## 1. Introduction

The interaction between marine ice sheets and the surrounding ocean currents has received increased attention in recent years due to its potential importance for the overall mass balance of ice sheets and an associated rapid sea level rise. The system in which these interactions occur can take on the form of tidewater glaciers with a near-vertical edge terminating in the ocean or floating ice shelves attached to the grounded ice sheet. The latter case is especially important for the Antarctic Ice Sheet (AIS), which is buttressed by a vast number of floating ice shelves along its margin. As shown in recent studies (Pritchard et al. 2012; Depoorter et al. 2013; Rignot et al. 2013; Golledge et al. 2015; DeConto and Pollard 2016), subshelf basal melting is a major factor in the mass loss of the AIS and its potential contribution to future sea level rise, particularly in the warmer waters of the Amundsen Sea sector (Rignot et al. 2014).

These aspects demonstrate the need for accurate models of the interaction between ice shelves and ocean. Traditionally, the main mechanism behind this interaction has been described by a buoyancy-driven overturning circulation beneath the floating ice shelf (MacAyeal 1985; Hellmer and Olbers 1989; Jenkins 1991). Fresh meltwater, generated either directly at the ice–ocean interface or at the grounding line in the form of subglacial discharge, is positively buoyant in the saline environment of the ice-shelf cavity and moves upward under the ice shelf base, creating a turbulent plume. Due to entrainment of the ambient water within the cavity or the possible inflow of warmer ocean waters, the plume can generate more basal melt. In stratified environments, the plume can also detach from the ice-shelf base when reaching a level of neutral buoyancy, leading to more complicated circulation patterns and different melting modes (Jacobs et al. 1992). However, applying these notions in the context of large-scale climate simulations with ice-sheet models and/or ocean general circulation models remains problematic (Asay-Davis et al. 2017).

From an ocean-modeling perspective, basal melt rates can be calculated by explicitly resolving the cavity circulation, using a parameterization of heat exchange to the ice shelf. This is currently only feasible in sufficient detail for single ice-shelf cavities for which the geometry is well known (e.g., Thoma et al. 2015; Asay-Davis et al. 2016; De Rydt and Gudmundsson 2016; Seroussi et al. 2017; Timmermann and Goeller 2017). Several experiments have been done with high-resolution models on continental scale to simulate the subshelf circulation and basal melt rates (e.g., Timmermann and Hellmer 2013; Dinniman et al. 2015; Mathiot et al. 2017; Naughten et al. 2018), but they are computationally expensive, especially if coupled to a dynamical ice-sheet model that captures geometry changes over long time scales. From an ice-modeling perspective with standalone ice-sheet models (e.g., De Boer et al. 2015), basal melt is usually described by highly simplified expressions based on the local ice–ocean flux (Beckmann and Goosse 2003), which by themselves do not explicitly account for the cavity circulations. The feedback between basal melt and the cavity circulation can be partially captured by a simple quadratic temperature dependence, as described by Holland et al. (2008) and applied by DeConto and Pollard (2016), which, however, still lacks important geometry-dependent effects.

Instead of fully resolving the ice-shelf cavity circulation, it would have great computational advantages if its dynamical features could be included in a straightforward way in a basal melt parameterization, which is the aim of this paper. The starting point is the quasi-one-dimensional plume model by Jenkins (1991) that, although simplified, describes the basic physics of the aforementioned buoyant meltwater plumes driving the cavity circulation. Jenkins (2011) already showed how under certain conditions the results of this plume model scale to a rather universal relation for the basal melt rates caused by subglacial discharge at the grounding line. However, it was shown that the length scale over which melting is directly influenced by this freshwater input at the grounding line (convection-driven melting) is typically small for ice shelves. Beyond this small distance from the grounding line, the dominant mechanism for plume dynamics is caused by the basal melt itself (melt-driven convection). A governing length scale for this mechanism was found by Lane-Serff (1995), depending on the ambient ocean temperature and the local freezing point. This regime, in which the depth-dependent freezing point dominates the cavity circulation, is central in the current study. It should be noted, however, that other processes remain important for basal melting of ice shelves, including tidal forcing (e.g., Mueller et al. 2012), the aforementioned subglacial discharge (e.g., Jenkins 2011; Slater et al. 2017), stratification of the ambient cavity water (e.g., Magorrian and Wells 2016), and the impact of frazil ice formations on the plume dynamics (e.g., Smedsrud and Jenkins 2004).

To obtain a practical parameterization describing subshelf basal melt rates in the dominant regime, Jenkins (2014) performed a second *empirical* analysis of the plume model results, again leading to a universal scaling for the melt rate that extends the analysis of Lane-Serff (1995). This parameterization unites the influence of geometry (basal slope and depth) with the dependence on ambient ocean properties. A key feature of the parameterization is a dimensionless melt-rate curve that contains a positive peak near the grounding line and a transition to refreezing further away. Lazeroms et al. (2018) applied the new parameterization to all Antarctic ice shelves and showed an improvement in modeled melt rates compared with simpler parameterizations (e.g., Beckmann and Goosse 2003), especially in terms of spatial variations and temperature sensitivity. By parameterizing the melt rates using the dynamical features of the plume model, one implicitly accounts for important effects of the cavity circulation without explicitly resolving it. Note that an alternative approach with similar behavior exists in the form of the box model by Olbers and Hellmer (2010), which was successfully applied to Antarctic ice shelves by Reese et al. (2018).

However, the empirical melt-rate curve used in Lazeroms et al. (2018) does not immediately provide insight in the underlying physical assumptions from which it was derived. It is essentially a polynomial fit of the scaled numerical data from the plume model. Lazeroms et al. (2018) provided a quick calculation from simplified equations that could retrieve the correct scaling factors, but not the dimensionless melt-rate curve itself. Moreover, a major practical drawback of this curve is its formulation in terms of a polynomial of degree 11, which is very sensitive to the numerical values of its coefficients and prone to implementation errors. Therefore, a more robust formulation in terms of a systematically derived analytical expression is desirable.

In this paper, we attempt to formalize the empirical analysis and the resulting basal-melt parameterization of Jenkins (2014). Starting with the plume model by Jenkins (1991), we show how an appropriate scaling of the plume model equations leads to a dimensionless system that can be solved analytically under certain conditions. This yields an analytical expression for the basal melt rate that is nearly identical to the empirical melt-rate curve of Jenkins (2014) and allows simple use in ice-sheet models. Knowing how these expressions are derived systematically from the underlying physical equations is important for understanding both the potential and the drawbacks of the parameterization for use in ice-sheet models, and it sheds light on a possible extension including more complex physics.

In the next section, we present the derivation of the basal melt rate starting with a brief description of the underlying plume model. The plume model equations are simplified step by step, by first assuming a constant basal slope and uniform ambient ocean properties and then applying an asymptotic analysis for small values of the slope and the thermal driving of the plume, which we show to be typical for (Antarctic) ice shelves. In section 3, we show numerical results comparing the analytical expression with the full plume model for various cases both within and beyond the assumptions of the formal derivation. Section 4 provides concluding remarks and a brief discussion of the remaining practical issues for numerical ice-sheet models.

## 2. Derivation of the model

In this section we derive the new analytical expression for the basal-melt rate beneath a floating ice shelf, based on the plume model by Jenkins (1991, 2011). Although the original model is designed for a general quasi-one-dimensional geometry of the ice-shelf base and general temperature and salinity profiles for the ambient ocean water, we will assume a constant basal slope and constant ambient properties to simplify the analysis. Using a suitable scaling and perturbation methods, this analysis leads to a dimensionless melt-rate curve that can be applied in more general cases, as shown in section 3.

### a. Plume model equations

The plume model of Jenkins (1991, 2011) describes the evolution of a buoyant meltwater plume beneath an ice shelf with a basal geometry that is uniform in the cross-flow direction (Fig. 1). This quasi-one-dimensional geometry can be described by a slope angle *α* that essentially depends on the basal depth *X*, where *z* values are defined as increasing upward with zero position at sea level. Furthermore, the ambient ocean water in the ice-shelf cavity has temperature

*D*and the (depth-averaged) plume velocity

*U*, plume temperature

*T*, and plume salinity

*S*as functions of

*X*are described by the following system:

*DU*as being determined by the melt rate

*L*is the latent heat of fusion for ice, and

*c*are the specific heat capacities of ice and ocean water, respectively. Finally, (4c) equates the interface temperature

The plume model in the form presented above can be solved numerically for any quasi-one-dimensional ice-shelf geometry, defined by the draft *D*, *U*, *T*, and *S* [(1)]. In general, one can impose an initial freshwater flux *melt-driven convection* where the primary source of buoyancy is the meltwater generated locally at the ice-shelf base. Therefore, we take

Constant parameters used in the plume model (Jenkins 1991, 2011) and the (derivation of the) melt-rate parameterization (28). The values for *c*,

### b. Simplified formulation and scaling

*L*. Compared to the three-equation formulation in (4), the two-equation formulation in (5) can underestimate the melt-rate maximum by up to 2% for the test cases considered in section 3, which, however, can increase up to 10% when the basal slope becomes near-vertical.

*constant*basal slope

*α*. Hence, the ice-shelf basal depth

*X*along the ice-shelf base as follows:

*D*,

*U*,

*X*. Formally, only four equations would be required to close the system, but keeping an equation for

*X*through the depth dependence of the freezing point. Equations (9) can now be nondimensionalized using the following scaling:

*τ*are the characteristic length, velocity, and temperature scales, respectively. These characteristic scales are defined as follows. The temperature difference between ambient ocean

*τ*to zero due to the pressure dependence of the freezing point, assuming uniform ambient water. In other words, it indicates the point where the ambient water would start freezing, which may be above or below sea level, depending on ambient conditions (Lane-Serff 1995). Finally, since the dynamics of the plume are determined by melt-induced buoyancy and not by an external flux, the velocity scale is fixed by taking the

*densimetric Froude number*equal to unity:

*η*and

*μ*are small. This will lead to an equation that can be solved analytically, yielding a closed expression for the dimensionless basal melt rate.

### c. Asymptotic analysis in terms of η and μ

*asymptotic expansion of order N*(or

*Poincaré expansion*; see, e.g., Nayfeh 1973; Eckhaus 1979; Holmes 1995; Mattheij et al. 2005) of a function

*ε*goes to zero [e.g.,

*O*(1) as

*ε*→ 0. This allows us to expand all dependent variables individually.

System (15) contains two parameters that are potentially small and that are dependent on external properties, namely *η* (depending on the basal slope) and *μ* (depending on the temperature difference *τ*). Since we disregard changes in the ambient salinity *η* and *μ* are indeed small for typical ice-shelf conditions and if they are of the same order of magnitude. With ambient salinity *τ* < 5°C and *η* and *μ* are indeed small parameters.

It appears that *μ* is potentially some orders of magnitude smaller than *η*, although one should keep in mind that small values of *overall* slope of a flowline of Filchner–Ronne Ice Shelf (FRIS), taking *μ* might approach the upper bound *η* is indeed one order of magnitude larger than *μ*, also considering that the slope near the grounding line is typically higher than the overall slope. This would justify the scaling assumption that terms of

*η*and

*μ*, that is, determine the scale function

*η*and

*μ*. From top to bottom, we then find the following leading-order balance of terms:

*μ*, while the temperature difference depends on the slope

*η*. As it turns out, all terms in (15) depending explicitly on the salinity difference

*s*will be of higher order in

*μ*and neglected in the following approximation. Consequently,

*s*and its corresponding primitive equation (15e) can be disregarded.

*μ*. Furthermore, we incorporate several algebraic factors that are essentially

*η*, which make the resulting equations more compact. This amounts to the following scaling:

We have now scaled the plume equations to a compact form containing only *δ* or *ε* to further simplify the analysis. Figure 2 shows these parameters as a function of *η* within a range of values deemed typical for ice shelves (the previously found values for FRIS and PIG are indicated in the figure, as well as a typical value for Ross ice shelf). It appears that *ε* is always larger than *δ* and larger than 0.1 over a significant range of values. Still, *ε* could be considered as a small parameter except for the larger values of *η*. Furthermore, although formally both *ε* and *δ* scale with *η* and their ratio is approximately constant, it appears that *δ* is one order of magnitude smaller than *ε* over a wide range of values except for very small

Equations (20) describe the behavior of the dimensionless quantities *x*. Since the melt rate *ε* and *μ*, (20) should be interpreted as follows: (20a) is the mass balance determined entirely by entrainment with meltwater input neglected; (20b) is a balance of buoyancy and drag forces, neglecting inertial accelerations up to *ε*. Note that the error in this approach with respect to the full plume model depends on the value of *ε* (i.e., the basal slope). The extent of the error in realistic test cases will become clear in section 3.

### d. Analytical expression for the melt rate

*φ*. In appendix A, we show how system (20) can be reduced to the following equation with

*δ*=

*ε*

^{2}:

*x*. Moreover, we have initial conditions

*φ*for the dimensionless melt rate:

*φ*, requiring only two boundary conditions. Hence, the boundary values for the plume temperature and plume salinity are automatically imposed by the solution of (23) and cannot be chosen independently. Physically, the initial meltwater at the grounding line undergoes a rapid adjustment as it mixes with the ambient water, a process also described by Jenkins (2011). By reducing the order of the system and the number of boundary conditions, we neglect this rapid adjustment and immediately start from the mixed conditions.

*φ*in terms of

*ε*. Restricting our attention to the leading-order behavior, the positive real-valued solution for

*η*and the thermal driving of the plume

*μ*. In this respect, it should be noted that (26) takes over the role of the dimensionless melt-rate curve applied by Lazeroms et al. (2018), which was described by a polynomial fit and found from a purely empirical study (Jenkins 2014). The expressions for the other plume variables

Figure 3a shows a plot of the analytical expression (26) as a function of the dimensionless coordinate *x*. The melt-rate curve indeed shows the desired behavior: the melt rate is zero at the grounding line (*ε*. Therefore, the errors with respect to the (numerical) solution of these original equations will depend on *ε*, and it is important to check how these errors are distributed over the entire domain of interest. Numerical solutions for *ε*. In fact, the analytical expression (26) is only valid for *ε* increases. Conversely, the endpoint moves toward

*ε*-dependent correction should provide the final scaling step that constrains the values of the along-slope coordinate between 0 and 1, where the latter corresponds to the aforementioned

*ε*-dependent endpoint of the plume. The theoretical discussion in appendix B suggests a correction of the following form:

*exactly*to a depth at which

These expressions should be compared with the basal-melt parameterization found empirically by Jenkins (2014) and applied to the Antarctic ice shelves by Lazeroms et al. (2018). In both formulations, the basal melt is calculated by multiplying a melt-rate scale by a dimensionless function ^{−1} °C^{−2}. Apart from this, (28a) contains the same dependence on

## 3. Numerical results

So far, the analytical expression for the melt rate given by (28a) has been directly compared with (23) from which it is derived. Since this equation is itself an approximation of the plume model, it is important to check how it compares to the full model. In the following, we shall evaluate the plume model as presented in section 2a and the parameterization given by (28) for various test cases. Both models are evaluated using the constant parameter values given in Table 1. We distinguish between effects of the geometry and the ambient ocean profiles, before applying the model to more realistic cases.

### a. Slope dependence

The main assumptions of the derivation as formulated in section 2b are a constant slope angle *α* (i.e., a linear ice-shelf draft) and uniform ambient properties *α* and extending up to sea level (

Figure 4a shows the results of both models for all of the aforementioned geometries. Clearly, in each case the parameterization closely follows the behavior of the plume model over the entire domain. All curves, except the vertical ice wall, show the same qualitative behavior outlined in section 2d: a region of melting (positive

The similarities between the first five cases become clear in Fig. 4b, where the plume model results with corresponding *X* coordinates have been scaled using the same dimensional factors as in (28a) and (28b) and directly compared with the dimensionless curve *ε* and might be improved by higher-order terms. Also note that this error depends on *ε* and will be smaller for smaller basal slopes. On the other hand, the errors near the end of the domain (

Among the results in Fig. 4, the vertical ice wall is a special case because it has an infinitely steep slope with *ε* in (21) approaches 1 and is no longer small. Hence, the analysis of the previous section is no longer valid for this case. This issue appears in Fig. 4 through the slightly higher discrepancies between the plume model and the parameterization. The relative error in this cases now reaches values between 20% and 30%. Nevertheless, both models are still reasonably close, which shows that the parameterization is useful even for high slopes and at the calving fronts of ice shelves and tidewater glaciers.

Since the constant-slope assumption will generally not hold for realistic ice-shelf geometries, the next step is to investigate the performance of the parameterization for varying slopes. Although this goes beyond the formal assumptions behind its derivation, it is possible to evaluate (28) by inserting a varying slope *α*), a concave shelf (decreasing *α*), a shelf with a periodically varying slope, and a shelf consisting of two line segments with different slopes (abrupt change in *α*). A comparison of the parameterization and the full plume model for these cases is shown in Fig. 5. In all cases, the parameterization still appears to give results close to the plume model, even though the scaled plume model results can differ considerably from the dimensionless curve

In particular, comparing Figs. 5a and 5d, we see that both the height of the melting peak and the error here are higher for the concave case, which starts with a high slope at the grounding line, than for the convex case. Due to the slope-dependent scaling in (28b), the melting peak is also closer to the grounding line in the concave case. Though the melting–freezing transition is almost perfectly predicted in both cases, a larger discrepancy remains in the refreezing region of Fig. 5d. This could be explained by considering that the discrepancies between the dynamically evolving plume model and the parameterization are typically higher in regions where the slope changes, because the plume model adapts more gradually to such changes. Having these slope changes in the refreezing region close to the inherent singularity at

An interesting example is shown in Fig. 5g because the parameterization remains close to the original plume model despite the rapid slope changes over the entire length of the ice shelf. This case clearly shows that high melt rates are obtained locally where the slope is relatively steep. Hence, although the dimensionless curve

The examples above show that the parameterization formulated in (28) agrees well with the original plume model, not only for simple constant-slope geometries but also for more complicated cases. Despite the presence of errors, which can be explained quite easily from the assumptions in the derivation, the agreement appears good enough to apply the parameterization to more realistic geometries.

### b. Thermal driving and stratification

Next, we investigate the effect of the temperature and the salinity of the ambient ocean water inside the ice-shelf cavity. The thermal driving, that is, the temperature difference

The preceding results confirm that the parameterization performs well compared with the plume model for different uniform values of

However, this does not hold when stratification (vertically varying

Formally, the current formulation of both the plume model and the parameterization only describes the first melting mode from the grounding line, but it is interesting to evaluate the models for a nonuniform ambient ocean and investigate their behavior. Figure 7 shows the results of this evaluation for the same case as Fig. 6, but with uniform

As far as the parameterization is concerned, Fig. 7 leads to the reassuring conclusion that it remains close to the plume model in most of the evaluated domain, except in the direct vicinity of the endpoint of the plume. This can be explained by noting again that the absolute value of

### c. Realistic flow line data

After investigating the effects of a changing slope and changing ambient temperature separately in the aforementioned idealized cases, we now turn to the evaluation of the melt-rate parameterization for more realistic geometries. Figure 8 compares the results from the plume model and the parameterization for three different ice-shelf geometries based on flow line data of FRIS (Bombosch and Jenkins 1995), Ross ice shelf (Shabtaie and Bentley 1987), and Pine Island Glacier (PIG; Crabtree and Doake 1982). These results should be compared with those shown in Lazeroms et al. (2018) for the same flow lines of FRIS and Ross ice shelf using the qualitatively similar parameterization of Jenkins (2014), as discussed in section 2d.

Note that all cases shown in Fig. 8 have uniform ambient ocean properties *effective ocean temperature* field was constructed by inverse modeling, causing the modeled basal melt rates to match area-averaged values from the observations by Rignot et al. (2013). Hence, we use this constructed field to obtain characteristic values for

The results in Fig. 8 essentially combine the effects of a varying slope shown in Fig. 5 into a much more complicated profile. For both FRIS and Ross, the parameterized melt rates closely agree with the results of the full plume model. In particular, the FRIS profile shows a transition from melting to refreezing which is again perfectly predicted. The Ross profile remains within the positive melt region and shows a slightly better agreement between the plume model and the parameterization, mostly due to the smaller slopes in the ice-shelf base. On the other hand, the PIG profile shows a considerable discrepancy between

To indicate how the modeled melt rates in Fig. 8 relate to observations for these realistic ice shelves, we show the averaged melt rates along the flow lines in Table 2, comparing again the values obtained with the full plume model and the analytical expression. Clearly, both models capture low melt rates for Ross, slightly higher melt rates for FRIS, and relatively high melt rates for PIG, at least for the chosen values of

Modeled basal melt rates (m yr^{−1}) averaged over the length of the flow lines shown in Fig. 8.

*U*and

*D*as determined from (10) and (19). Note that

*η*while the dependence on

*τ*disappears from the ratio

*η*given in Fig. 2 and for

^{−1}, the maximum value of

## 4. Conclusions

We provided a systematic derivation of the basal melt rate *α* and thermal driving

The advantage of the current derivation is twofold. First, it provides more insight into the main processes governing the plume dynamics and the resulting basal melt rate. The assumptions made to obtain the dimensionless function

Second, the expression for

Furthermore, the numerical results in section 3 show that the parameterization works well compared with the full plume model (including the three-equation model for the interface conditions) for various ice-shelf geometries and ambient ocean conditions in the regime where buoyancy dominates plume dynamics, even though theoretically its derivation is only valid for highly idealized cases. The largest discrepancies are visible where the basal slope is locally large or rapidly varying, due to the fact that higher-order terms in the (constant) slope were neglected in the derivation, but overall the parameterization remains close to the plume model as the latter responds more gradually to slope changes. Technically, the current formulation also breaks down in the vicinity of the plume endpoint, caused by the decrease of buoyancy and momentum close to

In the case of stratified ambient water, an extension of the current model to multiple uniform layers (Magorrian and Wells 2016) might be necessary to account for detached plumes and different melting modes, as briefly noted in section 3b. Such an extension can be formulated in terms of a stratification length scale

From a practical viewpoint, the current study only focuses on the quasi-one-dimensional plume dynamics along a single ice-shelf flow line with uniform ambient ocean properties. Other aspects need to be considered before the derived parameterization can be applied to realistic three-dimensional ice-shelf geometries for use in ice-sheet models. The two most important issues were discussed extensively in Lazeroms et al. (2018), namely the extension of the quasi-1D setting to 2D shelves and the required oceanic forcing field. For the first issue, Lazeroms et al. (2018) proposed a practical solution in the form of an algorithm that searches for multiple plume paths in each ice-shelf point and taking average, *effective* values for both the grounding-line depth *α* in order to calculate the basal melt rate in that point. Of course, this algorithm is not unique and it is still uncertain how sensitive the computed melt rates are to the method used to find effective plume paths.

The second issue of finding a suitable ocean forcing field might be more problematic, as observational data within ice-shelf cavities are sparse. For this reason, Lazeroms et al. (2018) constructed an *effective* temperature field from extrapolated ocean data by constraining the modeled basal melt rates to present-day observations from Rignot et al. (2013). The resulting forcing field contains horizontal variations in the ocean temperature (e.g., relatively warmer waters in the Amundsen Sea as in Figs. 8g–i), but it lacks information about seasonal variability and vertical profiles. Hence, the proper way to model the oceanic forcing needs to be investigated further. As a next step, horizontal variations in the ocean conditions could be incorporated by defining different coastal sectors, each with its own effective temperature and salinity. Vertical variations and stratification could then be included by using the multiple uniform layers mentioned previously. It should be pointed out that the method of constructing an effective temperature by inversion of the melt rates not only corrects for unknown temperature data, but also intrinsic biases in the melt-rate parameterization itself. The resulting temperature field should therefore be interpreted with care.

All in all, the current derivation of the basal melt parameterization is an important step in improving the description of ice–ocean interaction in ice-sheet models without fully resolving the ice-shelf cavity circulations. Its relatively simple formulation contains the minimal amount of physics needed to obtain the spatial variations in the melt rate between the grounding line and the ice-shelf front that cannot be captured by simpler models.

## Acknowledgments

Financial support for W.M.J. Lazeroms was provided by the Netherlands Organisation for Scientific Research (NWO-ALW-Open 824.14.003). Financial support for A. Jenkins was provided by the Natural Environment Research Council, Grant NE/N01801X/1. The first author wishes to acknowledge the hospitality of the WDY Group at the Department of Applied Physics, Eindhoven University of Technology, where part of the work was done. We gratefully acknowledge the constructive comments of three anonymous reviewers.

## APPENDIX A

### Solution in Terms of the Volume Flux *φ*

*φ*:

*x*derivatives of

*φ*. It should be clear now that the assumption

*φ*. Note that we are neglecting the term

*φ*:

*φ*and construct a solution. Equations (20c) and (20d) can be combined in the following way:

*φ*:

*ε*. We can then construct an (approximate) analytical solution to (A5) by using an asymptotic expansion:

Figure A1 shows *x*. These curves have several interesting properties. At *O*(1) prefactors]

*ε*. Finally, note that the approximate solution can potentially be improved by including the first-order term

*ε*. The expression for

## APPENDIX B

### Boundary Layer Solution around *x* = 1

In perturbation theory, boundary layers typically occur in higher-order problems with multiple boundary conditions, for which an asymptotic expansion [e.g., (16)] found by standard methods turns out to satisfy only a subset of these boundary conditions (see, e.g., Mattheij et al. 2005). The most well-known example of such behavior is fluid flow close to a solid wall or object, where a boundary layer region close to the surface is required to adapt the essentially inviscid outer flow to the viscous boundary conditions at the surface. In other words, a straightforward asymptotic expansion in the limit of small viscosity will fail close to the surface because viscosity becomes dominant here.

A similar situation occurs in our asymptotic approximation of (20) and (23): in the limit of small *ε*, we are essentially neglecting the temperature gradient in (20d), but this terms turns out to be dominant in a region around

*ξ*can have either positive or negative values. A first guess of the behavior of the solution can be found by substituting the rescaled

*x*in (25) and expanding it in a Taylor series around

*φ*in the boundary layer:

In principle, one can now solve (B4) for *overlap region*. Unfortunately, (B4) has no analytical solution, so this exercise can only be done numerically. Fig. B1a shows an example of a numerical matching for *ε* goes to zero, the endpoint will move closer to

The previous discussion provides more theoretical insight in the solution of (23), especially around *x* to values between 0 and 1, which indeed improves the agreement between the curves in Fig. 3. Not directly captured in this way is the behavior of the melt rate in the boundary layer, where it decreases until reaching zero at the endpoint of the plume (Fig. B1b).

## REFERENCES

Asay-Davis, X. S., S. L. Cornford, B. K. Galton-Fenzi, R. M. Gladstone, G. H. Gudmundsson, D. M. Holland, P. R. Holland, and D. F. Martin, 2016: Experimental design for three interrelated marine ice sheet and ocean model intercomparison projects: MISMIP v. 3 (MISMIP+), ISOMIP v. 2 (ISOMIP+) and MISOMIP v. 1 (MISOMIP1).

,*Geosci. Model Dev.***9**, 2471–2497, https://doi.org/10.5194/gmd-9-2471-2016.Asay-Davis, X. S., N. C. Jourdain, and Y. Nakayama, 2017: Developments in simulating and parameterizing interactions between the Southern Ocean and the Antarctic Ice sheet.

,*Curr. Climate Change Rep.***3**, 316–329, https://doi.org/10.1007/s40641-017-0071-0.Beckmann, A., and H. Goosse, 2003: A parameterization of ice shelf-ocean interaction for climate models.

,*Ocean Modell.***5**, 157–170, https://doi.org/10.1016/S1463-5003(02)00019-7.Bo Pedersen, F., 1980: Dense bottom currents in rotating ocean.

,*J. Hydraul. Div.***106**, 1291–1308.Bombosch, A., and A. Jenkins, 1995: Modeling the formation and deposition of frazil ice beneath Filchner-Ronne Ice Shelf.

,*J. Geophys. Res.***100**, 6983–6992, https://doi.org/10.1029/94JC03224.Crabtree, R. D., and C. S. M. Doake, 1982: Pine Island Glacier and its drainage basin: Results from radio echo-sounding.

,*Ann. Glaciol.***3**, 65–70, https://doi.org/10.1017/S0260305500002548.De Boer, B., and Coauthors, 2015: Simulating the Antarctic ice sheet in the Late-Pliocene warm period: PLISMIP-ANT, an ice-sheet model intercomparison project.

,*Cryosphere***9**, 881–903, https://doi.org/10.5194/tc-9-881-2015.DeConto, R. M., and D. Pollard, 2016: Contribution of Antarctica to past and future sea-level rise.

,*Nature***531**, 591–597, https://doi.org/10.1038/nature17145.Depoorter, M. A., J. L. Bamber, J. A. Griggs, J. T. M. Lenaerts, S. R. M. Ligtenberg, M. R. Van den Broeke, and G. Moholdt, 2013: Calving fluxes and basal melt rates of Antarctic ice shelves.

,*Nature***502**, 89–92, https://doi.org/10.1038/nature12567.De Rydt, J., and G. H. Gudmundsson, 2016: Coupled ice shelf-ocean modeling and complex grounding line retreat from a seabed ridge.

,*J. Geophys. Res. Earth Surf.***121**, 865–880, https://doi.org/10.1002/2015JF003791.Dinniman, M. S., J. M. Klinck, L.-S. Bai, D. H. Bromwich, K. M. Hines, and D. M. Holland, 2015: The effect of atmospheric forcing resolution on delivery of ocean heat to the Antarctic floating ice shelves.

,*J. Climate***28**, 6067–6085, https://doi.org/10.1175/JCLI-D-14-00374.1.Eckhaus, W., 1979:

*Asymptotic Analysis of Singular Perturbations*. North-Holland, 286 pp.Golledge, N. R., D. E. Kowalewski, T. R. Naish, R. H. Levy, C. J. Fogwill, and E. G. W. Gasson, 2015: The multi-millennial Antarctic commitment to future sea-level rise.

,*Nature***526**, 421–425, https://doi.org/10.1038/nature15706.Hellmer, H. H., and D. J. Olbers, 1989: A two-dimensional model for the thermohaline circulation under an ice shelf.

,*Antarct. Sci.***1**, 325–336, https://doi.org/10.1017/S0954102089000490.Holland, P. R., A. Jenkins, and D. M. Holland, 2008: The response of ice shelf basal melting to variations in ocean temperature.

,*J. Climate***21**, 2558–2572, https://doi.org/10.1175/2007JCLI1909.1.Holmes, M. H., 1995:

*Introduction to Perturbation Methods*. Springer, 356 pp.Jacobs, S. S., H. H. Helmer, C. S. M. Doake, A. Jenkins, and R. M. Frolich, 1992: Melting of ice shelves and the mass balance of Antarctica.

,*J. Glaciol.***38**, 375–387, https://doi.org/10.1017/S0022143000002252.Jenkins, A., 1991: A one-dimensional model of ice shelf-ocean interaction.

,*J. Geophys. Res.***96**, 20 671–20 677, https://doi.org/10.1029/91JC01842.Jenkins, A., 2011: Convection-driven melting near the grounding lines of ice shelves and tidewater glaciers.

,*J. Phys. Oceanogr.***41**, 2279–2294, https://doi.org/10.1175/JPO-D-11-03.1.Jenkins, A., 2014: Scaling laws for the melt rate and overturning circulation beneath ice shelves derived from simple plume theory. Geophysical Research Abstracts, Vol. 16, Abstract EGU2014-13755, https://meetingorganizer.copernicus.org/EGU2014/EGU2014-13755.pdf.

Lane-Serff, G. F., 1995: On meltwater under ice shelves.

,*J. Geophys. Res.***100**, 6961–6965, https://doi.org/10.1029/94JC03244.Lazeroms, W. M. J., A. Jenkins, G. H. Gudmundsson, and R. S. W. van de Wal, 2018: Modelling present-day basal melt rates for Antarctic ice shelves using a parametrization of buoyant meltwater plumes.

,*Cryosphere***12**, 49–70, https://doi.org/10.5194/tc-12-49-2018.MacAyeal, D. R., 1985: Evolution of tidally triggered meltwater plumes below ice shelves.

*Oceanology of the Antarctic Continental Shelf*, S. Jacobs, Ed., Antarctic Research Series, Vol. 43, Amer. Geophys. Union, 133–143.Magorrian, S. J., and A. J. Wells, 2016: Turbulent plumes from a glacier terminus melting in a stratified ocean.

,*J. Geophys. Res. Oceans***121**, 4670–4696, https://doi.org/10.1002/2015JC011160.Mahrt, L., 1982: Momentum balance of gravity flows.

,*J. Atmos. Sci.***39**, 2701–2711, https://doi.org/10.1175/1520-0469(1982)039<2701:MBOGF>2.0.CO;2.Mathiot, P., A. Jenkins, C. Harris, and G. Madec, 2017: Explicit and parametrised representation of under ice shelf seas in a

*z** coordinate ocean model NEMO 3.6.,*Geosci. Model Dev.***10**, 2849–2874, https://doi.org/10.5194/gmd-10-2849-2017.Mattheij, R. M. M., S. W. Rienstra, and J. H. M. ten Thije Boonkkamp, 2005: Partial Differential Equations: Modeling, Analysis, Computation. Society for Industrial and Applied Mathematics, 665 pp.

McPhee, M. G., 1992: Turbulent heat flux in the upper ocean under sea ice.

,*J. Geophys. Res. Oceans***97**, 5365–5379, https://doi.org/10.1029/92JC00239.McPhee, M. G., C. Kottmeier, and J. H. Morison, 1999: Ocean heat flux in the central Weddell sea during winter.

,*J. Phys. Oceanogr.***29**, 1166–1179, https://doi.org/10.1175/1520-0485(1999)029<1166:OHFITC>2.0.CO;2.Mueller, R. D., L. Padman, M. S. Dinniman, S. Y. Erofeeva, H. A. Fricker, and M. A. King, 2012: Impact of tide-topography interactions on basal melting of Larsen C Ice Shelf, Antarctica.

,*J. Geophys. Res.***117**, C05005, https://doi.org/10.1029/2011JC007263.Naughten, K. A., K. J. Meissner, B. K. Galton-Fenzi, M. H. England, R. Timmermann, H. H. Hellmer, T. Hattermann, and J. B. Debernard, 2018: Intercomparison of Antarctic ice-shelf, ocean, and sea-ice interactions simulated by MetROMS-iceshelf and FESOM 1.4.

,*Geosci. Model Dev.***11**, 1257–1292, https://doi.org/10.5194/gmd-11-1257-2018.Nayfeh, A. H., 1973:

*Perturbation Methods*. John Wiley and Sons, 425 pp.Olbers, D., and H. Hellmer, 2010: A box model of circulation and melting in ice shelf caverns.

,*Ocean Dyn.***60**, 141–153, https://doi.org/10.1007/s10236-009-0252-z.Pritchard, H. D., S. R. M. Ligtenberg, H. A. Fricker, D. G. Vaughan, M. R. Van den Broeke, and L. Padman, 2012: Antarctic ice-sheet loss driven by basal melting of ice shelves.

,*Nature***484**, 502–505, https://doi.org/10.1038/nature10968.Reese, R., T. Albrecht, M. Mengel, X. Asay-Davis, and R. Winkelmann, 2018: Antarctic sub-shelf melt rates via PICO.

,*Cryosphere***12**, 1969–1985, https://doi.org/10.5194/tc-12-1969-2018.Rignot, E., S. Jacobs, J. Mouginot, and B. Scheuchl, 2013: Ice-shelf melting around Antarctica.

,*Science***341**, 266–270, https://doi.org/10.1126/science.1235798.Rignot, E., J. Mouginot, M. Morlighem, H. Seroussi, and B. Scheuchl, 2014: Widespread, rapid grounding line retreat of Pine Island, Thwaites, Smith, and Kohler glaciers, West Antarctica, from 1992 to 2011.

,*Geophys. Res. Lett.***41**, 3502–3509, https://doi.org/10.1002/2014GL060140.Sergienko, O. V., 2013: Basal channels on ice shelves.

,*J. Geophys. Res. Earth Surf.***118**, 1342–1355, https://doi.org/10.1002/jgrf.20105.Seroussi, H., Y. Nakayama, E. Larour, D. Menemenlis, M. Morlighem, E. Rignot, and A. Khazendar, 2017: Continued retreat of Thwaites Glacier, West Antarctica, controlled by bed topography and ocean circulation.

,*Geophys. Res. Lett.***44**, 6191–6199, https://doi.org/10.1002/2017GL072910.Shabtaie, S., and C. R. Bentley, 1987: West Antarctic ice streams draining into the Ross Ice Shelf: configuration and mass balance.

,*J. Geophys. Res.***92**, 1311–1336, https://doi.org/10.1029/JB092iB02p01311.Slater, D., P. Nienow, A. Sole, T. Cowton, R. Mottram, P. Langen, and D. Mair, 2017: Spatially distributed runoff at the grounding line of a large Greenlandic tidewater glacier inferred from plume modelling.

,*J. Glaciol.***63**, 309–323, https://doi.org/10.1017/jog.2016.139.Smedsrud, L. H., and A. Jenkins, 2004: Frazil ice formation in an ice shelf water plume.

,*J. Geophys. Res.***109**, C03025, https://doi.org/10.1029/2003JC001851.Thoma, M., J. Determann, K. Grosfeld, S. Goeller, and H. H. Hellmer, 2015: Future sea-level rise due to projected ocean warming beneath the Filchner Ronne Ice Shelf: A coupled model study.

,*Earth Planet. Sci. Lett.***431**, 217–224, https://doi.org/10.1016/j.epsl.2015.09.013.Timmermann, R., and H. H. Hellmer, 2013: Southern Ocean warming and increased ice shelf basal melting in the twenty-first and twenty-second centuries based on coupled ice-ocean finite-element modelling.

,*Ocean Dyn.***63**, 1011–1026, https://doi.org/10.1007/s10236-013-0642-0.Timmermann, R., and S. Goeller, 2017: Response to Filchner–Ronne Ice Shelf cavity warming in a coupled ocean–ice sheet model–Part 1: The ocean perspective.

,*Ocean Sci.***13**, 765–776, https://doi.org/10.5194/os-13-765-2017.Zweng, M. M., and Coauthors, 2013:

*Salinity*. Vol. 2,*World Ocean Atlas 2013*, NOAA Atlas NESDIS 74, 39 pp.