Modeling Earth Systems

Inside the Earth motion of heat by radiation is very ineffective because rocks are opaque to the near infrared and visible areas of the electromagnetic spectrum, conduction is extremely slow process in the mantle because rocks are poor thermal conductors, taking billions of years for heat to go through even a fraction of the mantle's thickness. So, the only mechanism that can efficiently evacuate heat from the Earth's interior is convection. Convection results from the fact that, when heated, most solids and fluids expand, thus decreasing in density which makes them buoyant. Inside the Earth it might appear that convection would not be possible since the mantle is solid. However, although the mantle appears solid to us and it is indeed solid on short time scales such as those of seismic waves, in a scale of millions of years the mantle behaves like a viscous fluid, able to flow at relatively high speeds (tens of centimeters per year) making convection possible. Not so the lithosphere, which is nearly rigid and therefore releases its heat mostly by conduction. The flow of the mantle over geologic time is driven by gravity, which acts on the density differences created by the loss of heat at the Earth's surface.

Click on the icon below to see a 2-D simulation of convection in the mantle. Notice the upsurging columns of hot material and the splitting of the continent (the green slab on the top right). This animation is from Caltech's Seismological Laboratory ("Supe rcontinent Aggregation and Dispersal",1988; by M. Gurnis, B. H. Hager, and A. Raefsky)

The Bénard-Rayleigh experiment.

Thermal convection is the main process by which the Earth's heat engine works. It is the most effective way for an opaque mantle to transmit heat, and the simplest illustration of thermal convection is the classic Bénard (also called Rayleigh-Bénard) experiment, which consists of a thin layer of fluid (e.g., water) continuously heated from below. The temperature at the bottom can be varied, and at the top the fluid is in contact with the atmosphere so that heat can be conducted through the layer and out to space. Let us call the bottom temperature Tb and the temperature at the top Tt. Then Tb - Tt = DT is the temperature difference across the layer. We define the temperature gradient as the ratio b = DT / h, where h is the layer thickness.

The figure below is from E. Holbecher from IGB-Berlin and shows the transient development of five cells in a simulation of the Benard-Raleigh experiment. The system starts as a conduction only experiment, then two disturbances at the vertical boundaries introduce the convection cells. The streamlines (light blue) indicate the flow of fluid and the magenta lines indicate the position of the isotherms. As the cells grow the central region develops new cells. The simulation is produced using FAST-C(2D) code developed by Dr. Holzbecher.

The mathematics of convection is complex, but some aspects of the physics can be understood from first principles. Consider first a small parcel of fluid at some intermediate point in the layer and assume b is close to critical. If the parcel is then disturbed from its equilibrium position and moved slightly upwards, it goes into a cooler environment, where it is relatively lighter or buoyant (we assume that the pressure difference is negligible and that heat transfer with the environment is also negligible) so it will tend to raise more, which it will do, reaching a still colder place which will make it more buoyant still. This is a positive feedback process that eventually ends in the parcel being accelerated upwards.

A similar positive feedback occurs if the initial displacement is downwards. In this case, the parcel will be heavier than its surroundings and will tend to sink, and as it sinks it encounters increasingly lighter fluid, so it sinks ever more rapidly.

The increasing disturbance makes convection increase in vigor, but because of this increased efficiency in transporting heat and thus dissipating the energy put into the system, the temperature will drop, the gradient will decrease and the convection will slow down, stop, or reach a dynamic equilibrium or steady state in which cells are established, heat moves out but the entire pattern is fixed in time. Thus the amplifying positive feedback is kept on check by a negative feedback, an automatic regulation which opposes the original deviation and that is triggered automatically by the simple reason that once convection is established heat is transported more efficiently, which cools the system.