The Turbulence Budget: What Alpha Must Do to Drain a Disc in Three Million Years, and Why Measurements Refuse

If discs evolve viscously, a single number — the turbulence parameter alpha — must simultaneously set the disc lifetime, the accretion rate onto the star, and the turbulent velocities that observations measure. This paper prices all three demands on the reference disc, and reports where they agree and where they refuse to. The lifetime demand: the viscous time at 50 AU is 7 Myr at alpha = 1e-3 and 72 Myr at alpha = 1e-4, so draining the outer disc within the observed 3-Myr characteristic lifetime requires alpha of 2.4e-03 at 50 AU and 4.8e-03 at 100 AU. The accretion demand is milder: matching a typical T Tauri accretion rate of 1e-8 solar masses per year at 1 AU requires alpha of only 7.9e-04, and alpha = 1e-3 delivers 1.3e-08 solar masses per year — the inner disc's accretion is the one job a 1e-3 turbulence can do. The measurements, however, sit lower still: line broadening bounds alpha at or below about 1e-3, and dust settling prefers 1e-4 to 1e-3 — and at alpha = 1e-4 the accretion rate falls to 1.3e-09 and the outer disc would live 72 Myr. The budget therefore balances nowhere: it is short by a factor of about 2 to 5 against the most generous reading of the turbulence bounds, and by a factor of 10 to 25 against the settling-preferred values. Accretion driven substantially by magnetic winds rather than by the measured turbulence is a conclusion this arithmetic forces, with the size of the gap stated honestly rather than rounded up. Computed from stated assumptions, not measured.