the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 25 Apr 2025
Thermobaric circulation in a deep freshwater lake
Abstract. Numerical lake models are a powerful tool to optimize water management and mitigate changes due to climate change. Hence, detailed implementation of lake specific processes is crucial to ensure optimal results. However, common numerical lake models have so far omitted the effect of thermobaricity despite its significant influence on deep water circulation in deep lakes. The thermobaric effect is based on the temperature dependence of the compressibility of water. As a consequence, deep water can be significantly colder than 4 °C and deep water renewal becomes complex. For a proper investigation, numerical models can be appropriate tools to display and understand such processes better. Inspired by Lake Shikotsu, which is an excellent example for the influence of thermobaricity, we developed a simplified 1D model for thermobaric effects. Here, we used in situ density to replace potential density for stability considerations such as the Brunt-Väisälä frequency. To prevent any competing influences and isolate thermobaric effects, we excluded any external forcing except for the surface temperature input. Accordingly, we excluded salinity, chose a cylindrical bathymetry without shallow areas, and omitted any inflows. Therefore, the model reproduced deep water circulation solely based on thermal forcing at the surface. We were able to identify key features of the deep water renewal events as well as different phases of the mixing period. Additionally, we investigated the influence of previous deep water renewal events and the current surface temperature on the deep water circulation. Our results emphasize the feasibility and necessity of the implementation of thermobaricity in numerical lake models.
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RC1: 'Comment on egusphere-2025-1195', Anonymous Referee #1, 21 May 2025
This work is one of the very few attempts to understand and characterize the sequence of events leading to circulation—manifested as the cooling or warming of very deep water—in thermobaric, deep freshwater lakes using a simplified 1D model. The philosophy behind using this simplified 1D approach is to isolate the effects of thermobaricity and cabbeling, rather than focusing on wind-driven energy input or the complex hydrodynamics associated with realistic 2D or 3D bathymetry. That being said, the model successfully identified how the variation of the temperature of maximum density (Tmd) with depth under significant pressure alone (thermobaricity) can drive mixing in a deep lake. The model was applied to a deep, cold Japanese caldera lake (Lake Shikotsu), where the hydrodynamics are believed to be predominantly vertical. It also demonstrates how using potential density at the surface may lead to completely different results compared to using stability criteria.
- The abstract would benefit from additional concluding sentences that elaborate on the key outcomes of the model, particularly the main physical features identified.
- A clear distinction between thermobaric instability, thermobaricity, and cabbeling is needed, as these concepts are often confused. This clarification should be addressed consistently throughout the manuscript, including the conclusion. It would also be valuable to highlight that, in this case, cabbeling appears to result from eddy diffusion across Tmd at different depths—a particularly novel observation that, to my knowledge, has not been previously reported. As I understand it, this process involves the diffusion between a parcel of water already at a warmer temperature of maximum density (Tmd) and colder water, ultimately producing water at a lower Tmd. This mechanism deserves emphasis given its potential implications for deep mixing processes and how is it compared with “thermobaric instability”.
- There has been brief but noteworthy scientific debates regarding the appropriate criteria for evaluating stability, which merit mention. For instance, Georg Wüst (1932) and V. W. Ekman (1934) discussed the use of potential density—specifically, surface-referenced potential density—as a means of assessing stability. However, it is important to clarify that potential density referenced to an intermediate depth has since been recognized as a more reliable indicator. This approach closely resembles what is being applied here, but at a common depth corresponding to the lower parcel, and is supported by studies including Peeters et al. (1996), which also deserves mention. Finally, when considering which density measure to adopt, it may be useful to briefly reference the concept of quasi-density and explain why it has been excluded from the present analysis to contribute to the ongoing knowledge on the topic! It is very satisfying to see a comparison done with potential density at the surface, which I also believe one of the novel parts of this work.
- Why is the stability criterion being expressed in terms of density rather than simply using potential temperature, especially since salinity is excluded? (Gill, 1982; Imboden and Wüest, 1995). This approach might avoid the complications of selecting an appropriate density reference. On that note, as mentioned in your manuscript (line 202), in-situ density is largely dominated by pressure, and there has been a brief scientific debate on the validity of using in-situ density for stability evaluation (A.H. Lee and G.K. Rodgers, 1972; Thomas Osborn and Paul LeBlond, 1974), ultimately ruling out its use. I believe what you are referring to in this publication is potential density at a common reference depth (at the lower parcel depth, not at the surface), which is conceptually like using an intermediate depth. It is not in-situ density, otherwise potential density at the surface is also in-situ density but the in-situ density at P2=0. An important consideration is what happens when this comparison crosses the Tmd line, as this transition is critical in our case: the compensation depth, which is defined relative to Tmd, governs the overall flow structure. Also, I believe more justification is needed for the choice of evaluating density using the speed of sound (which is not measured, or maybe you have measurements not mentioned?), rather than the TEOS-10 approach utilizing potential temperature and salinity? As mentioned, it is mentioned that TEOS-10 “which includes the effect” compared to potential density, but still, potential density “at the surface”.
- I think it is worth defining the compensation depth. You later refer (line 71) to the intersection of the temperature profile with Tmd, which could be described as the compensation depth. It may help with clarity to introduce and use this term consistently throughout the manuscript.
- It is mentioned that the temperature profile remains isothermal throughout. Is this monitored using thermistors or a CTD, and what is the measurement accuracy of this isothermal profile? For example, is the variability within 0.1 °C or 0.5 °C? Clarifying this would help assess the significance of the observed isothermal conditions compared to the observed cooling/warming of the bottom water and also compare with other lakes. Additionally, where is the surface water temperature (model forcing) measured, and at what exact depth? In other lakes I believe it is usually at least 3-5 m deep in the surface mixed layer (I mean the shallowest thermistor).
- Can you provide a specific analysis or statistic isolating how much of the observed changes are driven by diffusion leading to “cabbeling” or “thermobaricity”? Additionally, how would changing the diffusion coefficient affect the overall process, since it seems like the main driver? It is also unclear how the surface layer remains stable while convection occurs just beneath it that is (I believe) driven by cabbeling induced through diffusion? Clarifying this mechanism would help improve the physical interpretation.
- Why are some profiles perfectly following Tmd, and are they considered stable according to the used stability criterion? Because I would think that maybe again turbulent diffusion perturbations might deem these profiles unstable. That would be interesting to think about.
- I think you need to clarify more the particular use of ±0.4 °C for different climate scenarios, the selection of a three-year spin-up period, and the chosen value for the diffusion coefficient. Also, the method of mixing during the 1hour time step, is it sweeping downwards? When does the algorithm stop?
- The discussion needs more comparison with previous studies especially with the closest model (Piccolroaz 1D model in 2013).
Specific notes:
Line 60: “Admittedly” I am confused from the structure of this sentence, what is being admitted?
Line 76: Potential density “at the surface “. I think it is worth stating this whenever mentioned.
Line 105: So, this is the oscillation frequency using potential density at a common depth, not using in-situ density as it appears. Because in-situ means in its place, but you are evaluating both at P2, so it is confusing. Using actual in-situ density gradient to evaluate N2 would give a misleading sign as it is always dominated by pressure, hence again it is worth noting that this is not the in-situ density gradient, but the potential density or the density at a common reference that is the lower parcel depth.
Citation: https://doi.org/10.5194/egusphere-2025-1195-RC1 - AC1: 'Reply on RC1', Joshua Marks, 03 Jul 2025
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RC2: 'Comment on egusphere-2025-1195', Anonymous Referee #2, 22 May 2025
Thermobaric circulation in deep freshwater lake by Marks et. al.
In this work, the authors undertake a numerical process study to demonstrate the effects of a thermobaric circulation in a cylindrical domain. This domain is inspired by Lake Shikoku, which has been previously observed to undergo thermobaric circulation. The authors employ a 1D column model to explore the vertical transport of heat over several simulated years. The main crux of the argument is that by considering the in-situ density (as opposed to the potential density which excludes thermobaric effects a priori), the authors identify the process by which thermobaric effects effectively mix the water column. Overall, I thought this article was put together well, and interesting. I have a few concerns, however, that should be addressed prior to publication.
Main points
- I am certainly empathetic to the process study approach utilized by the authors, and I’m happy to read work that uses an idealized approach to learn about different process in isolation. I am, however, wondering about the relative importance of thermobaric effects to other, potentially more vigorous, dynamical effects, especially those ignored in this study. I think a discussion on this topic by the authors would help the framing of the work.
- Related to the above point, on line 111, the authors comment “this kind of deep water renewal is suspected to have a significant influence in this lake”, and I’m wondering if they could clarify if they think this based on the observational data, or for some other reason, such as the depth.
- While I was reading this work, I kept asking myself “What is the specific thing that thermobaricity is doing that’s different”? It wasn’t until I read section 3.5 that things (sort of) cleared up for me, though I’m still not quite sure.
- In my opinion, the argument the authors try to make could be strengthened by first using section 3.5 as a straw man, and then discussing the new results (i.e. the results WITH thermobaricity). I think the authors even have their conclusions laid out this way already. Related to this, I encourage the authors to add a picture similar to figure 4, but for the “non-thermobaric” case. I think that would strengthen their argument for “what thermobaricity does”.
- I sort of understand what the authors are getting at in the “Convective Mixing” section, but I think some sort of schematic explaining the convective cell detached from the surface looks like, or maybe an arrow placed on figure 4(b) describing what they mean. (This would certainly aid in my understanding).
Minor Points
There are typos in a few places (eg lines 73, 77, 112, 118, and a few more). Please carefully check the manuscript
Line 36: Can the authors clarify what they mean? This sentence beginning with “Ultimately…” is confusing and I’m not sure what the authors mean.
Can the authors provide a little more info on how they arrived at equation (6). It’s not so clear to me, but I think they’ve taken the derivative of rho_pot (rearranged from equation (3)), and then made the approximation that rho_pot \approximation rho_in-situ in the denominator of equation (6)?
The authors mention that p=0 corresponds to atmospheric pressure at the surface (line 140), but this convention is employed (equation (4)) before it is mentioned in (section 3). Please mention this convention upon first usage.
Lines 94-97: it's not clear to me what you're saying here. Is this maximal deviation the deviation that occurs over 360 m, or between 3 and 4 deg(C), or something else? The sentence in line 96 seems to imply it’s something else.
Line 133: Can the authors clarify what they mean by this sentence? I’m getting confused by the use of the words in the parentheses.Line 203: “Stability frequency” is used. Is this standard? “Brunt-Väisälä frequency” is used in the abstract. I would standardize the usage throughout the paper.
In figure 1, pressure on the vertical axis is positive, but on the subsequent figures, it’s negative. I would suggest that it be standardized to one or the other, or clarified in the text.The authors model convection in a phenomenological way (i.e. all heat is exchanged between adjacent layers instantaneously). For the purposes of this work, I think it's probably fine, but I don't really know. Can the authors comment on whether they think that their approach is actually a good representation of the true convective processes going on in a lake? I.e. are the timescales appropriate? Is there evidence of a lake-wide circulation?
Citation: https://doi.org/10.5194/egusphere-2025-1195-RC2 - AC2: 'Reply on RC2', Joshua Marks, 03 Jul 2025
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RC3: 'Comment on egusphere-2025-1195', Anonymous Referee #1, 03 Jun 2025
This manuscript uses a vertical 1D idealized model to capture thermobaric effects on the seasonal evolution of thermal stratification. The aim of the manuscript is to highlight physical processes the dominate the seasonal cycle. The model implements a novel estimate for gravitational stability, as well as simplified vertical diffusion and surface thermal forcing. Using these three simple concepts they are able to reproduce the basic characteristics of observed thermal stratification in Lake Shikotsu, Japan, a caldera lake whose thermal dynamics are believed to also be mostly vertical 1D.
Much is made of the novel implementation of gravitational stability, but little is said of what previous modellers have done, eg Killworth et al (1996) and Piccolroaz and Toffolon (2013). How is the formulation derived here different and what are its advantages over other, existing formulations?
The distinction between instability induced by vertical displacement of a stable profile to a depth where it becomes unstable (“forced plume downwelling”) and mixing of waters (“cabbelling”) is an important one. While it shows up in the introduction (Line 40-50), it gets a bit lost in the rest of the text. For example, line 50 seems to equate cabelling with simply “thermobaricity”. I recommend clearly and consistently delineating and labelling these two processes throughout the ms. To me, the most interesting result of this work is the focus on how surface convection interacts with the Tmd line subject to cyclical surface forcing.
I encourage the authors to say more about the surface forcing. You use an hourly timestep to resolve the diurnal evolution. Cite or specify some details about the surface measurements (eg depth, sampling interval, instrument details). Why was it important to resolve the diurnal cycle? Do the results change if you use daily averages?
I would like the authors to say more about the two mixing processes built into the model (convective readjustment and diffusion) and how they interact. Currently the manuscript focuses on calculation of stability and subsequent convective readjustment, but says little about the effects of what appears to effectively be a background constant diffusivity set to a rather high value, especially for the deep waters of a deep lake with relatively small surface area. How sensitive is the model to the chosen value of diffusivity (or (time step)/(grid size) ratio)? How does the diffusivity interact with convective instability correction? Why did you even add diffusivity? Presumably the results are very different without it.
I find the use of “in situ density” to describe the stability model to be misleading. The formulation for stability developed in this ms seems to be a discrete approximation of potential density using a reference pressure at the lower of the two grid cells being compared. Put another way rho(T1,p2) can be said to be potential density from cell one evaluated at a reference pressure of cell two. I would be more comfortable saying either stability was estimated by “accounting for compressibility effects using local temperature and pressure”, or “using potential density with local reference pressure”, or something like that. I appreciate that the authors have written the formulation of stability in terms of density and drho/dp (or c or 1/bulk modulus) rather than temperature and alpha (thermal expansion coefficient), and there isn’t really a word for “compressibility effects” in this novel density formulation in the same way there is a thermal expansion coefficient (ie alpha) for a temperature formulation.
Minor/editorial comments
Title: should include words like model and 1D.
Abstract: The abstract includes a lot of introductory and methodology, but no results. This reads more like an aspirational conference abstract, rather than a complete work published in a journal.
Line 29: “The effect deriving from this property is called thermobaric effect”. This sentence is not very helpful in defining what you mean by “thermobaric effect” or “thermobaricity”. This is a good place to clearly define it, especially if you plan to use it to differentiate from “cabbeling” (Lines 39-44) or “forced plume downwelling” (line 50)
Line 40: “Cabbeling originates from thermal bars…” seems misleading and not very helpful. One might also say thermal bars originate from cabbeling. I recommend simply saying “Cabbeling occurs where …”
Line 43: “Although deep water renewal in some lakes is controlled only by thermobaricity, also cabbeling may be involved in the deep mixing…”. Without a definition of thermobaricity it is not clear what you mean by “some lakes”. Which lakes? What are their properties? Give an example of one that is controlled only by thermobaricity and not cabbeling.
Line 47: Define “compensation depth”. Also, “proceed” where?
Line 44: state explicitly the “convenient property of potential density” you are referring to.
Line 49 and 50: These two sentences together are very confusing. You are contrasting deep water mixing from wind forced downwelling under conditions of thermobaricity (ie “forced plume downwelling”) with “thermobaricity”. What is the difference? How are these not both “thermobaric effects”?
Line 50: Who are “them”
Line 67: “temperatures” should be “water temperatures”
Line 75: “my” should be “by”
Line 90: Tell us why it is ok to ignore the effects of local limnic chemistry that “must be included”
Eqn 6: Highlight in the text that rho_1 is evaluated at p_2. This is key to the whole scheme and could easily be missed by the reader. This might also be a good place to say something about rho_(in situ) evaluated at p_2 isn’t really “in situ” anymore, but effectively potential density using a grid specific reference pressure.
Line 118: “May” is misspelt
Line 125-135: More information about the numerical scheme is warranted to help understand the results. What is the order of operations? From the text it looks like the surface boundary condition is updated first, then diffusion occurs, then stability is considered. If this is the order, say so. Are the diffusion and stability calculations done in an upward or downward sweep? Also, this would be a good place to explain why diffusion is needed in the model. What are the implications of neglecting it? How sensitive is the model to time step and layer thickness, which controls the effective diffusion, e.g. why is half the volume exchanged each hour?
Line 170: It is worth pointing out that “summer warming” occurs over 25 hours.
Line 173: “WS2” I think should be “SW2”
Line 188: Who are “They”?
Line 195: I think you mean “SW3 and SW4” here
Line 225: If breakdown of prior strong summer stratification is important, then results will be sensitive to the linear interpolation of summer temperatures from May through October. In particular the summer peak will be missed. Would it make a difference if you interpolated linear to an estimated summer peak surface water temperature?
Line 225: “Strong winter period” is unclear
Line 233: I don’t understand what you mean by “every transition of maximum rho_pot”
Line 248: “similar lakes” Similar how?
Line 266: What is the difference between “diffusion and vertical mixing”?
Line 277: “the depth of the crossing” is unclear
Citation: https://doi.org/10.5194/egusphere-2025-1195-RC3 - AC3: 'Reply on RC3', Joshua Marks, 03 Jul 2025
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