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 $|R\nabla\Omega|$ of order the angular frequency $\Omega$ for slowly-rotating systems with $\Omega \ll |N|$, where $N$ is the \brvs\ frequency, and find that it declines as a power-law in $\Omega$ for rapidly-rotating systems with $\Omega \gg |N|$. 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.