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A Doubling of the Sun's Coronal Magnetic Field during the Last 100 Years

M. Lockwood, R. Stamper, and M.N. Wild

World Data Centre C-1 for STP, Rutherford Appleton Laboratory, Chilton, England, UK

NATURE Vol. 399, 3 June 1999. Pages 437-439

NATURE Vol. 399, 3 June 1999. Pages 437-439

A Doubling of the Sun's Coronal Magnetic Field during the Last 100 Years

M. Lockwood, R. Stamper, and M.N. Wild

World Data Centre C-1 for STP, Rutherford Appleton Laboratory, Chilton, England, UK

The solar wind, because it is an extended ionized gas of very high
electrical conductivity, drags some magnetic flux out of the Sun, thereby
filling the heliosphere with the weak interplanetary magnetic field ^{7,
24}. Magnetic reconnection - the merging of oppositely-directed magnetic
fields such that they become connected to each other - between the
interplanetary field and the Earth's magnetic field, allows energy from the
solar wind to enter the near-Earth environment. The Sun's properties, such as
its luminosity, are related to its magnetic field, though the connections are
as yet not well understood ^{15, 16}. Moreover, changes in the
heliospheric magnetic field have been linked with changes in total cloud cover
over the Earth, which may influence global climate change ^{17}. Here
we report that the measurements of the near-Earth interplanetary magnetic
field reveal that the total magnetic field leaving the sun has risen by a
factor 1.4 since 1964. Using surrogate interplanetary measurements, we find
that the rise since 1901 is by a factor of 2.3. This change may be related to
chaotic changes in the dynamo that generates the solar magnetic field. We do
not yet know quantitatively how such changes will influence the global
environment.

The *aa* index has been compiled from the range of variations in the
geomagnetic field during three-hourly periods, recorded since 1868 by pairs of
near-antipodal magnetometers in England and Australia^{1}. Figure 1 demonstrates
that the annual means
<*aa*> show a marked variation with the sunspot cycle, but have
also drifted upward throughout most of this century. These changes are almost
entirely due to variations in near-Earth interplanetary
space^{2}. Several attempts have been made to use the *aa* data
to deduce the interplanetary and solar conditions before the space age^{3,
4, 5}. The success of these extrapolations depends critically on the
quality of the correlation found between *aa* and the combination of the
interplanetary parameters (the empirical "coupling function" ^{6
}) used to quantify the controlling influence of the solar wind and the
interplanetary magnetic field (IMF). Recently, an unprecedentedly high and
significant correlation coefficient of 0.97 has been obtained
^{2}.

The method section describes a novel procedure by which we derive
information on the magnetic field in the solar atmosphere (corona) from the
<*aa*> data. *F*_{s} is the magnetic flux that
threads a roughly spherical "source" surface in the corona where the
sun's field becomes purely radial: it quantifies the amount of flux leaving
the sun and entering the heliosphere. The method employs three correlations of
extremely high significance for the period 1964-1996. Figure 2 shows the good agreement
between observed yearly
averages and their best-fit predicted values, derived using these
correlations. Figure 2a shows the observed and
predicted annual means of *aa* (<*aa*> and
*aa*_{p}, respectively), 2b is the same for the function *f*
(see method section), and 2c is for the magnitude of the radial component of
the IMF |*B*_{r}|.

The orientation of the IMF in annual averages is as predicted by Parker
Spiral theory^{2, 7}. This theory also predicts that the rise in the
magnitude of the mean radial component, seen in Figure
2c, reflects a corresponding change in the coronal source field
*B*_{o} (equation 5). A least-squares linear fit to
<|*B*_{r}|> for 1964-1996 (the green line in 2c) yields a
percent change (defined as 100 times the change, divided by the initial value)
of 41% (± 13%). In other words, there has
been a rise by a factor of 1.41 over the last three solar cycles. This rise is
present, but not commented on, in previously published coronal source field
estimates, modelled from the measured solar photospheric
field^{8}. Cosmic rays are shielded from Earth by both the IMF and the
solar wind flow and the observed decay in cosmic ray fluxes (by 3.7% since
1964) ^{2, 9} is, at least qualitatively, consistent with the rise in
the IMF.

The results of the extrapolation to before 1964 are shown in Figure 3. The values of
*F*_{s} derived from
the *aa* data are shown in grey and compare well with those from the
observed annual means of the IMF radial component
<|*B*_{r}|> for 1964-1996 (thick blue line). The coronal
source flux rises and falls in each solar cycle, lagging only slightly behind
the sunspot numbers *R*, shown in purple. The main differences between
*F*_{s} and <*aa*> arise because the effects of the
recurrent fast solar wind streams (in the declining phase of each
cycle^{9}) have effectively been removed by our procedure. For data at
all phases of the solar cycle, *F*_{s} has a correlation
coefficient of 0.75 with the simultaneous *R* (giving a significance
level of effectively 100%). To eliminate the solar cycle variations, we have
studied the 11-year running means and those for* F*_{s} and
*R* vary in a very similar way. In 1901, the 11-year running mean of
*F*_{s} was a minimum of 2.308 ´
10^{14} Wb, but rose to a peak value of 5.325 ´ 10^{14} Wb in
1992. Thus in the
intervening 91 years (covering roughly 8.5 sunspot cycles) there was a rise in
the average solar source flux of 131% (i.e. a rise by a factor of 2.31).

These changes in the solar magnetic field should be seen in the context of
longer-term changes in the sun, as inferred from historical sunspot and
auroral observations^{10} and from the terrestrial abundances of
isotopes such as ^{14}C and ^{10}Be (produced by cosmic ray
bombardment and deposited and stored, for example, in the polar
icecaps)^{11, 12}. The isotope data show solar activity can largely
disappear for periods of 50-100 years, such as the Maunder minimum
(1650-1700), although there is evidence that a weak and cyclic magnetic field
still emerged from the sun^{12}. By comparing the phase of the 88-year
oscillation prior to and after the Maunder minimum, it has been inferred that
the dynamo generating the solar field may be chaotic rather than
quasi-periodic^{13}: such behaviour may be relevant to the sudden
changes in *F*_{s} around 1900 and 1960. Recent studies have
linked changes in solar activity and *aa* with terrestrial climate
change^{14, 15, 16, 17}. The variation found here stresses the
importance of understanding the connections between the sun's output and
its magnetic field^{15, 16} and between terrestrial global cloud
cover, cosmic ray fluxes and the heliospheric field^{17}.

**Method**

We employ the optimum energy coupling function between the solar wind and
the Earth's magnetosphere derived by Stamper et al.^{2} using the
dimensional analysis proposed by Vasyliunas et al.^{18}. The solar
wind kinetic energy density dominates over the energy densities of both
thermal motions and the IMF. This is incident on the geomagnetic field, which
presents a roughly circular cross section to the flow. A fraction of the
incident energy is extracted, the power transferred to the magnetosphere
being^{2} :

$\begin{array}{cccc}{P}_{\alpha}& =& \{k\pi /2{\mu}_{0}^{(\mathrm{1/3}-\alpha )}\}{M}_{''E''}^{\mathrm{2/3}}{m}_{''SW''}^{(\mathrm{2/3}-''\alpha '')}{N}_{''SW''}^{(\mathrm{2/3}-''\alpha '')}{v}_{''SW''}^{(\mathrm{7/3}-''2\alpha '')}{B}_{''SW''}^{2''\alpha ''}{\mathrm{sin}}^{4}\mathrm{(\theta /2)}={\mathrm{aa}}_{''P''}/{s}_{''a''}& \text{(1)}\end{array}$

where ${m}_{''SW''}$ is the mean ion mass, ${N}_{''SW''}$ the concentration and ${v}_{''SW''}$ the speed of the solar wind. ${B}_{''SW''}$ is the IMF magnitude, $\mathrm{\theta}$ is the IMF orientation "clock angle"
^{19} , ${M}_{''E''}$ is the magnetic moment of the Earth
(taken from the IGRF model^{20 }), ${s}_{a}$ and $k$ are constants and ${\mathrm{aa}}_{''P''}$ is the best-fit prediction of
$\mathrm{aa}$. From annual means for 1964-1996, the best-fit "coupling
exponent" α is found to be 0.386,^{2
} and ${s}_{a}$ is obtained from a linear regression fit of
<*aa*> against ${P}_{\alpha}$. The
largest factor contributing to the rise in <*aa*> since 1964 is an
upward drift in ${B}_{''SW''}$, with significant rises in
${N}_{''SW''}$ and ${v}_{''SW''}$; however the mean $\theta $
has grown somewhat less favourable for
increasing <*aa*>.^{2} The dependence is sufficient to
allow derivation of ${B}_{''SW''}$ from <*aa*>.
In order to separate the effect of ${B}_{''SW''}$ from that of
the other interplanetary variables, we define a parameter $f$ :

$\begin{array}{cccc}f& =& {N}_{''SW''}^{(\mathrm{2/3}-\mathrm{\alpha )}}{v}_{''SW''}^{(\mathrm{7/3}-''2\alpha '')}{\mathrm{sin}}^{4}\mathrm{(\theta /2)}& \text{(2)}\end{array}$

the variation of which is dominated by that in the solar wind speed
${v}_{''SW''}$. The annual mean of ${v}_{''SW''}$ rises in the
declining phase of solar cycles^{9} because the Earth repeatedly
intersects fast solar wind streams from low-latitude extensions of coronal
holes^{21}. These occur every 27 days and so also raise the
geomagnetic recurrence index, $I$ (see Figure
1)^{22}. Hence we expect $f$ and $I$ to increase
together in the declining phase of the sunspot cycle. However, $I$ can
remain high at sunspot minimum (whereas ${v}_{''SW''}$ is lower)
because *aa* values are low and relatively constant^{23}. Hence
we adopted a relationship for a predicted $f$ of the form:

$\begin{array}{cccc}{f}_{''P''}& =& {s}_{f}{I}^{\beta}{\mathrm{aa}}^{\lambda}+{c}_{f}& \text{(3)}\end{array}$

where the exponents β and λ give the optimum correlation and the constants ${s}_{f}$ and ${c}_{f}$ are then found from a linear regression fit. The primary justification for the use of (3) is that it yields a correlation which is comparable (in magnitude and significance) to the other two shown in Figure 2. Note that ${f}_{''p''}$ reproduces both the drift and 22-year cycle in $f$. From (1)-(3) we can obtain a formula for estimating ${B}_{''SW''}$ from the $\mathrm{aa}$ index data series:

$\begin{array}{cccc}{B}_{''SW''}& =& {[\{2\mathrm{aa}{\mu}_{''o''}^{(\mathrm{1/3}-\alpha )}\}/\{{s}_{''a''}k\pi {m}_{''SW''}^{(\mathrm{2/3}-\alpha )}{M}_{''E''}^{\mathrm{2/3}}({s}_{f}{I}^{\beta}{\mathrm{aa}}^{\lambda}+{c}_{f})\}]}^{\mathrm{1/(2\alpha )}}& \text{(4)}\end{array}$

Parker spiral theory successfully predicts the radial and latitudinal
variations of the annual means of the heliospheric field^{2,7 }:

$\begin{array}{cccc}{B}_{''SW''}& =& {\{{B}_{''r''}^{2}+{B}_{''\phi ''}^{2}+{B}_{''\psi ''}^{2}\}}^{\mathrm{1/2}}& \\ & =& {B}_{''r''}{\{1+{tan}^{2}\gamma \}}^{\mathrm{1/2}}& \text{(5)}\\ & =& {B}_{0}{({R}_{0}/r)}^{2}{\{1+{(\omega rcos\psi /{v}_{''SW''})}^{2}\}}^{\mathrm{1/2}}& \end{array}$

where ${B}_{0}$ is the coronal source field at ${R}_{0}$ from the centre of the sun^{8}, ω is the equatorial angular solar rotation velocity
and ψ is the heliographic latitude (see Figure 4). In
annual means, the modulus of the out-of-ecliptic IMF component $<|{B}_{\psi}|>$ is well correlated with ${B}_{''SW''}$ and the mean $<{B}_{''y''}>$ is close to zero. The "garden hose angle" g of the IMF in
the ecliptic plane (equal to tan^{-1}*Bf * /*B*_{r} )
remains close to 45° and so the radial heliospheric field component
*B*_{r }is roughly proportional to *B*_{sw}, i.e. *B*_{r} =
*s*_{B}* B*_{sw} (Figure 2c). In addition, recent observations
by the Ulysses satellite have shown that latitudinal variations in the heliospheric field are small
(*B*_{r} is independent of y )^{24}. This result has been
used to derive the coronal source field *B*_{o} from photospheric field measurements and good
agreement found with observations of *B*_{r } near Earth at all phases of the solar
cycle^{8}. The total magnetic flux of the sun that threads the source surface (radius
*R*_{o}), *F*_{s}, is:

$\begin{array}{cccc}{F}_{''S''}& =& \mathrm{(1/2).4p}{R}_{0}^{2}{B}_{0}& \\ & =& 2''p''{r}^{2}{B}_{''r''}& \text{(6)}\\ & =& 2''p''{r}^{2}{s}_{''B''}{B}_{''SW''}& \end{array}$

where *r* = 1AU for observations near Earth^{ 8}. The factor of one half arises because half the
field threading the source surface is inward, the other half outward. We can compute the solar flux
${F}_{''s''}$ from the *aa* data using equations (4) and (6).

In order to extrapolate to before 1964, we assume that all three correlations (derived from the data for after 1964) were valid at all times since 1868. Specifically, we assume that the empirical functions ${''sin''}^{4}\mathrm{(q/2)}$ and ${f}_{''p''}$ behave as they did after 1964. We also assume that the empirical coupling exponent a and the mean mass of the solar wind are constant, that Parker spiral theory applied and that latitudinal variations in the heliospheric field were small then as now.

*Acknowledgements.* The data used are stored and made available via
World Data Centre, C1 for STP at RAL, which is funded by the UK Particle
Physics and Astronomy Research Council and, until 1 April 1999, by the
National Radio Propagation Programme of the UK Radiocommunications Agency. We
also thank the many scientists who have contributed data to the WDC.

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