## Abstract

A simple box model of the circulation into and inside the ocean cavern beneath an ice shelf is used to estimate the melt rates of Antarctic glaciers and ice shelves. The model uses simplified cavern geometries and includes a coarse parameterization of the overturning circulation and vertical mixing. The melting/freezing physics at the ice shelf/ocean interface are those usually implemented in high-resolution circulation models of ice shelf caverns. The model is driven by the thermohaline inflow conditions and coupling to the heat and freshwater exchanges at the sea surface in front of the cavern. We tune the model for Pine Island Glacier and then apply it to six other major caverns. The dependence of the melting rate on thermohaline conditions at the ice shelf front is investigated for this set of caverns, including sensitivity studies, alternative parameterizations, and warming scenarios. An analytical relation between the melting rate and the inflow temperature is derived for a particular model version, showing a quadratic dependence of basal melting on small values of the temperature of the inflow, which changes to a linear dependence for larger values. The model predicts melting at all ice shelf bases in agreement with observations, ranging from below a meter per year for Ronne Ice Shelf to about 25 m/year for the Pine Island Glacier. In a warming scenario with a one-degree increase of the inflow temperature, the latter glacier responds with a 1.4-fold increase of the melting rate. Other caverns respond by more than a tenfold increase, as, e.g., Ronne Ice Shelf. The model is suitable for use as a simple fast module izn coarse large-scale ocean models.

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## Notes

The common factor

*ρc*_{ p }is canceled in all “heat” balances.In the simulations discussed in this study, the unstable regime only occurred in transient states. The ultimate steady states are determined by the low

*κ*.The parameter

*C*has the dimension of cubic meters per second per density unit, i.e., m^{6}kg^{ − 1}s^{ − 1}. In the following, we give values for*C*leaving out the units and the factor 10^{6}.

## References

Braun M, Humbert A, Moll A (2008) Changes of Wilkins ice shelf over the past 15 years and inferences on its stability. The Cryosphere Discuss 2:341–382

De Angelis H, Skvarca P (2003) Glacier surge after ice shelf collapse. Science 299(5612):1560–1562

Dinniman MS, Klinck JM, Smith WO (2007) Influence of sea ice cover and icebergs on circulation and water mass formation in a numerical circulation model of the Ross Sea, Antarctica. Journal of Geophysical Research–Oceans, p 112

Grosfeld K, Schröder M, Fahrbach E, Gerdes R, Mackensen A (2001). How iceberg calving and grounding change the circulation and hydrography in the Filchner Ice Shelf-Ocean System. Journal of Geophysical Research–Oceans 106(C5):9039–9055

Grosfeld K, Hellmer HH, Jonas M, Sandhäger H, Schute M, Vaughan DG (1998) Marine ice beneath Filcher Ice Shelf: evidence from a multi-disciplinary approach. In: Jacobs SS, Weiss R (eds) Ocean, ice and atmosphere: interactions at Antarctic Continental margin, vol 75 of Antarctic Research Series. American Geophysical Union, Washington, DC

Hellmer HH (2004) Impact of Antarctic ice shelf melting on sea ice and deep ocean properties. Geophys Res Lett 31:L10307. doi:10.1029/2004GL19506

Hellmer HH, Jacobs SS, Jenkins A (1998) Oceanic erosion of a floating Antarctic glacier in the Amundsen Sea. In: Jacobs SS, Weiss R (eds) Ocean, ice and atmosphere: interactions at Antarctic Continental margin, vol 75 of Antarctic research series. American Geophysical Union, Washington, DC, pp 83–100

Hellmer HH, Olbers D (1989) A two-dimensional model for the thermohaline circulation under an ice shelf. Antarct Sci 1:325–336

Holland PR, Jenkins A, Holland DM (2008) The response of ice shelf basal melting to variations in ocean temperature. J Climate 21(11):2558–2572

IPCC, Contribution of working group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change(2007). Climate change 2007: The physical science basis (summary for policymakers). IPCC Secretariat, Geneva

Jacobs S, Giulivi CF (1998) Interannual ocean and sea ice variability in the Ross Sea. In: Jacobs SS, Weiss R (eds) Ocean, ice and atmosphere: Interactions at Antarctic Continental margin, vol 75 of Antarctic research series. American Geophysical Union, Washington, DC

Jacobs SS, Hellmer HH, Doake CSM, Jenkins A, Frolich RM (1992) Melting of ice shelves and the mass balance of Antarctica. J Glaciol 38:375–387

Jenkins A, Bombosch A (1995) Modeling the effect of frazil ice crystals on the dynamics and thermodynamics of ice shelf water plumes. J Geophys Res 100:6067–6981

Jenkins A, Hellmer HH, Holland DM (2001) The role of meltwater advection in the formulation of conservative boundary conditions at an ice–ocean interface. J Phys Oceanogr 31(1):285–296

Joughin I, Padman L (2003) Melting and freezing beneath Filchner-Ronne ice shelf, Antarctica. Geophys Res Lett 30(9):1477

McPhee MG (1992) Turbulent heat flux in the upper Ocean under Sea ice. J Geophys Res 97:5365–5379

Nicholls KW, Abrahamsen EP, Buck JJH, Dodd PA, Goldblatt C, Griffiths G, Heywood KJ, Hughes NE, Kaletzky A, Lane-Serff GF, McPhail SD, Millard NW, Oliver KIC, Perrett J, Price MR, Pudsey CJ, Saw K, Stansfield K, Stott MJ, Wadhams P, Webb AT, Wilkinson JP (2006). Measurements beneath an Antarctic ice shelf using an autonomous underwater vehicle. Geophys Res Lett 33(8):L08612

Nicholls KW, Padman L, Schröder M, Woodgate RA, Jenkins A, Osterhus S (2003) Water mass modification over the continental shelf north of Ronne ice shelf, Antarctica. Journal of Geophysical Research–Oceans 108(C8):3260

Olbers D, Zhang J (2008) The global thermohaline circulation in box and spectral low-order models. Part 1: single basin models. Ocean Dyn 58(3–4):311–334

Payne AJ, Holland PR, Shepherd AP, Rutt IC, Jenkins A, Joughin I (2007) Numerical modeling of ocean-ice interactions under Pine Island Bay’s ice shelf. Journal of Geophysical Research–Oceans 112(C10):C10019

Rignot E (2008) Changes in West Antarctic ice stream dynamics observed with ALOS PALSAR data. Geophys Res Lett 35(12):L12505

Rignot E, Jacobs SS (2002) Rapid bottom melting widespread near Antarctic ice sheet grounding lines. Science 296(5575):2020–2023

Skvarca P, Rack W, Rott H, Donangelo TIY (1999) Climatic trend and the retreat and disintegration of ice shelves on the Antarctic Peninsula: an overview. Polar Res 18(2):151–157

Smedsrud LH, Jenkins A, Holland DM, Nost OA (2006) Modeling ocean processes below Fimbulisen, Antarctica. Journal of Geophysical Research–Oceans 111(C1):C01007

Stommel H (1961) Thermohaline convection with two stable regimes of flow. Tellus 13:224–230

Thoma M, Jenkins A, Holland D, Jacobs S (2008) Modelling Circumpolar Deep Water intrusions on the Amundsen Sea continental shelf, Antarctica. Geophys Res Lett 35:L18602. doi:10.1029/2008GL034939

Williams MJM, Grosfeld K, Warner RC, Gerdes R, Determann (2001) Ocean circulation and ice-ocean interaction beneath the Amery Ice Shelf, Antarctica. Journal of Geophysical Research–Oceans 106(C10):22383–22399

Wingham DJ, Shepherd A, Muir A, Marshall GJ (2006) Mass balance of the Antarctic ice sheet. Philos Trans Royal Soc Math Phys Eng Sci 364(1844):1627–1635

Wong APS, Bindoff NL, Forbes A (1998) Ocean-ice shelf interaction and possible bottom water formation in Prydz Bay, Antarctica. In: Jacobs SS, Weiss, R (eds) Ocean, ice and atmosphere: interactions at Antarctic Continental Margin, vol 75 of Antarctic Research Series. American Geophysical Union, Washington, DC, pp 173–187

Wright DG, Stocker TF, Mercer D (1998). Closures used in zonally averaged ocean models. J Phys Oceanogr 28:791–804

## Acknowledgement

We appreciate the very useful comments and critiques of Adrian Jenkins.

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Responsible Editor: Joerg-Olaf Wolff

## Appendix

### Appendix

In Section 2, we have computed the melting rate as a function of the ambient cavern temperature *T*
_{
w
} and salinity *S*
_{
w
}. Here, we make the attempt to express it as a function of the temperature *T*
_{0} and salinity *S*
_{0} of the water flowing into the cavern at the front. For simplicity, we neglect the diffusive terms in the thermohaline balances. Summing the steady state balances for the boxes 2, *d*, and *w*, we arrive at

where Eq. 6 has been used. Inserting now *q* = *C*(*ρ*
_{2} − *ρ*
_{
w
}) with *ρ*
_{2} = *ρ*
_{0} and the three-equations relation Eq. 7, we arrive at a set of equations for *T*
_{
w
} and *S*
_{
w
} that defies analytical treatment. Some reasonable approximations, however, lead to a manageable problem. First, *T*
_{
bw
} − *T*
_{
w
} ≪ *λ* and (1 − *ν*)*S*
_{
bw
} ≪ *S*
_{
w
} are valid. Secondly, the melting physics can be linearized, as proposed by McPhee (1992). The freezing law Eq. 5 is applied with the salinity *S*
_{
bw
} in the turbulent layer replaced by the ocean salinity *S*
_{
w
} outside the layer, i.e., *T*
_{
bw
} = *a*
*S*
_{
w
} + *b* − *c*
*p*
_{
w
}. At the same time, a slightly modified coefficient \(\gamma_T^\star\) is used in the first equation of Eq. 6 to fit the nonlinear laws of the three-equations model. The melting rate becomes

With the abbreviations *x* = *T*
_{0} − *T*
_{
w
}, *y* = *S*
_{0} − *S*
_{
w
}, \( g_1=A_w\gamma_T^\star/C\rho_*, g_2=g_1/\nu\lambda, T^\star = a S_0 + bc p_w - T_0\), Eq. 22 becomes, after the mentioned approximations,

implying *y* = *x*
*S*
_{0} /(*νλ* + *x*) ≈ *x*
*S*
_{0}/*νλ* because *νλ* ≫ *x*. A quadratic problem is obtained for *x*, namely,

with *s* = *S*
_{0}/*νλ*. Furthermore, *a*
*s* ≪ 1 so that \({\left( \beta s -\alpha\right)} x^2 + g_1 {\left( T^\star + x\right)} =0\). Proper expansion reveals that *T*
_{
w
} = *T*
_{0} − *x* is quadratic for small *T*
_{0} and linear for large *T*
_{0}. The results, computing *T*
_{
w
}, *S*
_{
w
}, and *m*
_{
w
} from Eq. 26 as function of *T*
_{0}, are displayed in Fig. 9 for four values of the exchange coefficient \(\gamma_T^\star\). The configuration of RON has been used and the performance can be checked by comparison with the upper left panel (green curve) of Fig. 7. Obviously, a suitable choice for \(\gamma_T^\star\) lies between the value of the blue and red curves.

We may proceed to solve for the *i*-box properties *T*
_{
i
}, *S*
_{
i
}, and *m*
_{
i
}, which is even simpler because the overturning strength *q* is known now from the *w*-box solution. In fact, an analytical solution of the complete model can thus be given, even with the diffusion terms retained but based on the above described simplified freezing law.

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Olbers, D., Hellmer, H. A box model of circulation and melting in ice shelf caverns.
*Ocean Dynamics* **60**, 141–153 (2010). https://doi.org/10.1007/s10236-009-0252-z

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DOI: https://doi.org/10.1007/s10236-009-0252-z

### Keywords

- Antarctic ice shelves
- Melt rate
- Box model