Astronomy Reports, Vol. 48, No. 1, 2004, pp. 52–57. Translated from Astronomicheskiı Zhurnal, Vol. 81, No. 1, 2004, pp. 58–63.Original Russian Text Copyright c© 2004 by Rubinov.

The Nature of Sectorial–Hemispheric Asymmetry in the Distributionof Photospheric Structures on the Quiet Sun

V. P. MikhaılutsaKislovodsk Institute, Academy of Defensive Industries of the Russian Federation, Kislovodsk, Russia

Received April 22, 2003; in final form, July 26, 2003

Abstract—The poorly studied, spatially coherent large-scale structures on the solar surface characterizedby sectorial–hemispheric asymmetry may be produced by ultralow-frequency nonlinear oscillations of theSun. This requires that (i) the wavevector of the oscillation be normal to the rotational axis of the Sun,(ii) the radial component of the wavevector be about a factor of five greater than the component tangentialto the spherical surface, (iii) the Brunt–Vaisala frequency in the oscillation-trapping region correspond toa period of ≈ 22 yr, and (iv) the sectorial number of the spatially periodic component of the oscillation beequal to three (m = 3). c© 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION

During solar-activity minima, the photospheremapped in the Bartels coordinates appears as apattern of steady, spatially coherent structures inthe distribution of bright points of sizes 1–3′′ or so.These structures cover the entire solar surface, andthe pattern of their coherent distribution is known asthe sectorial–hemispheric asymmetry [1]. Figure 1shows this distribution for the minimum of 1964 as anexample. The solar surface is depicted in cylindricalcoordinates, and the contours of the distribution ofphotospheric bright points delineate regions of in-creased density. The sectorial–hemispheric asymme-try of such regions can clearly be seen in both equa-torial and polar regions of the Sun. The structuresin the equatorial region and the diametrically opposedpolar regions are linked like the fingers of left and righthands. Moreover, we can trace some correspondencein the distribution of structures between the northernpolar and the southern equatorial region, and betweenthe southern polar and the northern equatorial region[2]. A 22-yr recurrence cycle was detected in theBartels-longitude distribution of these structures,based on data for four successive solar minima(1964–1996) [1]. No physical interpretation has beenfound for the sectorial–hemispheric asymmetry in thedistribution of such features over the solar surface.

Here, we suggest an explanation for this asym-metry based on the possibility that an ultralow-frequency nonradial oscillation with a specific di-rection for its wavevector is present on the Sun.The modeled pattern of the distribution of surfaceelements includes the above features of the sectorial–hemispheric asymmetry.

1063-7729/04/4801-0052$26.00 c©

2. NONRADIAL OSCILLATIONSON THE SUN

The physics of nonradial oscillations is describedby Unno [3]. Such oscillations can be detected on astellar surface if they have the form of standing waves;i.e., there is a region of trapped nonradial oscillationson the stellar atmosphere. The Coriolis force, grav-itational force, and pressure play the major roles inexciting such waves.

The presence of such waves on the Sun remainsan open question [4, 5], although some experimentalevidence for their existence has recently been found[6, 7].

The dispersion relation for a low-frequency nonra-dial oscillation has the form [3]

ω2 ≈ N2k2H + (2ΩΩΩ k)2

k2, (1)

where N is the Brunt–Vaisala frequency (N2 =

∇ρ

(∇ ln ρ− 1

Г1∇ ln p

)ρ

; Г1 =(

∂ ln p

∂ ln ρ

)S

), ΩΩΩ the

rotational angular velocity of the star, k the wavevec-tor, kH the horizontal (tangential to the surface)component of the wavevector, p the gas pressure,and ρ the gas density. It follows from (1) that theoscillation frequencies are generally “tied” to therotational angular velocity of the star (the second termin the numerator).

When applied to the Sun, relationship (1) leads totwo postulates.

(1) If the sectorial–hemispheric asymmetry in thedistribution of surface structures is a manifestationof trapped nonradial oscillations, the second term in

2004 MAIK “Nauka/Interperiodica”

SECTORIAL–HEMISPHERIC ASYMMETRY IN PHOTOSPHERIC STRUCTURES 53

80°

70

60

50

40

30

20

10

80

70

60

50

40

30

20

10°

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360°Longitude

Lat

itude

N

S

04812162024283236404448>

Fig. 1. Distribution of the density of photospheric bright points over the solar surface during the activity minimum of 1964. Acylindrical projection of the northern and southern hemispheres is shown. The density scale in relative units is indicated to theright.

(1) should vanish due to their 22-yr periodicity, i.e.,k ⊥ ΩΩΩ, or kr cos θ − kθ sin θ = 0.

(2) According to postulate (1), the frequency ofthe trapped waves will be determined by the Brunt–

Vaisala frequency, and the factorkH

kshould be con-

stant throughout the trapping region (kH

k=

1α).

Postulates (1) and (2), together with the fact that atrapped standing wave must form an integer numberof periods (m) along any latitude circle on the spher-ical surface, enable the derivation of the followingexpressions for the wavevector components of thetrapped nonradial oscillation:

kr =

√α− 1

αk; kθ =

√α− 1

αkcotθ; (2)

kϕ =

√1− (α− 1)cot2θ

αk; kϕ =

m

r sin θ;

ASTRONOMY REPORTS Vol. 48 No. 1 2004

(kr) =m

sin θ

√α

1− (α− 1)cot2θ.

Saio and Lee [8, 9] proposed the introduction ofthe displacement vector for an element of themedium,ξξξ(dr, rdθ, r sin θdϕ), to describe the effect of nonra-dial oscillations on the medium in which they propa-gate, so that the linearized equations of motion for theelementary volume assume the form

−ρω2ξr − 2iρωΩ sin θξϕ = −∂p′

∂r− ρ′

∂Ψ∂r

,

−ρω2ξθ − 2iρωΩ cos θξϕ = −1r

∂p′

∂θ, (3)

−ρω2ξϕ + 2iρωΩ cos θξθ + 2iρωΩ sin θξr

= − 1r sin θ

∂p′

∂ϕ,

where Ψ is the gravitational potential. The displace-ment vector ξξξ is very convenient for detecting regionswith concentrations of surface elements on the Sun.

54 MIKHAILUTSA

The magnitude and direction of this vector directlyindicate the locations of such regions. In Eqs. (3), thetime dependence of the displacement vector has theform eiωt, as it should for a standing wave, while thespatial orientation is determined by the componentsξr, ξθ, and ξϕ.

Since the trapped nonradial oscillation is a spa-tially periodic wave, it is reasonable to assume thatthe perturbed pressure and density of the medium, p′

and ρ′, are also spatially periodic. We represent theseperturbations in the form⟨

p′

ρ′

⟩=

⟨p′0

ρ′0

⟩e±i(k·r) =

⟨p′0

ρ′0

⟩e±i(krr). (4)

We use krr from (2) to find that the gradient of thepressure perturbation in (3) has the components

∂p′

∂r= 0;

∂p′

∂ϕ= 0; (5)

∂p′

∂θ= p′0im

√(α− 1)

(α− 1) cos 2θ − sin 2θ[2 sin2 θ − (α− 1) sin 2θ]

.

To locate regions with concentrations of surfaceelements on the Sun, it is sufficient to determine thevector component ξθ from the system (3) using (5):

ξθ

N

↓

S

=

1

ρω2

[∂Ψ∂r

ρ′0 cos

(m

√2(α − 1)

2 sin2 θ − (α− 1) sin 2θ

)sin 2θ

− p′0m√

α− 1r

(α− 1) cos 2θ − sin 2θ[2 sin2 θ − (α− 1) sin 2θ]3/2

sin

(m

√2(α − 1)

2 sin2 θ − (α− 1) sin 2θ

)], (6)

ξθ

N

↑

S

=

1

ρω2

[−∂Ψ

∂rρ′0 cos

(m

√2(α− 1)

2 sin2 θ + (α− 1) sin 2θ

)sin 2θ

+ p′0m√

α− 1r

(α− 1) cos 2θ + sin 2θ[2 sin2 θ + (α− 1) sin 2θ]3/2

sin

(m

√2(α − 1)

2 sin2 θ + (α− 1) sin 2θ

)].

The first and second formulas in (6) represent themeridional components of the displacement vectorsin waves whose wavevectors are directed from northto south and vice versa, respectively. The azimuthalcomponent of the displacement vector ξϕ does notplay a determining role in this case, since the numberof regions with concentrations of surface elementsalong a latitude circle will ultimately be determinedby the sectorial number m. In contrast, the positionsand number of these regions along a meridional di-rection are important. Formulas (6) indicate that themeridional component of the displacement vector iscomposed of two parts. One results from the gradi-ent of the gravitational field and the other from thegradient of the pressure field. The ratio of these partsis not known, and we will assume this ratio to bea free parameter. For this reason, we will measurethe displacement on an absolute, rather than relative,scale.

Thus, we obtain a two-parameter model in whichone parameter is the quantity α specifying the degreeto which the wavevector of the nonradial oscillationis radial, and the other is the ratio of the contribu-tions of the gravitational and pressure fields to thedisplacement vector. The displacements of the surfaceelements in two adjacent longitudinal sectors on thesolar surface are determined by the quantities ξθ(N→S) and ξθ(N← S) in (6). Thus, the model describingthe degree of concentration of the surface elements isqualitative.

3. RESULTS OF THE MODELING

We first determine the azimuthal number m fromthe observed pattern (Fig. 1). There are three max-ima in the distribution of the density of photosphericbright points in both equatorial and polar regions; i.e.,m = 3. The parameter α can be estimated from the

ASTRONOMY REPORTS Vol. 48 No. 1 2004

SECTORIAL–HEMISPHERIC ASYMMETRY IN PHOTOSPHERIC STRUCTURES 55

Polar angle,

θ180°

–10–8

0

1008060 120 140 160

–6–4

–2

24

68

10

displacement

ξ

θ

, arb. units

G5/p1G1/p1G5/p5

Fig. 2. The meridional displacement ξθ(N → S) for m =3 and α = 5 as a function of the ratio of the contributionsof the gravitational (G) and pressure (p) fields. Shown arethe cases when the contribution of gravity is a factor offive greater than that of the pressure (squares), the twocontributions are equal (circles), and the contribution ofthe pressure is a factor of five greater than that of gravity(triangles).

details of the pattern of hemispheric asymmetry inthe distribution of regions with high concentrationsof points. If 1 < α < 2, formulas (6) indicate that ξθ

is distributed nearly symmetrically about the equa-tor, irrespective of the relative contributions of thegravitational and pressure fields. In turn, this resultsin a nearly symmetrical distribution of the regionswith concentrations of surface elements. Indeed, ifthe component ξθ is, for example, positive in thenorthern and negative in the southern hemisphere,the surface elements will move toward the equator inboth hemispheres. Otherwise, the displacement willbe poleward in both hemispheres.

However, the observations reveal a different pat-tern: the displacement appears to be toward the polein the northern hemisphere and toward the equator inthe southern hemisphere within a given sector, andvice versa in a contiguous sector (Fig. 1). In view ofthis, preliminary modeling yielded the estimate 5±1for α. The hemispheric asymmetry of the displace-ments is well defined starting from α = 3.

The relative contributions of the gravitational andpressure fields to ξθ can be judged from Fig. 2, wherethe meridional displacement ξθ(N→ S) is shown asa function of the polar angle for three relative contri-butions of these two fields. The squares correspondto the case when the contribution of the gravitationalforce to the displacement is a factor of five largerthan that of the pressure, the triangles to the oppositesituation, and the circles to equal contributions ofgravity and pressure.

ASTRONOMY REPORTS Vol. 48 No. 1 2004

Polar angle,

θ180°

1008060 120 140 160

displacement

ξ

θ

, arb. units

–10–8

0

–6–4–2

2468

10

40200

α

= 5; N

→

S

α

= 5; S

→

NG/P = 5/1

Fig. 3. Meridional displacement of surface elements ξθ

as a function of the polar angle for waves propagating inopposite directions along the meridian. The parameters ofthe waves are indicated at the top right.

We can see that a predominance of the gravita-tional force mainly results in a “depletion” at middlelatitudes. The surface elements are carried from thereto the polar region 20 from the pole. The predomi-nance of the pressure mainly depletes the equatorialbelt. The surface elements drift from the equatorialbelt to regions located 15 from the equator. Figure 1indicates a definite depletion at middle latitudes. Wetake this to indicate the predominance of the gravi-tational force in the generation of ultralow-frequencynonradial oscillations.

A graphical summary of the relative values of themeridional component of the displacement vector asa function of the polar angle for waves traveling overthe Sun from both north to south and in the oppo-site direction is given in Fig. 3 for certain parametervalues. We can see that the wave N→ S originatesin the equatorial region of the northern hemisphere,intersects the equator, and reaches the southern pole.The wave (N← S) emerges in the equatorial regionof the southern hemisphere, intersects the equator,and reaches the northern pole. Each wave shouldpropagate within its own longitudinal sector. Thislinks the surface structures in a polar region of theSun to structures in the equatorial region of the op-posite hemisphere. Such a coherence of structurescan clearly be seen in Fig. 1. The wave N→ S formsregions with concentrations of surface elements in theequatorial region of the northern hemisphere (wherethe sign of the displacement ξθ changes from positiveto negative) and in the polar region of the south-ern hemisphere. The wave N← S, which propagatesin the adjacent sector, forms high-concentration re-gions in the equatorial region of the southern hemi-

56 MIKHAILUTSA

10°

45°

90°

135°

170°0° 90° 180° 270° 360°

Longitude

Pol

ar a

ngle

02481632

Fig. 4.Modeled distribution of the regions with concentrations of surface elements produced by ultralow-frequency nonradialoscillations in cylindrical coordinates. The sectorial number is m = 3, the wavevector k is mainly radial (k/kH = 5), and thegravitational force dominates over the pressure (G/p = 5). The scale for the concentration (in relative units) is shown at thebottom right.

sphere and the polar region of the northern hemi-sphere.

We used the dependence presented in Fig. 3 toconstruct a two-dimensional map of the distributionof regions with concentrations of surface elements.

To this end, we computed the derivative∆ξθ

∆θ. We

can easily see that negative values of this deriva-tive mark regions with concentrations of surface el-ements, while positive values mark regions with a de-pletion of surface elements. The modeled distributionof high-concentration regions is shown in Fig. 4.

4. DISCUSSION

Under the assumptions made, the modeled pat-tern of the concentration of surface elements clearlydemonstrates the main features of sectorial–hemi-spheric asymmetry. Thus, we have shown that ultra-low-frequency nonradial oscillations can, in principle,produce regions with concentrations of surface ele-ments on the Sun, and the distribution of these re-gions exhibits a spatially coherent asymmetry. How-ever, some details of the observed distributions remainunaccounted for.

In particular, our model relates the numbers ofequatorial and polar regions with concentrations ofbright photospheric points in the two hemispheres.The numbers of such points should be equal and,in a true realization of hemispheric asymmetry, odd.The distribution observed in 1964 (Fig. 1) truly sat-isfies this requirement. However, the distributions forother epochs (1975, 1985–1986, 1996) [1] are not so

consistent. Distributions with m = 3 can be detectedin the equatorial region of the Sun at these epochs,whereas up to five high-concentration zones can befound in the polar region. The reason for this remainsunknown.

Another peculiarity is related to the overall patternof the distributions near the equator. They are some-times well-defined (1964, 1985–1986), and some-times smeared to some extent (1975, 1996). Thesefeatures suggest that the real situation is complicatedby some effects that have not been taken into account.

The solar surface rotates differentially. This wasnot included in the model for two reasons. First, thecoherent asymmetry in the distribution of regionswith concentrations of surface structures was man-ifest in a rigidly rotating system of longitudinal co-ordinates with a rotational period of 27.0d. Second,physical arguments suggest that the region of trap-ping of the traveling waves of pulsations cannot rotatedifferentially, and should be located in rigidly rotatinglayers of the Sun.

To conclude, our modeling has demonstrated thatlow-frequency nonradial oscillations can exert an or-ganizing effect on spatially coherent surface struc-tures on the Sun. This is possible if the wavevectorof the oscillation is essentially normal to the rotationalaxis of the Sun. The radial component of thewavevec-tor must dominate in terms of its magnitude. The sec-torial number in the spatially periodic component ofthe wave must be equal to three. The Brunt–Vaisalafrequency in the trapping region should correspond tothe observed period for variation of the spatial coher-ence (≈ 22 yr).

ASTRONOMY REPORTS Vol. 48 No. 1 2004

SECTORIAL–HEMISPHERIC ASYMMETRY IN PHOTOSPHERIC STRUCTURES 57

The presence of ultralow-frequency nonradial os-cillations enriches the spectrum of models describingthe cyclic magnetic activity of the Sun. In particular,we cannot rule out the possibility that the cyclic for-mation of sunspots is controlled by these oscillations.

REFERENCES1. V. P. Mikhailutsa, Solar Phys. 199, 13 (2001).2. V. P. Mikhailutsa and V. V. Makarova, Astron. Zh. 77,

554 (2000) [Astron. Rep. 44, 487 (2000)].

ASTRONOMY REPORTS Vol. 48 No. 1 2004

3. W. Unno et al., Nonradial Oscillations of Stars(Univ. Tokyo, Tokyo, 1989).

4. C. I. Wolf and J. B. Blizard, Solar Phys. 105, 1 (1986).5. M. Woodard, Astrophys. J. 347, 1176 (1989).6. V. P. Mikhailutsa, A. D. Wittmann, and M. Bianda,

Astron. Astrophys. Trans. 19, 79 (2000).7. V. P. Mikhailutsa and V. V. Makarova, Izv. Ross. Akad.

Nauk, Ser. Fiz. 63, 2152 (1999).8. H. Saio, Astrophys. J. 256, 717 (1982).9. U. Lee and H. Saio, Astrophys. J. 491, 839 (1997).

Translated by A. Getling