The Effect of the Convection Zone on Solar Spin-Down

A gradual reduction in the rotation of the solar surface layers induces a slow circulation in the convec- tion zone. This circulation pumps fluid into the interior and gives rise to spin-down currents which are much more intense than those resulting from a solar Ekman layer. The process is calculated here with a simple model representing the convection zone by a porous medium and neglecting density stratifica- tion. When the angular velocity of the boundary of a rigidly contained, rotating fluid is altered, surface stresses are created which drive circullations in the interior of the fluid, thus modifying its internal angular velocity (Einstein 1949; Bondi and Lyttleton 1948; Charney and Eliassen 1949; Prandtl 1952; Greenspan and Howard 1963). It has been suggested that this process, called "spin-down" by Greenspan and Howard (1963), operates in the Sun (Howard, Moore, and Spiegel 1967) as a result of the braking of the surface angular velocity by the solar-wind torque. (The braking process is discussed by Schatzman 1962; Brandt 1966; Weber and Davis 1967; and Mestel 1967.) In the picture of Howard el al., the convection zone is treated as an essentially rigid container which, in slowing down, exerts stress on the core boundary, thus setting up an Ekman boundary layer, as in the usual spin-down process. Indeed, the possibility was raised that the base of the convection zone can exert stress on the radiative core even more effectively than a rigid container can, because it would entrain fluid from the core. On the other hand, Dicke (1967) has argued that "an Ekman layer would not form in the Sun." In any case, an Ekman layer in the Sun might be as thin as 30 m (Howard et cii. 1967) and would be rather irregular, so that it seems desirable to inquire further into the driving mechanism for the solar spin-down process. The purpose of this Letter is to point out that spin-down can occur in the Sun even if no Ekman layer forms at all, since the convection zone itself can drive a circulation analogous to the Ekman circu- lation. We discuss here a model in which a spherical cavity is surrounded by a rotating shell of porous solid, and the whole system is filled with a homogeneous, incompressible fluid. Viscosity is neglected within the cavity, and Darcy's law is satisfied in the porous medium; i.e., in the porous shell, (1