The depth of the seafloor on the flanks of a mid-ocean ridge is determined mainly by the age of the oceanic lithosphere; older seafloor is deeper. During seafloor spreading, lithosphere and mantle cooling, contraction, and isostatic adjustment with age cause seafloor deepening. This relationship has come to be better understood since around 1969 with significant updates in 1974 and 1977. Two main theories have been put forward to explain this observation: one where the mantle including the lithosphere is cooling; the cooling mantle model, and a second where a lithosphere plate cools above a mantle at a constant temperature; the cooling plate model. The cooling mantle model explains the age-depth observations for seafloor younger than 80 million years. The cooling plate model explains the age-depth observations best for seafloor older that 20 million years. In addition, the cooling plate model explains the almost constant depth and heat flow observed in very old seafloor and lithosphere. In practice it is convenient to use the solution for the cooling mantle model for an age-depth relationship younger than 20 million years. Older than this the cooling plate model fits data as well. Beyond 80 million years the plate model fits better than the mantle model.
Background
The first theories for seafloor spreading in the early and mid twentieth century explained the elevations of the mid-ocean ridges as upwellings above convection currents in Earth's mantle.
The next idea connected seafloor spreading and continental drift in a model of plate tectonics. In 1969, the elevations of ridges was explained as thermal expansion of a lithospheric plate at the spreading center. This 'cooling plate model' was followed in 1974 by noting that elevations of ridges could be modeled by cooling of the whole upper mantle including any plate. This was followed in 1977 by a more refined plate model which explained data that showed that both the ocean depths and ocean crust heat flow approached a constant value for very old seafloor. These observations could not be explained by the earlier 'cooling mantle model' which predicted increasing depth and decreasing heat flow at very old ages.
Seafloor topography: cooling mantle and lithosphere models
The depth of the seafloor (or the height of a location on a mid-ocean ridge above a base-level) is closely correlated with its age (i.e. the age of the lithosphere at the point where depth is measured). Depth is measured to the top of the ocean crust, below any overlying sediment. The age-depth relation can be modeled by the cooling of a lithosphere plate or mantle half-space in areas without significant subduction. The distinction between the two approaches is that the plate model requires the base of the lithosphere to maintain a constant temperature over time and the cooling is of the plate above this lower boundary. The cooling mantle model, which was developed after the plate model, does not require that the lithosphere base is maintained at a constant and limiting temperature. The result of the cooling mantle model is that seafloor depth is predicted to be proportional to the square root of its age.
Cooling mantle model (1974)
In the cooling mantle half-space model developed in 1974, the seabed (top of crust) height is determined by the oceanic lithosphere and mantle temperature, due to thermal expansion. The simple result is that the ridge height or seabed depth is proportional to the square root of its age. In all models, 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 , which is assumed large compared to 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 . The system is assumed to be at a quasi-steady state, so that the temperature distribution is constant in time, i.e.
By calculating in the frame of reference of the moving lithosphere (velocity ), which has spatial coordinate and the heat equation is:
where is the thermal diffusivity of the mantle lithosphere.
Since T depends on x''' and t only through the combination :
Thus:
It is assumed that is large compared to other scales in the problem; therefore the last term in the equation is neglected, giving one-dimensional diffusion equation:
with the initial conditions
The solution for is given by the error function:
.