For example, the Gulf Stream, by which we refer to the full western boundary current near southern Florida, has two branches: the Florida Current and the Antilles Current, which flows offshore of the Bahama Banks (Figure 2). While the Florida Current carries a larger mean transport (about 32 Sv compared with about 5 Sv in the Antilles Current), both exhibit comparable variability (Lee et al., 1996). Thus, the total western boundary current flow could be constant, but its effect on coastal DSL would vary depending upon the respective contributions of the Florida and Antilles currents. In addition, assessing trends in the volume transport of complex, evolving, western boundary currents is challenging. This difficulty underlies the debate surrounding Ezer et al.’s (2013) conclusion that a Gulf Stream decline was responsible for accelerated sea level rise in the mid-Atlantic Bight (Ezer, 2015; Rossby et al., 2014). The deviation in Gulf Stream transport calculations found across studies is perhaps not surprising, given longitudinally varying changes in the Gulf Stream velocity, width, and position (Dong et al., 2019), and the presence of Gulf Stream meanders, eddies, and recirculation gyres.
Understanding the timescales over which the AMOC indicators discussed in section 5 (e.g., the Gulf Stream, and gyre densities) and AMOC strength variations are coherent is critical to their use as proxies of AMOC. There is little evidence for seasonal and interannual variability of the Florida Current or the Gulf Stream (characteristic over the timescales of many studies cited in section 5) to be related to AMOC. Using evidence that Sverdrup balance holds on multiannual to decadal timescales in the interior subtropics (Gray & Riser, 2014; Thomas et al., 2014; Wunsch, 2011), it can be demonstrated that (subtropical) AMOC variability must be mirrored by changes in the western boundary current at these timescales (de Boer & Johnson, 2007; Thomas et al., 2012). The Gulf Stream can therefore be expected to concentrate decadal-period changes in both the wind-driven and the thermohaline circulations, both of which are predicted to weaken in the 21st century (Beadling et al., 2018; Lique & Thomas, 2018; Thomas et al., 2012). However, this finding only applies southward of approximately 35°N, since the ocean to the north is not in Sverdrup balance (Gray & Riser, 2014; Thomas et al., 2014). Furthermore, there is no satisfactory way of defining the boundary between a western boundary current and the ocean interior when the ocean is dominated by mesoscale eddies (Wunsch, 2008). Models and observations also reveal a strong gyre dependence of AMOC changes, with interannual variability dominating in the subtropical gyre and decadal variability in the subpolar gyre (e.g., Bingham et al., 2007; Wunsch, 2011; Wunsch & Heimbach, 2013; Zhang, 2010). Lozier et al. (2010) used a data-assimilating numerical model to further demonstrate that gyre-dependent AMOC changes might be important on up to multidecadal periods.
Relatedly, there is evidence that property changes in the subpolar and subtropical gyres may not reflect changes in AMOC over certain timescales. Processes governing ocean density changes in this region on decadal timescales remain unclear (Williams et al., 2015; Buckley & Marshall, 2016; Menary et al., 2015; Piecuch, Ponte, et al., 2017; Robson et al., 2016); remote Rossby wave signals, local atmospheric forcing, changes in deep convection and water mass formation, mean flow advection, and gyre circulation “wobbles” all potentially play a role (Buckley & Marshall, 2016). Although data collected in the subpolar and subtropical gyres suggest southward propagation of deep hydrographic properties on advective (multiannual to decadal) timescales in the Labrador Current and Deep Western Boundary Current of the subtropical gyre (e.g., Molinari et al., 1998; Talley & McCartney, 1982; van Sebille et al., 2011), tracer studies have identified that the majority of water in the Labrador Current does not pass southwards into the subtropical gyre but instead cyclonically recirculates back around within the subpolar gyre (e.g., Bower et al., 2009; Rhein et al., 2002; Zou & Lozier, 2016). Of the deep subpolar water that is advected into the subtropical gyre, the intergyre pathway is not principally via the Deep Western Boundary Current but rather through the interior ocean (Bower et al., 2009; Lozier, 2010; Zhang, 2010), which is compensated by slow upper ocean advective pathways northwards out of the subtropical gyre that reach the greatest transport at depths of approximately 700 m (Burkholder & Lozier, 2011, 2014).
Implications for Along-Coast Variations and Across-Model Differences
Collectively, sections 6.1 to 6.3 indicate that U.S. East Coast continental shelf bathymetry, and the evolution of western boundary and coastal currents under local- and large-scale forcing, will influence the local coastal sea level expression associated with a given change in AMOC. The importance of these processes should be expected to vary regionally (e.g., north and south Cape Hatteras, but also within each region); future studies might probe the influence of these smaller scale along-coast variations on local sea level gradients (see section 7).
Focusing on time-mean sea level on the shelf, Higginson et al. (2015) suggest that coarse resolution models may exhibit errors in the representation of coastal sea level due to inadequate horizontal resolution, the form of the coastal boundary condition, poor representation of processes in shallow water, and/or unresolved continental shelf atmospheric forcing. Sections 6.1 and 6.2 support the importance of the representation of these coastal processes, and imply that differences in the model resolution may underlie some of the spread shown in Figure 5.
Over the global coastal ocean, Becker et al. (2016) find that climate models have a wide range of success in reproducing the spectral characteristics of observed tide gauge sea level variability. Little et al. (2017) specifically tested the ability of an initial condition ensemble of Community Earth System Model simulations to represent interannual U.S. East Coast DSL variability, finding that Community Earth System Model agrees well with observed tide gauge data along the Northeast U.S. coast, but poorly represents the time-mean and variability of DSL south of Cape Hatteras. The Minobe et al. (2017) framework (section 6.1) also exhibits disagreement with CMIP5 US east coast DSL changes south of ~35°N (see their Figure 10). This suggests that large-scale models might be particularly limited in the South Atlantic Bight. Here, in addition to complex shelf bathymetry, DSL variability may also be influenced by incoherence between the Gulf Stream and AMOC, the complex vertical and horizontal structure of western boundary currents, the potential effect of rapid western boundary current flow against the prevailing propagation of information in the direction of boundary waves, and the Antilles Current (section 6.3).
Penduff et al. (2010) find that higher-resolution models (as fine as 0.25°) show improved representations of variability and time-mean Sea Surface Height (SSH), especially in the eddy rich regions, in comparison to altimetry. Coastal sea level variability also appears improved with finer resolution, and DSL change under strong external forcing appears to be moderated near the coastline in models of higher resolution (Liu et al., 2016). Other high-resolution simulations show substantial modification of the coastal sea level signal (e.g., the two MPI models in Figure 5). Such resolution effects deserve more investigation as simulations become available (see, e.g., Haarsma et al., 2016).
In addition to the varied, resolution-dependent, representation of coastal processes and shelf bathymetry in models, which might be expected to disproportionately affect coastal DSL, the spatial variability in the “interior” DSL change in CMIP5 models implies that more complex changes in the 2-D overturning, or in the 3-D structure of the North Atlantic circulation, are relevant for determining patterns of DSL change. Bouttes et al. (2014) suggest that the underlying driver of differences in large-scale DSL change is related to locations of deep convection. Support for dependence on forcing is also evident in Kienert and Rahmstorf (2012), who find a substantially different DSL response to AMOC changes associated with different forcing (freshwater hosing, CO2 increases, Southern Ocean wind stress changes) within the same climate model.
Perspective and Future Directions


Penduff et al. (2010) find that higher-resolution models (as fine as 0.25°) show improved representations of variability and time-mean Sea Surface Height (SSH), especially in the eddy rich regions, in comparison to altimetry. Coastal sea level variability also appears improved with finer resolution, and DSL change under strong external forcing appears to be moderated near the coastline in models of higher resolution (Liu et al., 2016). Other high-resolution simulations show substantial modification of the coastal sea level signal (e.g., the two MPI models in Figure 5). Such resolution effects deserve more investigation as simulations become available (see, e.g., Haarsma et al., 2016).
In addition to the varied, resolution-dependent, representation of coastal processes and shelf bathymetry in models, which might be expected to disproportionately affect coastal DSL, the spatial variability in the “interior” DSL change in CMIP5 models implies that more complex changes in the 2-D overturning, or in the 3-D structure of the North Atlantic circulation, are relevant for determining patterns of DSL change. Bouttes et al. (2014) suggest that the underlying driver of differences in large-scale DSL change is related to locations of deep convection. Support for dependence on forcing is also evident in Kienert and Rahmstorf (2012), who find a substantially different DSL response to AMOC changes associated with different forcing (freshwater hosing, CO2 increases, Southern Ocean wind stress changes) within the same climate model.
Perspective and Future Directions
An antiphase relationship between large-scale North Atlantic meridional volume transport and U.S. East Coast DSL is broadly evident across a range of numerical simulations and observational analyses. This relationship can be interpreted using the simple geostrophic framework introduced in section 3. However, such a framework is insufficient to explain the widely differing along-coast AMOC-DSL scalings derived in models and observations, or variation across climate models. Furthermore, such an interpretation limits causal attribution: Geostrophy cannot provide information about the forces that drive sea level changes.
In this review, we have noted some possible origins for regional, model, timescale, and forcing dependence (section 6). However, we are unable to assess the degree to which each is responsible for variations in local scaling coefficients. Explanations for these deviations are essential to improve confidence in reconstructions of North Atlantic variability derived from tide gauge observations or paleoproxies and projections of coastal sea level change from current-generation climate models.
We thus encourage the sea level research community to pursue the following near-term goals: (1) an understanding of the relationship between AMOC and other North Atlantic currents; (2) an understanding of the vertical structure of the AMOC and its variation with respect to local bathymetry; (3) an assessment of the importance of ageostrophic processes to AMOC and related currents; and (4) an effort to connect these research results, including their region (latitude-), model, and timescale dependence, to their origins in heat, momentum, and buoyancy forcing. Such efforts should include new sea level studies, as well as the incorporation of existing and new findings from outside the sea level realm.
A simple step toward the first and second goals involves broadening the features of the ocean circulation analyzed in models beyond a single AMOC metric (e.g., the basin-wide maximum overturning stream function). Modeled and observed DSL changes have often been compared to AMOC changes at a different latitude, which involves an implicit or explicit assumption that such changes are synchronous and meridionally coherent, which is not supported by the literature cited in section 6.3. Indeed, such a coarse characterization of AMOC may underlie some of the difference in scaling coefficients shown in Figure 5.
As noted in section 4, another critical ambiguity of relevance, particularly important to the interpretation of observational analyses, is the coherence of AMOC and western boundary currents. Other important relationships include those between the GSNW and AMOC; Labrador Sea and subpolar gyre steric changes; and subpolar and subtropical gyre steric changes. Higher-resolution simulations can now represent the mean state and variability of coastal currents and indicate that climate-driven changes in these currents may differ from those in the large-scale (e.g., Saba et al., 2016). Although evidence in section 6.3 suggests that many components of AMOC, and subpolar and Nordic Seas buoyancy variability, may be coherent over multidecadal time frames (Pillar et al., 2016), there is evidence that interannual to decadal variability is not, particularly across the intergyre boundary. Modeling studies examining AMOC-DSL relationships can easily include metrics of some of these other AMOC components and indicators (possibly over different timescales), which would improve the scope of their results, and the ability to reconcile with observations.
The direct observational record of AMOC variability is limited; in this review, we have focused on the longer observed record of AMOC components. However, the ever extending record of AMOC at 26°N is now complemented by the OSNAP array, providing some perspective on gyre dependence, the meridional coherence of AMOC, and the relationship with other AMOC components. These AMOC records are complemented by new observational campaigns over the U.S. East coast continental shelf and slope (e.g., Andres et al., 2018, Gawarkiewicz et al., 2018). In addition to these instrumental records, proxy records of both coastal sea level and AMOC are available that are able to resolve decadal-centennial fluctuations (Engelhart & Horton, 2012; Kemp et al., 2017, 2018; Rahmstorf et al., 2015; Thornalley et al., 2018). Complemented by model results, these proxy observations could provide valuable constraints on multidecadal to centennial AMOC-DSL covariability.
With respect to the assessment of ageostrophic processes, we note that many modeling centers have begun to provide the output required to compute closed momentum budgets offline (Gregory et al., 2016; Wunsch & Heimbach, 2009; Yeager, 2015). Such budgets, both zonally integrated, and local, would clearly indicate the importance of ageostrophic processes, their time and latitude dependence, and (if possible) differences across a set of models. They could also include the effect of terms, including nonlinearity (Hughes et al., 2019), and, in higher-resolution models, eddy variability (Grégorio et al., 2015; Sérazin et al., 2015), that are not discussed in this review. High-resolution models also offer promise for better resolving the shelf and shelf processes, and they may constitute a means for testing the theories of coastal modulation of interior signals (section 6.1), under a wider range of conditions, forcing, and timescales.
Such analyses also move beyond the purely diagnostic, degenerate, statement of force balance supplied by geostrophy, allowing an understanding of the local, regional, basin, and global scale forcing responsible for coastal sea level changes. The incomplete interpretation provided by geostrophy is evident in Goddard et al. (2015), who linked an “extreme” interannual sea level rise event in the northeast US with an abrupt 30% AMOC weakening. However, this event occurred coincident with an anomalously negative North Atlantic Oscillation (NAO) associated with atmospheric pressure and wind anomalies. Piecuch and Ponte (2015) and Piecuch et al. (2016) demonstrated that 50% of this event could be explained by the inverse barometer effect and the remainder could be partly explained by local winds. The 30% drop in the AMOC itself was observed in the Gulf Stream transport (Ezer, 2015) and was explained by wind forcing (Zhao & Johns, 2014). It is thus more appropriate to view the sea level anomaly as driven by all of the forcings (local and remote) associated with the extreme NAO anomaly. Over longer timescales, causality often remains unclear: for example, differences in observed sea level changes along the US east coast have been attributed to changes in Gulf Stream position and strength, AMOC strength, and steric changes. While these changes may be coupled, and serve as indicators of AMOC, they do not identify causal drivers.
Even if causality can be established under certain forcing and timescales (e.g., interannual, driven by NAO), it does not imply that the same processes and AMOC components (and sea level signatures) are always relevant (e.g., on centennial timescales in the past or future). For example, Kenigson et al. (2018) find that the relationship between DSL and NAO is nonstationary, echoing the results of Andres et al. (2013). Looking farther into the future, 21st century changes in AMOC strength in climate models are principally forced by greenhouse gas-associated heat and buoyancy fluxes in the North Atlantic (Beadling et al., 2018; Bouttes et al., 2014; Slangen et al., 2015), rather than NAO-associated wind stress.


Such analyses also move beyond the purely diagnostic, degenerate, statement of force balance supplied by geostrophy, allowing an understanding of the local, regional, basin, and global scale forcing responsible for coastal sea level changes. The incomplete interpretation provided by geostrophy is evident in Goddard et al. (2015), who linked an “extreme” interannual sea level rise event in the northeast US with an abrupt 30% AMOC weakening. However, this event occurred coincident with an anomalously negative North Atlantic Oscillation (NAO) associated with atmospheric pressure and wind anomalies. Piecuch and Ponte (2015) and Piecuch et al. (2016) demonstrated that 50% of this event could be explained by the inverse barometer effect and the remainder could be partly explained by local winds. The 30% drop in the AMOC itself was observed in the Gulf Stream transport (Ezer, 2015) and was explained by wind forcing (Zhao & Johns, 2014). It is thus more appropriate to view the sea level anomaly as driven by all of the forcings (local and remote) associated with the extreme NAO anomaly. Over longer timescales, causality often remains unclear: for example, differences in observed sea level changes along the US east coast have been attributed to changes in Gulf Stream position and strength, AMOC strength, and steric changes. While these changes may be coupled, and serve as indicators of AMOC, they do not identify causal drivers.
Even if causality can be established under certain forcing and timescales (e.g., interannual, driven by NAO), it does not imply that the same processes and AMOC components (and sea level signatures) are always relevant (e.g., on centennial timescales in the past or future). For example, Kenigson et al. (2018) find that the relationship between DSL and NAO is nonstationary, echoing the results of Andres et al. (2013). Looking farther into the future, 21st century changes in AMOC strength in climate models are principally forced by greenhouse gas-associated heat and buoyancy fluxes in the North Atlantic (Beadling et al., 2018; Bouttes et al., 2014; Slangen et al., 2015), rather than NAO-associated wind stress.
Separation of local and remote wind-driven changes in circulation and sea level from remote buoyancy/deep water driven AMOC changes remains a key challenge. Such work will have to illuminate the timescales and climate forcing under which wind and buoyancy forcing are coupled. For example, Woodworth et al. (2014) indicate that wind forcing alone is largely responsible for decadal timescale sea level variability. However, since this study used a standalone ocean model, it is not clear what processes produce low-frequency wind variability. Furthermore, the large spatial scales of atmospheric forcing challenge efforts to isolate the AMOC-forced or remotely forced component of sea level change. Adjoint analyses or perturbation experiments (Heimbach et al., 2011; Pillar et al., 2016; Yeager & Danabasoglu, 2014) may help isolate the roles of wind and buoyancy forcing and elucidate the relevant pathways, state variables, and adjustment processes mediating connections between the open ocean and observed and projected US east coast sea level changes.
To conclude, there are many productive areas of research that can help refine our understanding of the relationship between the large-scale climate, AMOC, and coastal sea level. Given their importance to future sea level changes on the U.S. East Coast, and reconstruction of preinstrumental ocean circulation and climate variability, we anticipate the research community will pursue them with vigor.


Gregory et al. (2019)
Title:
Concepts and Terminology for Sea Level: Mean, Variability and Change, Both Local and Global
Key Points:
Describes concepts and terminology related to sea level, its variability, and changes both locally and globally.
Clarifies language and definitions for better communication in sea-level science
Discusses the inconsistencies and ambiguities in sea-level terminology across different disciplines.
Proposes new terminology and recommend replacing certain terms that are unclear or confusing.
Keywords:
sea level, concepts, terminology
Corresponding author:
Jonathan M. Gregory
Citation:
Gregory, J. M., Griffies, S. M., Hughes, C. W., Lowe, J. A., Church, J. A., Fukimori, I., et al. (2019). Concepts and Terminology for Sea Level: Mean, Variability and Change, Both Local and Global. Surveys in Geophysics, 40(6), 1251–1289. doi:10.1007/s10712-019-09525-z
URL:
https://link.springer.com/article/10.1007/s10712-019-09525-z
Abstract
Changes in sea level lead to some of the most severe impacts of anthropogenic climate change. Consequently, they are a subject of great interest in both scientific research and public policy. This paper defines concepts and terminology associated with sea level and sea-level changes in order to facilitate progress in sea-level science, in which communication is sometimes hindered by inconsistent and unclear language. We identify key terms and clarify their physical and mathematical meanings, make links between concepts and across disciplines, draw distinctions where there is ambiguity, and propose new terminology where it is lacking or where existing terminology is confusing. We include formulae and diagrams to support the definitions.
Introduction and Motivation
Changes in sea level lead to some of the most severe impacts of anthropogenic climate change (IPCC 2014). Consequently, they are a subject of great interest in both scientific research and public policy. Since changes in sea level are the result of diverse physical phenomena, there are many authors from a variety of disciplines working on questions of sea-level science. It is not surprising that sea-level terminology is inconsistent across disciplines (for example, “dynamic sea level” has different meanings in oceanography and geodynamics), as well as unclear or ambiguous even within a single discipline. (For instance, “eustatic” is ambiguous in the climate science literature.) We sometimes experience difficulty in finding correct and precise terms to use when writing about sea-level topics, or in understanding what others have written. Such communication problems hinder progress in research and may even confuse discussions about coastal planning and policy.
This situation prompted us to revisit the meaning of key sea-level terms, and to recommend definitions along with their rationale. In so doing, we aim to clarify meanings, make links between concepts and across disciplines and draw distinctions where there is ambiguity. We propose new terminology where it is lacking and recommend replacing certain terms that we argue are unclear or confusing. Our goal is to facilitate communication and support progress within the broad realm of sea-level science and related engineering applications.
In the next section, we outline the conventions and assumptions we use in our definitions and mathematical derivations. The following three sections (Sects. 3–5) contain the definitions, with a subsection for each major term defined, labelled with “N” and numbered consecutively throughout. In Sect. 3 we define five key surfaces: reference ellipsoid, sea surface, mean sea level, sea floor and geoid. We consider the variability and differences in these surfaces in Sect. 4, and quantities describing changes in sea level in Sect. 5. In Sect. 6, we show how relative sea-level change is related to other quantities in various ways. In Sect. 7 we describe how observational data are interpreted using the concepts we have defined. To facilitate sequential reading of this paper, the material of Sects. 3–7 is arranged to minimize forward references, though we were unable to avoid all.
We give a list of deprecated terms with recommended replacements in Sect. 8, and a list in Sect. 9 of all terms defined, referring to the subsections where they are defined, thus providing an index that also includes our notation. The appendices contain further discussion of some aspects at greater length.
The complexity of sea-level science is evident in the detail of the definitions and discussions in this paper. It may therefore be helpful to keep in mind that from the point of view of coastal planning and climate policy there are three quantities of particular interest. Extreme sea level along coasts (i.e., extreme coastal water level) or around offshore marine infrastructure (such as drilling platforms) is of great practical importance because of the enormous damage it can cause to human populations and their built environment and to ecosystems. The expected occurrence of extreme sea level under future climates is therefore relevant to decision-making on a range of time-horizons.
The dominant factor in changes of future local extremes is relative sea-level change (RSLC), i.e., the change in sea level with respect to the land (Lowe et al. 2010; Church et al. 2013). Where there is relative sea-level rise, coastal defences have to be raised to afford a constant level of protection against extremes, and low-lying areas are threatened with permanent inundation. Although RSLC depends on many local and regional influences, the majority of coastlines are expected to experience RSLC within a few tens of per cent of global-mean sea-level rise (GMSLR) (Church et al. 2013). Projections of GMSLR are therefore of interest to global climate policy, with both adaptation and mitigation in mind.
In general, greater GMSLR is projected for scenarios with higher rates of carbon dioxide emission and on longer timescales. Particular attention is paid, especially by risk-intolerant users, to the probability or possibility of future large changes in sea level (Hinkel et al. 2019). We recommend referring to these as high-end scenarios or projections of RSLC or GMSLR (rather than “extreme” scenarios), to avoid confusion with projections of extreme sea level.
Conventions and Assumptions
We here summarize the conventions and assumptions employed in the text and formulae of our definitions.
First Appearance of Terms
Within each definition subsection of Sects. 3–5, we use bold font for the first appearance of a term whose definition is the subject of another subsection. When we first define a term that does not have its own subsection, it appears with a slanted font. The reader can locate the definitions of terms marked in these ways by looking them up in Sect. 9. In the PDF of this paper, each bold term in Sects. 3–8 is a hyperlink to the relevant definition subsection.
Time-Mean and Changes
The sea surface varies on all the timescales of the Earth system, associated with sea-surface waves and tides, meteorological variability (from gustiness of winds to synoptic phenomena such as mid-latitude depressions and tropical cyclones), seasonal, interannual and longer-term internally generated climate variability (e.g., El Niño and the Interdecadal Pacific Oscillation), anthropogenic climate change and naturally forced changes (e.g., by volcanic eruptions, glacial cycles and tectonics). For various purposes of understanding and planning for sea-level variations it is helpful to draw a distinction between a time-mean state and fluctuations within that state. The definition of a “state” depends on the scientific interest or application. For sea level, the state might be defined by a time-mean long enough to remove tidal influence (about 19 years), or which characterizes a climatological state (conventionally 30 years), but it could be shorter, for example, if interannual variability were regarded as altering the state.
Thus, the time-mean state cannot be absolutely defined, but the concept is necessary. In this paper, mean sea level refers to a time-mean state whose precise definition should be specified when the term is used, and which is understood to be long enough to eliminate the effect of meteorological variations at least. We use symbols with a tilde and time-dependence, e.g., 𝑋̃(𝑡) for time-varying quantities, and symbols without any distinguishing mark and no time-dependence, e.g., X for time-mean quantities that characterize the state of the system. On longer timescales, the state itself may change, for example, due to anthropogenic influence. We use the symbol Δ and the word “change” to refer to the difference between any two states; thus, Δ𝑋 is “change in X”, e.g., change in relative sea level between the time-mean of 1986–2005 and the time-mean of 2081–2100. Anthropogenic sea-level change comes mostly through climate change, but there are other influences too, such as impoundment of water on land in reservoirs.
Local and Regional
By a local quantity, we mean one which is a function of two-dimensional geographical location 𝐫, specified by latitude and longitude. For some applications, it is important to consider variations of local quantities over distance scales of kilometres or less. Other quantities do not have such pronounced local variation and are typically considered as properties of regions, with distance scales of tens to hundreds of kilometres.
Global Mean Over the Ocean Surface Area


Thus, the time-mean state cannot be absolutely defined, but the concept is necessary. In this paper, mean sea level refers to a time-mean state whose precise definition should be specified when the term is used, and which is understood to be long enough to eliminate the effect of meteorological variations at least. We use symbols with a tilde and time-dependence, e.g., 𝑋̃(𝑡) for time-varying quantities, and symbols without any distinguishing mark and no time-dependence, e.g., X for time-mean quantities that characterize the state of the system. On longer timescales, the state itself may change, for example, due to anthropogenic influence. We use the symbol Δ and the word “change” to refer to the difference between any two states; thus, Δ𝑋 is “change in X”, e.g., change in relative sea level between the time-mean of 1986–2005 and the time-mean of 2081–2100. Anthropogenic sea-level change comes mostly through climate change, but there are other influences too, such as impoundment of water on land in reservoirs.
Local and Regional
By a local quantity, we mean one which is a function of two-dimensional geographical location 𝐫, specified by latitude and longitude. For some applications, it is important to consider variations of local quantities over distance scales of kilometres or less. Other quantities do not have such pronounced local variation and are typically considered as properties of regions, with distance scales of tens to hundreds of kilometres.
Global Mean Over the Ocean Surface Area
By global mean, we mean the area-weighted mean over the entire connected surface area of the ocean, i.e., excluding the land. The ocean includes marginal seas connected to the open ocean such as the Mediterranean Sea, Black Sea and Hudson Bay, but excludes inland seas such as the Caspian Sea, the African Great Lakes and the North American Great Lakes. It includes areas covered by sea ice and ice shelves, where special treatment is needed to define the level of the sea surface. We note that observational estimates of the global mean are often made from systems which lack complete coverage.
For centennial timescales, we can assume the ocean surface area A is constant, with A = 3.625×10^{14} m^2 (Cogley 2012). It is altered substantially by global-mean sea-level changes of many metres, such as on glacial–interglacial timescales or possibly over future millennia due to ice-sheet changes, and on geological timescales due to plate tectonics. The formulae we give for some quantities describing global-mean changes are not exactly applicable under those circumstances.
Sea-Water Density
Many of the formulae in this paper involve sea-water density. Although sea-water density is a local quantity, we treat it in many contexts as a globally uniform constant with a representative value 𝜌∗ (e.g., 1028  kg  m −3). For the density of freshwater added at the sea surface we use a constant 𝜌f=1000  kg  m −3 for convenience, neglecting the variation of <1% in freshwater density due to temperature.
Vertical Direction and Distance
Before considering the vertical location of surfaces, or the local vertical distance between two surfaces, we need to specify the meaning of vertical. Geodesy is concerned with horizontal and vertical distances measured relative to the reference ellipsoid, which is a surface fixed with respect to the solid Earth. Geophysical fluid dynamics, including ocean circulation dynamics, is concerned with horizontal distances on surfaces of constant geopotential, and vertical distances measured perpendicular to such surfaces, especially the geoid. We discuss the two frames of reference (one relative to the reference ellipsoid and the other to the geoid) in the subsections describing those two surfaces. The distinction between the two frames is relevant only to the real world, because numerical ocean circulation models implicitly assume an idealized effective gravity field in which the geoid and the reference ellipsoid are identical (often spherical rather than ellipsoidal). In reality, the geoid has an irregular shape, whose vertical separation from the reference ellipsoid is ±100 m and varies over horizontal length scales of 100s km.
Most of our formulae involve the local vertical coordinate of surfaces such as 𝑋(𝐫), for which we use the vertical distance above the reference ellipsoid (negative if below). We make this choice in order to give our formulae a well-defined interpretation. The choice of reference frame (with respect either to the reference ellipsoid or to the geoid) does not affect the geophysical definition of a surface, but the numerical value of its vertical coordinate at any given location is not the same in the two frames, because of the substantial difference between their reference surfaces. However, at a given location, there is negligible difference between the two frames regarding the local vertical direction. Hence, we can ignore the difference between the two definitions of “vertical” in evaluating a vertical gradient, the vertical distance 𝑌(𝐫)−𝑋(𝐫) between two surfaces, or the change in height Δ𝑋(𝐫) of a surface.
Surfaces
We here define five key surfaces used in sea-level studies. The reference ellipsoid and the associated terrestrial reference frame (depicted in Fig. 1) are geometrical constructions, chosen by convention. The other four surfaces (compared with the reference ellipsoid in Fig. 2) are geophysically defined and established with some uncertainty from observational data. These and other surfaces, such as datums defined by tides (e.g., mean lower-low water level), are located relative to the reference ellipsoid (Sect. 2.6), by their geodetic height as a function of geodetic location.
Fig. 1: The reference ellipsoid, which is used to locate other surfaces in a terrestrial reference frame, whose origin is the centre of the Earth. The figure shows the construction which defines the geodetic coordinates of an arbitrary point 𝐱 in 3D space. The line between 𝐱 and 𝐫 is normal to the reference ellipsoid, on which 𝐫 lies. The equator is intersected at 𝐩 by the meridian through 𝐫, and at 𝐩0 by the prime meridian, which defines the zero of longitude
N1 Reference ellipsoid:
The surface of an ellipsoidal volume of revolution chosen to approximate the geoid.
A reference ellipsoid is a conventional geometric construction used to specify locations in a terrestrial reference frame, i.e., relative to the solid Earth. Many reference ellipsoids have been defined by geodesists, and some are intended only for use over limited portions of the globe. A given specification of the reference ellipsoid is time-independent.
For purposes relating to global sea level we make the following requirements of the reference ellipsoid.
Its centre is the time-mean centre of mass of the Earth.
Its semi-major axis lies in the equatorial plane and its semi-minor axis along the rotation (polar) axis of the Earth.
Its axis of revolution is the rotation axis.
It is fixed with respect to the solid Earth, and it rotates with the Earth.
The International Earth Rotation and Reference Systems Service (www.iers.org) defines the International Terrestrial Reference Frame (ITRF). They recommend the GRS80 ellipsoid.
For more precise geodetic purposes, the ITRF defines the coordinates and their rates of change of a set of stations on the Earth’s surface. The coordinates are time-dependent because of tectonic motions and true polar wander, i.e., the time-dependence of the Earth’s rotation axis with respect to the solid Earth. The latter phenomenon is neglected in the above specification of the ellipsoid. If the rotation axis is invariant, the last point in our specification above is not necessary because, being a volume of revolution, the reference ellipsoid is symmetrical with respect to rotation about the axis.
To locate a point 𝐱 in 3D space in a reference frame based on the reference ellipsoid, we construct a straight line that passes through 𝐱 and is normal to the ellipsoid, which it intersects at 𝐫. The geodetic height of 𝐱 above the ellipsoid is the distance from 𝐫 to 𝐱 along this line, positive outwards. In our formulae, the vertical coordinate is the geodetic height, which is sometimes called ellipsoidal height. This is not the usual vertical coordinate for models of atmosphere and ocean circulation, which is instead defined relative to the geoid.
The geodetic latitude, commonly referred to simply as latitude, is the angle between the equatorial plane and the normal to the ellipsoid. It is different from the geocentric latitude, which is the angle between the equatorial plane and the line from the centre of the Earth to 𝐱. Geodetic and geocentric latitudes are the same for the poles and the equator, but elsewhere geodetic latitude is larger (as can be appreciated from Fig. 1), by up to about 0.2˚.


For purposes relating to global sea level we make the following requirements of the reference ellipsoid.
Its centre is the time-mean centre of mass of the Earth.
Its semi-major axis lies in the equatorial plane and its semi-minor axis along the rotation (polar) axis of the Earth.
Its axis of revolution is the rotation axis.
It is fixed with respect to the solid Earth, and it rotates with the Earth.
The International Earth Rotation and Reference Systems Service (www.iers.org) defines the International Terrestrial Reference Frame (ITRF). They recommend the GRS80 ellipsoid.
For more precise geodetic purposes, the ITRF defines the coordinates and their rates of change of a set of stations on the Earth’s surface. The coordinates are time-dependent because of tectonic motions and true polar wander, i.e., the time-dependence of the Earth’s rotation axis with respect to the solid Earth. The latter phenomenon is neglected in the above specification of the ellipsoid. If the rotation axis is invariant, the last point in our specification above is not necessary because, being a volume of revolution, the reference ellipsoid is symmetrical with respect to rotation about the axis.
To locate a point 𝐱 in 3D space in a reference frame based on the reference ellipsoid, we construct a straight line that passes through 𝐱 and is normal to the ellipsoid, which it intersects at 𝐫. The geodetic height of 𝐱 above the ellipsoid is the distance from 𝐫 to 𝐱 along this line, positive outwards. In our formulae, the vertical coordinate is the geodetic height, which is sometimes called ellipsoidal height. This is not the usual vertical coordinate for models of atmosphere and ocean circulation, which is instead defined relative to the geoid.
The geodetic latitude, commonly referred to simply as latitude, is the angle between the equatorial plane and the normal to the ellipsoid. It is different from the geocentric latitude, which is the angle between the equatorial plane and the line from the centre of the Earth to 𝐱. Geodetic and geocentric latitudes are the same for the poles and the equator, but elsewhere geodetic latitude is larger (as can be appreciated from Fig. 1), by up to about 0.2˚.
To define the longitude of 𝐱 (“geodetic” and “geocentric” are the same for longitude), consider the meridian passing through 𝐫, which intersects the equator at point 𝐩, and the prime meridian (the Greenwich meridian), which intersects the equator at 𝐩0. Viewing the Earth from above, the longitude is the anticlockwise angle between the lines from the centre of the Earth to 𝐩0 and to 𝐩.
Fig. 2: Relationship between surfaces relating to sea level. The normal to the reference ellipsoid defines the vertical in the terrestrial reference frame. The normal to the geoid is the vertical coordinate (z) for geophysical fluid dynamics, and anti-parallel to the local effective acceleration g due to gravity. The difference between these two definitions of the vertical direction is greatly exaggerated in this diagram; it is negligible in reality. The local vertical coordinates of mean sea level 𝜂, the geoid G and the sea floor 𝐹 are relative to the reference ellipsoid, while dynamic sea level 𝜁̃ is relative to the geoid. The local time-mean thickness of the ocean 𝐻 is the vertical distance between mean sea level and the sea floor. The deviation of atmospheric pressure 𝑝′𝑎 from its global mean causes the depression 𝐵 in sea level by the inverse barometer effect
N2 Sea surface 𝜂̃:
The time-varying upper boundary of the ocean. The sea-surface height is the geodetic height of the sea surface above the reference ellipsoid (a negative value if below).
In ocean areas without floating ice (sea ice, ice shelves or icebergs), the liquid sea surface is the bottom boundary of the atmosphere. In such areas, the existence of a well-defined sea-surface height (SSH)𝜂̃(𝐫,𝑡), that can be represented by a continuous and single-valued mathematical expression, presupposes a space–time averaging, because the instantaneous surface is ill-defined in the presence of some short-timescale phenomena that produce foam and sea spray, such as breaking surface waves and conditions of intense wind. We assume such averaging when speaking about the sea surface.
In ocean areas with floating ice, the liquid surface boundary is the bottom of the ice. For those areas, we define the SSH 𝜂̃ as the liquid-water equivalent sea surface𝜂̃LWE which the liquid would have if the ice were replaced by an equal mass of sea water of the density 𝜌s of the surface water in its vicinity. Following Archimedes’ principle,
𝜂̃LWE = 𝜂̃𝑠 + 𝑤𝑖𝑔𝜌s, (1)
where 𝜂̃𝑠 is the geodetic height of the liquid sea-water surface (beneath the ice) and 𝑤𝑖 is the weight per unit area of floating ice. (The depression of 𝜂̃𝑠 relative to 𝜂̃ is called the “ inverse barometer effect of sea ice” by Griffies and Greatbatch 2012, and 𝜂̃LWE is their “effective sea level”.) Although the liquid-water equivalent sea-surface height is not directly measurable, it is a convenient construct for many practical purposes of sea-level studies, and dynamically justifiable because the hydrostatic pressure and gravity beneath the ice are largely unaffected by the replacement of ice with liquid water.
The sea surface varies periodically with various frequencies due to tides . It varies also on all timescales due to sea-surface waves , atmospheric pressure, surface flux exchanges (with the atmosphere), river inflow, variability that is internally generated by ocean dynamics, motion of the sea floor and changes in the mass distribution within the ocean and solid Earth (discussed in many following entries).
Note that ocean dynamic sea level and ocean dynamic topography are distinct concepts from sea-surface height and from each other.
N3 Mean sea level (MSL) 𝜂: The time-mean of the  sea surface .
The period for the time-mean must be long enough to eliminate the effects of waves and other meteorologically induced fluctuations (as discussed in Sect. 2.2). Predicted tidal variations are subtracted if the period is not long enough to remove time-dependent tides , but permanent tides are included in MSL. For a precise definition of MSL, the period of the time-mean should be specified, and it could be described either with or without dependence on the time of year.
MSL is located by its geodetic height 𝜂(𝐫) above the reference ellipsoid (a negative value if below). In ocean models which regard the geoid and the reference ellipsoid as coincident, 𝜂 is equally the orthometric height of MSL above the geoid. MSL is sometimes called “mean sea surface”. We recommend against using this term, in order to make a clear distinction from “sea-surface height”.
N4 Sea floor 𝐹: The lower boundary of the ocean, its interface with the solid Earth.
The sea floor is the part of the surface of the solid Earth (whether bedrock or consolidated sediment, and lying beneath any unconsolidated sediment, e.g., Webb et al. 2013) that is always or sometimes submerged under sea water. The level of the sea floor varies due to solid-Earth tides , accumulation of sediment (with eventual compaction) and vertical land movement on a range of timescales.
We specify the instantaneous level of the sea floor by its geodetic height 𝐹̃(𝐫,𝑡) (negative over most of the ocean) relative to the reference ellipsoid . The local instantaneous thickness of the ocean (its vertical extent, the depth of the sea floor measured from a ship, a positive quantity sometimes called the depth of the water column) is given by
𝐻̃(𝐫,𝑡)=𝜂̃(𝐫,𝑡)−𝐹̃(𝐫,𝑡)≥0, (2)
i.e., the vertical distance between the sea surface and the sea floor. The choice of reference surface for vertical coordinates does not affect the value of 𝐻̃, because it is the difference between two vertical coordinates; 𝐻̃ would be the same if SSH and the sea floor were located by heights relative to the geoid rather than the reference ellipsoid. The time-mean thickness of the ocean𝐻
𝐻(𝐫)=𝜂(𝐫)−𝐹(𝐫)≥0 (3)
is likewise related to MSL 𝜂.


MSL is located by its geodetic height 𝜂(𝐫) above the reference ellipsoid (a negative value if below). In ocean models which regard the geoid and the reference ellipsoid as coincident, 𝜂 is equally the orthometric height of MSL above the geoid. MSL is sometimes called “mean sea surface”. We recommend against using this term, in order to make a clear distinction from “sea-surface height”.
N4 Sea floor 𝐹: The lower boundary of the ocean, its interface with the solid Earth.
The sea floor is the part of the surface of the solid Earth (whether bedrock or consolidated sediment, and lying beneath any unconsolidated sediment, e.g., Webb et al. 2013) that is always or sometimes submerged under sea water. The level of the sea floor varies due to solid-Earth tides , accumulation of sediment (with eventual compaction) and vertical land movement on a range of timescales.
We specify the instantaneous level of the sea floor by its geodetic height 𝐹̃(𝐫,𝑡) (negative over most of the ocean) relative to the reference ellipsoid . The local instantaneous thickness of the ocean (its vertical extent, the depth of the sea floor measured from a ship, a positive quantity sometimes called the depth of the water column) is given by
𝐻̃(𝐫,𝑡)=𝜂̃(𝐫,𝑡)−𝐹̃(𝐫,𝑡)≥0, (2)
i.e., the vertical distance between the sea surface and the sea floor. The choice of reference surface for vertical coordinates does not affect the value of 𝐻̃, because it is the difference between two vertical coordinates; 𝐻̃ would be the same if SSH and the sea floor were located by heights relative to the geoid rather than the reference ellipsoid. The time-mean thickness of the ocean𝐻
𝐻(𝐫)=𝜂(𝐫)−𝐹(𝐫)≥0 (3)
is likewise related to MSL 𝜂.
The shape of the sea floor is sometimes called the bottom topography or the bathymetry, for example in describing it as “rough” or “smooth”. These two synonymous terms are also used as names for the quantities −𝐹, 𝐺−𝐹 and 𝜂−𝐹(=𝐻), i.e., the time-mean vertical distance of the sea floor beneath the reference ellipsoid, the geoid or MSL, respectively. In order to be precise, it should be stated which of these alternatives is intended, since G and the reference ellipsoid differ by ±100 m (Sect. 2.6), and G and MSL differ by ±1 m, following time-mean ocean dynamic sea level .
N5 Geoid G: A surface on which the geopotentialΦ has a uniform value, chosen so that the volume enclosed between the geoid and the sea floor is equal to the time-mean volume of sea water in the ocean (including the liquid-water equivalent of floating ice).
The geopotential is a field of potential energy per unit mass, accounting for the Newtonian gravitational acceleration due to the mass of the Earth plus the centrifugal acceleration arising from the Earth’s rotation. We define the sign of the geopotential such that work is required to move a sea-water parcel from a lower geopotential (deeper in the ocean) to a higher geopotential (shallower in the ocean). Note that this sign convention for the geopotential is opposite to that used in geodesy.
The vertical gradient of the geopotential is equal to the local effective gravitational acceleration, g, 𝑔(𝐫,𝑡)=∂Φ/∂𝑧, usually referred to as “gravitational acceleration” in geophysical fluid dynamics. Hence, the effective gravitational acceleration is normal to the geoid, because the geoid is an equipotential surface, i.e., one on which the geopotential is constant. The Newtonian gravitational acceleration is time-dependent because the distribution of mass in the ocean (liquid and solid), on land (including the land-based cryosphere) and within the solid Earth is generally changing. The centrifugal acceleration is time-dependent because the Earth’s rotation rate and rotation axis are variable.
For models of atmosphere and ocean circulation, the vertical unit vector is directed anti-parallel to the effective gravitational acceleration g (or equivalently it is parallel to the local gradient of the geopotential). The height above the geoid of some point 𝐱 is the distance z, measured along the local vertical unit vector, from the geoid to 𝐱. The coordinate z (Fig. 2) is also called the orthometric height of 𝐱. Strictly, the orthometric height is measured along a plumb line, which is always normal to equipotential surfaces, but this distance differs negligibly from that measured along the perpendicular to the reference ellipsoid .
We define z such that 𝑧=0 is the geoid, 𝑧>0 is above the geoid, and 𝑧<0 is below. By horizontal we mean aligned with a surface of constant z. This is not strictly an equipotential surface, but the difference is locally negligible. It is, however, very different from a surface of constant geodetic height 𝑧′=𝑧+𝐺, where 𝐺(𝐫) is the geoid height above the reference ellipsoid (𝐺<0 where the geoid is below the ellipsoid).
The sea surface would coincide with the geoid if the ocean were in a resting steady state in the rotating frame of the earth. Although defining the geoid in this way is conceptually attractive, it is not realistic or practically useful. (See Appendix 1.)
The sea surface does not really coincide with the geoid because the ocean is not at rest. (See ocean dynamic sea level .) For example, mean sea level (MSL) north of the Antarctic Circumpolar Current (ACC) is at a higher geopotential than MSL south of the ACC; with respect to the geoid, MSL on the north side is roughly 2 m higher than on the south side.
Referring to its definition, we choose the geoid as the equipotential surface (out of the infinite set of them) which satisfies
∫(𝐺−𝐹)d𝐴 = 𝑉 = ∫𝐻 d𝐴 = ∫(𝜂−𝐹) d𝐴, (4)
using Eq. (3) for 𝐻, and where V is the volume of the global ocean and 𝐴=∫d𝐴 is its surface area. It follows from Eq. (4) that
∫𝜂 d𝐴 = ∫𝐺 d𝐴, (5)
i.e., MSL and geoid height above the reference ellipsoid have equal global means.
We define the geoid in terms of MSL 𝜂, rather than the sea-surface height 𝜂̃, in order to restrict changes in G and V to those occurring on the timescales of global-mean sea-level rise , rather than on shorter timescales related to meteorological, seasonal and interannual fluctuations. Our definition of the geoid treats it as a geophysical quantity which changes as the Earth system evolves. In some applications, the geoid is defined in a time-independent way as a particular geopotential surface within a particular model of the Earth’s gravity field.
We define the geoid to include the permanent ocean tide. With this choice, time-mean ocean dynamic sea level 𝜁 is determined solely by ocean dynamics and density. With the zero-tide convention, which is common in gravity-field models, 𝜁 would include the permanent ocean tide, which is almost +0.1 m at the equator and −0.2 m at the poles.
We define the geoid as 𝐺(𝐫)=𝐸(𝐫,Φ𝐺), with a choice of Φ𝐺 such that Eq. (4) is satisfied, where 𝐸(𝐫,Φ) is the geodetic height of the equipotential surface for geopotential Φ. The shapes of the equipotential surfaces, including the geoid, depend on the geographical distribution of mass over the Earth. According to Eq. (4), the global-mean geoid height must change by
1𝐴∫Δ𝐺 d𝐴 = 1𝐴∫Δ𝐹 d𝐴 + Δ𝑉 𝐴 (6)
if there is global-mean vertical land movement Δ𝐹 affecting the sea floor, or a change Δ𝑉 in the volume of the global ocean, whether due to change in density or in mass. Consequently, Φ𝐺 must change such that


∫𝜂 d𝐴 = ∫𝐺 d𝐴, (5)
i.e., MSL and geoid height above the reference ellipsoid have equal global means.
We define the geoid in terms of MSL 𝜂, rather than the sea-surface height 𝜂̃, in order to restrict changes in G and V to those occurring on the timescales of global-mean sea-level rise , rather than on shorter timescales related to meteorological, seasonal and interannual fluctuations. Our definition of the geoid treats it as a geophysical quantity which changes as the Earth system evolves. In some applications, the geoid is defined in a time-independent way as a particular geopotential surface within a particular model of the Earth’s gravity field.
We define the geoid to include the permanent ocean tide. With this choice, time-mean ocean dynamic sea level 𝜁 is determined solely by ocean dynamics and density. With the zero-tide convention, which is common in gravity-field models, 𝜁 would include the permanent ocean tide, which is almost +0.1 m at the equator and −0.2 m at the poles.
We define the geoid as 𝐺(𝐫)=𝐸(𝐫,Φ𝐺), with a choice of Φ𝐺 such that Eq. (4) is satisfied, where 𝐸(𝐫,Φ) is the geodetic height of the equipotential surface for geopotential Φ. The shapes of the equipotential surfaces, including the geoid, depend on the geographical distribution of mass over the Earth. According to Eq. (4), the global-mean geoid height must change by
1𝐴∫Δ𝐺 d𝐴 = 1𝐴∫Δ𝐹 d𝐴 + Δ𝑉 𝐴 (6)
if there is global-mean vertical land movement Δ𝐹 affecting the sea floor, or a change Δ𝑉 in the volume of the global ocean, whether due to change in density or in mass. Consequently, Φ𝐺 must change such that
1𝐴∫Δ𝐺 d𝐴 = ΔΦ 𝐺 𝐴∫∂𝐸(𝐫,Φ) ∂Φd𝐴 = ΔΦ 𝐺 𝑔, (7)
if we approximate g as globally uniform.
Variations and Differences in Surfaces
In this section we define terms for time-dependent variations in surfaces (on timescales shorter than those of mean sea-level change) and differences between surfaces.
N6 Tides: Periodic motions within the ocean, atmosphere and solid Earth due to the rotation of the Earth and its motion relative to the moon and sun. Ocean tides cause the  sea surface to rise and fall.
The astronomical tide is the dominant constituent of the ocean tides. It is caused by periodic spatial variations in local gravity. Tidal motion of the land surface and sea floor is due to elastic deformation of the solid Earth by gravitational tidal forces. The diurnal and annual cycles of insolation produce periodic variations in atmospheric pressure and winds (sea breezes), which cause the radiational tide in the atmosphere and ocean (e.g., Williams et al. 2018). The predicted tide is the sum of the astronomical and radiational constituents. Because the ocean and atmosphere are fluids, tidal forces within them cause tidal currents as well as displacements.
Sea-surface height (SSH) can be greatly elevated during a storm by a storm surge , and the consequent extreme sea level is sometimes called a storm tide. The tidal height is the vertical distance of the SSH due to the predicted tide above a local benchmark or a surface which is fixed with respect to the terrestrial reference frame. Often, this surface is a tidal datum, defined by a extremum of the periodic tide, such as mean lower-low water level.
The velocity of the ocean tidal currents depends on water depth. Therefore, relative sea-level change (RSLC) affects the tides. In most coastal locations, this interaction alters the tidal variations of the sea surface with respect to mean sea level by less than 10% of the RSLC (Pickering et al. 2017).
In most locations, the constituent of the ocean tide with the largest amplitude is the lunar semi-diurnal tide. The orbit of the moon around the Earth modulates the semi-diurnal tide to produce a large amplitude (spring tide) at new and full moon, and a small amplitude (neap tide) at half-moon. There are many smaller periodic constituents associated with the sun and moon. The precession of the plane of the moon’s orbit causes tidal variations with an 18.6-year cycle (the nodal period), affecting extreme sea level on this timescale. There are longer tidal periods.
The pole tide is caused by variations of the Earth’s rotation axis relative to the solid Earth, altering the centrifugal acceleration and local gravity. The two largest components of the pole tide have periods of 1 year and about 433 days. The latter is due to the Chandler wobble, which is not strictly periodic and arises from the mechanics of the Earth’s rotation alone (it is a free nutation), rather than being caused by the gravitation of other bodies in the solar system.
The time-means of the tidal forces of the moon and sun are nonzero. Hence, in addition to the periodic constituents, the tides have a constant constituent called the permanent tide, which tends to make the Earth and sea surface more oblate. Our definitions of mean sea level and the geoid use the mean-tide convention, including the permanent tide. In gravity models, the zero-tide convention is more usual, in which the permanent ocean tide is subtracted, but the permanent elastic tidal deformation of the solid Earth is retained; an estimate of the latter too is subtracted in the tide-free convention used by GNSS measurements.
N7 Inverse barometer (IB)𝐵: The time-dependent hydrostatic depression of the sea surface by atmospheric pressure variations, also called inverted barometer.
The ocean is almost incompressible. (A uniform change of 1 hPa over the ocean causes a global-mean sea-level rise of roughly 0.16 mm.) Therefore changes in atmospheric pressure have a negligible effect on the total volume of the ocean. However, they do move sea water around, and the effect on the sea surface depends on the deviation of sea-level pressure 𝑝̃𝑎(𝐫) from its global (ocean) mean, given by
𝑝̃′𝑎(𝐫,𝑡) = 𝑝̃𝑎(𝐫,𝑡) − 1𝐴∫𝑝̃𝑎(𝐫,𝑡) d𝐴. (8)
For timescales longer than a few days, we can assume the ocean to be in hydrostatic balance. Therefore, the depression of the sea-surface height (SSH) 𝜂̃ by IB is 𝐵̃=𝑝̃′𝑎/(𝑔𝜌s) where 𝑔(𝐫) is the acceleration due to gravity and 𝜌s(𝐫,𝜂̃) the surface sea-water density. That is, when 𝑝̃′𝑎>0 then sea level is depressed locally by 𝐵̃(𝐫), and it is raised when 𝑝̃′𝑎<0. The latter effect is an important contribution to storm surge . In a storm or cyclone, 𝑝̃𝑎 may fall by several 10 hPa, causing SSH to rise by several 100 mm.
The global mean of 𝑔𝜌s is approximately 9.9×10−5 m  Pa −1≡9.9 mm  hPa −1, with spatial and temporal variations of about 1% around this value. Hence, for most purposes of sea-level studies we can neglect the spatial variations in g and 𝜌s, and replace them with constants; thus,
𝐵̃=𝑝̃′𝑎𝑔𝜌∗. (9)
Hence, the global-mean IB correction is zero,
∫𝐵̃(𝐫,𝑡)d𝐴=0, (10)
which follows by definition of 𝑝̃′𝑎.


𝑝̃′𝑎(𝐫,𝑡) = 𝑝̃𝑎(𝐫,𝑡) − 1𝐴∫𝑝̃𝑎(𝐫,𝑡) d𝐴. (8)
For timescales longer than a few days, we can assume the ocean to be in hydrostatic balance. Therefore, the depression of the sea-surface height (SSH) 𝜂̃ by IB is 𝐵̃=𝑝̃′𝑎/(𝑔𝜌s) where 𝑔(𝐫) is the acceleration due to gravity and 𝜌s(𝐫,𝜂̃) the surface sea-water density. That is, when 𝑝̃′𝑎>0 then sea level is depressed locally by 𝐵̃(𝐫), and it is raised when 𝑝̃′𝑎<0. The latter effect is an important contribution to storm surge . In a storm or cyclone, 𝑝̃𝑎 may fall by several 10 hPa, causing SSH to rise by several 100 mm.
The global mean of 𝑔𝜌s is approximately 9.9×10−5 m  Pa −1≡9.9 mm  hPa −1, with spatial and temporal variations of about 1% around this value. Hence, for most purposes of sea-level studies we can neglect the spatial variations in g and 𝜌s, and replace them with constants; thus,
𝐵̃=𝑝̃′𝑎𝑔𝜌∗. (9)
Hence, the global-mean IB correction is zero,
∫𝐵̃(𝐫,𝑡)d𝐴=0, (10)
which follows by definition of 𝑝̃′𝑎.
The inverse-barometer response of the sea surface compensates for the effect of 𝑝̃′𝑎 on hydrostatic pressure within the ocean, and the subsurface ocean does not feel the fluctuations in atmospheric pressure. Consequently, the ocean behaves dynamically as if the sea-surface height were 𝜂̃+𝐵̃, which is called IB-corrected sea-surface height. Its time-mean is 𝜂+𝐵, the IB-corrected mean sea level. In most climate models, atmospheric pressure variations are not communicated to the ocean. In these models 𝐵̃ must be subtracted from the simulated SSH to produce a quantity that varies with 𝑝̃𝑎 like the observed 𝜂̃ does.
N8 Extreme sea level: The occurrence or the level of an exceptionally high or low local sea-surface height.
Extremely high sea-surface height (SSH) is caused by meteorological conditions as a storm surge , by sea-surface waves due to various causes and by exceptionally high or low (although predictable) tidal height. When considering coastal impacts, extreme sea level may be called extreme coastal water level. For decadal timescales, the main influence on changes in the frequency distribution of extreme sea level is relative sea-level change (RSLC), whose effect outweighs that of changes in meteorological forcing (Lowe et al. 2010; Church et al. 2013; Vousdoukas et al. 2018). To avoid confusion, we recommend the phrase high-end sea-level change to describe projections of very large RSLC, instead of using the word “extreme” for such projections.
N9 Storm surge: The elevation or depression of the sea surface with respect to the predicted tide during a storm.
Storm surges are caused during tropical cyclones and deep mid-latitude depressions by low atmospheric pressure, by strong winds pushing water towards the shore (or away from the shore, causing a negative surge) and by sea-surface waves breaking at the coast. Wave effects are usually excluded or underestimated by tide-gauges. If the actual sea-surface height (SSH) at location 𝐫 and time t due to tide and surge combined (sometimes called the storm tide) is 𝜂̃(𝐫,𝑡), and the predicted SSH due to the tide alone is 𝜂̃tide(𝐫,𝑡), the storm-surge heightσ is
σ(𝐫,𝑡)=𝜂̃(𝐫,𝑡)−𝜂̃tide(𝐫,𝑡), (11)
also called the surge residual or non-tidal residual.
The storm-surge height σ is the sum of three components: the inverse barometer (IB) effect of low atmospheric pressure, the wind setup caused by the wind-driven current, and the wave setup, which is the elevation of the sea surface due to breaking waves. All three effects are normally present, but intensified by storms. IB and wind setup tend to be more important on wide continental shelves, but wave setup can dominate in some cases (Pedreros et al. 2018), especially in areas of steep sea floor slope.
The swash is the uprush and backwash of water over the solid surface (e.g., sand or pebbles) generated by each wave. During the uprush, the swash extends above the wave setup. Its maximum height above the predicted tide, called the wave runup, gives the highest water level of the storm surge.
Particularly high SSH 𝜂̃=σ+𝜂̃tide occurs when the storm surge coincides with high tide. Without the meteorological forcing, storm-surge height σ would be zero, but since the tide level influences the propagation of the storm-forced signal, σ and 𝜂̃tide are not independent (Horsburgh and Wilson 2007).
The skew-surge height is the elevation of the highest sea surface that occurs within a single tidal cycle above the predicted level of the high tide within that cycle. If the actual SSH is 𝜂̃(𝐫,𝑡) and the predicted SSH due to the tide alone is 𝜂̃tide(𝐫,𝑡), the skew-surge height is
σ𝑘(𝐫,𝑡)= max 𝑡(𝜂̃(𝐫,𝑡))− max 𝑡(𝜂̃tide(𝐫,𝑡)), (12)
where  max 𝑡(𝑋(𝐫,𝑡)) means the maximum value of X that occurs at location 𝐫 during the interval of time t from one low tide to the next. For extreme-value analysis, the skew-surge height is preferable to the storm-surge height as a measure of the effect of the meteorological forcing alone in regions where skew-surge height is uncorrelated with tidal height (Williams et al. 2016).
N10 Sea-surface waves: Waves on the surface of the ocean, usually surface gravity waves caused by winds.
The amplitude of a wind wave depends on the strength of the wind, and the time and the distance of open ocean, called the fetch, over which the wind has blown. The sea surface typically exhibits a superposition of many waves of different amplitudes, velocities, frequencies and directions. A swell wave is a wind wave of low frequency which was generated far away.
A tsunami or seismic sea wave is an extreme sea level event caused by an earthquake, volcano, landslide or other submarine disturbance that suddenly displaces a volume of water. The displacement propagates as a long-wavelength surface gravity wave, but is not a tidal phenomenon, despite it sometimes being called a “tidal wave”.
The wave height is the vertical distance from the crest to the trough of a wave, respectively its highest and lowest points. The wave period is the interval of time between the passage of repeated features on the waveform such as crests, troughs or upward crossings of the mean level. The significant wave height is a statistic computed from wave measurements, defined as either the mean of the largest one-third of the wave heights, or four times the standard deviation of wave heights. (These statistics are approximately equal.) The significant wave period is the mean period of the largest one-third of the waves.
N11 Ocean dynamic sea level 𝜁: The local height of the sea surface above the geoid G, with the inverse barometer correction 𝐵 applied.
Instantaneous ocean dynamic sea level is defined by
𝜁̃(𝐫,𝑡)=𝜂̃(𝐫,𝑡)+𝐵̃(𝐫,𝑡)−𝐺(𝐫). (13)
It is determined jointly by ocean density and circulation. The time-mean ocean dynamic sea level is
𝜁(𝐫)=𝜂(𝐫)+𝐵(𝐫)−𝐺(𝐫), (14)
whose global mean
1𝐴∫𝜁(𝐫)d𝐴=0, (15)
in view of Eqs. (5) and (10).


N10 Sea-surface waves: Waves on the surface of the ocean, usually surface gravity waves caused by winds.
The amplitude of a wind wave depends on the strength of the wind, and the time and the distance of open ocean, called the fetch, over which the wind has blown. The sea surface typically exhibits a superposition of many waves of different amplitudes, velocities, frequencies and directions. A swell wave is a wind wave of low frequency which was generated far away.
A tsunami or seismic sea wave is an extreme sea level event caused by an earthquake, volcano, landslide or other submarine disturbance that suddenly displaces a volume of water. The displacement propagates as a long-wavelength surface gravity wave, but is not a tidal phenomenon, despite it sometimes being called a “tidal wave”.
The wave height is the vertical distance from the crest to the trough of a wave, respectively its highest and lowest points. The wave period is the interval of time between the passage of repeated features on the waveform such as crests, troughs or upward crossings of the mean level. The significant wave height is a statistic computed from wave measurements, defined as either the mean of the largest one-third of the wave heights, or four times the standard deviation of wave heights. (These statistics are approximately equal.) The significant wave period is the mean period of the largest one-third of the waves.
N11 Ocean dynamic sea level 𝜁: The local height of the sea surface above the geoid G, with the inverse barometer correction 𝐵 applied.
Instantaneous ocean dynamic sea level is defined by
𝜁̃(𝐫,𝑡)=𝜂̃(𝐫,𝑡)+𝐵̃(𝐫,𝑡)−𝐺(𝐫). (13)
It is determined jointly by ocean density and circulation. The time-mean ocean dynamic sea level is
𝜁(𝐫)=𝜂(𝐫)+𝐵(𝐫)−𝐺(𝐫), (14)
whose global mean
1𝐴∫𝜁(𝐫)d𝐴=0, (15)
in view of Eqs. (5) and (10).
In the Coupled Model Intercomparison Project (CMIP), 𝜁̃ is stored in the diagnostic named zos, which is defined to have zero mean (Equation H14 of Griffies et al. 2016). However, some models supply it with a nonzero time-dependent mean. If the global mean of the zos diagnostic is found to be nonzero, the global mean should be subtracted uniformly.
N12 Ocean dynamic topography: An estimate of ocean dynamic sea level computed from the ocean density structure above a reference level where the velocity is either known or assumed to be zero.
On any horizontal (see geoid for definition) level z within the ocean, the hydrostatic pressure is given by
𝑝̃(𝑧) = 𝑝̃𝑎 + 𝑔∫𝜁̃−𝐵̃𝑧𝜌̃(𝑧′)d𝑧′, (16)
which is the sum of the atmospheric pressure 𝑝̃𝑎 on the sea surface and the weight per unit horizontal area of sea water between z and the sea surface. The coordinate of the sea surface in this case is not 𝜂̃ but 𝜂̃−𝐺=𝜁̃−𝐵̃ by Eq. (13), the height of the sea surface above the geoid, because we are using the orthometric vertical coordinate z, which is the natural choice for ocean dynamics. Equation (9) leads to the horizontal gradient of the atmospheric pressure ∇𝑝̃𝑎 = ∇𝑝̃′𝑎 = 𝑔𝜌∗∇𝐵̃. Consequently, the horizontal gradient of pressure within the ocean is given by
∇𝑝̃ = ∇(𝑝̃𝑎 + 𝑔∫𝜁̃−𝐵̃𝑧𝜌̃(𝑧′)d𝑧′) (17a)
= 𝑔𝜌∗∇∇𝐵̃+(𝑔𝜌∗∇∇𝜁̃−𝑔𝜌∗∇∇𝐵̃)+𝑔∫𝜁̃−𝐵̃𝑧∇∇𝜌̃(𝑧′)d𝑧′ (17b)
= 𝑔𝜌∗∇∇𝜁̃+𝑔∫𝜁̃−𝐵̃𝑧∇∇𝜌̃(𝑧′)d𝑧′. (17c)
In the first step of this derivation, we used Eq. (9) for the inverse barometer correction 𝐵̃, approximated g and sea-surface 𝜌̃=𝜌∗ as constants, and applied Leibniz’s rule to differentiate the integral, which yields the two terms in parentheses in Eq. (17b), but no term for 𝜌̃ at z because ∇∇𝑧=0. From Eq. (17c) we obtain
∇𝜁̃(𝐫) = 1𝑔𝜌∗∇𝑝̃(𝐫,𝑧) − 1𝜌∗∫𝜁̃−𝐵̃𝑧∇𝜌̃(𝐫,𝑧′)d𝑧′, (18)
which relates the horizontal gradient of ocean dynamic sea level 𝜁̃ to the horizontal hydrostatic pressure gradient at a reference level z and the horizontal density gradient above that level. The second term on the right-hand side is the horizontal gradient of the dynamic heightD
𝐷 = −1𝜌∗∫𝜁̃−𝐵̃𝑧𝜌̃𝑑𝑧′ (19)
of the sea surface relative to z.
In much of the ocean interior (below the boundary layer and away from coastal and other strong currents), and taking a time-mean sufficient to eliminate tidal currents, geostrophy is a reasonable approximation, meaning that there is a balance (no net acceleration) between the pressure gradient and Coriolis forces, and all other forces are negligible. Therefore, 𝐯̃≃𝐯̃g, with the geostrophic velocity 𝐯̃g defined by
𝑓𝐤×𝜌̃𝐯̃g=−∇∇𝑝̃, (20)
where f is the Coriolis parameter and 𝐤 the vertical unit vector. If we can measure 𝐯̃ at some z and assume it is geostrophic, we arrive at
∇𝜁̃(𝐫)=−𝑓𝑔𝜌∗𝐤×𝜌̃𝐯̃(𝐫,𝑧)−1𝜌∗∫𝜁̃−𝐵̃𝑧∇𝜌̃(𝐫,𝑧′)d𝑧′, (21)
from Eq. (18).
Alternatively, if we do not know 𝐯̃ at any z, we assume there exists a level of no motion𝑧=−𝐿, which is a geopotential (horizontal) surface on which 𝐯̃=𝐯̃g=0, requiring the horizontal hydrostatic pressure gradient to vanish (∇∇𝑝̃=0) by Eq. (20). Therefore,
∇𝜁̃(𝐫)=−1𝜌∗∫𝜁̃ − 𝐵̃ − 𝐿∇∇𝜌̃(𝐫,𝑧)d𝑧, (22)


𝐷 = −1𝜌∗∫𝜁̃−𝐵̃𝑧𝜌̃𝑑𝑧′ (19)
of the sea surface relative to z.
In much of the ocean interior (below the boundary layer and away from coastal and other strong currents), and taking a time-mean sufficient to eliminate tidal currents, geostrophy is a reasonable approximation, meaning that there is a balance (no net acceleration) between the pressure gradient and Coriolis forces, and all other forces are negligible. Therefore, 𝐯̃≃𝐯̃g, with the geostrophic velocity 𝐯̃g defined by
𝑓𝐤×𝜌̃𝐯̃g=−∇∇𝑝̃, (20)
where f is the Coriolis parameter and 𝐤 the vertical unit vector. If we can measure 𝐯̃ at some z and assume it is geostrophic, we arrive at
∇𝜁̃(𝐫)=−𝑓𝑔𝜌∗𝐤×𝜌̃𝐯̃(𝐫,𝑧)−1𝜌∗∫𝜁̃−𝐵̃𝑧∇𝜌̃(𝐫,𝑧′)d𝑧′, (21)
from Eq. (18).
Alternatively, if we do not know 𝐯̃ at any z, we assume there exists a level of no motion𝑧=−𝐿, which is a geopotential (horizontal) surface on which 𝐯̃=𝐯̃g=0, requiring the horizontal hydrostatic pressure gradient to vanish (∇∇𝑝̃=0) by Eq. (20). Therefore,
∇𝜁̃(𝐫)=−1𝜌∗∫𝜁̃ − 𝐵̃ − 𝐿∇∇𝜌̃(𝐫,𝑧)d𝑧, (22)
using Eq. (18). There is no motion on 𝑧=−𝐿 so long as there is a compensation between undulations of dynamic sea level 𝜁̃ (on the left-hand side of Eq. 22), and variations of the density structure above 𝑧=−𝐿 (on the right-hand side). Such exact compensation does not generally occur in the ocean, and the level of no motion does not exist. However, it is a useful approximation in many situations. For example, an anomalous sea-surface high is associated with a depression of the pycnocline in the interior of subtropical gyres (e.g., Figure 3.3 of Tomczak and Godfrey 1994), thus leading to relatively weak flow beneath the pycnocline. In some regions, the approximation is not useful; in particular, sizeable 𝐯̃ occurs at all depths in the Southern Ocean.
The ocean dynamic topography is the estimate of ocean dynamic sea level made using Eqs. (21) or (22). Since Eqs. (21) and (22) are unaffected by adding a constant to 𝜁̃, the method provides only the difference in 𝜁̃ between any two points (i.e., the gradient); it cannot give 𝜁̃ for individual points relative to the geoid. Furthermore, it is not applicable in regions where the reference level for motion is below the sea floor , nor for differences between points in basins which are separated by sills that are shallower than the reference level.
Changes in Sea Level
The relationships between quantities determining changes in sea level are summarized in Fig. 3. The phrases “sea-level change” (SLC) and “sea-level rise” (SLR) are often used in the literature. These make sense when referring to the phenomenon in general, but more specific terms such as relative sea-level change and global-mean sea-level rise should be preferred where relevant.
Fig. 3: Relationships between quantities, defined in Sect. 5, that determine changes in sea level. The lengths of the arrows do not have any significance—they are only illustrative—and the dotted horizontal lines serve only to indicate alignment. All of the quantities are differences between two states, and all except h, ℎ𝜃 and ℎ𝑏 are functions of location 𝐫. Any closed circuit gives an equality, in which a term has a positive sign when traversed in the direction of its arrow, and a negative sign if in the opposite direction to its arrow. For example, Δ𝑅−Δ𝜂+Δ𝐹=0 (Eq. 24) is the circuit marked in red, Eq. (23) in orange, Eq. (38) in blue, and Eq. (54) in green
N13 Ocean dynamic sea-level change Δ𝜁:
The change in time-mean ocean dynamic sea level, i.e., the change in IB-corrected mean sea level relative to the geoid.
For the difference between two time-mean states of the climate, Eq. (13) gives
Δ𝜁(𝐫) = Δ𝜂(𝐫) + Δ𝐵(𝐫) − Δ𝐺(𝐫). (23)
Since the time-mean ocean dynamic sea level 𝜁(𝐫) always has a zero global mean by Eq. (15), so does Δ𝜁 i.e., global-mean sea-level rise is excluded from ocean dynamic sea-level change. This property depends on Eq. (5) and thus requires a different choice of geopotential to define the geoid in the two states, if there is any change in global-mean sea level.
N14 Geocentric sea-level change Δ𝜂:
The change in local mean sea level with respect to the terrestrial reference frame.
Geocentric sea-level change is the change in the height 𝜂(𝐫) of MSL relative to the reference ellipsoid . IB-corrected geocentric sea-level change is Δ𝜂 + Δ𝐵, i.e., the same with the inverse barometer correction added. Geocentric sea-level change must be distinguished from relative sea-level change .
N15 Relative sea-level change (RSLC) Δ𝑅: The change in local mean sea level relative to the local solid surface, i.e., the  sea floor . Relative sea-level change is also called “relative sea-level rise” (RSLR). (See Sect. 6 for an exposition of the relationship of RSLC to other quantities.)
Both the MSL height 𝜂 and the sea floor height 𝐹 may change and thus alter RSL. Hence, RSLC is geodetically expressed as
Δ𝑅(𝐫) = Δ𝜂(𝐫) − Δ𝐹(𝐫), (24)
the difference between geocentric sea-level change Δ𝜂 and vertical land movement Δ𝐹 (VLM). IB-corrected relative sea-level change is Δ𝑅+Δ𝐵, i.e., RSLC with the inverse barometer correction. Relative sea-level change is the quantity registered by a tide-gauge, which measures sea level relative to the solid surface where it is attached.
Since climate models do not include VLM, they do not distinguish between geocentric and relative sea-level change. In climate models where atmospheric pressure changes Δ𝑝𝑎 are not applied to the ocean, −Δ𝐵 must be added to include the effect of Δ𝑝𝑎 simulated by the atmosphere model. (Note that this adjustment should not be made to ocean dynamic sea-level change Δ𝜁, which by definition is IB-corrected; see Eq. 23.)
The term “relative sea level” is not employed in an absolute sense, but only in conjunction with “change”, because 𝜂−𝐹 (the analogue of Eq. 24) is simply the depth of the sea floor below MSL, equal to the time-mean thickness of the ocean 𝐻 (Eq. 3).
In view of Eq. (3), we may also write RSLC as
Δ𝑅(𝐫)=Δ𝐻(𝐫), (25)
i.e., the change in local ocean thickness, making it obvious that RSLC is not meaningful at locations which change from land to sea (transgression) or vice versa (regression), since 𝐻 is undefined on land.


Both the MSL height 𝜂 and the sea floor height 𝐹 may change and thus alter RSL. Hence, RSLC is geodetically expressed as
Δ𝑅(𝐫) = Δ𝜂(𝐫) − Δ𝐹(𝐫), (24)
the difference between geocentric sea-level change Δ𝜂 and vertical land movement Δ𝐹 (VLM). IB-corrected relative sea-level change is Δ𝑅+Δ𝐵, i.e., RSLC with the inverse barometer correction. Relative sea-level change is the quantity registered by a tide-gauge, which measures sea level relative to the solid surface where it is attached.
Since climate models do not include VLM, they do not distinguish between geocentric and relative sea-level change. In climate models where atmospheric pressure changes Δ𝑝𝑎 are not applied to the ocean, −Δ𝐵 must be added to include the effect of Δ𝑝𝑎 simulated by the atmosphere model. (Note that this adjustment should not be made to ocean dynamic sea-level change Δ𝜁, which by definition is IB-corrected; see Eq. 23.)
The term “relative sea level” is not employed in an absolute sense, but only in conjunction with “change”, because 𝜂−𝐹 (the analogue of Eq. 24) is simply the depth of the sea floor below MSL, equal to the time-mean thickness of the ocean 𝐻 (Eq. 3).
In view of Eq. (3), we may also write RSLC as
Δ𝑅(𝐫)=Δ𝐻(𝐫), (25)
i.e., the change in local ocean thickness, making it obvious that RSLC is not meaningful at locations which change from land to sea (transgression) or vice versa (regression), since 𝐻 is undefined on land.
When considering sea-level change on geological timescales, in the absence of information about ocean dynamic sea level 𝜁̃ or the inverse barometer effect, we might approximate Δ𝜁̃≃0 and Δ𝐵≃0, in which case Δ𝜂≃Δ𝐺 from Eq. (23), and Δ𝑅≃Δ𝐺−Δ𝐹 from Eq. (24). This quantity is defined everywhere and thus gives an approximate meaning to RSLC in regions of transgression and regression.
N16 Steric sea-level changeΔ𝑅𝜌: The part of  relative sea-level change   which is due to the change Δ𝜌 in ocean density, assuming the local mass of the ocean per unit area does not change. It is composed of thermosteric sea-level change Δ𝑅𝜃, which is the part due solely to the change Δ𝜃 in in-situ temperature, and halosteric sea-level changeΔ𝑅𝑆, which is the part due solely to the change Δ𝑆 in salinity.
The time-mean local mass of the ocean per unit area is
𝑚 = ∫𝜂𝐹 𝜌 d𝑧 = 𝐻 ρ̄ with ρ̄ ≡ 1𝐻 ∫𝜂𝐹 𝜌 d𝑧, (26)
where the first factor 𝐻 = 𝜂 − 𝐹 is the local time-mean thickness of the ocean (Eq. 3), and the second factor ρ̄ is the local vertical-mean time-mean density. If we change the density while keeping 𝑚 fixed, the thickness of the ocean changes, because
0 = Δ𝑚 = Δ𝐻|𝑚ρ̄ + H Δρ̄∣𝑚 (27)
(by making a linear approximation). Therefore,
Δ𝐻|𝑚 =−(𝐻/ρ̄)Δρ̄ with Δρ̄ = 1𝐻 ∫𝜂𝐹Δ𝜌d𝑧. (28)
This would exactly define steric sea-level change in a situation where mass did not move horizontally. But in reality there are horizontal transports, making it impossible to separate density changes due to local changes in properties from those due to the movement of mass. For convenience, we approximate ρ̄ with the constant 𝜌∗.
Since the RSLC is given by Δ𝑅 = Δ𝐻 (without inverse barometer correction, Eq. 25), steric sea-level change is
Δ𝑅𝜌 = −1𝜌∗∫𝜂𝐹 Δ𝜌d𝑧 = Δ𝑅𝜃 + Δ𝑅𝑆 (29)
with the density increment decomposed into thermal and haline components (by making a linear approximation)
Δ𝜌 = ∂𝜌∂𝜃Δ𝜃 + ∂𝜌∂𝑆Δ𝑆, (30)
and with the corresponding thermosteric and halosteric contributions
Δ𝑅𝜃 = −1𝜌∗∫𝜂𝐹∂𝜌∂𝜃Δ𝜃d𝑧Δ𝑅𝑆 = −1𝜌∗∫𝜂𝐹∂𝜌∂𝑆Δ𝑆d𝑧. (31)
Thermosteric sea-level change is often called thermal expansion, because ∂𝜌/∂𝜃<0, so increasing the temperature gives Δ𝑅𝜃>0. (See Appendix 2 regarding the dependence of density on salinity.) Relative sea-level change (without inverse barometer correction) is the sum of steric and manometric sea-level change (Eq. 35).
N17 Global-mean thermosteric sea-level riseℎ𝜃: The part of   global-mean sea-level rise   (GMSLR) which is due to thermal expansion.
This quantity is the global mean of local thermosteric sea-level change Δ𝑅𝜃 (due to temperature change, Eq. 31); thus,
ℎ𝜃 = 1𝐴∫Δ𝑅𝜃d𝐴 = −1𝜌∗𝐴∫∫𝜂𝐹∂𝜌∂𝜃Δ𝜃(𝐫,𝑧)d𝑧d𝐴. (32)
It is the change in global ocean volume due to change in temperature alone, divided by the ocean surface area. The CMIP variable zostoga is ℎ𝜃 calculated with respect to a fixed reference state. (Griffies et al. 2016 define the reference to be the initial state of the experiment for CMIP6.) Hence, differences in zostoga between two states give the global-mean thermosteric sea-level rise between those states.
Although halosteric sea-level change Δ𝑅𝑆 (due to salinity change, Eq. 31) can be locally of the same order of magnitude as thermosteric, global-mean halosteric sea-level change is practically zero. In Appendix 2 we detail the physical arguments leading to this conclusion. Salinity change should be excluded when calculating ℎ𝜃, to avoid including a spurious global-mean halosteric sea-level change. (See Appendix 2 here as well as Appendix H9.5 of Griffies et al. 2016.) However, salinity change must of course be included when calculating Δ𝑅𝑆. It follows that global-mean steric sea-level change, which equals ℎ𝜃 because global-mean halosteric sea-level change is zero, cannot be calculated as the global mean of local steric sea-level change. This apparent contradiction is due to the inaccuracy of the approximations made following Eq. (28).
N18 Manometric sea-level changeΔ𝑅𝑚: Definition A: The part of  relative sea-level change  (RSLC) which is not steric, or alternatively Definition B: The part of RSLC which is due to the change Δ𝑚(𝐫) in the time-mean local mass of the ocean per unit area, assuming the density does not change. In the following, we show that the two definitions are approximately the same.


ℎ𝜃 = 1𝐴∫Δ𝑅𝜃d𝐴 = −1𝜌∗𝐴∫∫𝜂𝐹∂𝜌∂𝜃Δ𝜃(𝐫,𝑧)d𝑧d𝐴. (32)
It is the change in global ocean volume due to change in temperature alone, divided by the ocean surface area. The CMIP variable zostoga is ℎ𝜃 calculated with respect to a fixed reference state. (Griffies et al. 2016 define the reference to be the initial state of the experiment for CMIP6.) Hence, differences in zostoga between two states give the global-mean thermosteric sea-level rise between those states.
Although halosteric sea-level change Δ𝑅𝑆 (due to salinity change, Eq. 31) can be locally of the same order of magnitude as thermosteric, global-mean halosteric sea-level change is practically zero. In Appendix 2 we detail the physical arguments leading to this conclusion. Salinity change should be excluded when calculating ℎ𝜃, to avoid including a spurious global-mean halosteric sea-level change. (See Appendix 2 here as well as Appendix H9.5 of Griffies et al. 2016.) However, salinity change must of course be included when calculating Δ𝑅𝑆. It follows that global-mean steric sea-level change, which equals ℎ𝜃 because global-mean halosteric sea-level change is zero, cannot be calculated as the global mean of local steric sea-level change. This apparent contradiction is due to the inaccuracy of the approximations made following Eq. (28).
N18 Manometric sea-level changeΔ𝑅𝑚: Definition A: The part of  relative sea-level change  (RSLC) which is not steric, or alternatively Definition B: The part of RSLC which is due to the change Δ𝑚(𝐫) in the time-mean local mass of the ocean per unit area, assuming the density does not change. In the following, we show that the two definitions are approximately the same.
If we change the local mass 𝑚 per unit area while keeping density fixed, by Eq. (26) the thickness of the ocean changes by Δ𝐻|𝜌=Δ𝑚/𝜌⎯⎯⎯, where 𝜌⎯⎯⎯ is the local vertical mean of 𝜌. (In reality, if the local mass per unit area changes, the density will probably change as well, since mass which is converging horizontally or through the sea surface is unlikely to have 𝜌=𝜌⎯⎯⎯ exactly.)
Since RSLC Δ𝑅=Δ𝐻 (without inverse barometer correction, Eq. 25), if we approximate 𝜌⎯⎯⎯ with the constant 𝜌∗, we obtain
Δ𝑅𝑚 ≃ Δ𝑚𝜌∗. (33)
This is Definition B of “manometric sea-level change”. The local change in mass Δ𝑚 can be estimated from the gravity field, or from the bottom pressure𝑝𝑏, i.e., the hydrostatic pressure at the sea floor , according to Δ𝑚=(Δ𝑝𝑏−Δ𝑝𝑎)/𝑔. (See Appendix 3.) Because of its relationship to 𝑝𝑏, manometric sea-level change is sometimes referred to as the “bottom pressure term” in sea-level change.
According to Definition B, the global mean of Δ𝑅𝑚 vanishes if the mass of the global ocean is constant, since 1/(𝐴𝜌∗)∫Δ𝑚d𝐴=0. However, Δ𝑅𝑚 may still be locally nonzero, due to rearrangement of the existing mass of the ocean. If the mass of the global ocean changes, the global mean of Δ𝑅𝑚 is nonzero and equals the barystatic sea-level rise (equality is approximate with Definition B of Δ𝑅𝑚, exact with Definition A), which is part of global-mean sea-level rise (GMSLR). Despite the correspondence between (local) manometric and (global) barystatic sea-level rise, we argue that these two concepts are sufficiently different to need distinct terms. (See the subsection for barystatic sea-level rise.)
If mass and density are both allowed to change, Eq. (26) gives
Δ𝑚 = Δ𝐻 ρ̄ + 𝐻 Δρ̄ => Δ𝐻 = (1/ρ̄ )(Δ𝑚 − ∫𝜂𝐹Δ𝜌d𝑧), (34)
using the expression for Δρ̄ from Eq. (28). Again approximating ρ̄ as 𝜌∗ and substituting from Eqs. (25), (29) and (33), we obtain
Δ𝑅 = Δ𝐻 ≃ Δ𝑅𝜌 + Δ𝑅𝑚, (35)
i.e., RSLC (without inverse barometer correction) is the sum of steric sea-level change and manometric sea-level change, which are, respectively, the parts due to change in density and change in mass per unit area. Since 𝐻 is defined only in ocean areas, the formulae are not valid for locations which change from land to sea or vice versa.
With Δ𝑅𝑚 defined by Eq. (33), Eq. (35) is only approximate, because of the replacement of 𝜌⎯⎯⎯ with 𝜌∗. We can make Eq. (35) exact if we retain the definition of Eq. (29) for steric sea-level change involving 𝜌∗ and adopt Definition A of “manometric sea-level change”, as
Δ𝑅𝑚 ≡ Δ𝑅 − Δ𝑅𝜌, (36)
i.e., Δ𝑅𝑚 is the part of RSLC that is not steric.
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)