The Effect of Magnetic Flux Distribution on Individual m
Frequencies
R. Howe, R. W. Komm, D. H. Landy and F. Hill
National Solar Observatory, P.O. Box 26732, Tucson AZ 85723, USA
Introduction
The Magnetic Flux Distribution
Shifts at selected m/l values
The GONG peakfinding algorithm (Hill et al. 1996) returns a
frequency for each n,l,m mode in an n,l multiplet. For many
analysis purposes, these frequencies are used to derive the
coefficients of polynomial series,
where the aj are known as ‘a coefficients’. The odd order
coefficients describe the differential rotation and the even
ones reflect departures from spherical symmetry in the
structure of the Sun.
We know that the even a coefficients correlate strongly
with the latitudinal distribution of the net magnetic flux on
the solar surface, such that the a2j coefficient is correlated
with the B2j component of a Legendre decomposition of
the flux. (Howe et al. 1999, Antia et al 2001). Now, the
even coefficients reflect the symmetric part of the
perturbation of the νnlm frequencies from the central
frequency νnl , so it is of interest to look at the shifts of
individual νnlm over the solar cycle. With about 4.5y of
GONG data analysed, covering the period from before
solar minimum almost to the current solar maximum, we
are well placed to study this effect.
Above we show the evolution of the net magnetic flux from
the Kitt Peak synoptic magnetogram data over the period of
the GONG observations. The colour scale corresponds to
the base 10 logarithm of the flux in Gauss.
Predicting the shifts as a function of m/l
The available data consist of around 150,000 Individual
frequencies up to l=150 in each of 45 overlapping 108-day
sets, making up around 1500 multiplets.
Sample Multiplet Fits
Above we show mean relative frequency shifts for
different m/l values over the period of the GONG
observations. For each multiplet of a given l value, only
the single value of |m| giving the closest approach to the
chosen m/l is used. The data are further restricted to l
values between 20 and 100 and frequency values
between 2.8 and 3.2 mHz. The number of such
frequencies common to all time periods is around 200 for
m/l values other than 0. The data are weighted by the
inverse mode inertia before being averaged. The solid
curves show the prediction from the flux distribution,
scaled by a linear fit to the data.
Discussion and Conclusions
The plots above show the results of attempting to fit a loworder polynomial to the l=81, n=8 multiplet for a lowactivity (top) and high activity (bottom) GONG data set.
The left-hand panels show the frequency as a function of
m, the central panels the frequency after subtraction of the
dominant linear term, and the right panels the residuals
after the successful fit. The red curve shows the first guess
frequencies used in PEAKFIND and the gold curve the
best fit to the frequencies. Notice how the m=±l
frequencies are below the curve in the high-activity set,
giving the sharp ‘hook’ on the m=-l end of the curve which
the polynomial fit has not reproduced.
This work utilizes data obtained by the Global Oscillation Network Group
(GONG) project, managed by the National Solar Observatory, which is operated
by AURA, Inc. under a cooperative agreement with the National Science
Foundation. The data were acquired by instruments operated by the Big Bear
Solar Observatory, High Altitude Observatory, Learmonth Solar Observatory,
Udaipur Solar Observatory, Instituto de Astrofísico de Canarias, and Cerro Tololo
Interamerican Observatory. RWK, and RH in part, were supported by NASA
contract S-92698-F. NSO/Kitt Peak data used here are produced cooperatively by
NSF/NOAO, NASA/GSFC, and NOAA/SEL.
The effects of the magnetic flux distribution on the shifts of
individual νnlm frequencies can clearly be seen in the GONG
data. The m=l case, for which the modes sample only the
region close to the equator where little activity has so far
penetrated, looks very different from the other cases
considered, with a smaller amplitude and a different phase to
the variation.
We expect the activity-related frequency shift of a given
νnlm frequency at time t to be proportional to the integral
over latitude of the kernel for that m and l multiplied by the
flux distribution B, so that the shift varies as:
where x is sin(latitude).
This is illustrated above for the flux distributions
corresponding to a selection of 108-day GONG time
periods. The shifts, shown here for l = 60, 70 and 80, are
mainly a function of m/l. The sharp downturn in the shift for
m close to ±l can be seen for the more recent, higheractivity time periods. The structure at small m/l is probably
due to numerical problems with evaluating the Ylm term.
At some epochs around solar maximum, these effects are
large enough to render unreliable attempts to characterize
the frequencies within a multiplet using only a small number
of coefficients. The greatest difference between the sectoral
and non-sectoral mode behaviour occurs around mid-1999,
but a consideration of the flux migration pattern from previous
cycles suggests that shortly after solar maximum, when flux
is concentrated close to the equator where it will
preferentially affect the sectoral modes, the differences will
again be large but of the opposite sign. We suggest that at
least 15 coefficient orders may be necessary to model the
data properly under such conditions.
References
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