Heat Flow

For today's group meeting, I assigned three papers for discussion.  These were


Blackwell, D. D., Bowen, R. G., Hull, D. A., Riccio, J. & Steele, J. L. Heat flow, arc volcanism, and subduction in northern Oregon. J. Geophys. Res. 87, 8735-8754 (1982).


Pollack, H. N. & Chapman, D. S. On the regional variation of heat flow, geotherms, and lithospheric thickness. Tectonophys. 38, 279-296 (1977).


Syracuse, E. M., Van Keken, P. & Abers, G. A. The global range of subduction zone thermal models. Phys. Earth Planet. Int. 183, 73-90 (2010).


The common theme to these papers is heat flow, or more accurately, heat flux, that is the amount of energy passing through a surface per unit time, per unit area.  Pollack and Chapman (1977) is a classic paper that shows how surface heat flux varies around the Earth. The surface heat flux represents the flux of heat passing through the surface of the Earth.  In ocean basins it can range from values greater than 100 mW/m^2 to 50 mW/m^2 or so, the former at the mid-ocean ridge and the latter in old ocean basins.  In continents, heat flux varies from values around 80-100 mW/m^2 in areas undergoing active extension like the Basin and Range Province in California or in volcanically active areas, such as the arc front in subduction zones.  But it can also range to very low values, down to 40 mW/m^2 in the stable interiors of cratons. To place this all in context, the solar flux of energy hit our planet is 300 W/m^2, some 10,000 times higher than the heat coming out of the Earth.  Solar radiation can affect our planet's climate, but clearly, the geothermal heat flux is miniscule (this is not to say that we can harness Earth's heat for energy, e.g., geothermal energy).


Before proceeding, it's worth making sure we know how heat flux is measured.  Heat flux is a very difficult thing to measure directly.  To do it directly, you would have to have an apparatus that captures the heat coming out of the Earth's surface. Because so little heat is coming out at any given time, this is a very difficult direct measurement to make.  So what is done instead is to determine the heat flux indirectly.  Assuming heat flows through the crust primarily by conduction, then the heat flux out of the Earth's surface, e.g., the surface heat flux qs, is equivalent to


qs = k*dT/dz


where k is the thermal conductivity and dT/dz is the geothermal gradient at the surface of the Earth.  The geothermal gradient is determined by measuring the temperature as a function of depth in a well or deep borehole, whereas k is determined in general by taking samples of the rock back to the lab and measuring k (in the oceans, this is done by dropping a thermistor into soft sediment and k is determined indirectly by tracking the rate at which the thermal halo relaxes around the thermistor).  The temperature-depth profile that defines the temperature gradient is typically based on wells that go down a few hundred meters, occasionally boreholes down to 1-2 km.  Typical continental crust is ~35 km thick.  Continental lithosphere might be 100 km thick or thicker, and the radius of the Earth is 6730 km!  As you can see, we are basically making a measurement on the very outer skin of the Earth.  How do we extrapolate this to depth?


To do so, we make use of the thermal diffusion equation.


rho*c*dT/dt = k d^2T/dz^2 + A       Eq 1


The left hand side represents the rate at which a given box of material in the Earth increases in heat content, where c is the specific heat, rho is density and dT/dt is the change in temperature per unit time.  The first term on the right is a measure of the imbalance between heat flowing in one side of my box and flowing out the other side.  If they are in balance, then there is not change in temperature of the box. If they are in balance, then temperature changes.  This is basically a 1-D version of the divergence of the gradient in temperature for those of you who are versed in vector calculus.  The second term on the right is the internal heat production term.  Earth's rocks have some amount of U, Th and K, which are radioactive and consequently generate heat every time they decay.  This decay is of course slow, so on human timescales we don't have to worry about it. But on geologic timescales, they play an important role.    So to summarize, Equation 1 basically states that the change in temperature of a given box is simply the result of imbalances in heat fluxes in and out of the box (first term on right) and internal heat production (second term on right).  Make sense?


Now, one of the simplifying assumptions we can make (and in some situations, it is a horrifying simplification) is that the thermal state of the region of interest is at steady state.  This means that dT/dt = 0.  There are some places where this is probably okay to first order, for example, in stable continental interiors where the underlying lithosphere thickness has not changed too much with time.  If the lithosphere thickness has remained constant over 100 My timescales and if the temperature of the underlying convecting mantle remains constant on such timescales, this is a fine assumption.  There are places where this is not the case, such as in the Basin and Range, which is rapidly extending.  So keep this in mind.


The other assumption that we're going to make is that advective processes, such as convection, can be ignored, at least in the shallower, colder parts of the lithosphere (which includes the crust).  In reality, Earth loses its heat by convection via mobile thermal boundary layers. Convection cells are defined a the surface by thermal boundary layers, through which vertical heat loss occurs through conduction.  So we can play with the conductive simplification if we're worried only about the upper part of the thermal boundary layer.  If we want to think about deeper depths, the conductive models used here will be egregiously wrong (before plate tectonics, many people made this mistake, including the great Lord Kelvin).


 So with all those caveats in mind, this is how we can use our surface heat flow measurements to infer the thermal state at greater depths.  We first integrate Eq. 1 and assume that kdT/dz at z = 0 is qs.  This gives


0 = k dT/dz + Az - qs        (Eq 2)


We can then integrate this steady state equation again with the boundary condition that T at z=0 is To.  This gives, upon rearrangement


T = To + qs*z/k - A(z^2)/2         (Eq 3)


So let's take a look at the above equation.  Let's first assume that heat production A = 0.  Then we simply have


T = To + qs*z/k                 (Eq 4)


which is linear versus k.  We use the thermal gradient at the surface, which defines the slope that we will use to extrapolate temperature with depth.  The resulting temperature profile will be a straight line that will just keep on going (question: where do you stop extrapolating?).


If we put A back in, what we find is that this term is negative and scales quadratically with z.  Accounting for heat production will cause the geotherm to curve in upon itself at depth, giving parabolic shaped geotherms. Knowing what A is, however, is not trivial.  What one has to do is measure the amount of U, Th, and K in the rocks, but in general, the only rocks available to you are at the surface or perhaps in your well.  You can't get much from the lower crust as you just have no way of getting there.  If A were constant with depth, the calculation would be simple. But it's not constant.


Most likely, A decreases with depth to first order.  The reason is that magmatic differentiation and re-working of crust tends to drive melts and fluids towards the surface because they are low density.  And because U, Th and K prefer to be in melts and fluids, this segregation processes gradually concentrates these heat producers towards the upper crust, depleting the lower crust in these elements as a consequence.  


Now, one of the most amazing observations in the Earth sciences came about back in the 60s if I recall (I wasn't born yet then, but I'm recalling from my readings).  The Harvard group, including Francis Birch, had measured heat flux across different parts of a batholith. At the same time, they measured the heat production A at the surface (remember, heat flow is different from heat production). They found a remarkable relationship. There was a linear relationship between qs and A, giving an empirical equation


qs = qo + b*A


where b is the slope of this correlation and qo is the y-intercept equivalent.  Simple.  When you get a straight line in nature, you have got to explain it.  The genius behind the Harvard group was their ability to come up with a simple, elegant solution to a fundamental observation.  What if A decreased exponentially with depth?  


A = Ao*exp(-z/zx)


where zx is some sort of characteristic lengthscale over which the heat production drops e-fold.  If you put the above equation back into the earlier equations I showed, you will end up with the following


qs = qo + zx [Ao*exp(-z/zx)]


The part in brackets is essentially A.  So on a qs versus A plot, zx is then equivalent to the slope of the line, or b!  So the only way to get a linear correlation between qs and A is if A actually decreases exponentially with depth.  The slope of the qs versus A curve is essentially the characteristic e-fold decay length, b, that is


qs = qo + b*[Ao*exp(-z/b)]


If we let z = 0, then qs = qo + b*Ao, which makes sense.  If we let z go to infinite, then qs = qo and there is no contribution from heat production.  This analysis led to the concept of reduced heat flow.  The idea is that most of the heat production is in the crust, and if you can subtract that contribution out, the remaining heat flow must come from the mantle.  After that seminal paper, many subsequent heat flow papers would simply calculate the reduced heat flow using the above equation and assuming some plausible b.


Now, if you have been astute, you might be wondering whether this exponential decay profile actually exists everywhere.  The answer is probably not, but more work is needed to evaluate this. Certainly, the crust is highly heterogeneous. And in some places, particularly places undergoing thrusting, it's likely that there are local layers of high heat production.  


Anyway, this is basically conductive geotherm 101.  There are plenty of caveats, but I outlined how this field is understood to make sure you appreciate what has been done in the past.


I will continue my discussion on the above papers in later blog posts as they are quite important.  But I just realized that I'm tired and want to do something else, so my brain has suddenly said, "stop". Can't fight that. Stay tuned.