Sea Floor Global Topography: Half-space Model
To first approximation, sea floor global topography in areas without significant subduction can be estimated by the half-space model. In this model, the seabed height is determined by the oceanic lithosphere temperature, due to thermal expansion. Oceanic lithosphere is continuously formed at a constant rate at the mid-ocean ridges. The source of the lithosphere has a half-plane shape (x = 0, z < 0) and a constant temperature T1. Due to its continuous creation, the lithosphere at x > 0 is moving away from the ridge at a constant velocity v, which is assumed large compared other typical scales in the problem. The temperature at the upper boundary of the lithosphere (z=0) is a constant T0 = 0. Thus at x = 0 the temperature is the Heaviside step function . Finally, we assume the system is at a quasi-steady state, so that the temperature distribution is constant in time, i.e. T=T(x,z).
By calculating in the frame of reference of the moving lithosphere (velocity v), which have spatial coordinate x' = x-vt, we may write T = T(x',z,t) and use the heat equation: where is the thermal diffusivity of the mantle lithosphere.
Since T depends on x' and t only through the combination, we have:
Thus:
We now use the assumption that is large compared to other scales in the problem; we therefore neglect the last term in the equation, and get a 1-dimensional diffusion equation: with the initial conditions .
The solution for is given by the error function :
.
Due to the large velocity, the temperature dependence on the horizontal direction is negligible, and the height at time t (i.e. of sea floor of age t) can be calculated by integrating the thermal expansion over z:
where is the effective volumetric thermal expansion coefficient, and h0 is the mid-ocean ridge height (compared to some reference).
Note that the assumption the v is relatively large is equivalently to the assumption that the thermal diffusivity is small compared to, where L is the acean width (from mid-ocean ridges to continental shelf) and T is its age.
The effective thermal expansion coefficient is different than the usual thermal expansion coefficient due to isostasic effect of the change in water column height above the lithosphere as it expands or retracts. Both coefficients are related by:
where is the rock density and is the density of water.
By substituting the parameters by their rough estimates: m2/sec, °C−1 and T1 ~1220 °C (for the atlantic and Indian oceans) or ~1120 °C (for the eastern pacific), we have:
for the eastern pacific ocean, and:
for the atlantic and Indian ocean, where the height is in meters and time is in millions of years. To get the dependence on x, one must substitute t = x/v ~ Tx/L, where L is the distance between the ridge to the continental shelf (roughly half the ocean width), and T is the ocean age.
Read more about this topic: Seafloor Spreading
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