Two types of marine geoid exist with the first type being the average level
of sea surface height (SSH) if the water is at rest (classical
definition), and the second type being satellite-determined with the
condition that the water is usually not at rest. The differences between
the two are exclusion (inclusion) of the gravity anomaly and non-measurable
(measurable) in the first (second) type. The associated 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 water being at
rest on the first type of marine geoid. Horizontal gradients of the second
type of DOT represent the surface geostrophic currents relative to flow on
the second type of marine geoid. Difference between the two is quantitatively
identified in this technical note through comparison between the first type
of DOT and the mean second type of DOT (MDOT). The first type of DOT is
determined by a physical principle that the geostrophic balance takes the
minimum energy state. Based on that, a new elliptic equation is derived for
the first type of DOT. The continuation of geoid from land to ocean leads to
an inhomogeneous Dirichlet boundary condition with the boundary values taking
the satellite-observed second type of MDOT. This well-posed elliptic equation
is integrated numerically on 1

Let the coordinates (

Two types of marine geoid and DOT:

The vertical deflection is the slope of the geoid

Equation (

Before satellites came into operation,

After satellites came into operation, SSH has been
observed with uniquely sampled temporal and spatial
resolutions by high-precision
altimetry above a reference ellipsoid (not geoid; Fu and Haines, 2013). Two
Gravity Recovery and Climate Experiment (GRACE) satellites, launched in 2002,
provide data to compute the marine geoid (called the GRACE Gravity Model,
GGM; see website:

The oceanic conditions at

A question arises: do the horizontal gradients of the second type of MDOT
(

The rest of the paper is outlined as follows. Section 2 describes the change in DOT due to the change in marine geoid from the first to the second type. Section 3 describes geostrophic currents and energy related to the first type of DOT. Section 4 presents the governing equation of the first type of DOT with the boundary condition at the coasts. Section 5 shows the numerical solution for the world oceans. Section 6 evaluates the change in global DOT from first to second type with oceanographic implications. Section 7 concludes the study.

The second type of MDOT (

Correspondingly, the change in DOT is given by

Conservation of potential vorticity for a dissipation-free fluid does not
apply precisely to sea water where the density is a function not only of
temperature and pressure but also of the dissolved salts. The effect of
salinity on density is very important in the distribution of water
properties. However, for most dynamic studies the effect of the extra state
variable is not significant and the conservation of potential vorticity is
valid (Veronis, 1980). Based on the conservation of the potential vorticity,
the geostrophic current reaches the minimum energy state (Appendix A). Due to
the minimum energy state, an elliptic partial differential equation for

If

Equation (

Using the first type of marine geoid

The volume-integrated total energy, i.e., sum of kinetic energy of the
geostrophic currents and the available potential energy (Oort et al., 1989),
for an ocean basin (

Substitution of Eqs. (

Derivatives in the

For a given density field, the second integration in the right side of
Eq. (

The three-dimensional integration Eq. (

The Euler–Lagragian equation of the functional Eq. (

Substitution of Eq. (

Let

Here,

The well-posed elliptic Eq. (

Derivatives in the

The first-type global DOT (

Histograms for

Horizontal gradients of the DOT, (

Similarly, the difference in global

The change in marine geoid from classically defined (first type, stand-alone
concept in oceanography) to satellite-determined (second type, stand-alone
concept in marine geodesy) largely affects oceanography. With the
classically defined marine geoid (average level of SSH if the water is at
rest), the horizontal gradients of the first type of DOT represent the absolute
surface geostrophic currents. With the satellite-determined (second type)
marine geoid by Eq. (

With conservation of potential vorticity, geostrophic balance represents the
minimum energy state in an ocean basin where the mechanical energy is
conserved. A new governing elliptic equation of the first type of DOT is derived
with water depth (

Difference between the two types of DOT is evident with a RRMS difference of
38.6 %. Horizontal gradients (representing geostrophic currents) of the
two types of DOTs are different with much smaller-scale structures in the
second type absolute DOT. The RRMS difference is near 1.0 in
both (

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

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
increase the accuracy of the calculation of the second type of DOT. However,
such a replacement does not solve the fundamental problem presented here,
i.e., incompatibility between satellite-determined marine geoid using the
gravity anomaly (

Finally, the mathematical framework described here (i.e., the elliptic
Eq.

The datasets used for this research are all from open
sources listed as follows: (1) Satellite-determined DOT data:

In large-scale motion (small Rossby number) with the Boussinesq
approximation, the linearized potential vorticity (

The two terms of

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)

Differentiation of Eq. (A9) with respect to

Let the three axes (

The parameters (

The governing Eq. (

The iteration method is used to solve the algebraic Eq. (B6) with a large value
of

With the given boundary condition (Eq.

Such iteration continues until the RRMS difference
reaches the criterion,

PCC discovered the problem, formulated the theory, calculated the two types of DOT, and prepared the manuscript.

The author declares that he has no conflict of interest.

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