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The Tibetan Plateau is a collage of continental fragments that were added successively to the Eurasian plate during the Paleozoic and Mesozoic eras. Paleomagnetic data indicate that these fragments were at southern latitudes during the Paleozoic. The sutures between these microplates are marked by scattered occurrences of ophiolitic material caught up between the crustal  blocks during accretion. From north to south, the main Tibetan crustal blocks are the Kunlun, Songban-Ganzi, Qiangtang, and Lhasa terranes. It is underlain by continental crust about 65 km thick, compared with usual thicknesses of about 30 km. Uplift of the plateau began in the early Miocene and it probably reached its present elevation by about 8 Ma (million years). Three major theories have been proposed for the origin of this immense thickness with many additional minor variations upon them.

In the first model, the crust is thickened by the folding, faulting and subsequent movement of large masses of rocks that are stacked one on top of another like cordwood. The process is like squeezing a block of clay by its ends: what happens is controlled by the rate of squeezing and mechanical behaviour of the clay. At sufficiently high rates of deformation, the clay will break and the resulting multitude of fractures will cause it to thicken in the middle. At slower rates of squeezing, the clay flows plastically, thickening by folding without fractures.

The second theory, continental subduction, entails the wholesale underthrusting of the Indian continental crust beneath the Tibetan Plateau and subsequent uplift.
This process is reminiscent of taking a block of ice and pushing it beneath another ice slab, causing the latter to rise upwards. However, it is difficult to imagine (but not impossible) how the buoyant Indian crust could be kept deep enough to get far beneath the plateau before bobbing to the surface. Perhaps the great speed at which India is colliding to Eurasia allowed this to happen.

The third proposal, continental injection, involves the introduction of Indian crust beneath Tibet as melted rock, called magma. Granitic melts derived from the subducting Indian crust rise into the overlying Eurasian plate and transfer heat into the base of the Tibetan Plateau. The resulting thickened crust is heated by radioactive decay of the elements potassium, uranium, and thorium that are preferentially concentrated in the magmas. Like a hot-air balloon, the heated crust is buoyant and rises with the addition of light granitic material at the bottom of the Eurasian crust increasing the height of the plateau.
The presence of a partially molten zone at the base of the Tibetan Plateau has been documented by seismic experiments. In addition to providing heat to cause uplift, the partially molten zone at the base of Tibet also inhibits the rise of basaltic melts. The ascent of these magmas is driven by differences in density between basaltic magma and the surrounding rocks. While basaltic magmas are lighter than the upper mantle in which they are produced and rise like droplets of oil in water, they tend to stall out when the density difference becomes too small.
Usually, the lower crust is cold and dense, thus promoting the ascent of basaltic magmas, but the hot Tibetan crust acts as a density "filter," stopping the rise of these mafic melts.
This mechanism may explain the high heat flow observed on the plateau and relative depth of mafic volcanic rocks.
 

Suggested readings:

Fielding et al., 1994, How flat is Tibet?: Geology, v. 22, p. 163-167.
Harrison et al., 1992, Raising Tibet: Science, v. 255, p. 1663-1670.
Kong et al., 1997, Evaluating the role of preexisting weaknesses and topographic distributions in the Indo-Asian collision by use of a thin-shell numerical model: Geology, v. 25, p. 527-530.
Molnar, 1989, The geological evolution of the Tibetan Plateau: American Scientist, v. 77, p. 350-359.
Rothery & Drury, 1984, The neotectonics of the Tibetan Plateau: Tectonics, v. 3, p. 19-26.
Tapponier et al., 1982, Propogating extrusion tectonics in Asia: New insights from simple experiments with plasticine: Geology, v. 10, p. 611-616.
Zhao & Morgan, 1985, Uplift of Tibetan Plateau: Tectonics, v. 4, p. 359-369.