Increased Melting of Marine‐Terminating Glaciers by Sediment‐Laden Plumes

This paper summarizes the results of the first investigation into the effect of particle‐laden plumes on glacier melting using laboratory experiments. We find that the melt rate, when the ice is exposed to a particle‐laden plume, can be larger than when exposed to an equivalent plume without particles. The increased melt rate is linked to an increase in the plume velocity in response to the presence of suspended particles. Including this increased velocity in a plume model improves melt rate predictions from the “three‐equation model” by approximately 45% for the range of particle concentrations used in this study.


Introduction
Ice loss from the Greenland Ice Sheet is currently a major contributor to global sea level rise.Although the rate of mass loss has been increasing over recent years (Mouginot et al., 2019), there remains a large uncertainty in predictions of future melt rates.In addition to sea level rise, the added freshwater released from Greenland has been linked to the observed freshening of the North Atlantic (Bamber et al., 2012), which could impact global and regional circulation patterns.A review of processes occurring within Greenland's glacial fjords, and their global importance, is provided in Straneo and Cenedese (2015).
For ocean induced melting of the ice sheet, the interaction between the ocean and the ice face is crucial.The ocean provides a source of heat and salt to the ice, the transport of which directly determines the melt rate of the ice face (Jenkins, 2011;Kerr & McConnochie, 2015).However, the transport process itself is highly complex and involves both relatively large-scale turbulent processes and small-scale molecular processes.
A commonly used parameterization, developed to predict the melt rate of an ice face based on the bulk temperature, salinity and velocity close to the ice, is known as the "three-equation model" (e.g., Jenkins, 2011) ṁ(( − ) + ) = Γ  1∕2  ( − ), (1) (3) • Laboratory experiments show that melting of an ice face can be increased by the presence of sediment in a subglacial discharge plume • The increased melting is linked to an increase in the plume velocity in response to the presence of suspended particles • Accounting for the increased plume velocity within the three-equation model significantly improves predictions of the melt rate Supporting Information: Supporting Information may be found in the online version of this article. 10.1029/2023GL103736 2 of 9 the product with the square root of the drag coefficient will be referred to as a Stanton number,  St ≡ C 1∕2 d ΓT,S .The Stanton number is the ratio of heat (or salt) transport to the thermal capacity of the fluid.In ice-ocean interaction modeling, it is used to provide a dimensionless measure of the rate of heat (or salt) transport from the fluid to the ice face (Jenkins, 2011).Malyarenko et al. (2020) provides a detailed review of this, and other, ice-ocean interaction parameterizations.
Due to the relatively warm air temperatures in Greenland over summer, a significant amount of surface melting of the ice sheet occurs.The surface meltwater percolates to the base of the ice sheet and flows beneath glaciers to the grounding line-the location where a glacier becomes afloat (Nienow et al., 2017).At the grounding line, the meltwater is released into the ocean and forms highly vigorous turbulent plumes that rises along the ice face (Fried et al., 2015;Straneo & Cenedese, 2015).The dynamics of subglacial plumes have received a large amount of attention over recent years as they are often associated with elevated melt rates (e.g., Hewitt, 2020;Straneo & Cenedese, 2015).
As the surface meltwater flows beneath the ice sheet, it can erode significant amounts of sediment (Cowton et al., 2012).As a result, subglacial plumes contain high sediment concentrations and can often be observed from surface photographs when they reach the surface (Mankoff et al., 2016).Suspended sediment can either enhance or reduce the turbulent intensity of a flow based on the particular parameter space (Balachander & Eaton, 2010).Recent experimental work has shown that the entrainment of ambient water into an axisymmetric turbulent plume is increased by up to 40% when the plume contains suspended dense sediment (McConnochie et al., 2021), suggesting an enhancement of the turbulent intensity.Based on standard plume models (Morton et al., 1956) an increased entrainment coefficient would be expected to decrease the plume velocity and increase the plume radius.From Equations 1 and 2, suspended sediment should therefore reduce the melt rate of an adjacent ice face compared to a plume without suspended sediment.However, such an effect has never been investigated.
In this paper we present laboratory experiments of a vertical ice face melting in contact with a particle-laden plume.Section 2 contains a description of the experimental apparatus and procedure.Experimental results are presented in Section 3 before a model that explains the experimental results, based upon the three-equation model, is developed in Section 4.

Methods
Experiments were conducted in a glass walled tank that was 150 cm long, 15 cm wide and 30 cm high and is shown schematically in Figure 1a.The tank was located in a temperature controlled room that was kept at approximately 3°C.The tank was initially filled to a depth of 25 cm with oceanic salt water that had been left in the room to thermally equilibrate for at least 24 hr prior to the experiment.
In all experiments, the ambient fluid temperature and salinity were 3.2 ± 0.2°C and 33.5 ± 0.5 g/kg, respectively.The plume fluid is supplied to the base of the ice block with a source volume flux Q s of 6.0 ± 0.1 cm 3 s −1 and fluid is removed from the tank at the same volume flux from the opposite end.The buoyancy flux due to the subglacial discharge is approximately 100 times that due to the melting of the ice face such that the buoyancy flux of the plume can be assumed to be constant with height and dominated by the source buoyancy flux (see Table S1 in Supporting Information S1 for exact values of the source buoyancy flux ratio and McConnochie and Kerr (2017) for further details).
The ice used in the experiments was made from fresh water with a small amount of blue food dye added for visualization.The fresh water was left at room temperature for at least 48 hr prior to being frozen so that the ice was free of air bubbles.The water was frozen in a mold in a freezer for at least 48 hr until it came to a uniform 10.1029/2023GL103736 3 of 9 temperature.The mold was constructed such that the width of the ice block was slightly smaller than the width of the tank so the ice block could be smoothly placed in, and removed from, the tank.Although this resulted in a very thin layer of fluid being trapped between the ice block and the tank wall, this did not appear to result in further convection and the melting against the sides of the tank was negligible.Prior to the experiment the ice was removed from the mold and placed in the temperature controlled room for 90 min.During this time the ice began to warm up, but didn't start melting, which ensured that almost all of the heat flux to the ice during an experiment resulted in melting of the ice rather than warming, and that the measured melt rate was approximately proportional to the total heat flux to the ice.Immediately before placing the ice in the tank it was dried and weighed on a scale.
During an experiment, a fresh water line plume was produced at the base of the ice block.A line plume, rather than an axisymmetric plume as in McConnochie et al. (2021), was used to ensure that the melt rate across the ice face was uniform across the width of the tank.The line plume was produced from an array of 10 point sources equally spaced across the width of the tank.Each source was designed to generate a turbulent plume at the exit location (Kaye & Linden, 2004) and the individual plumes quickly merged to form a line plume.The plume parameter (Parker et al., 2020) is given in Table S1 in Supporting Information S1 and shows that the source conditions were close to a pure plume and, thus, the source momentum flux is expected to be unimportant.The melt rate of the ice block was observed to be higher above the source locations along the bottom 1-2 cm of the ice block.Above this height the melt rate was spatially uniform, suggesting that the velocity of the plume was also spatially uniform-that is, that the plume was, in fact, two-dimensional.
The plume source fluid was fresh water that had been stored in the cold room for 24 hr.Prior to the experiment, small ice cubes were added to the source fluid to reduce the temperature of the fluid to between 0.5 and 1.0°C.The exact temperature of the source fluid was measured to within ±0.1°C with an alcohol thermometer before an experiment started.Immediately before the start, the ice cubes were removed, food dye was added to the fresh water for visualization and the desired mass of particles was added to the fluid.These particles were kept in suspension during an experiment by continually stirring the source fluid.
The particles used were solid glass microspheres with a density of 2.5 g/cm 3 and diameters of 38-53 μm and had a settling velocity, v s , which is approximately 10-100 times smaller than the velocity of the rising plume next to the ice.Representative Stokes and particle Reynolds numbers for the experiments were St ∼ 10 −3 and Re p ∼ 10 0 , respectively.
During an experiment, the plume fluid containing the particles rose vertically along the ice face before spreading horizontally as a buoyant surface current.As the plume fluid spread along the surface, particles settled from the surface current and were drawn back toward the ice face due to the entrainment of ambient fluid into the plume (Figure 1).
Each experiment was run for approximately 5 min.After this time, the face of the ice block started to retreat behind the position of the source and the height where the plume became attached to the ice face began to shift upwards.At the conclusion of an experiment the ice was removed from the tank, dried, and weighed.The loss of mass during an experiment was used to estimate the melt rate based on a density of ice of 0.92 g cm −3 and assuming that the melting was uniformly distributed over the ice face that was exposed to the particle-laden plume.The scale used to weigh the ice block had a resolution of ±2 g which resulted in an error in melt rate of ±5%-7% depending on the melt rate for a given experiment.
Figure 1b shows a photo from a qualitative experiment.A freshwater subglacial line plume (dyed green) was produced at the base of an ice block (dyed blue) which rose vertically and spread horizontally once it reached the free surface.Experiments were carried out with different concentrations of particles added to the subglacial plume.Similarly to the experiments presented in B. R. Sutherland et al. (2020), particles are expected to settle from the horizontally flowing surface current before being drawn back toward the ice face by entrainment of ambient fluid into the plume (dashed lines in Figures 1a and 1b).

Experimental Results and Discussion
Following McConnochie et al. (2021), we attempt to understand the effect of suspended particles on the melting of an ice face by characterizing the particle concentration by a buoyancy flux ratio, P. The buoyancy flux ratio compares the component of the source buoyancy flux that is due to the particles, B part , with the buoyancy flux due to the temperature and salinity induced density differences between the plume and the ambient fluid, B fluid Here the buoyancy flux is defined as where Q s is the plume volume flux at the source, g is the acceleration due to gravity, ρ a is the ambient fluid density, and ρ s is the source fluid density.
The appropriate value of density, ρ, will depend on which buoyancy flux is being considered (B part or B fluid ) such that only the considered component is included (i.e., when calculating B part , the particle contribution to the density anomaly is included in (ρ − ρ a ) while the salt contribution is not).The total source buoyancy flux is given by B part + B fluid , and P = 0 corresponds to a plume with no particles while P = −1 corresponds to a plume where the particle buoyancy is exactly balanced by the fluid buoyancy.Note that P ≤ 0 since the buoyancy flux due to the particles is downwards while the buoyancy flux due to fluid density differences is upwards.
On Figure 2 we show measurements of the melt rate, plotted against the buoyancy flux ratio.On the top axis, we also show an estimate of the particle concentration, ϕ.We use a simplified relationship of ϕ = −2.36P,based on a linear regression between the buoyancy flux and the particle concentration.However, we note that there is not a direct relationship between particle concentration and buoyancy flux ratio, due to small variations in the fluid densities affecting B fluid in each experiment.For the melting experiments, the concentrations are accurate to within 2%, and for the entrainment experiments, the concentrations are accurate to within 3%.An equivalent figure where the melt rates are plotted directly against the particle concentration is provided as Figure S1 in Supporting Information S1.
The estimated uncertainty is the difference between two repeated experiments without particles (±0.44 μm/s ≡ ±9% for the experiment without particles).The uncertainty in melt rate is similar to, but slightly larger than, the uncertainty in mass change measurements (±5%-7%) due to smaller uncertainties in other experimental measurements such as the fluid depth, temperatures, and source volume flux.Figure 2 shows that the melt rate of the ice block is increased by up to 60% by the presence of suspended particles in the adjacent plume.The melt rate appears to increase in an approximately linear way for −P ⪅ 0.3 but the dependence on buoyancy flux ratio for larger values of −P is negligible, with a possible slight decrease for −P ⪆ 0.6.With the limited data available, it is not possible to further constrain the relationship between melt rate and the buoyancy flux ratio, but we do not require any particular relationship for the analysis that follows.Estimates of the entrainment coefficient with an empirical fit through the data, both from McConnochie et al. (2021), are also shown on Figure 2. Results from McConnochie et al. (2021) are only shown for those experiments that had a particle size equal to that used in the present study.The entrainment coefficient is defined as α = U e /U, where U e is the velocity with which ambient fluid is entrained into the plume and U is the characteristic plume velocity (Morton et al., 1956).
Although the present experiments were conducted with a single total buoyancy flux and a single particle size, we expect the results to hold provided the particle size is sufficiently small, and the plume velocity is sufficiently large, such that individual particles do not settle out from the turbulent plume.McConnochie et al. (2021) showed that the entrainment coefficient is only a function of the particle concentration and does not depend on the total buoyancy flux or the particle size (at least within a range of particle sizes).However, as previously observed, increasing a single-component plume's buoyancy flux will increase the velocity of the plume, and hence, increase the melt rate (McConnochie & Kerr, 2017).This dependence of the melt rate on the total buoyancy flux is separate from the dependence on the particle buoyancy flux ratio illustrated in Figure 2 and is expected to be unchanged with little interaction between the two processes.

Modeling the Melt Rate
The entrainment coefficient was determined in McConnochie et al. ( 2021) by measuring the plume volume flux as a function of height in an axisymmetric plume and fitting the data to a standard, axisymmetric plume model (i.e., Q ∝ α 4/3 z 5/3 ).As such, the reported entrainment coefficients do not correspond directly with the two-dimensional line plume considered in this study.In addition, the plume velocity, which is expected to control the melt rate of the ice face, was not measured by McConnochie et al. (2021).The latter point is particularly important, especially given the experimental observation that the plume radius increased with distance from the source at the same rate, irrespective of the particle concentration.This result was not originally reported in McConnochie et al. (2021) but is demonstrated in Supporting Information S1. Figure S2 in Supporting Information S1 shows that the radius of an axisymmetric plume increases linearly with height and Figure S3 in Supporting Information S1 shows the spreading rate of the plume as a function of the buoyancy flux ratio.Figure S3 in Supporting Information S1 shows that the spreading rate of the plume is unaffected by the buoyancy flux ratio.The observation that the spreading rate is unaffected by particle concentration implies that the plume in McConnochie et al. ( 2021) did not follow the classical self-similar power laws of axisymmetric plumes, and that the plume velocity increases with increasing buoyancy flux ratio.
van Reeuwijk and Craske (2015) provide a theoretical explanation for the observation that the plume spreading rate is unaffected by suspended particles, despite the volumetric entrainment rate being higher.By considering conservation of mean kinetic energy, in addition to the usual conservation laws contained within Morton et al. (1956), they developed an energetic constraint on the entrainment coefficient which they termed an entrainment relation.Using this entrainment relation, they showed that entrainment is caused by three independent quantities: the ratio of turbulence production to energy flux, the net effect of buoyancy, and streamwise changes in the velocity profiles.For a self-similar flow, the third of these processes is zero, and for a turbulent jet, the second is zero.The same three terms could, in principle, cause the flow to spread but the second term (effects due to buoyancy), never affects the spreading rate.Therefore, assuming self-similar Gaussian profiles, the spreading rate is purely determined by the non-dimensional rate of turbulence production, which is equal for turbulent jets and plumes.Thus, if suspended particles change the net effect of buoyancy, either by increasing the growth in momentum flux or decreasing the growth in kinetic energy (see Equation 4.2 of van Reeuwijk & Craske, 2015), the entrainment coefficient of the plume would be increased while the spreading rate would be unchanged, consistent with our observations.We are unable to directly assess how the mean kinetic energy or momentum fluxes change within the plume.However, we note that the net effect of buoyancy in a turbulent plume is normally to increase the entrainment coefficient (van Reeuwijk & Craske, 2015), so we are simply proposing an enhancement to this effect, arising from the presence of suspended particles.
Accounting for both the two-dimensional geometry and the observed effects of suspended particles (i.e., increased volume entrainment but unchanged spreading rate), we model the two-dimensional wall plume in these experiments by where b is the width of the plume, z is the distance from the source, q is the volume flux per unit length, f is the buoyancy flux per unit length, and U is the plume velocity.z v is a virtual origin correction such that the volume flux at z = 0 is equal to the volume flux at the plume source.α 0 is the entrainment coefficient of a wall plume without suspended particles (Parker et al., 2020), α P is the particle concentration dependent entrainment coefficient, and the coefficient 0.714 assumes the same relative increase in entrainment coefficient as was observed for axisymmetric plumes in  2021), the spreading rate of the plume is unaffected by suspended particles while the volume flux increases more rapidly with increasing particle concentration.Under this formulation, the entrainment velocity, U e , can be defined as and the ratio between the entrainment velocity and the plume velocity is equal to α 0 , as in a plume without suspended particles.This highlights the consistency with standard plume models (Morton et al., 1956).
The temperature, T p , and salinity, S p , in the plume can be calculated based on the volume flux, the source properties (denoted with subscript s), and the ambient fluid properties (denoted with subscript a) T s q s gives the heat flux at the source of the plume and T a (q − q s ) gives the total heat flux entrained into the plume, with equivalent terms for the plume salinity.Here we have ignored the addition of meltwater and the heat flux into the ice as both are very small compared to the initial and entrained heat and salt fluxes.
The melt rate can be calculated by combining Equations 2, 8 and 13 to give The salinity at the boundary is found by solving the following rearranged form of Equations 1-3 under the assumption that there is no heat flux into the ice (T i = T b ) and that the depth dependence on melting point is negligible.To solve Equation 15 for the interface salinity, we assume that the ratio of transfer coefficients (Γ S /Γ T ) in the laboratory experiments is the same as that given in Jenkins (2011), and that the plume salinity quickly reaches the ambient value of 33.5 g/kg due to entrainment.Combined with Equation 3, this gives conditions at the boundary of S b = 12.8 g/kg and T b = −0.63°.
We use the experimentally measured melt rates, with Equations 14 and 15 to infer a value of the haline Stanton number, where H is the fluid depth and  Ṁ is the area averaged melt rate measured in the experiments.This approach is consistent with previous idealized studies conducted at small scale (e.g., Vreugdenhil & Taylor, 2019) where transfer coefficients have been calculated based on observed or simulated melt rates.In general, transfer coefficients and, in this case, the drag coefficient, would not be expected to have the same value in small scale experiments as in observational studies due to the radically different flow scales.This makes comparison of the measured melt rates with those predicted by the three-equation model impossible as there is no a priori estimate of the various parameters.Instead, we judge the accuracy of the model based on whether the implied haline Stanton number is constant, as would be practically helpful if the model were to be implemented in a general situation.
Figure 3 shows the haline Stanton number as a function of the buoyancy flux ratio, P. The blue data points are calculated as described above while the red data points follow the same process, but with α P = α 0 -that is, not including the effect of particle concentration on the entrainment coefficient.The blue dashed line shows the average of all data points based on the particle dependent entrainment coefficient, α P = α 0 − 0.714α 0 P, and the red dashed line shows the average based on the particle independent entrainment coefficient, α P = α 0 .Although the inferred coefficients are not expected to be directly comparable to geophysical scale values, it is noteworthy that the approximate value of ∼9.85 × 10 −5 is similar to that given in Jenkins (2011) of   1∕2  ΓS = 3.1 × 10 −5 .We note that the above described model makes no assumption on the relationship between melt rate and the buoyancy flux ratio, only that there is a linear relationship between the entrainment coefficient and the buoyancy flux ratio.As such, the observation shown on Figure 2 that the melt rate is constant for −P ⪆ 0.3 does not impact the model predictions.
Ideally, for model simplicity, the haline Stanton numbers should be constant.Otherwise, they would need to be defined as a function of the buoyancy flux ratio, or some other measure of particle concentration.Figure 3 shows that if the effect of suspended particles on the rate of entrainment is included, the haline Stanton number has an average value of 9.85 × 10 −5 with a standard deviation of 1.00 × 10 −5 .If the effect of suspended particles is not included, the average value is 11.38 × 10 −5 with an 80% larger standard deviation of 1.80 × 10 −5 .
Figure 3 shows that including the effects of suspended particles, as proposed here, could improve melt rate predictions in the presence of particle laden plumes by approximately 45%, based on the reduction in standard deviation.Since a plume model is typically coupled to the three-equation model in current numerical simulations, replacing the plume model with the particle dependent plume model given in Equations 6-10 is relatively simple.However, it will require more detailed knowledge of typical sediment concentrations at subglacial discharges, and confirmation that the entrainment coefficient behaves similarly at geophysical scales to laboratory scales.
Finally, the experimental measurements show a significant variability in the haline Stanton number (Figure 3).This suggests that additional processes may need to be considered, possibly relating to the observed independence of melt rate on buoyancy flux ratio for −P ⪆ 0.3 shown on Figure 2.For an axisymmetric plume, there is strong evidence that the entrainment coefficient increases linearly with buoyancy flux ratio for −P ⪆ 0.3 and that the spreading rate is constant.However, it is possible that a line plume behaves differently at extreme buoyancy flux ratios, or that the transfer coefficients themselves develop a dependence on the buoyancy flux ratio.At present, we do not believe that there is sufficient empirical evidence or theoretical understanding to support either of these possibilities, or to make further adjustments to the above described model.

Conclusion
Novel laboratory experiments have shown, for the first time, that submarine melting of marine-terminating glaciers can be enhanced by up to 60% by the presence of sediment in a subglacial discharge plume.Based on experimental results from McConnochie et al. (2021), the increased melt rate has been linked to a larger plume velocity in the presence of suspended particles.Accounting for the increased velocity, based on the buoyancy flux ratio, improves melt rate predictions from the three-equation model by approximately 45% for the range of particle concentrations used in this study.
A number of questions remain about the most appropriate way of applying these results to a geophysical scale.First, the sediment concentration (and hence, the buoyancy flux ratio) in subglacial discharge plumes is poorly constrained observationally.We are not aware of any observations of sediment concentration from the discharge point of a subglacial plume, but Cowton et al. (2012) provide measurements from the meltwater of a land-terminating Greenland glacier.They observed sediment concentrations of approximately 5 kg m −3 (ϕ ≈ 0.5%, − P ≈ 0.2) throughout summer with peaks of up to 15 kg m −3 (ϕ ≈ 1.5%, −P ≈ 0.6) which are comparable to those used in these experiments.Satellite observations can be used to estimate the sediment concentration at the surface expression of subglacial plumes (e.g., Schild et al., 2017) but these observations are difficult to relate to subsurface properties near the discharge location due to entrainment of ambient fluid and potential sedimentation of larger particles.Similarly, the particle size distribution in subglacial discharge plumes is highly uncertain.From a proglacial stream, particle sizes ranging from 5 to 228 μm have been observed (Hasholt et al., 2012).However, larger particles were likely present in the initial discharge but had sedimented before the measurement location.As such, we would expect a wide range of particle sizes within a subglacial discharge plume, with larger particles settling from the plume over a range of depths and smaller particles rising to the surface.Particles within a subglacial discharge plume could also have the potential to scour the ice face if they had a sufficiently large kinetic energy before the collision with the ice.Although the mechanism is thought to be a negligible contribution to the overall ablation rate, further investigation is needed to quantify its importance.
Second, experiments on the effect of suspended particles on turbulent entrainment have only considered an axisymmetric plume, whereas line and point source wall plumes are more relevant to subglacial discharge plumes.There is no reason to expect wall plumes to respond qualitatively different to axisymmetric plumes but the quantitative similarity should be investigated.
Finally, it remains to be demonstrated that the effects seen at a laboratory scale can be directly applied to a geophysical scale.Again, we do not see any reason why the behavior would change, particularly since McConnochie et al. (2021) observed no effect of changing the total buoyancy flux of the plume or the particle Stokes number (within a limited range), but confirmation at larger scales would be beneficial.
Despite these open questions, it is clear that the effects of suspended sediment should be accounted for when modeling the melt induced by subglacial plumes.Ignoring these effects will underestimate the melt rate of the ice face, possibly by significant amounts.

Figure 1 .
Figure 1.(a) A schematic of the tank used in the experiments.The tank is filled to a depth H with ambient fluid of temperature T a and salinity S a and an ice block is placed against one end wall.The plume fluid is supplied to the base of the ice block with a volume flux Q s and drawn out of the tank with the same volume flux from the opposite side.Particles fall from the surface buoyant current with a settling velocity v s which is significantly smaller than the velocity of the rising plume next to the ice.(b) A photograph taken from a qualitative experiment of the top-left section of the tank.The source fluid was dyed green and a layer of fluid close to the surface at the start of the experiment was dyed red.The light red region in the bottom half of the tank indicates particles settling and being drawn back to the ice face due to turbulent entrainment.The light red color is caused by the transport of some ambient water by the settling particles.

Figure 2 .
Figure 2. Measured melt rate (blue circles, left axis) and the entrainment coefficient estimated by McConnochie et al. (2021) (black crosses, right axis) plotted against the buoyancy flux ratio (bottom axis).An estimate of the particle concentration is also shown (top axis).Also shown is an empirical fit to the entrainment coefficient (black line, right axis) from McConnochie et al. (2021) given by α = 0.105 − 0.75P.

Figure 3 .
Figure 3.The haline Stanton number inferred from experimentally measured melt rates.Data points shown in blue account for the increased entrainment with buoyancy flux ratio (i.e., Equation10) while data points in red assume that the entrainment coefficient is independent of the buoyancy flux ratio.The dashed lines show the average of the similarly colored data points with black data points (P = 0) included in both averages.The average for the blue and red data sets are   1∕2  Γ = 9.85 × 10 −5 and   1∕2  Γ = 11.38 × 10 −5 , respectively.