Seafloor depth versus age

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 ${\displaystyle v}$, 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 ${\displaystyle T_1\cdot \mathrm{\Theta }(-z)}$. The system is assumed to be at a quasi-steady state, so that the temperature distribution is constant in time, i.e. ${\displaystyle T=T(x,z).}$

By substituting the parameters by their rough estimates into the solution for the height of the ocean floor ${\displaystyle h(t)}$:

${\displaystyle \begin{array}{cccc}\kappa & \sim 8\cdot {10}^{-7}{\mathrm{m}}^2\cdot {\mathrm{s}}^{-1}& & \text{for the thermal diffusivity}\\ \alpha & \sim 4\cdot {10}^{-5}{}^{\circ }{\mathrm{C}}^{-1}& & \text{for the thermal expansion coefficient}\\ T_1& \sim 1220{}^{\circ }\mathrm{C}& & \text{for the Atlantic and Indian oceans}\\ T_1& \sim 1120{}^{\circ }\mathrm{C}& & \text{for the eastern Pacific}\end{array}}$

we have:

${\displaystyle h(t)\sim \{\begin{array}{ll}{h}_0-390\sqrt{t}& \text{for the Atlantic and Indian oceans}\\ h_0-350\sqrt{t}& \text{for the eastern Pacific}\end{array}}$

where the height is in meters and time is in millions of years. To get the dependence on x, one must substitute t = x/${\displaystyle v}$ ~ Ax/L, where L is the distance between the ridge to the continental shelf (roughly half the ocean width), and A is the ocean basin age.

Rather than height of the ocean floor ${\displaystyle h(t)}$ above a base or reference level ${\displaystyle h_b}$, the depth of the seabed ${\displaystyle d(t)}$ is of interest. Because ${\displaystyle d(t)+h(t)=h_b}$(with ${\displaystyle h_b}$ measured from the ocean surface) we can find that:

${\displaystyle d(t)=h_b-h_0+350\sqrt{t}}$; for the eastern Pacific for example, where ${\displaystyle h_b-h_0}$ is the depth at the ridge crest, typically 2500 m.

Cooling plate model (1977)

The depth predicted by the square root of seafloor age found by the 1974 cooling mantle derivation is too deep for seafloor older than 80 million years. Depth is better explained by a cooling lithosphere plate model rather than the cooling mantle half-space. The plate has a constant temperature at its base and spreading edge. Derivation of the cooling plate model also starts with the heat flow equation in one dimension as does the cooling mantle model. The difference is in requiring a thermal boundary at the base of a cooling plate. Analysis of depth versus age and depth versus square root of age data allowed Parsons and Sclater to estimate model parameters (for the North Pacific):

~125 km for lithosphere thickness

${\displaystyle T_1\sim 1350{}^{\circ }\mathrm{C}}$ at base and young edge of plate

${\displaystyle \alpha \sim 3.2\cdot {10}^{-5}{}^{\circ }{\mathrm{C}}^{-1}}$

Assuming isostatic equilibrium everywhere beneath the cooling plate yields a revised age-depth relationship for older sea floor that is approximately correct for ages as young as 20 million years:

${\displaystyle d(t)=6400-3200\exp (-t/62.8)}$ meters

Thus older seafloor deepens more slowly than younger and in fact can be assumed almost constant at ~6400 m depth. Their plate model also allowed an expression for conductive heat flow, q(t) from the ocean floor, which is approximately constant at ${\displaystyle 1\cdot {10}^{-6}\mathrm{c}\mathrm{a}\mathrm{l}{\mathrm{c}\mathrm{m}}^{-2}{\mathrm{s}\mathrm{e}\mathrm{c}}^{-1}}$ beyond 120 million years:

${\displaystyle q(t)=11.3/\sqrt{t}}$

Parsons and Sclater concluded that some style of mantle convection must apply heat to the base of the plate everywhere to prevent cooling down below 125 km and lithosphere contraction (seafloor deepening) at older ages. Morgan and Smith showed that the flattening of the older seafloor depth can be explained by flow in the asthenosphere below the lithosphere.

The age-depth-heat flow relationship continued to be studied with refinements in the physical parameters that define ocean lithospheric plates.

Impacts

The usual method for estimating the age of the seafloor is from marine magnetic anomaly data and applying the Vine-Matthews-Morley hypothesis. Other ways include expensive deep sea drilling and dating of core material. If the depth is known at a location where anomalies are not mapped or are absent, and seabed samples are not available, knowing the seabed depth can yield an age estimate using the age-depth relationships.

Along with this, if the seafloor spreading rate in an ocean basin increases, then the average depth in that ocean basin decreases and therefore its volume decreases (and vice versa). This results in global eustatic sea level rise (fall) because the Earth is not expanding. Two main drivers of sea level variation over geologic time are then changes in the volume of continental ice on the land, and the changes over time in ocean basin average depth (basin volume) depending on its average age.