In the two-layer model of the fjord, the melting of the ice tongue is the only local source/sink of freshwater/heat. This can be used to formulate conservation relations for volume, salt, and heat . At the sill, conservation of volume is given by 1Q=QA+M, where Q and QA are the volume outflow and inflow of glacially modified and Atlantic waters, respectively, and M the freshwater input due to melting. The ice consists of pure freshwater, implying that the meltwater input M does not affect the salinity budget. Hence conservation of salt is given as 2SQ=SAQA. Combining Eqs. () and () yields Knudsen's relation for salt conservation: 3ΔSQ=SAM,ΔS=defSA-S.

The heat budget involves a balance between advective heat transport towards the glacier and basal melt. The advective heat flux is 4H=c(TAQA+TfM-TQ), where c is the heat capacity (per unit volume) of seawater and the melt freshwater input M is assumed to have the salinity-dependent freezing temperature Tf. Note that Tf will be taken as a constant set by the grounding-line pressure and SA. Using Eqs. () and (), we obtain 5H=c[ΔTQ+(Tf-TA)M],ΔT=defTA-T. The heat flux is related to the ice melt (M) as 6H=M[L+ci(Tf-Ti)], where L is the latent heat of freezing, ci the heat capacity of ice, and Ti the ice temperature.

By combining Eqs. () and (), we obtain 7ΔTQ=M[L/c+ci/c(Tf-Ti)+(TA-Tf)]. Here, L/c≈75 ∘C and in northern Greenlandic fjords TA-Tf is typically 3 ∘C. Hence, the terms involving TA-Tf can to a good approximation be neglected in Eqs. () and (). It is convenient to define a generalized Gade temperature : 8TG=defL/c+ci/c(Tf-Ti), which would be the decrease in temperature of a unit volume of water from which sensible heat is extracted to melt ice corresponding to a unit volume of liquid water.

Note that the equivalent ice temperature defined by is approximately -TG⋅ρwρi, where ρwρi≈1.1 is the density ratio between water and ice.

We will use a parametrization of the basal melt (M), which depends on the difference between the ocean water temperature (TC) and the freezing point temperature of seawater (Tf) at the grounding line . We denote the thermal forcing as 13T=defTC-Tf. If AW reaches the grounding line, then TC=TA, and the thermal forcing is denoted as 14TA=defTA-Tf. However, mixing between in- and out-flowing waters over a sill can lower TC relative to TA, which implies that T can be lower than TA.

We model the area-integrated basal melt as 15M=γ1Tn1, where γ1 is a coefficient that depends on features such as the ice-tongue geometry and subglacial discharge and n1>0 is an exponent. We assume that the plume volume flow QP is also a function of the thermal forcing T and given by 16QP=γ2Tn2, where γ2 is a constant and n2>0 an exponent.

Several studies have used models of varying complexity to examine the relationship between thermal forcing and the area-integrated melt on Greenlandic ice tongues and Antarctic ice shelves (e.g.,  ). These investigations report values of n1 in the range from 1 to 2, with a preference for n1 values of around 1.5 to 2.0 . The reported range of n1 is likely to reflect both different ice–ocean interaction regimes and model assumptions on the boundary conditions at the ice–ocean boundary. There are fewer studies that specifically comment on the relationship between the volume transport in the plume and the thermal forcing, but reported a linear dependence of QP on the thermal forcing; i.e., n2≈1.

Primarily, the heat flux to the ice and the associated melting depend on the product of the thermal forcing and the plume velocity (say u), which in turn is related to the plume buoyancy . If the plume buoyancy is proportional to T and the buoyancy force is balanced by a linear basal friction, then u∝T. This gives n1=2, corresponding to a quadratic relation between melt and thermal forcing . If the basal friction is quadratic, i.e.,  proportional to u2, on the other hand, scaling analyses suggest that u∝T1/2 , which gives n1=1.5. applied the plume model of to the Ryder ice tongue, and their results suggest that n1≈1.7 and n2≈0.7. Notably, if M∝uT and QP∝u, then M/QP∝T. In view of Eqs. () and () this implies that 17n1-n2=1. This constraint on the exponents leads to some attractive simplifications of the dynamics, which will be used in the analyses.

It is worth noting that theoretical considerations (e.g.,  , and references therein) indicate that on marine glaciers where the buoyancy source is dominated by the subglacial discharge near the grounding line rather than by the distributed melt along the submerged glacier, the melt is approximately proportional to the thermal forcing and the plume volume transport is essentially independent of the thermal forcing. This limit of high subglacial discharge, characteristic of summer conditions at Greenlandic tidewater glaciers (e.g.,  ), is described by the case of n1=1 and n2=0, which satisfies the constraint of Eq. (). Some aspects of the case with high subglacial discharge are discussed in Appendix .

In summary, the literature reports a range of values for n1 and n2. However, n1=2 and n2=1 appear as one reasonable choice for the exponents for qualitatively examining the dynamics of large Greenlandic ice tongues such as the 79∘ N, Petermann, and Ryder glaciers. This will be the baseline case when we examine the interplay between melt dynamics and hydraulic control in Sect. 3. In Sect. , we will consider how variations in the values of n1 and n2 affect the results, including the case of n1=1 and n2=0. When we derive general results below, however, we will allow n1 and n2 to be arbitrary positive numbers but subject to the constraint n1>n2.

Consider a situation in which the fjord geometry, via frictional resistance or hydraulic control, does not limit the exchange flow and its associated heat transport towards the ice tongue. We assume that unmodified AW reaches the glacier (TC=TA), and the melt processes create a plume volume flow QP that sets the exchange flow: if TA increases, the exchange flow also increases at the rate given by Eq. (). This regime, in which the strength of the exchange flow is controlled locally by the glacier basal melt, will be referred to as the melt-controlled regime, and a contrasting hydraulically controlled exchange flow regime will be presented in Sect. .

In the melt-controlled regime, where Q=QP, we can use the heat conservation relation (Eq. ) together with Eqs. () and () to obtain 18ΔTTG=γ1γ2Tn1-n2. This relationship, which is equal to M/Q, shows that ΔT as well as the ratio M/Q increase with T; i.e., the meltwater fraction in the plume increases with thermal forcing. Note that the condition n1-n2=1 yields a linear relation between ΔT as well as M/Q and the thermal forcing.

By dividing Eq. () with T/TG, we obtain 19ΔTT=TGγ1γ2Tn1-n2-1. Since T=TA in the melt-controlled regime, the left-hand side in this expression equals (TA-T)/(TA-Tf), which is less or equal to 1 since T≥Tf. When n1-n2=1, the right-hand side becomes independent of T and equals 20σ=defTGγ1γ2. This parameter is a non-dimensional measure of the temperature of the outflowing glacially modified water (T): when σ=1, T=Tf, and when σ=0, T=TA. (This interpretation of the Eq. () applies also when n1-n2≠1, but then ΔT/T is no longer constant in the melt-controlled regime.) As will be shown below, σ influences aspects of the hydraulically controlled regime.

To summarize, the flow in the melt-controlled regime is specified by a knowledge of the AW properties TA and SA, which determine the thermal forcing T=TA. In turn, this yields M, Q, and ΔT (Eqs. , , ), and ΔS is obtained from Eq. ().

We will now go on to examine how a hydraulically controlled exchange flow affects the melt dynamics. For this purpose, it is useful to write QP as a function of the temperature difference. By using Eq. (), we obtain 21QP=γ2γ2ΔTγ1TGn2n1-n2. Since n1>0 and n2>0, this shows that the melt-controlled exchange flow increases with ΔT when n1-n2>1.

Fjord and sill geometries may impose limits on the exchange flow, which in turn can potentially alter the basal melt dynamics. In particular, hydraulic control of a two-layer exchange flow over a sill sets an upper bound for the exchange flow (say QH), which is determined by the upstream height of the AW layer above the sill (h) and the layer density difference . Exchange flow strengths below the critical value QH are unconstrained by the geometry and are referred to as subcritical flows. Thus, it is conceivable that a sufficiently strong melt-driven exchange flow or a high sill can cause a transition from a subcritical flow to a critical, hydraulically controlled flow .

How the flow evolves as the melt-driven exchange flow (or the sill height) is gradually increased and approaches the hydraulically controlled limit is complex and depends on fjord and sill geometry . Observations from the Ryder and 79∘ N glaciers show that the inflow at the sills in front of the ice tongues are hydraulically controlled and that the thickness of the inflow layer is thin compared to the upper outflowing layer . This implies that the flow can be approximated by a one-layer hydraulic model representing the inflowing AW layer. Two additional features allow for simplifications of the hydraulic model. First, the depth of the AW layer on the seaward side (upstream) of the sill is much larger than at the sill, which implies that the upstream AW inflow velocity is negligible. Second, the inflow over the sill is confined in a channel that is small compared to the internal Rossby radius, which allows the Earth's rotation to be neglected. In this situation, the maximum hydraulically controlled volume flow is given by 22QH=Wh3/2233/2gΔρρ01/2, where W is the cross-sectional width of the lower layer on the sill,

For simplicity, we assume a rectangular cross section, which implies that W does not depend on h. Note that the lower-layer width W may be smaller than the fjord width if the inflow is confined in a deeper channel crossing the sill, which is the case for Ryder; see Fig. 1 in .

The results presented above suggest that there can exist two different flow regimes in a fjord with basal ice-tongue melting: one where the exchange flow Q is determined locally by the basal melt processes on the glacier and one where it is hydraulically controlled. The two regimes have the following characteristics.

In the melt-controlled regime, the volume flow of the meltwater plume is smaller than the upper limit set by hydraulic control; i.e.,  QP<QH. Accordingly, the sill does not constrict the exchange flow, which is specified by Eq. () (or Eq. ). The flow at the sill is subcritical , and as result there is limited mixing between inflowing and outflowing waters. Essentially unmodified AW reaches the grounding line of the ice tongue (Fig. a), which implies that the thermal forcing is given by T=TA. Since the freezing temperature (Tf) is set by the grounding-line depth (and SA, which to a good approximation can be taken as constant here), the thermal forcing at a specific glacier is externally determined by the AW temperature (TA).

At the transition between the two flow regimes T=TA and QP=QH, which implies that the transports given by Eqs. () and () are equal. By using this and Eq. (), which applies at the transition, after some rearrangements we obtain 27γ11/3kH2/3hγ2T3n2-n13=1. This represents the condition QP/QH=1 and gives the relationship between T and h at the regime transition. The height of the AW above the sill at the transition (say hL) as a function of TA is given by 28hL=kH-2/3γ1-1/3γ2TA3n2-n13. When h>hL, the flow is in the melt-controlled regime, and when h<hL, the exchange flow becomes hydraulically controlled. Alternatively, Eq. () gives the thermal forcing at the regime transition as a function of h 29TL=kH2h3γ1γ2313n2-n1. Using this and Eq. (), the temperature difference at the regime transition can be written as 30ΔTLTG=γ1TLn1-n2γ2. If n1-n2=1, then ΔTL=σTL; see Eq. ().

Lowering the AW height always brings the flow towards the hydraulic regime. If h is fixed, then TL is also fixed, which implies that changes in the AW temperature can cause a transition between the melt-controlled and the hydraulically controlled regimes. As illustrated in Fig. , the nature of the transition depends on the value of the exponent 3n2-n1 in Eq. (). If 3n2-n1>0, increasing TA will bring the flow towards the hydraulically controlled regime. On the other hand if 3n2-n1<0, decreasing TA will brings the flow towards the hydraulically controlled regime. This behavior follows from the fact that QP∝ΔTn2n1-n2 and QH∝ΔT1/2. In the case of 3n2-n1=0, the regime transition height hL is independent of TA. This is because QP and QH then have the same dependence on ΔT.

Figure  illustrates the relation between exchange flow and temperature difference (Eqs.  and ) for the case of n1=2 and n2=1, which implies that QP∝ΔT. Here, the flow is in the melt-controlled regime if ΔT<ΔTL or from Eq. () equivalently if TA<TL. By increasing ΔT, the flow increases and is shifted towards and into hydraulic control.

In the case where 3n2-n1<0, hydraulic control – for a fixed h – occurs for weak thermal forcing and exchange flow. In a real fjord, additional exchange flows driven by winds and tides may be larger than a model-predicted weak hydraulic flow . This can prevent the establishment of hydraulic control and in effect yield an exchange flow in the melt-controlled regime. We will briefly discuss this case in Sect.  .

It is possible that transitions between melt-controlled and hydraulically controlled regimes can also be caused by seasonal variations in subglacial discharge, even if the AW features remain unchanged. The reason is that the basal melt M increases with the subglacial discharge , which has a pronounced seasonal cycle that tracks the surface melt on the glaciers (e.g., ). Thus, it is possible that in some fjords the exchange circulation can be stronger and hydraulically controlled in summer when the subglacial discharge peaks and weaker and melt-controlled in winter. The details of such seasonal regime transitions will depend, among other things, on how the ratio between exchange flow and melt (M/Q) depends on the subglacial discharge. However, we will not pursue this topic further.

To examine some concrete aspects of the model, we will now apply it in a qualitative way to the ice tongues of the Ryder, 79∘ N, and Petermann glaciers. We consider scenario 2, assuming n1=2 and n2=1 and try to crudely estimate model parameters characterizing the melt–flow dynamics. We use observations of flow rates and melt rates (Q and M) and hydrography from the three glaciers reported in the literature . We recall that observations show that the sill exchange flows of the Ryder and 79∘ N glaciers are hydraulically controlled but that the exchange flow over the relatively deep and wide sill in Petermann Fjord is not. Note that there are uncertainties in the observations and in model assumptions, and the present exercise is primarily an illustration of how the model can be applied.

Guided by the model physics, we estimate model variables and parameters as follows: the hydrographic observations give TC as the near-bottom temperatures inside the sill and TA as the sill-depth temperature outside the sill. Note that outside the sills, the temperatures are nearly constant below the sill depths; see Fig. . The outflow model temperature T, which represents a flow-weighted mean over of an outflow distributed vertically over a range of temperatures, is less straightforward to determine from hydrography. Here, we use the relation M/Q=ΔT/TG=ΔS/SA (which follows from Eqs. () and ()) to find values of T and S that roughly satisfy these conditions and at the same time characterize outflowing water. This allows Φ to be determined from Eq. ().

The model parameters γ1, γ2, and kH are estimated as follows: from Eq. (), we obtain γ1≈M/T2. By using Eqs. (), (), and (), we obtain γ2≈Q/[T(1-Φ)]. This provides the estimate 46σ=TGγ1γ2≈TGMTQ(1-Φ). By using Eqs. () and (), we obtain kHh3/2≈Q3/2/M1/2, and an estimate of h then gives kH. Note that kH can also be determined from a knowledge of the cross-sectional sill width W.

The estimates of Q from Ryder and Petermann are more uncertain than the ones from 79∘ N reported by , which are based on a 1-year moored time series of velocity. The Ryder estimate of Q is based on a single instantaneous current measurement on the inner sill in Sherard Osborn Fjord . Our Petermann estimate of Q is based on hydrography and geostrophic velocities presented by : using their Fig. 7, we estimate the outflow of glacially modified water (in the depth range of 150 to 250 m) to be on the order of 50×103 m3 s-1.

Table  summarizes observational features and estimated model parameters. Notably, the estimates of σ, around 0.1–0.2, are similar for the three glaciers. This reflects that the cooling due to ice melt of the meltwater plume is small compared to the upper limit (σ=1), in which the outflow temperature (T) approaches the freezing point. For the Ryder and 79∘ N glaciers, which have hydraulically controlled exchange flows, the estimated h/hL is 0.7 and 0.4, respectively. The lower value of h/hL at 79∘ N Glacier implies a higher sensitivity of the melt to changes in h/hL; see Fig. . Further, the reduction factors (R=T/TA), which are directly inferred from the hydrography, are only slightly below unity: R is 0.96 and 0.86 at Ryder and 79∘ N, respectively. If as assumed here M∝T2, this implies that, relative to the situation where unmodified AW reaches the grounding line, the melt rates are reduced by about 10 % and 30 % at Ryder and 79∘ N, respectively. Taken together, this suggests that hydraulic control and associated entrainment reduce the basal melt on both ice tongues but that currently this effect is more pronounced at the 79∘ N Glacier.

Figure  shows model-predicted melt rates M and R factors as a function of TA for the Ryder and 79∘ N glaciers when h is kept at observed values. The figure also shows the melt rates that would occur if unmodified AW reached the grounding lines: the melt rates in the hydraulically controlled regime are lower and their dependence on TA is weaker. The model results show that a 1 ∘C increase in TA from present values increases M by about 75 % and 50 % at Ryder and 79∘ N, respectively. The basal melt is more sensitive to the same change in TA at Ryder than at 79∘ N simply because TA is larger at 79∘ N. For fractional changes in TA, the response in basal melt is more similar: a 10 % increase in TA yields an increase in M of about 15 % at both glaciers. (Note that a 10 % increase in TA corresponds to an increase in ∼0.3 and ∼0.4 ∘C of TA at Ryder and 79∘ N, respectively). In the absence of hydraulic control, where M∝TA2, the corresponding increase in M would be 20 %. Figure  shows the model-predicted dependence of Ryder basal melt on TA and h. Currently, the AW interface is about 30 m below the transition depth hL at which hydraulic control ceases; i.e., a lifting of the AW interface by about 30 m would bring the flow into the melt-controlled regime.

At Petermann Glacier, the exchange flow is not hydraulically controlled because the height of the AW above the sill is larger than the transition height hL (Eq. ). Note that hL decreases with the cross-sectional width of the sill (hL∝W-2/3). Ryder and Petermann have similar values of TA, and it is primarily the wide sill in Petermann Fjord that yields a small value of hL (see Fig. 5 in ).