Geotherms in the boundary layer undulate over the width of the convection cell in Figure 1. The ?20 km of scalloped relief for the T? = 40 K model is potentially detectable by seismic methods. Other effects may indirectly be detectable. The flow pattern may produce mineral fabric. The time variation of temperature along a flow path of a rock is potentially detectable by xenolith studies. I obtain scaling relationships for relief, with forethought beginning with penetration of an upwelling into the lithosphere.
cuatro.step one. Upwelling On Lithosphere
The thermal lithosphere-asthenosphere boundary in the model is gradational. Tractions and velocities normal to horizontal planes vary continuously with depth. I begin by obtaining scaling relationships for vertical velocity with an upwelling (Figure 2). ?.
Importantly, the laterally averaged (root mean square) vertical velocity in the rheological boundary layer VZ scales with Vup. Lateral velocities of rapid flow in the active part of the rheological boundary layer and velocities within downwellings scale as T? ?2 , while the average vertical velocity within the rheological boundary layer and the velocity of sluggish flow at the top of the rheological boundary layer scale as VZ ? T? ?1 .
The newest scaling relationship during the (10) gives a reasonable approximation having requirement for T
I obtain calibrated scaling relationships for relief at the base of the lithosphere, beginning with the inference that laterally average heat flow is constrained. That is, I let model results guide scaling analysis, as a rigorous simple scaling relationship is not evident. From (8), the asthenosphere viscosity at constant heat flow is proportional to T? 4 . My models have the predicted heat flow in (8) of mW m ?2 . I present models with T? = , 50, 65, 80, 100, 120, 150 K. The half-space viscosity of these models retaining extra digits used in the calculations is 0.0405, 0.128, 0.3125, 0.8925, 2.048, 5, , and ? 10 19 Pa s, respectively.
The ratio TL/T? conveniently delineates the domain of stagnant-lid convection. [ Solomatov and Moresi, 2000 ]. I briefly discuss low values of this parameter, as significant scalloping occurs at high values. Figure 4 has TL/T? = () = 10.8, which is within the stagnant-lid regime. The laterally averaged heat flow oscillates before converging to a limit slightly higher than the theoretical prediction (Figure 5). The downwelling is not centered in the model and the dipping streamlines in the rightmost upwelling indicate that convection is time dependent. Otherwise, geotherms are relatively flat away from downwellings. The time dependent behavior of the model with T? of 150 K is more complicated. I discuss it in Appendix B.
I concentrate sub-regime of stagnant-lid convection of low values of T? where significant scalloping occurs. This region of parameter space is inconvenient in physical and numerical experiments. The linear viscosity models in Table 4 of Solomatov and Moresi had TL/T? between and , which is equivalent to T? = 106–67 K for TL = 1300°C. Davaille and Jaupart [1993a] had laboratory TL/T? < which implies T? > 94 K for my parameters.
Figure 1 illustrates a basic feature of stagnant-lid convection: planform with small T? = 40 K (TL/T? = 32.5) differs from that at large T? = 100 K. Isotherms dip more from upwelling to downwelling for T? = 40 K than the célibataires athées latter case. Streamlines for T? = 40 K that penetrate the rheological boundary layer reach their minimum depths near the upwelling and cross dipping isotherms at small angles. Streamlines for T? = 100 K reach their minimum depths near the downwelling and cross isotherms at moderate angles. In both cases, the minimum temperature on a streamline occurs at the lip of the downwelling.