Simple Thermal Isostasy

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1 GS 388 Handout on Thermal Isostasy 1 SIMPLE THERMAL ISOSTASY...1 COOLING OF SUB-OCEANIC LITHOSPHERE...3 APPENDIX: BASIC HEAT FLOW EQUATION...9 Simple Thermal Isostasy This section develops the very simplest view of thermal isostasy, and illustrates the application of the coefficient of thermal expansion. The next section explains the application of one dimensional heat diffusion to the cooling of the sub-oceanic lithosphere. The final section, as an appendix, gives a brief derivation of the heat diffusion equation. In the highly simplified diagram below the 1300 C isotherm is elevated by a vertical distance Z. The geotherms at the two locations 1 and 2 indicated in the section are shown below. The geotherm is approximated as a simple linear conduction profile through the "lithosphere" and a constant temperature in the "asthenosphere". profile 1 profile 2 h crust Moho mantle lithosphere 1300 C asthenosphere Z = amount the lithosphere is "thinned" Z

2 GS 388 Handout on Thermal Isostasy 2 The differences in temperature between the two profiles imply a thermal expansion of the material in the plateau section. This can be calculated for a small element, dz, of a unit area vertical column which has its temperature increased by T, as follows: or, integrating over the whole column, dh = α T dz z=zo h= α T 2 -T 1 z=0 dz where T2and T1 are the two temperature profiles shown in the figure below, α is the coefficient of thermal expansion and Ta is the temperature of the asthenosphere. The integration can be broken up to two simpler ones, equivalent to computing the thermal expansion of the two profiles separately relative to a zero thickness lithosphere (Ta at all depths), so that z = Zo- Z h= α T 2 z=0 z=zo dz α T 1 z=0 dz and evaluated to give h=α Z (Ta -To) 2 profile 1 profile 2 T2(Z) T1(Z) Z depth depth

3 GS 388 Handout on Thermal Isostasy 3 Note that variations of α with depth are ignored. Estimates of α the upper mantle/crust of 3 x 10-5 cgs, Ta = 1300 C, and To = 0 give h ~ (0.02) Z where h and z are measured in the same units. Thus a 100 km rise in the 1300*C isotherm, equivalent to a 100 km thinning of the lithosphere, implies an uplift of 2 km. The basis for this very simple view of thermal isostasy is developed in the next section. Cooling of Sub-oceanic Lithosphere Sub-oceanic lithosphere begins to form as it moves away on either side of the spreading ridge. The plate loses heat, cools and thickens as it moves away from the spreading ridge. The cooling can be approximated by a very simple one dimensional model of a uniform half-space with a fixed surface temperature. This one-dimensional model has only vertical heat flow and vertical temperature variation. We start with the half space at a uniform temperature, Ta, and then expose the suface to a lower temperature, To, which will then be held fixed in time. This starting temperature throughout the medium will correspond to the asthenosphere temperature, while the fixed surface temperature will approximate the temperature of the ocean bottom. We thus model the cooling of a vertical column of material as it moves away from the oceanic ridge. The model neglects the lateral flow of heat from one column to the next, which is assumed small in comparison to the vertical flow. spreading ridge T = To t = 0 t > 0 T = To T = Ta T = T(z) As the column cools, it contracts by an amount governed by the coefficient of thermal expansion, α, of the material. The thermal contraction will deepen the ocean and thus increase the thickness of the water column. The isostatic effect of the added water will further deepen the

4 GS 388 Handout on Thermal Isostasy 4 ocean. The combined effects give a relation between ocean depth and the time since the column started to cool, i.e. the age of the ocean floor. Let T(z,t) = temperature as a function of depth, z, beneath the ocean floor, and time, t;. To = T(z=0) = fixed surface temperature at t = 0 and all times later. Zero time is when the column appeared along the ocean ridge system. Ta is the temperature of the asthenosphere, taken in the simple half-space model to be the temperature throughout the half space at time t = 0. The temperature is constant through the half space only at t = 0. The one-dimensional heat diffusion equation must be solved to find out how the temperatures vary with depth and time after zero time as a given column moves away from the ridge. The surface temperature To (< Ta) is suddenly applied at zero time and then held fixed. The solution (see Turcotte and Schubert for derivation) for T(z,t) is as follows: T(z,t) = T 0 +(T a -T 0 )erf z 2 βt The "erf" function is an abbreviation for the error function (also called the probability integral, although our problem has nothing to do with probability) and is defined as follows: erf (x) = 2 π u=x e u2 du u=0 In the solution for temperature, the argument of the erf function is the combination z 2 βt which involves both z and t. The quantity in the denominator has the dimensions of z, i.e. depth, and is a time-dependent scaling parameter for the plot of T versus z. Three profiles at times of 0, 5 and 50 MY are shown below (coefficient of thermal diffusivity, β, is taken as 25 km 2 /MY, a reasonable value for the earth). The next figure is a normalized plot of the shape of the profile, which is simply a plot of the "erf" function.

5 GS 388 Handout on Thermal Isostasy 5 Temperature, C Ta = 1300 C Zo = 0 km t = 0 MY t = 5 MY Zo = 25.3 km 50 depth, km t = 50 MY T = 0.89 Ta Zo = 79.9 km

6 GS 388 Handout on Thermal Isostasy normalized temperature = (T-To)/(Ta-To) linear approximation error function solution 1.5 z normalized depth = 2 βt 2.0 Suppose we define the thermal "thickness" of the lithosphere, Zo, as the depth at which the temperature reaches a certain large fraction of the temperature of the asthenosphere (Ta), say 89%; that is, T(z=Zo)/(Ta-To)=0.89. This means that the argument of the erf function must have the specific value There is nothing magic about 89% in terms of the mechanical properties of the material, but it will allow us to make a simplification of the temperature profile that preserves its essential characteristics, as discussed below. The transition from the strong material carried along with the lithosphere and the weak, fluid-like material in the asthenosphere is thought to occur at a temperature which is a large fraction of Ta, but the exact value is not very well known. 89% is probably not a bad guess. The main point is that by choosing a particular value for this ratio, we then set the value for the argument of the erf function and so define a particular relationship between Zo and t. Zo will thus be a measure of the "thickness of the lithosphere" as the thickness of the layer whose bottom reaches 89% of the asthenospheric temperature, Ta. Thus, Z o = βt = 2.26 βt gives a convenient measure of the "thickness" of the lithosphere, Z o, as a function of time, t.

7 GS 388 Handout on Thermal Isostasy 7 The thermal contraction due to the cooling during the time t is obtained by integrating the effect in a small depth range, dz, from the surface on down through the column. We did this in the handout on "thermal isostasy". The change of the height of an element of the column, dh', due to a change in temperature in the cooled section compared to the initial section will be given by: and dh' = [Ta-T(z,t)] (α) (dz) h' = z= (Ta - T(z,t)) α dz z=0 where α = coefficient of thermal expansion and h' is the decrease in height of the column due to thermal contraction alone. Now we can simplify this integration if we replace the curvilinear form of the erf function by a simple linear function as shown in in the figure. This replacement preserves certain important characteristics of the problem, even though it over simplifies the exact shape of the temperature profile (note that the linear approximation is not a bad one, though). If we define Zo as we did above to give an isotherm at the "base of the lithosphere" at 89% of the asthenosphere temperature, and draw the linear profile as in the figure above so that it reaches the asthenosphere temperature at the depth Z o, then the integration of this linear function will give the exact same answer for h' as we would get by integrating the "right" expression for T(z) with the erf function. This is in fact the rationale for choosing the seemingly abitrary value of 89%. It provides a simple but quantitatively accurate way to replace the erf function with a simple linear profile, one that is easy to plot and think about, as demonstrated in the handout on thermal isostasy. The linear relationship between temperature and depth is given by T(z) = To + (z/zo)(ta-to) T(z) = Ta for z < Zo, and for z > Zo and the integral is easy to do, giving the contraction, h', as h' = α (Ta - To) (Zo/2) Note that the information from the erf function solution is contained in the definition of Zo, which also relates depth, time and the coefficient of thermal diffusivity, β, so the essential physics is preserved in this approach. A result of the thermal contraction, however, is that additional water is added to the column. Let h* be the height change due only to the isostatic response of the added water, and let D o be the depth of the ocean at the oceanic ridge ( i.e., the column at t = 0 ). The mass (per unit area) added to a column located off the ridge at a depth D > Do is (D-Do)ρw. This added mass must be compensated by removal of an appropriate thickness, h*, of asthenospheric material, resulting in an isostatic subsidence h*. The mass balance gives h* ρm = (D-Do) ρw. where ρ m is the density of the asthenosphere and ρ w is the density of the ocean water. The depth of the ocean is thus given by the combination of h', the thermal contraction effect, and h*, the isostatic effect of the extra water: D-Do = h' + h* = h' + (D-Do)(ρw/ρm), or, solving for D-Do,

8 GS 388 Handout on Thermal Isostasy 8 D-D o = h' 1- ρ w ρ m = α(ta - To) Z o 2 1- ρ w ρ m This is the equation which gives ocean depth, D, as a function of age; remember that Zo has the square root of time in it. Estimates of the parameters which are reasonable from what we know about the physical properties of mantle material, and which together fit the observed relationship between ocean depth versus age, are as follows: To = 0 C, Ta=1300 C, β=25 km2 /MY, ρm=3.3 gm/cm 3, Do=2.5 km, and α = cgs. The situation looks like the figure below. spreading ridge ρ w Do D z o ρ m Curves of lithospheric thickness and ocean depth for these values are shown below as functions of time intepreted as age of the lithosphere OCEAN DEPTH VERSUS TIME ocean depth, km -5.5 time, MY

9 GS 388 Handout on Thermal Isostasy lithospheric thickness, km LITHOSPHERE THICKNESS VERSUS AGE 10 age, MY

10 GS 388 Handout on Thermal Isostasy 10 Appendix: Basic Heat Flow equation Consider a small box with sides dx, dy and dz. Heat flows parallel to the z axis only, and temperatures vary only as a function of z. Q in = q(z 1 ) dxdydt z = z 1 Note: positive q is pointing downwards towards the positive z direction z = z 2 Q out = q(z 2 ) dxdydt The net heat gain in the box during the time dt is the difference between the amount of heat flowing into the box and the amount flowing out of the box. The inflow is Q in = q(z 1 ) dx dy dt, where q(z 1 ) is the heat flow per unit area per unit time in the positive z direction, and taken across a unit area that is oriented perpendicular to the z axis and is located at z = z 1. Likewise, the amount of heat flowing out of the box is Q out = q(z 2 ) dx dy dt. The net gain in heat will be related to an increase in temperature T given by the expression {heat gain per unit mass per unit temp.} {mass per unit volume} {volume} { T} or {C} {ρ} {dx dy dz} {( T/ t) dt} where C is the specific heat capacity in calories per gram per degree, ρ is the density, and T is the

11 GS 388 Handout on Thermal Isostasy 11 temperature. Thus or Q in - Q out = [q(z 1 ) - q(z 2 )] dx dy dt = Cρ ( T/ t) dx dy dz dt q(z 1 ) - q(z 2 ) = Cρ ( T/ t) dz Now q(z 2 ) can be evaluated from the differential expression q(z 2 ) = q(z 1 ) + ( q/ z)dz, and then the difference q(z 1 ) - q(z 2 ) substituted for in the preceding equation to give, after cancelling the dz, dx and dy terms, - q/ z = Cρ ( T/ t). Now we need to add a further relation between heat flow and temperature. This is the basic conduction relation for one dimensional heat flow, q = - κ ( T/ z), where κ is now the coefficient of thermal conduction (don't confuse this with the other parameters that we have considered and unfortunately denoted by the same greek letter!). κ, like C and ρ, are material constants. This equation is the basis of heat flow deteminations: measurements of temperature as a function of depth in a piston core, borehole or mine are used to estimate T/ z, and κ is measured for samples of the rock in which the temperature measurements were made. Substituting for q, we have now the basic heat flow equation or where κ 2 T z 2 =Cρ T t β 2 T z 2 = T t β = κ C ρ β is called the coefficient of thermal diffusivity, a composite material constant with dimensions of length squared divided by time. The square root of this constant scales the relation between time and space in the movement or diffusion of heat. For the seismic wave equation we had a constant whose square root scaled the movement of stress or strain as distance proportional to time (the seismic wave velocity). Here, the square root of β gives us distance proportional to the square root of time, which is the fundamental characteristic of diffusion. Typical values of β estimated for mantle material are km 2 /MY where MY = million years (it so turns out that

12 GS 388 Handout on Thermal Isostasy 12 these are convenient units). Thus for a time of 1 MY a characteristic distance of heat diffusion is about 5-6 km; for 10 MY, km; for 100 MY, km; and for a billion years, only km! Conductive heat flow in the earth is a very sluggish process. The heat flow equation developed above does not include sources of heat within the rock itself. This can be included with a parameter, A, which is the amount of heat generated per unit volume per unit time. Heat is generated by radioactive decay of certain elements found in rocks, and is especially important in the upper crust of continental regions where the radioactive isotopes are concentrated. We put this heat producing factor on the left side of the heat flow equation as another addition to the rate of heat addition to our little unit volume of material. This source of heat, together with the difference between the heat flowing in and the heat flowing out, produces the change in temperature seen on the right side of the equation: A+κ 2 T z 2 =Cρ T t Note that the terms each give heat accumulated per unit volume per unit time. This equation can be rewritten in terms of only κ and β as A κ + 2 T z 2 = 1 β If there is no change of temperature with time a "steady state" situation the right side is zero, and we have the very simple differential equation d 2 T dz 2 =-A κ T t Note that all this discussion is for temperature varying only in the z direction, i.e. the one dimensional heat flow problem.

GEOL5690 Class notes #2: Passive Margins & Thermal Subsidence

GEOL5690 Class notes #2: Passive Margins & Thermal Subsidence GEOL569 Class notes #2: Passive Margins & Thermal Subsidence Reference: Turcotte and Schubert, sec. 4-15, 4-16, 4-23, 4-25, last is most relevant, earlier readings provide background. [Alternative: Stüwe,