The dynamics of marine-terminating outlet glaciers are of fundamental interest in glaciology and affect mass loss from ice sheets in a warming climate. In this study, we analyze the response of outlet glaciers to different sources of climate forcing. We find that outlet glaciers have a characteristically different transient response to surface-mass-balance forcing applied over the interior than to oceanic forcing applied at the grounding line. A recently developed reduced model represents outlet-glacier dynamics via two widely separated response timescales: a fast response associated with grounding-zone dynamics and a slow response of interior ice. The reduced model is shown to emulate the behavior of a more complex numerical model of ice flow. Together, these models demonstrate that ocean forcing first engages the fast, local response and then the slow adjustment of interior ice, whereas surface-mass-balance forcing is dominated by the slow interior adjustment. We also demonstrate the importance of the timescales of stochastic forcing for assessing the natural variability in outlet glaciers, highlighting that decadal persistence in ocean variability can affect the behavior of outlet glaciers on centennial and longer timescales. Finally, we show that these transient responses have important implications for attributing observed glacier changes to natural or anthropogenic influences; the future change already committed by past forcing; and the impact of past climate changes on the preindustrial glacier state, against which current and future anthropogenic influences are assessed.

Marine-terminating outlet glaciers drain large portions of the Greenland and Antarctic ice sheets, conveying ice from interior catchments to the ocean. Their dynamic response to a changing climate is a critical component of projections of sea-level rise

Yet despite the clear signals of mass loss from the Greenland and Antarctic ice sheets

In this study, we focus on a principle common to all outlet glaciers: climate forcing predominantly comes either from the atmosphere, via changes to the surface mass balance in the interior catchment, or from the ocean, via changes to ice discharge at the marine margin. Our approach is to conduct idealized model experiments that isolate key physical principles of transient glacier responses to climate. We use a recently developed reduced model

We will focus exclusively on stable geometries. The marine-ice-sheet instability

Before describing the dynamic models, it is useful to begin with the geometry and basic flux-balance arguments of the system we are investigating. We consider an idealized stable outlet glacier, a schematic of which is presented in Fig.

Ice flux across the grounding line (

In this study, we consider the grounding line the boundary of the outlet-glacier system. That is, we consider floating ice a part of the boundary condition for the system's output flux (

To simulate the dynamics of this outlet-glacier system, we begin with a 1-D (flowline) version of the ice-sheet model developed by Pollard and Deconto (

A crucial simplification in the PD12 model is that flux across the grounding line is parameterized in the form of Eq. (

Figure

We begin by comparing the flowline model's response to forcing from either surface-mass-balance changes in the interior or ocean forcing at the terminus. In all model experiments throughout this study, “interior forcing” and “ocean forcing” will refer to perturbations applied in the following manner. Interior surface-mass-balance anomalies are assumed to be spatially uniform. We represent ocean forcing very simply by perturbing the grounding-line-flux coefficient,

Figure

As an alternative to an impulse forcing, we also consider the glacier's response to stochastic variability in either

Variability is intrinsic to climate, arising from the fundamentally chaotic nature of the natural system. The associated glacier fluctuations are a crucial part of characterizing glacier dynamics, and the implications have been extensively explored for mountain glaciers

Parameters used for flowline and two-stage models. Values for

Model schematic and responses to ocean forcing (via

Robel, Roe, and Haseloff (

The RRH model reduces the outlet glacier to a system with two degrees of freedom:

The model dynamics are derived by balancing ice fluxes through these linked reservoirs (Fig.

Two coupled equations capture the transient adjustment of the two degrees of freedom,

Because achieving a steady state requires adjustment of both

RRH also linearized these equations for fluctuations

Here we generalize the RRH linearization to simultaneously include perturbations to interior surface mass balance (

RRH showed that the two-stage model emulated the response of a flowline model forced with surface-mass-balance anomalies. Their flowline model

Figure

Figure

Comparing the transient response of flowline and two-stage models to ocean forcing (via

The two-stage model's agreement with the flowline model suggests that it captures the essential dynamics responsible for the contrasting transient responses to interior and grounding-line-flux anomalies. At this point, it is useful to discuss some of the interpretations enabled by this reduced model.

The linearized two-stage model (Eq.

These modes can also be conceptualized as a two-stage low-pass filter on any forcing time series (e.g., Fig.

While these mathematical interpretations may seem abstract, it is helpful to remember that the two-stage model was derived from mass conservation, and that the linear response times (eigenmodes) reflect the large-scale glacier geometry. Because a glacier is a Stokes (i.e., noninertial) fluid driven by potential-energy gradients, the large-scale glacier geometry must reflect the aggregate dynamics by which the system seeks flux balance. This relationship between geometry and dynamics is another way to interpret why

To help illustrate this, Fig.

Although the flowline model captures a more realistic and spatially distributed response, the two-stage model contains enough geometric information to emulate the basic sequence of flux anomalies. Note that more discretized interior reservoirs could be added to the two-stage model, which would eventually approach the form of the flowline model. However, a single reservoir could not capture the essential transient response. The distinct stages of the response to a perturbation in

Flux changes following interior vs. ocean forcing.

In the previous sections, we imposed stochastic variability with a flat power spectrum (i.e., white noise) because it allowed us to identify the influence of forcing type and model physics across all relevant frequencies. In reality, surface-mass-balance and ocean variability may exhibit different power spectra. Climate variables associated with the atmosphere (such as surface mass balance) often have little interannual memory

We consider four types of synthetic forcing with different persistence characteristics: (1) white noise; (2) and (3) first-order autoregressive (AR-1) processes with persistence timescales

We generate time series following

We normalize the forcing time series so that they all have the same variance. This ensures that, for a given choice of

Figure

The power spectra of the forcings and responses show why persistence has such a strong effect. The response to white-noise forcing reveals the system's sensitivity across all timescales (Fig.

The practical takeaway is that persistence in climate forcing increases the total variance of length fluctuations. When combined with the finding that terminus-flux anomalies excite the fast mode of response more than surface-mass-balance anomalies, the implication is that ocean variability – which tends to have persistence – may drive much larger terminus fluctuations than surface-mass-balance variability.

Figures 1–3 demonstrate that marine-terminating outlet glaciers have different transient responses to interior and ocean forcing, because of how the fast and slow modes respond in each case. In the next sections, we examine the consequences for three key issues: the committed response to forcing, the attribution of an observed change, and a glacier's memory of past climate changes. The relative roles of ocean and interior forcing will, of course, vary widely among individual glaciers. Rather than conducting simulations of specific settings, we will use our simplified model framework to outline the general implications and assess how the combination of fast and slow dynamics applies to each question.

The effect of climatic persistence on glacier fluctuations.

Any system with a nonzero response time will lag applied forcing. “Committed change” refers to the total response such a system would need to undergo to attain equilibrium with the current level of forcing. In the context of a warming climate, committed change is an important lower bound on future change that is independent of future emission scenarios. It has long been recognized that surface temperatures lag

To illustrate the current committed change in outlet glaciers, we use the two-stage model and follow the framework of

We consider ramp forcing scenarios where

The most basic result is that, in all cases, the transient response as of 2020 is a small fraction (

The slow response (interior thinning

The slow mode also means that there can be large uncertainties in committed change if the magnitude of forcing is uncertain. Figure

Parameters varied between three idealized glaciers (top) and the resulting steady-state values (bottom). The steady state is calculated by the nonlinear two-stage model, and the linearized response times are given by Eqs. (

Glacier responses to idealized climate forcing over the industrial era.

We now turn to the topic of attributing outlet-glacier retreats to natural or anthropogenic forcing. The attribution of an observed change to a particular cause (i.e., an external forcing) can be a challenge because of factors specific to individual glaciers, such as complexities in bed geometry, regional climate, or the local collection of ice–ocean interactions. It can also be a challenge because of factors intrinsic to the transient ice dynamics, which affect the amount of the forced response that can be expressed over a given time. We focus here on this latter set, and in particular on the contrasting implications of ocean vs. interior forcing.

Attribution is often framed in terms of the signal-to-noise ratio (SNR) of an observed trend. For a variable

For outlet glaciers, however, two types of forcing (ocean and interior) and much longer response times are at play. To understand how these factors interact on the timescales of historical anthropogenic forcing, we consider three idealized scenarios of stationary variability plus a trend (Fig.

Figure

In case 1, noisy surface mass balance drives small length fluctuations, but the very slow response to the trend in

These idealized cases are useful for understanding factors that affect the detectability of a length trend, but they also demonstrate that the SNR may be a problematic metric in a practical sense. Because most of the natural glacier variability is expressed at very low frequencies,

An alternative approach is to evaluate an observed

For variability in

As

The widely separated dynamical timescales characteristic of outlet glaciers pose a unique challenge for understanding their modern changes. The slow mode means that the overall variance (i.e.,

Detecting the response to a climate trend in the presence of natural variability. Three types of natural variability are considered in each panel. The top row corresponds to interannual variability in

We have thus far considered outlet-glacier fluctuations due to stationary interannual to multidecadal climate variability. However, long response times also imply some memory of climate variations throughout the Holocene. Ice-sheet models are often spun up using paleoclimate proxy data precisely for this reason

First, we consider a climate scenario with idealized representations of three events: a deglacial warming at

We use the two-stage model, linearized with respect to the Holocene climate with parameters for glacier 1 (see Table 2;

Figure

The idealized LIA is a much more discrete “event” than is supported by paleoclimate records and ignores other variations in the Holocene. Thus, we also consider the response to a more realistic forcing time series. Again, this is not a reconstruction of actual terminus changes, but it is useful to see how glacier memory integrates the continuum of variations found in paleoclimate records. We use a time series of temperatures for Disko Bay, Greenland, from the regional reconstruction of

The glacier responses in Fig.

Response to long-term climate variations.

Marine-terminating outlet glaciers are sensitive to two fundamental types of forcing: changes in surface mass balance, which are distributed over a large interior catchment, and changes in their flux at the grounding line, which are typically driven by the ocean. We have used two models of different complexity to explore and contrast the dynamic responses to these two categories of forcing. Our key findings are as follows:

Ocean forcing (via the grounding-line-flux coefficient,

The two-stage model

Persistence in stochastic climate forcing amplifies natural grounding-line fluctuations (Fig.

Despite contrasting responses on short timescales, slow (i.e., millennial) dynamics dominate the full system response for both types of forcing. This implies a large committed response (Fig.

Given the rapid observed changes linked to ocean forcing in the past few decades, it is useful to discuss points (3) and (4) further. Increases in discharge from Greenland outlet glaciers have been linked to regional climate variability and warming of the subpolar North Atlantic

Our results show that climate and glacier fluctuations over the historical record also have a longer-timescale context. The widely separated glacier response timescales mean that fast fluctuations are essentially superimposed on millennial fluctuations. This is clear even in the flowline model, which does not make an explicit approximation of only two timescales (e.g., Fig.

A number of simplifying assumptions throughout our study warrant some discussion. First of all, we have assumed a constant, prograde bed slope. In reality, variations in bed topography can have a strong effect on retreat rates and sensitivity, both at the terminus

A related issue is the instability associated with retrograde slopes

Another simplification is that we have focused on spatially uniform interior forcings, whereas in reality, surface-mass-balance anomalies are likely to be greater near the marine margin. We conducted several experiments with the flowline model in which

Finally, we chose to impose ocean forcing via the grounding-line-flux coefficient, which adjusts the parameterized relationship between ice thickness and grounding-line flux. This allowed us to compare flux perturbations in a very general way, but it would be more realistic to directly force a dynamic ice shelf or to perturb calving processes. The analytical flux conditions of

Previous studies have employed a variety of forcing strategies for glaciers without buttressing ice shelves. Some have directly perturbed stresses at the grounding line

Regardless of how forcing is implemented, our analyses show that the coupling between grounding-zone and interior dynamics is a key part of an outlet glacier's response to ocean forcing. In particular, increased discharge is what eventually precipitates the slow (but large) second stage of grounding-line retreat. Our idealized ocean forcing has limitations, of course, but the mechanism for a second stage of retreat is physically robust: elevated interior fluxes must eventually fall as the interior drains, creating a tendency towards further grounding-line retreat (Fig.

At the ice-sheet scale, ocean forcing must often be simplified considerably, and the optimal strategy remains to be determined. The Ice Sheet Model Intercomparison Project for CMIP6

The impact that the choice of parameterization would have on projections of near-term sea-level rise is not immediately clear, however.

Here we present the linearized two-stage model in more detail, and derive a discrete autoregressive form. Again, a full model derivation can be found in RRH; we start here from the linearized equations with perturbations in

Code used for the analyses in this study is available as a public repository at

All authors contributed to the study design. JEC carried out the analyses and wrote the manuscript, with input from all authors.

The authors declare that they have no conflict of interest.

We are very grateful to David Pollard (Penn State University) for model code and a generous introduction to using the flowline model. We also thank Ginny Catania, Fiamma Straneo, and Donald Slater for insightful conversations on outlet-glacier dynamics. Finally, we thank Martin Lüthi and the anonymous reviewer for comments that helped clarify the manuscript, and we thank the editor Olivier Gagliardini. John Erich Christian was supported by the NSF Graduate Research Fellowship Program.

This research has been supported by the National Science Foundation, Division of Graduate Education (grant no. DGE-1256082) and National Science Foundation, Office of Polar Programs (grant no. OPP-1542756).

This paper was edited by Olivier Gagliardini and reviewed by Martin Lüthi and one anonymous referee.