# Modelling the two-way coupling of tidal sand waves and benthic organisms: a linear stability approach

## Abstract

We use a linear stability approach to develop a process-based morphodynamic model including a two-way coupling between tidal sand wave dynamics and benthic organisms. With this model we are able to study both the effect of benthic organisms on the hydro- and sediment dynamics, and the effect of spatial and temporal environmental variations on the distribution of these organisms. Specifically, we include two coupling processes: the effect of the biomass of the organisms on the bottom slip parameter, and the effect of shear stress variations on the biological carrying capacity. We discuss the differences and similarities between the methodology used in this work and that from ‘traditional’ (morphodynamics only) stability modelling studies. Here, we end up with a \(2\times 2\) linear eigenvalue problem, which leads to two distinct eigenmodes for each topographic wave number. These eigenmodes control the growth and migration properties of both sand waves and benthic organisms (biomass). Apart from hydrodynamic forcing, the biomass also grows autonomously, which results in a changing fastest growing mode (FGM, i.e. the preferred wavelength) over time. As a result, in contrast to ‘traditional’ stability modelling studies, the FGM for a certain model outcome does not necessarily have to be dominant in the field. Therefore, we also analysed the temporal evolution of an initial bed hump (without perturbing biomass) and of an initial biomass hump (without perturbing topography). It turns out that these local disturbances may trigger the combined growth of sand waves and spatially varying biomass patterns. Moreover, the results reveal that the autonomous benthic growth significantly influences the growth rate of sand waves. Finally, we show that biomass maxima tend to concentrate in the region around the trough and lee side slope of sand waves, which corresponds to observations in the field.

## Keywords

Tidal sand waves Benthic organisms Process-based modelling Linear stability analysis Two-way coupling Logistic growth## 1 Introduction

Knowledge of tidal sand waves is of practical interest, as they tend to endanger a broad range of applications for coastal shelf seas [26, 28, 32, 39]. For instance, migrating sand waves may form obstacles for shipping and infrastructure (e.g. oil and gas platforms, cables and wind farms). Also, due to an increased pressure on the coastal environment [54], as well as the increased awareness towards ecological sustainability, knowledge about the interaction of morphological processes with benthic species is gaining importance [2].

The presence of benthic species in coastal areas has been extensively studied [30, and references therein]. Although benthic species may influence the sediment dynamics in many ways, they are commonly classified in two functional groups, namely stabilisers and destabilisers [55]. For example, the sea urchin *Echinocardium cordatum* (see Fig. 1b) is able to stabilise the bed by reworking the top sediment layer [27]. In addition, benthic species that protrude from the bed into the water column (e.g. the tube-building worm *Lanice conchilega* [19, 37]) may affect water motion as well [22]. Nevertheless, this general classification makes it possible to capture the complex interactions among biology and hydro- and sediment dynamics in mathematical models [8, 34].

Various aspects regarding sand wave formation have been studied for over 2 decades using process-based morphodynamic models [4]. Many of these model studies involve a method called linear stability analysis [20], where the stability of a sandy seabed, subject to tidal motion, is analysed. For a symmetrical tide, while only incorporating bed load sediment transport, Hulscher [24] used this method to explain the formation of tidal sand waves. She showed that small bed perturbations distort the tidal flow in such a way that small tide-averaged vertical recirculation cells appear in the water column. The presence of these cells results in a near-bed flow from trough to crest, which in turn induces a net transport of sediments in the same direction. On the other hand, gravitational effects favour sediment transport in a down-slope direction. It is the competition between these two processes that eventually leads to sand wave growth or decay. The typical result of this method is a set of modes with either a positive or a negative growth rate, where the mode with the largest positive growth rate is termed the fastest growing mode (FGM). The FGM is characterised by a wavelength, an orientation and a growth and migration rate (the latter being zero in case of symmetrical forcing). Other studies have shown that the properties of the FGM are comparable to those of sand waves observed in the field [25, 52].

Other model studies have extended the approach of Hulscher [24] and identified additional processes affecting the initial growth of sand waves. For instance, migration due to wind [14, 31] and tidal asymmetry [5], suspended sediment transport and turbulence formulation [6, 7], varying grain sizes [49] and sand scarcity [36]. Moreover, it has been shown that the presence of benthic organisms can influence the initial growth of sand waves [10]. In addition, other researchers used non-linear models to investigate equilibrium sand wave shapes [33, 43, 47], storm effects [13], the role of turbulence [11] and suspended sediment [12].

Whereas the majority of the idealised model studies into morphological features only consider the morphological evolution of the bed, some studies included a coupling between bed topography and multiple grain size fractions in order to explain observed phase differences between them [21, 41, 50, 51]. Moreover, for riverine environments a similar approach has been used, here focussing on the coupling between vegetation and morphodynamics [3, 16]. In these latter studies, a coupled model has been successfully employed to explain bar formation patterns in rivers.

Although previous research has provided evidence of two-way interactions between benthos and coastal morphology, present modelling tools are lacking the ability to investigate these effects. In this paper we use a linear stability approach to develop a fully two-way coupled model to study the feedbacks between sand waves and benthic organisms. In particular, we include the effects of benthos on the bottom roughness; and benthic habitat variations are represented through the biological carrying capacity. The main purpose of this work is to determine whether disturbances in the spatial distribution of benthic organisms may trigger the development of phase-related bed patterns, and vice versa. To this end, we show that the outcome of the two-way coupled methodology is essentially different from that of ‘traditional’ (uncoupled) stability methods. Moreover, we zoom in on the effect of the interaction among three different time scales (hydrodynamical, morphological and biological) within the coupled system.

This paper is organised as follows. The coupled biogeomorphological model is presented in Sect. 2. Next, the solution method, involving a scaling procedure and a linear stability analysis, is given in Sect. 3. In Sect. 4 the results are given, followed by the discussion and conclusions, presented in Sects. 5 and 6, respectively.

## 2 Model formulation

### 2.1 Geometry

### 2.2 Hydrodynamics

### 2.3 Sediment transport

### 2.4 Bottom evolution

### 2.5 Biomass evolution

### 2.6 Coupling coefficients

- \(f_{\mathrm {slip}}\)
- In this work we focus on species that protrude out of the bed (e.g.
*L. conchilega*). They influence the water motion by increasing the bottom roughness through their tubes [22, 35] or created mounds [23]. As defined in Eq. (4), bottom roughness is represented by the bottom slip parameter. To describe the behaviour of these species, we define \(f_\text {slip}\) (see Fig. 4a) by an exponentially decreasing function of biomasswith \(S_\text {bio}^*\) as the actual slip parameter, \(\mu _\text {slip} = S_\text {bio,max}^*/S^*\) as the ratio between the highest possible bottom slip parameter due to benthic biomass \(S_\text {bio,max}^*\) and the abiotic bottom slip parameter \(S^*\), and \(\kappa _\text {slip}^*\) as the rate of increase due to an increasing biomass density.$$\begin{aligned} f_\text {slip}=\frac{S_\text {bio}^*}{S^*} = \left( 1-\mu _\text {slip}\right) \exp \left( -\kappa _\text {slip}^*\phi ^*\right) +\mu _\text {slip}, \end{aligned}$$(8) - \(f_{\mathrm {eq} }\)
- In order to represent the influence of morphological processes on the benthic biomass, we let the carrying capacity be a function of the bottom shear stress. It is widely accepted that the bottom shear stress is a suitable predictor for the distribution of benthic organisms (e.g., de Jong et al. [18]). Therefore, the correction factor for the carrying capacity in Eq. (7) is written as$$\begin{aligned} f_\text {eq}=\frac{\phi _\text {eq,bio}^*}{\phi _\text {eq}^*} = {\left( \tau ^*_\text {ref}-\left| \tau ^*\right| \right) }\kappa ^*_\text {eq}+1, \end{aligned}$$(9)
where \(\phi ^*_\text {eq,bio}\)is the actual carrying capacity, \(\kappa _\text {eq}^*\) is the rate of change due to an increasing density of biomass and \(\tau ^*_\text {ref}\) is a reference value for the critical shear stress. Our choice for this linear function (see Fig. 4c) is justified by the fact that only the derivative of \(f_\text {eq}\) in \(\tau ^*_\text {ref}\) matters in the linear stability analysis.

Overview of the model parameters and their values used in this work

Parameter | Symbol | Values | Unit |
---|---|---|---|

Tidal frequency \((M_2)\) | \(\sigma ^*\) | \(1.41\,\times \,10^{-4}\) | \(\hbox {s}^{-1}\) |

Tidal velocity amplitude \((M_2)\) | \(U^*\) | 0.5 | \(\hbox {m s}^{-1}\) |

Average depth | \(H^*\) | 30 | m |

Gravitational acceleration | \(g^*\) | 9.81 | \(\hbox {m s}^{-2}\) |

Vertical eddy viscosity | \(A^*_{\mathrm {v}}\) | 0.04 | \(\hbox {m}^{2}\,\hbox {s}^{-1}\) |

Water density | \(\rho ^*\) | 1020 | \(\hbox {kg m}^{-3}\) |

Slip parameter | \(S^*\) | 0.01 | \(\hbox {m s}^{-1}\) |

Bed load coefficient | \(\alpha ^*\) | \(1.56\,\times \,10^{-5}\) | \(\hbox {m}^{7/2}\hbox {s}^{2}\,\hbox {kg}^{-3/2}\) |

Slope correction factor | \(\lambda ^*\) | 1.5 | – |

Sediment diameter | \(d^*\) | 350 | \(\upmu\)m |

Logistic growth rate | \(\alpha ^*_{\mathrm {g}}\) | 0.1, 0.5, 1 | \(\hbox {m kg}^{-1}\,\hbox {year}^{-1}\) |

Undisturbed carrying capacity | \(\phi ^*_{\mathrm {eq}}\) | 1 | \(\hbox {kg m}^{-1}\) |

Biological dispersal rate | \(D^*\) | 100 | \(\hbox {m}^{2}\,\hbox {year}^{-1}\) |

Highest possible slip parameter | \(S^*_{\mathrm {bio,max}}\) | \(2\cdot S^*\) | \(\hbox {m s}^{-1}\) |

Rate of increase (slip parameter) | \(\kappa ^*_{\mathrm {slip}}\) | 0.5 | \(\hbox {m kg}^{-1}\) |

Rate of change (carrying capacity) | \(\kappa ^*_{\mathrm {eq}}\) | 0.5 | \(\hbox {m s}^{2}\,\hbox {kg}^{-1}\) |

Topographic wave number | \(k^*\) | 0–0.04 | \(\hbox {m}^{-1}\) |

Residual current velocity \((M_0)\) | \(U_{M_0}^*\) | 0.1–1 | \(\hbox {m s}^{-1}\) |

Domain length | \(D^*\) | 10,000 | m |

Initial height bed hump | \(h_\text {gaus}^*\) | 0.05 | m |

Half-width initial hump | \(\delta ^*\) | 50, 100 | m |

Initial height biomass hump | \(\phi _\text {gaus}^*\) | 0.03, 0.05 | \(\hbox {kg m}^{-1}\) |

## 3 Solution method

### 3.1 Outline

First we will present a scaling procedure, after which we describe the forcing. Next, we introduce the linear stability analysis, in which the basic and perturbed state solutions are given. Finally, we describe the individual process contributions.

### 3.2 Scaling procedure

*x*,

*z*,

*t*) and quantities \((u, w, h, \zeta , \tau , q, \phi )\) according to

*t*and \(t_\text {long}\), such that two time scales are identified, i.e. the tidal time scale \(1/\sigma ^*\) and a yet to be determined biogeomorphological time scale \(T^*_\text {long}\). The biogeomorphological time scale is given by

### 3.3 Forcing

*M*is the number of tidal constituents, which in turn are represented by the complex amplitudes of the Fourier components \(\hat{F}_m = \overline{\hat{F}_{-m}}\). The actual forcing is hence real-valued and chosen such that—in the case of a flat bed—a prescribed depth-averaged flow is attained. In the results shown in this paper, \(M = 8\) and has been chosen based on numerical experiments, such that the contribution of harmonics \(\pm \left( M+1\right)\) to the solution are negligible. This value is determined by the inclusion of the critical shear stress in the sediment transport formulation.

### 3.4 Linear stability analysis

*c*.

*c*. . Here we further distinguish between contributions proportional to the complex amplitudes \(\breve{h}_1\) and \(\breve{\phi }_1\), respectively.

### 3.5 Basic state

The basic state describes the solution of the system obtained over a flat bed. For the sake of brevity, the solution to the basic flow problem is given in "Flow solution" of “Appendix 1”. Furthermore, the basic bed load sediment transport solution is given in “Sediment transport" of “Appendix 1”, and the basic coupling coefficients are described in “Coupling coefficients" of “Appendix 1”.

Finally, in the basic state there are no spatial variations in sediment transport, hence the bed evolution equals zero and the flat bed remains flat. However, as can be seen in “Biomass evolution" of “Appendix 1”, the basic benthic biomass does increase (spatially uniform) due to autonomous logistic growth.

### 3.6 Perturbed state

## 4 Results

First, we present and analyse the obtained linear eigenvalue problem and its properties, which consists of two distinct eigenmodes. Then we will describe a reference case, where no coupling is present. The results of the coupled system follows after that, for both symmetrical and asymmetrical forcing. Finally, we show bed and biomass evolution in case of an initial hump of either topography (without perturbing biomass), or biomass (without perturbing topography).

An important note for the following results is that we differentiate between fixed and variable values for the benthic basic state. As an evolving benthic basic state complicates the interpretation of the results, in Sects. 4.1–4.4 we first describe the results for a fixed basic biomass. Second, in Sect. 4.5 we focus on the effect of the temporal evolution of the benthic basic state.

### 4.1 Linear eigenvalue problem

### 4.2 Solution properties

### 4.3 Reference case: no coupling

In order to better interpret the outcome of this methodology, we first present a reference case with a fixed benthic basic state of \(\phi ^*_0=0.25\ \hbox {kg m}^{-1}\) (Fig. 5a) and \(\phi ^*_0=0.75\ \hbox {kg m}^{-1}\) (Fig. 5b), corresponding to the increasing and reducing growth regions of the logistic growth profile (Fig. 3), respectively. Here the coupling coefficients \(f_\text {eq}\) and \(f_\text {slip}\) are set to 1, such that \(\omega _\text {flow,biotic}\) and \(\omega _\text {eq,abiotic}\) turn out to be zero. The resulting evolution equations [see Eq. (36)] are thus characterised by a diagonal matrix, and hence, no coupling.

This is plotted in Fig. 5, where for the two resulting eigenvalues \(\varGamma _1, \varGamma _2\) the growth rate is given as a function of the wave number. As can be seen from the corresponding eigenvectors \(\vec {\chi }_1,\vec {\chi }_2\), the solid line corresponds to topography, while the dashed-dotted line corresponds to biomass. From now on—whenever applicable—we will therefore refer to these eigenmodes as the ‘morphological’ and ‘biological’ eigenmode, respectively. However, as we will see further on, this distinction is not always justified.

Interestingly, around \(k^*=0\ \hbox { m}^{-1}\) we observe non-zero growth rates for the ‘biological’ eigenmode, unlike the ‘morphological’ eigenmode, which has a tendency towards zero growth. Indeed, we see in Eq. (77) that the logistic growth contribution does not depend on the wave number, such that there is no damping effect.

For larger values of the benthic basic state, the growth rate of the ‘biological’ eigenmode uniformly decreases, which again can be ascribed to the logistic growth contribution. Moreover, for the ‘biological’ eigenmode we see a tendency towards lower growth rates for larger wave numbers (smaller wavelengths) due to the biological dispersal effect.

### 4.4 Interpretation of the two-way coupled system

#### 4.4.1 Symmetrical forcing

To facilitate interpretation of the solution of the fully two-way coupled model, we first focus on a symmetrical forcing, with a (fixed) benthic basic state of \(\phi ^*_0=0.75\ \hbox {kg m}^{-1}\).

The result for this case is plotted in Fig. 6, where the (real-valued) eigenvalues and associated eigenvectors, moduli and phase shifts are presented. Figure 6a shows the growth rate of the two eigenmodes. The FGM for this case is denoted by the black dots, which corresponds to the highest growth rate of \(\varGamma _1\).

Also, in Fig. 6a the eigenvectors \(\vec {\chi }_1,\vec {\chi }_2\) associated to the FGM are shown, which correspond to the result illustrated in Fig. 6b. Here, the solid lines, related to \(\varGamma _1\), clearly show that this eigenvalue influences both bed topography as well as benthic biomass. Unlike the first eigenvector, \(\vec {\chi }_2\) has only a minor influence on the topographic perturbations. This becomes more clear in Fig. 6c, where the moduli are given. Here, an amplitude ratio of value one indicates that the contribution of the eigenvector to the amplitude is equal for both topography as well as biomass, and that an amplitude ratio close to zero indicates that either bed or biomass is dominant. Based on these results \(\varGamma _2\) can thus be referred to as the eigenvalue of the ‘biological’ eigenmode, whereas \(\varGamma _1\) can be ascribed to the ‘mixed’ eigenmode.

Next, we will focus again on a symmetrical forcing, but now for \(\phi ^*_0=0.25\ \hbox {kg m}^{-1}\). The corresponding model results are plotted in Fig. 7, where panel (b) now shows the imaginary part of the eigenvalues, presented as the migration rate. Furthermore, the modulus \(\varTheta _2^\text {h}\) (Fig. 7c) shows that the relative topography amplitude is larger in this case, such that it is not justified any more to refer to this eigenmode as the ‘biological’ eigenmode.

- \(\varGamma _1 < \varGamma _2\)
The first is for small wave numbers where \(\varGamma _2\) is dominant. Here it stands out that the \(180^{\circ }\) phase shift, which was present in Fig. 6d, is not visible any more. It turns out that for the range of modes where the eigenvalue of the ‘mixed’ eigenmode dominates the ‘biological’, no phase shift occurs. However, this behaviour can only be observed when the morphological amplitude of the ‘biological’ eigenmode is small compared to the amplitude of the biomass, i.e. when the modulus \(\varTheta _2^\text {h}\) is close to zero.

- \(\varGamma _1 > \varGamma _2\)
Second, in the part where \(\varGamma _1\) is dominant, both eigenmodes show a phase shift of \(180^{\circ }\), similar to what was observed in Fig. 6d.

- \(\varGamma _1 = \overline{\varGamma _2}\)
The third range we observe is where both eigenmodes show the exact same growth rates, while their imaginary parts are non-zero. The latter results in migration rates for this range of modes, which might seem counter-intuitive for a symmetrical forcing. Moreover, from the moduli we see that the amplitude ratio for both eigenvalues is equal as well. It turns out that the eigenvectors for this situation (not shown here) are each other’s complex conjugate. Consequently, it follows that in this particular region of wave numbers, the perturbations can be interpreted as two identical travelling waves migrating in opposite direction, such that both topography and biomass behave like a standing wave. Furthermore, the phase difference between the two standing waves (bed and biomass) ranges between \(0^{\circ }\) and \(180^{\circ }\). As pointed out in Sect. 4.3, these results only hold for a specific moment in time, since the benthic basic state is fixed. It is thus possible that this standing wave will not fully develop in the field, as this behaviour can only be observed in case of increasing biological growth \((\phi ^*_0 <0.5\ \hbox {kg m}^{-1}\), see Fig. 3). This type of behaviour is also observed in other morphodynamic stability studies, as will be further discussed in Sect. 5.

#### 4.4.2 Asymmetrical forcing

We will now focus on the effects of an asymmetrical forcing, specifically due to the presence of a residual current. In the results below, an \(M_0\) residual current of \(0.01\hbox { m s}^{-1}\) has been superimposed upon the \(M_2\) tidal forcing. Compared to the symmetrical reference case (Fig. 5a) the growth rates of both sand waves and biomass are hardly influenced by the residual current (not shown here). Furthermore, the ‘morphological’ eigenmode shows an increasing positive migration rate for an increasing wave number (whereas the ‘biological’ eigenmode shows no migration at all, also not shown here).

Figure 8 shows the results for the full two-way coupling and a benthic basic state of \(\phi ^*_0=0.25\ \hbox {kg m}^{-1}\). Compared to the uncoupled result described above, the migration rates show an overall increase for the first eigenmode, while a small migration in opposite direction can be observed for the second. In addition, we see that, compared to the reference case, for the first eigenmode the growth rate increases and the FGM shifts towards a shorter wavelength, in opposition to \(\varGamma _2\), which is hardly influenced.

When looking at the phase shifts between the amplitudes of the eigenmodes (Fig. 8d), we see that in particular in the region where \(\varGamma _1\) is dominant, the phase shifts change compared to symmetrical case (Fig. 7d). For the dominant (‘mixed’) eigenmode, it thus follows that the crests of the biomass perturbations are concentrated on the stoss side of the topography perturbations. If we increase the residual current even further (not shown here) we observe that this eigenmode has a tendency towards a phase shift of around \(-\,90^{\circ }\).

### 4.5 Hump evolution: initial value problem

#### 4.5.1 Topography versus biomass hump

#### 4.5.2 Logistic growth rate

#### 4.5.3 Residual current

Although not distinctly visible from Fig. 11, the preferred phase difference between biomass and topography in this situation is slightly less than \(180^{\circ }\). For an increasing residual current strength, this phase difference decreases even more. To illustrate this, we present in Fig. 12a the bed and biomass profile after \(8\ \hbox {year}\) in case of a residual current strength of \(U^*_{M_0} =0.05\ \hbox {kg m}^{-1}\). It can be clearly seen that the biomass crests are now concentrated on the lee side of the developing sand waves. Additionally, in Fig. 12b the phase difference between biomass and topography \(\left( \theta ^\phi \right)\) is plotted as a function of the residual current strength. It shows that for an increasing residual velocity, the phase shift gradually decreases and tends towards a value of \(90^{\circ }\).

## 5 Discussion

### 5.1 Comparison to ‘traditional’ stability analysis

In ‘traditional’ stability modelling studies (i.e. morphodynamics only, uncoupled), it is common to unravel the individual contributions to the growth and migrations rates (e.g., Campmans et al. [14]). Also for the two-way coupled model presented in this work, these contributions can be specified (as in “Appendix 3”). However, within the current methodology the interpretation of these contributions is somewhat different, since the resulting system of equations forms a \(2\times 2\) eigenvalue problem [Eq. (36)], leading to two distinct complex growth rates. As a result, the growth (and migration) rate does not depend proportionally on its associated entries any more. For instance, a change of a certain individual contribution related to bed perturbations \(\big (\omega ^{\text {h}}_{{h}_1},\ \omega ^\phi _{{{h}}_1}\big )\) does not solely affect the associated topography growth rate—as is the case for ‘traditional’ stability analysis—but now also that of biomass. Consequently, to fully understand the magnitude of these contributions, the resulting eigenvectors of the eigenvalue problem have to be taken into account as well. These eigenvectors describe the relative contribution of the entries to the associated growth rates. We would like to stress that our approach still implies a linear problem, and thus can be analytically solved by means of linear algebra.

In a study into current-generated sorted bed forms, van Oyen et al. [50] found that the resulting eigenmodes could be assigned to either roughness or topography. In our model, we observe that the eigenmodes can be related to either biomass or topography, but only to a certain extent. It turns out that the classification of these eigenmodes is bound by a certain range of parameter settings. Particularly for smaller values of the benthic basic state, both eigenmodes have the tendency to influence both perturbations (here referred to as the ‘mixed’ eigenmode). However, using this classification still contributes to our general understanding of the system, such that we can effectively use this methodology for a systematic process analysis.

### 5.2 Autonomous benthic growth and the role of the biological time scale

In contrast to ‘traditional’ stability analysis, the FGM for a given parameter setting does not have to be the eventual mode (wavelength) observed in the field. The role of the autonomous benthic growth here is crucial; as the resulting eigenmodes are only valid for a certain moment in time, the FGM (and its properties) may vary due to an evolving biomass. In particular, the largest effect of the autonomous benthic growth on the perturbations enters the system through the logistic growth contribution (\(\omega _\text {logistic}\)). It appears that this contribution leads to positive growth rates for the ‘biological’ eigenmode if the benthic basic state is below the inflection point of the logistic growth profile (Fig. 3), while the opposite holds for values larger than the inflection point. Moreover, we can conclude from the moduli that if the benthic basic state is above the inflection point, the growth rate of the topography perturbation is almost solely determined by the ‘mixed’ eigenmode, whereas for values lower than the inflection point, both eigenmodes influence the topographical growth rate. On the other hand, the biomass growth is determined by both eigenmodes regardless the benthic basic state. Furthermore, the results showed that the growth rate of the ‘biological’ mode is almost spatially uniform (equal for all \(k^*\)), whereas the ‘mixed’ mode shows a distinct FGM. As a consequence, the ‘mixed’ mode is mainly responsible for the eventually occurring wavelength in the field.

Although we gain much insight from the cases where \(\phi ^*_0\) is fixed, one should continue to study the temporal behaviour of the system to fully understand the outcome. Imposing a hump (biomass or topography, without perturbing the other) shows us that only a small disturbance may trigger growth of rhythmic patterns of both sand waves and biomass. It appears that the biological time scale is an important factor for the initial evolution of the coupled system. If the biological time scale were much shorter than the morphological time scale, it would be justified to only look at the FGM for the situation of fully developed biomass \(( \phi ^*_0=1\ \hbox {kg m}^{-1})\). Conversely, if both time scales were of the same order, the FGM would constantly be subject to change. Important to note here is that a slow biological growth does not only affect the biomass evolution, but also the initial growth of the topographic perturbations. Although linear stability models are not able to describe finite-amplitude behaviour of sand waves, non-linear models, used for this particular purpose, determine the FGM (preferred wavelength) to bypass the initial growth stage, and hence, speed up computational time [47]. If biological processes would be included in these non-linear models, it should thus be noted that the FGM might vary over time.

From the results it follows that phase shifts of various magnitudes may occur between the crests of bed and biomass perturbations. In particular for the cases where the benthic basic state is fixed, rich behaviour is visible with regard to phase shifts. In the situation where a topography hump is imposed, an anti-phase develops between bed and biomass. Remarkably, when a biomass hump is imposed, this phase difference only starts to evolve after the autonomous biomass evolution has passed the inflection point of logistic growth. Moreover, for an increasing residual current strength, the phase difference decreases and the biomass crests are concentrated on the lee side of the sand waves \(\left( \theta ^\phi = 90^{\circ }{-}180^{\circ }\right)\). This contrasts the result from Fig. 8, where an opposite (i.e. negative) phase shift is present for the dominant ‘mixed’ eigenmode. Indeed, from the crest and trough development in Fig. 11 it can be seen that before both perturbations have developed a steady phase difference, crests and troughs may show negative migration rates, resulting in phase shifts that significantly differ from the eventual phase difference. Again, this emphasizes the importance of taking into account the evolving basic state, rather than only looking at a fixed value for \(\phi ^*_0\).

For a limited range of modes, and given that the benthic basic state is below the inflection point, we showed that standing waves for both sand waves and biomass can occur. In this case both eigenmodes are each others complex conjugate, each displaying migration in opposite direction. The temporal evolution of the superimposed bed and biomass patterns would thus be oscillatory, hence a standing wave. This type of behaviour is observed in other morphodynamic stability studies as well [38, 48]. The latter showed that for waves approaching the coast perpendicularly, surf zone patterns may behave as standing waves. Moreover, they showed that for oblique wave incidence this behaviour vanishes. These findings correspond to our results where complex conjugate eigenmodes only occur for symmetrical tidal forcing. Considering that under field conditions a pure symmetrical tide does not occur—and taking into account the evolution of the benthic basic state—it is highly unlikely that this standing wave pattern will develop in nature.

### 5.3 Comparison to field data

Our model shows that the biomass of benthic organisms and sand waves develop in anti-phase (or close to), which is supported by observations in the field. Baptist et al. [1], and more recently, Damveld et al. [17] observed that organisms living on top of the seabed as well as within, occur much more frequently in sand waves troughs compared to the crests. Moreover, preliminary results from a recent field campaign show that various abiotic parameters, which are good predictors for the occurrence of benthic organisms, such as silt content and permeability, also show phase related patterns over sand waves [15]. This study showed that silt content, for example, is much higher in the sand wave troughs, but also on the lee slope, compared to the crests and stoss slope of the sand wave. Although other processes influence the habitat of benthic organisms as well, the observed patterns from this field study generally agree with the phase differences presented in this work.

To enable further comparison of our model results to field observations, the next step is to gather more information about local environmental and biological conditions. We are particularly interested in the biomass values over sand wave fields. In addition, data from different locations enables us to quantitatively relate these biological parameters to the wavelengths and growth and migration rates of the local bed forms. Furthermore, the parametrisations used in this model need to be fitted to more realistic parameter settings, although information on the effect of benthic organisms on the hydrodynamic and morphological processes in subtidal areas is scarce. Nevertheless, Borsje et al. [9] already proposed several biological parametrisations, which could serve as a starting point for future work. For instance, this two-way coupled model allows for the inclusion of a biologically influenced critical shear stress. Finally, in order to be flexible towards the available experimental data, we would like to point out that our modelling approach is not restricted to the use biomass as an indicator for benthic organisms, but that other indicators can be used as well (e.g., abundance, biodiversity).

## 6 Conclusions and outlook

In this paper we developed a fully two-way coupled model between sand waves and benthic organisms. With this model we are able to systematically investigate the processes that are leading to the formation of sand waves and the spatial and temporal distribution of benthos over these bed forms. Although the parametrisations used for the two-way coupling were not yet fitted to local environmental data, we showed that a local biomass disturbance leads to the growth of sand waves, and vice versa. Furthermore, we observed that phase shifts may occur between sand waves and biomass perturbations, similar to field observations. For a symmetrical forcing, we observed that they are in anti-phase. Furthermore, a residual current leads to a phase shift where the biomass maxima are concentrated on the lee slope of the sand waves.

This is the first study including the two-way coupling between sand waves and benthic organisms in a process-based morphodynamic model. In doing so, we recognised that the methodology of this work differs from ‘traditional’ morphodynamic stability modelling studies. Here, we ended up with a linear eigenvalue problem that eventually results in two distinct eigenmodes, instead of one eigenmode for the ‘traditional’ stability analysis. Moreover, the benthic basic state displays autonomous growth, such that the temporal evolution of the bed and biomass profile has to be taken into account to fully understand the results of this methodology. Also, the growth of this benthic basic state plays an important role in this study, as its stage relative to the inflection point of the logistic growth determines whether the eigenmodes can be related to either biology, morphology, or both. Also, if the benthic basic stage is below the inflection point an in-phase pattern may initially develop. This is especially relevant in case of slow biological growth, i.e. a long biological time scale. Finally, we have shown that the biological time scale significantly influences the morphological evolution. For slow biological growth, sand waves also tend to develop on a slower rate, in contrast to a fast biological growth.

A next step is to investigate these processes using parametrisations which are better fitted to experimental data. A sensitivity analysis into a realistic biological parameter range would then give insight in their influence on the model results. Using these insights, this model can eventually be used for predicting the formation properties of tidal sand waves combined with biological evolution in shallow sandy seas.

## Notes

### Acknowledgements

This work is part of the NWO funded SANDBOX project. Also, the financial support of Royal Boskalis Westminster N. V. and the Royal Netherlands Institute for Sea Research (NIOZ) is much appreciated. The authors further wish to thank dr.ir. G. H. P. Campmans for providing a first version of the model script. Finally, we would like to thank an anonymous reviewer for the input on an earlier draft of this manuscript.

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