These musings relate to some very nice Monte Carlo simulations performed by David Hansen relating to the long-term 2-D thermal structure of sedimentary basins, relating to the potential for thermally modulated inversion.
From the point of view of modeling the long-term thermal consequences of basin formation, the value of the mantle heat flow is crucial because, along with the upper mantle thermal conductivity, it determines the steady state thermal gradient at the Moho. Consequently, the Moho cooling associated with shallowing is a direct function of qm. This is highlighted by the Sandiford (1999) figure
The question the is how do we estimate qm? There is a common, but incorrect, view that the value of hr extracted from regression of surface heat flow versus heat production data (e.g., Balling, 1995) provides a proxy for qm However, hr always overestimates overestimates the value of qm for two reasons:
Firstly, the linear relation implies that all the variation in surface heatflow is contained in an upper crustal layer of thickness ~hr. The value of qr then represents all the deeper sources, including the mantle heat flow as well as the crustal sources at depths greater than ~hr. For average lower crustal heat production rates of 0.25-0.5 microWm-3 we might expect a lower crustal contribution, qlc, = 5 -10 mWm-2. In which case hr must overestimates qm by at least the value of qlc.
More importantly, it is inappropriate to view the surface heat flow - heat production field in terms of a 1-d (vertical) conduction problem, as is implicitly assumed in the regression approach. This is because lateral heat transfer will always case a significant rotation of the heat flow - heat production relation as shown below:
In the bottom figure, the dashed line represents the vertically-integrated contributions that would apply if there was no lateral heat transfer. The open symbols show how the surface heat flow that applies as a consequence lateral heat transfer. There will always be a rotation such that regressions of observed data yield a y-intercept that overestimates the 1-d reduced heat flow and the slope underestimate the thickness/length-scale of the layer responsible for the observed variation.
How significant is this effect (which I call the "jaupart effect", after Jaupart's classic 1983 paper). To evaluate this I have used some of the Scandinavian data from the NGDC (1993) database as shown below:
This data yields regression shown below. This looks a bit different from Balling (1995) in part because I do not have all his data, and also because I have not applied any corrections. Balling gets a qr of 30-40 mWm-2 by applying the corrections and using an updated dataset.
Note that the regression only uses the red circled data.
Using this data as a guide to the range in surface heat production parameters and characteristic heat flow field, I have modeled the relations expected for various different characteristic horizontal length-scales for heat production variation, as shown below.
In these models, I apply a qm of 15 mWm-2 and qlc of 5 mWm-2 such that the real qr is 20 mWm-2. An upper crustal radiogenic layer of thickness 15 kms contains all the variation in heat production required to explain the observed variation in surface heat flow. The characteristic length-scale for variation in heat production distribution in this layer is given by hx. The computed temperature field (and the extracted surface heat flow - heat production relation as shown by circles) are then used to compute hr and qr as shown by the solid line regression. The rotation induced by the effects of lateral heat flow can be compared to the theoretical red dashed line for 1-d conduction.
These models show that it is quite possible to generate the Scandinavian heat flow -heat production relations (i.e.qr. 30 mWm-2) using a qm as low as 15 mWm-2, provided that the characteristic length-scale for the horizontal variation in heat production is less than about 100 kms. Note that even if the variation is 300 kms, qr overestimates the real reduced heat flow by 5 mWm-2 (and, for these models, qm by 10 mWm-2).
While more detailed analysis is required, the case can be made that the mantle heat flow in the Southern Scandinavia region is as low as 20 mWm-2. Following Sandiford (1999) we can specify the ratio of qm= qm/qc, which for an surface heatflow of 50-60 mWm-2 in southern Scandinavia, may therefore be as low as 0.5. This contrasts with the assumed parameters in the Hansen & Nielsen manuscript where qm would seem to typically be greater than 1 (qm = 30-50 mWm-2, qc 10-50 mWm-2). For a 5 km thick basin this will lead to an underestimate of the rifted Moho temperature by at least 20 C (see bottom left figure below, from Sandiford, 1999). Of course, for finite width basins lateral conduction will modulate this.
These musing relate to some computations by David Hansen on the thermal structure of ancient rift basins. Lateral conduction combined with variable Moho depth beneath ancient rifted basins conspire to produce patterns of thermal structure which lead to relative weakening of rifted margins. Here are some of my calculations/figures.
The following shows a cartoon of the general structure of my models:
The figure below shows the result of a model which favours long-term cooling and strengthening beneath the basin centre, as a function of the basin width. The top panel shows the Moho temperature and Tz=35 km for various width basins. The Moho temperature at various points (lower left) and measure of weakening (lower right)is normalised against the far field. h2 is the basin margin
The following figures are variants on the above.
The figure below shows the normalisation against the center of the basin
the run parameters are summarised below
(Having said all that about qm and now done the simulations following Hansen's approach, I don't seem to get much dependence on qm - just as Hansen reports).
The following show my Monte Carlo simulations (> 20,000 individual FEM solutions) -very much motivated by Hansens modlling. The figures showing temperature changes over a range of parameter that might conceivably vary from basin to basin (note the wide range in parameters).
The first figure shows the Moho temperature at the center of the basin minus the far field moho temperature. About half the runs show long-term heating of the Moho beneath the basin.
It is clear that the majopr controls are the width of the basin, and (as Hansen reports) the density and the conductivity of the basin fill. Second order controls are provided by mantle heat flow (qm), and the thickness (zuc) of the pre-existing heta producing layer in the crust.
The following shows the Moho T at the basin centre minus the Moho T at the basin edge. The basin edge shows a statistically- significant increase in Moho temperature compared with the basin center, as documented by Hansen & Neilson
Histogram plots emphasize the above conclusions .
A crucial factor is the basin width, here plotted against the the ratio of the observed Moho temperature change at thebasin centre to the predicted 1-d Moho temperature change Tm (eg T(4) in Sandiford, 1999). The left figures ishows the solutions where the 1d Moho temperature, Tm, increases (as predicted for > 60 % of all runs). In these cases Tz (or T(3) in sandiford 1999) must also be positive -i.e., there is an increase in temperature at any given depth benath the basin (see inversion applet for another view). In these cases lateral heat flow away from the basin diminishes the amplitude of the the observed heating (Tp < 1) and for about 10% of cases actually results in long-term Moho cooling (such that Tp < 0 as shown by blue dots). Note that the Moho beneath the basin still must be hotter than equivalent depths outside the basin. The right hand shows cases where 1-d cooling of the Moho is predicted (Tm < 0) but Tz > 0. In these cases the lateral heat flow away form the basin centre augments the theoretical cooling such that Tp > 1.
Note that basins with half widths of 40 kms (a typical North Sea scenario) show about 50% of the theoretical heating. At half widths > 100 km, appropirate to very wide basins, most cases show ~ 80-90 % of the theoretical long-term heating. Such wide basins are clearly much more prone to suffer the Sandiford (1999) effect of long-term thermally controlled weakening.
These conclusions are supported by 2-D thero-mechancial models implemented by Susann Frederickson, as shown below. For details see Sandiford et al, cited below.
More illustrations of these simulation results can be found in my figures library under the Basins_2D and Hansen sections of the Miscellaneous section, while an applet demonstrating some of the long-term 1-d thermal consequences of basin formation can be found by following the link inversion applet.
Sandiford, M., 1999, Mechanics of basin inversion, Tectonophysics, 305,109-120.