Intro to Earth Sciences

Introduction to Earth Sciences I

3.2 Mantle convection

If needs to be ~2000 for convection to occur it is reasonable to ask whether the Earth's interior, given what we have inferred about it in various ways from analysis of external measurements, can convect heat and hence experience convective motion. The answer is yes.

First, we know from our studies in Topics 1 and 2 that the Earth is surprisingly soft in the interior. The mantle is best thought of as a viscous fluid capable of flow - albeit very slowly. Thus, if the Earth has a sufficiently strong temperature gradient, meaning that if it is heated enough from below, and making a reasonable guess at the expansion coefficient and diffusivity, convection should take place.

The two major sources of heat generation in the deep Earth are the liquid outer core and radiogenic heat production in the mantle itself. The mantle contains uranium, thorium and potassium, each of which is radiogenic and produces heat as a by-product of radioactive decay. Many of these radiogenic elements are extracted from the mantle during melting and reside in the continental crust where they are in concentrations more than 200 times greater than the mantle. Nevertheless, because of the great extent of the mantle, its radiogenic heat production is very significant. The Earth's liquid outer core is also a major contributor to heat production at the surface. The exact temperature profile of the Earth is, of course, quite difficult to determine and is generally inferred from melting experiments on materials that we believe are present in the deep Earth. The illustration shows a typical proposed temperature profile in the Earth which comprises three main parts -

Figure 3.2.1

The upper part shows a very steep gradient in the crust where heat is transported mostly by conduction.
Beneath that the gradient is much more shallow in the mantle.
A steep gradient between the lowermost mantle and outer core.

Although fairly shallow, the mantle temperature gradient is in excess of adiabatic - the temperature gradient that is due only to the weight of the overlying material which will cause it to compress. That is, the adiabat can be thought of as the temperature gradient due to "self-compression" in the mantle. Because of radiogenic heat production and heating from the core beneath, the mantle's temperature gradient is super adiabatic.

Given this temperature profile and reasonable estimates of the other physical quantities we can calculate an average value of Rayleigh number for the mantle to be about 20,000 to 30,000, and hence we can expect vigorous convection in the mantle.

Also, in the mantle the Prandlt number is about and is as good as infinite for most purposes. So we really expect that convection should be happening in the mantle.

Mantle convection is not simple. Because heating occurs both from below and from within [core heat and radiogenic heat production] and the heat sources are more uniformly distributed. When a heat source is more distributed the convection often becomes multi-celled and a type of convection known as Rayleigh-Bernard convection takes place.

Figure 3.2.2

Figure 3.2.3

Figure 3.2.4

These animations show numerical simulations of multi-celled convection. The upper one takes a simple case and shows how the patter will break up into several cells as convection evolves. The lower simulates a more complex(and more realistic) case. The colors are as before (reds are hot, blues are cool) and we see several centers of upwelling and downwelling and that individual centers move laterally as convection evolves.

Mpeg movies -- These should play with Windows Media Player

Figure 3.2.5	Figure 3.2.6
Figure 3.2.7	Figure 3.2.8

From: http://www.geof.ruu.nl/~trompert/html/mantle.html

In fact, the actual patterns of flow may be more complex still with sinuous patterns of fluid movement that migrate around throughout the whole volume of the convecting region as this model shows.

Figure 3.2.9

For an infinite Prandlt number, fluid at high Rayleigh numbers convection becomes time dependent. That is, various cells of changing size will form, stir the fluid for awhile and then disappear. The changing pattern of convective cells in such a fluid cannot be predicted, and is certainly associated with a non-linear behavior. Fluid flow becomes turbulent rather than regular. Whether such a flow structure can be characterized as chaotic is less clear. We know, however, that the behavior of the system is sensitive to small changes in critical parameters like the Rayleigh number, and that it will evolve from a simple state of single-celled convection to one of multiple-celled convection then to a highly disordered state as such changes occur. These are certainly aspects of behavior that are like chaotic systems.