mean molecular mass, $m_{\rm H}$ is the mass of the hydrogen atom, and $$h \equiv H/r = c_{\rm s}/(\Omega r) \simeq 0.054 \left( \frac{r}{{\rm 10 \, au}} \right)^{1/4} \,.$$ We fix $\alpha = 10^{-3}$ for the results in this section. Given these inputs, $\nu = \nu(r) \propto r^1$.

Apart from the mass sink, equation (\[eq:dsig\_dt\]), which combines the 1D mass and momentum equations, is identical to the diffusion equation governing an isolated viscous disc as derived by @lp_1974. What is missing is an explicit accounting for the repulsive Lindblad torque exerted by the planet. Many studies include the planetary torque by introducing, into the momentum equation, a term for the torque per unit radius that scales as ${\rm sgn}(x)/x^4$, where $x \equiv r - r_{\rm p}$ (e.g., @lp_1986; @ld_2006). Compared against 2D hydrodynamical simulations, this $1/x^4$ prescription has been shown in 1D studies to reproduce the azimuthally averaged surface density profiles of repulsive gaps near their peripheries (at $x \gtrsim 4 H$) but not near gap centers (at $x \lesssim 4 H$; @fsc_2014, their section 4.3). In particular the 1D torque density prescription, which assumes angular momentum is deposited locally and neglects wave propagation, fails to recover the flat bottoms 