Reynolds stress and heat flux in spherical shell convection

Context. Turbulent fluxes of angular momentum and heat due to rotationally affected convection play a key role in determining differential rotation of stars. Aims. We compute turbulent angular momentum and heat transport as functions of the rotation rate from stratified convection. We compare results from spherical and Cartesian models in the same parameter regime in order to study whether restricted geometry introduces artefacts into the results. Methods. We employ direct numerical simulations of turbulent convection in spherical and Cartesian geometries. In order to alleviate the computational cost in the spherical runs and to reach as high spatial resolution as possible, we model only parts of the latitude and longitude. The rotational influence, measured by the Coriolis number or inverse Rossby number, is varied from zero to roughly seven, which is the regime that is likely to be realised in the solar convection zone. Cartesian simulations are performed in overlapping parameter regimes. Results. For slow rotation we find that the radial and latitudinal turbulent angular momentum fluxes are directed inward and equatorward, respectively. In the rapid rotation regime the radial flux changes sign in accordance with earlier numerical results, but in contradiction with theory. The latitudinal flux remains mostly equatorward and develops a maximum close to the equator. In Cartesian simulations this peak can be explained by the strong 'banana cells'. Their effect in the spherical case does not appear to be as large. The latitudinal heat flux is mostly equatorward for slow rotation but changes sign for rapid rotation. Longitudinal heat flux is always in the retrograde direction. The rotation profiles vary from anti-solar (slow equator) for slow and intermediate rotation to solar-like (fast equator) for rapid rotation. The solar-like profiles are dominated by the Taylor-Proudman balance.

Local numerical modelling of magnetoconvection and turbulence - implications for mean-field theories

During the last decades mean-field models, in which large-scale magnetic fields and differential rotation arise due to the interaction of rotation and small-scale turbulence, have been enormously successful in reproducing many of the observed features of the Sun. In the meantime, new observational techniques, most prominently helioseismology, have yielded invaluable information about the interior of the Sun. This new information, however, imposes strict conditions on mean-field models. Moreover, most of the present mean-field models depend on knowledge of the small-scale turbulent effects that give rise to the large-scale phenomena. In many mean-field models these effects are prescribed in ad hoc fashion due to the lack of this knowledge. With large enough computers it would be possible to solve the MHD equations numerically under stellar conditions. However, the problem is too large by several orders of magnitude for the present day and any foreseeable computers. In our view, a combination of mean-field modelling and local 3D calculations is a more fruitful approach. The large-scale structures are well described by global mean-field models, provided that the small-scale turbulent effects are adequately parameterized. The latter can be achieved by performing local calculations which allow a much higher spatial resolution than what can be achieved in direct global calculations. In the present dissertation three aspects of mean-field theories and models of stars are studied. Firstly, the basic assumptions of different mean-field theories are tested with calculations of isotropic turbulence and hydrodynamic, as well as magnetohydrodynamic, convection. Secondly, even if the mean-field theory is unable to give the required transport coefficients from first principles, it is in some cases possible to compute these coefficients from 3D numerical models in a parameter range that can be considered to describe the main physical effects in an adequately realistic manner. In the present study, the Reynolds stresses and turbulent heat transport, responsible for the generation of differential rotation, were determined along the mixing length relations describing convection in stellar structure models. Furthermore, the alpha-effect and magnetic pumping due to turbulent convection in the rapid rotation regime were studied. The third area of the present study is to apply the local results in mean-field models, which task we start to undertake by applying the results concerning the alpha-effect and turbulent pumping in mean-field models describing the solar dynamo.

Open and closed boundaries in large-scale convective dynamos

Earlier work has suggested that large-scale dynamos can reach and maintain equipartition field strengths on a dynamical time scale only if magnetic helicity of the fluctuating field can be shed from the domain through open boundaries. To test this scenario in convection-driven dynamos by comparing results for open and closed boundary conditions. Three-dimensional numerical simulations of turbulent compressible convection with shear and rotation are used to study the effects of boundary conditions on the excitation and saturation level of large-scale dynamos. Open (vertical field) and closed (perfect conductor) boundary conditions are used for the magnetic field. The contours of shear are vertical, crossing the outer surface, and are thus ideally suited for driving a shear-induced magnetic helicity flux. We find that for given shear and rotation rate, the growth rate of the magnetic field is larger if open boundary conditions are used. The growth rate first increases for small magnetic Reynolds number, Rm, but then levels off at an approximately constant value for intermediate values of Rm. For large enough Rm, a small-scale dynamo is excited and the growth rate in this regime increases proportional to Rm^(1/2). In the nonlinear regime, the saturation level of the energy of the mean magnetic field is independent of Rm when open boundaries are used. In the case of perfect conductor boundaries, the saturation level first increases as a function of Rm, but then decreases proportional to Rm^(-1) for Rm > 30, indicative of catastrophic quenching. These results suggest that the shear-induced magnetic helicity flux is efficient in alleviating catastrophic quenching when open boundaries are used. The horizontally averaged mean field is still weakly decreasing as a function of Rm even for open boundaries.

Key Factors Influencing Economic Relationships and Communication in Finnish Food Chains

The aim of this report is to discuss the role of the relationship type and communication in two Finnish food chains, namely the pig meat-to-sausage (pig meat chain) and the cereal-to-rye bread (rye chain) chains. Furthermore, the objective is to examine those factors influencing the choice of a relationship type and the sustainability of a business relationship. Altogether 1808 questionnaires were sent to producers, processors and retailers operating in these two chains of which 224 usable questionnaires were returned (the response rate being 12.4%). The great majority of the respondents (98.7%) were small businesses employing less than 50 people. Almost 70 per cent of the respondents were farmers. In both chains, formal contracts were stated to be the most important relationship type used with business partners. Although for many businesses written contracts are a common business practice, the essential role of the contracts was the security they provide regarding the demand/supply and quality issues. Relative to the choice of the relationship types, the main difference between the two chains emerged especially with the prevalence of spot markets and financial participation arrangements. The usage of spot markets was significantly more common in the rye chain when compared to the pig meat chain, while, on the other hand, financial participation arrangements were much more common among the businesses in the pig meat chain than in the rye chain. Furthermore, the analysis showed that most of the businesses in the pig meat chain claimed not to be free to choose the relationship type they use. Especially membership in a co-operative and practices of a business partner were mentioned as the reasons limiting this freedom of choice. The main business relations in both chains were described as having a long-term orientation and being based on formal written contracts. Typical for the main business relationships was also that they are not based on the existence of the key persons only; the relationship would remain even if the key people left the business. The quality of these relationships was satisfactory in both chains and across all the stakeholder groups, though the downstream processors and the retailers had a slightly more positive view on their main business partners than the farmers and the upstream processors. The businesses operating in the pig meat chain seemed also to be more dependent on their main business relations when compared to the businesses in the rye chain. Although the communication means were rather similar in both chains (the phone being the most important), there was some variation between the chains concerning the communication frequency necessary to maintain the relationship with the main business partner. In short, the businesses in the pig meat chain seemed to appreciate more frequent communication with their main business partners when compared to the businesses in the rye chain. Personal meetings with the main business partners were quite rare in both chains. All the respondent groups were, however, fairly satisfied with the communication frequency and information quality between them and the main business partner. The business cultures could be argued to be rather hegemonic among the businesses both in the pig meat and rye chains. Avoidance of uncertainty, appreciation of long-term orientation and independence were considered important factors in the business cultures. Furthermore, trust, commitment and satisfaction in business partners were thought to be essential elements of business operations in all the respondent groups. In order to investigate which factors have an effect on the choice of a relationship type, several hypotheses were tested by using binary and multinomial logit analyses. According to these analyses it could be argued that avoidance of uncertainty and risk has a certain effect on the relationship type chosen, i.e. the willingness to avoid uncertainty increases the probability to choose stable relationships, like repeated market transactions and formal written contracts, but not necessary those, which require high financial commitment (like financial participation arrangements). The probability of engaging in financial participation arrangements seemed to increase with long-term orientation. The hypotheses concerning the sustainability of the economic relations were tested by using structural equation model (SEM). In the model, five variables were found to have a positive and statistically significant impact on the sustainable economic relationship construct. Ordered relative to their importance, those factors are: (i) communication quality, (ii) personal bonds, (iii) equal power distribution, (iv) local embeddedness and (v) competition.

On global solar dynamo simulations

Global dynamo simulations solving the equations of magnetohydrodynamics (MHD) have been a tool of astrophysicists who try to understand the magnetism of the Sun for several decades now. During recent years many fundamental issues in dynamo theory have been studied in detail by means of local numerical simulations that simplify the problem and allow the study of physical effects in isolation. Global simulations, however, continue to suffer from the age-old problem of too low spatial resolution, leading to much lower Reynolds numbers and scale separation than in the Sun. Reproducing the internal rotation of the Sun, which plays a crucual role in the dynamo process, has also turned out to be a very difficult problem. In the present paper the current status of global dynamo simulations of the Sun is reviewed. Emphasis is put on efforts to understand how the large-scale magnetic fields, i.e. whose length scale is greater than the scale of turbulence, are generated in the Sun. Some lessons from mean-field theory and local simulations are reviewed and their possible implications to the global models are discussed. Possible remedies to some of the current issues of the solar simulations are put forward.

Dependence of the large-scale vortex instability on latitude, stratification and domain size

In an earlier study, we reported on the excitation of large-scale vortices in Cartesian hydrodynamical convection models subject to rapid enough rotation. In that study, the conditions for the onset of the instability were investigated in terms of the Reynolds (Re) and Coriolis (Co) numbers in models located at the stellar North pole. In this study, we extend our investigation to varying domain sizes, increasing stratification, and place the box at different latitudes. The effect of the increasing box size is to increase the sizes of the generated structures, so that the principal vortex always fills roughly half of the computational domain. The instability becomes stronger in the sense that the temperature anomaly and change in the radial velocity are observed to be enhanced. The model with the smallest box size is found to be stable against the instability, suggesting that a sufficient scale separation between the convective eddies and the scale of the domain is required for the instability to work. The instability can be seen upto the colatitude of 30 degrees, above which value the flow becomes dominated by other types of mean flows. The instability can also be seen in a model with larger stratification. Unlike the weakly stratified cases, the temperature anomaly caused by the vortex structures is seen to depend on depth.