Seafloor spreading

In the mantle half-space model,[28] the seabed height is determined by the oceanic lithosphere and mantle temperature, due to thermal expansion. The simple result is that the ridge height or ocean depth is proportional to the square root of its age.[28] 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 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 ${\displaystyle T_{1}\cdot \Theta (-z)}$. 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 calculating in the frame of reference of the moving lithosphere (velocity v), which has spatial coordinate ${\displaystyle x'=x-vt,}$ ${\displaystyle T=T(x',z,t).}$ and the heat equation is:

${\displaystyle {\frac {\partial T}{\partial t}}=\kappa \nabla ^{2}T=\kappa {\frac {\partial ^{2}T}{\partial ^{2}z}}+\kappa {\frac {\partial ^{2}T}{\partial ^{2}x'}}}$

where ${\displaystyle \kappa }$ is the thermal diffusivity of the mantle lithosphere.

Since T depends on x' and t only through the combination ${\displaystyle x=x'+vt,}$:

${\displaystyle {\frac {\partial T}{\partial x'}}={\frac {1}{v}}\cdot {\frac {\partial T}{\partial t}}}$

Thus:

${\displaystyle {\frac {\partial T}{\partial t}}=\kappa \nabla ^{2}T=\kappa {\frac {\partial ^{2}T}{\partial ^{2}z}}+{\frac {\kappa }{v^{2}}}{\frac {\partial ^{2}T}{\partial ^{2}t}}}$

It is assumed that ${\displaystyle v}$ is large compared to other scales in the problem; therefore the last term in the equation is neglected, giving a 1-dimensional diffusion equation:

${\displaystyle {\frac {\partial T}{\partial t}}=\kappa {\frac {\partial ^{2}T}{\partial ^{2}z}}}$

with the initial conditions

${\displaystyle T(t=0)=T_{1}\cdot \Theta (-z).}$

The solution for ${\displaystyle z\leq 0}$ is given by the error function:

${\displaystyle T(x',z,t)=T_{1}\cdot \operatorname {erf} \left({\frac {z}{2{\sqrt {\kappa t}}}}\right)}$.

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:

${\displaystyle h(t)=h_{0}+\alpha _{\mathrm {eff} }\int _{0}^{\infty }[T(z)-T_{1}]dz=h_{0}-{\frac {2}{\sqrt {\pi }}}\alpha _{\mathrm {eff} }T_{1}{\sqrt {\kappa t}}}$

where ${\displaystyle \alpha _{\mathrm {eff} }}$ is the effective volumetric thermal expansion coefficient, and h₀ 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 ${\displaystyle \kappa }$ is small compared to ${\displaystyle L^{2}/A}$, 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 ${\displaystyle \alpha _{\mathrm {eff} }}$ is different from the usual thermal expansion coefficient ${\displaystyle \alpha }$ due to isostasic effect of the change in water column height above the lithosphere as it expands or retracts. Both coefficients are related by:

${\displaystyle \alpha _{\mathrm {eff} }=\alpha \cdot {\frac {\rho }{\rho -\rho _{w}}}}$

where ${\displaystyle \rho \sim 3.3\ \mathrm {g} \cdot \mathrm {cm} ^{-3}}$ is the rock density and ${\displaystyle \rho _{0}=1\ \mathrm {g} \cdot \mathrm {cm} ^{-3}}$ is the density of water.

By substituting the parameters by their rough estimates:

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

we have:[28]

${\displaystyle h(t)\sim {\begin{cases}h_{0}-390{\sqrt {t}}&{\text{for the Atlantic and Indian oceans}}\\h_{0}-350{\sqrt {t}}&{\text{for the eastern Pacific}}\end{cases}}}$

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 ${\displaystyle h(t)}$above a base or reference level ${\displaystyle h_{b}}$, the depth of the ocean ${\displaystyle d(t)}$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 2600 m.

The depth predicted by the square root of seafloor age derived above is too deep for seafloor older than 80 million years.[27] Depth is better explained by a cooling lithosphere plate model rather than the cooling mantle half-space.[27] The plate has a constant temperature at its base and spreading edge. Analysis of depth versus age and depth versus square root of age data allowed Parsons and Sclater[27] to estimate model parameters (for the North Pacific):

~125 km for lithosphere thickness

${\displaystyle T_{1}\thicksim 1350\ {}^{\circ }\mathrm {C} }$ at base and young edge of plate

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

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:

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

Thus older seafloor deepens more slowly than younger and in fact can be assumed almost constant at ~6400 m depth. 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.[27] Their plate model also allowed an expression for conductive heat flow, q(t) from the ocean floor, which is approximately constant at ${\displaystyle 1\cdot 10^{-6}\mathrm {cal} \,\mathrm {cm} ^{-2}\mathrm {sec} ^{-1}}$ beyond 120 million years:

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