by Dan Hughes
This post challenges the conventional framework for simulating meltwater flows on glaciers and ice sheets.
Increased melting rates due to potential increases in temperature would add liquid water directly into the oceans. An additional aspect is that the meltwater, on reaching the base of glaciers, might lead to increased sliding of the glaciers and the consequent calving at the terminus.
The World Resources Institute (WRI) has summarized the IPCC AR6 results regarding melting of Greenland and Antarctica ice:
Should warming reach between 2 degrees C (3.6 degrees F) and 3 degrees C (5.4 degrees F), for example, the West Antarctic and Greenland ice sheets could melt almost completely and irreversibly over many thousands of years, causing sea levels to rise by several meters.
| Temperature Increase | 1.5 C (2.7 F) | 2.0 C (3.6 F) | 3.0 C (5.4 F) |
| Global mean sea level rise by 2100 | 0.28 – 0.55 m(0.92 – 1.80 ft) | 0.33 – 0.61 m(1.08 – 2.00 ft) | 0.44 – 0.76 m(1.44 – 2.40 ft) |
Meltwater flows on the surface of and with glaciers and other ice sheets are important relative to the addition of liquid water into Earth’s oceans, and to bulk motions of the glaciers and ice sheets. Glacial meltwater might flow along the surface like a stream or river, accumulate in surface lakes, flow downward into open crevasses or moulins, accumulate as lakes interior to the ice mass, flow as a sheet of liquid between the ice bottom and bedrock, or flow enclosed in channels partially or completely embedded within the ice mass.
Flows that reach the boundary of the ice sheet deplete the ice mass balance and can contribute to sea level rise if the flow reaches the sea. Meltwater remaining on the surface of the glacier or ice sheet can refreeze and have no impact on the glacier mass balance. Flows reaching the base of the glaciers by way of crevasse and moulins are considered to provide potential lubrication and flotation that enhances bulk ice motions.
How solid is the foundation for simulating glacial meltwater flows that are included in projections of ice sheet melting?
Glacial meltwater flows have been modeled for more than four decades using thermal-hydraulic modeling. The widely used Springer-Hutton formulation is based on principles of continuum mechanics, and detailed mathematical reduction to the standard 1-dimensional channel- average form for engineering applications. A steady-state energy balance equation is applied to flow of liquid water in ice channels embedded in large ice masses. The Spring-Hutter system considers the case of evolution in time and space of the flow area of the channel. Changes in flow area are caused by ice melting and dynamics of the ice in which channels are located. There have been numerous studies providing clarifications, modifications and applications of Spring-Sutter framework.
New paper
I have conducted a detailed analysis of the Spring-Sutter equations and their solutions in this paper [EDHmelt]
The paper clarifies and improves calculations of the role of viscous dissipation of kinetic energy into thermal energy as this physical process appears in models of meltwater flows embedded in and at the boundaries of glaciers and ice sheets.
Meltwater flows on the surface of and within glaciers and other ice sheets are important relative to the addition of liquid water into Earth’s oceans, and to bulk motions of the glaciers and ice sheets. Glacial meltwater might flow along the surface like a stream or river, accumulate in surface lakes, flow downward into open crevasses or moulins, accumulate as lakes interior to the ice mass, flow as a sheet of liquid between the ice bottom and bedrock, or flow enclosed in channels partially or completely embedded within the ice mass.
Flows that reach the boundary of the ice sheet deplete the ice mass balance and can contribute to sea level rise if the flow reaches the sea. Meltwater remaining on the surface of the glacier or ice sheet can refreeze and have no impact on the glacier mass balance. Flows reaching the base of the glaciers by way of crevasse and moulins are considered to provide potential lubrication and flotation that enhances bulk ice motions.
A dimensionless form for steady-state energy balance for the liquid, accounting for effects of meltwater on the bulk liquid, is developed and solved. Analytical solutions of the temperature distribution along the channel are developed. The solutions explicitly illustrate effects of viscous dissipation of kinetic energy into heat, and the consequence effects on melting ice at the liquid-ice interface.
The paper shows that:
- Letting viscous dissipation of kinetic energy go directly into melting is not correct
- The energy equations are not complete because they do not account for meltwater entering the bulk liquid
- The Spring-Hutter accounting for meltwater entering the bulk liquid is not correct.
________________________________________________________________________
Stopped reading right there.
A while back, there was a whole discussion about why that claim isn’t true. Maybe a search will turn that WUWT post up. Here
Here’s the start of the discussion LINK
The general idea, true even at Ross Ice Shelf scales, is that these ice bottoms are not smooth. Translation, solid rocky bottom eventually emerges by ice ‘scraping’. So bottom melt cannot accelerate anything by very much. Glaciers proceed at the rate of their bottom ice deformation, not bottom melt.
You should have read further.
At that point in the article, Doug Hughes was describing the models he analyzed.
How solid is the foundation for simulating glacial meltwater flows that are included in projections of ice sheet melting?
The paper shows that:
Letting viscous dissipation of kinetic energy go directly into melting is not correctThe energy equations are not complete because they do not account for meltwater entering the bulk liquidThe Spring-Hutter accounting for meltwater entering the bulk liquid is not correct.
Could, should, might ….. .
If ifs and ands were pots and pans etc.
I’m rather sure that Newton, Einstein and other eminent scientists seldom if every used those words.
Ice sheets are already accounted for in sea level. If an ice sheet melts, the sea level would drop. I understand that if a glacier enters the water, that is an addition. But ice sheets are floating. Fill a glass with ice cubes, then fill with water to the top and watch while it melts. The level will drop. So, water off an ice shelf does not add to sea level, at all.
A Whisky drinker should know that. But Greens are not used to drink ardent spirits 😀
This climate activist is ardent to extreme insanity.
https://rumble.com/v5i78k5-climate-activist-throws-a-hissy-fit-because-people-are-starting-to-question.html?e9s=src_v1_ucp
What a wonderful rant,
I think he’s blown a head gasket as well as his main fuse board.
I note HE happily uses all the products that produce ‘The Devil Gas’ CO2;
Shirley, this has to be a put-up, an act to show the totally deranged nature of the far-left idea-logjam.
Either way, it is a wonderful piece of slap-stick comedy !
Surely you can stop calling me Shirley.
As an ex-Burt, I like to be known as Penelope !! (;-))
Mr Penelope of Miss Penelope?
Or perhaps, Them Penelope.
Please let me know so I can totally ignore it. 🙂
Being called an effing idiot is a wonderful way to convince me of the “science” behind climate change.
A serious whisky drinker knows that you don’t put ice in good whisky, it just masks the taste.
Whiskey and other copies are different things altogether of course.
A glass filled with ice to begin with models a grounded ice situation. Add water to a glass to some level, add a few ice cubes, note the level. That’s a floating ice situation. No change after the ice has melted.
Correct, when water freezes, it expands, so when the floating ice melts, the displaced water surrounding the floating ice shrinks to replace the floating ice. Hence, water level decreases.
I am no expert on the meltwater modeling topic of this referenced EDHmelt paper. But I am sure its conclusions are correct for a completely different reason, based on three readily findable observational facts and two small bits of logic.
The post begins by referring to the alarming WRI summary of IPCC AR6, slightly paraphrased here for brevity and clarity:
’2-3C increase could melt Greenland and Antarctic ice sheets almost completely.’
NOPE! The post’s complicated math and my simple facts agree.
We know from the Greenland NEEM ice core that the peak Eemian temperature there then was about +8C over present.
We know from the Antarctic Epica Dome C ice core that the peak Eemian temperature there then was +6-8C above present.
Considering two different poles with very different regional geographic conditions, that is remarkably good paleoproxy agreement. So a robust observational ice core ‘fact’.
And we know that the Eemian sea level high stand was ‘only’ about 2-2.5 meters above present (the uncertainty is both locational and tectonic). There are several readily accessible robust papers on this topic. See essay ‘One if by Land’ in ebook Blowing Smoke for good references and two easily refuted counter-examples.
So it is IMPOSSIBLE that the WRI cited IPCC AR6 melt alarm could ever be true even in a 20 millennia (Eemian duration) timeframe. Let alone in the next century IPCC climate alarm time frame. So the melt models upon which it is based MUST be false. This post technically explains why—fine for Judith, but complicated for the skeptical layman to communicate. My alternative confirmation is much simpler.
Rud, agree with you. I suggest that modelling of glacier melting is a waste of time. In the last hundred years there has been no measurable increase in sea levels. We are in a slight warming period of a longer term decline in world temperatures. The warming has nothing to do with CO2. It has come from some reduction in cloud cover. Many are predicting that come 5 to 30 years time the cooling trend will continue and the Earth might again be like the little iceage. I will probably be dead by then and will not care. Australia where I live will likely not be affected. The continent is moving north to the tropics (by a few metres/yr) and that will benefit all living here.
Just a minor nit:
Modelling is not a waste of time when it is used to gain a better understanding of a phenomenon.
When it is used to “predict” then it is useless. When it ignores error stack up and uncertainty bands in the outputs, it is totally useless.
Modelling and simulation are tools only, not data, not science.
I use modelling and simulation all the time. Sometimes new insights are gained. Often it gives reasonable distributions of potential results. On rare occasion it reveals something not previously known. Real world testing is always performed and the results are compared to the model. If there is a mismatch, the model is at fault.
Sparta, I did not say all modelling is a waste of time. All engineering designers and innovators model to scale up or down, and make improvements. I said “I suggest that modelling of glacier melting is a waste of time” which it is because there a better things to research.
The warmunists’ claim that meltwater makes it all the way to the bottom of an ice sheet thousands of feet thick, all of that ice pack at less than the freezing point, is ludicrous on its face. Not gonna happen. Equally ludicrous is that crevasses would extend thousands of feet vertically to the very bottom of an ice pack despite hundreds if not thousands of psi internal pressure and a highly plastic medium like ice.
Then they claim that the heat of friction between ice and underlying rock as the ice moves is what causes the meltwater at the bottom of the ice sheet, which is equally ludicrous when considering the extremely slow velocity of glaciers, measured in cm per hour or micrometers per second!
Finally, the notion that such liquid water at the bottom of the glacier, even if it existed, would cause glaciers to be lubricated and thus slide faster, and then somehow then end up in the ocean and melt, is even more ludicrous.
Friction is proportional to both relative velocity of two contacting surfaces, as well as the relative roughness of the surfaces. Think of machined solid surfaces in contact with and moving against each other, like inside rotating machinery. This friction is a micro property of the surfaces, where surface irregularity is measured in microns. Lubricants serve to separate the surfaces to reduce or eliminate direct contact and thus reduce friction. The more effective the lubrication the less frictional wear occurs.
Ice in glaciers most certainly is in direct contact with rocky surfaces, and is well known to grind away the rocky surfaces, creating glacial valleys as can be seen in typical mountain glaciers, or former glacial valleys as in Yosemite NP, and in deposits of glacial till and loess (rock flour) left behind when the glaciers melt.
So obviously glaciers most certainly do directly contact and erode the underlying rocky surfaces which means little to no practical lubrication effect occurs. Indeed, considering “macro friction” where glaciers move on top of folded, uplifted, eroded and weathered rock with surface roughness measured in tens of feet if not much more, the glaciers are forced to plastically deform as they move downhill, rather than just slide along like an ice cube on a smooth surfaced countertop.
The references I have now included here are generally a chronological record of review papers that cover the big picture. Very likely I have not followed the lubrication aspect in sufficient detail, but I find it difficult to reconcile my mental vision of lubrication with my mental vision of the ice-land interface at the base of glaciers.
Lubrication ‘is consistent with’ observed significant bulk glacier motions with observed significant increases in meltwater outflows at the terminus. The amount of field data is monotonically increasing as the focus on climate change continues unabated. In general however many of the necessary basic aspects remain unknowable.
Drawings/sketches/photographs of the variety of potential flow fields, both on top of and internal to ice, are available on the Web.
Correct on all points.
Here is introductory material that did not make it into the post:
References
Joseph S Walder (1982). Stability of sheet flow of water beneath temperate glaciers and implications for glacier surging. Journal of Glaciology, Vol. 28, No. 99, 1982.
Abstract. A mathematical model is presented for the stability of sheet flow of water beneath a temperate glacier. Enhanced viscous heat dissipation in thick parts of the sheet tends to make sheet flow unstable. the instability increasing as sheet thickness and pressure gradient increase. However, incipient channels may be destroyed as the glacier slides over protuberances on its bed. Quasi-stable sheet flow may be possible for sheets up to several millimeters in thickness, especially beneath glaciers that have relatively gentle surface slopes and slide at moderate to high speeds. Such water sheets may somewhat reduce the effective roughness of glacier beds. but probably not enough to allow surge initiation. Furthermore, the presence of numerous water-filled cavities at the glacier bed will tend to reduce the sheet thickness and lessen the degree of ” lubricatio n” of the glacier bed by the water sheet.
Joseph S. Walder (2010). Röthlisberger channel theory: its origins and consequences, Journal of Glaciology, Vol. 56, No. 200, 2010.
Abstract. The theory of channelized water flow through glaciers, most commonly associated with the names of Hans Rothlisberger and Ron Shreve and their 1972 papers in the Journal of Glaciology, was developed at a time when interest in glacier–bed processes was expanding, and the possible relationship between glacier sliding and water at the bed was becoming of keen interest. The R-channel theory provided for the first time a physically based conceptual model of water flow through glaciers. The theory also marks the emergence of glacier hydrology as a glaciological discipline with goals and methods distinct from those of surface-water hydrology.
G.E. Flowers (2015). Modelling. water flow under glaciers and ice sheets. Proceedings of the Royal Society A. 471: 20140907. http://dx.doi.org/10.1098/rspa.2014.0907
Abstract. Recent observations of dynamic water systems beneath the Greenland and Antarctic ice sheets have sparked renewed interest in modelling subglacial drainage. The foundations of today’s models were laid decades ago, inspired by measurements from mountain glaciers, discovery of the modern ice streams and the study of landscapes evacuated by former ice sheets. Models have progressed from strict adherence to the principles of groundwater flow, to the incorporation of flow ‘elements’ specific to the subglacial environment, to sophisticated two dimensional representations of interacting distributed and channelized drainage. Although presently in a state of rapid development, subglacial drainage models, when coupled to models of ice flow, are now able to reproduce many of the canonical phenomena that characterize this coupled system. Model calibration remains generally out of reach, whereas widespread application of these models to large problems and real geometries awaits the next level of development.
More
Vena W. Chu (2013). Greenland ice sheet hydrology: A review, Progress in Physical Geography, 2014, Vol. 38, No. 1, pp. 19–54 DOI: 10.1177/0309133313507075
Abstract. Understanding Greenland ice sheet (GrIS) hydrology is essential for evaluating response of ice dynamics to a warming climate and future contributions to global sea level rise. Recently observed increases in temperature and melt extent over the GrIS have prompted numerous remote sensing, modeling, and field studies gauging the response of the ice sheet and outlet glaciers to increasing meltwater input, providing a quickly growing body of literature describing seasonal and annual development of the GrIS hydrologic system. This system is characterized by supraglacial streams and lakes that drain through moulins, providing an influx of meltwater into englacial and subglacial environments that increases basal sliding speeds of outlet glaciers in the short term.
Sarah L. Greenwood, Caroline C. Clason, Christian Helanow, and Martin Margold (2016). Theoretical, contemporary observational and palaeo-perspectives on ice sheet hydrology: Processes and products. Earth-Science Reviews, Volume 155, pp. 1-27, 2016.
Abstract. Meltwater drainage through ice sheets has recently been a key focus of glaciological research due to its influence on the dynamics of ice sheets in a warming climate. However, the processes, topologies and products of ice sheet hydrology are some of the least understood components of both past and modern ice sheets. This is to some extent a result of a disconnect between the fields of theoretical, contemporary observational and palaeo glaciology that each approach ice sheet hydrology from a different perspective and with different research objectives. With an increasing realisation of the potential of using the past to inform on the future of contemporary ice sheets, bridging the gaps in the understanding of ice sheet hydrology has become paramount. Here, we review the current state of knowledge about ice sheet hydrology from the perspectives of theoretical, observational and palaeo glaciology. We then explore and discuss some of the key questions in understanding and interpretation between these research fields, including: 1) disagreement between the palaeo record, glaciological theory and contemporary observations in the operational extent of channelised subglacial drainage and the topology of drainage systems; 2) uncertainty over the magnitude and frequency of drainage events associated with geomorphic activity; and 3) contrasts in scale between the three fields of research, both in a spatial and temporal context. The main concluding points are that modern observations, modelling experiments and inferences from the palaeo record indicate that drainage topologies may comprise a multiplicity of forms in an amalgam of drainage modes occurring in different contexts and at different scales. Drainage under high pressure appears to dominate at ice sheet scale and might in some cases be considered efficient; the sustainability of a particular drainage mode is governed primarily by the stability of discharge. To gain better understanding of meltwater drainage under thick ice, determining what drainage topologies are reached under high pressure conditions is of primary importance. Our review attests that the interconnectivity between research subdisciplines in progressing the field is essential, both in interpreting the palaeo record and in developing physical understanding of glacial hydrological processes and systems.
And more:
Christine F. Dow (2022), The role of subglacial hydrology in Antarctic ice sheet dynamics and stability: a modelling perspective, Annals of Glaciology, Vol. 63, Issue 87-89, September 2022, pp. 49-54. Published online by Cambridge University Press: 23 March 2023 DOI: https://doi.org/10.1017/aog.2023.9
Abstract. Subglacial hydrology is an important component of the ice dynamic system in Antarctica but is challenging to investigate due to the large spatial scales of the catchment systems, the ice thickness, and remote location. Here I discuss key discoveries about Antarctic subglacial hydrology from the Glacier Drainage System (GlaDS) model, including the presence of long, often high-pressure, subglacial channels. These channels pump tens of cubic metres per second of freshwater into ice-shelf cavities and directly affect melt rates at the critical grounding zone regions. Future ice dynamics and ice-shelf cavity models should take subglacial hydrology into account if they are to accurately predict future behaviour of the Antarctic Ice Sheet.
Amy Jenson, Mark Skidmore, Lucas Beem, Martin Truffer, and Scott McCalla (2023). Modeling saline fluid flow through subglacial ice-walled channels and the impact of density on fluid flux, EGUsphere preprint repository Preprint. Discussion started: 28 April 2023, https://doi.org/10.5194/egusphere-2023-792
Abstract. Subglacial hydrological systems have impacts on ice dynamics, as well as, nutrient and sediment transport. There has been extensive effort to understand the dynamics of subglacial drainage through numerical modeling. These models, however, have focused on freshwater in warm ice and neglected the consideration of fluid chemistry such as salts. Saline fluid can exist in cold-based glacier systems where freshwater cannot and understanding the routing of saline fluid is important for understanding geochemical and microbiological processes in these saline cryospheric habitats. A better characterization of such terrestrial environments may provide insight to analogous systems on other planetary bodies. We present a model of channelized drainage from a hypersaline subglacial lake and highlight the impact of salinity on melt rates in an ice-walled channel. The model results show that channel walls grow more quickly when fluid contains higher salt concentrations which lead to higher discharge rates. We show this is due to a higher density fluid moving through a gravitational potential. This model provides a framework to assess the impact of fluid chemistry and properties on the spatial and temporal variation of fluid flux.
Malena Andernach, Marie-Luise Kapsch, and Uwe Mikolajewicz (2024). Impact of Greenland Ice Sheet Disintegration on Atmosphere and Ocean Disentangled, Earth System Dynamics Discussions, European Geosciences Union, Preprint. Discussion started: 22 August 2024. https://doi.org/10.5194/esd-2024-24
Abstract. We analyze the impact of a disintegrated Greenland Ice Sheet (GrIS) on the global climate through steady-state simulations with the MPI-ESM (Max Planck Institute for Meteorology Earth System Model). This advances our understanding of the intricate feedback between the GrIS and the full climate system. Sensitivity experiments enable the quantification of the individual contributions of altered Greenland surface elevation and properties (e.g., land cover) to the atmospheric and oceanic climate response. Removing the GrIS results in reduced mechanical atmospheric blocking, warmer air temperatures over Greenland and thereby changes in the atmospheric circulation. The latter alters the wind stress on the ocean, which controls the ocean-mass transport through the Arctic Gateways. Without the GrIS, the upper Nordic Seas are fresher, attenuating deep-water formation. In the Labrador Sea, deep-water formation is weaker despite a higher upper-ocean salinity, as the inflow of dense overflow from the Denmark Strait is reduced. Our sensitivity experiments show that the atmospheric response is primarily driven by the lower surface elevation, whereas altered Greenland surface properties mostly amplify but also counteract few of the changes. The lower Greenland elevation dominates the ocean response through wind-stress changes. Only in the Labrador Sea, altered Greenland surface properties dominate the ocean response, as this region stores excessive heat from the Greenland warming. The main drivers vary vertically: The elevation effect controls upper-ocean densities, while surface properties are important for the intermediate and deep ocean. Despite the confinement of most responses to the Arctic, a disintegrated GrIS also influences remote climates. The altered climate in response to a GrIS disintegration also constrains a potential ice-sheet regrowth to high-bedrock eastern Greenland.
This seems to be a complete waste of effort. Looking through the paper the only place where any mention is made of the disagreement between this work and that of Spring in Hutter is in the appendix where there are the two sentences:
“Meltwater effects are measured by the difference between the bulk liquid and ice temperature, and not by the ice temperature alone. The former approach is not correct, while the latter is.”
Now if you want to claim that there is an error in a previous paper then that is fine but you need more justification than simply stating “the former approach is not correct”. What he is referring to is the statement in Spring and Hutter that the rate of ice melting is proportion to the temperature difference between the liquid temperature and the ice temperature. While Mr. Hughes claims that the rate of ice melting depends only on the temperature of the ice. And if you think Mr. Hughes is correct then imagine pouring water at 95 degrees over some ice compared with pouring water at 1 degree over the same ice and asking yourself which will melt the ice faster.
Then there is the weird result whereby he writes the solution to a real differential equation for the temperature in terms of complex numbers. All of which could be rewritten in terms of real valued
functions using the simple fact that tan(i x)= i sinh(2x)/(1+cosh(2 x) which makes it clear that asympotic solution is.
Equation (10) gives the liquid-to-ice energy exchange, Equation (11) gives the associated mass exchange, the last RHS term of Equation (9) gives the meltwater effect on the ;liquid temperature.
If you cannot identify explicitly the terms in those equations that are not correct, and at the same time offer corrections, the equations will be taken to be correct.
There’s a typo, I think, in the tan[ i x ] transformation. A closing ) is missing in the denominator. The transformation with which I am familiar is tan[ i x ] = i tanh[ x ], and these of course have the same limit as z->Infinity.
Dan,
in your appendix you give what you say is the original and incorrect equation for the
temperature. Comparing that equation to your one the difference is the presence of the
delta T^2 term. This comes from the m (delta T) term where m is the rate at which mass
is melted and enters the flow. And both you and Spring and Hutter agree on that. So the disagreement comes about because Spring and Hutter claim that m is proportional to
the temperature difference while you claim that it depends only on the temperature of the ice i.e.:
“Meltwater effects are measured by the difference between the bulk liquid and ice temperature, and not by the ice temperature alone. “
So the entire paper boils down to whose definition of m is correct. And you do not give
any argument for why you claim that the meltwater doesn’t depend on the temperature of
the bulk liquid. Again if you imagine pouring hot water over a block of ice compared with pouring cold water over ice and asking in which case the ice will melt faster it would be
obvious that the hotter the water the quicker the ice will melt. Hence Spring and Hutter’s inclusion of a delta T^2 squared term in the equation.
So what is your argument for why the rate of melting ice does not depend on the temperature of the liquid and only on the ice temperature alone. Unless you can justify that statement there is no reason to believe your paper.
And apologies for the missing bracket in the equation. So we agree that the limit as z->infinity of tanh(z) is one which makes the asymptotic solution obvious.
I’ll add that to my top 1,000 things to worry about- and put it at the bottom.
Irreversibly?
I seem to recall at points in the past both were Ice free and they are not glaciated.
I believe- I’m sure a geologist will correct me- that the continental glaciers have advanced and retreated dozens of times during the Pleistocene. The times of retreat were the good times.
I believe you are correct.
“Should warming reach between 2 degrees C (3.6 degrees F) and 3 degrees C (5.4 degrees F), for example, the West Antarctic and Greenland ice sheets could melt almost completely”
BS Alert
Most of Antarctica has a permanent temperature inversion that results in a negative greenhouse effect. See pink area on chart below’
A tiny amount of melting of two ice shelves and the tiny peninsula is most likely too small to measure: Statistically insignificant. If the melting estimates are correct, then the melting is irrelevant
Antarctica holds 90% of all ice on land. The melting there does not depend on greenhouse gases, which make most of the peninsua COLDER due to the permanent temperature inversion.
The author is ignorant of Antarctica climate science and should be ignored. The Climate Etc. website has deteriorated into a waste of time.
Reading comprehension alert:
These are not the words of the author. And that is blindly obvious in the text.
oops, I forgot.
The authors of those words are the IPCC. Does The IPCC is ignorant of Antarctica climate science and should be ignored. The IPCC website has deteriorated into a waste of time, hold?
Speculation about meltwater is not important when almost all of Antarctica is not melting.
If a false IPCC statement is included in an article, it should be thoroughly refuted in that article.
Climate Etc. has had only six articles since July 5, 2024, over three months ago, and most of them were mediocre to poor climate science theories. The batting average at WUWT is MUCH higher with 540 articles every three months.
A note on nomenclature.
I made an unfortunate spur of the moment decision regarding nomenclature. The RHS of Eq.(14) where I use Ec_subscript0 is better represented by T_bar with subscript zbar-infinity. The former is too easily confused with the Eckert Number, which I represent by Ec, without a subscript.
Eckert Number
The response of the Eckert Number for the case with the value at the entrance being less than 1 is interesting. For all cases, the Eckert Number, given a sufficiently long channel always approaches 1.0. When Ec<1 at the inlet, the response follow the Logistic Curve, which is also known as the Sigmoid Function, I think. There seems to be some differences of opinion along this line.
That curve represents the introductory, rapid growth, and saturation phases of, say, a new/different product or process. Like for example introduction of coal during times when wood was the primary source of heat. Or steam when water power was the primary source of power. Or introduction of cell phones when land-lines were the primary source. Or electricity during times when we didn’t have any.
In the case of Ec<1 viscous dissipation plays a minor role, compared to liquid-to-ice energy exchange, near the inlet, the introductory phase; and increases to be the dominate role, the raid growth phase; given a sufficiently long channel, at which liquid-to-ice energy exchange is set by viscous dissipation, the saturated phase.
Ran across this paper that might help with flow resistance at ice base and terrain interface:
Włodzimierz Czernuszenko (2011). Spatially Averaged Log-Law for Flows over Rough Bed in Zero- and Non-Zero-Pressure Gradient Boundary Layers, Archives of Hydro-Engineering and Environmental Mechanics, Vol. 58, No. 1–4, pp. 65–86 DOI: 10.2478/v10203-011-0004-7
Focus on big roughness elements.
30,000 years ago the sea level was 125 metres lower than it currently is. The polar ice sheets were grounded at that point.
Surely the upward pressure on those ice sheets is adding to the stresses and strains leading to breakages and an increase in the rate at which glaciers move to the sea.
It does not require any melt water, just the metres and metres of sea water under the edges of the ice sheet. Every day the sea moves up and down with tides, and every now and then there will be storm surges.
What on earth do you think will happen to the ice sheet given these pressures?