= Solution
Use characteristic, vertically averaged quantities at fixed radius. Let $H$ be a disk thickness, $\rho\sim\Sigma/H$, and let the <alpha disk> prescription give $\mu\sim\alpha p/\Omega$, up to a convention-dependent constant. From the preceding <radiation pressure> and constant-<opacity> balance, $\mu\sim c/\kappa$ is independent of $\Sigma$; thus
$$
p\sim\frac{c\Omega}{\alpha\kappa}.
$$
Vertical <hydrostatic equilibrium> gives $p\sim\rho\Omega^2H^2\sim\Sigma\Omega^2H$, hence
$$
H\sim\frac{c}{\alpha\kappa\Omega\Sigma}\propto\Sigma^{-1}.
$$
The vertically averaged <kinematic viscosity> satisfies $\bar\nu\Sigma=\int\mu\,dz\sim\mu H$. Consequently
$$
\boxed{\bar\nu\Sigma\sim\frac{c^2}{\alpha\kappa^2\Omega}\,\Sigma^{-1}.}
$$
At fixed radius the omitted factors are independent of $\Sigma$, so this is the requested inverse-density scaling of the <viscous torque in an accretion disk>.
This is an order-of-magnitude vertical model, as requested. Imposing both exact pointwise $\mu=\text{constant}$ and exact pointwise $\mu\propto p(z)$ would force $p$ to be constant, contrary to nontrivial vertical <hydrostatic equilibrium>. The characteristic-pressure prescription supplies a scaling, not an exact simultaneous vertical solution to all those idealized equations.
Back to article page
