The viscoelastic solid-Earth and sea level response to the redistribution of surface mass in the cryosphere, ocean, terrestrial water and sediments has a numerical modeling history of nearly six decades . This response is typically expressed in terms of bedrock deformation, changes in relative sea level, gravitational potential, geocentric motion, and Earth's rotation parameters. These are collectively referred to as the Gravity, Rotation and Deformation (GRD) signals . GRD models have been used in problems where the loading time scale ranges from thousands of years often referred to as Glacial Isostatic Adjustment or GIA, to sub-seasonal e.g., and applied on regional e.g. to global scales . As such, GRD models have proven instrumental in better interpreting the signals recorded in geologic and geodetic data, such as past sea level indicators, tide gauges, Global Navigation Satellite Systems (GNSS), terrestrial and space gravimetry, and satellite altimetry. Outstanding differences between GRD models generally stem from three distinct sources e.g.: (1) the selection and curation of constraining datasets; (2) treatment of solid Earth structure and mantle rheology; and (3) various refinements to the loading history. The presented Love number solver enables systematic investigation of model disparities related to the radial solid-Earth structure and mantle rheology.

The emergence of modeling approaches that couple ice sheet dynamics with GRD processes has revealed important feedback mechanisms impacting the grounding line migration and marine ice sheet instability on decadal and longer timescales e.g.. Our main motivation is to enable the Ice-sheet and Sea-level System Model (ISSM) to tackle such coupled problems and enhance its sea-level projections framework . Our capability is also adapted to lower complexity models wherein the ice history is prescribed a priori e.g. using the ICE-6G model from, which may be considered as a one-way coupling problem.

In this paper, we document the recently-coded Love number solver in the ISSM framework available at 10.5281/zenodo.13984073, capable of accounting for a range of linear viscoelastic rheologies for the mantle and lithosphere. We report our numerical approach and optimization strategy. By employing higher-order linear viscoelasticity that predicts rapid deformation at short time scales e.g., this new ISSM capability may be used to examine rapid ice sheet-solid Earth interactions on time scales of months to decades. The capability also facilitates studying general interactions between the Hydrosphere and Solid Earth as well as GRD phenomena in other planetary bodies. It may also be adapted to investigate a host of problems involving tidal dispersion and dissipation in Earth and elsewhere.

New model approaches that deviate from the surface load Love number formulation are used to include a laterally variable Earth rheology . These new methods combine prior information on the radial dependence of mantle rheology with lateral viscosity contrast estimated from seismic velocity anomalies. While poised to become the standard in GRD modeling at sometime in the future, 3D Earth GRD models are computationally costly compared to approaches that approximate the Earth structure to be only radially dependent, a necessary ingredient of the Love number approach. This currently limits our ability to evaluate uncertainty on Earth parameters. We note that potential solutions are explored in the community in the form of emulators and sensitivity kernels . Love number approaches still remain a powerful method to explore the background rheology profile, and to establish a robust prior for its parameters owing to their low computational cost. This is particularly beneficial for frameworks that rely on ensemble modeling to comprehensively approach interdisciplinary problems such as projections of sea-level change . Recent improvements to the theory and numerical implementation of the Love number problem include new approaches to time-dependent response to tidal and surface loading as well as rigorously accounting for the adiabaticity of mantle thermal structure . Furthermore, a new formulation of a linear rheology, the Extended Burgers rheology (EBM), featuring both transient and steady-state relaxation, have been documented by , , , . Work parallel to our new ISSM core has recently been published by who also solves the Love number problem with a higher-order linear viscoelasticity. The latter work does not pursue EBM rheology, but rather a structurally similar formulation developed by based upon experiments with granular borneol. , and the work presented here, represent a new and exciting direction for radially stratified models of GRD processes that occur during the Little Ice Age and Anthropocene and that should be included in solid Earth-ice interaction models of future sea-level change. As these new rheological models emerge, we believe that it is first important to understand what observational constraints can be put on transient relaxation parameters at the lowest level of model complexity.

One of the traditional goals of GIA modeling is to determine the spatial dependence of the effective Newtonian viscosity that characterizes the mantle circulation time scales t≥10 million years, e.g.. While have recently pointed out that an improved reconciliation among various inferences of lithospheric thicknesses over ten thousand to hundred million year time scales may be achieved by invoking frequency dependent rheology, there is general consensus on the internal consistency of sub-lithospheric viscosity structure based on both GIA and mantle convection. However, geodetic observations of load-related vertical motions of the crust following load changes during the past 100 years suggest much lower values of the effective viscosity e.g. and this may be related to the fact that this short time-scale is not recording information about long-term viscosity e.g.. More convincing evidence for transient rheology comes from the geodetic observations of time-dependent, widespread crustal motions following large earthquakes. The later observations have been modeled using the simple Burgers rheology, from which a value of the amplitude factor Δ¯ may be deduced. By using the linear constitutive EBM equations that have been proposed to explain laboratory experimental data e.g. we may employ parameter values that are consistent with either post-seismic studies of high temperature and pressure laboratory studies, or to both. In Table  we present some of the values of the transient relaxation strength parameter from tidal, post-seismic and laboratory experiments for both Δ¯ and Δ.

Transient rheology may also be necessary to explain time-varying features of geodesy that are global in scale, such as the ongoing changes in the Earth shape, J2˙ e.g.. At a global scale there exist abundant information associated with the response of the Earth to body tides. Theories for the frequency-dependent dissipative processes interior to planet-moon systems have long been sought to reconcile the astronomically based necessity of low Q/high dispersion with high temperature and pressure laboratory data e.g.. Over the past 50 years a frequency-dependent Andrade rheology has been employed in the study of planetary Q e.g.. However, it has become clear that EBM models are also applicable to the study of tidal dispersion and dissipation e.g.. Therefore, our new ISSM Love number computational capability can also support the development of planetary interior models that satisfy tidal observations and simultaneously provide prediction of a long-term viscosity.

A basic assertion of the Andrade model is that the creep function takes a form that includes a term with time-dependence, βtn, with n≈1/3 and β a coefficient e.g.. Hence, developing models consistent with a long-term viscosity are excluded. One of our future goals will be to couple ice sheet grounding line evolution to motions of the bedrock topography, as the latter influence ice sheet mass balance projections, and as such, the coupled ice sheet model may require updating solid Earth viscoelastic response on a sub annual time scale. In such models it will be important to have a reference long-term viscosity incorporated into the model. In summary, EBM models find support from models of both post-seismic geodetic observations , terrestrial and planetary tides and may additionally be connected to long-term viscosity inferences e.g..

Our main focus in this paper is to develop a toolset that enables the coupling of surface mass transport models (e.g. ice flow models) and GRD models with a variety of linear rheologies. Thus, our GRD model uses a Love number framework with radially symmetric Earth representation. Specifically, we target the following key capabilities for our solver:

The Love numbers are expressed as the time-dependent response to a Heaviside forcing function, thus facilitating the coupling with surface mass transport model through an explicit time-stepping framework.

In the following sections we lay out the theory underlying the Love number system, explain our approach to both implementing and optimizing the numerical resolution of that system, and assess the efficiency and accuracy of our solver with respect to the community benchmark of .

At the heart of the Love number problem is an ordinary differential equation system first described in its viscoelastic form by . It is derived by combination of equations governing the conservation of mass and linear momentum and Poisson's equation: 1∂ρ∂t=-∇⋅(ρu¯)2ρ∂2u¯∂t2=∇⋅Σ¯¯+ρ∇(ΦE+S)3ΔΦE=-4πGρ Here u¯ represents the displacement vector of solid-Earth material at any location, ΦE the Earth's gravity potential (excluding any surface loading), ρ is the solid-Earth material density, Σ¯¯ is the stress tensor, S is the gravity potential of the external forcing, G is Newton's gravitational constant and and t the time coordinate (see Table  for the list of symbols used in this paper). In these balance equations we include both the perturbed and unperturbed states, respectively indicated with subscripts 1 and 0. The unperturbed state balance is: 4∇P0(r)=-ρ0(r)g0(r)e¯r for the spherically symmetric static Earth, with P0, ρ0 and g0 representing the hydrostatic pressure, density and gravity, r the radius variable and e¯r the unit vector in the radial direction. The model static potential is 5∇2ΦE0(r)=-4πGρ0(r) where we define the separation of total potential by 6ΦE=ΦE0(r)+ΦE1(t,r,θ,φ) and density by 7ρ=ρ0(r)+ρ1(t,r,θ,φ). Here perturbed variables ΦE1 and ρ1 represent small anomalies to the unperturbed state induced by the forcing. Note that Eqs. () and () may be used to construct viscoelastic models for any terrestrial-like planetary body. We also note that the unperturbed gravity may be constructed from: 8g0(r)=4πGr2∫0rr′2ρ0(r′)dr′, with G Newton's gravitational constant. Furthermore, the perturbed density is also recoverable from the mass conservation equation as 9ρ1=-ρ0∇⋅u¯-u¯⋅∂ρ0∂re¯r. For brevity we shall drop the subscript “1” from ρ1 in what follows. For completeness we also note that the perturbed pressure field is 10P1=ρ0(r)g0(r)u¯⋅e¯r. Note that we have yet to specify the relationship between Σ¯¯ and u¯ as it depends on the choice of rheological model employed for solid Earth material.

The problem is simplified by two approximations. First, the deformation regime is considered to be limited to small pertubations of the system, i.e. cross-terms δuδv can be neglected in expressions with the form u.v=(u0+δu)(v0+δv)=u0v0+u0δv+v0δu+δuδv, wherein u0 and v0 refer to variables (displacement, gravity potential, density, etc) in the initial state of the system. Second, the Earth structure is assumed to be radially symmetric (i.e. with no lateral variations) and the material to be isotropic.

The full system of perturbation equations is fully derived by . The resulting system involves 6 radially-dependent variables (and their first radial derivatives) conventionally noted yi(r) for 1<i<6 : 11u¯(r,θ,φ,t)=a∑n=1nmax⁡∑m=0ny1,n(r,t)Ynm(θ,φ)e¯r+y3,n(r,t)∂Ynm(θ,φ)∂θe¯θ+∂Ynm(θ,φ)sin⁡θ∂φe¯φ,12Σ¯¯⋅e¯r=μ̃∑n=1nmax⁡∑m=0ny2,n(r,t)Ynm(θ,φ)e¯r+y4,n(r,t)∂Ynm(θ,φ)∂θe¯θ+∂Ynm(θ,φ)sin⁡θ∂φe¯φ,13ΦE1(r,θ,φ,t)=gSa∑n=1nmax⁡∑m=0ny5,n(r,t)Ynm(θ,φ),14gr(r,θ,φ,t)=gS∑n=1nmax⁡∑m=0ny6,n(r,t)Ynm(θ,φ), where θ and φ are the colatitude and longitude, respectively. The yi variables represent the spherical harmonic coefficients of the non-dimensionalized vertical displacement (y1), vertical traction (y2), tangential displacement (y3), tangential traction (y4), gravitational potential (y5) and radial gravity (y6). The scaling constants associated with the yi variables are a, μ̃, a, μ̃, gSa and gS, respectively, with μ̃ an arbitrary shear modulus value. Note that throughout this manuscript we employ spherical harmonics functions, Ynm(θ,φ), with n and m the spherical harmonic degree and order, respectively. The harmonic truncation degree nmax⁡ depends on desired applications, and often ranges from 2 (tidal applications) to over 500 (high-resolution regional loading applications). For the derivation of the y-system of equations these may either be defined with complex or real forms. The real form is: 15Ynm(θ,φ)=NnmPnm(θ)[cos(mφ)+sin(mφ)], commonly used in geodesy. Here the normalization factor Nnm is as defined in , 16Nnm≡(2n+1)(n-m)!(2-δ0m)(n+m)!, with associated Legendre polynomials Pnm(θ) as defined in (see Table 12.3 therein), and δ0m=1 if m=0, δ0m=0 otherwise. These harmonics have the advantage that they are also used in contemporary GRACE and GRACE-FO Level 2 gravity releases e.g.. Complex harmonics that are generally more useful for wave analysis are also a viable option e.g. for deriving the y-system of ordinary differential equations, the same as those derived by .

Essential to the decomposition is the orthogonality of the spherical harmonics. The expansion: 17N(θ,φ,t)ur(θ,φ,t)uE(θ,φ,t)uN(θ,φ,t)=∑n=1nmax⁡∑m=0nSnm(t)gS∗δL+kn(t)hn(t)ln(t)1sin⁡(θ)∂∂φ-ln(t)∂∂θYnm(θ,φ), allows the solution of the partial differential equations system to be reduced to a numerical problem for ordinary differential equations. Here, ∗ indicates a temporal convolution, N the geoid, ur, uE and uN are the radial, East and North surface displacement components, and h, l and k are Love numbers. For common applications on the Earth system, we typically expect δL=0 for tidal forcing; δL=1 for surface loading: the geoid N reflects the gravity field of both the solid-Earth and the surface load; however, for tidal deformation, excluding the direct term δL means that the forcing (from the Moon, Sun or other celestial bodies) is not itself counted as the Earth's gravity field.

The Love numbers h(t), k(t) and l(t) are the time-variable coefficients that describe this linear relationship, directly obtained from y1(t,a), y5(t,a), y3(t,a), respectively. Our end goal is to determine their values for a given structure and rheology of the solid-Earth. To this end, it is convenient to resolve the system in the Laplace domain as opposed to the time domain. There, for viscoelastic models such as Maxwell, Burgers and the Extended Burgers Material, the rheology equation can be expressed as a linear relationship between the stress and strain tensors, analogous to the elastic problem see the correspondence principle, : 18Σ¯¯(s)=λ(s)∇⋅u¯(s)+2μ(s)ε¯¯(s), where ε¯¯ is the strain tensor obtained from u¯ and its spatial derivatives. Table lists the expression of μ(s) for different rheology models, and the bulk modulus κ is assumed to be independent of frequency , thus λ(s)=κ-23μ(s).

Here we list the equations governing the yi system at the Earth's center, surface and interfaces between layers.

At the Earth surface (r=a), Ly4=0, Ly1 and Ly3 are free (free surface hypothesis), and Ly2, Ly5, Ly6 are related to the surface forcing S by the following equations. For n>1 and if the forcing type is surface loading (δL=1): 19y2(a)=-2n+13ρEμ̃Snm20y6(a)+n+1ay5(a)=2n+1gsaSnm with ρE the mean Earth density and a the Earth surface radius. For n>1 and if the forcing type is tidal (δL=0), there is no radial traction related to the weight of the load, thus: 21y2(a)=022y6(a)+n+1ay5(a)=2n+1gSaSnm

When n=1, in both cases, the system degenerates and we must impose y5=0 as an additional boundary condition to fix the center of the frame at the center of mass of the system (including the whole solid-Earth Earth and any surface load). This is because the degree one term of the gravity potential has the unique property to be directly linked to the position of the center of mass in the frame.

Considering that the Love numbers represent the proportionality coefficients between the forcing S and the resulting deformation, we set Snm=1 in the above equations and subsequently in our approach. This yields: 230001000100000000n+1a1Ly¯(a)=0-δL2n+13ρEμ̃δ1n2n+1gSa

We use the incompressible homogenous solid sphere solution (hereafter referred to as the inner sphere solution) from to effectively create boundary conditions for y3,y4,y5 at r=rinner (the other 3 variables must be null within the inner sphere because they follow terms in r-n that would otherwise diverge at r=0). We employ this method because the numerical approach used for the propagation in other layers leads to divergent terms.

We employ this method because the numerical approach used for the propagation in other layers leads to divergent terms. At r=rinner: 27raA1n+3n(n+1)ra2μμ̃n+2n+104πρIGrgS1raA21nra2μμ̃n-1nr204πρIGrgS0-ρIμ̃001gSangS1y¯(r)=0¯, with ρI is the average density below rinner, A1=2μμ̃(n-1-3n)+4πρI2G3μ̃r2 and A2=2μμ̃n-1r2+4πρI2G3μ̃.

In a solid layer, the following relation exists between yi variables and their radial derivatives : 28y¯˙=D¯¯y¯ with the derivative coefficient matrix:

29D¯¯=-2λx(λ+2μ)μ̃λ+2μn(n+1)λx(λ+2μ)…-4ρgax-12μ(λ+23μ)λ+2μ1μ̃x2-4μx(λ+2μ)ρgax-6μ(λ+23μ)λ+2μn(n+1)μ̃x2…-1x01x…1μ̃x2ρgax-6μ(λ+23μ)λ+2μ-λλ+2μμμ̃22μn(n+1)-1+λ2n(n+1)-1(λ+2μ)x2…4πρGagS00…00-4πρGagSn(n+1)x……000…n(n+1)x0-ρρ̃…μ̃μ00…-3x-ρρ̃x0…001…0n(n+1)x2-2x where μ=μ(s) and λ=λ(s) are parameterized via their rheological expressions in Table . In this step, we aim to use the above expression to derive the relationship between the yi values at the bottom and the top of layer j: 30jy¯(rtop)=jB¯¯jy¯(rbottom)

Matrix B¯¯ sometimes referred to as the fundamental or propagation matrix, e.g. is a 6-by-6 coefficient matrix that can be used to propagate yi values from rbottom to rtop within the layer, discretized along a radial grid with step Δr. The propagation matrix is computed numerically via the following method:

Let r1:=rbottom at the bottom of the layer, q:=1 and yi=q(r1):=1 and yi≠q(r1):=0.

The outer core material is taken to be an inviscid fluid, therefore the radial traction y4=0 (no shear stress is supported), and y3 is undetermined. Under the hypothesis that the shape of the fluid core interfaces is described by gravity equipotential surfaces , y1 is no longer independent from y5 and y6, and y2 only reflects the change in lithostatic pressure which can be computed from y1. As a result, the system is reduced to a 2 independent variable system based on y5 and y6 (Eq. ), as in . A similar method to the previous section is then employed to propagate y5 and y6 through the outer core, the rest of the variables being recovered via boundary conditions laid out in Eqs. () and ().

Within the fluid core, the effective yi system is therefore: 312y5˙2y6˙=D¯¯2y52y6

with: 32D¯¯=4πρGag1-16πρGag+n(n+1)x1x-2x-4πρGag

In this step we are looking for a method to compute time-dependent Love numbers, h(t), k(t), and l(t) from the estimates of frequency-dependent Love numbers h(s), k(s), and l(s). Our approach here is motivated by our goal to employ Love numbers alongside mass transport models, where the coupled model alternatively computes the change of ice thickness, water column height, ocean pressure, etc., and the ensuing solid-Earth response. In particular, it has to be possible to incrementally compute the ongoing and evolving Earth response before the complete time series of surface mass changes is known. Our choice is to assume that the temporal form of the forcing potential is a unit Heaviside function such that H(t):=1 for t≥0, H(t)=0 for t<0. The forcing potential can then be discretized as follows: 33Snm(t)=∑tL=0tSnm(tL)H(t-tL) And thus with Heaviside Love numbers, the temporal convolution in Eq. () becomes: 34N(θ,φ,t)ur(θ,φ,t)uE(θ,φ,t)uN(θ,φ,t)=∑tL=0t∑n,mSnm(tL)gSH(t-tL)×δL+kn(t-tL)hn(t-tL)ln(t-tL)1sin(θ)∂∂φ-ln(t-tL)∂∂θYnm(θ,φ)

Several methods are available to perform this inverse Laplace transform, and the respective advantages and shortcomings of the main approaches used in GIA are discussed in . Here, we choose to employ a method known as the Post-Widder formulation, that has been successfully employed by for Love number computation. This method is a purely numerical approach to computing the inverse Laplace transform, based on the evaluation of successive derivatives of the spectral Love numbers . It does not require the exploration of the complex frequency plane, nor a priori of the relaxation modes. This makes it the candidate of choice to compute complex rheology structures we are interested in, and in particular non-Maxwellian rheologies such as the Extended Burgers Material. This is because we expect an infinite number of relaxation modes from the distribution governing the transient relaxation, as it corresponds to a continuous distribution of viscosities with associated time scales ranging from τL to τH .

On the other hand, this method is known to have drawbacks for compressible models. identify two types of gravitational instability affecting Maxwell compressible regimes: (1) dilatation modes, which depend on the ratio between Lamé parameters, can most affect low degrees and horizontal deformation; and (2) Rayleigh-Taylor instabilities, related to density stratification, impact models with few mantle layers the most. Using traditional normal mode approaches , one is normally able to identify and exclude unstable modes based on the sign of their relaxation time. This is not possible with the Post-Widder method as individual modes are not identified, and as a result, there is a chance that particular choices of frequency samples lead to numerical artifacts in the results if the system is solved near the frequency of an unstable mode. Mitigation strategies can be employed to reduce both the likelihood of encountering such modes and their amplitude. indicates that the input model for radial density and elastic parameter profiles such as PREM, may be checked to detect instabilities in the initial state and emphasizes the stabilizing role of an elastic lithosphere; note that finer layering of the Earth model increases the timescales of the instable modes, such that their contributions may be lessened at or below post-glacial deformation time scales.

The Post-Widder formulation is implemented as follows: 35f(t)M=∑p=12MξMpf(sp), where f is the function we wish to perform the inverse Laplace transform of any of the Love numbers, M indicates the maximum number of derivations of f considered, and the frequency samples necessary to estimate f at time t are given by sp=pln⁡(2)t. ξMp are the coefficients for the frequency samples to get the proper derivatives: 36ξMp=(-1)p+Mp∑q=Floor(p+12)min⁡(p,M)qM+1M!CqMCq2qCp-qq, with Cpq the binomial coefficient. This method is based on the Saltzer summation for inverse Laplace transform, through Eqs. (2), (6), (7) in , and is similar to the accelerated Gaver sequence employed by .

The main downside of this approach is its numerical stability, as repeated derivatives of estimated Love numbers is prone to the propagation of errors via catastrophic cancellation. point out that multiprecision environments, supporting up to 128 digits variables, can be helpful in mitigating this effect for larger values of M. Our implementation can be run with either double (16 digits) or quadruple (32 digits) precision. However, we find that the level of systematic errors associated with our method in the propagation step does not allow quadruple precision to improve the accuracy of our solutions with respect to the benchmark solutions. We also found that catastrophic cancellation was more susceptible to happen in near-elastic or near-fluid regimes where the frequency dependence is small. We have found that the ideal value of M is not the same for different values of t. Our approach is thus as follows. If Mmax⁡<4, we compute f(t)Mmax⁡ straightforwardly from Eq. (), as there are not enough iterations of M to apply our error growth detection method. Otherwise, for each value of t:

Compute f(t)1, f(t)2, f(t)3 and f(t)4. Assume M=4.

The polar motion transfer function (hereafter PMTF) M is the analog of Love numbers for the movement of the Earth spin axis. This displacement occurs in response to the same forcing and produces additional changes to the Earth gravity potential and bedrock motion known as the rotational feedback . Here we consider only the secular polar motion, neglecting the Chandler wobble as it produces no long-term trend. Following we have: 37M(s)=1+k2(s)1-k2T(s)/ks. with k2 the degree 2 loading Love number, k2T the degree 2 tidal Love number, ks the secular degree 2 tidal Love number. This function can then be expressed in the time domain as a Heaviside response M(t) via an inverse Laplace transform, by the same process highlighted above for Love numbers. In Fig.  we compare the results from our computation with respect to the benchmark of . We find satisfactory agreement with relative errors within 10-5 for t<100000 years (which is usually the maximum scale employed in GIA modeling). Therefore, in subsequent comparisons to the benchmark result we focus on reporting the relative error rather than the direct output in our figures, as differences in the left panel are not visible.