1. Introduction
Loss of ice from the Greenland Ice Sheet contributed ~8 mm to global sea level between 1992 and 2012 (Shepherd et al. 2012; Vaughan et al. 2013), with the rate of loss accelerating over the same period (Rignot et al. 2011), such that between 2009 and 2012, Greenland contributed ~1 mm yr−1 to global sea level (Enderlin et al. 2014). Driven by a period of tidewater glacier acceleration and retreat (Moon et al. 2012; Jiskoot et al. 2012), ice flux into the ocean from tidewater glaciers accounted for approximately half of Greenland’s mass balance deficit in the early 2000s (van den Broeke et al. 2009). Tidewater glacier dynamics can therefore have a significant impact on Greenland Ice Sheet mass balance.
The dramatic changes observed at many of Greenland’s outlet glaciers occurred during a period of atmospheric and oceanic warming (Mernild et al. 2014; Rignot et al. 2012); however, a conclusive attribution of the response of tidewater glaciers to one of these factors remains elusive. Indeed, many of the processes affecting tidewater glaciers are promoted by both of these forcings, such that a full understanding of the dynamics likely requires consideration of both atmospheric and oceanic factors (Straneo and Cenedese 2015). One such process is submarine melting of the calving front, on which this paper is focused.
Submarine melting may contribute to mass loss either directly via melting of submerged ice (Motyka et al. 2013; Bartholomaus et al. 2013; Inall et al. 2014) or indirectly by controlling calving style and rate (O’Leary 2011; Chauché et al. 2014) or grounded ice flux (Holland et al. 2008). Submarine melting is thought to be promoted by both the presence of warm water of subtropical origin in Greenlandic fjords (Straneo et al. 2010; Mortensen et al. 2011) and the emergence of subglacial discharge at the grounding line of the glacier (Jenkins 2011). Presence of the latter results in proglacial plumes as the discharge rises buoyantly, and the high ice-adjacent water velocities generated increase the turbulent transfer of heat to the ice. These proglacial plumes are the focus of this study.
Given the difficulties of directly measuring submarine melt rates, estimates have to date relied either on modeling or on hydrographic data taken some distance from the glacier terminus. The hydrographic data can in theory be used to calculate a net toward-glacier heat flux. This method has been applied around Greenland to obtain submarine melt rates ranging from 0.7 to 10 m day−1 (Rignot et al. 2010; Sutherland and Straneo 2012; Xu et al. 2012; Inall et al. 2014). Such estimates should, however, be viewed with caution, as fjord dynamics can display significant short-term variability (Jackson et al. 2014; Straneo and Cenedese 2015) such that a calculated melt rate may not be indicative of a longer-term mean. There may also be considerable loss of heat between the flux gate and calving front because of the melting of submerged proglacial ice mélange (Inall et al. 2014).
An alternative approach uses high-resolution numerical modeling to predict ice-adjacent water velocities and temperatures, which are then converted to a submarine melt rate using a melt parameterization (Holland and Jenkins 1999). This has been undertaken in both two (Xu et al. 2012; Sciascia et al. 2013, 2014) and three dimensions (Xu et al. 2013; Kimura et al. 2014; Slater et al. 2015; Carroll et al. 2015). These models have facilitated investigation of the spatial distribution of submarine melting and of how submarine melting responds to variations in subglacial discharge, fjord temperature, and near-terminus subglacial hydrology. A key result from these studies is that per plume, submarine melt rate responds sublinearly to increasing subglacial discharge. Specifically, these studies suggest a relation 
A final method of investigating submarine melt rates—and the approach taken in this study—is buoyant plume theory (BPT). BPT has been successfully applied to a wide range of environmental phenomena, traceable back to the classic paper by Morton et al. (1956). BPT describes the evolution of a source of buoyancy as it rises through an ambient fluid. In this glacial application we use BPT to describe ice-adjacent water velocity and temperature within a proglacial plume. This approach avoids recourse to computationally expensive numerical simulations but is limited to idealized geometries. BPT has previously been applied to tidewater glaciers. MacAyeal (1985) and Jenkins (1991) were pioneering papers on the glacial application of BPT. Wells and Worster (2008) developed the theoretical basis of coupling plume theory with submarine melting while Jenkins (2011) used BPT to propose, in advance of the numerical studies reported above, a cube root dependence of submarine melt rate on subglacial discharge and a linear dependence on fjord temperature. O’Leary (2011) applied BPT to obtain submarine melt rates for three glaciers in West Greenland, and Cenedese and Linden (2014) have used laboratory experiments to investigate plume dynamics with a glacial motivation. Most recently, Cowton et al. (2015) used BPT to force the glacier boundary of an ocean general circulation model adapted to a fjord and Carroll et al. (2015) employed BPT to investigate plume outflow depth in a fjord.
In spite of the increasing body of research on submarine melting, there remain significant gaps in our understanding, both observationally (there has yet to be a direct measurement of submarine melting at a tidewater glacier) and in the modeling (for example, the wide range of melt discharge exponent γ values present in the literature). Given the potential importance of submarine melting for tidewater glacier dynamics, there is a need for further investigation of ice–ocean interaction, of which this study is an example.
This study uses BPT to investigate the dynamics of plumes in contact with the vertical calving fronts of tidewater glaciers with a focus on the submarine melt induced by the plume. We investigate both point and line plume sources and consider a fjord that is uniformly or linearly stratified. We also explore under what conditions a plume will reach the fjord surface. In undertaking these investigations we aim to (i) explain the variation in the value of the melt discharge exponent γ found in the literature, (ii) suggest under what conditions a certain melt exponent may apply, and (iii) facilitate assessment of the likely variation in submarine melt at tidewater glaciers, both in recent decades and under future climate scenarios.
2. Methods
a. Introduction to the model
In this paper we consider mainly a half-conical geometry for proglacial plumes (Fig. 1), which we believe to be appropriate for plumes arising from channelized subglacial drainage. The glacier terminates in a fjord of depth h and is assumed to have a vertical calving front in contact with the flat side of the plume. The plume has radius b(z), vertical velocity u(z), temperature T(z), and salinity S(z), assumed uniform across the radius of the plume. It experiences drag (coefficient Cd) and induces submarine melt 
The half-conical plume considered in this study. The plume emerges into the fjord at the grounding line of the glacier and rises buoyantly, growing through entrainment of ambient fjord water.
Citation: Journal of Physical Oceanography 46, 6; 10.1175/JPO-D-15-0132.1
The plume is initiated at the glacier grounding line by a source of cold and fresh subglacial discharge. With a Greenlandic application in mind, the proglacial fjord is saline. Thus, the density ρ(z) of the plume—defined through an equation of state as a function of its temperature and salinity—is initially less than that of the ambient water ρa(z), and the plume rises buoyantly. We assume that the plume is turbulent at the source. Turbulence causes the plume to entrain ambient water so that it grows as it rises. Following Morton et al. (1956) and numerous other successful applications of BPT, we assume that the rate of entrainment into the plume is proportional to plume velocity u, with a constant of proportionality α.
The entrainment of ambient water means that the temperature and salinity (and therefore density) of the plume are diluted toward the ambient conditions, thereby altering the plume buoyancy. If as a result the plume density exceeds the ambient water density, the plume is then negatively buoyant, will slow down, and may not reach the fjord surface.
b. Defining equations
With a nonlinear equation of state (e.g., Fofonoff and Millard 1983), Eqs. (1a)–(2c) are referred to in this paper as the full model (FM), the solution of which is achieved by numerical integration. To develop scalings for plume variables and submarine melt under variation in the inputs, we require the model to be analytically tractable, and we thus now describe some simplifications.
c. Analytical model
d. Initial conditions
Solution of Eqs. (1a)–(1d) requires initial conditions for plume radius b0, velocity u0, temperature T0, and salinity S0. Since the plume is initiated by fresh subglacial discharge at the pressure melting point, we take T0 = λ2 + λ3h and S0 = 0 psu. It is less clear how to choose b0 and u0.
The simplicity of this model requires that the subglacial discharge emerges into the fjord vertically, a situation that is unlikely to occur at a tidewater glacier. Horizontal emergence may be more realistic, as the subglacial channel feeding the plume presumably lies along the ice–bed interface prior to reaching the grounding line. Horizontal emergence has been implemented in previous models, with choice of channel size and flow velocity based on the Manning equation (Mugford and Dowdeswell 2011) or balance of wall melt and creep closure (Slater et al. 2015). Numerical models suggest that discharge emerging horizontally quickly transitions to vertical flow, after which point our model should be applicable. We should therefore treat our results near the grounding line with caution.
To solve the model we now require only ambient conditions Ta(z) and Sa(z) and subglacial discharge Q0. For two classes of ambient stratification (uniform and linear), we proceed with the analytical model to obtain fundamental scalings for plume properties and submarine melt under variation in the inputs. Values of the physical parameter values used are provided in Table S1.
3. Uniform stratification
a. Solution of defining equations
The simplest ambient conditions we can consider are those of a uniformly stratified fjord, where temperature and salinity do not vary with depth. This situation is relevant typically to proglacial fjords in Alaska (Bartholomaus et al. 2013) and also at depth in Greenland (Chauché et al. 2014). In addition plume dynamics in a uniform stratification provide a good approximation to the initial rise of a plume in a linear stratification (Morton 1959), and many of the results regarding submarine melting carry over to the linearly stratified case.
Effect of change in (a)–(e) Sa, (f)–(j) Ta, and (k)–(o) Q0 on the half-conical plume in a uniform stratification. Unless being varied, the forcing parameters take values Q0 = 50 m3 s−1, Ta = 3°C, and Sa = 33 psu. Results shown are for the analytical model.
Citation: Journal of Physical Oceanography 46, 6; 10.1175/JPO-D-15-0132.1
b. Local submarine melt rates
Local submarine melt rates are calculated by substituting Eqs. (6a)–(6c) into Eqs. (2a)–(2c) and are plotted in Figs. 2e, 2j, and 2o. Immediately above the grounding line, melt increases quickly with height as the plume warms through entrainment of ambient water. Far from the grounding line, when plume temperature is close to the ambient, variation in melt with height is dominated by velocity and decays as the inverse cube root of distance above the virtual source. The maximum melt rate is located at depth between these two regions. We now discuss the effect of Q0, Ta, and Sa on melt rates. Qualitatively, melt appears insensitive to Sa (Fig. 2e), linearly sensitive to Ta (Fig. 2j), and sublinearly sensitive to Q0 (Fig. 2o).
Consider first the dependence of melt on Sa. Plume velocity is weakly affected and plume salinity responds approximately linearly to Sa [Eqs. (6) and Fig. 2]. The submarine melt parameterization does respond to change in plume salinity through Sb and Tb; however, percentage changes in Sa in glacial settings are typically small (e.g., Straneo and Cenedese 2015) and therefore do not result in significant variability in melt rates (Figs. 2e, 4d). Turning to the response of submarine melt rate to change in Ta, Eqs. (6) and Fig. 2 show that the only plume variable that responds significantly to Ta is plume temperature, and that it does so in an approximately linear fashion. The submarine melt parameterization is also close to linear in plume temperature (Holland and Jenkins 1999), and it therefore follows that melt rates respond linearly to Ta.
We can use the melt rate parameterization Eqs. (2a)–(2c) to show that dTb/du = 0, 0 < dTb/dT < 1, and dTb/dS < 0 (supplemental material). Therefore, if Ta > T0, the second term on the right-hand side of Eq. (7a) is negative (i.e., the effect of increasing subglacial discharge on plume thermal forcing causes a decrease in local melt rate). Conversely, the first term in Eq. (7a) is positive, which says that the effect of increasing subglacial discharge on plume velocity causes an increase in local melt rate.
In the point source limit (z ≫ z0) of Eq. (7a), the second term tends to zero and the first term tends to ⅓. This is the region in which the effect of initial volume flux on plume temperature has been forgotten, and therefore 
Taking Ta = 3°C and Sa = 33 psu, we plot the value of Eq. (7a) as a function of z and Q0 in Figs. 3a and 3c where h = 300 and 1000 m, respectively. We see that 0.30 < γ < 0.35 for 
(a),(c) Plots of local melt rate discharge exponent (i.e., γ in the relationship 
Citation: Journal of Physical Oceanography 46, 6; 10.1175/JPO-D-15-0132.1
c. Total submarine melt rates
We seek last a value for the exponent in the relationship 
Comparison of total melt from the full model (x axes) with total melt from Eq. (11) (y axes). Each data point represents a particular choice of parameters from the ranges indicated (thus, there are 750 data points on each plot). We distinguish data points by (a) subglacial discharge Q0, (b) calving front height h, (c) fjord temperature Ta, and (d) fjord salinity Sa.
Citation: Journal of Physical Oceanography 46, 6; 10.1175/JPO-D-15-0132.1
We next consider a linearly stratified fjord, where plumes may not reach the fjord surface and the relationship between submarine melt and subglacial discharge is modified.
4. Linear stratification
a. Defining equations
b. Stratification in temperature but not salinity
Plume velocity and temperature in the linear stratification cases described in the text: (a), (b) dTa/dz = −0.005°C m−1 and dSa/dz = 0 psu m−1; (c),(d) dTa/dz = 0°C m−1 and dSa/dz = −0.005 psu m−1; and (e),(f) dTa/dz = −0.005°C m−1 and dSa/dz = −0.005 psu m−1. Note that in (c)–(f) we plot nondimensionalized z on the y axis. Gray lines show the ambient temperature; in (f) the plotting of nondimensional z on the y axis leads to two ambient temperature lines: the lighter gray applies for Q0 = 10 m3 s−1, the darker gray for Q0 = 100 m3 s−1. Ambient values at the grounding line are Ta,0 = 3°C and Sa,0 = 33 psu.
Citation: Journal of Physical Oceanography 46, 6; 10.1175/JPO-D-15-0132.1
Since local melt rates scale linearly with plume temperature, it follows from Eq. (14) that local and therefore total melt rates decrease linearly as |dTa/dz| increases (with ambient temperature at the grounding line held fixed). The second term on the right-hand side of Eq. (14) is an increasing but weak function of Q0. Indeed, in the point source limit we obtain Ta − T ≈ (3/8)zdTa/dz, which is independent of Q0. Therefore, the presence of stratification in temperature slightly increases the sensitivity of plume temperature to subglacial discharge, but temperature is independent of Q0 in the point source limit. The exponent γ in the relationship 
Total melt rate discharge exponent (i.e., γ in the relationship 
Citation: Journal of Physical Oceanography 46, 6; 10.1175/JPO-D-15-0132.1
c. Stratification in salinity but not temperature
These scalings assume a point source for the plume located at z = 0; however, plumes at tidewater glaciers are initiated by finite sources. As motivated above, there is a region 
Evolution of plume temperature contrast is given by Eq. (14) with dTa/dz = 0. The response of plume temperature to change in subglacial discharge (Fig. 5d) is therefore similar to the uniform stratification case. It follows that in the point source limit z ≫ z0 (if the plume reaches this far) the response of melt rates to varying subglacial discharge is dominated by the effect on plume velocity.
Assuming then that, under change in Q0, local melt rates are controlled by plume velocity, it follows from Eqs. (15) that local melt rates would scale with 
The exponent in the relationship 
d. Stratification in both temperature and salinity
We now compare this to previous sections. With stratification in salinity but not temperature, Ta − T tends to 0 far from the grounding line (section 4c, Fig. 5d). With stratification in temperature but not salinity, Ta − T became independent of Q0 far from the grounding line (section 4b, Fig. 5b). The critical difference with stratification in both temperature and salinity is that at the furthest point from the grounding line, Ta − T does not tend to 0 and retains sensitivity to Q0 [Eq. (18), Fig. 5f]. Therefore, when considering the effect of change in Q0 on submarine melt, we must consider the effect on plume temperature. Equation (18) and Fig. 5f show that as Q0 is increased, and at the point where the plume reaches its maximum height, plume temperature may be significantly decreased. This effect is less pronounced for small Q0, small 
The total melt rate discharge exponent may therefore be substantially reduced relative to the case considered in section 4c. For dTa/dz = −0.005°C m−1 and dSa/dz = −0.01 psu m−1 (Fig. 6c), the exponent is only slightly reduced. However, if we increase stratification in temperature and decrease stratification in salinity (Fig. 6d), the exponent may be reduced to ~½ and is only unaffected at low Q0. Given the complexity of the relationship between total melt rate and subglacial discharge in the presence of stratification, it is not generally possible to obtain an equivalent of Eq. (11) when stratification is present.
5. Line plumes
For completeness, we now briefly consider the alternative geometry of line plumes. In the line plume case discharge is distributed uniformly across the glacier grounding line and is therefore appropriate for distributed subglacial drainage or low and very wide channels. The results are qualitatively similar to the half-conical geometry, and we therefore only highlight areas with interesting differences. The subglacial discharge Q0 is regarded as a discharge per unit width of glacier. The plume is wedge shaped with the vertical side against the glacier and the inclined face in contact with the fjord (Fig. S2). Plume width b(z) is taken as the thickness of the wedge at height z. Note that this is now the same model as considered in Jenkins (2011).
For a uniform stratification, an equivalent solution to Eqs. (6a)–(6c) is easily found (Linden et al. 1990; Jenkins 2011; Straneo and Cenedese 2015; see also supplemental material). The point source correction distance is given by 
Plots of local melt rate discharge exponent (i.e., γ in the relationship 
Citation: Journal of Physical Oceanography 46, 6; 10.1175/JPO-D-15-0132.1
In a linear stratification, line plume variables scale as 
The character of submarine melting in a linear stratification is analogous to the half-conical case. With stratification in temperature but not salinity, uniform stratification results may be applied (Fig. 8a). With stratification in salinity but not temperature, and provided local melt rate follows plume velocity, we have total melt rates scaling with 
Line plume equivalent of Fig. 6. Total melt rate discharge exponent (i.e., γ in the relationship 
Citation: Journal of Physical Oceanography 46, 6; 10.1175/JPO-D-15-0132.1
6. Discussion
In the following we discuss first the relevance of characteristic plume heights to tidewater glaciers. We then consider the relationship between submarine melt rate and subglacial discharge and use our results to estimate variation in melt rates in recent decades and under projected atmospheric and oceanic warming. We finally discuss the implications for glacier dynamics.
In a linearly stratified fjord, the height of plume buoyancy reversal zbr and maximum plume rise zmh scale approximately with subglacial discharge raised to the power ¼ for a point source plume and ⅓ for a line source plume. We can use these scalings to comment on the visibility of a plume on the fjord surface. Consider a fjord of depth h. If the stratification and subglacial discharge are such that zbr > h, the plume will be less dense than the ambient fjord water when it reaches the surface and will thus flow downfjord at the fjord surface, as is often observed in Greenland (e.g., Tedstone and Arnold 2012). If instead zmh > h > zbr, the plume will reach the fjord surface adjacent to the glacier but will sink before flowing downfjord, likely undergoing some mixing as it sinks. The final possibility is h > zmh, in which case the plume does not reach the fjord surface and there may be no visible evidence of discharge emerging at the glacier grounding line.
The scalings can also be used to suggest the magnitude of subglacial discharge required for each of these transitions. To illustrate this, we consider an example from Store Glacier, West Greenland. Application of Eq. (16a) with h = 500 m, N2 ~ 2 × 10−5 s−2, and 
Considering now submarine melt, we find that no single scaling for melt with subglacial discharge and stratification can be applied universally. Rather, the appropriate scaling depends on a combination of the subglacial discharge, stratification, and calving front height. We give here a qualitative summary of our results. In a uniform stratification, local submarine melt rates close to the grounding line may decrease with increasing subglacial discharge when the resulting decrease in plume thermal forcing outweighs the increase in velocity. However, it remains the case that total submarine melt rate scales with subglacial discharge raised to the power ⅓ regardless of plume geometry provided discharge does not exceed critical values as discussed in the results. Once linear stratification in salinity is introduced, the exponent may be as large as ¾ (⅔) when a half-conical (line) plume does not reach the fjord surface. As subglacial discharge is increased and temperature stratification is introduced, this exponent is reduced.
Turning to previous work on the melt discharge exponent, Xu et al. (2012) and Sciascia et al. (2013) considered line plumes and found total melt rate exponents consistent with ⅓. Sciascia et al. (2013) had a two-layer stratification, but melt rates were dominated by the thicker lower layer that was unstratified. Xu et al. (2012) did not use a uniform stratification, but because they generated few data points when the plume did not reach the surface, they may not have been able to identify the ⅔ exponent predicted here.
Our half-conical plume results can be compared to three-dimensional numerical studies. In a uniform stratification, Kimura et al. (2014) found that a total melt rate exponent of ⅓ fitted their results until high initial plume velocities forced the plume away from the ice. With an observed stratification from in front of Store Glacier, Xu et al. (2013) suggested an exponent of 0.85 (0.5) at low (high) discharge. We believe the transition between these two values is similar to that which we observe in our results; total melt becomes less sensitive to subglacial discharge once the plume-ice contact area can no longer significantly increase. The slight differences in the exponent between Xu et al. (2013) and this paper might be explained by parameterization of turbulence in the numerical model (Slater et al. 2015) or the geometry of the plume source (Kimura et al. 2014). We therefore believe that our results regarding the melt discharge exponent are consistent with previous work and indeed can offer some explanation for the range of values reported.
Our study is most comparable to that of Jenkins (2011), which considered a line plume in the uniform stratification limit, finding a local melt rate discharge exponent of ⅓. As noted in section 2c, Jenkins (2011) focused on the region where buoyancy input from submarine melting is comparable to the initial buoyancy flux and was therefore able to neglect the region where plume temperature is not close to the ambient value. Our focus in this paper on the region in which the submarine melt feedback is negligible means we need to make explicit consideration of the evolution of plume temperature. For sufficiently small discharges, our results are in agreement with Jenkins (2011). However, for Greenland-relevant parameters, we suggest that the local melt rate discharge exponent can be significantly reduced from ⅓ and may even be negative (Fig. 7a).
With the above understanding, we consider under what circumstances a certain exponent should apply. The larger exponents [⅔ for a distributed (line) input, ¾ for a localized (point) input] apply for total melt rate when plumes do not reach the surface, although these exponents can be reduced by temperature stratification. It is probably necessary to solve the full equations to find an exact exponent. These exponents are likely relevant to glaciers terminating in deep fjords (e.g., Rink Isbrae) or glaciers with a distributed drainage system such that the discharge is split over many channels and is therefore less likely to reach the surface. An exponent of ⅓ applies for glaciers in weakly stratified or shallow fjords when plumes reach the surface (e.g., Svalbard or Alaska) or at large glaciers with high subglacial discharge. In these cases we derived simple expressions—Eqs. (11) and (20)—for estimating total submarine melt volume.
We can also use our scalings to assess the likely variation in submarine melting in recent decades and in the future. Assuming, for example, a warming of fjord water from 2° to 3°C (Holland et al. 2008), Eq. (11) suggests an increase in submarine melt of ~29%. Supposing runoff increased over the same period by ~25% (Hanna et al. 2011), a melt discharge exponent of ⅓ (¾) gives an increase in submarine melting of 8% (18%). In combination, one can suggest that in recent decades submarine melt rates may have increased by up to ~50% in response to atmospheric and ocean warming. By the end of the century, under a doubling of subglacial discharge (Fettweis et al. 2013) and additional ocean warming of 2°C (Yin et al. 2011), we can estimate an 80% (140%) increase in submarine melting. Such estimates are of course simplistic in that they take no account of possible changes in fjord circulation or subglacial hydrology and rely on uncertain predictions of atmospheric and ocean warming.
When spatially averaged over a glacier terminus, predicted submarine melt rates (e.g., ~3 m day−1; Slater et al. 2015) are generally much smaller than large Greenland tidewater glacier velocities (e.g., ~20 m day−1 at Helheim Glacier; Bevan et al. 2015). It should be noted that this study has focused on regions of the calving front affected by significant subglacial discharge; regions unaffected by subglacial discharge are still expected to melt (e.g., Sciascia et al. 2013) and may contribute to the spatially averaged melt rate. The above estimates nevertheless suggest that submarine melting would be unable to solely account for the recently observed retreat of such glaciers. If ocean forcing has been the primary driver of tidewater glacier behavior in recent decades, we therefore need to invoke a sensitive coupling between submarine melting and glacier dynamics. There is not yet a consensus in the literature on whether this coupling exists.
In a recent model of Store Glacier, Todd and Christoffersen (2014) found that terminus position was insensitive to an increase in submarine melt rate of up to 100%, an observation that they attributed to the particular bed and lateral topography at Store. At Helheim Glacier, Cook et al. (2014) found an order of magnitude increase in submarine melt was required to make the modeled glacier retreat. These studies therefore suggest that our estimated changes in submarine melting in recent decades would be unable to drive significant glacier retreat. In contrast, other studies (Weertman 1974; Nick et al. 2009; Enderlin et al. 2013) propose that glaciers with beds that deepen inland can respond dramatically to terminus perturbation through the marine ice sheet instability, and O’Leary and Christoffersen (2013) advocate a sensitive coupling between submarine melting and calving rate that may not be fully captured in models to date. Therefore, even with estimates of variation in melt rates in recent decades, the role of submarine melting in the dynamics of Greenland’s tidewater glaciers remains ambiguous.
We note one final point regarding our results. The submarine melt rate parameterization Eqs. (2a)–(2c) should be used with caution, as it has thus far only been validated beneath an Antarctic ice shelf (Jenkins et al. 2010), and therefore, there is significant uncertainty in the value of the heat and salt transfer coefficients ΓT and ΓS. However, provided the form of the melt rate parameterization does not change [i.e., 
7. Conclusions
In this paper we have used buoyant plume theory to investigate the dynamics of proglacial plumes arising from the input of subglacial discharge at the grounding line of tidewater glaciers, focusing on the induced submarine melting of the calving front. In particular we have aimed to derive scalings for variation in submarine melt rates in terms of subglacial discharge, fjord properties, and calving front height.
We find that no simple relationship exists between submarine melt rate, subglacial discharge, and fjord stratification. We suggest that the relationship between subglacial discharge and submarine melt rate prevalent in the literature (i.e., submarine melt rate scales with subglacial discharge raised to the ⅓ power) is appropriate for local or total melt rates in a uniformly stratified fjord regardless of plume source geometry provided discharge does not exceed a critical value. In these cases, it is possible to formulate simple equations for total melt induced [Eqs. (11), (20)]. However, once linear stratification is introduced, the total melt rate discharge exponent may be as large as ¾ (⅔) for a point (line) source plume, though the exponent is complicated by stratification in temperature that may reduce the exponent somewhat. These higher exponents are likely representative for large glaciers terminating in deep water in Greenland where plumes are rarely seen, and where submarine melt rates could therefore be more sensitive to the magnitude of subglacial discharge than previously thought. Our findings are also able to explain the range of values of the exponent found in the literature.
We used our melt rate scalings to estimate that submarine melt rates may have increased by ~50% in recent decades, driven by a combination of atmospheric and ocean warming. Whether this is sufficient to explain the observed dynamic changes at tidewater glaciers in Greenland over the same time period remains uncertain; if it is, this would indicate a sensitive coupling between submarine melting and calving dynamics. Since submarine melting is likely to increase in response to predicted atmospheric and ocean warming, it is clear there is the potential for future dynamic response of tidewater glaciers to submarine melting and thus the need for further research into ice–ocean interaction in Greenland.
Acknowledgments
Donald Slater is supported by a NERC PhD studentship. Tom Cowton is supported by NERC Grant NE/K014609/1 awarded to Peter Nienow and Andrew Sole.
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