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# Simple Thermal Isostasy

Simple Thermal Isostasy

## GS 388 Handout on

GS 388 Handout on Thermal Isostasy 10Appendix: Basic Heat Flow equationConsider a small box with sides dx, dy and dz. Heat flows parallel to the z axis only, andtemperatures vary only as a function of z.Q in = q(z 1 ) dxdydtz = z 1Note: positive q is pointingdownwards towards thepositive z directionz = z 2Q out = q(z 2 ) dxdydtThe net heat gain in the box during the time dt is the difference between the amount of heat flowinginto the box and the amount flowing out of the box. The inflow isQ 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 acrossa unit area that is oriented perpendicular to the z axis and is located at z = z 1 . Likewise, the amountof heat flowing out of the box isQ 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

GS 388 Handout on Thermal Isostasy 11temperature. ThusorQ in - Q out = [q(z 1 ) - q(z 2 )] dx dy dt = Cρ (∂T/∂t) dx dy dz dtq(z 1 ) - q(z 2 ) = Cρ (∂T/∂t) dzNow q(z 2 ) can be evaluated from the differential expressionq(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, aftercancelling 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 basicconduction 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 parametersthat we have considered and unfortunately denoted by the same greek letter!). κ, like C and ρ, arematerial constants. This equation is the basis of heat flow deteminations: measurements oftemperature 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 equationorwhereκ ∂2 T∂z 2 =Cρ∂T ∂tβ ∂2 T∂z 2 = ∂T∂tβ =κC ρβ is called the coefficient of thermal diffusivity , a composite material constant withdimensions of length squared divided by time. The square root of this constant scales the relationbetween time and space in the movement or diffusion of heat. For the seismic wave equation wehad a constant whose square root scaled the movement of stress or strain as distance proportionalto time (the seismic wave velocity). Here, the square root of β gives us distance proportional to thesquare root of time , which is the fundamental characteristic of diffusion. Typical values of βestimated for mantle material are 25-40 km 2 /MY where MY = million years (it so turns out that

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