it. Secondly, the correlation length parallel to the field turns out to be greater than the one perpendicular to it, indicating a tendency towards two-dimensional motion. Finally, the possibility of special situations is briefly discussed in which the turbulent velocity is enhanced by the magnetic field, and the anisotropies of the velocity components and the correlation lengths are opposite to those above.     

On the relation between the peak frequency and the corresponding rise time of solar microwave impulsive bursts and the height dependence of magnetic fields

In the present paper we report the results of a correlation analysis for 57 microwave impulsive bursts observed at six frequencies in which we have obtained a regression line between the peak frequency and the corresponding rise time of microwave impulsive bursts: {ie361-01} (with a correlation coefficient of - 0.43). This can be explained in the frame of a thermal model. The magnetic field decrease with height has to be much slower than in a dipole field in order to explain the weak dependence of f _p on t _r . This decrease of magnetic field with height in burst sources is based on the relationship between f _p and t _r found by assuming a thermal flare model with a collisionless conduction front.On leave from Beijing Observatory, Academia Sinica, Beijing, China.     

Evaluating Mean Magnetic Field in Flare Loops

We analyze multiple-wavelength observations of a two-ribbon flare exhibiting apparent expansion motion of the flare ribbons in the lower atmosphere and rising motion of X-ray emission at the top of newly-formed flare loops. We evaluate magnetic reconnection rate in terms of V _r B _r by measuring the ribbon-expansion velocity (V _r) and the chromospheric magnetic field (B _r) swept by the ribbons. We also measure the velocity (V _t) of the apparent rising motion of the loop-top X-ray source, and estimate the mean magnetic field (B _t) at the top of newly-formed flare loops using the relation 〈V _t B _t〉≈〈V _r B _r〉, namely, conservation of reconnection flux along flare loops. For this flare, B _t is found to be 120 and 60 G, respectively, during two emission peaks five minutes apart in the impulsive phase. An estimate of the magnetic field in flare loops is also achieved by analyzing the microwave and hard X-ray spectral observations, yielding B=250 and 120 G at the two emission peaks, respectively. The measured B from the microwave spectrum is an appropriately-weighted value of magnetic field from the loop top to the loop leg. Therefore, the two methods to evaluate coronal magnetic field in flaring loops produce fully-consistent results in this event.     

Generation of acoustic and gravity waves by turbulence in an isothermal stratified atmosphere

Lighthill's method of calculating the aerodynamic emission of sound waves in a homogeneous atmosphere is extended to calculate the acoustic and gravity-wave emission by turbulent motions in a stratified atmosphere. The acoustic power output is P _ac 10³ _o u _o ³ /l _o M ⁵ ergs/cm³ sec, and the upward gravity wave flux is F _zgr 10² _o U _o ³ /l _o (l _o ergs/cm³ sec. Here u ₀ is the turbulence velocity scale, l ₀ is its length scale, and H the scale height at the atmosphere. M = u ₀/c ₀ is the Mach number of the turbulence. The acoustic power output is proportional to the maximum value of the turbulence spectrum, and inversely to its rate of falloff at high frequencies. The stratification cuts off the acoustic emission at low Mach numbers. The gravity emission occurs near the critical angle to the vertical _c = cos^–1 / ₂, where ₂ ² = ( - 1)/ ² (c ₀/H), and at very short wavelengths. It is proportional to the large wave number tail of the turbulence spectrum. On the sun, gravity-wave emission is much more efficient than acoustic, but can occur only from turbulent motions in stable regions, whereas acoustic waves are produced by turbulence in the convection zone.     

Studies of magnetic fields in the solar photosphere using the line-ratio method

The longitudinal magnetic field measured using the Fe I λ 525 and Fe I λ 524.7 nm lines and global magnetic field of the sun differ depending on the observatory. To study the cause of these discrepancies, we calculate the H _‖(525)/H _‖(524.7) ratios for various combinations of magnetic elements and compare them with the corresponding observed values. We use the standard quiet model of the solar photosphere suggesting that there are magnetic fields of different polarities in the range between zero and several kilogauss. The magnetic element distribution is found as a function of magnetic field strength and the parameters of this distribution are determined for which the calculated H _‖(525)/H _‖(524.7) ratio agrees with the observed one. The sigma-components are found to be shifted differently for various points of the Fe I λ 525 nm profile calculated for the inhomogeneous magnetic field. The farther the point is from the line center, the larger the sigma-components shift. Such a peculiarity of the profiles may be responsible for the discrepancies in the measured values of the global magnetic field obtained at different observatories. The increase in modulus of the global magnetic field during the maxima of solar activity can be due to a larger fraction of magnetic elements with kilogauss magnetic fields.