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Technical Note: Two Types of Absolute Dynamic Ocean Topography

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Technical Note: Two Types of Absolute Dynamic Ocean Topography Ocean Sci., 14, 947–957, 2018 https://doi.org/10.5194/os-14-947-2018 © Author(s) 2018. This work is distributed under the Creative Commons Attribution 4.0 License. Technical note: Two types of absolute dynamic ocean topography Peter C. Chu Naval Ocean Analysis and Prediction Laboratory, Department of Oceanography Naval Postgraduate School, Monterey, CA 93943, USA Correspondence: Peter C. Chu ([email protected]) Received: 17 April 2018 – Discussion started: 23 May 2018 Revised: 28 July 2018 – Accepted: 15 August 2018 – Published: 4 September 2018 Abstract. Two types of marine geoid exist with the first type for the second type (satellite-determined marine geoid with being the average level of sea surface height (SSH) if the gravity anomaly) than for the first type (geostrophic balance water is at rest (classical definition), and the second type without gravity anomaly). Such a difference needs further at- being satellite-determined with the condition that the wa- tention from oceanographic and geodetic communities, es- ter is usually not at rest. The differences between the two pecially the oceanographic representation of the horizontal are exclusion (inclusion) of the gravity anomaly and non- gradients of the second type of MDOT (not the absolute sur- measurable (measurable) in the first (second) type. The as- face geostrophic currents). sociated absolute dynamic ocean topography (referred to as DOT), i.e., SSH minus marine geoid, correspondingly also has two types. Horizontal gradients of the first type of DOT represent the absolute surface geostrophic currents due to 1 Introduction water being at rest on the first type of marine geoid. Hor- izontal gradients of the second type of DOT represent the Let the coordinates (x, y, z) be in zonal, latitudinal, and surface geostrophic currents relative to flow on the second vertical directions. The absolute dynamic ocean topography O type of marine geoid. Difference between the two is quan- (hereafter referred to as DOT) D is the sea surface height titatively identified in this technical note through compari- (SSH, waves and tides filtered out) relative to the marine son between the first type of DOT and the mean second type geoid (i.e., the equipotential surface), of DOT (MDOT). The first type of DOT is determined by O O a physical principle that the geostrophic balance takes the D D S − N; (1) minimum energy state. Based on that, a new elliptic equa- O tion is derived for the first type of DOT. The continuation of where S is the SSH; N is the marine geoid height above O geoid from land to ocean leads to an inhomogeneous Dirich- the reference ellipsoid (Fig. 1). D is an important signal in O let boundary condition with the boundary values taking the oceanography and N is of prime interest in geodesy. Equa- satellite-observed second type of MDOT. This well-posed el- tion (1) is also applicable if defined relative to the center O liptic equation is integrated numerically on 1◦ grids for the of the Earth. The geoid height N.x;y/ and other associated world oceans with the forcing function computed from the measurable quantities such as gravity anomaly 1g.x, y/ are World Ocean Atlas (T , S) fields and the sea-floor topogra- related to the anomaly of the gravitational potential V .x, y, phy obtained from the ETOPO5 model of NOAA. Between z/ to a first approximation by the well-known Brun’s formula the first type of DOT and the second type of MDOT, the (e.g., Hofmann-Wellenhof and Moritz, 2005), relative root-mean square (RRMS) difference (versus RMS V .x;y;0/ of the first type of DOT) is 38.6 % and the RMS difference N.x;y/O D ; (2) in the horizontal gradients (versus RMS of the horizontal g gradient of the first type of DOT) is near 100 %. The stan- −2 dard deviation of horizontal gradients is nearly twice larger where g D 9:81 m s , is the globally mean normal gravity, which is usually represented by g0 in geodesy. The gravity Published by Copernicus Publications on behalf of the European Geosciences Union. 948 P. C. Chu: Technical Note: Two types of absolute dynamic ocean topography when the water at rest) without using the gravity anomaly. In this framework, the geostrophic balance 1 @pO 1 @pO u D − ; v D (7) g f ρO @y g f ρO @x and hydrostatic balance @pO D −ρg O (8) @z are used for large-scale (i.e., scale > 100 km) processes. Here (ug, vg/ are geostrophic current components; f is the Corio- lis parameter; (pO, ρ/O are in situ pressure and density, respec- tively, which can be decomposed into Figure 1. Two types of marine geoid and DOT: (a) the first type with N being the average level of SSH if water is at rest (classical defi- ρO D ρ C ρ.z/ C ρ; pO D −ρ gz C p.z/ C p: (9) nition), and (b) represents the second type with satellite-determined 0 0 N∗ (water in motion on N∗/. −3 Here, ρ0 D 1025 kg m is the characteristic density; (ρ, p/ are horizontally uniform with ρ vertically increasing with anomaly is the vertical derivative of the potential depth (stable stratification) @V .x;y;0/ @ρ=@z ≡ −ρ .n.z//2=g; (10) 1g.x;y/ D − ; (3) 0 @z where n(z) is the buoyancy frequency (or called the Brunt– where the anomaly of the gravity potential V satisfies the Vaisala frequency); (p, ρ/ are anomalies of pressure and den- Laplace equation sity. Near the ocean surface, it is common to use the charac- @2V @2V @2V teristic density and corresponding pressure (p0;ρ0/ to repre- C C D 0: (4) sent (pO, ρ/O . Vertical integration of Eq. (8) from N to S after @x2 @y2 @z2 replacing (pO, ρ/O by (p0;ρ0/ in Eqs. (7) and (8) leads to The vertical deflection is the slope of the geoid g @D g @D u .S/ − u .N/ D − ; v .S/ − v .N/ D ; (11) g g f @y g g f @x @NO 1 @V @NO 1 @V D ; D ; (5) @x g @x @y g @y where which connects to the gravity anomaly by D D S − N (12) ! @.1g/ @2NO @2NO D g C : (6) is the first type of DOT. Since the first type of marine geoid @z @x2 @y2 (N) is defined as the average level of SSH if the water is at rest, Equation (6) links the vertical gravity gradient to the hor- O izontal Laplacian of the marine geoid height N and serves ug.N/ D 0; vg.N/ D 0; (13) as the basic principle in the satellite marine geodesy. Since DO is the difference of the two large fields S and NO (2 orders the horizontal gradient of D represents the absolute surface of magnitude larger than D/O , it is extremely sensitive to any geostrophic currents. error in either S or NO – even a 1 % error in either field can After satellites came into operation, SSH has been ob- lead to error in DO that is of the same order of magnitude as DO served with uniquely sampled temporal and spatial resolu- itself (Wunsch and Gaposchkin, 1980; Bingham et al., 2008). tions by high-precision altimetry above a reference ellipsoid Before satellites came into operation, S was measured (not geoid; Fu and Haines, 2013). Two Gravity Recovery and from sparse surveying ships and tide gauge stations located Climate Experiment (GRACE) satellites, launched in 2002, along irregular local coastlines. However, NO was not easy to provide data to compute the marine geoid (called the GRACE observe. Without satellite measurements, the marine geoid is Gravity Model, GGM; see website: http://www.csr.utexas. defined as the average level of SSH if the water is at rest and edu/grace/ (last access: 31 July 2018); Tapley et al., 2003; denoted here by N, which is called the classical marine geoid Shum et al., 2011). In addition, the European Space Agency’s (or first type of marine geoid, Fig. 1a). The first type of ma- GOCE mission data, along with the GRACE data, have pro- rine geoid can be taken as a stand-alone concept in oceanog- duced the best mean gravity field or the geoid model at a raphy since it is on the basis of the hypothesis (mean SSH spatial scale longer than 67 km half-wavelength (or spherical Ocean Sci., 14, 947–957, 2018 www.ocean-sci.net/14/947/2018/ P. C. Chu: Technical Note: Two types of absolute dynamic ocean topography 949 harmonics completed to degree 300). This marine geoid is get-data/dynamic-ocean-typography/. This dataset is a sub- the solution of Eq. (6), traction of a second type of marine geoid of GRACE (Bing- ham et al., 2011) from a mean (1993 to 2006) altimetric sea 2 2 @ N∗ @ N∗ 1 @.1g/ C D ; surface. The change in marine geoid from first (N/ to second 2 2 @x @y g @z (N ∗/ type is represented by where N∗ is the satellite-determined marine geoid from the 1N D N ∗ − N: (16) measurable gravity anomaly 1g, and called the second type of marine geoid (Fig. 1b), which is different from N, defined Correspondingly, the change in DOT is given by by Eq. (13). Correspondingly, the second type of DOT is de- D − D − fined by 1D D∗ D 1N; (17) where Eqs. (12) and (15) are used. 1D is of interest in D∗ D S − N∗.t/; (14) oceanography. 1N is of interest in geodesy.
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