We propose “manometric” as a new term because in the existing literature there is no unambiguous and generally used term for Δ𝑅𝑚. It may be described as the “mass effect on”, “mass contribution to”, “mass component of” or “mass term in” sea level or sea-level change, but these descriptions could equally well refer to GRD -induced sea-level change (the effects of a change in the geographical distribution of mass) or barystatic sea-level rise, so they can be confusing. (“Manometric” is an existing word, referring to the measurement of hydrostatic pressure using a column of liquid, a concept that is closely related to bottom pressure.)
N19 Barystatic sea-level riseℎ𝑏: The part of  global-mean sea-level rise  (GMSLR) which is due to the addition to the ocean of water mass that formerly resided within the land area (as land water storage or land ice) or in the atmosphere (which contains a relatively tiny mass of water), or (if negative) the removal of mass from the ocean to be stored elsewhere. It is also called “barystatic sea-level change”.
Land water storage, also called terrestrial water storage, is water on land that is stored as groundwater, soil moisture, water in reservoirs, lakes and rivers, seasonal snow and permafrost. Land ice means ice sheets, glaciers, permanent snow and firn. Barystatic sea-level rise includes contributions from changes in all of these.
It does not include changes in the parts of ice shelves and glacier tongues whose weight is supported by the ocean rather than resting on land. (These floating parts constitute the majority of the mass of ice shelves and glacier tongues, but near the grounding line on the seaward side some part of the weight may be supported by the land-based ice.) Where land ice rests on a bed which is below mean sea level, it is already displacing sea water. Therefore, the land ice contribution to barystatic sea-level rise excludes the mass whose liquid-water equivalent volume equals the volume of sea water already displaced. The remainder, which is not currently displacing sea water, is often referred to as the ice mass or volume above flotation in glaciology.
We define barystatic sea-level rise as
ℎ𝑏 = Δ𝑀𝜌f𝐴, (37)
i.e., the change in mass Δ𝑀 of the global ocean from added freshwater, converted to a change in global ocean volume and divided by the ocean surface area A. Because global-mean halosteric change is negligible, the salinity of the existing sea water does not affect ℎ𝑏. Any contribution 𝛿𝑀 to barystatic sea-level rise can be expressed as its sea-level equivalent (SLE)𝛿𝑀/(𝜌f𝐴), using the same formula.
The formula provides a convenient method of quantifying the changes in the mass of the ocean if A is constant. However, ℎ𝑏 and SLE may not accurately indicate the contribution of added mass to global-mean ocean thickness if there is a substantial change to A, as for example in the transition from glacial to interglacial.
Calculating the global mean of manometric sea-level change Δ𝑅𝑚 from its Definition B (Eq. 33) gives 1/𝐴∫Δ𝑅𝑚d𝐴=Δ𝑀/(𝜌∗𝐴)≃ℎ𝑏, i.e., approximately equal to the barystatic sea-level change, but not exactly since 𝜌∗≠𝜌f. With Definition A, the global mean of Δ𝑅𝑚 exactly equals ℎ𝑏. (See global-mean sea-level rise .) Despite this relationship between manometric sea-level change and barystatic sea-level rise, we argue that we need distinct terms for them, rather than referring to the latter as the global mean of the former, for two reasons.
First, barystatic sea-level rise is well defined by conservation of water mass on Earth and can be evaluated from the change in mass of other stores of water, e.g., ice sheets and glaciers, without considering the ocean. This has been the usual approach in observational studies of the budget of global-mean sea-level rise, and is the only possibility for diagnosing ℎ𝑏 from the majority of climate models whose ocean component is Boussinesq or has a linear free surface, and therefore does not conserve water mass. Secondly, the partitioning of RSLC into steric and manometric (Eq. 35) is somewhat arbitrary, because it depends on the choice of 𝜌∗ as a reference density.
Neither reason for the distinction of Δ𝑅𝑚 and ℎ𝑏 applies to thermosteric sea-level change; its contribution to global-mean sea-level rise can only be conceived or evaluated as the global mean of the local Δ𝑅𝜃, whose definition by Eq. (31) is well defined.
In recent literature, “eustatic” is often used as a synonym for “barystatic”, whereas in geological literature eustatic sea-level change means either global-mean sea-level rise or global-mean geocentric sea-level rise . Because of this confusion of meaning, we deprecate the term “eustatic”, following the last three assessment reports of the Intergovernmental Panel on Climate Change (Church et al. 2001; Meehl et al. 2007; Church et al. 2013).
N20 Sterodynamic sea-level changeΔ𝑍: Relative sea-level change  due to changes in ocean density and circulation, with  inverse barometer  (IB) correction.
This term is the sum of ocean dynamic sea-level change (which includes the IB correction) and global-mean thermosteric sea-level rise ,
Δ𝑍(𝐫)=Δ𝜁(𝐫)+ℎ𝜃. (38)
It can be diagnosed from ocean models (even those that do not conserve mass as per the commonly used Boussinesq models) as the sum of the changes in zos and zostoga. Sterodynamic sea-level change is the part of relative sea-level change that can be simulated with such models. (As discussed above for ocean dynamic sea-level change Δ𝜁, the change in zos calculated from CMIP data should have zero global mean.)
“Sterodynamic” is a term which is newly introduced in this paper. We propose it because in the existing literature there is no clear, simple or generally used term for Δ𝑍. It is a concept that appears in the literature, where it is referred to by various cumbersome phrases, such as “the oceanographic part of sea-level change”, “steric plus dynamic sea-level change” or “sea-level change due to ocean density and circulation change”.
N21 Vertical land movement (VLM)Δ𝐹: The change in the height of the  sea floor  or the land surface.
VLM has several causes, including isostasy, elastic flexure of the lithosphere, earthquakes and volcanoes (due to tectonics). All of these involve a change in height of the existing solid surface. In contrast, landslides and sedimentation alter the solid surface and its height by transport of materials; some authors count them as VLM. Extraction of groundwater and hydrocarbons may cause subsidence (sinking of the solid surface) by compaction (the reduction in the liquid fraction in the sediment). These anthropogenic effects can be locally large, e.g., in Manila, and can exceed the natural effects by orders of magnitude. Where VLM occurs near the coast, it may cause emergence or submergence of land and thus alter the coastline.
Isostasy or isostatic adjustment is the process of adjustment of the lithosphere (the crust and the rigid upper part of the mantle) towards a hydrostatic equilibrium in which it is regarded as floating in the asthenosphere (the underlying viscous mantle, which is of higher density than the lithosphere), with an equal pressure everywhere at some notional horizontal level beneath the lithosphere. On geological timescales, isostatic adjustment occurs in response to changes in the mass load of the lithosphere upon the mantle beneath (the asthenosphere and lower mantle), due to erosion, sedimentation or emplacement of igneous rocks.
On climate timescales there are large changes in load due to the varying mass of ice on land during glacial–interglacial cycles. (See glacial isostatic adjustment .) Isostatic adjustment occurs over multi-millennial timescales determined by the viscous flow of the mantle beneath the lithosphere. An elastic response of the lithosphere, on annual timescales, occurs in response to changes in load. Although it is small compared with the eventual isostatic response, it is much more rapid, and hence responsible for significant VLM due to contemporary and recent historical changes in land ice, for instance in West Antarctica.
N22 GRD: Changes in Earth  Gravity, Earth Rotation (and hence centrifugal acceleration) and viscoelastic solid-Earth Deformation.


N21 Vertical land movement (VLM)Δ𝐹: The change in the height of the  sea floor  or the land surface.
VLM has several causes, including isostasy, elastic flexure of the lithosphere, earthquakes and volcanoes (due to tectonics). All of these involve a change in height of the existing solid surface. In contrast, landslides and sedimentation alter the solid surface and its height by transport of materials; some authors count them as VLM. Extraction of groundwater and hydrocarbons may cause subsidence (sinking of the solid surface) by compaction (the reduction in the liquid fraction in the sediment). These anthropogenic effects can be locally large, e.g., in Manila, and can exceed the natural effects by orders of magnitude. Where VLM occurs near the coast, it may cause emergence or submergence of land and thus alter the coastline.
Isostasy or isostatic adjustment is the process of adjustment of the lithosphere (the crust and the rigid upper part of the mantle) towards a hydrostatic equilibrium in which it is regarded as floating in the asthenosphere (the underlying viscous mantle, which is of higher density than the lithosphere), with an equal pressure everywhere at some notional horizontal level beneath the lithosphere. On geological timescales, isostatic adjustment occurs in response to changes in the mass load of the lithosphere upon the mantle beneath (the asthenosphere and lower mantle), due to erosion, sedimentation or emplacement of igneous rocks.
On climate timescales there are large changes in load due to the varying mass of ice on land during glacial–interglacial cycles. (See glacial isostatic adjustment .) Isostatic adjustment occurs over multi-millennial timescales determined by the viscous flow of the mantle beneath the lithosphere. An elastic response of the lithosphere, on annual timescales, occurs in response to changes in load. Although it is small compared with the eventual isostatic response, it is much more rapid, and hence responsible for significant VLM due to contemporary and recent historical changes in land ice, for instance in West Antarctica.
N22 GRD: Changes in Earth  Gravity, Earth Rotation (and hence centrifugal acceleration) and viscoelastic solid-Earth Deformation.
These three effects are all caused by changes in the geographical distribution of ocean and solid mass over the Earth. They are often considered together because they occur simultaneously and may interact. Changes in gravitation and rotation alter the geopotential field and hence the geoid 𝐺(𝐫), while deformation of the solid Earth changes the sea floor topography 𝐹(𝐫) through vertical land movement . By altering G and 𝐹, GRD induces relative sea-level change (e.g., Tamisiea and Mitrovica 2011; Kopp et al. 2015), which redistributes but does not change the global ocean volume and thus causes no global-mean sea-level rise . GRD-induced relative sea-level changeΔ𝛤 is defined as
Δ𝛤 = Δ𝐺′ − Δ𝐹′ (39)
(derived in Sect. 6 as Eq. 51) where Δ𝐺′ and Δ𝐹′ are the deviations of the changes in the geoid and in the sea floor from their respective global (ocean) means. By construction, the global (ocean) means of 𝐺′ and 𝐹′ are each zero; hence, the global (ocean) mean of Δ𝛤 is zero.
Whatever the cause, redistribution of the ocean mass itself has GRD effects, and thereby the ocean affects its own mass distribution and mean sea level (MSL). Thus, MSL, the geoid and the sea floor must all be related in a self-consistent solution, which in the context of glacial isostatic adjustment (GIA) is expressed by the sea-level equation (Farrell and Clark 1976).
The ocean GRD effects are called self-attraction and loading (SAL), where “loading” means the weight on the solid Earth. SAL is caused by climatic change in ocean density and circulation (Gregory et al. 2013), which do not involve any change in the mass of the ocean. SAL is also a component of GRD which is caused by changes in land ice and in the solid Earth; thus, SAL contributes to the sea-level effects of GIA, contemporary GRD and mantle dynamic topography as well.
We propose the new term “GRD” in the absence of any existing single term to describe this frequently discussed group of effects. GRD-induced relative sea-level change may be described as the “mass effect”, “mass contribution”, “mass component” or “mass term”, but these labels could equally well refer to manometric sea-level change if local, or barystatic sea-level rise if global, so they can be confusing. Moreover, “GRD” is helpful as a label for a concept which unifies SAL, GIA, contemporary GRD and mantle dynamic topography.
N23 Glacial isostatic adjustment (GIA): GRD  due to ongoing changes in the solid Earth caused by past changes in land ice.
GIA is caused by the viscous adjustment of the mantle to changes in the load on the lithosphere that occurred when mass was transferred from land ice into the ocean, or the reverse. It is dominated by the ongoing effects of the deglaciation following the Last Glacial Maximum. Due to the reduction in the mass load on land, areas that were beneath former ice sheets are generally rising. This process is sometimes called post-glacial rebound, but that term is unsatisfactory because GIA involves remote vertical land movement as well, both upward and downward. Areas adjacent to the former ice sheets are subsiding as mantle material moves towards the areas of uplift, while land near to the coast is rising and the sea floor is generally subsiding as a result of the increase in the mass of the ocean. The ongoing widespread redistribution of mass also affects the geoid . Together, the changes in geoid and sea floor cause GIA-induced relative sea-level changeΔ𝛤GIA.
Previous changes in land ice during the Holocene contribute to GIA as well, but GIA does not include the contributions from any ongoing change in land ice or ocean mass, whose effects we call contemporary GRD .
N24 Contemporary GRD: GRD  due to ongoing changes in the mass of water stored on land as ice sheets, glaciers and land water storage.
Such transfers of mass cause instantaneous changes in the geoid , and vertical land movement (VLM) on annual timescales due to elastic deformation of the solid Earth, which causes further change to the geoid. Together, these effects produce relative sea-level change (RSLC). There are also slower responses, both VLM and geoid, due to viscous deformation of the asthenosphere. Note that contemporary GRD excludes GIA; the former arises from ongoing change in the mass of water on land, and the latter from past change.
The elastic deformation and associated geoid contributions to contemporary GRD-induced relative sea-level change are separately proportional to the mass Δ𝑀 which has been added to the ocean. Hence, their sum is
Δ𝛤𝑝(𝐫)=Δ𝑀𝛾𝑝(𝐫), (40)
where 𝛾𝑝(𝐫) is a geographically dependent constant of proportionality, independent of Δ𝑀. Since the barystatic sea-level rise is Δ𝑀/𝜌f𝐴 (Eq. 37), the RSLC due to the combination of these three effects is
Δ𝑅𝑝(𝐫) = Δ𝑀𝜙(𝐫) where 𝜙(𝐫) = 𝛾𝑝(𝐫) + 1/𝜌f𝐴. (41)
The addition of freshwater to the ocean will induce sterodynamic sea-level change as well (e.g., Agarwal et al. 2015).
The barystatic–GRD fingerprint 𝜙 is a constant geographical pattern, often called a sea-level fingerprint, or sometimes a static-equilibrium fingerprint to contrast it with the patterns of ocean dynamic sea-level change . “Fingerprint” without qualification can be easily confused with climate detection and attribution studies where the same word refers to the patterns caused by particular climate change forcing agents such as greenhouse gases. “Static-equilibrium” is not informative about the processes concerned. The part of contemporary GRD-induced RSLC due to viscous deformation and associated geoid change cannot be represented by a constant pattern because it depends on convolving the history of mass addition with the time-dependent solid-Earth response.
N25 Mantle dynamic topography: GRD  due to ongoing changes in the solid Earth caused by mantle convection and plate tectonics.


The elastic deformation and associated geoid contributions to contemporary GRD-induced relative sea-level change are separately proportional to the mass Δ𝑀 which has been added to the ocean. Hence, their sum is
Δ𝛤𝑝(𝐫)=Δ𝑀𝛾𝑝(𝐫), (40)
where 𝛾𝑝(𝐫) is a geographically dependent constant of proportionality, independent of Δ𝑀. Since the barystatic sea-level rise is Δ𝑀/𝜌f𝐴 (Eq. 37), the RSLC due to the combination of these three effects is
Δ𝑅𝑝(𝐫) = Δ𝑀𝜙(𝐫) where 𝜙(𝐫) = 𝛾𝑝(𝐫) + 1/𝜌f𝐴. (41)
The addition of freshwater to the ocean will induce sterodynamic sea-level change as well (e.g., Agarwal et al. 2015).
The barystatic–GRD fingerprint 𝜙 is a constant geographical pattern, often called a sea-level fingerprint, or sometimes a static-equilibrium fingerprint to contrast it with the patterns of ocean dynamic sea-level change . “Fingerprint” without qualification can be easily confused with climate detection and attribution studies where the same word refers to the patterns caused by particular climate change forcing agents such as greenhouse gases. “Static-equilibrium” is not informative about the processes concerned. The part of contemporary GRD-induced RSLC due to viscous deformation and associated geoid change cannot be represented by a constant pattern because it depends on convolving the history of mass addition with the time-dependent solid-Earth response.
N25 Mantle dynamic topography: GRD  due to ongoing changes in the solid Earth caused by mantle convection and plate tectonics.
The dynamics of the interior of the Earth cause vertical land movement , such as the uplift of mid-ocean ridges by upwelling material and the formation of oceanic trenches due to subduction. At the same time, material with different density is redistributed within the Earth, altering the geoid . The consequent GRD-induced relative sea-level change can be very large on geological timescales, amounting to hundreds of metres. Mantle dynamic topography does not include glacial isostatic adjustment (although that is also due to ongoing changes in the solid Earth).
Mantle dynamic topography is often called “dynamic topography” in the solid-Earth literature and also refers to changes in topography on land. We deprecate “dynamic topography” in a sea-level context because it could be confused with ocean dynamic topography .
N26 Global-mean sea-level rise (GMSLR)h: The increase Δ𝑉 in the volume of the ocean divided by the ocean surface area A, also called “global-mean sea-level change” (GMSLC). Observational estimation of h is described in Sect. 7.
By definition, GMSLR is
ℎ = Δ𝑉/𝐴 = 1/𝐴Δ(∫(𝜂−𝐹)d𝐴) = 1/𝐴∫Δ𝑅(𝐫)d𝐴 = 1/𝐴∫Δ𝐻(𝐫)d𝐴, (42)
which follows from Eqs. (4), (24) and (25). Hence, GMSLR is the global mean of relative sea-level change Δ𝑅 and equals the global mean of the change Δ𝐻 in the thickness (or “depth”) of the ocean. Note that GMSLR differs from global-mean geocentric sea-level rise because GMSLR is unaltered by a global-mean change (1/𝐴)∫Δ𝐹d𝐴 in the level of the sea floor 𝐹, provided the global ocean volume V does not change.
The global ocean volume can change due to changes in ocean density or due to changes in ocean mass. Hence, GMSLR is the sum of global-mean thermosteric sea-level rise and barystatic sea-level rise ,
ℎ=ℎ𝜃+ℎ𝑏. (43)
A satisfactory explanation of historical observed GMSLR in terms of thermosteric and barystatic contributions has been achieved in recent years thanks to improvements in both observations and models (Church et al. 2011; Gregory et al. 2013; Chambers et al. 2017).
Equation (43) is implied by the Definition A of manometric sea-level change , as the part of relative sea-level change which is not steric (Eq. 36), whose global mean
1𝐴∫Δ𝑅𝑚d𝐴=1𝐴∫Δ𝑅d𝐴−1𝐴∫Δ𝑅𝜌d𝐴=ℎ−ℎ𝜃, (44)
using Eq. (42) and recalling that global-mean steric sea-level change is purely thermosteric. By definition, the part of h which is not steric is the part which is due to addition of mass, so it must be the case that
ℎ𝑏=1𝐴∫Δ𝑅𝑚d𝐴, (45)
i.e., barystatic sea-level rise equals the global mean of manometric sea-level change by Definition A.
The added mass is unlikely to have exactly the temperature of the existing water to which it is added, implying that changes will probably occur to temperature and hence to density. Because of the nonlinearity of the dependence of 𝜌 on 𝜃, there may be a nonzero contribution to ℎ𝜃 in consequence. Since this is a steric effect, by definition it is not part of ℎ𝑏.
From Definition B (Eq. 33) we obtain an approximate expression for global-mean manometric sea-level change as (1/𝐴)∫Δ𝑚/𝜌∗d𝐴=Δ𝑀/(𝜌∗𝐴), where Δ𝑀 is the added mass. This is the same as the expression (Eq. 37) for ℎ𝑏 except that 𝜌f is replaced by 𝜌∗. This difference is the result of the approximation in Eq. (33) that 𝜌∗ ≃ ρ̄. Physically, it is because manometric sea-level change Δ𝑅𝑚 (a local quantity) is dominated by redistribution of existing sea water, for which 𝜌∗ is a good choice of representative density, whereas barystatic sea-level rise (a global quantity) is due only to addition or subtraction of freshwater of density 𝜌f, since the redistributive effect is zero in the global mean.
On glacial–interglacial and geological timescales, the variation of ocean area cannot be neglected, so GMSLR is ill-defined. However, it is still meaningful to consider global-mean relative sea-level changeℎ𝑅, which is the change in global-mean ocean thickness
ℎ𝑅 = 𝑉+Δ𝑉/(𝐴+Δ𝐴) − 𝑉/𝐴 = 𝑉/(𝐴+Δ𝐴) (Δ𝑉/𝑉−Δ𝐴/𝐴). (46)
If A is constant, ℎ𝑅 = ℎ. An increase in A (Δ𝐴>0) gives a negative contribution to ℎ𝑅, counteracting the positive contribution from a concomitant increase in V.
In geological literature, global-mean sea-level rise is sometimes called “eustatic sea-level change”. Following the last three assessment reports of the Intergovernmental Panel on Climate Change (Church et al. 2001; Meehl et al. 2007; Church et al. 2013), we deprecate “eustatic” because it has become a confusing term, which is also used to mean global-mean geocentric sea-level rise or barystatic sea-level rise .
N27 Global-mean geocentric sea-level riseℎ𝐺: The global-mean change in  mean sea level  with respect to the terrestrial reference frame.
This quantity is the global mean of Δ𝜂, the change in MSL relative to the reference ellipsoid . From Eqs. (4) and (5) we have
ℎ𝐺 = 1/𝐴∫Δ𝜂d𝐴 = 1/𝐴∫Δ𝐺d𝐴 = 1/𝐴(Δ𝑉 + ∫Δ𝐹d𝐴) = ℎ + 1/𝐴 ∫Δ𝐹 d𝐴, (47)


On glacial–interglacial and geological timescales, the variation of ocean area cannot be neglected, so GMSLR is ill-defined. However, it is still meaningful to consider global-mean relative sea-level changeℎ𝑅, which is the change in global-mean ocean thickness
ℎ𝑅 = 𝑉+Δ𝑉/(𝐴+Δ𝐴) − 𝑉/𝐴 = 𝑉/(𝐴+Δ𝐴) (Δ𝑉/𝑉−Δ𝐴/𝐴). (46)
If A is constant, ℎ𝑅 = ℎ. An increase in A (Δ𝐴>0) gives a negative contribution to ℎ𝑅, counteracting the positive contribution from a concomitant increase in V.
In geological literature, global-mean sea-level rise is sometimes called “eustatic sea-level change”. Following the last three assessment reports of the Intergovernmental Panel on Climate Change (Church et al. 2001; Meehl et al. 2007; Church et al. 2013), we deprecate “eustatic” because it has become a confusing term, which is also used to mean global-mean geocentric sea-level rise or barystatic sea-level rise .
N27 Global-mean geocentric sea-level riseℎ𝐺: The global-mean change in  mean sea level  with respect to the terrestrial reference frame.
This quantity is the global mean of Δ𝜂, the change in MSL relative to the reference ellipsoid . From Eqs. (4) and (5) we have
ℎ𝐺 = 1/𝐴∫Δ𝜂d𝐴 = 1/𝐴∫Δ𝐺d𝐴 = 1/𝐴(Δ𝑉 + ∫Δ𝐹d𝐴) = ℎ + 1/𝐴 ∫Δ𝐹 d𝐴, (47)
where h is global-mean sea-level rise (GMSLR) defined by Eq. (42). Thus, global-mean geocentric sea-level rise ℎ𝐺 differs from GMSLR because the latter is unaltered by a global-mean change (1/𝐴)∫Δ𝐹d𝐴 in the level of the sea floor 𝐹, provided the volume of the ocean does not change. On geological timescales, when the area of the ocean may change, the global-mean change in level of the sea floor is Δ((1/𝐴)∫𝐹d𝐴).
Relationships Determining Relative Sea-Level Change
Relative sea-level change (RSLC) is Δ𝑅 = Δ𝜂 − Δ𝐹 (Eq. 24). By applying the inverse barometer correction, we obtain IB-corrected RSLC
Δ𝑅 + Δ𝐵 = Δ𝜂 + Δ𝐵 − Δ𝐹 (48a)
= Δ[𝜂−𝐺+𝐵] + Δ[𝐺−𝐹] (48b)
= Δ𝜁 + Δ[𝐺−𝐹], (48c) where in Eq. (48c) we used the definition of ocean dynamic sea-level change Δ𝜁 (Eq. 23) to rewrite the first term. From the definition of the geoid (Eq. 4) we obtain
1/𝐴 ∫ Δ(𝐺−𝐹)d𝐴 = Δ𝑉/𝐴 = ℎ, (49)
the global-mean sea-level rise . Let us write Δ𝐺(𝐫)=Δ𝐺′(𝐫)+(1/𝐴)∫Δ𝐺d𝐴 and similarly for Δ𝐹, thus defining Δ𝐺′,Δ𝐹′ as the local deviations of Δ𝐺,Δ𝐹 from their respective global (ocean) means. Therefore,
Δ(𝐺−𝐹) = Δ(𝐺′−𝐹′) + 1/𝐴 ∫Δ(𝐺−𝐹)d𝐴 = Δ(𝐺′−𝐹′)+ℎ. (50)
This leads to our expression for GRD -induced relative sea-level change (Eq. 39) as
Δ𝛤(𝐫)≡Δ[𝐺′(𝐫)−𝐹′(𝐫)]=Δ[𝐺(𝐫)−𝐹(𝐫)]−ℎ. (51)
Substituting Eq. (51) in Eq. (48c) gives IB-corrected RSLC as
Δ𝑅(𝐫)+Δ𝐵(𝐫)=Δ𝜁(𝐫)+ℎ+Δ𝛤(𝐫), (52)
the sum of ocean dynamic sea-level change Δ𝜁, global-mean sea-level rise h and GRD-induced RSLC Δ𝛤. Using Eqs. (43) and (38) we obtain
Δ𝜁(𝐫)+ℎ=Δ𝜁(𝐫)+ℎ𝜃+ℎ𝑏=Δ𝑍(𝐫)+ℎ𝑏, (53)
Hence, IB-corrected RSLC is
Δ𝑅(𝐫)+Δ𝐵(𝐫)=Δ𝑍(𝐫)+ℎ𝑏+Δ𝛤(𝐫), (54)
the sum of sterodynamic sea-level change Δ𝑍(𝐫), barystatic sea-level rise ℎ𝑏 and GRD-induced RSLC. The contemporary GRD -induced RSLC due to a change 𝛿𝑀𝑖 in any of the stores of water on land (as land water storage or land ice, e.g., in a lake or an ice sheet) has both a barystatic and a GRD-induced effect on sea level, which are related and may interact (e.g., Gomez et al. 2012). Provided they are both proportional to 𝛿𝑀𝑖, we can rewrite Eq. (54) as
Δ𝑅(𝐫) + Δ𝐵(𝐫) = Δ𝑍(𝐫) + ∑𝑖𝛿𝑀𝑖𝜙𝑖 + Δ𝛤GIA + Δ𝛤𝑍, (55)
where 𝜙𝑖 is the barystatic–GRD fingerprint (Eq. 41) of store i of water, Δ𝛤GIA is GIA-induced RSLC, and Δ𝛤𝑍 is the GRD-induced RSLC of ocean mass redistribution (self-attraction and loading) associated with sterodynamic sea-level change. The last term is typically neglected.
Equation (55) is the means by which MSL projections are derived from coupled atmosphere–ocean general circulation models (AOGCMs). These models do not simulate GRD-induced RSLC (because they have time-independent geoid and sea floor) and are not generally used to compute barystatic sea-level rise (because they do not include adequate representations of land ice or land water storage). RSLR projections are therefore obtained by combining sterodynamic sea-level change simulated by an AOGCM with separately calculated projections of barystatic sea-level rise and GRD-induced RSLC using climate change simulations from the AOGCM applied to models of glaciers, ice sheets and the solid Earth (Church et al. 2013; Kopp et al. 2014; Slangen et al. 2014).


the sum of sterodynamic sea-level change Δ𝑍(𝐫), barystatic sea-level rise ℎ𝑏 and GRD-induced RSLC. The contemporary GRD -induced RSLC due to a change 𝛿𝑀𝑖 in any of the stores of water on land (as land water storage or land ice, e.g., in a lake or an ice sheet) has both a barystatic and a GRD-induced effect on sea level, which are related and may interact (e.g., Gomez et al. 2012). Provided they are both proportional to 𝛿𝑀𝑖, we can rewrite Eq. (54) as
Δ𝑅(𝐫) + Δ𝐵(𝐫) = Δ𝑍(𝐫) + ∑𝑖𝛿𝑀𝑖𝜙𝑖 + Δ𝛤GIA + Δ𝛤𝑍, (55)
where 𝜙𝑖 is the barystatic–GRD fingerprint (Eq. 41) of store i of water, Δ𝛤GIA is GIA-induced RSLC, and Δ𝛤𝑍 is the GRD-induced RSLC of ocean mass redistribution (self-attraction and loading) associated with sterodynamic sea-level change. The last term is typically neglected.
Equation (55) is the means by which MSL projections are derived from coupled atmosphere–ocean general circulation models (AOGCMs). These models do not simulate GRD-induced RSLC (because they have time-independent geoid and sea floor) and are not generally used to compute barystatic sea-level rise (because they do not include adequate representations of land ice or land water storage). RSLR projections are therefore obtained by combining sterodynamic sea-level change simulated by an AOGCM with separately calculated projections of barystatic sea-level rise and GRD-induced RSLC using climate change simulations from the AOGCM applied to models of glaciers, ice sheets and the solid Earth (Church et al. 2013; Kopp et al. 2014; Slangen et al. 2014).
According to Eqs. (35) or (36), Δ𝑅=Δ𝑅𝜌+Δ𝑅𝑚, the sum of steric sea-level change Δ𝑅𝜌 and manometric sea-level change Δ𝑅𝑚, which are the parts due, respectively, to change in density and change in mass per unit area. In general, Δ𝑅𝑚≠0 even if ℎ𝑏=0, because ocean mass may be redistributed. In particular, because Δ𝑅𝑚=−1/𝜌s∫𝜂𝐹Δ𝜌d𝑧 is small on the continental shelves (where the ocean is shallow), but ocean dynamics will not permit a strong gradient in 𝜁 to develop across the shelf break, global-mean thermosteric sea-level rise demands a redistribution of ocean mass onto the shelves (Landerer et al. 2007; Yin et al. 2010), with consequent ocean GRD (Gregory et al. 2013).
Observations of Sea-Level Change
Estimates of global-mean sea-level rise (GMSLR) for the last century depend mainly on records from tide-gauges. These instruments register coastal relative sea-level change (RSLC) Δ𝑅=Δ𝜂−Δ𝐹 (Eq. 24), which is affected by local vertical land movement (VLM) Δ𝐹. VLM is large in some places, with strong geographical gradients.
GMSLR is calculated as the global mean of RSLC, ℎ=(1/𝐴)∫Δ𝑅d𝐴 (Eq. 42). However, tide-gauges measure Δ𝑅 only at points on the coast and thus give a sparse, non-uniform and unrepresentative sampling of the global ocean area. The calculation therefore depends on physically based methods for extrapolation. Considering Eq. (52) in the form
ℎ = Δ𝑅 + Δ𝐵 − Δ𝜁 − Δ𝛤 (56)
we see that in principle h can be calculated from Δ𝑅 from any tide-gauge by applying the inverse barometer (IB) correction Δ𝐵, and subtracting local ocean dynamic sea-level change Δ𝜁 and local GRD -induced RSLC Δ𝛤. The global mean of each of these three adjustments is zero, so Eq. (42) is satisfied. In practice, using historical records, it is necessary to combine many tide-gauges in order to reduce the influence of unforced variability in 𝜁̃.
The IB adjustment is fairly small and can be made accurately from atmospheric pressure records. Various methods are used to allow for the spatial pattern of Δ𝜁, for example by calculating the mean over sets of gauges presumed to be representative of large regions (e.g., Jevrejeva et al. 2008), or by using spatial patterns of 𝜂̃ variation observed by satellite altimetry during its shorter period of availability (e.g., Church and White 2011). Because glacial isostatic adjustment (GIA) is the only part of GRD (including VLM) for which a global field is available, most estimates of GMSLR exclude all tide-gauges where GIA is not the only significant contribution to VLM (those affected by earthquakes, anthropogenic subsidence, sediment compactions, etc.). At the tide-gauges which are retained, we adjust for GIA-induced RSLC Δ𝛤GIA(𝐫) (e.g., Figure 3a of Tamisiea and Mitrovica 2011), estimated by combining solid-Earth models, the sea-level equation and reconstructed histories of deglaciation.
Alternatively, tide-gauge records may be corrected for VLM using vertical motion calculated from collocated GNSS (e.g., GPS) receivers. Effectively, this transforms RSLC to geocentric sea-level change Δ𝜂=Δ𝑅+Δ𝐹 (Eq. 24). Geoid adjustments must be applied to GNSS-corrected tide-gauge records just as for satellite altimetry, as described in the next paragraph.
Geocentric sea-level change Δ𝜂(=Δ𝜁−Δ𝐵+Δ𝐺, Eq. 23) has been measured over most of the global ocean since the early 1990s by satellite radar altimetry, using instruments which are located in a terrestrial reference frame (equivalent to the reference ellipsoid), and measure their vertical distance from the sea surface. To study the contemporary causes of observed geocentric sea-level change we must subtract Δ𝐺GIA(𝐫), the effect of GIA on the geoid (e.g., Figure 3b of Tamisiea and Mitrovica 2011), from IB-corrected geocentric sea-level change, thus:
Δ𝜂 + Δ𝐵 − Δ𝐺 GIA = Δ𝜁 + Δ𝐺NGIA, (57)
where Δ𝐺NGIA = Δ𝐺 − Δ𝐺GIA is due to ongoing redistribution of water mass on the Earth’s surface.
To convert global-mean geocentric sea-level rise ℎ𝐺=(1/𝐴)∫Δ𝜂d𝐴 to GMSLR h requires an adjustment for the global mean of Δ𝐹, according to Eq. (47). Although several processes can produce large local VLM, the only large global-mean effect is GIA. There is no contemporary GMSLR associated with GIA, so Eq. (49) gives (1/𝐴)∫Δ𝐹GIAd𝐴=(1/𝐴)∫Δ𝐺GIAd𝐴⇒(1/𝐴)∫Δ𝐺NGIAd𝐴=ℎ+(1/𝐴)∫Δ𝐹NGIAd𝐴. Hence, the global mean of Eq. (57) becomes
ℎ𝐺 = ℎ + 1/𝐴∫Δ𝐹GIA d𝐴 + 1/𝐴∫Δ𝐹NGIA d𝐴, (58)


Geocentric sea-level change Δ𝜂(=Δ𝜁−Δ𝐵+Δ𝐺, Eq. 23) has been measured over most of the global ocean since the early 1990s by satellite radar altimetry, using instruments which are located in a terrestrial reference frame (equivalent to the reference ellipsoid), and measure their vertical distance from the sea surface. To study the contemporary causes of observed geocentric sea-level change we must subtract Δ𝐺GIA(𝐫), the effect of GIA on the geoid (e.g., Figure 3b of Tamisiea and Mitrovica 2011), from IB-corrected geocentric sea-level change, thus:
Δ𝜂 + Δ𝐵 − Δ𝐺 GIA = Δ𝜁 + Δ𝐺NGIA, (57)
where Δ𝐺NGIA = Δ𝐺 − Δ𝐺GIA is due to ongoing redistribution of water mass on the Earth’s surface.
To convert global-mean geocentric sea-level rise ℎ𝐺=(1/𝐴)∫Δ𝜂d𝐴 to GMSLR h requires an adjustment for the global mean of Δ𝐹, according to Eq. (47). Although several processes can produce large local VLM, the only large global-mean effect is GIA. There is no contemporary GMSLR associated with GIA, so Eq. (49) gives (1/𝐴)∫Δ𝐹GIAd𝐴=(1/𝐴)∫Δ𝐺GIAd𝐴⇒(1/𝐴)∫Δ𝐺NGIAd𝐴=ℎ+(1/𝐴)∫Δ𝐹NGIAd𝐴. Hence, the global mean of Eq. (57) becomes
ℎ𝐺 = ℎ + 1/𝐴∫Δ𝐹GIA d𝐴 + 1/𝐴∫Δ𝐹NGIA d𝐴, (58)
recalling that the global means of Δ𝐵 and Δ𝜁 are zero. In response to the shift of mass from the land (as ice) into the ocean since the Last Glacial Maximum, and the consequent mantle adjustment, the sea floor is subsiding on average, giving a trend in (1/𝐴)∫Δ𝐹GIAd𝐴 of about −0.3  mm  year −1 (Tamisiea and Mitrovica 2011). Thus, ℎ𝐺<ℎ due to GIA. Contemporary changes in land ice cause elastic deformation of the sea floor. This gives a negative (1/𝐴)∫Δ𝐹NGIAd𝐴 which reduces ℎ𝐺 by about 8% of the barystatic sea-level rise (Frederikse et al. 2017).
Deprecated Terms and Recommended Replacements
Deprecated term
Recommended replacement
Eustatic sea-level change
Barystatic sea-level rise or barystatic sea-level change
for global-mean sea-level rise due to change in the mass of
the ocean, but not its density
Global-mean sea-level rise for the global mean of
relative sea-level change, due to the change in the volume
of the ocean
Global-mean geocentric sea-level rise for the global
mean of change in mean sea level relative to the terrestrial
reference frame, due to the combined effects of change in
the volume of the ocean and change in the level of the sea
floor
Dynamic topography
Ocean dynamic sea level for mean sea level above the
geoid due to ocean dynamics
Ocean dynamic topography for ocean dynamic sea level
estimated from ocean density
Sea-level change due to mantle dynamic topography for
GRD-induced relative sea-level change due to solid-Earth
dynamics
Mean sea surface
Mean sea level (MSL)
Mean sea-level change
or local sea-level change
Relative sea-level change (RSLC) or relative sea-level
rise (RSLR) for the change in mean sea level relative to the
land
Geocentric sea-level change for the change in mean sea
level relative to the terrestrial reference frame
Global sea-level change
(GSLC)
Gobal-mean sea-level rise (GMSLR) or global-mean sea-
level change (GMSLC) for the global mean of relative sea-
level change, due to the change in the volume of the ocean
Global-mean geocentric sea-level rise for the global-mean
change in mean sea level relative to the terrestrial
reference frame
Extreme sea level
Extreme sea level for the occurrence of exceptionally
high local sea surface due to short-term phenomena, or
extreme coastal water level when considering coastal impacts.
High-end sea-level change for projections or scenarios
of very large RSLR or GMSLR
Sea-level change due to
thermal expansion
Thermosteric sea-level change for contribution to
relative sea-level change
Global-mean thermosteric sea-level rise for contribution
to global-mean sea-level change
Sea-level change due to
oceanographic processes or
steric plus dynamic sea-
level change
Sterodynamic sea-level change for the change in relative
sea-level due to change in ocean density and circulation
Sea-level fingerprint or
static-equilibrium
fingerprint
Barystatic–GRD fingerprint for the sum of the RSLC from
GRD (elastic and geoid) and the barystatic sea-level rise due
to the addition of a unit mass of water to the global ocean
Post-glacial rebound (PGR)
Glacial isostatic adjustment (GIA)
Mass effect on, mass term
in, mass component of, or
mass contribution to sea
level or to sea-level
change
Barystatic sea-level rise for the contribution to global-
mean sea-level rise from the change of mass of the global
ocean (associated with changes in mass of water and ice on
land), as opposed to global-mean thermosteric sea-level rise
Manometric sea-level change for the contribution to
relative sea-level change due to change in the local mass of
the ocean per unit area, as opposed to steric sea-level
change
GRD-induced relative sea-level change for the effects on
relative sea level from geoid change and vertical land
movement, as opposed to steric, ocean dynamic and barystatic
sea-level change
Bottom pressure term in
sea-level change
Manometric sea-level change
List of Defined Terms and Notations
This table gives the entry number or section (not the page number) in which each term is defined. In the PDF, each term is a hyperlink to the relevant text. The rows for which there is no entry number are included only to define notation.
Above flotation
N19
Altimetry
Section 7
A
Area of the global ocean
Asthenosphere
N21
Astronomical tide
N6
𝑝𝑎
Atmospheric pressure at the
sea surface
ℎ𝑏
Barystatic sea-level rise
N19
𝜙
Barystatic–GRD fingerprint
N24
Bathymetry
N4
Bottom pressure
N18
Bottom topography
N4
Chandler wobble
N6
Compaction
N21
Contemporary GRD
N24
𝜌
Density of water
Depth of the water column
N4
D
Dynamic height
N12
g
Effective gravitational acceleration
N5
Ellipsoidal height
N1
Equipotential surface
N5
Eustatic sea-level change
N19
Extreme coastal water level
N8
Extreme sea level
N8
Fetch
N10
Free nutation
N6
Geocentric latitude
N1
Δ𝜂
Geocentric sea-level change
N14
Geodetic height
N1
Geodetic latitude
N1
G
Geoid
N5
Geoid height
N5
Φ
Geopotential
N5
Geostrophy
N12
vg
Geostrophic velocity
N12
Δ𝛤GIA
GIA-induced relative sea-level change
N23
Glacial isostatic  adjustment (GIA)
N23
Global mean
Section 2.4
ℎ𝐺
Global-mean geocentric sea-level rise
N27
ℎ𝑅
Global-mean relative sea-level change
N26
h
Global-mean sea-level rise (GMSLR)
N26
ℎ𝜃
Global-mean thermosteric sea-level rise
N17
Gravitational acceleration
N5
GRD
N22
Δ𝛤
GRD-induced relative sea-level change
N22
Greenwich meridian
N1
Halosteric sea-level change Δ𝑅𝑆
N16
High-end
N8
Horizontal
N5
IB-corrected geocentric  sea-level change
N14
IB-corrected mean sea level
N7


N19
Altimetry
Section 7
A
Area of the global ocean
Asthenosphere
N21
Astronomical tide
N6
𝑝𝑎
Atmospheric pressure at the
sea surface
ℎ𝑏
Barystatic sea-level rise
N19
𝜙
Barystatic–GRD fingerprint
N24
Bathymetry
N4
Bottom pressure
N18
Bottom topography
N4
Chandler wobble
N6
Compaction
N21
Contemporary GRD
N24
𝜌
Density of water
Depth of the water column
N4
D
Dynamic height
N12
g
Effective gravitational acceleration
N5
Ellipsoidal height
N1
Equipotential surface
N5
Eustatic sea-level change
N19
Extreme coastal water level
N8
Extreme sea level
N8
Fetch
N10
Free nutation
N6
Geocentric latitude
N1
Δ𝜂
Geocentric sea-level change
N14
Geodetic height
N1
Geodetic latitude
N1
G
Geoid
N5
Geoid height
N5
Φ
Geopotential
N5
Geostrophy
N12
vg
Geostrophic velocity
N12
Δ𝛤GIA
GIA-induced relative sea-level change
N23
Glacial isostatic  adjustment (GIA)
N23
Global mean
Section 2.4
ℎ𝐺
Global-mean geocentric sea-level rise
N27
ℎ𝑅
Global-mean relative sea-level change
N26
h
Global-mean sea-level rise (GMSLR)
N26
ℎ𝜃
Global-mean thermosteric sea-level rise
N17
Gravitational acceleration
N5
GRD
N22
Δ𝛤
GRD-induced relative sea-level change
N22
Greenwich meridian
N1
Halosteric sea-level change Δ𝑅𝑆
N16
High-end
N8
Horizontal
N5
IB-corrected geocentric  sea-level change
N14
IB-corrected mean sea level
N7
IB-corrected relative sea-level change
N15
IB-corrected sea-surface height
N7
𝜃
In-situ temperature of ocean water
B
Inverse barometer (IB)
N7
Inverted barometer
N7
Isostasy
N21
Isostatic adjustment
N21
Land ice
N19
Land water storage
N19
Latitude
N1
Level of no motion
N12
Liquid-water equivalent sea
surface
N2
Lithosphere
N21
Local
Section 2.3
Longitude
N1
Δ𝑅𝑚
Manometric sea-level change
N18
Mantle dynamic topography
N25
M
Mass of the global ocean
𝜂
Mean sea level (MSL)
N3
Mean-tide
N6
Neap tide
N6
Nodal period
N6
Ocean
Section 2.4
𝜁
Ocean dynamic sea level
N11
Δ𝜁
Ocean dynamic sea-level
change
N13
Ocean dynamic topography
N12
Ocean GRD
N22
m
Ocean mass per unit area
Orthometric height
N5
Permanent tide
N6
Pole tide
N6
Post-glacial rebound
N23
Predicted tide
N6
Prime meridian
N1
Radiational tide
N6
Reference ellipsoid
N1
Regression
N15
Δ𝑅
Relative sea-level change
(RSLC)
N15
𝜌∗
Representative density of
ocean water
S
Salinity of ocean water
F
Sea floor
N4
Sea surface
N2
Sea-level equation
N22
Sea-level equivalent (SLE)
N19
Sea-level fingerprint
N24
Sea-surface height (SSH)
N2
Sea-surface waves
N10
Seismic sea wave
N10
Self-attraction and loading
(SAL)
N22
Significant wave height
N10
Significant wave period
N10
Skew-surge height
N9
Spring tide
N6
State
Section 2.2
Static-equilibrium
fingerprint
N24
Δ𝑅𝜌
Steric sea-level change
N16
Δ𝑍
Sterodynamic sea-level
change
N20
Storm surge
N9
Storm tide
N6
σ
Storm-surge height
N9
Subsidence
N21
Surge residual
N9
Swash
N9
Swell wave
N10
Terrestrial reference frame
N1
Terrestrial water storage
N19
Thermal expansion
N16
Thermosteric sea-level
change Δ𝑅𝜃
N16
H
Thickness of the ocean
N4
Tidal currents
N6
Tidal datum
N6
Tidal height
N6
Tide-free
N6
Tide-gauge
N15
Tides
N6
Transgression
N15
True polar wander
N1
Tsunami
N10
Vertical
N1
Δ𝐹
Vertical land movement
(VLM)
N21
V
Volume of the global ocean
Wave height
N10
Wave period
N10
Wave runup
N9
Wave setup
N9
Wind setup
N9
Wind wave
N10
Zero-tide
N6
Appendix 1: No Resting Steady State Exists for a Realistic Ocean
A state of rest requires zero acceleration parallel to the surface, which must therefore be an equipotential. This condition is satisfied by the geoid because it is an equipotential surface by definition. However, zero acceleration at the surface is not a sufficient condition for the ocean to remain at rest. If there are any horizontal density gradients within the ocean, there will be pressure gradients beneath the horizontal surface, producing forces that will set the ocean into motion. So another necessary condition for zero ocean circulation is the absence of density gradients along horizontal surfaces, i.e., density is a function of depth only. This configuration is quite unlike the real state of the ocean.
Appendix 2: Why We Can Ignore Global Halosteric Sea-Level Change
When freshwater enters the ocean, such as from melting continental ice sheets, it adds to the ocean mass and in turn increases global-mean sea level (barystatic sea-level rise). Ocean salinity also changes due to the dilution of sea water, thus suggesting a role for a global halosteric sea-level change (Munk 2003; Levitus et al. 2005). However, the net effect on global-mean sea level is almost entirely barystatic since the global halosteric effect is negligible (Lowe and Gregory 2006). We can understand why this is so by recognizing that freshwater entering the ocean sees its salinity increase while the ambient sea water is itself freshened. These compensating salinity changes (which are often ignored, as by Munk 2003 and Levitus et al. 2005) have corresponding compensating sea-level changes, thus bringing the global halosteric effect to near zero. We demonstrate this effect in the following subsections, by considering a two-bucket thought experiment where one bucket holds freshwater (bucket-1) and the other holds sea water (bucket-2). We ask how the total water volume changes upon homogenizing the water in the two buckets, while conserving the masses of freshwater and salt. As we will see, the total volume of homogenized water is very nearly equal to the sum of the volume initially in the two separate buckets (to within 0.1%).
Conservation of Mass for Freshwater and Salt
Let the two buckets contain water of mass 𝑀𝑛, volume 𝑉𝑛, salinity 𝑆𝑛, and density 𝜌𝑛, 𝑛=1,2, and assume they have equal Conservative Temperature and equal pressure. Now homogenize the water from the two buckets into a single larger bucket, and assume no change in pressure nor any heat of mixing so that Conservative Temperature also remains unchanged. The total mass of freshwater and salt is unchanged upon homogenizing, so that
𝑀 = 𝑀1 + 𝑀2; 𝑀𝑆 = 𝑀1𝑆1 + 𝑀2𝑆2, (59)
where M is the total mass and S is the salinity of the homogenized water. Return the homogenized water to the original buckets, placing the same mass 𝑀1 back into the first bucket and mass 𝑀2 into the second bucket.
Dependence of Density on Salinity


Conservation of Mass for Freshwater and Salt
Let the two buckets contain water of mass 𝑀𝑛, volume 𝑉𝑛, salinity 𝑆𝑛, and density 𝜌𝑛, 𝑛=1,2, and assume they have equal Conservative Temperature and equal pressure. Now homogenize the water from the two buckets into a single larger bucket, and assume no change in pressure nor any heat of mixing so that Conservative Temperature also remains unchanged. The total mass of freshwater and salt is unchanged upon homogenizing, so that
𝑀 = 𝑀1 + 𝑀2; 𝑀𝑆 = 𝑀1𝑆1 + 𝑀2𝑆2, (59)
where M is the total mass and S is the salinity of the homogenized water. Return the homogenized water to the original buckets, placing the same mass 𝑀1 back into the first bucket and mass 𝑀2 into the second bucket.
Dependence of Density on Salinity
The dimensionless coefficients 𝛼≡𝜌−1∂𝜌/∂𝜃 and 𝛽≡𝜌−1∂𝜌/∂𝑆, which are used to compute steric sea-level change (Eq. 31), measure the relative change in the in-situ density as a function of temperature and salinity. Because ∂𝜌/∂𝜃 is generally negative, the volume of a given parcel of sea water increases as its temperature rises; this phenomenon is called thermal expansion. By analogy, since ∂𝜌/∂𝑆>0, the corresponding effect for salinity is sometimes called “haline contraction”. This is a misleading analogy, because if the salinity has increased but the mass has not changed, some freshwater must have been replaced by salt, so the parcel is materially altered, unlike in the case of adding heat. If salt is added to the parcel but no mass is removed, the salinity, mass and volume of the parcel will all increase. The notion of “haline contraction” has led some previous authors to draw incorrect conclusions about the effect on sea level from adding freshwater to the ocean. We here aim to clarify this situation.
The change in the volume of water due to homogenization depends on the value of 𝛽. For the surface ocean, representative values are 𝜌=1028  kg  m −3 and 𝑆=0.035, while for freshwater 𝜌=𝜌f=1000  kg  m −3 and 𝑆=0. Hence, a representative 𝛽≈(1/1000)(1028−1000)/(0.035−0)=28/35=0.8. This coefficient has a roughly 5% relative variation across the ocean, with most of that variation determined by temperature rather than salinity. (See Roquet et al. 2015, as well as Figure 1 in Griffies et al. 2014.) Since our concern is with salinity changes, we take 𝛽 to be constant in the following.
Computing the Change in Total Volume
The change in total volume upon homogenization is the sum of the changes in the two buckets,
𝛿𝑉=𝛿𝑉1+𝛿𝑉2. (60)
Since the mass of water in the two buckets remains the same before and after homogenization, the volume in the two buckets is altered only due to changes in their respective densities
𝛿𝜌𝑛=𝛿(𝑀𝑛/𝑉𝑛)=−(𝜌𝑛/𝑉𝑛)𝛿𝑉𝑛⇒𝛿𝜌𝑛/𝜌𝑛=−𝛿𝑉𝑛/𝑉𝑛, (61)
i.e., the relative density increases as the relative volume decreases. Because the buckets have the same temperature in our experiment, relative density changes occur only through salinity changes, according to
𝛿𝜌𝑛/𝜌𝑛=𝛽𝛿𝑆𝑛=−𝛿𝑉𝑛/𝑉𝑛, (62)
in which case the change in total volume is
𝛿𝑉=𝛿𝑉1+𝛿𝑉2=−(𝑉1𝛽𝛿𝑆1+𝑉2𝛽𝛿𝑆2). (63)
Mass conservation for salt means that
𝛿(𝑀𝑆)=𝛿(𝑀1𝑆1)+𝛿(𝑀2𝑆2)=0. (64)
Furthermore, since mass in the two buckets is unchanged, salt conservation leads to
𝑀1𝛿𝑆1+𝑀2𝛿𝑆2=0, (65)
so that salinity in one bucket rises while that in the other falls. Making use of this result in Eq. (63) leads to our desired expression for total volume change
𝛿𝑉=−𝛽𝛿𝑆1(𝑉1−𝑉2𝑀1/𝑀2)=−𝛽𝑉1𝛿𝑆1(1−𝜌1/𝜌2). (66)
Connecting to Thickness Changes
Equation (66) provides an expression for the change in total volume upon homogenizing two buckets of water with equal Conservative Temperatures, equal pressures, but with differing salinities. To connect to sea level, assume an equal cross-sectional area, A, for the buckets, so that the volume of water is given by 𝑉𝑛=𝐴ℎ𝑛, where ℎ𝑛 is the thickness of the water in the bucket. Equation (66) then says that upon homogenization, the thickness of water changes by
𝛿ℎ=−𝛽ℎ1𝛿𝑆1(1−𝜌1/𝜌2),(67)
and that the total thickness of the homogenized water is given by
ℎnew=ℎ1+ℎ2+𝛿ℎ=ℎ2+ℎ1[1−𝛽𝛿𝑆1(1−𝜌1/𝜌2)]. (68)
As expected, we see that 𝛿ℎ=0 only when 𝛽=0 or 𝜌1=𝜌2. The first is never true, and the second is not true in the general case of differing temperatures (in which case there are thermosteric changes ignored in our discussion).
An Ocean Example
To explore the oceanic implications of Eq. (68), assume bucket-1 initially has freshwater with density 𝜌1=𝜌f, whereas bucket-2 initially has sea water with density 𝜌2=𝜌s=𝜌f+𝜌′. The salinity change for bucket-1 is 𝛿𝑆1=𝑆, since this bucket went from its original freshwater concentration to the homogenized sea water with salinity S. The halosteric-induced thickness change (Eq. 67) is thus given by
𝛿ℎ=−ℎ1𝛽𝑆(𝜌′/𝜌s)<0. (69)
How large is this effect? For the case of an upper ocean with salt concentration 𝑆=0.035, sea-water density 𝜌s=1028 kg  m −3⇒𝜌′=28 kg  m −3 and 𝛽=0.8, we have
𝛿ℎ=−ℎ1×0.8×0.035×(28/1028)≈−ℎ1×7.6×10−4. (70)
To within roughly 8 parts in 104, the change in thickness of the ocean column is nearly identical to the thickness of freshwater added to the ocean. For example, if we add one metre of freshwater into the upper ocean (ℎ1=1 m ), then the change in sea level is equal to one metre minus the tiny amount 0.76 mm . Hence, as emphasized by Lowe and Gregory (2006), we can generally ignore the contribution to global-mean sea level from global halosteric effects.
Appendix 3: Bottom Pressure
The hydrostatic pressure at the ocean sea floor is commonly referred to as the ocean bottom pressure 𝑝̃𝑏, usually calculated as
𝑝̃𝑏=𝑝̃𝑎+𝑔𝑚̃, (71)


An Ocean Example
To explore the oceanic implications of Eq. (68), assume bucket-1 initially has freshwater with density 𝜌1=𝜌f, whereas bucket-2 initially has sea water with density 𝜌2=𝜌s=𝜌f+𝜌′. The salinity change for bucket-1 is 𝛿𝑆1=𝑆, since this bucket went from its original freshwater concentration to the homogenized sea water with salinity S. The halosteric-induced thickness change (Eq. 67) is thus given by
𝛿ℎ=−ℎ1𝛽𝑆(𝜌′/𝜌s)<0. (69)
How large is this effect? For the case of an upper ocean with salt concentration 𝑆=0.035, sea-water density 𝜌s=1028 kg  m −3⇒𝜌′=28 kg  m −3 and 𝛽=0.8, we have
𝛿ℎ=−ℎ1×0.8×0.035×(28/1028)≈−ℎ1×7.6×10−4. (70)
To within roughly 8 parts in 104, the change in thickness of the ocean column is nearly identical to the thickness of freshwater added to the ocean. For example, if we add one metre of freshwater into the upper ocean (ℎ1=1 m ), then the change in sea level is equal to one metre minus the tiny amount 0.76 mm . Hence, as emphasized by Lowe and Gregory (2006), we can generally ignore the contribution to global-mean sea level from global halosteric effects.
Appendix 3: Bottom Pressure
The hydrostatic pressure at the ocean sea floor is commonly referred to as the ocean bottom pressure 𝑝̃𝑏, usually calculated as
𝑝̃𝑏=𝑝̃𝑎+𝑔𝑚̃, (71)
where the first term is the atmospheric pressure at the liquid-water equivalent sea surface and the second term is the weight of the mass per unit area 𝑚̃(𝐫) of sea water, given by Eq. (26). This formula makes two approximations. First, hydrostatic pressure does not exactly equal the weight per unit area of the fluid above it because of the curvature of the Earth (Ambaum 2008). Second, g should appear within the vertical integral of density to obtain m (Eq. 26), because g depends on z. However, these approximations are entirely adequate for sea-level studies.
The difference in bottom pressure between two states
Δ𝑝𝑏=Δ𝑝𝑎+𝑔Δ𝑚, (72)
that is, the sum of the change in local atmospheric pressure and the change in the weight of the local ocean. Using Eqs. (26) and (25) to replace Δ𝑚,
Δ𝑝𝑏=Δ𝑝𝑎+𝑔𝜌∗Δ𝐻+𝑔∫𝜂𝐹Δ𝜌(𝑧)d𝑧, (73)
where 𝐻=𝜂−𝐹 is the time-mean thickness of the ocean (Eq. 3). This form separates the change in pressure due to sea water into one term (the second) due to the change in the local thickness of the ocean, and another (the third) which is proportional to the local vertical-mean change in sea-water density.


Horton et al. (2018)
Title:
Mapping Sea-Level Change in Time, Space, and Probability
Keywords:
sea level, climate change, Holocene, Last Interglacial, Mid-Pliocene Warm Period, sea-level rise projections
Corresponding author:
Horton
Citation:
Horton, B. P., Kopp, R. E., Garner, A. J., Hay, C. C., Khan, N. S., Roy, K., & Shaw, T. A. (2018). Mapping Sea-Level Change in Time, Space, and Probability. Annual Review of Environment and Resources, 43(1), 481–521. doi:10.1146/annurev-environ-102017-025826
URL:
https://www.annualreviews.org/doi/10.1146/annurev-environ-102017-025826
Abstract
Future sea-level rise generates hazards for coastal populations, economies, infrastructure, and ecosystems around the world. The projection of future sea-level rise relies on an accurate understanding of the mechanisms driving its complex spatio-temporal evolution, which must be founded on an understanding of its history. We review the current methodologies and data sources used to reconstruct the history of sea-level change over geological (Pliocene, Last Interglacial, and Holocene) and instrumental (tide-gauge and satellite altimetry) eras, and the tools used to project the future spatial and temporal evolution of sea level. We summarize the understanding of the future evolution of sea level over the near (through 2050), medium (2100), and long (post-2100) terms. Using case studies from Singapore and New Jersey, we illustrate the ways in which current methodologies and data sources can constrain future projections, and how accurate projections can motivate the development of new sea-level research questions across relevant timescales.
Introduction
As recorded instrumentally and reconstructed from geological proxies, sea levels have risen and fallen throughout Earth’s history, on timescales ranging from minutes to millions of years. Sea-level projections depend on establishing a robust relationship between sea level and climate forcing, but the vast majority of instrumental records contain less than 60 years of data, which are from the late twentieth and early twenty-first centuries (1-3). This brief instrumental period captures only a single climate mode of rising temperatures and sea level within a baseline state that is climatically mild by geological standards. Complementing the instrumental records, geological proxies provide valuable archives of the sea-level response to past climate variability, including periods of more extreme global mean surface temperature (e.g., 4-7). Ultimately, information from the geological record can help assess the relationship between sea level and climate change, providing a firmer basis for projecting the future (8), but current ties between past changes and future projections are often vague and heuristic. Greater interconnections between the two sub-disciplines are key to major progress.
The linked problems of characterizing past sea-level changes and projecting future sea-level rise face two fundamental challenges. First, regional and local sea-level changes vary substantially from the global mean (9). Understanding regional variability is critical to both interpreting records of past changes and generating local projections for effective coastal risk management (e.g., 9, 10). Second, uncertainty is pervasive in both records of past changes and in the physical and statistical modeling approaches used to project future changes (e.g., 11), and it requires careful quantification and statistical analysis (12). Quantification of uncertainty becomes particularly important for decision analysis related to future projections (e.g., 13).
Here, we review the mechanisms that drive spatial variability, as well as their contributions to the uncertainty in mapping sea level on different timescales. We describe methodologies and data sources for piecing together lines of evidence related to past and future sea level to map changes in space, time, and probability. We review the sources and statistical methods applied to proxy [Pliocene, Last Interglacial (LIG) and Holocene] and instrumental (tide gauges and satellites) data, and the statistical and physical modeling approaches used to project future sea-level changes. Finally, we highlight two case study regions - Singapore and New Jersey - to illustrate the way in which proxy and instrumental data can improve future projections, and conversely how future projections can guide the development of new sea-level research questions to further constrain projections.
Mechanisms for global, regional, and local relative sea-level changes
Relative sea level (RSL) is defined as the difference in elevation between the sea surface and the land. Global mean sea level (GMSL) is defined as the areal mean of either RSL or sea-surface height over the global ocean. It is often approximated by taking various forms of weighted means of individual RSL records, sometimes with corrections for specific local processes. Over the twentieth century, GMSL trends (14) were dominated by increases in ocean mass due to melting of land-based glaciers (e.g., 15) and ice sheets (e.g., 16), and by thermal expansion of warming ocean water. Changes in land water storage due to dam construction and groundwater withdrawal also made a small contribution (e.g., 17). Over a variety of timescales, RSL differs from GMSL, because of key driving processes such as atmosphere/ocean dynamics, the static-equilibrium effects of ocean/cryosphere/hydrosphere mass redistribution on the height of the geoid and the EarthÕs surface, glacio-isostatic adjustment (GIA), sediment compaction, tectonics, and mantle dynamic topography (MDT). The driving processes are spatially variable and cause RSL change to vary in rate and magnitude among regions (Figure 1).
Figure 1: Mapping uncertainty of sea-level drivers on different timescales based on available estimates. The length of colored bars along the x-axis represents the characteristic timescale over which a process may occur, rather than the total time duration over which the process has been active. The color scale represents the range in magnitude of relative sea-level change driven by a process over an event or observed/predicted timescale. It does not imply a specific relationship of the change in amplitude with timescale, given the nonlinear nature of many of these processes. The color scheme for glacial eustasy is also scaled to encompass predicted changes in global mean sea level of decimeters in the next several decades to meters over the next several centuries. (b) The uncertainty of instrumental and proxy recorders of sea level. The x (age) axis represents the time span over which the proxy may be used (given the temporal range of the dating method used to determine its age), rather than the proxy’s temporal uncertainty. To estimate the contribution of a given process, the vertical and temporal resolution of a chosen instrument or proxy cannot exceed the magnitude and rate of sea-level change driven by that process.
Atmosphere/ocean dynamics are the dominant driver of spatial heterogeneity in RSL on annual and multidecadal timescales (18-21), as well as a significant driver on longer timescales during periods with limited land-ice changes, such as the Common Era (22-25). The highest rates of RSL rise over the past two decades (greater than 15 mm/year) have occurred in the western tropical PaciÞc (18, 26), although the pattern appears to have reversed since 2011 (27). Observations and numerical model simulations (18, 28) conÞrm that the intensification of trade winds, which occurs when the PaciÞc Decadal Oscillation (PDO) exhibits a negative trend, accounts for the amplitude and spatial pattern of RSL rise in the western tropical PaciÞc. In the western North Atlantic Ocean, changes in the strength and/or position of the Gulf Stream impact RSL trends differently north and south of North Carolina, where the Gulf Stream separates from the US Atlantic coast and turns toward northern Europe (19, 22, 23, 29). In fact, there is a .30-cm difference in sea-surface height between New Jersey and North Carolina (29). Climate models project that by the late twenty-Þrst century, associated with a decline in the Atlantic Meridional Overturning Circulation (AMOC), ocean dynamic sea-level rise of up to 0.2 to 0.3 m could occur along the western boundary of the North Atlantic (30). However, coastal ocean dynamic variability in the western North Atlantic has been largely driven over the past few decades by local winds, with limited evidence for coupling to AMOC strength (21, 31).


Atmosphere/ocean dynamics are the dominant driver of spatial heterogeneity in RSL on annual and multidecadal timescales (18-21), as well as a significant driver on longer timescales during periods with limited land-ice changes, such as the Common Era (22-25). The highest rates of RSL rise over the past two decades (greater than 15 mm/year) have occurred in the western tropical PaciÞc (18, 26), although the pattern appears to have reversed since 2011 (27). Observations and numerical model simulations (18, 28) conÞrm that the intensification of trade winds, which occurs when the PaciÞc Decadal Oscillation (PDO) exhibits a negative trend, accounts for the amplitude and spatial pattern of RSL rise in the western tropical PaciÞc. In the western North Atlantic Ocean, changes in the strength and/or position of the Gulf Stream impact RSL trends differently north and south of North Carolina, where the Gulf Stream separates from the US Atlantic coast and turns toward northern Europe (19, 22, 23, 29). In fact, there is a .30-cm difference in sea-surface height between New Jersey and North Carolina (29). Climate models project that by the late twenty-Þrst century, associated with a decline in the Atlantic Meridional Overturning Circulation (AMOC), ocean dynamic sea-level rise of up to 0.2 to 0.3 m could occur along the western boundary of the North Atlantic (30). However, coastal ocean dynamic variability in the western North Atlantic has been largely driven over the past few decades by local winds, with limited evidence for coupling to AMOC strength (21, 31).
Gravitational, rotational, and elastic deformational effects - also called static-equilibrium effects - reshape sea level nearly instantaneously in response to the redistribution of mass be.tween the cryosphere, the ocean, and the terrestrial hydrosphere (32-35). These effects are linked to the change in self-gravitation of the ice sheets and liquid water, the response of the Earth’s rotational vector to the redistribution of mass at the Earth’s surface, and the elastic response of the solid Earth surface to changing surface loads (Figure 2b,c). Unique RSL change geometries, sometimes called “fingerprints,” can be associated with the melting of different ice sheets and glaciers, and this response scales linearly with the magnitude of a marginal ice-mass change (32, 34). The dominant self-gravitation signal will result in a RSL fall near a shrinking land ice mass, which will be compensated by a RSL rise in the far Þeld that will be greater than the GMSL signal expected from the water mass inßux. The exact spatial pattern of RSL change depends on the geometry of the melting undergone by the ice reservoir. Recent studies (36, 37) have examined how mass loss centered in different portions of an ice sheet or glacial region will affect RSL differently. For example, New Jersey experiences a RSL fall in response to mass loss in southern.most Greenland, even though it experiences a modest (approximately 50% of the global mean) RSL rise in response to uniform melting across Greenland (Figure 2b).
Figure 2: (a) Dynamic sea-level contribution to sea surface height (millimeter/year) from 2006-2100 under the RCP8.5 experiment of the Community Earth System Model, as archived by Coupled Model Intercomparison Project Phase 5 (30). Elastic fingerprints of projections of (b) Greenland ice-sheet mass loss and (c) West Antarctic ice-sheet mass loss, presented as ratios of RSL change to GMSL change (140). (d) Contribution of glacio-isostatic adjustment (GIA) to present-day relative sea level (RSL) change (millimeter/year), calculated using the ICE-5G ice loading history (52), combined with a maximum-likelihood solid-Earth model identified through a Kalman smoother tide-gauge analysis (119). ModiÞed from Kopp et al. (24).
Pliocene:
epoch in the geologic timescale that extends from 5.3 million to 1.8 million years ago, during which the Earth experienced a transition from relatively warm climates to the prevailing cooler climates of the Pleistocene; includes the Mid-Pliocene Warm Period (.3.2 to 3.0 million years ago), which is the most recent period in geologic time with temperatures comparable to those projected for the twenty-first century.
Last Interglacial:
the interglacial stage prior to the current Holocene interglacial (an interglacial is a geological interval of warmer global average temperature, characterized by the absence of large ice sheets in North America and Europe); the Last Interglacial extends from approximately 129,000 to 116,000 years ago, corresponds to Marine Isotope Stage5e and is also known as the Eemian.
Holocene:
current geological epoch, beginning approximately 11,650 years ago, after the last glacial period; the start of the Holocene is formally defined by chemical (delta^{18}O) shifts in an ice core from northern Greenland that reflect climate warming.
Relative sea level (RSL):
height of the sea surface at a specific location, measured with respect to the height of the surface of the solid Earth.
Dynamic sea level:
sea-surface height variations produced by oceanic and atmospheric circulation and by temperature and salinity distributions.
Global mean sea level (GMSL):
areal average height of relative sea level (or, in some uses, sea-surface height) over the Earth’s oceans combined; influenced primarily by the volume of seawater and the size and shape of the ocean basins; in the geological literature, GMSL is classically referred to as “eustatic sea level”.
Static-equilibrium effects:
gravitational, elastic, and rotational effects of mass redistribution at the Earth surface, which lead to changes in both sea-surface height and the height of the solid Earth; these combined effects give rise to what are known as “sea-level fingerprints,” or the geographic pattern of sea-level change following the rapid melting of ice sheets and glaciers.
Over longer timescales, GIA arises as the viscoelastic mantle responds to the transfer of mass between land-based ice sheets and the global ocean during a glacial cycle. GIA induces deformation of the solid Earth, as well as changes in the EarthÕs gravitational Þeld and its rotational state (38-43). After a change in surface load, the elastic component of the deformation is recovered nearly instantaneously, but the viscoelastic properties of the underlying mantle determine the characteristics of the recovery over longer timescales. In general, this recovery takes place over thousands of years, although in localized regions underlain by low-viscosity mantle, it can take place over decades to centuries (e.g., 44). On a global scale, at least in the Quaternary period, the system does not reach isostatic equilibrium, because it is interrupted by the initiation of another glacial cycle. The deformation that is observed today is the overprint of a series of glacial cycles that extend from the Pliocene through the Pleistocene glacial cycles and into the Holocene (7, 45, 46).
GIA models simulate the evolution of the solid Earth as a function of the rheological structure and ice-sheet history (42, 47, 48). During the glacial phase of a glaciation-deglaciation cycle, the depression of land beneath ice sheets causes a migration of mantle material away from ice-load centers. This migration results in the formation of a forebulge in regions adjacent to ice sheets (e.g., the mid-Atlantic coast of the United States). Following the ice-sheet retreat, mantle material flows toward the former load centers. These centers experience postglacial rebound, while the forebulge retreats and collapses. In regions located beneath the centers of Last Glacial Maximum ice sheets (e.g., Northern Canada and Fennoscandia), postglacial uplift has resulted in RSL records characterized by a continuous fall; rates of present-day uplift greater than 10 mm/year occur in these near-Þeld locations (42). In the former forebulges, land is subsiding at a rate that varies with distance from the former ice centers. Along the US Atlantic coast, rates of present-day subsidence reach a maximum amplitude of close to 2 mm/year (49). In regions distal from the former ice sheet, the GIA signal is much smaller (50). These regions are characterized by present-day GIA-induced rates of RSL change that are near constant or show a slight fall (<0.3 mm/year), due to hydro-isostatic loading (continental levering) and to a global fall in the ocean surface linked to the hydro-and glacio-isostatic loading of the bottom of the Earth’s ocean basins (equatorial ocean syphoning) (Figure 2d, 51).


GIA models simulate the evolution of the solid Earth as a function of the rheological structure and ice-sheet history (42, 47, 48). During the glacial phase of a glaciation-deglaciation cycle, the depression of land beneath ice sheets causes a migration of mantle material away from ice-load centers. This migration results in the formation of a forebulge in regions adjacent to ice sheets (e.g., the mid-Atlantic coast of the United States). Following the ice-sheet retreat, mantle material flows toward the former load centers. These centers experience postglacial rebound, while the forebulge retreats and collapses. In regions located beneath the centers of Last Glacial Maximum ice sheets (e.g., Northern Canada and Fennoscandia), postglacial uplift has resulted in RSL records characterized by a continuous fall; rates of present-day uplift greater than 10 mm/year occur in these near-Þeld locations (42). In the former forebulges, land is subsiding at a rate that varies with distance from the former ice centers. Along the US Atlantic coast, rates of present-day subsidence reach a maximum amplitude of close to 2 mm/year (49). In regions distal from the former ice sheet, the GIA signal is much smaller (50). These regions are characterized by present-day GIA-induced rates of RSL change that are near constant or show a slight fall (<0.3 mm/year), due to hydro-isostatic loading (continental levering) and to a global fall in the ocean surface linked to the hydro-and glacio-isostatic loading of the bottom of the Earth’s ocean basins (equatorial ocean syphoning) (Figure 2d, 51).
Global models of the GIA process have contributed to our knowledge of solid Earth geodynamics through the constraints they provide on the effective viscosity of the mantle. They also provide a means to constrain the evolution of land-based ice sheets and ocean bathymetry over a glacial cycle (e.g., 42, 52), which can in turn be used to provide boundary conditions for tests of global climate models under paleoclimate conditions (e.g., 53). Global GIA models have traditionally relied on a simple Maxwell representation of the EarthÕs rheology and on spherically symmetric models of the EarthÕs mantle viscosity. Research is currently underway to include complex rheologies and lateral heterogeneity in mantle viscosity in GIA models, features that may be of substantial importance in regions with a complex geological structure, such as West Antarctica (e.g., 44).
Locally, RSL can also change in response to sediment compaction driven by natural processes and by anthropogenic groundwater and hydrocarbon withdrawal (e.g., 54, 55). Many coastlines are located on plains composed of unconsolidated or loosely consolidated sediments, which com.pact under their own weight as the pressure of overlying sediments leads to a reduction in pore space (54, 56). Sediment compaction can occur over a range of depths and timescales. In the Mississippi Delta, Late Holocene subsidence due to shallow compaction has been estimated to be as high as 5 mm/year (57). Anthropogenic groundwater or hydrocarbon withdrawal can accelerate sediment compaction. For example, from 1958-2006 CE, subsidence in the Mississippi Delta was 7.6 ± 0.2 mm/year, with a peak rate of 9.8 ± 0.3 mm/year prior to 1991 that corresponded to the period of maximum oil extraction (58). Many other deltaic regions, including the Ganges, Chao Phraya, and Pasig Deltas, experience high rates of subsidence, linked to a combination of natural and anthropogenic sediment compaction, and they exhibit some of the highest rates of present-day RSL rise (59). Deeper, often poorly understood, processes also contribute to coastal subsidence, including thermal subsidence and fault motion (60).
On some coastlines, deformation caused by tectonics can be an important driver of RSL change. Indeed, reconstructions of RSL can be used to estimate the presence and rate of vertical land motion caused by coastal tectonics at regional spatial scales (e.g., 61). Coastlines may have near-instantaneous or gradual rates of uplift or subsidence due to coseismic movement associated with earthquakes or longer-term post-or interseismic deformation. Geodetic measurements of the 2011 Tohoku-oki and the 2004 Indian Ocean megathrust earthquakes revealed several meters of near-instantaneous coseismic vertical land motion (e.g., 62), which were followed by postseismic recovery that quickly exceeded the amount of coseismic change (63, 64). Based on a collection of bedrock thermochronometry measurements (65), gradual rates of vertical land motion from tectonics vary from <0.01 mm/year to 10 mm/year. Stable, cratonic regions (e.g., central and western Australia, central North America, and eastern Scandinavia) exhibit negligible vertical land motion rates of <0.01 mm/year. Rates are higher (0.01-0.1 mm/year) along passive margins (e.g., southeastern Australia, Brazil, and the US Atlantic coast) and the highest vertical land motion rates (1- 10 mm/year) are found in several tectonically active mountainous areas (e.g., the Coastal Mountains of British Columbia, Papua New Guinea, and the Himalayas). Most of these rate estimates are integrated over several millions to tens of millions of years [and may include influence from other low-frequency signals such as MDT or karstification (66)], and therefore have insufficient resolution to reveal temporal variations on shorter timescales (65).
A common approach to calculate long-term tectonic vertical land motion uses LIG shore.lines, often inappropriately assuming that, once tectonically corrected, the elevation of GMSL at the LIG was .5 m above present. However, when calculating long-term tectonics from LIG shorelines, uncertainty in LIG GMSL and departures from GMSL due to GIA in response to glacial-interglacial cycles and excess polar ice-sheet melt relative to present-day values must be considered (67, 68). Instrumental observations (e.g., global positioning system, interferometric synthetic-aperture radar) of vertical land motion can provide the resolution to decipher temporal variations in rates, although the observation period is too short to capture the full period over which these processes operate.
MDT refers to the surface undulations induced by mantle ßow (69, 70). One of the most important consequences of MDT studies is their influence on estimates of long-term GMSL and RSL change (69). Several geophysical approaches have been developed to model global and regional MDT. Husson & Conrad (71) proposed that the dynamic effect of longer-term (~10^8 years) change in tectonic velocities on GMSL could be up to ~80 m from a model based on boundary layer theory. Conrad & Husson (72) used a forward model of mantle flow based on the present-day mantle structure and plate motions to estimate that the current rate of GMSL rise induced by MDT is <1 m/million years. An increasing body of evidence suggests that MDT can contribute to regional RSL change at rates of >1 m/100 kyr (e.g., 73, 74). The uncertainties due to MDT in RSL reconstructions become increasingly large further back in time (73, 75). Observationally, MDT is essentially indistinguishable from long-term tectonic change.
Glacial isostatic adjustment (GIA):
response of the solid Earth to mass redistribution during a glacial cycle; isostasy refers to a concept whereby deformation takes place in an attempt to return the Earth to a state of equilibrium; GIA refers to isostatic deformation related to ice and water loading during a glacial cycle
Sediment compaction:
reduction in the volume of sediments caused by a decrease in pore space, which has the effect of lowering the height of the solid Earth surface; can occur naturally or due to the anthropogenic extraction of fluids (such as water and fossil fuels) from the pore space.
Mantle dynamic topography (MDT):
differences in the height of the surface of the solid Earth caused by density-driven flow within the Earth’s mantle.
Past and current observations of sea-level change
Reconstructions of Relative Sea Levels from Proxy Data
Geological reconstructions of RSL are derived from sea-level proxies, the formation of which was controlled by the past position of sea level (76). Sea-level proxies, which have a systematic and qunatifiable relationship with contemporary tides (77), include sedimentary, geomorphic, archeological, and fixed biological indicators, as well as coral reefs, coral microatolls, salt-marsh flora, and salt-marsh fauna (Figure 1b). The relationship of a proxy to sea level is deÞned by its “indicative meaning,” which describes the central tendency (reference water level) and vertical range (indicative range) of its relationship with tidal level(s).


Glacial isostatic adjustment (GIA):
response of the solid Earth to mass redistribution during a glacial cycle; isostasy refers to a concept whereby deformation takes place in an attempt to return the Earth to a state of equilibrium; GIA refers to isostatic deformation related to ice and water loading during a glacial cycle
Sediment compaction:
reduction in the volume of sediments caused by a decrease in pore space, which has the effect of lowering the height of the solid Earth surface; can occur naturally or due to the anthropogenic extraction of fluids (such as water and fossil fuels) from the pore space.
Mantle dynamic topography (MDT):
differences in the height of the surface of the solid Earth caused by density-driven flow within the Earth’s mantle.
Past and current observations of sea-level change
Reconstructions of Relative Sea Levels from Proxy Data
Geological reconstructions of RSL are derived from sea-level proxies, the formation of which was controlled by the past position of sea level (76). Sea-level proxies, which have a systematic and qunatifiable relationship with contemporary tides (77), include sedimentary, geomorphic, archeological, and fixed biological indicators, as well as coral reefs, coral microatolls, salt-marsh flora, and salt-marsh fauna (Figure 1b). The relationship of a proxy to sea level is deÞned by its “indicative meaning,” which describes the central tendency (reference water level) and vertical range (indicative range) of its relationship with tidal level(s).
Under the uniformitarian assumption that the indicative meaning is constant in time, the indicative meaning can be determined empirically by direct measurement of modern analogs. The reconstruction begins with Þeld measurement of the elevation of a paleo sea-level proxy with respect to a common datum (e.g., mean tide level). The vertical uncertainty of a RSL reconstruction is primarily related to the indicative range of the sea-level proxy (Figure 1b). For proxies that form in intertidal settings, including most sedimentary, fixed biological, and geomorphic indicators, vertical uncertainties are proportional to the magnitude of local tidal range. In contrast, the vertical distribution of corals varies among species and is driven by light attenuation, along with a host of other factors (78). Although vertical RSL uncertainties are not systematically larger for older reconstructions, paleo-tidal range change (see, e.g., 79) or variations in the relationship of a proxy to sea level over time may introduce unquantifiable uncertainties.
Paleo sea-level proxies are dated to provide a chronology for past RSL changes. The method used to date a proxy dictates the age range over which it may be used (Figure 1b). Sea-level proxies can be dated directly using radiometric methods. Radiocarbon dating is used to obtain chronologies over decades to the last .50,000 years, whereas methods such as U-series or luminescence dating can be used over hundreds of thousands of years (80). Sea-level proxies may also be dated by correlation with marine oxygen isotope stages, magnetic reversals, or other chronologies using bio-or chemostratigraphy. The age uncertainties of RSL reconstructions increase over time (e.g., 8, 68).
Three geological intervals have been the particular focus of attention for sea-level reconstructions, because they provide analogues for future predicted changes: the Mid-Pliocene Warm Period (MPWP), the LIG stage, and the Holocene.
Sea-level proxy:
any physical, biological, or chemical feature with a systematic and quantifiable relationship to sea level at its time of formation.
Mid-Pliocene Warm Period (~3.2 to 3.0 million years ago)
The MPWP is the most recent period in geologic time with temperatures comparable to those projected for the twenty-first century (e.g., 81). The period is characterized by a series of 41-kyr, orbitally paced climate cycles marked by three abrupt shifts in the stacked oxygen isotope record (82, 83). During this time, atmospheric CO2 ranged between approximately 350 and 450 ppm, and the configuration of oceans and continents was similar to todayÕs, which permits feasible comparison of oceanic and atmospheric conditions in models of Pliocene and modern climate (84, 85). Global climate model simulations estimate peak global mean surface temperatures between 1.9-3.6 C above preindustrial (86).
Both modeling and field evidence suggest that polar ice sheets were smaller during the MPWP, but constraints on the magnitude of GMSL maxima are highly uncertain. Early reconstructions of Pliocene sea levels have been derived from estimates of global ice volume from the temperature-corrected oxygen isotope composition of foraminifera and ostracods (e.g., 87), although the analytical uncertainties (~±10 m) and the uncertainties associated with separating the ice volume, temperature, and diagenetic signals may be greater than the estimated magnitude of GMSL change from the present (8). Therefore, attention has focused on geomorphic proxy reconstructions from paleoshoreline deposits (e.g., 88, 89) or erosional features caused by sea-level fluctuations such as disconformities on atolls (e.g., 90). Estimating GMSL during the Pliocene is complicated by the amount of time elapsed since the formation of sea-level proxies, which allows processes operating over long timescales, such as tectonic uplift or MDT, to create differences between RSL and GMSL of up to tens of meters (e.g., 74). The uncertain contribution of RSL change from MDT makes reconstructions from this period highly challenging, although increasing the number and spatial distribution of RSL reconstructions from the Pliocene may help to derive a GMSL estimate consistent with model predictions of GIA and MDT that use a unique and internally consistent set of physical conditions (7).
Last Interglacial (~129,000-116,000 years ago)
During the LIG, global mean surface temperature was comparable to or slightly warmer than present, although the peak LIG CO2 concentration of .285 ppm (e.g., 91) was considerably less than that of the present (.400 ppm). Global mean sea-surface temperature was .0.5 ± 0.3.C above its late nineteenth-century level (92). Model simulations indicate little global mean surface temperature change during the Last Inter.glacial stage, whereas combined land and ocean proxy data imply .1.C of warming, but with possible spatio-temporal sampling biases (93). Due to higher orbital eccentricity during the LIG, polar warming was more extreme (94). Greenland surface temperatures peaked .5-8.C above preindustrial levels (95), and Antarctic temperatures were .3-5.C warmer (96), both comparable to late twenty-first-century projections under Representative Concentration Pathway (RCP) 8.5 (97).
LIG RSL reconstructions are much more abundant than those for the MPWP (e.g., 4, 6, 68, 78, 98; see also Figure 3c). Geomorphic (marine terraces, shore platforms, beachrock, beach deposits and ridges, abrasion and tidal notches, and cheniers), sedimentary (lagoonal deposits), coral reef, and geochemical sea-level proxies are used to reconstruct RSL changes during this period (e.g., 68). Compilations of RSL data combined with spatio-temporal statistical and GIA modeling indicate that peak GMSL was extremely likely >6 m but unlikely >9 m above present (6, 99). These estimates are in agreement with site-speciÞc, GIA-corrected coastal records in the Seychelles at 7.6 ± 1.7 m (4) and in Western Australia at 9 m (100) above present (Figure 3c).
GMSL during the LIG may have experienced multiple peaks, possibly associated with orbitally driven, asynchronous land-ice minima at the two poles (e.g., 8, 99). A significant fraction of the Greenland ice sheet remained intact throughout the LIG period, with recent alternative reconstructions limiting the peak Greenland GMSL contribution to .2 m (95) or 4-6 m (101) of the ice sheetÕs total 7-m sea-level equivalent mass. Thermal expansion and the melting of mountain glaciers together likely contributed .1 m (102, 103). This implies a significant contribution to LIG GMSL from Antarctic ice melt. However, there is little direct observational evidence of mass loss from the Antarctic region. Additional constraints from RSL reconstructions in mid-to high-latitude regions may help to partition contributions from Greenland and Antarctica (104).
Estimates of LIG GMSL have not yet incorporated the effects of MDT, which may contribute as much as 4 ± 7m(1.) to RSL change in some regions (e.g., Southwestern Australia) (73). In contrast to MPWP, the magnitudes of RSL change due to GIA and MDT are roughly the same order, although the spatial pattern associated with the two processes should be distinct (e.g., 4, 73). Whether a formal accounting for MDT would significantly alter the estimated height of the LIG highstand is unknown.
Representative Concentration Pathway (RCP):


GMSL during the LIG may have experienced multiple peaks, possibly associated with orbitally driven, asynchronous land-ice minima at the two poles (e.g., 8, 99). A significant fraction of the Greenland ice sheet remained intact throughout the LIG period, with recent alternative reconstructions limiting the peak Greenland GMSL contribution to .2 m (95) or 4-6 m (101) of the ice sheetÕs total 7-m sea-level equivalent mass. Thermal expansion and the melting of mountain glaciers together likely contributed .1 m (102, 103). This implies a significant contribution to LIG GMSL from Antarctic ice melt. However, there is little direct observational evidence of mass loss from the Antarctic region. Additional constraints from RSL reconstructions in mid-to high-latitude regions may help to partition contributions from Greenland and Antarctica (104).
Estimates of LIG GMSL have not yet incorporated the effects of MDT, which may contribute as much as 4 ± 7m(1.) to RSL change in some regions (e.g., Southwestern Australia) (73). In contrast to MPWP, the magnitudes of RSL change due to GIA and MDT are roughly the same order, although the spatial pattern associated with the two processes should be distinct (e.g., 4, 73). Whether a formal accounting for MDT would significantly alter the estimated height of the LIG highstand is unknown.
Representative Concentration Pathway (RCP):
one of a set of four standardized pathways, developed by the global climate modeling and integrated assessment modeling communities, describing possible future pathways of climate forcing over the twenty-first century; Extended Concentration Pathways extended the RCPs to the end of the twenty-third century
Figure 3: Reconstructions of global mean sea level (GMSL) and relative sea level (RSL) (6, 25, 119) from (a) the instrumental period, (b)the Holocene, and (c) the Last Interglacial. (a) GMSL estimates during the Instrumental Period from Dangendorf et al. (2017) (118) (green), Hay et al. (2015; Gaussian process regression (GPR)) (119) (dark blue), Hay et al. (2015; Kalman smoothing (KS)) (119) (light blue), Jevrejeva et al. (2014) (117) (orange), and Church & White (2011) (116) (purple), as well as individual RSL records from tide gauges obtained from the Permanent Service for Mean Sea Level (PSMSL). (b) GMSL (and sea-level equivalent signal) derived from far-field data (purple: Lambeck et al. (2014) (46), with 2-sigma uncertainty range) and from GIA models [ICE-5G (Peltier 2004) (52) (red); ICE-6G (Peltier et al. 2015) (42) (blue); Bradley et al. 2016 (111) (black)]. RSL data from Southern Disko Bugt (175, 176); Arisaig, Scotland (177); West Guangdong, southern China (178); Northeastern Brazil (179); South Shetland Island, Antarctica (180); and Langebaan Lagoon, South Africa (181). (c) GMSL estimate from Kopp et al. (6) and RSL reconstructions from Barbados (182-192) interpreted using coral depth distributions presented in Hibbert et al. (78), the Netherlands (193) as interpreted by Kopp et al. (6), the Red Sea continuous delta-18O record (green crosses) with probabilistic model and 95% confidence interval shown (green shading) (194), the Seychelles (195), and Western Australia (100, 196-206) where coral data are interpreted as a lower limit on RSL. Unless otherwise indicated, model uncertainties are 1-sigma, whereas data uncertainties are 2-sigma. Error bars cross at the median of the vertical probability distribution and midpoint of the age distribution of each data point. U-series data are screened following guidelines presented in Hibbert et al. (2016) (78) or Dutton et al. (2015) (195). With the exception of the Red Sea dataset, RSL reconstructions have not been corrected for tectonic motion. To demonstrate the potential inßuence of tectonics on RSL reconstructions, we plot data from Barbados (C1) using uncorrected elevations (solid lines) and elevations corrected for tectonic uplift (dashed lines).
The Holocene (11,700 years ago to present)
A global temperature reconstruction for the early to middle Holocene (from ~9.5-5.5 ka), derived from both marine and terrestrial proxies, suggests a global mean surface temperature ~0.8C higher than preindustrial temperatures (105). This estimate, however, conflicts with climate models that simulate a warming trend through the Holocene. The discrepancy may be due to uncertainties in both the seasonality of proxy reconstructions and the sensitivity of climate models to orbital forcing (106). Recent evidence suggests that estimates of global temperatures may be biased by sub-seasonal sensitivity of marine and coastal temperature estimates from the North Atlantic, with pollen records from North America and Europe instead suggesting a later period of peak warmth from .5.5-3.5 ka and temperatures .0.6.C warmer than the late nineteenth century (107).
The Holocene has more abundant and highly resolved RSL reconstructions than previous interglacial periods (Figure 3b, 45). These reconstructions are sourced from sedimentary (wet.land, deltaic, estuarine, lagoonal facies), geomorphic (beachrock, tidal and abrasional notches), archaeological, coral reef, coral microatoll, and other biological sea-level proxies. The abundance of RSL data from this period, combined with the preservation of near-Þeld glacial deposits, has contributed to the development of a relatively well-constrained history of ice-sheet retreat, particularly in the Northern Hemisphere (108). However, questions remain about Antarctic and Greenland contributions to GMSL (e.g., 109, 110), the timing of when (prior to the twentieth century rise) land-ice contributions to GMSL ceased (e.g., 46, 111, 112), and the magnitude and timescale of internal variability in glaciers, ice sheets, and the ocean.
Given the resolution of Holocene data, detailed sea-level reconstructions from this period are important for constraining local to regional processes and providing estimates of rates of change. GIA is the dominant process driving spatial variability during the Holocene (45). Records from locations formerly covered by ice sheets (near-field regions such as Antarctica, Greenland, Canada, Sweden, and Scotland) reveal a complex pattern of RSL fall from a maximum marine limit due to the net effect of inputs from melting of land-based ice sheets and glacio-isostatic uplift. Rates of RSL fall in near-field regions during the early Holocene were up to -69 ± 9 m/ka (45). Regions near the periphery of ice sheets (intermediate-field locations such as the mid-Atlantic and Pacific coasts of the United States, and northwestern Europe and the Caribbean) display fast rates of RSL rise (up to 10 ± 1 m/ka) in the early Holocene in regions near the center of forebulge collapse. Regions far from ice sheets exhibit a mid-Holocene highstand, the timing (between 8 and 4 ka) and magnitude (between <1 and 6 m) of which vary among South American, African, Asian, and Oceania regions. With diminishing contributions from GIA and melting of land-based ice sheets during the past several millennia, lower amplitude local-to regional-scale processes, such as steric effects or ocean dynamics, are manifested in RSL records (22).
Although the past two millennia (the Common Era) may not be a direct analog for future changes, semi-empirical relationships between high-resolution RSL reconstruction can be paired with temperature reconstructions (e.g., 113) that show periods of both warming (e.g., Medieval Climate Anomaly) and cooling (e.g., Little Ice Age) to show climate forcing on timescales (multi.decadal to multicentennial) and magnitudes relevant to future climate and sea-level scenarios (e.g., 25). The more complete geologic record in the Common Era permits the reconstruction of continuous time series of decimeter-scale RSL change over this period using salt-marsh sequences and coral microatolls (5). The resolution of reconstructions is comparable to future sea-level changes over the next decades to centuries, which enables an examination of regional dynamic variability in sea level that is not possible for earlier periods (24). In addition, the GIA signal is approximately linear over the Common Era and, therefore, it is easier to quantify its contribution to RSL.
Tide gauge:
device for measuring the height of the sea surface with respect to a reference height fixed to the solid Earth.
Reconstructions of Relative Sea Level from Instrumental Data