the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 11 Dec 2025
Estuarine mixing
Abstract. This review paper presents, explains and discusses major aspects of estuarine mixing which is defined as the destruction of salinity variance. Due to the large amounts of brackish water in estuaries produced by mixing of fresh river discharge and salty ocean water, mixing is one major characteristic of what is an estuary. In this review, mixing is quantified locally as well as on estuary-wide scales. Diagnostics of integrated mixing are given for estuarine volumes bounded by transects as well as isohalines (surfaces of constant salinity) moving with the flow. It is shown how entrainment across a moving isohaline surface depends on gradients of turbulent salt flux and mixing per salinity class. Various relations are derived that link estuarine salt mixing to other estuarine properties such as the freshwater discharge and the bulk estuarine circulation. For estuaries bounded towards the ocean by a fixed transect, the Knudsen mixing law is explained, where estuarine mixing is the product of the Knudsen salinities of inflowing and outflowing water masses and the river discharge. When the estuarine volume is bounded by a moving isohaline surface of salinity S, mixing inside the estuary is simply the product of S2 and the river discharge. Major processes that drive estuarine mixing are presented on various time scales (tidal, fortnightly, weather and discharge time scales) and spatial scales (channel-shoal interaction, mixing fronts). As underlying methods for the quantification of mixing, observational concepts, as well as numerical modelling methods such as consistent turbulence closure modelling and numerical mixing analyses are presented. As an outlook, some future perspectives are sketched. Many of the concepts presented in this review are illustrated using simulation results from a numerical model setup of the Elbe River estuary.
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2025-6062', Anonymous Referee #1, 03 Mar 2026
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RC2: 'Comment on egusphere-2025-6062', Robert Hetland, 20 Mar 2026
This review of new methods to quantify and understand mixing in estuaries is timely and important, and will be very useful for the community. I have a number of comments that are primarily oriented around changing emphasis. I think the topics covered and structure of the paper are reasonable. I also think the choice to focus on the Elbe estuary is fine as it is representative of an important class of estuaries. Thus, I recommend publication after (somewhat) major revisions to the text, but do not recommend restructuring or any additional analysis. It should also be noted that my recommendations are mainly just opinions that the authors may implement as they choose. As it stands the text is accurate and more or less complete.
Major comments:
The theory discussed in this paper is kinematic description based on the distribution of salt in the estuary, but of course the distribution of salt depends on estuarine dynamics. I don’t think that the paper needs to change focus to include a long section on dynamics, there are other reviews that cover this, but I do think that this relationship should be discussed early in the paper. Right now, for example, the Simpson number is first introduced about halfway through the manuscript. I would suggest to review the big concepts in estuarine dynamics in section 3, and relate to them throughout where it makes sense.
I think that there are some assumptions built into the analysis of the Elbe and similar estuaries that make this review useful for a particular set of estuaries but not all. An example of this bias is on line 624, where the authors state “fluxuations at the open boundary are typically small.” This is not true for ‘short’ tidal estuaries, as described by Chen et al, (2012, JPO). Specifically, while the analysis presented in the paper can be performed on Merrimack, other metrics are probably better suited for characterizing that small, tidal estuary. For this reason, I suggest being more explicit about this bias in the text, and changing the title to be more specific than the overly broad ‘Estuarine Mixing’. Perhaps ‘Estuarine Mixing in Large Estuaries’ or similar.
I don’t like the analogy to mixing in tea, which I find too pedantic, and suggest replacing that with a more mathematically based discussion oriented around Figure 1. The rest of the paper is quite mathematically heavy, so no need to present mixing from an undergraduate perspective. More generally, I think that the discussion of stirring vs mixing needs to be more explicitly related to estuarine circulation. In estuaries, gradients are generated through strain, but estuaries typically do not contain an unresolved submesoscale eddy field which would ‘stir’ the salinity gradients. Orienting the discussion around the actual unresolved circulation features that cause stirring in estuaries will be important to differentiate estuaries from other oceanic flows. Another example of this is in the discussion of numerical mixing. Schichting et al, (2023, 2024, JAMES) find much higher fractions of numerical to physical mixing, I suspect because of the character of the flow. The submesoscale field creates strong convergence at the fronts, and this drives numerical mixing. In estuaries, this kind of circulation does not exist, and to the extent there are convergent flows, e.g., at shoals, these are determined by topographic interactions. This is likely also related to the success of numerically diffusive unstructured grid models in estuaries – the large numerical mixing is swamped by the even larger physical mixing. Finally, I think the discussion of mixing in section 2.2 could be shortened, as this is not necessarily particular to estuaries.
I would like to see the discussion of TEF expanded. I think this section could use more examples or analogies to make the ideas more clear. TEF is central to modern thinking about estuarine exchange, and I think that a review paper on this topic should lay a solid foundation for people new to the field.
The discussion of the universal law of mixing would benefit from some comparisons to my work on river plumes, based on earlier work by Garvine (1999). Garine noted that mixing and plume area were inversely correlated, which I later quantified as fresh water mixing through plume isohalines; the universal law is seen in the steady state version of equation 4 of Hetland (2005). I think examples like this will help to connect estuarine mixing to related shelf processes. One place this is already hinted at in the manuscript is the paragraph starting on line 604, where high flows push the mixing out into the Changjiang plume. This might just be seen in the context of the universal mixing law, where the estuary is fresher, and so mixes less (i.e., smaller S at the mouth in Q_R S^2, but the total mixing in the estuary/plume system still needs to approach Q_R S_{ocean}^2).
I found the discussion in the paragraph starting on line 614 interesting. It seems that the result of Broatch and MacCready can be explained by the universal law of mixing, since for the (large) Puget Sound, S at the mouth will not change much, but Q_R can vary by an order of magnitude. Generally, even though S is squared in the mixing term, increasing its importance in mixing, Q_R often has more variability in large estuaries.
Minor suggestions:
In section 3.3 I think I would prefer to see simpler notation. Specifically, I think the {dia,z} subscripts are cumbersome and should be simplified. Perhaps only u_dia needs to be differentiated from the general Eulerian u; it seems f and j are always diahaline variables.
Line 511: Delete second ‘many others’.
Line 534: Perhaps ‘vertical mixing’ is not intended here, but rather vertical turbulent diffusivity? If vertical mixing was intended, rewrite the sentence to be more clear.
Figure 11: The bottom panel is hard to read. Suggest to separate into two panels.
Line 567: change ‘not suppressed’ to ‘not completely suppressed’?
Signed, Rob Hetland, PNNL
Citation: https://doi.org/10.5194/egusphere-2025-6062-RC2
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This manuscript is a detailed discussion of the use of the Salinity variance method to quantify mixing in estuarine models. This technique was described by Burchard and Rennau (2008) primarily as a method to quantify numerical mixing in models and was exploited by Li and Geyer (2018) to quantify the spatial and temporal structure of mixing (defined as the explicit destruction of vertical salinity variance and assess the role of straining and advection in the process. The paper does an excellent job describing in somewhat excruciating detail the development and use of this method, it’s relationship to the estuarine exchange flow and follows with a discussion of various mixing process in estuarine systems. Overall , the paper will be of great interest to the estuarine community, and the detailed description of the development and use of this method is commendable. However, I feel like the paper is way too long and this will limit its impact. I must admit that I am somewhat torn by this in that the length of the paper allows for detail that appears on one publication, and this would be useful for the community but if the authors could reduce the length of the paper by 20% or so I think the impact of this review would be increased.
I really don’t have any specific criticisms regarding the material covered and as far as I can tell the authors have done an excellent job providing an overview of this topic. If the editor agrees with my assessment that shortening the paper is warranted- I leave it to the authors to decide what to omit in the revision.