the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 22 Jan 2026
Coupling between sub-mesoscale eddies, internal waves, and turbulence in the deep Mediterranean: A spectral investigation
Abstract. Interaction between energy-abundant (sub-)mesoscale eddies and internal waves can lead to turbulence generation and may prove important for replenishment of nutrients for deep-sea life and circulation. However, observational evidence of such interaction is scarce and precise energy transfer is unknown. In this paper, an extensive spectral study is reported using mooring data from nearly 3000 high-resolution temperature sensors in about half-a-cubic hectometer of seawater above a deep flat Northwestern-Mediterranean seafloor. The number of independent data records partially improves statistics for better determination of spectral slopes, which however do not show a roll-off to the viscous dissipation range of turbulence. The spectra hardly show power-laws ωp having exponent p = -5/3 representing an inertial subrange that evidences shear-induced isotropic turbulence. Instead, they are dominated by p = -7/5 representing a buoyancy subrange, which evidences convection-induced anisotropic turbulence. In contrast with p = -5/3 that indicates a downgradient cascade of energy, p = -7/5 characterizes by an ambiguous cascade direction. At height h < 50 m above seafloor, p = -7/5 is found adjacent to instrumental noise. The p = -7/5 is also found in the sub-mesoscale/internal wave band that is elevated in variance by one order of magnitude. It is reasoned that this sub-inertial range cannot represent isotropic motions, hence p ≠ -5/3 at all heights, and a new deep-sea energy cascade is proposed between mesoscales and turbulence dissipation. Only higher up in more stratified waters an inertial subrange is formed. The transition from internal waves into large-scale turbulence follows p = -2, while a higher-frequency transition from 0 to π phase change reflects overturns of slanted convection or standing-wave breaking leading to isotropic turbulence.
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Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2026-193', Anonymous Referee #1, 25 Feb 2026
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AC1: 'Reply on RC1', Hans van Haren, 27 Feb 2026
>>>My replies are behind>>>
The paper describes unique observations of temperature in a 70x70x128 m volume above the bottom somewhere in the “western Mediterranean.” This is an amazing dataset, but the analysis presented here is deeply flawed. This is made more difficult by the obtuse and verbose writing style.
>>>Thank you for the appreciation of the enormous amount of work that went into obtaining this data set. The exact location in the WMed is given in the paper. The review is not at all helpful, but destructive commenting without any evidence. A pity.
The analysis tools consist of a variety of frequency autospectra and cross-spectra. Dynamical interpretation is done by looking at spectral slopes. For example, traditional three-dimensional isotropic homogeneous Kolmogorov turbulence is diagnosed by a frequency spectrum with a -5/3 slope. Kolmogorov theory, however, predicts a wavenumber (not frequency) spectrum of -5/3. Such a transposition is commonly done in flows with a mean velocity that is much larger than the turbulent velocity fluctuations; a few percent is typical for boundary layers. Under these conditions the ‘Taylor Hypothesis’ assumes that frequency can be converted to wavenumber using the mean velocity, that is, a frequency of 1 Hz converts to 1 cpm for a mean velocity of 1 m/s. For this dataset, the low frequency (‘submesoscale’ or ‘inertial’ in Figure 2, 3) velocities vary from 0.5 to about 5 cm/s on time scales of a day or less so the conversion factor from frequency to wavenumber varies with time. The effect of this broadband advection velocity on the frequency spectra is to smear a given wavenumber over a wide band of frequencies and distort the spectral slopes.
>>>Indeed Kolomgorov posed the theory in terms of distance, which however is commonly transferred to frequency under Taylor hypothesis of ‘frozen turbulence’. Commonly the ‘mean’ or large scale velocity is twice the fluctuating velocity, not a few percent (e.g., https://courses.ems.psu.edu/meteo300/node/737). The data presented in the manuscript have little to do with a frictional boundary layer flow, and turbulence is either induced by geothermal heating from below or by internal wave action from above, mainly. The measurements are truely Eulerian measurements, and the remark about bandsmearing is not relevant.
To say this differently, the array is 70 m across and a typical advection velocity is 2 cm/s. Thus it takes 3500s, an hour, for a water parcel to cross the array, corresponding to a wavenumber of roughly 24 cpd, right in the middle of the frequency band analyzed. Much lower frequencies will be dominated by advection, and might be interpretable as a wavenumber spectrum if the advective velocity were changed over time. Much higher frequencies might be interpreted as an actual temporal change. However, when advection velocity is 0.5 cm/s, the transition frequency would be 3 cpd; when it is 5 cm/s, 60 cpd. If this effect is ignored, a given wavenumber will be smeared by a factor of 20 in frequency, with the amount of smearing, and thus the spectral slopes, varying as a function of frequency. I note that the transition from vertically coherent to vertically incoherent in Figure 3 occurs in this range, so it may be due to this effect.
>>>The reviewer confuses wavenumber with frequency by stating ‘...a wavenumber of roughly 24 cpd...’. The speculative number-naming is irrelevant, because the data are not averaged over the 70 m of the mooring, but the statistics are. Turbulent motions do not have a fixed scale of 70 m, but many different, usually smaller, scales. The 70-m horizontal length has nothing to do with the (in)coherence, which is not only vertically by the way, as the transition is already noticeable in single line (even single sensor data) in Fig. 3, as was also observed in North-Atlantic data from a mooring with 5 lines 4 (and 8) m apart (van Haren et al., 2016).
More fundamentally, scalar spectral slopes alone are a blunt tool to identify dynamics, particularly the subtle differences between -5/3 (1.6) and 7/5 (1.4). Stronger tests on the temperature spectra, such as their scaling with the dissipation rates of both kinetic energy and temperature, are necessary to justify the strong dynamical claims about ‘scaling’ found the the paper’s summary.
There are no statistical confidence levels on any of the spectra or co-spectra, raising the possibility that many of the features discussed are just statistical noise.
>>>Statistical confidence levels are given in all power spectra and in l.362 for coherence. Hence, irrelevant remark about statistical noise. Confusing remark about scaling with (single value?) dissiaption rates.
I conclude that in their current form, the spectral plots are dynamically uninterpretable due to their unknown mix of temporal and spatial measurements. It may be difficult to correct for these effects.
There’s lots more to say about presentation and wording, but there’s little point until these big issues are resolved. However, the same analysis could be described in manuscript that was 1/3 of the current size.
In contrast, the snippet of data shown in the top panel of the movie is fascinating, showing internal waves in the top stratified parts, and a bottom boundary layer with a large overturn near the interface. I’d think that an analysis based on this type of phenomenology would be far more productive than trying to interpret the data in terms of idealized spectral models. With careful and thoughtful analysis, I think that both scalar and kinetic energy dissipation rates could be estimated from such images and related to the structures seen. The current analysis is like being presented with a map of the human genome and trying to interpret it using Aristotle’s philosophy of the origins of life.
>>>I disagree, the reviewer apparently has no feeling for Eulerian measurements. Not a single example comment on the presentation is given, and thus not a single improvement can be made. Alas. The review should be given on the works presented, not on things not presented (such as dissipation rates; yes these can be estimated from the data, and have been done, but these are not presented here). Like in Galilei’s times, new findings arise awkward reactions.
According the data statement, the author is unwilling to release the cleaned temperature data. Clearly it exists- that’s what’s in the movie. The excuse given, that a released data set would be imperfect, is not sufficient; all data is imperfect. Given the richness and uniqueness of this dataset and the wealth of analyses that could come from a community analysis, with holding the data is entirely unsatisfactory and should prevent publication.
>>>The data of NIOZ T-sensors are truely raw data from custom-made instrumentation. If the reviewer has worked with these data he would not have made above comment, which is based on no evidence at all. After months of preparations for setting movie-making up, movie-making on a routine basis still takes about twice real-time, due to all tuning and manual checking during post-processing. So, yes the half-day movie data can be made available but it is impossible to do all data. Even then, all temperature data are relative to a fixed reference, which is not absolutely calibrated and may lead to jumps between short period data.
Citation: https://doi.org/10.5194/egusphere-2026-193-AC1
-
AC1: 'Reply on RC1', Hans van Haren, 27 Feb 2026
-
AC2: 'Comment on egusphere-2026-193', Hans van Haren, 08 Apr 2026
Clarifying comment,
I hereby withdraw the manuscript above from submission for publication in Ocean Science.
Hans van Haren
Citation: https://doi.org/10.5194/egusphere-2026-193-AC2
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2026-193', Anonymous Referee #1, 25 Feb 2026
The paper describes unique observations of temperature in a 70x70x128 m volume above the bottom somewhere in the “western Mediterranean.” This is an amazing dataset, but the analysis presented here is deeply flawed. This is made more difficult by the obtuse and verbose writing style.
The analysis tools consist of a variety of frequency autospectra and cross-spectra. Dynamical interpretation is done by looking at spectral slopes. For example, traditional three-dimensional isotropic homogeneous Kolmogorov turbulence is diagnosed by a frequency spectrum with a -5/3 slope. Kolmogorov theory, however, predicts a wavenumber (not frequency) spectrum of -5/3. Such a transposition is commonly done in flows with a mean velocity that is much larger than the turbulent velocity fluctuations; a few percent is typical for boundary layers. Under these conditions the ‘Taylor Hypothesis’ assumes that frequency can be converted to wavenumber using the mean velocity, that is, a frequency of 1 Hz converts to 1 cpm for a mean velocity of 1 m/s. For this dataset, the low frequency (‘submesoscale’ or ‘inertial’ in Figure 2, 3) velocities vary from 0.5 to about 5 cm/s on time scales of a day or less so the conversion factor from frequency to wavenumber varies with time. The effect of this broadband advection velocity on the frequency spectra is to smear a given wavenumber over a wide band of frequencies and distort the spectral slopes.
To say this differently, the array is 70 m across and a typical advection velocity is 2 cm/s. Thus it takes 3500s, an hour, for a water parcel to cross the array, corresponding to a wavenumber of roughly 24 cpd, right in the middle of the frequency band analyzed. Much lower frequencies will be dominated by advection, and might be interpretable as a wavenumber spectrum if the advective velocity were changed over time. Much higher frequencies might be interpreted as an actual temporal change. However, when advection velocity is 0.5 cm/s, the transition frequency would be 3 cpd; when it is 5 cm/s, 60 cpd. If this effect is ignored, a given wavenumber will be smeared by a factor of 20 in frequency, with the amount of smearing, and thus the spectral slopes, varying as a function of frequency. I note that the transition from vertically coherent to vertically incoherent in Figure 3 occurs in this range, so it may be due to this effect.
More fundamentally, scalar spectral slopes alone are a blunt tool to identify dynamics, particularly the subtle differences between -5/3 (1.6) and 7/5 (1.4). Stronger tests on the temperature spectra, such as their scaling with the dissipation rates of both kinetic energy and temperature, are necessary to justify the strong dynamical claims about ‘scaling’ found the the paper’s summary.
There are no statistical confidence levels on any of the spectra or co-spectra, raising the possibility that many of the features discussed are just statistical noise.
I conclude that in their current form, the spectral plots are dynamically uninterpretable due to their unknown mix of temporal and spatial measurements. It may be difficult to correct for these effects.
There’s lots more to say about presentation and wording, but there’s little point until these big issues are resolved. However, the same analysis could be described in manuscript that was 1/3 of the current size.
In contrast, the snippet of data shown in the top panel of the movie is fascinating, showing internal waves in the top stratified parts, and a bottom boundary layer with a large overturn near the interface. I’d think that an analysis based on this type of phenomenology would be far more productive than trying to interpret the data in terms of idealized spectral models. With careful and thoughtful analysis, I think that both scalar and kinetic energy dissipation rates could be estimated from such images and related to the structures seen. The current analysis is like being presented with a map of the human genome and trying to interpret it using Aristotle’s philosophy of the origins of life.
According the data statement, the author is unwilling to release the cleaned temperature data. Clearly it exists- that’s what’s in the movie. The excuse given, that a released data set would be imperfect, is not sufficient; all data is imperfect. Given the richness and uniqueness of this dataset and the wealth of analyses that could come from a community analysis, with holding the data is entirely unsatisfactory and should prevent publication.
Citation: https://doi.org/10.5194/egusphere-2026-193-RC1 -
AC1: 'Reply on RC1', Hans van Haren, 27 Feb 2026
>>>My replies are behind>>>
The paper describes unique observations of temperature in a 70x70x128 m volume above the bottom somewhere in the “western Mediterranean.” This is an amazing dataset, but the analysis presented here is deeply flawed. This is made more difficult by the obtuse and verbose writing style.
>>>Thank you for the appreciation of the enormous amount of work that went into obtaining this data set. The exact location in the WMed is given in the paper. The review is not at all helpful, but destructive commenting without any evidence. A pity.
The analysis tools consist of a variety of frequency autospectra and cross-spectra. Dynamical interpretation is done by looking at spectral slopes. For example, traditional three-dimensional isotropic homogeneous Kolmogorov turbulence is diagnosed by a frequency spectrum with a -5/3 slope. Kolmogorov theory, however, predicts a wavenumber (not frequency) spectrum of -5/3. Such a transposition is commonly done in flows with a mean velocity that is much larger than the turbulent velocity fluctuations; a few percent is typical for boundary layers. Under these conditions the ‘Taylor Hypothesis’ assumes that frequency can be converted to wavenumber using the mean velocity, that is, a frequency of 1 Hz converts to 1 cpm for a mean velocity of 1 m/s. For this dataset, the low frequency (‘submesoscale’ or ‘inertial’ in Figure 2, 3) velocities vary from 0.5 to about 5 cm/s on time scales of a day or less so the conversion factor from frequency to wavenumber varies with time. The effect of this broadband advection velocity on the frequency spectra is to smear a given wavenumber over a wide band of frequencies and distort the spectral slopes.
>>>Indeed Kolomgorov posed the theory in terms of distance, which however is commonly transferred to frequency under Taylor hypothesis of ‘frozen turbulence’. Commonly the ‘mean’ or large scale velocity is twice the fluctuating velocity, not a few percent (e.g., https://courses.ems.psu.edu/meteo300/node/737). The data presented in the manuscript have little to do with a frictional boundary layer flow, and turbulence is either induced by geothermal heating from below or by internal wave action from above, mainly. The measurements are truely Eulerian measurements, and the remark about bandsmearing is not relevant.
To say this differently, the array is 70 m across and a typical advection velocity is 2 cm/s. Thus it takes 3500s, an hour, for a water parcel to cross the array, corresponding to a wavenumber of roughly 24 cpd, right in the middle of the frequency band analyzed. Much lower frequencies will be dominated by advection, and might be interpretable as a wavenumber spectrum if the advective velocity were changed over time. Much higher frequencies might be interpreted as an actual temporal change. However, when advection velocity is 0.5 cm/s, the transition frequency would be 3 cpd; when it is 5 cm/s, 60 cpd. If this effect is ignored, a given wavenumber will be smeared by a factor of 20 in frequency, with the amount of smearing, and thus the spectral slopes, varying as a function of frequency. I note that the transition from vertically coherent to vertically incoherent in Figure 3 occurs in this range, so it may be due to this effect.
>>>The reviewer confuses wavenumber with frequency by stating ‘...a wavenumber of roughly 24 cpd...’. The speculative number-naming is irrelevant, because the data are not averaged over the 70 m of the mooring, but the statistics are. Turbulent motions do not have a fixed scale of 70 m, but many different, usually smaller, scales. The 70-m horizontal length has nothing to do with the (in)coherence, which is not only vertically by the way, as the transition is already noticeable in single line (even single sensor data) in Fig. 3, as was also observed in North-Atlantic data from a mooring with 5 lines 4 (and 8) m apart (van Haren et al., 2016).
More fundamentally, scalar spectral slopes alone are a blunt tool to identify dynamics, particularly the subtle differences between -5/3 (1.6) and 7/5 (1.4). Stronger tests on the temperature spectra, such as their scaling with the dissipation rates of both kinetic energy and temperature, are necessary to justify the strong dynamical claims about ‘scaling’ found the the paper’s summary.
There are no statistical confidence levels on any of the spectra or co-spectra, raising the possibility that many of the features discussed are just statistical noise.
>>>Statistical confidence levels are given in all power spectra and in l.362 for coherence. Hence, irrelevant remark about statistical noise. Confusing remark about scaling with (single value?) dissiaption rates.
I conclude that in their current form, the spectral plots are dynamically uninterpretable due to their unknown mix of temporal and spatial measurements. It may be difficult to correct for these effects.
There’s lots more to say about presentation and wording, but there’s little point until these big issues are resolved. However, the same analysis could be described in manuscript that was 1/3 of the current size.
In contrast, the snippet of data shown in the top panel of the movie is fascinating, showing internal waves in the top stratified parts, and a bottom boundary layer with a large overturn near the interface. I’d think that an analysis based on this type of phenomenology would be far more productive than trying to interpret the data in terms of idealized spectral models. With careful and thoughtful analysis, I think that both scalar and kinetic energy dissipation rates could be estimated from such images and related to the structures seen. The current analysis is like being presented with a map of the human genome and trying to interpret it using Aristotle’s philosophy of the origins of life.
>>>I disagree, the reviewer apparently has no feeling for Eulerian measurements. Not a single example comment on the presentation is given, and thus not a single improvement can be made. Alas. The review should be given on the works presented, not on things not presented (such as dissipation rates; yes these can be estimated from the data, and have been done, but these are not presented here). Like in Galilei’s times, new findings arise awkward reactions.
According the data statement, the author is unwilling to release the cleaned temperature data. Clearly it exists- that’s what’s in the movie. The excuse given, that a released data set would be imperfect, is not sufficient; all data is imperfect. Given the richness and uniqueness of this dataset and the wealth of analyses that could come from a community analysis, with holding the data is entirely unsatisfactory and should prevent publication.
>>>The data of NIOZ T-sensors are truely raw data from custom-made instrumentation. If the reviewer has worked with these data he would not have made above comment, which is based on no evidence at all. After months of preparations for setting movie-making up, movie-making on a routine basis still takes about twice real-time, due to all tuning and manual checking during post-processing. So, yes the half-day movie data can be made available but it is impossible to do all data. Even then, all temperature data are relative to a fixed reference, which is not absolutely calibrated and may lead to jumps between short period data.
Citation: https://doi.org/10.5194/egusphere-2026-193-AC1
-
AC1: 'Reply on RC1', Hans van Haren, 27 Feb 2026
-
AC2: 'Comment on egusphere-2026-193', Hans van Haren, 08 Apr 2026
Clarifying comment,
I hereby withdraw the manuscript above from submission for publication in Ocean Science.
Hans van Haren
Citation: https://doi.org/10.5194/egusphere-2026-193-AC2
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The paper describes unique observations of temperature in a 70x70x128 m volume above the bottom somewhere in the “western Mediterranean.” This is an amazing dataset, but the analysis presented here is deeply flawed. This is made more difficult by the obtuse and verbose writing style.
The analysis tools consist of a variety of frequency autospectra and cross-spectra. Dynamical interpretation is done by looking at spectral slopes. For example, traditional three-dimensional isotropic homogeneous Kolmogorov turbulence is diagnosed by a frequency spectrum with a -5/3 slope. Kolmogorov theory, however, predicts a wavenumber (not frequency) spectrum of -5/3. Such a transposition is commonly done in flows with a mean velocity that is much larger than the turbulent velocity fluctuations; a few percent is typical for boundary layers. Under these conditions the ‘Taylor Hypothesis’ assumes that frequency can be converted to wavenumber using the mean velocity, that is, a frequency of 1 Hz converts to 1 cpm for a mean velocity of 1 m/s. For this dataset, the low frequency (‘submesoscale’ or ‘inertial’ in Figure 2, 3) velocities vary from 0.5 to about 5 cm/s on time scales of a day or less so the conversion factor from frequency to wavenumber varies with time. The effect of this broadband advection velocity on the frequency spectra is to smear a given wavenumber over a wide band of frequencies and distort the spectral slopes.
To say this differently, the array is 70 m across and a typical advection velocity is 2 cm/s. Thus it takes 3500s, an hour, for a water parcel to cross the array, corresponding to a wavenumber of roughly 24 cpd, right in the middle of the frequency band analyzed. Much lower frequencies will be dominated by advection, and might be interpretable as a wavenumber spectrum if the advective velocity were changed over time. Much higher frequencies might be interpreted as an actual temporal change. However, when advection velocity is 0.5 cm/s, the transition frequency would be 3 cpd; when it is 5 cm/s, 60 cpd. If this effect is ignored, a given wavenumber will be smeared by a factor of 20 in frequency, with the amount of smearing, and thus the spectral slopes, varying as a function of frequency. I note that the transition from vertically coherent to vertically incoherent in Figure 3 occurs in this range, so it may be due to this effect.
More fundamentally, scalar spectral slopes alone are a blunt tool to identify dynamics, particularly the subtle differences between -5/3 (1.6) and 7/5 (1.4). Stronger tests on the temperature spectra, such as their scaling with the dissipation rates of both kinetic energy and temperature, are necessary to justify the strong dynamical claims about ‘scaling’ found the the paper’s summary.
There are no statistical confidence levels on any of the spectra or co-spectra, raising the possibility that many of the features discussed are just statistical noise.
I conclude that in their current form, the spectral plots are dynamically uninterpretable due to their unknown mix of temporal and spatial measurements. It may be difficult to correct for these effects.
There’s lots more to say about presentation and wording, but there’s little point until these big issues are resolved. However, the same analysis could be described in manuscript that was 1/3 of the current size.
In contrast, the snippet of data shown in the top panel of the movie is fascinating, showing internal waves in the top stratified parts, and a bottom boundary layer with a large overturn near the interface. I’d think that an analysis based on this type of phenomenology would be far more productive than trying to interpret the data in terms of idealized spectral models. With careful and thoughtful analysis, I think that both scalar and kinetic energy dissipation rates could be estimated from such images and related to the structures seen. The current analysis is like being presented with a map of the human genome and trying to interpret it using Aristotle’s philosophy of the origins of life.
According the data statement, the author is unwilling to release the cleaned temperature data. Clearly it exists- that’s what’s in the movie. The excuse given, that a released data set would be imperfect, is not sufficient; all data is imperfect. Given the richness and uniqueness of this dataset and the wealth of analyses that could come from a community analysis, with holding the data is entirely unsatisfactory and should prevent publication.