On the Reynolds stresses in mean-field hydrodynamics II. Two-dimensional turbulence and the problem of negative viscosity

The methods proposed in a foregoing paper are used for the derivation of the eddy viscosity of a two-dimensional homogeneous isotropic turbulence. In contrast to three-dimensional isotropic turbulence is shown that for the two-dimensional case the eddy viscosity (i) has no definite sign, (ii) tends to zero if the molecular viscosity tends to zero, (iii) is negative for special cases. However, the modes with large wave numbers decay in any case as is shown by investigating a dispersion relation.     

Two-dimensional stochastic motions and the problem of differential rotation

This paper deals with the spatial dependence of the angular velocity in a rotating turbulent fluid sphere. The original turbulence unaffected by the global rotation is assumed to be two-dimensional where the stochastic force field producing the turbulence does not possess a radial component. By using results of earlier papers we proceed to the treatment of a rotational rate, , no longer small compared to _c (frequency of turbulent mode). It is shown that for _c the angular velocity increases with increasing radius but no latitudinal dependence exists. Contrary to this, for 2 _c an equatorial acceleration is possible and related to negativity of the two-dimensional eddy viscosity. Furthermore, the outer layers rotate faster than the inner ones. These findings coincide with Gilman's numerical results. Ward's observations, as well as the characteristic scales of supergranulation and giant cells, suggest the presence of negative two-dimensional eddy viscosity on the Sun.     

The maintenance of solar differential rotation by two-dimensional turbulence

A numerical model has been made to test the theory that solar differential rotation is maintained by the Countergradient transport of energy peculiar to two-dimensional turbulence. After a brief discussion of this turbulent process, the numerical methods employed and their application to the sun are reviewed. The results of one problem are presented, indicating that this model can represent the observed large-scale nature of the sun's surface. The reader is referred to the author's dissertation for complete details of the methods and calculations.     

On the differential rotation of the solar photosphere

It is shown that sunspots as tracers can give the same results for the differential rotation of the solar photosphere as the Doppler-shift measurements, if the sunspots used have only insignificant motion relative to their immediate photospheric surroundings.     

On the Sun's differential rotation and pole-equator temperature difference

The Sun's differential rotation can be understood in terms of a preferential stabilization of convection (by rotation) in the polar regions of the lower part of the convection zone (where the Taylor number is large). A significant pole-equator difference in flux () can develop deep inside the convection zone which would be unobservable at the surface, because can be very efficiently reduced by large scale meridional motions rising at the poles and sinking at the equator. This is the sense of circulation needed to produce the observed equatorial acceleration of the Sun. Differential rotation is generated, therefore, in the upper part of the convection zone (where the interaction of rotation with convection is small) and results as the convection zone adjusts to a state of negligible Taylor number.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

On the differential rotation with height in the solar atmosphere

Spectroscopic measurements of solar rotation having good height discrimination show no change in angular velocity through the photosphere layers but an increase of 8% for the Hα chromosphere (epoch 1968.9). Spectroscopic results in general are compared with measures made with tracers, i.e. sunspots, filaments, etc., and it is seen that the spectroscopic method always shows increased differential rotation with height, while tracers indicate none. A westward flowing wind is proposed that increases in velocity with height, but produces negligible movement to magnetic regions associated with tracers. Kitt Peak National Observatory Contribution No. 450. Operated by The Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.     

On the interaction between differential rotation and magnetic fields in the Sun

We have performed 3-D numerical simulations of compressible convection under the influence of rotation and magnetic fields in spherical shells. They aim at understanding the subtle coupling between convection, rotation and magnetic fields in the solar convection zone. We show that as the magnetic Reynolds number is increased in the simulations, the magnetic energy saturates via nonlinear dynamo action, to a value smaller but comparable to the kinetic energy contained in the shell, leading to increasingly strong Maxwell stresses that tend to weaken the differential rotation driven by the convection. These simulations also indicate that the mean toroidal and poloidal magnetic fields are small compared to their fluctuating counterparts, most of the magnetic energy being contained in the non-axisymmetric fields. The intermittent nature of the magnetic fields generated by such a turbulent convective dynamo confirms that in the Sun the large-scale ordered dynamo responsible for the 22-year cycle of activity can hardly be located in the solar convective envelope.     

Surface Reynolds stresses determined from the analysis of facular motions and the maintenance of the Sun's differential rotation

From the analysis of the motions of faculae for a four year period from 1967 to 1970, the latitudinal angular momentum transport by Reynolds stresses at the Sun's surface is calculated. The result agrees fairly well with the one obtained by Ward analyzing spot motions. A comparison with theory suggests that the Sun's differential rotation could be maintained only by Reynolds stresses at the surface. The implications of this result are discussed.     

On the time dependence of differential rotation in young late-type stars

A model for the angular momentum transfer within the convection zone of a rapidly rotating star is introduced and applied to the analysis of recent observations of temporal fluctuations of the differential rotation on the young late-type stars AB Doradus (AB Dor) and LQ Hydrae (LQ Hya). Under the hypothesis that the mean magnetic field produced by the stellar dynamo rules the angular momentum exchanges and that the angular velocity depends only on the distance s from the rotation axis and the time, the minimum azimuthal Maxwell stress  | B_sB _φ|  , averaged over the convection zone, is found to range from ∼0.04 to  ∼0.14 T²  . If the poloidal mean magnetic field   B _s   is of the order of 0.01 T, as indicated by the Zeeman–Doppler imaging maps of those stars, then the azimuthal mean field   B _φ  can reach an intensity of several teslas, which significantly exceeds equipartition with the turbulent kinetic energy. Such strong fields can account also for the orbital period modulation observed in cataclysmic variables and RS Canum Venaticorum systems with a main-sequence secondary component. Moreover, the model allows us to compute the kinetic energy dissipation rate during the maintenance of the differential rotation. Only in the case of the largest surface shear observed on LQ Hya may the dissipated power exceed the stellar luminosity, but the lack of a sufficient statistic on the occurrence of such episodes of large shear does not allow us to estimate their impact on the energy budget of the convection zone.     

On the differential rotation of the polar caps of neutron stars

The model of a magnetized rotating neutron star with an electric current in the region of its fluid polar magnetic caps is considered. The presence of an electric current leads to differential rotation of the magnetic caps. The rotation structure is determined by the electric current density distribution over the surface. In the simplest axisymmetric configuration, the current flows in one direction near the polar cap center and in the opposite direction in the outer ring (the total current is zero for the neutron star charge conservation). In this case, two rings with opposite directions of rotation appear on the neutron star surface, with the inner ring always lagging behind the star’s main rotation. The differential rotation velocity is directly proportional to the electric current density gradient along the polar cap radius. At a width of the region of change in the electric current from 1 to 10² cm and a period ～1 s and a magnetic field B ～ 10¹² G typical of radio pulsars, the linear differential rotation velocity is ～10^?2–10^?4 cm s^?1 (corresponding to a revolution time of ～0.1–10 yr).     

Rotating convection and the solar differential rotation

We discuss the implication of a numerical experiment on rotating convection and its relevance to the construction of a model for the solar differential rotation.     

Modelling the differential rotation of F stars

We model stellar differential rotation based on the mean-field theory of fluid dynamics. DR is mainly driven by Reynolds stress, which is anisotropic and has a non-diffusive component because the Coriolis force affects the convection pattern. Likewise, the convective heat transport is not strictly radial but slightly tilted towards the rotation axis, causing the polar caps to be slightly warmer than the equator. This drives a flow opposite to that caused by differential rotation and so allows the system to avoid the Taylor-Proudman state. Our model reproduces the rotation pattern in the solar convection zone and allows predictions for other stars with outer convection zones. The surface shear turns out to depend mainly on the spectral type and only weakly on the rotation rate. We present results for stars of spectral type F which show signs of very strong differential rotation in some cases. Stars just below the mass limit for outer convection zones have shallow convection zones with short convective turnover times. We find solar-type rotation and meridional flow patterns at much shorter rotation periods and horizontal shear much larger than on the solar surface, in agreement with recent observations. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

22-Year periodicity in the solar differential rotation

Using the data on sunspot groups compiled during 1879–1975, we determined variations in the differential rotation coefficientsA andB during the solar cycle. The variation in the equatorial rotation rateA is found to be significant only in the odd numbered cycles, with an amplitude ∼ 0.01 μ rads^-1. There exists a good anticorrelation between the variations of the differential rotation rateB derived from the odd and even numbered cycles, suggesting existence of a ‘22-year’ periodicity inB. The amplitude of the variation ofB is ∼ 0.05 μ rad s^-1.     

On the hydrodynamics of accretion discs

The widely used boundary condition on the tangential stress in accretion discs is discussed and found wanting on a number of counts. In particular, it is shown that the value of the viscous energy dissipation calculated from it is unphysical. An alternative boundary condition is suggested. Using this, it is found that the energy balance in the disc places certain restrictions on the rotational velocity in the disc. It seems unlikely that the latter could be of the Keplerian form.     

On the oscillatory behaviour of kinematic mean-field dynamos

The paper considers the question after the circumstances in which kinematic mean-field dynamos can have oscillatory magnetic field modes. The conducting fluid body is allowed to be of almost arbitrary shape; its surroundings are vacuum. A general relation for the frequency of oscillation is derived. This relation is discussed more closely for models with pure α²-mechanism. Proof is given that no oscillations can occur for constant α. The investigations published so far on spherical models with pure α²-mechanism call up the question whether there is a chance for axisymmetric modes to be oscillatory. For both spherical models and disk models the possibility of oscillatory axisymmetric modes is demonstrated by examples.     

On the REYNOLDS Stresses in Mean‒Field Hydrodynamics I. Incompressible Homogeneous Isotropic Turbulence

We consider an incompressible fluid undergoing turbulent motions. Using the methods of mean-field electrodynamics the relations between the REYNOLDS stresses and the mean velocity field are deduced in linear approximation. The homogeneous isotropic case is treated in more detail and we show that the effects can be described by a turbulent viscosity which proved to be positive in any case.     

Magnetic field and differential rotation in the radiative zone of the Sun

The problem of the interaction between magnetic fields and differential rotation in the radiative zone of the Sun is investigated. It is demonstrated that effects of magnetic buoyancy can be neglected in the analysis of this interaction. It is shown that hydromagnetic torsional waves propagating from the solar core cannot be responsible for the 22-year solar cycle. A possible geometry of the magnetic field that conforms with stationary differential rotation is considered. A verifying method for hypotheses on the structure of the magnetic field and torsional oscillations in the radiative zone of the Sun is proposed based on helioseismic data.     

Radial variation of differential rotation in the solar electron corona

Long-lived brightness structures in the solar electron corona persist over many solar rotation periods and permit an observational determination of coronal magnetic tracer rotation as a function of latitude and height in the solar atmosphere. For observations over 1964–1976 spanning solar cycle 20, we compare the latitude dependence of rotation at two heights in the corona. Comparison of rotation rates from East and West limbs and from independent computational procedures is used to estimate uncertainty. Time-averaged rotation rates based on three methods of analysis demonstrate that, on average, coronal differential rotation decreases with height from 1.125 to 1.5 R _S. The observed radial variation of differential rotation implies a scale height of approximately 0.7 R _S for coronal differential rotation.Model calculations for a simple MHD loop show that magnetic connections between high and low latitudes may produce the observed radial variations of magnetic tracer rotation. If the observed tracer rotation represents the rotation of open magnetic field lines as well as that of closed loops, the small scale height for differential rotation suggests that the rotation of solar magnetic fields at the base of the solar wind may be only weakly latitude dependent. If, instead, closed loops account completely for the radial gradients of rotation, outward extrapolation of electron coronal rotation may not describe magnetic field rotation at the solar wind source. Inward extrapolations of observed rotation rates suggest that magnetic field and plasma are coupled a few hundredths of a solar radius beneath the photosphere.