An Analytical Derivation of Ice-Shelf Basal Melt Based on the Dynamics of Meltwater Plumes

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 or, equivalently, the along-slope coordinate X, where corresponds to the grounding-line depth . Note that all 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 and salinity , which in principle can be depth dependent as well.

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Schematic picture of the plume model of Jenkins (1991, 2011) for a typical ice-shelf geometry with basal slope . The plume is described by a thickness D, speed U, temperature T, and salinity S and forced by entrainment of ambient ocean water and basal melting .

Citation: Journal of Physical Oceanography 49, 4; 10.1175/JPO-D-18-0131.1

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The dynamics of the plume under the ice shelf in Fig. 1 can be modeled by a two-layer system, in which the plume thickness 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:

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corresponding to conservation of mass, momentum, heat, and salt, respectively. The first equation describes the plume volume flux DU as being determined by the melt rate (input of meltwater from the ice–ocean interface) and the entrainment rate (input of ambient ocean water), which in turn is modeled by (Bo Pedersen 1980):

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where is a dimensionless constant. As already explained by Jenkins (1991), this entrainment parameterization is chosen for its simple dependence on plume velocity and basal slope, giving good agreements with laboratory studies for small slopes, but we note that more general 2D parameterizations without an explicit slope dependence exist (see, e.g., Sergienko 2013).

The second equation [(1b)] provides the balance between the driving force (buoyancy), determined by the density contrast of the plume with respect to the ambient ocean, and the drag force with dimensionless drag coefficient . The density contrast is found from a linearized equation of state:

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where is the haline contraction coefficient and the thermal expansion coefficient. Note that Coriolis forces have been neglected in (1b), as well as along-stream variations in the buoyancy (see, e.g., Sergienko 2013). Coriolis forces are known to have a significant effect on larger ice shelves, but Jenkins (2011) provides an estimate of rotational length scales, on which we will briefly comment in section 3c. A more detailed analysis of the momentum equation was performed by Mahrt (1982), showing that the along-stream variations in buoyancy can generally be neglected. This assumption most notably breaks down on the outer parts of ice shelves, where the plume decelerates due to either a loss of buoyancy or a reduction in the basal slope.

Equations (1c) and (1d) describe the input of heat and salt due to entrainment and melting , as well as turbulent exchange through the sub-ice-shelf boundary layer with exchange coefficients (Stanton numbers) and . Furthermore, and are the temperature and salinity at the ice–ocean interface. To close the system, we require three additional conditions at the ice–ocean interface (Jenkins 1991, 2011):

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Equations (4a) and (4b) describe the balance of heat and salt fluxes at the interface, where and are the temperature and salinity of the ice, L is the latent heat of fusion for ice, and and c are the specific heat capacities of ice and ocean water, respectively. Finally, (4c) equates the interface temperature with the local pressure freezing point for salinity at depth , given by the (linearized) liquidus condition with constant coefficients , , and .

The plume model in the form presented above can be solved numerically for any quasi-one-dimensional ice-shelf geometry, defined by the draft and slope angle , and any vertical profile for the ambient ocean properties and . The constant parameter values used in this study are summarized in Table 1. Furthermore, the plume model requires four initial conditions at the grounding line () for the quantities D, U, T, and S [(1)]. In general, one can impose an initial freshwater flux at the grounding line with zero salinity and temperature equal to the local pressure freezing point, as discussed in Jenkins (2011). However, in contrast to the aforementioned paper, we are mainly interested in the regime of melt-driven convection where the primary source of buoyancy is the meltwater generated locally at the ice-shelf base. Therefore, we take as initial condition at the grounding line throughout this paper. The numerical solution of the plume model will be revisited in section 3, where it is compared with the analytical formulation derived in the following.

Table 1.

Constant parameters used in the plume model (Jenkins 1991, 2011) and the (derivation of the) melt-rate parameterization (28). The values for c, , , , , and are for potential temperature and practical salinity units. Note that the values of , , and technically depend on the ambient salinity ; the presented values are for psu.

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b. Simplified formulation and scaling

To facilitate the analytical derivation, we shall apply a number of simplifications to the original plume model. First of all, following Jenkins (2011), the interface conditions in (4) can be replaced by a simpler formulation (McPhee 1992; McPhee et al. 1999) consisting of only two equations:

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where is the pressure freezing point of the plume, taking over the role of the interface temperature , and is an effective heat exchange coefficient. This simplification also requires replacing by and by in the plume heat balance in (1c). Note that the interface salinity can be eliminated directly by substituting (4b) in the salinity equation (1d), where the ice salinity is assumed to be zero. Furthermore, the ice temperature is eliminated by neglecting the conductive heat flux [the last term on the right-hand side in (5a)], considering that . Alternatively, one could assume the heat conduction term to be constant, as in Jenkins (2011), which would not change the algebra because this term can be incorporated as a small correction to the value of 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.

Next, we consider an ice-shelf geometry with constant basal slope α. Hence, the ice-shelf basal depth is related to the coordinate X along the ice-shelf base as follows:

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where the grounding-line depth corresponds to . Moreover, we also take uniform ambient ocean properties and (this assumption and its consequences will be discussed further in section 3b). In particular, we can now define the characteristic freezing point at the grounding line as follows:

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which is also constant for a fixed geometry. Note that we have:

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where is the salinity contrast between plume and ambient ocean.

Taking into account these assumptions, the plume equations can now be written in a more compact way, expressed only in terms of D, U, , , , and independent variable X. Formally, only four equations would be required to close the system, but keeping an equation for will simplify the analysis later on. The resulting system has the following closed form:

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Note that these equations only have constant coefficients, as well as some terms that are explicit in X through the depth dependence of the freezing point. Equations (9) can now be nondimensionalized using the following scaling:

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where , , and τ are the characteristic length, velocity, and temperature scales, respectively. These characteristic scales are defined as follows. The temperature difference between ambient ocean and the freezing point at the grounding line is considered to be the main external parameter that drives the initial melting at the grounding line and, hence, the initial buoyancy flux. Therefore, we take

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Neglecting the effects of stratification, Coriolis forces, and initial freshwater fluxes at the grounding line, the main length scale governing the plume dynamics is associated with (cf. Lane-Serff 1995; Jenkins 2011; Lazeroms et al. 2018):

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This length scale denotes the vertical distance from the grounding line over which the thermal driving of the plume decreases from the initial value 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:

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where is the characteristic density scale defined in (10). In addition to these physical scales, we define two dimensionless parameters that determine the entrainment rate and the basal melt rate, respectively:

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noting that the factor is equivalent to the factor in Jenkins (2011). Using (10)–(14), the plume equations in (9) can be written in the following way:

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where we introduced a set of new parameters , , and , which are defined in Table 1 along with their typical values. The dimensionless system in (15) will now be simplified further using the observation that the parameters 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

A central notion in perturbation theory is the asymptotic expansion of order N (or Poincaré expansion; see, e.g., Nayfeh 1973; Eckhaus 1979; Holmes 1995; Mattheij et al. 2005) of a function in terms of a small problem parameter :

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in which each consecutive term decreases strictly faster as ε goes to zero [e.g., ]. Ideally, this allows the asymptotic expansion to converge to the function for . Such an expansion can be used to systematically approximate problems that are too complicated to solve analytically. As we shall see below, it is often convenient to scale the equations by the leading-order asymptote so that the first term in the expansion is 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 , the remaining parameters in (15) are model constants with values given in Table 1. Before we proceed, it is worthwhile investigating if η and μ are indeed small for typical ice-shelf conditions and if they are of the same order of magnitude. With ambient salinity psu and a typical grounding-line depth m, a lower bound for the freezing point is −3°C. With typical ambient temperatures between −2° and +2°C, this yields upper bounds τ < 5°C and , which is indeed small. A typical estimate of the basal slope is more difficult to find, as the slope can vary considerably along the ice shelf. As cannot exceed 1, a strict upper bound is . Hence, both 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 are very common for most ice shelves, even close to the grounding line. For example, we can consider the overall slope of a flowline of Filchner–Ronne Ice Shelf (FRIS), taking m and a horizontal distance of 700 km, yielding and [see, e.g., the data of Bombosch and Jenkins (1995) used in Lazeroms et al. (2018) and the test cases in Fig. 8]. Taking °C as a typical ambient temperature, we obtain °C and . Similarly, we could estimate the slope of Pine Island Glacier (PIG, also shown in Fig. 8) as , giving , while μ might approach the upper bound for warm cavities mentioned above. In these cases, 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 in (15) could be neglected with respect to terms of . Physically, this means neglecting the direct effect of the melt rate on the volume flux in (15a) with respect to the entrainment rate. Note that this condition might be violated locally for relatively warm ice shelf cavities with small slopes.

The next step is to determine how the dependent variables scale with η and μ, that is, determine the scale function in analogy to (16) for each variable. This can be done by inspection of (15), requiring that every equation retains a meaningful balance of terms in the limits and . A crucial observation here is that by construction, since otherwise would not be the governing length scale. This allows us to estimate the magnitudes of the terms in (15a) as , , and . Assuming that and as explained earlier, we see that the final term, the melt rate, is indeed neglected at leading order. Similar arguments can be applied to the other terms in (15), where the constant parameters such as are considered because they are independent of η and μ. From top to bottom, we then find the following leading-order balance of terms:

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which yields

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Physically, this means that both the density and salinity differences scale with the melt rate 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.

We can now extract the scales in (18) and expand to leading order in μ. Furthermore, we incorporate several algebraic factors that are essentially functions of η, which make the resulting equations more compact. This amounts to the following scaling:

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By substituting these expression in (15) and dividing by common factors, we then obtain the following system, valid up to :

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with

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where we have also used the mass equation (20a) to expand the derivatives in the momentum and heat equations (20b) and (20d). Note that this system is indeed uncoupled from the salinity equation, so that it can be disregarded in the remainder of this study. This also clarifies the reason for initially keeping the salinity equation, as mentioned in section 2a, because it simplifies the scaling analysis done so far. Furthermore, we see that the advection terms in the mass and buoyancy equations (20a) and (20c) are of leading order, while those in the momentum and heat equations are , which will have important consequences further on.

We have now scaled the plume equations to a compact form containing only terms except for the velocity and temperature derivatives, which have new factors and . What remains is the question of whether and are still small parameters and if we can use an asymptotic expansion in either 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 . This would justify a second assumption , making the momentum advection term in (20b) one order smaller than the heat advection term in (20d).

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Logarithmic plots of the parameters δ and ε defined in (21) and their ratio as a function of the entrainment parameter η. The vertical dashed lines indicates the typical value for Ross (), FRIS (), and PIG ().

Citation: Journal of Physical Oceanography 49, 4; 10.1175/JPO-D-18-0131.1

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Equations (20) describe the behavior of the dimensionless quantities as a function of the dimensionless coordinate x. Since the melt rate is essentially the product [see (5a)], the scaled product describes the dimensionless melt rate. To leading order in 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 ; (20c) is the buoyancy flux determined entirely by the melt rate, neglecting higher-order changes in plume temperature and plume salinity; and (20d) determines the plume temperature contrast from the depth-dependent freezing point, entrainment, and the ice–ocean interface heat flux, neglecting higher-order changes in plume salinity and melt rate, while downstream changes in temperature are . In the next subsection, we show how system (20) can be solved with an asymptotic expansion in 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

The structure of the solution of (20) is most easily understood by writing it as a single equation. It turns out that the volume flux, defined as

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is the most convenient variable to use, for example, because (20a) then becomes , directly expressing the dimensionless velocity in terms of φ. In appendix A, we show how system (20) can be reduced to the following equation with δ = ε²:

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where the primes indicate derivatives with respect to x. Moreover, we have initial conditions . The first condition simply corresponds to zero initial mass flux, that is, at the grounding line, while the condition corresponds to , which is implied by (20b). This is the most compact formulation of the problem. Correspondingly, we find an expression in terms of φ for the dimensionless melt rate:

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Note that the problem is now reduced to a second-order differential equation for 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.

An analytical expression for the melt rate can be obtained from (24) after finding the solution of (23). As explained in detail in appendix A, an analytical solution will be constructed using an asymptotic expansion of φ in terms of ε. Restricting our attention to the leading-order behavior, the positive real-valued solution for is found to be

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Note that the function has an important physical meaning: it is the scaled volume flux of ambient water in the plume that determines the strength of the overturning circulation inside the cavity, making it a key quantity for parameterizing the effects of ice-shelf melting in ocean models. Furthermore, as explained in appendix A, this leading-order solution is equivalent to the assumption that the volume fraction of meltwater in the plume falls linearly as , which can be seen by taking in (23). From (25) and (24), we can directly obtain an expression for the melt rate:

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We now have the main result of the derivation, namely an explicit analytical expression for the dimensionless melt rate. Physically, it describes the basal melt rates along the plume path after applying an appropriate scaling in terms of the basal slope 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 , , , and , as well as interesting points about their physical interpretation, are given in appendix A.

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 () and increases to a positive peak at before transitioning to negative melt (refreezing) around . This transition point agrees with the empirical value found by Lane-Serff (1995). The same qualitative behavior is seen in the empirical melt-rate curve applied in Lazeroms et al. (2018), also shown in Fig. 3a. However, since (26) represents only a leading-order asymptotic approximation of (23) and (24), it is independent of 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 and are shown in Fig. 3a and directly compared with the analytical expression. Clearly, the analytical expression approximates the numerical curves well close to the grounding line (), but the discrepancies become larger further away from the grounding line and increase with ε. In fact, the analytical expression (26) is only valid for , while the numerical solutions continue until the points and , respectively. Hence, the “endpoint” of the plume moves further away from as ε increases. Conversely, the endpoint moves toward for , in which case the numerical solution converges to expression (26) for .

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Comparison of the analytical solution for calculated from (26) (solid line) with numerical solutions calculated from (23) and (24) (dashed lines). Also shown is the polynomial fit from Lazeroms et al. (2018) scaled with a constant factor 0.124 (dotted line). Each panel shows two numerical solutions for and , respectively. (a) Solutions as functions of the original dimensionless coordinate x. (b) Solutions as functions of the new coordinate containing the slope correction of (27). Note that the analytical solution and the polynomial fit are valid on the domain , while the numerical solutions are valid until the plume endpoints indicated by circles.

Citation: Journal of Physical Oceanography 49, 4; 10.1175/JPO-D-18-0131.1

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So far, we have only considered the zeroth-order terms in the asymptotic expansion. One might expect to improve the approximation by adding higher-order terms in our asymptotic expansion. However, this turns out not to be possible for technical reasons explained in appendix B. On the other hand, some of the aforementioned discrepancies might be overcome by adding an additional slope-dependent correction, similar to the ad hoc scaling used by Jenkins (2014) and Lazeroms et al. (2018). Intuitively, such an additional 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:

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where turns out to give the best match between the analytical expression and the numerical solution of (23) for . Figure 3b shows how the results from Fig. 3a change when the additional correction is applied. The three curves now agree well in most of the domain, except for a small region near . Note, however, that this region may not be important in practice for the following reasons. The point (i.e., the original coordinate) corresponds exactly to a depth at which , that is, where is equal to the local ambient freezing point. It appears unphysical to have ambient waters colder than the local freezing point, except for, for example, locally supercooled plumes with possible nucleation of frazil ice crystal, which is not captured by the plume model equations in (1). Therefore, the point can only be reached at the surface when is equal to the ambient freezing temperature at atmospheric pressure. For higher , the point occurs above the sea surface and is clearly nonphysical. Moreover, the point corresponds to a corrected value for positive slopes , making it more unlikely for the model to approach the point , and increasingly so for steeper slopes. The results in section 3 indeed show that the true endpoint of the plume always has a value and the discrepancies remain small or outranked by other effects such as varying slopes or stratification. Most importantly, the correction in (27) ensures that the transition point between melting and refreezing is almost perfectly predicted for .

To summarize, we have systematically derived an analytical expression (26) for the dimensionless basal melt rate as a function of the along-slope coordinate under an ice-shelf with constant slope and constant ambient properties. We can trace back the derivation to the dimensional quantities and arrive at the final form of the proposed basal melt parameterization, including dimensional factors:

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where is the analytical function in (26) with input given by

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Similarly, we can express the dimensional volume flux of the cavity circulation in terms of the function in (25):

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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 of a dimensionless coordinate . In fact, the melt-rate scale in (28a) is almost equal to the empirical scale shown in Lazeroms et al. (2018), being a product of geometric factors involving and a quadratic dependence on . A notable difference is the addition of some new constants that only have a very small effect on the final value of (e.g., the constant with respect to ). Moreover, the constant prefactor in (28a) is now explicitly given in terms of other model constants, whereas Lazeroms et al. (2018) simply used a parameter m yr⁻¹ °C⁻². Apart from this, (28a) contains the same dependence on with the same exponents as in the empirical model. Similarly, the dimensionless coordinate in (28b) has the same basic form as in the empirical model: the depth difference scaled by the temperature difference and multiplied by a slope-dependent factor. This factor differs slightly from the one in Lazeroms et al. (2018), but in both cases it is an ad hoc correction with the purpose of constraining the coordinate values between 0 and 1 for different values of the slope. Finally, the function given by (26) now plays the same role as the dimensionless polynomial fit in Lazeroms et al. (2018). Note, however, that the analytical expression (26) and the polynomial fit differ by a factor of 0.124 (as shown in Fig. 3a) due to the aforementioned difference in the constant dimensional prefactor. In the next section, we show how the basal-melt parameterization given by (28) performs for various ice-shelf geometries and ambient ocean properties.

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 and . First, we investigate how well the analytical expression (28) agrees with the full plume model [including the three-equation formulation in (4)] for such simplified cases. In the following, both models are evaluated for simple geometries defined by the grounding-line depth and a slope angle α and extending up to sea level (). The reference geometry has m and , so that the plume path extends up to km. By order of magnitude, this is comparable to the larger Antarctic ice shelves such as FRIS and Ross ice shelf. Moreover, we consider steep and flat geometries [ and , respectively] and shallow and deep geometries ( m and m, respectively), as well as the special case of a vertical ice wall [ and m]. In each case, the plume model is evaluated up to the ice-shelf front at , though in section 3b we will show examples where the plume path ends before reaching the end of the domain.

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 ) closer to the grounding line and a transition to refreezing (negative ) further away from the grounding line. A steeper slope causes an increase in melt near the grounding line, as well as stronger refreezing further away, due to stronger volume flux and entrainment, while the flatter slope has the opposite effect. Likewise, a deeper grounding line causes both increased melting and refreezing due to the lower freezing point and an associated increase of thermal driving at greater depths, with the shallow case showing the opposite. In all cases, relatively small discrepancies between the models appear both at the melting peaks closer to the grounding line and near the end of the domain in the area of refreezing, while the transition point between melting and refreezing is almost perfectly predicted by the parameterization.

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Comparison of the basal melt rates obtained from (28) (solid lines) and the full plume model (dashed lines) for different idealized ice-shelf geometries with constant slope and uniform ambient properties ( psu, °C). (a) Unscaled results. (b) Scaled plume model results compared with the dimensionless melt-rate curve . The used geometries are reference geometry (black) with m and , steep geometry (blue) with , flat geometry (cyan) with , shallow geometry (red) with m, deep geometry (green) with m, and [inset in (a)] vertical ice wall (gray) with m and . Note that in all cases the curves extend up to the end of the ice-shelf base at sea level.

Citation: Journal of Physical Oceanography 49, 4; 10.1175/JPO-D-18-0131.1

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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 . The plume model results collapse onto nearly the same curve, which is again slightly underpredicted by the parameterization at the melting peak (). The typical relative error in the melting region for these five cases is around 10% (disregarding the small regions where the melt rate goes to zero). This particular error can be traced back to the fact that we only use a zeroth-order approximation in 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 () are related to the singularities discussed in appendixes A and B and are not easily improved, though they appear relatively small with relative values between 0% and 10%. Note that the good agreement around the melting-freezing transition point () is partly the result of the correction term (27) that rescales the plume-path coordinate.

Among the results in Fig. 4, the vertical ice wall is a special case because it has an infinitely steep slope with . For such high values of the slope, the parameter 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 directly in the equations. Here, we consider four geometries: a convex shelf (increasing 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 .

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Comparison of the basal melt rates obtained from (28) (solid lines) and the full plume model (dashed lines) for different idealized ice-shelf geometries with varying slope and uniform ambient properties ( psu, °C). (left) Unscaled results. (center) Scaled plume model result compared with the dimensionless melt-rate curve . (right) Draft of the used geometries: (a)–(c) convex, (d)–(f) concave, (g)–(i) periodically varying slope, and (j)–(l) abrupt change in slope.

Citation: Journal of Physical Oceanography 49, 4; 10.1175/JPO-D-18-0131.1

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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 , as shown in the concave case, will also contribute to these errors.

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 has only one positive melting peak, the unscaled parameterization yields many melting peaks. These peaks are located at the same positions as in the plume model, but the plume model again adapts more gradually to the slope changes. Therefore, the relative error in is highest in regions where the slope flattens, though this has only a minor impact because the melt rates are small here. A similar behavior is shown in Fig. 5j for the abruptly changing slope. As the slope suddenly increases, the parameterization predicts a discontinuous jump in the melt rate whereas the plume model adapts more gradually.

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 at the grounding line, is perhaps the most important input variable determining the absolute value of the melt rates in (28a). Figure 4 already showed that a deeper grounding line leads to a higher melting peak because the freezing point decreases with depth. By varying the (still uniform) ambient temperature , keeping the geometry and other variables fixed, we obtain the results shown in Fig. 6 for a constant basal slope, as in Fig. 4. The results show the same qualitative behavior and good agreement between the parameterization and the plume model as in the previous section. The effect of increasing is immediately clear: the melt rates increase in the whole domain, while the position of the melting peak (in the dimensional sense) appears to shift to the right. When is sufficiently high (in this case −1.4°C), the region of refreezing disappears and melting occurs in the entire domain. The deeper reasons for this behavior are illustrated by the scaled curves in Fig. 6b. All plume model results for the different values of lie along the same curve, but increasing causes the endpoint of the plume (indicated by circles and corresponding to km) to shift to the left. Hence, the effect of increasing in the parameterization is twofold: in the dimensionless sense the range of decreases because a larger depth change would be required to compensate the higher thermal driving at the grounding line (causing the refreezing region to disappear), but in the dimensional sense the melt-rate scale increases quadratically according to (28a). It also follows that the position of the ice-shelf front can correspond to any value of between 0 and 1 depending on the ambient temperature, where can only be approached if is close to the surface freezing point (section 2d).

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Comparison of the basal melt rates obtained from (28) (solid lines) and the full plume model (dashed lines) for varying but uniform ambient temperature , uniform salinity psu, and a fixed ice-shelf geometry with constant slope. (left) Unscaled results. (center) Scaled plume model result compared with the dimensionless melt-rate curve , with circles indicating the endpoint of the plume at km. (right) Draft of the used geometry.

Citation: Journal of Physical Oceanography 49, 4; 10.1175/JPO-D-18-0131.1

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The preceding results confirm that the parameterization performs well compared with the plume model for different uniform values of . We now briefly comment on the effect of uniform values of salinity . This quantity appears in the parameterization in the first factor of (28a), as well as indirectly through the freezing point and the (small) parameter . In this sense, it determines the initial strength of buoyancy on the meltwater plume. Since absolute values of salinity vary only slightly within a given region (e.g., a few percent around 34 psu in Antarctica; see, e.g., Zweng et al. 2013), the value of used in either the parameterization or the plume model will only have a very small effect on the results, as long as it is assumed uniform. In other words, the current, unstratified formulation of the parameterization can be used in regional simulations [e.g., Lazeroms et al. (2018) for Antarctica] without a spatially varying field for , as the absolute values in this field will not significantly affect the melt rates.

However, this does not hold when stratification (vertically varying and ) is taken into account. In fact, stratification might be the most important phenomenon that is absent in the derivation of section 2. In reality, stratification causes plumes to detach from the ice-shelf base when reaching levels of neutral buoyancy, leading to new plumes that are uncoupled from the grounding-line conditions. These form the well-known subshelf melting modes described by Jacobs et al. (1992). As shown by Jenkins (2011), the strength of the stratification can be described by a stratification length scale , and determining the relative importance of stratification boils down to a comparison of with the currently used freezing-point length scale . The numerical results of Jenkins (2011) reveal an interesting difference between Antarctic ice shelves () and Greenland fjords (), indicating that stratification is typically less important for Antarctica than for Greenland.

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 and vertically varying, stably stratified . First, note that all curves show roughly the same behavior close to the grounding line, because the grounding-line conditions (thermal driving and slope) are equal. However, as the plume moves up toward levels of lower , it eventually reaches a point of neutral buoyancy and the evaluation of the plume model is stopped before the buoyancy changes sign. These endpoints are again indicated in Fig. 7 by circles (note that the true endpoints should be at when extrapolated from the numerical results).

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As in Fig. 6, but for nonuniform ambient salinity (stable stratification) and uniform ambient temperature °C. The circles now indicate the variable endpoint of the plume model, which is integrated until buoyancy changes sign. In all cases, the salinity equals 34.65 psu at and decreases with vertical gradients shown in the legend.

Citation: Journal of Physical Oceanography 49, 4; 10.1175/JPO-D-18-0131.1

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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 does not change significantly and has a negligible impact on the parameterization output. Near the endpoint, the formulation of section 2 breaks down in a way similar to the case of uniform ambient ocean (cf. Fig. 3b). Nevertheless, it appears that the essentially unstratified parameterization (28) can still be applied in stratified environments if the position of the endpoint can be identified and parameterized. A detailed discussion of such an extension is beyond the scope of this work, but estimates of the stratification length scale and its impact on the plume model were given by Jenkins (2011). In practical simulations, one might also consider defining different uniform layers in the ice-shelf cavity corresponding to the melting modes described by Jacobs et al. (1992) and applying the current formulation separately to each layer, similar to what was done by Magorrian and Wells (2016) for near-vertical glaciers terminating in a strongly stratified ocean.

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.

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As in Fig. 5, but for three realistic ice-shelf geometries along the flow lines of (a)–(c) FRIS, (d)–(f) Ross ice shelf, and (g)–(i) PIG. Note that we used a different value for the ambient ocean temperature beneath PIG (°C), deemed characteristic for the warmer waters in the Amundsen Sea. Circles indicate the endpoint of the plume.

Citation: Journal of Physical Oceanography 49, 4; 10.1175/JPO-D-18-0131.1

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Note that all cases shown in Fig. 8 have uniform ambient ocean properties and . Although it is technically possible to use depth-dependent profiles for these quantities, such data is usually only available from observations near the ice-shelf front and not within the ice-shelf cavity, especially for the larger ice shelves. Therefore, it is unclear if using such observed profiles for and would give more realistic results. The lack of ocean data within the cavities and the associated modeling issues were discussed extensively in Lazeroms et al. (2018). In that study, a two-dimensional 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 for each ice shelf, yielding °C for FRIS and Ross and °C for PIG. Furthermore, we take psu as in the previous sections. It is important to stress that these ocean temperatures are only effective values that yield a plausible order of magnitude for the basal melt within the current modeling framework, as many more details of the cavity circulation (e.g., stratification, discussed in section 3b) and the bathymetry are required to realistically model and and the associated melt-rate profiles.

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 km and km, partly caused by the very steep basal slope at km (cf. the gray curve in Fig. 4) and partly by the rather abrupt change in the slope, giving an abrupt change in the parameterized melt rate as opposed to the more gradual behavior of the plume model (cf. Figs. 5j–l). However, the parameterized melt rate appears to “catch up” with the plume model further along the path and the overall order of magnitude remains comparable.

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 . The same trend can be seen in the area-averaged data of Rignot et al. (2013), although it is problematic to give a direct comparison of these area averages with the line averages presented here. Furthermore, we have assumed a monotonically increasing ice-shelf base along the flow line, which is not always valid for realistic geometries. For a more thorough discussion on applying the parameterization to realistic 3D ice shelf geometries, see Lazeroms et al. (2018).

Table 2.

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

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Finally, we briefly comment how the region of influence of the neglected Coriolis forces can be estimated. As discussed by Jenkins (2011), the validity of the plume equations (1) is limited by the Ekman number rather than the Rossby number, that is, friction remains more important than rotation as long as the plume thickness is smaller than the Ekman length. This was estimated by Jenkins (2011) as . In this inequality, we can now substitute the analytical solutions for and (appendix A), which leads to

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where and are the dimensional scales of U and D as determined from (10) and (19). Note that still depends on the slope parameter η while the dependence on τ disappears from the ratio . A quick analysis of this inequality shows that for the typical values of η given in Fig. 2 and for s⁻¹, the maximum value of for which the inequality holds ranges from 0.2 (smaller slopes) to 0.6 (higher slopes). Comparing this to Fig. 8, we can conclude that rotational effects likely become important for larger shelves such as FRIS and Ross, but less important for smaller shelves such as PIG. However, close to the grounding line up to there is a general region in which the Coriolis effect might be neglected.