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Dpt. of Civil and Environmental Engineering University of Trento (Italy) Channel competition in tidal flats Marco Toffolon & Ilaria Todeschini
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1/12 Channel competition in tidal flats Tidal channels on a tidal flat, Coos Bay, Oregon Tidal channels on a tidal flat, Venice Lagoon
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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
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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
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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
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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
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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
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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)
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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
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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:
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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
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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
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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
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