The Stationary State of Magnetic Field Near theSun

Gesesew Reta HabtieAIMS Ghana,Cape Coast, Ghana

Department of Physics, Debre Berhan University, Debre Berhan, Ethiopia

Dr. Teresa Del Rio (Supervisor)UPV/EHU, Department of Applied science, Bilbao, Spain

ABSTRACTCorona, the Sun’s outer atmosphere, has a varying structure closely related with its magnetic field. Its most promi-nent features are helmet streamers and coronal holes. For a long time, these streamers were easily viewed onlyduring sun’s total eclipses. But nowadays coronographer are able to monitor the structure of the corona continu-ously. Helmet streamers are related to closed magnetic field lines and coronal holes to the presence of open magneticfield lines. In this project we model analytically the magnetic field of a stationary isothermal corona, showing howopen magnetic field lines arise if the plasma flows a a solar wind. We then use a numerical model to compare ourresults with a more complicated situation in which the interaction of the plasma with the field is analyzed withinthe approximation of magnetohydrodynamics.

1 IntroductionThe Sun is the star at the center of the Solar System. It is by far the most important source of energy for life on Earth. TheSun is a nearly perfect spherical ball of hot plasma . Its interior is optically thick, and that means that all electromagneticradiations escaping from the Sun comes from the photosphere (a thin layer defining the surface of the Sun at a distanceof 513380.7 km form its center), or the solar atmosphere (chromosphere and corona). The corona is very dynamic, and itchanges from day to day. Nowadays, it is known that the structure of the corona in the visible light is directly related withthe magnetic field of the Sun, and the dynamic behaviour of the corona is related to the dynamic behaviour of the underlyingmagnetic field. Many papers have been dedicated to the magnetic field of the Sun and its relation to the Corona. The generalobjective of this project is to get an expression for stationary state magnetic field of solar corona and its relation to thestructures appearing in a quiet Sun.

Solar corona is always expanding. This atmosphere is around ten billion times less dense than earth’s atmosphere. Thebrightness of solar corona is less than by a factor of million from that of the surface of the Sun or photosphere. Since thesolar corona is always expanding into space, it has no specific boundaries, but its shape is determined by the magnetic field ofsun. During a solar eclipse or with instruments operating in different wavelengths it is possible to see the various structuresof the solar corona, like coronal streamers, helmet streamers, solar wind, coronal holes, coronal mass ejections etc.

1.1 Coronal StreamersCoronal streamers were the first large structure observed around the Sun during total eclipses [7]. These streamers are createddue to the stretching of magnetic field because of the influence of solar wind. Figure 1 was taken during the total eclipseswhich happened on June, 1973 and it shows the coronal structure clearly.

Fig. 1. Image of the first coronal streamer taken during solar eclipses [7].

1.2 Solar WindThe suns atmosphere is not stationary like that of earth’s atmosphere. Since the temperature of the sun is extremely highand the the solar atmosphere is not stable, charged particles, mostly electrons and protons, flow out from solar atmosphereto space continuously. This continuous flow is called solar wind. It is a supersonic flow, with a speed much larger than thespeed of propagation of sound or Alfven waves in the medium. At 1 astronomical unit from the Sun, the speed of the solarwind ranges from 250 to 1000 km/s.

Studies of the properties of the solar wind have shown that there are two types solar wind, the so-called fast and slow solarwind. Fast solar wind is more regular, less dense and as its name indicates, faster. On the other hand, slow solar wind is denserand much more irregular in composition. The coronal magnetic field has a complex structure, particularly when sunspots andactive regions are present. Here I will concentrate on the two main types of coronal magnetic fields structures, associated toopen and closed magnetic field lines. These different configurations result in different solar wind and interplanetary magneticfield properties. As we have pointed out above, the bottom of helmet streamers corresponds to closed magnetic field linesand the coronal holes to open field lines.

1.3 Solar CycleThe magnetic field of the Sun changes polarity with an approximate periodicity of 11 years. These changes in the mag-netic field manifest themselves in many aspects, such as level of radiation emitted from solar atmosphere, ejection of solarmaterials like mass ejection, number of sunspots on its surface, and other phenomena.

When the sun is at solar minima it is almost spotless and there are less solar flares and other eruptive phenomena. Thereis a lower level of emitted radiation which is emitted from the corona. The magnetic field configuration at solar minima issimilar to that of dipole at the surface of the Sun, and as we move away form the surface we find helmet streamers near theequator and coronal hole at higher solar latitudes. As the solar cycle evolves, the number of Sun spots increases, reflectinga much more complicated magnetic structure [16]. The maximum number of sunspot in the cycle is called solar maxima.During this situation, there is a much higher level of radiation emitted from the sun to space, and eruptive phenomena likecoronal mass ejections and flares are much more common. At solar maximum, helmet streamers and coronal holes appearat mid latitudes, reflecting a very complicated magnetic field structure at the interior of the Sun. The two images placed sideby side in figure 2 show the solar cycle taken by different telescopes.

The image at left hand side of figure 2 was taken with the Vacuum Telescope of the National Solar Observatory (NSO), andit gives an information about solar cycle occurred between 8 January 1992 and 25 July 1999. The image found at the righthand side of 2 was taken with the Solar X-ray telescope (SXT) on the Yohkoh spacecraft (courtesy of Yohkoh Team). Thisimage also gives a clear information how the sun’s surface varies during a solar cycles.

Fig. 2. Maps of ten photospheric magnetograms (left hand side) and Soft X-ray maps of the sun (right hand side) spanning a period ofalmost a full solar cycle.

In the above two images it is possible to see the solar minimum and solar maximum clearly. The suns at the bottom side ofthe figure 2 are at solar maxima because the number of sun spots are maximum, but at the top of each images the suns areat solar minima because the suns are spotless. In X ray images, regions with groups of spots, called active regions, appearbrighter, reflecting the presence of denser plasmas trapped in the closed magnetic field lines.

1.4 MethodologyIn this work I have reviewed some of the magnetohydrodynamics (MHD) needed to explain some of the features of themagnetic field of the solar corona. Then I have reproduced some results of a research paper, filling some of the gaps inthe calculation of solutions of differential equations and using Scipy/Python to reproduce the results, including figures.Finally I have analyzed the results of some more sophisticated models available at the Community Coordinated ModellingCenter ( CCMC) at NASA to study or model the different configurations of the coronal magnetic field and other physicalparameters like, density of particles in the corona , coronal temperature and velocity of particles in solar corona at solarmaximum and solar minimum. To do this model I used MAS model whit run numbers Lionel BIREE 050709 SH 10 andanna chulaki 031214 SH 1 for solar minimum and solar maximum respectively.

2 Literature ReviewIn their study of an stationary corona, G.W.Pneuman and R.A.Koppa [1970] [3] realized that the helmet streamer type ofconfiguration with its associated neutral point and sheet current is of central importance, and they were able to proof that anschematic picture of a typical helmet streamer configuration which is can be predicted by the equations of magnetohydro-dynamics. They solved the complete set equations numerically for the first time in an iterative way, for a stationary coronawith a dipole magnetic field at the photosphere as boundary conditions.

Parker’s Model has become a standard first approximation to describe the solar wind, and it is discussed is most reviews ofsolar physics. In here we have followed the approach by Kallenrode [5].

Serge Koutchemy and Mossel Livshiths[1992] [7] were able to reproduce some of the characteristics of the magnetic fieldof the corona in the same configuration (stationary and with dipolar boundary conditions) solving the convection-diffusionequation in an analytical way for various Reynolds number. As it is possible to show from their model, open magneticfield lines appear at high latitudes as the Reynolds number grows, and the appear at lower latitudes as the Reynolds numberincreases. For sufficiently low number, as expected, the field approaches the field of a dipole.

Nowadays, the approach of Pneuman and Koppa has been generalized to include more realistic boundary conditions. Inparticular, Linker et al [8], have developed MAS model, that uses real magnetographs of the photosphere as boundarycondition. This model is nowadays available at the CCMC at NASA, and in this project I have used the results of some runsof this model.

3 Introduction to Magnetohydrodynamics (MHD)Magnetohydrodynamics is a powerful tool for representing dynamics of many different types of plasma phenomena, at leastin a first approximation [10]. It is particularly useful in modelling large scale, relatively low frequency plasma phenomenafor which the plasma can be treated as a single conducting field in the presence of a magnetic field. The main approximationsof MHD are that the plasma is considered to be everywhere neutral and that the velocity of the plasma is everywhere muchsmaller than the speed of light.

3.1 The equations of magnetohydrodynamicsMass continuity:In fluid dynamics, the continuity equation states that, in any steady state process, the rate at whichmass enters a system is equal to the rate at which mass leaves the system [17]. That is:

∂n∂t

+ ~5· (n~u) = 0. (1)

where n is the number density and~u is the velocity of fluid.

Conservation of Momentum continuity: Another MHD equation can be derived from conservation of momentum.Considering the magnetic, pressure and gravitational forces and applying Newtons equation of motion for parcels ofmass4m and volume4V , that is

mn(~u

∂~u∂r

+∂~u∂t

)=−~5P+ ~J×~B+ρg. (2)

where m is the average mass of particles , ρ is the mass density, g is the gravitational acceleration and P is the momentum.

Thermodynamics: If we consider an ideal plasma flow with no heating , conduction and transport process, then changesin fluid’s element like pressure and volume is called adiabatic process, which can be governed by equation,

T1

T2=

(V2

V1

)(γ−1)

. (3)

or, similarly possible to show that

T1

T2=

(P2

P1

)( γ−1γ

)

. (4)

WhereV1 and V2 are the volume of state 1 and 2 respectively, T1 and T2 are the temperatures for state 1 and 2 respectively,P1 and P2 are the pressures for state 1 and 2 respectively and γ = 5

3 .

Generalized Ohm’s Law: It is given by

~J = σ~E ′ (5)

⇒ ~J = σ

(~E +~V ×~B

)(6)

where (~E ′ = ~E +~V ×~B), is the change of the electric field in a change of reference frame, σ is the plasma conductivity,J is the current density, ~V is the velocity of plasma and ~B is the magnetic field field.

Maxwell equations: These are

~5·~E =ρ

ε0(7)

~5·~B = 0 (8)

~5×~E = −∂~B∂t

(9)

~5×~B = µ0~J (10)

In the approximations of MHD, the displacement current in Ampere-Maxwell equations is neglected, because it is verysmall compared to conduction current.

Convection Diffusion Equation: Starting with the Generalized Ohms Law and some manipulations gives;

∂~B∂t

= ~5× (~V ×~B)+η~52~B (11)

This equation is called induction-diffusion equation, which expresses how the magnetic field of a plasma varies withtime within the approximations of MHD. The first component ( ~5× (~V × ~B)) is called convection or flow term and

the second component (η~52~B) is called diffusion term, where η = 1

σµ0, is called magnetic diffusivity and can be safely

taken as a constant, although the conductivity (σ) depends on the temperature.

We can write the variables in the way that the product of the magnitude of the vector and its direction as;

~B = B0~b′, t = T~t ′, ~x,~y,~z = L(~x′,~y′,~z′),

~V =LT~v′, ~5=

~5′L.

substituting these in the above equation gives ; and some rearrangements gives;

⇒ ∂b∂t ′

= ~5′× (~v′×~b)+η

~5′T~bL2

= ~5′× (~v′×~b)+~5′2~bRn

. (12)

Where

Rn =

(T η

L2

)−1

=L2

T η=

LVη

.

Rn is a dimensionless constant and it is called the magnetic Reynolds number. This number is a measure of the sizeof the advection term, compared with the size of diffusion term, and it is very important to know whether the systemis diffusion or flow dominated. In other words, it is possible to do the following modification on induction equationdepending on the value of Reynolds number;

If Rn >> 1, then the diffusion term of induction equation can be neglected and the evolution of the magnetic fieldwill be described by:

∂~B∂t

= ~5× (~V ×~B).

In this case the evolution of plasma and field verify the so-called frozen flux theorem.

If Rn << 1, then the advection term of induction equation becomes neglected and will have the form:

∂~B∂t

= η~52~B.

In this case, the plasma can be considered to be essentially at rest, and the magnetic field diffuses tending tobecome more uniform.

The magnetic Reynolds number is high in plasmas with high conductivity, high velocities and large scales for the changesof the magnetic field. In higher regions of the corona, it is large and the frozen field theorem is well satisfied. Closerto the surface, both terms have similar importance and have to be taken into account to understand both the equilibriumconfiguration of the plasmas and the time evolution of the plasmas and magnetic fields.

4 The Magnetic Field of Solar CoronaThe coronal magnetic field has a complex structure, particularly when sunspots and active regions are present. Here I willconcentrate on the two main types of coronal magnetic fields structures, associated to open and closed magnetic field lines.These different configurations result in different solar wind and interplanetary magnetic field properties. As we have pointedout above, the bottom of helmet streamers corresponds to closed magnetic field lines and the coronal holes to open fieldlines.

Due to the Lorentz force acted by magnetic field, charged particles in corona are forced to move essentially following themagnetic field lines and therefore, particles bounded to closed field lines have difficulties to escape. This means that theparticles in the regions with closed magnetic field lines cannot escape easily outside the helmet streamer regions, and theyoscillate in a closed loop in lower regions of the corona. As we have mentioned, this is the reason why the density in theregions with closed field lines is higher, and these regions look brighter.

In regions where the magnetic field lines are open, particles escape more easily from the gravitation of the Sun because themagnetic fields are lined in outward direction of solar corona, so that the particles have a good opportunity to go far fromcorona and disappear there. Since many particles are escaping from this region and they are not coming back, the numberof particles is small when compared with the area of the regions, and this tells us that the density is lower in this region ofcorona.

Along the solar cycle, the global configuration of the magnetic field of the Sun and the corona changes in a very significantway. During solar minimum the magnetic field near the photosphere is close to the field of a dipole. The solar windexpansion distorts this magnetic field in the corona, and two different regions appear, one close to the solar equator in whichthe magnetic field lines are closed, where the corona is denser, and helmet streamers appear, and one at higher latitudeswhere the field lines are open. In the highest parts of a coronal streamer, there are very close parallel but opposite field lines,associated to a current sheet that encircles the solar equator. At solar maximum, the structure of the magnetic field line ismuch more complicated. Helmet streamers appear at higher latitudes, and there may be coronal holes, sources of fast solarwind at low latitudes. In this case, the current sheet has a much more complicated structure.

The main purpose of the project is to give an approximation of these structures of the magnetic field in the simplest situation:a stationary situation that is a good approximation of the corona at solar minima. We will be able to see how open andclose lines arise in this simple case, and give an estimate of the latitude at which close and open field lines separate at solarminimum. In the case of the solar maximum, we will show with the aid of a numerical model available at the CommunityCoordinated Modelling Center at NASA how the structure of the magnetic field changes and other properties of the solarcorona change in solar maximum and solar minimum.

The solution of MHD equations for the Solar Corona involves the solution of a complicated set of coupled differentialequations. The first solution for this problem was presented by Pneumma and Kopp [3], where they solved the numericalproblem in an iterative way for a stationary isothermal and axially symmetric corona, using a dipole field as a boundarycondition at the surface of the Sun. In this chapter, we will adopt a simpler approach, and we will show how some of thefeatures of a high temperature stationary corona can be explained analytically. Although this simple model described is ableto predict some of the features of the corona, such as the presence of open field lines and a latitude separating open andclosed field lines, some other features, such as the current sheet and the actual shape of the helmet streamer are beyond themodel.

In this simple model we will assume that the high temperature corona gives rise to solar wind, and to simplify our equationswe will consider this wind as radial and constant in magnitude. Assuming a constant solar wind, and a dipolar field asboundary condition, we will solve the convection diffusion equations, and we will show that the magnetic field is carriedaway in an almost radial way, except in the region near the solar equation. Since the magnetic field at the surface is dipolar,this is a first approximation valid for solar minimum. The latitudinal extent of the region with closed magnetic field lineswill depend on the magnetic Reynolds number, and therefore, on the conductivity of the plasma and the velocity of the solarwind.

Since the boundary conditions that we wil use are the magnetic field of the dipole, we will start this chapter with a shortreview of the magnetic dipole. In a second section we will discuss our method of representation of the magnetic field linesand finally we will solve the convection-difussion equation and show the magnetic field lines for different values of theReynolds number, comparing them with the field of a magnetic dipole.

4.1 Magnetic field lines: axially symmetric fieldsThe magnetic field equation for dipole is

⇒ ~B(r,θ) = r̂(

2acosθ

r3

)+ θ̂

(asinθ

r3

). (13)

where the radial and angular component of the magnetic fields are given as below;

⇒ ~Br =2acosθ

r3 ,

~Bθ =asinθ

r3 and,

~Bϕ = 0.

To show the complex structure of solar corona magnetic fields it is useful to represent the field by means of magneticfield lines, that is, lines that are everywhere parallel to the magnetic field vector. Dipolar geometry underlies many examplesof coronal magnetic fields; so that in this project it is discussed how to visualize the magnetic field lines.

The field lines verify that the the resultant unit vector (d~l) is parallel to the magnetic field ~B at every point. In the caseof an axially symmetric field we have;

d~l = d~rr̂+~rdθθ̂. (14)

If these equations are parallel then, one can say that;

drBr

=rdθ

Bθ

,

And possible to show that;

⇒ lnr = lnsin2θ+ c,

where c is integration constant. Then the radius r is expressed as;

r = r0 sin2θ, (15)

where r0 is a constant indicating the distance of the magnetic field line to the center of the dipole in the equatorial plane.By taking x,y plane and start from x = r0 and y = 0, that corresponds to r = r0 and θ = π

2 , it is possible to plot the magneticfield lines for a dipole. Then by using these conditions it is possible to plot the magnetic field lines. Obviously the magneticfield lines for a dipole are closed lines. We notice that the magnetic field lines of a dipole verify that the field ψ is constant

along this lines. This statement can be generalized to any axial magnetic field derived from a vector potential of the formgiven by:

−→A =ψ(r,θ)r sinθ

ϕ̂. (16)

For a dipole ψ is a constant and easy to proof.

4.1.1 A first approximation to the magnetic field of the Solar coronaThe magnetic field of the Sun is transported by solar wind into the interplanetary medium. The Solar wind deforms

the magnetic field of the Sun, giving rise to a field which is almost everywhere radial, with open field lines. In a firstapproximation, the evolution of the magnetic field can be deduced from the induction equation, where the rate of change ofmagnetic field is expressed as the summation of diffusion term and some other convection term, mathematically;

∂~B∂t

= ~5× (~V ×~B)+η~52~B. (17)

The solution of this equation is only possible if the velocity of the plasma is known, and to deduce the value of the velocity wewould need the complete set of MHD equations. To get a first approximation to the problem, we will consider a stationarysolution, where the magnetic field does not change in time. Moreover, we will consider a constant velocity, as obtainedfrom the solution of Parker’s model of the Solar wind. This is a rough approximation, since the velocity near the Sun isnot constant, and it depends strongly on the magnetic fields. Nevertheless this approximation is able to reproduce someimportant characteristic of the Solar magnetic field, in particular the presence of open radial field lines.

Since the magnetic field is considered as stationary, the left hand side of the equation

∂~B∂t

= 0⇒ ~5× (~V ×~B)+η~52~B = 0. (18)

Then to get the magnetic field equation at the Solar corona, it is reasonable to assume the solar surface is spherical withradius R, since the Sun is almost a perfect sphere. The flow of plasma in the corona is always in radial direction. We assumethat the velocity is constant and radial:

~V = vr̂.

Then we take the assumption that the magnetic field at the surface of the Sun is a dipolar magnetic field, and that theaxial symmetry will not be broken by a radial velocity of the plasma. So, In similar way with magnetic dipolar field themagnetic field, we assumed a vector potential is of the form:

~A =ψ(r,θ)r sinθ

ϕ̂,

where the function ψ(r,θ) must be determined. To calculate the curl of ~A we use spherical coordinates, then we have;

~B(r,θ,ϕ) = r̂1

r sinθ

[∂

∂θsinθ~Aϕ−

∂

∂ϕ

~Aθ

]+ θ̂

[1

r sinθ

∂

∂ϕ

~Ar−1r

∂

∂rr~Aϕ

]+ ϕ̂

1r

[∂

∂rr~Aθ−

∂

∂θ

~Ar

]. (19)

Since: Ar = Aθ = 0, the vector potential has only one component and the magnetic field can be written as:

~B(r,θ,ϕ) = r̂1

r2 sinθ

[∂

∂θψ(r,θ,ϕ)

]+ θ̂

[1

r sinθ

∂

∂rψ(r,θ,ϕ)

]. (20)

This is a general expression for magnetic field of solar corona, but still we don’t know the vector field ψ(r,θ,ϕ). This canttell us to deduce the components of magnetic field are:

~Br =1

r2 sinθ

∂

∂θψ(r,θ), (21)

~Bθ =1

r sinθ

∂

∂rψ(r,θ), (22)

~Bϕ = 0. (23)

To determine the stationary state magnetic field for solar corona in this approximation, we have to determine the vectorpotential first. Bringing the condition for stationary state magnetic field from equation then, we can analyse each termsindividually:

∂~B∂t

= 0 = ~5× (~V ×~B)+η~52~B.

And the curl of the magnetic field becomes;

~5×~B = r̂1

r sinθ

[∂

∂θsinθ~Bϕ−

∂

∂ϕ~Bθ

]+ θ̂

[1

r sinθ

∂

∂ϕ~Br−

1r

∂

∂rr~Bϕ

]+ ϕ̂

1r

[∂

∂rrBθ−

∂

∂θ~Br

]. (24)

⇒ ~5×~B = −ϕ̂

(1

r sinθ

∂2

∂r2 ψ(r,θ)+1r3

∂

∂θ

[1

sinθ

∂

∂θψ(r,θ)

]).

and

⇒~V ×~B =−ϕ̂v

r sinθ

∂

∂rψ(r,θ). (25)

Thus we can transform our equation to:

~V ×~B−η~5×~B = 0, (26)

⇒ 0 = −ϕ̂

(v

r sinθ

∂

∂rψ(r,θ)

)− ϕ̂η

(1

r sinθ

∂2

∂r2 ψ(r,θ))+

(1r3

∂

∂θ

[1

sinθ

∂

∂θψ(r,θ)

]), (27)

This is a differential equation for the field ψ(r,θ). Solving this equation enables us to determine the vector field ~A(r,θ) andthe magnetic field as well. This equation is a second order separable equation which can be solved by using separation ofvariables. Suppose the solution ψ(r,θ) is in the form:

ψ(r,θ,ϕ) = ψrψθ. (28)

The angular term of the proposed solution must verify ;

ψθ =asin2

θ

r

∣∣r=R =

asin2θ

R. (29)

and the radial term is :

⇒ ψr =1

Rn +2

(Rn +2

Rr

), (30)

where Rn is the magnetic Reynolds number Rn =vRη

. Thus the combined solution would be:

ψ(r,θ) =asin2

θ

R(Rn +2)

(Rn +2

Rr

). (31)

As we have shown above,the magnetic field lines are given by ψ constant. Writing this constant as (asin2θ0/R) the magnetic

field lines can be written as;

sin2θ0

sin2θ

=1

(Rn +2)

(Rn +2

Rr

). (32)

So that equation represents a line which crosses the surface of the Sun at R,θ0.If the angle θ0verifies

sin2θ0 >

Rn

Rn +2.

The distance of the field line to the center cannot reach infinity, which physically tells us that the magnetic field lines areclosed, and the possible values of the angle θ are in the range from θ0 and π/2

But if

sin2θ0 <

Rn

Rn +2.

The magnetic field lines will reach r→ ∞, and will therefore be open field lines going to infinity. The angle θl separatingthese two families will be given by:

sin2θl =

Rn

Rn +2,

θl = arcsin

(√Rn +2

Rn

). (33)

When the magnetic Reynolds number is large this limiting angle is close to π

2 and most lines are open. In the limit of verysmall Reynolds angle, we recover the magnetic field lines of a dipole. The following model shows the structure of magneticfield lines.

Fig. 3. Model of magnetic field lines for Rn = 0 and Rn = 2

Fig. 4. Dipolar magnetic field lines for Rn = 0 and Rn = 10

This simple model describes in a correct way the open field lines, but it predicts very elongated closed field lines that arenot observed in nature. This is mainly due to the fact that we have ignored the effect of the magnetic field in the deductionof the solar wind, and we have considered a motion that is independent of the magnetic field (in this model, the velocity ofthe plasma is constant and radial everywhere). As we will see in next chapter, a more complete solution shows that near themagnetic equator the velocity of the plasma gets a non-radial component, that changes significantly the shape of the closedfield lines, giving rise to the characteristic shape of a helmet streamer. In fact, above a certain height, magnetic field linescorresponding to radial fields of opposite sign get so closed together that a sheet current develops. The shape of current sheetreally depends on magnetic field on the Sun and it changes during the solar cycle.

5 Numerical Models of Solar CoronaThe solar coronal magnetic field, the cause of the most spectacular phenomena in the heliosphere, plays a crucial role indetermining the structure of the solar corona, such as the shapes, positions, and sizes of the coronal streamers and thecoronal holes [9].

In this chapter we discuss a more complete physical model of magnetic fields of the corona. What we have done assumesthat solar wind velocity is constant, ignoring the effect of magnetic forces in the evolution of the plasma in the corona. Aswe have mentioned, in order to improve the approximation the complete MHD equations must be solved, and this can onlybe done in a numerical way. For this purpose we have chosen the National Aeronautics and Space Administration (NASA)’sCommunity Coordinate Modelling center(CCMC). In this modelling centre there are five coronal models [18]. Among thesemodel for this project we have chosen the MHD Around a Sphere(MAS) model. Because, these model is a full MHD whichcalculates the coronal temperature, density , velocity in addition to magnetic field [18] . And since calculating the thesephysical parameters was our primary objective, we chosen MAS model.

This global MHD model is a model used to study solar corona without taking any assumptions about the nature of coronalcurrents or other properties. in this model the non linear interactions between the plasma and magnetic field in the sub-Alfvenic solar corona, where the velocity of the plasma is lower than the velocity of propagation of waves, and the super-Alfvenic interplanetary space can be self-consistently treated. Currently, most of MHD models of the global solar corona andsolar wind employ the measured or synthetic data of the solar photospheric magnetic field as the bottom boundary conditions,and so does the MAS model. In addition, other MHD simulations use observational data of density, temperature and velocityon the solar surface for a specified Carrington Rotation (CR) besides the solar photospheric magnetic field to prescribe theplasma parameters on the bottom boundary. In the case of the MAS model, the values of these magnitudes are provided bythe user.

5.1 Synoptic maps of the magnetic field at the surface of the SunObservations of the Sun at a given moment allow us to measure the magnetic field at the surface with using Zeeman effect.Since not all the Sun is visible at a given time, so-called synoptic maps are constructed for each Carrington rotation (afull rotation of the Sun) These synoptic maps are available from different solar observatories, and can be used as boundaryconditions for more realistic models of the evolution of the solar corona.

In a synoptic map, the differences between the Sun at solar minimum and solar maximum are apparent. The magnetic field at

the surface of the Sun is very regular at solar minimum, and presents characteristic regions of high intensity, correspondingto Sun spots or active regions during solar maximum. Since in this chapter we will describe the magnetic field of thecorona in two carrington rotations, CR2081 (March 2009), corresponding to solar minimum, and CR2143 (October 2013),corresponding to solar maximum.

Fig. 5. Synoptic map of solar surface during solar minimum and solar maximum.

Figure 5 (a) shows the synoptic map of sun at solar minimum, but (b) is the map during solar maxima. As it is possibleto see from figure 5 in panel (a) the surface of the sun is uniform, that means there is a minimum number of sunspot on thesurface of the sun and that is the reason to say the sun is solar minima. But in figure 5 (b) there are a lot of sun spots, that isthe reason why we said the sun is at solar maximum.

5.2 3D coronal magnetic fieldThe typical solar minimum corona exhibits a well defined streamer belt with coronal holes in the polar regions [8].

During the solar maximum, the corona is highly structured and extends far outwards. During the solar minima, only fewstructures are visible and the corona appears smaller. However, the corona does not have a sharp outer boundary, but insteadshows structures which extend into different heights and then fade into the background .The corona can be seen in visiblelight during a solar eclipse as a structured, irregular ring of rays around the solar disk.

The following two plots show the results of a the MAS model retrieved from the Community Coordinate Modelling cen-ter(CCMC) in NASA. These models shows the structure of magnetic field lines for the suns outer atmosphere, as predictedfor Carrington rotation 2081 (figure 6 ) and Carrington rotation 2143 (Carrington rotation 2143) .

Fig. 6. Solar magnetic field model, Model: MAS, Model type: (a)during solar minima, (b) During solar maxima.

The different aspect of the magnetic field at solar minimum and solar maximum is apparent. Figure 6(a) is a model formagnetic field lines of the Sun at solar minima. We can see in this example that at solar minima case magnetic field lines arewell structured, and we retrieve a similar structure to that described in previous sections. For the solar minimum almost allopen magnetic fields lines originate at higher latitudes the sun, while closed, but distorted field lines appear in the equatorialregion. This clearly indicates that during solar minimum, most coronal holes are at higher solar latitudes, and that helmetstreamers concentrate in the equatorial region of the Sun. Looking at figure 6(b) we see that the structure of the magneticfield at solar maximum is on the contrary very complex. In this case we find closed field lines at higher latitudes, and openfield lines originating at equatorial latitudes. This explains the complex aspect of the corona at solar maximum, with helmetstreamers appearing at different latitudes, and more importantly, the presence of open field lines (corresponding to coronalholes) at lower equatorial regions which might give rise to the so-called Coronal Interaction Regions (CIRS).

5.3 2D plots of density, temperature, velocity and magnetic fieldsThe MAS model solves the complete MHD equations for the solar corona, including thermodynamics. As a consequence itpredicts not only the structure of the magnetic, but also other properties such as the density and temperature of the plasmaat different regions, and the velocity of the solar wind. In this section I will include some 2-dimensional plots of thosemagnitudes, comparing solar minimum and solar maximum, using the runs described above.

5.3.1 Speed vs magnetic field vs velocity plotIn figure 7 the Continuous lines represent magnetic field lines, open in black and closed in red. Arrows represent velocity ofthe plasma and colour contours its modulus in a logarithmic scale. Panel (a) show the results for CR2081 as an example ofsolar minimum and panel (b) the results for CR2143 as an example of solar maximum.

Fig. 7. Simulation of the Solar Corona. Model:MAS, Model type:solar.(a)Solar Minima case,(b) Solar maxima case.

As we can see from figure 7, the solid flow lines which represents the structures of magnetic field line plotted in a meridianplane for solar minimum panel (a) and solar maximum panel (b). We see how the magnetic field at solar minimum resemblesour results in chapter 4, with solar polar regions as sources of open magnetic field lines and equator as a source of for closedfield lines. On the contrary the magnetic field at solar maximum has a much more complicated structure.

In figure 7, the speed is indicated by a color code, but the direction is also plotted as a vector field representing the velocityof particles. Both in solar minimum and solar maximum, the solar wind velocity is essentially radial at these relatively lowdistances from the Sun, even if the magnetic field line is much more complicated in the case of the solar maximum. In bothcases it can be seen that the solar wind gets faster as we go farther from the surface, in direct relation with what we havediscussed on Parker’s solar wind model. Finally it is clear from the plots that in coronal holes (regions corresponding to openfield lines, plotted in black), higher velocities are reached at closer distances to the Sun.

5.3.2 Density vs magnetic field vs velocity plot.As figure 7 the continuous lines in figure 8 represent magnetic field lines, open in black and closed in red. Colour contoursrepresent the number density in a logarithmic scale. Panel (a) show the results for CR2081 as an example of solar minimumand panel (b) the results for CR2143 as an example of solar maximum.

Fig. 8. Simulation of the Solar Corona. Model:MAS, Model type:solar.(a).Solar Minima case,(b) solar maxima case

On figure 8 the colour code represents the density distribution in solar corona. From this plot we can see that the density ofparticles in corona is maximum at the equatorial region of the sun which is very near to photosphere and is decreases fast aswe move away from the surface of the Sun. In the image corresponding to solar minimum (panel a) the density of particles incorona decreases in a slower way in the equatorial region, and this is related with the presence of a brighter helmet streamerat those regions. Another way of looking at it is to see that the density of particles at the same radial distance varies as wemove from the equator to the poles. The density at solar poles is lower than that of the solar equator. This statement alsoclearly fitted with the theory we developed in the first chapter. Panel b represent a similar diagram for the solar maximum.We see how the density pattern reproduces the presence of open and closed field lines. In the regions with closed field lines,the density remains higher at higher altitudes, and the density is clearly lower in the coronal hole near the equator (which isrelated to the corresponding open field lines).

5.3.3 Temperature vs magnetic field vs velocity plotAs in previous figures, continuous lines represent magnetic field lines, open in black and closed in red. Colour contoursrepresent the temperature in a linear scale. Panel (a) show the results for CR2081 as an example of solar mimimun and panel

(b) the results for CR2143 as an example of solar maximum.

Fig. 9. Simulation of the Solar Corona. Model:MAS, Model type:solar.(a).Solar Minima case,(b) solar maxima case

The last parameter is the temperature which is given by the colour code in figure 9 a and b. The simulations tell us thetemperature distribution in solar corona is not radially uniform, but it is possible to conclude the temperature is continuouslydecreasing radially. This statement is only works for regions out of photosphere.

6 ConclusionIn this project we have discussed the basic features of solar corona like helmet streamers, solar wind and effects of Sun’smagnetic field on structure of solar corona. We have done it first with analytical approximations and then with the aid ofa numerical model. In our analytical approximations, the basic assumptions we used in this project were that the coronais expanding in spherical symmetric way, that the temperature of corona is isothermal and the magnetic field of corona isstationary, that mean the magnetic field is invariant with time.

Paker’s solar model ignores the effect of the magnetic field of the Sun, but in lower regions of the corona, the magnetic fieldcannot be ignored. Magnetohydrodynamics is very useful in representing dynamics of different types of plasma subject tomagnetic field. In our case, we have used one of the equations of MHD, the convection-diffusion equation, that models theevolution of the magnetic field line if the velocity of the plasma is known, as a summation of convection and diffusion terms.The relative importance of these two terms is given by the magnetic Reynolds number. In our case, within our approximationof a stationary magnetic field, assuming a dipolar configuration at the photosphere, and assuming also a constant radial flowfo the solar wind, weare able to show that at higher latitudes the magnetic field lines are open.

Nevertheless, the magnetic field of the corona affects the flow of plasma inside it, and the approximation of a radial flow isnot consistent. Since the magnetic field lines structures of solar corona are different from surface to surface of the sun, thedirection of the forces acted by the magnetic field is also in different direction. Magnetic field around the solar poles aredirected radially outward for that matter the particles also move to that direction with high speed but for the case of equatorial

regions the magnetic field lines are closed, so that the particle faced a resistive force and results low speed of the particles.The pressure, density and temperatures are other physical parameters vary with latitude in the solar corona, and this can onlybe modelled numerically. We have used numerical models provided by CCMC to illustrate how density, temperature andwind velocity are affected by the magnetic field. Since in the numerical model we use utilizes the photospheric magneticfield as a boundary condition, in this last section we have also compared the solar mimimun with the solar maximum.

AcknowledgementsFirst of all I would like to thank Dr. Teresa Del Rio for her unflagging support by taking more time editing and clarifyingthings via skype and email, with out her this paper might not be done. I would like to give my gratitude for AIMS Ghana forproviding me all necessary conditions and facilities to do this research.

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