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An Oceanographer's Guide to GOCE and the Geoid

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An Oceanographer's Guide to GOCE and the Geoid An Oceanographer’s Guide to GOCE and the Geoid C. W. Hughes, R. J. Bingham To cite this version: C. W. Hughes, R. J. Bingham. An Oceanographer’s Guide to GOCE and the Geoid. Ocean Science, European Geosciences Union, 2008, 4 (1), pp.15-29. hal-00298352 HAL Id: hal-00298352 https://hal.archives-ouvertes.fr/hal-00298352 Submitted on 24 Jan 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Ocean Sci., 4, 15–29, 2008 www.ocean-sci.net/4/15/2008/ Ocean Science © Author(s) 2008. This work is licensed under a Creative Commons License. An Oceanographer’s Guide to GOCE and the Geoid C. W. Hughes and R. J. Bingham Proudman Oceanographic Laboratory, 6 Brownlow St., Liverpool L3 5DA, UK Received: 5 September 2006 – Published in Ocean Sci. Discuss.: 20 September 2006 Revised: 28 November 2007 – Accepted: 14 December 2007 – Published: 24 January 2008 Abstract. A review is given of the geodetic concepts nec- essary to understand the time dependent gravity field and its essary for oceanographers to make use of satellite gravity relationship to mass movements in the earth system. data to define the geoid, and to interpret the resulting prod- The primary geodetic quantity of interest to oceanogra- uct. The geoid is defined, with particular attention to sub- phers is the geoid. This is the level surface which would co- tleties related to the representation of the permanent tide, and incide with sea level if the ocean was in a static equilibrium. the way in which the geoid is represented in ocean models. It is the surface relative to which slopes must be calculated The usual spherical harmonic description of the gravitational to determine geostrophic currents (with a correction for at- field is described, together with the concepts required to cal- mospheric pressure gradients). The geoid can be determined culate a geoid from the spherical harmonic coefficients. A from space by measuring the Earth’s gravity field via its ef- brief description is given of the measurement system in the fect on the motion of satellites and of control masses within GOCE satellite mission, scheduled for launch shortly. Fi- those satellites. nally, a recipe is given for calculation of the ocean dynamic This note starts by defining the geoid, and noting some topography, given a map of sea surface height above a ref- subtleties to its definition. This is followed by a description erence ellipsoid, a set of spherical harmonic coefficients for of the spherical harmonic representation of the geoid and the gravitational field, and defining constants. some aspects of that which must be accounted for in inter- preting the data. A description of the GOCE measurement system is then given, followed by a recipe for how to calcu- late the ocean dynamic topography given a mean sea surface 1 Introduction and a set of spherical harmonic coefficients for the gravita- tional field. Satellite gravity measurements are becoming an important tool in physical oceanography, with the success of the GRACE mission and the imminent launch of GOCE. Ac- 2 Definition of the geoid cordingly, it is becoming important for oceanographers to understand satellite gravity. This is not as straightforward The geoid is a “horizontal” or “level” surface, a surface as might be thought, since there are a number of subtleties which is everywhere perpendicular to the local direction of of geodesy associated with the interpretation of gravity data, gravity. If there were no waves or currents in the ocean, it is and the usual product takes the form of a set of spherical har- where the sea surface would eventually settle in equilibrium. monic coefficients. Oceanographers are generally not used Since dynamics in the ocean make it possible for sea level to to working with either of these, so the purpose of this note depart from the geoid, the actual vertical distance of sea sur- is to describe the basics of the relevant geodetic issues, with face height above the geoid is known as the ocean’s dynamic particular reference to GOCE and its measurement system. topography. The aim is to describe the static (time mean) component of The actual shape of the geoid includes structure at all the gravity field, without going into the additional detail nec- length scales. To a first approximation it is a sphere with ra- dius about 6371 km. A closer approximation is an ellipsoid, Correspondence to: C. W. Hughes with equatorial radius about 21.4 km longer than the polar ra- ([email protected]) dius. Relative to this ellipsoid, the geoid undulates by up to Published by Copernicus Publications on behalf of the European Geosciences Union. 16 C. W. Hughes and R. J. Bingham: GOCE and the Geoid 100 m on the largest scales. On relatively short length scales along a geopotential) of the geopotential Wp on a pressure (a few km to a few hundred km) the geoid is closely related surface, considered as mapped onto the horizontal surface. to topography as the gravitational attraction of, for example, This is often written as hWp g hZ, making the approx- a seamount will pull water towards it leading to a bump in imation that g is constant,∇ and=− hence∇ that the geopotential on the sea surface above it (although gravity is stronger imme- the pressure surface can be represented as a geometric height diately above the seamount, this does not lead to a depres- Z of the pressure surface above a geopotential surface. sion in sea level. Rather, it is the lateral gravitational force Both of these equations involve geometric approximations which pulls water from either side of the seamount, leading of the order of the aspect ratio of the flow or of the slope of to a raised level above the seamount). This is the principle the pressure surface, but both retain their essential form when behind using sea level measurements from satellite altimetry generalized to account for these approximations. It is clear to help map the sea floor, as used for example by Smith and that it is the gradient of geopotential along a constant pres- Sandwell (1997). At longer length scales, topography does sure surface which is important in (2), and that geopotentials not have such a large influence as the weight of mountains are important as the surface along which pressure gradients is balanced by low density anomalies beneath them (rather are calculated (defining the direction of k) in (3). ˆ like the compensation of sea level anomalies by movements The concept of dynamic topography is most clearly inter- of the thermocline often observed in the ocean), as the moun- preted as in (2) as the geopotential on a constant pressure sur- tains “float” like icebergs on the mantle beneath. face. In the absence of a spatially-varying atmospheric pres- The geoid is not, however, simply a gravitational equipo- sure, the sea surface would be a surface of constant pressure, tential surface. The Earth is rotating, and in the rotating ref- and hence the geopotential on the sea surface would be a dy- erence frame we feel a centrifugal force which must be added namic topography. With variable atmospheric pressure we to the gravitational attraction to give what is usually termed must instead calculate geopotential on the inverse-barometer “gravity”. corrected sea surface, as described below. To summarise this in mathematical terms, if we write the acceleration due to gravity as the gradient of a potential Alternatively, the concept of dynamic topography can be conceived in terms of pressure as in (3). As long as sea level g W, (1) is close to the geoid, it can be used with hydrostatic balance = ∇ plus an “inverse barometer” correction for atmospheric pres- then the geoid is a surface of constant W (note the sign in sure, to calculate pressure on the geoid (this is something of this equation: the geodetic convention is, counterintuitively, a fiction where the geoid is above sea level, but is sufficiently that greater height and energy corresponds to lower potential, accurate for most calculations). unlike electrostatic theory, for example). We are therefore interested in mapping either the geopo- There are an infinite number of surfaces of constant W tential along the IB-corrected sea level surface, or the height (geopotential surfaces), which results in the question of of the IB-corrected sea level above the geoid (which leads to which one to define as “the” geoid. Although loosely de- a fictional pressure on the geoid). Note that, in the former fined as the geopotential closest to observed sea level, it is in case, we do not actually need to calculate the geoid, only practice usually calculated as the geopotential corresponding the geopotential at a known set of positions. This makes the to the value at the surface of a fictional reference ellipsoidal calculation simpler (but see Sect. 6.3), and avoids some am- earth with approximately the same mass, radius, and flatten- biguities (such as choices of reference surfaces). ing (i.e. equatorial bulge) as the real Earth. The main reason for oceanographic interest in the geopo- If η is the height of the sea surface above the geoid, tential lies in its special role in the primary dynamical bal- then the height of the IB-corrected sea surface is given by η ρgpa, where ρ is the density of seawater at the surface, g ance of large-scale ocean currents: geostrophy.
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