This applet shows the contribution of crustal heat sources to the geotherm for the exponential and homogeneous models respectively (i.e., the mantle heat flow contribution has been removed. Note that for identical length scales the exponential model produces twice the effect of the homogeneous model. For explanation, see discussion below.
The temperature field for the exponential model is given by :
In Eqn 1 qr is the reduced heat flow that applies at deep levels within the lithosphere, beneath all significant heat production. One physical interpretation of qr is that it is the heat flux provided by convective processes in the deeper mantle to the base of the lithosphere.
In the homogeneous model the temperature field in the heat-producing layer is
Note that in both Eqn 1 and Eqn 2, the first term on the right represents the component of the temperature field due to the heat flow from beneath the heat producing parts of the lithosphere. The second term on the right represents the contribution due to heat sources in the crust. This term can be used to define the quantity DT. the temperature contribution due to crustal heat production in the crust. DT reaches its maximum value (DTmax) at the depth at which heat production becomes negligible (i.e. at the base of the heat-producing layer). The appropriate expressions for DTmax for the exponential and homogeneous models are, respectively :
In Eqn 3 zc is the depth at which the exponential heat production distribution is terminated, which we approximate as the Moho depth.
One attribute of the length scales for these two models is that, for such distributions it can be evaluated from the regression of surface heat production and heat flow data. Typically, such regressions yield a value of ~10 km. An alternative, more general, formulation of the length-scale, which makes no explicit assumption about the form of the heat production distribution, can be made directly in terms of DTmax. Defined in this way the length-scale, h, is simply:
where qc is the depth integrated heat production :
This parameterization also highlights the fact that
the temperature deviation, DTmax,
is very sensitive to the depth at which the heat production is located.
For both the homogeneous and exponential heat production models outlined
above qc = hrHs, these distributions
allow h to be expressed in terms of hr. For the
which reduces to h = hr for zc>> hr, while for the homogeneous model
These scaling relations imply that for identical values of hr and qc the exponential heat production distribution results in DTmax exactly two times that of the homogeneous heat production distribution.