Ocean Sci., 14, 1–11, 2018https://doi.org/10.5194/os-14-1-2018© Author(s) 2018. This work is distributed underthe Creative Commons Attribution 4.0 License.

Technical note: Two types of absolute dynamic ocean topographyPeter C. ChuNaval Ocean Analysis and Prediction Laboratory, Department of Oceanography Naval Postgraduate School,Monterey, CA 93943, USA

Correspondence: Peter C. Chu (pcchu@nps.edu)

Received: 17 April 2018 – Discussion started: 23 May 2018Revised: 28 July 2018 – Accepted: 15 August 2018 – Published:

Abstract. Two types of marine geoid exist with the first typebeing the average level of sea surface height (SSH) if thewater is at rest (classical definition), and the second typebeing satellite-determined with the condition that the wa-ter is usually not at rest. The differences between the twoare exclusion (inclusion) of the gravity anomaly and non-measurable (measurable) in the first (second) type. The as-sociated absolute dynamic ocean topography (referred to asDOT), i.e., SSH minus marine geoid, correspondingly alsohas two types. Horizontal gradients of the first type of DOTrepresent the absolute surface geostrophic currents due towater being at rest on the first type of marine geoid. Hor-izontal gradients of the second type of DOT represent thesurface geostrophic currents relative to flow on the secondtype of marine geoid. Difference between the two is quan-titatively identified in this technical note through compari-son between the first type of DOT and the mean second typeof DOT (MDOT). The first type of DOT is determined bya physical principle that the geostrophic balance takes theminimum energy state. Based on that, a new elliptic equa-tion is derived for the first type of DOT. The continuation ofgeoid from land to ocean leads to an inhomogeneous Dirich-let boundary condition with the boundary values taking thesatellite-observed second type of MDOT. This well-posed el-liptic equation is integrated numerically on 1◦ grids for theworld oceans with the forcing function computed from theWorld Ocean Atlas (T , S) fields and the sea-floor topogra-phy obtained from the ETOPO5 model of NOAA. Betweenthe first type of DOT and the second type of MDOT, therelative root-mean square (RRMS) difference (versus RMSof the first type of DOT) is 38.6 % and the RMS differencein the horizontal gradients (versus RMS of the horizontalgradient of the first type of DOT) is near 100 %. The stan-dard deviation of horizontal gradients is nearly twice larger

for the second type (satellite-determined marine geoid withgravity anomaly) than for the first type (geostrophic balancewithout gravity anomaly). Such a difference needs further at-tention from oceanographic and geodetic communities, es-pecially the oceanographic representation of the horizontalgradients of the second type of MDOT (not the absolute sur-face geostrophic currents).

1 Introduction

Let the coordinates (x, y, z) be in zonal, latitudinal, andvertical directions. The absolute dynamic ocean topography(hereafter referred to as DOT) D̂ is the sea surface height(SSH, waves and tides filtered out) relative to the marinegeoid (i.e., the equipotential surface),

D̂ = S− N̂, (1)

where S is the SSH; N̂ is the marine geoid height abovethe reference ellipsoid (Fig. 1). D̂ is an important signal inoceanography and N̂ is of prime interest in geodesy. Equa-tion (1) is also applicable if defined relative to the centerof the Earth. The geoid height N̂(x,y) and other associatedmeasurable quantities such as gravity anomaly 1g(x, y) arerelated to the anomaly of the gravitational potential V (x, y,z) to a first approximation by the well-known Brun’s formula(e.g., Hofmann-Wellenhof and Moritz, 2005),

N̂(x,y)=V (x,y,0)

g, (2)

where g = 9.81 m s−2, is the globally mean normal gravity,which is usually represented by g0 in geodesy. The gravity

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2 P. C. Chu: Technical Note: Two types of absolute dynamic ocean topography

Figure 1. Two types of marine geoid and DOT: (a) the first type withN being the average level of SSH if water is at rest (classical defi-nition), and (b) represents the second type with satellite-determinedN∗ (water in motion on N∗).

anomaly is the vertical derivative of the potential

1g(x,y)=−∂V (x,y,0)

∂z, (3)

where the anomaly of the gravity potential V satisfies theLaplace equation

∂2V

∂x2 +∂2V

∂y2 +∂2V

∂z2 = 0. (4)

The vertical deflection is the slope of the geoid

∂N̂

∂x=

1g

∂V

∂x,∂N̂

∂y=

1g

∂V

∂y, (5)

which connects to the gravity anomaly by

∂(1g)

∂z= g

(∂2N̂

∂x2 +∂2N̂

∂y2

). (6)

Equation (6) links the vertical gravity gradient to the hor-izontal Laplacian of the marine geoid height N̂ and servesas the basic principle in the satellite marine geodesy. SinceD̂ is the difference of the two large fields S and N̂ (2 ordersof magnitude larger than D̂), it is extremely sensitive to anyerror in either S or N̂ – even a 1 % error in either field canlead to error in D̂ that is of the same order of magnitude as D̂itself (Wunsch and Gaposchkin, 1980; Bingham et al., 2008).

Before satellites came into operation, S was measuredfrom sparse surveying ships and tide gauge stations locatedalong irregular local coastlines. However, N̂ was not easy toobserve. Without satellite measurements, the marine geoid isdefined as the average level of SSH if the water is at rest anddenoted here byN , which is called the classical marine geoid(or first type of marine geoid, Fig. 1a). The first type of ma-rine geoid can be taken as a stand-alone concept in oceanog-raphy since it is on the basis of the hypothesis (mean SSH

when the water at rest) without using the gravity anomaly. Inthis framework, the geostrophic balance

ug =−1f ρ̂

∂p̂

∂y, vg =

1f ρ̂

∂p̂

∂x(7)

and hydrostatic balance

∂p̂

∂z=−ρ̂g (8)

are used for large-scale (i.e., scale > 100 km) processes. Here(ug, vg) are geostrophic current components; f is the Corio-lis parameter; (p̂, ρ̂) are in situ pressure and density, respec-tively, which can be decomposed into

ρ̂ = ρ0+ ρ(z)+ ρ, p̂ =−ρ0gz+p(z)+p. (9)

Here, ρ0 = 1025 kg m−3 is the characteristic density; (ρ, p)are horizontally uniform with ρ vertically increasing withdepth (stable stratification)

∂ρ/∂z≡−ρ0(n(z))2/g, (10)

where n(z) is the buoyancy frequency (or called the Brunt–Vaisala frequency); (p, ρ) are anomalies of pressure and den-sity. Near the ocean surface, it is common to use the charac-teristic density and corresponding pressure (p0,ρ0) to repre-sent (p̂, ρ̂). Vertical integration of Eq. (8) from N to S afterreplacing (p̂, ρ̂) by (p0,ρ0) in Eqs. (7) and (8) leads to

ug(S)− ug(N)=−g

f

∂D

∂y, vg(S)− vg(N)=

g

f

∂D

∂x, (11)

where

D = S−N (12)

is the first type of DOT. Since the first type of marine geoid(N ) is defined as the average level of SSH if the water is atrest,

ug(N)= 0, vg(N)= 0, (13)

the horizontal gradient of D represents the absolute surfacegeostrophic currents.

After satellites came into operation, SSH has been ob-served with uniquely sampled temporal and spatial resolu-tions by high-precision altimetry above a reference ellipsoid(not geoid; Fu and Haines, 2013). Two Gravity Recovery andClimate Experiment (GRACE) satellites, launched in 2002,provide data to compute the marine geoid (called the GRACEGravity Model, GGM; see website: http://www.csr.utexas.edu/grace/ (last access: 31 July 2018); Tapley et al., 2003;Shum et al., 2011). In addition, the European Space Agency’sGOCE mission data, along with the GRACE data, have pro-duced the best mean gravity field or the geoid model at aspatial scale longer than 67 km half-wavelength (or spherical

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P. C. Chu: Technical Note: Two types of absolute dynamic ocean topography 3

harmonics completed to degree 300). This marine geoid isthe solution of Eq. (6),

∂2N∗

∂x2 +∂2N∗

∂y2 =1g

∂(1g)

∂z,

where N∗ is the satellite-determined marine geoid from themeasurable gravity anomaly 1g, and called the second typeof marine geoid (Fig. 1b), which is different from N , definedby Eq. (13). Correspondingly, the second type of DOT is de-fined by

D∗ = S−N∗(t), (14)

where N∗(t) changes with time due to temporally varyinggravity anomaly 1g. Thus, comparison between the first-type and second-type geoids should be conducted betweenN and N∗. Here, N∗ is the temporally mean of N∗(t). As forDOT, the first type of DOT (D) should be compared to thesecond type mean DOT (MDOT),

D∗ = S−N∗. (15)

The oceanic conditions at N and N∗ are different: wa-ter is at rest on N (see Eq. 13), but in motion on N∗. Theoceanographic community ignores such a difference, alsotreating horizontal gradients of the second type of DOTas the absolute surface geostrophic currents. For example,the second type of MDOT (D∗) data are posted at theNASA/JPL website: https://grace.jpl.nasa.gov/data/get-data/dynamic-ocean-typography/ (last access: 31 July 2018); itshorizontal gradients are also taken as the absolute surfacegeostrophic currents.

A question arises: do the horizontal gradients of the sec-ond type of MDOT (D∗) represent the absolute surfacegeostrophic currents? This paper will answer the question us-ing the temporally averaged SSH and marine geoid satellitealtimetric and gravimetric measurements from NASA (i.e.,the second type of MDOT (D∗)), and solving a new ellipticequation of D numerically. Given (S, N∗, D) leads to theanswer of the question.

The rest of the paper is outlined as follows. Section 2describes the change in DOT due to the change in marinegeoid from the first to the second type. Section 3 describesgeostrophic currents and energy related to the first type ofDOT. Section 4 presents the governing equation of the firsttype of DOT with the boundary condition at the coasts. Sec-tion 5 shows the numerical solution for the world oceans.Section 6 evaluates the change in global DOT from first tosecond type with oceanographic implications. Section 7 con-cludes the study.

2 Change in DOT from first to second type

The second type of MDOT (D∗) data are downloadedfrom the NASA/JPL website: https://grace.jpl.nasa.gov/data/

get-data/dynamic-ocean-typography/. This dataset is a sub-traction of a second type of marine geoid of GRACE (Bing-ham et al., 2011) from a mean (1993 to 2006) altimetric seasurface. The change in marine geoid from first (N) to second(N∗) type is represented by

1N =N∗−N. (16)

Correspondingly, the change in DOT is given by

1D =D∗−D =−1N, (17)

where Eqs. (12) and (15) are used. 1D is of interest inoceanography. 1N is of interest in geodesy. Equation (17)shows that the key issue to evaluate 1D is to determine D(i.e., first type of DOT).

Conservation of potential vorticity for a dissipation-freefluid does not apply precisely to sea water where the den-sity is a function not only of temperature and pressure butalso of the dissolved salts. The effect of salinity on density isvery important in the distribution of water properties. How-ever, for most dynamic studies the effect of the extra statevariable is not significant and the conservation of potentialvorticity is valid (Veronis, 1980). Based on the conservationof the potential vorticity, the geostrophic current reaches theminimum energy state (Appendix A). Due to the minimumenergy state, an elliptic partial differential equation for Dis derived with coefficients containing sea-floor topographyH , and forcing function containing temperature and salinityfields.

If1D is negligible in comparison toD, the change in ma-rine geoid from N to N∗ does not change the oceanographicinterpretation of absolute DOT, i.e., the horizontal gradientsof D∗ also represent the absolute surface geostrophic cur-rents. If 1D is not negligible, the horizontal gradient of D∗does not represent the absolute surface geostrophic currents.

3 Geostrophic currents and energy

Equation (9) implies,

∂ρ̂

∂x=∂ρ

∂x,∂ρ̂

∂y=∂ρ

∂y, (18)

∂p̂

∂x=∂p

∂x,∂p̂

∂y=∂p

∂y. (19)

Using the first type of marine geoid N , the horizontal gra-dients of D lead to the absolute surface geostrophic currents(see Eqs. 11 and 13). Integration of the thermal wind relation,

∂ug

∂z=

g

f ρ0

∂ρ

∂y,∂vg

∂z=−

g

f ρ0

∂ρ

∂x, (20)

from the ocean surface to depth z leads to depth-dependentgeostrophic currents,

ug(z)= ug(S)+ uBC(z), vg(z)= vg(S)+ vBC(z), (21)

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4 P. C. Chu: Technical Note: Two types of absolute dynamic ocean topography

where

uBC(z)=−g

f ρ0

0∫z

∂ρ

∂ydz′, vBC(z)=

g

f ρ0

0∫z

∂ρ

∂xdz′, (22)

are the baroclinic geostrophic currents. Here, f = 2�sin(ϕ)is the Coriolis parameter; �= 2π/(86400s) is the meanEarth rotation rate; and ϕ is the latitude.

The volume-integrated total energy, i.e., sum of kinetic en-ergy of the geostrophic currents and the available potentialenergy (Oort et al., 1989), for an ocean basin (W ) is given by

E =

∫ ∫ ∫W

[12(u2

g+ v2g)+

g2ρ2

2ρ20n

2

]dxdydz. (23)

Substitution of Eqs. (21) and (22) into Eq. (23) leads to

E(Dx ,Dy ,ρ)=g2

2

∫ ∫ ∫W[

(−Dy +f uBCg

)2/f 2+ (Dx +

f vBCg

)2/f 2+

ρ2

ρ20n

2

]dxdydz

=g2

2

∫ ∫ ∫W

[D2x/f

2+D2

y/f2+ 2Dx

vBCfg− 2Dy

uBCfg

]dxdydz

+12

∫ ∫ ∫W

[u2

BC+ v2BC+

g2ρ2

ρ20n

2

]dxdydz (24)

4 Governing equation of D

For a given density field, the second integration in the rightside of Eq. (24) is known. The geostrophic currents takingthe minimum energy state provides a constraint for D,

G(Dx,Dy)≡

∫ ∫ ∫W[(

D2x +D

2y + 2Dx

f vBCg− 2Dy

f uBCg

)/f 2

]dxdydz→min. (25)

The three-dimensional integration Eq. (25) over the oceanbasin is conducted by

∫ ∫ ∫W

[. . .]dxdydz=∫ ∫R

0∫

−H

[. . .]dz

dxdy, (26)

where R is the horizontal area of the water volume, H is thewater depth. Thus, Eq. (25) becomes

G(Dx,Dy)=

∫ ∫R

L(Dx,Dy)dxdy→min, (27)

L(Dx,Dy)≡[H(D2

x +D2y)+ 2DxY − 2DyX

]/f 2, (28)

where the parameters (X, Y ) are given by

X(x,y)≡f

g

0∫−H

uBCdz=−1ρ0

0∫−H

0∫z

∂ρ̂

∂ydz′dz (29)

Y (x,y)≡f

g

0∫−H

vBCdz=1ρ0

0∫−H

0∫z

∂ρ̂

∂xdz′dz, (30)

which represent vertically integrated baroclinic geostrophiccurrents scaled by the factor f/g (unit: m). Here, Eq. (18) isused (i.e., horizontal gradient of in situ density is the same asthat of density anomaly).

The Euler–Lagragian equation of the functional Eq. (27)is given by

∂L

∂D−∂

∂x

(∂L

∂Dx

)−∂

∂y

(∂L

∂Dy

)= 0. (31)

Substitution of Eq. (28) into Eq. (31) gives an elliptic par-tial differential equation (i.e., the governing equation) for thefirst type of DOT (i.e., D),

f 2∇

[(H/f 2

)∇D

]=−F,

or

H

[∇

2D+ r(x)∂D

∂x+ r(y)

∂D

∂y− 2(β/f )

∂D

∂y

]=−F, (32)

where

F ≡

(∂Y

∂x−∂X

∂y

), ∇ ≡ i

∂

∂x+ j

∂

∂y(33)

r(x) ≡1H

∂H

∂x, r(y) ≡

1H

∂H

∂y, β =

2�a

cos(ϕ), (34)

where a = 6370km, is the mean Earth radius. Thegeostrophic balance does not exist at the Equator. The Cori-olis parameter f needs some special treatment for low lati-tudes. In this study, f is taken as 2�sin(5π/180) if latitudeis between 10◦ N and 0◦; and as −2�sin(5π/180) if latitudeis between 0◦ and 10◦ S.

Let 0 be the coastline of the ocean basin. Continuation ofthe geoid from land to oceans gives

N |0 =Nl|0, N∗|0 =Nl|0, (35)

which leads to

N |0 =N∗|0. (36)

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P. C. Chu: Technical Note: Two types of absolute dynamic ocean topography 5

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) sec-ond 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) histogramof 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 difference1(∂D/∂x)= ∂D∗/∂x− ∂D/∂x, (d) histogram of global ∂D/∂x, and (e) histogram of global ∂D∗/∂x.

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6 P. C. Chu: Technical Note: Two types of absolute dynamic ocean topography

Here, Nl is the geoid over land. The boundary conditionEq. (36) can be rewritten as

D|0 = (S−N)|0 = (S−N∗)|0 =D∗|0 (37)

which is the boundary condition of D.

5 Numerical solution of D

The well-posed elliptic Eq. (32) is integrated numerically on1◦× 1◦ grids for the world oceans with the boundary val-ues (i.e., Eq. 37) taken from the MDOT (1993–2006) field(i.e., D∗), at the NASA/JPL website: https://grace.jpl.nasa.gov/data/get-data/dynamic-ocean-typography/ (last access:31 July 2018) (0.5◦ interpolated into 1◦ resolution). The forc-ing function F is calculated on a 1◦×1◦ grid from the WorldOcean Atlas 2013 (WOA13) temperature and salinity fields,which was downloaded from the NOAA National Centers forEnvironmental Information (NCEI) website: https://www.nodc.noaa.gov/OC5/woa13/woa13data.html (last access: 31July 2018). The three-dimensional density was calculated us-ing the international thermodynamic equation of seawater –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 wasdownloaded from the NCEI 5-Minute Gridded Global ReliefData Collection at the website: https://www.ngdc.noaa.gov/mgg/global/etopo5.HTML (last access: 31 July 2018). Dis-cretization of the elliptic Eq. (32) and numerical integrationare given in Appendix B.

6 Difference between the two DOTs

The first-type global DOT (Di,j ; Fig. 2a) is the numericalsolution of the elliptic Eq. (32) with the boundary conditionEq. (37). The second type global MDOT (D∗i,j ; Fig. 2b) isdownloaded from the NASA/JPL website: https://grace.jpl.nasa.gov/data/get-data/dynamic-ocean-typography/ (last ac-cess: 31 July 2018). The difference between the two DOTs,

1Di,j =D∗i,j −Di,j , (38)

is evident in the world oceans (Fig. 2c). Here, (i, j ) de-note the horizontal grid point. The relative root-mean square(RRMS) of 1D is given by

RRMS(1D)=

√1M

∑i

∑j

(1Di,j )2√1M

∑i

∑j

(Di,j )2= 0.386, (39)

where M = 38877 is the number of total grid points. BothD and D∗ have positive and negative values. The arith-metic mean values (0.524 and −3.84 cm) are much smallerthan the RMS mean values. They are 1 order of magnitude

smaller than the corresponding standard deviations (54.9 and71.2 cm; see Fig. 2d, e). The magnitude ofD andD∗ are rep-resented by their root-mean squares, which are close to theirstandard deviations.

Histograms for Di,j (Fig. 2d) and D∗i,j (Fig. 2e) are bothnon-Gaussian and negatively skewed. The major differencebetween the two is the single modal form ofDi,j with a peakat around 20 cm and the bimodal form of D∗i,j with a highpeak at around 30 cm and a low peak at −140 cm. The sta-tistical parameters are different, e.g., mean value and stan-dard deviation are 0.524 and 54.9 cm for Di,j , and −3.84and 71.2 cm for D∗i,j . Skewness and kurtosis are −0.83 and3.01 for Di,j , and −0.87 and 2.80 for D∗i,j .

Horizontal gradients of the DOT, (∂Di,j/∂x, ∂Di,j/∂y)and (∂D∗i,j/∂x,∂D∗i,j/∂y), have oceanographic signifi-cance (related to the geostrophic currents). They are calcu-lated using the central difference scheme at inside-domaingrid points and the first order forward/backward differencescheme at grid points next to the boundary. The differencein global ∂Di,j/∂x (Fig. 3a) and ∂D∗i,j/∂x (Fig. 3b) is ev-ident with much smaller-scale structures in ∂D∗i,j/∂x. Thedifference between the two gradients (Fig. 3c),

1(∂Di,j/∂x)= ∂D∗i,j/∂x− ∂Di,j/∂x, (40)

has the same order of magnitude as the gradients themselveswith the relative RRMS of 1(∂D/∂x),

RRMS[1(∂D/∂x)

]=√

1M

∑i

∑j

[1(∂Di,j/∂x)

]2√

1M

∑i

∑j

(∂Di,j/∂x)2= 1.04, (41)

which implies that the non-surface latitudinal geostrophiccurrent component of the second type of MDOT has the sameorder of magnitude as the surface latitudinal geostrophic cur-rent component of the first type of DOT. Histograms for∂Di,j/∂x (Fig. 3d) and ∂D∗i,j/∂x (Fig. 3e) are near sym-metric with mean values around (−1.29, −0.78)×10−8 andstandard deviations (2.69, 4.95)×10−7. The standard devia-tion of ∂D∗i,j/∂x is almost twice that of ∂Di,j /∂x.

Similarly, the difference in global ∂Di,j/∂y (Fig. 4a) and∂D∗i,j/∂y (Fig. 4b) is evident with much smaller-scale struc-tures in ∂D∗i,j/∂y. The difference between the two gradients(Fig. 4c),

1(∂Di,j/∂y)= ∂D∗i,j/∂y− ∂Di,j/∂y, (42)

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P. C. Chu: Technical Note: Two types of absolute dynamic ocean topography 7

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) thedifference 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 themselveswith the relative root-mean square (RRMS) of 1(∂D/∂y),

RRMS[1(∂D/∂y)

]=√

1M

∑i

∑j

[1(∂Di,j/∂y)

]2√

1M

∑i

∑j

(∂Di,j/∂y)2= 0.98, (43)

which implies that the non-surface zonal geostrophic currentcomponent of the second type of MDOT has the same orderof magnitude as the surface zonal geostrophic current com-ponent of the first type of DOT. Histograms for ∂Di,j /∂y(Fig. 4d) and ∂D∗i,j/∂y (Fig. 4e) are also near symmetricwith the mean values around (2.32, 1.18)×10−7 and stan-dard deviations (1.20, 2.44)×10−6 . The standard deviationof ∂D∗i,j/∂y is twice that of ∂Di,j/∂y. The denominators ofEqs. (41) and (43) represent the magnitude of the horizontalgradients of the first type of DOT.

7 Conclusions

The change in marine geoid from classically defined (firsttype, stand-alone concept in oceanography) to satellite-determined (second type, stand-alone concept in marinegeodesy) largely affects oceanography. With the classicallydefined marine geoid (average level of SSH if the water is at

rest), the horizontal gradients of the first type of DOT rep-resent the absolute surface geostrophic currents. With thesatellite-determined (second type) marine geoid by Eq. (6),the horizontal gradients of the second type of MDOT donot represent the absolute surface geostrophic currents. Thedifference between the two types of DOT represents an ad-ditional component to the absolute surface geostrophic cur-rents.

With conservation of potential vorticity, geostrophic bal-ance represents the minimum energy state in an ocean basinwhere the mechanical energy is conserved. A new governingelliptic equation of the first type of DOT is derived with wa-ter depth (H ) in the coefficients and the three-dimensionaltemperature and salinity in the forcing function. This gov-erning elliptic equation is well posed. Continuation of thegeoid from land to ocean leads to an inhomogeneous Dirich-let boundary condition.

Difference between the two types of DOT is evident witha RRMS difference of 38.6 %. Horizontal gradients (repre-senting geostrophic currents) of the two types of DOTs aredifferent with much smaller-scale structures in the secondtype absolute DOT. The RRMS difference is near 1.0 in both(x, y) components of the DOT gradient, which implies thatthe non-absolute surface geostrophic currents identified fromthe second type has the same order of magnitude as the ab-solute surface geostrophic currents identified by the first typeof DOT.

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8 P. C. Chu: Technical Note: Two types of absolute dynamic ocean topography

The notable difference between the two types of DOTraises more questions in oceanography and marine geodesy:is there any theoretical foundation to connect the classicalmarine geoid (stand-alone concept in oceanography usingthe principle of surface geostrophic currents without 1g) tothe satellite-determined marine geoid (stand-alone conceptin marine geodesy using 1g without the principle of sur-face geostrophic currents)? How can the satellite-determinedmarine geoid using the gravity anomaly (1g) be conformedto the basic physical oceanography principle of surfacegeostrophic currents? What is the interpretation of the hor-izontal gradients of the second type of MDOT (D∗)? Is thereany evidence or theory to show (ug(N∗)= 0, vg(N∗)= 0)similar to Eq. (13)? More observational and theoretical stud-ies are needed in order to solve those problems. The mainchallenge for oceanographers is how to use the satellite al-timetry observed SSH such as the Surface Water and OceanTopography (SWOT, https://swot.jpl.nasa.gov/, last access:31 July 2018) to infer the ocean general circulations atthe surface. A new theoretical framework rather than thegeostrophic constraint needs to be established.

The GOCE satellite-determined data-only geoid modelis 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 ofDOT. However, such a replacement does not solve the funda-mental problem presented here, i.e., incompatibility betweensatellite-determined marine geoid using the gravity anomaly(1g) and the classical marine geoid (mean SSH when thewater at rest) on the basis of the basic physical oceanographyprinciple of surface geostrophic currents.

Finally, the mathematical framework described here (i.e.,the elliptic Eq. 32 with boundary condition Eq. 37) may leadto a new inverse method for calculating three-dimensionalabsolute geostrophic velocity from temperature and salinityfields since the surface absolute geostrophic velocity is thesolution of Eq. (32). This will be a useful addition to theexisting β-spiral method (Stommel and Schott, 1977), boxmodel (Wunsch, 1978), and P -vector method (Chu, 1995;Chu et al., 1998, 2000).

Data availability. The datasets used for this research areall 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 ac-cess: 31 July 2018), (2) World Ocean Atlas 2013 (WOA13):https://www.nodc.noaa.gov/OC5/woa13/woa13data.html (NOAANational Centers for Environmental Information; last access: 31July 2018), and (3) NCEI 5- 5-Minute Gridded Global Relief DataCollection: https://www.ngdc.noaa.gov/mgg/global/etopo5.HTML(NOAA National Centers for Environmental Information; lastaccess: 31 July 2018).

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P. C. Chu: Technical Note: Two types of absolute dynamic ocean topography 9

Appendix A: Geostrophic balance as a minimum energystate in an energy conserved basin

In large-scale motion (small Rossby number) with theBoussinesq approximation, the linearized potential vorticity(5) is given by

5≈

[f +

(∂v

∂x−∂u

∂y

)]∂ρ̂

∂z≈ f

(−ρ0n

2

g+∂ρ

∂z

)−ρ0n

2

g

(∂v

∂x−∂u

∂y

), (A1)

where, ρ0 = 1025 kg m−3 is the characteristic density. With-out the frictional force and 0 horizontally integrated buoy-ancy flux at the surface and bottom, the energy (including ki-netic and available potential energies) is conserved in a three-dimensional ocean basin (V )

E =

∫ ∫ ∫V

Jdxdydz, J ≡12

(u2+ v2+w2

)

+g2ρ2

2ρ20n

2, (A2)

dEdt= 0. (A3)

The two terms of J are kinetic energy and available poten-tial energy.

To show the geostrophic balance taking the minimum en-ergy state for a given linear PV (see Eq. A1), the constraint isincorporated by extremizing the integral (see also in Vallis,1992; Chu, 2018)

I ≡

∫ ∫ ∫V

{12

(u2+ v2+w2

)+g2ρ2

2ρ20n

2+µ(x,y,z)

[f

(−ρ0n

2

g+∂ρ

∂z

)−ρ0n

2

g

(∂v

∂x−∂u

∂y

)]}dxdydz,

(A4)

where µ(x,y,z) is the Lagrange multiplier, which is a func-tion of space. If it were a constant, the integral would merelyextremize energy subject to a given integral of PV, and re-arrangement of PV would leave the integral unaltered. Ex-tremization of the integral Eq. (A4) gives the three Euler–

Lagrange equations,

∂K

∂ρ−∂

∂z

∂K

∂ρz= 0, (A5)

∂K

∂u−∂

∂y

∂K

∂uy= 0, (A6)

∂K

∂v−∂

∂x

∂K

∂vx= 0, (A7)

where K is in the integrand appearing in Eq. (A4). Substitu-tion of K into Eqs. (A5), (A6), (A7) leads to

g2

ρ20n

2ρ = f

∂µ

∂z, (A8)

u=ρ0n

2

g

∂µ

∂y, v =−

ρ0n2

g

∂µ

∂x. (A9)

Differentiation of Eq. (A9) with respect to z and use ofEq. (A8) leads to

∂u

∂z=

g

f ρ0

∂ρ

∂y=∂ug

∂z,∂v

∂z=−

g

f ρ0

∂ρ

∂x=∂vg

∂z, (A10)

which shows that (u, v)= (ug, vg) have the minimum energystate.

Appendix B: Numerical solution of the Eq. (32)

Let the three axes (x, y, z) be discretized into local rectangu-lar grids in horizontal and non-uniform grids in vertical (xi,j ,yi,j , zk) with cell sizes (1◦× 1◦),

1y =π

360rE, 1xj =1y cosφj , 1zk = zk − zk+1,

i = 1,2, . . ., I ; j = 1,2, . . .,J ; k = 1,2, . . .,Ki,j , (B1)

where k =1 for the surface, k =Ki,j for the bottom; φj is thelatitude of the grid point; rE = 6371km, is the Earth radius;I = 360; and J = 1804. The subscripts in Ki,j in Eq. (B1)indicates non-uniform water depth in the region.

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10 P. C. Chu: Technical Note: Two types of absolute dynamic ocean topography

The parameters (Xi,j , Yi,j ) in Eqs. (29) and (30) (inSect. 4) are calculated by

Xi,j ≡1

4ρ0

Ki,j∑k=2

k∑l=1

ρ̂i,j+1,l − ρ̂i,j,l

1y+ρ̂i+1,j+1,l − ρ̂i+1,j,l

1y

+ρ̂i,j+1,l+1− ρ̂i,j,l+1

1y+ρ̂i+1,j+1,l+1− ρ̂i+1,j,l+1

1y

1zk(1zl+1zl+1

2

)](B2)

Yi,j ≡1

4ρ0

Ki,j∑k=2

k∑l=1

ρ̂i+1,j,l − ρ̂i,j,l

1xj+ρ̂i+1,j+1,l − ρ̂i,j+1,l

1xj

+ρ̂i+1,j,l+1− ρ̂i,j,l+1

1xj+ρ̂i+1,j+1,l+1− ρ̂i,j+1,l+1

1xj

1zk(1zl+1zl+1

2

)], (B3)

which gives the discretized forcing function

Fi,j =Yi+1,j −Yi−1,j

21xj−Xi,j+1−Xi,j−1

21y. (B4)

The governing Eq. (32) is discretized by

Di+1,j − 2Di,j +Di−1,j

(1xj )2+Di,j+1− 2Di,j +Di,j−1

(1y)2

+ r(x)i,j

Di+1,j −Di−1,j

21xj+

(r(y)i,j −

2βjfj

)Di,j+1−Di,j−1

21y=−

Fi,j

Hi,j, (B5)

which is reorganized by

2(

1+ cos2φj

)Di,j =

(1+

12r(x)i,j 1y cosφj

)Di+1,j

+

(1−

12r(x)i,j 1y cosφj

)Di−1,j + cos2φj[

1+(r(y)i,j −

2cotφjrE

)1y

2

]Di,j+1

+ cos2φj

[1−

(r(y)i,j −

2cotφjrE

)1y

2

]Di,j−1

+Fi,j

Hi,j(1y)2cos2φj . (B6)

The iteration method is used to solve the algebraicEq. (B6) with a large value of I × J . It starts from the 0thstep,

D(0)i,j = 0, i = 1,2, . . ., I ; j = 1,2, . . .J. (B7)

With the given boundary condition (Eq. 37, see Sect. 4)and forcing function (Eq. B4), the first type of DOT at thegrid points can be computed from steps n to n+1,

2(

1+ cos2φj

)D(n+1)i,j =

(1+

12r(x)i,j 1y cosφj

)D(n)i+1,j

+

(1−

12r(x)i,j 1y cosφj

)D(n)i−1,j + cos2φj[

1+(r(y)i,j −

2cotφjrE

)1y

2

]D(n)i,j+1

+ cos2φj

[1−

(r(y)i,j −

2cotφjrE

)1y

2

]D(n)i,j−1

+Fi,j

Hi,j(1y)2cos2φj . (B8)

Such iteration continues until the RRMS differencereaches the criterion,

r =

√1M

I∑i=1

J∑j=1

[D(n+1)i,j −D

(n)i,j

]2

√1M

I∑i=1

J∑j=1

[D(n)i,j

]2< 10−6, (B9)

where M = 38877, is the total number of the grid points onthe ocean surface.

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P. C. Chu: Technical Note: Two types of absolute dynamic ocean topography 11

Author contributions. PCC discovered the problem, formulatedthe theory, calculated the two types of DOT, and prepared themanuscript.

Competing interests. The authors declare that they have no conflictof interest.

Acknowledgements. The author thanks Chenwu Fan for invaluablecomments and computational assistance, NOAA/NCEI for theWOA-2013 (T, S) and ETOPO5 sea-floor topography data, andNASA/JPL (second type) MDOT data.

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

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