## 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}.

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## Acknowledgement

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

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## 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