= Solution
Use a characteristic midplane <temperature> $T$, density $\rho$ and <disk scale height> $H$. For an <ideal gas> of fixed composition, dominant <gas pressure> gives $c_s^2\propto T$, where $c_s$ is the <sound speed>. Vertical <hydrostatic equilibrium> and the <alpha disk> prescription give
$$
H\sim\frac{c_s}{\Omega}\propto\frac{T^{1/2}}\Omega,\qquad
\rho\sim\frac\Sigma H\propto\Sigma\Omega T^{-1/2},\qquad
\nu\sim\alpha c_sH\propto\frac{\alpha T}\Omega.
$$
Numerical factors in the vertical averaging do not affect the exponents. The stipulated <opacity> becomes
$$
\kappa\propto(\Sigma\Omega T^{-1/2})^{-3/2}T^{5/4}
=\Sigma^{-3/2}\Omega^{-3/2}T^2.
$$
For an optically thick, radiatively efficient disk, <radiative diffusion> gives a cooling rate per unit area proportional to $T^4/(\kappa\Sigma)$. The viscous heating rate is proportional to $\nu\Sigma\Omega^2$. Thus
$$
Q^-\propto\Sigma^{1/2}\Omega^{3/2}T^2,\qquad
Q^+\propto\alpha\Sigma\Omega T.
$$
Equating heating and cooling gives $T\propto\alpha\Sigma^{1/2}\Omega^{-1/2}$, so
$$
\nu\propto\alpha^2\Sigma^{1/2}\Omega^{-3/2}.
$$
At fixed $\alpha$ and central mass, $\Omega\propto R^{-3/2}$. Therefore the <gas-pressure alpha disk with inverse-density opacity> satisfies
$$
\boxed{\nu\propto\Sigma^{1/2}R^{9/4}.}
$$
For <thermal stability of an accretion disk>, perturb $T$ at fixed $R$ and $\Sigma$, since the thermal adjustment is faster than radial mass redistribution. The logarithmic heating and cooling slopes are respectively one and two. At equilibrium, a fractional <temperature> increase therefore makes the excess heating $Q^+-Q^-$ decrease, and the disk cools back. Thus \b[the local thermal equilibrium is stable] within the assumed gas-pressure, radiative-diffusion closure.
For <viscous stability of an accretion disk>, use the thermally equilibrated relation between <viscosity> and <surface density>. It gives
$$
\nu\Sigma\propto\Sigma^{3/2}R^{9/4},\qquad
\left.\frac{\partial(\nu\Sigma)}{\partial\Sigma}\right|_R>0.
$$
Locally freezing the background radius in the <Keplerian viscous diffusion equation>, a short radial <Fourier mode> has growth rate $-3k_R^2\partial_\Sigma(\nu\Sigma)<0$. Density enhancements diffuse rather than grow, so \b[the disk is locally viscously stable] as well.
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