Calculating gravitationally self-consistent sea level changes driven by dynamic topography

Austermann, J.; Mitrovica, J. X.

doi:10.1093/gji/ggv371

Abstract

We present a generalized formalism for computing gravitationally self-consistent sea level changes driven by the combined effects of dynamic topography, geoid perturbations due to mantle convection, ice mass fluctuations and sediment redistribution on a deforming Earth. Our mathematical treatment conserves mass of the surface (ice plus ocean) load and the solid Earth. Moreover, it takes precise account of shoreline migration and the associated ocean loading. The new formalism avoids a variety of approximations adopted in previous models of sea level change driven by dynamic topography, including the assumption that a spatially fixed isostatic amplification of ‘air-loaded’ dynamic topography accurately accounts for ocean loading effects. While our approach is valid for Earth models of arbitrary complexity, we present numerical results for a set of simple cases in which a pattern of dynamic topography is imposed, the response to surface mass loading assumes that Earth structure varies only with depth and that isostatic equilibrium is maintained at all times. These calculations, involving fluid Love number theory, indicate that the largest errors in previous predictions of sea level change driven by dynamic topography occur in regions of shoreline migration, and thus in the vicinity of most geological markers of ancient sea level. We conclude that a gravitationally self-consistent treatment of long-term sea level change is necessary in any effort to use such geological markers to estimate ancient ice volumes.

Sea level change, Mantle processes, Dynamics of lithosphere and mantle

1 INTRODUCTION

The earliest discussions of plate tectonics focused on horizontal motions at the Earth's surface, reflecting a preoccupation with the concepts of continental drift and seafloor spreading. However, the driving force for these motions—thermochemical convection within the Earth's mantle—is also responsible for vertical deflections of the Earth's surface (McKenzie 1977) that have come to be known as dynamic topography (e.g. Hager et al.1985). The earliest simulations of time-varying dynamic topography connected the process to epeirogenic motions of continents and long-term regional- and global-scale sea level changes, and typically focused on subduction-controlled dynamics (e.g. Mitrovica et al.1989; Gurnis 1990, 1992, 1993; Holt & Stern 1994; Coakley & Gurnis 1995; Mitrovica et al.1996; Lithgow-Bertelloni & Gurnis 1997; Pysklywec & Mitrovica 1998; Gurnis et al.1998; Pysklywec & Mitrovica 1999). However, several studies also demonstrated a clear link between upwelling mantle flow and large-scale fluctuations in the topography of continents (e.g. Lithgow-Bertelloni & Silver 1998; Gurnis et al.2000; Forte et al.2010).

Predictions of present-day, global-scale dynamic topography have been derived from viscous flow modelling constrained by seismic inferences of mantle heterogeneity (e.g. Hager et al.1985; Ricard & Wuming 1991; Forte et al.1993; Wen & Anderson 1997; Steinberger 2007). As the resolution of seismic imaging has progressively improved, viscous flow modelling of time-varying dynamic topography has shown concomitant advances in resolution, allowing applications to the detailed, regional-scale geological record of continents (e.g. Daradich et al.2003; Moucha et al.2009; Rowley et al.2013; Liu et al.2014). These studies have had fundamental implications for our understanding of long-term sea level change. For example, they have highlighted the fact that dynamic topography is present globally, not only at plate boundaries, and that it can significantly contaminate records of long-term sea level change at all locations, even passive continental margins (e.g. Moucha et al.2008; Müller et al.2008; Conrad & Husson 2009; Spasojevic & Gurnis 2012; Rowley et al.2013). Moreover, these analyses have shown that the globally averaged change in the shape of ocean basins is governed by the integrated effect of dynamic uplift and subsidence throughout these basins, not just changes in elevation linked to fluctuations in the rate of plate creation.

While the connection between dynamic topography and regional and global sea level changes has been established, a rigorous mathematical and numerical formalism for accurately mapping viscous flow predictions of time-dependent dynamic topography into geographically variable sea level change is lacking. Previous efforts to compute this mapping have several shortcomings:

A change in dynamic topography will redistribute water, and the loading (or unloading) associated with this redistribution will, in turn, drive displacements of the crust and hence cause sea level changes. This loading effect is generally treated in an approximate sense through isostatic amplification of the ‘air-loaded’ dynamic topography. In this regard, some studies (e.g. Moucha et al.2008; Rowley et al.2013) assume water cover everywhere and apply a global amplification to the air-loaded dynamic topography obtained from their viscous flow modelling. In contrast, Müller et al. (2008) applied no amplification to the dynamic topography change for their predictions along the U.S. East Coast. A hybrid approach only amplifies the dynamic topography at sites with ocean cover (Steinberger 2007; Flament et al.2013), however, it and the other approaches neglect water loading within any region of shoreline migration. This assumption is problematic given that the error introduced into the prediction will be largest in the vicinity of an ancient sea level marker.
The above studies also ignore the possibility of flexural effects in the response to water loading, that is, they assume local hydrostatic equilibrium in the crustal response. Flexure has a direct effect on the computed sea level change via its impact on crustal elevation (e.g. the crustal displacement will extend outside the region of loading). Moreover, it has indirect effects on sea level change because any departures from hydrostatic equilibrium will drive gravitational and rotational effects that feedback into sea level.
Studies of dynamic topography are often only concerned with the change in crustal elevation due to mantle stresses. Sea level, however, is the difference between the elevation of both the gravitational equipotential defining the sea surface and the crust, and therefore changes in the gravitational field that arise from mantle convection will perturb sea level, an effect only rarely taken into account (Spasojevic & Gurnis 2012; Conrad & Husson 2009).
A further shortcoming involves the assumption of a local application of isostasy (e.g. Müller et al.2008; Conrad & Husson 2009; Spasojevic & Gurnis 2012). Any loading-induced uplift (subsidence) of the crust must be matched by subsidence (uplift) elsewhere to conserve mass of the solid Earth. Local treatments of isostasy (whether they incorporate flexure or not) ignore this global-scale compensation.
Finally, predictions of long-term sea level change should also conserve the mass of the ocean (or ice plus ocean), but this constraint is not always applied in such calculations (e.g. Flament et al.2013).

In this article, we present a gravitationally self-consistent mathematical treatment of global-scale sea level change driven by time-varying dynamic topography, convection-driven geoid perturbations, ice mass changes and sediment redistribution. Our approach is based on a formalism developed in studies of ice age sea level change (Farrell & Clark 1976; Mitrovica & Milne 2003; Kendall et al.2005; Dalca et al.2013) and it conserves mass of both the surface (ice plus ocean) load and the solid Earth while taking precise account of the migration of shorelines. In its most general form, and under the assumption that the dynamic topography, convection-driven geoid perturbations, ice mass and sediment histories are known, the method is valid for viscoelastic Earth models of arbitrary complexity; one only needs to have a numerical method for computing the response of the Earth model to an evolving surface mass load. Our approach may be used to consider dynamic topography in isolation; however, we have included the possibility of contemporaneous ice mass changes and sediment fluxes to provide a framework that avoids errors that may be incurred by assuming that the total sea level change can be computed by treating each of the driving processes in isolation. Finally, we also outline the extension necessary when the timescale being considered is sufficiently long that plate motions must be taken into account.

In addition, we present successive, special cases that assume: (1) isostatic equilibrium is maintained throughout the evolution of the system and (2) the Earth model varies with depth alone. The first assumption simplifies the numerical calculations by ignoring time-dependent viscous adjustments and this allows one to evolve the system backwards in time starting from a known (present-day) topography. Taken together, the two assumptions allow us to make use of fluid Love number theory to compute the response of the Earth model to the surface mass loading. In this case, we adopt profiles of density and elastic moduli from the PREM seismic model (Dziewonski & Anderson 1981) and the thickness of the elastic lithosphere is a free parameter in the modelling. To illustrate the new, gravitationally self-consistent treatment of sea level change driven by dynamic topography, we present a series of numerical predictions based on these special cases.

2 CALCULATING LONG-TERM SEA LEVEL CHANGE

2.1 Calculating dynamic topography from mantle convection

A variety of approaches have been adopted to compute dynamic topography in mantle convection simulations. The most rigorous but computationally expensive approach is to allow the surface to deform freely. These free surface calculations allow the upper boundary of the computational domain to respond dynamically to evolving stresses in the mantle (Gurnis et al.1996; Zhong et al.1996; Kaus et al.2010; Kramer et al.2012). An alternate, fixed surface approach introduces a top layer in the computational domain with a comparatively low viscosity and low density, which serves as a proxy for air and yields quasi-free surface deformation. This approach is often referred to as the ‘sticky-air’ method (Schmeling et al.2008; Crameri et al.2012). The third and most common approach models mantle convection within a fixed domain and computes the radial stresses at the top of the domain. The dynamic topography (DT) is then calculated as the compensation height that balances these radial stresses (McKenzie 1977; Hager et al.1985; Zhong et al.1993; Flament et al.2013):

\begin{equation} {\it DT}=\frac{\sigma _{{\rm rr}}}{\Delta \rho \cdot g} \end{equation}

(1)

where σ_rr is the radial stress at the upper surface of the mantle flow model, Δρ is the density contrast between the crust and the overlying material (e.g. air, water or sediment) and g is the gravitational acceleration. This approach can be improved by including self-gravitation associated with the disturbed topography (Zhong et al.2008) or flexural effects of an elastic lithosphere in response to the internal deformation (Golle et al.2012). The three distinct approaches yield comparable predictions for the dynamic topography (Crameri et al.2012); the last approach is characterized by a slight shift backwards in time, compared to the others, because it assumes an instantaneous isostatically adjusted response of the surface to internal loading.

As is clear from the basic principles of isostasy that underlie eq. (1), the amplitude of predicted dynamic topography arising from viscous stresses associated with mantle flow increases as the density difference between the crust and overlying material decreases. Thus, the choice of an appropriate density contrast is a key issue in comparing predictions of dynamic topography with the geological record. Regardless of the method employed for modelling dynamic topography, one has to choose an appropriate density contrast between the crust and the overlying material. As described in Section 1, this is a shortcoming of existing models of dynamic topography and it can be overcome using the framework discussed below.

2.2 Sea level changes in the presence of dynamic topography

The derivation presented below is based on the ice age sea level theory developed by Mitrovica & Milne (2003), Kendall et al. (2005) and Dalca et al. (2013), following the canonical treatment of Farrell & Clark (1976). Farrell & Clark (1976) derived the so-called ‘sea level equation’ governing gravitationally self-consistent sea level changes on a viscoelastic Earth driven by global ice mass variations. Their derivation assumed that the position of shorelines was fixed in time, which is equivalent to assuming that all shorelines are characterized by steep vertical cliffs such that any local changes in sea level produce no transgression or regression (i.e. no shoreline migration). Their calculations were based on spherically symmetric, Maxwell viscoelastic and non-rotating Earth models, and in this case they adopted viscoelastic Love number theory (Peltier 1974; Wu & Peltier 1982) to compute time-dependent perturbations in the elevation of the solid surface and gravitational field. Mitrovica & Milne (2003) extended the sea level equation to accurately incorporate shoreline migration and Kendall et al. (2005) presented an algorithm for solving the equation in the case of 1-D and 3-D Earth models of arbitrary complexity. The Mitrovica & Milne (2003) treatment allows for the feedback of perturbations in Earth rotation into sea level, and in the case of a 1-D, rotating Earth model, Kendall et al. (2005) derived the necessary equations making use of viscoelastic Love number theory. The Dalca et al. (2013) study extended these treatments to incorporate sediment redistribution into the gravitationally self-consistent sea level calculation. Their form of the theory, which defined the topography as including ice and sediment, is advantageous to our treatment of dynamic topography, and so it forms the basis of our derivation.

We adopt the following symbolism:

R - Earth's bedrock elevation;

H - Sediment thickness;

I - Thickness of land-based and grounded marine-based ice;

G - Height of the equipotential surface that defines the sea surface.

All these quantities have a dependence on position, expressed in terms of colatitude θ and east longitude φ, and time t. We will henceforth suppress these explicit dependencies to simplify the expressions. In this regard, we abbreviate a generic function χ(θ, φ, t_j) by χ, or χ_j when it is necessary to make the time dependence explicit.

Sea level is defined as the difference between the heights of two bounding surfaces: the sea surface equipotential and the solid surface (including the height of ice and/or sediments), where the latter is given by R + H + I:

\begin{equation} SL = G - (R + H + I). \end{equation}

(2)

In this case, topography is simply defined as the negative of sea level:

\begin{equation} T = -SL = (R + H + I) - G. \end{equation}

(3)

Moreover, the ocean height is simply the projection of sea level onto the location of oceans:

\begin{equation} S = SL \cdot C, \end{equation}

(4)

where C is the ocean function defined by:

\begin{equation} C = \left\lbrace \begin{array}{r@{\quad}l}1 & { if\quad SL> 0} \\ 0 & { if \quad SL \le 0} \end{array} \right. \end{equation}

(5)

The total surface mass load L is comprised of water (oceans), ice and sediments:

\begin{equation} L = \rho _W \cdot S + \rho _I \cdot I + \rho _H \cdot H, \end{equation}

(6)

where, ρ_W, ρ_I and ρ_H are the densities of water, ice and sediment, respectively.

Geological markers record local changes in sea level, as defined in eq. (2), and thus we are interested in how changes in grounded ice, sediments, water and dynamic topography, perturb the elevation of the bedrock and sea surface height from some initial state. Any time-dependent field χ at some time t_j can be expressed as a sum of the initial value at the onset of loading, at time t₀, and a perturbation from this state:

\begin{equation} \chi _j = \chi _0 + \Delta \chi _j. \end{equation}

(7)

This form may be applied to any of the quantities R, H, I, G, SL, T, S, C and L. As an example, the changing bedrock elevation may be written as:

\begin{equation} R_j = R_0 + \Delta R_j, \end{equation}

(8)

where the term R₀ includes the background signal from dynamic topography. The perturbation of the bedrock elevation can furthermore be decomposed into a time-dependent perturbation in air-loaded dynamic topography plus a perturbation due to the response of the bedrock to the changing surface mass (ice plus ocean plus sediment) load. If we denote these terms by ΔDT_A and ΔR^L, respectively, we have:

\begin{equation} \Delta R_j = \Delta {\it DT}_{A,j} + \Delta R^L_j. \end{equation}

(9)

It is important to emphasize that it is air-loaded dynamic topography that should appear in this equation. The amplification from air-loaded to water-loaded dynamic topography at sites under water, as indicated by eq. (1), is naturally incorporated into eq. (9) (as well as the broader sea level calculation) via the term |$\Delta R^L_j$|⁠. That is, an increase in the air-loaded dynamic topography (i.e. an uplift) at sites within the oceans will displace water and this unloading will act to amplify the uplift, as required. Similarly a dynamic subsidence computed in the air-loaded case at such sites will increase the local water load and amplify the subsidence. To be consistent, the air-loaded dynamic topography change and the load-induced perturbation to the bedrock elevation (the two terms on the right-hand-side of eq. (9)) should be computed using the same Earth structure model.

Using the above expressions, the change in sea level is given by:

\begin{eqnarray} \Delta SL_j = SL_j \!-\! SL_0 = \Delta G_j \!-\! (\Delta R_j^L + \Delta H_j + \Delta I_j + \Delta {\it DT}_{A,j}).\nonumber \\ \end{eqnarray}

(10)

The change in ocean height S cannot simply be written as a combination of perturbations in G, R, I and H because it involves the projection onto the ocean function given by eq. (4), which is itself time-dependent. Using this equation, together with the general form (7) and the simple relationship between sea level and topography (3) yields:

\begin{equation} \Delta S_j = \Delta SL_j \cdot C_j - T_0 \cdot \Delta C_j . \end{equation}

(11)

Eqs (3), (5), (10) and (11) represent the generalized form of the sea level equation valid for any timescale of forcing and Earth models of arbitrary complexity. Physically, eq. (11) indicates that the ocean height change at time t_j is equal to the change in sea level projected onto the ocean function, both at time t_j, plus a correction term that accounts for shoreline migration (Mitrovica & Milne 2003). The latter involves a projection of the initial topography onto a field (ΔC_j) that is only non-zero in regions experiencing shoreline migration between times t₀ and t_j (i.e. ΔC_j is only non-zero when C₀ ≠ C_j).

It is also useful to have an expression for the change in the surface mass load from its initial state. Using eqs (6) and (7), this change is given by

\begin{equation} \Delta L_j = \rho _W \Delta S_j + \rho _I \Delta I_j + \rho _H \Delta H_j . \end{equation}

(12)

The change in the water load driven by dynamic topography is embedded in both fields on the right-hand-side of eq. (4). Sea level, SL, is dependent on dynamic topography via eq. (2) (and eqs 8 and 9), and this dependence also alters the ocean function C via eq. (5). The latter demonstrates that ocean mass redistribution driven by dynamic topography will, in general, lead to shoreline migration, even at shorelines which experience no local change in dynamic topography. The sea level equation incorporates this physics while maintaining gravitational self-consistency.

The sea surface equipotential G is perturbed by both the surface mass loading and by mantle convection. We could have incorporated this second contribution into our definition for dynamic topography; however, models of mantle flow-induced dynamic topography typically define dynamic topography as an absolute measure of bedrock height (or as being relative to the centre of mass of the Earth), not as a height relative to the geoid (or sea surface equipotential). The perturbation in the height of the sea-surface may be expressed as:

\begin{equation} \Delta G_j = \Delta G^{mc}_j + \Delta G^L_j, \end{equation}

(13)

where the first term on the right-hand-side denotes the perturbation due to mantle convection and the second term is the sea surface equipotential due to surface mass loading. The former is the perturbation to the geoid that mantle flow modellers often compute in tandem with dynamic topography.

Eqs (10)–(12) highlight the integral nature of the sea level eq. (11). In particular, from eqs (9), (10), (11) and (13), the change in ocean height, ΔS_j, depends on surface load-induced perturbations to the height of the bedrock (⁠|$\Delta R^L_j$|⁠) and sea surface potential (⁠|$\Delta G^L_j$|⁠). However, both |$\Delta R^L_j$| and |$\Delta G^L_j$| are, in turn, dependent on the ocean height change since the latter forms a component of the total surface mass load given by eq. (12).

It is traditional in gravitationally self-consistent ice age sea level theory to decompose the right-hand side of eq. (13) into geographic variable and globally uniform terms (Farrell & Clark 1976). To this end, we can write:

\begin{equation} \Delta G_j = \Delta \mathcal {G}^{mc}_j + \Delta \mathcal {G}^L_j + \frac{\Delta \Phi _j}{g}, \end{equation}

(14)

where |$\Delta \mathcal {G}^{mc}$| and |$\Delta \mathcal {G}^L$| are the geographically variable components of ΔG^mc and ΔG^L, respectively. While the sea surface must remain an equipotential surface, the specific equipotential that defines the sea surface need not remain constant over time (Farrell & Clark 1976; Dahlen 1976). Indeed, changes in the mass of the ocean in response to variations in global ice volumes, as well as changes in the shape of the ocean basin associated with load-induced deformation or the redistribution of sediments, will perturb the equipotential that defines the sea surface. In eq. (14), this time-dependent, globally uniform perturbation is denoted by ΔΦ_j, and dividing this value by the surface gravitational acceleration, g, converts the perturbation into a geographically uniform perturbation in the height of the initial sea surface equipotential.

Using eq. (14) into eq. (10) allows us to decompose the sea level change as follows:

\begin{equation} \Delta SL_j = \Delta \mathcal {SL}_j +\frac{\Delta \Phi _j}{g}, \end{equation}

(15)

where

\begin{equation} \Delta \mathcal {SL}_j = \Delta \mathcal {G}^L_j + \Delta \mathcal {G}^{mc}_{j} - (\Delta R^L_j + \Delta H_j + \Delta I_j + \Delta {\it DT}_{A,j}) \end{equation}

(16)

and thus the sea level equation (11) can be rewritten as

\begin{equation} \Delta S_j = \Delta \mathcal {SL}_j \cdot C_j + \left(\frac{\Delta \Phi _j}{g}\right) C_j - T_0 \cdot \Delta C_j, \end{equation}

(17)

where |$\Delta \mathcal {SL}_j$| is given by eq. (16) and C_j by eq. (5).

The term ΔΦ_j/g can be solved for by invoking conservation of mass of the ocean plus ice reservoirs, which requires that

\begin{equation} \rho _I \int \!\!\!\! \int _{\Omega }\Delta I_j {\rm d}\Omega + \rho _W \int \!\!\!\! \int _{\Omega }\Delta S_j {\rm d}\Omega = 0, \end{equation}

(18)

where Ω spans the full surface area of the Earth. Integrating eq. (17) over the Earth's surface, using eq. (18) and solving for ΔΦ_j/g yields:

\begin{eqnarray} \frac{\Delta \Phi _j}{g} &=& - \frac{1}{A_j} \left[ \frac{\rho _I}{\rho _W} \int \!\!\!\! \int _{\Omega }\Delta I_j {\rm d}\Omega + \int \!\!\!\! \int _{\Omega } \Delta \mathcal {SL}_j \cdot C_j {\rm d}\Omega \right.\nonumber \\ &&\left.-\, \int \!\!\!\! \int _{\Omega } T_0 \cdot \Delta C_j {\rm d} \Omega \right]. \end{eqnarray}

(19)

In this expression,

\begin{equation} A_j=\int \!\!\!\! \int _{\Omega }C_j {\rm d}\Omega . \end{equation}

(20)

The sea level eq. (17), together with the set of eqs (3), (5), (16), (19) and (20), governs gravitationally self-consistent sea level changes in response to mantle-convection-induced deformation of the Earth's solid surface (dynamic topography) and gravitational field, ice mass variations and sediment redistribution.

The time-dependent fields ΔDT_A, |$\Delta \mathcal {G}^{mc}$|⁠, ΔI, and ΔH are provided on input, and the full solution yields the fields ΔR^L, |$\Delta \mathcal {G}^L$| and ΔC, the (time-dependent) scalar ΔΦ and, both the global sea level change ΔSL (via eq. 15) and the ocean height change ΔS. The global sea level change and the ocean height change are related through the sea level eq. (17), and the fact that the former is in general non-zero over land has an important physical interpretation, as elucidated by Dahlen (1976). In particular, at some specific site (θ, φ) on land at time t, the value ΔSL(θ, φ, t) represents the change in the local depth of water that would be measured if one were to dig a deep, infinitesimally thin canal connecting the site to the ocean.

As we noted above, the sea level equation is an integral equation since the quantity being solved for, ocean height change, is a component of the surface mass loading. In practice, this complication is generally addressed numerically by solving the sea level equation (17) iteratively, whereby at each time step an initial guess of the ocean height change is input into the right-hand-side of the equation, which is then solved to yield an updated estimate of ΔS_j, and the process is repeated until convergence. One then moves to the next time step and the process is repeated. A reasonable first guess to the solution is to assume that the load-induced perturbation in the elevation of the bedrock and sea-surface are equal to zero. In this case:

\begin{equation} \Delta \mathcal {SL}^{i=0}_j = \Delta \mathcal {G}^{mc}_{j} - (\Delta H_j + \Delta I_j + \Delta {\it DT}_{A,j}) \end{equation}

(21)

and

\begin{equation} \Delta S^{i=0}_j = \Delta \mathcal {SL}^{i=0}_j \cdot C^{i=0}_j + \left(\frac{\Delta \Phi ^{i=0}_j}{g}\right) C^{i=0}_j - T_0 \cdot \Delta C^{i=0}_j, \end{equation}

(22)

with

\begin{eqnarray} \frac{\Delta \Phi ^{i=0}_j}{g} &=& - \frac{1}{A^{i=0}_j} \left[ \frac{\rho _I}{\rho _W} \int \!\!\!\! \int _{\Omega }\Delta I_j {\rm d}\Omega + \int \!\!\!\! \int _{\Omega }\Delta \mathcal {SL}^{i=0}_j \cdot C^{i=0}_j {\rm d}\Omega\right.\nonumber\\ &&\left. -\, \int \!\!\!\! \int _{\Omega } T_0 \cdot \Delta C^{i=0}_j {\rm d} \Omega \right] \end{eqnarray}

(23)

and

\begin{equation} A^{i=0}_j=\int \!\!\!\! \int _{\Omega }C^{i=0}_j {\rm d}\Omega . \end{equation}

(24)

In these equations, the superscript i denotes the iteration counter.

The fields I_j and ΔI_j represent grounded ice height. In solving the sea level equation, a check is made at each time step to ensure that the ice thickness in marine sectors is sufficient to ensure that the ice is grounded, and if it is not, the ice model is revised to remove any floating ice (Kendall et al.2005). If we denote the input model ice history as |$I^{{\rm model}}_j$|⁠, then this check is based on the following criterion:

\begin{eqnarray} I_j = \left\lbrace \begin{array}{ll}I^{{\rm model}}_j \quad\mbox{ if $SL_j + I^{{\rm model}}_j < 0$} \\ I^{{\rm model}}_j \quad {\rm if}\, SL_j + I^{{\rm model}}_j > 0\,{\rm and}\\ \qquad\qquad I^{{\rm model}}_j \,\rho _I > (SL_j + I^{{\rm model}}_j)\, \rho _W \\ 0 \hspace{19.91684pt} \quad\mbox{otherwise.} \end{array} \right. \end{eqnarray}

(25)

The sea level equation (17) requires that the initial topography T₀ is specified. In traditional ice age applications, T₀ is computed through a second iterative procedure. In particular, one begins with a first guess for T₀ and this is used to compute sea level changes from the beginning of the simulation to the present day. The difference between the computed present-day topography and the observed topography is then used to update the estimate of T₀ and the process is repeated until convergence, that is, until the computed present-day topography matches the observed topography to within a specified tolerance. This ‘outer’ iteration, which loops over the full time span of the simulation, should, in general, be adopted in the present application, however, we note an important exception below.

The set of equations we have derived here are valid for arbitrarily complex Earth models. Regardless of the level of complexity, the main requirement is a method for computing perturbations in the elevation of the Earth's bedrock surface and sea surface, ΔR^L and |$\Delta \mathcal {G}^L$| respectively, driven by surface mass loading. For models with 3-D mantle structure, ΔR^L and |$\Delta \mathcal {G}^L$| can be computed using a variety of numerical techniques developed for problems in glacial isostatic adjustment (e.g. Wu & van der Wal 2003; Zhong et al.2003; Latychev et al.2005).

2.2.1 Some special cases

There are a variety of special cases one might consider in computing sea level changes driven by dynamic topography. For example, if one adopts a spherically symmetric Earth model, then the equations can be cast in the spectral domain making use of viscoelastic Love number theory (Peltier 1974; Wu & Peltier 1982). The relevant equations are given in Dalca et al. (2013).

For long timescale applications that are generally of interest when considering changes in dynamic topography, one might additionally assume that the Earth system reaches full isostatic equilibrium across each time step in which the dynamic topography is updated. That is, one may assume that these time steps are long enough that all viscous stresses in response to the surface loading have relaxed and isostatic equilibrium has been reached. Departures from isostatic equilibrium may arise, for example, if the viscoelastic lithosphere has a relaxation time comparable to the timescale of the convective forcing. Furthermore, internal density discontinuities contribute buoyancy in response to the convective flow for timescales of order several million years (Piromallo et al.1997). Isostatic adjustment in response to a surface mass load is also impacted by these internal discontinuities, and thus reaching complete isostatic equilibrium would require a similarly long timescale. However, the departure from the equilibrium state would be relatively small for shorter timescales since the viscous modes associated with the internal discontinuities only contribute at the percent level to the load-induced surface displacement.

The assumption of isostatic equilibrium further simplifies the algorithm by making the ‘outer’ iteration (to find T₀) unnecessary. In this case, since time-dependent viscous effects are ignored, the simulation is reversible in time and T₀ can be interpreted as the present-day topography, which is known. The simulation then steps backwards in time through the history of ice, sediment, dynamic topography and gravity fields. The assumption of isostatic equilibrium also allows one to evolve the system directly from t = 0 to t_j without calculating intermediate time steps.

In the examples discussed below, we assume isostatic equilibrium and a 1-D Earth model with a purely elastic lithosphere of specified thickness. In this case, the viscoelastic Love number theory appropriate for a 1-D Earth can be simplified to use the long-term, fluid limit of the Love numbers (Wu & Peltier 1982). The relevant equations for this case are provided in the Appendix. The Earth's viscoelastic lithosphere is characterized by a suite of relaxation times. Thus, the assumption that the lithosphere retains any elastic strength is only valid if the timescale of convective flow is much shorter than the longest of these relaxation times (in this case, the lithospheric thickness we are adopting in the numerical model should be interpreted as an effective elastic thickness associated with the unrelaxed lithospheric modes). Furthermore, the assumption of isostatic equilibrium requires that all other viscous relaxation is complete (i.e. that the timescale of flow is much longer than the other relaxation times associated with the viscoelastic lithosphere). Finally, the adoption of a 1-D Earth model ignores important complexity associated with lateral variations in lithospheric strength, including the presence of weak zones at plate boundaries.

While we make the assumption that the adoption of fluid Love number theory is valid in the results presented below, we emphasize that a time-forward and iterative procedure is necessary in calculations in which viscous effects cannot be ignored, as is the case for Pleistocene ice age calculations where changes in ice and ocean loading occur on timescales such that glacial isostatic adjustment must be modelled as a viscoelastic phenomenon.

2.3 Schematic illustration of sea level physics in the presence of dynamic topography

Fig. 1 provides a schematic illustration of the physics of sea level change near a shoreline in the presence of the combined effects of a local perturbation in dynamic topography and a lowering of the sea-surface height (neglecting for this illustration the flexural effects of the elastic lithosphere). The latter may arise from a variety of processes, including, for example, the growth of ice cover at some location distant from the shoreline. Figs 1(a)–(c) decompose the total sea level change into a two-step evolution, the first associated with the perturbation in dynamic topography and the second from the lowering of the sea surface height. If the perturbation in dynamic topography and sea surface height occurs in tandem (i.e. in one step), then the sea level change is given by the evolution from Fig. 1(a) to (directly) Fig. 1(d).

Figure 1.

Schematic illustration of the physics of sea level change near a shoreline subject to dynamic topography and a change in sea surface height (neglecting lithospheric flexure).

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In the first time step of the two-step evolution (t₀ to t₁ or frame a to b), a relatively constant perturbation to the air-loaded dynamic topography occurs producing an uplift of ΔDT_A,1 (Fig. 1b). In the region landwards of the original shoreline, there is no amplification in the dynamic topography since the region begins and ends under air cover. As one moves from left to right between the shoreline position at t₀ and t₁, progressively more water is displaced by the dynamic topography and the unloading leads to a crustal uplift (or amplification from the air-loaded dynamic topography) that increases linearly (⁠|$\Delta R_1^L$|⁠) up to the new shoreline because at this point the crustal uplift has raised the bedrock to the sea-surface. To the right of this shoreline, the water unloading associated with the air-loaded dynamic topography is constant since the amount of water that is displaced is the same everywhere right of the shoreline at t₁, and hence the response to this changing load is the same.

In the second time step (t₁ to t₂ or frame b to c), the sea surface height drops to the level G₂, and the unloading produces crustal uplift at all locations to the right of the shoreline position at t₁ (Fig. 1c). The uplift of the crust increases linearly as one moves from this shoreline to the location where the newly uplifted crust intersects the new sea surface height (⁠|$\Delta R_2^L$|⁠), and this intersection defines the shoreline at the end of the second time step. To the right of the shoreline at t₂ the crustal uplift due to unloading is constant. The final position of the solid surface is given by the line R₂.

The two step processes merges into one step in the evolution from Fig. 1(a) directly to Fig. 1(d). The drop in the sea surface from G₀ to G₁ and the contemporaneous perturbation in air-loaded dynamic topography, DT_A,1, leads to a crustal uplift from R₀ to R₁ and a migration of the shoreline from t₀ in Fig. 1(a) to its position at t₁ in Fig. 1(d). To the left of the original shoreline the crust remains subaerial throughout the evolution and there is no ocean unloading effect. To the right of the new shoreline position, the crust remains under ocean cover and the unloading-induced uplift, |$\Delta R_1^L$|⁠, is constant. Between the old and new shoreline this crustal uplift increases monotonically towards the latter.

This simple example illustrates the equivalence between the water unloading and the amplification of air-loaded dynamic topography as discussed in the context of eq. (1). We emphasize, however, that as mentioned above, these schematics miss an important component of the sea level response, namely elastic flexure. If one includes an elastic crust/lithosphere layer in the physical response, as we do in the calculations described below, then the crustal uplift will extend laterally beyond the region of unloading, and this will impact both the amplitude of the sea level change and the predicted migration of the shoreline.

2.4 Including plate motion

We have thus far ignored the signal associated with the horizontal motion of tectonic plates, which would alter the geometry of the surface mass load and move any given geographic site through the field of mantle convection induced dynamic topography and geoid fluctuations. On timescales of 10³–10⁵ yr, this effect can generally be neglected. However, on longer timescales, the impact of plate motions needs to be considered.

Let us assume, in the following, that the Earth remains in a state of isostatic equilibrium in response to any change in the surface mass load associated with plate motions. In this case, the impact of changes in ice and sediment loads, dynamic topography and the gravity field on sea level can, for a given site, be most simply expressed in a reference frame fixed to the site (i.e. a Lagrangian frame of reference) rather than in a fixed mantle reference frame (e.g. hot spot reference frame).

As described in eq. (7), a perturbation in a general time-dependent quantity χ from the onset of loading, at time t₀, to a time t_j is simply the difference between the initial value of the quantity and its value at time j. If, across this time interval, a specific point on a given plate travels from (θ₀, φ₀) to (θ_j, φ_j) as a consequence of plate motion, then the response of the Earth at this site only depends on the net difference in the load and dynamic topography that the site experiences across this time interval, and not on the full time history from t₀ to t_j:

\begin{equation} \Delta \chi = \chi _j(\theta _j,\varphi _j) - \chi _0(\theta _0,\varphi _0). \end{equation}

(26)

If we consider, for example, the air-loaded dynamic topography, then ΔDT_{A, j} expressed in eq. (26) will change if the dynamic topography due to mantle convection evolves, or if the plate moves through a gradient in dynamic topography. The change in the parameter χ can be positioned in a coordinate system fixed to either the initial geography (i.e. Δχ = Δχ(θ₀, φ₀)) or the final geography (i.e. Δχ = Δχ(θ_j, φ_j)). In either case, there will be regions for which Δχ is undefined; areas of plate destruction (e.g. subduction zones) for the case Δχ(θ₀, φ₀), or areas of plate creation (e.g. mid-ocean ridges) for Δχ(θ_j, φ_j).

Within the sea level formalism, changes in parameters such as ice, sediments, the gravity field and dynamic topography must be defined globally to guarantee mass conservation. Therefore, any undefined regions have to be interpolated while preserving the conservation of water plus ice mass and solid Earth mass (or volume) between t₀ and t_j. This is a relatively easy procedure for changes in ice volume because ice sheets are generally located away from plate boundaries. For all other quantities, an appropriate interpolation scheme must be chosen.

3 APPLICATIONS

In the following section, we apply the gravitationally self-consistent sea level methodology derived above to a series of case studies that assume a spherically symmetric (i.e. depth varying) Earth that maintains isostatic equilibrium at all times (i.e. the case considered in the Appendix). Our goal is to highlight the accuracy of the new approach relative to previous, approximate treatments of sea level changes driven by dynamic topography. We use the density and elastic structure of the seismic model PREM (Dziewonski & Anderson 1981) in all calculations. Furthermore, while our standard results will adopt an elastic lithosphere of thickness 90 km, we will also consider a thickness of 130 km to investigate the sensitivity of the results to this parameter. While the calculations further assume a globally constant crustal density of 3300 kg/m³, the method developed in this paper does allow to include spatially varying crustal densities. Finally, to emphasize the underlying physics of sea level changes, all dynamic topography fields discussed below are simple, hypothetical geometries that are imposed rather than computed from a realistic simulation of mantle convection.

3.1 Example 1: isostatic amplification of dynamic topography

To illustrate the results of the new sea level methodology, we begin with a simple scenario in which an axisymmetric air-loaded dynamic topography perturbation, centred near Norfolk, West Virginia, is applied along the U.S. East Coast. The air-loaded perturbation is characterized by a peak uplift of 160 m. Figs 2(a)–(c) show the sea level change that one would predict by simply scaling the input air-loaded dynamic topography using different density contrast models, as discussed in Section 2.1. Specifically, we consider three cases in which we apply: (1) no amplification (air cover); (2) an amplification assuming a water cover; and (3) no amplification in zones of present-day land cover and water cover amplification in present-day ocean regions. Fig. 1(d) is the sea level change predicted using the full, gravitationally self-consistent treatment described in the last section. In each frame of the figure we superimpose the predicted shoreline prior to the perturbation in dynamic topography. Note that the shoreline after the perturbation is applied corresponds to the modern coastline and the shoreline prior to the perturbation is the ancient coastline. In practice, we calculate this by going backwards in time (as discussed in Section 2.2), starting at the known present-day topography and calculating the unknown ancient topography by applying the negative change in dynamic topography and in the case of Fig. 2(d), solving the sea level equation.

Figure 2.

Change in sea level near the U.S. East Coast in response to a synthetic perturbation in dynamic topography. Frame (d) is the sea level change based on the gravitationally self-consistent sea level methodology described in the text, while frames (a)–(c) are based on the following assumptions: (1) air load everywhere, (2) water load everywhere and (3) air load over continents and water load over oceans. The white lines are the predicted shorelines prior to the change in dynamic topography.

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To explore the error in the three approximate solutions, Figs 3(a)–(c) show differences between the prediction in Fig. 2(d) and each of the fields in Figs 2(a)–(c). The second column on Fig. 3 shows the same fields plotted in the first column, except that we zoom in on the U.S. East Coast. In all frames of Fig. 3, the white line is reproduced from the ancient shoreline position predicted from the gravitationally self-consistent calculation (Fig. 2d). The reason for this choice is that the frames of Fig. 3 represent errors in the prediction incurred by using an approximate treatment of sea level change, and our intent is to demonstrate how these errors will map onto the elevation of the “true”location of the ancient shoreline.

Figure 3.

(a–c) Difference between the predicted sea level change calculated using a fixed amplification of an imposed, air-loaded dynamic topography (Figs 2a–c) and the sea level change calculated using our gravitationally self-consistent sea level methodology (Fig. 2d). (d–f) Close up of the frames (a–c), respectively.

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As illustrated in the schematic of Fig. 1(b), the gravitationally self-consistent simulation accurately accounts for water mass unloading driven by dynamic topography. This unloading tends to increase as one moves from the ancient to modern shoreline and it remains relatively constant over the present-day ocean. The unloading associated with the simulation in Fig. 2(d) is more complex than in Fig. 1(b) because the initial bedrock topography (R₀) is more variable, and because the former incorporates elastic flexure.

In regard to the approximate calculations of sea level change, Fig. 2(a) does not incorporate any water unloading and therefore the air-loaded dynamic topography is not amplified in this case. The error in the calculation of the sea level change (Figs 3a and d) is largely limited to regions oceanwards of the ancient shoreline (i.e. |$\Delta R_1^L$| in Fig. 1b is not captured). This is not strictly true because elastic flexure extends the effect of the water unloading landwards of the ancient shoreline in the accurate sea level calculation (Fig. 2d). In contrast to this case, Fig. 2(b) assumes that water unloading occurs everywhere. This leads to a significant error landwards of the ancient shoreline and an error that decreases as one moves from the ancient to modern shoreline (Figs 3b and e). The error in Figs 3(b) and (d) remains non-zero over a localized region oceanwards of the modern shoreline because flexural effects associated with water unloading are not accounted for. Finally, the calculation that assumes a water-amplified dynamic topography oceanwards of the modern shoreline, and no amplification landwards of this shoreline (Fig. 2c), neglects the effects of ocean unloading, and associated flexure, within the region of shoreline migration. As a consequence, the error is largest in the zone between the two shorelines, and it tapers to zero on either side of this region (Figs 3c and f). The geometries of the errors associated with the approximate treatments is dependent on whether the region is subject to a regression or transgression, but these plots nevertheless provide a sense of the scale and amplitude of these errors.

Fig. 4(a) shows the sea level change predicted for sites along the ancient shoreline in the gravitationally self-consistent calculation (blue; see also Fig. 2d). For the purpose of comparison, we also show predictions along the same profile for the three approximate treatments of sea level change in Figs 2(a)–(c). (The results for the case where no isostatic amplification is applied, Fig. 2(a), and the hybrid case treated in Fig. 2(c), are indistinguishable on Fig. 4(a) and therefore simply plot on top of each other.) All calculations predict a significant drop in sea level across the profile, as expected given the dynamic uplift of the region in this simulation (see Fig. 2). However, the air-loaded and hybrid calculations predict a sea level fall ∼15 per cent less than the gravitationally self-consistent calculation, while the water load approximation yields an amplitude ∼25 per cent greater than the calculation based on our new methodology.

Figure 4.

Change in sea level calculated using an air load approximation (Fig. 2a), a water load approximation (Fig. 2b), a hybrid approximation (Fig. 2c) and the full gravitationally self-consistent methodology (Fig. 2d) with a lithospheric thickness of 90 km (solid blue) and 130 km (dashed blue). (a) Prediction along the location of the ancient shoreline (see the inset) as calculated with the full methodology. The air load approximation and hybrid approximation are identical over the modern continents and therefore these two approximations yield the same values. (b) Prediction perpendicular to the shoreline (see the inset). The hybrid approximation changes from the air load approximation to the water load approximation at the position of the modern shoreline. A thicker lithosphere leads to more pronounced flexure that extends further into the continent and oceans.

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The sign of the error associated with the air load and hybrid calculations can be understood with reference to Fig. 1(b), which illustrates the cross-sectional profile expected from a gravitationally self-consistent solution of sea level change. In contrast to this case, the approximate solutions neglect the water unloading in the region between the ancient and modern shoreline (and the flexural effects of this unloading) because they assume air loading for all sites landwards of the modern shoreline (corresponding to the shoreline at t₁). Thus, within the region of shoreline migration the approximate solution does not accurately capture the uplift of the crust and thus it predicts a sea level fall of smaller amplitude than the gravitationally self-consistent solution in the vicinity of this region. In contrast, the water load approximation overestimates the change in load (and hence crustal uplift) in all regions landwards of the modern shoreline and therefore predicts a sea level fall of greater amplitude than the self-consistent calculation.

Fig. 4(b) shows the computed sea level change perpendicular to the shoreline. The water load calculation overestimates the sea level change over continents while the air load calculation underestimates the change over oceans. The self-consistent calculation is characterized by a smooth transition between the two regions across the zone of shoreline migration and slightly beyond. The modern shoreline is marked by the transition of the hybrid approximation from the air to water load prediction. Assuming a thicker lithosphere (dashed blue line) results in broader flexural effects and a wider zone of transition.

3.2 Example 2: can independent signals be summed?

The gravitationally self-consistent sea level methodology described above can be used to consider the impact of dynamic topography in isolation (as in the example of the last section), however, it also admits the possibility of contemporaneous dynamic topography fluctuations, ice mass changes and sediment fluxes. To date, studies have computed these effects independently and summed them up (e.g. Müller et al.2008; Rowley et al.2013). This assumes the processes are independent of one another, an assumption that does not necessarily hold. To consider the potential error introduced in the latter approach, we return to, and extend, the numerical example treated in the last section. In particular, we apply the same change in air-loaded dynamic topography as shown in Fig. 2(a) but in addition we assume a contemporaneous growth of an ice sheet in the far field of the U.S. East Coast that leads to a fall in sea level over the oceans of approximately 30m. These tandem effects are analogous to the example illustrated schematically in Fig. 1.

To begin, we perform a single, gravitationally self-consistent calculation that considers both processes, a change in dynamic topography and an increase in ice volume, together. The change in sea level computed for this case using our new sea level methodology is given in Fig. 5(a). Next, we perform two independent calculations. The first involves the perturbation in dynamic topography and no far-field change in ice volume, and the second includes the change in ice volume in the absence of a signal from dynamic topography. Both calculations are also based on our gravitationally self-consistent sea level methodology. We then sum the sea level changes obtained from these two calculations. Fig. 5(b) shows the difference between the latter ‘superposition’ procedure and the full calculation shown in Fig. 5(a).

Figure 5.

(a) Change in sea level near the U.S. East Coast in response to a perturbation in both dynamic topography and ice volume in the far field of the region, as computed using the gravitationally self-consistent sea level methodology described in the text. The white line shows the predicted location of the ancient shoreline. (b) Difference between the calculation shown in frame (a) and an approximate approach in which the sea level change due to both perturbations is computed separately using the same sea level method and the total response is calculated by summing the two effects.

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The error incurred in the approximate solution is greatest in areas of shoreline migration, but it extends beyond this region as a consequence of flexural effects. We can understand this trend by once again considering Fig. 1. The full solution accurately accounts for the total migration of the shoreline as a consequence of the tandem effects of a dynamic topography change and far-field drawdown of the sea surface (i.e. Figs 1a–d). However, the approximate solution yields a different initial shoreline for each process. In the example of Fig. 5, both processes act to move the shoreline oceanwards, and thus the approximate solution errs by not accounting for the full lateral extent of the shoreline migration (i.e. the migration of the shoreline at t₀ in Fig. 1a to the position at t₁ in Fig. 1d) and thus underestimates the water loading. The sea level change computed using the approximate treatment will predict a sea level fall of lower magnitude than the full solution, and this leads to the sign of the error in Fig. 5(b).

The amplitude of the error in the sea level prediction along the predicted location of the ancient shoreline is about 10 per cent of the sea level fall over the oceans due to far field ice growth (30 m). This is the level of error one would incur if the geological record were corrected for dynamic topography (using a gravitationally self-consistent calculation) and the residual used to infer the ice volume change. The error would of course be much larger if one had used the approximate treatments of sea level change due to dynamic topography (as illustrated in Fig. 4). Finally, differences in the location of the ancient shoreline, as predicted using the gravitationally self-consistent treatment of sea level change and the approximation in which the effects are computed independently, differ negligibly (compare the white lines in both frames of Fig. 5). We note, however, that this agreement largely reflects the relatively high topographic slope in the region of the ancient shoreline. A shallow slope would, of course, result in a greater discrepancy in the predicted shoreline migration, and this would in turn influence the amplitude of the error incurred in the approximate solution.

4 SUMMARY

In this study we have outlined a gravitationally self-consistent formalism for including dynamic topography and perturbations in the gravity field due to mantle convection in calculations of sea level change. Our formalism is based on ice age sea level theory (Mitrovica & Milne 2003; Kendall et al.2005; Dalca et al.2013) and indeed includes the possibility of ice mass changes and sediment redistribution. It further accurately accounts for shoreline migration, conservation of the mass of ice plus water and, of course, the mass of the solid Earth. (The latter is enforced by ensuring that the globally averaged radial deformation of the crust in response to the total surface mass load is zero.)

Our formulation is valid for arbitrary Earth models but we also consider the special case of a 1-D (depth varying) Earth model that maintains isostatic equilibrium throughout the simulation. In this case, the relevant response equations are based on fluid Love number theory. Numerical examples based on this special case have allowed us to explore the level of error incurred in previous predictions of sea level change that (1) assume a simplified amplification of air-loaded dynamic topography to account for the influence of water cover and (2) consider the impact of ice mass changes and dynamic topography on sea level in isolation. These demonstrate that the error associated with the simplified treatment of the amplification of dynamic topography can vary from ∼−15 per cent to ∼25 per cent of the predicted sea level change. Superimposing ice mass and dynamic topography effects introduces an additional error of 10 per cent in the ice volume estimate. Rowley et al. (2013) have argued for a dynamic topography change in water-loaded dynamic topography of up to 50 m along the U.S. East Coast since the mid-Pliocene climate optimum. In this case, the two sources of error we have identified might lead to an error of ∼10 m in any effort to use geological data from this region to infer sea level equivalent changes in ice mass flux over the last 3 Myr. This level of error is significant and comparable to the total eustatic sea level change associated with the collapse of the Greenland and West Antarctic Ice Sheets.

As interest in ice age sea level change and ice sheet stability broadens from the usual focus on Holocene and modern sea level to encompass records of early Pleistocene and Pliocene age, traditional glacial isostatic adjustment analyses must account for the sea level signal associated with mantle dynamics. The methodology presented here is an attempt to incorporate the latter signal, and in particular dynamic topography and associated geoid fluctuations, into the gravitationally self-consistent theory of ice age sea level change that dates to the pioneering work of Farrell & Clark (1976).

We thank two anonymous reviewers for their constructive comments on this manuscript, as well as Alessandro Forte, Rob Moucha and Dave Rowley for many discussions related to the ideas in this paper. We acknowledge funding from Harvard University and the Harvard University Center for the Environment.

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32

APPENDIX A: EXPRESSIONS FOR |$\Delta \mathcal {SL}$| IN THE CASE OF SPHERICALLY SYMMETRIC EARTH MODELS

This appendix derives the relevant equations, based on Love number theory, for a prediction of the change in global sea level due to the surface mass loading of a spherically symmetric Earth model. The derivation expands the theory described in Kendall et al. (2005) and Dalca et al. (2013) to include dynamic topography and prescribed perturbations in the geoid. However, we only treat the isostatic equilibrium response, that is, we assume that the viscous response has fully decayed. We begin with the case of a non-rotating Earth.

A1 Non-rotating Earth model

The deformational and gravitational response of a spherically symmetric Earth model to a surface mass load can be written as a space–time convolution of the appropriate Green's function with the surface load. In particular, the response Δχ can be expressed as:

\begin{equation} \Delta \chi = \int _{-\infty }^t \! \int \!\!\!\! \int _{\Omega }\Delta L(\theta ^{\prime }, \varphi ^{\prime }, t^{\prime }) \cdot {\rm GF}(\gamma , t-t^{\prime }){\rm d}\Omega ^{\prime } {\rm d}t^{\prime }, \end{equation}

(A1)

where GF denotes the Green's function, θ and φ represent the colatitude and east longitude, and γ is the angle between the observation point (θ, φ) and the load point (θ^′, φ^′). This angle can be calculated as:

\begin{equation} {\cos ({\gamma}) = \cos (\theta) \cos ({\theta} ^{\prime}) + \sin (\theta) \sin (\theta ^{\prime}) \cos ({\varphi} - {\varphi}^ \prime).} \end{equation}

(A2)

If we denote the Green's function for the radial displacement of the solid surface and the gravitational potential perturbation by Γ(γ, t) and ϕ(γ, t), respectively, then these responses can be expressed as:

\begin{equation} \Delta \mathcal {G} = \int _{-\infty }^t \! \int \!\!\!\! \int _{\Omega }\Delta L(\theta ^{\prime }, \varphi ^{\prime }, t^{\prime }) \cdot \frac{\phi (\gamma ,t-t^{\prime })}{g} {\rm d}\Omega ^{\prime } {\rm d}t^{\prime } \end{equation}

(A3)

\begin{equation} \Delta R = \int _{-\infty }^t \! \int \!\!\!\! \int _{\Omega }\Delta L(\theta ^{\prime }, \varphi ^{\prime }, t^{\prime }) \cdot \Gamma (\gamma ,t-t^{\prime }) {\rm d}\Omega ^{\prime } {\rm d}t^{\prime }, \end{equation}

(A4)

where ΔL is given by eq. (12). Using these expressions in eq. (16) gives

\begin{eqnarray} \Delta \mathcal {SL}_j &=& \int _{-\infty }^t \! \int \!\!\!\! \int _{\Omega }\Delta L(\theta ^{\prime }, \varphi ^{\prime }, t^{\prime }) \cdot \left[ \frac{\phi (\gamma ,t-t^{\prime })}{g} - \Gamma (\gamma ,t-t^{\prime }) \right]\nonumber \\ &&\times\, {\rm d}\Omega ^{\prime } {\rm d}t^{\prime } - \Delta H_j - \Delta DT_{A,j} - \Delta I_j + \Delta \mathcal {G}^{mc}_j. \end{eqnarray}

(A5)

The Green's functions in the above equations can be written in terms of the viscoelastic k and h Love numbers at spherical harmonic degree l (Peltier 1974; Tromp & Mitrovica 1999):

\begin{equation} k_l(t) = k_l^E \delta (t) + \sum _{k=1}^K r^{\prime l}_k e^{-s^l_k t} \end{equation}

(A6)

\begin{equation} h_l(t) = h_l^E \delta (t) + \sum _{k=1}^K r^l_k e^{-s^l_k t}, \end{equation}

(A7)

where the response is decomposed into an instantaneous elastic contribution and a discrete set of viscous decays.

As we have noted, our calculations will assume that the viscous response has fully decayed. In this case, our expression for the perturbation in global sea level can be simplified by expressing the surface mass load as having a single Heaviside step function time dependence, and performing the time convolution between the step function and the viscoelastic Green's functions analytically (where the latter is expressed in terms of the viscoelastic Love numbers). At infinite time, this convolution yields the following time-independent Green's functions:

\begin{equation} \phi (\gamma ) = \frac{a g}{M_e} \sum _{l=0}^{\infty } [1 + k_l^{f} ] P_l (\cos \gamma ) \end{equation}

(A8)

\begin{equation} \Gamma (\gamma ) = \frac{a}{M_e} \sum _{l=0}^{\infty } h_l^{f} P_l (\cos \gamma ), \end{equation}

(A9)

where a and M_e are the radius and mass of the Earth, respectively, and P_l is the Legendre polynomial at spherical harmonic degree l. In these expressions, |$h_l^{f}$| and |$k_l^{f}$| are the so-called fluid Love numbers that have the form (Peltier 1974, 1976):

\begin{equation} k_l^f = k_l^E + \sum _{k=1}^K {r^{\prime l}_k} /{s^l_k} \end{equation}

(A10)

\begin{equation} h_l^f = h_l^E + \sum _{k=1}^K {r^l_k}/{s^l_k}. \end{equation}

(A11)

Using these equilibrium forms, and the expression (12) for the change in surface mass load, yields the following version of eq. (A5) for the change in global sea level:

\begin{eqnarray} \Delta \mathcal {SL} &=& \frac{a}{M_e} \sum _{l=0}^{\infty } E_l \int \!\!\!\! \int _{\Omega }[\rho _W \Delta S(\theta ^{\prime },\varphi ^{\prime }) + \rho _I \Delta I (\theta ^{\prime },\varphi ^{\prime }) \nonumber \\ &&+\, \rho _H \Delta H(\theta ^{\prime },\varphi ^{\prime })] \cdot P_l(\cos \gamma ) {\rm d} \Omega ^{\prime } - \Delta H \nonumber \\ &&-\,\Delta I - \Delta DT_{A}+ \Delta \mathcal {G}^{mc}, \end{eqnarray}

(A12)

where

\begin{equation} E_l=1+k_l^{f}-h_l^{f}. \end{equation}

(A13)

Note that under the assumption of isostatic equilibrium, computing the perturbation in sea level between any two time steps requires only the total difference in the load across the interval and thus, with no loss of generality, we consider only a single time step and drop the subscript j.

We can simplify the spatial convolution in eq. (A12) by introducing spherical harmonic basis functions. A general spherical harmonic decomposition of a scalar field χ can be written as:

\begin{equation} \chi (\theta ,\varphi ) = \sum _{l=1}^{\infty } \sum _{m=-l}^l \chi _{lm} Y_{lm}(\theta ,\varphi ) \end{equation}

(A14)

where the basis functions are normalized in the following way

\begin{equation} {\int \!\!\!\! \int _{\Omega}Y_{l^{\prime},m^{\prime}}(\theta ,\varphi ) Y^*_{lm} (\theta ,{\varphi}) \sin {\theta} {\rm d}\theta {\rm d}\varphi = 4 {\pi} {\delta} _{l^{\prime },l} \delta _{m^{\prime},{m}}.} \end{equation}

(A15)

In this equation, the asterisk denotes the complex conjugate. In this case one can show that the following relationship holds (Mitrovica & Peltier 1991)

\begin{equation} {\int \!\!\!\! \int _{\Omega }\chi (\theta ^{\prime },\varphi ^{\prime }) P_l (\cos \gamma ) {\rm d}\Omega ^{\prime } = \frac{4 \pi a}{2l+1} \sum _{m=-l}^l \chi _{lm} Y_{lm}(\theta ,{\varphi}). } \end{equation}

(A16)

We may now use the spherical harmonic formulation from eqs (A14) and (A16) to rewrite eq. (A12):

\begin{eqnarray} \Delta \mathcal {SL} &=& \sum _{lm} T_l E_l [ \rho _W \Delta S_{l,m} + \rho \Delta I_{l,m} + \rho _H \Delta H_{l,m}] \cdot Y_{l,m}(\theta ,\varphi )\nonumber \\ &&- \sum _{lm} \Delta H_{l,m} Y_{l,m} (\theta ,\varphi ) - \sum _{lm} \Delta I_{l,m} Y_{l,m} (\theta ,\varphi )\nonumber \\ && - \sum _{lm} \Delta DT_{A;l,m} Y_{l,m} (\theta ,\varphi ) + \sum _{lm} \Delta \mathcal {G}^{mc}_{l,m} Y_{l,m} (\theta ,\varphi ), \end{eqnarray}

(A17)

where the subscript l, m refers to a spherical harmonic coefficient of degree l and order m,

\begin{equation} T_l=\frac{4 \pi a^3}{(2l+1) M_e}, \end{equation}

(A18)

and we adopt the short form:

\begin{equation} {\sum _{lm} = \sum _{l=0}^{\infty} {\sum}_{m=-l}^l. } \end{equation}

(A19)

The fully spectral form of eq. (A17) is

\begin{eqnarray} \Delta \mathcal {SL}_{l,m} &=& T_l E_l [\rho _W \Delta S_{l,m} + \rho _I \Delta I_{l,m} + \rho _H \Delta H_{l,m}] - \Delta H_{l,m}\nonumber \\ && -\, \Delta I_{l,m} - \Delta DT_{A;l,m} + \Delta \mathcal {G}^{mc}_{l,m}. \end{eqnarray}

(A20)

This equation describes the spatially variable component to sea level change. The spatially invariant component that follows from volume conservation (19) can also be written in terms of a spherical harmonic expansion:

\begin{equation} \frac{\Delta \Phi }{g} = \frac{1}{C_{0,0}} (- \frac{\rho _I}{\rho _W}\Delta I _{0,0}- RO_{0,0} +TO_{0,0}). \end{equation}

(A21)

Here we have made use of the following property

\begin{equation} {\int \!\!\!\! \int _{\Omega} \chi (\theta ,{\varphi}) {\rm d} {\Omega} = {\chi}_{00}. } \end{equation}

(A22)

We also used the following projections

\begin{equation} RO = \Delta \mathcal {SL} \cdot C = \sum _{lm} RO_{l,m} Y_{l,m}(\theta ,\varphi ) \end{equation}

(A23)

and

\begin{equation} TO = T_0[C - C_0] = \sum _{lm} TO_{l,m}Y_{l,m}(\theta ,\varphi ), \end{equation}

(A24)

where C represents the final ocean function.

A2 Rotating Earth model

Changes in the Earth's surface mass load (ice, ocean, sediments) and convective motions in the interior perturb the inertia tensor of the Earth, and thus the magnitude and orientation of the Earth's rotation axis. A change in the rotation vector, and the perturbation in the centrifugal potential that this change implies, impacts global sea level (Han & Wahr 1989; Milne & Mitrovica 1996, 1998; Mound & Mitrovica 1998). In this section, we review how this rotational feedback into global sea level can be incorporated into the Love number theory outlined in Appendix A1.

Convective perturbations to Earth rotation can be computed from the same viscous flow models of mantle convection used to compute dynamic topography and geoid anomalies, and a variety of approximate methods have been derived for this purpose (Ricard et al.1993; Steinberger & O'Connell 2002; Chan et al.2011). In ice age research, considerations of rotational stability deepen the integral nature of the sea level calculation since perturbations in the rotation vector are impacted by sea level changes and act, in turn, to perturb sea level. This rotational feedback can be included by augmenting eq. (A20) to include a term associated with the sea level response to the perturbed centrifugal potential (Milne & Mitrovica 1998). In the present case, this additional term will include a signal associated with convection, which is prescribed at the outset, and a surface mass loading term that will be successively improved through the iteration procedure described above. The latter requires a method for predicting the perturbation to the rotation vector associated with a surface mass loading (Mitrovica et al.2005). To be consistent with the assumption applied in Appendix A1, the ice age component of this calculation would treat the case of isostatic equilibrium.

The time-varying centrifugal potential Λ(θ, φ, t) can be decomposed into an initial value prior to the onset of loading and a perturbation from this state due to loading:

\begin{equation} \Lambda (\theta ,\varphi ,t) = \Lambda (\theta ,\varphi ,t_0) + \Delta \Lambda (\theta ,\varphi ,t). \end{equation}

(A25)

The perturbation to the centrifugal potential, also known as the rotational driving potential, may be described as the difference between two ellipsoidal forms with distinct amplitude and orientation, and it will involve only spherical harmonic coefficients of degree zero and two. This term can therefore be written as (e.g. Milne & Mitrovica 1996):

\begin{equation} \Delta \Lambda (\theta ,\varphi ,t) = \Delta \Lambda _{0,0} Y_{0,0} (\theta ,\varphi ) + \sum ^2_{m=-2} \Delta \Lambda _{2,m} Y_{2,m}(\theta ,\varphi ). \end{equation}

(A26)

In analogy to the case of surface mass loading, fluid Love numbers exist that describe the deformation of the solid surface and the gravitational equipotential surface due to a changing rotational driving potential. These have the form (Peltier 1974, 1976):

\begin{equation} k_l^{f,T} = k_l^{T,E} + \sum _{k=1}^K r^{\prime l,T}_k /{s^l_k} \end{equation}

(A27)

\begin{equation} h_l^{f,T} = h_l^{T,E} + \sum _{k=1}^K r^{l,T}_k /{s^l_k}. \end{equation}

(A28)

Using these expressions, eq. (A20) is revised in the following manner to include rotational feedback:

\begin{eqnarray} \Delta \mathcal {SL}_{l,m} &=& T_l E_l [\rho _W \Delta S_{l,m} + \rho _I \Delta I_{l,m} + \rho _H \Delta H_{l,m}]\nonumber \\ && +\, \frac{1}{g} E_l^T\Delta \Lambda _{l,m} - \Delta H_{l,m} - \Delta I_{l,m} - \Delta DT_{A;l,m} + \Delta \mathcal {G}^{mc}_{l,m},\nonumber \\ \end{eqnarray}

(A29)

where

\begin{eqnarray} E_l^T=1+k_l^{f,T}-h_l^{f,T} \end{eqnarray}

(A30)

All other equations described in Appendix A1 remain unchanged.

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Article Contents

Calculating gravitationally self-consistent sea level changes driven by dynamic topography

Abstract

1 INTRODUCTION

2 CALCULATING LONG-TERM SEA LEVEL CHANGE

2.1 Calculating dynamic topography from mantle convection

2.2 Sea level changes in the presence of dynamic topography

2.2.1 Some special cases

2.3 Schematic illustration of sea level physics in the presence of dynamic topography

2.4 Including plate motion

3 APPLICATIONS

3.1 Example 1: isostatic amplification of dynamic topography

3.2 Example 2: can independent signals be summed?

4 SUMMARY

REFERENCES

APPENDIX A: EXPRESSIONS FOR |$\Delta \mathcal {SL}$| IN THE CASE OF SPHERICALLY SYMMETRIC EARTH MODELS

A1 Non-rotating Earth model

A2 Rotating Earth model

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Article Contents

Calculating gravitationally self-consistent sea level changes driven by dynamic topography

Abstract

1 INTRODUCTION

2 CALCULATING LONG-TERM SEA LEVEL CHANGE

2.1 Calculating dynamic topography from mantle convection

2.2 Sea level changes in the presence of dynamic topography

2.2.1 Some special cases

2.3 Schematic illustration of sea level physics in the presence of dynamic topography

2.4 Including plate motion

3 APPLICATIONS

3.1 Example 1: isostatic amplification of dynamic topography

3.2 Example 2: can independent signals be summed?

4 SUMMARY

REFERENCES

APPENDIX A: EXPRESSIONS FOR |$\Delta \mathcal {SL}$| IN THE CASE OF SPHERICALLY SYMMETRIC EARTH MODELS

A1 Non-rotating Earth model

A2 Rotating Earth model

Citations

Views

Altmetric

Email alerts

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