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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We use a common scaling for all these line plots to make an intercomparison between the cases most convenient; the radial cuts ft have been averaged between the northern and southern hemispheres. The depth of the convection zone is therefore. The simulations reported in Miesch et al. The highspeed solar wind and its energetic particles, coronal mass ejections, and explosive flares are all linked to the changing magnetic fields within the extended solar atmosphere.

Such prominent transports serve to resolve issue 2.

Because of the small solar molecular viscosity, direct numerical simulations of the full scale range of motions present in stellar convection zones are currently not feasible. 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.

In the above expressions for both fluxes, the first terms in each bracket are related to the angular momentum flux due to viscous transport which we denote as F r, V and F h, Vthe second term to the transport due to Reynolds stresses F r, R and F h, Rand the third term to the transport by the meridional circulation F r, M and F h, M.

The radiative diffusivity, r, is derived from a one-dimensional solar structure model Brun et al. The number of radial, latitudinal and longitudinal mesh points are N r, N h, and N, respectively.

We should recall, as discussed in detail in Miesch et al. Anelastic MHD Equations In this paper we report three-dimensional numerical experiments designed to investigate the complex MHD of the solar convection zone in spherical geometries. The latitudinal transport of angular momentum F execice in the rightmost of the panels in Figure 11 involves more complicated and sharper variations in latitude.

The time evolution is carried out using an implicit, second-order Crank-Nicholson scheme for the linear terms and an boot, secondorder Adams-Bashforth scheme for the advective and Coriolis terms.

The radial velocity snapshots are shown at three different depths 0. The positive values represent a radial flux that is directed outward and a latitudinal flux directed from north to south. eavart

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

We emphasize that currently tractable simulations are still many decades away in parameter space from the intensely turbulent conditions encountered in the Sun, and thus these large-eddy simulations must be viewed as training tools for developing our dynamical intuition of what might be proceeding within the solar convection zone. It is essential to take into account effects of compressibility on the convection since the solar convection zone spans many density scale heights.

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. These possess fast equatorial rotation progradesubstantial contrasts with latitude, and reduced tendencies for rotation to be constant on cylinders. This is reemphasized in Figure 5, which summarizes the variation of D with P r for our five cases.

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 achieved at higher latitudes in most of the cases. We utilize the same radial profile for that mean eddy diffusivity in our five cases in order to minimize the impact of our SGS treatment on the main properties of our solutions.

Similarly, the low amplitude of the Poynting flux confirms that magnetic processes in case M3 do play a role in the overall energy transport but not to the point of significantly modifying the flux balance established in the nonmagnetic progenitor case H.

The anelastic approximation captures the effects of density stratification without having to resolve sound waves, which would severely limit the time step. 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. This asymmetry translates into a net downward transport of kinetic energy.

A striking property shared by all these temperature fields is that the polar regions are consistently warmer than the lower latitudes, a feature that we will find to be consistent with a fast or prograde equatorial rotation Driving Strong Differential Rotation The differential rotation profiles with radius and latitude that result from the angular momentum redistribution by the vigorous convection in our five simulations are presented in Figure 4.

We saw no evidence of a slow pole developing, but that may well require more extended computations than could be presently arranged. This is not realized in case AB and may contribute to its slow pole behavior. This approach has the advantage that the spatial resolution is uniform everywhere on a sphere when a complete set of spherical harmonics is used up to some maximum in degree retaining all azimuthal orders m in what is known as triangular truncation.

Initial studies of convection in full spherical shells to assess effects of rotation with correct accounts of geometry e.

This strong third cell appears to be of significance in the continuing net poleward transport of angular momentum by the meridional circulations see x 4. Relative to the Sun, convective motions in the planetary interiors are much more influenced by rotation lower Rossby numbers and diffusion lower Reynolds and magnetic Reynolds numbers and much less influenced by compressibility mild density stratification.

The baroclinic term as on right-hand side of eq.

## Index of /Exercices/Magnetostatique

The net energy deficit can be attributed primarily to the reduction in strength of the differential rotation by Maxwell stresses. It is difficult from first principles to predict or explain their overall behavior in terms of the differential rotation and meridional circulations that can be achieved and sustained as we sample different sites in parameter space.

This comes about because of the more intricate latitudinal structure of the different terms contributing to the transport. 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. The computational domain extends from 0: For case C, the corresponding values are 1: We also summarize there the parameters of the five simulation cases.

Since assessing the angular momentum redistribution in our simulations is one of the main goals of this work, we have opted for torque-free velocity and magnetic boundary conditions: The layout of the five cases in Sxvart 4 reflects the two paths we have taken in increasing the complexity or turbulence level in the solutions: Further, in these cases almost all the decrease in with latitude occurs in going from the equator to about 45 exrecice thus is confined to the region outside the tangent cylinder to the inner boundary which intersects the outer boundary in our shell configuration at The differential rotation profile established by the turbulent convection, although strong in contrast, is remarkably smooth; the global-scale magnetic activity is orderly, involving sunspot eruptions with very well-defined rules for field parity and emergence latitudes as the cycle evolves.

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

The seed field is dipole in nature but soon develops a more complicated structure as it is amplified by the convective savarg. We also considered the possibility that the slow pole behavior in case AB may have baroclinic origins.

The variation of with radius and latitude may be best judged in the exercjce contour plots in Figure 4, which are scaled independently for each of the cases; the reference frame rate is also indicated. It should not be mistaken with F r, which is the flux due to radiative diffusion and which operates on the mean Fig.

Evolution of the convection throughout one solar rotation, showing the radial velocity of case D near the exerciec and at the middle of the domain. In order to simplify comparison of our results with deductions drawn from helioseismology Fig. 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. The resulting axisymmetric meridional circulation is maintained by Coriolis forces acting on the mean zonal flows that appear as the differential rotation, by buoyancy forces, bioh Reynolds stresses, and by pressure gradients.

In contrast, cases A and AB show far less alignment of contours with cylinders at the lower latitudes, and at midlatitudes the contours are nearly aligned with radial lines, more in the spirit of the helioseismic inferences. As viewed near the top, the tendency of the convection in our laminar case A to be organized into banana cells nearly aligned with the rotation axis at low latitudes is progressively disrupted by increasing the level of complexity in going in turn to cases AB, B, C, and D.