The solar cycle
Prof. Lidia van Driel-Gesztelyi
Post-graduate lecture series, MSSL, 19 January 2016
The Sun is a variable star
… and we live in its
ever-changing corona.
The discovery of sunspots
• First sunspot sightings: >2000 years
ago in China
• Aristotle: the Sun is perfect and
blemish-free
• Mid-fourth century B.C.: 1st
reasonably certain sunspot
sighting by Theophrastus
(Aristotle’s student)
• 1608: discovery of the telescope
→ 1610: discovery of sunspots
by several independent observers
e.g. Galileo and Christoph Scheiner.
Galileo Galilei’s sunspot drawing from the early
17th century, from his manuscript Delle Machie
Solari
Discovery of sunspots - England
Sunspots, independently of Galileo’s sunspot sightings, were also discovered
in England in the XVII century by Thomas Harriot (1560-1621), who was born
and educated in Oxford (St. Mary Hall). He was a mathematician and natural
philosopher.
•  1609: he bought a telescope in The Netherlands
•  28 November 1611: his first recorded observation of a sunspot.
Discovery of magnetic fields in sunspots
• The first step towards physical understanding of sunspots became
possible when George Ellery Hale of Mt. Wilson Observatory detected
magnetic fields in sunspots in 1908 (Hale, ApJ 28, 315, 1908).
• Hale made this discovery by observing splitting of several spectral lines
originating from sunspots and attributing it to the Zeeman effect
(discovered by Peter Zeeman in 1896 - Nobel prize in 1902).
• Hale proved the magnetic origin of the spectral line splitting by showing
the presence of circular polarisation of the light.
Hα fibril structure around
sunspots is similar to the
distribution of iron filing
around a bar magnet - this
inspired Hale to look for
magnetic signals.
DOT image of AR10786
taken on July 8, 2005
Zeeman splitting - example
Spectrograph slit position
Sunspot image
spectrum
circular polarization
Position of V-λ profile
V-λ profile
Zeeman splitting and circular polarization of the Fe I line at 1564.8 nm. The
correlation tracker is used to scan the sunspot image across the entrance of the
spectrograph.
Observation on 9 Nov. 1999 at the German Vacuum Tower Telescope on Tenerife, with the
infrared polarimater TIP (Schlichenmaier and Collados, 2002).
2-w HMI
Magnetic fields are
ubiquitous on the
Sun, even in the socalled “quiet-sun”
regions, and they are
in continuous
evolution. The movie
shows two weeks of
magnetic field
observations by
SDO’s HMI
instrument.
Magnetic vector field of emerging flux
SDO/HMI vector magnetic field AR 12119, 18 July 2014 03:20 – 14:00 UT
Credit: Sally Dacie, Dave Long, Sudong Sun
Emergence of AR11158 – 11-15 February 2011
Pre-emergence signatures are seen at 60000 km depth as acoustic travel-time
perturbations with helioseismology (movie: Thomas Hartlep and Scott Winegarden)
http://sun.stanford.edu/~thartlep/movies/ar1158.mp4
Solar active region tomography
Images of the Sun
from the
photosphere to the
corona
on 12-01-2016
Note that strong
magnetic
concentrations are
dark in the
photosphere,
surrounded by
dispersed fields
(bright faculae in the
chromosphere) and
connected by bright
loops in the corona.
Movie of 30 years of magnetic data
Movie by
D. Hathaway
Blue: negative; yellow: positive magnetic polarity. Kitt Peak Obs., MDI, HMI magnetograms
Solar cycle - history
•  1776: Having observed sunspots in 1761 and 1764 to 1776, Christian Horrebow
(1718-1776) Danish astronomer mentions their possible periodic variation in
his diary.
•  1844: The 11-year cyclic variation in the number of sunspots was discovered by
Heinrich Schwabe (Astron. Nachr. 21, 233, 1944) based on his 18 years of
sunspot observations - he reported a ~10-year cycle.
D.H. Hathaway, LRSP, 2015-4.
Number of sunspot groups observed by H. Schwabe between 1826-43
Solar cycle - history
•  1849: (Johann) Rudolf Wolf started to tabulate daily sunspot numbers
+ started to reconstruct sunspot behaviour back to the beginning of the
17th century using a few long-term observers and filling gaps with
geomagnetic activity measurements as proxies for the sunspot number (!).
Wolf’s sunspot index or Zürich relative sunspot number is defined as:
R = k(10g + n)
Where n is the number of individual spots visible on the disk, g is the number
of sunspot groups (ARs with spots) and k is a correction factor to adjust for
observers, telescopes and site conditions.
This became the base of the Wolf, Zürich, and now International Sunspot Number
series.
The ‘11-year’ cycle
Colours: Months with observation every day; months with
1-10 days observation missing; months with 11-20 days
observation missing; months with >20 days observation
missing. Hathaway, LRSP (2015)
•  The International Sunspot
Number Series is the longest
available record of solar
activity, and the single most
analysed dataset.
•  Averages (daily, monthly,
yearly) are used.
•  Standard smoothing is a 13month running mean
centred on the month in
question and using halfweights for the months at the
start and end.
•  The times of solar cycle
minima and maxima are
usually given in terms of these
smoothed numbers.
The ‘11-year’ cycle
• amplitude variations
with factor of 3
• length: 8-14 yr
• mean length: 11.1 yr
• asymmetry in
rise and decline strongest for
high-amplitude cycles
(Waldmeier effect)
Colours: Months with observation every day; months with
1-10 days observation missing; months with 11-20 days
observation missing; months with >20 days observation
missing. Hathaway, LRSP (2015)
Group Sunspot Number
Devised by Hoyt and Schatten (Solar Phys. 181, 491, 1998), counts only the number of sunspot groups,
averaging observations from multiple observers:
12.08 N
RG =
kiGi
∑
N i=1
N is the number of observers, ki is the i-th observer’s correction factor, Gi is the number of sunspot
groups observed by observer I, and 12.08 normalises the number to the International Sunspot Number.
Its primary use is to extend sunspot number series back to the earliest telescopic observations of
sunspots in 1610.
1879-1995
For missing days from 1750 to 1818 (Cycles 1-6; red points) in
the SN count RG is used, and it probably represents an
overestimate.
RG
= 1.01
R
D.H. Hathaway, LRSP, 2015-4.
Data gaps and uncertainty
Sunspots observed in the same month
6 July 2014 (left) and 7 July 2014 (right).
Having only a few observations per
month can seriously bias the SN, which
adds great uncertainty to the early
historical cycles.
Solar Dynamic Observatory
Helioseismic and Magnetic Imager
(SDO/HMI) continuum data (top), and
Atmospheric Imaging Assembly
(SDO/AIA) 1700 Å continuum data
Revisions of the SN
Over the centuries errors have crept into the record, degrading its value for long-term studies. New data
and discoveries now allow scientists to detect and correct errors. The SN series has not been adjusted
since Rudolf Wolf created it over 160 years ago…
Recent years, a huge international group effort has been made to eliminate past errors and
inconsistencies and provide a more reliable sunspot record (with error estimates!) from 1610 to the
present.
Sunspot numbers, mainly the uncertain early records, have been revised based on new information, as
new sources came to light (principally due to work by José Vaquero and Reiner Arlt).
Leif Svalgaard (JSWSC,3, A24, 2013) noted that from 1946 Max Waldmeier (observer 1945-79)
changed the SN calculation method, by giving greater counts (2-5) to larger spots. This practice has
continued up to the present and went unnoticed by the community. Svalgaard estimates that this inflated
the modern SN by ~20% !
The revised series has replaced the original SN data at SILSO (Brussels) on 1 July 2015 (the earlier data
remains available).
The 400-year sunspot record is the longest continuously recorded daily measurement made in science.
The revision of SN has far-reaching impact on other areas (e.g. the impact of solar activity on the
terrestrial climate, or the galactic cosmic rays and the radioisotopes they produce in the Earth’s
atmosphere).
Revisions of the SN
The SN
revision is
still work in
progress…
In the revised SN series
•  The so-called “Modern Maximum” disappears
•  Sunspot activity shows no upward trend over the last 250 years (but a 80-100
year long-term cycle, the Gleissberg cycle, remains)
•  Three detected “inhomogeneities” in method since 1880 have been corrected
•  Cycle 24 will become the weakest in 200 years
Clette, Svalgaard, Vaquero, Cliver: SSRv, 186, 35, 2014
The Maunder minimum
Early records of sunspots indicate that the Sun went through a period of inactivity in
the late 17th century. Very few sunspots were seen on the Sun from about 1645 to
1715. Although the observations were not as extensive as in later years, the Sun was
in fact well observed during this time and this lack of sunspots is well documented.
There is evidence that the Sun has had similar periods of inactivity in the more distant
past. More recent low-activity period was the Dalton minimum (1790-1830).
Sunspot area
As sunspot area is proportional with the magnetic flux of the
sunspot, sunspot area are considered more physical measures of
solar activity than the SN.
Cho et al., ApJ 811, 49 (2015)
1874-1976: sunspot areas were measured at the Royal Greenwich
Observatory (RGO) and corrected for foreshortening, in the
millionths of the solar hemisphere (MHS, µHem).
Pores
Small forming spots
Mature sunspots
SN and sunspot area data are well
correlated at 99.4% level with a
proportionality constant of 16.7, and an
offset of 4 (for a single sunspot with a
correction factor of 0.6, ISN = 6.6).
D.H. Hathaway, LRSP, 2015-4.
Since 1977 the Debrecen Heliophysical
Observatory continues the program of sunspot
area (and position) measurements which started
at the RGO.
Sunspot area
Credit: D.H. Hathaway, LRSP, 2015-4.
Sunspot area as a function of latitude and time. Sunspots form in two bands, one in each hemisphere, which
start at ~25° from the equator at the start of a cycle and migrate toward the equator as the cycle progresses.
This was called “Spörer’s law of zones” by Maunder (MNRAS, 50, 251, 1890) and illustrated by his (made in
collaboration with his wife Annie Maunder) “Butterfly Diagram” (Maunder, MNRAS, 64, 747, 1904).
The 10.7 cm solar flux
Disc-integrated emission at λ=10.7 cm, ν=2800 MHz taken from 1946 close to Ottawa, and
since 1990 in Penticton, Canada. Six months overlap ensured continuity and cross-calibration.
Measurements are taken several times/day to avoid flaring times.
Advantage: objective, and virtually independent of weather conditions, so it is a continuous
series.
Its correlation with the RI is a bit complicated, as its slope changes for RI >30.
Holland and Vaughn (JGR, 89,11, 1984) formulated this as:
F10.7 = 67+0.97 RI + 17.6(e-0.035R -1)
and the 10.7 flux may lag behind the RI by ~1 month.
The relationship seemed to have changed after 1997.
Ÿ
The cause of the change may be a lower number of
small spots in cycle 23 than before (Tapping and Valdés, (
(2011), Lefèvre and Clette (2011).
D.H. Hathaway, LRSP, 2015-4.
¢
1947-1997
1998-
Total irradiance
The total solar irradiance (TSI) is the radiant energy emitted by the Sun at all wavelengths m2 s-1 at 1 AU
outside of the Earth’s atmosphere è space-borne observations. It was named un-aptly “solar constant”.
PMOD composite
(v. d41-62-1204) –
ISN
Different behaviour
for cycle 23 (20%
lower ISN, but the
same TSI as for
cycles 21 & 22)
Daily measurements of the TSI from
instruments on different satellites.
The systematic offsets among measurements
taken with different instruments complicate
determinations of the long-term behavior.
ACRIM composite –
ISN
Weaker correlation,
if any – is the
irradiance tied to
magnetic activity?
D.H. Hathaway, LRSP, 2015-4.
Total solar irradiance – the PMOD composite
• Irradiance changes are due
to the competition between
dark sunspots and bright
faculae in which the faculae
win: they over-compensate
the effect of sunspots.
max
max
3 solar cycles
max
• The total solar irradiance
changes by 0.2% from cycle
maximum to minimum.
Total solar irradiance – the PMOD composite
max
max
3 solar cycles
max
Flares and CMEs
Ÿ
data before cycle 23
¢ data after 1997
D.H. Hathaway, LRSP, 2015-4.
Occurrence of big flares (X-flares) during the
cycle – they can occur at any time.
There is a tendency for flares to occur more
frequently during the declining phase of the
cycle.
Carrington’s super-flare on 1 Sept. 1859
occurred during the rising phase at a SN~70
during the rising phase of cycle 10.
Monthly number of M- and X-class flares vs.
the SN for 03-1976 – 12-2013. Proportionality
is weaker for high SN (>100).
Geomagnetic activity
Credit: D.H. Hathaway, LRSP,
2015-4.
Solar activity increases the
baseline level of geomagnetic
activity.
•
•
•
•
Smoothed monthly geomagnetic index aa (red) and the SN (black). The aa index extends back to
1868, the longest geomagnetic time series. It is derived from two antipodal stations at ~±50°.
The minima in geomagnetic activity tend to occur just just after the SN minima, and it tends to
remain high during the declining phase of the sunspot cycle. The latter is due to high-speed solarwind streams from low-latitude coronal holes.
There is a multi-cycle trend in geomagnetic activity, which follows the long-term trend in the SN.
There are two components of geomagnetic variability, one is in phase, another is out of phase with
the solar cycle (Feynmann, JGR, 87, 6153, 1982).
Cosmic rays
Monthly averaged cosmic ray flux
from the Climax Neutron Monitor
(1951-2006) and rescaled
sunspot number (multiplied by
five and offset by 4500).
SN
19
20
GCR
•
•
•
21
22
23
D.H. Hathaway, LRSP, 2015-4.
The flux of galactic cosmic rays (GCRs) are electrons and nuclei accelerated to ≥GeV energies by
supernova explosions. They are scattered by tangled heliospheric magnetic field and fluctuations in it
(CMEs, interfaces of solar-wind streams with different solar-wind velocity).
Cosmic ray variations are anti-correlated with solar activity but with differences depending upon the
Sun’s global magnetic field polarity (A+ indicates periods with positive polarity north pole, while A–
indicates periods with negative polarity).
GCR modulations lag (2m or 10-14m) SN variations, and the lag depends on the Sun’s polarity (even
- odd cycle variation), on which the shape of the GCR variation also depends.
Radioisotopes in tree rings and ice cores
•
•
•
•
•
The radioisotopes 14C and 10Be are produced in Earth’s
stratosphere by the impact of GCRs on 14N and 16O.
The 14C gets oxidized to form CO2, which is absorbed by
plants è remains traceable in tree rings.
The 10Be gets oxidized and becomes attached to aerosols
that can precipitate in snow è remains traceable in annual
ice layers.
GCR modulation by the solar cycle è atmospheric
abundance variation of 14C (Stuiver and Quay, Science, 207,
11, 1980) and 10Be (Beer et al., Nature, 347,164, 1990).
The production rates are function of magnetic latitude, which
changes as the Earth’s magnetic dipole wanders and varies
in strength. The production/transport/storage/deposition
process is complicated and makes direct comparison
between 14C and the solar cycle difficult (Usoskin, LRSP,
2013-1).
Long-term trends: The Maunder Minimum
•
•
Walter Maunder (MNRAS, 50, 251, 1890), reporting on the work of Spörer, noted that for a
seventy-year period from 1645 – 1715 the course of the sunspot cycle was interrupted.
Annie Maunder closely collaborated with her husband Walter in the follow-up investigation
of the historic sunspot records. They also noted a possible correlation between sunspot
activity and Earth's climate, including a period of low activity and cold weather much later
referred to as the "Maunder Minimum."
J.A. Eddy (Science, 192, 1189, 1976) provided additional references to the lack of activity
during this period and referred to it as the Maunder Minimum. He noted that many
observers prior to 1890 had noticed this lack of activity and that both he and Maunder were
simply pointing out an overlooked aspect of solar activity.
Recent Grand Minima & Maxima
Sunspot
index
Reconstruction of solar activity over the past millennium
• From telescopic sunspot observations (light solid line, starts ~ 1650)
• Proxy of sunspot number from 14C data (heavy solid line left, index c)
• Northern hemisphere aurorae (circles, in sightings per decade, index a)
Charateristics:
• Gleissberg cycle (presence of a superposed 80-100 year periodicity)
• Grand minima (Wolf, Spörer, Maunder)
• High activity period in the 12th-early 13th century
Long-term cycle data
10Be in
ice cores and 14C in tree
rings data provide10 000-year
reconstruction of past solar activity
These records indicate about 15
Grand Maxima and 25 Grand
Minima over this time period (one in
every 400 years). Note that
presently we see the end of a Grand
Maximum.
Cycle characteristics
Cycles 19-23
Ÿ
•  The Waldmeier effect
(Waldmeier Astron. Mitt. Zurich, 1935, 1939)
Can be used for predictions after a cycle has started.
Rise Time (in months) ≈ 35 + 1800/ Amplitude (in Sunspot Number).
•  Double peaks – the Gnevishev gap (Gnevishev, 1963)
There is no consensus on the origin
– signature of a quasi-biannual oscillation
– effect of the N-S phase shift in activity?
Norton & Gallagher (Solar Phys. 261, 193, 2010)
show that each hemisphere has its
double-peaked activity.
•  N-S asymmetry / phase shift
The two hemispheres are never get out of phase by
more than ~10 months.
D.H. Hathaway, LRSP, 2015-4.
SN
¢ 10.7 cm
The deep 23/24 cycle minimum
There was a long, unexpected delay in the start of cycle 24 left behind a solar cycle minimum
unlike any seen in living memory, indicating the end of the Modern Grand Maximum.
•
•
•
•
In December of 2008, the 13-month smoothed SN
dropped to 1.7 – its lowest value since July of 1913,
and the smoothed number of spotless days in a
month reached its highest value since August of
1913. In September of 2009, the geomagnetic aaindex dropped to its lowest value on record (since
1868), while the galactic cosmic ray flux reached
record highs (since 1953).
Since that minimum, we have seen cycle 24 rise
slowly through one peak and then another to a
maximum smoothed sunspot number of 81.9 in April
2014.
While this behavior is not exceptional in terms of
the historical record, it is exceptional when
considering that the last time this was seen was 100
years ago.
It is likely that solar activity is going through a
Dalton-minimum type phase, which would mean
2-3 consecutive low-activity cycles.
Movie of 30 years of magnetic data
Movie by
D. Hathaway
Blue: negative; yellow: positive magnetic polarity. Kitt Peak Obs., MDI, HMI magnetograms
Hale’s law
Cycle 23, 19 January 2002
Cycle 24, 19 January 2012
N hemisphere: - +
S hemisphere: + -
N hemisphere: + S hemisphere: - +
Magnetic polarities of sunspot pairs located in the northern and southern solar hemispheres
are reversed; in one hemisphere the negative magnetic polarity sunspot leads the positive
polarity sunspot (with respect to the westward apparent motion due to solar rotation), on the
other one the reverse is true. Polarities reverse at the beginning of a new solar cycle.
Bipole tilt
8o
6o
4o
~6-7° at 30°
~1-2° at the equator
0o
10o
20o
30o
2o
Observation:
•  Bipolar active regions show a tilt relative to the equator: the leading spot is closer to the
equator than the following spot. The tilt increases with latitude. This is Joy’s law (Hale et
al., 1919).
•  The tilt increases with decreasing sunspot size/magnetic flux content -> effect of
convective turbulence.
•  The tilt may vary inversely with the latitude of the cycle.
Cause:
•  Coriolis forces acting on the rising expanding flux tubes leads to clockwise rotation of the
flux tube on the Northern hemisphere and counter-clockwise rotation on the South.
•  Direction of the flux strand in the sub-photospheric layers (Ω effect).
Butterfly diagram
DAILY SUNSPOT AREA AVERAGED OVER INDIVIDUAL SOLAR ROTATIONS
90N
SUNSPOT AREA IN EQUAL AREA LATITUDE STRIPS (% OF STRIP AREA)
> 0.0%
> 0.1%
> 1.0%
30N
EQ
30S
90S
1870
0.5
1880
1890
1900
1910
1920
1930
1940
1950
DATE
1960
1970
1980
1990
2000
2010
2020
AVERAGE DAILY SUNSPOT AREA (% OF VISIBLE HEMISPHERE)
0.4
0.3
0.2
0.1
0.0
1870
12
1880
13
1890
14
1900
http://solarscience.msfc.nasa.gov/images/BFLY.PDF
15
1910
1920
16
1930
17
18
1940
1950
DATE
19
1960
20
1970
21
1980
22
1990
23
2000
2010
2020
HATHAWAY NASA/ARC 2016/01
The sunspot “butterfly diagram”, showing the
fractional coverage of sunspots as a function of
solar latitude and time (D.Hathaway).
•  Sunspots are restricted to latitudinal bands
~30º wide, symmetric about the equator.
•  Sunspots emerge closer and closer to the
equator in the course of a cycle, peaking in
coverage at about ±15º of latitude.
•  Note the absence of sunspots at high latitudes ( > 40º) at any time during the cycle, and the
equatorward drift of the sunspot distribution as the cycle proceeds from maximum to minimum
are particularly striking here.
•  Note how the latitudinal distribution of sunspots is never exactly the same, and how for
certain cycles (for example cycle 20, 1965---1976) there exists a pronounced North--South
asymmetry in the hemispheric distributions.
•  Note also how, at solar minima, spots from each new cycle begin to appear at mid-latitudes
while spots from the preceding cycle can still be seen near the equator, and how sunspots are
almost never observed within a few degrees in latitude of the equator.
•  Sunspot maximum (1991, 1980, 1969,...) occurs about midway along each butterfly, when
sunspot coverage is maximal at about 15 degrees latitude.
Butterfly diagram
DAILY SUNSPOT AREA AVERAGED OVER INDIVIDUAL SOLAR ROTATIONS
90N
SUNSPOT AREA IN EQUAL AREA LATITUDE STRIPS (% OF STRIP AREA)
> 0.0%
> 0.1%
> 1.0%
30N
EQ
30S
90S
1870
0.5
1880
1890
1900
1910
1920
1930
1940
1950
DATE
1960
1970
1980
1990
2000
2010
2020
2010
2020
AVERAGE DAILY SUNSPOT AREA (% OF VISIBLE HEMISPHERE)
0.4
0.3
0.2
0.1
0.0
1870
12
1880
13
1890
14
1900
http://solarscience.msfc.nasa.gov/images/BFLY.PDF
15
1910
1920
16
1930
17
18
1940
1950
DATE
19
1960
20
1970
21
1980
22
1990
23
2000
HATHAWAY NASA/ARC 2016/01
The width and slope of the butterfly
Top: Latitude positions of the
sunspot area centroid in each
hemisphere for each Carrington
Rotation as functions of time from
cycle start.
Bottom: The centroids of the
centroids in 6-month intervals are
shown for large, medium, and small
cycles. The exponential fit to the
active latitude positions is shown
with 2𝜎 error bars.
D.H. Hathaway, LRSP, 2015-4.
The activity bands widen during the
rise to maximum and narrow during
the decline to minimum. This width is
primarily a direct function of the
sunspot number or area with little, if
any, further dependence on cycle
size or phase.
Magnetic butterfly diagram - polar field reversal
Each
pixel
column
is a fullsun
synoptic
map the sum
of
magnetic
fields
along a
given
latitude
Both polar caps (θ > +/-60°) are occupied by faculae of a dominant polarity. They change polarity
at about the solar maximum to the following polarity of the given hemisphere of the actual cycle.
They are most extended and have the highest total magnetic flux just before the sunspot
minimum.
The red arrows indicate the change of magnetic polarity of the poles.
Hale’s law is also seen, due to leading-polarity magnetic fields being more concentrated and
dominant, alternating on opposite hemispheres and in successive cycles.
Observations
Magnetic connectivity
before polarity reversal
Magnetic connectivity
after polarity reversal
Main rules of solar activity
Hale(-Nicholson)’s law (Hale, 1924):
active regions on opposite hemispheres
have opposite leading magnetic polarities,
alternating between successive sunspot cycles.
Spörer’s butterfly diagram (Carrington, 1858):
spot groups tend to emerge at progressively
lower latitudes as a cycle progresses.
Joy’s law (Hale et al., 1919):
the centre of gravity of the leading polarity in
bipolar sunspot groups tends to lie closer to the
equator than that of the following polarity; the tilt
increases with latitude.
~6-7° at 30°
~1-2° at the equator
Surface flows
• Surface flows are dominated by the
basic rotation of the Sun and cellular
flows.
• But they also include differential
rotation and an axisymmetric
meridional flow.
• These flows can be measured by
direct Doppler imaging, feature
tracking and helioseismic inversions.
• These surface flows, and their
extensions into the interior, play
significant roles in the solar cycle.
Flow components
The meridional circulation has been particularly difficult to characterize
because of its weakness and masking by other velocity signals. There
have been conflicting measurements since it was first measured by
Beckers in 1978.
Differential rotation
Sub-surface
solar differential rotation
from helioseismology
Solar differential rotation: Ω(φ)=A+Bsin2φ+Csin4φ
For magnetic features:
Ω(φ)=14.38-1.95sin2φ-2.17sin4φ
Meridional circulation
• The axisymmetric flow in the meridional plane is generally
known as the meridional circulation.The meridional
circulation in the solar envelope is much weaker than the
differential rotation, making it relatively difficult to measure.
Although it can in principle be probed using global
helioseismology, the effect of meridional circulation on global
acoustic oscillations is small and may be difficult to
distinguish from rotational and magnetic effects. Thus, we
must currently rely on surface measurements and local
helioseismology.
• The meridional circulation is largely poleward from the equator at about 20 m/s
and extends deep into the convection zone from the surface. Evidence is mounting
that the flow is time-dependent.
•  The equatorward return flow deep down in the convection zone has not been
measured. However, it should exist (mass conservation). However, since ρ
increases with depth, a 20 m/s poleward flow would be balanced by a ~ 4 m/s
equatorward flow in the lower half of the convection zone…
Meridional circulation - cycle variation
(a)
Hernandez Gonzalez 2010
(b)
Spatial and temporal variation of the meridional circulation in the surface layers of the Sun.
(a)  The meridional flow as a function of the solar cycle and depth. In the shallow layers (3-8 Mm
depth) a “bump” is present in the activity belt during the high-activity years, while there is no
significant cycle variation at greater depth (11-14 Mm). This is attributed to a geostrophic
flow due to the lower sub-surface temperature of active regions (Spruit, 2003).
(b)  The meridional flow as a function of latitude and time at a depth of 5.8 Mm, averaged for the
two hemispheres – note the fluctuations and cycle dependence.
Dynamo
Dynamo refers to the complex of mechanisms that cause the magnetic
phenomena in the Sun. Usually, it is broken down into three components:
(1)  Generation of strong, large-scale fields of periodically reversing polarity
(2)  The rise of these fields to the photosphere
(3)  The processing, spreading across, and removal from the photosphere of
magnetic flux
After decades of magnetic measurements on the Sun which included the
weaker facular fields besides the strong ones, inspired by the observed
time-dependent behaviour of the photospheric magnetic fields, in 1961
Babcock put forward a conceptual model, which describes the 22-year cycle
in five stages.
Babcock’s model
:
1st stage
2nd stage
1st stage:
About 3 years before the onset of the new sunspot cycle, the new field to be involved in the new
cycle is approximated by a dipole field symmetric about the rotation axis.
That poloidal field is identified with observed field in the polar caps.
Low-latitude regions are residuals of the preceding cycle.
All the internal poloidal field lines are arbitrarily taken to lie in a relatively thin sheet at depth in the
order of 0.1 R that extends in latitude from -55o to +55o having flux of 8x1021 Mx.
2nd stage:
The originally poloidal field is pulled into a helical spiral in the activity belts, by diff. rot., with the
result that the field becomes nearly toroidal and gets amplified - after 3 years the equator will
have gained in its rotation about 5.6 turns on the latitude circles +/- 55o, which gives the same
number of turns for the field lines on both hemispheres (0.0001→0.03T). Further amplification
was reached by twisting by the roller bearing effect of the radial differential rotation shear: αeffect (→0.3T)
Third stage: Formation of active regions
• Provides a natural explanation of the Hale-Nicolson polarity
law. Each Ω-shaped loop erupting through the surface will
produce a bipolar active region with preceding and following
magnetic polarity, and p and f polarities switch at the equator.
• Explains Spörer’s law: given the sin2φ (φ=latitude) term in the
solar differential rotation, the intensification proceeds most
rapidly at φ=+/-30o, so the 1st ARs of the cycle expected to
erupt there, and only later at lower latitudes.
It takes 8 years between the 1st spots at φ=+/-30o and the last
low-latitude spots to appear using the differential rotation curve.
Solar differential rotation: Ω(φ)=A+Bsin2φ+Csin4φ
For magnetic features:
Ω(φ)=14.38-1.95sin2φ-2.17sin4φ
Fourth stage:
Describes the neutralization and subsequent reversal of the general poloidal field
as a result of the systematic inclination of active regions (i.e. that the following
polarity tends to lie at higher latitude than the leading polarity; Joy’s law). When
opposite polarities are brought together, equal amount of opposite flux cancel.
~99% of AR flux cancels against remnant flux from neighbour ARs, less than 1% of
following polarity makes it to the nearest pole, first neutralizing the existing field and
then replacing them by flux of opposite polarity. The same fraction of leading
polarities of the two hemispheres cancel in the equatorial strip. The cancellations
are helped by a presumed meridional flow pattern (pole/equatorward at high/low
latitudes).
Fifth stage:
Babcock assumed that ~11 years after the beginning of the 1st stage the polar fields
correspond to a purely poloidal field opposite to that during the 1st phase. From here
on, the dynamo process has been suggested to continue for the second half of the
22-yr activity cycle, now with all polarities reversed with respect to the first 11-yr.
In Babcock’s model, the migration of the f polarity towards the nearest pole was
included simply as an observed fact.
Leighton (1969) interpreted the mean flux transport as the combined effect of
• the dispersal of magnetic elements by a random walk process
• and (like Babcock) the asymmetry in the flux emergence (tilt - Joy’s law)
Leighton included this flux transport in a quantitative, closed kinematic model for the
solar cycle, dealing with zonal and radial averages of the magnetic field, which are
assumed to vary only with latitude and time.
→ Babcock-Leighton semiempirical kinematic model
Limitations of the B-L model
• Heuristic and semi-quantitative, assumes a highly simplified initial state.
• Does not explain why the sunspot zones drift towards the equator.
• Kinematical - velocity field v is assumed, while the dynamics of the magnetic phenomena are
much less understood; Assumes a meridional circulation pattern (low-latitudes towards the
equator, high-latidudes toward the pole), which does not agree with modern observations.
• In the original Babcock model the internal poloidal field lines were arbitrarily taken to lie in a
relatively thin sheet at a depth of 0.1 R€, while now it is commonly accepted that the toroidal
flux cannot reside in the convection zone proper, but it is stored in the thin boundary layer
below the convection zone called the tachocline.
• It neglects the diffusive effects of the convective motions on the magnetic field (convective
motions were not known at that time).
• The intensification of the field is produced by both latitudinal and radial gradients in the Sun’s
angular rotation rate. The latter impart a further twist to the flux tubes, adjusting the α effect to
the required level in order to reproduce Spörer’s law. However, the radial gradient in the
differential rotation (in the convection zone) is not confirmed by helioseismology - however,
there is radial gradient in the tachocline.
Recent models
Since then, dynamo models have aimed fully
dynamical solutions of the induction equation
together with the coupled mass, momentum and
energy relations for the plasma
→ dynamo equation.
ω-effect
α-effect
All dynamo models rely on the differential rotation
to pull the field in the horizontal direction (ωeffect).
The main problem of the solar dynamo is how to
generate the properly cyclic poloidal field.
Parker (1955): rising plasma blobs (in the CZ)
expand laterally → they rotate due to the Coriolis
force → helical turbulence in convective motions
→ rising magnetic elements carry a poloidal field
component correct for the next cycle.
With helioseismology establishing the strong differential rotation shear at the base
of the convection zone in the tachocline, there is little doubt that the dynamo
operates there, and the magnetic field produced there has a magnitude of the order
of 10 T.
However, such strong field can not be twisted by helical turbulence, so for the
generation of the poloidal field the BL idea is revived - i.e. that it is produced by
the decay of bipolar regions on the surface.
It is the meridional circulation which will carry this field poleward and then down to
the base of the convection zone, where it is stretched by the differential rotation to
produce a strong toroidal field.
→ advective dynamo model (Choudhuri et al., 1995).
Though the details are far from being clear, it seems that solar magnetic fields
are generated and maintained by the dynamo process.
Cycle prediction – polar fields
Out of the many methods which are based on cycle statistics, the method based on polar fields (Schatten
et al., JRL, 5, 411, 1978), one of the most “physical” method is based on the polar field strength at the
start of the cycle, in accordance with Babcock’s dynamo model.
The method works for the few cycles the data span allows prediction to be made.
Svalgaard et al. (GRL 32, L01104, 2005) gave a prediction of RMax(24)=75±8. Actual peak value: 81.9! This
is about half of RMax(23), with a prior polar field being half as strong. Simple!
Cycles 15-24
Wilcox Solar Observatory data (1976-)
D.H. Hathaway,
LRSP, 2015-4.
Using Mount Wilson Observatory-observed counts of polar faculae
as a proxy and calibrating it with Wilcox and SOHO/MDI magnetic
measurements for field strength and flux, Munoz-Jaramillo et al.
(ApJL, 767, L25, 2012) created a century-long new polar magnetic
flux database for cycle predictions. The measurements taken
within 2 years of cycle minimum give the best prediction.
Squares (circles) - North (south) hemisphere
Munoz-Jaramillo et al., 2012.
Cycle prediction – flux transport dynamos
3D (latitude, depth, and time) flux-transport dynamo models have been developed to include the
kinematic effects of the Sun’s meridional circulation, finding that it can play a significant role in the
magnetic dynamo (Dikpati and Charbonneau, 1999). In these models the speed of the meridional
circulation sets the cycle period and influences both the strength of the polar fields and the amplitudes
of following cycles. Two predictions were made based on flux transport dynamos with assimilated
data, with very different results.
Dikpati et al. (2006 ) predicted an amplitude for cycle 24 of 150 – 180. They used:
(i) Rotation profile and a near surface meridional flow based on helioseismic observations.
(ii) Two source terms for the poloidal field – one at the surface due to the Joy’s Law tilt of the emerging
active regions, and one in the tachocline due to hydrodynamic and MHD instabilities.
(iii) They drove the model with a surface source of poloidal field that depends upon the sunspot areas
observed since 1874. The surface poloidal source term drifted linearly from 30° to 5° over each cycle
with an amplitude that depended on the observed sunspot areas.
(iv) The prediction was based on the strength of the toroidal field produced in the tachocline. They
found excellent agreement between this toroidal field strength and the amplitude of each of the last
eight cycles, but failed predicting the small amplitude of Cycle 24.
D.H. Hathaway, LRSP, 2015-4.
Cycle prediction – flux transport dynamos
Choudhuri et al. (2007) predicted an amplitude for cycle 24 of 80 using a similar flux-transport
dynamo, but using the surface poloidal field at minimum as the assimilated data.
They used a similar axisymmetric model for the poloidal and toroidal fields, but with a meridional
flow that extends below the base of the convection zone and a diffusivity that remains high
throughout the convection zone. In their model, the toroidal field in the tachocline produces flux
eruptions when its strength exceeds a given limit. The number of eruptions is proportional to the
sunspot number and was used as the predicted quantity. They assimilate data by instantaneously
changing the poloidal field at minimum throughout most of the convection zone to make it match the
dipole moment obtained from the Wilcox Solar Observatory observations. They found an excellent fit
to the last three cycles (the full extent of the data) and found 𝑅max (24) ∼ 80, in agreement with the
polar field prediction of Svalgaard et al. (2005).
Problems:
(i)  Flux-transport models assume that the meridional flow sinks inward at the poles and returns
toward the equator at the bottom of the tachocline. However, most recent helioseismology
measurements indicate a shallower return depth, and a two-cell structure (Zhao et al., 2013),
which are incompatible with these dynamo models.
(ii)  The meridional flow is time-dependent.
Tobias et al. (2006) and Bushby and Tobias (2007) note that even weak stochastic perturbations to the
parameters of flux transport dynamos produce substantial changes to the activity cycles. They
conclude that the solar dynamo is deterministically chaotic and thus inherently unpredictable.
D.H. Hathaway, LRSP, 2015-4.
Cycle prediction – flux transport dynamos
Choudhuri et al. (2007) predicted an amplitude for cycle 24 of 80 using a similar flux-transport
dynamo, but using the surface poloidal field at minimum as the assimilated data.
They used a similar axisymmetric model for the poloidal and toroidal fields, but with a meridional
flow that extends below the base of the convection zone and a diffusivity that remains high
throughout the convection zone. In their model, the toroidal field in the tachocline produces flux
eruptions when its strength exceeds a given limit. The number of eruptions is proportional to the
sunspot number and was used as the predicted quantity. They assimilate data by instantaneously
changing the poloidal field at minimum throughout most of the convection zone to make it match the
dipole moment obtained from the Wilcox Solar Observatory observations. They found an excellent fit
to the last three cycles (the full extent of the data) and found 𝑅max (24) ∼ 80, in agreement with the
polar field prediction of Svalgaard et al. (2005).
Problems:
(i)  Flux-transport models assume that the meridional flow sinks inward at the poles and returns
toward the equator at the bottom of the tachocline. However, most recent helioseismology
measurements indicate a shallower return depth, and a two-cell structure (Zhao et al., 2013),
which are incompatible with these dynamo models.
(ii)  The meridional flow is time-dependent.
Tobias et al. (2006) and Bushby and Tobias (2007) note that even weak stochastic perturbations to the
parameters of flux transport dynamos produce substantial changes to the activity cycles. They
conclude that the solar dynamo is deterministically chaotic and thus inherently unpredictable.
D.H. Hathaway, LRSP, 2015-4.
Summary - solar cycle
11-year cycle:
• amplitude variations with factor of 3
• length: 8-14 yr
• mean length: 11.1 yr ± 14 months
• asymmetry in rise and decline - strongest for high-amplitude cycles
Rules:
• Hale(-Nicolson)’s law: active regions on opposite hemispheres have opposite leading
magnetic polarities, alternating between successive sunspot cycles.
• Spörer’s butterfly diagram: spot groups tend to emerge at progressively lower altitude
as a cycle progresses.
• Joy’s law: the centre of gravity of the leading polarity in bipolar sunspot groups tends to
lie closer to the equator than that of the following polarity: the tilt increases with latitude.
Longer-term cycle modulations:
• 80-100 yr (Gleissberg cycle)
• number of grand minima (Wolf, Spörer, Maunder)
• grand maximum in the 12th-early 13th century
Summary: B-L dynamo model
The five stages of the development of the cycle as described by
the Babcock-Leighton model:
(i) low activity phase – pure poloidal magnetic field
(ii) winding up and intensification of the field by differential rotation
(iii)  emergence of active regions – sense of positive and
negative polarities are equivalent of Hale’s law
(iv)  neutralisation and reversal of general poloidal field due to
flux cancellation with flux diffused out of following polarity spots
which are closer to the poles (Joy’s law)
(v)  renewed winding up by differential rotation with polar field
polarity reversed. – 22-year cycle is required to restore original
polarity.
The solar dynamo
The solar dynamo: Toroidal and radial magnetic fields.
http://sdo.gsfc.nasa.gov/gallery/animations/item/266