A&A 448, 433-438 (2006)
DOI: 10.1051/0004-6361:20042245
P. J. Käpylä^{1,2} - M. J. Korpi^{3} - M. Ossendrijver^{2} - I. Tuominen^{1,3}
1 - Astronomy Division, Department of Physical Sciences,
PO Box 3000, 90014 University of Oulu, Finland
2 - Kiepenheuer-Institut für Sonnenphysik,
Schöneckstrasse 6, 79104 Freiburg, Germany
3 - Observatory, PO Box 14, 00014 University of Helsinki,
Finland
Received 25 October 2004 / Accepted 9 November 2005
Abstract
Aims. The Strouhal number (St), which is a nondimensional measure of the correlation time, is determined from numerical models of convection. The Strouhal number arises in the mean-field theories of angular momentum transport and magnetic field generation, where its value determines the validity of certain widely used approximations, such as the first order smoothing (hereafter FOSA). More specifically, the relevant transport coefficients can be calculated by means of a cumulative series expansion if
.
Methods. We define the Strouhal number as the ratio of the correlation and turnover times, which we determine separately, the former from the autocorrelation of velocity, and the latter by following test particles embedded in the flow.
Results. We find that the Strouhal numbers are, generally, of the order of 0.1 to 0.4 which is close to the critical value above which deviations from FOSA become significant. Increasing the rotational influence tends to shorten both timescales in such a manner that St decreases. However, we do not find a clear trend as a function of the Rayleigh number for the parameter range explored in the present study.
Key words: hydrodynamics
In mean-field theories these problems are circumvented by relating
the Reynolds stresses and electromotive force to the mean quantities
(the rotation vector
and the mean magnetic field
,
respectively) and their gradients by
means of a cumulative series expansion (see van Kampen
1974a,b). In the best known and
most often used approach, the first order smoothing approximation
(FOSA), only the first terms of these expansions are taken into
account. This approach can be shown to be valid if either the
relevant Reynolds number,
,
or the Strouhal number
It is well-known that the requirement for a small Reynolds number is not satisfied in stellar environments. However, the question of the Strouhal number is not settled. On account of the observations of the solar surface granulation, one can estimate the correlation and turnover times to be roughly equal, indicating that at the solar surface (see e.g. Chap. 6 of Stix 2002). Similar values can also be estimated for supergranulation for which typical numbers are m s^{-1}, m, and s. However, even for granulation the precise value has, to our knowledge, not been established, and nothing is known about St in the deeper layers. Furthermore, recent results from forced turbulence calculations indicate that if the higher order correlations in the equations of the passive scalar flux (Brandenburg et al. 2004, hereafter BKM) and electromotive force (Brandenburg & Subramanian 2005) are taken into account via the so-called minimal -approximation (Blackman & Field 2002, 2003), the Strouhal number can be seen to substantially exceed unity and roughly equal to unity, respectively.
Although forced turbulence is rather different in comparison to convection, the aforementioned studies still raise the question whether the results of the convection calculations can be interpreted within the framework of the standard mean-field theory as has been done in numerous studies during recent years (e.g. Brandenburg et al. 1990; Pulkkinen et al. 1993; Ossendrijver et al. 2001, 2002; Käpylä et al. 2004; Rüdiger et al. 2005). Motivated by the unknown status of the Strouhal number for convection and the previous studies on the subject in different contexts, we set out to calculate St from numerical calculations of convection. In order to do this, we calculate the correlation time from the autocorrelation of velocity and determine the turnover time by following test particles embedded into the flow and define the Strouhal number as the ratio of the two.
The remainder of the paper is organised as follows: Sect. 2 summarises briefly the numerical model used and in Sect. 3 the results of the study are discussed. Finally, Sect. 4 gives the conclusions.
A detailed description of the convection model can be found in Käpylä et al. (2004, hereafter Paper I) and the convection calculations are made with a setup identical to that used in Paper I. See Table 1 for a summary of the main parameters.
For the purposes of the present study we have added the possibility
to follow the trajectories of Lagrangian test particles in the
model. In order to integrate the trajectory we need the
velocity at the position of the test particle at each integration
step. This is done by finding the grid points
(n_{x},n_{y},n_{z}) next
to the test particle and using linear interpolation to obtain the
velocity at the correct position
(2) |
The calculations were made with a modified version of the numerical method described in Caunt & Korpi (2001). The calculations were carried out on the KABUL and BAGDAD Beowulf clusters at the Kiepenheuer-Institut für Sonnenphysik, Freiburg, Germany, and on the IBM eServer Cluster 1600 supercomputer hosted by CSC Scientific Computing Ltd., in Espoo, Finland.
Table 1: Summary of the calculations and the main parameters. From left to right: the Rayleigh, Reynolds, Taylor, and Coriolis numbers, the latitude, and the grid size.
Figure 1: Horizontally averaged correlations of with respect to the snapshot at t = 100 for the run Co0. Each curve is separated by in time units of . | |
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Figure 1 gives an example from run Co0, showing the horizontally averaged correlations of the vertical velocity u_{z} from twelve snapshots, each separated by one time unit, with respect to the snapshot at t_{0} = 100. The correlation diminishes monotonically as a function of time and for the eleventh snapshot the correlation is below 0.5 in the whole convection zone. Using the procedure described above, we find the correlation time to be for this snapshot (average over the correlation times with respect to 206 different snapshots gives a value 9.4, see the third column of Table 2). If the correlation is calculated for one of the horizontal velocity components, is similar in magnitude, except near the surface and in a layer immediately below the convection zone where it is somewhat longer than the one calculated from the vertical velocity. This effect can be understood to arise from the persistent horizontal flows near the boundaries of the convection zone where the up- and downflows diverge to the horizontal directions. In what follows we estimate the correlation time from the vertical velocity, for which remains more or less constant within the convection zone as indicated by Fig. 1. As stated above, our final result is an average over snapshots and the corresponding standard deviations of the correlation times are given in Table 2.
Figure 2: Velocity autocorrelations for u_{z} as functions of r and rotation for the runs indicated in the legend. | |
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Figure 3: Vertical positions of two test particles in the run Co0. The dotted vertical lines denote the times where the particle denoted by the solid line changes direction from downward motion to upward motion. The dash-dotted line shows a particle which is stuck in the lower overshoot layer for the whole duration of the calculation. | |
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A way to improve the estimate given by Eq. (4) is to follow the trajectories of Lagrangian test particles in the flow. This is done by finding the times where a particle changes its direction, i.e. turns over. Thus, one turnover time would be the time between two consecutive turns to the same direction, e.g. from downward motion to upward motion. The advantage of this method is that the assumption of the vertical scale of convection is removed. However, the danger with this method is that the smallest scales begin to dominate due to, for example, contributions from particles stuck in the stable layer. This problem is discussed below.
Table 2: From left to right: rms-velocity averaged over the convectively unstable region and time, correlation time from the autocorrelation of the vertical velocity, turnover time from test particle trajectories, and the Strouhal number. The last column states the number of snapshots with respect to which the correlation times and lengths were calculated.
Figure 3 shows the trajectories of two test particles in the run Co0. The solid line shows a particle which is carried by the convective flow throughout the calculation (the particles were introduced in the flow at t = 60). In the figure we only denote the turnovers from downward to upward motion, but including also the opposite changes of direction a total of 21 turnovers are registered. Figure 4 shows a histogram of the registered turnover times from run Co0. In total, the thousand particles make 21 300 individual turnovers in the course of the calculation. The distribution is centered near 20 having an exponential tail towards longer times. Arithmetic average over the distributions shown in Fig. 4 gives a turnover time , which indicates that the Strouhal number to be significantly less than 0.8 which was obtained with the simple estimate of the turnover time.The test particle method takes into account the variable spatial scale of convection but it also picks up the small scale turnovers in the stable layer. A small number of particles get stuck in the lower overshoot layer for a long time and some even stay there for the complete duration of the calculation (the dash-dotted line in Fig. 3) and contribute a large number of small turnover times. These particles, however, have only a small effect on the estimate of for the nonrotating and slowly rotating cases where the turnover time might be slightly underestimated.
Figure 4: Distribution of turnover times from run the Co0 with the test particle method. | |
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From Fig. 2 and the second column of Table 2 we can determine that whilst decreases as a function of rotation, a similar trend is visible in . Thus we would expect that the ratio not to vary much as a function of Co. This is indeed the trend that we observe for which seems to support the conjecture that .
Figure 5: The correlation time ( left) and the Strouhal number ( right) functions of the Coriolis number. Powerlaws and are plotted. The stars, diamonds, triangles, and squares represents the calculations at latitudes (equator), , , and (south pole), respectively. | |
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For the slowest rotation, we find little differences to the nonrotating case Co0 which was discussed above. Also the variation of the timescales as function of latitude is small in comparison to the variation within individual runs. For the Co1 set the correlation time is markedly shorter (about 6-7 time units as opposed to 9-10 in the run Co0 and the Co01 set). This can be explained by the deflection of the vertical flows by the Coriolis force, leading to smaller vertical scale of convection which tends to shorten the correlation time. Support for this conjecture is given by the decrease of the turnover time from the test particles. A similar decrease in correlation time is noted to occur also if it is estimated from the horizontal velocities.
For the most rapidly rotating case, Co10, the trend of decreasing spatial scales continues. This is manifested in the clearly shorter turnover time as opposed to the more slowly rotating runs discussed above. Here, it is also important to note the misleading value of the turnover time given by the simple method, Eq. (4). Whereas increases with rotation due to the smaller velocities in general, the actual turnover time decreases due to the smaller spatial scale of convection. As for the correlation time, the strong Coriolis forces tend to disrupt the cellular structure of the convection rapidly resulting in a shorter . Considering the Strouhal number, the decrease in overweights the decrease of the turnover time so that the actual value of St decreases as well.
We find that is more or less consistent with a powerlaw as a function of the Coriolis number for moderate and rapid rotation (see Fig. 5).
The Strouhal number arises in the mean-field theories of angular momentum transport and hydromagnetic dynamos where its value determines the validity of certain widely used approximations, such as the first order smoothing. These approximations are based on a cumulative series expansion of the relevant turbulent correlation, e.g. the electromotive force in the dynamo theory. Essentially, the higher order terms in this expansion are proportional to the Strouhal number. Thus the value of St determines whether or not the expansion converges.
The main results can be summarised as follows:
Acknowledgements
P.J.K. acknowledges the financial support from the Finnish graduate school of astronomy and space physics and travel support from the Kiepenheuer intitute. P.J.K. thanks NORDITA and its staff for their hospitability during his visit. M.J.K. acknowledges the hospitality of LAOMP, Toulouse and the Kiepenheuer-Institut, Freiburg during her visits, and the Academy of Finland project 203366. Travel support from the Academy of Finland grant 43039 is acknowledged. The authors thank Axel Brandenburg and Michael Stix for their useful comments on the manuscript and Wolfgang Dobler for illuminating discussions. The anonymous referee is acknowledged for the helpful comments on the manuscript.