Mixing of a tracer s (for which we use salinity here as an example) occurs at the micro-scale when tracer gradients are reduced by molecular diffusion, following the Fickian law, 2∂sˇ∂t+∂∂xjuˇjsˇ-κ∂sˇ∂xj=0, where the Einstein summation convention ajbj=∑j=13ajbj=a1b1+a2b2+a3b3 has been applied. In Eq. (), sˇ is the instantaneous tracer concentration sˇ=[s]+s̃ with the Reynolds-averaged tracer concentration [s] and the fluctuating component s̃, κ is the molecular diffusivity of salinity, and uˇ=uˇ1 and vˇ=uˇ2 are the horizontal and wˇ=uˇ3 is the vertical velocity component. The terms in the brackets on the left hand side of Eq. () are the advective and the molecular diffusive fluxes, the divergence of which determines the change of the salinity distribution. This transport equation determines the salinity distribution on all scales ranging from the sub-millimetre scales of molecular diffusion to the global scales of meridional overturning circulation, including scales of estuarine mixing.

In turbulence theory, the Reynolds average (also called ensemble average) is defined as the average of an infinite number of macroscopically identical but microscopically different flow realisations, where the turbulent random fluctuations are averaged out . Consequently, [s̃]=0. In estuarine physics, and similarly in most fields of larger-scale oceanography, the Reynolds-averaged rather than the instantaneous properties of the flow are considered. Field observations of tracer concentrations (e.g., from Conductivity-Temperature-Depth (CTD) probes) as well as numerical model results are supposed to represent Reynolds-averaged quantities. A continuity equation (incompressibility condition) is used in most ocean models: 3∂uˇi∂xi=0,∂[ui]∂xi=0,∂ũi∂xi=0, applying to instantaneous and thus to Reynolds-averaged and fluctuating velocity fields. Using Eq. (), a dynamic equation for the Reynolds-averaged salinity can be derived: 4∂[s]∂t+∂∂xj[uj][s]+[ũjs̃]-κ∂[s]∂xj=0, with the advective tracer flux [uj][s], the turbulent tracer flux [ũjs̃] and the diffusive tracer flux -κ∂[s]/∂xj. Based on Eq. (), it is also possible to derive an equation for the micro-structure tracer variance [s̃2]: 5∂s̃2∂t+∂∂xjujs̃2+ũjs̃2-κ∂[s̃2]∂xj=-2[ũjs̃]∂[s]∂xj︸Ps-2κ∂s̃∂xj2︸χs see, e.g., Eq. (5) by. In Eq. (), Ps quantifies the production of micro-structure variance due to turbulent stirring (with the Reynolds-averaged tracer gradient vector ∂[s]/∂xj) and χs represents destruction of microstructure variance due to molecular mixing.

Multiplying Eq. () by 2[s] gives a transport equation for the square of the Reynolds-averaged tracer: 6∂[s]2∂t+∂∂xj[uj][s]2+2[ũjs̃][s]-κ∂[s]2∂xj=2[ũjs̃]∂[s]∂xj︸-Ps-2κ∂[s]∂xj2, where on the right-hand side the stirring term Ps appears as a sink term besides a destruction term due to molecular diffusivity. In contrast to Eq. (), where the destruction of micro-structure variance occurs due to molecular diffusivity acting on micro-structure gradients ∂s̃/∂xj, in Eq. () the molecular diffusivity acts on the much smaller Reynolds-averaged gradient ∂[s]/∂xj such that this term is generally negligible. This means that variance is first transferred from the Reynolds-averaged regime of Eq. () to the turbulent regime Eq. (), where it is then dissipated. In turbulence closure modelling typically Ps=χs see, e.g., Eq. (31) by is applied such that stirring equals mixing, following a local equilibrium assumption. More details on turbulence closure modelling suitable for estuaries are given in Sect. .

In short: micro-structure tracer variance is produced by stirring Ps (increase of local micro-structure gradients due to turbulent eddies) and dissipated by mixing χs, while the divergence term on the left-hand side just spatially redistributes the micro-structure tracer variance. Note that the stirring term is twice the product of the turbulent flux times the Reynolds-averaged tracer gradient. The stirring term Ps is typically positive, since the Reynolds-averaged salinity gradient and the turbulent salt flux have opposite signs due to the generally down-gradient property of turbulent fluxes classical exceptions occur in convective boundary layers, see e.g..

This can be explained by a laboratory experiment with salt mixing. A simple idealised model of this is given in Appendix and results are shown in Fig. . After having carefully pumped saltwater of 30 g kg⁻¹ underneath freshwater (with some continuous mixing), the local salinity variance is mostly low: sufficiently small control volumes would contain water with a small salinity range (Fig. a). Introduction of turbulence by means of a spoon will lead to stirring (increase of Ps), such that local control volumes (marked as the area between the two black bars in Fig. ) will contain streaks of saltwater at various ranges, with sharp gradients between them, such that the local variance s̃2 is increased. This is demonstrated as initial conditions for moderate stirring (Fig. b) and strong stirring (Fig. c). Now, mixing χs will be moderately or strongly enhanced due to the small (and constant) molecular diffusivity κ acting on the strong micro-scale gradients ∂s̃/∂z (which are squared in the mixing term). At the end of this process, the salt will be almost fully diluted in the water such that local variance becomes small, with s̃→0 and sˇ→[s]=15 g kg⁻¹. Further introduction of turbulence by a spoon will not lead to further stirring (and thus not to further mixing), because the tracer gradients have vanished.

In real estuaries, stirring typically occurs due to vertical shear instabilities driven by tidal flow, generating large eddies as shown in Figs.  and in Sect. . Via the classical turbulence downward cascade, smaller and smaller eddies are generated such that finally mixing is enhanced at the smallest scales. Since estuaries are typically narrow and friction-dominated, horizontal instabilities on the submesoscale play a minor role for stirring. An exception would be large fjord-type estuaries with weak tides such as the Baltic Sea .

Direct in-situ measurements of salinity mixing χs are difficult to obtain due to the small value of the molecular salinity diffusivity of κ=O(10-9 m² s-1) and the consequently strong gradients at small scales, but successful attempts have been reported by for locations on the continental shelf. According to these authors the salinity-gradient spectrum peaks at dissipative scales ten times smaller than the temperature-gradient spectrum, such that most salinity variance decay occurs in the sub-millimetre range. Therefore, and because of estuaries having generally higher levels of turbulence than continental shelves, direct observations of χs in estuaries are not feasible, and indirect observations are needed (see Sect. ). Instead of using turbulence observations, mixing in estuaries is mostly studied by means of well-calibrated fine-resolution numerical models equipped with accurate numerical discretisations and physically based turbulence closures (Sect. ).

Whereas the irreversible process of mixing happens at very small scales, the quantification of mixing is accomplished both in observations and in models through the application of turbulence closure assumptions . On the level of numerical ocean models, the turbulent fluxes are typically parameterised by means of the eddy diffusivity assumption, resulting in down-gradient turbulent tracer fluxes: 7[ũs̃]=-Kh∂s∂x;[ṽs̃]=-Kh∂s∂y;[w̃s̃]=-Kv∂s∂z, now using salinity s=[s] as Reynolds-averaged tracer concentration. In Eq. (), Kh is the horizontal eddy diffusivity and Kv is the vertical eddy diffusivity. With this the Reynolds-averaged salinity budget equation Eq. () becomes: 8∂s∂t︸change+∂(us)∂x+∂(vs)∂y+∂(ws)∂z︸advection-∂∂xKh∂s∂x-∂∂yKh∂s∂y-∂∂zKv∂s∂z︸diffusion=0, showing that salinity changes are exclusively determined by the divergence of advective and turbulent fluxes. Note that in Eq. () molecular tracer fluxes have been neglected. With Eq. () the production of micro-structure variance due to stirring becomes 9Ps=2Kh∂s∂x2+2Kh∂s∂y2+2Kv∂s∂z2=!χs, where in the last step stirring and mixing of micro-structure salinity variance are set equal, which is a typical assumption in turbulence closure modelling (see Sect. ). The local variance decay χs is used as a local measure for the mixing of Reynolds-averaged salinity . This local equilibrium assumption is generally valid on the temporal and spatial scales that are resolved by numerical ocean models. In detail, χs appears as a sink term in both the salinity variance and salinity-square budgets. The corresponding derivation is shown in Appendix .

In estuaries, the vertical term of the salinity variance decay Eq. () typically dominates over the horizontal terms due to the dominance of tidally-driven vertical shear , such that we obtain 10χs≈χs,v=2Kv∂s∂z2. This is the case because in estuaries horizontal turbulent transports and their divergences are small compared to the vertical transports, with the consequence that in estuarine models the horizontal diffusion is often neglected in the parameterised salinity budget equation Eq. () as well as in the salinity variance equation Eq. (). Due to this dominance of vertical processes in estuaries, it is instructive to study the balance of the vertical variance, 11sv′2=s-1η+H∫-Hηsdz2=s-s¯v2, for the vertical integral of which the following budget equation can be derived from Eq. (), see for details: 12∂∂t∫-Hηsv′2dz︸rate of change+∂∂x∫-Hηusv′2dz+∂∂y∫-Hηvsv′2dz︸ advection=-2∫-Hηuv′sv′∂s¯v∂xdz-2∫-Hηvv′sv′∂s¯v∂ydz︸horizontal straining-∫-Hηχs,vdz︸vertical mixing, where η is the surface elevation. Equation () shows that the vertical variance balance is time-dependent and spatially variable. In contrast to the total salinity variance budget, the vertical salinity variance budget has source terms, the so-called horizontal straining terms, representing the conversion of horizontal variance (sh′2, where sh′=stot′-sv′=s¯v-s¯tot) associated with the horizontal salinity gradient ∂s¯v/∂x to vertical variance . Note that horizontal straining is split into longitudinal straining and lateral straining (first and second term of horizontal straining, respectively) and can be a source (mainly during ebb straining) and a sink (mainly during flood straining) of vertical variance, whereas the effect of vertical mixing is always to reduce the vertical variance. According to , estuarine mixing is driven in a three-step process: First, horizontal variance is provided to the estuary by means of boundary variance transports from the river and the ocean, through the boundary transport term in Eq. (). Then the horizontal straining term in Eq. () converts horizontal variance into vertical variance, which is then in a third step mixed away by the vertical mixing term in Eq. ().

Although this review paper mainly focusses on estuarine salt mixing, a short introduction into the hydrodynamics determining the salt distribution and the turbulence available for salinity mixing will be given here. More detailed reviews can be found in and .

Estuaries are characterised by a longitudinal salinity gradient extending from the region of the estuarine mouth with values typically close to ocean salinity (somewhat below 35 g kg⁻¹) to values of river salinity (typically below 0.5 g kg⁻¹) in the freshwater range. Since saline water is more dense than fresh water, this salinity gradient causes a longitudinal density gradient which results in a longitudinal pressure gradient in the momentum balance, with a barotropic and a baroclinic term. While the baroclinic term is zero at the surface and increases continuously towards the bottom (driving water in up-estuarine direction, with intensification at the bottom), the barotropic pressure-gradient term is independent of the vertical position and drives water out of the estuary. In addition oscillating tides provide small-scale turbulence resulting in vertical shear stress divergence and consequently in diffusion of the longitudinal velocity profiles. The combination of these forces results in a gravitationally-driven tidally averaged exchange flow, with near-bottom flow directed in up-estuarine direction and near-surface flow directed in down-estuarine direction, the so-called gravitational circulation. The strength of the stratifying gravitational circulation depends on the ratio of the gravitational forces due to the longitudinal density gradient and the de-stratifying vertical shear stress divergence. This ratio is expressed as the Simpson number 13Si=D2|bx|u*2 with the water depth D, the horizontal buoyancy gradient bx=∂b/∂x, the buoyancy b=-g/ρ0(ρ-ρ0), the potential density ρ, the reference density ρ0, the gravitational acceleration g, and the bottom friction velocity u*. There are several other hydrodynamic processes contributing to estuarine exchange flow. One essential process is tidal straining : During flood saltier ocean water is sheared over less salty estuarine water, such that the water column becomes statically unstable and thus highly turbulent such that properties are vertically homogenised. During ebb the opposite occurs, resulting in stable stratification and suppression of turbulence. This asymmetry of turbulence does not only affect the salt distribution, but also the velocity profiles: During flood up-estuarine momentum is transported downwards in a much stronger amount than down-estuarine momentum is transported downwards during ebb, which in a tidal average leads to an up-estuarine residual flow near the bottom . This process has also been named ESCO (eddy-viscosity – shear covariance) by . In an idealised model study showed that in tidally energetic flows the contribution of ESCO to estuarine circulation could be stronger than that of gravitational circulation. Other important hydrodynamic processes generating estuarine circulation are lateral circulation , estuarine convergence and wind straining . It has been shown that gravitational circulation, ESCO, and lateral circulation are strongly scaling with Si . Since Si is larger during neap tide than during spring tide (due to smaller u*), the estuarine exchange flow is expected to be stronger during neap tide. The estuarine circulation in concert with vertical mixing is a major process carrying salt into the estuary, against the river discharge. Often, this so-called shear dispersion process is parameterised by a horizontal diffusion term in the longitudinal salt balance equation .

The Simpson number Si does also have a strong influence on the stratification in an estuary. For Si<0.2 (tidally energetic), the water column would be mixed throughout the tidal cycle. For Si>1 (weak tidal energy), stratification should be maintained during the entire tidal cycle. For intermediate situations with 0.2≤Si≤1, however, the water column should stratify during ebb and destratify during flood, leading to strain-induced periodic stratification SIPS,. Since mixing is typically proportional to the square of the vertical salinity gradient, see Eq. (), stratification has a major impact on estuarine mixing. Specific examples of physical drivers of estuarine mixing are discussed in Sect. .

The estuarine exchange flow of water masses defined by salinity, i.e., the net inflow of high salinity ocean waters and the net outflow of low salinity estuarine waters, can be best quantified in terms of time-averaged transports in fixed salinity classes. The resulting Total Exchange Flow (TEF) provides an analysis framework based on salinity coordinates rather than geopotential (z-)coordinates which is consistently linked to the Knudsen theory as well as to estuarine mixing. As shown by several authors , the Eulerian (z-coordinate) framework could be mapped back to time-averaged salinities, but the resulting exchange flow profiles would significantly underestimate the exchange flow. Therefore, the TEF framework has developed into a major research tool for analysing estuarine dynamics. For the Baltic Sea, approaches similar to TEF had already been developed earlier . Here, we briefly explain the theoretical framework for TEF and refer to the literature for the details . Given a fixed transect T across an estuary, the time-averaged volume, salt and salt-squared transports across the transect for all salinities >S are defined as 14Q(S)=∫A(S)udA,Qs(S)=∫A(S)usdA,Qs2(S)=∫A(S)us2dA, where triangular brackets denote temporal averaging, u is the velocity normal to the transect (positive when directed into the estuary) and A(S) is the part of the transect area with instantaneous salinities >S. It should be noted that Q(S) is the streamfunction of the estuarine circulation in salinity space . When defining Smax⁡ and Smin⁡ as the maximum and minimum salinities occurring on the transect during the averaging period, respectively, then sufficiently long averaging results in Q(Smax⁡)=Qs(Smax⁡)=QS2(Smax⁡)=0, Q(Smin⁡)=-Qr (total volume transport equals river discharge) and Qs(Smin⁡)=0 (total salt transport vanishes under long-term averaging). The link to mixing is given by Qs2(Smin⁡)=M (total salinity squared transport equals bulk mixing as defined in Eq. 20), see details in . These properties of Q(S), Qs(S), and Qs2(S) are demonstrated for a cross-channel transect near the mouth of the Elbe River estuary in Fig. a,c,d, where nearly balanced conditions are given such that the respective deviations from the expected values at S=Smin⁡ are small.

Taking the S-derivative of Eq. () results in the volume, salinity and salinity-squared transport per salinity class (the Total Exchange Flow, TEF), 15q(S)=-∂Q(S)∂S,qs(S)=-∂Qs(S)∂S,qs2(S)=-∂Qs2(S)∂S, where the minus sign ensures that inflow at high salinities is positive to highlight the character of the exchange flow. The connection to the Knudsen relations Eqs. (), () and () is given by separately integrating the positive and negative contributions (denoted by superscripts ⁺ and ⁻) of the transport per salinity class, 16Qin=∫Smin⁡Smax⁡q+(S)dS,Qout=∫Smin⁡Smax⁡q-(S)dS,Qs,in=∫Smin⁡Smax⁡qs+(S)dS,Qs,out=∫Smin⁡Smax⁡qs-(S)dS,Qs2,in=∫Smin⁡Smax⁡qs2+(S)dS,Qs2,out=∫Smin⁡Smax⁡qs2-(S)dS, and by deriving transport-weighted inflow and outflow salinities and squared salinities, 17sin=Qs,inQin,sout=Qs,outQout,(s2)in=Qs2,inQin,(s2)out=Qs2,outQout.

An exemplary TEF profile is found in Fig. b for the Elbe River estuary. Zero values for q(S) occur for extreme values of Q(S), in consistency with Eq. (). For the two-layer exchange flow shown here, there is one unique maximum of Q(S), such that the salinity S at which this occurs is the dividing salinity sdiv between inflow and outflow . Note that the dividing salinity sdiv can also be used to calculate the Knudsen parameters Qin, Qout, sin, and sout, providing a numerically more robust method compared to the direct computation via Eq. () . For multi-layer flows, multiple dividing salinities may occur .

The TEF analysis framework has been applied for a variety of estuarine studies, such as for tidal estuaries , tidal bays , non-tidal estuaries , inverse estuaries , fjords , and regional seas .

To assess how entire estuaries quantitively act as mixing machines, the local relations derived in Sect.  are now integrated over estuarine volumes. For this, the continuity equation Eq. () and the salinity equation Eq. () are first integrated over the estuarine volume V which is separated from the ocean by means of a fixed vertical transect: 18Qin+Qout+Qr=dVdt,Qinsin+Qoutsout=ds¯Vdt, (see Fig. a) with the inflow transport and representative salinity Qin≥0 and sin (the ocean water inflow), outflow transport and representative salinity Qout≤0 and sout (the brackish water outflow), as defined in Eqs. () and (). Further quantities in Eq. () are the river run-off Qr≥0 (assuming zero river salinity for simplicity), the average salinity in the estuary s¯, and the volume-integrated salinity s¯V. Details of the derivation of Eq. () are given in . The conservation laws Eq. () have already been formulated by and have become the basis for analyses of exchange flow in many estuaries see e.g.,. The positioning of the transect that separates the estuarine volume from the ocean is arbitrary, but often the geographical location of the river mouth is chosen. In Eq. () freshwater transports across the surface (evaporation or precipitation), the bottom (submarine groundwater discharge) as well as horizontal diffusive salt transports across the transect are neglected. Relations including freshwater transport through the sea surface can be found in . Under long-term averaged conditions, the volume and salt storage terms on the right-hand side of Eq. () would vanish. Under such circumstances, sout≤sin and Qin≤-Qout must hold, as illustrated for the Elbe estuary in Fig. a, b, where balanced conditions with a nearly vanishing volume storage dVdt are given. Under such circumstances, inflow from the ocean occurs at higher salinities than outflow towards the ocean due to mixing with riverine water.

The bulk mixing of an estuary is then obtained by averaging the integrated salinity variance budget Eq. () in time: 19ddt∫Vs′tot2dV≈Qrs¯tot2+Qin(s¯tot-sin)2+Qout(s¯tot-sout)2-M, with the estuarine bulk mixing 20M=∫VχsdV, see the derivations by and . In Eq. (), the approximations (s2)in≈sin2 and (s2)out≈sout2 have been made for simplicity, where (s2)in and (s2)out are inflowing and outflowing salinity squares, respectively, as defined in Eq. (). Assuming long-term averaging such that the temporal derivatives vanish and using Eq. (), we finally obtain 21M≈Qinsin2+Qoutsout2=sinsoutQr=(sin-sout)sinQin, , relating the Knudsen parameters directly to estuarine mixing. While had mentioned the role of mixing for the estuarine exchange flow qualitatively, Eq. () gave the first quantitative estimate of estuarine mixing as a function of the Knudsen parameters. An accurate bulk mixing estimate allowing (s2)in=sin2 and (s2)out=sout2 has been derived by : 22M=sout(s2)in-sin(s2)outsin-soutQr. For the special case of (s2)in=sin2 and (s2)out=sout2, Eq. () is identical to Eq. (). These equalities are only exact when the inflowing and outflowing salinities are constant in time and space during the averaging interval . For estuaries with strongly fluctuating salinities at the mouth such as for the short Merrimack estuary, see relation Eq. () has to be used instead of Eq. () to obtain an accurate estimate for the mixing.

Relation Eq. () is demonstrated in Fig. c for a numerical simulation of the Elbe estuary. It can be seen that the estimate is near-exact for almost the entire length of the estuary, proving its value for the study of bulk mixing in realistic estuaries, as also tested in studies by and .

The principle of salt mixing inside an estuary bounded by a fixed transect is sketched in Fig. a. The first relation of Eq. () shows that the mixing does also balance the exchange of squared salinity with the ocean, such that mixing can also be defined as the reduction of squared salinity integrated over the estuary which often simplifies the calculations, see, as it is expressed locally in Eq. (). The second relation of Eq. () shows that estuarine mixing can be estimated simply by knowing inflowing and outflowing salinities and the river run-off . The third relation of Eq. () demonstrates the relation between estuarine circulation quantified as strength of Qin, see and mixing, a topic that is expanded on in Sect. .

Using the Knudsen relations Eq. (), yet another useful reformulation of Eq. () has been derived by for estuaries in which the riverine inflow has non-zero salinity: 23M≈Qr(sr-sout)2+Qin(sin-sout)2, which is identical to Eq. () for a river salinity of sr=0. In Eq. (), the right-hand side is split into two terms representing the mixing pathways from the inflows to the outflows, with the first one leading from the river inflow to the brackish water outflow and the second one leading from the seawater inflow to the brackish water outflow (with source-water salinities put first in the brackets).

The Knudsen mixing relation Eq. () has been extended by for the case of non-zero freshwater fluxes through the surface, i.e., precipitation and evaporation: 24M≈sinsout(Qr+Qsurf)-ssurf2Qsurf, with the surface freshwater transport Qsurf (positive for net precipitation) and the representative surface salinity ssurf (square root of surface salinity variance transport divided by -Qsurf). Since ssurf>(sinsout)1/2 for evaporation and ssurf<(sinsout)1/2 for precipitation, both evaporation and precipitation are sources of mixing, in addition to the exchange flow. The example of the Persian Gulf as an inverse estuary with strong evaporation is briefly discussed in Sect. .

For the interpretation of the mixing relations it is instructive to consider the mixing completeness Mc by non-dimensionalising Eq. () using Qr and sin: 25Mc=Msin2Qr≈soutsin, where the river run-off Qr and inflowing salinity sin (sometimes equated to the ocean salinity) can be considered as the external forcing of the estuary. Mixing completeness in estuaries can cover the full range of theoretically possible values of 0≤Mc≤1. Table  gives a number of examples for estuarine systems with low, medium and high mixing completeness. It should be noted that the mixing completeness is always calculated with respect to a fixed transect and that for each estuarine system the mixing completeness varies strongly with discharge and tidal intensity (e.g., during the spring-neap cycle).

In the extreme case of no mixing, the riverine freshwater would flow out at the surface with no modification and no ocean water entering the system (sout=0 and Qin=0), such that the mixing completeness would be zero. In estuaries with low mixing, brackish water of only low salinity is produced, with sout≪sin and Qin≪Qr, such that the mixing completeness is sout/sin≪1, see sketch in Fig. a. This would theoretically be the case for deep fjords with low tidal energy . Mostly, fjords do however have large water bodies and low discharge such that the freshwater is strongly diluted by tidal mixing as, for example, for the Puget Sound where the mixing completeness is as large as about 0.97 seeand Table . Low mixing values of about Mc=0.18 have been observed for the tidally intense Merrimack estuary during high discharge see. Under these conditions, the salt intrusion length shortens considerably such that high-salinity ocean water and low-salinity river water are in close contact at the mouth of this estuary discharging directly into the coastal ocean, leading to low mixing. Other low values of mixing completeness are also observed for the Hudson river estuary during neap tide Mc=0.36, see and the Elbe River estuary at high discharge Mc=0.37, see. In both cases, the water is relatively strongly stratified in the region of the transect such that sout≪sin. The large variability that estuaries have due to changes in discharge and tidal intensity is demonstrated by the fact that the Elbe for low discharge, the Merrimack for low discharge, and the Hudson for spring show values of mixing completeness as high as Mc=0.75, Mc=0.86 and Mc=0.87, respectively. Almost complete mixing is obtained for the shallow Wadden Sea of the German Bight, as sketched in Fig. b. In the Sylt-Rømø-Bight, where the freshwater run-off is low and tidal mixing is high, calculated values of inflowing and outflowing salinity both >30 g kg⁻¹ with only about 1 g kg⁻¹ difference, such that the mixing compleness is about Mc=sout/sin≈30/31≈0.97. These are values comparable to Puget Sound, see above. For the Baltic Sea, a non-tidal semi-enclosed sea in northern Europe, estimated sin=17.4 g kg⁻¹ and sout=8.7 g kg⁻¹, such that the mixing completeness is Mc=0.5, a value that could be confirmed by a multi-decadal model simulation .

Often, it is instructive to consider dynamics of estuaries in an isohaline framework, i.e., to evaluate transports, mixing and other properties relative to moving surfaces of constant salinity (isohalines) instead of the Eulerian framework with fixed spatial coordinates. With this, a quasi-Lagrangean perspective is added to the analysis with reference to the moving flow. In the isohaline analysis, geographical features such as a fixed transect at the mouth of the estuary do not play a central role. Since isohalines can move inside and outside the estuarine water body and extend over large areas covering parts of the estuary and the river plume, the isohaline analysis treats estuary and river plume as a dynamic continuum. This isohaline view of estuarine dynamics was first proposed by , with specific reference to the Baltic Sea with its isohalines extending over up to 1000 km from the Central Baltic Sea to its Western reaches . Later, the isohaline concept was applied to tidal estuaries and river plumes . Here, we first introduce a local diahaline analysis, before we discuss the bulk analysis of estuarine dynamics across isohaline surfaces in Sect. .

Local mixing can move isohaline surfaces vertically such that a diahaline mass transport occurs relative to the moving isohaline surface. When normalised to isohaline unit surface and unit mass, this results in a so-called entrainment velocity. Starting for explanation with the one-dimensional salinity budget equation Eq. (), a coordinate transformation from geopotential z to salinity coordinates s (assuming a stable salinity stratification with ∂s/∂z<0), after time averaging in salinity coordinates, 〈⋅〉S, a formulation for the vertical entrainment velocity udia,z(S) (vertical velocity relative to the vertically moving isohaline) is obtained: 26udia,z(S)=w-∂z∂tS=∂∂SKv∂S∂zS=-∂jdia,z(S)∂S , where jdia,z(S) is the time-averaged upward salinity flux through the moving isohaline. The vertical velocity of the isohaline S due to both advection and turbulent diffusion is given by ∂z/∂t (with z being the vertical position of the isohaline). Details of the derivation of Eq. () are given in Eqs. ()–(). The meaning of Eq. () is sketched in Fig.  a: there is a maximum of vertical turbulent salinity flux jdia,z in the entrainment layer that caps the turbulent bottom boundary layer. This maximum results from a large vertical salinity gradient |∂s/∂z| at a still high level of turbulence originating from the boundary layer, expressed as eddy diffusivity Kv. Below this maximum, vertical salinity flux is divergent, thus lowering the local salinity which in time-average leads to an upward entrainment velocity udia,z. Above the entrainment layer, the opposite happens, resulting in a downward salinity flux. A similar process has been described and sketched by , using density fluxes near the bottom of the ocean. The exchange flow in the bottom boundary layer itself with upwelling near the bottom and downwelling above has already been described by . It should be noted that the total diahaline salt flux consists of two contributions, with one advective contribution and one diffusive contribution: 27fdia,z(S)=udia,z(S)S+jdia,z(S)=-∂jdia,z(S)∂SS+jdia,z(S), with the consequence that volume flux and salt flux are not proportional to each other and that the distribution of the diffusive salt flux in salinity coordinates entirely determines the total diahaline salt flux.

To relate jdia,z to mixing χs, defined the local mixing per salinity class which for a vertical water column with a monotone salinity profile reads as 28m(S)=-∂z∂SχsS=-∂z∂S2Kv∂S∂z2S=2-Kv∂S∂zS=2jdia,z(S). which can be seen as a thickness-weighted time-average of the local mixing χs, see also , and . Combining Eqs. () and () results in a key relation between entrainment velocity and mixing, 29udia,z(S)=-12∂m(S)∂S, which could be called the diahaline mixing-entrainment relation. Note that here upward velocities udia,z and fluxes jdia,z are denoted as positive quantities. Details of the derivation of Eq. () for non-monotone salinity distributions in three dimensions can be found in . The principle of Eq. () is sketched in Fig. b: as given by Eq. (), m has local maxima in the same locations as jdia,z, i.e., in the entrainment layers. For mixing per salinity class increasing with height, ∂m/∂z>0⇔∂m/∂S<0 (for stable salinity stratification), a positive entrainment velocity udia,z>0 is expected. For mixing decreasing with height, the opposite occurs. This leads to a typical pattern of diahaline exchange flow in estuaries with positive (upward, towards lower salinities into the estuary) entrainment through an isohaline near the bottom, and a negative (downward, towards higher salinities out of the estuary) entrainment through the same isohaline near the surface further seawards. For realistic estuaries this has been shown for the Hudson River estuary , the Pearl River estuary , the Elbe River estuary , the Changjiang River estuary and the Baltic Sea . In particular, and calculated both sides of Eq. () independently to demonstrate their equality (aside from small numerical differences) in real-world estuarine systems. The advantage of Eq. () over Eq. () is given by the fact that χs and thus m can be split into physical and numerical contributions (see Sect. ), such that numerically generated spurious entrainment can be calculated, as shown by for the Baltic Sea.

The relationship between diahaline mixing m and entrainment velocity udia,z across the isohaline of 11 g kg⁻¹ is shown in Fig.  for the Elbe River estuary in northern Germany. It is clearly visible that as stated in Eq. (), entrainment requires mixing since hotspots of the two quantities align well. In the up-estuary reach of the isohaline surface, where it is close to the bottom, upwelling (red) dominates, whereas at the down-estuary near-surface reaches of the isohaline downwelling (blue) dominates.

Local diahaline mixing as introduced in Sect.  can be expanded to estuarine volumes . The local relation Eq. () for the total diahaline salt flux fdia,z(S) can be integrated over the entire isohaline surface to result in 30Fdia(S)=Qdia(S)S+Jdia(S)=-∂Jdia(S)∂SS+Jdia(S), where Fdia is the total salt transport, Qdia<0 is the diahaline volume transport, and Jdia>0 is the diffusive salt transport across the isohaline surface see Fig. 8a by. If instead of a fixed transect T a moving isohaline of salinity S is considered as the seaward boundary of the estuary (see Fig.  b), the volume and salt budget is of this form: 31Qdia,in(S)+Qdia,out(S)+Qr=dV(S)dt,Qdia,in(S)+Qdia,out(S)S+Jdia(S)=d(s¯totV(S))dt, with the isohaline volume V(S), the average salinity inside this volume s¯tot, the net advective inflow through the isohaline, Qdia,in>0, and the the net advective outflow through the isohaline, Qdia,out<0. Transformations of Eq. () show that long-term averaged mixing inside the estuarine volume bounded by an isohaline S is 32M(S)=∫V(S)χsdV=S2Qr, which can be seen as a special case of M≈sinsoutQr from Eq. () with sin=sout=S . The relation Eq. () is exact for long-term averaging and zero freshwater transports through the surface and bottom of the estuary. A mixing relation for non-zero river salinity is shown in Eq. ().

With M(S) being a continuous function of S and assuming that Qr is independent of S, we can take the derivative of M(S) with respect to S: 33m(S)=∂M(S)∂S=2SQr, where m(S) is the salt mixing per salinity class. It should be noted that m(S) can also be obtained by integrating the local mixing per salinity class m(S) from Eq. () over the projection of the isohaline surface to the horizontal . A discrete version of Eq. () is sketched in Fig. c in order to explain the infinitesimal property of the mixing per salinity class. The linear dependence of m(S) on salinity S has been formulated and derived as the universal law of estuarine mixing .

The relation Eq. () can be explained by first stating that the volume transport across the isohaline must for long-term averaged conditions equal the river runoff Qr. Furthermore, the advective salt transport across the isohaline, -S(Qdia,in+Qdia,out)=SQr, must equal the diahaline diffusive salt transport, such that Jdia(S)=SQr, see the second equation in Eq. (). With 34m(S)=2Jdia(S), which can be derived by integration of Eq. () over the horizontal projection of the isohaline surface, relation Eq. () is obtained . A relation equivalent to the combination of Eqs. () and () had been derived by , based on earlier work of , to quantify the turbulent salt transport into river plumes due to entrainment, see the steady-state version of his Eq. (4).

To accurately reproduce the universal law by means of models of realistic estuaries such as the Pearl River estuary , the Changjiang River estuary and the Elbe River estuary , averaging over one spring-neap cycle is typically sufficient. In a numerical model, the mixing which a salinity field experiences is the sum of the parameterised physical mixing mphy(S) and spurious numerical mixing mnum(S) due to the discretisation of the advection operator, see Sect.  for details. Both, relations Eqs. () and (), are tested for the Elbe estuary in Fig. . There, it can be seen that the directly computed total mixing quantities M(S) and m(S), each consisting of the sum of numerical and physical mixing, closely follow their respective relation up to the point where the isohaline surfaces partly leave the model domain through the open boundary (grey-shaded areas in the Figure). Here, we first find an underestimation of the predicted mixing by relations Eqs. () and () which can likely be related to left-over stratification in the German Bight from earlier high-discharge periods entering the model domain via the open boundary and then being mixed away. For very high salinity classes, the mixing in the model is much weaker than the predicted mixing since substantial parts of the isohaline surfaces are outside of the model domain such that most of the potential mixing is not covered by the model. While the Knudsen mixing law Eq. () is only valid when salinity fluctuations at the open boundary are limited, the universal law of estuarine mixing, Eqs. () and (), is exact for all estuaries (without relevant freshwater fluxes across the sea surface).

One interesting consequence of the universal law of estuarine mixing can be seen by reformulating Eq. () as 35m(S)=v(S)χ¯s(S)=2SQr, where v(S) is the volume per salinity class and χ¯s(S) is the salinity mixing averaged over the salinity class S. Since for long-term averages v(S)χ¯s(S) is fixed, Eq. () means that at low mixing rates χ¯s(S) the volumes per salinity class should be large, with the consequence that at low mixing rates, isohaline spacing should be wide. When comparing for example the salinity fields for the Hudson River estuary for neap tide and for spring tide , the isohaline spacing at springs is wider than at neaps. Assuming that storage terms do not play an essential role for these situations, the conclusion could be drawn that average mixing is smaller at springs than at neaps. This is consistent with the findings of who find that maximum mixing occurs during late neap tides, see also Sect. . This is also in line with the study of who showed that increased (background) diffusivity would reduce the area of a river plume see also the river plume study bywho showed that additional mixing due to islands in the plume region can reduce the plume area and volume.

In his famous abyssal recipes fitted a vertical one-dimensional advection-diffusion equation to hydrographic observations in the central Pacific Ocean and concluded that turbulent mixing with an effective vertical diffusivity of 1.3×10-4 m² s⁻¹ would be needed to explain the global overturning circulation. In later studies, the decomposition of the underlying mixing processes into wind and tidal mixing and their regional distribution had been further specified see e.g.,. On the much smaller scales of estuaries, the same must be postulated: estuarine circulation requires mixing and vice versa. Here, we discuss different concepts of this duality.

Let us first summarise what we have discussed about this issue until now. In his fundamental paper, already stated that estuarine circulation is associated with mixing (Sect. ). The general process is that salty ocean water entering the estuary is first mixed with fresh river water inside the estuary and then ejected as brackish water towards the ocean, making estuaries mixing machines . When there is no mixing inside the estuary, then no further salty water can enter the estuary in the long term (Sect. ). In that sense, the volume transport of salty water flowing into the estuary, Qin, is a good measure for the estuarine circulation . A first quantitative estimate for the tight relationship between estuarine circulation and mixing has been established in the third relation of Eq. () by , showing a proportionality between the bulk mixing M and Qin, with (sin-sout)sin as factor of proportionality.

The streamfunction Q(S) of the estuarine circulation as defined in Eq. () is the time-averaged volume transport into the estuary across a fixed transect for all salinities above S and therefore contains the information about the Total Exchange Flow, see Sect.  for details. It can be directly linked to mixing, as derived already by to quantify the overturning circulation of the Baltic Sea: 36Q(S)=Qdia,est(S)=-∂Jdia,est(S)∂S=-12∂mest(S)∂S, where the subscript _est means that diahaline transport Qdia, diahaline salt flux Jdia and diahaline mixing per salinity class m are only considered for the part of the isohaline that is located on the estuarine side of the transect, see Fig. . The first equality in Eq. () is demonstrated in Fig. a: under long-term averaged conditions the volume transport Q(S) into the subvolume bounded by the transect T, the isohaline S and the bottom must be equal to the volume transport across the isohaline S, Qdia,est(S). The second equality in Eq. () results from the integration of the entrainment relation Eq. () over the projection of the isohaline part situated upstream of the transect T. The third relation of Eq. () which had not been proposed by is simply the S-derivative of Eq. () restricted to the upstream part of the estuary. stated about this relation (his Eq. 7): Equation () represents in the most simple way how the deep water supply is related to the overall vertical (i.e. cross-isohaline) mixing properties in the basin. What specifically calls the deep water supply to the Central Baltic Sea is more generally the up-estuarine part of the estuarine circulation. Note that independently of , used the first two equalities of Eq. () to calculate exchange flow accumulated between two estuarine segments.

When choosing the dividing salinity S=sdiv for relation Eq. () with Q(sdiv)=Qin (see Sect. ), then the quantification of the estuarine circulation is directly linked to mixing: 37Qin=-∂Jdia,est(sdiv)∂S=-12∂mest(sdiv)∂S, see details in and Fig. b. The significance of Eq. () is that it is a direct quantification of the estuarine circulation (defined as Qin) by means of the (S-gradient of the) diahaline mixing. This relation is directly applicable to simple estuaries with a typical two-layer exchange flow, but has also been extended to multi-layer exchange flow . For the Elbe River estuary, relation Eq. () is demonstrated in Fig. .

This review focuses on mixing of salt. The reason is that salinity is the defining constituent of estuaries, continuously ranging from minimum values near zero in the river water towards ocean salinity values near the mouth of the estuary or in the river plume. Therefore, salinity can be used as a coordinate in estuaries substituting spatial coordinates. Also, salinity is conservative with basically no inner sinks and sources, and also bottom and surface fluxes of salt are negligible. However, many other constituents are mixing in estuaries, such as heat, oxygen, nutrients, and pollutants. Based on the work of salinity coordinates and temperature coordinates, for larger ocean scales, theoretical Water Mass Transformation (WMT) frameworks have been developed to analyse mixing of constituents other than salt . To evaluate non-conservative behaviour of estuarine tracers due to sources or sinks, such tracers are often represented as function of salinity . By doing so, non-conservative tracer mixing is identified by a non-linear relation between tracer concentration and salinity. However, as shown by , such non-linear behaviour could also be caused by tracer variability in the freshwater source of the estuary. There is a large body of literature about effects of estuarine mixing of tracers other than salt on ecosystem dynamics. For example, proposes differential vertical mixing of suspended particulate matter (SPM) as a mechanism of creating Estuarine Turbidity Maxima. Tidal covariance between longitudinal velocity and concentration of SPM due to vertical SPM mixing can lead to up-estuary SPM transport . In a similar way, explain the generation of local maxima of carbon dioxide partial pressure in estuaries, the so-called Estuarine Gas Exchange Maxima. Nitrogen-to-phosphate ratios in estuaries has been shown to critically depend on tidal mixing . These are just a few examples which show the essential role of mixing of tracers other than salt in estuaries. However, a general theory for such tracer mixing has not yet been proposed.

In estuaries, various time scales are relevant, including semi-diurnal tidal time scales and the fortnightly spring-neap cycle as well as times scales of weather and river-run-off (days to months). In the subsequent sections, the most relevant processes on these time scales are discussed.

If estuaries had a flat bottom, their dynamics could be largely explained by means of one-dimensional or two-dimensional models without lateral variations. However, most estuaries are characterised by one or more deep channels (often deepened due to dredging) in longitudinal direction which carry most of the tidal flow. Shoals at the sides and between the channels lead to a typical channel-shoal structure where the channel-shoal transition leads to dynamic processes crucial to estuarine circulation and mixing (Sect. ). But also in longitudinal direction, estuaries are not smooth. Channels often show a strong along-channel variability, e.g., due to constrictions in width and depth leading to local fronts and enhanced mixing (Sect. ).

Direct field measurements of the mixing of salt in estuaries is impractical, due to the microscopic scales at which the dissipation of salinity variance occurs, see Sect. . The turbulent motions that drive mixing can be resolved however, and numerous field investigations have quantified mixing based on measurements of turbulent motions at scales from metres to centimetres, then using theoretical arguments to relate the observable characteristics of the turbulence with either the mixing χs or the diahaline turbulent salt flux jdia,z. A common approach is to use the relationship between the eddy diffusion coefficient for mass Kv and turbulent kinetic energy dissipation rate ε, 38Kv=Rf1-RfN2ε as proposed by , where Rf is the flux Richardson number and N is the buoyancy frequency. With an estimate of Kv and a local measure of the vertical salinity gradient ∂s/∂z, the vertical turbulent salt flux jdia,z and the salinity variance dissipation χs are readily estimated, see Eqs. () and (). pioneered the use of a free-falling shear probe in an estuary for the purpose of quantifying mixing; since then this method has been followed in a variety of estuarine settings . This method is challenging for measuring turbulence near the bottom, due to the risk of smashing the delicate shear probe into the bottom. An alternative method is the estimation of the dissipation rate ε based on the inertial-subrange velocity spectrum measured by a turbulence-resolving current meter . A related methodology pioneered by , and applied by , and among others is to use an Acoustic Doppler Current Profiler (ADCP) to obtain a direct measurement of the Reynolds stress [ũw̃], then in combination with a measure of the vertical shear to estimate eddy viscosity Av, and then to infer the eddy diffusivity Kv and the mixing rate χs.

These methods are most effective in weak stratification conditions, when dissipation rates tend to be high and the turbulent length scale is readily resolved by the sensor. As stratification gets stronger, however, the scales of turbulent motions decrease, as scaled by the Ozmidov scale 39LO=εN31/2 making it harder to obtain a reliable estimate of ε. More problematical is that the estimation of mixing χs depends on the square of the local salinity gradient, see Eq. (), which itself is a challenging quantity to measure at the small vertical scales relevant to turbulence within a stratified environment.

Micro-conductivity sensors provide a means of resolving the salinity gradient and associated overturns at scales relevant to the characterization of turbulent motions . The Thorpe overturn scale LT provides an alternative means of estimating the turbulent dissipation rate: 40ε≈LT2N3, as shown by in his turbulence measurements in the Hudson River estuary. It is particularly useful in the stratified interior, where the small vertical scales of turbulence make other methods of estimating ε more difficult .

All of the above methods depend on a turbulence closure assumption as well as an estimate of mixing efficiency to link the dissipation rate ε to the actual mixing of salt, χs. The use of micro-conductivity sensors offers a more direct approach to estimating the mixing of salt. Following , high-frequency micro-conductivity time series measurements can resolve the inertial subrange and the viscous-convective subrange of the salinity variance spectrum 41Sss(k)=b0χsε-1/3K-1min⁡(K,Kη)-2/3, where Sss is the power spectral density of salinity variance, b0=0.4 is the Kolmogorov constant for scalar variance, K is the wave number and Kη=0.04(ε/ν3)1/4 is the Kolmogorov wave number, with the molecular viscosity ν. Note that the last term in Eq. () resolves the transition from the inertial subrange to the viscous-convective subrange at length-scales of Kη-1. As long as the height of the spectrum within either the inertial or viscous-convective subrange can be estimated, Eq. () can be used in combination with an estimate of ε to estimate mixing, i.e., the dissipation of salinity variance χs, without relying on turbulence closure assumptions. Moreover, the formula depends only weakly on the estimate of ε, which is often hard to estimate accurately in the stratified turbulent regime. used this method to estimate mixing rates in the highly stratified Connecticut River estuary. Their analysis demonstrated peak values of χs in the pycnocline in association with intense shear instability during ebb tide (Fig. ).

Acoustic backscatter also provides a more direct means of estimating mixing, as demonstrated by . Using a broadband array of echo sounders, demonstrated that the observed scattering in the pycnocline of the Connecticut River estuary has a spectral slope consistent with the viscous-convective subrange. Other scatterers, such as fish, bubbles and sediment, have distinctly different spectral slopes. If the stratification is strong enough and other scatterers do not overwhelm the signal, the acoustic backscatter intensity can be used as a nearly direct measure of χs. While the signal-to-noise ratio of the acoustic amplitude does not offer the same precision as microstructure measurements, it produces remarkable spatial resolution, as demonstrated by acoustic measurements within a train of Kelvin-Helmholz instabililities in the Connecticut River estuary (Fig. ). Based on the evidence provided by the analysis of , echo-sounding imagery can be used to identify regions of intense mixing, even if the actual magnitude of χs cannot be quantified.

Since direct observations of turbulent properties in estuaries are very tedious and noisy (see the discussion in Sect. ), the analysis of mixing in estuaries largely relies on numerical models. The advantage of numerical models is certainly their coverage of the entire four-dimensional estuarine space (three spatial directions and time), whereas observations can only sparsely cover this space. However, to ensure that the numerical model results sufficiently represent real estuaries, several measures need to be taken: besides realistic input data into the model (such as bathymetry, open boundary conditions, meteorological forcing) and a thorough validation using observational data, the numerical model itself must be physically sound and numerically accurate. There is an extensive body of literature addressing these two topics . Here, we will present in more detail two aspects which are key to the proper assessment of mixing in estuaries: turbulence closure modelling (Sect. ) and numerical mixing analysis (Sect. ).

It is expected that the trend of ever-increasing computational resources will continue. As an important development, the ongoing replacement of the more traditional Central Processing Units (CPUs) by modern Graphics Processing Units (GPUs) will improve the overall computational efficiency. As computational codes for coastal ocean models will become available for GPUs, this increase in computational power can be used for finer resolution models, longer simulation periods or larger model domains. Finer resolution models would be able to resolve further processes of estuarine mixing with more accuracy, such as smaller-scale topographic effects. Specifically, consequences of coastal engineering such as dredging and dam-building, which are commonly under-resolved in contemporary numerical simulations, could be reproduced more realistically. Today, multi-decadal simulations of well-resolved estuaries are hardly feasible. Such longer simulation periods could, however, be used to better reproduce interannual variability and consequences of long-term trends such as sea-level rise and changes in precipitation and evaporation patterns. Larger model domains could help to include more of the tidal or non-tidal parts of rivers or larger portions of the adjacent ocean including the river plume. It has been shown that the universal law of estuarine mixing Eq. () is only valid for isohaline surfaces that do not leave the model domain. On the other hand, isohaline theory does not distinguish between estuarine and river plume mixing, and often the transition between the two cannot be seen from model topography and coastlines. Therefore, it is desirable to simulate the entire river-estuary-plume region within a single model setup. Similarly, the confluence of multiple estuaries and river plumes could be simulated at fine resolution when computational resources further improve.

More powerful computer resources could also allow for models with improved model physics, such as using higher-order turbulence parameterisations or the application of Large-Eddy Simulation (LES) models. Both improvements would have a direct effect on the computation of mixing. One example for higher-order turbulence closure models could be the use of non-local models that do not enforce the down-gradient assumption of turbulent fluxes see the recent study by. Furthermore, parameterisations of Langmuir Turbulence which potentially affects estuarine mixing in the presence of wind waves could be added . The application of LES models to estuaries would mean that the most energetic turbulent eddies could be resolved instead of being parameterised by turbulence closure models. Then, only the small-scale mixing would need to be parameterised, for which generally relatively simple closures should be sufficient. A further advantage of LES models over classical coastal ocean models is their non-hydrostatic pressure calculation which would allow for the reproduction of non-hydrostatic effects such as internal lee wave generation , generation of interfacial waves at pycnoclines when the surface and the bottom boundary layers interact or Langmuir Circulation effects in shallow coastal waters . Two specific LES model applications have been reported by and who calculated a tidal water-column setup with periodic boundary conditions in both horizontal directions. So far, no further estuarine LES applications have been published. Specifically efficient LES model codes such as Oceananigans that can be executed on GPUs open vast possibilities of coastal see the recent study by and estuarine LES applications in the future, starting with idealised setups, but also with the future potential to simulate estuarine dynamics over more realistic topographies.

The concept of Water Mass Transformation (WMT) has been strongly extended in recent years see, e.g.,. Such multi-dimensional WMT concepts using other constituents in addition to salinity, such as temperature or biogeochemical tracer concentrations, are typically applied to large-scale ocean problems. Estuarine applications have yet to be created, but could give insight where, for example, temperature plays an important role in addition to salinity, such as shown for the Arctic Ocean , the Persian Gulf or the Baltic Sea .

Finally, applications of Machine Learning (ML) have entered all fields of oceanography, including coastal and estuarine research. Typical estuarine applications would comprise the calculation of river discharge from meteorological data or the estimation of the salt intrusion length inside estuaries . It is not yet obvious in which way fast, efficient and well-trained ML algorithms could be exploited to support research on estuarine mixing, but it can certainly be expected that ML applications will soon extend our toolkit in addition to numerical modelling and observational methods.