Convective differential rotation in stars and planets - I. Theory

We derive the scaling of differential rotation in both slowly and rapidly rotating convection zones using order of magnitude methods. Our calculations apply across stars and fluid planets and all rotation rates, as well as to both magnetized and purely hydrodynamic systems. We find shear vertical bar R del Omega vertical bar of order the angular frequency Omega for slowly rotating systems with Omega << vertical bar N vertical bar, where N is the Brunt-Vaisala frequency, and find that it declines as a power law in Omega for rapidly rotating systems with Omega >> vertical bar N vertical bar. We further calculate the meridional circulation rate and baroclinicity and examine the magnetic field strength in the rapidly rotating limit. Our results are in general agreement with simulations and observations and we perform a detailed comparison with those in a companion paper.