We use a process-based width-averaged model that solves for the equations for water motion, sediment mass conservation, and dynamics of the sediment bottom pool. The model domain is described by Cartesian (x, z) coordinates. The estuary is assumed to be a single channel running from the mouth (x=0) to the upstream limit (x=L; see Fig. ). The vertical axis runs from an x-dependent bed level z=-H(x) to the surface z=R(x)+ζ(x,t). Here, R(x) is a subtidal reference level caused by the river set-up and ζ(x,t) is the (tidal) surface variation. The width of the estuary is also x-dependent and denoted by a function B(x). The equations for the water motion are the width-averaged continuity and momentum equations. Assuming hydrostatic pressure and the Boussinesq approximation, these equations are read as 1(Bu)x+Bwz=0,2ut+uux+wuz=-gζx+gβ∫zR+ζsxdz′+(Aνuz)z. Here, subscripts x, z, and t denote derivatives with respect to these dimensions. The functions u and w are the velocity components in the x and z directions, g is the acceleration of gravity, ρ0 is a reference density, and Aν is the vertical eddy viscosity. The function s(x,z,t) denotes salinity, and β is the haline contraction coefficient; it is assumed that density differences are dominated by salinity. Specific choices for modelling salinity are discussed in Sect. . At the surface, a no-stress condition and kinematic condition are prescribed. At the bed, we prescribe the partial sip law Aνuz=sfu, where sf is a partial slip parameter, and a non-permeability condition. The partial slip parameter and eddy viscosity, both representing effects of turbulence, are assumed constant over the semi-diurnal cycle for simplicity (but potentially varying over the spring–neap cycle). The eddy viscosity is assumed to be vertically uniform.

The sediment mass balance is described by the equation 3ct+ucz+(w-ws)cz=(Kνcz)z+(Khcx)x, where c is the suspended sediment concentration, ws is the settling velocity, and Kν and Kh are the vertical and horizontal eddy diffusivities. Both Kν and Kh are vertically uniform and constant over the tidal cycle. At the surface, a no-flux condition is used. At the bottom, the re-suspension of sediment is considered according to 4Kνcz=M|τb|fat z=-H, where M is an erosion parameter, τb is the bed shear stress, and f is the erodibility. The erodibility is a value between 0 and 1 indicating the amount of sediment available on the bed averaged over the tidal cycle, from permanently starved conditions at f=0 to abundant supply at f=1 .

The amount of sediment at the bed is modelled using the Exner equation, stating that the change in sediment mass on the bed equals deposition minus re-suspension: 5Sbed,t=wsc|z=-H-M|τb|f, where Sbed is the amount of easily erodible sediment in the bottom pool or bottom sediment stock (in kgm-2). It is assumed that the thickness of the bottom pool does not affect the water depth. It is useful to also define the total tidally averaged sediment stock S, which describes the amount of easily erodible sediment in the bed and water column, i.e. 6S=Sbed+∫-HR+ζcdz, where 〈⋅〉 denotes averaging over a typical semi-diurnal period (formally defined in Sect. ). We relate the erodibility f to the stock S using the relation derived by . For small values of S, this relation states that f increases with S (i.e. supply-limited conditions). For sufficiently large values of S, the erodibility equals unity, meaning that adding more sediment to the bottom pool will not affect the amount of sediment in suspension (i.e. erosion-limited conditions).

The hydrodynamic forcing for this model consists of an M2, M4, S2, and S4 tide at the mouth (see also Sect. ) and a constant discharge at the head of the estuary. For the sediment model a constant tidally averaged and depth-averaged sediment concentration is prescribed at the mouth, and we impose no inflow of sediment from the watershed. We do not consider any initial condition here, since we will only study the model in dynamic equilibrium, i.e. the state reached after a long time with concentrations varying over the semi-diurnal and spring–neap timescale but not changing on sub-spring–neap timescales.

The solution method consists of several steps. First using a scaling and perturbation approach, the model equations are approximated following work by and . Next, a scaling of the tidal and spring–neap dynamics will justify the use of a two-timescale perturbation method to further approximate and simplify the temporal dynamics. The use of this method to analyse spring–neap dynamics is new, and it will be demonstrated that it not only leads to numerically efficient solutions, but also facilitates understanding of the dynamics. The resulting equations are solved numerically.

The vertical velocity and sediment profiles are resolved using semi-analytical methods (i.e. analytical supplemented by numerical quadrature rules where needed; see ). The along-channel dimensions are discretised using a finite difference method of first order for advective terms and second order for dispersive terms on an equidistant grid. In the fast time variable t1, the model is solved in terms of harmonic components. As the forcing consists only of residual, D2, and D4 components, the solution consists of residual, D2, D4, and over-tidal components. Using harmonic components means that the model is directly solved for dynamic equilibrium in the variable t1; i.e. no time-stepping or spin-up is needed. In the slow t2 time variable, the model is solved using time-stepping, starting from an initial condition for S. In this study, we start from S=0 and time integrate until a dynamic equilibrium on the spring–neap cycle is reached. We used 200 time steps per spring–neap cycle, implying a time step of approximately 1.8 h. Note that this time step is much coarser than what would usually be needed if time-stepping were also used for solving for the semi-diurnal tides. Hence, the application of the two-timescale approach results in much smaller computation times than would be needed otherwise.

In order to analyse the sediment dynamics, we study three processes: the D2-tide-averaged horizontal transport, bottom pool movement, and re-suspension in the water column, as sketched conceptually in Fig. . These processes, of course, are coupled, yet it is useful to study them separately to better understand the SPM concentration.

First we consider the D2-tide-averaged horizontal transport. We analyse this using the sediment transport capacity: the transport that would occur if a small uniform layer of sediment were added on the bed. We look specifically for the locations where transport capacity converges, i.e. locations where the transport capacity changes from positive (i.e. importing) to negative (i.e. exporting). This means sediment tends to accumulate there. To further understand why the transport capacity changes over the spring–neap cycle, we will make a decomposition into contributions from various physical components. Here we will use the decomposition as implemented in iFlow, of which a selection of relevant terms is discussed below:

river – transport related to river flushing of tidally re-suspended sediment and effects of tide–river interaction on sediment re-suspension;

Second, the global D2-tide-averaged transport leads to changes in the sediment stock S (i.e. sediment in suspension and at the bottom) at each location in the estuary. Associated with each stock is an erodibility f used as a measure for the bottom pool thickness. Changes in the stock occur on the slow timescale, and hence it is needed to consider the history of the system. We compare the bottom pool with the convergence points. If the system responds fast, the bottom pool will be most thick at the convergence points, while a delay may be present in slower systems.

Third, knowing the spatial distribution of sediment on the bottom, we may focus on each location of the estuary in isolation. The SPM concentrations at each location follow from the local balance between re-suspension and settling dynamics in the water column and interaction with the bottom pool. This can be expressed as (adapted from ) 13c(t1,t2)=c^(t1,t2)f(t2), where c^ is the sediment capacity. The sediment capacity is the maximum concentration one could attain at any moment given there is sufficient sediment available (i.e. f=1). It depends on hydrodynamic quantities (i.e. bed shear stress, turbulence) and sediment parameters (settling velocity, erosion parameter). We hence see that the SPM concentration follows from understanding both the sediment capacity and erodibility. Under supply-limited conditions, f<1, the bottom pool is so thin that all sediment is suspended from it for at least part of the semi-diurnal cycle (illustrated in the left water column in Fig. ). Hence, both changes in the stock and changes in sediment capacity affect the SPM concentration. Under erosion-limited conditions, f=1, there is always sediment present on the bed (illustrated in the right water column in Fig. ). Sediment added to the stock on a tide-averaged basis will go to the bottom pool without affecting f. During times of erosion-limited conditions, the observed SPM concentration is therefore not affected by horizontal transport but only by changing sediment capacity.

Overall, our framework consists of analysing how transports distribute the sediment stock along the estuary, after which we can look at the slowly varying bottom pool (via f) at each location. The SPM concentration then follows from an instantaneous balance between the bottom pool and the water column depending on the sediment capacity. We argue that this is a useful way of thinking about the SPM dynamics but also acknowledge that transport, bottom pool, and SPM concentration are mutually dependent and this analysis does not imply the dynamics are linear.

We specifically want to highlight the importance of the slow timescale adaptation of the sediment stock. To this end, we compare the model results to their quasi-stationary equivalent. The model as described above accounts for the time it takes for the sediment stock in the estuary to adapt to the changing hydrodynamic conditions and is referred to as the dynamic computation. This is compared to a case where the sediment stock is assumed to instantaneously adapt to the hydrodynamic conditions, referred to as quasi-stationary. The quasi-stationary case is a useful reference because the sediment distribution is a direct consequence of the present hydrodynamic conditions without any memory effects. The mathematical procedure for computing quasi-stationary conditions is discussed in Appendix A. The analysis framework presented in the previous section will be applied to both the dynamic and quasi-stationary model runs to help highlight the importance of memory effects.

Figure  shows results of the idealised case study, comparing the quasi-stationary results with the dynamic results. Figure a shows the spring–neap-averaged along-channel near-bed concentration for the quasi-stationary case (blue line) and dynamic case (red line). The shaded areas indicate the range of variation in the semi-diurnally averaged (i.e. D2-averaged) near-bed concentration over the spring–neap cycle. Looking at the mean concentrations, ETM zones are observed near km 8 and 55 in both the quasi-stationary and dynamic cases. Moreover, in this specific case it turns out that mean concentrations are of a similar magnitude. This is not generally the case, as will be illustrated in the following sections. A generally valid result is that the spring–neap variation in the D2-averaged concentration is smaller in the dynamic case than in the quasi-stationary case. Figure b and c show the near-bed concentration as a function of x and t. In the quasi-stationary case, the highest concentrations are attained after spring tide near km 55 (marked by × in the figure), and the ETM moves gradually downstream towards neap. The ETM is absent in the time between neap and spring tide. In the dynamic case, the ETM near km 55 remains present during the entire spring–neap cycle and moves by about 10 km during this time. Interestingly, another peak in concentration is observed some time after neap tide near km 45.

To better understand the dynamics we apply the analysis framework developed in Sect. . Hence, we first look at the transport capacity. Since there is no feedback of the sediment stock or concentration to the water motion and sediment capacity in this simple model, the transport capacity depends on the instantaneous flow conditions and hence is the same in the quasi-stationary and the dynamic case. The transport capacity scales with the bed friction, which, in this model, scales linearly with the D2 velocity amplitude near the bed. To facilitate the comparison of the transport capacity during spring and neap tide, we therefore divide the transport capacity by the near-bed velocity amplitude at x=0. Figure  shows this rescaled transport capacity for the default case at neap tide (Fig. a) and spring tide (Fig. b). First looking at the total transport capacity, we see two sediment convergence zones (i.e. transitions from importing capacity to exporting capacity): at km 15 and 41 during neap tide and km 5 and 56 during spring tide. During neap and spring tide, the transport capacities due to the river flow are equal after scaling by the near-bed velocity amplitude. The transport capacity due to the external D4 tide is almost the same after scaling, while the contributions related to the internally generated D4 tide and spatial settling lag are much larger during spring than neap. These dependencies of the transport components on the stage of the spring–neap cycle can be explained through an analysis of the model equations. Results are presented in Table . The transport capacity depends on the tidal velocity, which (due to linearity) scales with the tidal surface elevation amplitude |AD2| and |AD4| (i.e. D2 and D4 tidal elevation) and a function of the local D2-D4 phase difference f(ϕD4-ϕD2) (see , for an explicit expression of this function). Since the transport capacity due to river discharge scales linearly with the D2 amplitude (and hence the D2 velocity), the rescaled transport capacity due to river discharge in Fig.  is constant over the spring–neap cycle. The transport capacity by the externally generated D4 tide is related to the asymmetry between the linearly propagating parts of the D2 and D4 tides and also scales with the D2 amplitude; additionally the variation in the D4 amplitude and D2-D4 phase difference is small between spring and neap in this particular case. Hence also the rescaled transport capacity of this component is similar during both phases. The internally generated D4 tide is caused by the quadratic interaction of the D2 tide due to processes including advection and Stokes drift. Hence, the transport capacity due to the internally generated D4 tide relates to the asymmetry between this non-linearly generated D4 tide and the D2 tide and scales with the tidal amplitude cubed. Similarly, spatial settling lag follows from non-linear interactions and scales with the tidal amplitude cubed. This explains the difference between neap and spring transport capacities.

Secondly, we study the variation in the bottom pool. Figure d and e show the erodibility in the quasi-stationary and dynamic cases. The black lines indicate the location of the sediment convergence points in the quasi-stationary case (leading to either a local maximum in erodibility or f=1) and the local maxima in erodibility in the dynamic case. The dashed lines indicate areas where erosion-limited conditions (f=1) are found. In the quasi-stationary case, convergence points are found during part of the cycle between km 0 and 10 as well as further upstream. At spring and neap tide (dashed yellow lines), the convergence points match the downward zero crossings of the transport capacity in Fig. . Erosion-limited conditions are found going from spring to neap tide directly downstream from a sediment convergence point. In the dynamic case, local maxima in erodibility are found near km 5–8 and 45–55. An erosion-limited zone is found from late spring to neap near km 50. Comparing the dynamic and quasi-steady case, it is observed that erosion-limited conditions are not immediately attained at spring tide in the dynamic case, since it takes time to build up the bottom pool. This is confirmed by the transport (Fig. g), which shows that sediment is imported towards the bottom pool around spring. Similarly, erosion-limited conditions do not immediately disappear after neap tide. The transport shows sediment is exported, but it takes time for sediment to leave the bottom pool. Also, the bottom pool moves much less in the dynamic case compared to the quasi-stationary case. Just before neap in the quasi-stationary case, the ETM moves downstream in a short time, while this downstream movement is only weakly present in the dynamic case. After neap in the quasi-stationary case, the ETM disappears. In the dynamic case we do observe the exporting of sediment, but this is not fast enough for the ETM to disappear entirely. Note that either transports in the quasi-stationary case are positive, indicating there are erosion-limited conditions with a growing bottom pool, or transport is zero.

Finally, we will look at the SPM concentration, which is the product of the sediment capacity and erodibility. In this idealised model, sediment capacity c^ changes only through variations in the bed shear stress over the spring–neap cycle. Hence, the subtidal sediment capacity is highest during spring tide and smallest during neap tide. In the quasi-stationary case (Fig. b), the highest concentrations are found just after spring tide. There are erosion-limited conditions, and hence concentration depends on the bed shear stress. Whereas the bed shear stress is maximum at spring tide, the bottom pool is slightly further downstream after spring tide, where bed shear stresses are higher than upstream. During the spring–neap cycle, the ETM follows the location of the maximum erodibility. In the dynamic case (Fig. c), the location of the highest concentrations follows the location of the bottom pool. Now, the highest concentrations occur some time after spring tide not only because of the bed shear stress, but also because the bottom pool still needs time to build up. Another peak in concentration is found after neap tide, as the bottom pool is still present for some time, while the estuary is still exporting sediment and bottom shear stresses increase again.

The timing and location of the ETM resulting from spring–neap variations depend strongly on whether or not a bottom pool with erosion-limited conditions (f=1) can form. If such a bottom pool forms, sediment keeps accumulating and the sediment concentration is at capacity conditions. To show the effect of the bottom pool on the SPM concentration we compare cases with different values of the erosion parameter M=1×10-5 and M=1×10-3 sm-1. Within this model, the sediment transport capacity simply scales linearly with the erosion parameter, while the spatial distribution remains the same. Hence, the underlying sediment transport tendency remains the same as in the default case (cf. Fig. ).

Figure a, c, e, and g show the sediment concentration, erodibility, and transport for small M. All results correspond to the dynamic case, with the only comparison with the quasi-stationary case in panel a. Due to the small erodibility, maximum concentrations are quite small, but since the transport capacity is the same, ETM zones are still found in the same locations (near km 5 and 50). The mean concentration in the dynamic case is bigger than that in the quasi-stationary case (panel a), while the variation is smaller. The mean concentration is higher because the sediment concentration is at capacity conditions (f=1, panel e) in areas at the entrance and near km 50 during the entire spring–neap cycle. Sediment is still imported and exported to and from the bottom pool (panel g), but erosion-limited conditions persist. Hence, the ETM does not vanish after neap as in the quasi-steady case (cf. Fig. c). Consistent with the capacity conditions, the highest concentrations in both ETM zones are found during spring tide, when bed shear stresses are at their maximum.

Figure b, d, f, and h show the results for the larger value of M, again corresponding to the dynamic case. The ETM near km 5 is still present but only very weakly, while there is still a clear ETM near km 50. Mean concentrations in the dynamic case are significantly smaller than in the quasi-stationary case, while variations over the cycle remain smaller in the dynamic case (panel b). The estuary never reaches capacity conditions (f<1), and maximum erodibility in both ETM zones is reached close to neap tide (Fig. f). Maximum concentrations are now found between spring and neap tide, when erodibility is approaching its maximum, while the bed shear stress is not yet at its minimum. After neap tide, the ETM does not disappear as in the quasi-steady case, but concentrations do decrease after neap tide. This is because the bottom pool is not big enough to remain present for long after neap tide, preventing increasing concentrations even though the bed shear stress increases.

Comparing the situations with high and low M, it is observed that there is a significant dynamic effect of the spring neap cycle in both cases, compared to the quasi-stationary state. In both cases, it takes significant time for the estuary to move the sediment stock around and adapt to the changing conditions. Under erosion-limited conditions, an important dynamic effect is the build-up and break-down of the bottom pool. During times of export, this means that erosion-limited conditions can be sustained for a significant amount of time or even the entire spring–neap cycle as it takes time before the bottom pool is drained. This is reflected in a different timing of maxima and minima in the sediment concentrations. Indeed, comparing Fig. c and d, maximum concentrations for small M occur at the time when minimum concentrations are attained for large M and vice versa. Since the underlying transport capacity is identical in both cases, this is due to the bottom pool thickness relative to the possible re-suspension by the tide.

In previous studies of the effects of spring–neap tides on salt distributions in estuaries, the dynamic effects of the spring–neap cycle are often quantified by measuring the delay between the salinity distribution and the tidal amplitude (e.g. ). For sediment, even in our highly simplified model, such a relationship is not possible. Figure  already shows the complicated relationship between the tidal amplitude and transport capacity. As the various physical contributions scale differently with the tidal amplitudes and D2-D4 phase difference, the spring–neap variation in total sediment transport capacity is not a monotone function of the tidal amplitude. This total transport capacity determines the tendency for the estuary to redistribute sediment over time which then results in an erodibility f and finally the sediment concentration. As there is no monotone relation between the spring–neap forcing and the transport capacity, there is no monotone response of the sediment concentration to the tidal amplitude, and hence an analysis in terms of delay is not possible.

We may, however, still estimate a timescale for the adjustment of the estuary to changing forcing conditions. If the adjustment time is much shorter than the spring–neap period, the dynamic SPM concentration will be similar to the quasi-steady result. If the adjustment time is similar to or bigger than the spring–neap period, results may deviate significantly from the quasi-stationary results, as shown in the previous sections. A simple expression to estimate the order of magnitude of the adjustment time of the estuary Tadj could be 15Tadj=max⁡t∫0LBSdx-min⁡t∫0LBSdx〈|T|〉x=0, i.e. the difference between the maximum and minimum total amount of sediment in the estuary divided by the mean (〈⋅〉) absolute sediment transport at the mouth. Note that this timescale is diagnostic – i.e. it requires knowledge of the sediment balance to compute it. By this definition, Tadj in the default case is approximately 23 d. This is longer than the spring–neap cycle duration, consistent with the results showing significant dynamic effects.

Next, we discuss the application of the model to the Loire configuration , in which the model includes various additional physical processes (see Sect. ). The additional processes mean that the sediment concentration now influences the flow (through its effect on turbulence), sediment capacity, and transport capacity. The dynamic effects of the spring–neap cycle are therefore visible not only in the erodibility and sediment concentration, but also in the sediment capacity and transport capacity.

Figure a shows near-bed average concentration (lines) and variability over the spring–neap cycle (shaded areas) in the quasi-stationary and dynamic cases. The ETM zones are located around km 8 and 30 with mean concentrations of around 30 gL-1. Figure b and c show the time dependence of the near-bed concentration. The ETM zones only move a few kilometres over the spring–neap cycle for both the quasi-stationary and dynamic cases. In the quasi-stationary case there is a small time interval after neap tide when the ETM disappears from the estuary. In the dynamic case, however, the ETM mostly remains present during the spring–neap cycle. An exception is a small time interval near km 8 when stratification briefly collapses leading to somewhat lower concentrations. The along-channel maximum near-bed concentrations vary between 26 and 52 gL-1 over the spring–neap cycle, with higher near-bed concentrations at neap. Surface concentrations (Fig. d and e) have along-channel maxima which vary between 1 and 3 gL-1, with the maximum occurring after spring tide.

To better understand these results, we will again apply our analysis framework. Figure  shows the transport capacity scaled by the local near-bed velocity amplitude in the dynamic case at neap and spring tide. First concentrating on the total transport capacity, the downward zero crossings, indicating the locations of maximum erodibility, are located at approximately the same locations. This helps to explain the quite minor qualitative differences between spring and neap in the erodibility and sediment concentration noted earlier. The transport capacity at spring is larger than at neap tide, even relative to the near-bed velocity. During spring tide, the eddy viscosity is bigger mainly due to bigger shear stresses, and the effective settling velocity is smaller due to hindered settling. Hence, sediments are mixed up higher in the water column, and the water column can accommodate and transport more sediment. This effect of mixing causes all contributions to the transport capacity to increase at spring. In the km 8 ETM, the baroclinic contribution is most important, closely followed by the effect of the externally generated D4 tide. In the km 40 ETM, the external tidal asymmetry is dominant at neap, while spatial settling lag is dominant at spring.

The various contributions to the transport capacity no longer scale with the tidal amplitude exactly as in Table  due to the added physical processes. Table  presents the relative differences between spring and neap magnitude of the various transport contributions (not scaled by the near-bed velocity) in the areas 0–10 km (first column) and 20–40 km (second column). The relative spring–neap differences are much larger than the scaling of Table  (third column) suggests, especially in the first 10 km. Relatively, however, we still notice that the spatial settling lag and internal asymmetry are much more strongly amplified at spring due to their greater sensitivity to the tidal amplitude.

Next looking at the bottom pool dynamics, it is observed that the erodibility (Fig. f and g) is largest around the locations of the ETM zones, consistent with the convergence of transport capacity. The erodibility is highest around neap, with erosion-limited conditions in km 0–8. The amount of sediment in the bottom pool remains of the same order of magnitude during spring and neap (not shown). Erodibility is nevertheless highest during neap because the sediment capacity is low due to small bed shear stress and strong stratification. The quasi-stationary and dynamic cases show quite similar results, except for some time after neap tide, when the quasi-stationary case shows flushing of the bottom pool. In the dynamic case, the erodibility remains significantly larger than zero. Sediment is transported (Fig. i) to the bottom pool between km 0–10 around neap tide (∼0.5<t/T<1) and to the bottom pool at km 40 around spring tide (∼0<t/T<0.5). Hence, even though trapping occurs at approximately the same locations during the entire spring–neap cycle, sediment is transported between the two bottom pools between spring and neap.

Finally considering the SPM concentration in the dynamic case, the location of high near-bed concentrations (Fig. c) does not move much due to the stable locations of the bottom pools. The highest near-bed concentrations occur at neap because the erodibility is high and strong stratification keeps sediment confined close to the bed. Surface concentrations (Fig. e) correspondingly are lowest during neap. Maximum surface concentrations occur some time after spring tide due to larger mixing.

Summarising, we find that the transport capacity indicates similar trapping locations over the entire spring–neap cycle, with sediment transported between the two trapping areas. A combination between high erodibility and low bottom shear stress at neap and lower erodibility but higher bed shear stress at spring means quite high concentrations can be attained during the entire cycle. Observed peaks in bottom and surface concentrations are mainly related to the stratification, determining whether sediment can be mixed or is confined to the bed. The differences between the dynamic and quasi-stationary cases are quite small, except for a window of a few days after neap, indicating a rapid adjustment of the estuary. This is supported by the (diagnostic) adjustment scale (Sect. ), which in this case is approximately 4 d.

Our framework of analysis of the dynamics of SPM extends beyond the scope of this study. Firstly, such an analysis may be extended to any variation of the forcing parameters that is much slower than the D2 tidal timescale, such as seasonal discharge variations (see, for example, ). The analysis framework can partly be used to interpret results from numerical simulation models. The inclusion of spatially varying viscosity and diffusivity through more complex turbulence closures does not affect the interpretation framework. Non-linear interactions between water motion, salinity, SPM, and mixing within such turbulence closures do complicate a cause-and-effect analysis in the same way as we found in the Loire case but otherwise still allow the use of the analysis framework. Furthermore, it is possible to compute the bottom pool thickness, erodibility, and total sediment transport capacity from such models (see , Chap. 7 for an example in the Delft3D model). Decompositions of the transport capacity, such as the ones used in this study, can usually not be computed from simulation models.

The aim of this study was to provide insights into various effects of the spring–neap cycle on SPM dynamics, but it ignored various potentially important aspects. Some aspects that were not considered in the idealised case, such as the influence of velocity, salinity, and sediment concentration on turbulence parameters, were added in the Loire case study. Still, D2 tidal variations in the salinity and turbulence parameters were not considered, thus excluding effects of tidal straining and eddy viscosity-shear covariance (ESCO). These processes would affect the trapping and re-suspension dynamics in a complex way. Some other aspects that were ignored are worth mentioning specifically for further study. Firstly, the width averaging means the lateral dynamics are ignored. In estuaries it is quite possible that sediment accumulates on shallow flanks under erosion-limited conditions, while supply-limited conditions dominate in the deeper channel. The effects of spring–neap variations in such contexts still need further investigation. Also, it is assumed that all sediment in the bottom pool remains easily erodible without a critical threshold for the bed shear stress. The consolidation of sediments on the spring–neap timescale is not taken into account and may additionally significantly affect the dynamics. Thirdly, the present model only considers changes in the bottom pool over the spring–neap cycle as a consequence of the tidally averaged transport. This ignores the intratidal variations in the bottom pool thickness.

The two-timescale formalism as applied here ignores some dynamics that are assumed to be unimportant. For example, it was assessed that the interaction between the tidal and spring–neap cycle in the sediment balance is negligible, but this assumption still needs a posteriori verification. Also, it was assumed that the water motion adapts quickly to spring–neap conditions. p. 75 argue that the overall storage of water in the estuary can potentially be larger during spring tide than neap tide due to a different subtidal water set-up. Hence there is a subtidal flow of water in and out of the estuary over the spring–neap cycle. Our scaling arguments indicate this effect to be negligible, but to the authors' knowledge, this has not been systematically verified.

Quasi-stationary model results are frequently used to gain understanding of estuarine sediment dynamics (e.g. ). Since quasi-stationary results do not depend on the forcing history but are consistent with the actual (tidal) forcing, this greatly reduces complexity. In cases where the dynamics of, for example, the spring–neap cycle are important, some caution needs to be taken in the interpretation of such results. First considering the case where quasi-stationary results are used to study spring–neap-averaged results, it is clear from our results that spring–neap mean sediment concentrations in the dynamic and quasi-stationary cases are not the same. By calibration using, for example, the erosion parameter, similar concentrations between the dynamic mean and quasi-stationary case may be nevertheless obtained, and hence one should be careful that the value of such a calibration parameter is highly context-dependent. As the contributions to the sediment transport capacity scale non-linearly with the tidal amplitude and D2-D4 phase difference, using an average spring–neap amplitude and phase difference results in an error in the relative importance of processes, which may result in a shift in the sediment trapping location. As long as similar processes are important during spring and neap tide, the dominant processes are still identified by taking a spring–neap average. Therefore, the quasi-stationary simulations should be more regarded as qualitatively than quantitatively correct when dynamics are important.

Secondly, we consider the use of quasi-stationary reasoning to understand the dynamics during spring or neap tidal phases. From our results it is concluded this should be done with caution, since the dynamic sediment concentrations may be quite different from the quasi-stationary ones, as there are significant memory effects. Hence, one would use these results best to indicate a range of the dynamics occurring over the spring–neap cycle, acknowledging that the dynamic spring–neap variability in SPM concentration is smaller.