Editing Isostasy (section)

== Models ==
Three principal models of isostasy are used:<ref name=Watts2001/>{{sfn|Kearey|Klepeis|Vine|2009|pp=42-45}}
# The Airy–Heiskanen model – where different topographic heights are accommodated by changes in [[Crust (geology)|crustal]] thickness, in which the crust has a constant density
# The Pratt–Hayford model – where different topographic heights are accommodated by lateral changes in [[Rock (geology)|rock]] [[density]].
# The Vening Meinesz, or flexural isostasy model – where the [[lithosphere]] acts as an [[Elasticity (physics)|elastic]] plate and its inherent rigidity distributes local topographic loads over a broad region by bending.

Airy and Pratt isostasy are statements of buoyancy, but flexural isostasy is a statement of buoyancy when deflecting a sheet of finite elastic strength. In other words, the Airy and Pratt models are purely hydrostatic, taking no account of material strength, while flexural isostacy takes into account elastic forces from the deformation of the rigid crust. These elastic forces can transmit buoyant forces across a large region of deformation to a more concentrated load.

{{anchor|Isostatic anomaly}}
Perfect isostatic equilibrium is possible only if mantle material is in rest. However, [[Mantle convection|thermal convection]] is present in the mantle. This introduces viscous forces that are not accounted for the static theory of isostacy. The '''isostatic anomaly''' or IA is defined as the Bouger anomaly minus the gravity anomaly due to the subsurface compensation, and is a measure of the local departure from isostatic equilibrium. 
At the center of a level plateau, it is approximately equal to the [[free air anomaly]].{{sfn|Kearey|Klepeis|Vine|2009|pp=45-48}}  Models such as deep dynamic isostasy (DDI) include such viscous forces and are applicable to a dynamic mantle and lithosphere.<ref name="Czechowski2019">{{cite journal | title=Mantle Flow and Determining Position of LAB Assuming Isostasy | last=Czechowski | first=L. | journal=Pure and Applied Geophysics | year=2019 | volume=176 | issue=6 | pages=2451–2463 | doi=10.1007/s00024-019-02093-8| bibcode=2019PApGe.176.2451C | doi-access=free }}</ref> Measurements of the rate of [[isostatic rebound]] (the return to isostatic equilibrium following a change in crust loading) provide information on the viscosity of the upper mantle.{{sfn|Kearey|Klepeis|Vine|2009|p=45}}

=== Airy ===
[[File:Airy Isostasy.jpg|thumb|right|Airy isostasy, in which a constant-density crust floats on a higher-density mantle, and topography is determined by the thickness of the crust.]]
[[File:Backstripping and eustasy correction.jpg|thumb|Airy isostasy applied to a real-case basin scenario, where the total load on the mantle is composed by a crustal basement, lower-density sediments and overlying marine water]]

The basis of the model is [[Pascal's law]], and particularly its consequence that, within a fluid in static equilibrium, the hydrostatic pressure is the same on every point at the same elevation (surface of hydrostatic compensation):<ref name=Watts2001/>{{sfn|Kearey|Klepeis|Vine|2009|p=43}}

h<sub>1</sub>⋅ρ<sub>1</sub> = h<sub>2</sub>⋅ρ<sub>2</sub> = h<sub>3</sub>⋅ρ<sub>3</sub> = ... h<sub>n</sub>⋅ρ<sub>n</sub>

For the simplified picture shown, the depth of the mountain belt roots (b<sub>1</sub>) is calculated as follows:

:<math> (h_1+c+b_1)\rho_c = (c\rho_c)+(b_1\rho_m) </math>
:<math> {b_1(\rho_m-\rho_c)} = h_1\rho_c </math>
:<math> b_1 = \frac{h_1\rho_c}{\rho_m-\rho_c} </math>

where <math> \rho_m </math> is the density of the mantle (ca. 3,300&nbsp;kg m<sup>−3</sup>) and <math> \rho_c </math> is the density of the crust (ca. 2,750&nbsp;kg m<sup>−3</sup>). Thus, generally:
<br />
:''b''<sub>1</sub> ≅ 5⋅''h''<sub>1</sub>

In the case of negative topography (a marine basin), the balancing of lithospheric columns gives:

:<math> c\rho_c = (h_2\rho_w)+(b_2\rho_m)+[(c-h_2-b_2)\rho_c] </math>
:<math> {b_2(\rho_m-\rho_c)} = {h_2(\rho_c-\rho_w)} </math>
:<math> b_2 = (\frac{\rho_c-\rho_w}{\rho_m-\rho_c}){h_2} </math>

where <math> \rho_m </math> is the density of the mantle (ca. 3,300&nbsp;kg m<sup>−3</sup>), <math> \rho_c </math> is the density of the crust (ca. 2,750&nbsp;kg m<sup>−3</sup>) and <math> \rho_w </math> is the density of the water (ca. 1,000&nbsp;kg m<sup>−3</sup>). Thus, generally:
<br />
:''b''<sub>2</sub> ≅ 3.2⋅''h''<sub>2</sub>

=== Pratt ===
For the simplified model shown the new density is given by: <math> \rho_1 = \rho_c \frac{c}{h_1+c} </math>, where <math>h_1</math> is the height of the mountain and c the thickness of the crust.<ref name=Watts2001/>{{sfn|Kearey|Klepeis|Vine|2009|pp=43-44}}

=== Vening Meinesz / flexural ===
[[File:Local-regional isostasy - flexure, elastic thickness.jpg|thumb|Cartoon showing the isostatic vertical motions of the lithosphere (grey) in response to a vertical load (in green)]]
This hypothesis was suggested to explain how large topographic loads such as [[seamounts]] (e.g. [[Hawaiian Islands]]) could be compensated by regional rather than local displacement of the lithosphere. This is the more general solution for [[lithospheric flexure]], as it approaches the locally compensated models above as the load becomes much larger than a flexural wavelength or the flexural rigidity of the lithosphere approaches zero.<ref name=Watts2001/>{{sfn|Kearey|Klepeis|Vine|2009|pp=44-45}}

For example, the vertical displacement ''z'' of a region of ocean crust would be described by the [[differential equation]]

:<math>D\frac{d^4z}{dx^4}+(\rho_m-\rho_w)zg = P(x)</math>

where <math>\rho_m</math> and <math>\rho_w</math> are the densities of the aesthenosphere and ocean water, ''g'' is the acceleration due to gravity, and <math>P(x)</math> is the load on the ocean crust. The parameter ''D'' is the ''flexural rigidity'', defined as

:<math>D=ET^3_c/12(1-\sigma^2)</math>

where ''E'' is [[Young's modulus]], <math>\sigma</math> is [[Poisson's ratio]], and <math>T_c</math> is the thickness of the lithosphere.  Solutions to this equation have a characteristic wave number

:<math>\kappa=\sqrt[4]{(\rho_m-\rho_w)g/4D}</math>

As the rigid layer becomes weaker, <math>\kappa</math> approaches infinity, and the behavior approaches the pure hydrostatic balance of the Airy-Heiskanen hypothesis.{{sfn|Kearey|Klepeis|Vine|2009|p=45}}