Two hundred marine-terminating Greenland outlet glaciers deliver more than half of the annually accumulated ice into the ocean and have played an important role in the Greenland ice sheet mass loss observed since the mid-1990s. Submarine melt may play a crucial role in the mass balance and position of the grounding line of these outlet glaciers. As the ocean warms, it is expected that submarine melt will increase, potentially driving outlet glaciers retreat and contributing to sea level rise. Projections of the future contribution of outlet glaciers to sea level rise are hampered by the necessity to use models with extremely high resolution of the order of a few hundred meters. That requirement in not only demanded when modeling outlet glaciers as a stand alone model but also when coupling them with high-resolution 3-D ocean models. In addition, fjord bathymetry data are mostly missing or inaccurate (errors of several hundreds of meters), which questions the benefit of using computationally expensive 3-D models for future predictions. Here we propose an alternative approach built on the use of a computationally efficient simple model of submarine melt based on turbulent plume theory. We show that such a simple model is in reasonable agreement with several available modeling studies. We performed a suite of experiments to analyze sensitivity of these simple models to model parameters and climate characteristics. We found that the computationally cheap plume model demonstrates qualitatively similar behavior as 3-D general circulation models. To match results of the 3-D models in a quantitative manner, a scaling factor of the order of 1 is needed for the plume models. We applied this approach to model submarine melt for six representative Greenland glaciers and found that the application of a line plume can produce submarine melt compatible with observational data. Our results show that the line plume model is more appropriate than the cone plume model for simulating the average submarine melting of real glaciers in Greenland.

Since the 1990s the decadal loss of ice mass by the Greenland ice sheet
(GrIS) has quadrupled

Model parameters of the LP and CP model with typical fjord default values. Note that the values may differ for specific experiments (as explicitly stated in the corresponding descriptions).

A plume model describes buoyancy-driven rise of subglacial meltwater after it
exits subglacial channels, until it reaches neutral buoyancy near the
surface. Two counteracting processes control its evolution: (a) submarine melting of the ice–ocean interface under the floating tongue (if
any) and upwards along the calving front and (b) turbulent entrainment and
mixing of surrounding fjord water. These processes act to maintain or
reduce plume buoyancy, respectively. Subglacial meltwater discharge

Both models are formulated in one dimension,

The LP model after

We do not use this approximation in our calculation, but this is nevertheless
helpful to interpret some of the results presented in our paper, in
particular in quantifying the amount of melt rate and simplifying the melt
rate dependence on temperature and subglacial discharge (Appendix

The second plume model investigated in this paper is the CP model

For the differential equation system of Eqs. (

In the rest of the paper, for simplicity, we refer to the boundary
condition at

For both LP and CP models, initial dimensions (radius or thickness)

It turns out that for a given subglacial discharge, simulated velocity
rapidly adjusts to a “balance” velocity, regardless of the initial velocity
(Fig.

Different runs of the CP model for different initial velocities.
Panel

Our sensitivity tests show that initial velocities higher than

A direct comparison between LP (defined per unit width of the grounding line)
and CP (point-wise) models requires an assumption about a length scale

Sensitivity of cumulative melt rate to different initial velocities
for both plume models. Melt rate (black) is in percent of the cumulative melt
achieved with initial balance velocity

Results in Fig.

We shall see later in this paper (Sect.

Melt rate profiles in a well-mixed fjord simulated by the CP model
(black) and LP model (blue) for a width

In the next sections we perform a number of sensitivity studies with respect
to key parameters. To that end we choose a default experimental setting as a
benchmark. Unless otherwise stated, we consider a 500 m deep, well-mixed
fjord with ambient temperature

It is known that melt rate depends strongly on subglacial discharge. In
agreement with previous studies

To explain this change of power law we undertook a dimensional analysis to
obtain theoretical solutions for the LP in a well-mixed fjord (Appendix

Cumulative melt rate of the LP model for

Note, however, that this analysis holds when the plume reaches a dynamic
equilibrium.

In addition, as mentioned in the introduction, stratification can change this
power law. We also performed experiments with stratification as in

Entrainment is the mechanism through which the volume flux of the plume
increases with distance from its source, as warmer, saltier fjord water mixes
into the plume. This leads to more heat availability for melting, but also to decreased buoyancy – and velocity – as the plume gets
saltier. Reduced velocity in turn reduces melting (Eq.

In both plume models, entrainment depends on an entrainment rate parameter
and on glacier slope, as

Determination of the power law

Rearranging the LP equations (Eqs.

As a result, for the LP model, an increase in

Numerical determination of the power law

Thus, this leads to the statement that cumulative CP melt rate increases with
increasing entrainment factor

Cumulative melt rates of the different plume models as a function of
entrainment parameter

Cumulative melt rate in the LP as a function of entrainment
parameter

Different fjords are characterized by different temperature and salinity
profiles. Since the temperature of the ocean is projected to increase with
global warming, dependence of melt rate on ocean temperature is crucial to
study glacier response to global warming. Previous experiments with 2-D and
3-D ocean models, as well as analytical solutions
(

Cumulative melt rate per glacier width

The glacier front angle

Influence of stratification and discharge on the melt rate profile
of the LP

However, for small

Melt profile

Studies of turbulent plumes caused by subglacial discharge and their effect
on submarine glacier melting have been performed using 2-D and 3-D nonhydrostatic general circulation ocean models (GCM)

These models typically parameterize unresolved, subgrid-scale turbulence with
a turbulent diffusivity.

We compare the melt rate profiles obtained in the experiments by

For the channelized subglacial discharge the most recent numerical
experiments by

Comparison between LP, CP and GCM simulations.

From this comparison of simple models with physically based model it appears
that the LP model needs to be scaled down (except for

Few studies exist in which submarine melt has been calculated directly based on
field measurements. We used here the available data to test the LP and CP
models against observations. However, the results have to be observed with
caution since a single temperature profile does not necessarily represent
monthly or even annual temperature profile. As

For the years 2002–2006

Simulated cumulative melt rate (%) of empirical estimated cumulative melt rate for different entrainment rates for three West Greenland glaciers.

Melt rate of

In a small fjord in West Greenland the melt rate of four glaciers was
determined by measuring the fjord salinity, temperature and velocity close to
the glacier fronts

Kangerlussuup Sermia average undercut profile at the
terminus

Another well-documented glacier is Store Glacier.

Three vertical melt rate profiles of the LP model

Comparison of the melt rate calculated with the LP model and the
empirical data obtained with the Gade and Motyka model

Estimated subglacial discharge

Measured versus simulated melt rate for a number of glaciers, for
data given in Table

We tested both line and cone plume models against available empirical data
for melt rate, and the line plume was best suited to reproduce observations
(Table

We presented two simple models for simulation of the submarine melt rate of
marine-terminating glaciers, the so-called cone plume and line plume models,
and studied sensitivity of these two models to different forcings (fjord
temperature, stratification, subglacial discharge) and model parameters
(entrainment parameter

Our analysis demonstrates that for small subglacial discharge, typical for
winter conditions, cumulative melt does not depend on the discharge. For high
discharge typical of summer conditions we found a power dependence of

We investigated the sensitivity of the melt rate to the entrainment parameter

Our comparison of the CP and LP models to results of 3-D GCM experiments showed qualitatively similar melt rate profiles.

In most cases, the LP model overestimates the results of the GCM by
approximately a factor of 2, while the CP model underestimates melt rate from
GCMs. More importantly, we find the same power law dependence of melt rate on
subglacial water discharge as in

In the case of the long floating tongue, like the Petermann Glacier, the LP model significantly overestimates the melt rate outside of the narrow zone along the grounding line, which is probably due to the missing Coriolis force in the plume models.

Although it is known that in summer a part of the subglacial meltwater is delivered in the fjord through several channels, we found that the submarine melt rate associated with the discharge through the channels and better described by the CP model makes up only a small amount of the empirically estimated total melt rate of a glacier front. Furthermore the total number of channels for every summer is unknown for different glaciers. When we compare the LP model to empirical data, it is evident that the LP model is more appropriate than the CP model for simulation of both winter and summer melt of real Greenland glaciers. However, the model has to be adjusted for individual glaciers since the scaling parameter is not the same for different glaciers. Thus, for the future we will use the tuned LP model coupled to a 1-D ice flow model to determine the importance of submarine melt rate to glacier dynamics.

The code for the line and cone plume, written in Python, is available as Supplement.

Data were either received via personal communication (see acknowledgments) or digitized with

In this Appendix, we analyze the LP model equations in order to derive
approximate analytical solutions. This in turn helps to interpret the results
of the numerical experiments presented in this paper, performed with the more
complete plume models from

We restrict the analysis to the typical conditions of a 500 m deep Greenlandic
fjord (

After

Investigation of melting proportion in the plume equations for
different LP experiments. The plume model was run in a well-mixed environment
for different parameter settings:

In Fig.

Temperature and salinity Eqs. (

A similar reasoning as in the previous section (using Eq.

The balance velocity is more conveniently expressed as a function of plume's
volume flux

Summary of Appendix variables. Illustrative value provided for

Evolution of

Evolution of

With Eqs. (

The error of Eq. (

We can identify two limiting cases for Eq. (

where the critical discharge

We also note from Eqs. (

and AG designed the study and conducted the analysis. JB implemented the numerical models and performed the experiments. MP and JB derived the analytical solutions. JB prepared the manuscript with contribution from all authors.

The authors declare that they have no conflict of interest.

This work was funded by Leibniz-Gemeinschaft, WGL Pakt für Forschung SAW-2014-PIK-1. We thank David Sutherland for the CTD data of Kangerlussuup Sermia, Fiamma Stranero and Dustin Carroll for the CTD measurement data of Helheim, and Donald Slater for providing us with the data of his experiment. Furthermore we thank Adrian Jenkins for providing us with his plume code to compare our simulated melt rate profiles among the models. Thanks to Reinhard Calov for support in numerical procedures. Edited by: G. Hilmar Gudmundsson Reviewed by: four anonymous referees