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The depth of the seafloor on the flanks of a
mid-ocean ridge A mid-ocean ridge (MOR) is a seafloor mountain system formed by plate tectonics. It typically has a depth of about and rises about above the deepest portion of an ocean basin. This feature is where seafloor spreading takes place along a diverge ...
is determined mainly by the ''age'' of the
oceanic lithosphere A lithosphere () is the rigid, outermost rocky shell of a terrestrial planet or natural satellite. On Earth, it is composed of the crust (geology), crust and the portion of the upper mantle (geology), mantle that behaves elastically on time sca ...
; older seafloor is deeper. During
seafloor spreading Seafloor spreading or Seafloor spread is a process that occurs at mid-ocean ridges, where new oceanic crust is formed through volcanic activity and then gradually moves away from the ridge. History of study Earlier theories by Alfred Wegener an ...
, 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 Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convec ...
in
Earth's mantle Earth's mantle is a layer of silicate rock between the crust and the outer core. It has a mass of 4.01 × 1024 kg and thus makes up 67% of the mass of Earth. It has a thickness of making up about 84% of Earth's volume. It is predominantly sol ...
. The next idea connected seafloor spreading and
continental drift Continental drift is the hypothesis that the Earth's continents have moved over geologic time relative to each other, thus appearing to have "drifted" across the ocean bed. The idea of continental drift has been subsumed into the science of pla ...
in a model of
plate tectonics Plate tectonics (from the la, label=Late Latin, tectonicus, from the grc, τεκτονικός, lit=pertaining to building) is the generally accepted scientific theory that considers the Earth's lithosphere to comprise a number of large ...
. In 1969, the elevations of ridges was explained as
thermal expansion Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions. Temperature is a monotonic function of the average molecular kinetic ...
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 The upper mantle of Earth is a very thick layer of rock inside the planet, which begins just beneath the crust (at about under the oceans and about under the continents) and ends at the top of the lower mantle at . Temperatures range from appro ...
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 Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
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 Oceanic crust is the uppermost layer of the oceanic portion of the Plate tectonics, tectonic plates. It is composed of the upper oceanic crust, with pillow lavas and a dike (geology), dike complex, and the lower oceanic crust, composed of troct ...
, 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 Subduction is a geological process in which the oceanic lithosphere is recycled into the Earth's mantle at convergent boundaries. Where the oceanic lithosphere of a tectonic plate converges with the less dense lithosphere of a second plate, the ...
. 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 A lithosphere () is the rigid, outermost rocky shell of a terrestrial planet or natural satellite. On Earth, it is composed of the crust (geology), crust and the portion of the upper mantle (geology), mantle that behaves elastically on time sca ...
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 ridge A mid-ocean ridge (MOR) is a seafloor mountain system formed by plate tectonics. It typically has a depth of about and rises about above the deepest portion of an ocean basin. This feature is where seafloor spreading takes place along a diverge ...
s. The source of the lithosphere has a half-plane shape (''x'' = 0, ''z'' < 0) and a constant temperature ''T''1. 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''0 = 0. Thus at ''x'' = 0 the temperature is the
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
T_1\cdot\Theta(-z). The system is assumed to be at a quasi-
steady state In systems theory, a system or a Process theory, process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those p ...
, 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 has spatial coordinate x' = x-vt, T=T(x',z, t). and the
heat equation In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
is: :\frac = \kappa \nabla^2 T = \kappa\frac + \kappa\frac where \kappa is the
thermal diffusivity In heat transfer analysis, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure. It measures the rate of transfer of heat of a material from the hot end to the cold end. It has the SI ...
of the mantle lithosphere. Since ''T'' depends on ''x and ''t'' only through the combination x = x'+vt,: :\frac = \frac\cdot\frac Thus: :\frac = \kappa \nabla^2 T = \kappa\frac + \frac \frac It is assumed that v is large compared to other scales in the problem; therefore the last term in the equation is neglected, giving one-dimensional diffusion equation: :\frac = \kappa\frac with the initial conditions :T(t=0) = T_1\cdot\Theta(-z). The solution for z\le 0 is given by the
error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a special (non-elementary ...
: :T(x',z,t) = T_1 \cdot \operatorname \left(\frac\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'': :h(t) = h_0 + \alpha_\mathrm \int_0^ (z)-T_1z = h_0 - \frac\alpha_\mathrmT_1\sqrt where \alpha_\mathrm is the effective volumetric
thermal expansion Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions. Temperature is a monotonic function of the average molecular kinetic ...
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 \kappa is small compared to L^2/A, where ''L'' is the ocean width (from mid-ocean ridges to
continental shelf A continental shelf is a portion of a continent that is submerged under an area of relatively shallow water, known as a shelf sea. Much of these shelves were exposed by drops in sea level during glacial periods. The shelf surrounding an island ...
) and ''A'' is the age of the ocean basin. The effective thermal expansion coefficient \alpha_\mathrm is different from the usual thermal expansion coefficient \alpha due to isostasic effect of the change in water column height above the lithosphere as it expands or contracts. Both coefficients are related by: : \alpha_\mathrm = \alpha \cdot \frac where \rho \sim 3.3 \ \mathrm\cdot \mathrm^ is the rock density and \rho_0 = 1 \ \mathrm \cdot \mathrm^ is the density of water. By substituting the parameters by their rough estimates into the solution for the height of the ocean floor h(t): :\begin \kappa &\sim 8\cdot 10^ \ \mathrm^2\cdot \mathrm^&& \text \\ \alpha &\sim 4\cdot 10^ \ ^\mathrm^&& \text \\ T_1 &\sim 1220 \ ^\mathrm && \text \\ T_1 &\sim 1120 \ ^\mathrm && \text \end we have: :h(t) \sim \begin h_0 - 390 \sqrt & \text \\ h_0 - 350 \sqrt & \text \end 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 A continental shelf is a portion of a continent that is submerged under an area of relatively shallow water, known as a shelf sea. Much of these shelves were exposed by drops in sea level during glacial periods. The shelf surrounding an island ...
(roughly half the ocean width), and ''A'' is the ocean basin age. Rather than height of the ocean floor h(t)above a base or reference level h_b, the depth of the seabed d(t)is of interest. Because d(t)+h(t)=h_b(with h_b measured from the ocean surface) we can find that: :d(t)=h_b-h_0+350\sqrt; for the eastern Pacific for example, where h_b-h_0 is the depth at the ridge crest, typically 2600 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 :T_1\thicksim1350\ ^\mathrm at base and young edge of plate :\alpha\thicksim3.2\cdot 10^ \ ^\mathrm^ 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: :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. Their plate model also allowed an expression for conductive heat flow, ''q(t)'' from the ocean floor, which is approximately constant at 1\cdot 10^\mathrm\, \mathrm^ \mathrm^ beyond 120 million years: :q(t)=11.3/\sqrt 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 The asthenosphere () is the mechanically weak and ductile region of the upper mantle of Earth. It lies below the lithosphere, at a depth between ~ below the surface, and extends as deep as . However, the lower boundary of the asthenosphere is not ...
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 In geophysics, a magnetic anomaly is a local variation in the Earth's magnetic field resulting from variations in the chemistry or magnetism of the rocks. Mapping of variation over an area is valuable in detecting structures obscured by overlying ...
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 Mean sea level (MSL, often shortened to sea level) is an average surface level of one or more among Earth's coastal bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical datuma standardised g ...
sea level rise Globally, sea levels are rising due to human-caused climate change. Between 1901 and 2018, the globally averaged sea level rose by , or 1–2 mm per year on average.IPCC, 2019Summary for Policymakers InIPCC Special Report on the Ocean and Cry ...
(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.


See also

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Sea level Mean sea level (MSL, often shortened to sea level) is an average surface level of one or more among Earth's coastal bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical datuma standardised g ...
*
Sea-level curve The sea-level curve is the representation of the changes of the sea level throughout the geological history. The first such curve is the Vail curve or Exxon curve. The names of the curve refer to the fact that in 1977 a team of Exxon geologist ...
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Sea level equation Post-glacial rebound (also called isostatic rebound or crustal rebound) is the rise of land masses after the removal of the huge weight of ice sheets during the last glacial period, which had caused isostatic depression. Post-glacial rebound a ...
*
Sea level rise Globally, sea levels are rising due to human-caused climate change. Between 1901 and 2018, the globally averaged sea level rose by , or 1–2 mm per year on average.IPCC, 2019Summary for Policymakers InIPCC Special Report on the Ocean and Cry ...


References


Further reading

{{Cite journal, last=McKenzie, first=Dan, date=2018-05-30, title=A Geologist Reflects on a Long Career, journal=Annual Review of Earth and Planetary Sciences, language=en, volume=46, issue=1, pages=1–20, doi=10.1146/annurev-earth-082517-010111, bibcode=2018AREPS..46....1M, issn=0084-6597, doi-access=free Coastal and oceanic landforms Physical oceanography Basalt Geological processes Plate tectonics Volcanic landforms Oceanographical terminology