20 Apr 2021
20 Apr 2021
Flow separation, dipole formation and water exchange through tidal strait
 ^{1}Akvaplanniva AS, 7462 Trondheim, Norway
 ^{2}The University of Oslo, Department of Geosciences, 0315 Oslo, Norway
 ^{3}Nord University, 8026 Bodø, Norway
 ^{4}Akvaplanniva AS, 9296 Tromsø, Norway
 ^{1}Akvaplanniva AS, 7462 Trondheim, Norway
 ^{2}The University of Oslo, Department of Geosciences, 0315 Oslo, Norway
 ^{3}Nord University, 8026 Bodø, Norway
 ^{4}Akvaplanniva AS, 9296 Tromsø, Norway
Abstract. We investigate the formation and evolution of dipole vortices and their contribution to water exchange through idealized tidal straits. Selfpropagating dipoles are important for transporting and exchanging water properties through straits and inlets in coastal regions. In order to obtain a robust dataset to evaluate flow separation, dipole formation and evolution and the effect on water exchange, we conduct 164 numerical simulations, varying the width and length of the straits as well as the tidal forcing. We show that dipoles are formed and start propagating at the time of flow separation, and their vorticity originates in the velocity front formed by the separation. We find that the dipole propagation velocity is proportional to the tidal velocity amplitude, and twice as large as the dipole velocity derived for a dipole consisting of two point vortices. We analyse the processes creating a net water exchange through the straits and derive a kinematic model dependent on dimensionless parameters representing strait length, dipole travel distance and dipole size. The net tracer transport resulting from the kinematic model agrees closely with the numerical simulations and provide understanding of the processes controlling net water exchange.
Ole Anders Nøst and Eli Børve
Status: final response (author comments only)

RC1: 'Comment on os202130', Anonymous Referee #1, 11 May 2021
The manuscript investigates formation and propagation of dipoles at tidal straits via numerical simulation. The authors derived equations to calculate dipole propagation velocity and to estimate net exchange through the strait. These equations show good agreements with the simulation results. This study advances the knowledge on transport mechanisms at tidal straits. The topic matches the scope of Ocean Science. It could be used as a basis for further studies on more realistic and more complex strait geometry and flow conditions. Thus, this manuscript is recommended to be accepted after some minor revisions listed below.
 Line 234: Here r is defined as the distance from the vortex center, a is the vortex radius. However, in Fig. 17 and Eq. 31, if I understand it right, r becomes the vortex radius. I recommend to make the variable names consistent.
 Figure 14: The tracked velocity is mostly lower than the theoretical values in (a), but higher than the theoretical values in (b). What might be the reasons?
 Line 374: The authors should provide, or at least discuss the valid range of Eq. 31. By ignoring the high order terms, Eq. 31 could be inaccurate when r is large (how to define “large”?). On the other hand, r=0 is not a realistic situation either. Moreover, r/Ld seems to be unknown before numerical simulations are completed. So the question is, given so many limitations, how useful is Eq. 31 in estimating the net transport? Why not simply calculate net transport through numerical modeling? I think the authors should provide more explanation on the significance and the applicability of the kinematic model.
 Figure 19: This figure shows that Sd and St are similar, so why do we need Sd? The authors state that St is “difficult to understand”, but this is a very subjective statement. I encourage the authors to further explain the difference between Sd and St, as well as the advantage of using Sd (rather than St).
Overall, the language used in this manuscript is a little bit verbose. Unfortunately, as a nonnative speaker, I cannot provide detailed recommendations on grammar. I suggest the authors try to make the language more succinct. A few typos I caught are:
 Line 386: Long straits produce less dipoles than “short” straits?
 Line 414: Our results “suggest” that …?
 Line 460: Equation 15 and 18 “do” not have…?
The figures are generally informative. Some suggestions are:
 Figure 2: The figure labels “a” and “b” look similar to the variable names. The authors might consider separating them. For example, put labels outside the box, add parenthesis or change fonts.
 Figure 16, 18 and 19: Some markers are with grey halos and some are not. The difference between these two types of markers is not explained in the caption or in the legend.

AC2: 'Reply on RC1', Ole Anders Nøst, 04 Jun 2021
Thanks for the recommendation to accept the paper with minor revisions.
We appreciate the comments on variable names, language, figure labels and captions. These will be taken very seriously when writing a revised manuscript, but in this reply we focus on the more scientific questions.
Figure 14: Why is the tracked velocities lower than the theoretical values in a, but higher than theoretical values in b?
It is hard to know the exact answer to this question, and it could be a coincidence since Figure 14 only shows two examples. However, the explanation may be the mesh resolution. Figure 14a shows dipole propagation velocity for a narrow strait where the dipole travels a large distance from the strait (it is the same simulation as presented in Figure 4). Figure 14b shows dipole propagation velocities for a wider strait where the dipole travels a much shorter distance (see Figure 5). Figure 1 illustrates how the mesh is defined for narrow and wide straits. The area of fine mesh resolution depends on the strait width and a narrow strait has a smaller area of fine mesh resolution than a wide strait. So, in the narrow strait shown in Figure 14a, the dipole moves far away from the region of fine resolution, while in Figure 14b, the dipole stays in the fine resolution much longer. Generally, the dipole in 14a travels in a coarser grid than the dipole in 14b. Although we cannot prove that this is the reason for tracked velocities being lower than theoretical values in 14a and higher in 14b, it is a likely explanation.
This question is interesting also when considering sensitivity of the results to mesh resolution. We have run simulations in 82 different geometries, and for the reason explained above, the dipoles move in different mesh resolution dependent on their travel speed and the strait width. Despite this, the propagation velocity agrees well with the theoretical values (Figure 15) and the tracer transport agrees with the kinematic model for all straits (Figure 18). In addition to how we have argued in our reply to reviewer #2, this is another indication that our results are not affected by mesh resolution.Line 374: Equation 31 is important because it provides understanding to the processes. The agreement with the simulation results strongly indicates that the kinematic model contains the main processes at play. So, the main point of equation 31 is to provide understanding and not estimate transport. However, it can also be used to make rough estimates of transport if no simulations are present. Estimates on transport can be made from the strait velocity and geometry, estimating dipole propagation velocity from Equation 18. This requires an estimate on the aspect ratio of the vortices, which largely depends on the strait width.
The main simplification leading to Equation 31 is that we estimate the fraction of the tracer inside the sink radius by assuming the dipole and jet has the form of a rectangle. This is of course not true, but it gives a simple expression for the effective transport. The improvement in the agreement with simulations resulting from adjusting r/Ld tells us that it is important to include in the model that only a fraction of the dipole escapes the return flow.
When it comes to ignoring the higher order terms, we see when going through this again, that this is not necessary to do. In the manuscript we use a constant value of r/Ld=0.4, for all simulations, which corresponds to (1r/Ld) = 0.6. If not ignoring the higher order terms the expression for qe (Equation 31) becomes
qe = (1SL)*(1Sd / (1+r/Ld)).
Setting 1/(1+r/Ld)=0.6, will give the exact same fit to the simulation data as in the manuscript. So, ignoring the higher order terms actually has no effect, and including all terms also gives a rather simple expression for qe. In a revised manuscript we will use the full expression for qe without ignoring the higher order terms. Thanks for setting focus on this and apologize for making this a little too complicated in the manuscript. Figure 19: The point of this work is not to come up with a parameter that works better than the Strouhal number. The point of showing that Sd is similar to St is that the understanding provided by this work can also be used to understand St.
St is more like a scaling, using values of strait velocity, strait width and tidal period, while Sd comes from a kinematic model of tracer transport. However, it is hard to know which value of St which will give net transport. Figure 19 shows Sd plotted against St/Stc, where Stc is the threshold value of St, such that St<Stc is gives nonzero tracer transport. In making the plot we have picked Stc from the simulated tracer transports. A threshold value of 0.13 is given by several authors, but this is likely to change with geometry. Then Sd is a simpler parameter, because it depends on sink radius and dipole travel distance. If these can be estimated, we know that Sd<1 may give transport if also SL<1.

AC2: 'Reply on RC1', Ole Anders Nøst, 04 Jun 2021

RC2: 'Comment on os202130', Anonymous Referee #2, 13 May 2021
General overview
This is an interesting contribution that is within the remit of the Ocean Science journal. The authors discuss the dynamics of vortex dipole formation due to tidal flow. In general, it is an ambitious contribution that touches on a range of physics expected through the flow separation effects triggered through tidal straits
Specific comments
 On the definition of the case study  An idealised setup is explored and yet some practical elements seem to unnecessarily be included. For example, why would Coriolis effects be included in this case? Elements such as symmetry are also not exploited which could be interesting.
 "The water column is divided into two layers in the vertical"  does this provide sufficient vertical resolution? The depth the authors use could be used to argue that frictional effects are not so significant , (which is still not fully accurate as indicated later), but surely the flow separation (particularly in constricted straits) will lead to upwelling that will affect the vortex evolution. How is vorticity calculated? is it averaged across the 2 layers or is the top layer considered in the analysis
 Mesh discretisation  The analysis is based on spatially defined metrics such as vorticity and circulation. These are extremely sensitive to the resolution of the model. The fine resolution close to the strait bounds can lead to higher vorticity peaks that will influence the conclusions of the study  while coarse resolution will dissipate the vortices. It is challenging to conduct this analysis using unstructured models, and some earlier studies included a mesh sensitivity, and a normalisation based on the element length to check whether the key dynamics are accurately preserved independently of the mesh resolution (see e.g. Vouriot et al, Env. Fl. Mech., 19, 328348 (2019)). It is understood that the same issue was indicated in an earlier submission of this manuscript so this must be addressed. For example, Eq. 12 seems heavily affected by the mesh resolution and must proven that it is mesh independent.
 Increasing length of the strait will increase the effects of friction, and thus influence u and Q through the strait. This seems to be the reason for the monopolo/dipole trends of Fig.3. However, this is the reason why I suspect that Section 8.1 is invalid as friction becomes more and more important as the strait length increases.
 How was Γ determined for each vortex? How was Ts estimated and is uθ the propagation velocity of the vortex?

AC1: 'Reply on RC2', Ole Anders Nøst, 01 Jun 2021
Many thanks for good and challenging comments from the reviewer. We have had special focus on investigating the effect of mesh resolution and done some new simulations with finer resolution. This has led to interesting results and deeper insight into the processes. We have answered the questions and comments of the reviewer below. Figures is presented in a supplementary pdf.
Specific comments:
The Coriolis effect is vital for our setup to work, as the forcing comes from a Kelvin wave propagating with the coast to the right. A Kelvin wave cannot exist without the Coriolis effect. We have chosen this setup for two reasons: 1) It is an idealized model of the tides through straits in the Lofoten peninsula in Norway (I see that we have not mentioned this in the manuscript, but this will be corrected in a revision). 2) We wanted a setup where the pressure difference across the strait is not dependent on the flow through the strait. In our setup the pressure difference is mostly caused by the phase difference between the northern and southern part of the peninsula. This is setup by the Kelvin wave propagating around the peninsula, and less dependent on the flow through the strait. This is in contrast to, for instance, the entrance into a fjord, where the water flow through the entrance determines the water level in the fjord. The idea is that for theoretical considerations we can assume that the pressure difference across the strait is independent on strait flow. This is important when discussing friction and strait length (see below), where we can assume that the pressure difference over the strait (deta) is independent of the tidal velocity in the strait.
Vertical layering:
We regard this as a 2D barotropic study. The actual reason for having two vertical levels in the model is that our tracer model did not work within a 2D model and the simplest solution to this was to include two levels. However, all our analysis is done using vertically averaged velocities.Mesh discretization:
Vorticity is extremely sensitive to mesh resolution, and it is clear that the processes of separation and vortex formation is affected by mesh resolution. In our case the spatial scale of the initial vortices is close to the smallest scale the model can resolve. Will this affect our conclusions of tracer transport, dipole propagation velocity and separation time?Vorticity is created in the velocity front formed by flow separation. The simulated vorticity in the velocity front will probably depend strongly on model resolution. However, the total production of vorticity with time is probably less dependent on resolution. This can be shown by integration the vorticity over an area containing a segment of the velocity front. During a time t, a velocity front with length U * t is formed, where U is the tidal velocity in the strait. An area integral of the vorticity in the front segment equals U^2 * t. This is obtained using Stokes theorem and assuming that the velocity on one side of the front is U while it is zero on the other side of the front. This result suggests that if the model resolution is sufficient to correctly represent the strait velocity and a flow separation, the vorticity production is likely to be correct. Since the vortices are formed from segments of the front, the total vorticity in the vortices and the circulation are likely to be similar between models of different resolution. Based on this analysis, we will argue that local vorticity is extremely sensitive to mesh resolution, but the circulation is less sensitive to resolution as long as the model produces a flow separation, and the strait velocity is correct.
To study the effect of resolution, we have repeated a number of the simulations using finer mesh resolution. In the new simulations, the resolution at the coast is set to 10 m inside the strait. The simulations presented in the manuscript has 50 m resolution at the coastline. We have selected 7 strait configurations which are simulated with higher resolution. These are the three simulations shown in figures 46 in the manuscript plus four others of different strait width and length.
The figures that we refer to in this text is presented in the supplementary pdf. In the new fine resolution simulations, we see that the size of the initial vortices is smaller than in the coarser simulations presented in the manuscript. Figure 1 shows the core radius at time t=Ts, for the old (50 m resolution) and new (10 m resolution) simulations. A similar plot of the time Ts is presented in Figure 2. Vorticity is very sensitive to resolution and as a consequence of this, the method of finding the separation time from the maximum vorticity within the strait exist does not work with the new simulations. From Figure 3 we see that the separation time does not correspond to the maximum vorticity for all the three straits presented in the figure. Strait velocity is slightly lower in the new simulations but corresponds well to the old simulation results (Figure 4). The dipole velocities are not significantly affected by the resolution of the simulations (Figure 5), and as a result the effective transports are also very similar in the old and new simulations. Thus, the main conclusions of the paper are not affected by model resolution.
Friction and strait length:
Section 8.1 is not invalid as it stands now, because it is not only the effect of friction that will increase with strait length. In our case it is the linear acceleration term that is the main cause of the length effect seen in Figure 3. However, we agree that, for straits with shallower depth, friction may play an important role with increasing strait length.The relation between the different terms in the momentum balance can be illustrated by scaling:
2U/T + U^2/L = g deta/L + Cd U^2/H
The first term is the linear acceleration, where the timescale used equals T/2, and T is the M2 tidal period. The second term is the nonlinear acceleration, the third term is the pressure force (deta is the sea surface height difference over the strait) and the last term is bottom friction. U is a velocity scale, H is depth and L is strait length. From this it is clear the pressure force and nonlinear acceleration terms decreases with strait length, while the linear acceleration and friction are both independent of length. If it is friction or linear acceleration that determines the length effect seen in Figure 3, depends on the relation between these two terms. In our case, where H=100m, Cd~0.001 and T~45000, the acceleration is about 4 times larger than the friction term for U=1m/s. For L<10 km, the nonlinear acceleration dominates the linear and frictional terms.
When nonlinear acceleration dominates, this will balance the pressure term which gives a velocity scale, U~sqrt(g deta), which is independent of length (here we assume that deta is independent of strait dynamics, see discussion on importance of Coriolis effect). However, if it is the linear acceleration or friction that balances the pressure force, the result is a velocity scale that decreases with length. The relation between linear acceleration and friction does not depend on length. In our case, where H=100 m, it is mainly the linear acceleration that leads to the length effect seen in Figure 3. For shallower depth, it is likely that friction will cause a similar effect and we will include this in the analysis in a revised manuscript.
Determination Γ
Γ was determined by the LambOseen equations (Equation 11 in the manuscript) to the simulated vortices. Through the fitting, we find the core radius a, and the maximum vorticity. When these two variables are known Γ can be determined from Equation 11b.
uθ (in Equation 11a) is the azimuthal velocity of the vortex. In the manuscript we determine Ts by finding the maximum (or minimum) vorticity within the strait exit. We see by visual inspection that this corresponds well with the time of separation. However, in the new simulation with finer resolution presented above, the maximum (minimum) vorticity method does not work, and Ts is then determined by visual inspection of the simulation results.

AC1: 'Reply on RC2', Ole Anders Nøst, 01 Jun 2021

RC3: 'Comment on os202130', Anonymous Referee #3, 24 May 2021
The manuscript describes the generation, dynamics, and transport properties of a pair vortices through tidal straits, which is a key mechanisms in water exchanges. A set of 164 simulations with varying strait width, length, and tidal amplitude is investigated. Along with comments already posted by the two referees, a few more suggestions are provided below.
The paper is within the scope of Ocean Science.
The manuscript is well written and the physical mechanisms at play are extensively described. However, it somehow lacks of conciseness, and the reader can get lost in long descriptions that could be replaced, or complemented by illustrative sketches (or simulation outputs) for the sake of clarity. An example is the description of the origin of the asymmetry leading to net exchange (l1830).
Introduction:
The authors quote a few experimental studies. Recent work by Albagnac et al (i.e. A threedimensional experimental investigation of the
structure of the spanwise vortex generated by a shallow vortex dipole. (2014) Environmental Fluid Mechanics, vol. 14 (n° 5). pp. 957970. ISSN 15677419, and later articles) provide quantitative description of the 3D dynamics of vortex dipoles, including in stratified environment.Sec 3: an illustration of the effect of tidal amplitude would have been interesting.
Fig 13: S1 and S2 are not described in in the caption.
l412: phenomena  > phenomenon.
Conclusion: Multitidal forcings are often present at straits, an interesting perspective would be to investigate their effect on the overall dynamics and the impact on water exchanges.
As suggested by the authors, comparisons with realistic configurations would be of interest to evaluate the parameters controlling the dipole dynamics. A relevant configuration could be the one of the Gibraltar strait, where the variety of finescale structures and the water exchange through the strait are actively studied combining LES simulations and sea campaigns (Numerical modelling of hydraulic control, solitary waves and primary instabilities in the Strait of Gibraltar, Hilt et al. Ocean Modelling, 2020).

AC3: 'Reply on RC3', Ole Anders Nøst, 01 Jul 2021
The reviewer has no direct criticism of the scientific content of the manuscript but has some comments/suggestions on the written presentation, topics that could be further illustrated, references to include and cases of realistic configurations that could be explored. We will answer the comments one by one:
Lack of conciseness: When writing the revised manuscript, we will keep this in mind and do our best to be as concise as possible. When it comes to describing the asymmetry leading to exchange (line 1830), we will consider including an illustrative sketch to make this clearer.
Quoting experimental studies: Thanks for making us aware of the paper by Albagnac et al. We will incorporate this into the overview of the field in the introduction.
Illustration of tidal amplitude: Figure 3 also shows the effect of tidal amplitude. Basically, lower amplitude gives lower velocities resulting in less dipoles being formed. The three examples shown in Figure 46 is chosen because they show three typical patterns summarizing what we see in all straits and for both tidal amplitudes used. In our opinion Figure 3 is a good illustration of the effect of tidal amplitude, but we will make sure that this is also clearly stated in the text.
Fig 13 and line 412: This will be corrected in a revised manuscript.
Multitidal forcing and comparison with realistic configurations: Yes, multitidal forcing will impact the water exchange through tidal straits. We do not include this here, but we work on two other manuscripts on tidal straits in a more realistic setting. One of these manuscripts is also submitted to Ocean Science (https://doi.org/10.5194/os202141) and considers tidal transports in the Lofoten region in Northern Norway. Here we use a barotropic model of the tides in the regions and study tidal pumping through straits and rectified transports around islands. A third paper presenting a 3D study of the same region is underway. Here we examine the role of tidal transports in a model that also includes atmospheric forcing and river runoff. In this work, we clearly see that multitidal forcing is important.

AC3: 'Reply on RC3', Ole Anders Nøst, 01 Jul 2021
Ole Anders Nøst and Eli Børve
Ole Anders Nøst and Eli Børve
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