Surveys in Geophysics
https://doi.org/10.1007/s10712-019-09525-z(0123456789().,-volV)(0123456789().,-volV)

Concepts and Terminology for Sea Level: Mean, Variability
and Change, Both Local and Global
Jonathan M. Gregory1,2 • Stephen M. Grifﬁes3 • Chris W. Hughes4 •Jason A. Lowe2,14 • John A. Church5 • Ichiro Fukimori6 • Natalya Gomez7 •Robert E. Kopp8 • Felix Landerer6 • Gonéri Le Cozannet9 • Rui M. Ponte10 •Detlef Stammer11 • Mark E. Tamisiea12 • Roderik S. W. van de Wal13
Received: 1 October 2018 / Accepted: 28 February 2019Ó The Author(s) 2019

Abstract
Changes in sea level lead to some of the most severe impacts of anthropogenic climatechange. Consequently, they are a subject of great interest in both scientific research andpublic policy. This paper defines concepts and terminology associated with sea level andsea-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 andacross 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.
Keywords Sea level  Concepts  Terminology

1 Introduction and Motivation
Changes in sea level lead to some of the most severe impacts of anthropogenic climatechange (IPCC 2014). Consequently, they are a subject of great interest in both scientificresearch and public policy. Since changes in sea level are the result of diverse physicalphenomena, 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 acrossdisciplines (for example, ‘‘dynamic sea level’’ has different meanings in oceanography andgeodynamics), as well as unclear or ambiguous even within a single discipline. (Forinstance, ‘‘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.
& Jonathan M. Gregory
j.m.gregory@reading.ac.uk
Extended author information available on the last page of the article

123

Surveys in Geophysics

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 isambiguity. 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 thedefinitions, 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 differencesin 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 wehave defined. To facilitate sequential reading of this paper, the material of Sects. 3–7 isarranged 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 listin 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 anddiscussions in this paper. It may therefore be helpful to keep in mind that from the point ofview 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 becauseof the enormous damage it can cause to human populations and their built environment andto ecosystems. The expected occurrence of extreme sea level under future climates istherefore 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 toafford a constant level of protection against extremes, and low-lying areas are threatenedwith 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 percent 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 inmind.
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 riskintolerant 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 projectionsof extreme sea level.

2 Conventions and Assumptions
We here summarize the conventions and assumptions employed in the text and formulae ofour definitions.

123Surveys in Geophysics

2.1 First Appearance of Terms
Within each definition subsection of Sects. 3–5, we use bold font for the first appearance ofa term whose definition is the subject of another subsection. When we first define a termthat 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.

2.2 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 andlonger-term internally generated climate variability (e.g., El Niño and the Interdecadal
Pacific Oscillation), anthropogenic climate change and naturally forced changes (e.g., byvolcanic 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 timemean state and fluctuations within that state. The definition of a ‘‘state’’ depends on thescientific interest or application. For sea level, the state might be defined by a time-meanlong 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 bespecified when the term is used, and which is understood to be long enough to eliminate theeffect of meteorological variations at least. We use symbols with a tilde and time-de~ for time-varying quantities, and symbols without any distinguishingpendence, e.g., XðtÞmark and no time-dependence, e.g., X for time-mean quantities that characterize the stateof the system. On longer timescales, the state itself may change, for example, due toanthropogenic influence. We use the symbol D and the word ‘‘change’’ to refer to thedifference between any two states; thus, DX 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 otherinfluences too, such as impoundment of water on land in reservoirs.

2.3 Local and Regional
By a local quantity, we mean one which is a function of two-dimensional geographicallocation r, specified by latitude and longitude. For some applications, it is important toconsider 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.

2.4 Global Mean Over the Ocean Surface Area
By global mean, we mean the area-weighted mean over the entire connected surface areaof the ocean, i.e., excluding the land. The ocean includes marginal seas connected to theopen ocean such as the Mediterranean Sea, Black Sea and Hudson Bay, but excludes inland123Surveys in Geophysics

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 isneeded 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, withA ¼ 3:625  1014 m 2 (Cogley 2012). It is altered substantially by global-mean sea-levelchanges 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.

2.5 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 arepresentative value q (e.g., 1028 kg m 3 ). For the density of freshwater added at thesea surface we use a constant qf ¼ 1000 kg m 3 for convenience, neglecting thevariation of \1% in freshwater density due to temperature.

2.6 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 withhorizontal and vertical distances measured relative to the reference ellipsoid, which is asurface fixed with respect to the solid Earth. Geophysical fluid dynamics, including ocean
circulation dynamics, is concerned with horizontal distances on surfaces of constantgeopotential, and vertical distances measured perpendicular to such surfaces, especially thegeoid. We discuss the two frames of reference (one relative to the reference ellipsoid andthe 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 XðrÞ, forwhich 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 ofreference frame (with respect either to the reference ellipsoid or to the geoid) does notaffect the geophysical definition of a surface, but the numerical value of its verticalcoordinate 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 YðrÞ  XðrÞ between two surfaces, or the change inheight DXðrÞ of a surface.

123Surveys in Geophysics

3 Surfaces
We here define five key surfaces used in sea-level studies. The reference ellipsoid and theassociated terrestrial reference frame (depicted in Fig. 1) are geometrical constructions,chosen by convention. The other four surfaces (compared with the reference ellipsoid inFig. 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.
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 ina 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 portionsof 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 thereference ellipsoid.
1. Its centre is the time-mean centre of mass of the Earth.
2. Its semi-major axis lies in the equatorial plane and its semi-minor axis along therotation (polar) axis of the Earth.
3. Its axis of revolution is the rotation axis.

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 ofan arbitrary point x in 3D space. The line between x and r is normal to the reference ellipsoid, on which rlies. The equator is intersected at p by the meridian through r, and at p0 by the prime meridian, which
defines the zero of longitude123

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Fig. 2 Relationship between surfaces relating to sea level. The normal to the reference ellipsoid defines thevertical 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 isnegligible in reality. The local vertical coordinates of mean sea level g, the geoid G and the sea floor F are
relative to the reference ellipsoid, while dynamic sea level f~ 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 p0a from its global mean causes the depression B in sea level by the inverse barometer
effect4. 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 ofchange 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 theabove 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, weconstruct a straight line that passes through x and is normal to the ellipsoid, which itintersects 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 formodels 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 theequatorial 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 tox. Geodetic and geocentric latitudes are the same for the poles and the equator, but123Surveys in Geophysics

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 theprime 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 thecentre of the Earth to p0 and to p.
N2 Sea surface g~: The time-varying upper boundary of the ocean. The sea-surface
height is the geodetic height of the sea surface above the reference ellipsoid (anegative 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-definedsea-surface height (SSH) g~ðr; tÞ, 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 g~ as the liquid-water equivalent sea surface g~LWE which theliquid would have if the ice were replaced by an equal mass of sea water of the density qs
of the surface water in its vicinity. Following Archimedes’ principle,
wi;
g~LWE ¼ g~s þ
ð1Þ
gqs
where g~s is the geodetic height of the liquid sea-water surface (beneath the ice) and wi isthe weight per unit area of floating ice. (The depression of g~s relative to g~ is called the‘‘inverse barometer effect of sea ice’’ by Griffies and Greatbatch 2012, and g~LWE is their‘‘effective sea level’’.) Although the liquid-water equivalent sea-surface height is notdirectly measurable, it is a convenient construct for many practical purposes of sea-levelstudies, and dynamically justifiable because the hydrostatic pressure and gravity beneaththe 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 oceanand 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) g: The time-mean of the sea surface.
The period for the time-mean must be long enough to eliminate the effects of waves andother 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 thetime-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 gðrÞ above the reference ellipsoid (a negativevalue if below). In ocean models which regard the geoid and the reference ellipsoid as123

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coincident, g 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 aclear 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 isalways or sometimes submerged under sea water. The level of the sea floor varies due tosolid-Earth tides, accumulation of sediment (with eventual compaction) and vertical land
movement on a range of timescales.
~ tÞ
We specify the instantaneous level of the sea floor by its geodetic height Fðr;(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 aship, a positive quantity sometimes called the depth of the water column) is given by~ tÞ ¼ g~ðr; tÞ  Fðr;~ tÞ  0;
Hðr;

ð2Þ

i.e., the vertical distance between the sea surface and the sea floor. The choice of reference~ because it is the difference
surface for vertical coordinates does not affect the value of H,~
between two vertical coordinates; H would be the same if SSH and the sea floor werelocated by heights relative to the geoid rather than the reference ellipsoid. The time-mean
thickness of the ocean HHðrÞ ¼ gðrÞ  FðrÞ  0

ð3Þ

is likewise related to MSL g.
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 arealso used as names for the quantities F, G  F and g  Fð¼ HÞ; i.e., the time-meanvertical 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 isintended, 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 geopotential U has a uniform value, chosen sothat the volume enclosed between the geoid and the sea floor is equal to the timemean volume of sea water in the ocean (including the liquid-water equivalent offloating 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 ahigher geopotential (shallower in the ocean). Note that this sign convention for thegeopotential is opposite to that used in geodesy.
The vertical gradient of the geopotential is equal to the local effective gravitational
acceleration, g, gðr; tÞ ¼ oU=oz, usually referred to as ‘‘gravitational acceleration’’ ingeophysical fluid dynamics. Hence, the effective gravitational acceleration is normal to thegeoid, because the geoid is an equipotential surface, i.e., one on which the geopotential isconstant. The Newtonian gravitational acceleration is time-dependent because the distribution of mass in the ocean (liquid and solid), on land (including the land-based123Surveys in Geophysics

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 antiparallel 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 ismeasured along a plumb line, which is always normal to equipotential surfaces, but thisdistance differs negligibly from that measured along the perpendicular to the reference
ellipsoid.
We define z such that z ¼ 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 asurface of constant geodetic height z0 ¼ z þ G, where GðrÞ is the geoid height above thereference 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 steadystate 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; withrespect 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 theinfinite set of them) which satisfies
ZZZðG  FÞ dA ¼ V ¼
H dA ¼ ðg  FÞ dA;
ð4Þ
using Eq. (3) for H, and where V is the volume of the global ocean and A ¼
surface area. It follows from Eq. (4) thatZZ
g dA ¼
G dA;

RdA is its

ð5Þi.e., MSL and geoid height above the reference ellipsoid have equal global means.
We define the geoid in terms of MSL g, rather than the sea-surface height g~, in order torestrict changes in G and V to those occurring on the timescales of global-mean sea-levelrise, 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 f is determined solely by ocean dynamics and density. With thezero-tide convention, which is common in gravity-field models, f would include the permanent ocean tide, which is almost þ 0:1 m at the equator and  0:2 m at the poles.
We define the geoid as GðrÞ ¼ Eðr; UG Þ, with a choice of UG such that Eq. (4) issatisfied, where Eðr; UÞ is the geodetic height of the equipotential surface for geopotential
U. The shapes of the equipotential surfaces, including the geoid, depend on the123Surveys in Geophysics

geographical distribution of mass over the Earth. According to Eq. (4), the global-mean
geoid height must change by
Z
Z
1
1
DV
ð6Þ
DG dA ¼
DF dA þAA
A
if there is global-mean vertical land movement DF affecting the sea floor, or a change DV
in the volume of the global ocean, whether due to change in density or in mass. Consequently, UG must change such that
Z
Z
1
DUG
oEðr; UÞDUG
ð7Þ
DG dA ¼
dA ¼
;
A
oU
A
g
if we approximate g as globally uniform.

4 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 therotation 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 byperiodic spatial variations in local gravity. Tidal motion of the land surface and sea floor isdue to elastic deformation of the solid Earth by gravitational tidal forces. The diurnal andannual 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, andthe consequent extreme sea level is sometimes called a storm tide. The tidal height is thevertical 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 thelunar 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 thesun and moon. The precession of the plane of the moon’s orbit causes tidal variations withan 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 componentsof the pole tide have periods of 1 year and about 433 days. The latter is due to the123Surveys in Geophysics

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 ofother 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 atmosphericpressure have a negligible effect on the total volume of the ocean. However, they do movesea water around, and the effect on the sea surface depends on the deviation of sea-levelpressure p~a ðrÞ from its global (ocean) mean, given byZ1
p~a ðr; tÞ dA:
ð8Þ
p~0a ðr; tÞ ¼ p~a ðr; tÞ A
For timescales longer than a few days, we can assume the ocean to be in hydrostaticbalance. Therefore, the depression of the sea-surface height (SSH) g~ by IB is B~ ¼
p~0a =ðg qs Þ where gðrÞ is the acceleration due to gravity and qs ðr; g~Þ the surface sea-water~
density. That is, when p~0a [ 0 then sea level is depressed locally by BðrÞ,and it is raised
when p~0a \0. The latter effect is an important contribution to storm surge. In a storm or
cyclone, p~a may fall by several 10 hPa, causing SSH to rise by several 100 mm.
The global mean of gqs 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 qs , and replacethem with constants; thus,0p~
B~ ¼ a :g q
Hence, the global-mean IB correction is zero,Z~ tÞ dA ¼ 0;
Bðr;

ð9Þ

ð10Þ

which follows by definition of p~0a .
The inverse-barometer response of the sea surface compensates for the effect of p~0a onhydrostatic 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~ which is called IB-corrected sea-surface height. Its time-meansurface height were g~ þ B,
is g þ B, the IB-corrected mean sea level. In most climate models, atmospheric pressure
variations are not communicated to the ocean. In these models B~ must be subtracted fromthe simulated SSH to produce a quantity that varies with p~a like the observed g~ does.

123Surveys in Geophysics

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 astorm 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 levelmay be called extreme coastal water level. For decadal timescales, the main influence onchanges 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 ofusing the word ‘‘extreme’’ for such projections.
N9 Storm surge: The elevation or depression of the sea surface with respect to thepredicted 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 theshore, 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 thestorm tide) is g~ðr; tÞ, and the predicted SSH due to the tide alone is g~tide ðr; tÞ, the stormsurge height r isrðr; tÞ ¼ g~ðr; tÞ  g~tide ðr; tÞ;ð11Þ

also called the surge residual or non-tidal residual.
The storm-surge height r is the sum of three components: the inverse barometer (IB)effect of low atmospheric pressure, the wind setup caused by the wind-driven current, andthe wave setup, which is the elevation of the sea surface due to breaking waves. All threeeffects 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 thehighest water level of the storm surge.
Particularly high SSH g~ ¼ r þ g~tide occurs when the storm surge coincides with hightide. Without the meteorological forcing, storm-surge height r would be zero, but since thetide level influences the propagation of the storm-forced signal, r and g~tide 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 isg~ðr; tÞ and the predicted SSH due to the tide alone is g~tide ðr; tÞ, the skew-surge height is
rk ðr; tÞ ¼ max t ð~gðr; tÞÞ  max t ð~
gtide ðr; tÞÞ;ð12Þ

where max t ðXðr; tÞÞ means the maximum value of X that occurs at location r during theinterval of time t from one low tide to the next. For extreme-value analysis, the skew-surgeheight is preferable to the storm-surge height as a measure of the effect of the meteorologicalforcing alone in regions where skew-surge height is uncorrelated with tidal height (Williams
et al. 2016).

123

Surveys in Geophysics

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 ofwater. 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, orfour 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 f: 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~ tÞ ¼ g~ðr; tÞ þ Bðr;~ tÞ  GðrÞ:
fðr;

ð13Þ

It is determined jointly by ocean density and circulation. The time-mean ocean dynamic
sea level isfðrÞ ¼ gðrÞ þ BðrÞ  GðrÞ;ð14Þ

whose global mean
1
A

ZfðrÞ dA ¼ 0;

ð15Þ

in view of Eqs. (5) and (10).
In the Coupled Model Intercomparison Project (CMIP), f~ 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 ofthe 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 computedfrom 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 hydrostaticpressure is given by
Z f
~ B~
ð16Þ
~ 0 Þ dz0 ;qðzp~ðzÞ ¼ p~a þ gz123Surveys in Geophysics

which is the sum of the atmospheric pressure p~a on the sea surface and the weight per unithorizontal area of sea water between z and the sea surface. The coordinate of the sea
surface in this case is not g~ but g~  G ¼ f~  B~ by Eq. (13), the height of the sea surface
above the geoid, because we are using the orthometric vertical coordinate z, which is thenatural choice for ocean dynamics. Equation (9) leads to the horizontal gradient of the~ Consequently, the horizontal gradient of
p0a ¼ gq rB.
atmospheric pressure r~
pa ¼ r~pressure within the ocean is given by
!
Z f
~ B~
0
0
~ Þ dzr~
p ¼ r p~a þ g
ð17aÞ
qðz
z

~ þg
¼ gq rB~ þ ðgq rf~  gq rBÞ

Z f
~ B~

~ 0 Þ dz0
rqðz

ð17bÞ

z

¼ gq rf~ þ g

Z f
~ B~

~ 0 Þ dz0 :
rqðz

ð17cÞ

z

~
In the first step of this derivation, we used Eq. (9) for the inverse barometer correction B,
approximated g and sea-surface q~ ¼ q as constants, and applied Leibniz’s rule to differentiate the integral, which yields the two terms in parentheses in Eq. (17b), but no termfor q~ at z because rz ¼ 0. From Eq. (17c) we obtain
Z ~ ~
1 fB
~ ¼ 1 r~ð18Þ
~ z0 Þ dz0 ;pðr; zÞ 
rqðr;
rfðrÞ
gq
q z
which relates the horizontal gradient of ocean dynamic sea level f~ 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 height D
Z ~ ~
1 fB
ð19Þ
q~ dz0D¼
q zof the sea surface relative to z.
In much of the ocean interior (below the boundary layer and away from coastal andother 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, v~ ’ v~g , with the geostrophic velocity v~g defined by~vg ¼ r~p;
f k  q~

ð20Þ

where f is the Coriolis parameter and k the vertical unit vector. If we can measure v~ atsome z and assume it is geostrophic, we arrive at123Surveys in Geophysics

1~ ¼  f k  q~
~vðr; zÞ 
rfðrÞ
gq
q

Z f
~ B~

~ z0 Þ dz0 ;
rqðr;

ð21Þ

z

from Eq. (18).
Alternatively, if we do not know v~ at any z, we assume there exists a level of no motionz ¼ L, which is a geopotential (horizontal) surface on which v~ ¼ v~g ¼ 0, requiring thehorizontal hydrostatic pressure gradient to vanish (r~p ¼ 0) by Eq. (20). Therefore,
Z f
~ B~
~ ¼ 1
ð22Þ
~ zÞ dz;
rfðrÞ
rqðr;
q L
using Eq. (18). There is no motion on z ¼ L so long as there is a compensation between
undulations of dynamic sea level f~ (on the left-hand side of Eq. 22), and variations of thedensity structure above z ¼ 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 thepycnocline. In some regions, the approximation is not useful; in particular, sizeable v~
occurs at all depths in the Southern Ocean.
The ocean dynamic topography is the estimate of ocean dynamic sea level made using~ theEqs. (21) or (22). Since Eqs. (21) and (22) are unaffected by adding a constant to f,
method provides only the difference in f~ between any two points (i.e., the gradient); it
cannot give f~ for individual points relative to the geoid. Furthermore, it is not applicable inregions 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.

5 Changes in Sea Level
The relationships between quantities determining changes in sea level are summarized inFig. 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.
N13 Ocean dynamic sea-level change Df: The change in time-mean ocean
dynamic sea level, i.e., the change in IB-corrected mean sea level relative to thegeoid.
For the difference between two time-mean states of the climate, Eq. (13) givesDfðrÞ ¼ DgðrÞ þ DBðrÞ  DGðrÞ:ð23Þ

Since the time-mean ocean dynamic sea level fðrÞ always has a zero global mean by
Eq. (15), so does Df i.e., global-mean sea-level rise is excluded from ocean dynamic sealevel change. This property depends on Eq. (5) and thus requires a different choice ofgeopotential to define the geoid in the two states, if there is any change in global-mean sea
level.

123

Surveys in Geophysics

Fig. 3 Relationships between quantities, defined in Sect. 5, that determine changes in sea level. The lengthsof 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, hh andhb are functions of location r. Any closed circuit gives an equality, in which a term has a positive sign whentraversed in the direction of its arrow, and a negative sign if in the opposite direction to its arrow. For
example, DR  Dg þ DF ¼ 0 (Eq. 24) is the circuit marked in red, Eq. (23) in orange, Eq. (38) in blue, andEq. (54) in green

N14 Geocentric sea-level change Dg: The change in local mean sea level withrespect to the terrestrial reference frame.
Geocentric sea-level change is the change in the height gðrÞ of MSL relative to thereference ellipsoid. IB-corrected geocentric sea-level change is Dg þ DB, i.e., the samewith the inverse barometer correction added. Geocentric sea-level change must be
distinguished from relative sea-level change.
N15 Relative sea-level change (RSLC) DR: 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 g and the sea floor height F may change and thus alter RSL. Hence,
RSLC is geodetically expressed asDRðrÞ ¼ DgðrÞ  DFðrÞ;ð24Þ

the difference between geocentric sea-level change Dg and vertical land movement DF(VLM). IB-corrected relative sea-level change is DR þ DB; 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 Dpa
are not applied to the ocean, DB must be added to include the effect of Dpa simulated by
the atmosphere model. (Note that this adjustment should not be made to ocean dynamic
sea-level change Df, which by definition is IB-corrected; see Eq. 23.)123

Surveys in Geophysics

The term ‘‘relative sea level’’ is not employed in an absolute sense, but only in conjunction with ‘‘change’’, because g  F (the analogue of Eq. 24) is simply the depth of thesea 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 asDRðrÞ ¼ DHðrÞ;ð25Þ

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 f~ or the inverse barometer effect, we might
approximate Df~ ’ 0 and DB ’ 0, in which case Dg ’ DG from Eq. (23), and DR ’
DG  DF 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 change DRq : The part of relative sea-level change which isdue to the change Dq in ocean density, assuming the local mass of the ocean per unit
area does not change. It is composed of thermosteric sea-level change DRh , whichis the part due solely to the change Dh in in-situ temperature, and halosteric sealevel change DRS , which is the part due solely to the change DS in salinity.
The time-mean local mass of the ocean per unit area is
Z
Z g
1 g
q dz ¼ Hq with q
q dz;
m¼
H F
F

ð26Þ

where the first factor H ¼ g  F is the local time-mean thickness of the ocean (Eq. 3), andthe second factor q 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 ¼ Dm ¼ DHjm q þ HDqjm

ð27Þ

(by making a linear approximation). Therefore,
DHjm ¼ ðH=qÞ Dq

with

Dq ¼

1
H

Z g
Dq dz:

ð28Þ

F

This would exactly define steric sea-level change in a situation where mass did not movehorizontally. 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 ofmass. For convenience, we approximate q with the constant q .
Since the RSLC is given by DR ¼ DH (without inverse barometer correction, Eq. 25),steric sea-level change isZ
1 g
DRq ¼ Dq dz ¼ DRh þ DRS
ð29Þ
q F
with the density increment decomposed into thermal and haline components (by making alinear approximation)
Dq ¼

oq
oq
Dh þ
DS;
oh
oS

ð30Þ

123

Surveys in Geophysics

and with the corresponding thermosteric and halosteric contributions
ZZ
1 g oq
1 g oq
Dh dz
DRS ¼ DS dz:
DRh ¼ q F ohq F oS

ð31Þ

Thermosteric sea-level change is often called thermal expansion, because oq=oh\0, so
increasing the temperature gives DRh [ 0. (See Appendix 2 regarding the dependence ofdensity on salinity.) Relative sea-level change (without inverse barometer correction) is thesum of steric and manometric sea-level change (Eq. 35).
N17 Global-mean thermosteric sea-level rise hh : The part of global-mean sealevel rise (GMSLR) which is due to thermal expansion.
This quantity is the global mean of local thermosteric sea-level change DRh (due totemperature change, Eq. 31); thus,
Z
Z Z g
1
1
oq
DRh dA ¼ 
Dhðr; zÞ dz dA:
hh ¼
ð32Þ
A
q A
F oh
It is the change in global ocean volume due to change in temperature alone, divided by theocean surface area. The CMIP variable zostoga is hh calculated with respect to a fixedreference state. (Griffies et al. 2016 define the reference to be the initial state of theexperiment for CMIP6.) Hence, differences in zostoga between two states give theglobal-mean thermosteric sea-level rise between those states.
Although halosteric sea-level change DRS (due to salinity change, Eq. 31) can belocally of the same order of magnitude as thermosteric, global-mean halosteric sea-levelchange is practically zero. In Appendix 2 we detail the physical arguments leading to thisconclusion. Salinity change should be excluded when calculating hh , 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 DRS . It follows that global-mean steric sea-level change, whichequals hh 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 theinaccuracy of the approximations made following Eq. (28).
N18 Manometric sea-level change DRm : Definition A: The part of relative sealevel change (RSLC) which is not steric, or alternatively Definition B: The part of
RSLC which is due to the change DmðrÞ in the time-mean local mass of the ocean
per unit area, assuming the density does not change. In the following, we show thatthe 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 DHjq ¼ Dm=q, where q is the local vertical mean of q.
(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 haveq ¼ q exactly.)
Since RSLC DR ¼ DH (without inverse barometer correction, Eq. 25), if weapproximate q with the constant q , we obtain
DRm ’

123

Dm
:
q

ð33Þ

Surveys in Geophysics

This is Definition B of ‘‘manometric sea-level change’’. The local change in mass Dm canbe estimated from the gravity field, or from the bottom pressure pb ; i.e., the hydrostaticpressure at the sea floor, according to Dm ¼ ðDpb  Dpa Þ=g: (See Appendix 3.) Becauseof 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 Rglobal mean of DRm vanishes if the mass of the global
ocean is constant, since 1=ðAq Þ Dm dA ¼ 0. However, DRm 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 DRm is nonzero and equals the barystatic sea-level rise(equality is approximate with Definition B of DRm , exact with Definition A), which is partof 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-levelrise.)
If mass and density are both allowed to change, Eq. (26) gives
Z g
Dq dz ;
ð34Þ
Dm ¼ DH q þ H Dq ) DH ¼ ð1=qÞ Dm Fusing the expression for Dq from Eq. (28). Again approximating q as q and substituting
from Eqs. (25), (29) and (33), we obtainDR ¼ DH ’ DRq þ DRm ;ð35Þ

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 indensity and change in mass per unit area. Since H is defined only in ocean areas, theformulae are not valid for locations which change from land to sea or vice versa.
With DRm defined by Eq. (33), Eq. (35) is only approximate, because of the replacementof q with q . We can make Eq. (35) exact if we retain the definition of Eq. (29) for stericsea-level change involving q and adopt Definition A of ‘‘manometric sea-level change’’,asDRmDR  DRq ;ð36Þ

i.e., DRm is the part of RSLC that is not steric.
We propose ‘‘manometric’’ as a new term because in the existing literature there is nounambiguous and generally used term for DRm . It may be described as the ‘‘mass effecton’’, ‘‘mass contribution to’’, ‘‘mass component of’’ or ‘‘mass term in’’ sea level or sealevel change, but these descriptions could equally well refer to GRD-induced sea-levelchange (the effects of a change in the geographical distribution of mass) or barystatic sealevel rise, so they can be confusing. (‘‘Manometric’’ is an existing word, referring to themeasurement of hydrostatic pressure using a column of liquid, a concept that is closely
related to bottom pressure.)
N19 Barystatic sea-level rise hb : 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’’.

123Surveys in Geophysics

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 andpermafrost. Land ice means ice sheets, glaciers, permanent snow and firn. Barystatic sealevel rise includes contributions from changes in all of these.
It does not include changes in the parts of ice shelves and glacier tongues whose weightis supported by the ocean rather than resting on land. (These floating parts constitute themajority of the mass of ice shelves and glacier tongues, but near the grounding line on theseaward 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 ¼

DM
;
qf A

ð37Þ

i.e., the change in mass DM of the global ocean from added freshwater, converted to achange in global ocean volume and divided by the ocean surface area A. Because globalmean halosteric change is negligible, the salinity of the existing sea water does not affecthb . Any contribution dM to barystatic sea-level rise can be expressed as its sea-level
equivalent (SLE) dM=ðqf AÞ, using the same formula.
The formula provides a convenient method of quantifying the changes in the mass of theocean if A is constant. However, hb and SLE may not accurately indicate the contributionof added mass to global-mean ocean thickness if there is a substantial change to A, as forexample in the transition from glacial to interglacial.
Calculating the global meanof manometric sea-level change DRm from its DefiniRtion B (Eq. 33) gives 1=A DRm dA ¼ DM=ðq AÞ ’ hb ; i.e., approximately equal to thebarystatic sea-level change, but not exactly since q 6¼ qf . With Definition A, the global
mean of DRm exactly equals hb : (See global-mean sea-level rise.) Despite this relationshipbetween manometric sea-level change and barystatic sea-level rise, we argue that we needdistinct 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 Earthand can be evaluated from the change in mass of other stores of water, e.g., ice sheets andglaciers, 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, thepartitioning of RSLC into steric and manometric (Eq. 35) is somewhat arbitrary, because it
depends on the choice of q as a reference density.
Neither reason for the distinction of DRm and hb applies to thermosteric sea-levelchange; its contribution to global-mean sea-level rise can only be conceived or evaluated as
the global mean of the local DRh , whose definition by Eq. (31) is well defined.
In recent literature, ‘‘eustatic’’ is often used as a synonym for ‘‘barystatic’’, whereas ingeological 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 the123Surveys in Geophysics

Intergovernmental Panel on Climate Change (Church et al. 2001; Meehl et al. 2007;
Church et al. 2013).
N20 Sterodynamic sea-level change DZ: 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,DZðrÞ ¼ DfðrÞ þ hh :ð38ÞIt can be diagnosed from ocean models (even those that do not conserve mass as per thecommonly 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 simulatedwith such models. (As discussed above for ocean dynamic sea-level change Df, 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 DZ. 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 sealevel change’’ or ‘‘sea-level change due to ocean density and circulation change’’.
N21 Vertical land movement (VLM) DF: 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 existingsolid surface. In contrast, landslides and sedimentation alter the solid surface and its heightby transport of materials; some authors count them as VLM. Extraction of groundwater andhydrocarbons may cause subsidence (sinking of the solid surface) by compaction (thereduction in the liquid fraction in the sediment). These anthropogenic effects can be locallylarge, 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 crustand the rigid upper part of the mantle) towards a hydrostatic equilibrium in which it isregarded as floating in the asthenosphere (the underlying viscous mantle, which is ofhigher 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 onland 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 tocontemporary 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.

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These three effects are all caused by changes in the geographical distribution of ocean andsolid 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 GðrÞ, while deformation of the solid Earth changes the sea floor
topography FðrÞ 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 change DC is defined as
DC ¼ DG0  DF 0ð39Þ(derived in Sect. 6 as Eq. 51) where DG0 and DF 0 are the deviations of the changes in thegeoid and in the sea floor from their respective global (ocean) means. By construction, theglobal (ocean) means of G0 and F 0 are each zero; hence, the global (ocean) mean of DC 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, thegeoid 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 theocean. SAL is also a component of GRD which is caused by changes in land ice and in thesolid Earth; thus, SAL contributes to the sea-level effects of GIA, contemporary GRD andmantle 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 bedescribed 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 aconcept 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 thelithosphere that occurred when mass was transferred from land ice into the ocean, or thereverse. 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 beneathformer 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 coastis rising and the sea floor is generally subsiding as a result of the increase in the mass ofthe 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 change DCGIA .
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.

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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, whichcauses further change to the geoid. Together, these effects produce relative sea-levelchange (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 GRDinduced relative sea-level change are separately proportional to the mass DM which has
been added to the ocean. Hence, their sum is
DCp ðrÞ ¼ DM cp ðrÞ;

ð40Þ

where cp ðrÞ is a geographically dependent constant of proportionality, independent of DM.
Since the barystatic sea-level rise is DM=qf A (Eq. 37), the RSLC due to the combinationof these three effects is
DRp ðrÞ ¼ DM /ðrÞwhere/ðrÞ ¼ cp ðrÞ þ 1=qf A:

ð41Þ

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 / is a constant geographical pattern, often called a sealevel fingerprint, or sometimes a static-equilibrium fingerprint to contrast it with thepatterns of ocean dynamic sea-level change. ‘‘Fingerprint’’ without qualification can beeasily confused with climate detection and attribution studies where the same word refers
to the patterns caused by particular climate change forcing agents such as greenhousegases. ‘‘Static-equilibrium’’ is not informative about the processes concerned. The part ofcontemporary GRD-induced RSLC due to viscous deformation and associated geoidchange cannot be represented by a constant pattern because it depends on convolving thehistory of mass addition with the time-dependent solid-Earth response.
N25 Mantle dynamic topography: GRD due to ongoing changes in the solid Earthcaused by mantle convection and plate tectonics.
The dynamics of the interior of the Earth cause vertical land movement, such as the upliftof mid-ocean ridges by upwelling material and the formation of oceanic trenches due tosubduction. 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 bevery large on geological timescales, amounting to hundreds of metres. Mantle dynamic
topography does not include glacial isostatic adjustment (although that is also due toongoing changes in the solid Earth).
Mantle dynamic topography is often called ‘‘dynamic topography’’ in the solid-Earthliterature 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 DV in the volume of theocean divided by the ocean surface area A, also called ‘‘global-mean sea-levelchange’’ (GMSLC). Observational estimation of h is described in Sect. 7.

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By definition, GMSLR is
Z

Z
Z
DV 1
1
1
¼ D
ðg  FÞ dA ¼
DRðrÞ dA ¼
DHðrÞ dA;
h¼
A
A
A
A

ð42Þ

which follows from Eqs. (4), (24) and (25). Hence, GMSLR is the global mean of relative
sea-level change DR and equals the global mean of the change DH in the thickness (or‘‘depth’’) of the ocean. Note that GMSLR differs from global-mean
R geocentric sea-level
rise because GMSLR is unaltered by a global-mean change ð1=AÞ DF dA in the level ofthe 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 andbarystatic sea-level rise,h ¼ hh þ hb :ð43ÞA satisfactory explanation of historical observed GMSLR in terms of thermosteric andbarystatic contributions has been achieved in recent years thanks to improvements in bothobservations 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 thepart of relative sea-level change which is not steric (Eq. 36), whose global mean
Z
Z
Z
1
1
1
DRm dA ¼
DR dA DRq dA ¼ h  hh ;
ð44Þ
A
A
A
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 additionof mass, so it must be the case thatZ
1
DRm dA;ð45Þ
hb ¼
Ai.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 q on h, there may be a nonzero contribution to hh in consequence. Since this is a steric effect, by definition it is not part of hb .
From Definition B (Eq. 33) we obtainan approximate expression for global-meanRmanometric sea-level change as ð1=AÞ Dm=q dA ¼ DM=ðq AÞ, where DM is the added
mass. This is the same as the expression (Eq. 37) for hb except that qf is replaced by q .
This difference is the result of the approximation in Eq. (33) that q ’ q. Physically, it isbecause manometric sea-level change DRm (a local quantity) is dominated by redistributionof existing sea water, for which q 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 qf , 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 globalmean relative sea-level change hR , which is the change in global-mean ocean thicknessV þ DV V
V
DV DA
 ¼

:
ð46Þ
hR ¼
A þ DA A A þ DA V
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Surveys in Geophysics

If A is constant, hR ¼ h. An increase in A (DA [ 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 sealevel change’’. Following the last three assessment reports of the Intergovernmental Panelon 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 rise hG : The global-mean change in meansea level with respect to the terrestrial reference frame.
This quantity is the global mean of Dg, the change in MSL relative to the reference
ellipsoid. From Eqs. (4) and (5) we have


Z
Z
Z
1
1
1
Dg dA ¼
DG dA ¼
DV þ
DF dA
hG ¼
A
A
A
ð47Þ
Z
1
DF dA;¼hþ
A
where h is global-mean sea-level rise (GMSLR) defined by Eq. (42). Thus, global-meangeocentric sea-level rise hGR differs from GMSLR because the latter is unaltered by aglobal-mean change ð1=AÞ DF dA in the level of the sea floor F, provided the volume ofthe ocean does not change. On geological timescales, when the area
R of the ocean may
change, the global-mean change in level of the sea floor is Dðð1=AÞ F dAÞ.

6 Relationships Determining Relative Sea-Level Change
Relative sea-level change (RSLC) is DR ¼ Dg  DF (Eq. 24). By applying the inverse
barometer correction, we obtain IB-corrected RSLCDR þ DB ¼ Dg þ DB  DF

ð48aÞ

¼ D½g  G þ B þ D½G  F

ð48bÞ

¼ Df þ D½G  F ;ð48cÞ

where in Eq. (48c) we used the definition of ocean dynamic sea-level change Df (Eq. 23)
to rewrite the first term. From the definition of the geoid (Eq. 4) we obtain
Z
1
DV
ð49Þ
DðG  FÞ dA ¼
¼ h;
A
A
R
the global-mean sea-level rise. Let us write DGðrÞ ¼ DG0 ðrÞ þ ð1=AÞ DG dA andsimilarly for DF, thus defining DG0 ; DF 0 as the local deviations of DG; DF from theirrespective global (ocean) means. Therefore,Z1DðG  FÞ dA ¼ DðG0  F 0 Þ þ h:ð50ÞDðG  FÞ ¼ DðG0  F 0 Þ þAThis leads to our expression for GRD-induced relative sea-level change (Eq. 39) as123

Surveys in Geophysics

DCðrÞD½G0 ðrÞ  F 0 ðrÞ ¼ D½GðrÞ  FðrÞ  h:

ð51ÞSubstituting Eq. (51) in Eq. (48c) gives IB-corrected RSLC asDRðrÞ þ DBðrÞ ¼ DfðrÞ þ h þ DCðrÞ;ð52Þ

the sum of ocean dynamic sea-level change Df, global-mean sea-level rise h and GRDinduced RSLC DC. Using Eqs. (43) and (38) we obtainDfðrÞ þ h ¼ DfðrÞ þ hh þ hb ¼ DZðrÞ þ hb ;ð53ÞHence, IB-corrected RSLC isDRðrÞ þ DBðrÞ ¼ DZðrÞ þ hb þ DCðrÞ;ð54Þ

the sum of sterodynamic sea-level change DZðrÞ, barystatic sea-level rise hb and GRDinduced RSLC. The contemporary GRD-induced RSLC due to a change dMi in any of thestores 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 dMi , we can rewrite
Eq. (54) asX
dMi /i þ DCGIA þ DCZ ;DRðrÞ þ DBðrÞ ¼ DZðrÞ þ
ð55Þ
iwhere /i is the barystatic–GRD fingerprint (Eq. 41) of store i of water, DCGIA is GIAinduced RSLC, and DCZ is the GRD-induced RSLC of ocean mass redistribution (selfattraction and loading) associated with sterodynamic sea-level change. The last term istypically neglected.
Equation (55) is the means by which MSL projections are derived from coupledatmosphere–ocean general circulation models (AOGCMs). These models do not simulate
GRD-induced RSLC (because they have time-independent geoid and sea floor) and are notgenerally 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 obtainedby combining sterodynamic sea-level change simulated by an AOGCM with separately
calculated projections of barystatic sea-level rise and GRD-induced RSLC using climatechange simulations from the AOGCM applied to models of glaciers, ice sheets and thesolid Earth (Church et al. 2013; Kopp et al. 2014; Slangen et al. 2014).
According to Eqs. (35) or (36), DR ¼ DRq þ DRm , the sum of steric sea-level change
DRq and manometric sea-level change DRm , which are the parts due, respectively, tochange in density and change in mass per unit area. In general, DRm 6¼ 0 even if hb ¼ 0,
Rgbecause ocean mass may be redistributed. In particular, because DRm ¼ 1=qs F Dq dz issmall on the continental shelves (where the ocean is shallow), but ocean dynamics will not
permit a strong gradient in f to develop across the shelf break, global-mean thermostericsea-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).

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7 Observations of Sea-Level Change
Estimates of global-mean sea-level rise (GMSLR) for the last century depend mainly onrecords from tide-gauges. These instruments register coastal relative sea-level change(RSLC) DR ¼ Dg  DF (Eq. 24), which is affected by local vertical land movement(VLM) DF. VLM is large in some places, with strong geographical gradients.
R
GMSLR is calculated as the global mean of RSLC, h ¼ ð1=AÞ DR dA (Eq. 42).
However, tide-gauges measure DR only at points on the coast and thus give a sparse, nonuniform 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 ¼ DR þ DB  Df  DC

ð56Þ

we see that in principle h can be calculated from DR from any tide-gauge by applying theinverse barometer (IB) correction DB, and subtracting local ocean dynamic sea-levelchange Df and local GRD-induced RSLC DC. The global mean of each of these threeadjustments is zero, so Eq. (42) is satisfied. In practice, using historical records, it isnecessary to combine many tide-gauges in order to reduce the influence of unforced~
variability in f.
The IB adjustment is fairly small and can be made accurately from atmospheric pressurerecords. Various methods are used to allow for the spatial pattern of Df, for example bycalculating 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 g~ 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 adjustfor GIA-induced RSLC DCGIA ðrÞ (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 Dg ¼ DR þ DF (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 Dg(¼ Df  DB þ DG, Eq. 23) has been measured overmost 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 DGGIA ðrÞ, theeffect of GIA on the geoid (e.g., Figure 3b of Tamisiea and Mitrovica 2011), from IBcorrected geocentric sea-level change, thus:
Dg þ DB  DGGIA ¼ Df þ DGNGIA ;ð57Þ

where DGNGIA ¼ DG  DGGIA is due to ongoing redistribution of water mass on the
Earth’s surface.
R
To convert global-mean geocentric sea-level rise hG ¼ ð1=AÞ Dg dA to GMSLR
h requires an adjustment for the global mean of DF, according to Eq. (47).

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Although several processes can produce large local VLM, the only large globalmean effect is GIA. There Ris no contemporaryR GMSLR associatedR with GIA,
so Eq. (49) gives ð1=AÞ DFGIA dA ¼ ð1=AÞ DGGIA dA ) ð1=AÞ DGNGIA dAR¼ h þ ð1=AÞ DFNGIA dA. Hence, the global mean of Eq. (57) becomes
Z
Z
1
1
DFGIA dA þ
DFNGIA dA;
hG ¼ h þ
ð58Þ
AA
recalling that the global means of DB and Df are zero. In response to the shift of mass fromthe land (as ice) into the ocean since the Last Glacial Maximum, and the consequent
mantleRadjustment, the sea floor is subsiding on average, giving a trend in ð1=AÞ DFGIA dA ofabout  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 aRnegative ð1=AÞ DFNGIA dA which reduces hG by about 8% of the barystatic sea-level
rise (Frederikse et al. 2017).

8 Deprecated Terms and Recommended Replacements

Deprecated term

Recommended replacement

Eustatic sea-level change

Barystatic sea-level rise or barystatic sea-levelchange for global-mean sea-level rise due tochange in the mass of the ocean, but not its density
Global-mean sea-level rise for the global mean ofrelative sea-level change, due to the change in thevolume 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 theocean 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-levelchange 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 sealevel rise (RSLR) for the change in mean sea level
relative to the land
Geocentric sea-level change for the change in meansea level relative to the terrestrial reference frame

Global sea-level change (GSLC)

Gobal-mean sea-level rise (GMSLR) or globalmean sea-level change (GMSLC) for the global
mean of relative sea-level change, due to thechange in the volume of the ocean
Global-mean geocentric sea-level rise for the
global-mean change in mean sea level relative tothe terrestrial reference frame

123Surveys in Geophysics

Deprecated term

Recommended replacement

Extreme sea level

Extreme sea level for the occurrence of
exceptionally high local sea surface due to shortterm 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 torelative sea-level change
Global-mean thermosteric sea-level rise forcontribution to global-mean sea-level change

Sea-level change due to oceanographic processes orsteric 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 thebarystatic sea-level rise due to the addition of aunit 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-levelchangeBarystatic sea-level rise for the contribution to
global-mean sea-level rise from the change ofmass 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 thelocal mass of the ocean per unit area, as opposed tosteric sea-level change
GRD-induced relative sea-level change for theeffects on relative sea level from geoid change andvertical land movement, as opposed to steric,ocean dynamic and barystatic sea-level change

Bottom pressure term in sea-level change

Manometric sea-level change

9 List of Defined Terms and Notations
This table gives the entry number or section (not the page number) in which each term isdefined. 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.

A

Above flotation

N19

Altimetry

Section 7

Area of the global ocean
Asthenosphere

N21

Astronomical tideN6

pa

Atmospheric pressure at the sea surface

hb

Barystatic sea-level rise

N19

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Surveys in Geophysics/

Barystatic–GRD fingerprint

N24

Bathymetry

N4

Bottom pressure

N18

Bottom topography

N4

Chandler wobble

N6

Compaction

N21

Contemporary GRD

N24q

Density of water
Depth of the water column

N4

D

Dynamic height

N12g

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

Dg

Geocentric latitude

N1

Geocentric sea-level change

N14

Geodetic height

N1

Geodetic latitude

N1

G

Geoid

N5

Geoid height

N5

U

Geopotential

N5

Geostrophy

N12

v~g

Geostrophic velocity

N12

DCGIA

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

N26hGlobal-mean sea-level rise (GMSLR)

N26

hh

Global-mean thermosteric sea-level rise

N17

Gravitational acceleration

N5

DC

h

123

GRD

N22

GRD-induced relative sea-level change

N22Greenwich meridian

N1

Halosteric sea-level change DRS

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

In-situ temperature of ocean water

Surveys in Geophysics

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

DRm

Manometric sea-level change

N18

Mantle dynamic topography

N25

M

Mass of the global oceangMean sea level (MSL)

N3

Mean-tide

N6

Neap tide

N6

Nodal period

N6

Ocean

Section 2.4

f

Ocean dynamic sea level

N11

Df

Ocean dynamic sea-level change

N13

Ocean dynamic topography

N12

Ocean GRD

N22m

Ocean mass per unit area
Orthometric height

N5

Permanent tide

N6

Pole tideN6

Post-glacial rebound

N23

Predicted tideN6

Prime meridian

N1

Radiational tide

N6

Reference ellipsoid

N1

Regression

N15

DR

Relative sea-level change (RSLC)

N15

q

Representative density of ocean water

S

Salinity of ocean waterFg~

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

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Significant wave height

N10

Significant wave period

N10

Skew-surge height

N9

Spring tide

N6

State

Section 2.2

Static-equilibrium fingerprint

N24

DRq

Steric sea-level change

N16

DZ

Sterodynamic sea-level change

N20

Storm surge

N9

r

H

Storm tide

N6

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 DRh

N16

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

TsunamiN10

Vertical

N1

DF

Vertical land movement (VLM)

N21

V

Volume of the global ocean
Wave height

N10

Wave period

N10

Wave runup

N9

Wave setup

N9Wind setup

N9Wind wave

N10

Zero-tide

N6

Acknowledgements We thank Cathy Raphael from NOAA/GFDL for kindly drafting the figures, and John
Krasting, Jianjun Yin, Sophie Nowicki, Angélique Melet, Carl Wunsch, Unnikrishnan, Chris Merchant andDéborah Idier for useful comments. We are grateful to Don Chambers and the other (anonymous) reviewer
for their encouraging remarks and helpful suggestions for improvement. Rui Ponte acknowledges support
from NASA Grant NNH16CT01C. John Church was partially supported by the Centre for Southern
Hemisphere Oceans Research, a joint research centre between QNLM and CSIRO. Jonathan Gregory’s work
was supported by Grant NE/R000727/1 from the UK Natural Environment Research Council on123

Surveys in Geophysics‘‘Addressing the Grand Challenge of regional sea level change prediction’’. The Grand Challenge steeringcommittee provided a forum for discussions during the preparation of this paper. We are grateful to
Palanimuthu and colleagues for their careful typesetting of this complicated manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate if changes were made.

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 theocean 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 theocean into motion. So another necessary condition for zero ocean circulation is the absenceof density gradients along horizontal surfaces, i.e., density is a function of depth only. Thisconfiguration 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 theocean 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 isnegligible (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 themasses 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 qn ,n ¼ 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 nochange 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 ¼ M1 þ M2

M S ¼ M1 S1 þ M2 S2 ;

ð59Þwhere M is the total mass and S is the salinity of the homogenized water. Return thehomogenized water to the original buckets, placing the same mass M1 back into the firstbucket and mass M2 into the second bucket.

123Surveys in Geophysics

Dependence of Density on Salinity
The dimensionless coefficients a q1 oq=oh and b q1 oq=oS, which are used tocompute steric sea-level change (Eq. 31), measure the relative change in the in-situ density
as a function of temperature and salinity. Because oq=oh 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 oq=oS [ 0, the corresponding effect for salinity issometimes 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 bysalt, so the parcel is materially altered, unlike in the case of adding heat. If salt is added tothe 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 b.
For the surface ocean, representative values are q ¼ 1028 kg m 3 and S ¼ 0:035, while
for freshwater q ¼ qf ¼ 1000 kg m 3 and S ¼ 0. Hence, a representativeb ð1=1000Þð1028  1000Þ=ð0:035  0Þ ¼ 28=35 ¼ 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 b to be constant in thefollowing.

Computing the Change in Total Volume
The change in total volume upon homogenization is the sum of the changes in the twobuckets,
dV ¼ dV1 þ dV2 :ð60ÞSince 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
dqn ¼ dðMn =Vn Þ ¼ ðqn =Vn Þ dVn ) dqn =qn ¼ dVn =Vn ;ð61Þ

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 throughsalinity changes, according to
dqn =qn ¼ b dSn ¼ dVn =Vn ;ð62Þ

in which case the change in total volume isdV ¼ dV1 þ dV2 ¼ ðV1 b dS1 þ V2 b dS2 Þ:

ð63Þ

Mass conservation for salt means thatdðM SÞ ¼ dðM1 S1 Þ þ dðM2 S2 Þ ¼ 0:
Furthermore, since mass in the two buckets is unchanged, salt conservation leads to

123

ð64Þ

Surveys in Geophysics

M1 dS1 þ M2 dS2 ¼ 0;

ð65Þ

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 changedV ¼ b dS1 ðV1  V2 M1 =M2 Þ ¼ b V1 dS1 ð1  q1 =q2 Þ:

ð66Þ

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 withdiffering 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 ¼ A hn , where hn is the thickness of thewater in the bucket. Equation (66) then says that upon homogenization, the thickness ofwater changes bydh ¼ b h1 dS1 ð1  q1 =q2 Þ;ð67Þ

and that the total thickness of the homogenized water is given by
hnew ¼ h1 þ h2 þ dh ¼ h2 þ h1 ½1  b dS1 ð1  q1 =q2 Þ :ð68ÞAs expected, we see that dh ¼ 0 only when b ¼ 0 or q1 ¼ q2 . The first is never true, andthe second is not true in the general case of differing temperatures (in which case there arethermosteric changes ignored in our discussion).

An Ocean Example
To explore the oceanic implications of Eq. (68), assume bucket-1 initially has freshwaterwith density q1 ¼ qf , whereas bucket-2 initially has sea water with density
q2 ¼ qs ¼ qf þ q0 . The salinity change for bucket-1 is dS1 ¼ S, since this bucket wentfrom its original freshwater concentration to the homogenized sea water with salinity
S. The halosteric-induced thickness change (Eq. 67) is thus given bydh ¼ h1 b S ðq0 =qs Þ\0:

ð69ÞHow large is this effect? For the case of an upper ocean with salt concentration S ¼ 0:035,
sea-water density qs ¼ 1028 kg m 3 ) q0 ¼ 28 kg m 3 and b ¼ 0:8, we havedh ¼ h1  0:8  0:035  ð28=1028Þh1  7:6  104 :ð70Þ

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 ¼ 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 andGregory (2006), we can generally ignore the contribution to global-mean sea level fromglobal halosteric effects.

123Surveys in Geophysics

Appendix 3: Bottom Pressure
The hydrostatic pressure at the ocean sea floor is commonly referred to as the ocean bottompressure p~b , usually calculated as~p~b ¼ p~a þ gm;

ð71Þ

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 mðrÞof sea water, given by
Eq. (26). This formula makes two approximations. First, hydrostatic pressure does notexactly 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 toobtain 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
Dpb ¼ Dpa þ gDm;

ð72Þthat 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 Dm,
Z g
ð73Þ
Dpb ¼ Dpa þ gq DH þ gDqðzÞ dz;
F

where H ¼ g  F is the time-mean thickness of the ocean (Eq. 3). This form separates thechange in pressure due to sea water into one term (the second) due to the change in thelocal thickness of the ocean, and another (the third) which is proportional to the local
vertical-mean change in sea-water density.

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Afﬁliations
Jonathan M. Gregory1,2 • Stephen M. Grifﬁes3 • Chris W. Hughes4 •Jason A. Lowe2,14 • John A. Church5 • Ichiro Fukimori6 • Natalya Gomez7 •Robert E. Kopp8 • Felix Landerer6 • Gonéri Le Cozannet9 •Rui M. Ponte10 • Detlef Stammer11 • Mark E. Tamisiea12 • Roderik S. W. van de Wal13
1

National Centre for Atmospheric Science, University of Reading, Reading, UK

2

Met Office Hadley Centre, Exeter, UK

3

NOAA Geophysical Fluid Dynamics Laboratory and Princeton University Program in Atmosphericand Oceanic Sciences, Princeton, USA

4

University of Liverpool and National Oceanography Centre, Liverpool, UK

5

University of New South Wales, Sydney, Australia

6

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, USA

7

McGill University, Montreal, Canada

8

Rutgers University, New Brunswick, USA

9

BRGM, French Geological Survey, Orléans, France123Surveys in Geophysics
10

Atmospheric and Environmental Research, Inc., Lexington, USA

11

University of Hamburg, Hamburg, Germany

12

Center for Space Research, University of Texas at Austin, Austin, USA

13

Institute for Marine and Atmospheric Research Utrecht and Geosciences, Physical Geography,
Utrecht University, Utrecht, The Netherlands

14

Priestley Centre, University of Leeds, Leeds, UK123