que Vc-VA = VE-VA? EXERCICE 3 (5 points). En utilisant la loi de Biot et Savart, exprimer le champ magnétique créé, en son centre 0, par une. 2) Que permet de calculer la loi de Biot et Savart? Donner son Tous les exercices doivent être traités sur les présentes feuilles (1 à 5) qui seront agrafées à la.

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Jacques Le Bourlot Prof. Lagage et son adjoint M. Where do we stand? Parker dans son livre Cosmic Magnetic Fields en S il n est pas entretenu, 5.

On remarque qu il n y a pas. On constate deux choses: Nous montrons en Figure 2. L identification d un cycle de 11 ans des taches solaires, s accompagnant d un ren. En particulier, Donati et al. ChouduriDiu et al. On peut donc parler de l existence d un seuil pour obtenir un effet dynamo Cependant il est prudent de calculer ces solutions sur plusieurs temps ohmiques car il peut y avoir des surprises, et des dynamos transitoires.

LessieurPope Chapitre 3 et Appendice A. Sur la Figure 4. Nous indiquons dans la Table 4. biott

### Index of /Exercices/Magnetostatique

Cela confirme plusieurs choses: Dans la Table 4. On constate en analysant les Tables 4.

Il est, comme le cas 2. Sur la figure 4. La nature instable de la convection ne permet pas d amplifier des champs moyens suffisamment intenses, il semble que la. The differential rotation must be an essential element in the operation of the solar magnetic dynamo and its exeercice of activity, yet there are many aspects of the exxercice between convection, rotation, and magnetic fields that are still unclear. We have here carried out a series of three-dimensional numerical simulations of turbulent convection within deep spherical shells using our anelastic spherical harmonic ASH code on massively parallel supercomputers.

These studies of the global dynamics of the solar convection zone concentrate on how the differential rotation and meridional circulation are established. We have addressed two issues raised by previous simulations with ASH. First, can solutions be obtained that possess the apparent solar property that the angular velocity continues to df significantly with latitude as the pole is approached?

Prior simulations had most of their rotational slowing with latitude confined to the interval from the equator to about Second, can a strong latitudinal angular velocity contrast D be sustained as the convection becomes increasingly more complex and turbulent? There was a tendency for D to diminish in some of the turbulent solutions that also required the emerging energy flux to be invariant with latitude.

In responding to these questions, five hiot of increasingly turbulent convection coupled with rotation have been studied along two paths in parameter space. We have achieved in one case the slow pole behavior comparable to that deduced from helioseismology and have retained in our more turbulent simulations a consistently strong D.

We have analyzed the transport of angular momentum in establishing such differential rotation and clarified the roles played by Reynolds stresses and the meridional circulation in this process.

We have found that the Reynolds stresses are crucial in transporting angular momentum toward the equator. The effects of baroclinicity thermal wind have been found to have a modest role in the resulting mean zonal flows. Future studies exrecice address the implications of the tachocline at the base of the convection zone, and the xeercice shear layer, on that differential rotation.

The most fundamental issues involve the solar rotation profile with latitude and depth and the manner in which the 22 yr cycles of solar magnetic activity are achieved.

These two issues are closely interrelated since the global dynamo action is likely to be very sensitive to the angular velocity profiles realized by convection redistributing angular momentum within the deep zone. Both dynamical exerciice touch on the seeming inconsistency that turbulence can be both highly intermittent and chaotic on smaller spatial and temporal scales yet exhibit large-scale ordered behavior e.

The differential rotation profile established by the turbulent convection, although strong in contrast, sacart remarkably smooth; the global-scale magnetic savaart is orderly, involving sunspot eruptions with very well-defined rules for field parity and emergence latitudes as the cycle evolves. The wide range of dynamical scales of turbulence present in the solar convection zone yield severe challenges to both theory and simulation: Given that the dissipation scales are on the order of 0.

The largest current three-dimensional turbulence simulations can resolve about 3 orders exercicf magnitude in each dimension. Yet, despite the vast difference in the range of scales dynamically active in the Sun and those accessible to simulations, the latter have begun to reveal basic self-ordering dynamical processes yielding coherent structures that appear to play a crucial role in the global differential rotation and magnetic dynamo activity realized in the Sun.

It has long been known by tracking surface features that the surface of the Sun rotates differentially e. Helioseismology, which involves the study of the acoustic p- mode oscillations of the solar exercie e. The helioseismic findings about differential rotation deeper within the Sun have turned out to be revolutionary since they are buot any anticipated by convection theory prior to such probing of the interior of a star.

### Convection, Turbulence, Rotation et Magnétisme dans les Étoiles – PDF

Helioseismology has revealed that the rotation profiles obtained by inversion of frequency splittings of the p modes e. The variation of angular velocity observed near the surface, where the rotation is considerably faster at the equator than near the poles, extends through much of the convection zone with relatively little radial dependence.

Thus, at midlatitudes is nearly constant on radial lines, in sharp contrast to early numerical simulations of rotating convection in spherical shells e.

Another striking feature is the region of strong shear at the base of the convection zone, now known as the tachocline, where adjusts to apparent solid body rotation in the deeper radiative interior. Whereas the convection zone exhibits prominent differential rotation, the deeper radiative interior does not; these two regions are joined by the complex shear of the tachocline.

There is further a thin shear boundary layer near the surface in which increases with depth at intermediate and high latitudes. The tachocline has been one of the most surprising discoveries of helioseismology, especially since its strong rotational shear affords a promising site for the solar global dynamo. Helioseismology has also recently detected prominent variations in the rotation rate near the base of the convective envelope, with a period of 1.

These are the first indications of dynamical changes close to the presumed site of the global dynamo as the cycle advances. Such a succession of developments from helioseismology provides both a challenge and a stimulus to theoretical work on solar convection zone dynamics.

## Convection, Turbulence, Rotation et Magnétisme dans les Étoiles

Seeking to understand solar differential rotation and magnetism requires three-dimensional simulations of convection in the correct full spherical geometry. However, the global nature of such solutions represents a major computational problem given that the largest scale is pinned, and only 3 orders of magnitude smaller in scale can be represented. Much of the small-scale dynamics in the Sun dealing with supergranulation and granulation are, by necessity, then largely omitted.

The alternative is to reduce the fixed maximum scale by studying smaller localized domains within the full shell and utilizing the 3 orders of magnitude to encompass the dynamical range of turbulent scales. There are clear trade-offs: Both approaches are needed, and the efforts are complementary, as reviewed in detail by Gilman and Miesch Highly turbulent but localized threedimensional portions of a convecting spherical shell are being studied to assess transport properties and topologies of dynamical structures e.

The surface shear layer and the tachocline at the base of the convective zone are indicated as well as the zone covered by our computational domain gray area adapted from Howe et al. Without recourse to direct simulations, the angular momentum and energy transport properties of turbulent convection have also been considered using mean-field approaches to derive second-order correlations the Reynolds stresses and anisotropic heat transport under the assumption of the separability of scales.

Although such procedures involve major uncertainties, the resulting angular momentum transport, which is described by mechanisms such as the so-called effect, have served to reproduce the solar meridional circulation e. Various other states can be achieved by adjusting parameters. Initial studies of convection in full spherical shells to assess effects of rotation with correct accounts of geometry e. We here report on our continuing studies with the anelastic spherical harmonic ASH code Clune et al.

The simulations reported in Miesch et al. Most of the resulting angular velocity profiles in the seven simulations considered have begun to make substantial contact with the helioseismic deductions within the bulk of the solar convection zone.

These possess fast equatorial rotation progradesubstantial contrasts with latitude, and reduced tendencies for rotation to be constant on cylinders. The simulations with ASH have not yet sought to deal with questions of the near-surface rotational shear layer nor with the formation of a tachocline near the base of the convection zone.

These studies have revealed that to achieve fast equators, it is essential that parameter ranges be considered in which the convection senses strongly the effects of rotation, which translates into having a convective Rossby number less than unity for large Taylor numbers.

Such rotationally constrained convection exhibits downflowing plumes that are tilted away from savarf local radial direction, resulting in velocity correlations and thus Reynolds stresses that are found to have a significant role in the redistribution of angular momentum. This seems to provide execice to realize solar-like profiles. Further, it is desirable to impose thermal boundary conditions at the top of the domain that enforce the constancy of emerging flux with latitude in order to be consistent with what appears to be observed.

We wish to focus on two outstanding blot raised by the prior simulations with ASH that need particular savvart concerning the differential rotation established within the bulk of the solar convection zone.

As issue 1, the helioseismic inferences in Figure 1 emphasize that in the Sun appears to decrease significantly with latitude even at midand high latitudes, a property that has been difficult to attain in the prior seven simulations. The substantial latitudinal decrease in angular velocity, say D, in the models is primarily achieved in going from the equator to about 45, with little further decrease in niot at higher latitudes in most of the cases. Whereas the overall latitudinal contrasts from equator to pole in the models and the Sun are roughly of the same order, the angular velocity in the Sun continues to slow down much more as the pole is approached.

Two models, designated as LAM in Miesch et al. Thus, in confronting issue 1, we will search in parameter space for solutions that can achieve profiles in which the decrease with latitude does not taper off at midlatitudes and for which the contrast D is at least comparable to the helioseismic findings. As issue 2, with the convection becoming more turbulent, achieved by decreasing either the thermal or viscous diffusivities, there is a tendency for the latitudinal contrast D in the solutions to diminish or even decrease very prominently, thus being at variance with D deduced from helioseismology.

This behavior appears to blot from increasing complexity leading to a weakening of nonlinear velocity correlations eexrcice have a crucial role in angular momentum redistribution. These Reynolds stress terms are strong in the laminar solutions that involve tilted columnar convection cells banana cells aligned with the rotation axis; they weaken as the flows become more intricate but would be expected to become again significant once coherent structures develop at higher levels of turbulence.

For example, the model TUR in Miesch et al.

## Index of /Exercices/Magnetostatique

As a result, TUR has a fairly interesting angular momentum transport attributed to the nonlinear correlations that sustain a exerfice of differential rotation slightly weaker than LAM, but it too has a considerable variation of heat flux with latitude. The model T2 in Elliott et al.

Absent those features, T2 yielded profiles with a small D and even a slightly slower equatorial rotation rate than that in the midlatitudes. Thus, in confronting issue 2, we seek turbulent solutions that possess profiles with fast equators and strong exwrcice contrasts D and emerging heat fluxes that vary little with latitude.

To achieve this, we have considered two paths in parameter space that yield more turbulent solutions by either varying the Prandtl number or keeping it fixed while maintaining the same rotational constraint as measured by a convective Rossby number.

We describe briefly in x 2 the ASH code and the set of parameters used for the simulations studied here. In x 3 we discuss the properties of rotating turbulent convection and the resulting differential rotation and meridional circulation for the five cases A, AB, B, C, and D.

In x 4 we analyze the transport of angular momentum by several processes and the influence of baroclinic savary in establishing the mean. In x 5 we reflect on the significance of our findings, especially in terms of dealing with the two issues raised by the prior simulations with ASH.

In brief overview, solar values are taken for the heat flux, rotation rate, mass, and radius, and a perfect gas is assumed since the upper boundary of the shell lies below the H and He ionization zones; contact is made with a real solar structure model for the radial stratification being considered.