A displacement of the heliospheric current sheet (HCS) south of the helioequator by approx.10deg was proposed by Simpson et al. (1996) as a possible explanation of the north-south asymmetry in the galactic cosmic rays observed by Ulysses during its first fast transit in 1994-1995. The idea was not supported by magnetic field measurements on Ulysses and, on this ground, was dismissed by Simpson et al. (1996). In addition, Erdos & Balogh (1998) argued that any north-south symmetry was unlikely as there should be flux balance between the magnetic sectors of opposite polarity. Nonetheless, many in the scientific community have accepted the original suggestion of Simpson et al. (1996) that a displacement of the HCS was responsible for the cosmic ray asymmetry. In this paper, using a magnetohydrodynamic model of the solar corona and solar wind that includes both dipole and quadrupole magnetic source terms, we show that a north-south asymmetry of the magnetic field on the Sun does not give rise to a displacement of the HCS. The lack of displacement of the HCS results from a latitudinal redistribution of magnetic flux near the Sun where the plasma beta much  1), the HCS can be displaced if there is a difference in total pressure between the hemispheres

Goldstein, Melvyn L.

Usmanov, Arcadi V.

NASA Technical Reports Server

Numerical Modeling of Space Plasma Flows: ASTRONUM-2012
ASP Conference Series, Vol. 474
N. V. Pogorelov, E. Audit, and G. P. Zank, eds.
c©2013 Astronomical Society of the Paciﬁc
The North-South Asymmetry of the Heliospheric Current Sheet:
Results of an MHD Simulation
Arcadi V. Usmanov1,2 and Melvyn L. Goldstein2
1Department of Physics and Astronomy, University of Delaware, Newark,
Delaware, USA
2NASA Goddard Space Flight Center, Greenbelt, Maryland, USA
Abstract. A displacement of the heliospheric current sheet (HCS) south of the he-
lioequator by ∼10◦ was proposed by Simpson et al. (1996) as a possible explanation
of the north-south asymmetry in the galactic cosmic rays observed by Ulysses during
its ﬁrst fast transit in 1994-1995. The idea was not supported by magnetic ﬁeld mea-
surements on Ulysses and, on this ground, was dismissed by Simpson et al. (1996).
In addition, Erdo¨s & Balogh (1998) argued that any north-south symmetry was un-
likely as there should be ﬂux balance between the magnetic sectors of opposite polarity.
Nonetheless, many in the scientiﬁc community have accepted the original suggestion
of Simpson et al. (1996) that a displacement of the HCS was responsible for the cos-
mic ray asymmetry. In this paper, using a magnetohydrodynamic model of the solar
corona and solar wind that includes both dipole and quadrupole magnetic source terms,
we show that a north-south asymmetry of the magnetic ﬁeld on the Sun does not give
rise to a displacement of the HCS. The lack of displacement of the HCS results from a
latitudinal redistribution of magnetic ﬂux near the Sun where the plasma β  1. The
latitudinal redistribution is a direct consequence of the magnetic ﬁeld gradient between
pole and equator. Near the Sun, the latitudinal gradient in magnetic ﬁeld generates
meridional ﬂows directed equatorward that tend to relax the gradient in the magnetic
ﬁeld (to make it more latitude-independent) as heliocentric distance increases. If there
is an asymmetry between north and south magnetic ﬁeld strength then the meridional
ﬂows are also asymmetric (i.e., stronger in the hemisphere of stronger magnetic ﬁeld).
Because the magnetic ﬂuxes (positive and negative) in the hemispheres must be equal,
the redistribution shifts the HCS in the direction of the hemisphere with a weaker ﬁeld
and brings the ﬁeld strength on both sides of the HCS into balance by ∼16 R. At
larger distances, where the magnetic ﬁeld is relatively weak (β  1), the HCS can be
displaced if there is a diﬀerence in total pressure between the hemispheres.
1. Introduction
The apparent displacement of the symmetry plane of the cosmic-ray nuclei and anoma-
lous helium by ∼10◦ south of the heliographic equator was a surprising result of Ulysses
observations during its ﬁrst fast latitude transit from -80◦ to +80◦ in heliographic lati-
tude in 1994-1995 (Simpson et al. 1996). In contrast to the cosmic ray ﬂux, the radial
magnetic ﬁeld observed concurrently by Ulysses was largely north-south symmetric
(Smith & Balogh 1995; Smith et al. 1995; Forsyth et al. 1996), which led Simpson et al.
(1996) to dismiss the idea that the displacement or “coning” of the HCS was the cause
of the asymmetry in the cosmic ray modulation and to conclude that the “cosmic-ray
179
https://ntrs.nasa.gov/search.jsp?R=20140010318 2019-08-31T19:54:12+00:00Z
180 Usmanov and Goldstein
intensity oﬀset is not a local phenomenon”. Nevertheless, the idea that the asymme-
try in the cosmic ray ﬂuxes reﬂected an asymmetry in the heliospheric magnetic ﬁeld
gained acceptance in the scientiﬁc community.
This interpretation appeared to be supported by observations of the solar mag-
netic ﬁeld taken at the Wilcox Solar Observatory that indicated that the magnetic ﬁeld
strength within the polar caps of the Sun was, indeed, signiﬁcantly diﬀerent during the
Ulysses transit: the southern polar ﬁeld was noticeably stronger than the northern polar
ﬁeld (Smith et al. 2000). These data led Smith et al. (2000) to suggest that the sym-
metry of the heliospheric magnetic ﬁeld in Ulysses observation might be a coincidence
resulting from Ulysses’ movement in latitude together with temporal changes in the
polar ﬁelds.
Mursula & Hiltula (2003) plotted the polarity of the heliospheric magnetic ﬁeld
as observed at 1 AU separately for fall and spring seasons (when Earth is located,
respectively, above and below the helioequator) from 1965 to 2001. They noted that
the correspondence of the polarity to the Sun’s polar ﬁeld in the hemisphere of Earth’s
excursion (the Rosenberg & Coleman (1969) eﬀect) was more prominent during fall
seasons. Mursula & Hiltula (2003) suggested that this asymmetry indicated that the
southward shift of the HCS was a persistent pattern. However, the estimated magnitude
of the shift they found was only “a few degrees,” i.e., much smaller than the shift needed
to explain the cosmic ray asymmetry observed by Ulysses.
Erdo¨s & Balogh (1998) noted that any north-south asymmetry in the sector struc-
ture was unlikely because of the requirement of ﬂux balance between sectors with oppo-
site polarities and the observed heliolatitude independence of the radial magnetic ﬁeld
as observed by Ulysses during its ﬁrst latitude transit. More recently, Erdo¨s & Balogh
(2010) extended their analysis to Ulysses’ 3rd fast transit in 2006 and concluded that “a
southward displacement of the HCS by 2-3◦ is possible.” Virtanen & Mursula (2010)
arrived at a similar conclusion. Although none of the authors indicated a statistical er-
ror of the estimated displacement, it appears to be small compared with the observed
asymmetry of the cosmic rays.
During solar minimum, one can approximate the Sun’s magnetic ﬁeld as a dipole
nearly aligned with the solar rotation axis. The HCS is correspondingly aligned with the
helioequatorial plane (e.g., Hoeksema & Scherrer 1986; Sanderson et al. 2003). Higher
harmonics are typically small, but a quadrupole contribution can be signiﬁcant and has
been considered as a source of the north-south asymmetry (see, e.g., Osherovich et al.
1999; Bravo & Gonza´lez-Esparza 2000; Mursula & Hiltula 2004).
In this paper, we simulate the observed north-south asymmetry of the solar mag-
netic ﬁeld using a superposition of dipole and quadrupole contributions. We then use a
magnetohydrodynamic model of the solar corona and solar wind to compute the plasma
and magnetic ﬁeld parameters from the coronal base out to 100 AU. We show that the
asymmetry that results from the quadrupole contribution does not produce an oﬀset of
the HCS. Near the Sun, where the plasma β (the ratio of thermal and magnetic pressure)
is  1, the latitudinal redistribution of magnetic ﬂux by meridional ﬂows eliminates
the displacement of the HCS and brings the magnetic ﬁeld strength on both sides of
the HCS into balance by ∼16 R, where R is the solar radius. At larger distances,
where the magnetic ﬁeld is relatively weak (β  1), the HCS can still deviate from the
helioequator if the total pressure in the hemispheres is diﬀerent, but any such oﬀset is
small.
North-South Asymmetry of the HCS 181
2. Simulation Model
We use the numerical three-dimensional model of the solar corona and solar wind de-
scribed in Usmanov (1996); Usmanov et al. (2000); Usmanov & Goldstein (2003);
Usmanov et al. (2012). In the most recent version of that model, the computational do-
main is divided into three sub-regions: the inner or “coronal” region that extends from
the coronal base to 20 R, the “intermediate” region from 20 R to 0.3 AU, and the
“outer” region from 0.3 to 100 AU. The governing equations in the inner and interme-
diate regions include the usual set of mass, momentum, magnetic induction, and energy
conservation equations coupled with an evolution equation for the energy density of
Alfve´n waves in the WKB approximation (Jacques 1978; Usmanov 1996; Usmanov
et al. 2000; Usmanov & Goldstein 2003). In the outer region, the mean-ﬁeld solar-
wind equations that account for pickup proton eﬀects and turbulent heating are solved
simultaneously with turbulence transport equations (Usmanov et al. 2009, 2011, 2012).
We obtain steady-state solutions in the frame of reference rotating with the Sun using
the time relaxation method in the inner and outer regions and a forward integration
along the radial coordinate in the intermediate region. We use the third-order Cen-
tral Weighted Essentially Non-Oscillatory (CWENO) numerical scheme (Kurganov &
Levy 2000) in combination with a Runge-Kutta third-order time discretization (Gottlieb
et al. 2001).
We take the boundary conditions at the coronal base to be axisymmetric with a
magnetic ﬁeld consisting of both a dipole and a quadrupole aligned with the rotation
axis of the Sun. The intensity of the dipole is set by the value of the radial ﬁeld on
Sun’s poles of 13 G and the quadrupole contribution is speciﬁed to make the ﬁeld
strongly asymmetric so that the northern ﬁeld is 1/3 the strength of the southern ﬁeld
(8 and 23.5 G, respectively). The neutral line is correspondingly shifted southward by
Δθ ∼ 15◦. The reader is referred to Usmanov et al. (2012) for a description of the other
boundary conditions and model parameters.
3. Simulation Results
Figure 1 shows radial proﬁles of the radial magnetic ﬁeld Br, which deﬁnes the sector
structure and dominates over other components in the inner heliosphere, at a number of
heliocentric distances from 1 to 215 R (1 AU). The initially strongly asymmetric (with
respect to the equatorial plane) proﬁle (Figure 1a) becomes increasingly symmetric as
heliocentric distance increases (Figure 1b–1d) and is virtually symmetric by r = 32 R
(Figure 1e–1f). The latitudinal oﬀset Δθ of the neutral line decreases even faster and
disappears by r = 16 R. Note that beyond 32 R, the magnetic ﬁeld is only weakly
dependent of latitude (except for the region around the heliospheric current sheet where
Br changes polarity) in agreement with the Ulysses observations that found no evidence
of a signiﬁcant latitudinal gradient in Br (Smith & Balogh 1995).
In Figure 2 we show a contour plot of plasma and magnetic ﬁeld parameters in
the meridional plane in the inner region 1-20 R. While the distributions of radial (ur)
and meridional (uθ) velocities are signiﬁcantly asymmetric about the equatorial plane,
the magnetic ﬁeld parameters B and r2Br become more symmetric and the neutral line
(the heavy blue line in Figure 2d) approaches the helioequator as the distance increases.
Obviously, the redistribution of magnetic ﬂux is associated with the meridional ﬂows
generated by the equatorward gradients of magnetic pressure. Because the magnetic
182 Usmanov and Goldstein
-20
-10
0
10
20
-0.2
-0.1
0.0
0.1
0.2
-0.02
0.00
0.02
90 60 30 0 -30 -60 -90
Heliolatitude, deg.
Br, G Br, nT
r = 1 Rʘ
r = 4 Rʘ
r = 8 Rʘ
-400
       0
   400
r = 16 Rʘ
-100
-50
0
50
100 r = 32 Rʘ
-3
-2
-1
0
1
2
3
90 60 30 0 -30 -60 -90
Heliolatitude, deg.
r = 215 Rʘ
(a)
(f)
(e)
(d)
(c)
(b)
Δθ 
Figure 1. Latitudinal proﬁles of the radial magnetic ﬁeld Br at heliocentric dis-
tances r = 1, 4, 8, 16, 32, and 215 R. Δθ is the latitudinal oﬀset of the neutral
line.
0
153
305
458
610
-148
-93
-37
19
74
6•10-4
9•10-3
1•10-1
2•100
2•101
1
102 4 6 102 4 6102 4 6 102 4 6
ur, km s-1 uθ, km s-1 B, G
-23.2
-15.4
-7.7
0.0
7.7
(a) (b) (c) (d)
r2Br, Rʘ
2
G
Figure 2. Contour plots of (a) the radial velocity ur, (b) meridional velocity uθ,
(c) magnetic ﬁeld magnitude B, and (d) radial magnetic ﬁeld Br scaled as r2 in the
meridional plane from 1 to 20 R. The heavy blue line in (d) is the neutral line.
North-South Asymmetry of the HCS 183
ur, km s-1 uθ, km s-1 r
2Br, AU
2
B, nT
20 40 60 80 20 40 60 80 20 40 60 80 20 40 60 80
303
426
550
673
796
-14.4
-8.7
-3.0
2.7
8.3
1.7•10 -4
3.6•10 -3
7.7•10 -2
1.6•10 0
3.5•101
-3.3
-1.7
0.0
1.7
3.3
(a) (b) (c) (d)
nT
Figure 3. Contour plots in the meridional plant of (a) the radial velocity ur, (b)
meridional velocity uθ, (c) magnetic ﬁeld magnitude B, and (d) radial magnetic ﬁeld
Br scaled as r2 in the region from 0.3 to 100 AU. The heavy blue line in (d) is the
neutral line.
ﬁeld and its gradient are larger in the southern hemisphere, the equatorward ﬂows in
that hemisphere are correspondingly stronger. The latitudinal redistribution of magnetic
ﬂux in the solar corona and solar wind has been discussed by Suess & Smith (1996);
Suess et al. (1996); Usmanov et al. (2000) in the context of the lack of a signiﬁcant
latitude gradient in the radial magnetic ﬁeld as observed by Ulysses.
Figure 3 shows same parameters as in Figure 2 in the outer region from 0.3 to
100 AU. Unlike the inner region, where B is deﬁned mostly by Br (Figure 2c), the
dominant component outside of 1 AU is the azimuthal ﬁeld Bφ. The pattern of merid-
ional velocities in the outer region is also signiﬁcantly diﬀerent due to the interplay
of forces in the meridional direction (see Usmanov et al. 2000). The solar wind ﬂow
is notably asymmetric about the equatorial plane with lower velocities in the north-
ern hemisphere. Correspondingly, the magnetic ﬁeld winding and the ﬁeld itself are
stronger in this hemisphere and the neutral line deviates northward from the helioequa-
tor by several degrees. Even in the present case of a highly asymmetric ﬁeld at the Sun,
the deviation of the neutral line from the helioequator is still relatively small.
4. Conclusions
We have presented simulation results from a magnetohydrodynamic solar wind model
with a magnetic ﬁeld on the Sun composed of a dipole and a quadrupole. The quadru-
pole contribution is a source of north-south asymmetry and we studied how the asym-
metry maps into the heliosphere. We have shown that
(1) The north-south asymmetry of Sun’s magnetic ﬁeld is not translated directly
into an asymmetry of the heliospheric magnetic ﬁeld.
(2) The latitudinal ﬂows redistribute magnetic ﬂux in latitude. This leads to a
relaxation of latitudinal gradients in the magnetic ﬁeld, except for a relatively narrow
band where those gradients are concentrated (the heliospheric current sheet). This eﬀect
was observed by Ulysses.
184 Usmanov and Goldstein
(3) Near the Sun, where plasma β  1, the meridional redistribution of magnetic
ﬂux eliminates the north-south asymmetry so that the magnetic ﬁeld strengths on both
sides of the HCS come into balance by ∼16 R.
(4) At larger distances, where the magnetic ﬁeld is relatively weak (β  1), the
HCS can be displaced due to diﬀerences in total pressure between the hemispheres.
Although a displacement of “a few degrees” appears to be possible, one of order 10◦ is
unlikely.
Acknowledgments. The work of AVU was supported by the NSF/DOE Partner-
ship in Basic Plasma Science and Engineering Program grant AST-1004035, and by
NASA grant NNX09AH79G to the University of Delaware. Resources supporting this
work were provided by the NASA High-End Computing (HEC) Program through the
NASA Advanced Supercomputing (NAS) Division at the Ames Research Center and
the NASA Center for Climate Simulation (NCCS) at the Goddard Space Flight Center.
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The North-South Asymmetry of the Heliospheric Current Sheet: Results of an MHD Simulation

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