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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The magnetic field topology generally does not exhibit any clear symmetries about the equator, although some of the B patches at low latitudes do have an antisymmetric counterpart. The effects of baroclinicity thermal wind have been found to have a modest role in the resulting mean zonal sabart.

Thus, breaking the Taylor-Proudman constraint that requires rotation to be constant on cylinders, equal to zero, can be achieved by establishing a latitudinal entropy gradient. We have investigated the impact of such magnetic energy leakage on the dynamo action by computing one case in which the magnetic field was required to be purely radial at the boundaries, yielding.

Further, since the fluid is electrically conducting, currents will flow and magnetic fields must be built. LessieurPope The interface dynamo paradigm is thus based on the following underlying processes or building blocks: This is a well-known difficulty in dynamo simulations within astrophysical or geophysical contexts see, e.

ESAGilman, P. Figure 3 provides an overview of radial velocity snapshots in our five simulations at three depths near the top, middle, and bottom accompanied by the fluctuating temperature fields at middepth.

This first process is often referred to as the! Thus, here too the Reynolds stress terms are significant players in the overall balance. Our five simulations have shown that there is some variety in the meridional circulations achieved, all of which involve multicelled structures.

The single-cell behavior there for case AB appears to enable more effective extraction of angular momentum by Reynolds stresses from the high to the low latitudes, thereby yielding a distinctive rotational slowing of the high latitudes. The typical amplitudes in these large-scale circulations are about 20 m s 1 and are thus comparable to the values deduced from local domain helioseismic probing of the uppermost convection zone based on either time-distance.

These general properties are shared by our five cases, all of which have achieved good overall flux balance with radius, as can be assessed by examining F t.

## Index of /Exercices/Magnetostatique

Others believe that the poloidal field is regenerated by the cumulative action of many small-scale cyclonic turbulent motions on the field throughout the convection zone, rather than just close to the surface e. As stated earlier, we have introduced a seed magnetic poloidal field into our hydrodynamical case H for three different values of the magnetic diffusivity, corresponding to cases M1, M2, and M3 Table 1.

There is some evidence of a latitudinal variation in the photospheric temperature of at least a few kelvins with the same sense obtained from observations of the solar limb see, e. This may be key in the monotonic decrease of with latitude of case AB extending into the polar regions and provides our first clue for how issue 1 is resolved within this case.

This leads to a nonzero electromagnetic flux through the boundaries. This overall dynamical system of turbulent convection is sufficiently complex that we are uncertain whether there may exist multiple basins of attraction leading to different classes of differential rotation. 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.

This would be expected since the buoyancy driving has strengthened relative to the dissipative mechanisms as measured by the increasing Rayleigh number R a Table 1. The baroclinicity in our solutions, resulting from latitudinal heat transport that sets up a pole-to-equator temperature and entropy contrast, contributes to not being constant on cylinders, but it is not the dominant player as envisioned in some discussions of mean-field models of solar differential rotation e.

### Index of /Exercices/Magnetostatique

However, even this path yields a turbulent solution eercice D whose D is still close to the helioseismic contrast, thus largely resolving issue 2. This is reemphasized in Figure 5, which summarizes the variation of D with P r for our five cases. 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.

Match to an external potential magnetic field at top and bottom: The horizontal structure of the radial and longitudinal buot field is also shown in Figure 4. It is appropriate to consider if these represent truly spun-up solutions in a statistical sense and further biott several distinctive profiles could be achieved for the same control parameters.

In this spirit, there has been substantial theoretical progress recently in trying to understand how the differential rotation profiles deduced from helioseismology may be established in the bulk of the convection zone. The dynamo-generated magnetic fields exhibit a complex structure and evolution, with radial fields concentrated in downflow lanes and toroidal fields organized into twisted ribbons that are extended in longitude and achieve field strengths of up to G.

Highly turbulent but localized threedimensional portions of a convecting spherical shell are being studied to assess transport properties savsrt topologies of dynamical structures e. However, Lorentz forces in localized regions of case M3 do have a noticeable dynamical effect, particularly with regard to the evolution of strong downflow lanes where magnetic tension forces can inhibit vorticity generation. Fluctuations about this mean state are denoted by, P, T, and S.

These differences between temperature and entropy are accounted for by effects of the pressure field necessary to drive exercicd meridional circulation. As to the relative maturity of the spun-up states shown in Figure 4, these vary from case to case because of the rapidly increasing computational expense in dealing with the finer spatial resolution required by the more complex simulations.

The latitudinal entropy and temperature gradients established within our simulations should li examined further. The highspeed solar wind and its energetic particles, coronal mass ejections, and explosive flares are all linked to the changing magnetic fields within lki extended solar atmosphere. Our principal clues come from Figure 11, where we find that only in case AB is the Reynolds stress component of the net radial angular momentum flux F r, R through shells at various radii uniformly directed outward.

With these new models, we have focused on two fundamental issues raised in comparing the solar differential rotation deduced from helioseismology with the profiles achieved in the prior threedimensional simulations of turbulent convection with the ASH code Miesch et al.

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

For very large problems, the Legendre transformations dominate lki work load, and as a result, great care has been taken to optimize their performance on cache-based architectures. As issue 2, there was a tendency for D to diminish or even decrease sharply within the prior simulations as the convection became more turbulent, yielding values of D that were becoming small compared to the helioseismic deductions.

This can result from suitable correlations in velocity and thermal structures yielding a latitudinal heat flux that may produce substantial entropy variations at the higher latitudes, biof leading to greater baroclinic contributions in equation 11 that define the variation of mean zonal velocity.

Case AB in Figure 4 is unique in having the monotonic decrease of with latitude continue onward to high latitudes, which is also the trend deduced from helioseismic measurements.

At low Reynolds numbers the transition between equatorial modes and polar modes occurs near the tangent cylinder. We accomplish this through a series of threedimensional numerical simulations of MHD convection within rotating spherical shells using our anelastic spherical harmonic ASH code on massively parallel supercomputers.

Dark tones in turn represent downflow, inward, and westward fields, with the ranges for each color table indicated. These yield both v r v and v v correlations necessary for the Reynolds stress contributions to the radial and latitudinal angular momentum fluxes analyzed in Figure These studies of the global dynamics of the solar convection zone concentrate on how the differential rotation and meridional circulation are established.

Thus, issues concerning the tachocline are not dealt with, including the downward transport of magnetic fields -effect into this region where strong toroidal fields may be stretched into existence. Such downward transport of angular momentum is well compensated by the two other terms F r, R and F r, M, having reached a statistical equilibrium of nearly no net radial flux, as can be seen by noting that the solid curve F r is close to zero.