RETRACTED ARTICLE: A Complete Formula of Ocean Surface Absolute Geostrophic Current

Abstract

Absolute geostrophic current at the ocean surface (S) contains three components: (1) absolute geostrophic current at the geoid undulation (N), (2) geostrophic shear (between S and N) due to horizontal gradient of dynamic ocean topography, and (3) geostrophic shear (between S and N) due to horizontal gradient of density. However, only the second component is used to represent the surface absolute geostrophic current in physical oceanography and satellite geodesy. Four public datasets are used to calculate each component. Evident difference between the surface absolute geostrophic current and the second component (i.e., geostrophic shear due to horizontal gradient of dynamic ocean topography) was found by the statistical characteristics with the relative root mean square difference of 0.605. It shows that the common practice in physical oceanography and satellite geodesy is not correct. The horizontal gradient of dynamic ocean topography alone cannot be claimed as the surface absolute geostrophic current.

Introduction

Ocean surface absolute geostrophic current (AGC) is the linkage between oceanography and satellite geodesy through dynamic ocean topography D, which is the sea surface height (S) (waves and tides filtered out) relative to the geoid undulation (N), D = S − N. Before satellites coming into practice, the two disciplines (oceanography and geodesy) were almost no connection. The gravity measurements over the ocean were very sparse. The marine geoid undulation N would coincide with the average level of sea surface height if the water were at rest. With such a theoretical hypothesis (no motion at N), it is a common practice in oceanography that the horizontal gradient of S relative to the level of no motion (to be considered as D) represents the AGC at S. At that time, the geodetic community did not use the surface AGC.

After satellites coming into practice, S is measured by satellite altimetry (relative to ellipsoid)^1,2; N (also relative to ellipsoid) is determined by gravity models with satellite measurements, such as Gravity field and steady-state Ocean Circulation Explorer (GOCE)³, Gravity Recovery and Climate Experiment (GRACE)⁴, and combined GOCE-GRACE⁵. The mean dynamic ocean topography (MDOT) is obtained through subtracting N from mean S. The MDOT calculated from satellite measurements is different from the classical conception (no motion at N)⁶.

Now, the two communities have common interest on MDOT through the surface AGC. The oceanographic community uses the satellite determined MDOT to get mean general ocean circulation, i.e., surface AGC. The geodetic community uses the surface AGC to verify the static gravity models. Questions arise: Is water at rest at the given N in the oceans? Does horizontal gradient of MDOT represent the mean surface AGC? To answer these questions, a complete formula of ocean surface AGC is presented on the base of geostrophic and hydrostatic balances. It contains three parts: (1) AGC at N, (2) geostrophic shear from N to S due to horizontal gradient of MDOT, and (3) geostrophic shear from N to S due to horizontal gradient of density. The general practice in oceanography and geodesy only uses the second part to represent the surface AGC.

When N is given, it is very easy for an oceanographer to identify if N is a level of no motion. In fact, satellite measurements, such as altimetry for S and gravimetry from GOCE and GRACE for N, determine MDOT with a minimum value of −1.9981 m and a maximum value of 1.9538 m (see website: https://grace.jpl.nasa.gov/data/get-data/dynamic-ocean-typography/). It is hard to imagine that a level (i.e., the marine geoid N) within 2 m above and below the ocean surface (S) would be a level of no motion. The AGC at N can be determined from temperature and salinity profiles using the existing inverse methods such as the β-spiral⁷, box model⁸, and P-vector^9,10,11,12. The geostrophic shear between S and N due to density can be obtained from temperature and salinity profiles. Thus, the three parts of the surface AGC can be evaluated, and so as the errors of using horizontal gradient of MDOT only for the surface AGC.

A Complete Formula

The global oblate reference ellipsoid is used in geodesy with the semi-major axis (used for the Earth radius) as the equatorial reference and the semi-minor axis as the distance between the Earth center and either pole. The relative difference between the semi-major and semi-minor axes (called the flattening) closes to 1/300. The gravity field potential is the sum of the potential of the Earth’s gravitational attraction and the potential of the centrifugal force due to the Earth’s rotation. The centrifugal force varies on latitude with a maximum at the equator and a minimum (zero) at both poles. The gravity (g), most important variable in geodesy, is generally 0.5% more at the poles than on the equator due to the flattening (less distance to the center of the Earth at the pole than on the equator) and the centrifugal force. Various global gravity field models have been developed to describe the three-dimensional Earth’s gravity field potential, and in turn to identify the geoid undulation (N)¹³. However, the flattening of the Earth (1/300) and variation of g (0.5%) are neglected in oceanography.

The geostrophic and hydrostatic balances in the ocean are represented by

$$-(2\Omega \,\sin \,\varphi )v=-\,\frac{1}{\rho Rcos\varphi }\frac{\partial p}{\partial \lambda },\,(2\Omega \,\sin \,\varphi )u=-\,\frac{1}{\rho R}\frac{\partial p}{\partial \varphi }$$

(1)

$$\frac{\partial p}{\partial z}=-\,\rho {g}_{0}$$

(2)

where g₀ = 9.81 m/s², is the globally averaged gravity; (φ, λ, z) are the latitude, longitude, and vertical coordinate (positive upward); (u, v) are zonal and latitudinal components of the AGC; p is the pressure; Ω = 2π/(86400 s), is the Earth rotation rate; and R = 6.3781364 × 10⁷ m, is the Earth radius. The equatorial zone (5°S–5°N) is not included in the analysis because the geostrophic balance (1) does not exist. Let the pressure field at N and S be represented by ${p}_{N}(\lambda ,\varphi )$ and ${p}_{S}(\lambda ,\varphi )$. Vertical integration of hydrostatic balance Eq. (2) from N to S gives the pressure at S

$${p}_{S}(\lambda ,\varphi )={p}_{N}(\lambda ,\varphi )-{\int }_{N(\lambda ,\,\varphi )}^{S(\lambda ,\,\varphi )}\rho (\lambda ,\varphi ,z){g}_{0}dz$$

(3)

Horizontal pressure gradients at S are given by

$$\frac{\partial {p}_{S}(\lambda ,\varphi )}{\cos \,\varphi \partial \lambda }=\,\frac{\partial {p}_{N}(\lambda ,\varphi )}{\cos \,\varphi \partial \lambda }-\left[\frac{\partial S\left(\lambda ,\varphi \right)}{\cos \,\varphi \partial \lambda }\rho \left(\lambda ,\varphi ,S\right){g}_{0}-\frac{\partial N\left(\lambda ,\varphi \right)}{\cos \,\varphi \partial \lambda }\rho \left(\lambda ,\varphi ,N\right){g}_{0}\right]-{\int }_{N\left(\lambda ,\varphi \right)}^{S\left(\lambda ,\varphi \right)}{g}_{0}\frac{\partial \rho \left(\lambda ,\varphi ,z\right)}{\cos \,\varphi \partial \lambda }dz,\,$$

(4a)

$$\frac{\partial {p}_{S}\left(\lambda ,\varphi \right)}{\partial \varphi }=\frac{\partial {p}_{N}\left(\lambda ,\varphi \right)}{\partial \varphi }-\left[\frac{\partial S\left(\lambda ,\varphi \right)}{\partial \varphi }\rho \left(\lambda ,\varphi ,S\right){g}_{0}-\frac{\partial N\left(\lambda ,\varphi \right)}{\partial \varphi }\rho \left(\lambda ,\varphi ,N\right){g}_{0}\right]-{\int }_{N(\lambda ,\varphi )}^{S(\lambda ,\varphi )}{g}_{0}\frac{\partial \rho (\lambda ,\varphi ,z)}{\partial \varphi }dz$$

(4b)

The second terms in (4a and 4b) are re-written by

$$-{g}_{0}\left[\frac{\partial S\left(\lambda ,\varphi \right)}{\cos \,\varphi \partial \lambda }\rho \left(\lambda ,\varphi ,S\right)-\frac{\partial N\left(\lambda ,\varphi \right)}{\cos \,\varphi \partial \lambda }\rho \left(\lambda ,\varphi ,N\right)\right]=-\,{\rho }_{S}{g}_{0}\left[\frac{\partial S\left(\lambda ,\varphi \right)}{\cos \,\varphi \partial \lambda }-\frac{{\rho }_{N}}{{\rho }_{S}}\frac{\partial N\left(\lambda ,\varphi \right)}{\cos \,\varphi \partial \lambda }\right]$$

(5a)

$$-{g}_{0}\left[\frac{\partial S\left(\lambda ,\varphi \right)}{\partial \varphi }\rho \left(\lambda ,\varphi ,S\right)-\frac{\partial N\left(\lambda ,\varphi \right)}{\partial \varphi }\rho \left(\lambda ,\varphi ,N\right)\right]=-\,{\rho }_{S}{g}_{0}\left[\frac{\partial S\left(\lambda ,\varphi \right)}{\partial \varphi }-\frac{{\rho }_{N}}{{\rho }_{S}}\frac{\partial N\left(\lambda ,\varphi \right)}{\partial \varphi }\right]\,$$

(5b)

Here, ${\rho }_{S}=\rho (\lambda ,\varphi ,S),\,{\rho }_{N}=\rho (\lambda ,\varphi ,N)$, are the sea-water density at S and N. The density ratio ${\rho }_{N}/{\rho }_{S}\cong 1$ is due to the Boussinesq approximation. Substitution of (4a and 4b) into (1) leads to a complete formula for the surface AGC,

$$({u}_{S},\,{v}_{S}\,)=({u}_{1},{v}_{1})+({u}_{2},{v}_{2})+({u}_{3},{v}_{3}),$$

(6)

where (u₁, v₁) is the AGC at N;

$${u}_{2}(\lambda ,\varphi )=-\,\frac{{g}_{0}}{2\Omega \,\sin \,\varphi }\frac{\partial D(\lambda ,\varphi )}{R\partial \varphi },\,{v}_{2}(\lambda ,\varphi )=\frac{{g}_{0}}{2\Omega \,\sin \,\varphi }\frac{\partial D(\lambda ,\varphi )}{R\,\cos \,\varphi \partial \lambda }$$

(7)

is the geostrophic shear from N to S due to horizontal gradient of D; and

$${{\rm{u}}}_{3}({\rm{\lambda }},{\rm{\varphi }})=-\,\frac{{g}_{0}}{2\Omega \,\sin \,\varphi }{\int }_{N(\lambda ,\varphi )}^{S(\lambda ,\varphi )}\frac{\partial \rho (\lambda ,\varphi ,z)}{R\partial \varphi }dz,\,{{\rm{v}}}_{3}({\rm{\lambda }},{\rm{\varphi }})=\frac{{g}_{0}}{2\Omega \,\sin \,\varphi }{\int }_{N(\lambda ,\varphi )}^{S(\lambda ,\varphi )}\frac{\partial \rho (\lambda ,\varphi ,z)}{R\,\cos \,\varphi \partial \lambda }dz\,$$

(8)

is the geostrophic shear from N to S due to horizontal gradient of density. The formula (6) is not the mode decomposition of the surface AGC. It is just a horizontal differentiation of (3) with the righthand side containing the definite integral with variable upper limit $S(\lambda ,\varphi )$ and lower limit $N(\lambda ,\varphi )$, and shows the geostrophic shear from N to S containing the horizontal gradients of DOT (v₂) and density (v₃). The part of v₃ is not included in v₂.

Before satellites coming into practice, classical geodesy uses a theoretical hypothesis that the marine geoid is the average level of sea surface height if the water is at rest, i.e., vanish of AGC at N, (u₁, v₁) = 0. After ignoring horizontal density gradient between S and N, (u₃, v₃) = 0, the AGC at the ocean surface is determined solely by the horizontal MDOT gradients,

$${u}_{S}(\lambda ,\varphi )=-\,\frac{{g}_{0}}{2\Omega \,\sin \,\varphi }\frac{\partial D(\lambda ,\varphi )}{R\partial \varphi },\,{v}_{S}(\lambda ,\varphi )=\frac{{g}_{0}}{2\Omega \,\sin \,\varphi }\frac{\partial D(\lambda ,\varphi )}{R\,\cos \,\varphi \partial \lambda }$$

(9)

Similar idea [existence of a reference level of no motion (z₀)] prevails in oceanography for a long time in determination of AGC using hydrographic data, with various z₀ values (−500, −550, −650, −1000, −1200, −1800, −2000, −4000 m, …) by different people for different regions¹⁴. If the same formula (6) is used for the surface AGC [represented by $(\tilde{u},\tilde{v})$] with replacement of N by z₀, the AGC on z₀ is zero,

$$({\tilde{{\rm{u}}}}_{1},\,{\tilde{{\rm{v}}}}_{1})=0.$$

(10)

The second term in (6) becomes the horizontal gradient of sea surface height S since the reference level z₀ is flat,

$${\tilde{u}}_{2}(\lambda ,\varphi )=-\,\frac{{g}_{0}}{2\Omega \,\sin \,\varphi }\frac{\partial S(\lambda ,\varphi )}{R\partial \varphi },\,{\tilde{v}}_{2}(\lambda ,\varphi )=\frac{{g}_{0}}{2\Omega \,\sin \,\varphi }\frac{\partial S(\lambda ,\varphi )}{R\,\cos \,\varphi \partial \lambda }$$

(11)

$${\tilde{{\rm{u}}}}_{3}({\rm{\lambda }},{\rm{\varphi }})=-\,\frac{{g}_{0}}{2\Omega \,\sin \,\varphi }{\int }_{{z}_{0}}^{S(\lambda ,\varphi )}\frac{\partial \rho (\lambda ,\varphi ,z)}{R\partial \varphi }dz,\,{\tilde{v}}_{3}({\rm{\lambda }},{\rm{\varphi }})=\frac{{g}_{0}}{2\Omega \,\sin \,\varphi }{\int }_{{z}_{0}}^{S(\lambda ,\varphi )}\frac{\partial \rho (\lambda ,\varphi ,z)}{R\,\cos \,\varphi \partial \lambda }dz\,$$

(12)

Despite the uncertainty of z₀, Eqs. (10)–(12) are still used in oceanography such as (z₀ = −1000 m)¹⁵ and (z₀ = −500 m)¹⁶.

To eliminate the uncertainty in determining z₀, the geoid undulation N was suggested to replace z₀ as the reference level¹⁷, i.e., use of (9) for the surface AGC. It becomes the common belief in marine geodesy and oceanography. Numerous papers have been published using the incomplete formula (9) to represent the surface AGC^{18,19,20,21,22,23,24}. Note that the theoretical hypothesis of no motion at N hasn’t been tested. Since satellite altimetry and gravimetry coming into practice, high-resolution observational sea surface height and gravity data were collected. The geoid undulation N was calculated by various gravity models. Now, it is ready to test this theoretical hypothesis (no motion at N) using existing three-dimensional climatological geostrophic current data. In addition, the error due to use of incomplete formula (9) can also be estimated.

Data sources

Four existing datasets were used in this study: (1) EIGEN-6C4 for N (http://dataservices.gfz-potsdam.de/icgem/showshort.php?id=escidoc:1119897)^13,25; (2) world ocean geostrophic velocity inverted from World Ocean Atlas 2013 (WOA13) with the P-Vector method (https://data.nodc.noaa.gov/cgi-bin/iso?id=gov.noaa.nodc:0121576) for the AGC at N, i.e., (u₁, v₁)^26,27; (3) MDOT (https://grace.jpl.nasa.gov/data/get-data/dynamic-ocean-typography/) for calculating (u₂, v₂)²⁸, and (4) WOA13 temperature and salinity fields (https://www.nodc.noaa.gov/OC5/woa13/) for calculating (u₃, v₃).

EIGEN-6C4

EIGEN-6C4, elaborated jointly by GFZ Potsdam and GRGS Toulouse, is a static global combined gravity field model up to degree and order 2190. The model is on the website http://icgem.gfz-potsdam.de/home for public use^13,25. The author ran the model with 1° × 1° resolution for 17 seconds to get the global N (Fig. 1). Look at Fig. 1, no one can claim no geostrophic currents on N.

Figure 1

Digital data for EIGEN-6C4 geoid undulation (N) with 1° × 1°, computed online at the website http://icgem.gfz-potsdam.de/home.

World ocean geostrophic velocity inverted from WOA13

This dataset comprises 3D gridded climatological fields of AGC inverted from World Ocean Atlas-2013 (WOA13) temperature and salinity fields using the P-vector method. It provides a climatological AGC field that is dynamically compatible to the WOA2013 (T, S) fields²⁶. The dataset has the same spatial resolution and temporal variation (annual, monthly, seasonal) as WOA13 (T, S) fields, but does not cover the equatorial zone (5°S–5°N) due to the geostrophic balance being the theoretical basis for the P-vector inverse method²⁹. This dataset can be downloaded from the NCEI website: https://data.nodc.noaa.gov/cgi-bin/iso?id=gov.noaa.nodc:0121576. This three-dimensional AGC dataset is in the z-coordinate system from z = 0 (top) to z = −8,900 m (bottom). Therefore, the AGC at N in the oceans can be obtained from this dataset with only non-positive N values. Positive N is out of the range of z [0 ≥ z ≥ −8,900 m], no data would be available from the climatological AGC field²⁶. Altogether, we have 15,481 total data points. Figure 2b shows AGC vectors (u₁, v₁) at N.

Figure 2

Velocity vectors for (a) AGC at the surface S, v_S, (b) AGC at the geoid (N), v₁, (c) geostrophic shear due to MDOT, v₂, and (d) geostrophic shear due to density, v₃. Note that areas in the equatorial (5°S to 5°N) and with N > 0 are not included. The total number of grid points are 15,481.

MDOT

The MDOT data can be downloaded from NASA Jet Propulsion Laboratory web site: https://grace.jpl.nasa.gov/data/get-data/dynamic-ocean-typography/, with −1.9981 m as the minimum and 1.9538 m as the maximum. A tilt of S relative to N (i.e., MDOT) measures a part of the long-term averaged strength of geostrophic shear, (u₂, v₂) (Fig. 2c), rather than commonly thought AGC at the surface⁶. Also, water moves on S, how could the water not move on N (within 2 m above or below S)?

WOA13 temperature and salinity fields

The WOA13 climatological annual mean temperature and salinity fields with 1° × 1° resolution were downloaded from the website https://www.nodc.noaa.gov/OC5/woa13/. The three-dimensional density $\rho (\lambda ,\varphi ,z)$ was calculated using the international thermodynamic equation of seawater-2010, which was downloaded from the website: http://unesdoc.unesco.org/images/0018/001881/188170e.pdf. Substitution of $\rho (\lambda ,\varphi ,z)$ into (8) leads to the long-term averaged geostrophic shear from N to S due to horizontal density gradient, (u₃, v₃) (Fig. 2d). If $S(\lambda ,\varphi )$ takes positive values, the density between z = S(λ,φ) and z = 0 take the value at z = 0. This is justifiable since the computation in this paper is limited for negative values of N, and the MDOT varies between −1.9981 m and 1.9538 m. The maximum positive value for $S(\lambda ,\varphi )$ is less than 4 m, which is thinner than upper ocean mixed layer depth.

Statistical characteristics

Histograms of (u_S, v_S), (u₁, v₁), (u₂, v₂), and (u₃, v₃) (Fig. 3) and associated statistical parameters (Table 1) show evident difference between (u_S, v_S), and (u₂, v₂). Here, the statistical parameters (mean, standard deviation, skewness, and kurtosis) were calculated from 15,481 data points of each velocity component. All the velocity components have small mean values, small skewness for (u_S, v_S), (u₁, v₁), (u₂, v₂) (i.e., nearly symmetric) and large negative skewness (−3,72, −2.4) for (u₃, v₃). The standard deviation is much larger in u_S (13.60 cm/s) than in u₂ (8.44 cm/s), but comparable between v_S (7.51 cm/s) and v₂ (7.15 cm/s). This is due to small variabilities in v₁ (1.60 cm/s) and (0.32 cm/s). The kurtosis is much larger than 3 in all the components. The differences in the Kurtosis between u_S (4.10), and u₂ (7.24), and between v_S (8.84), and v₂ (7.55) are quite evident.

Figure 3

Histograms of (a) u_S, (b) v_S, (c) u₁, (d) v₁, (e) u₂, (f) v₂, (g) u₃, and (h) v₃. Evident difference of histograms between u_S and u₂ and v_S and v₂ indicates inaccuracy of using v₂ only to represent v_S (i.e., current practice in oceanography and satellite geodesy).

Table 1 Statistical characteristics of the AGC at the surface S (v_S), AGC at the geoid undulation N (v₁), geostrophic shear from N to S due to the horizontal gradient of the MDOT (v₂), and due to the horizontal gradient of the density (v₃), with the symbol ‘*’ to represent the current practice in physical oceanography and satellite geodesy. Here, the statistical parameters were calculated from 15,481 data points of each velocity component.

Relative root mean square difference

Summation of the three component vectors v₁ = (u₁, v₁), v₂ = (u₂, v₂), and v₃ = (u₃, v₃) gives ocean surface AGC vector v_S = (u_S, v_S). Figure 4 shows the relationships between each component v_K (K = 1, 2, 3) and as well as v₁ + v₂ versus v_S. For each case, the relative root mean square difference (E) is computed by

$${\rm{E}}({{\bf{v}}}_{{\rm{S}}},{\bf{v}})=\sqrt{\sum _{i}\sum _{j}[{({u}_{S}-u)}_{i,j}^{2}+{({v}_{S}-v)}_{i,j}^{2}]}/\sqrt{\sum _{i}\sum _{j}[{({u}_{S})}_{i,j}^{2}+{({v}_{S})}_{i,j}^{2}]}$$

(13)

where v is v₁, v₂, v₃, or v₁ + v₂; (i, j) are the horizontal grid indices with the total grid points of 15,481. Table 2 lists the values of E for all the three components: 0.866, 0.605, and 0.851. It is noted that v_S is poorly correlated to v₁, and v₃, with large values of E (0.866, 0.851). This is due to large amount data of (u₁, v₁) and (u₃, v₃) near zero (also see Fig. 3c,d,g,h), with no connection to the change of (u_S, v_S). Dynamical mechanisms causing such poor correlations need to be explored further.

Figure 4

Scatter diagrams for (a) u₁ and u_S, (b) v₁ and v_S, (c) u₂ and u_S, (d) v₂ and v_S, (e) u₃ and u_S, (f) v₃ and v_S, (g) (u₁ + u₂) and u_S, and (h) (v₁ + v₂) and v_S. The relative root mean square difference between the component vectors and v_S are presented on the left panels.

Table 2 Relative root mean square difference between each component and complete form of surface AGC with the symbol ‘*’ to represent the current practice in physical oceanography and satellite geodesy.

Evidence presented here shows that the relative root mean square difference between the surface AGC (v_S) and the geostrophic shear due to horizontal gradient of MDOT (v₂) is 0.605. Large differences in E and statistical parameters (mean, standard deviation, skewness, and kurtosis) show that the current practice in physical oceanography and satellite geodesy (representing v_S by v₂) is questionable. It is important to add the absolute geostrophic current at the geoid (v₁) and the geostrophic shear due to inhomogeneous density (v₃).