Sea Level Equation
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Sea Level Equation

The sea level equation (SLE) is the linear, integral equation that describes the sea-level variations associated with the Glacial Isostatic Adjustment (GIA).

The basic idea of the SLE dates back to 1888, when Woodward published his pioneering work on the form and position of mean sea level,[1] and only later has been refined by Platzman [2] and Farrell [3] in the context of the study of the ocean tides. In the words of Wu and Peltier,[4] the solution of the SLE yields the space- and time-dependent change of ocean bathymetry which is required to keep the gravitational potential of the sea surface constant for a specific deglaciation chronology and viscoelastic earth model. The SLE theory was then developed by other authors as Mitrovica & Peltier,[5] Mitrovica et al.[6] and Spada & Stocchi.[7] In its simplest form, the SLE reads

${\displaystyle S=N-U,}$

where ${\displaystyle S}$ is the sea-level change, ${\displaystyle N}$ is the sea surface variation as seen from Earth's center of mass, and ${\displaystyle U}$ is vertical displacement.

In a more explicit form the SLE can be written as follow:

${\displaystyle S(\theta ,\lambda ,t)={\frac {\rho _{i}}{\gamma }}G_{s}\otimes _{i}I+{\frac {\rho _{w}}{\gamma }}G_{s}\otimes _{o}S+S^{E}-{\frac {\rho _{i}}{\gamma }}{\overline {G_{s}\otimes _{i}I}}-{\frac {\rho _{w}}{\gamma }}{\overline {G_{o}\otimes _{o}S}},}$

where ${\displaystyle \theta }$ is colatitude and ${\displaystyle \lambda }$ is longitude, ${\displaystyle t}$ is time, ${\displaystyle \rho _{i}}$ and ${\displaystyle \rho _{w}}$ are the densities of ice and water, respectively, ${\displaystyle \gamma }$ is the reference surface gravity, ${\displaystyle G_{s}=G_{s}(h,k)}$ is the sea-level Green's function (dependent upon the ${\displaystyle h}$ and ${\displaystyle k}$ viscoelastic load-deformation coefficients - LDCs), ${\displaystyle I=I(\theta ,\lambda ,t)}$ is the ice thickness variation, ${\displaystyle S^{E}=S^{E}(t)}$ represents the eustatic term (i.e. the ocean-averaged value of ${\displaystyle S}$), ${\displaystyle \otimes _{i}}$ and ${\displaystyle \otimes _{o}}$ denote spatio-temporal convolutions over the ice- and ocean-covered regions, and the overbar indicates an average over the surface of the oceans that ensures mass conservation.

## References

1. ^ Woodward, R. S., 1888. On the form and position of mean sea level. United States Geol. Survey Bull., 48, 87170.
2. ^ Platzman , G. W., 1971. Ocean tides. In Lectures in Applied Mathematics, 14, part 2, pp. 239292, American Mathematical Society, Providence, RI.
3. ^ Farrell, W. E., 1973. Earth tides, ocean tides and tidal loading. Phil. Trans. R. Soc. Lond. A, 274, 253259.
4. ^ Wu, P., and W. R. Peltier. Glacial isostatic adjustment and the free-air gravity anomaly as a constraint on deep mantle viscosity. Geophys. J. R. Astron. Soc., 74, 377449, 1983.
5. ^ Mitrovica, J. X. & Peltier, W. R., 1991. On postglacial geoid subsidence over the equatorial ocean. J. geophys. Res., 96, 20,05320,071.
6. ^ Mitrovica, J. X., Davis, J. L. & Shapiro, I. I., 1994. A spectral formal- ism for computing three-dimensional deformations due to surface loads. J. geophys. Res., 99, 70577073.
7. ^ Spada G. & Stocchi, P., 2006. The Sea Level Equation, Theory and Numerical Examples. ISBN 88-548-0384-7, 96 pp., Aracne, Roma.