Origin of the polar magnetic field reversals

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    E. E. BENEVOLENSKAYA Pulkovo Observatory, St. Petersburg, 196140, Russia

    (Received 31 May, 1995; in final form 8 February, 1996)

    Abstract. The polar magnetic field on the Sun changes its sign during the maximum of solar cycles. It is known that the phenomenon of three-fold reversal of the polar magnetic field occurred in solar cycle 20. Using the magnetograph data of the Mount Wilson Observatory from 1967 to 1993, we confirm a previously suggested topological model of the three-fold magnetic-field reversal (Benevolenskaya, 1991). From the data set we have found that cycles with three-fold polar magnetic field reversals are characterized by a pronounced high-frequency component of the magnetic field compared with cycles with single polar magnetic-field reversals.

    1. Introduction

    Babcock (1959) was the first to observe a polar magnetic-field reversal. During cycle 19, the polarity of the high-latitude dipolar magnetic field of the Sun was opposite to that of the Earth from 1953 to 1957. About the middle of 1957, the polarity of the magnetic field near the south heliographic pole reversed. The reversal of the field near the north pole occurred in November, 1958. The Sun's polar field then became parallel to that of the Earth.

    The following reversals have occurred in cycle 20. According to Waldmeier, cycle 20 has shown an anomaly (Waldmeier, 1973). While the southern polar zone of prominences behaved regularly, two polar zones (instead of one) appeared in the northern hemisphere. The first was regular in every respect and reached the pole shortly after sunspot maximum. The second, irregular and weaker, appeared at mean latitudes only when the regular zone was already disappearing. This second zone drifted towards the pole and disappeared at the beginning of 1971. Compared to the first zone the second had a delay of about 2 yr. Howard (1974) pointed out that the 'anomalous' filament motion found by Waldmeier in the north referred to a true field reversal. Later Makarov and Fatianov (1982) identified a three-fold magnetic field reversal in cycle 20 using Ho~ synoptic charts. Other three-fold reversals of the solar polar magnetic field in one of the hemispheres are found to have happened during the last 120 yr (Makarov and Sivaraman, 1989a, b). These three-fold reversals took place in the southern hemisphere alone in cycles 12 and 14 and in the northern hemisphere alone in solar cycles 16, 18, and 20. The single reversals took place in the odd cycles. It should be noted that there are also 'under- reversals'. They correspond to the situation when zones of alternating polarity appearing at middle and high latitudes do not extend to the poles. This situation occurred in cycle 21 in the southern hemisphere (Figure l(c)).

    Solar Physics 167: 47-55, 1996. (~) 1996 KIuwer Academic Publishers. Printed in Belgium.


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    J I ~ I [ I I I I I I I I J I I I d i ] ] I I

    .--= . 4 0 ~ i l o ,,1~, ~ . - . : : : : , " " ," . - . ' - " : ' =," ':~ -'" '0 ) ! - " ' " - ' : ' " " " - . . . . . . . . . 2:=.::" " ._ -20m, -.,~.?., . /


    2. Analysis of the Experimental Data

    The Mount Wilson magnetograph data of the line-of-sight magnetic flux (BII) covers the range between 67 ~ southern latitude and 67 ~ northern latitude in steps of 2 ~ .

    The technique of smoothing the original data was developed by H. Snodgrass as follows. The field values are taken from a strip 4-6.8 degrees in longitude centered at the central meridian of each day's observation. For each averaging period (about two months) the average field found by summing over all 68 latitudes has been subtracted from the data. The resulting array for each given time thus has zero average. This subtraction is based on the assumption that the magnetograph- measured fields sum to zero over the solar surface, which introduces some error into the data, in particular since the central portion of the solar disk is favored for line-of-sight magnetic field measurements, and since the magnetograph does not register all of the magnetic flux because it saturates.

    The values of the Br(O, t) component were then smoothed by 11 Carrington rotations.

    There is a linear connection between Bll, Br , and Bo given by

    BII = Br sin 0 + Bo cos 0. (1)

    Assuming that the true average field direction is radial and Bo ~- O, the B~ component can be easily found by

    B~ = BII (2) sin 0 '

    where 0 is the colatitude. The values of the calculated radial and observed line-of-sight components are

    represented in Figures l(b) and l(c). The relative sunspot number is shown in Figure l(a).

    Using diffusion Equation (3), describing the evolution of the background mag- netic field, we can determine the source of the background magnetic field,

    OB~ Ot -- F (Br ) -}- T IV2Br ' (3)

    where F(Br) is the source of the poloidal field and rj is the diffusivity. From Equation (1) the derivative OB~/Ot ~- ABe~At is proportional to the

    source of the poloidal field plus the diffusion term. The values of ABe~At using different At (6, 12, and 18 Carrington periods) were computed. Contours of the ABr/At latitude-time space for At = 9 and 12 rotations are shown in Figures l(d) and l(e).

    During the solar cycle the zones of increasing and decreasing surface magnetic field appear in both the N and S hemisphere (see Figures l(d) and l(e)). These results agree with those based on Stanford magnetograms (Benevolenskaya, 1994).




    P- 100 E


    0 0




    li 0


    Solar Cycle 20 J

    5 10 15 Frequency Number

    5 10 15 Frequency Number

    Figure 2. (a) Power spectrum Pyv for B,, and (b) for ABe~At for cycle 20. The frequency number represents the frequency in units of 1/128P~ -1 (solid line = N hemisphere, dotted line = S hemisphere).

    Further, using the values of Br (0, t), a Fourier analysis for each of cycles 20, 21, and 22 was done separately. The time series consists of 128 Calxington rotations, and the frequency resolution is A f = 1/128 p~l .

    The power spectrum Pyy (0, t) was calculated:

    Pyy(O, f) = 2y*y/T, (4)


    y(O, f )= f Br(O, t)exp -j2~It dr, 0


    where f is the frequency and 0 is the colatitude. Then the power spectrum Puu (0, t) was averaged over latitude for the northern

    and southern hemispheres separately. The first frequency point (fl) corresponds to T1 = 128 pc1, approximately 10 yr. The results of the spectral analysis for cycles 20, 21, and 22 are shown in Figures 2(a), 3(a), 4(a). Frequency numbers (in units of f l) are marked on the axis 'Frequency Number'.

    Table I gives the values ofPvy(fl ) with T1 = 10 yr and Pyy(fs) = (Puu(f4) + Puu(fs) + Puu(f6))/3.0 with 775 = 2.0 yr for each hemisphere and cycle.

    Using the values of Pry(f) from Table I, the ratio of the amplitudes is






    0 0



    Solar Cycle 21 i


    5 10 15 Frequency Number


    i I \\

    5 10 Frequency Number

    0 - - '

    0 15

    Figure 3. (a) Same as Figure 2(a) but for cycle 21. (b) The same as Figure 2(b) but for cycle 21.


    .~ 15o

    E 100




    ~ 2



    0 0

    Solar Cycle 22 l g

    5 10 15 Frequency Number

    ', b 1 \

    5 10 Frequency Number


    Figure 4. (a) Same as Figure 2(a) but for cycle 22. (b) Same as Figure 2(b) but for cycle 22.


    Table I The values of Pyu(f) for the N and S hemi- spheres

    N cycle T1 = 10 yr T5 = 2.0 yr N S N S

    20 186 186 31 18 21 229 229 34 50 22 130 130 8 19

    Table II The values of the ratio A for the N and S hemispheres

    Ncycle 20 21 22

    AN 2.2 2.6 4.0 3-fold single single reversal reversal reversal

    As 3.2 2.2 2.6 single under- single reversal reversal reversal

    A = I ( f l ) / I ( fs) = ~/Pyy ( f l ) /Pyy(fs)

    for each hemisphere and cycle. The values of the ratio A are represented in Table II, where the type of the reversal of the polar magnetic field is shown. From Table II it follows that the ratio between the low-frequency and the high-frequency harmonics was lower in the northern hemisphere in cycle 20, when a 3-fold reversal occurred, but also in the southern hemisphere in cycle 21, when under-reversal occurred.

    The power spectrum Pyy(f) for ABr /At is shown in Figures 2(b), 3(b), 4(b). From these figures we can see that the frequency with T = 1.5-2.5 yr is dominant in both hemispheres.

    3. Discussion

    From the analysis of magnetographic data we conclude that the three-fold magnetic field reversals are related to the power of the high-frequency component. These results agree with the previous topological model for the three-fold magnetic field reversal (Benevolenskaya, 1991; Benevolenskaya and Makarov, 1990, 1992).

    Solar magnetic fields are generated by helicity and differential rotation in the solar convection zone (Parker, 1979). The linear dynamo process may generate one carrier frequency, a:c. Within the framework of linear stochastic theory, a spectrum


    of frequencies is generated for dynamo numbers D > De. In the nonlinear theory, stable frequencies with a period shorter than that of the cycle period may occur (Hoyng, 1990). The toroidal fields (By component) are formed in the convection zone and eventually emerge at the surface as active regions. The background magnetic fields (Br component) are in turn transported poleward due to diffusion and meridional circulation (Wang, Nash, and Sheeley, 1989). Let us assume that

    B~ = B~ + Be, (6)

    where Br 1 is a low-frequency component with a period on the order of 22 yr and B6 is a high-frequency component.

    According to Kleeorin and Ruzmaikin (1988), the B~ field can be represented as a dynamo wave:

    = 0)cos [ ct + 0)- 2] , (7)

    where br (r, 0) is the amplitude of the radial field in the convection zone, g(r, O) is the phase function, g(Ro, O) = O, ~ = 27r/Tr is the frequency of the solar cycle. Similarly, the Be field can be represented by

    Be = b6(r, O)cos [we(t)~ + gl(r, O) - 21 9 (8)

    If cJ6(t) is slowly varying with time, w~(t) ~_ cJ~ = constant and 91(r, O) = 9(r, O) + ~, b~(r, O) = Abe(r, 0). At some fixed latitude 0c,

    B~(t, O) = b~(O) sina~r + b6(O) sin(wet + c?). (9)

    Consider the domain B~ = 0. The function

    Br(t, O) = br(O) sinc~ct + b~(~O A ) sin(cJet + ~) (10)

    becomes zero depending on the relation between the phase, ~, frequencies, cJc and cJe, and the ratio of the amplitudes, A. The expression for the source of the B~ field can be written as follows:

    zxs (t, o) At

    - b~(O)c~r - b~)c~ecos(cJ6t + ~) . (11)

    When cJ~ >> cJc and A > 1, the low-frequency term is dominant because br (0) > br (O)/A in Equation (10), while the high-frequency term prevails in space and time in Equation (11) because b~ (O)cJ~/A > b~ (0)cJc. This result agrees with the result of spectral analysis for/3~ and ABe~At.

    Assuming T~ -~ 20 yr and Te ~- 2.0 yr, k = ~e/w~ = 10.



    .:t v , , , a)



    ~ ~ ~ ~ ~ , c)


    Figure 5. Simulated distribution of poloidal magnetic field on the Sun: (a) for k = ~o6/wc = 10, A = 2.2, and b = 7 G in the northern hemisphere, and (b) for the southern hemisphere; (c) for k = 10, A = 10, and b = 7 G in the N hemisphere, and (d) in the S hemisphere.

    It has been mentioned before that three-fold polar magnetic-field reversals occur during the even cycles in one of the hemispheres when the north polar magnetic field changes polarity from - to + and the south polar magnetic field from to - . This defines the phase to be ~; = 7r.

    The simulated magnetic field, BT, at some fixed latitude, Oc, is illustrated in Figure 5 in the case (a) ~ = 7r, A = 2.2, and k = 10, and (b) ~ = 7r, A = 6.0, and k = 10. From this figure we note that during the low-frequency change of polarity from - to + in the northern hemisphere and from + to - in the southern hemi- sphere, the high-frequency wave can appear with zones of alternating polarities. These results do not contradict the investigation of Hoeksema (1994), according to which the leading polarity region generally expands toward the equator, while the following polarity region expands toward the poles. The boundary between the two remains approximately constant. The polar field reverses when the following polarity region reaches high latitudes. The reversal resulted from two components in cycle 21: a gradually expanding, antisymmetric flux pattern, and a series of fast-moving surges that appear to have the same sign in the northern and southern hemispheres. In cycle 22 the gradually expanding term seems to dominate.

    With our model of the three-fold polar magnetic-field reversals (Benevolen- skaya, 1991) combined with our analysis of magnetograph data we can make the following conclusion: when the amplitude ratio between the low-frequency and high-frequency components is lower, a three-fold magnetic field reversal is more likely. Further, the N-S asymmetry in the polar magnetic field reversal is related


    to the N-S asymmetry in the high-frequency component, as follows from Table II. Cycles with three-fold polar field reversals are characterized by a pronounced high- frequency component of the magnetic field with a period of about two years, as compared with cycles in which single polar magnetic-field reversals take place.


    The author is grateful to Prof. Herschel Snodgrass for providing her with the Mount Wilson magnetograph data and for useful discussions, and to the referee for suggested improvements.


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