Surface elevation

In fluid dynamics, surface elevation, also called surface height or free-surface elevation, is the height of a free surface relative to a specified reference level. Values are positive above the reference and negative below it. The symbol η is used for surface elevation in descriptions of water waves and shallow-water flow.

Sources contain slighly different definitions of surface elevation, primarily varying the choice of reference. For example, sea-surface height can be meaqsured above a geoid or above a reference ellipsoid.

Definitions

Height above a fixed reference level

The British Oceanographic Data Centre's vocabulary defines surface elevation as a signed vertical distance from a fixed reference point. In ocean data, the CF Standard Name Table distinguishes sea-surface height above mean sea level, above the geoid, and above a reference ellipsoid. In that table, mean sea level is a local time average taken over a period long enough to remove tidal variations.

Displacement from still-water level

Sinusoidal wave parameters: depth d, amplitude a, wave height H, still water level SWL

In linear water-wave theory, surface elevation is measured from the still-water level (SWL). As a result, it is positive at a wave crest and negative at a wave trough. For a small-amplitude sinusoidal wave, the crest elevation and the depth of the trough below that level both equal the amplitude. Wave height is measured from a trough to the crest that follows it. In a sinusoidal wave it equals twice the amplitude.

Height and displacement in shallow-water models

In the shallow water equations, the height of the free surface can be distinguished from both water depth and displacement from a mean level. In Geoffrey K. Vallis's notation, η is the height of the water surface and ηb is the height of the bottom, both measured from the same arbitrary origin. The water depth h is thus their difference:

h=η−ηb.

If the origin is chosen as an average bottom height:

η=ηb+h=H+Δη,

where H is the mean water depth and Δη is the surface displacement from its mean level. Thus, η is a height measured from the origin, and Δη is how far the surface sits above or below its mean.

Sea-surface height and anomalies

Copernicus Marine Service uses two definitions of sea-surface height (SSH):

  • For ocean models, SSH is the height above the geoid, the reference surface set by Earth's gravity field.
  • For satellite altimetry, SSH is the height above a reference ellipsoid, a mathematical approximation to Earth's shape.

The model quantity corresponds to absolute dynamic topography (ADT) in altimetry.

A sea-level anomaly (SLA) is the difference between the sea-surface height and a mean sea surface (MSS):

SLA=SSH−MSS.

The mean surface is calculated over a specified reference period, so the anomaly depends on that period.

See also

References

  1. ↑ Hearn 2008, p. 33.
  2. ↑ U.S. Army Corps of Engineers 2008, p. II-1-42.
  3. 1 2 British Oceanographic Data Centre n.d.
  4. ↑ World Meteorological Organization 2018, p. 14.
  5. ↑ Vallis 2018, p. 34.
  6. ↑ Anais P. n.d., Sea Surface Height.
  7. ↑ CF Conventions 2026, sea_surface_height_above_mean_sea_level.
  8. ↑ U.S. Army Corps of Engineers 2008, p. II-1-9.
  9. ↑ U.S. Army Corps of Engineers 2008, p. II-1-4.
  10. ↑ World Meteorological Organization 2018, p. 2.
  11. ↑ Vallis 2018, pp. 33–34.
  12. ↑ Vallis 2018, p. 64.
  13. ↑ Anais P. n.d., Sea Surface Height; Absolute Dynamic Topography.
  14. ↑ Anais P. n.d., Sea Level Anomaly.

Sources

  • Anais P. (n.d.). "Sea Surface Height variables and their definitions". Copernicus Marine Help Center. Retrieved 2026-10-12.
  • "Surface elevation (S0600259)". NERC Vocabulary Server, collection S06. British Oceanographic Data Centre. n.d. Retrieved 2026-10-12.
  • "CF Standard Name Table, version 95". CF Conventions. 2026-09-16. sea_surface_height_above_mean_sea_level. Retrieved 2026-10-12.
  • Hearn, Clifford J. (2008). The Dynamics of Coastal Models (PDF). Cambridge: Cambridge University Press. ISBN 978-0-521-80740-1.
  • U.S. Army Corps of Engineers (2008-08-01). "Chapter II-1: Water Wave Mechanics". Coastal Engineering Manual, Part II: Coastal Hydrodynamics (PDF). EM 1110-2-1100 (Change 2 ed.). U.S. Army Corps of Engineers.
  • Vallis, Geoffrey K. (2018). Fluid Dynamics of the Atmosphere and Ocean (PDF) (Lecture notes, ECMM719). University of Exeter.
  • World Meteorological Organization (2018). Guide to Wave Analysis and Forecasting (PDF). WMO-No. 702 (2018 ed.). Geneva: World Meteorological Organization. ISBN 978-92-63-10702-2.

Public Domain This article incorporates text from this source, which is in the public domain: U.S. Army Corps of Engineers (2008-08-01). "Chapter II-1: Water Wave Mechanics". Coastal Engineering Manual, Part II: Coastal Hydrodynamics (PDF). EM 1110-2-1100 (Change 2 ed.). U.S. Army Corps of Engineers. pp. II-1-4, II-1-9.