Simple Thermal Isostasy
GS 388 Handout on Thermal Isostasy 60.0 0.5 1.00.0normalized temperature = (T-To)/(Ta-To)0.890.5linear approximation1.01.13error function solution1.5znormalized depth =2 βt2.0Suppose we define the thermal "thickness" of the lithosphere, Zo, as the depth at whichthe temperature reaches a certain large fraction of the temperature of the asthenosphere (Ta), say89%; that is, T(z=Zo)/(Ta-To)=0.89. This means that the argument of the erf function must havethe specific value 1.13. There is nothing magic about 89% in terms of the mechanical properties ofthe material, but it will allow us to make a simplification of the temperature profile that preserves itsessential characteristics, as discussed below. The transition from the strong material carried alongwith the lithosphere and the weak, fluid-like material in the asthenosphere is thought to occur at atemperature which is a large fraction of Ta, but the exact value is not very well known. 89% isprobably not a bad guess. The main point is that by choosing a particular value for this ratio, wethen set the value for the argument of the erf function and so define a particular relationshipbetween Zo and t. Zo will thus be a measure of the "thickness of the lithosphere" as the thicknessof the layer whose bottom reaches 89% of the asthenospheric temperature, Ta. Thus,Z o = 1.13 2 βt = 2.26 βtgives a convenient measure of the "thickness" of the lithosphere, Z o , as a function of time, t.
GS 388 Handout on Thermal Isostasy 7The thermal contraction due to the cooling during the time t is obtained by integrating theeffect in a small depth range, dz, from the surface on down through the column. We did this in thehandout on "thermal isostasy". The change of the height of an element of the column, dh', due toa change in temperature in the cooled section compared to the initial section will be given by:anddh' = [Ta-T(z,t)] • (α) • (dz)h' =z=∞(Ta - T(z,t)) α dzz=0where α = coefficient of thermal expansion and h' is the decrease in height of the column due tothermal contraction alone. Now we can simplify this integration if we replace the curvilinearform of the erf function by a simple linear function as shown in in the figure. This replacementpreserves certain important characteristics of the problem, even though it over simplifies the exactshape of the temperature profile (note that the linear approximation is not a bad one, though). If wedefine Zo as we did above to give an isotherm at the "base of the lithosphere" at 89% of theasthenosphere temperature, and draw the linear profile as in the figure above so that it reaches theasthenosphere temperature at the depth Z o , then the integration of this linear function will give theexact same answer for h' as we would get by integrating the "right" expression for T(z) with theerf function. This is in fact the rationale for choosing the seemingly abitrary value of 89%. Itprovides a simple but quantitatively accurate way to replace the erf function with a simple linearprofile, one that is easy to plot and think about, as demonstrated in the handout on thermalisostasy. The linear relationship between temperature and depth is given byT(z) = To + (z/Zo)(Ta-To)T(z) = Tafor z < Zo, andfor z > Zoand the integral is easy to do, giving the contraction, h', ash' = α (Ta - To) (Zo/2)Note that the information from the erf function solution is contained in the definition of Zo, whichalso relates depth, time and the coefficient of thermal diffusivity, β, so the essential physics ispreserved in this approach.A result of the thermal contraction, however, is that additional water is added to thecolumn. Let h* be the height change due only to the isostatic response of the added water , and letD o be the depth of the ocean at the oceanic ridge ( i.e., the column at t = 0 ). The mass (per unitarea) added to a column located off the ridge at a depth D > Do is (D-Do)ρw. This added massmust be compensated by removal of an appropriate thickness, h*, of asthenospheric material,resulting in an isostatic subsidence h*. The mass balance givesh* • ρm = (D-Do) • ρw.where ρ m is the density of the asthenosphere and ρ w is the density of the ocean water. The depthof the ocean is thus given by the combination of h', the thermal contraction effect, and h*, theisostatic effect of the extra water:D-Do = h' + h* = h' + (D-Do)(ρw/ρm) , or, solving for D-Do,