GEO 5/6690 Geodynamics 24 Oct 2014 © A.R. Lowry 2014 Read for Wed 29 Oct: T&S Last Time: Brittle-field rheology The “Seismogenic Zone” is observed from:  Seismology (depth distribution of earthquakes) Limited by horizontal load stress if integral of stress is < integral of strength Limited also by rate-state stability transition which occurs at temperatures significantly above the brittle-ductile transition.  Geodesy, from strain that builds where fault patches are coupled or “locked”, and from slow slip events that associate with transitional frictional state  Geology including structural and textural information from exhumed faults  Information derived from lab experiments

Next Journal Article Reading: For Monday Oct 27: Lowry et al. (2000) Dynamic elevation of the Cordillera, western United States, J. Geophys. Res ,371–23,390. Important to think about: Emphasize the parts having to do with flexural strength and rheology (but read the whole thing). This paper is fifteen years old now! What have we learned since and what could we do differently?

Flexural Isostasy Recall from T&S (and maybe a half-dozen other classes you’ve taken) that isostasy is a stress balance resulting in uniform pressure at depth: E.g., this balance of vertical stress: This balance of only the vertical stress is called “Airy isostasy” (after the Airy, 1855, paper on Himalaya) In reality, blocks are not disconnected and we must account also for horizontal stress in the lithosphere. Approximate as flexure of a thin plate with vertical stress “loads” :

The governing equation for flexural isostasy is expressed in terms of a flexural rigidity D and vertical deflection w as where P is a horizontal (in-plane or “tectonic”) stress,  is the difference in density at the surface and in the asthenosphere, g is gravitational acceleration, and q is the vertical stress load. Note (1): This equation assumes a “thin-plate approximation”, in which we neglect depth-dependent compression within the plate resulting from the vertical load stress. This is a reasonable approximation except when wavelengths are short relative to the plate thickness (Comer, JGRAS, 1983) and errors introduced are small relative to those associated with neglecting failure/relaxation).

Note (2): This equation works for either a Cartesian or a spherical geometry (but the eigenfunctions are different). Note (3): If it is a truly perfectly elastic plate over an inviscid fluid, D relates to elastic plate thickness T e as where E is Young’s modulus, is Poisson’s ratio Note (4): If there are two elastic plates with no stress coupling between them (the “leaf-spring” approximation), then D tot = D 1 + D 2 or equivalently

Note (5) If D is uniform (and neglecting P ), solution is simple if the loads and flexure are sines and cosines. Example: let where H I is initial amplitude of height of a topographic load (i.e., before it is reduced by flexure; k = 2  / is wavenumber (& is wavelength);  0 is surface density Which after evaluating derivatives turns out to be: (after we divide both sides by sin(kx).) So letting H T = H I + W T, for every 2D k = (k x, k y ) in FT domain, (Note this is Cartesian!)

Problem: What if loads are both surface and internal? Total isostatic balance includes surface (topographic) mass plus internal mass variations plus lithospheric stress Surface loads are under- compensated by subsurface mass because of flexural strength of the lithosphere Internal loads are under- compensated by surface topographic response If rigidity D and mean profile density of the lithosphere are known, can solve for two unknowns (surface and internal load mass) from two observations (gravity and topography fields)

Separation of loads is useful for: Estimation of lithospheric strength and rheology (parameterized by effective elastic thickness T e ) Understanding processes of mass redistribution in/on the Earth Surface loading processes:Internal loading processes: Erosion/exhumation Deposition Normal faulting (footwall uplift) Reverse faulting (hanging-wall thrust) Volcanic construction Thermal mass variations Compositional mass variations Crustal thickening or thinning by lower crustal flow Cooled igneous intrusions