= Solution
<Stellar rotation> is a problem of coupled force balance, heat transport and <angular momentum transport>. A <star> need not rotate as a rigid body: in the <Sun>, the convective envelope rotates faster near the equator than near the poles, while much of the radiative interior rotates more nearly uniformly. The <tachocline> is the thin transition near the base of the convection zone. Surface tracking and <Doppler effect> measurements constrain the exterior flow; <solar rotational mode splitting> in <helioseismology> constrains interior <stellar rotation> through integrals of the form $\delta\omega_{n\ell m}\simeq m\int K_{n\ell m}\Omega\,dV$. Multiple kernels are needed, so surface rotation is not a measurement of every depth.
A useful mean <velocity> decomposition is $\mathbf U=\varpi\Omega\mathbf e_\phi+\mathbf v_m$, where $\varpi=r\sin\theta$ is distance from the axis and $\mathbf v_m$ is <stellar meridional circulation>. In a frame of constant angular velocity $\boldsymbol\Omega_0$, use the relative <velocity> $\mathbf V=\mathbf U-\boldsymbol\Omega_0\times\mathbf r$. Its momentum equation includes
$$
\frac{D\mathbf V}{Dt}+2\boldsymbol\Omega_0\times\mathbf V
=-\frac{\nabla p}{\rho}-\nabla\left(\Phi-\frac12\Omega_0^2\varpi^2\right)
+\frac{(\nabla\times\mathbf B)\times\mathbf B}{\mu_0\rho}
+\frac{1}{\rho}\nabla\cdot\boldsymbol\tau.
$$
The <Coriolis force> changes the direction of a parcel's motion but does no work. Thermal driving, release of differential rotational <kinetic energy>, contraction, or external torques supply the energy of the flows. Solid-body <stellar rotation> in a suitable <hydrostatic equilibrium> does not by itself require a meridional flow.
For slowly circulating, approximately axisymmetric material, the leading mechanical balance in an inertial frame is $\nabla p/\rho=-\nabla\Phi+\varpi\Omega^2\mathbf e_\varpi$. Taking its <curl> gives the <stellar thermal-wind balance>
$$
\varpi\partial_z\Omega^2=\frac{(\nabla p\times\nabla\rho)_\phi}{\rho^2}
\simeq\frac{g}{rc_p}\partial_\theta S.
$$
Here $z$ is the axial coordinate, $S$ is specific <entropy> and $c_p$ is the specific heat at fixed <pressure>. The last form uses approximately radial <hydrostatic equilibrium> and uniform composition. If the material is <barotropic>, the right-hand side vanishes and angular velocity is constant on cylinders. Small latitudinal <entropy> differences can instead support the observed noncylindrical <differential rotation>. Stresses and inertial terms modify this leading balance. The equation relates shear to <baroclinicity>; it does not give the circulation speed or explain how the shear is maintained.
That second question requires an angular-momentum budget. With $j=\varpi^2\Omega$ and density-weighted mean quantities, <conservation of angular momentum> can be written schematically as
$$
\partial_t(\rho j)+\nabla\cdot(\rho\mathbf v_m j+\mathbf F_J)=0,
$$
$$
\mathbf F_J=\rho\varpi\langle u'_\phi\mathbf u'_m\rangle
-\frac{\varpi}{\mu_0}\langle B_\phi\mathbf B_m\rangle
-\rho\nu\varpi^2\nabla\Omega.
$$
The three contributions are <Reynolds stress>, magnetic <Maxwell stress tensor> and <viscosity> transport. Appropriate density weighting replaces the simple correlations in a strongly compressible average. Since $\nabla\cdot(\rho\mathbf v_m)=0$ in a steady star, the stationary equation gives <gyroscopic pumping in a star>:
$$
\boxed{\rho\mathbf v_m\cdot\nabla j=-\nabla\cdot\mathbf F_J.}
$$
A torque that maintains <differential rotation> therefore drives a meridional response unless another stress cancels it. Conversely the circulation transports <angular momentum> and feeds back on the rotation. An assumed meridional pattern without its driving torque is not a closed dynamical model.
In the convection zone, outward heat transport drives <convection>. Rotation deflects the rising and falling parcels and makes their correlations anisotropic. <Nondiffusive convective angular momentum transport> can drive shear even from an initially uniform rotation rate; treating every <Reynolds stress> as a positive eddy <viscosity> would only smooth shear and miss its source. A limiting parcel that conserves $j$ when it moves outwards slows relative to uniform rotation, whereas rapid rotational constraint and correlated horizontal motions change the sign and direction of the net transport. Thus the solar equator-fast pattern requires the actual convective correlations, not a universal parcel argument. The <Rossby number> $u_c/(2\Omega\ell_c)$ compares convective inertia with rotational deflection: deep, slower, larger-scale <convection> is more rotationally constrained than the fastest photospheric motions. Turnover times $\ell_c/u_c$ range from minutes near the visible surface to weeks or longer at greater depth. On these times the <Reynolds stresses> can redistribute angular momentum and drive circulation, while the star's total spin changes much more slowly.
In a radiative region, rotation distorts <equipotential surfaces>. Under uniform rotation and uniform composition, leading <hydrostatic equilibrium> makes pressure, density and temperature constant on each such surface, but effective gravity varies along it. <Radiative diffusion> then produces a flux $\mathbf F=f(\Phi_{\rm eff})\nabla\Phi_{\rm eff}$. Its divergence contains $f'|\nabla\Phi_{\rm eff}|^2$, which varies over a surface and generally cannot match a nuclear source constant there. This is the <radiative-equilibrium obstruction in a rotating barotropic star>. Thermal imbalance drives <Eddington-Sweet circulation>, with heat advection entering
$$
\rho T\left(\partial_t S+\mathbf v_m\cdot\nabla S\right)
=\rho\epsilon_{\rm nuc}-\nabla\cdot\mathbf F+\text{dissipative heating}.
$$
Its global thermal estimate is
$$
\epsilon_\Omega\sim\frac{\Omega^2R^3}{GM},\qquad
\boxed{t_{\rm ES}\sim\frac{t_{\rm KH}}{\epsilon_\Omega},\quad t_{\rm KH}\sim\frac{GM^2}{RL}.}
$$
A rotational thermal imbalance of order $\epsilon_\Omega L$ processes the thermal reservoir on this longer time. For the slowly rotating <Sun>, $t_{\rm KH}$ is of order $3\times10^7$ years and $\epsilon_\Omega$ of order $2\times10^{-5}$, giving $t_{\rm ES}\sim10^{12}$ years, much longer than the solar age. Stable composition gradients impede the flow further. Thus classical radiative <Eddington-Sweet circulation> cannot by itself establish the Sun's interior rotation rapidly. In a rapidly rotating star the same estimate is shorter and rotational mixing can compete with its evolutionary lifetime. Torque-driven convection-zone circulation and radiative thermal circulation have different controlling balances.
<Magnetic fields> introduce a further coupling. A <poloidal magnetic field> linking different layers is wound by differential rotation into a <toroidal magnetic field>, roughly $\partial_tB_\phi\simeq\varpi\mathbf B_p\cdot\nabla\Omega$. The resulting <Maxwell stress tensor> transport angular momentum. Communication along a field occurs on an <Alfvén speed> crossing time $t_A\sim\ell\sqrt{\mu_0\rho}/B$, which can be far shorter than molecular diffusion if a coherent connecting field exists. In steady ideal axisymmetric induction, absence of continuing winding requires $\mathbf B_p\cdot\nabla\Omega=0$; this is constancy along the field, not automatically solid-body rotation throughout the star. The topology and strength of an interior field matter. <Solar dynamo> action, especially the interaction of shear with poloidal-field regeneration, couples <stellar rotation> to the magnetic cycle. Magnetic feedback can change shear and circulation on dynamical or cycle times; an approximately eleven-year activity cycle and twenty-two-year magnetic polarity cycle are distinct from secular spin-down.
A magnetized <stellar wind> also supplies an external torque. It enforces approximate corotation out to an effective <Alfvén radius> $R_A$, giving <wind-driven magnetic braking of a solar-type star> $\dot J\sim-\dot M_w\Omega R_A^2$. Its spin-down time is $t_J\sim I/(\dot M_wR_A^2)$, much longer than a convective turnover time. For slowly changing moment of inertia and an unsaturated field scaling that gives $\dot J=-K\Omega^3$, integration gives $\Omega^{-2}=\Omega_0^{-2}+2Kt/I$, the <Skumanich rotation law>. A growing or contracting star also changes $I$ and can spin up or down even without torque. Differential spin-down can create shear between layers unless internal transport couples them.
Finally, shear can excite <Kelvin-Helmholtz instability>, and convection can launch <internal gravity waves> into the stable radiative interior. Wave damping deposits angular momentum; the net transport depends on excitation and selective absorption and is not fixed by a wave-crossing time alone. Such stresses, magnetic coupling and circulation are candidate agents for the nearly uniform radiative rotation and the narrow <tachocline>. Molecular <kinematic viscosity> has a diffusion time $\ell^2/\nu$ generally too long to couple the entire solar interior over its age, whereas a turbulent effective <kinematic viscosity> can act much faster but is not appropriate to every stable layer. A circulation has an advection time $\ell/U_m$, thermal diffusion has time $\ell^2/\chi$, and magnetic diffusion has time $\ell^2/\eta_m$; these must be compared with the local <stellar rotation> and evolutionary times rather than conflated with them. The solar dynamical time $\sqrt{R^3/(GM)}$ is about half an hour, its rotation period is of order a month, and secular magnetic braking takes an evolutionary time. \b[Rapid force adjustment, convective maintenance of shear, cyclic magnetic feedback and slow global spin evolution are separate processes.] Their coupled conservation equations and timescale hierarchy are the basis of a consistent account of solar rotation.
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