The effect of the equatorially symmetric zonal winds of Saturn on its gravitational field

The penetration depth of Saturn’s cloud-level winds into its interior is unknown.A possible way of estimating the depth is through measurement of the effect of the winds on the planet’s gravitational field.We use a self-consistent perturbation approach to study how the equatorially symmetric zonal winds of Saturn contribute to its gravitational field.An important advantage of this approach is that the variation of its gravitational field solely caused by the winds can be isolated and identified because the leading-order problem accounts exactly for rotational distortion,thereby determining the irregular shape and internal structure of the hydrostatic Saturn.We assume that（i）the zonal winds are maintained by thermal convection in the form of non-axisymmetric columnar rolls and（ii）the internal structure of the winds,because of the Taylor-Proundman theorem,can be uniquely determined by the observed cloud-level winds.We calculate both the variation △J_n,n=2,4,6...of the axisymmetric gravitational coefficients J_n caused by the zonal winds and the non-axisymmetric gravitational coefficients △J_（nm） produced by the columnar rolls,where m is the azimuthal wavenumber of the rolls.We consider three different cases characterized by the penetration depth 0.36 R_S,0.2 R_S and 0.1 R_S,where R_S is the equatorial radius of Saturn at the 1-bar pressure level.We find that the high-degree gravitational coefficient （ J_（12）＋△J_（12）） is dominated,in all the three cases,by the effect of the zonal flow with ｜△J_（12）/J_（12）｜〉100%and that the size of the non-axisymmetric coefficientsdirectly reflects the depth and scale of the flow taking place in the Saturnian interior.