Page 3 - Sea-level change_2004

ARTICLE IN PRESS
K. Lambeck, A. Purcell / Quaternary Science Reviews 24 (2005) 1969–1988 1971

concomitant change in water load. The two terms grounding line (Lambeck et al., 2003). Thus, the ocean
include contributions from the deformation of the crust geometry andthe water depth must be known at each
under the surface load and from the change in the epoch andthis is achieved through introducing high-
gravitational potential of the earth–ice–water system. resolution topography (including bathymetry) models
Because the redistribution of melt water is a function of into the solution of (1) and(2) and estimating the
the deformation of the ocean basins and of the change in palaeo-topography/bathymetry
gravitational attraction by the ice, the two isostatic
(3)
DðtÞ¼ Dðt P Þ Dz rsl
terms are coupledwith Dz I-h a function of Dz esl , Dz I-g
and Dz I-h itself. Thus, the explicit separation of the two
as part of the iterative process. As the ice heights locally
isostatic terms in (1b) is artificial andis done here only
change linearly between integration time steps (see
for illustrative purposes.
below), so does the ocean boundary evolve linearly
A key assumption made in formulating the isostatic
between successive time steps.
terms is that the response of the earth to a surface load,
In the first approximation, the glacio-isostatic term is
of gravitational potential U, is linear. Thus, if the loadis evaluatedfrom the integrals of the ice loaddefined for
decomposed into potential harmonic constituents U n , each epoch. In subsequent iterations, because the depth
then for each harmonic both the deformation and of the water column near the ice margins is a function of
change in gravitational potential are proportional to the the gravitational attraction of the ice, andtherefore of
corresponding harmonic U n , with the proportionality
the ice itself, the evaluation of Dz I-g , is not independent
coefficients defined by time-dependent load Love
of the water loadand this interaction is includedin the
numbers. The evaluation of the response therefore
subsequent iterations. For illustrative purposes only,
reduces to the evaluation of the potential integrals of
this is ignoredin the next section and the first-iteration
the ice and water loads on the deformable earth, with
solution for Dz I-g is usedto provide an approximate
the requirement that ocean-ice mass is conservedand
estimate of the effects of the ice loadon sea level in the
that the ocean surface remains an equipotential surface
Mediterranean. The first-iteration water-load correction
at all times. These assumptions have formedthe base of
assumes that the melt water is distributed uniformly
most models for global isostatic reboundandassociated
over the oceans whose time dependence of area is
sea-level change (e.g. Cathles, 1975; Peltier and defined by the boundaries Dðt P Þ Dz esl ¼ 0. In subse-
Andrews, 1976; Nakada and Lambeck, 1987; Tromp quent iterations, the water load is distributed according
andMitrovica, 1999a, b), andthe principal complexity to the predicted sea-level change of the previous
is the evaluation of the Love numbers for realistic earth
iteration andthis allows the full coupling between the
models and ensuring that the above-mentioned coupling
various components to be considered. Five iterations are
is properly treated. Finally, rotational effects (Milne and
foundto be sufficient for present purposes but for some
Mitrovica, 1998) are included in the full formulation of
high-resolution analyses higher iterations are appropri-
Eq. (1).
ate (e.g. Lambeck et al., 2004b). Here, again for
The start time of the glacial loading history is taken
illustrative purposes only, the water-loadcontribution
either as the Last Interglacial (LIg) or an earlier is evaluatedfrom
interglacial, depending on whether the predictions are
limitedto the periodsince maximum glaciation or Dz I-h Dz rsl ðDz I-g þ Dz esl Þ, (4)
whether they include the full last cycle. The ocean
boundary Oðt o Þ anddepth Dðt o Þ at this start time is where Dz rsl is the fully coupled, multi-iteration solution
initially defined by the present geometry and the basin and Dz I-g is the first approximation glacio-isostatic term.
andcoastline geometry is computed through time for a The a priori information requiredfor predicting the
specifiedice history and earth rheology up to the present glacially-driven sea-level change includes the definition
time t P . Because of earth-memory effects, this geometry of the earth’s rheology, anda knowledge of the ice
Oðt P Þ, Dðt P Þ may not be the same as the actual present- sheets through time. In reality, neither is necessarily
day values and the geometry through time is adjusted by known with adequate accuracy from ab initio con-
subtracting out the difference between the starting siderations and both are evaluated in part from the
geometry at LIg andthe final geometry predicted for analysis of glacial reboundresponses. By analyzing sea-
t P . At each epoch, the ice thickness on the shelves is level responses from different geographic localities and
comparedwith the predicted depth of water to establish at different times, and by combining the results with
if the ice is grounded. If it is, the volume in (2) includes those from different responses (e.g. surface displacement
all ice grounded on the shelf. If it is not, the floating ice measurements andmoments of inertia and gravity
forms part of the water column along with the water change) some effective separation of ice- andearth-
layer between the base of the ice andthe sea floor. When model parameters can be achieved. In particular, specific
the shelf ice has thinnedsufficiently for it to float, the combinations of assumedice sheet histories and
ocean boundary is defined by the new location of the rheological models can provide successful predictive