Editing Seafloor spreading (section)

===Cooling mantle model===
In the mantle half-space model,<ref name=":0" /> the oceanic crust height is determined by the [[oceanic lithosphere]] and mantle temperature, due to thermal expansion. Note that the seabed height is not the same as the oceanic crust height, as there is also a layer of sediments over the oceanic crust, typically few hundreds meter thick (but can also be few kilometers thick and up to 20 kilometers thick on few near-shore locations).<ref>[https://www.ngdc.noaa.gov/mgg/sedthick/ Total Sediment Thickness of the World's Oceans and Marginal Seas ]</ref> The simple result is that the ridge height or ocean depth is proportional to the square root of its age.<ref name=":0" /> Oceanic lithosphere is continuously formed at a constant rate at the [[mid-ocean ridge]]s. The source of the lithosphere has a half-plane shape (''x'' = 0, ''z'' < 0) and a constant temperature ''T''<sub>1</sub>. 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 to other typical scales in the problem. The temperature at the upper boundary of the lithosphere (''z'' = 0) is a constant ''T''<sub>0</sub> = 0. Thus at ''x'' = 0 the temperature is the [[Heaviside step function]] <math>T_1\cdot\Theta(-z)</math>. The system is assumed to be at a quasi-[[steady state]], so that the temperature distribution is constant in time, i.e. <math>T=T(x,z).</math>

By calculating in the frame of reference of the moving lithosphere (velocity ''v''), which has spatial coordinate <math>x' = x-vt,</math>  <math>T=T(x',z, t).</math> and the [[heat equation]] is:

:<math>\frac{\partial T}{\partial t} = \kappa \nabla^2 T = \kappa\frac{\partial^2 T}{\partial^2 z} + \kappa\frac{\partial^2 T}{\partial^2 x'}</math>

where <math>\kappa</math> is the [[thermal diffusivity]] of the mantle lithosphere.

Since ''T'' depends on ''x''' and ''t'' only through the combination <math>x = x'+vt,</math>:

:<math>\frac{\partial T}{\partial x'} = \frac{1}{v}\cdot\frac{\partial T}{\partial t}</math>

Thus:

:<math>\frac{\partial T}{\partial t} = \kappa \nabla^2 T = \kappa\frac{\partial^2 T}{\partial^2 z} + \frac{\kappa}{v^2} \frac{\partial^2 T}{\partial^2 t}</math>

It is assumed that <math>v</math> is large compared to other scales in the problem; therefore the last term in the equation is neglected, giving a 1-dimensional diffusion equation:

:<math>\frac{\partial T}{\partial t} = \kappa\frac{\partial^2 T}{\partial^2 z}</math>

with the initial conditions

:<math>T(t=0) = T_1\cdot\Theta(-z).</math>

The solution for <math>z\le 0</math> is given by the [[error function]]:

:<math>T(x',z,t) = T_1 \cdot \operatorname{erf} \left(\frac{z}{2\sqrt{\kappa t}}\right)</math>.

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'':

:<math>h(t) = h_0 + \alpha_\mathrm{eff} \int_0^{\infty} [T(z)-T_1]dz = h_0 - \frac{2}{\sqrt{\pi}}\alpha_\mathrm{eff}T_1\sqrt{\kappa t} </math>

where <math>\alpha_\mathrm{eff}</math> is the effective volumetric [[thermal expansion]] coefficient, and ''h<sub>0</sub>'' is the mid-ocean ridge height (compared to some reference).

The assumption that ''v'' is relatively large is equivalent to the assumption that the thermal diffusivity <math>\kappa</math> is small compared to <math>L^2/A</math>, where ''L'' is the ocean width (from mid-ocean ridges to [[continental shelf]]) and ''A'' is the age of the ocean basin.

The effective thermal expansion coefficient <math>\alpha_\mathrm{eff}</math> is different from the usual thermal expansion coefficient <math>\alpha</math> due to isostasic effect of the change in water column height above the lithosphere as it expands or retracts. Both coefficients are related by:

:<math> \alpha_\mathrm{eff} = \alpha \cdot \frac{\rho}{\rho-\rho_w}</math>

where <math>\rho \sim 3.3 \ \mathrm{g}\cdot  \mathrm{cm}^{-3}</math> is the rock density and <math>\rho_0 = 1 \ \mathrm{g} \cdot \mathrm{cm}^{-3}</math> is the density of water.

By substituting the parameters by their rough estimates:

:<math>\begin{align}
\kappa &\sim 8\cdot 10^{-7} \ \mathrm{m}^2\cdot \mathrm{s}^{-1} \\ 
\alpha &\sim 4\cdot 10^{-5} \ {}^{\circ}\mathrm{C}^{-1} \\
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{align}</math>

gives:<ref name=":0" />

:<math>h(t) \sim \begin{cases} h_0 - 390 \sqrt{t} & \text{for the Atlantic and Indian oceans} \\ h_0 - 350 \sqrt{t} & \text{for the eastern Pacific} \end{cases}</math>

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'' ~ ''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 <math>h(t)</math> above a base or reference level <math>h_b</math>, the depth of the ocean <math>d(t)</math> is of interest. Because <math>d(t)+h(t)=h_b</math>(with <math>h_b</math> measured from the ocean surface):

:<math>d(t)=h_b-h_0+350\sqrt{t}</math>; for the eastern Pacific for example, where <math>h_b-h_0</math> is the depth at the ridge crest, typically 2600 m.