Gyroscopic pumping in a star ($\rho {\mathbf{v}}_m\cdot \nabla ({\varpi }^2\Omega )=-\nabla \cdot {\mathbf{F}}_J$)

In a steady axisymmetric star, a torque due to Reynolds stress, Maxwell stress tensor or viscosity must balance advection of specific angular momentum by stellar meridional circulation. Writing the nonadvective angular momentum transport flux as ${\mathbf{F}}_J$ gives the displayed balance. Thus stresses that maintain differential rotation also drive a meridional response. The stellar thermal-wind balance supplies a separate leading force constraint, not a replacement for this torque equation.