DISK WINDS, JETS, AND OUTFLOWS: THEORETICAL AND COMPUTATIONAL FOUNDATIONS Ralph E. Pudritz 1 , Rachid Ouyed 2 , Christian Fendt 3 , and Axel Brandenburg 4 1 Dept. of Physics & Astronomy, McMaster University, Hamilton, ON L8S 4M1, Canada 2 Dept. of Physics & Astronomy, University of Calgary, Calgary, AB T2N 1N4, Canada 3 Max Planck Institute for Astronomy, K¨ onigstuhl 17, D-69117 Heidelberg, Germany 4 Nordita, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark We review advances in the theoretical and computational studies of disk winds, jets and outflows including: the connection between accretion and jets, the launch of jets from magnetized disks, the coupled evolution of jets and disks, the interaction of magnetized young stellar objects with their surrounding disks and the relevance to outflows, and finally, the link between jet formation and gravitational collapse. We also address the important predictions of the theory on jet kinematics, collimation, and rotation, that have recently been confirmed by high spatial and spectral resolution HST observations. Finally, we show that disk winds have a universal character that can account for jets and outflows during the formation of massive stars as well as brown dwarfs. 1. INTRODUCTION The close association of jets and outflows with protostel- lar accretion disks is one of the hallmarks of the accretion picture of low mass star formation. The most energetic out- flow phase occurs during gravitational collapse of a molec- ular cloud core - the so-called Class 0 phase - when much of its envelope is still raining down onto the forming proto- stellar disk and the disk accretion rate is high. Later, in the T-Tauri star (TTS) stage, when most of the original core has been accreted and the young stellar object (YSO) is being fed by lower accretion rates through the surrounding Kep- lerian accretion disk, the high-speed jet becomes optically visible. When the disk disappears in the weak-lined TTS (WTTS) phase, the jet goes with it. The most comprehensive theoretical picture that we have for these phenomena is that jets are highly collimated, hy- dromagnetic disk winds whose torques efficiently extract disk angular momentum and gravitational potential energy. Jets also sweep up ambient molecular gas and drive large scale molecular outflows. A disk wind was first suggested as the origin of jets from accretion disks around black holes in the seminal paper by Blandford and Payne (1982, BP82), and was soon proposed as the mechanism for protostellar jets (Pudritz and Norman, 1983, 1986). Several major observational breakthroughs have taken place in the study of jets and outflows since PPIV (held in 1998). Direct, high resolution spectro-imaging and adap- tive optics methods discovered the rotation of protostellar jets (e.g. Bacciotti et al., 2003). These observations also re- vealed that jets have an onion-like, velocity structure (with the highest speeds being closest to the outflow axis). This body of work provides strong support for the idea that jets originate as centrifugally driven MHD winds from extended regions of their surrounding disks (see chapter by Ray et al.). Recently, outflows and disks have also been discovered around massive stars (see chapter by Arce et al.) as well as brown dwarfs (eg. Bourke et al. 2005) implying that the mechanism is of importance across the entire stellar mass spectrum. Major advances in the theoretical modeling of these sys- tems have also occurred, due primarily to a variety of MHD computational studies. Simulations can now resolve the global evolution of disks and outflows, track the interaction of disks with central magnetized stars, and even follow the generation of outflows during gravitational collapse. This body of work shows that jets and disks are closely coupled and that the basic dynamics of jets scales to YSOs of all masses. Our review examines the theory of the central engine of jets and its exploration through the use of computer simu- lations. We focus mainly on developments since PPIV and refer to the review by K¨ onigl and Pudritz (2000, KP00) for a discussion of the earlier literature, as well as Pudritz (2003) and Heyvaerts (2003) for more technical background. We first discuss the basic theory of disk winds and their kine- matics (§2). We review computational studies of jets from accretion disks treated as boundary conditions (§3), as well as global simulations including the disk (§4). We then ex- amine the innermost regions of the disk where the stellar magnetosphere interacts with the disk, as well as the sur- face of the star that may drive an accretion-powered outflow (§5). Finally, we discuss how outflows are generated during the early stages of the gravitational collapse (§6). 2. THEORY OF DISK WINDS An important insight into the nature of the engine for jets can be gleaned from the observed ratio of the momen- tum transport rate (or thrust) carried by the CO molecular 1