Turbulence‐driven angular momentum transport in modulated Kepler flows

The velocity fluctuations in a spherical shell arising from sinusoidal perturbations of a Keplerian shear flow with a free amplitude parameter ε are studied numerically by means of fully 3D nonlinear simulations. The investigations are performed at high Reynolds numbers, i.e. 3000 < Re < 5000. We find Taylor‐Proudman columns of large eddies parallel to the rotation axis for sufficiently strong perturbations. An instability sets in at critical amplitudes with ε_crit ∝ Re^—1. The whole flow turns out to be almost axisymmetric and nonturbulent exhibiting, however, a very rich radial and latitudinal structure. The Reynolds stress 〈u′_ru′_ϕ〉 is positive in the entire computational domain, from its Gaussian radial profile a positive viscosity‐alpha of about 10^—4 is derived. The kinetic energy of the turbulent state is dominated by the azimuthal component 〈u^′2_ϕ whereas the other components are smaller by two orders of magnitude. Our simulations reveal, however, that these structures disappear as soon as the perturbations are switched off. We did not find an “effective” perturbation whose amplitude is such that the disturbance is sustained for large times (cf. Dauchot & Daviaud 1995) which is due to the effective violation of the Rayleigh stability criterion. The fluctuations rapidly smooth the original profile towards to pure Kepler flow which, therefore, proves to be stable in that sense.     

Helical magnetorotational instability of Taylor‐Couette flows in the Rayleigh limit and for quasi‐Kepler rotation

The magnetorotational instability (MRI) of differential rotation under the simultaneous presence of axial and azimuthal components of the (current‐free) magnetic field is considered. For rotation with uniform specific angular momentum the MHD equations for axisymmetric perturbations are solved in a local short‐wave approximation. All the solutions are overstable for B_z · B_ϕ ≠ 0 with eigenfrequencies approaching the viscous frequency. For more flat rotation laws the results of the local approximation do not comply with the results of a global calculation of the MHD instability of Taylor‐Couette flows between rotating cylinders. – With B_ϕ and B_z of the same order the traveling‐mode solutions are also prefered for flat rotation laws such as the quasi‐Kepler rotation. For magnetic Prandtl number Pm → 0 they scale with the Reynolds number of rotation rather than with the magnetic Reynolds number (as for standard MRI) so that they can easily be realized in MHD laboratory experiments. – Regarding the nonaxisymmetric modes one finds a remarkable influence of the ratio B_ϕ/B_z only for the extrema. For B_ϕ ≫ B_z and for not too small Pm the nonaxisymmetric modes dominate the traveling axisymmetric modes. For standard MRI with B_z ≫ B_ϕ, however, the critical Reynolds numbers of the nonaxisymmetric modes exceed the values for the axisymmetric modes by many orders so that they are never prefered. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A theory of differntial rotation based on the discussion of turbulent transport of angular momentum

This paper deals with the theory of the solar rotational law. We assume the turbulence to be of the largest influence compared with the momentum flux caused by molecular viscosity and meridional circulation. Firstly we use heuristical forms for the needed cross correlations Qrφ (turbulent radial momentum flux) and Qνφ (turbulent latitudinal momentum flux): Qrφ = −α₀r ϑΩ/ϑr · sin ν + Q₀ sin ν + Q₂ sin³ ν, Qνφ = −δ₀ ϑΩ/ϑν· sin ν + P₂ sin²θ cos θ. It is shown that a radial dependence of the angular velocity Ω is given by Q₀. Furthermore, the observed equatorial acceleration occurs in the case of non-negativity of Q₂ and/or P₂. Because of the spatial dependence of the solar angular velocity the coefficients of Q and P are unfortunately not to be measured. Secondly, we determine the coefficients with a theory founded upon the hypothesis that a rotating stochastical force field — independent from Ω — maintains an anisotropic turbulence. The global fast rotation produces, indeed, finite cross correlations Q₂ and P₂. It is suggested that horizontally directed turbulent motions with not too small radial correlations lengths and time scales of about 2 weeks could be responsible for the solar differential rotation. Finally, we show that also short-living turbulent horizontal modes provide the observed equatorial acceleration if they occur preferably at the equatorial region.     

Hydrodynamic processes of angular momentum transport in the interior of a rotating massive hydrogen-burning star

The evolution of a rotating star with a mass of 16M _⊙ at the hydrogen burning phase is considered together with the hydrodynamic processes of angular momentum transport in its interior. Shear turbulence is shown to limit the amplitude of the latitudinal variations in mean molecular weight on a surface of constant pressure in a layer with variable chemical composition. The resulting nonuniformity in the mean molecular weight distribution and the turbulent energy transport along the surface of constant pressure reduce the absolute value of the meridional circulation velocity. Nevertheless, meridional circulation remains the main mechanism of angular momentum transport in the radial direction in a layer with variable chemical composition. The intensity of the processes of angular momentum transport by meridional circulation and shear turbulence is determined by the angular momentum of the star. At a fairly high angular momentum, more specifically, at J = 3.69 × 10⁵² g cm² s^?1, the star during the second half of the hydrogen-burning phase in its convective core has characteristics typical of classical early Be stars.     

Über turbulente Bewegungen,die von Temperaturschwankungen hervorgerufen werden – eine kritische Betrachtung zur Boussinesq-Approximation

This paper deals with fluctuating motions which are caused by a given stochastical temperature field acting in a gas with gravitation and T¯ = const. It results that the often used BOUSSINESQ approximation much underestimates the horizontal motions in case wide-spread temperature fluctuations occur. For sufficiently large scales the horizontal motion exceeds the vertical ones even in the case of the temperature field fluctuating completely isotropically. Scales of 1000 km and 1 day in the Earth atmosphere lead to the observed value u′_horiz/|u′_vertic ≈︁ 10. Finally besides the relation between density correlation and pressure correlation we determine the expression for the turbulent mass transport vanishing with the molecular viscosity.     

Effect of partial mixing of matter on the hydrodynamic angular momentum transport processes in massive main-sequence stars

We consider the evolution of a rotating star with a mass of 16M _⊙ and an angular momentum of 3.25 × 10⁵² g cm² s^?1, along with the hydrodynamic transport of angular momentum and chemical elements in its interiors. When the partial mixing of matter of the turbulent radiative envelope and the convective core is taken into account, the efficiency of the angular momentum transport by meridional circulation in the stellar interiors and the duration of the hydrogen burning phase increase. Depending on the Schmidt number in the turbulent radiative stellar envelope, the ratio of the equatorial rotational velocity to the circular one increases with time in the process of stellar evolution and can become typical of early-type Be stars during an additional evolution time of the star on the main sequence. Partial mixing of matter is a necessary condition under which the hydrodynamic transport processes can increase the angular momentum of the outer stellar layer to an extent that the equatorial rotational velocity begins to increase during the second half of the evolutionary phase of the star on the main sequence, as shown by observations of the brightest stars in open star clusters with ages of 10–25 Myr. When the turbulent Schmidt number is 0.4, the equatorial rotational velocity of the star increases during the second half of the hydrogen burning phase in the convective core from 330 to 450 km s^?1.     

Nonlinear simulations of magnetic Taylor‐Couette flow with currentfree helical magnetic fields

Themagnetorotational instability (MRI) in cylindrical Taylor‐Couette flow with external helical magnetic field is simulated for infinite and finite aspect ratios. We solve the MHD equations in their small Prandtl number limit and confirm with timedependent nonlinear simulations that the additional toroidal component of the magnetic field reduces the critical Reynolds number from O (10⁶) (axial field only) to O (10³) for liquid metals with their small magnetic Prandtl number. Computing the saturated state we obtain velocity amplitudes which help designing proper experimental setups. Experiments with liquid gallium require axial field ∼50 Gauss and axial current ∼4 kA for the toroidal field. It is sufficient that the vertical velocity u_z of the flow can be measured with a precision of 0.1 mm/s.We also show that the endplates enclosing the cylinders do not destroy the traveling wave instability which can be observed as presented in earlier studies. For TC containers without and with endplates the angular momentum transport of the MRI instability is shown as to be outwards. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sun's inertial motion and luminosity

Fast Fourier analysis of the detrended record of solar irradiance obtained by the Nimbus-7 cavity pyrheliometer shows a rich spectrum of significant frequencies between about 30 and 850 nHz (periods between 13 and 400 days). Wolff and Hickey (1987a, b), elaborating on a model developed by Wolff (1974a, b, 1976, 1983, 1984), suggest that many of these peaks arise due to interference of rigidly rotating global solar oscillations (r- and g-modes). Their model fit is quite good in the region above about 135 nHz, but less satisfactory below this threshold. We note that the FFT spectrum of d² L/d² t, the second derivative of angular momentum of the solar inertial motion, contains peaks matching the large peaks in the irradiance spectrum below 400 nHz with periods near 0.08, 0.24, 0.65, and about 1 yr. We discuss the origins of the peaks in the d² L/d²t spectra and review some previous studies bearing on the question of a possible relationship of solar motion and solar activity. The future persistence of the observed spectral peaks of irradiance with periods near 0.24 and 0.65 yr will provide a key test for this hypothesis.     

Evolutionary status of the primary component of YZ Cassiopeiae

The models of non-rotating and rotating 2.31M _\ stars of Population I composition have been calculated, starting at the threshold of stability. A 2.31M _\ star was chosen to compare the results with the observational parameters of the primary component of the well-known detached binary YZ Cassiopeiae. The effects of rotation on the internal structure during the evolution of the star were studied by constructing sequences of axisymmetric rotating models under the assumption that angular momentum was conserved according to a predetermined angular velocity distribution depending on the structure of the star.The first section of this paper deals with effects of rotation on the evolutionary behaviours of the 2.31M _\ star through the pre-Main-Sequence evolution as well as the zero-age Main Sequence.In the second section of this paper, the evolutionary studies have been extended up to near-hydrogen exhaustion phase in order to obtain a theoretical model corresponding to the given mass and radius of the primary component of YZ Cassiopeiae. The theoretical models were found to be in a good agreement with observational parameters. The computed rotating models of the primary of YZ Cassiopeiae indicates that its evolutionary age is 6.01×10⁸ years; and the central hydrogen content 0.183 — which means that about 75% of its original value was depleted.     

Evolution of the Oort cloud angular momentum: Numerical simulation

This paper considers the evolution of a flat svarm of cometary bodies (under the effect of the passage of stars), initially moving in one direction along the circular orbits with radii 1.4×10⁴<r<2×10⁴ AU and along elliptic orbits with semi-major axes 5×10³<a<1×10⁴ AU and with perihelia within 50<q<100 AU. Numerical simulation shows that the original flat belt of comets is thermalizing. Its root-mean-squarez-coordinate grows withr. A cometary cloud forms with a dense flattened inner core and a rarefied halo (the Oort cloud proper). The value =N _core/N _halo varies within a wide range (up to the order of magnitude) depending on the model used (N _core andN _halo are the numbers of comets in the core and the halo, respectively).The hypothesis of a massive Oort cloud (Marochniket al., 1988) implies that the Oort cloud should have a large angular momentum. This paper employs numerical simulation to calculate Oort cloud models to which the initially flat located at the periphery of the solar nebula rotating cometary swarms is evolving in time. The loss of the initial angular momentum over the time of the Oort cloud evolution is not large.     

Pre-Main-Sequence evolution of rotating low-mass stars

The evolutionary behaviour of rotating low-mass stars in the mass range 0.2 and 0.9M has been investigated during the pre-Main-Sequence phase. The angular momentum is conserved locally in radiative regions and totally in convective regions, according to a predetermined angular velocity distribution depending on the structure of the star. As the stars contract toward the zero-age Main Sequence, they spin up under the assumption that the angular momentum is conserved during the evolution of the stars. When the stars have differential rotations, their inner regions rotate faster than the outer regions. The effective temperatures and luminosities of rotating low-mass stars are obtained lower than those of non-rotating stars. They have lower central temperature and density values compared to those of non-rotating stars.     

Temporal behaviour of global perturbations in compressible axisymmetric flows with free boundaries

The dynamics of small global perturbations in the form of a linear combination of a finite number of non‐axisymmetric eigenmodes is studied in the two‐dimensional approximation. The background flow is assumed to be an axisymmetric perfect fluid with adiabatic index γ = 5/3 rotating with a power law angular velocity distribution Γ ∝ r^–q , 1.5 < q < 2.0, confined by free boundaries in the radial direction. The substantial transient growth of acoustic energy of optimized perturbations is discovered. An optimal energy growth G is calculated numerically for a variety of parameters. Its value depends essentially on the perturbation azimuthal wavenumber m and increases for higher values of m. The closer the rotation profile to the Keplerian law, the larger growth factors can be obtained but over a longer time. The highest acoustic energy increase found numerically is of order ∼10² over ∼6 typical Keplerian periods. Slow neutral eigenmodes with corotation radius beyond the outer boundary mostly contribute to the transient growth. The revealed linear temporal behaviour of perturbations may play an important role in angular momentum transfer in toroidal flows near compact relativistic objects (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solar wind and spin of small interplanetary particles

The action of the solar corpuscular radiation on the rotational properties of small interplanetary dust particles is investigated. It is shown that the solar wind increases the angular momentum (spin) of the particle. Analytic solutions are presented for dominant terms in which quantities of the orders (v/u) ⁿ ,n 1, are neglected (v is the orbital velocity of dust particle around the Sun andu is the speed of the solar wind particles).     

A statistical study of the subsurface structure and eruptivity of solar active regions

A statistical study of 77 solar active regions (ARs) is conducted to investigate the existence of identifiable correlations between the subsurface structural disturbances and the activity level of the active regions. The disturbances examined in this study are 〈|δΓ ₁/Γ ₁|〉, 〈|δc ²/c ²|〉, and 〈|δc ²/c ²?δΓ ₁/Γ ₁|〉, where Γ ₁ and c are the thermodynamic properties of first adiabatic index and sound speed modified by magnetic field, respectively. The averages are over three depth layers: 0.975–0.98R _⊙, 0.98–0.99R _⊙ and 0.99–0.995R _⊙ to represent the structural disturbances in that layer. The level of the surface magnetic activity is measured by the Magnetic Activity Index (MAI) of active region and the relative and absolute MAI differences (rdMAI and dMAI) between the active and quiet regions. The eruptivity of each active region is quantified by its Flare Index, total number of coronal mass ejections (CMEs), and total kinetic energy of the CMEs. The existence and level of the correlations are evaluated by scatter plots and correlation coefficients. No definitive correlation can be claimed from the results. While a weak positive trend is visible between dMAI and 〈|δΓ ₁/Γ ₁|〉 and 〈|δc ²/c ²|〉 in the layer 0.975–0.98R _⊙, their correlation levels, being approximately 0.6, are not sufficiently high to justify the correlation. Some subsurface disturbances are seen to increase with eruptivity indices among ARs with high eruptivity. The statistical significance of such trend, however, cannot be ascertained due to the small number of very eruptive ARs in our sample.     

Angular momentum transport by thermal emission in black hole accretion disks

We calculate the amount of angular momentum that thermal photons carry out of a viscous black hole accretion disk, due to the strong Doppler shift imparted to them by the high orbital velocity of the radiating disk material. While thermal emission can not drive accretion on its own, we show that along with disk heating it does nonetheless result in a loss of specific angular momentum, thereby contributing to an otherwise viscosity‐driven accretion flow. In particular, we show that the fraction of the angular momentum that is lost to thermal emission at a radius r in a standard, multi‐color disk is ∼0.4r_s/r, where r_s is the Schwarzschild radius of the black hole. We briefly highlight the key similarties between this effect and the closely related Poynting‐Robertson effect (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Equilibrium structure of stars obeying a generalized differential rotation law

In the present paper we have considered the problem of determining the equilibrium structure of differentially rotating stars in which the angular velocity of rotation varies both along the axis of rotation and in directions perpendicular to it. For this purpose, a generalized law of differential rotation of the type ² =b ₀+b ₁ s ²+b ₂ s ⁴+b ₃ z ²+b ₄ z ⁴+b ₅ z ² s ² (here is a nondimensional measure of the angular velocity of a fluid element distants from the axis of rotation andz from the plane through the centre of the star perpendicular to the axis of rotation, andb's are suitably chosen parameters) has been used. Whereas Kippenhahn and Thomas averaging approach has been used to incorporate the rotational effects in the stellar structure equations, Kopal's results on Roche equipotentials have been used to obtain the explicit form of the stellar structure equations, which incorporate the rotational effects up to second order of smallness in the distortion parameters. The method has been used to compute the equilibrium structure of certain differentially rotating polytropes. Certain differentially rotating polytropes. Certain differentially rotating models of the Sun have also been computed by using this approach.     

Effects of variations of parallel angular velocity and vorticity on the oscillations of compressible homogeneous rotating ellipsoids

Earlier work on the oscillations of an ellipsoid is extended to investigate the behaviour of a nonequilibrium compressible homogeneous rotating gaseous ellipsoid, with the components of the velocity field as linear functions of the coordinates, and with parallel angular velocity and uniform vorticity. The dynamical behaviour of the ellipsoid is obtained by numerically integrating the relevant differential equations for different values of the initial angular velocity and vorticity. This behaviour is displayed by the (a ₁,a ₂) and (a ₁,a ₃) phase plots, where thea _i's (i = 1, 2, 3) are the semi-diameters, and by the graphs ofa ₁,a ₂,a ₃, the volume, and the angular velocity as functions of time.The dynamical behaviour of the nonequilibrium ellipsoid depends on the deviation of the angular momentum from its equilibrium value; for larger deviations, the oscillations are more nonperiodic with larger amplitudes.An initially ellipsoidal configuration always remains ellipsoidal, but it cannot become spheroidal about its rotation axis, though it may become spheroidal instantaneously about either one of the other two principal axes.For an ellipsoid approaching axisymmetry about its axis of rotation, the angular velocity can suddenly increase by a large amount. Thus if an astrophysical object can be modelled by a nonequilibrium ellipsoid, it may occasionally undergo sudden large increases of angular velocity.     

Satellites and Riemannian geometry

In an axially symmetric three-dimensional Riemann-spaceg _ik(u ¹,u ²)?u ³ represents the cyclic parameter-, a gravitational potential ?(u ¹,u ²) is given. For all masspoints with equal total energy and equal angular momentum there exists a function Ψ(u ¹,u ²) by means of which the equations of motion can be reduced to a simple ordinary second-order differential equation. The function ? can be interpreted as the velocity with which the masspoint moves in the two-dimensional spaceu ¹,u ². Of particular interest is the case where the spaceu ¹,u ²,u ³ is Euclidean. Ifu ¹,u ² are Cartesian coordinates in a planeu ³=const., and if the tangent vector of the trajectoryu ¹(t)u ²(t) has the components cosω, sinω it is shown that the triple integral $$\smallint \smallint \smallint \psi du^1 du^2 d\omega $$ is an invariant integral in Cartan's sense, in other words, if the integral is extended over a domain in a meridian plane at timet=0, it keeps its value at any time.