# Tidal Flats Morphodynamics: A new Conceptual Model to Predict Their Evolution over a Medium-Long Period

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Tidal Flat Morphodynamics: Some Preliminary Considerations

_{w}computed at the bottom.

_{w}is the friction factor and U

_{w}the maximum wave bottom orbital velocity. The friction factor is evaluated by means of the Soulsby formulation [65]:

_{w}T/2π being one half of the horizontal orbital excursion, T the wave period, z

_{0}= K

_{N}/30, and K

_{N}the equivalent Nikuradse roughness. This last parameter is generally taken as a function of the median diameter if the bed is composed of coarse grains, while for fine cohesive grains or mud-beds, K

_{N}can be set equal to a few millimeters as an approximation of the mud protrusions height [66].

_{N}= 0.005 m, tends to be quite flat and homogeneous for the analyzed depths. These trends are very different from the non-monotonic curve suggested by Fagherazzi et al. [47] for a wind speed of 8 m/s, as already discussed by Pascolo et al. [45]. The reason could be attributable to the assumptions of a K

_{N}value equal to 5 × 10

^{−5}m and above all monochromatic wind waves, generated with an imposed constant wave period.

## 3. Conceptual Model

_{m}and the current contribution τ

_{c}are respectively equal to:

_{c}being the absolute value of the current velocity, φ the angle between the wave motion direction and the current, g the gravity acceleration, n the Manning coefficient, and d the water depth.

_{w}, is computed through the Equations (1) and (2), where the wave friction factor derives from an equivalent Nikuradse roughness of 0.005 m and the wave bottom velocity is the result of the numerical generation process achieved performing SWAN in stationary mode on a regular grid. In Table 1, a summary has been reported of the main applied source terms and their relative formulation references and set parameters. The wind speed range, chosen for the wave generation, is from 6 m/s to 14 m/s as deduced from data collected inside the Marano and Grado lagoon [51]. This range is coherent with values observed inside the Venice lagoon [47,48,54], the Virginia Coast Reserve in Virginia [55] and the Willapa Bay in Washington State [46], taken as examples.

^{1/3}. This value has been calibrated in a previous hydrodynamic modeling study of the Marano and Grado lagoon [51]. The current velocity is assumed variable in an appropriate range, taking into account that generally it rarely exceeds 0.25 m/s on the tidal flats in shallow and confined basins [3,9,46,48].

_{c}*, the wind speed U

_{wind}* and the angle φ* between the resultant wind waves and tidal current, have been established. The depth d* corresponds to the value for which this maximum shear stress is equal to the critical erosion threshold τ

_{crit}*. In this manner, a limit depth separates two conditions: an erosion condition for depths lower than d* and a no-erosion condition for depths greater than d*. The conceptual model provides the theoretical curve of this limit depth d

_{lim}(U

_{c}, U

_{wind}*, φ*, τ

_{crit}*) at varying current velocities and for different assigned values of U

_{wind}*, φ*, τ

_{crit}*.

_{c}*. Therefore, the continuous arrows direction shows the tendency of the flat bed to deepen, while the dotted arrows suggest the depth can be reduced by a potential deposition.

_{lim_exp}and d

_{lim_lin}being respectively the limit depth given by the exponential and the linear trends, for which the relative coefficients A

_{i}and B

_{i}with i = 1, 2 can be computed by means of the least squares method.

## 4. Study Site

^{2}and it is bordered by the delta systems of the two main rivers that cross Friuli Venezia Giulia: the Tagliamento on the west and the Isonzo on the east. On the sea side, the lagoon area is bounded by a series of barrier islands alternating with six tidal inlets. The shoreline develops from the western tidal inlet of Lignano to the eastern inlet of Primero for an overall length of about 32 km.

## 5. Results

_{1964}and d

_{2009}. If the difference d

_{1964}− d

_{2009}is negative, an erosion trend will have been verified over the 45 years. The limit depths provided by the conceptual model have been compared to both the measured depth in 1964 and that surveyed in 2009. Similarly, a negative difference d

_{1964}− d

_{lim}means that a deepening trend should have happened over the 45 years, while the difference d

_{2009}− d

_{lim}suggests if the depth measured in 2009 is still under an erosion trend or not. In all these cases, the symbol “x” is used to identify an erosion condition.

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Venice lagoon and Marano and Grado lagoon depicted in the (

**a**) European context and (

**b**) in the Northern Adriatic Sea.

**Figure 2.**(

**a**) Root mean square value of the maxima of the bottom orbital velocity and (

**b**) wave bed shear stress computed on different depths d by means SWAN [61]. The equivalent roughness height is 0.05 m. The dashed line indicates the critical erosion threshold adopted by Fagherazzi et al. [47] and is evaluated as a mean value for the tidal flats of the Venice lagoon [63,64], while the continuous line is the non-monotonic curve that the same authors proposed for a wind speed equal to 8 m/s.

**Figure 3.**Comparison between the erosion critical threshold (dashed red line) and the maximum total bed shear stress due to the combination of the wave contribution obtained from a wind speed of 10 m/s and different values of current velocity as a function of depth d. The angle between waves and current direction is respectively equal to: (

**a**) 0°, (

**b**) 45°and (

**c**) 90°.

**Figure 4.**Scheme of the conceptual model, which provides the limit depth value d

_{lim}between an erosion condition and a no-erosion condition. The limit depth is a function of the current velocity U

_{c}, the wind speed U

_{wind}, the angle φ between generated wind waves and the current flow, and the critical erosion shear stress τ

_{crit}. The asterisked variables inside the rectangles represent the values assigned as input to the model.

**Figure 5.**(

**a**) Limit water depth obtained by the conceptual model with a wind speed of 10 m/s and the critical erosion shear stress is 0.7 Pa. (

**b**) Case of collinearity between waves and current. The dashed lines are regression curves. The arrows indicate the tidal flat tendency to deepen or to be filled if sediments supply is available.

**Figure 6.**Water depth d corresponding to the maximum bed shear stress equal to the critical threshold of 0.7 Pa obtained for different values of the current velocity U

_{c}and angles between the wave motion and the flow direction. The wave field is generated by a wind speed of (

**a**) 8 m/s and (

**b**) 12 m/s. The dashed lines are regression curves.

**Figure 7.**Water depth d corresponding to the maximum bed shear stress equal to different critical thresholds and current velocities U

_{c}. The wave field is generated by a wind speed of 10 m/s. The angles between the wave motion and the flow direction are specified above each graph. (

**a1**) Original curves in the condition of collinearity; (

**a2**) shifted curves in the condition of collinearity; (

**b1**) original curves in the middle condition; (

**b2**) shifted curves in the middle condition; (

**c1**) original curves in the condition of orthogonality; (

**c2**) shifted curves in the condition of orthogonality.

**Figure 8.**Marano and Grado lagoon, Italy. The six tidal inlets, the main channels and the seagrass distribution are highlighted. The contour refers to the more recent bathymetric data surveyed in 2009. Circles represent tidal flats that have been considered for the application of the developed conceptual model.

**Figure 9.**Representative sequence of winds and water levels acting over an annual period, as deduced from data collected inside the Marano and Grado lagoon and subsequent analyses [51]. The tidal level variation is the sum of three main harmonic components which are defined by the respective amplitude A

_{i}, period T

_{i}and phase φ

_{i}.

**Figure 10.**(

**a**) Density frequency area distributions as a function of bottom elevation d obtained from the two available bathymetries respectively in 1964 and 2009. Depth accuracy is about 0.05 m [20,71,72]. (

**b**) Characterization of the points taken inside the selected areas in term of maximum current velocities U

_{c}during wind events of 10 m/s blowing from respectively 90° N and 180° N, and actual depth d.

**Table 1.**Summary of the source terms used in SWAN simulation for the wind wave generation process and their relative formulation references and set parameters.

Source Term | Formula | Parameters | |
---|---|---|---|

Wind input: linear growth | Cavaleri and Malanotte-Rizzoli [68] | - | |

Wind input: exponential growth | Janssen [69] | - | |

White capping | Janssen [69] | cds1 = 4.5 δ = 0.5 | rate of dissipation dependency on wave number |

Bottom friction dissipation | Madsen [62] | kn = 0.05 m | equivalent roughness height |

Surf breaking | Battjes and Janssen [70] | γ = 0.78 | breaker index |

**Table 2.**Summary of the parameters entering in the Equations (6) and (7) for the specified wind speed and angles between the flow direction and the wave motion.

Wind Speed | Parameters | Collinearity 0° | Middle Condition 45° | Orthogonality 90° | |
---|---|---|---|---|---|

8 m/s | exponential | A_{1} (m) | 0.0203 | 0.0163 | 0.0047 |

B_{1} (s/m) | 18.213 | 17.841 | 19.522 | ||

linear | A_{2} (m) | 29.966 | 30.441 | 25.115 | |

B_{2} (s/m) | −5.5993 | −6.2116 | −5.9576 | ||

10 m/s | exponential | A_{1} (m) | 0.1245 | 0.1137 | 0.0348 |

B_{1} (s/m) | 13.674 | 12.551 | 13.776 | ||

linear | A_{2} (m) | 28.526 | 28.699 | 27.688 | |

B_{2} (s/m) | −3.7127 | −4.374 | −6.0564 | ||

12 m/s | exponential | A_{1} (m) | 0.4853 | 0.5103 | 0.3933 |

B_{1} (s/m) | 11.338 | 9.2039 | 6.4084 | ||

linear | A_{2} (m) | 24.685 | 22.104 | 21.241 | |

B_{2} (s/m) | −1.0692 | −1.1855 | −3.0313 |

**Table 3.**Summary of the parameters within the Equations (8) for the specified wind speed and angle between the flow direction and the wave motion.

Wind Speed | Parameters | Collinearity 0° | Middle Condition 45° | Orthogonality 90° |
---|---|---|---|---|

10 m/s | A_{1} (m) | 0.1245 | 0.1137 | 0.0348 |

B_{1} (s/m) | 13.674 | 12.551 | 13.776 | |

C (s·m^{−1}·Pa^{−2}) | 0.2759 | 0.3750 | 0.4625 | |

D (s·m^{−1}·Pa) | −0.7851 | −0.9602 | −1.0703 | |

E (m/s) | 0.4149 | 0.4893 | 0.5234 |

**Table 4.**The list of the preliminary parameters for the conceptual model within each area, referring to the wind condition which leads to the maximum value of the bed shear stress.

Lagoon Portion | Area | τ_{crit} (Pa) | U_{c} (m/s) | Wind Direction | Wave-Current Angle |
---|---|---|---|---|---|

West | A1 | 0.7 | 0.097 | 90° N | 0° |

A2 | 0.7 | 0.092 | 90° N | 0° | |

A3 | 0.7 | 0.177 | 90° N | 0° | |

A4 | 0.5 | 0.122 | 90° N | 0° | |

A5 | 0.5 | 0.145 | 90° N | 0° | |

A6 | 0.7 | 0.178 | 90° N | 0° | |

A7 | 0.7 | 0.132 | 180° N | 45° | |

A8 | 0.7 | 0.177 | 180° N | 0° | |

Central | A9 | 0.7 | 0.117 | 180° N | 0° |

A10 | 0.7 | 0.132 | 90° N | 45° | |

A11 | 0.7 | 0.125 | 180° N | 0° | |

A12 | 0.7 | 0.187 | 90° N | 0° | |

A13 | 0.7 | 0.149 | 90° N | 0° | |

A14 | 0.7 | 0.131 | 90° N | 45° | |

A15 | 0.7 | 0.130 | 180° N | 0° | |

A19 | 0.7 | 0.144 | 180° N | 0° | |

A20 | 0.7 | 0.112 | 180° N | 0° | |

East | A16 | 0.5 | 0.053 | 180° N | 0° |

A17 | 0.5 | 0.056 | 180° N | 45° | |

A18 | 0.5 | 0.051 | 180° N | 45° |

**Table 5.**Validation of the conceptual model. Symbol x is used when the difference between the depths are negative, suggesting an erosion state. The compared depths are those measured in 1964 (d

_{1964}) and 2009 (d

_{2009}) and the limit depth (d

_{lim}) provided by the conceptual model.

Area | Depth (m) | Erosion State | ||||
---|---|---|---|---|---|---|

d_{1964} | d_{2009} | d_{lim} | d_{1964} − d_{2009} | d_{1964} − d_{lim} | d_{2009} − d_{lim} | |

A1 | 0.83 | 0.84 | 0.47 | |||

A2 | 0.83 | 0.77 | 0.44 | |||

A3 | 0.79 | 1.20 | 1.42 | x | x | x |

A4 | 0.75 | 0.90 | >1.5 | x | x | x |

A5 | 0.83 | 0.83 | >1.5 | x | x | |

A6 | 1.05 | 1.17 | 1.42 | x | x | x |

A7 | 1.05 | 1.07 | 0.61 | |||

A8 | 0.91 | 1.08 | 1.41 | x | x | x |

A9 | 0.82 | 0.83 | 0.62 | |||

A10 | 0.96 | 0.98 | 0.60 | |||

A11 | 0.94 | 0.83 | 0.69 | |||

A12 | 0.80 | 1.08 | >1.5 | x | x | x |

A13 | 0.77 | 0.89 | 0.96 | x | x | x |

A14 | 0.63 | 0.85 | 0.60 | x | ||

A15 | 0.54 | 0.79 | 0.74 | x | x | |

A19 | 0.45 | 0.92 | 0.90 | x | x | |

A20 | 0.67 | 0.96 | 0.58 | x | ||

A16 | 0.69 | 0.86 | 0.90 | x | x | x |

A17 | 0.23 | 0.81 | 0.84 | x | x | x |

A18 | 0.26 | 0.46 | 0.78 | x | x | x |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Petti, M.; Pascolo, S.; Bosa, S.; Bezzi, A.; Fontolan, G.
Tidal Flats Morphodynamics: A new Conceptual Model to Predict Their Evolution over a Medium-Long Period. *Water* **2019**, *11*, 1176.
https://doi.org/10.3390/w11061176

**AMA Style**

Petti M, Pascolo S, Bosa S, Bezzi A, Fontolan G.
Tidal Flats Morphodynamics: A new Conceptual Model to Predict Their Evolution over a Medium-Long Period. *Water*. 2019; 11(6):1176.
https://doi.org/10.3390/w11061176

**Chicago/Turabian Style**

Petti, Marco, Sara Pascolo, Silvia Bosa, Annelore Bezzi, and Giorgio Fontolan.
2019. "Tidal Flats Morphodynamics: A new Conceptual Model to Predict Their Evolution over a Medium-Long Period" *Water* 11, no. 6: 1176.
https://doi.org/10.3390/w11061176