Technical Note: Two Types of Absolute Dynamic Ocean Topography

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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 ˆ 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 ˆ ˆ a physical principle that the geostrophic balance takes the D = S − N, (1) minimum energy state. Based on that, a new elliptic equa- ˆ tion is derived for the first type of DOT. The continuation of where S is the SSH; N is the marine geoid height above ˆ geoid from land to ocean leads to an inhomogeneous Dirich- the reference ellipsoid (Fig. 1). D is an important signal in ˆ 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 ˆ 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)ˆ = , (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 = 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 ∂pˆ 1 ∂pˆ u = − , v = (7) g f ρˆ ∂y g f ρˆ ∂x and hydrostatic balance

∂pˆ = −ρg ˆ (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; (pˆ, ρ)ˆ 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- ρˆ = ρ + ρ(z) + ρ, pˆ = −ρ gz + p(z) + p. (9) nition), and (b) represents the second type with satellite-determined 0 0 N∗ (water in motion on N∗). −3 Here, ρ0 = 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) = − , (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- + + = 0. (4) sent (pˆ, ρ)ˆ . Vertical integration of Eq. (8) from N to S after ∂x2 ∂y2 ∂z2 replacing (pˆ, ρ)ˆ 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) = − , v (S) − v (N) = , (11) g g f ∂y g g f ∂x ∂Nˆ 1 ∂V ∂Nˆ 1 ∂V = , = , (5) ∂x g ∂x ∂y g ∂y where which connects to the gravity anomaly by D = S − N (12) ! ∂(1g) ∂2Nˆ ∂2Nˆ = g + . (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- ˆ izontal Laplacian of the marine geoid height N and serves ug(N) = 0, vg(N) = 0, (13) as the basic principle in the satellite marine geodesy. Since Dˆ is the difference of the two large fields S and Nˆ (2 orders the horizontal gradient of D represents the absolute surface of magnitude larger than D)ˆ , it is extremely sensitive to any geostrophic currents. error in either S or Nˆ – even a 1 % error in either field can After satellites came into operation, SSH has been ob- lead to error in Dˆ that is of the same order of magnitude as Dˆ 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, Nˆ 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) + = , 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 = 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- = − = − fined by 1D D∗ D 1N, (17) where Eqs. (12) and (15) are used. 1D is of interest in D∗ = S − N∗(t), (14) oceanography. 1N is of interest in geodesy. Equation (17) where N∗(t) changes with time due to temporally varying shows that the key issue to evaluate 1D is to determine D gravity anomaly 1g. Thus, comparison between the first- (i.e., first type of DOT). type and second-type geoids should be conducted between Conservation of potential vorticity for a dissipation-free N and N ∗. Here, N ∗ is the temporally mean of N∗(t). As for fluid does not apply precisely to sea water where the den- DOT, the first type of DOT (D) should be compared to the sity is a function not only of temperature and pressure but second type mean DOT (MDOT), also of the dissolved salts. The effect of salinity on density is very important in the distribution of water properties. How- D∗ = S − N ∗. (15) ever, for most dynamic studies the effect of the extra state variable is not significant and the conservation of potential The oceanic conditions at N and N ∗ are different: wa- vorticity is valid (Veronis, 1980). Based on the conservation ter is at rest on N (see Eq. 13), but in motion on N ∗. The of the potential vorticity, the geostrophic current reaches the oceanographic community ignores such a difference, also minimum energy state (Appendix A). Due to the minimum treating horizontal gradients of the second type of DOT energy state, an elliptic partial differential equation for D as the absolute surface geostrophic currents. For example, is derived with coefficients containing sea-floor topography the second type of MDOT (D∗) data are posted at the H, and forcing function containing temperature and salinity NASA/JPL website: https://grace.jpl.nasa.gov/data/get-data/ fields. dynamic-ocean-typography/ (last access: 31 July 2018); its If 1D is negligible in comparison to D, the change in ma- horizontal gradients are also taken as the absolute surface rine geoid from N to N ∗ does not change the oceanographic geostrophic currents. interpretation of absolute DOT, i.e., the horizontal gradients A question arises: do the horizontal gradients of the sec- of D∗ also represent the absolute surface geostrophic cur- ond type of MDOT (D∗) represent the absolute surface rents. If 1D is not negligible, the horizontal gradient of D∗ geostrophic currents? This paper will answer the question us- does not represent the absolute surface geostrophic currents. ing the temporally averaged SSH and marine geoid satellite altimetric and gravimetric measurements from NASA (i.e., the second type of MDOT (D∗)), and solving a new elliptic 3 Geostrophic currents and energy equation of D numerically. Given (S, N ∗, D) leads to the answer of the question. Equation (9) implies, The rest of the paper is outlined as follows. Section 2 ∂ρˆ ∂ρ ∂ρˆ ∂ρ describes the change in DOT due to the change in marine = , = , (18) geoid from the first to the second type. Section 3 describes ∂x ∂x ∂y ∂y ∂pˆ ∂p ∂pˆ ∂p geostrophic currents and energy related to the first type of = , = . (19) DOT. Section 4 presents the governing equation of the first ∂x ∂x ∂y ∂y type of DOT with the boundary condition at the coasts. Sec- Using the first type of marine geoid N, the horizontal gra- tion 5 shows the numerical solution for the world oceans. dients of D lead to the absolute surface geostrophic currents Section 6 evaluates the change in global DOT from first to (see Eqs. 11 and 13). Integration of the thermal wind relation, second type with oceanographic implications. Section 7 con- cludes the study. ∂ug g ∂ρ ∂vg g ∂ρ = , = − , (20) ∂z f ρ0 ∂y ∂z f ρ0 ∂x 2 Change in DOT from first to second type from the ocean surface to depth z leads to depth-dependent geostrophic currents, The second type of MDOT (D∗) data are downloaded from the NASA/JPL website: https://grace.jpl.nasa.gov/data/ ug(z) = ug(S) + uBC(z), vg(z) = vg(S) + vBC(z), (21) www.ocean-sci.net/14/947/2018/ Ocean Sci., 14, 947–957, 2018 950 P. C. Chu: Technical Note: Two types of absolute dynamic ocean topography where where R is the horizontal area of the water volume, H is the water depth. Thus, Eq. (25) becomes 0 0 ZZ Z Z g ∂ρ 0 g ∂ρ 0 G(Dx,Dy) = L(Dx,Dy)dxdy → min, (27) uBC(z) = − dz , vBC(z) = dz , (22) f ρ0 ∂y f ρ0 ∂x R z z h 2 2 i 2 L(Dx,Dy) ≡ H (Dx + Dy) + 2DxY − 2DyX /f , (28) are the baroclinic geostrophic currents. Here, f = 2sin(ϕ) where the parameters (X, Y ) are given by is the Coriolis parameter; = 2π/(86400s) is the mean Earth rotation rate; and ϕ is the latitude. 0 0 0 Z Z Z The volume-integrated total energy, i.e., sum of kinetic en- f 1 ∂ρˆ 0 X(x,y) ≡ uBCdz = − dz dz (29) ergy of the geostrophic currents and the available potential g ρ0 ∂y − − z energy (Oort et al., 1989), for an ocean basin (W) is given by H H 0 0 0 f Z 1 Z Z ∂ρˆ ≡ = 0 " # Y (x,y) vBCdz dz dz, (30) ZZZ 1 g2ρ2 g ρ0 ∂x E = (u2 + v2) + dxdydz. (23) −H −H z g g 2 2 2 2ρ0 n W which represent vertically integrated baroclinic geostrophic currents scaled by the factor f/g (unit: m). Here, Eq. (18) is Substitution of Eqs. (21) and (22) into Eq. (23) leads to used (i.e., horizontal gradient of in situ density is the same as that of density anomaly). The Euler–Lagragian equation of the functional Eq. (27) g2 ZZZ is given by E(Dx,Dy,ρ) = 2 ∂L ∂ ∂L ∂ ∂L W − − = 0. (31) " 2 # f uBC 2 2 f vBC 2 2 ρ ∂D ∂x ∂Dx ∂y ∂Dy (−Dy + ) /f + (Dx + ) /f + dxdydz g g ρ2n2 0 Substitution of Eq. (28) into Eq. (31) gives an elliptic par- 2 ZZZ g 2 2 2 2 vBC uBC tial differential equation (i.e., the governing equation) for the = D /f + D /f + 2Dx − 2Dy dxdydz 2 x y fg fg W first type of DOT (i.e., D), " # 1 ZZZ g2ρ2 2 h 2 i + u2 + v2 + dxdydz (24) f ∇ H/f ∇D = −F, 2 BC BC 2 2 ρ0 n W or ∂D ∂D ∂D H ∇2D + r(x) + r(y) − 2(β/f ) = −F, (32) ∂x ∂y ∂y 4 Governing equation of D where ∂Y ∂X ∂ ∂ For a given density field, the second integration in the right F ≡ − , ∇ ≡ i + j (33) side of Eq. (24) is known. The geostrophic currents taking ∂x ∂y ∂x ∂y 1 ∂H 1 ∂H 2 the minimum energy state provides a constraint for D, r(x) ≡ , r(y) ≡ , β = cos(ϕ), (34) H ∂x H ∂y a ZZZ where a = 6370km, is the mean Earth radius. The ≡ G(Dx,Dy) geostrophic balance does not exist at the Equator. The Cori- W olis parameter f needs some special treatment for low lati- 2 2 f vBC f uBC 2 tudes. In this study, f is taken as 2sin(5π/180) if latitude D + D + 2Dx − 2Dy /f dxdydz → min. (25) x y g g is between 10◦ N and 0◦; and as −2sin(5π/180) if latitude is between 0◦ and 10◦ S. The three-dimensional integration Eq. (25) over the ocean Let 0 be the coastline of the ocean basin. Continuation of basin is conducted by the geoid from land to oceans gives

N|0 = Nl|0, N ∗|0 = Nl|0, (35)  0  ZZZ ZZ  Z  which leads to [...]dxdydz = [...]dz dxdy, (26)   | = | W R −H N 0 N ∗ 0. (36)

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Figure 2. (a) First type of DOT (i.e., D) which is the solution of Eq. (32) with boundary condition of Eq. (37, unit: cm), (b) second type of MDOT (1993–2006, i.e., D∗, unit: cm) downloaded from the NASA/JPL website: https://grace.jpl.nasa.gov/data/get-data/ dynamic-ocean-typography (31 July 2018), (c) difference between the two DOTs (i.e., 1D), (d) histogram of global D, and (e) histogram of global D∗.

Figure 3. Derivatives in the x direction of (a) the ﬁrst-type DOT (i.e., ∂D/∂x), (b) the second-type MDOT (i.e., ∂D∗/∂x), (c) the difference 1(∂D/∂x) = ∂D∗/∂x − ∂D/∂x, (d) histogram of global ∂D/∂x, and (e) histogram of global ∂D∗/∂x. www.ocean-sci.net/14/947/2018/ Ocean Sci., 14, 947–957, 2018 952 P. C. Chu: Technical Note: Two types of absolute dynamic ocean topography

Here, Nl is the geoid over land. The boundary condition smaller than the corresponding standard deviations (54.9 and Eq. (36) can be rewritten as 71.2 cm; see Fig. 2d, e). The magnitude of D and D∗ are represented by their root-mean squares, which are close to their D|0 = (S − N)|0 = (S − N ∗)|0 = D∗|0 (37) standard deviations. Histograms for D (Fig. 2d) and D∗ (Fig. 2e) are both D i,j i,j which is the boundary condition of . non-Gaussian and negatively skewed. The major difference between the two is the single modal form of Di,j with a peak 5 Numerical solution of D at around 20 cm and the bimodal form of D∗i,j with a high peak at around 30 cm and a low peak at −140 cm. The sta- The well-posed elliptic Eq. (32) is integrated numerically on tistical parameters are different, e.g., mean value and stan- ◦ ◦ 1 × 1 grids for the world oceans with the boundary val- dard deviation are 0.524 and 54.9 cm for Di,j , and −3.84 ues (i.e., Eq. 37) taken from the MDOT (1993–2006) field and 71.2 cm for D∗i,j . Skewness and kurtosis are −0.83 and (i.e., D∗), at the NASA/JPL website: https://grace.jpl.nasa. 3.01 for Di,j , and −0.87 and 2.80 for D∗i,j . gov/data/get-data/dynamic-ocean-typography/ (last access: Horizontal gradients of the DOT, (∂Di,j /∂x, ∂Di,j /∂y) ◦ ◦ 31 July 2018) (0.5 interpolated into 1 resolution). The forc- and (∂D∗i,j /∂x,∂D∗i,j /∂y), have oceanographic signifi- ing function F is calculated on a 1◦ ×1◦ grid from the World cance (related to the geostrophic currents). They are calcu- Ocean Atlas 2013 (WOA13) temperature and salinity fields, lated using the central difference scheme at inside-domain which was downloaded from the NOAA National Centers for grid points and the first order forward/backward difference Environmental Information (NCEI) website: https://www. scheme at grid points next to the boundary. The difference nodc.noaa.gov/OC5/woa13/woa13data.html (last access: 31 in global ∂Di,j /∂x (Fig. 3a) and ∂D∗i,j /∂x (Fig. 3b) is ev- July 2018). The three-dimensional density was calculated us- ident with much smaller-scale structures in ∂D∗i,j /∂x. The ing the international thermodynamic equation of seawater – difference between the two gradients (Fig. 3c), 2010, which is downloaded from the website: http://unesdoc. unesco.org/images/0018/001881/188170e.pdf (last access: 31 July 2018). The ocean bottom topography data H was 1(∂Di,j /∂x) = ∂D∗i,j /∂x − ∂Di,j /∂x, (40) downloaded from the NCEI 5-Minute Gridded Global Relief Data Collection at the website: https://www.ngdc.noaa.gov/ mgg/global/etopo5.HTML (last access: 31 July 2018). Dis- has the same order of magnitude as the gradients themselves cretization of the elliptic Eq. (32) and numerical integration with the relative RRMS of 1(∂D/∂x), are given in Appendix B.

6 Difference between the two DOTs RRMS1(∂D/∂x) = r 1 PP 2 The ﬁrst-type global DOT (Di,j ; Fig. 2a) is the numerical M 1(∂Di,j /∂x) i j solution of the elliptic Eq. (32) with the boundary condition = r 1.04, (41) Eq. (37). The second type global MDOT (D∗i,j ; Fig. 2b) is 1 PP 2 M (∂Di,j /∂x) downloaded from the NASA/JPL website: https://grace.jpl. i j nasa.gov/data/get-data/dynamic-ocean-typography/ (last access: 31 July 2018). The difference between the two DOTs, which implies that the non-surface latitudinal geostrophic = − 1Di,j D∗i,j Di,j , (38) current component of the second type of MDOT has the same order of magnitude as the surface latitudinal geostrophic cur- is evident in the world oceans (Fig. 2c). Here, (i, j) de- rent component of the ﬁrst type of DOT. Histograms for note the horizontal grid point. The relative root-mean square ∂D /∂x (Fig. 3d) and ∂D /∂x (Fig. 3e) are near sym- (RRMS) of 1D is given by i,j ∗i,j metric with mean values around (−1.29, −0.78)×10−8 and −7 r 1 PP 2 standard deviations (2.69, 4.95)×10 . The standard devia- M (1Di,j ) i j tion of ∂D∗i,j /∂x is almost twice that of ∂Di,j /∂x. = = RRMS(1D) r 0.386, (39) Similarly, the difference in global ∂Di,j /∂y (Fig. 4a) and 1 PP(D )2 M i,j ∂D∗i,j /∂y (Fig. 4b) is evident with much smaller-scale struc- i j tures in ∂D∗i,j /∂y. The difference between the two gradients where M = 38877 is the number of total grid points. Both (Fig. 4c), D and D∗ have positive and negative values. The arith- metic mean values (0.524 and −3.84 cm) are much smaller than the RMS mean values. They are 1 order of magnitude 1(∂Di,j /∂y) = ∂D∗i,j /∂y − ∂Di,j /∂y, (42)

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Figure 4. Derivatives in the y direction of (a) the first-type DOT (i.e., ∂D/∂y), (b) the second-type MDOT (i.e., ∂D∗/∂y), and (c) the difference 1(∂D/∂y) = ∂D∗/∂y − ∂D/∂y; (d) histogram of global ∂D/∂y and (e) histogram of global ∂D∗/∂y. has the same order of magnitude as the gradients themselves rest), the horizontal gradients of the first type of DOT rep- with the relative root-mean square (RRMS) of 1(∂D/∂y), resent the absolute surface geostrophic currents. With the satellite-determined (second type) marine geoid by Eq. (6), RRMS 1(∂D/∂y) = the horizontal gradients of the second type of MDOT do r 1 PP 2 not represent the absolute surface geostrophic currents. The M 1(∂Di,j /∂y) i j difference between the two types of DOT represents an ad- = 0.98, (43) ditional component to the absolute surface geostrophic cur- r 1 PP 2 M (∂Di,j /∂y) rents. i j With conservation of potential vorticity, geostrophic bal- which implies that the non-surface zonal geostrophic current ance represents the minimum energy state in an ocean basin component of the second type of MDOT has the same order where the mechanical energy is conserved. A new governing of magnitude as the surface zonal geostrophic current com- elliptic equation of the first type of DOT is derived with water depth (H) in the coefficients and the three-dimensional ponent of the first type of DOT. Histograms for ∂Di,j /∂y temperature and salinity in the forcing function. This gov- (Fig. 4d) and ∂D∗i,j /∂y (Fig. 4e) are also near symmetric with the mean values around (2.32, 1.18)×10−7 and stan- erning elliptic equation is well posed. Continuation of the dard deviations (1.20, 2.44)×10−6 . The standard deviation geoid from land to ocean leads to an inhomogeneous Dirich- let boundary condition. of ∂D∗i,j /∂y is twice that of ∂Di,j /∂y. The denominators of Eqs. (41) and (43) represent the magnitude of the horizontal Difference between the two types of DOT is evident with gradients of the first type of DOT. a RRMS difference of 38.6 %. Horizontal gradients (repre- senting geostrophic currents) of the two types of DOTs are different with much smaller-scale structures in the second 7 Conclusions type absolute DOT. The RRMS difference is near 1.0 in both (x, y) components of the DOT gradient, which implies that The change in marine geoid from classically defined (first the non-absolute surface geostrophic currents identified from type, stand-alone concept in oceanography) to satellite- the second type has the same order of magnitude as the ab- determined (second type, stand-alone concept in marine solute surface geostrophic currents identified by the first type geodesy) largely affects oceanography. With the classically of DOT. defined marine geoid (average level of SSH if the water is at www.ocean-sci.net/14/947/2018/ Ocean Sci., 14, 947–957, 2018 954 P. C. Chu: Technical Note: Two types of absolute dynamic ocean topography

The notable difference between the two types of DOT raises more questions in oceanography and marine geodesy: is there any theoretical foundation to connect the classical marine geoid (stand-alone concept in oceanography using the principle of surface geostrophic currents without 1g) to the satellite-determined marine geoid (stand-alone concept in marine geodesy using 1g without the principle of surface geostrophic currents)? How can the satellite-determined marine geoid using the gravity anomaly (1g) be conformed to the basic physical oceanography principle of surface geostrophic currents? What is the interpretation of the horizontal gradients of the second type of MDOT (D∗)? Is there any evidence or theory to show (ug(N ∗) = 0, vg(N ∗) = 0) similar to Eq. (13)? More observational and theoretical studies are needed in order to solve those problems. The main challenge for oceanographers is how to use the satellite altimetry observed SSH such as the Surface Water and Ocean Topography (SWOT, https://swot.jpl.nasa.gov/, last access: 31 July 2018) to infer the ocean general circulations at the surface. A new theoretical framework rather than the geostrophic constraint needs to be established. The GOCE satellite-determined data-only geoid model is more accurate and with higher resolution than GRACE. The change of GRACE to the GOCE geoid model may in- crease the accuracy of the calculation of the second type of DOT. However, such a replacement does not solve the funda- mental problem presented here, i.e., incompatibility between satellite-determined marine geoid using the gravity anomaly (1g) and the classical marine geoid (mean SSH when the water at rest) on the basis of the basic physical oceanography principle of surface geostrophic currents. Finally, the mathematical framework described here (i.e., the elliptic Eq. 32 with boundary condition Eq. 37) may lead to a new inverse method for calculating three-dimensional absolute geostrophic velocity from temperature and salinity ﬁelds since the surface absolute geostrophic velocity is the solution of Eq. (32). This will be a useful addition to the existing β-spiral method (Stommel and Schott, 1977), box model (Wunsch, 1978), and P -vector method (Chu, 1995; Chu et al., 1998, 2000).

Data availability. The datasets used for this research are all from open sources listed as follows: (1) Satellite- determined DOT data: https://grace.jpl.nasa.gov/data/get-data/ dynamic-ocean-typography/ (Tapley et al,. 2003; last access: 31 July 2018), (2) World Ocean Atlas 2013 (WOA13): https://www.nodc.noaa.gov/OC5/woa13/woa13data.html (NOAA National Centers for Environmental Information; last access: 31 July 2018), and (3) NCEI 5- 5-Minute Gridded Global Relief Data Collection: https://www.ngdc.noaa.gov/mgg/global/etopo5.HTML (NOAA National Centers for Environmental Information; last access: 31 July 2018).

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Appendix A: Geostrophic balance as a minimum energy Lagrange equations, state in an energy conserved basin

∂K ∂ ∂K In large-scale motion (small Rossby number) with the − = 0, (A5) Boussinesq approximation, the linearized potential vorticity ∂ρ ∂z ∂ρz ∂K ∂ ∂K (5) is given by − = 0, (A6) ∂u ∂y ∂uy ∂K ∂ ∂K − = 0, (A7) ∂v ∂u ∂ρˆ ρ n2 ∂ρ ∂v ∂x ∂vx 5 ≈ f + − ≈ f − 0 + ∂x ∂y ∂z g ∂z ρ n2 ∂v ∂u − 0 − , (A1) where K is in the integrand appearing in Eq. (A4). Substitu- g ∂x ∂y tion of K into Eqs. (A5), (A6), (A7) leads to where, ρ = 1025 kg m−3 is the characteristic density. With- g2 ∂µ 0 ρ = f , (A8) out the frictional force and 0 horizontally integrated buoy- 2 2 ρ0 n ∂z ancy ﬂux at the surface and bottom, the energy (including ki- ρ n2 ∂µ ρ n2 ∂µ netic and available potential energies) is conserved in a three- u = 0 , v = − 0 . (A9) dimensional ocean basin (V) g ∂y g ∂x

Differentiation of Eq. (A9) with respect to z and use of ZZZ 1 E = J dxdydz, J ≡ u2 + v2 + w2 Eq. (A8) leads to 2 V g2ρ2 + , (A2) ∂u g ∂ρ ∂ug ∂v g ∂ρ ∂vg 2 2 = = , = − = , (A10) 2ρ0 n ∂z f ρ0 ∂y ∂z ∂z f ρ0 ∂x ∂z dE = 0. (A3) dt which shows that (u, v) = (ug, vg) have the minimum energy state. The two terms of J are kinetic energy and available potential energy. To show the geostrophic balance taking the minimum energy state for a given linear PV (see Eq. A1), the constraint is incorporated by extremizing the integral (see also in Vallis, 1992; Chu, 2018) Appendix B: Numerical solution of the Eq. (32)

Let the three axes (x, y, z) be discretized into local rectangu- ( ZZZ 1 g2ρ2 lar grids in horizontal and non-uniform grids in vertical (xi,j , I ≡ u2 + v2 + w2 + + µ(x,y,z) ◦ ◦ 2 2 yi,j , zk) with cell sizes (1 × 1 ), 2 2ρ0 n V ρ n2 ∂ρ ρ n2 ∂v ∂u f − 0 + − 0 − dxdydz, g ∂z g ∂x ∂y π 1y = r , 1xj = 1y cosφj , 1zk = zk − zk+ , (A4) 360 E 1 i = 1,2,...,I; j = 1,2,...,J ; k = 1,2,...,Ki,j , (B1) where µ(x,y,z) is the Lagrange multiplier, which is a function of space. If it were a constant, the integral would merely where k =1 for the surface, k = Ki,j for the bottom; φj is the extremize energy subject to a given integral of PV, and re- latitude of the grid point; rE = 6371km, is the Earth radius; arrangement of PV would leave the integral unaltered. Ex- I = 360; and J = 1804. The subscripts in Ki,j in Eq. (B1) tremization of the integral Eq. (A4) gives the three Euler– indicates non-uniform water depth in the region. www.ocean-sci.net/14/947/2018/ Ocean Sci., 14, 947–957, 2018 956 P. C. Chu: Technical Note: Two types of absolute dynamic ocean topography

The parameters (Xi,j , Yi,j ) in Eqs. (29) and (30) (in With the given boundary condition (Eq. 37, see Sect. 4) Sect. 4) are calculated by and forcing function (Eq. B4), the first type of DOT at the grid points can be computed from steps n to n+1, Ki,j k 1 XX X ≡ i,j 2 (n+1) 1 (x) (n) 4ρ0 2 1 + cos φj D = 1 + r 1y cosφj D k=2 l=1 i,j 2 i,j i+1,j  ρˆ −ρ ˆ ρˆ −ρ ˆ  i,j+1,l i,j,l + i+1,j+1,l i+1,j,l 1 + − (x) (n) + 2  1y 1y  1 ri,j 1y cosφj Di− ,j cos φj  1zk 2 1 ρî,j+1,l+1 −ρ î,j,l+1 ρî+1,j+1,l+1 −ρ î+1,j,l+1  + +  2cotφj 1y 1y 1y 1 + r(y) − D(n) i,j i,j+1 1zl + 1zl+1 rE 2 (B2) 2cotφ 1y 2 + 2 − (y) − j (n) cos φj 1 ri,j Di,j−1 Ki,j k rE 2 1 XX Y ≡ Fi,j i,j + (1y)2cos2φ . (B8) 4ρ0 = = j k 2 l 1 Hi,j  ρî+1,j,l −ρ î,j,l ρî+1,j+1,l −ρ î,j+1,l  + Such iteration continues until the RRMS difference  1xj 1xj   ρˆ −ρ ˆ ρˆ −ρ ˆ 1zk reaches the criterion,  + i+1,j,l+1 i,j,l+1 + i+1,j+1,l+1 i,j+1,l+1  1xj 1xj s I J h (n+1) (n)i2 1z + 1z + 1 P P − l l 1 M Di,j Di,j , (B3) i=1 j=1 2 = −6 r s < 10 , (B9) I J h i2 which gives the discretized forcing function 1 P P (n) M Di,j i=1 j=1 Yi+1,j − Yi−1,j Xi,j+1 − Xi,j−1 Fi,j = − . (B4) 21xj 21y where M = 38877, is the total number of the grid points on the ocean surface. The governing Eq. (32) is discretized by

D − 2D + D D − 2D + D i+1,j i,j i−1,j + i,j+1 i,j i,j−1 2 2 (1xj ) (1y) D − D 2β + (x) i+1,j i−1,j + (y) − j ri,j ri,j 21xj fj Di,j+ − Di,j− Fi,j 1 1 = − , (B5) 21y Hi,j which is reorganized by 2 1 (x) 2 1 + cos φj Di,j = 1 + r 1y cosφj Di+ ,j 2 i,j 1 1 (x) 2 + 1 − r 1y cosφj Di− ,j + cos φj 2 i,j 1 2cotφ 1y + (y) − j 1 ri,j Di,j+1 rE 2 2cotφ 1y + 2 − (y) − j cos φj 1 ri,j Di,j−1 rE 2 Fi,j 2 2 + (1y) cos φj . (B6) Hi,j The iteration method is used to solve the algebraic Eq. (B6) with a large value of I × J . It starts from the 0th step, (0) = = ; = Di,j 0, i 1,2,...,I j 1,2,...J. (B7)

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 957

Author contributions. PCC discovered the problem, formulated Wunsch, C.: The general circulation of the North Atlantic west the theory, calculated the two types of DOT, and prepared the of 50◦ W determined from inverse methods, Rev. Geophys., 16, manuscript. 583–620, 1978. Wunsch, C. and Gaposchkin, E. M.: On using satellite altimetry to determine the general circulation of the oceans with application Competing interests. The author declares that he has no conﬂict of to geoid improvement, Rev. Geophys., 18, 725–745, 1980. interest.

Acknowledgements. The author thanks Chenwu Fan for invaluable comments and computational assistance, NOAA/NCEI for the WOA-2013 (T, S) and ETOPO5 sea-ﬂoor topography data, and NASA/JPL (second type) MDOT data.

Edited by: John M. Huthnance Reviewed by: C. K. Shum and one anonymous referee

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