MAGNETOCENTRIFUGALLY DRIVEN FLOWS FROM YOUNG STARS AND DISKS .2. FORMULATION OF THE DYNAMICAL PROBLEM

We formulate the dynamical problem of a cool wind centrifugally driven from the magnetic interface of a young star and an adjoining Kepleria n disk. We examine the situation for mildly accreting T Tauri stars th at rotate slowly as well as rapidly accreting protostars that rotate n ear break-up. In both cases a wind can be driven from a small X-region just outside the stellar magnetopause, where the field lines assume a n open geometry and are rooted to material that rotates at an angular speed equal both to the local Keplerian value and to the stellar angul ar speed. Assuming axial symmetry for the ideal magnetohydrodynamic fl ow, which requires us to postpone asking how the (lightly ionized) gas is loaded onto field lines, we can formally integrate all the governi ng equations analytically except for a partial differential equation t hat describes how streamlines spread in the meridional plane. Apart fr om the difficulty of dealing with PDEs of mixed type, finding the func tional forms of the conserved quantities along streamlines-the ratio b eta of magnetic field to mass flux, the specific energy H of the fluid in the rotating frame, and the total specific angular momentum J carr ied in the matter and the field-constitutes a standard difficulty in t his kind of (Grad-Shafranov) formalism. Fortunately, because the ratio of the thermal speed of the mass-loss regions to the Keplerian speed of rotation of the interface constitutes a small parameter epsilon, we can attack the overall problem by the method of matched asymptotic ex pansions. This procedure leads to a natural and systematic technique f or obtaining the relevant functional dependences of beta, H, and J. Mo reover, we are able to solve analytically for the properties of the fl ow emergent from the small transsonic region driven by gas pressure wi thout having to specify the detailed form of any of the conserved func tions, beta, H, and J. This analytical solution provides inner boundar y conditions for the numerical computation in a companion paper by Naj ita & Shu of the larger region where the main acceleration to terminal speeds occurs.