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iFlow-Inwerking en ontwikkeling SKN 21 Incorporation of intertidal zones in iFlow

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iFlow-Inwerking en ontwikkeling SKN 21 Incorporation of intertidal zones in iFlow

Kaptein, S.J.; Schramkowski, G.; Mostaert, F.

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Flanders hydraulics Research Ref.: WL2020R18_153_1 Idealized modelling, numerical simulations, Scheldt, intertidal zones 24 Appendices (p.): 3 ‫ ܈‬No ‫ ܈‬Available online

Author(s):

Kaptein, S.J.

Control Name Reviser(s):

Schramkowski, G.

Project leader:

Schramkowski, G.

Signature

Approval Getekend door: Frank Mostaert (Signature) Getekend op: 2020-03-15 21:09:31 +01:00 Reden: Ik keur dit document goed

Head of Division:

F-WL-PP10-2 Version 7 Valid as from 3/01/2017

Mostaert, F.

iFlow�Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow

Abstract In this report, we present the effect of incorporating tidal zones in the idealized, width�based model iFlow. It appears that incorporating intertidal zones affects the hydrodynamics (i.e. water levels and flow velocities) via four different mechanisms. Each mechanism has been implemented in the two existing solvers of iFlow, the semi�analytical solver and the numerical solver. An additional solver, the full analytical solver, was created and is applicable to estuary models with a horizontal bottom and an exponentially converging width. It was found that the differences in results between these solvers where overall converging quadratically with increasing grid resolution, emphasizing correct implementation. However, one of the terms, does only converge linearly for a horizontal bottom. In the case of a more realistic bottom, quadratic convergence is still obtained for two of the four terms, while a third term only converges linearly and the fourth term does not converge with increasing grid resolution at all. Although these convergence discrepancies require some additional attention, the error between the terms stays less than 1% giving confidence into the implementation. Overall, the incorporation of intertidal zones was found to have a significant effect on the prediction of the M4 tide. The quantification of this effect has not been realized, mainly because the unrealistic bathymetry used did not allow for a significant analysis. The investigation of the quantitative effect of intertidal zones on the Scheldt is left for a follow up study.

fields of knowledge: Idealized modelling, numerical simulations, Scheldt, intertidal zones

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iFlow�Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow

Contents Abstract................................................................................................................... III List of Figures ............................................................................................................ VI List of Tables ............................................................................................................. VII 1 Introduction .........................................................................................................

1

2 Reformulation of the equations including the effects of intertidal zones ....................................

2.1 Governing equations ...........................................................................................

2.2 Perturbation method and extension to the width and depth variables ..................................

2.3 Parameterization of depth and width variations............................................................

2 2 3 5

3 Analytical results ....................................................................................................

3.1 Analytical formulation, general case .........................................................................

3.2 Analytical formulation, steady component ..................................................................

3.2.1 Depth return flow .........................................................................................

3.2.2 Width return flow .........................................................................................

3.3 Analytical formulation, M4 component, general case .....................................................

3.4 Analytical solution for a horizontal bottom and an exponential width variation .......................

3.4.1 Equations ...................................................................................................

3.4.2 Homogeneous solution ...................................................................................

3.4.3 Particular solutions ........................................................................................

3.4.4 Total solution ...............................................................................................

3.4.5 Individual forcing terms ...................................................................................

6 6 7 8 8 9 10 10 11 11 11 12

4 Simulation results ................................................................................................... 15 4.1 Validation ........................................................................................................ 15 4.2 Effect of the incorporation of intertidal zones on the total M4 surface elevation (horizontal bottom) 20 4.3 Case of a varying bottom ...................................................................................... 22 4.4 Conclusion....................................................................................................... 23 A1 Intermediate steps for the computation of the width/depth dependent M4 ...............................

A1.1 Resolution method for the particular solution ..............................................................

A1.2 Integration constants of the homogeneous solution .......................................................

A1.3 Derivatives of the leading order solutions ...................................................................

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A1 A1 A2 A3

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iFlow‐Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow

List of Figures Figure 1

Evolution of the relative variation in width over the length of the estuary. .........................

Figure 2

Evolution of the differences in water levels computed by the semi‐analytical and the fully analytical solver (configuration of Wei et al., 2016). ...................................................

Evolution of the differences in water levels computed by the numerical and the fully analytical solver (configuration of Wei et al., 2016). ...............................................................

Evolution of the differences in water levels computed by the numerical and the semi‐analytical solver (configuration of Wei et al., 2016). ...............................................................

Evolution of the differences in water levels computed by the numerical and the fully analytical solver (configuration of Wei et al., 2016). ...............................................................

Leading order solutions for the water level and the bottom velocity for the Wei et al., 2016 configuration. ...............................................................................................

Leading order solutions for the water level and the bottom velocity for the Brouwer et al., 2017 configuration. ........................................................................................

First order surface elevation decomposed into separate width‐depth contributions (Wei et al., 2016) (only full analytical results). ...................................................................

First order surface elevation decomposed into separate width‐depth contributions (Wei et al., 2016) (only full analytical results). ...................................................................

Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9

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1 16 17 18 19 20 21 21 22

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List of Tables Table 1 Table 2

Table 3

Parameter values for an idealized Scheldt model (Wei et al., 2016). .................................. 15 Parameter values for a different idealized Scheldt model Brouwer et al., 2017. Note that these parameters also include a phase shift of ‐1.3∘ between the external M4 and the external M2 tides. .......................................................................................................... 20 Coefficients for the polynomial approximation of the water depth.................................... 22

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VII

iFlow‐Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow

1 Introduction Processed based, exploratory models such as iFlow (Dijkstra et al., 2017) have been used with increasing fre‐ quency over the last years, both in scientific research (Brouwer et al., 2018; Dijkstra et al., 2019a,b) or in studies following requests from customers (Brouwer et al., 2017). The benefits of these models with respect to clas‐ sical three‐dimensional models (such as Telemac), is (i) their speed and (ii) the possibility to decompose output variables (e.g. water‐level or flow velocity) into individual components related to specific physical mechanisms.

A disadvantage of these models is that they are highly idealized or simplified in terms of both geometry and governing equations. This implies that some crucial mechanisms influencing the hydrodynamics or sediment transport might be overlooked. Since iFlow is a width‐averaged model, the width of an estuary, as defined in the model, might vary over its length, but is always taken constant in time. Nevertheless, basic studies of bathymetry in the Scheldt show that the width of the Scheldt can vary with up to 40% on an intertidal time‐ scale at some location (see Fig. 1). This phenomenon is caused by large areas called intertidal zones, that are covered by water at high tide but fall dry during low‐tide. This feature implies that some major properties of Figure 1 – Evolution of the relative variation in width over the length of the estuary.

intertidal zones are not yet included in iFlow. An example of such a property is the additional volume offered by intertidal zones for storage of water, which has a direct impact on the water‐level. A possibility to circum‐ vent this shortcoming within the iFlow framework, is to modify its governing equations to allow the width of the estuary to vary with the water‐level. In the remaining part of the report, we first discuss the adaptation of the governing equations. Subsequently, we evaluate our approach via the investigation of the parameter space for a simplified geometry of an estuary and compare analytical solutions to the iFlow results. Finally, the model with intertidal zone effects is applied to a realistic geometry of the Scheldt river.

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2 Reformulation of the equations including the effects of intertidal zones 2.1

Governing equations

According to Dijkstra et al., 2017, the governing equations in iFlow consist of a widthâ€?averaged Shallowâ€?water equation for the conservation of momentum and two continuity equations. However, in the present study the vertical velocity does not play a role of interest. As a result, we limit ourselves to a widthâ€?averaged and depthâ€? integrated equation for the conservation of mass governing the surface elevation. The equations of interest are đ?‘…+đ?œ

đ?œ•đ?‘˘ đ?œ•đ?‘˘ đ?œ•đ?œ 1 đ?œ•đ?œŒ đ?œ• đ?œ•đ?‘˘ đ?œ•đ?‘˘ (đ?œˆđ?‘‡ ) , +đ?‘˘ +đ?‘¤ = −đ?‘” − âˆŤ đ?‘‘đ?‘§ Ěƒ + đ?œ•đ?‘Ą đ?œ•đ?‘Ľ đ?œ•đ?‘§ đ?œ•đ?‘Ľ đ?œŒref đ?œ•đ?‘Ľ đ?œ•đ?‘§ đ?œ•đ?‘§

(1a)

�

đ?‘…+đ?œ

đ??ľ

đ?œ•đ?œ đ?œ• ⎛ ⎞ ⎜ + ⎜đ??ľ âˆŤ đ?‘˘đ?‘‘đ?‘§ âŽ&#x; âŽ&#x;=0 đ?œ•đ?‘Ą đ?œ•đ?‘Ľ âŽ? −đ??ť âŽ

(1b)

in which đ?‘˘ is the horizontal velocity variable, đ?œ the surface elevation, đ?‘Ľ is the along estuary coordinate, đ?‘§ the vertical coordinate and đ?‘Ą time. The symbol represents đ?‘” the acceleration of gravity, đ?‘… the mean level of the river surface, đ?œŒref the reference density, đ?œŒ the density of the fluid, đ?œˆđ?‘‡ the eddyâ€?viscosity, đ??ť the depth and đ??ľ the width. It was chosen to highlight the width with a different color such that its role within the governing equations is more visible. The depth is also displayed in a different color because it is demonstrated in the Sec. 2.3 that width and depth are intrinsically linked via the widthâ€?averaging process.

A set of boundary conditions is associated to the governing equations. These boundary conditions comprise a partial slip condition at the bed (i.e. đ?‘§ = −đ??ť), a noâ€?stress condition at the surface (i.e. đ?‘§ = đ?‘… + đ?œ ), a timeâ€? dependent tidal forcing at the seaward boundary (i.e. đ?‘Ľ = 0) and an imposed river discharge at the landward boundary (i.e. đ?‘Ľ = đ??ż). The equations for these boundary conditions are đ?œ•đ?‘˘ = đ?‘ đ?‘“ đ?‘˘ đ?œ•đ?‘§ đ?œ•đ?‘˘ đ?œˆđ?‘‡ =0 đ?œ•đ?‘§ đ?œ = đ??´M2 cos(đ?œŽđ?‘Ą) + đ??´M4 cos(2đ?œŽđ?‘Ą + đ?œ‘) đ?œˆđ?‘‡

at đ?‘§ = −đ??ť,

(2a)

at đ?‘§ = đ?‘… + đ?œ ,

(2b)

at đ?‘Ľ = 0,

(2c)

at đ?‘Ľ = đ??ż,

(2d)

đ?‘…+đ?œ

đ??ľ âˆŤ đ?‘˘đ?‘‘đ?‘§ = −đ?‘„ −đ??ť

respectively. Several parameters appear: đ?‘ đ?‘“ , the friction coefficient, đ??´M2 and đ?œŽ, respectively the amplitude and the frequency of the M2 tide, and đ??´M4 , the amplitude of the M4 tide.

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2.2

Perturbation method and extension to the width and depth variables

The iFlow philosophy for solving the governing equations is based on the perturbation technique. The first assumption upon which this technique is based is that some of the variables are decomposable into a series of terms. In these series, each term is of order đ?œ€ = đ?œ /đ??ť with respect to the previous term and đ?œ€ ≪ 1. For example, a variable đ?œ‚ can be written as đ?œ‚ = đ?œ‚ 0 + đ?œ‚1 + đ?œ‚2 + â‹Ż with đ?œ‚1 /đ?œ‚0 = đ?‘‚(đ?œ€), đ?œ‚2 /đ?œ‚1 = đ?‘‚(đ?œ€), etc. Up to now, this decomposition technique has been applied to the unknown variables (i.e. đ?‘˘, đ?‘¤ and đ?œ ), as well as to some of the physical mechanisms (tidal elevation at the seaward boundary, river flux at the landward boundary), but not to the width đ??ľ or to the depth đ??ť. From now on, also these variables are split following đ??ľ = đ??ľ0 + đ??ľ1 + â‹Ż , đ??ť = đ??ť0 + đ??ť1 + â‹Ż .

The second assumption of the perturbation technique is based on a harmonic decomposition. The harmonic decomposition implies that each variable evolves periodically in time, according to a specific frequency that is characteristic of the variable’s order. At leading order, the variables will evolve according to the frequency of the M2 tide, while at first order, the variables will evolve according to the frequency of the M4 tide, but will also have a timeâ€?independent component referred to as the ’M0 frequency’. As a result of this assumption, it is very useful to define a complex function, denoted by â‹…,Ě‚ verifying ∗ đ?œ‚02 Ě‚ (đ?‘Ľ, đ?‘§) iđ?œŽđ?‘Ą đ?œ‚02 Ě‚ (đ?‘Ľ, đ?‘§) −iđ?œŽđ?‘Ą đ?‘’ + đ?‘’ 2 2 âˆ—Ě‚ (đ?‘Ľ, đ?‘§) đ?œ‚ Ě‚ (đ?‘Ľ, đ?‘§) 2iđ?œŽđ?‘Ą đ?œ‚14 đ?œ‚1 (đ?‘Ľ, đ?‘§, đ?‘Ą) = 14 + đ?‘’ đ?‘’−2iđ?œŽđ?‘Ą +đ?œ‚10 â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x; 2 2

đ?œ‚0 (đ?‘Ľ, đ?‘§, đ?‘Ą) =

(3) (4)

đ?œ‚14

where đ?œ‚ designates either đ?‘˘, đ?‘¤ or đ?œ . The first order solution consist of a signal evolving as 2đ?œŽđ?‘Ą, which has Ě‚ |, and a timeâ€?independent signal, of amplitude đ?œ‚10 . In these notations, the first a complex amplitude of |đ?œ‚14 subscript (hereafter also denoted đ?‘?) refers to the order and the second subscript (hereafter also denoted đ?‘˜) refers to the tidal component. The superscript ∗ denotes the complex conjugated associated to đ?œ‚.Ě‚ By defining đ?œ‚đ?‘?đ?‘˜ Ě‚ = âˆŁđ?œ‚đ?‘?đ?‘˜ Ě‚ âˆŁ đ?‘’đ?‘–đ?œ‘ , it is then easy to show that đ?œ‚0 = |đ?œ‚02 Ě‚ (đ?‘Ľ, đ?‘§)| cos (đ?œŽđ?‘Ą + đ?œ‘(đ?‘Ľ, đ?‘§)) ,

(5)

đ?œ‚14 = |đ?œ‚14 Ě‚ (đ?‘Ľ, đ?‘§)| cos (2đ?œŽđ?‘Ą + đ?œ‘(đ?‘Ľ, đ?‘§)) ,

(6)

in other words, the amplitudes of đ?œ‚đ?‘?đ?‘˜ and đ?œ‚đ?‘?đ?‘˜ Ě‚ are equal. This formulation has implications for the time derivâ€? atives đ?œ• đ?‘“đ?‘›Ě‚ = đ?‘›iđ?œŽđ?‘“đ?‘›Ě‚ . đ?œ•đ?‘Ą This second assumption directly implies that the parameterization of đ??ľ and đ??ť has be chosen such that the velocity and flow patterns they generate are consistent with Eqs (3) and (4). This condition will be further detailed in Sec. 2.3.

Applying the decomposition method to the set of governing equations Eqs.(1aâ€???) gives at leading order đ?œ•đ?‘˘0 đ?œ• đ?œ•đ?‘˘ đ?œ•đ?œ − (đ?œˆđ?‘‡ 0 ) = −đ?‘” 0 , đ?œ•đ?‘Ą đ?œ•đ?‘§ đ?œ•đ?‘§ đ?œ•đ?‘Ľ

(7a)

đ?‘…

đ?œ•đ?œ đ?œ• ⎛ ⎞ ⎜ đ??ľ0 0 + ⎜đ??ľ0 âˆŤ đ?‘˘0 đ?‘‘đ?‘§ âŽ&#x; âŽ&#x; = 0, đ?œ•đ?‘Ą đ?œ•đ?‘Ľ âŽ? −đ??ť0 âŽ

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(7b)

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iFlowâ€?Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow for the governing equations, and đ?œ•đ?‘˘0 = đ?‘ đ?‘“ đ?‘˘0 đ?œ•đ?‘§ đ?œ•đ?‘˘ đ?œˆđ?‘‡ 0 = 0 đ?œ•đ?‘§ đ?œ 0 = đ??´M2 cos(đ?œŽđ?‘Ą)

đ?œˆđ?‘‡

at đ?‘§ = −đ??ť0 ,

(8a)

at � = �,

(8b)

at đ?‘Ľ = 0,

(8c)

at đ?‘Ľ = đ??ż,

(8d)

đ?‘…

đ??ľ0 âˆŤ đ?‘˘0 đ?‘‘đ?‘§ = 0 −đ??ť0

for the boundary conditions. For the exact scaling assumptions, in particular for the convective terms and the baroclinic term, the reader is referred to Chernetsky et al., 2010; Dijkstra et al., 2017. The attentive reader will remark that at leading order no changes have occurred to the governing equations, which is in agreement with the hypothesis that width variations only appear at first order.

At first order, the governing equations Eqs.(1a�1b) become �

đ?œ• đ?œ•đ?‘˘1 đ?œ•đ?œ 1 đ?œ•đ?œŒ0 đ?œ•đ?‘˘ đ?œ•đ?‘˘ đ?œ•đ?‘˘1 − (đ?œˆ )=−đ?‘” 1 âˆ’âˆŤ đ?‘‘ đ?‘§ Ěƒ − đ?‘˘ 0 0 − đ?‘¤0 0 , đ?œ•đ?‘Ą đ?œ•đ?‘§ đ?‘‡ đ?œ•đ?‘§ đ?œ•đ?‘Ľ đ?œŒref đ?œ•đ?‘Ľ đ?œ•đ?‘Ľ đ?œ•đ?‘§

(9a)

�

đ?‘…

đ?œ•đ?œ đ?œ• ⎛ đ?œ• ⎞ ⎜ đ??ľ0 1 = − (đ??ľ đ?‘˘ | đ?œ ) ⎜đ??ľ0 âˆŤ đ?‘˘1 đ?‘‘đ?‘§ âŽ&#x; âŽ&#x;− đ?œ•đ?‘Ą đ?œ•đ?‘Ľ đ?œ•đ?‘Ľ 0 0 đ?‘… 0 âŽ? −đ??ť0 ⎠đ?‘…

đ?œ•đ?œ đ?œ• đ?œ• ⎛ ⎞ ⎜ (đ??ľ0 đ?‘˘0|−đ??ť0 đ??ť1 ) − đ??ľ1 0 − − âŽ&#x;, ⎜đ??ľ1 âˆŤ đ?‘˘0 đ?‘‘đ?‘§ âŽ&#x; đ?œ•đ?‘Ľ đ?œ•đ?‘Ą đ?œ•đ?‘Ľ â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x; ⎠âŽ? −đ??ť0 Additional depth forcing â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;

(9b)

Additional width forcing

for the governing equations, and đ?œˆđ?‘‡

đ?œ•đ?‘˘1 đ?œ•đ?‘˘ đ?œ• đ?œ•đ?‘˘ =đ?‘ đ?‘“ đ?‘˘1 − đ?‘ đ?‘“ đ??ť1 0 + đ??ť1 (đ?œˆđ?‘‡ 0 ) đ?œ•đ?‘§ đ?œ•đ?‘§ đ?œ•đ?‘§ đ?œ•đ?‘§ â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;â?&#x;

at đ?‘§ = −đ??ť0 ,

(10a)

at � = �,

(10b)

at đ?‘Ľ = 0,

(10c)

at đ?‘Ľ = đ??ż,

(10d)

Additional depth forcing

đ?œˆđ?‘‡

đ?œ•đ?‘˘1 đ?œ• đ?œ•đ?‘˘ = − đ?œ 0 (đ?œˆđ?‘‡ 0 ) đ?œ•đ?‘§ đ?œ•đ?‘§ đ?œ•đ?‘§ đ?œ 1 =đ??´M4 cos(2đ?œŽđ?‘Ą − đ?œ‘)

đ?‘…

đ?‘…

đ?‘„ đ??ľ âˆŤ đ?‘˘1 đ?‘‘đ?‘§ = − 1 − đ?‘˘0|đ?‘… đ?œ 0 â?&#x;â?&#x;â?&#x;â?&#x;â?&#x; − đ?‘˘0|−đ??ť0 đ??ť1 − 1 âˆŤ đ?‘˘0 đ?‘‘đ?‘§ đ??ľ0 đ??ľ0 −đ??ť0 −đ??ť0 â?&#x;â?&#x;â?&#x;â?&#x;â?&#x; Additional depth

Additional

forcing

width forcing

for the boundary conditions. It is only at the first order that the width variations start to influence the equations of motion through the appearance of new forcing terms denoted ’additional width forcing’ or ’additional depth forcing’. These two terms appear in the two continuity equations, in the bottom boundary condition and in the landward boundary condition, where the river flow rate is imposed.

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2.3

Parameterization of depth and width variations

As mentioned previously, the parameterization of depth and width variation cannot be taken arbitrarily. In order to be consistent with the ordering and harmonic decomposition of the variables, it is required that đ??ľ0 and đ??ť0 are constant, while đ??ľ1 and đ??ť1 vary linearly with the water level đ?œ 02 , the latter evolving with a đ?œŽâ€? frequency. A parameterization satisfying these conditions for đ??ľ is đ??ľ = đ??ľ0 +

Δđ??ľ đ?œ +â‹Ż, 2đ??´M2 02 â?&#x;

(11)

đ??ľ1

where Δđ??ľ is the difference in width of the estuary between high and low water.

Since iFlow is a widthâ€?averaged model, conservation of the crossâ€?section implies that changes in the estuary width also involve changes in the estuary depth. This implication is taken account for by defining variation of the crossâ€?section đ?’œ with respect to the water depth as đ?œ

đ?’œ = âˆŤ đ??ľ(đ?œ ′ )đ?‘‘đ?œ ′ .

(12)

−∞

Simultaneously, by definition đ?’œ = đ??ľ (đ?œ ) (đ??ť + đ?œ ) .

(13)

Differentiation Eqs (12) and (13) against đ?œ , and equating the result gives đ??ľ(đ?œ ) =

đ?œ•đ??ľ đ?œ•đ??ť (đ??ť + đ?œ ) + đ??ľ + đ??ľ (đ?œ ) . đ?œ•đ?œ đ?œ•đ?œ

(14)

Finally, by noticing that đ??ť ≍ đ?œ , a first order approximation gives đ?œ•đ??ť đ??ť đ?œ•đ??ľ =− . đ?œ•đ?œ đ??ľ đ?œ•đ?œ

(15)

Using Eq. (11) into Eq. (15) gives an ordering of đ??ť: đ??ť Δđ??ľ đ??ť (đ?œ ) = đ??ť0 − 0 đ?œ 02 2đ??´â?&#x; M2 â?&#x;â?&#x;đ??ľ â?&#x;0â?&#x; â?&#x;â?&#x;

(16)

đ??ť1

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3 Analytical results 3.1

Analytical formulation, general case

The new terms related to width (and depth) variations only appear at first order. Accordingly, the resolution will focus on the equations for the first order variables. Equations (9a�9b) can be re�written introducing new symbols for the forcing terms in order to highlight them. For simplicity, the horizontal density gradient and the eddy viscosity are assumed to be constant over the vertical and the river elevation is neglected (� = 0).

These assumptions give đ?œ•đ?‘˘1 đ?œ• 2 đ?‘˘1 đ?œ•đ?œ đ?œ•đ?œŒ − đ?œˆđ?‘‡ = − đ?‘” 1 − đ?œ‰(đ?‘Ľ, đ?‘§, đ?‘Ą) + đ?‘” â&#x;¨ â&#x;Š đ?‘§, đ?œ•đ?‘Ą đ?œ•đ?‘§ 2 đ?œ•đ?‘Ľ đ?œ•đ?‘Ľ

(17a)

0

đ?œ•đ?œ 1 đ?œ• 1 đ?œ•đ??ľ0 ⎛ 1 đ?œ• đ??ľ ⎞ đ?‘… đ??ť )⎜ =−( + đ?›ž (đ?‘Ľ, đ?‘Ą) − đ?œ đ??ľ (đ?‘Ľ, đ?‘Ą), ⎜ âˆŤ đ?‘˘1 đ?‘‘đ?‘§ + đ?›ž (đ?‘Ľ, đ?‘Ą) + đ?›ž (đ?‘Ľ, đ?‘Ą)âŽ&#x; âŽ&#x;− đ?œ•đ?‘Ą đ?œ•đ?‘Ľ đ??ľ0 đ?œ•đ?‘Ľ đ??ľ0 đ?œ•đ?‘Ľ âŽ?−đ??ť0 ⎠(17b) The associated boundary conditions, Eqs (10aâ€?10d), can be simplified to đ?œ•đ?‘˘1 đ?‘ đ?‘“ = đ?‘˘1 + đ?œ’đ??ť (đ?‘Ľ, đ?‘Ą) đ?œ•đ?‘§ đ?œˆđ?‘‡ đ?œ•đ?‘˘1 = − đ?œ’đ?‘… (đ?‘Ľ, đ?‘Ą) đ?œ•đ?‘§ đ?œ 1 =đ??´M4 cos(2đ?œŽđ?‘Ą + đ?œ‘) 0

âˆŤ đ?‘˘1 đ?‘‘đ?‘§ = − −đ??ť0

đ?‘„1 1 đ??ľ − đ?›ž đ?‘… (đ?‘Ľ, đ?‘Ą) − đ?›ž đ??ť (đ?‘Ľ, đ?‘Ą) − đ?›ž (đ?‘Ľ, đ?‘Ą) đ??ľ0 đ??ľ0

at đ?‘§ = −đ??ť0 ,

(18a)

at � = 0,

(18b)

at đ?‘Ľ = 0,

(18c)

at đ?‘Ľ = đ??ż.

(18d)

The newly introduced symbols, đ?œ‰, đ?›ž đ?‘… , đ?›ž đ??ť , đ?œ đ??ľ , đ?œ’đ?‘… , đ?œ’đ??ť and yield đ?œ•đ?‘˘0 đ?œ•đ?‘˘ + đ?‘¤0 (đ?‘Ľ, đ?‘§, đ?‘Ą) 0 , đ?œ•đ?‘Ľ đ?œ•đ?‘§ đ?›ž đ?‘… (đ?‘Ľ, đ?‘Ą) = đ?œ 0 (đ?‘Ľ, đ?‘Ą)đ?‘˘0 (đ?‘Ľ, đ?‘Ą)|đ?‘§=0 ,

(19b)

đ?›ž đ??ť (đ?‘Ľ, đ?‘Ą) = đ??ť1 (đ?‘Ľ, đ?‘Ą)đ?‘˘0 (đ?‘Ľ, đ?‘Ą)|đ?‘§=−đ??ť ,

(19c)

đ?œ‰(đ?‘Ľ, đ?‘§, đ?‘Ą) = đ?‘˘0 (đ?‘Ľ, đ?‘§, đ?‘Ą)

(19a)

0

0

đ?›ž (đ?‘Ľ, đ?‘Ą) = đ??ľ1 âˆŤ đ?‘˘0 đ?‘‘đ?‘§, đ??ľ

(19d)

−đ??ť0

đ??ľ1 đ?œ•đ?œ 0 , đ??ľ0 đ?œ•đ?‘Ą đ?‘ đ?œ•đ?‘˘0 đ?œ• 2 đ?‘˘0 âˆŁ + đ??ť1 âˆŁ , đ?œ’đ??ť (đ?‘Ľ, đ?‘Ą) = − đ?‘“ đ??ť1 đ?œˆđ?‘‡ đ?œ•đ?‘§ −đ??ť0 đ?œ•đ?‘§ 2 −đ??ť đ?œ đ??ľ (đ?‘Ľ, đ?‘Ą) =

(19e) (19f)

0

đ?œ’đ?‘… (đ?‘Ľ, đ?‘Ą) = đ?œ 0

đ?œ• 2 đ?‘˘0 âˆŁ , đ?œ•đ?‘§ 2 0

(19g) (19h)

6

WL2020R18_153_1

Final version

iFlow�Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow

3.2

Analytical formulation, steady component

Since the equations governing the steady state variables are linear, the solution consists of the sum of the solutions to the individual forcing terms. As a result, the forcing terms that are not related to width or depth variations will not be considered here. The equations governing the steady state component of the variables are obtained by replacing the different complex quantities by the formulation of Eqs (3), (4), which gives

đ?œˆđ?‘‡

đ?œ• 2 đ?‘˘10 đ?œ•đ?œ =đ?‘” 10 , 2 đ?œ•đ?‘§ đ?œ•đ?‘Ľ

(20a) 0

đ?œ• ⎛ ⎛ ⎞ ⎞ đ??ť đ??ľ ⎜ 0=− âŽ&#x; + đ?›ž10 (đ?‘Ľ)âŽ&#x; âŽ&#x; ⎜ đ??ľ0 ⎜ ⎜ âˆŤ đ?‘˘10 đ?‘‘đ?‘§ + đ?›ž10 (đ?‘Ľ)âŽ&#x; đ?œ•đ?‘Ľ âŽ? âŽ?−đ??ť0 ⎠âŽ

(20b)

and the associated boundary conditions đ?œ•đ?‘˘10 đ?‘ đ?‘“ = đ?‘˘10 + đ?œ’đ??ť 10 (đ?‘Ľ) đ?œ•đ?‘§ đ?œˆđ?‘‡ đ?œ•đ?‘˘10 =0 đ?œ•đ?‘§ đ?œ 10 =0 0 đ??ť âˆŤ đ?‘˘10 đ?‘‘đ?‘§ = − đ?›ž10 (đ?‘Ľ) −

−đ??ť0

1 đ??ľ đ?›ž (đ?‘Ľ) đ??ľ0 10

at đ?‘§ = −đ??ť0 ,

(21a)

at � = 0,

(21b)

at đ?‘Ľ = 0,

(21c)

at đ?‘Ľ = đ??ż..

(21d)

In these equations, the forcing terms of the steady state variables area đ??ťĚ‚ 12 đ?‘˘Ě‚∗02 + đ??ťĚ‚ ∗12 đ?‘˘Ě‚02 , 4 0 0 1 1 = đ??ľĚ‚ 12 âˆŤ đ?‘˘Ě‚∗02 đ?‘‘đ?‘Ľ + đ??ľĚ‚ ∗12 âˆŤ đ?‘˘Ě‚02 đ?‘‘đ?‘Ľ, 4 4 −đ??ť0 −đ??ť0

đ??ť = đ?›ž10

(22)

đ??ľ đ?›ž10

(23)

đ?œ’đ??ť 10 = −

1 đ?‘ đ?‘“ Ě‚ ∗ đ?œ• đ?‘˘Ě‚02 1 đ?œ• 2 đ?‘˘Ě‚∗02 1 đ?œ• 2 đ?‘˘Ě‚02 1 đ?‘ đ?‘“ Ě‚ đ?œ• đ?‘˘Ě‚∗02 đ??ť 12 âˆŁ − đ??ť 12 âˆŁ + đ??ťĚ‚ 12 âˆŁ + đ??ťĚ‚ ∗12 âˆŁ .

2 4 đ?œˆđ?‘‡ đ?œ•đ?‘§ −đ??ť 4 đ?œˆđ?‘‡ đ?œ•đ?‘§ −đ??ť 4 đ?œ•đ?‘§ −đ??ť 4 đ?œ•đ?‘§ 2 −đ??ť 0

0

0

(24)

0

Additionally, integrating Eq. (20b) along đ?‘Ľ between đ?‘Ľ and đ??ż, using Eq. (21d) gives 0 đ??ť (đ?‘Ľ) + âˆŤ đ?‘˘10 đ?‘‘đ?‘§ + đ?›ž10

−đ??ť0

1 đ??ľ đ?›ž (đ?‘Ľ) = 0 đ??ľ0 10

(25)

Depth induced partial slip The depth induced partial slip boundary condition velocity verifies đ?œ• 2 đ?‘˘dips đ?‘” đ?œ•đ?œ dips = , 2 đ?œ•đ?‘§ đ?œˆđ?‘‡ đ?œ•đ?‘Ľ đ?œ•đ?‘˘dips đ?‘ đ?‘“ = đ?‘˘dips + đ?œ’đ??ť 10 (đ?‘Ľ) đ?œ•đ?‘§ đ?œˆđ?‘‡ đ?œ•đ?‘˘dips =0 đ?œ•đ?‘§

(26a) at đ?‘§ = −đ??ť0 ,

(26b)

at � = 0.

(26c)

The solution to this set of equations is �dips = (

Final version

đ?œ•đ?œ dips đ?œˆđ?‘‡ đ??ť đ?‘§2 đ??ť 2 đ??ť − 0 − 0)đ?‘” − đ?œ’10 (đ?‘Ľ) 2đ?œˆđ?‘‡ 2đ?œˆđ?‘‡ đ?‘ đ?‘“ đ?œ•đ?‘Ľ đ?‘ đ?‘“ WL2020R18_153_1

(27)

7

iFlow�Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow From Eq. (25), we have that 0

âˆŤ đ?‘˘dips đ?‘‘đ?‘§ = 0,

(28)

−đ??ť0

so that

3.2.1

đ?œ•đ?œ dips đ??ť =− 0 đ?œ•đ?‘Ľ đ?‘ đ?‘“

đ?œˆđ?‘‡ đ?œ’đ??ť 10 (đ?‘Ľ) 1 đ??ť0 3 đ??ť0 2 ) đ?‘”( + 3 đ?œˆđ?‘‡ đ?‘ đ?‘“

(29)

Depth return flow

The depth return flow velocity verifies đ?œ• 2 đ?‘˘depth đ?‘” đ?œ•đ?œ depth = , 2 đ?œ•đ?‘§ đ?œˆđ?‘‡ đ?œ•đ?‘Ľ đ?œ•đ?‘˘depth đ?‘ đ?‘“ = đ?‘˘depth đ?œ•đ?‘§ đ?œˆđ?‘‡ đ?œ•đ?‘˘depth =0 đ?œ•đ?‘§

(30a) at đ?‘§ = −đ??ť0 ,

(30b)

at � = 0,

(30c) (30d)

The solution to this set of equations is �depth (�) = (

đ?œ•đ?œ depth đ?‘§2 đ??ť 2 đ??ť − 0 − 0)đ?‘” .

2đ?œˆđ?‘‡ 2đ?œˆđ?‘‡ đ?‘ đ?‘“ đ?œ•đ?‘Ľ

(31)

The depth return flow induced surface elevation follow from the condition 0 đ??ť âˆŤ đ?‘˘depth đ?‘‘đ?‘§ = −đ?›ž10 (đ?‘Ľ)

(32)

−đ??ť0

such that

3.2.2

đ?œ•đ?œ depth = đ?œ•đ?‘Ľ

đ??ť đ?›ž10 (đ?‘Ľ) 3 1 đ??ť0 đ??ť 2 đ?‘”( + 0 ) 3 đ?œˆđ?‘‡ đ?œˆđ?‘‡

(33)

Width return flow

The width return flow velocity verifies đ?œ• 2 đ?‘˘width đ?‘” đ?œ•đ?œ width = , 2 đ?œ•đ?‘§ đ?œˆđ?‘‡ đ?œ•đ?‘Ľ đ?œ•đ?‘˘width đ?‘ đ?‘“ = đ?‘˘width đ?œ•đ?‘§ đ?œˆđ?‘‡ đ?œ•đ?‘˘width =0 đ?œ•đ?‘§

(34a) at đ?‘§ = −đ??ť0 ,

(34b)

at � = 0,

(34c) (34d)

The solution to this set of equations is �width (�) = (

8

đ?‘§2 đ??ť 2 đ??ť đ?œ•đ?œ − 0 − 0 ) đ?‘” width .

2đ?œˆđ?‘‡ 2đ?œˆđ?‘‡ đ?‘ đ?‘“ đ?œ•đ?‘Ľ WL2020R18_153_1

(35)

Final version

iFlow�Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow The width return flow induced surface elevation is given by 0

âˆŤ đ?‘˘width đ?‘‘đ?‘§ = − −đ??ť0

such that

3.3

đ?œ•đ?œ width = đ?œ•đ?‘Ľ

1 đ??ľ đ?›ž (đ?‘Ľ) đ??ľ0 10

(36)

đ??ľ đ?›ž10 (đ?‘Ľ) 1 đ??ť0 3 đ??ť0 2 đ?‘”đ??ľ0 ( + ) 3 đ?œˆđ?‘‡ đ?œˆđ?‘‡

(37)

Analytical formulation, M4 component, general case

To find the equations governing the M4 tidal component, Eqs (4) and (3) are injected into the governing equations Eqs (17a) and (17b), and the associated boundary conditions (18a), (18bâ€?18d), and projected on đ?‘’2iđ?œŽđ?‘Ą . đ?œ• 2 đ?‘˘Ě‚14 đ?œ•đ?œ Ě‚ (38a) 2iđ?œŽđ?‘˘Ě‚14 − đ?œˆđ?‘‡ = − đ?‘” 14 2 đ?œ•đ?‘§ đ?œ•đ?‘Ľ 0

1 đ??ťâŽž 1 đ?œ• đ??ľ 1 đ??ľĚ‚ Ě‚ = − ( đ?œ• + 1 đ?œ•đ??ľ0 ) ⎛ ⎜ 2iđ?œŽđ?œ 14 đ?›ž14 Ě‚ − đ?œ 14 . Ě‚ âŽ&#x; ⎜ âˆŤ đ?‘˘Ě‚14 đ?‘‘đ?‘§ + đ?›ž14 âŽ&#x;− đ?œ•đ?‘Ľ đ??ľ0 đ?œ•đ?‘Ľ 2 2đ??ľ0 đ?œ•đ?‘Ľ 2 âŽ?−đ??ť0 âŽ

(38b)

The associated boundary conditions are đ?‘ đ?‘“ đ?œ• đ?‘˘Ě‚14 1 đ?‘˘Ě‚ + đ?œ’Ě‚ đ??ť = đ?œ•đ?‘§ đ?œˆđ?‘‡ (đ?‘Ľ) 14 2 14 đ?œ• đ?‘˘Ě‚14 =0 đ?œ•đ?‘§

at đ?‘§ = −đ??ť0 ,

(39a)

at � = 0

(39b) (39c)

with 1 1 đ??ť đ?›ž14 Ě‚ = đ??ťĚ‚ 12 (đ?‘Ľ, đ?‘Ą)đ?‘˘Ě‚02 (đ?‘Ľ, đ?‘Ą)âˆŁ đ?‘§=−đ??ť0 2 2

= 2 [đ?›ž đ??ť ]

(40)

= 2 [đ?›ž đ??ľ ]

(41)

Ě‚ 1 đ??ľĚ‚ 12 đ?œ• đ?œ 02 1 đ??ľĚ‚ đ?œ 14 = 2 2 đ??ľ0 đ?œ•đ?‘Ą

= 2 [đ?œ đ??ľ ]

(42)

đ?‘ 1 đ?œ• đ?‘˘Ě‚ đ?œ• 2 đ?‘˘Ě‚02 1 đ??ť âˆŁ ) + đ??ťĚ‚ 12 đ?œ’Ě‚ 14 = ( − đ?‘“ đ??ťĚ‚ 12 02 âˆŁ 2 2 đ?œˆđ?‘‡ đ?‘‘đ?‘§ −đ??ť đ?œ•đ?‘§ 2 −đ??ť

= 2 [đ?œ’đ??ť ]

(43)

đ?‘” đ?œ•đ?œ Ě‚ 1đ?œˆ (đ?›źM4 cosh (đ?‘&#x;M4 (đ?‘Ľ)đ?‘§) − 1) 14 − đ?‘‡ đ?›źM4 đ?œ’Ě‚ đ??ť cosh (đ?‘&#x;M4 (đ?‘Ľ)đ?‘§) 2iđ?œŽ đ?œ•đ?‘Ľ 2 đ?‘ đ?‘“

(44)

0

1 1 đ??ľ đ?›ž14 Ě‚ = đ??ľĚ‚ 12 âˆŤ đ?‘˘Ě‚02 đ?‘‘đ?‘§ 2 2 −đ??ť0

0

0

The general solution to Eq. (38a), satisfying the boundary conditions is đ?‘˘Ě‚14 (đ?‘Ľ, đ?‘§) = with đ?‘&#x;M4 (đ?‘Ľ) = √ đ?›źM4 =

Final version

2iđ?œŽ đ?œˆđ?‘‡ (đ?‘Ľ)

(45a)

đ?‘ đ?‘“ (đ?‘Ľ) . đ?‘&#x;M4 (đ?‘Ľ)đ?œˆđ?‘‡ (đ?‘Ľ) sinh(đ?‘&#x;M4 (đ?‘Ľ)đ??ť0 ) + đ?‘ đ?‘“ (đ?‘Ľ) cosh(đ?‘&#x;M4 (đ?‘Ľ)đ??ť0 )

WL2020R18_153_1

(45b)

9

iFlow�Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow Injecting Eq. (44) into Eq. (38b) gives the equation for the surface elevation, with four width or depth related forcing terms: �4

Ě‚ Ě‚ đ?œ• 2 đ?œ 14 đ?œ•đ?‘‡4 1 đ?œ•đ??ľ0 đ?œ• đ?œ 14 4đ?œŽ2 Ě‚ ) + ( + đ?‘‡ − đ?œ = đ??šdips + đ??šdepth + đ??šwidth + đ??šsurf đ?œ•đ?‘Ľ2 đ?œ•đ?‘Ľ đ??ľ0 đ?œ•đ?‘Ľ 4 đ?œ•đ?‘Ľ đ?‘” 14

(46)

with đ?›źM4 sinh (đ?‘&#x;M4 (đ?‘Ľ)đ??ť0 (đ?‘Ľ)) − đ??ť0 (đ?‘Ľ) đ?‘&#x;M4 iđ?œŽ đ?œ• đ?œˆđ?‘‡ đ?›źM4 đ??ť iđ?œŽ 1 đ?œ•đ??ľ0 đ?œˆđ?‘‡ đ?›źM4 đ??ť = ( đ?œ’Ě‚ sinh (đ?‘&#x;M4 đ??ť0 )) + đ?œ’Ě‚ sinh (đ?‘&#x;M4 đ??ť) đ?‘” đ?œ•đ?‘Ľ đ?‘ đ?‘“ đ?‘&#x;M4 đ?‘” đ??ľ0 đ?œ•đ?‘Ľ đ?‘ đ?‘“ đ?‘&#x;M4

�4 (�) =

(47a)

đ??šdips

(47b)

iđ?œŽ đ?œ• đ?›žĚ‚ đ??ť 1 đ?œ•đ??ľ0 đ??ť ( + đ?›žĚ‚ ) , đ?‘” đ?œ•đ?‘Ľ đ??ľ0 đ?œ•đ?‘Ľ iđ?œŽ 1 đ?œ• đ?›žĚ‚ đ??ľ =− , đ?‘” đ??ľ0 đ?œ•đ?‘Ľ iđ?œŽ = − đ?œ đ??ľĚ‚ . đ?‘”

đ??šdepth = −

(47c)

đ??šwidth

(47d)

đ??šsurf

(47e)

3.4

Analytical solution for a horizontal bottom and an exponential width variation

3.4.1

Equations

To find fully analytical solutions for Eq. (46), we assume a constant eddy viscosity, a horizontal bottom, a conâ€? stant bottom friction, and an exponentially decaying width đ??ľ0 (đ?‘Ľ) = đ??ľđ?‘ exp (−

đ?‘Ľ ) đ??żđ?‘?

such that

1 đ?œ•đ??ľ0 1 =− đ??ľ0 đ?œ•đ?‘Ľ đ??żđ?‘?

with đ??ľđ?‘ the width at the sea boundary and đ??żđ?‘? the estuarine convergence length. Additionally, the parametâ€? erization of đ??ľ0 and đ??ť0 are đ??ť1 = − đ??ľ1 =

đ??ť0 đ?œ đ?‘?′ 02

(48a)

đ??ľ0 đ?œ đ?‘?′ 02

(48b)

where đ?‘?′ is a constant. Note that this parameterization is in agreement with Eqs (11) and (16) as long as đ?‘?′ = 2đ??´M2

đ??ľ0 . Δđ??ľ

(49)

These assumptions simplify Eq. (46) to Ě‚ Ě‚ đ??šdips đ??šdepth đ??šwidth đ??šsurf 1 đ?œ• đ?œ 14 4đ?œŽ2 Ě‚ đ?œ• 2 đ?œ 14 − − đ?œ 14 = + + + .

2 đ?œ•đ?‘Ľ đ??żđ?‘? đ?œ•đ?‘Ľ đ?‘”đ?‘‡4 đ?‘‡4 đ?‘‡4 đ?‘‡4 đ?‘‡4 đ?œ• đ?œ’Ě‚ đ??ť 1 đ??ť iđ?œŽ đ?œˆđ?‘‡ đ?›źM4 − đ??šdips = sinh (đ?‘&#x;M4 đ??ť0 ) ( đ?œ’Ě‚ ) đ?‘” đ?‘ đ?‘“ đ?‘&#x;M4 đ?œ•đ?‘Ľ đ??żđ?‘?

(50b)

đ??šdepth = −

(50c)

đ??šwidth

(50d)

đ??šsurf

10

iđ?œŽ đ?œ• đ?›žĚ‚ đ??ť 1 đ??ť ( − đ?›žĚ‚ ) , đ?‘” đ?œ•đ?‘Ľ đ??żđ?‘? iđ?œŽ đ?‘Ľ đ?œ• đ?›žĚ‚ đ??ľ =− exp ( ) , đ?‘”đ??ľđ?‘ đ??żđ?‘? đ?œ•đ?‘Ľ iđ?œŽ = − đ?œ đ??ľĚ‚ . đ?‘”

(50a)

WL2020R18_153_1

(50e)

Final version

iFlow�Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow 3.4.2

Homogeneous solution

The general solution to Eqs 46 is đ?œ hĚ‚ (đ?‘Ľ) = đ??ž1 exp (

đ?‘˜M đ?‘˜M đ?‘Ľ đ?‘Ľ − 4 đ?‘Ľ) + đ??ž2 exp ( + 4 đ?‘Ľ) 2đ??żđ?‘?

2 2đ??żđ?‘?

2

(51)

where đ??ž1 and đ??ž2 are constants depending on the boundary conditions, and the complex wave number đ?‘˜M4 verifies đ?‘˜M4 = √

3.4.3

16đ?œŽ2 đ?‘&#x;M4 1 16đ?œŽ2 1 √ + + = đ?‘”đ?‘‡4 đ?‘”đ?›źM4 sinh(đ?‘&#x;M4 đ??ť0 ) − đ?‘”đ?‘&#x;M4 đ??ť0 đ??ż2đ?‘? đ??ż2đ?‘?

(52)

Particular solutions

To each forcing term corresponds a unique particular solution. It turns out that it always possible to write the forcing of each particular solution under the same form, which we name the canonical form. The equations for the four different width or depth dependent forcing terms will be written in this canonical form in order to facilitate the determination of each particular solution. A possibility of such a canonical form is đ?œ• 2 đ?œ đ?‘?Ě‚ 1 đ?œ• đ?œ đ?‘?Ě‚ 4đ?œŽ2 Ě‚ − − đ?œ =đ??šcanon , đ?œ•đ?‘Ľ2 đ??żđ?‘? đ?œ•đ?‘Ľ đ?‘”đ?‘‡4 đ?‘? đ?‘Ľ ) cosh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) đ??żđ?‘? đ?‘Ľ đ?‘Ľ + đ?‘†đ??š exp ( ) sinh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) + đ??¸đ??š exp ( ) , đ??żđ?‘? đ??żđ?‘?

đ??šcanon =đ??śđ??š exp (

(53)

where the canonical forcing term đ??šcanon refers either to đ??šdips , đ??šdepth , đ??šwidth or đ??šsurf , and the constants đ??śđ??š , đ?‘†đ??š and đ??¸đ??š can be filled in accordingly. The particular solution then takes the form Ě‚ = đ??śđ?‘† exp ( đ?œ part

đ?‘Ľ đ?‘Ľ đ?‘Ľ ) cosh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) + đ?‘†đ?‘† exp ( ) sinh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) + đ??¸đ?‘† exp ( ) , đ??żđ?‘? đ??żđ?‘? đ??żđ?‘?

(54)

where the constants đ??śđ?‘† , đ?‘†đ?‘† and đ??¸đ?‘† read

�� =

đ?‘˜M 4đ?œŽ2 2 2 đ??żđ?‘? đ??śđ??š − (đ?‘˜M2 − đ?‘”đ?‘‡4 ) đ?‘†đ??š 2 đ?‘˜2M 2 − (đ?‘˜2 − 4đ?œŽ2 ) M2 đ?‘”đ?‘‡4 đ??ż2đ?‘?

,

(55)

đ??śđ?‘† =

đ?‘˜M 2 đ??żđ?‘? đ?‘† đ??š đ?‘˜2M 2 đ??ż2đ?‘?

,

(56)

đ??¸đ?‘† = −

− (đ?‘˜M22 − − (đ?‘˜M22 −

4đ?œŽ2 đ?‘”đ?‘‡4 ) đ??śđ??š 2 4đ?œŽ2 đ?‘”đ?‘‡4 )

đ?‘”đ?‘‡4 đ??¸ .

4đ?œŽ2 đ??š

(57)

For details about the calculation method of the constants, the reader is referred to Appendix A1.1 3.4.4

Total solution

The total solution obviously consists of the sum of the homogeneous solution and the particular solution, Ě‚ (đ?‘Ľ) = đ?œ â„ŽĚ‚ (đ?‘Ľ) + đ?œ đ?‘?Ě‚ (đ?‘Ľ). đ?œ 14

Final version

WL2020R18_153_1

(58)

11

iFlowâ€?Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow Ě‚ = 0 and The integration constant đ??ž1 and đ??ž2 can now be determined using the boundary conditions đ?œ 14 Ě‚ /đ?œ•đ?‘Ľ(đ??ż) = 0. The computation of đ??ž1 and đ??ž2 is detailed in Appendix, and leads to đ?œ• đ?œ 14 đ?œ†3 − đ?œ† 2 đ?œ†4 , đ?œ†1 − đ?œ† 2 đ?œ† − đ?œ† 1 đ?œ†4 . đ??ž2 = 3 đ?œ†2 − đ?œ† 1 đ??ž1 =

(59a) (59b)

The parameters đ?œ†1 , đ?œ†2 , đ?œ†3 and đ?œ†4 were introduced for conciseness and read đ?‘˜M đ?‘˜M đ??ż 1 đ??ż − 4 ) exp ( − 4 ), 2đ??żđ?‘?

2 2đ??żđ?‘?

2 đ?‘˜M đ?‘˜M đ??ż 1 đ??ż đ?œ†2 = ( + 4 ) exp ( + 4 ), 2đ??żđ?‘?

2 2đ??żđ?‘?

2 đ??ż đ??ś + đ??¸đ?‘† đ?œ†3 = − exp ( ) ( đ?‘† + đ?‘†đ?‘† đ?‘˜M2 ) , đ??żđ?‘? đ??żđ?‘? đ?œ†4 = −đ?‘†đ?‘† sinh (−đ?‘˜M2 đ??ż) − đ??śđ?‘† cosh (−đ?‘˜M2 đ??ż) − đ??¸đ?‘† . đ?œ†1 = (

3.4.5

(60a) (60b) (60c) (60d)

Individual forcing terms

In the case of a horizontal bottom and an exponentially converging width, it is possible to have a fully analytical expression of the forcing terms. Since the forcing terms (between brackets, [â‹…]) depend on the leading order variables đ?‘˘0 and/or đ?œ 02 , an analytical expression of these quantities is required. Under the assumptions stated in the beginning of this section, the equations governing the leading order variables, Eqs (7aâ€?7b), admit fully analytical solutions. According to Wei et al., 2016, the analytical solution for đ?‘˘0 , đ?œ 02 and đ?œ•đ?œ 02 /đ?œ•đ?‘Ľ under the present assumptions are đ?‘” đ?œ•đ?œ (đ?›ź cosh (đ?‘&#x;M2 đ?‘§) − 1) 02 , iđ?œŽ M2 đ?œ•đ?‘Ľ đ?‘˜M đ?‘˜M đ?‘Ľ đ?œ 0Ě‚ = đ??śM2 exp ( ) (− sinh ( 2 (đ?‘Ľ − đ??ż)) + đ??żđ?‘? đ?‘˜M2 cosh ( 2 (đ?‘Ľ − đ??ż))) , 2đ??żđ?‘?

2 2

�̂0 =

(61a) (61b)

with, đ?‘&#x;M2 = √ đ?›źM2 =

(62a)

đ?‘ đ?‘“ , đ?œˆđ?‘‡ đ?‘&#x;M2 sinh (đ?‘&#x;M2 đ??ť0 ) + đ?‘ đ?‘“ cosh (đ?‘&#x;M2 đ??ť0 )

(62b)

4đ?œŽ2 đ?‘&#x;M2 1 + , đ??ż2đ?‘? đ?‘” (đ?›źM2 sinh (đ?‘&#x;M2 đ??ť0 ) − đ?‘&#x;M2 đ??ť0 )

(62c)

đ?‘˜M2 = √ đ??śM2 =

iđ?œŽ , đ?œˆđ?‘‡

đ??´M2 . đ?‘˜M2 đ??ż đ?‘˜M đ??ż ) + đ?‘˜M2 đ??żđ?‘? cosh ( 2 ) sinh ( 2 2

(62d)

đ??šdips

The starting point for the development of đ??šdips is đ??šdips iđ?œŽ đ?œˆđ?‘‡ đ?›źM4 đ?œ• đ?œ’Ě‚ đ??ť 1 đ??ť = sinh (đ?‘&#x;M4 đ??ť0 ) ( − đ?œ’Ě‚ ) đ?‘‡4 đ?‘”đ?‘‡4 đ?‘ đ?‘“ đ?‘&#x;M4 đ?œ•đ?‘Ľ đ??żđ?‘?

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iFlowâ€?Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow Using the definition of đ?œ’Ě‚ đ??ť , Eq. (43), this equation can be rewritten đ??šdips iđ?œŽ đ?œˆđ?‘‡ đ?›źM4 = đ?‘‡4 đ?‘”đ?‘‡4 đ?‘ đ?‘“ đ?‘&#x;M4 sinh (đ?‘&#x;M4 đ??ť0 ) (

2 đ?‘ đ?‘ đ?‘“ đ?œ• đ?œ• 2 đ?‘˘Ě‚0 1 đ?œ• đ?‘˘Ě‚ Ě‚1 đ?œ• đ?‘˘Ě‚0 âˆŁ Ě‚1 đ?œ• đ?‘˘Ě‚0 âˆŁ âˆŁ ) − )) ( − đ?‘“ đ??ťĚ‚1 0 âˆŁ + đ??ťĚ‚1 + đ??ť (− đ??ť đ?œ•đ?‘Ľ đ?œˆđ?‘‡ đ?‘‘đ?‘§ −đ??ť đ?œ•đ?‘§ 2 −đ??ť đ??żđ?‘? đ?œˆđ?‘‡ đ?‘‘đ?‘§ −đ??ť đ?œ•đ?‘§ 2 −đ??ť 0

0

0

0

(64) or

đ??šdips đ?‘ iđ?œŽ đ?œˆđ?‘‡ đ?›źM4 đ?œ• 1 đ?œ• đ?‘˘Ě‚ đ?œ• 2 đ?‘˘Ě‚0 = sinh (đ?‘&#x;M4 đ??ť0 ) ( − ) ( − đ?‘“ đ??ťĚ‚1 0 âˆŁ + đ??ťĚ‚1 âˆŁ ) đ?‘‡4 đ?‘”đ?‘‡4 đ?‘ đ?‘“ đ?‘&#x;M4 đ?œ•đ?‘Ľ đ??żđ?‘? đ?œˆđ?‘‡ đ?‘‘đ?‘§ −đ??ť đ?œ•đ?‘§ 2 −đ??ť 0

(65)

0

Finally, đ??šdips đ?‘ đ?œˆ đ?‘&#x;M đ?›ź M đ?›ź M đ??ť = − đ??śM22 đ?‘‡ 2 2 4 ′0 sinh (đ?‘&#x;M4 đ??ť0 ) ( đ?‘“ sinh (đ?‘&#x;M2 đ??ť0 ) + đ?‘&#x;M2 cosh (đ?‘&#x;M2 đ??ť0 )) đ?‘‡4 đ?‘ đ?‘“ đ?‘&#x;M4 đ?‘‡4 đ?‘? đ?œˆđ?‘‡ đ?‘˜M42 đ??ż2đ?‘? đ?‘˜M đ?‘˜M22 đ??żđ?‘? đ?‘˜M32 đ?‘Ľ ) cosh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) + (− + + 2 ) sinh(đ?‘˜M2 (đ?‘Ľ − đ??ż))) exp ( ) ((− đ??żđ?‘?

4 4 4 4đ??żđ?‘?

(66) The final expression for đ??šdepth can be found using the expressions of the derivatives of đ?œ 02 , given in Appendix A1.3, đ??šdepth

The starting point for the development of đ??šdepth is đ??šdepth iđ?œŽ đ?œ• đ?›žĚ‚ đ??ť 1 đ??ť =− ( − đ?›žĚ‚ ) . đ?‘‡4 đ?‘”đ?‘‡4 đ?œ•đ?‘Ľ đ??żđ?‘?

(67)

Using the definition of đ?›žĚ‚ đ??ť , Eq. (40), this equation can be rewritten đ??šdepth iđ?œŽ đ?œ• 1 Ě‚ =− ( (đ??ťĚ‚1 đ?‘˘Ě‚0 |đ?‘§=−đ??ť ) − đ??ť đ?‘˘Ě‚ | ).

0 đ?‘‡4 đ?‘”đ?‘‡4 đ?œ•đ?‘Ľ đ??żđ?‘? 1 0 đ?‘§=−đ??ť0 Injecting the expression for đ??ť1 , i.e. Eq. (48a), the formulation becomes đ??šdepth iđ?œŽ đ?œ• đ??ť0 Ě‚ 1 đ??ť0 Ě‚ (− ( ′ đ?œ 0 đ?‘˘Ě‚0 |đ?‘§=−đ??ť ) + ). đ?œ đ?‘˘Ě‚ | =− 0 đ?‘‡4 đ?‘”đ?‘‡4 đ?œ•đ?‘Ľ đ?‘? đ??żđ?‘? đ?‘?′ 0 0 đ?‘§=−đ??ť0 The combination with the formula for the leading order velocity đ?‘˘0 , i.e. Eq. (61a), yields Ě‚ Ě‚ đ??šdepth iđ?œŽ đ??ť0 đ?‘” đ?œ• Ě‚ đ?œ• đ?œ 0 ) − 1 đ?‘” (đ?›źM cosh(−đ?‘&#x;M đ??ť0 ) − 1) đ?œ 0Ě‚ đ?œ• đ?œ 0 ) ( − 1) đ?œ = ( (đ?›ź cosh(−đ?‘&#x; đ??ť ) M M 0 0 2 2 2 2 đ?‘‡4 đ?‘”đ?‘‡4 đ?‘?′ đ?œ•đ?‘Ľ iđ?œŽ đ?œ•đ?‘Ľ đ??żđ?‘? iđ?œŽ đ?œ•đ?‘Ľ 2

đ??šdepth đ??ť đ?œ•đ?œ Ě‚ đ?œ• 2 đ?œ 0Ě‚ 1 Ě‚ đ?œ• đ?œ 0Ě‚ ⎞ ⎜( 0 ) + đ?œ 0Ě‚ âŽ&#x;. = ′ 0 (đ?›źM2 cosh(−đ?‘&#x;M2 đ??ť0 ) − 1) ⎛ − đ?œ đ?‘‡4 đ?‘? đ?‘‡4 đ?œ•đ?‘Ľ đ?œ•đ?‘Ľ2 đ??żđ?‘? 0 đ?œ•đ?‘Ľ âŽ? ⎠The final expression for đ??šdepth can be found using the expressions of the derivatives of đ?œ 02 , given in Appendix A1.3, đ??šdepth đ??ť đ?‘Ľ =đ??śM22 ′ 0 (đ?›źM2 cosh (−đ?‘&#x;M2 đ??ť0 ) − 1) exp ( ) đ?‘‡4 đ?‘? đ?‘‡4 đ??żđ?‘?

(68) đ??ż2đ?‘? đ?‘˜M42 đ?‘˜M22 đ??żđ?‘? đ?‘˜M32 đ?‘˜M2 (( − ) cosh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) + (− + ) sinh (đ?‘˜M2 (đ?‘Ľ − đ??ż))) 4 4 4 4đ??żđ?‘?

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iFlowâ€?Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow đ??šwidth

The starting point for the development of đ??šwidth is đ??šwidth iđ?œŽ đ?‘Ľ đ?œ• đ?›žĚ‚ đ??ľ =− exp ( ) đ?‘‡4 đ?‘”đ??ľđ?‘ đ?‘‡4 đ??żđ?‘? đ?œ•đ?‘Ľ

(69)

Using the definition of đ?›žĚ‚ đ??ľ , Eq. (41), this equation can be rewritten 0

đ??šwidth iđ?œŽ đ?‘Ľ đ?œ• (đ??ľĚ‚ 1 âˆŤ đ?‘˘Ě‚0 đ?‘‘đ?‘§) =− exp ( ) đ?‘‡4 đ?‘”đ??ľđ?‘ đ?‘‡4 đ??żđ?‘? đ?œ•đ?‘Ľ −đ??ť0

(70)

Injecting the expression for đ??ľ1 , i.e. Eq. (48b), the formulation becomes 0 đ??šwidth iđ?œŽ đ?œ Ě‚ đ?œ• đ?œ•đ?œ Ě‚ = − ′ (− 0 + 0 + đ?œ 0Ě‚ ) âˆŤ đ?‘˘Ě‚0 đ?‘‘đ?‘§. đ?‘‡4 đ?‘”đ?‘? đ?‘‡4 đ??żđ?‘? đ?œ•đ?‘Ľ đ?œ•đ?‘Ľ −đ??ť0

(71)

After some math, we obtain 2

iđ?œŽ đ?‘‡ đ?œ•đ?œ Ě‚ đ?œ• 2 đ?œ 0Ě‚ 1 Ě‚ đ?œ• đ?œ 0Ě‚ ⎞ đ??šwidth ⎜( 0 ) + đ?œ 0Ě‚ âŽ&#x;, =− ′ 2⎛ − đ?œ đ?‘‡4 đ?‘”đ?‘? đ?‘‡4 đ?œ•đ?‘Ľ đ?œ•đ?‘Ľ2 đ??żđ?‘? 0 đ?œ•đ?‘Ľ âŽ? ⎠with đ?‘‡2 (đ?‘Ľ) =

(72)

đ?›źM2 sinh (đ?‘&#x;M2 (đ?‘Ľ)đ??ť0 (đ?‘Ľ)) − đ??ť0 (đ?‘Ľ). đ?‘&#x;M2

(73)

The final expression for đ??šdepth can be found using the expressions of the derivatives of đ?œ 02 , given in Appendix A1.3, đ??śM2 đ?‘‡ đ?‘˜M2 đ?‘˜M4 đ??ż2 đ??żđ?‘? đ?‘˜M32 đ?‘˜M đ??šwidth đ?‘Ľ + 2 ) sinh(đ?‘˜M2 (đ?‘Ľ − đ??ż))) = − ′2 2 exp ( ) ((− 2 + 2 đ?‘? ) cosh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) + (− đ?‘‡4 đ?‘? đ?‘‡4 đ??żđ?‘?

4 4 4 4đ??żđ?‘?

(74) đ??šsurf

The starting point for the development of đ??šsurf is đ??šsurf iđ?œŽ đ??ľĚ‚ =− đ?œ . đ?‘‡4 đ?‘”đ?‘‡4

(75)

Using the definition of đ?œ đ??ľĚ‚ , Eq. (42), this equation can be rewritten đ??šsurf iđ?œŽ đ??ľĚ‚ 1 đ?œ• đ?œ 0Ě‚ =− . đ?‘‡4 đ?‘”đ?‘‡4 đ??ľ0 đ?œ•đ?‘Ą

(76)

Expliciting the complex timeâ€?derivative of đ?œ đ??ľĚ‚ and the parameterization of the width variation, Eq. (48b), đ??šsurf đ?œŽ2 2Ě‚ = đ?œ đ?‘‡4 đ?‘”đ?‘‡4 đ?‘?′ 0

(77)

Finally, using Eq. (87e), đ??śM22 đ?œŽ2 2 đ??šsurf đ?‘Ľ = đ?œ 0Ě‚ exp ( ) ((đ??żđ?‘? đ?‘˜M2 )2 −1+(1+đ?‘˜M22 đ??ż2đ?‘? ) cosh (đ?‘˜M2 (đ?‘Ľ − đ??ż))−2đ??żđ?‘? đ?‘˜M2 sinh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) ) ′ đ?‘‡4 2 đ?‘”đ?‘‡4 đ?‘? đ??żđ?‘?

(78)

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4 Simulation results The new, width or depth dependent, forcing terms have been implemented in iFlows semi�analytical and nu� merical modules. In this report, we only focus on the resulting additional water elevation at M4 frequency.

These elevations are more difficult to implement as compared to the M0 water elevations, and are believed to be important contributors to the M4 tidal signal in the Scheldt. The depth and width dependent water el� evations at M0 frequency, the additional residual velocities and the additional M4 tidal velocities are analyzed in a follow up project. In the semi�analytical solver, the second order equation for the surface�elevation (i.e.

Eq. (46) is integrated numerically while the solution for the velocity is still given by Eq (44). In the numer� ical solver both the velocity and the surface elevation are computed numerically. Additionally to these two solvers, a fully analytical solver was created for estuary models with a horizontal bottom and an exponentially converging width.

4.1

Validation

We choose to take advantage of these three solvers, to verify the implementations of the forcing terms re� lated to width and depth variations, Eqs (47b�47e). The configuration used for the validation was one with a horizontal bottom and an exponential converging width. The parameter values are largely inspired by the parameter values for the Scheldt by Wei et al., 2016 and given in Table 1. The aim is to validate the implement�

Table 1 – Parameter values for an idealized Scheldt model (Wei et al., 2016).

đ??´M2 (m)

đ??´M4 (m)

đ??ż (km)

đ??ľđ?‘ (m)

2.00

0.20

200

4000

đ?‘?

(km) 50

đ??ť0 (m)

đ?‘„1 (m3 s−1 )

đ?œˆđ?‘‡ (m2 s−1 )

đ?‘ đ?‘“ (m s−1 )

10

90

0.0099

0.0085

ation of the width and depth dependent terms contributing to the first order surface elevation. Accordingly, the leading order surface elevation and velocity are computed with the same semi�analytical solver, regardless if the first order solver is numerical, semi�analytical or fully analytical. In this way, any possible discrepancies between the solutions at leading order is not influencing the solutions at first order.

The relative performances of the semi�analytical, analytical and numerical solvers are showed in Figs 2, 3 and 4. Quadratic convergence was checked by refining the grid 10 times in the horizontal and vertical directions.

The error between two solvers, defined as âˆŁ

đ?œ đ?‘ đ?‘œđ?‘™đ?‘Łđ?‘’đ?‘&#x;1 − đ?œ đ?‘ đ?‘œđ?‘™đ?‘Łđ?‘’đ?‘&#x;2 âˆŁ, đ?œ đ?‘ đ?‘œđ?‘™đ?‘Łđ?‘’đ?‘&#x;1

(79)

is reduced by two orders of magnitude for đ?œ depth , đ?œ width and đ?œ surf , when the grid is refined by one order of magnitude. The relative error between the đ?œ dips values computed by different solvers only reduces by one order of magnitude for a reason yet unknown.

The relative behavior of the solvers is similar when the phase is analyzed. As illustration, the phases computed with the fully analytical method and the numerical method are displayed in Fig. 5. The phases of đ?œ depth , đ?œ width and đ?œ surf converge quadratically while the phase of đ?œ dips converges linearly. These results give confidence in the correct implementation of the forcing terms in iFlow, even if the nonâ€?quadratic convergence for đ?œ dips needs to be clarified.

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iFlow‐Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow Figure 2 – Evolution of the differences in water levels computed by the semi‐analytical and the fully analytical solver (configuration of Wei et al., 2016).

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iFlow�Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow Figure 3 – Evolution of the differences in water levels computed by the numerical and the fully analytical solver (configuration of Wei et al., 2016).

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iFlow‐Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow Figure 4 – Evolution of the differences in water levels computed by the numerical and the semi‐analytical solver (configuration of Wei et al., 2016).

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iFlow�Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow Figure 5 – Evolution of the differences in water levels computed by the numerical and the fully analytical solver (configuration of Wei et al., 2016).

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4.2

Effect of the incorporation of intertidal zones on the total M4 surface elevation (horizontal bottom)

In the previous section, it is shown that the implementation of the forcing due to intertidal width and depth variations leads to converging results for all the width and depth dependent terms. As a result, we can now analyze with confidence the relative effect of depth and width dependent surface elevations with respect to the total M4 surface elevation. However, the first order forcing terms depend on the velocity and surface elevation at leading order. Additionally, particularly the bathymetry of the Scheldt is highly idealized in the current model and this property can significantly influence the leading order results. As a result, we will first study the surface elevation and bottom velocity at leading order, in the Wei et al., 2016 configuration and in the Brouwer et al., 2017 configuration (more details in Table 2).

Table 2 – Parameter values for a different idealized Scheldt model Brouwer et al., 2017. Note that these parameters also include a phase shift of �1.3∘ between the external M4 and the external M2 tides.

đ??´M2 (m)

đ??´M4 (m)

đ??ż (km)

đ??ľđ?‘ (m)

1.77

0.14

160

6000

đ?‘?

(km) 50

đ??ť0 (m)

đ?‘„1 (m3 s−1 )

đ?œˆđ?‘‡ (m2 s−1 )

đ?‘ đ?‘“ (m s−1 )

10

90

0.061

0.003

The leading order surface elevation and the leading order bottom velocity are displayed in Figs 6 and 7. The expected evolution of the M2 surface elevation, i.e. a steady increase of the M2 amplitude until km 120 followed by a drop of the amplitude towards the weir (see measurement data in Brouwer et al., 2017), is not reproduced by neither of the two models. Particularly, towards the end of the estuary, the amplitude of the M2 is about 3m with respect to a depth of 10m for the Wei et al., 2016 parameter settings. The M2 surface elevation obtained using the Brouwer et al., 2017 parameter settings is better at the upstream side of the estuary, with an elevation of about 1.75m. However, it also lacks the maximum around km 120. The amplitude of the bottom velocity predicted by Brouwer et al., 2017 are much smaller than the one produced

by Wei et al., 2016, which is explained by the large friction coefficient of the latter with respect to the former.

Both the phase of the surface elevation and the phase of the velocity seem to be of inverse sign with respect to the measured data.

Figure 6 – Leading order solutions for the water level and the bottom velocity for the Wei et al., 2016 configuration.

The leading order results show already quite some discrepancies with respect to the measurements. This feature implies that the first order results will probably show the same, if not a larger, difference with the measurements. However, we can still estimate the relative impact intertidal zones could have on the tidal signal. In this regard, it is for example particularly interesting to investigate the đ??šdips and the đ??šdepth term, since

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iFlow�Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow Figure 7 – Leading order solutions for the water level and the bottom velocity for the Brouwer et al., 2017 configuration.

they are strongly dependent on the bottom velocity.

Figure 8 – First order surface elevation decomposed into separate width�depth contributions (Wei et al., 2016) (only full analytical results).

The different width and depth dependent M4 surface elevation components are displayed in Fig. 8 for the Wei et al., 2016 configuration, in Fig. 9 for the Brouwer et al., 2017 configuration. Every figure also contains the total M4 signal, with and without intertidal terms. As expected, the đ?œ depth term is much smaller in the Wei et al., 2016 configuration than in the Brouwer et al., 2017 configuration due to larger friction. However, for the đ?œ dips term, the opposite is true, probably due to higher gradients close to the wall in case of higher friction.

Figure 9 and Fig. 8 also prove the phase is crucial for the damping or amplifying effects of the M4 components

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iFlow�Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow Figure 9 – First order surface elevation decomposed into separate width�depth contributions (Wei et al., 2016) (only full analytical results).

Table 3 – Coefficients for the polynomial approximation of the water depth.

đ?‘‘1

đ?‘‘2

đ?‘‘3

â€?2.9013 Ă— 10−24 m−4

1.4030 Ă— 10−18 m−3

â€?2.4218 Ă— 10−13 m−2

đ?‘‘4

đ?‘‘5

đ?‘‘6

1.7490 Ă— 10−8 m−1

â€?5.21410 Ă— 10−4

15.332m

generated by intertidal effects. Indeed, opposite phases but similar amplitudes cause the đ?œ depth term to damp the overall M4 tide in the Wei et al., 2016 case (see Fig. 8).

4.3

Case of a varying bottom

The case of a varying bottom has also been investigated. The depth đ??ť0 is now a polynomial function of the fifth order đ??ť0 (đ?‘Ľ) = đ?‘‘1 đ?‘Ľ5 + đ?‘‘2 đ?‘Ľ4 + đ?‘‘3 đ?‘Ľ3 + đ?‘‘4 đ?‘Ľ2 + đ?‘‘5 đ?‘Ľ + đ?‘‘6 (80) where the coefficent đ?‘?đ?‘– are taken from Brouwer et al., 2017 and given in Table 3 The comparison of the solutions obtained at first order by the semiâ€?analytical solver and the numerical solver shows that the đ?œ depth now converges linearly instead of quadratically and that the đ?œ dips term does not conâ€? verge with increasing grid resolution. However, the relative difference between the numerical solution and the semiâ€?analytical solution is of order 1% or lower. Nevertheless, quadratic convergence is expected and these discrepancies need to be elucidated

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4.4

Conclusion

Four width and dependent forcings of the M4 tide were implemented in iFlow’s numerical and semi‐analytical modules. An additional full analytical module was created for horizontal bottom and exponential width con‐ figurations. For the horizontal bottom configurations, the error between the solutions computed by different solvers converges for increasing grid resolution proving accurate implementation. In the non‐horizontal bot‐ tom configuration, one of the terms does not converge. However, the error is small (order 1%). A first analysis of the solutions shows that width and depth variations, or intertidal zones, are significant contributors to the M4 surface elevations. The model is now operational for an investigation with a more realistic bathymetry as well as a more realistic first order width and depth parameterisation. This investigation should take place in a follow‐up project and should clarify if the discrepancies between measured M4 amplitudes and modeled M4 amplitudes (in iFlow) can be eliminated by the incorporation of intertidal zones in iFlow.

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References Brouwer, R. L.; Schramkowski, G. P.; Dijkstra, Y. M.; Schuttelaars, H. M. (2018). Time Evolution of Estuar‐ ine Turbidity Maxima in Well‐Mixed, Tidally Dominated Estuaries: The Role of Availability‐and Erosion‐Limited Conditions. Journal of Physical Oceanography 48 (8): 1629–1650 Brouwer, T.; Schramkowski, G.; Mostaert, F. (2017). Geïdealiseerde processtudie van systeemovergangen naar hypertroebelheid. 2.0. WL Rapporten, 13_103_3. Flanders Hydraulics Research: Antwerp, Belgium Chernetsky, A. S.; Schuttelaars, H. M.; Talke, S. A. (2010). The effect of tidal asymmetry and temporal settling lag on sediment trapping in tidal estuaries. Ocean Dynamics 60 (5): 1219–1241 Dijkstra, Y. M.; Brouwer, R. L.; Schuttelaars, H. M.; Schramkowski, G. P. (2017). The iFlow modelling frame‐ work v2. 4: a modular idealized process‐based model for flow and transport in estuaries. Geoscientific Model Development 10 (7): 2691–2713 Dijkstra, Y. M.; Schuttelaars, H. M.; Schramkowski, G. P. (2019a). A Regime Shift From Low to High Sediment Concentrations in a Tide‐Dominated Estuary. Geophysical Research Letters 46 (8): 4338–4345 Dijkstra, Y. M.; Schuttelaars, H. M.; Schramkowski, G. P.; Brouwer, R. L. (2019b). Modeling the Transition to High Sediment Concentrations as a Response to Channel Deepening in the Ems River Estuary. 124 (3): 1578– 1594 Wei, X.; Schramkowski, G. P.; Schuttelaars, H. M. (2016). Salt dynamics in well‐mixed estuaries: Importance of advection by tides. Journal of Physical Oceanography 46 (5): 1457–1475

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A1 Intermediate steps for the computation of the width/depth dependent M4 A1.1 Resolution method for the particular solution The total solution of Eq. (50a) consists of the sum of a homogeneous solution, given by Eq. (51), and a particular solution for each forcing term. The particular solutions are assumed to be of the form given by Eq. (54). As a reminder, this solution is written đ?œ part = đ??śđ?‘† exp (

đ?‘Ľ đ?‘Ľ đ?‘Ľ ) cosh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) + đ?‘†đ?‘† exp ( ) sinh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) + đ??¸đ?‘† exp ( ) , đ??żđ?‘? đ??żđ?‘? đ??żđ?‘?

Accordingly, the derivatives of the assumed analytical solution are đ?œ•đ?œ đ?‘† đ?‘Ľ = ( đ?‘† + đ?‘˜M2 đ??śđ?‘† ) exp ( ) sinh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) đ?œ•đ?‘Ľ đ??żđ?‘? đ??żđ?‘? đ??śđ?‘† đ?‘Ľ đ??¸ đ?‘Ľ +( + đ?‘˜M2 đ?‘†đ?‘† ) exp ( ) cosh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) + đ?‘† exp ( ) , đ??żđ?‘? đ??żđ?‘? đ??żđ?‘? đ??żđ?‘?

2 đ?‘˜M 1 đ?œ• đ?œ đ?‘Ľ = (( 2 + đ?‘˜M22 ) đ?‘†đ?‘† + 2 2 đ??śđ?‘† ) exp ( ) sinh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) 2 đ?œ•đ?‘Ľ đ??żđ?‘? đ??żđ?‘? đ??żđ?‘? đ?‘˜M 1 đ?‘Ľ đ??¸ đ?‘Ľ + (( 2 + đ?‘˜M22 ) đ??śđ?‘† + 2 2 đ?‘†đ?‘† ) exp ( ) cosh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) + đ?‘†2 exp ( ) . đ??żđ?‘? đ??żđ?‘? đ??żđ?‘? đ??żđ?‘? đ??żđ?‘?

(81a)

(81b)

By injecting Eqs (54), (81a) and (81b) into Eq. (53), finding the constant đ??śđ?‘† , đ?‘†đ?‘† and đ??¸đ?‘† is equivalent to resolving đ?‘˜M 4đ?œŽ2 ) đ?‘† đ?‘† + 2 đ??śđ?‘† ) = đ?‘† đ??š , đ?‘”đ?‘‡4 đ??żđ?‘?

2 đ?‘˜ 4đ?œŽ M ) đ??śđ?‘† + 2 đ?‘†đ?‘† ) = đ??śđ??š , − đ?‘”đ?‘‡4 đ??żđ?‘?

4đ?œŽ2 − đ??¸ = đ??¸đ??š . đ?‘”đ?‘‡4 đ?‘†

((đ?‘˜M22 −

(82a)

((đ?‘˜M22

(82b) (82c)

such that finally

�� =

đ?‘˜M 4đ?œŽ2 2 2 đ??żđ?‘? đ??śđ??š − (đ?‘˜M2 − đ?‘”đ?‘‡4 ) đ?‘†đ??š 2 đ?‘˜2M 2 − (đ?‘˜2 − 4đ?œŽ2 ) M2 đ?‘”đ?‘‡4 đ??ż2đ?‘?

đ??śđ?‘† =

đ?‘˜M 2 đ??żđ?‘? đ?‘†đ??š đ?‘˜2M 2 đ??ż2đ?‘?

đ??¸đ?‘† = −

Final version

− (đ?‘˜M22 − − (đ?‘˜M22 −

4đ?œŽ2 đ?‘”đ?‘‡4 ) đ??śđ??š 2 4đ?œŽ2 đ?‘”đ?‘‡4 )

đ?‘”đ?‘‡4 đ??¸ 4đ?œŽ2 đ??š

WL2020R18_153_1

A1

iFlow�Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow

A1.2 Integration constants of the homogeneous solution The total analytical solution, i.e. the sum of the homogeneous and the particular solution, is đ?‘˜M đ?‘˜M đ?‘Ľ đ?‘Ľ − 4 đ?‘Ľ) + đ??ž2 exp ( + 4 đ?‘Ľ) 2đ??żđ?‘?

2 2đ??żđ?‘?

2 đ?‘Ľ đ?‘Ľ đ?‘Ľ + đ??śđ?‘† exp ( ) cosh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) + đ?‘†đ?‘† exp ( ) sinh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) + đ??¸đ?‘† exp ( ) . đ??żđ?‘? đ??żđ?‘? đ??żđ?‘?

Ě‚ =đ??ž1 exp ( đ?œ 14

(83)

In this solution, the constants đ??ž1 and đ??ž2 can be computed, using the boundary conditions. The imposed Ě‚ (0) = 0 gives surface elevation at đ?‘Ľ = 0, i.e. đ?œ 14 đ??ž1 + đ??ž2 + đ??śđ?‘† cosh (−đ?‘˜M2 đ??ż) + đ?‘†đ?‘† cosh (−đ?‘˜M2 đ??ż) + đ??¸đ?‘† = 0.

(84)

Ě‚ /đ?œ•đ?‘§ = 0.

The imposed flowâ€?rate at đ?‘Ľ = đ??ż translates into đ?œ• đ?œ 14 đ??ž1 (

đ?‘˜M đ?‘˜M đ?‘˜M đ??ż 1 đ??ż đ?‘˜M4 1 đ??ż) − 4 ) exp ( − 4 đ??ż) + đ??ž2 ( + 4 ) exp ( 2đ??żđ?‘?

2 2đ??żđ?‘?

2 2đ??żđ?‘?

2 2đ??żđ?‘? 2 đ??ś đ??¸ đ??ż + ( đ?‘† + đ?‘†đ?‘† đ?‘˜M2 + đ?‘† ) exp ( ) = 0 đ??żđ?‘? đ??żđ?‘? đ??żđ?‘?

(85)

By posing đ?‘˜M đ?‘˜M đ??ż đ??ż 1 − 4 ) exp ( − 4 ), 2đ??żđ?‘?

2 2đ??żđ?‘?

2 đ?‘˜M đ?‘˜M đ??ż 1 đ??ż đ?œ†2 = ( + 4 ) exp ( + 4 ), 2đ??żđ?‘?

2 2đ??żđ?‘?

2 đ??ś + đ??¸đ?‘† đ??ż đ?œ†3 = − exp ( ) ( đ?‘† + đ?‘†đ?‘† đ?‘˜M2 ) , đ??żđ?‘? đ??żđ?‘? đ?œ†4 = −đ?‘†đ?‘† sinh (−đ?‘˜M2 đ??ż) − đ??śđ?‘† cosh (−đ?‘˜M2 đ??ż) − đ??¸đ?‘† , đ?œ†1 = (

the two boundary conditions reduce to the system đ??ž 1 đ?œ†1 + đ??ž 2 đ?œ†2 = đ?œ† 3 ,

(86a)

đ??ž1 + đ??ž 2 = đ?œ† 4 .

(86b)

This system has for solution đ?œ†3 − đ?œ† 2 đ?œ†4 đ?œ†1 − đ?œ† 2 đ?œ†3 − đ?œ† 1 đ?œ†4 đ??ž2 = đ?œ†2 − đ?œ† 1 đ??ž1 =

A2

WL2020R18_153_1

Final version

iFlow�Inwerking en ontwikkeling SKN 21: Incorporation of intertidal zones in iFlow

A1.3 Derivatives of the leading order solutions The first order solutions depend on the leading order solutions and its derivatives. The leading order solutions are already given in the main body of this document (see Eqs (61a) and (61b)). Some additional derivatives are đ?œ•đ?‘˘0 đ?‘” = đ?›źM2 đ?‘&#x;M2 sinh(đ?‘&#x;M2 đ?‘§) đ?œ•đ?‘§ iđ?œŽ 2 đ?œ• đ?‘˘0 đ?‘” = đ?›źM2 đ?‘&#x;M22 cosh(đ?‘&#x;M2 đ?‘§) đ?œ•đ?‘§ 2 iđ?œŽ đ?‘˜M2 đ??żđ?‘? đ?‘˜M đ?‘Ľ đ?œ•đ?œ 02 1 =đ??śM2 exp ( ) (− + 2 ) sinh ( 2 (đ?‘Ľ − đ??ż)) , đ?œ•đ?‘Ľ 2đ??żđ?‘?

2đ??żđ?‘?

2 2 đ?œ• 2 đ?œ 02 đ?‘Ľ =đ??śM2 exp ( ) đ?œ•đ?‘Ľ2 2đ??żđ?‘? đ?‘˜M2 đ?‘˜M đ?‘˜M đ?‘˜M đ??żđ?‘? đ?‘˜M32 1 ((− 2 + 2 ) sinh ( 2 (đ?‘Ľ − đ??ż)) + (− 2 + ) cosh ( 2 (đ?‘Ľ − đ??ż))) 4 2 4đ??żđ?‘?

4 2 4đ??żđ?‘?

2 đ?œ 02 =

đ??śM22 đ?‘Ľ exp ( ) 2 đ??żđ?‘?

2

((đ??żđ?‘? đ?‘˜M2 ) − 1 + (1 +

(87a) (87b) (87c)

(87d)

(87e) 2

đ?‘˜M2 đ??ż2đ?‘? ) cosh (đ?‘˜M2 (đ?‘Ľ

− đ??ż)) − 2đ??żđ?‘? đ?‘˜M2 sinh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) )

2 đ?‘˜M2 đ?‘˜M22 đ??ż2 đ?‘˜M4 đ??ż2đ?‘? đ?‘˜M42 đ?‘Ľ 1 1 đ?œ•đ?œ 02 ) =đ??śM22 exp ( ) (( đ?‘? 2 − 2 + − + ) cosh(đ?‘˜ (đ?‘Ľ − đ??ż)) − ( )) ( M 2 2 đ?œ•đ?‘Ľ đ??żđ?‘?

8 4 8 4 8đ??żđ?‘?

8đ??ż2đ?‘?

(87f) đ?œ 02

đ?œ•đ?œ 02 đ?‘Ľ =đ??śM22 exp ( ) đ?œ•đ?‘Ľ đ??żđ?‘? ( (−

đ?‘˜M2 đ??żđ?‘? đ?‘˜M2 đ??żđ?‘?

1 1 + 2 )+( − 2 ) cosh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) 4đ??żđ?‘?

4 4đ??żđ?‘?

4 + (−

đ?œ 02

(87g)

đ?‘˜M2 đ?‘˜M32 đ??ż2đ?‘? ) sinh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) ) 4 4

đ?œ• 2 đ?œ 02 đ?‘Ľ =đ??śM22 exp ( ) 2 đ?œ•đ?‘Ľ đ??żđ?‘? đ?‘˜M2 đ??ż2đ?‘? đ?‘˜M42 đ??ż2 đ?‘˜M4 1 1 ( (− 2 + đ?‘? 2 + ) cosh (đ?‘˜ (đ?‘Ľ − đ??ż)) − + ) M2 4 8 8 8đ??ż2đ?‘?

8đ??ż2đ?‘?

(87h)

(87i) and 2 đ?‘˜M22 đ?‘˜M42 đ??ż2đ?‘? đ?‘Ľ đ?œ•đ?œ 02 1 đ?œ•đ?œ 02 đ?œ• 2 đ?œ 02 2 exp ( + ( ) − đ?œ =đ??ś ) ( (− + ) cosh (đ?‘˜M2 (đ?‘Ľ − đ??ż)) đ?œ 02 M 02 2 đ?œ•đ?‘Ľ2 đ?œ•đ?‘Ľ2 đ??żđ?‘? đ?œ•đ?‘Ľ đ??żđ?‘?

4 4

đ??żđ?‘? đ?‘˜M32 đ?‘˜M + (− + 2 ) sinh(đ?‘˜M2 (đ?‘Ľ − đ??ż))) 4 4đ??żđ?‘?

Final version

WL2020R18_153_1

(88a)

A3

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