Viscous dissipation heating by flows of melted ice on Greenland

by Dan Hughes

The contribution of viscous dissipation conversion of kinetic energy into thermal energy has been significantly over-estimated in three recent publications. The kinetic energy content of the macro-scale mean flow is assigned to be the heat dissipation into thermal energy. The estimate leads to temperature increases that make significant contributions to melting ice on Greenland.

A recent news release announced the findings of the research, and a video of a melt-lake draining into the glacier ice is in the news release and also at YouTube here.

A different estimate, in which the viscous dissipation is determined at the micro-scale of the flow, is calculated in these notes. This estimate, and the associated temperature increases in the flow, are significantly less than that based on the macro-scale. A PDF file with my analysis is here [BSLdissip02]

Comments, especially corrections for incorrectos, will be appreciated.

59 responses to “Viscous dissipation heating by flows of melted ice on Greenland

  1. I am a layman and I would only like to comment that your paper, Dan, could be more effective if there was an executive overview part of your abstract which simply laid out the consequences of this phenomena and your analysis. Why does the non-specialist care about this and why is your analysis more accurate? The way it is, people either need to be very familiar with this issue and/or depend on the referenced articles for a broad framing and context. Wishing you all the best!

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  3. Tomas Milanovic

    lnicholson the question is whether the glacier’s melt is “faster than previously thought”.
    Viscous heat dissipation is indeed a complex matter so that it is mostly readable only by people who have experience with hydraulic engineering and/or fluid mechanics.
    The question here is whether the specific viscous heat dissipation (in J/kg) is “greater than previously thought” or whether the total viscous heat dissipation (in W) is “greater than previously thought” .
    Dan shows that the answer on the first question is no.
    As the second number is obtained by multiplying the first by the flow (J/kg x kg/s=J/s=W) , the only way for it to be “greater than previously thought” is to find a flow much bigger than what has been found previously.

    Personnally I don’t believe that it is probable that after more than 1 century of research, scientist and engineers would underestimate the specific viscous heat dissipation by an order of magnitude and find Dan’s demonstration convincing.

    • Curious George

      Could the effect be measured at Niagara Falls, instead of at an inaccessible glacier in Greenland?

  4. Robert D Clark

    The ICE BLOCK, glacier to you, is being melted forming an overhang. The 34- degree salt-water on the bottom of the ocean is melting the ice forming an overhang. Eventually it will break off, melt, thus keeping a constant ocean height. and a constant surface temperature of the earth.
    The flow at the vertical edge is up. it is the lighter fresh water.
    This will continue for another 100,000 years.
    Then the ocean will drop for about 8,000 years.

  5. ‘The Medieval Warm Period was approximately 1 °C warmer than present, and the Little Ice Age 0.6 °C cooler than present, in central Greenland.’ K.M.Cuffey

    • Greenland’s ice sheet is 400,000 years old. Before that there was global warming which obviously had nothing to do with humanity’s contribution two atmospheric CO2 levels.

      • Robert D Clark

        You are looking at an ice core over land. I am talking about an ice block which is about 12,000 years since it was formed.

      • True, true, a free floating ice shelf will not raise sea levels if it melts.

      • Robert D Clark

        It is not floating. I call it an ice block because it was formed on frozen land at the beginning of this Ice Age when the ocean was about 400FT lower than present.

      • … over 2 million years ago?

      • Robert D Clark

        About 12,000 years ago.

      • Interestingly…

        ‘Approximately 125,000 years ago, the sea level was approximately 8 meters higher than it is today. This was during the Sangamonian’

      • Seems in arguable that most of the sea level rise over the last 12,000 years occurred before humanity ever took to the seas and wooden boats…

      • Robert D Clark

        You are looking at the high-water level of the last ice age at the end of the making of the ice blocks. I am talking about the low water level at the beginning of this ice age. That is when the ice blocks began to be made on the frozen land as the water level rose.

      • exactly… ‘
        Climate has always changed and is always changing. The last Ice Age, which covered places like what is now New York City with ice two miles deep ended between 17,000 and 12,500 years ago…’ ~Botkin

      • Robert D Clark

        Yoy are correct

      • Robert D Clark

        If I use your numbers, 17,000 years ago the ice blocks were completely gone. The earth was losing more heat than it retained. Nature was still removing heat from the oceans and dropping the frozen water on the frozen land of the continents.
        12,500-years ago the earth began retaining more heat than it lost. To keep a relatively constant surface temperature the ice on the continents began to melt.

  6. Clyde Spencer

    It seems that the calculation of kinetic energy, available for conversion to heat, of the falling water assumes that the water makes it to bedrock, which may not be the case. I don’t remember seeing any good evidence for the assumption in previously reading on this topic. It is more likely, in my judgement, the the water only makes it to around the region of transition from brittle to plastic flow of the ice, which is nominally ~60 m. I know from personal experience with work in the ice tunnel at Camp Tuto (Greenland), that there was a substantial flow of water that was below the surface, but above the tunnel and therefore above the bedrock.

  7. “Following equation (8) in that paper the authors explicitly state that it
    is assumed that the entire potential energy content of the mean flow is
    converted by viscous dissipation into thermal energy. An assumption that is
    not in agreement with the engineering literature.”

    Seems a reasonable assumption. Energy is conserved. If it doesn’t go into thermal energy, where does it go?

  8. I think nobody knows this finding on geothermal heat in Greenland:
    “Earth’s internal heat drives rapid ice flow and subglacial melting in Greenland”

  9. If the water is still moving when it hits the ocean, then there is kinetic energy left over. Not all the gravitational potential would be converted to heat.

    • At least it won’t be converted to heat in contact with the glacier. It will heat the water into which it empties. If that water moves as a result, there is still kinetic energy left over.

    • 1 kg water at 1000 m height has 10000 J potential energy.
      If it is still moving at 1 m/s at the bottom, it has 0.5 J kinetic energy. The rest was dissipated as heat.

      • So, this effect isn’t related to CO2? And it would occur to some extent on any glacier with active drainage, I suppose.

      • It appears the basal melt rates and vertical ice deformations aren’t radically greater during the melt season vs non-melt season. They appear quite a bit higher during the week or so of rainfall. So the effect is narrowly constrained in time annually.

      • Nick,

        “If it is still moving at 1 m/s at the bottom, it has 0.5 J kinetic energy. “

        I don’t understand the ‘if’ part of this statement. For steady flows of fluids that are essentially incompressible, in constant cross-sectional flow area channels, the velocity is everywhere the same value.

        When the flow is driven by gravity downward, how can it not be moving?

        The issue in this discussion is power, not work.

      • Dan,
        The “if” referred to the postulated 1 m/s. The point is that no feasible kinetic energy at the bottom can be more than a tiny faction of the potential energy. So where has the rest of it gone? It can only have been dissipated as heat. The paper correctly calculates the amount and points out that it is a significant addition.

      • It appears the effect on basal melt rates and vertical ice deformations occurs only when it’s raining, which is about a week out of the year in their data. Perhaps the warmth of the rain is a key element in the effect?

      • “The point is that no feasible kinetic energy at the bottom can be more than a tiny faction of the potential energy.”

        I’m still not following this idea. The data report the amount of liquid water that enters the channel. Ignoring the possible effects of thermal interactions between the liquid water and the ice walls of the channel, for steady-state conditions that amount of liquid water must, by mass conservation, come out the bottom of the channel. Given the flow area of the channel and a basically incompressible fluid, the velocity at the bottom can be calculated.

        Some possible thermal interactions include: 1) increased mass flow as the ice walls are melted by the water, 2) refreezing of the liquid by loss of energy to the walls, and 3) closure of the channel if sufficient water refreezes: a Stefan-Gratz problem.

        The process is clearly not steady state; as the melt-lake empties the gravitational head to drive the liquid into the channel changes. The hydrodynamic effects of potential water-ice thermal interactions will also affect the flow down the channel. I also consider it to be highly unlikely that the channel is flowing full. As has been mentioned in comments, it’s not like the massive volumetric flows that are assumed to correspond to 8.8 GW of power are underway 24/7/365.

      • Dan. Some heat will be generated by turbulent flow if the water starts flowing fast enough in the channels. The diameter and shape of the channel probably isn’t consistent which will also tend to induce turbulence. So I would expect some heat to be created on the way down. Also, when the water hits the bottom, it almost certainly will be turbulent there. Any heat created will be taken from the kinetic energy of the water.

      • Nowhere have I indicated that turbulent kinetic energy is not converted to thermal energy. On the contrary I gave a calculation of that phenomenon.

      • Dan,
        I’m still not following this idea. The data report the amount of liquid water that enters the channel. Ignoring the possible effects of thermal interactions between the liquid water and the ice walls of the channel, for steady-state conditions that amount of liquid water must, by mass conservation, come out the bottom of the channel. Given the flow area of the channel and a basically incompressible fluid, the velocity at the bottom can be calculated.

        I think Nick’s point is that if the water starts at an altitude of ~1000m and flows down to sea level, then if there is no dissipation of energy it would be travelling at an unrealistically large velocity when it reaches sea level (100s of km/hr, unless I’ve made a silly mistake). For any realistic velocity at the base of the flow, a vast majority of the kinetic energy must have been converted to thermal energy.

  10. This is not intended to be a criticism of the paper by Dan Hughes, but rather an observation of many papers in climate science. They proceed with an assumption that what is happening is unprecedented and without AGW the dynamics would not have occurred. In Greenland there was concern about melting of the AIS at least 80 years ago.

    Are we to assume that the basal melting only started in the last several decades? The Karlsson paper covers the last 30 years. Is there evidence that this the only period where such melting occurred. There is literature that Iceland glaciers were melting ~1900.

    Secondly, as noted in another comment above, and in the Karlsson paper, previous papers have found evidence of geothermal activity is impacting basal melting of the Ice Sheet and peripheral glaciers. I hope at some point there’s research into the relative contribution to GMSLR from each source of heat. The Karlsson paper, as well as other studies, make the assumption that the geothermal activity provides a constant level of heat. For decades maybe. But for longer periods is this true? In his study of Antarctica, Schroeder, 2014, noted temporal variability in the effects of geothermal activity.

    • Clyde Spencer

      I can assure you that there was subsurface meltwater in the Greenland Ice Sheet near Thule AFB when I was there in 1966.

  11. ‘The real world did follow a halfway predictable path, according to one interpretation of new field studies. In 1976, analysis of deep-sea cores revealed a prominent 100,000-year cycle in the ebb and flow of ice ages. That corresponded to a predictable astronomical cycle of variations in the Earth’s orbit. However, the cyclical changes of sunlight reaching the Earth seemed trivially small. The group of scientists who published the evidence thought the cycle of glacial periods must be almost self-sustaining, and the orbital changes only nudged it into the shifts between states.’

    You may decide that there is no risk – so risk management is not required. But that’s a message that hasn’t gained traction in 30 years.

  12. dougbadgero

    Now let’s do it for falling rain. This is a tempest in a teapot. Isn’t the implication of the second law that all energy ultimately becomes changes in entropy?

  13. Unfortunately this topic is a little more in-accessible and esoteric than I initially realized. Even the available Open Access materials that I have found do not shed a lot of light at a laypersons view.

    I think I have correctly presented the analysis as developed in Bird, Stewart, Lightfoot, and Bird, and I think I have correctly presented those equations. More importantly I think I have given sufficient information that the results of my calculations can be checked. None of this material has been mentioned.

    • To what end has Judith Curry published your critique here? You are critiquing a article about a climate science paper ( The math in the Methods and Materials section of the paper is pretty complex for this audience. It’s clear that most aren’t really paying attention to the math you present from the 1950/60s era texts of Bird et. al. either.

      If your analysis is justified, why not submit it to the Proceedings of the National Academy of Sciences?

      People that read this blog believe the science that they choose to believe, and because you are critiquing a paper that sounds an alarm about climate change, some will choose to believe your pdf, even if they aren’t qualified to provide the peer-review that you seem to be requesting.

      To those people, I say: read and fully understand the original paper before you accept the critique.

      To Judith Curry, I say: When you post a critique of some peer-reviewed science, you are giving your endorsement of that critique. Would you have submitted this yourself?

      • “When you post a critique of some peer-reviewed science, you are giving your endorsement of that critique. Would you have submitted this yourself?”

        Yes indeed. It’s long past time that Judith take some responsibility for what’s posted here.

      • Brower was published in 1999. The same equation is also in Gerhart, Hochstein and Gerhart, Fundamentals of Fluid Mechanics: SI version, published in 2021.

        So, the same formula has been publish in textbooks from 1940 by Vennard, 1960 by Bird, Stewart, and Lightfoot, through 2021 by the above.

        Strange how textbooks work, isn’t it.

        Thank you for your exceedingly insightful comment regarding the technical matter at hand.

      • It doesn’t take a genius to see what’s going on here. Have a look at Fig. 1, chart B. The basal melt rate is MUCH higher when it is raining. And that’s what, about 2 weeks out of the year?

      • To Dan Hughes.

        The problem with equations from a textbook written in 1960 is that you have to be very careful to use them in a way that applies to the physical situation being examined.

        In chapter 7 of Bird, Stewart and Lightfoot, the flow described is through a closed system where volume in = volume out — as you say in your pdf “a simple closed channel like a straight pipe”. The situation under a glacier is nothing like a simple closed channel. Much more water exits the bottom of the glacier than is melted on top (the paper suggests about 0.5 cubic km on the surface dropping down, whereas the melt rate of greenland is estimated to be over 200 cubic km).

        How exactly does the viscous dissipation along a simple closed channel relate to viscous dissipation when the water that has fallen combines with two orders of magnitude more water?

        To Nick Stokes energy conservation question, there is gravitational potential energy that must be accounted for. Where does it go? The authors of the paper can’t make direct measurements so it is right to question their methods, but their paper does an admirable job of doing some clear analysis given some well stated and reasonable assumptions.

        The assumptions you have made to apply the equations of Bird, Stewart and Lightfoot are not reasonable.

      • There is no doubt that water flowing under the force of gravity will cause some warming, especially if that flow it turbulent. This isn’t big news as far as I know. Not rocket science either.

  14. Ireneusz Palmowski

    Stratospheric intrusion in the east will bring a strong drop in surface temperatures in the eastern US.
    Record Cold with 40 °F below normal Temperatures to impact Southeast U.S. on Sunday, some areas even colder than Alaska.

  15. Ireneusz Palmowski

    If anyone doesn’t understand how thin the troposphere is in winter at mid-latitudes, they should be in the southeastern US right now. Then he will understand how a total freeze can occur in a few hours.

    • And it is messing with one of my favorite golf tournaments with all the rain coming in from the cold front.. It might drop to 29 F in Jacksonville Sunday morning.

  16. Ireneusz Palmowski

    Sergei Babkin sang his hit “I’m a Soldier” in the Ukrainian manner.

  17. To Dan Hughes

    The paper that you critique asserts that Greenland meltwater that falls a kilometer or more to the base of a glacier will shed lots of gravitational potential energy. It says that a large portion of this is converted to heat via viscous dissipation and that this heat contributes significantly to the basal melt rate.

    Your pdf asserts that, because a long-standing model of flow through “a simple closed channel like a straight pipe” suggests a low upper bound on viscous dissipation, the falling water cannot be responsible for generating as much heat as stated in the paper.

    In a comment above, I observed that the surface melt is estimated by the authors of the paper to be 0.5 cubic km annually whereas the total melt rate of Greenland is roughly 200 cubic km ( Obviously, the falling water will combine with a much larger sub-glacial body of melt water causing turbulence and increased outward pressure in that larger body. Furthermore, the dynamic melting and channels that characterize the sub-glacial flow is nothing like “a simple closed channel like a straight pipe”.

    It’s not your conclusions are incorrect, it is that you have not modeled the physical situation adequately to support making ANY conclusions.

    Moreover, this observation should be obvious to anyone with a sufficient background in math and physics who looks into the details of your model.

    The thing about a paper that is published in a scientific journal is that the authors are accountable. If someone challenges their findings through official channels, they must respond and publish appropriate corrections.

    Meanwhile, your pdf has probably succeeded in convincing several people that “yet another” peer-reviewed climate science paper is fundamentally flawed. I know that you don’t want your efforts to join the huge wealth of online misinformation about climate science. You should retract this post.

    • The focus of the post was the concept of dissipation of kinetic energy by turbulence and its conversion into thermal energy.

      Illustrative examples of applications of concepts are frequently presented by use of situations for which the concept is an integral part.

      That the examples might not reflect the totality of possible applications does not in anyway negate the validity of the concepts. No one has yet explicitly cited errors in the equations that I took from textbooks.

      I think that mathematical descriptions of viscous heat dissipation, by definition, should include physical phenomena and processes closely associated with turbulence and viscosity. Additionally, the description, again by definition, should be applicable to all flows; horizontal flows, flows downward with gravity, and flows upward against gravity.

      • Dan Hughes: “That the examples might not reflect the totality of possible applications does not in anyway negate the validity of the concepts.”

        Sure, the concepts that are represented by the math in your pdf are valid. You just haven’t demonstrated that they model the physical situation described in the paper you are criticising (Young et. al.).

        I will clarify what I mean.

        We know that water is descending say a kilometer from the top of a glacier to the bottom so we know that it sheds gravitational potential energy. We don’t know what the water does on the way down.

        Some water might be falling in a mostly empty shaft — a situation in which the water will tend to remain in contact with the walls of the shaft. The math in your pdf does not apply to this situation.

        Some water might be falling down a shaft that is regularly interrupted by shelves of ice. The math in your pdf does not apply to this situation.

        Some water might be falling through a full, straight channel with a hydraulic radius of 2m — the situation that you claim to have modeled. Let’s think a bit about that scenario, observing that the channel is 1km long and vertical. I didn’t see where you factored in the pressure gradient from the weight of the water. Seems to me that it would result in some pretty enormous pressure. Where does that pressure go? If it is transferred to sub-glacial flow then the math in your pdf does not apply to this situation. If it causes the ice to crack and water to flow in new fissures, the math in your pdf does not apply to this situation.

        Your math is not wrong, it just doesn’t apply to the physical situation described in Young et. al. There’s a reason that they didn’t attempt to determine the viscous dissipation at the micro-scale of the flow — not enough is known about the nature of the flow to make that determination.

        You tried, but you failed. Let it go.

        What CAN be said, however, is that water is descending from the top of the glacier to the bottom and shedding gravitational potential energy in the process. Insofar as increased flow velocity cannot account for that much energy, it is reasonable to assume that most of it is dissipated as heat.

    • Regarding future ice melt estimates. I’ll assume that GCMs are the source of estimates of future ice melt.

      I do not address any issues associated with those estimates and GCMs.

      I cannot because I do not have any ideas whatsoever about how those estimates are calculated in any GCM.

      I understand your comment to imply that you do know. Explicit citations to the equations would be helpful to clarify your comments.

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  21. To Dan Hughes:

    You seem to have moved discussion of this topic to a new thread — but I thought this comment to be more appropriate here.

    Take a 10 foot long tube. Put a nozzle on one end, fill it with water and hold up the open end with your thumb over the tip. There’s 6 feet of vertical tube, a bend and 4 feet of horizontal tube along the ground terminated at a nozzle.

    Take your thumb off the open end that you’re holding up, and open the nozzle so that water sprays out.

    If we use this as a qualitative analogy to meltwater descending from the top of a glacier, your math is an appropriate model of the vertical portion of the tube.

    As the water sprays out of the nozzle, how much viscous dissipation occurs in the vertical portion of the tube? Extremely little (your model confirms this — and that’s all it does).

    The gravitational potential energy of the water in the vertical portion of the tube is shed as it descends. But we must have conservation of energy, so where does it go? It is converted to kinetic energy and a little bit of heat at the nozzle.

    I’ll be 100% clear. I am not suggesting that you can model the sub-glacial flow by looking at a model of a horizontal tube and a nozzle. I’m pointing out that you have failed to identify how your model relates to sub-glacial flow with any quantitative analysis and so it is inappropriate for you to make quantitative assessments of the how much viscous dissipation is occurring.

    My point? Same as it always was: your math may be perfectly fine, but it does not model how gravitational potential energy is converted to kinetic energy and heat as water descends from the top of a glacier to the sub-glacial flow. Not even close.