Oblique convergence across the San Andreas Fault in southern California has lead to rapid uplift of the San Gabriel Mountains since 5 Ma. Tomographic models of this region include a sheet-like, high velocity anomaly, extending to depths >200 km. One proposed mechanism for the formation of this feature is gravitational instability of the lithosphere, resulting in a cold thermal downwelling. We present a systematic study of the sensitivity of lithospheric deformation (lithospheric downwelling, topography, deflection of the Moho, convergent velocity profile) to the viscosity structure in 2-D numerical models of lithospheric instability, including a pre-existing zone of weakness. Strike-parallel strain within a convergent region should create a local zone of weakness due to the non-Newtonian response of the lithosphere. Such a weak zone can focus convergent motion, increasing and localizing the growth-rate of a lithospheric instability. We find that the observed characteristics of deformation depend on the width (xw) and relative viscous weakening factor (fw), the ratio of crust to mantle-lithospheric viscosity (ηc/ηm), and the absolute viscosity of the lithosphere (ηm). For the San Gabriel Mountains, we find that a range of models is capable of reproducing the observed Moho deflection and depth extent of the downwelling within the short time since the onset of convergence. However, only a small subset of models, (xw = 20 km, fw = 50--100, ηm = 7.5 ¿ 1020--1.5 ¿ 1021 Pa s, ηc/ηm > 100) reproduce both the narrow, high topography and horizontal shortening profile.