1. Dpt. of Civil and Environmental Engineering University of Trento (Italy) Channel competition in tidal flats Marco Toffolon & Ilaria Todeschini

2. 1/12 Channel competition in tidal flats Tidal channels on a tidal flat, Coos Bay, Oregon Tidal channels on a tidal flat, Venice Lagoon

3. 2/12 TWO POSSIBLE DIFFERENT APPROACHES 1.THE PROBLEM OF THE INITIAL FORMATION OF TIDAL CHANNELS IN A SALT MARSH 2.THE PROBLEM OF THE STABILITY OF ALREADY DEVELOPED CHANNELS WITH RESPECT TO A PERTURBATION OF THEIR STATE Channel competition in tidal flats a. one single channel (Fagherazzi and Furbish, 2001) b. a network of channels (D’Alpaos et al.,2005) c. initiation of tidal channels VERY SIMPLIFIED APPROACH

4. 3/12 FORMULATION OF THE PROBLEM THE SYSTEM IS CONSTITUTED ONLY BY TWO ELEMENTS Channel competition in tidal flats SIMPLE CONCEPTUAL MODEL: 1.TIDAL FLAT 2.CHANNELS We suppose that the free surface level varies in the flat without drying

5. 4/12 HYPOTHESIS: The contribution of the flat to the total discharge is neglected the total discharge depends only on the planimetric surface S (quasi-static model) Simplified EROSION LAW typical of cohesive sediments The discharge in each channel is estimated using the usual equilibrium relationship  Tidally averaged (e.g. Fagherazzi and Furbish) Channel competition in tidal flats Only the altimetric evolution is considered, while the variation of the width is neglected

6. 5/12 STABILITY WITH CONSTANT ENERGY SLOPE If we assume a constant J along the transversal direction (e.g. Fagherazzi and Furbish, 2001) Channel competition in tidal flats J Perturbation analysis starting from the equilibrium configuration: d 1, d 2 perturbations of the two water depths UNKNOWNS: INSTABILITY except for the case d 01 =d 02 In the case of TWO CHANNELS: d 10 = 1 d 20 = 0 (only the first channel is perturbed) PERTURBATION OF THE FLOW DEPTH channel 1 channel 2

7. 6/12 BASIC IDEA: Every channel drains a portion of the tidal flat “COMPETENCY AREA” STABILITY WITH VARIABLE ENERGY SLOPE Channel competition in tidal flats TWO COUPLED MODELS: 1.TRANSVERSAL DRAINAGE AND WATERSHED DELIMITATION 2.LONGITUDINAL WAVE PROPAGATION ALONG THE CHANNELS to establish a relationship between Q and the free surface elevation in each channel, h 1 and h 2 to determine h 1 and h 2 as functions of d 1 and d 2 in the case of TWO INITIALLY IDENTICAL channels  Q 0 is the same

8. 7/12 1. WATERSHED DELIMITATION Poisson equation (Rinaldo et al. 1999) where: Channel competition in tidal flats f is the friction factor hypothesis: Rough salt marsh (from Lawrence et al., 2004) the longitudinal fluxes in the salt marsh are neglected the tidal oscillation is small with respect to the average depth D 0(f) in the flat  D m = D 0(f) +h m  D 0(f)

9. 8/12 In a tidal cycle in the generic section x: the absolute value is due to the fact that these are periodic functions The variation of the total discharge at the mouth: Channel competition in tidal flats 1. WATERSHED DELIMITATION

10. 9/12 2. WAVE PROPAGATION IN THE CHANNELS The free surface elevation in i th channel: following DRONKERS (1964) Channel competition in tidal flats SYMMETRIC CONFIGURATION WITH TWO INITIALLY IDENTICAL CHANNELS if we substitute (d 1 -d 2 ) from the previous expression and we integrate in time to have the average value in a tidal cycle if we integrate in the longitudinal direction to have the difference of the total discharge Linearizing, we obtain:

11. 10/12 Substituting the expression for Q into the differential equation system: Channel competition in tidal flats becomes EIGENVALUES: 1.(-7/3) 2.(4-7/3) positive for always negative Two channel separated by a distance L > L c can be considered INDEPENDENT Two channel separated by a distance L < L c influence each other and tend to form an unstable system INSTABILITY if where

12. 11/12 Channel competition in tidal flats THRESHOLD VALUE OF THE DISTANCE L L > L C  STABLE L < L C  UNSTABLE PERTURBATION OF THE FLOW DEPTH channel 1 channel 2 Physical interpretation: Increasing L, Q decreases Q 0 increases Mutual influence (Q/ Q 0 ) DECREASES

13. 12/12 Channel competition in tidal flats THRESHOLD DISTANCE FOR CHANNEL STABILITY Shallow and rough flat and channels Deep and smooth flat and channels L c is quite large: do channels in nature usually influence each other? Limitations of the model: the longitudinal flux has been neglected the wave propagation theory ignores finite-length effects