scholarly journals Convective differential rotation in stars and planets – I. Theory

2020 ◽  
Vol 498 (3) ◽  
pp. 3758-3781 ◽  
Author(s):  
Adam S Jermyn ◽  
Shashikumar M Chitre ◽  
Pierre Lesaffre ◽  
Christopher A Tout
Keyword(s):  
Differential Rotation ◽  
Detailed Comparison ◽  
Companion Paper ◽  
Angular Frequency ◽  
Circulation Rate ◽  
Rotation Rates ◽  
Rotating Systems ◽  
Order Of Magnitude ◽  
The Magnetic Field ◽  
Hydrodynamic Systems

ABSTRACT We derive the scaling of differential rotation in both slowly and rapidly rotating convection zones using order of magnitude methods. Our calculations apply across stars and fluid planets and all rotation rates, as well as to both magnetized and purely hydrodynamic systems. We find shear |R∇Ω| of order the angular frequency Ω for slowly rotating systems with Ω ≪ |N|, where N is the Brünt–Väisälä frequency, and find that it declines as a power law in Ω for rapidly rotating systems with Ω ≫ |N|. We further calculate the meridional circulation rate and baroclinicity and examine the magnetic field strength in the rapidly rotating limit. Our results are in general agreement with simulations and observations and we perform a detailed comparison with those in a companion paper.

scholarly journals Convective differential rotation in stars and planets – II. Observational and numerical tests

2020 ◽  
Vol 498 (3) ◽  
pp. 3782-3806
Author(s):  
Adam S Jermyn ◽  
Shashikumar M Chitre ◽  
Pierre Lesaffre ◽  
Christopher A Tout
Keyword(s):  
Gravity Waves ◽  
Differential Rotation ◽  
Internal Gravity Waves ◽  
Companion Paper ◽  
General Picture ◽  
Giant Stars ◽  
Numerical Tests ◽  
Rotating Systems ◽  
Order Of Magnitude ◽  
Red Giant

ABSTRACT Differential rotation is central to a great many mysteries in stars and planets. In part I, we predicted the order of magnitude and scaling of the differential rotation in both hydrodynamic and magnetohydrodynamic convection zones. Our results apply to both slowly and rapidly rotating systems, and provide a general picture of differential rotation in stars and fluid planets. We further calculated the scalings of the meridional circulation, entropy gradient, and baroclinicity. In this companion paper, we compare these predictions with a variety of observations and numerical simulations. With a few exceptions, we find that these are consistent in both the slowly rotating and rapidly rotating limits. Our results help to localize core–envelope shear in red giant stars, suggest a rotation-dependent frequency shift in the internal gravity waves of massive stars, and potentially explain observed deviations from von Zeipel’s gravity darkening in late-type stars.

scholarly journals Solar Rotation over Solar Cycle 21

1991 ◽  
Vol 130 ◽  
pp. 241-245
Author(s):  
Elisabeth Ribes ◽  
Istvan Vince ◽  
Pierre Mein ◽  
Eduardo Neto Ferreira
Keyword(s):  
Solar Cycle ◽  
Differential Rotation ◽  
Rotation Rate ◽  
Meridional Circulation ◽  
Solar Rotation ◽  
Circulation Pattern ◽  
Rotation Rates ◽  
Grand Average ◽  
Order Of Magnitude ◽  
The Mean

Abstract Having measured the rotation rate of sunspots through solar cycle 21, from 1977 to 1983, we have found that the mean differential rotation averaged over this seven year record is similar to the grand average differential rotation determined by Howard et al. (1984) over the period 1921-1982. However, the rotation rate does change from year to year. These changes are evidenced by a steepening or a flattening of the mean differential rotation profile, as well as significant changes in the equatorial rate. The presence of a time-dependent pattern of azimuthal rolls inferred from the meridional circulation pattern of the sunspots offers a qualitative explanation of the observed rotation rates. The amplitude of the changes is almost one order of magnitude larger than that of the torsional oscillations found by Howard and LaBonte (1981).

scholarly journals Rotational dependence of turbulent transport coefficients in global convective dynamo simulations of solar-like stars

2020 ◽  
Vol 642 ◽  
pp. A66
Author(s):  
J. Warnecke ◽  
M. J. Käpylä
Keyword(s):  
Magnetic Field ◽  
Differential Rotation ◽  
Turbulent Transport ◽  
Transport Coefficients ◽  
Slow Rotation ◽  
Test Field ◽  
Convective Envelope ◽  
Rapid Rotation ◽  
Rotation Rates ◽  
The Magnetic Field

Context. For moderate and slow rotation, the magnetic activity of solar-like stars is observed to strongly depend on rotation, while for rapid rotation, only a very weak or no dependency is detected. These observations do not yet have a solid explanation in terms of dynamo theory. Aims. We aim to find such an explanation by numerically investigating the rotational dependency of dynamo drivers in solar-like stars, that is, stars that have a convective envelope of similar thickness to that of the Sun. Methods. We ran semi-global convection simulations of stars with rotation rates from 0 to 30 times the solar value, corresponding to Coriolis numbers, Co, of 0 to 110. We measured the turbulent transport coefficients contributing to the magnetic field evolution with the help of the test-field method, and compared with the dynamo effect arising from the differential rotation that is self-consistently generated in the models. Results. The trace of the α tensor increases for moderate rotation rates with Co0.5 and levels off for rapid rotation. This behavior is in agreement with the kinetic α based on the kinetic helicity, if one takes into account the decrease of the convective scale with increasing rotation. The α tensor becomes highly anisotropic for Co ≳ 1. Furthermore, αrr dominates for moderate rotation (1 <  Co <  10), and αϕϕ for rapid rotation (Co ≳ 10). The effective meridional flow, taking into account the turbulent pumping effects, is markedly different from the actual meridional circulation profile. Hence, the turbulent pumping effect is dominating the meridional transport of the magnetic field. Taking all dynamo effects into account, we find three distinct regimes. For slow rotation, the α and Rädler effects are dominating in the presence of anti-solar differential rotation. For moderate rotation, α and Ω effects are dominant, indicative of αΩ or α2Ω dynamos in operation, producing equatorward-migrating dynamo waves with a qualitatively solar-like rotation profile. For rapid rotation, an α2 mechanism with an influence from the Rädler effect appears to be the most probable driver of the dynamo. Conclusions. Our study reveals the presence of a large variety of dynamo effects beyond the classical αΩ mechanism, which need to be investigated further to fully understand the dynamos of solar-like stars. The highly anisotropic α tensor might be the primary reason for the change of axisymmetric to non-axisymmetric dynamo solutions in the moderate rotation regime.

scholarly journals Flux expulsion with dynamics

2016 ◽  
Vol 791 ◽  
pp. 568-588 ◽  
Author(s):  
Andrew D. Gilbert ◽  
Joanne Mason ◽  
Steven M. Tobias
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Lorentz Force ◽  
Differential Rotation ◽  
Time Scale ◽  
Scaling Laws ◽  
Dynamical Regime ◽  
Kinematic Evolution ◽  
Flux Expulsion ◽  
The Magnetic Field

In the process of flux expulsion, a magnetic field is expelled from a region of closed streamlines on a $TR_{m}^{1/3}$ time scale, for magnetic Reynolds number $R_{m}\gg 1$ ($T$ being the turnover time of the flow). This classic result applies in the kinematic regime where the flow field is specified independently of the magnetic field. A weak magnetic ‘core’ is left at the centre of a closed region of streamlines, and this decays exponentially on the $TR_{m}^{1/2}$ time scale. The present paper extends these results to the dynamical regime, where there is competition between the process of flux expulsion and the Lorentz force, which suppresses the differential rotation. This competition is studied using a quasi-linear model in which the flow is constrained to be axisymmetric. The magnetic Prandtl number $R_{m}/R_{e}$ is taken to be small, with $R_{m}$ large, and a range of initial field strengths $b_{0}$ is considered. Two scaling laws are proposed and confirmed numerically. For initial magnetic fields below the threshold $b_{core}=O(UR_{m}^{-1/3})$, flux expulsion operates despite the Lorentz force, cutting through field lines to result in the formation of a central core of magnetic field. Here $U$ is a velocity scale of the flow and magnetic fields are measured in Alfvén units. For larger initial fields the Lorentz force is dominant and the flow creates Alfvén waves that propagate away. The second threshold is $b_{dynam}=O(UR_{m}^{-3/4})$, below which the field follows the kinematic evolution and decays rapidly. Between these two thresholds the magnetic field is strong enough to suppress differential rotation, leaving a magnetically controlled core spinning in solid body motion, which then decays slowly on a time scale of order $TR_{m}$.

Controlling Numerical Dispersion by Variably Timed Flux Updating in Two Dimensions

1982 ◽  
Vol 22 (03) ◽  
pp. 409-419 ◽  
Author(s):  
R.G. Larson
Keyword(s):  
Finite Difference ◽  
Pressure Gradient ◽  
Single Point ◽  
Companion Paper ◽  
Two Dimensions ◽  
Independent Component ◽  
Numerical Dispersion ◽  
Coarse Mesh ◽  
Fixed Concentration ◽  
Order Of Magnitude

Abstract The variably-timed flux updating (VTU) finite difference technique is extended to two dimensions. VTU simulations of miscible floods on a repeated five-spot pattern are compared with exact solutions and with solutions obtained by front tracking. It is found that for neutral and favorable mobility ratios. VTU gives accurate results even on a coarse mesh and reduces numerical dispersion by a factor of 10 or more over the level generated by conventional single-point (SP) upstream weighting. For highly unfavorable mobility ratios, VTU reduces numerical dispersion. but on a coarse mesh the simulation is nevertheless inaccurate because of the inherent inadequacy of the finite-difference estimation of the flow field. Introduction A companion paper (see Pages 399-408) introduced the one-dimensional version of VTU for controlling numerical dispersion in finite-difference simulation of displacements in porous media. For linear and nonlinear, one- and two-independent-component problems, VTU resulted in more than an order-of-magnitude reduction in numerical dispersion over conventional explicit. SP upstream-weighted simulations with the same number of gridblocks. In this paper, the technique is extended to two dimensional (2D) problems, which require solution of a set of coupled partial differential equations that express conservation of material components-i.e., (1) and (2) Fi, the fractional flux of component i, is a function of the set of s - 1 independent-component fractional concentrations {Ci}, which prevail at the given position and time., the dispersion flux, is given by an expression that is linear in the specie concentration gradients. The velocity, is proportional to the pressure gradient,. (3) where lambda, in general, can be a function of composition and of the magnitude of the pressure gradient. The premises on which Eqs. 1 through 3 rest are stated in the companion paper. VTU in Two Dimensions The basic idea of variably-timed flux updating is to use finite-difference discretization of time and space, but to update the flux of a component not every timestep, but with a frequency determined by the corresponding concentration velocity -i.e., the velocity of propagation of fixed concentration of that component. The concentration velocity is a function of time and position. In the formulation described here, the convected flux is upstream-weighted, and all variables except pressure are evaluated explicitly. As described in the companion paper (SPE 8027), the crux of the method is the estimation of the number of timesteps required for a fixed concentration to traverse from an inflow to an outflow face of a gridblock. This task is simpler in one dimension, where there is only one inflow and one outflow face per gridblock, than it is in two dimensions, where each gridblock has in general multiple inflow and outflow faces. SPEJ P. 409^

scholarly journals Comparison of the deep atmospheric dynamics of Jupiter and Saturn in light of the Juno and Cassini gravity measurements

2020 ◽  
Author(s):  
Yohai Kaspi ◽  
Eli Galanti ◽  
Adam Showman ◽  
David Stevenson ◽  
Tristan Guillot ◽  
...  
Keyword(s):  
Planetary Science ◽  
Gravity Inversion ◽  
Zonal Flows ◽  
Zonal Jets ◽  
Cassini Mission ◽  
Gravity Measurements ◽  
Order Of Magnitude ◽  
The Magnetic Field ◽  
Intrinsic Characteristic

&lt;p&gt;The nature and structure of the observed east-west flows on Jupiter and Saturn has been a long-standing mystery in planetary science. This mystery has been recently unraveled by the accurate gravity measurements provided by the Juno mission to Jupiter and the Grand Finale of the Cassini mission to Saturn. These two experiments, which coincidentally happened around the same time, allowed the determination of the overall vertical and meridional profiles of the zonal flows on both planets. In this talk, we discuss what has been learned about the zonal jets on the gas giants in light of the new data from these two experiments. The gravity measurements not only allow the depth of the jets to be constrained, yielding the inference that the jets extend to roughly 3000 and 9000 km below the observed clouds on Jupiter and Saturn, respectively, but also provide insights into the mechanisms controlling these zonal flows. Specifically, for both planets this depth corresponds to the depth where electrical conductivity is within an order of magnitude of 1 S/m, implying that the magnetic field likely plays a key role in damping the zonal flows. An intrinsic characteristic of any gravity inversion, as discussed here, is that the solutions might not be unique. We analyze the robustness of the solutions and present several independent lines of evidence supporting the inference that the jets reach these depths.&lt;/p&gt;

scholarly journals Biases in Thorpe-Scale Estimates of Turbulence Dissipation. Part I: Assessments from Large-Scale Overturns in Oceanographic Data

2015 ◽  
Vol 45 (10) ◽  
pp. 2497-2521 ◽  
Author(s):  
Benjamin D. Mater ◽  
Subhas K. Venayagamoorthy ◽  
Louis St. Laurent ◽  
James N. Moum
Keyword(s):  
Large Scale ◽  
Companion Paper ◽  
Open Ocean ◽  
Luzon Strait ◽  
Length Scale ◽  
Ensemble Averaging ◽  
Tidal Period ◽  
Thorpe Scale ◽  
Order Of Magnitude ◽  
A Site

AbstractOceanic density overturns are commonly used to parameterize the dissipation rate of turbulent kinetic energy. This method assumes a linear scaling between the Thorpe length scale LT and the Ozmidov length scale LO. Historic evidence supporting LT ~ LO has been shown for relatively weak shear-driven turbulence of the thermocline; however, little support for the method exists in regions of turbulence driven by the convective collapse of topographically influenced overturns that are large by open-ocean standards. This study presents a direct comparison of LT and LO, using vertical profiles of temperature and microstructure shear collected in the Luzon Strait—a site characterized by topographically influenced overturns up to O(100) m in scale. The comparison is also done for open-ocean sites in the Brazil basin and North Atlantic where overturns are generally smaller and due to different processes. A key result is that LT/LO increases with overturn size in a fashion similar to that observed in numerical studies of Kelvin–Helmholtz (K–H) instabilities for all sites but is most clear in data from the Luzon Strait. Resultant bias in parameterized dissipation is mitigated by ensemble averaging; however, a positive bias appears when instantaneous observations are depth and time integrated. For a series of profiles taken during a spring tidal period in the Luzon Strait, the integrated value is nearly an order of magnitude larger than that based on the microstructure observations. Physical arguments supporting LT ~ LO are revisited, and conceptual regimes explaining the relationship between LT/LO and a nondimensional overturn size are proposed. In a companion paper, Scotti obtains similar conclusions from energetics arguments and simulations.

Dynamically consistent magnetic fields produced by differential rotation

1987 ◽  
Vol 178 ◽  
pp. 521-534 ◽  
Author(s):  
D. R. Fearn ◽  
M. R. E. Proctor
Keyword(s):  
Magnetic Field ◽  
Velocity Field ◽  
Flow Field ◽  
Differential Rotation ◽  
Zonal Flow ◽  
Nonlinear Characteristic ◽  
Poloidal Magnetic Field ◽  
Self Consistent ◽  
Self Consistency ◽  
The Magnetic Field

We investigate the dynamical consequences of an axisymmetric velocity field with a poloidal magnetic field driven by a prescribed e.m.f. E. The problem is motivated by previous investigations of dynamically driven dynamos in the magnetostrophic range. A geostrophic zonal flow field is added to a previously described velocity, and determined by the requirement that Taylor's constraint (Taylor 1963) (guaranteeing dynamical self-consistency of the fields) be satisfied. Several solutions are exhibited, and it is suggested that self-consistent solutions can always be found to this ‘forced’ problem, whereas the usual α-effect dynamo formalism in which E is a linear function of the magnetic field leads to a difficult transcendentally nonlinear characteristic value problem that may not always possess solutions.

scholarly journals Magnetorotational supernovae with different equations of state

2011 ◽  
Vol 7 (S279) ◽  
pp. 357-358
Author(s):  
Sergey G. Moiseenko ◽  
Gennady S. Bisnovatyi-Kogan
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Angular Momentum ◽  
Differential Rotation ◽  
Equations Of State ◽  
Supernova Explosion ◽  
Explosion Energy ◽  
Iron Core ◽  
Wave Forms ◽  
The Magnetic Field

AbstractWe present results of the simulation of a magneto-rotational supernova explosion. We show that, due to the differential rotation of the collapsing iron core, the magnetic field increases with time. The magnetic field transfers angular momentum and a MHD shock wave forms. This shock wave produces the supernova explosion. The explosion energy computed in our simulations is 0.5-2.5 ċ 1051erg. We used two different equations of state for the simulations. The results are rather similar.

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