Modi inerziali all’interno di un guscio sferico in rotazione
Monday 20th December 2010
We investigate the properties of inertial modes in a rotating ﬂuid inside a spherical rotating shell as viscosity tends to 0. This problem is tackled linearizing the equations of motion for an incompressible ﬂuid around its static equilibrium, in the rotating reference frame. The study of such modes has applications in many ﬁelds, such as astrophysics (like planetary atmospheres and planetary cores) or geophysics (oceans), and it can also be of interest in engineering (like in the planning of the ﬂuid-ﬁlled spinning spacecrafts). An important parameter is the Ekman number, which is a dimensionless number that measures the ratio of viscous force to Coriolis force. When the Ekman number is very low, the viscous action is eﬀective only on extremely small scales; as a result, many length scales are present in the solution. From the point of view of a numerical resolution of the problem, the discretization must be very ﬁne in order to take into account the lowest scales involved in the solution, and this makes the numeric problem extremely diﬃcult. A graphic analysis of the numerical solutions shows that kinetic energy is concentrated along surfaces that follow the characteristic directions of the attractors of the inviscid hyperbolic problem; therefore, we note the existence of internal shear layers in addition to those of the border. Through numerical simulations we show that the damping rate of inertial modes scales with the cube root of the Ekman number. We study then the proﬁles of the velocity ﬁeld across and along the characteristics and, in particular, we show that the width of the shear layers around attractors also scales with the cube root of the Ekman number.