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## Generation of the solar wind

One gives here a summary of fluid models of the solar wind which originate from the first isothermal model by Parker (1958). The aim of these models is to predict the wind evolution as well as its transient behavior (CMEs, etc…) given the surface conditions (mainly the magnetic field topology).

R. Grappin

### The solar wind

The solar wind has been discovered indirectly by Biermann in 1951. He understood that the deviation of the comet tails opposite to the Sun’s direction could be explained only by assuming that the Sun is the source of a plasma emitted by the Sun.
This plasma comes from definite Solar regions, denoted as open’’. The plasma density is very small at the Earth’s orbit (some particles per cm^3). The plasma is very hot ($\simeq 10 000$ K) and its speed is $\simeq 600$ km/s (Mach $\simeq 10$), so that the transport of the Sun’s material from its upper atmosphere (corona’’) up to the Earth takes about four days. Parker in 1958 discovered that the solar wind seemed to be a particular solution of the gas equations when assuming a central gravity field. A supersonic solution like the solar wind flow is obtained at two conditions : (i) that the upper atmosphere of the Sun be very hot ($10^6$ K) ; (ii) that the gravity decreases with the distance towards the center of the star. These two conditions are correct.
Then, the remaining problem to explain the generation of the solar wind, is to explain the high coronal temperature.

### Observational data

Figure 1 Mesure de la vitesse du vent solaire à l’aide des scientillations radio.
Source Scott et al 1983

The acceleration range
Radio-astronomy has soon allowed to localize the acceleration region of the wind at all latitudes. Figure 1 shows the speed increase between the surface and 50 solar radii.
Sources of the Wind
In the corona, the magnetic energy dominates : hence, the plasma flow is guided by the topology of magnetic field lines.
The sources of the wind are identified with the surface regions where the magnetic field lines are opened, i.e. can allow the flow to escape solar gravity. Some important points are :
(i) a small fraction of solar surface is open ;
(ii) the open fraction varies within the solar cycle ;
(iii) in open regions, field lines do not follow radial lines : they expand more rapidly than that (Figure 2).

Figure 2. Champ magnétique dans la couronne, extrapolé à partir des mesures à la surface., pour un minimum d’activité solaire à gauche, pour un maximum à droite. Source : Wang 2009, communication personnelle.

Relation between coronal magnetic field expansion and long-distance speed : Arge-Wang-Sheeley law.

Figure3. Schéma montrant la sur-expansion des tubes de flux magnétique dans la couronne solaire.

The over-expansion f of the magnetic field lines (Figure 3) is defined from the cross-section A(r) of the local magnetic flux tube, after normalizing the cross-section by that of the reference "radial’’ cross-section varying proportionally to r2 :

f = \frac{A(r_{max})}{A(r_0)} (\frac{r_0}{r_{max}})^2

(rmax≈ 2.5 RS)
NB One uses also the index n = (1/A)dA/dr
The tube over-expansion is thus defined either by the index n >2, or by the nombrer f > 1. The comparision between the over-expansion parameter f and the solar wind speed at the Earth’s orbit has led to the empirical Arge-Wang-Sheeley law that we summarize as follows :
U_{1AU} \simeq 1/f^{1/3}

This law reproduces well measurements made by the Ulysses mission both at low and high latitudes (Figure 4).

Figure 4. Mesures par la sonde Ulysses (ESA/NASA) de la vitesse du vent solaire pendant plusieurs années (haut) ; la couleur indique la direction du champ magnétique. Les courbes sont superposées à des images du Soleil obtenues par la sonde SOHO (ESA/NASA). En dessous est tracé le nombre de tâches solaires, en fonction du temps pendant la même période, mesure du niveau d’activité solaire. On voit clairement des compurtements différents, dépendant de l’activité solaire.
Source MacComas et al (2003)

### Physics of the solar Wind

Figure 5. Stratification en densité et en température, avec références à quelques modèles numériques.

Parker (1958) has shown that a star could generate a transonic flow that could reach the sound speed at several solar radii.
To obtain this result, he had to simplify the dynamics, assuming the corona to be at a temperature of several millions Kelvin (Fig.5).
However, the thermal conduction must cool the corona very rapidly, so that, to maintain the high required coronal temperature,
one must permanently feed the corona with new thermal energy, in other words bring new mechanical energy (magnetic or kinetic) and dissipate it into heat.
The resulting high gas pressure then accelerates the wind.
The remaining mechanical energy that has not been dissipated in the corona is then passively transported by the wind and progressively dissipated into heat, thus slowing down the cooling of the plasma.

The initial energy source if of course the Sun, more precisely the magnetic field that emerges permanently from below the surface, due to the thermal convection that appears in the form of permanent surface boiling called solar granulation’’.
Figure 6 shows indeed that the solar X luminosity of magnetic regions and, more generally, of stars is proportional to the solar magnetic flux at the sollar/stellar surface.

Figure 6. Relation entre la luminosité X et le flux magnétique à la surface pour des structures solaires et pour des étoiles.
Source : Pevtsov et al, 2003

### Questions studied at LPP :

Defining a solar wind model in which heating and acceleration are obtained via the injection of mechanical energy (waves, involving magnetic and kinetic energy)
Defining models of turbulent dissipation into heat allowing to transform the previous model into a closed model allowing to predict properties of the wind at the Earth’s orbit, given the magnetic field at the surface.
The solar wind model, in a preminary "slow’’ version, is described in detail in the « VP  » web page.
For exospheric models of the solar wind (i.e., not easily describable in terms of macroscopic quantities as temperature speed pressure etc...), the reader is referred to the LESIA web page.

Solar Probe Plus and Solar Orbiter missions on which several researchers at LPP work will allow to give new constraints to theoretical models, as they will give missing informations on the plasma close to the Sun, both on average properties and their time-dependance.

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