P. Egorov - G. Rüdiger - U. Ziegler
Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany
Received 26 March 2004 / Accepted 9 June 2004
Box simulations of rotating convection are presented with the aim to study the divergence-vorticity correlation computed from the horizontal velocity components of the turbulent flow field. Our investigation is motivated by new observations of the solar velocity field in the upper convection zone using the techniques of so-called time-distance helioseismology (Duvall et al. 1993). From such observations can be derived directly and because of the close relation to the helicity it is an important quantity for solar dynamo theory. The divergence-vorticity correlation obtained from the simulations with Taylor number 103 is found to be in agreement with the observations of supergranulation. In particular, vanishes at the equator and is negative (positive) in the northern (southern) hemisphere. For larger Taylor numbers the amplitude of increases, which is consistent with the increase in strength of the Coriolis force acting on the turbulent flow. We also demonstrate that the correlation and the kinetic helicity of the flow field have very similar spatial profiles. The observed values of , therefore, yield a clear indication of the existence of negative (positive) kinetic helicity in the surface layers of the northern (southern) hemisphere.
Key words: turbulence - Sun: granulation - Sun: helioseismology
It is therefore natural to look for observational confirmations of this concept. A big step in this direction was the statistics of the current helicity
The kinetic helicity
In particular, the vertical component of the vorticity,
The candidates to study the rotational influence on the turbulence at the solar surface are mesogranules and supergranules due to their nonmagnetic character. The average cell sizes of supergranules are in the range 15-30 Mm with predominantly horizontal velocities. Recent observations (e.g., Hathaway et al. 2002) have shown that the typical rms horizontal flow speed is 360 m/s while the rms vertical flow speed is 30 m/s.
The rotational effect is estimated by the Coriolis number
We did not find observations for the magnitude of , but only maximum values of the divergence and vorticity were given (Brandt et al. 1988; Simon et al. 1988; Simon et al. 1994). Wang et al. (1995) report indications of a negative divergence-vorticity correlation for the northern hemisphere.
A clear detection of the influence of the Coriolis force is due to Duvall & Gizon (2000) and Gizon & Duvall (2003) using the method of time-distance helioseismology (Duvall et al. 1993). This technique provides both maps of the horizontal divergence of the flows and information about two individual horizontal components of the velocity. We shall use their results for comparison with our numerical simulations. The magnitude of the vorticity was found for the latitudinal interval of with zero at the equator and a value s-1 at the interval boundaries (see Fig. 7b in Gizon & Duvall 2003). A vorticity of 10-6 s-1 corresponds to an angular velocity of or a typical circular velocity of 10 m/s. Gizon & Duvall (2003) detected a "significant correlation of a few percent between the vertical vorticity and the horizontal divergence'' (their Fig. 7a). Positive (negative) divergence in the northern hemisphere is correlated with clockwise (counterclockwise) vorticity i.e. . changes its sign in the southern hemisphere and vanishes at the equator. Both the sign and the latitudinal variation of characterize the effect of the Coriolis force on the flow. The magnitude of the correlation is obtained for the range of latitudes where it varies from s-2to s-2.
Similar kinds of simulation, both the "local-box'' (e.g., Hathaway 1984; Brummell et al. 1996, 1998) and the full spherical shell (e.g., Miesch et al. 2000) have been used to study the effect of rotation on laminar/turbulent, incompressible/compressible convection. A correlation of counter-clockwise (cyclonic) flows with downdrafts and clockwise (anticyclonic) with updrafts as well as a switch in the direction of circulations in the bottom part of convection zone has been found for a wide range of Taylor number varying from 102 to 107. The second result in its turn yields a sign reverse of the kinetic helicity along the radial direction; this has also been confirmed by simulations of Brummell et al. (1998) and Miesch et al. (2000).
In this paper the influence of the Coriolis force on the turbulence is simulated for the solar surface flow pattern of supergranulation. We shall show that the results of the simulations are close to both the quasilinear theory and the observations. In the last section the relation between the DIV-CURL correlation and the helicity is discussed with the result that the observations indicate a negative (positive) kinetic helicity in the northern (southern) hemisphere.
The box rotates around the polar axis from west to east. The computational domain is discretized by grid points which are uniformly distributed in each coordinate direction.
The governing equations describing thermal convection in a
stratified medium involving the effect of
the Coriolis force are
The set (9)-(11) is closed through the
ideal gas equation of state
The initial distribution of the physical quantities corresponds to a 3-layer polytropic stratification. All quantities are periodic in the horizontal direction. At the bottom (z=-2) and the top (z=0) of the box impermeable conditions are imposed for the vertical velocity while the horizontal velocities satisfy a stress-free boundary condition. The temperature and density are fixed at the top of the domain, , and a constant heat flux is injected at the bottom.
The dimensionless parameters and are used to control the simulations. In our calculations and are fixed whereas characterize the magnitude of the angular velocity vector.
Equations (9)-(11) are solved with the finite-difference, fractional-step code NIRVANA (Ziegler 1998, 1999).
|Figure 1: The intensity ratio in the box simulations for different colatitudes. The colatitudes are (poles, solid), (dashed) and (dot-dashed), .|
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|Figure 2: The correlation in the box simulations for various latitudes after both horizontal and temporal averaging vs. depth for .|
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Figure 3 shows horizontal and time averages of
scaled by Eq. (15) vs. latitude plotted in comparison with observational results
indicated by the shaded area. In Fig. 3
), asterisks (
denote the computed values of at a depth z=-0.3 (see Fig. 2) and for different latitudes
|Figure 3: Comparison of computed with observational data for (triple-dot-dashed), (solid) and (dashed); the numerical values are taken near the top of the convection zone (z=-0.3). The shaded area covers the range of the observed data.|
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For slow rotation ( ) the correlation is very close to the observations. For higher Taylor numbers the amplitude of is about one order of magnitude greater than the observed value for supergranules. The properties of the turbulence are shown in Table 1 for different rotation rates and box inclinations. The rms value of normalized by the sound speed is given as characteristic for the flow field near the top of the convection zone. With increasing rotation rate the rms velocities change from almost uniform i.e. independent of in the case to a nonuniform distribution for .
The coincidence between the simulations and the observations illustrated in Fig. 3 is expected when comparing the Coriolis numbers in the simulations with those prevailing in the supergranulation. The slower the box rotates the smaller the influence of the Coriolis force on the velocity field and, as a consequence, would become smaller. Although the Sun has , the supergranules rotate more slowly than suggested by their Taylor number. Indeed, for we have a similar to the value estimated for the supergranulation flow field. This explains the good agreement of the computed with the observations. The obtained Coriolis numbers for the higher Taylor number simulations are more related to giant cells (also the typical cell size corresponds to giant cells), hence, the effect of rotation is at least one order of magnitude greater than the observed one. Extrapolation of our results to the mesogranulation pattern leads to the value of 10-11 s-2 for the maximum(poles) correlation.
|Figure 4: Horizontally- and time-averaged kinetic helicity vs. depth for different latitudes in the box simulations for . Note the close relation of this quantity to the simulated correlation given in Fig. 2. The shape of the helicity curve is very close to results of Miesch et al. (2000, their Fig. 22).|
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Table 1: Properties of turbulence for different Taylor numbers and colatitudes at z=-0.3.