### Abstract

- Top of page
- Abstract
- 1 Introduction
- 2 Advection-Dominated Ice Shelf Temperature
- 3 Model Formulation and Experiments
- 4 Results and Discussion
- 5 Conclusions
- Appendix A
- Appendix B
- Acknowledgments
- References
- Supporting Information

[1] Dynamic and thermodynamic regimes of ice shelves experiencing weak (1 m year^{−1}) to strong (~10 m year^{−1}) basal melting in cold (bottom temperature close to the in situ freezing point) and warm oceans (bottom temperature more than half of a degree warmer than the in situ freezing point) are investigated using a 1-D coupled ice/ocean model complemented with a newly derived analytic expression for the steady state temperature distribution in ice shelves. This expression suggests the existence of a basal thermal boundary layer with thickness inversely proportional to the basal melt rate. Model simulations show that ice shelves afloat in warm ocean waters have significantly colder internal ice temperatures than those that float in cold waters. Our results indicate that in steady states, the mass balance of ice shelves experiencing strong and weak melting is controlled by different processes: in ice shelves with strong melting, it is a balance between ice advection and basal melting, and in ice shelves with weak melting, it is a balance between ice advection and deformation. Sensitivity simulations show that ice shelves in cold and warm oceans respond differently to increase of the ocean heat content. Ice shelves in cold waters are more sensitive to warming of the ocean bottom waters, while ice shelves in warm waters are more sensitive to shallowing of the depth of the thermocline.

### 1 Introduction

- Top of page
- Abstract
- 1 Introduction
- 2 Advection-Dominated Ice Shelf Temperature
- 3 Model Formulation and Experiments
- 4 Results and Discussion
- 5 Conclusions
- Appendix A
- Appendix B
- Acknowledgments
- References
- Supporting Information

[2] Antarctic ice shelves exist in a variety of oceanographic thermal regimes: from “cold” regime exemplified by the Filchner-Ronne and Ross ice shelves, where the sub–ice shelf water is dominated by high-salinity shelf water (HSSW) at ~−1.8°C [e.g., *Jacobs et al*., 1979; *Nicholls et al*., 2001], to “warm” regime exemplified by the Pine Island Glacier (PIG) or George VI ice shelves, where the sub–ice shelf water is dominated by circumpolar deep water (CDW) (~1.2°C) [e.g., *Jacobs et al*., 2011; *Jenkins and Jacobs*, 2008]. Ice shelves in “cold” oceans generally experience a range of basal conditions, from freezing to weak melting (1 m year^{−1}), and typically melt near the grounding line [e.g., *Engelhardt and Determann*, 1987; *Jenkins and Doake*, 1991; *Nicholls et al*., 2001; *Joughin and Padman*, 2003; *Jenkins et al*., 2006]. In contrast, observations of ice shelves in “warm” ocean environments suggest widespread basal melting with average melt rates up to tens of meters per year [e.g., *Jenkins and Jacobs*, 2008; *Jacobs et al*., 2011]. Given that ice shelves in both types of oceanic environment are fed by ice streams with similar thermodynamics and dynamics, one is led to the question of to what extent is the state (geometric, dynamic, and thermodynamic) of an ice shelf determined by the oceanic environment in which it floats.

[3] Knowledge of the thermal state of today's ice shelves is primarily restricted to those in “cold” ocean conditions, where in situ borehole temperature measurements have been taken [*Zotikov et al*., 1980; *Orheim et al*., 1990]. This thermal state also has impact on ice shelf melting and flow, with flow being dependent on ice temperature through ice viscosity [e.g., *MacAyeal and Thomas*, 1986; *Humbert et al*., 2005]. However, it remains unclear what thermal regimes ice shelves should have in different ocean environments, and to what degree differences in basal melting rates can be attributed to differences in ocean environment or to differences in ice shelf thermal state.

[4] Basal mass balance remains one of the most difficult to determine unknowns in overall shelf mass balance. Its direct observation is technically and logistically challenging [*Jenkins et al*., 2006]. The majority of basal mass balance estimates come from either oceanographic measurements [e.g., *Jacobs et al*., 1979, 2011] or remote-sensing observations [e.g., *Joughin and Padman*, 2003; *Shepherd et al*., 2003, 2004]. The former provide bulk values (i.e., area averaged), and the latter rely on a set of assumptions (e.g., steady state) and other measurements (e.g., ice thickness and surface accumulation rates) that might have insufficient resolution and are often taken during different time periods. These indirect estimates lack details necessary to establish the effect of surrounding oceans on ice shelves, to attribute causes of observed ice shelf changes, and to make projections of the possible ice shelf changes under different climate conditions.

[5] This study aims to establish fundamental aspects of the ice shelf/sub–ice shelf cavity systems that are determined by differences in the ambient oceanographic conditions. We investigate the coupled geometric, dynamic, and thermodynamic behavior of ice shelves in the two “classical” oceanographic environments: one dominated by “cold” high-salinity shelf water (−1.8°C) and the other dominated by “warm” circumpolar deep water (1.2°C). The main question considered in this study is the following: if two identical ice streams flow into “cold” and “warm” oceans, how different are the ice shelves that they produce? We use a 1-D coupled ice shelf/ocean model and a newly derived analytic expression for the steady state temperature distribution in ice shelves experiencing basal melting to conduct our investigations. The coupled model includes an ice flow model and a plume ocean model. We investigate the following aspects of the ice shelf/sub–ice shelf cavity system: ice shelf morphology, melt rate distribution, ice shelf dynamic and thermodynamic states, and their mutual effects. Also, we consider the implications of this coupled interaction on modeling approaches. Finally, we explore sensitivities of this system to oceanic and grounded ice conditions. We investigate the effects of increase in the ocean heat content in two ways: through warming the bottom ocean water and through shallowing the depth of the thermocline.

### 2 Advection-Dominated Ice Shelf Temperature

- Top of page
- Abstract
- 1 Introduction
- 2 Advection-Dominated Ice Shelf Temperature
- 3 Model Formulation and Experiments
- 4 Results and Discussion
- 5 Conclusions
- Appendix A
- Appendix B
- Acknowledgments
- References
- Supporting Information

[6] Ice shelves are the fast-flowing components of ice sheets, moving at hundreds to thousands of meters per year. Consequently, their thermal state is dominated by heat advection rather than diffusion. We show this below by comparing characteristic scales of different terms in the advection-diffusion equation governing ice shelf temperature. We restrict our analysis to ice shelves in steady state, with fixed, time-independent geometry. Ablation/accumulation at the top surface, as well as variations in the surface temperature are disregarded to simplify the analysis and restrict it to basal melting; however, the analysis can easily be extended to account for these factors. A justification for these simplifications in the present study will be provided in section 4.

[7] For ice shelves that flow in one horizontal direction only, the steady state heat equation is as follows:

- (1)

where *x* and *z* are the horizontal and vertical coordinates, *T*(*x*,*z*) is ice temperature, *u* and *w* are ice horizontal and vertical velocity components, *κ*_{i} is the thermal diffusivity of ice (assumed to be independent of density and temperature), and subscripts *x* and *z* denote the partial derivatives with respect to *x* and *z*, respectively. Viscous heating and horizontal diffusion are disregarded due to their negligible effects [e.g., *MacAyeal and Thomas*, 1986]. Boundary conditions are as follows:

- (2a)

- (2b)

- (2c)

where *s* and *b* are the elevations of the top and bottom surfaces of the ice shelf, *T*_{s} is temperature at the ice shelf top surface (assumed to be uniform), *T*_{g}(*z*) is the ice shelf temperature profile at the grounding line, *x* = 0, and *T*∗(*x*) is the seawater freezing temperature that depends on in situ seawater salinity *S* and pressure *p*

- (3)

where *c*_{1}, *c*_{2}, and *c*_{3} are empirical constants.

[8] Other than within a few ice thicknesses of the grounding line and ice front, the horizontal ice shelf velocity components do not depend on the vertical coordinate *z* [*MacAyeal*, 1989]; therefore, the vertical ice shelf velocity component varies linearly with *z* as a result of ice incompressibility (firn densification is disregarded for simplicity). These facts, and the use of a stretched vertical coordinate

- (4)

where *H* = *s* − *b* is ice thickness, allow equation ((1)) to be written in the following form [e.g., *MacAyeal and Thomas*, 1986; *Hindmarsh*, 1999; *MacAyeal*, 1997, p. 270–273]:

- (5)

where is the surface accumulation rate (indicating negative for ablation) and is the basal melt rate (positive for melting). We assume that the surface ablation rate is negligible compared to basal melt rate ; therefore, the first term in the square brackets on the left-hand side is set to zero.

[9] Characteristic values for *u*, *H*, , and *L* are 300 m year^{−1}, 1000 m, 1 m year^{−1}, and 300 km, respectively. With the ice thermal diffusivity *κ*_{i} = 36 m^{2} year^{−1}, the right-hand side of equation ((5)) that represents heat diffusion is at least 2 orders of magnitude smaller than both terms in the left-hand side, and can therefore be neglected. The Peclet number, , for ice shelves with basal melting is much greater than 1, indicating that heat advection is the dominant process. The ice shelf temperature solution under these circumstances is as follows:

- (6)

where *T*_{g}[*ζ*] ≡ *T*_{g}(*ζ*), *T*∗(*x*) is determined by equation ((3)),

- (7a)

- (7b)

and where *q*(*x*) and *q*_{g} are the ice fluxes at a point *x* and at the grounding line, respectively. A derivation of this solution is presented in Appendix A. The characteristic thickness of the thermal boundary layer, represented by the exponential term on the right-hand side of equation ((6)), is , which is on the order of a few meters for basal melt rates on the order of a few meters per year (10 m for a 3.6 m year^{−1} melt rate), and less than a meter when basal melting is strong (~0.5 m for a 70 m year^{−1} melt rate).

[10] The mass and energy balance at the ice shelf bottom surface (the Stefan condition) that determines the melt rate is as follows [*Holland and Jenkins*, 1999]:

- (8a)

- (8b)

where *k*_{i} is heat conductivity of ice, *γ*_{S,T} are the salt and heat transfer coefficient at the ice-ocean interface (defined below), *ρ*_{w} is seawater density, *c*_{w} is the specific heat capacity of seawater, *S*_{o}(*x*) and *T*_{o}(*x*) are the ocean mixed layer salinity and temperature, respectively, and *L*_{i} is the ice latent heat of fusion. The first term of equation ((8b)) is the heat flux into ice above the bottom, and the second term is the heat flux from the ocean mixed layer. The expression for temperature in the ice shelf represented by equation ((6)) allows computation of the heat flux into ice. Substituting the heat flux expression into equation ((8b)) and rearranging terms, we arrive at the following expression:

- (9)

where *c*_{i} is the specific heat capacity of ice. This expression has a simple heat balance interpretation. Heat stored in the ocean mixed layer (the left-hand side) is available to do three things: (1) warm the ice shelf thermal boundary layer to the in situ melting point, *T**; (2) melt that ice that has reached the in situ melting point (terms in the first curly brackets on the right-hand side of equation ((9))); and (3) conduct into the colder ice above (the last term on the right-hand side of equation ((9))).

### 5 Conclusions

- Top of page
- Abstract
- 1 Introduction
- 2 Advection-Dominated Ice Shelf Temperature
- 3 Model Formulation and Experiments
- 4 Results and Discussion
- 5 Conclusions
- Appendix A
- Appendix B
- Acknowledgments
- References
- Supporting Information

[41] Using a coupled 1-D ice/ocean model and an analytic expression for ice shelf temperature, we have found that the oceanic environment in which ice shelves flow determines their states. Ice shelves flowing in cold, HSSW oceans differ from those in warm, CDW oceans in fundamental ways: their mass balance is controlled by different processes (ice advection and deformation in HSSW versus ice advection and melting in warm oceans), their thermal structure is different (warmer interior in cold oceans versus colder interior in the warm oceans), and they respond differently to different mechanisms that lead to the increase of the ocean heat content (higher sensitivity to warming of the cold bottom water versus higher sensitivity to shallowing the depth of the thermocline in the warm ocean). These findings suggest that ice shelves in different oceanic environments require treatments (e.g., modeling and observations) specific to each environment.

[42] In addition, we have established that the ice shelf/sub–ice shelf cavity systems are inherently coupled with strong feedbacks in geometry, melt rates, ice flow, and temperature. This fact has an important implication on the modeling treatments of these systems. Despite very large melt rates achieved near the grounding line in the warm, CDW ocean environment, simulations with the coupled model produce steady state configurations in which the effects of strong melting are compensated for by ice influx from the grounding line. However, such configurations cannot be simulated with uncoupled models where the ice shelf and cavity circulation components are treated separately. Assumptions of a static ice shelf in ocean-only simulations are justifiable only in circumstances where the ice shelf is close to steady state. A possible rule of thumb that determines whether an ice shelf/sub–ice shelf cavity system requires coupled treatment can be based on the leading order mass balance of an ice shelf: if it is between ice advection and deformation, then it is reasonable to apply ocean-only models; otherwise, a coupled ice/ocean model is required.