World ocean annual mean absolute geostrophic velocity on marine geoid of EIGEN‐6C4 from WOA13

This data set was established from two open sources: (a) geoid undulation (N) from EIGEN‐6C4, which is a static global combined gravity field model up to degree and order 2,190, and (b) ‘World ocean geostrophic velocity inverted from World Ocean Atlas 2013 with the P‐vector method’ (NCEI accession 0,121,576). With the given non‐positive values of N in the oceans, absolute geostrophic currents (u, v) are easily obtained on N with 1°×1° resolution from the second data set except the equatorial zone (5oS – 5oN) due to the non‐existence of the geostrophic balance. Altogether, the data set contains 15,868 (u, v) data pairs. The data set shows that the hypothetical situation that there would be no flow on the geoid does not exist.


| INTRODUCTION
The marine geoid (N) is an equipotential surface of the earth's gravity field to represent the mean sea level without the influence of tides, currents and winds. With a hypothetical situation that if the sea surface elevation was governed only by geostrophy and coincided one particular equipotential taken as the geoid, then there would be no flow on the geoid. This hypothesis has never been tested since the marine geoid is generally considered as an imaginary surface by the oceanographic community. After satellites coming into practice, the surface N has been determined by various gravity models using satellite observations such as Gravity field and steadystate Ocean Circulation Explorer (GOCE) (Johannessen et al., 2003), Gravity Recovery and Climate Experiment (GRACE) (Jet Propulsion Laboratory, 1998) and combined GOCE-GRACE (Bruinsma et al., 2013). Thus, the hypothesis can be tested through calculating the absolute geostrophic current (AGC) on N from temperature (T) and salinity (S) fields using existing inverse methods such as the β-spiral (Stommel & Schott, 1977), box model (Wunsch, 1978), minimum energy (Chu, 2018) and P-vector (Chu, 1995;Chu et al., 1998a;Chu & Li, 2000;Yuan et al., 2014).
where (λ, φ) are longitude and latitude; T is the static disturbing gravity potential and given by Kosteleckў et al., (2015) where G = 6.674 × 10 -11 m 3 kg -1 s -2 is the gravitational constant; M = 5.9736 × 10 24 kg is the mass of the earth; R = 6.3781364 × 10 6 m is the earth radius; r is the radial distance with z = r -R; P l,m (sin ) is the Legendre associated functions with (l, m) the degree and order of the harmonic expansion; (C l,m , C el l,m , S l,m ) are the harmonic geopotential coefficients (Stokes parameters) with C el l,m belonging to the reference ellipsoid (Stokes, 1849). The EIGEN-6C4 model was developed jointly by the GFZ Potsdam and GRGS Toulouse up to degree and order 2,190. Following the instruction in the website, the author ran the EIGEN-6C4 model in 1°×1° resolution for 17 s to get the global N ( Figure 1).

| P-vector inverse method
The P-vector inverse method was proposed in 1995 and described in detail in a book (Chu, 2006). Readers may ignore this section if there is no intention to use this method to calculate AGC from (T, S) data. This method is outlined as follows (Chu, 1995(Chu, , 2006Chu & Fan, 2015a). Let the earth-fixed local coordinates (x, y, z) be used in the P-vector calculation with x-axis in the zonal direction (eastward positive), y-axis in the meridional direction (northward positive), and z-axis in the vertical (upward positive). Let V be the velocity vector and ρ the density. Large-scale motions in the ocean interior are geostrophic and hydrostatic balanced, and density conserved, where f is the Coriolis parameter and ∇ is the three-dimensional gradient operator. Equations (3) and (4) show that the vector V is perpendicular to both ∇ and ∇q and therefore parallel to ∇ × ∇q. Thus, the velocity vector V is given by where the unit vector P is calculated from hydrographic data; is the speed parameter with | | the speed. After the unit vector P is determined, the speed parameters (k) = (x, y, z k ) and (m) = (x, y, z m ) satisfy the algebraic equations.
due to the thermal wind relation between z k and z m , If the determinant of the two linear algebraic equations (6a) and (6b) is non-zero, that is the Pvector spiral exists (Chu, 2000), the speed parameter at these two levels (k) and (m) can be determined after solving the linear algebraic equations (6a) and (6b), and in turn the horizontal velocity (i.e. AGC).
The traditional AGC calculation using the P-vector method employs the least squares fitting over the entire water column. This could introduce errors into the P-vector AGCs, because the potential density and potential vorticity are not conserved in the surface mixed layer, because the wholecolumn fitting could cause the fitting to be biased towards the large vectors. To test the accuracy, a second computation for AGC (u*, v*) was conducted with the P-vector calculation in the subsurface water column from z = −500 m to the bottom and calculate the upper ocean (z > −500 m) geostrophic currents using the dynamic height calculation. For each z level, the relative root mean square difference (RRMSD).
is calculated to represent the accuracy. Here, N k is the number of the grid points for the z k -level; N is the total number of the grid points for the three-dimensional ocean. Figure 2 shows that the RRMSD reduces rapidly with depth from 0.12 near the sea surface to 0.013 at z = −5,500 m. Besides, the number of observational data to build WOA13 (T, S) climatology is relatively low in high latitudes. This may cause large error bars there. Using the P-vector method, a global three-dimensional AGC F I G U R E 2 Vertical dependence of RRMSD between two AGC calculations 134 | CHU data set was established from WOA13 and published on the NCEI website (https://data.nodc.noaa.gov/cgi-bin/iso?id=gov. noaa.nodc:0121576) for public use (Chu & Fan, 2015a,b).

| Identification of AGC on N
The aforementioned P-vector method is simple and effective to compute AGC from (T, S) fields. If we use the existing inverted AGC field from the open data set on the NCEI website (https://data.nodc.noaa.gov/cgi-bin/iso?id=gov.noaa. nodc:0121576), the annual mean AGC data set (WOZV13_ annual.nc) should be downloaded since the static gravity model EIGEN-6C4 provides the climatological annual mean N field. This open AGC data set is in the z-coordinate system from z = 0 (top) to z = −8,900 m (bottom). Therefore, the AGC on N in the oceans can be obtained from this data set with only non-positive N values. Because positive N is out of the range of z [0 ≥ z ≥ −8,900 m], no data would be available from the climatological AGC field.

| DATA RECORDS
This global data set for the AGC on N is publicly available at the Flanders Marine Institute website (https://doi. org/10.14284/ 398) as text and NetCDF files, which include data citation, data set identifiers, metadata and ordering instructions. The 15,868 AGC vectors (u N ,v N ) (Figure 3), histograms of (u N , v N ) ( Figure 4) and associate statistical parameters (Table 1) clearly show that the water moves on N.
The gappy areas in Figure 3 include the area with (N > 0, i.e. above the sea surface) where no data exist in the annual mean AGC data set (WOZV13_annual.nc) and the equatorial area (5 o S -5 o N) where the geostrophic balance does not exist. A possible approach to determine AGC on the marine geoid for N > 0 is to let the N surface and the S surface not relative to the same ellipsoid, that is to submerge the N surface into the ocean and to calculate the AGCs on the entire N surface. This will lead to a classical problem in physical oceanography about the reference level, which is beyond the scope of this paper.

| TECHNICAL VALIDATION
The two models (EIGEN-6C4 and P-vector) are well established for many years. The EIGEN-6C4 is a geodetic community model after validation (Förste et al., 2014;Ince et al., 2019;Klokočnìk et al., 2020;Kosteleckў et al., 2015;Pal et al., 2016;Steffen et al., 2017). The P-vector method was evaluated (Chu et al., 1998b) and applied to calculate the absolute velocity from hydrographic data for the South China Sea (Chu et al., 1998a;Chu & Li, 2000), the Japan  (Chu et al., 2001a) and the western North Pacific (Chu et al., 2002(Chu et al., , 2003Yuan et al., 2014). In conjunction with the wind forcing, the P-vector method is also used to calculate the global volume transport (Chu & Fan, 2007). To reduce error, a variational P-vector method was developed (Chu et al., 2001b).

| DATA DOWNLOAD
The climatological annual mean AGC on the marine geoid (u N , v N ) (unit: cm/s) along with the EIGEN-6C4 geoid (N) data (unit: m) on 1°×1° resolution was published by the Flanders Marine Institute at the website https://doi. org/10.14284/ 398 for public use. Each data set has text and netCDF formats with (uvN.nc, uvN.txt) for the annual mean AGC on N and (EIGEN-6C4.nc, EIGEN-6C4.txt) for the geoid data. Readers are also encouraged to calculate the monthly mean AGC on the marine geoid from the two open sources, http://icgem.gfz-potsd am.de/home and https://data. nodc.noaa.gov/cgi-bin/iso?id=gov.noaa.nodc:0121576 with two MATLAB codes. The first code pv.m is used to calculate the AGC from (T, S) fields (Appendix 1). The second code getuvN.m is used to get AGC on the marine geoid undulation N [i.e. (u N , v N )].