Σ = \int_{- \infty}^{\infty} ρ d z, F_{M} = 2 π r \int_{- \infty}^{\infty} ρ u_{r} d z,

G = - 2 π r^{2} \int_{- \infty}^{\infty} (Π_{r ϕ} + \frac{B _{r} B _{ϕ}}{4 π}) d z, T = 2 π r [\frac{B _{ϕ} B _{z}}{4 π}]_{- \infty}^{\infty} .

G

positive for outward angular momentum transport in an ordinary Keplerian accretion disk. Vertical integration of mass conservation gives

2 π r \partial_{t} Σ + \partial_{r} F_{M} = 0.

h = r u_{ϕ} = r^{2} Ω

. Multiply the azimuthal equation by

r

, use the continuity equation to put its left-hand side in conservative form, and integrate over

z

. The assumed decay removes the vertical mass and viscous fluxes, whereas the magnetic surface stress remains:

\partial_{t} (2 π r Σ h) + \partial_{r} (F_{M} h) = - \partial_{r} G + r T .

Subtracting

h

times the integrated mass equation yields

F_{M} \frac{d h}{d r} = - \partial_{r} G + r T .

Since

F_{M} = - (d h / d r)^{- 1} (\partial_{r} G - r T)

, substitution in mass conservation gives the required one-dimensional advection-diffusion equation

\partial_{t} Σ = \frac{1}{2 π r} \partial_{r} [(\frac{d r}{d h}) (\partial_{r} G - r T)] .

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