next previous
Up: Solar activity cycle and


Subsections

4 Solar activity cycle and rotation rate

We studied solar cycles 19, 20, 21 and 22 separately, focusing on the ten most active years (Fig. 4). For each cycle, a frame is given which is composed of 17 partially overlapping bins of power spectra of 480 d, disposed in ordinate according to the dates of the observations. The x-axis shows the synodic rotation period. The contours represent the prominent maxima of the power spectra. For reference, we superimposed the radio flux variation in arbitrary units on each frame. "M" and "m" indicate respectively the Annual Mean Sunspot Numbers' (AMSN) maxima and minima (SGD 1997). The mark "CR" on the x-axis gives the position of the Carrington Rotation period.

4.1 Activity and cycle

Some points relating to the Fig. 4 can be emphasized:

$\bullet$
Solar rotation varies according to the solar cycle. For instance the most active cycle, number 19, clearly shows that the synodic rotation periods $(P^{\oplus})$ are shorter at maximum activity than before and after:


  \begin{figure}
\par\includegraphics[width=16.5cm,clip]{ms2771f5.eps}\end{figure} Figure 5: Superimposition of power spectra peaks of frequencies of cycles 19 to 22. Ordinate divisions are half-years. To distnguish the different cycles. (This figure is available in color in electronic form).

The time variation of the set of shortest rotation periods of each ordinates (bin) traces a C-shaped inferior limit of synodic rotation period, indicated by dotted lines. Frequencies with longer periods appear on the right hand side of this limit,untill periods of 33 d. The other cycles (20, 21 and 22) show the same general behavior. The right hand side end of the C-shaped limit (the highest rotation period) will be considered as the "minimum" of activity of the cycle. We consider periods higher than 31 d as "quiet sun".

$\bullet$
Table 1 displays the rotation characteristics resulting from the examination of the four cycles. The table gives the synodic rotation period $(P^{\oplus})$ in days and the sidereal rotation rate $ (\Omega ^{\star }) $ in degrees per day. For each cycle figures concerning rotation maxima are provided in Cols. 3 and 4, those for "minima" in Cols. 5 and 6, and those for quiet sun in Cols. 7 and 8.

$\bullet$
No correlation can be established between the Monthly Mean Sunspot Number (MMSN) maximum of the cycle and its minimum of rotation period. However the four cycles have a wide variety of activity levels (Table 2, Col. 2). The maximum of cycle 20 is only 54% of the maximum of cycle 19, whereas the rotation rate variation is only 0.2%. We also note that coronal rotation period maximum and sunspot number maximum do not necessarily coincide in time (Tables 1 and 2). Column 3 of Table 2 gives the time difference in months between these maxima, if any.

$\bullet$
During the period of maximum of activity, there are broader frequency ranges, whereas in "minimum" this range becomes narrower.

$\bullet$
For long periods around activity minima, no outstanding frequencies are sufficiently revealed, because at minimum the active regions are generally few, small and short-lived. Under these conditions, the absence of information does not mean lack of preferential longitudes; moreover Fig. 3 shows a outstanding frequency at $P^{\oplus} = 31\rm d $.
$\bullet$
The quiet sun is always present during the active period (frequencies 1/31 to 1/33) and does not show significant variation.

$\bullet$
Time variations of rotation periods (Table 3) are steeper in the increasing phase (average $\rm0.8^{\circ}/d/y$) than in the decreasing phase $\rm (- 0.5^{\circ}/d/y)$ of the activity cycle.

$\bullet$
The accuracy of the choice of 480 d is subsequently justified by the extension of the y-axis of frequency peaks. It is evident that lower time resolution analysis hides important results.

$\bullet$
Not all the frames of Fig. 4 show frequency concentration around the Carringtion Rate. Keeping only the most prominent frequencies in the four frames of Fig. 4, we superimpose them and build Fig. 5. This shows frequencies around the Carrington Rotation period, in agreement with Fig. 3. We see around the activity maximum period (approximately 4 to 5 years) a richness of frequencies with values higher than the Carrington Rate.

$\bullet$
Figure 5 shows a fish-bone structure, composed of a set of three or four C-shaped layers of frequencies, making rotation rate variation obvious throughout the cycle. All cycles participate in each C-shaped structure, showing that this is a general characteristic of magnetic flux related to rotation rate. All fish-bone components show the same curvature, i.e. the same variation during the cycle. In Fig. 4, we again find this fish-bone structure in each frame. Do these fish-bones correspond to the rotation rate of different components of the solar atmosphere, or are they due to magnetic tubes rooted at different depths in the convective zone?


 

 
Table 2: Sunspot activity maxima.
date AMSN $\Delta T/ {\rm month}$
1 2 3
1957 Oct. 190.2 -
1969 Mar. 105.9 -
1979 Sep. 155.4 -9
1989 Jun. 157.2 -12



 

 
Table 3: Acceleration of rotation period in $\rm ^{\circ }/d/y$.
cycle increasing phase decreasing phase
19 0.9 -0.8
20 1.1 -0.6
21 0.8 -0.3
22 0.6 -0.3


4.2 Rotation and magnetic flux

For frequencies in Fig. 4 we compute an activity index R15, a 15 month mean of MMSN. The 15 month period is the closest odd number to 480 d. Note that R15 is very close to AMSN. In Fig. 6 we plot the sidereal rotation rate $ (\Omega ^{\star }) $ and the activity index (R15). Solid signs mark the frequencies of the C-shaped limits of the cycles, frequencies jointed by dotted lines in Fig. 4. The set of solid points (cycles 19, 21 and 22) can be approximated by an empirical equation:

\begin{displaymath}R_{15}= 32 \times (\Omega^{\star} - 13)^{2}+25 \makebox [1cm]
{for}\Omega^{\star} \rm\geq 13^{\circ}/d.
\end{displaymath} (1)

The open signs show the frequency of the right hand side of the C-shape limits in Fig. 4. In the plane $\Omega^{\star} - R_{15}$ the frequency distribution also shows a lower limit of rotation rate, following the relation:

\begin{displaymath}R_{15} = 75 \times (\Omega^{\star} - 13)^{4}+25 \makebox [1cm]
{for}\Omega^{\star} \rm\leq 13^{\circ}/d.
\end{displaymath} (2)

Curves (1) and (2) are shown in Fig. 6.

As far as cycle 20 (the weakest of our sample) is concerned, the two limiting curves (1) and (2) are valid with a translation of $\rm +0.5^\circ/d.$ This shift indicates the necessity of a faster rotation to ensure magnetic field emergence. Using Eqs. (1) and (2) we can find a dependence of sunspot number on the rotation rate:

\begin{displaymath}200 \geq R_{15} \geq 32 \times (\Omega^{\star} - \Omega^{\sta...
...25
\makebox [1cm] {for}\Omega^{\star} \geq \Omega^{\star}_{0}
\end{displaymath} (3)


\begin{displaymath}200 \geq R_{15} \geq 75 \times (\Omega^{\star} - \Omega^{\sta...
...25
\makebox [1cm] {for}\Omega^{\star} \leq \Omega^{\star}_{0}
\end{displaymath} (4)


  \begin{figure}
\includegraphics[width=8.8cm,clip]{ms2771f6.eps}\end{figure} Figure 6: Relationship between the sidereal rotation rate $ (\Omega ^{\star }) $ and solar activity (R15). Cycle 19 - circles, cycle 20 - squares, cycle 21 - triangles, cycle 22 - ellipse. Solid points are limit of rotation period, linked by dotted lines in Fig. 4. The curves draw the upper and lower rotation rates limits at a given activity index, accordingly to Eqs. (1) and (2).

here $\Omega^{\star}_{0}= \rm 13^{\circ}/d$ for cycles 19, 21 and 22, and $\Omega^{\star}_{0}=\rm 13.5^{\circ}/d$ for cycle 20.

In order to express the activity index (R15) in units of total magnetic flux $(\Phi_{\mid B \mid})$, we determine the following empirical relation, based on cycle 21:

\begin{displaymath}\Phi_{\mid B \mid}=4\times 10^{21} \times (R_{15}+50) \; {\rm Mx}.
\end{displaymath} (5)

Figure 6 indicates that, for a given rotation rate $\Omega^{\star}$, only activity fluxes equal to or greater than that given by Eqs. (1) and (2) can emerge. Thus, the most likely rotation rate for the emergence of the magnetic field is $ \Omega^{\star}_{0} $. At high or low rates, only stronger magnetic flux emerges, the weak flux being scattered before it reaches the top of the convective zone.

4.3 Hysteresis

Following the time evolution of sidereal rotation of the C-shape limits for each cycle (solid points of Fig. 6), we may see that during the starting phase of the activity cycle, the rotation rate first grows rapidly, whereas later the magnetic flux grows rapidly untill maximum. This fact shows the important role of the Coriolis force in magnetic field emergence.

We note that cycles 20, 21 and 22 display hysteresis loops during their time evolution. No hysteresis was detected for cycle 19, the strongest one. Especially for cycles 21 and 22, the hysteresis shows that in the decreasing phase higher rotation is needed $(\Delta\Omega^{\star}=\rm0.3^{\circ}/d)$ for the emergence of the same magnetic flux than during the increasing phase.


next previous
Up: Solar activity cycle and

Copyright ESO 2002