= Solution
With $\mathcal G=0$ and $\mathcal T=2\pi v_mh'\Sigma$, the evolution equation reduces to the radial <continuity equation>
$$
\partial_t\Sigma=-\frac1r\partial_r(rv_m\Sigma),
$$
or, for the mass per unit radius $M_r=2\pi r\Sigma$,
$$
\partial_tM_r+\partial_r(v_mM_r)=0.
$$
The outward mass flux is $F_M=2\pi rv_m\Sigma$. If $\dot M>0$ denotes the inward <accretion rate>, steady mass conservation requires $F_M=-\dot M$. Hence
$$
\boxed{2\pi r v_m\Sigma=-\dot M},
\qquad
\boxed{\Sigma(r)=-\frac{\dot M}{2\pi r v_m(r)}}.
$$
This is positive because the magnetic drift velocity satisfies $v_m<0$.
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