Turbulent Transport Coefficients Research Articles

Web apps for profile fitting and power balance analysis at Wendelstein 7-X.

Two novel web apps for W7-X are introduced: Profile Cooker and Power House. They are designed to streamline the workflow of profile fitting and power balance analysis while offering a graphical user interface that works in any common browser. This allows us to compile a comprehensive database of experimental power balance results. All fitting functions available in Profile Cooker are presented and compared on the basis of example profiles. The power balance equation assumed in Power House is established and its individual terms are discussed. The main focus of the power balance analysis is on the turbulent transport coefficients. A model for quick calculation of neutral beam power deposition based on experimental profiles is presented. Neoclassical root transition poses an issue for power balance analysis due to the uncertainty of the radial electric field. A global, neoclassical simulation with the code EUTERPE is performed for a set of experimental profiles to gain an understanding of the neoclassical root transition.

Stability and transport of gyrokinetic critical pedestals

A gyrokinetic threshold model for pedestal width–height scaling prediction is applied to multiple devices. A shaping and aspect ratio scan is performed on National Spherical Torus Experiment (NSTX) equilibria, finding Δped=0.92A1.04κ−1.240.38δβθ,ped1.05 for the wide-pedestal branch with pedestal width Δped , aspect ratio A, elongation κ, triangularity δ, and normalized pedestal height βθ,ped . The width–transport scaling is found to vary significantly if the pedestal height is varied either with a fixed density or fixed temperature, showing how fueling and heating sources affect the pedestal density and temperature profiles for the kinetic-ballooning-mode (KBM) limited profiles. For an NSTX equilibrium, at fixed density, the wide branch is Δped=0.028(qe/Γe−1.7)1.5∼ηe1.5 and at fixed temperature Δped=0.31(qe/Γe−4.7)0.85 ∼ηe0.85 , where qe and Γe are turbulent electron heat and particle fluxes and ηe=∇ln⁡Te/∇ln⁡ne for an electron temperature Te and density ne . Pedestals close to the KBM limit are shown to have modified turbulent transport coefficients compared to the strongly driven KBMs. The role of flow shear is studied as a width–height scaling constraint and pedestal saturation mechanism for a standard and lithiated wide pedestal discharge. Finally, the stability, transport, and flow shear constraints are combined and examined for an NSTX experiment.

Transport-driven toroidal rotation with general viscosity profile

Using the assumption of a weak normalized turbulent viscosity, usually valid in practice, the modulated-transport model (Stoltzfus-Dueck 2012 Phys. Plasmas 19 055908) is generalized to allow the turbulent transport coefficient to vary in an arbitrary way on radial and poloidal position. The new approach clarifies the physical interpretation of the earlier results and significantly simplifies the calculation, via a boundary-layer asymptotic method. Rigorous detailed appendices verify the result of the simple boundary-layer calculation, also demonstrating that it achieves the claimed order of accuracy and providing a concrete prediction for the strong plasma flows in the immediate vicinity of the last closed flux surface. The new formulas are used to predict plasma rotation at the core-edge boundary, in cases with and without externally applied torque. Dimensional formulas and extensive discussion are provided, to support experimental application of the new model.

Relations among Turbulent Fluctuations, Zonal Flows, and Transport Coefficients in Time Series Data of Gyrokinetic Simulations

The relations among turbulent amplitude, zonal-flow amplitude, and transport level are discussed for the time series data of nonlinear gyrokinetic simulations for magnetized toroidal plasmas. Since it was shown that the transport coefficient can be expressed as a function of the time-averaged turbulent fluctuation level and the zonal flow amplitude [M. Nunami et al., Phys. Plasmas 20, 092307 (2013)], we apply the results to a model function for the turbulent plasma transport coefficient to extend to a functional relation which includes not the time-averaged data but the time-series data of gyrokinetic simulations. We obtain a new functional relation to the turbulent fluctuations, the zonal-flow amplitudes, and the transport coefficients as a function of the input parameters of the gyrokinetic simulations such as plasma temperature gradients. It is also confirmed that the obtained functional relation can reduce relative error which is compared with the original function with time-averages.

Modelling stellar convective transport with plumes – I. Non-equilibrium turbulence effect in double-averaging formulation

ABSTRACT Plumes in a convective flow are considered to be relevant to the turbulent transport in convection. The effective mass, momentum, and heat transports in the convective turbulence are investigated in the framework of time–space double averaging procedure, where a field quantity is decomposed into three parts: the spatiotemporal mean (spatial average of the time-averaged) field, the dispersion or coherent fluctuation, and the random or incoherent fluctuation. With this framework, turbulent correlations in the mean-field equations are divided into the dispersion/coherent and random/incoherent correlation part. By reckoning the plume as the coherent fluctuation, a transport model for the convective turbulence is constructed with the aid of the non-equilibrium effect, in which the change of turbulence characteristics along the mean stream is taken into account for the modelling of the turbulent transport coefficients. In this work, for the first time, change of turbulence properties along plume motions is incorporated into the expression of the turbulent transport coefficients. This non-equilibrium model is applied to a stellar convective flow. One of the prominent characteristics of a surface cooling-driven convection, the enhanced and localized turbulent mass flux below the surface layer, which cannot be reproduced at all by the usual eddy-diffusivity model with mixing length theory (MLT), is well reproduced by the present model. Our results show that the incorporation of plume motion into turbulent transport model is an important and very relevant extension of mean-field theory beyond the heuristic gradient transport model with MLT.

Non-locality of the turbulent electromotive force

ABSTRACTThe generation of large-scale magnetic fields ($\overline{\boldsymbol {B}}$) in astrophysical systems is driven by the mean turbulent electromotive force ($\overline{\rm{\boldsymbol {\cal E}} {}}$), the cross-correlation between local fluctuations of velocity and magnetic fields. This can depend non-locally on $\overline{\boldsymbol {B}}$ through a convolution kernel Kij. In a new approach to find Kij, we directly fit the time-series data of $\overline{\rm{\boldsymbol {\cal E}} {}}$ versus $\overline{\boldsymbol {B}}$ from a galactic dynamo simulation using singular value decomposition. We calculate the usual turbulent transport coefficients as moments of Kij, and show the importance of including non-locality over eddy length-scales to fully capture their amplitudes and that higher order corrections to the standard transport coefficients are small in this case.

Generation of mean flows in rotating anisotropic turbulence: The case of solar near-surface shear layer

Context.Results from helioseismology indicate that the radial gradient of the rotation rate in the near-surface shear layer (NSSL) of the Sun is independent of latitude and radius. Theoretical models using the mean-field approach have been successful in explaining this property of the NSSL, while global direct or large-eddy magnetoconvection models have so far been unable to reproduce this.Aims.We investigate the reason for this discrepancy by measuring the mean flows, Reynolds stress, and turbulent transport coefficients under conditions mimicking those in the solar NSSL.Methods.Simulations with as few ingredients as possible to generate mean flows were studied. These ingredients are inhomogeneity due to boundaries, anisotropic turbulence, and rotation. The parameters of the simulations were chosen such that they matched the weakly rotationally constrained NSSL. The simulations probe locally Cartesian patches of the star at a given depth and latitude. The depth of the patch was varied by changing the rotation rate such that the resulting Coriolis numbers covered the same range as in the NSSL. We measured the turbulent transport coefficient relevant for the nondiffusive (Λ-effect) and diffusive (turbulent viscosity) parts of the Reynolds stress and compared them with predictions of current mean-field theories.Results.A negative radial gradient of the mean flow is generated only at the equator where meridional flows are absent. At other latitudes, the meridional flow is comparable to the mean flow corresponding to differential rotation. We also find that the meridional components of the Reynolds stress cannot be ignored. Additionally, we find that the turbulent viscosity is quenched by rotation by about 50% from the surface to the bottom of the NSSL.Conclusions.Our local simulations do not validate the explanation for the generation of the NSSL from mean-field theory where meridional flows and stresses are neglected. However, the rotational dependence of the turbulent viscosity in our simulations agrees well with theoretical predictions. Moreover, our results agree qualitatively with global convection simulations in that an NSSL can only be obtained near the equator.

Investigating Global Convective Dynamos with Mean-field Models: Full Spectrum of Turbulent Effects Required

Abstract The role of turbulent effects for dynamos in the Sun and stars continues to be debated. Mean-field (MF) theory provides a broadly used framework to connect these effects to fundamental magnetohydrodynamics. While inaccessible observationally, turbulent effects can be directly studied using global convective dynamo (GCD) simulations. We measure the turbulent effects in terms of turbulent transport coefficients, based on the MF framework, from an exemplary GCD simulation using the test-field method. These coefficients are then used as an input into an MF model. We find a good agreement between the MF and GCD solutions, which validates our theoretical approach. This agreement requires all turbulent effects to be included, even those which have been regarded as unimportant so far. Our results suggest that simple dynamo models, as are commonly used in the solar and stellar community, relying on very few, precisely fine-tuned turbulent effects, may not be representative of the full dynamics of dynamos in global convective simulations and astronomical objects.

Scaling of Turbulent Viscosity and Resistivity: Extracting a Scale-dependent Turbulent Magnetic Prandtl Number

Abstract Turbulent viscosity ν t and resistivity η t are perhaps the simplest models for turbulent transport of angular momentum and magnetic fields, respectively. The associated turbulent magnetic Prandtl number Pr t ≡ ν t /η t has been well recognized to determine the final magnetic configuration of accretion disks. Here, we present an approach to determining these “effective transport” coefficients acting at different length scales using coarse-graining and recent results on decoupled kinetic and magnetic energy cascades. By analyzing the kinetic and magnetic energy cascades from a suite of high-resolution simulations, we show that our definitions of ν t , η t , and Pr t have power-law scalings in the “decoupled range.” We observe that Pr t ≈ 1–2 at the smallest inertial-inductive scales, increasing to ≈5 at the largest scales. However, based on physical considerations, our analysis suggests that Pr t has to become scale independent and of order unity in the decoupled range at sufficiently high Reynolds numbers (or grid resolution) and that the power-law scaling exponents of velocity and magnetic spectra become equal. In addition to implications for astrophysical systems, the scale-dependent turbulent transport coefficients offer a guide for large-eddy simulation modeling.

Bayesian analysis of turbulent transport coefficients in 2D interchange dominated ExB turbulence involving flow shear

While turbulent transport is known to dominate the radial particle and energy transport in the plasma edge, a self-consistent description of turbulent transport in mean-field transport codes remains lacking. Mean-field closure models based on the turbulent kinetic energy (k ⊥) and turbulent enstrophy (ζ⊥) have recently been proposed to self-consistently model this transport. This contribution analyses the diffusive particle transport relations of these models by means of the Bayesian framework for parameter estimation and model comparison. The reference data includes a set of isothermal simulations that not only include the scrape-off layer (SOL) but also the outer core region (in which a drift wave-like model is activated) and a set of anisothermal SOL simulations, both obtained with the TOKAM2D turbulence code. This analysis shows that the k ⊥(–ζ⊥) model does not appropriately capture the diffusion coefficients for these new data sets, presumably due to the strong flows in the diamagnetic direction that appear in these new cases. While flow shear is expected to quench the turbulence and the turbulent transport, its effect was not explicitly taken into account in the earlier k ⊥(–ζ⊥) transport models. As flow shear provides a new mechanism for the decorrelation of the turbulence, we propose to introduce an additional time scale in the diffusive transport relation as . Inspiration is drawn from shear decorrelation times reported in literature to propose several new candidate models, which are then analysed in a Bayesian setting. This allowed identifying irrelevant terms for certain models and to rank all models according to the Bayesian evidence. While the new models accounting for shear do improve the match to the data, significant errors still remain. Also, no single model could be identified that performs best for all data sets.

On the Existence of Shear-current Effects in Magnetized Burgulence

Abstract The possibility of explaining shear flow dynamos by a magnetic shear-current (MSC) effect is examined via numerical simulations. Our primary diagnostics is the determination of the turbulent magnetic diffusivity tensor η . In our setup, a negative sign of its component η yx is necessary for coherent dynamo action by the SC effect. To be able to measure turbulent transport coefficients from systems with magnetic background turbulence, we present an extension of the test-field method (TFM) applicable to our setup where the pressure gradient is dropped from the momentum equation: the nonlinear TFM (NLTFM). Our momentum equation is related to Burgers’ equation and the resulting flows are referred to as magnetized burgulence. We use both stochastic kinetic and magnetic forcings to mimic cases without and with simultaneous small-scale dynamo action. When we force only kinetically, negative η yx are obtained with exponential growth in both the radial and azimuthal mean magnetic field components. Using magnetokinetic forcing, the field growth is no longer exponential, while NLTFM yields positive η yx . By employing an alternative forcing from which wavevectors whose components correspond to the largest scales are removed, the exponential growth is recovered, but the NLTFM results do not change significantly. Analyzing the dynamo excitation conditions for the coherent SC and incoherent α and SC effects shows that the incoherent effects are the main drivers of the dynamo in the majority of cases. We find no evidence for MSC-effect-driven dynamos in our simulations.

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

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.

Karl‐Heinz Rädler (1935–2020)

Karl-Heinz Rädler died on the 9th of February 2020 at his home near Potsdam at the age of 85 years. He is known for his path-breaking contributions to the development of cosmic dynamo theory. Together with Max Steenbeck and Fritz Krause, he presented a rigorous derivation of the α effect in 1966. A few years later, after their work became known in the West, many scientists around the globe applied their theory and started working on models of solar and Galactic dynamos. The field of mean-field dynamo theory was born, and it started a new industry in astrophysics. The roots of what is now sometimes referred to as the Potsdam Dynamo School go back to an earlier episode that started in Jena. After graduating from the University of Leipzig, Karl-Heinz Rädler developed his lifelong interest in the origin of cosmic magnetic fields when he took up a position as an assistant at the Institute for Magnetohydrodynamics at the Akademie der Wissenschaften in Jena. He joined the group of Max Steenbeck, where the theory of averaged magnetic fields was formulated and where he defended his PhD thesis “Zur Elektrodynamik turbulent bewegter leitender Medien.” It marked the beginning of a new discipline of mathematical physics, dealing with the equations of magnetohydrodynamics in turbulent media. In the 1970s, the research activities were moved to the Zentralinstitut für Astrophysik at Potsdam. This activity significantly shaped the profile of the Institute and contributed to its high international standing. Until today, many of the theoretical, numerical, and experimental studies of magnetohydrodynamics are based on the monograph “Mean-field Magnetohydrodynamics and Dynamo Theory” by Krause and Rädler of 1980. After the Fall of the Iron Curtain, owing to Rädler's scientific competence and personal integrity, as well as his democratic way of thinking, he was elected as the spokesperson of the scientific council of the Institute. Professor Rädler was chosen to become the founding director of what became the new Astrophysical Institute of Potsdam on the premises of the old observatory at Babelsberg. During that time, the Institute started collaboration with the Large Binocular Telescope in Arizona, as well as the telescope systems GREGOR and STELLA at Tenerife. It was also during this time when K.H. Rädler was active as the Editor-in-Chief of the Astronomische Nachrichten/Astronomical News. Another scientific highlight was his involvement in the theoretical underpinning of the Karlsruhe dynamo experiment in the late 1990s. Using the mean-field approach, he and his team in Potsdam prepared detailed predictions for the excitation conditions and saturation behavior for the dynamo experiment. Karl-Heinz Rädler remained an acting master of the field until his last years. Particularly noteworthy is his work of 2005, when he devised a numerical procedure for computing the full set of turbulent transport coefficients. With that, mean-field dynamo theory became an accurate and predictive tool beyond the realm of quasilinear theory. In celebration of Rädler's 75th birthday, friends and colleagues presented their latest research in Stockholm on the “α effect and beyond.” Some of the contributions in the special issue of Geophysical and Astrophysical Fluid Dynamics provide a lasting memory of this. His lifelong accomplishments have been honored by the Astronomische Gesellschaft through its award of the Karl Schwarzschild medal of 2013. His very last big contribution to the community was as one of the guest editors of the Journal of Plasma Physics of 2018 with the title “50 years of Mean Field Electrodynamics,” which fittingly characterizes a subject of which he was part of since the very first hour.

Turbulent viscosity and magnetic Prandtl number from simulations of isotropically forced turbulence

Context.Turbulent diffusion of large-scale flows and magnetic fields plays a major role in many astrophysical systems, such as stellar convection zones and accretion discs.Aims.Our goal is to compute turbulent viscosity and magnetic diffusivity which are relevant for diffusing large-scale flows and magnetic fields, respectively. We also aim to compute their ratio, which is the turbulent magnetic Prandtl number, Pmt, for isotropically forced homogeneous turbulence.Methods.We used simulations of forced turbulence in fully periodic cubes composed of isothermal gas with an imposed large-scale sinusoidal shear flow. Turbulent viscosity was computed either from the resulting Reynolds stress or from the decay rate of the large-scale flow. Turbulent magnetic diffusivity was computed using the test-field method for a microphysical magnetic Prandtl number of unity. The scale dependence of the coefficients was studied by varying the wavenumber of the imposed sinusoidal shear and test fields.Results.We find that turbulent viscosity and magnetic diffusivity are in general of the same order of magnitude. Furthermore, the turbulent viscosity depends on the fluid Reynolds number (Re) and scale separation ratio of turbulence. The scale dependence of the turbulent viscosity is found to be well approximated by a Lorentzian. These results are similar to those obtained earlier for the turbulent magnetic diffusivity. The results for the turbulent transport coefficients appear to converge at sufficiently high values of Re and the scale separation ratio. However, a weak trend is found even at the largest values of Re, suggesting that the turbulence is not in the fully developed regime. The turbulent magnetic Prandtl number converges to a value that is slightly below unity for large Re. For small Re we find values between 0.5 and 0.6 but the data are insufficient to draw conclusions regarding asymptotics. We demonstrate that our results are independent of the correlation time of the forcing function.Conclusions.The turbulent magnetic diffusivity is, in general, consistently higher than the turbulent viscosity, which is in qualitative agreement with analytic theories. However, the actual value of Pmtfound from the simulations (≈0.9−0.95) at large Re and large scale separation ratio is higher than any of the analytic predictions (0.4−0.8).

Turbulent transport coefficients in galactic dynamo simulations using singular value decomposition

ABSTRACT Coherent magnetic fields in disc galaxies are thought to be generated by a large-scale (or mean-field) dynamo operating in their interstellar medium. A key driver of mean magnetic field growth is the turbulent electromotive force (EMF), which represents the influence of correlated small-scale (or fluctuating) velocity and magnetic fields on the mean field. The EMF is usually expressed as a linear expansion in the mean magnetic field and its derivatives, with the dynamo tensors as expansion coefficients. Here, we adopt the singular value decomposition (SVD) method to directly measure these turbulent transport coefficients in a simulation of the turbulent interstellar medium that realizes a large-scale dynamo. Specifically, the SVD is used to least-square fit the time series data of the EMF with that of the mean field and its derivatives, to determine these coefficients. We demonstrate that the spatial profiles of the EMF reconstructed from the SVD coefficients match well with that taken directly from the simulation. Also, as a direct test, we use the coefficients to simulate a 1D mean-field dynamo model and find an overall similarity in the evolution of the mean magnetic field between the dynamo model and the direct simulation. We also compare the results with those which arise using simple regression and the ones obtained previously using the test-field method, to find reasonable qualitative agreement. Overall, the SVD method provides an effective post-processing tool to determine turbulent transport coefficients from simulations.

Stellar Dynamos in the Transition Regime: Multiple Dynamo Modes and Antisolar Differential Rotation

Abstract Global and semi-global convective dynamo simulations of solar-like stars are known to show a transition from an antisolar (fast poles, slow equator) to solar-like (fast equator, slow poles) differential rotation (DR) for increasing rotation rate. The dynamo solutions in the latter regime can exhibit regular cyclic modes, whereas in the former one, only stationary or temporally irregular solutions have been obtained so far. In this paper we present a semi-global dynamo simulation in the transition region, exhibiting two coexisting dynamo modes, a cyclic and a stationary one, both being dynamically significant. We seek to understand how such a dynamo is driven by analyzing the large-scale flow properties (DR and meridional circulation) together with the turbulent transport coefficients obtained with the test-field method. Neither an αΩ dynamo wave nor an advection-dominated dynamo are able to explain the cycle period and the propagation direction of the mean magnetic field. Furthermore, we find that the α effect is comparable or even larger than the Ω effect in generating the toroidal magnetic field, and therefore, the dynamo seems to be of α 2Ω or α 2 type. We further find that the effective large-scale flows are significantly altered by turbulent pumping.

EDGE2D-EIRENE simulations of the influence of isotope effects and anomalous transport coefficients on near scrape-off layer radial electric field

EDGE2D-EIRENE (the ‘code’) simulations show that radial electric field, Er, in the near scrape-off layer (SOL) of tokamaks can have large variations leading to a strong local E × B shear greatly exceeding that in the core region. This was pointed out in simulations of JET plasmas with varying divertor geometry, where the magnetic configuration with larger predicted near SOL Er was found to have lower H-mode power threshold, suggesting that turbulence suppression in the SOL by local E × B shear can be a player in the L–H transition physics (Delabie et al 2015 42nd EPS Conf. on Plasma Physics (Lisbon, Portugal, 22–26 June 2015) paper O3.113 (http://ocs.ciemat.es/EPS2015PAP/pdf/O3.113.pdf), Chankin et al 2017 Nucl. Mater. Energy 12 273). Further code modeling of JET plasmas by changing hydrogen isotopes (H–D–T) showed that the magnitude of the near SOL Er is lower in H cases in which the H-mode threshold power is higher (Chankin et al 2017 Plasma Phys. Control. Fusion 59 045012). From the experiment it is also known that hydrogen plasmas have poorer particle and energy confinement than deuterium plasmas, consistent with the code simulation results showing larger particle diffusion coefficients at the plasma edge, including SOL, in hydrogen plasmas (Maggi et al 2018 Plasma Phys. Control. Fusion 60 014045). All these experimental observations and code results support the hypothesis that the near SOL E × B shear can have an impact on the plasma confinement. The present work analyzes neutral ionization patterns of JET plasmas with different hydrogen isotopes in L-mode cases with fixed input power and gas puffing rate, and its impact on target electron temperature, Te, and SOL Er. The possibility of a self-feeding mechanism for the increase in the SOL Er via the interplay between poloidal E × B drift and target Te is discussed. It is also shown that reducing anomalous turbulent transport coefficients, particle diffusion and electron and ion heat conductivities, leads to higher peak target Te and larger Er, suggesting the possibility of a positive feedback loop, under an implicitly made assumption that the E × B shear in the SOL is capable of suppressing turbulence.

Magnetic and rotational quenching of the Λ effect

Context. Differential rotation in stars is driven by the turbulent transport of angular momentum.Aims. Our aim is to measure and parameterize the non-diffusive contribution to the total (Reynolds plus Maxwell) turbulent stress, known as the Λ effect, and its quenching as a function of rotation and magnetic field.Methods. Simulations of homogeneous, anisotropically forced turbulence in fully periodic cubes are used to extract their associated turbulent Reynolds and Maxwell stresses. The forcing is set up such that the vertical velocity component dominates over the horizontal ones, as in turbulent stellar convection. This choice of the forcing defines the vertical direction. Additional preferred directions are introduced by the imposed rotation and magnetic field vectors. The angle between the rotation vector and the vertical direction is varied such that the latitude range from the north pole to the equator is covered. Magnetic fields are introduced by imposing a uniform large-scale field on the system. Turbulent transport coefficients pertaining to the Λ effect are obtained by fitting. The results are compared with analytic studies.Results. The numerical and analytic results agree qualitatively at slow rotation and low Reynolds numbers. This means that vertical (horizontal) transport is downward (equatorward). At rapid rotation the latitude dependence of the stress is more complex than predicted by theory. The existence of a significant meridional Λ effect is confirmed. Large-scale vorticity generation is found at rapid rotation when the Reynolds number exceeds a threshold value. The Λ effect is severely quenched by large-scale magnetic fields due to the tendency of the Reynolds and Maxwell stresses to cancel each other. Rotational (magnetic) quenching of Λ occurs at more rapid rotation (at lower field strength) in the simulations than in the analytic studies.Conclusions. The current results largely confirm the earlier theoretical results, and also offer new insights: the non-negligible meridional Λ effect possibly plays a role in the maintenance of meridional circulation in stars, and the appearance of large-scale vortices raises the question of their effect on the angular momentum transport in rapidly rotating stellar convective envelopes. The results regarding magnetic quenching are consistent with the strong decrease in differential rotation in recent semi-global simulations and highlight the importance of including magnetic effects in differential rotation models.

Calculating turbulent transport tensors by averaging single-plume dynamics and application to dynamos

ABSTRACT Transport coefficients in turbulence are comprised of correlation functions between turbulent fluctuations and efficient methods to calculate them are desirable. For example, in mean-field dynamo theories used to model the growth of large-scale magnetic fields of stars and galaxies, the turbulent electromotive force is commonly approximated by a series of tensor products of turbulent transport coefficients with successively higher order spatial derivatives of the mean magnetic field. One ingredient of standard models is the kinematic coefficient of the zeroth-order term, namely the averaged kinetic pseudo-tensor $\boldsymbol \alpha$, that converts toroidal to poloidal fields. Here we demonstrate an efficient way to calculate this quantity for rotating stratified turbulence, whereby the pre-averaged quantity is calculated for the motion of a single plume, and the average is then taken over an ensemble of plumes of different orientations. We calculate the plume dynamics in the most convenient frame, before transforming back to the lab frame and averaging. Our concise configuration space calculation gives essentially identical results to previous lengthier approaches. The present application exemplifies what is a broadly applicable method.

Fast H isotope and impurity mixing in ion-temperature-gradient turbulence

In ion-temperature-gradient (ITG) driven turbulence, the resonance condition leads to ion particle turbulent transport coefficients significantly larger than electron particle turbulent transport coefficients. This is shown in nonlinear gyrokinetic simulations and explained by an analytical quasilinear model. It is then illustrated by JETTO-QuaLiKiz integrated modelling. Large ion particle transport coefficients implies that the ion density profiles are uncorrelated to the corresponding ion source, allowing peaked isotope density profiles even in the absence of core source. This also implies no strong core accumulation of He ash. Furthermore, the relaxation time of the individual ion profiles in a multi-species plasma can be significantly faster than the total density profile relaxation time which is constrained by the electrons. This leads to fast isotope mixing and fast impurity transport in ITG regimes. In trapped-electron-mode (TEM) turbulence, in presence of electron heating about twice the ion heating, the situation is the inverse: ion particle turbulent transport coefficients are smaller than their electron counterpart.