Nature-Based Solutions for Coastal Management Formulas PDF

Nature-based solutions for coastal management 𝐻 1 = 𝐻! 1 + β𝑥 4𝑎𝑁𝐻! 𝑘 𝑠𝑖𝑛ℎ" 𝑘𝑙 + 3𝑠𝑖𝑛ℎ𝑘𝑙 β=...

Nature-based solutions for coastal management 𝐻 1 = 𝐻! 1 + β𝑥 4𝑎𝑁𝐻! 𝑘 𝑠𝑖𝑛ℎ" 𝑘𝑙 + 3𝑠𝑖𝑛ℎ𝑘𝑙 β= 𝐶# 9𝜋 𝑠𝑖𝑛ℎ2𝑘ℎ + 2𝑘ℎ 𝑠𝑖𝑛ℎ𝑘ℎ FLOW ENERGY ATTENUATION – ANALYTICAL FORMULATIONS Nature Based Solutions for Coastal Management Maria Maza, 2024  Wave energy attenuation Drag coefficient New approaches Nature Based Solutions for Coastal Management Maria Maza, 2024  Wave energy attenuation Over the years, energy dissipation induced by coastal ecosystems has been analyzed using different approaches. The first approaches calculated the energy dissipation of waves using an analytical formulation based on the conservation of energy equation assuming a steady flow. The energy gradient of the flow produced along the ecosystem is related to the dissipation induced by the flow considering the drag force exerted on the ecosystem. This drag force is calculated assuming a constant drag coefficient and a constant representative width of the ecosystem elements per unit height along the height of the ecosystem. Dalrymple et al. (1984) first presented this energy conservation approach for regular waves following linear wave theory and Méndez and Losada (2004) extended the formulation for irregular waves. This approach calculates the decay of the wave height as a function of an attenuation coefficient as follows: 𝐻 1 = 𝐻! 1 + 𝛽𝑥 Nature Based Solutions for Coastal Management Maria Maza, 2024  Wave energy attenuation 𝐻 1 = 𝐻! 1 + 𝛽𝑥 𝐻! incident wave height 𝐻 wave height at a distance x within the ecosystem 𝛽 wave attenuation coefficient Most studies in the literature fit this attenuation coefficient using experimental data. Nature Based Solutions for Coastal Management Maria Maza, 2024  Wave energy attenuation Higher density and biomass values lead to Example: Maza et al. 2015 higher attenuation rates for both species. biomechanical propertiesof the two real salt marshes used in the experiments are also higher beta ~ higher evaluated and related to wave damping attenuation revealing ahigher attenuation for stiffer vegetation. ro square? dot and line represent water depth? Nature Based Solutions for Coastal Management Maria Maza, 2024  Wave energy attenuation Example: Maza et al. 2015 New equation derived for wave-current conditions Nature Based Solutions for Coastal Management Maria Maza, 2024  Wave energy attenuation There are studies that give an analytical expression for this coefficient as a function of: Flow parameters (wave height and period, draft). Geometric characteristics of the ecosystem (representative width, density, height). Drag coefficient, 𝐶#. Dalrymple et al. (1984) obtained the first analytical formulation for 𝛽 for regular waves. Mendez and Losada (2004) presented an expression for irregular waves. Losada et al. (2016) derive the formulas for waves and current conditions. Nature Based Solutions for Coastal Management Maria Maza, 2024  Wave energy attenuation Paper Study Ecosystem Formulation Variables and considerations Dalrymple (1984) Analytical Generic 𝐻 1 𝐻: wave height at the meadow (m) = 𝐻! 1 + β𝑥 𝐻! : incident wave height (m) 𝛽: attenuation coefficient (1/m) 4𝑎𝑁𝐻! 𝑘 𝑠𝑖𝑛ℎ " 𝑘𝑙 + 3𝑠𝑖𝑛ℎ𝑘𝑙 𝑥: separation between positions (m) β= 𝐶 𝑎: stem diameter (m); 𝑁: number 9𝜋 𝑠𝑖𝑛ℎ2𝑘ℎ + 2𝑘ℎ 𝑠𝑖𝑛ℎ𝑘ℎ # stems per square meter (1/m2); ℎ: function of drag force water depth (m); 𝑘: wave number (1/m); 𝑙: vegetation length; 𝐶# : drag coefficient - Wave decay for regular waves Mendez and Lab Seagrass 𝐻$%& 1 𝐻$%& : root-mean-square wave height = Losada (2004) (polypropyle 𝐻$%&,! 1 + β𝑥 at the meadow (m) ne strips, 𝐻$%&,! : root-mean-square incident Asano et al. 𝑎𝑁𝐻$%&,! 𝑘 𝑠𝑖𝑛ℎ " 𝑘𝑙 + 3𝑠𝑖𝑛ℎ𝑘𝑙 wave height (m) 1988) β= 𝐶 β: attenuation coefficient (1/m) 3 𝜋 𝑠𝑖𝑛ℎ2𝑘ℎ + 2𝑘ℎ 𝑠𝑖𝑛ℎ𝑘ℎ # 𝑥: separation between positions (m) 𝑎: stem diameter (m); 𝑁: number stems per square meter (1/m2); ℎ: water depth (m); 𝑘: wave number (1/m); 𝑙: vegetation length; 𝐶# : drag coefficient - Wave decay for irregular waves Nature Based Solutions for Coastal Management Maria Maza, 2024  Paper Study Vegetation Formulation Variables and considerations Losada Lab Saltmarshe 𝐻 = 1 Regular waves 𝐻, 𝐻𝑟𝑚𝑠 : wave height and root et al. s (Spartina 𝐻𝐻0 1+𝛽𝑤𝑐 𝑥1 mean-square wave height at the 𝑟𝑚𝑠 (2016) anglica and 𝐻𝑟𝑚𝑠,0 = 1+𝛽𝑤𝑐 ′ 𝑥 Irregular waves meadow (m) Puccinellia 𝛽𝑤𝑐 𝐻0 , 𝐻𝑟𝑚𝑠,0 : incident wave height maritima) 3𝜋 3𝑘𝑐𝑜𝑠ℎ 3 𝑘ℎ and root-mean-square incident =# 8𝑔 ; 1 3 𝑠𝑖𝑛ℎ 3 𝑘𝑙𝐷 + 3𝑠𝑖𝑛ℎ𝑘𝑙𝐷 8 wave height (m) 𝑔𝑘 ′ 2𝑎𝑁 𝐻0 𝛽𝑤𝑐 , 𝛽𝑤𝑐 : attenuation coefficient 2 𝜎 − 𝑈0 𝑘 1 1 for regular and irregular waves 2𝑘ℎ 𝑔 2 𝑔 4𝑘ℎ 3𝑘 2 𝑔 2 98 + < 𝑡𝑎𝑛ℎ𝑘ℎ + 𝑈0 3 + + 𝑈 𝑐𝑜𝑡ℎ𝑘ℎ 𝑈0 respectively (1/m) 𝑠𝑖𝑛ℎ2𝑘ℎ 𝑘 8 𝑠𝑖𝑛ℎ2𝑘ℎ 8 0 𝑘 𝑥: separation between positions 1 −1 1 2𝑘ℎ 𝑔 2 (m) + 1+ 𝑡𝑎𝑛ℎ𝑘ℎ 9$ 𝐶𝐷𝑤𝑐 2 𝑠𝑖𝑛ℎ2𝑘ℎ 𝑘 𝑎: stem diameter (m); 𝑁: number stems per square meter 𝛽′𝑤𝑐 (1/m2); ℎ: water depth (m); 𝜎: 2 𝜋 3𝑘𝑐𝑜𝑠ℎ 3 𝑘ℎ wave frequency (1/s); 𝑈0 : =# 8𝑔 ; 1 3 𝑠𝑖𝑛ℎ 3 𝑘𝑙𝐷 + 3𝑠𝑖𝑛ℎ𝑘𝑙𝐷 8 current velocity (m/s); 𝑘: wave 𝑔𝑘 𝑎𝑁 2 𝜎 − 𝑈0 𝑘 𝐻𝑟𝑚𝑠,0 number (1/m); 𝑙𝐷 : deflected 1 1 vegetation length; 𝑔: gravity 2𝑘ℎ 𝑔 2 𝑔 4𝑘ℎ 3𝑘 2 𝑔 29 8 ′ 𝑈0 acceleration (m/s ); 𝐶𝐷𝑤𝑐 , 𝐶 𝐷𝑤𝑐 : 2 + < 𝑡𝑎𝑛ℎ𝑘ℎ + 𝑈0 3 + + 𝑈0 𝑐𝑜𝑡ℎ𝑘ℎ 𝑠𝑖𝑛ℎ2𝑘ℎ 𝑘 8 𝑠𝑖𝑛ℎ2𝑘ℎ 8 𝑘 drag coefficient for regular and −1 1 1 2𝑘ℎ 𝑔 2 irregular waves respectively. + 1+ 𝑡𝑎𝑛ℎ𝑘ℎ 9$ 𝐶 ′ 𝐷𝑤𝑐 2 𝑠𝑖𝑛ℎ2𝑘ℎ 𝑘 - Wave height decay for waves and current conditions and regular and irregular wave trains. Nature Based Solutions for Coastal Management Maria Maza, 2024  Wave energy attenuation Drag coefficient New approaches Nature Based Solutions for Coastal Management Maria Maza, 2024  Drag coefficient For these studies in which an expression of β is proposed that takes into account the characteristics of the ecosystem, 𝐶# is an unknown and is obtained by existing formulations in the literature based on experimental and field data, i.e., it is a parameter that must be calibrated/validated. These formulations have usually been presented as a function of the Reynolds number, 𝑅𝑒 = U*L/nu, or the Keulegan Carpenter number, 𝐾𝐶 = U*T/L, calculated from the mean diameter of the elements forming the ecosystem. Nature Based Solutions for Coastal Management Maria Maza, 2024  Insight: drag coefficient is site specific, it is not predictable. What can we design for pilot? Find a similar existing case different drag coefficient formulations - wide range! Paper Study Vegetation Formulation Considerations Kobayashi Lab Seagrasses (artificial 2.4 - 𝑅𝑒: Reynolds number considering 2200 et al. (1993) Zostera noltii from 𝐶𝐷 = 0.08 + vegetation diameter 𝑅𝑒 Asano et al. 1988) - 𝑅𝑒 range: 2000 – 18000 - Regular waves Mendez et Lab Seagrasses (artificial 2200 2.2 - Two drag coefficients: one considering 𝐶𝐷 = 0.08 + no movement al. (1999) Zostera noltii from 𝑅𝑒 vegetation motion by introducing relative 4600 2.9 Asano et al. 1988) 𝐶𝐷 = 0.40 + considering velocity between plants and fluid in the 𝑅𝑒 drag force calculation and another one movement without considering plant motion. - 𝑅𝑒: Reynolds number considering vegetation diameter - 𝑅𝑒 range: 200 – 15500 - Regular waves Mendez Lab Kelp (artificial 𝐶𝐷 = 0.47𝑒𝑥𝑝 −0.052𝐾𝐶 - Two formulations as a function of and Laminaria hyperborea 𝑒𝑥𝑝 −0.0138 𝐾𝐶L Keulegan Carpenter number (𝐾𝐶): ℎ𝑣 ⁄ℎ 0.76 Losada from Dubi and Torum, 𝐶𝐷 = second one accounts for relative 0.3 (2004) 1995; Lovas and 𝐾𝐶L vegetation height (ℎ𝑣 ⁄ℎ: ℎ𝑣 is vegetation Torum, 2001) ℎ𝑣 ⁄ℎ 0.76 height and ℎ water depth). - 𝐾𝐶: Keulegan Carpenter number considering vegetation diameter - 𝐾𝐶 range: 3 – 59; 𝐾𝐶 ℎ𝑣 /ℎ −0.76 range: 7 - 172 Nature Based Solutions for Coastal Management Maria Maza, 2024  Paper Study Vegetation Formulation Considerations Augustin Lab Artificial submerged 𝑙𝑠 𝑑 - Bulk drag coefficient per unit width 𝐶𝐷 = 𝐶′𝐷 et al. (2009) and emergent wooden Δ𝑆ℎ - 𝐶′𝐷 : individual stem drag coefficient dowels - 𝑙𝑠 : stem height (equal to ℎ for emergent conditions); 𝑑: stem diameter; Δ𝑆: stem spacing; ℎ: water depth. - Regular waves Myrhaug et Analytical Generic 𝑠𝐴 - Stochastic method to get root-mean- 𝐶𝐷,𝑟𝑚𝑠 = 𝑟 𝐵 1 + 𝑘𝑝 𝑎𝑟𝑚𝑠 al. (2009) 𝑟𝐵 square drag coefficient - 𝑘𝑝 : wave number associated with the spectral peak; 𝑎𝑟𝑚𝑠 : root-mean-square wave amplitude; 𝑠, 𝑟, 𝐴, 𝐵: variables function of wave parameters - Irregular waves Paul and Field Seagrass (Zostera 1.45 - 𝑅𝑒: Reynolds number considering 153 Amos noltii) 𝐶𝐷 = 0.06 + vegetation diameter 𝑅𝑒 (2011) - Formulation valid for significant wave height 𝐻𝑠 ≥ 0.1 m - 𝑅𝑒 range: 100 - 1000 Nature Based Solutions for Coastal Management Maria Maza, 2024  Paper Study Vegetation Formulation Considerations Jadhav Field Saltmarshes (Spartina 2600 - 𝑅𝑒: Reynolds number considering 𝐶𝐷 = 0.36 + and Chen alterniflora) 𝑅𝑒 vegetation diameter (2012) - 𝑅𝑒 range: 600 - 3200 Jadhav et Field Saltmarshes (Spartina 2 - Spectrally variable drag coefficient 𝐶𝐷,𝑗 = 𝐶𝐷 𝛼𝑛,𝑗 al. (2013) alterniflora) - 𝐶𝐷 : spectrally-averaged drag coefficient - 𝛼: normalized velocity attenuation parameter obtained as the ratio of vegetation-affected velocity and velocity in the absence of vegetation. - subindex 𝑗 represents the jth frequency component of wave spectrum. Maza et al. Lab Seagrasses (artificial 2200 0.88 - Two drag coefficients: one considering 𝐶𝐷 = 0.87 + no movement (2013) Posidonia oceanica 𝑅𝑒 vegetation motion by introducing relative 4600 1.9 fron Stratigaki et al. 𝐶𝐷 = 1.61 + considering velocity between plants and fluid in the 𝑅𝑒 2011) drag force calculation and another one movement without considering plant motion. - 𝑅𝑒: Reynolds number considering vegetation diameter - 𝑅𝑒 range: 2500 – 8500 - Regular waves Nature Based Solutions for Coastal Management Maria Maza, 2024  Paper Study Vegetation Formulation Considerations Pinsky et Review lab Seagrasses, 𝑙𝑜𝑔 𝐶𝐷 = 𝛽0 + 𝛽1 𝑙𝑜𝑔 𝑐 ∗ 𝑅𝑒 - Statistical analysis al. (2013) and field saltmarshes, - Fitted parameters: 𝛽0 = -1.72; 𝛽1 =-1.67; mangroves 𝑐 = 3*10-4 - 𝑅𝑒: Reynolds number considering vegetation diameter 0.817 Ozeren et Lab Saltmarshes (J. 55.2 - Drag coefficient introducing vegetation 𝐶𝐷 = Regular waves al. (2013) roemerianus) 𝐾𝐶 ℎ𝑣 /ℎ −2 height and water depth 58.5 0.641 𝐶𝐷 = Irregular waves − 𝐾𝐶: Keulegan Carpenter number as a 𝐾𝐶 ℎ𝑣 /ℎ −2 function of vegetation diameter − ℎ𝑣 : vegetation height; ℎ: water depth - 𝐾𝐶 ℎ𝑣 /ℎ −2 range: 5 - 80 for regular waves; 5 – 40 for irregular waves 0.977 Möller et al. Lab Saltmarshes (Elymun 305.5 - Drag coefficient for regular and 𝐶𝐷 = −0.046 + Regular (2014) athericus) 𝑅𝑒 irregular waves waves - 𝑅𝑒: Reynolds number considering 227.3 1.615 vegetation diameter 𝐶𝐷 = 0.159 + Irregular 𝑅𝑒 - 𝑅𝑒 range: 100 - 1100 waves Anderson Lab Saltmarshes (artificial @AA.B E.B@ - Drag coefficient as a function of 𝑅𝑒 or 𝐶? = 0.76 + Submerged and Smith Spartina made using CD 𝐾𝐶 for submerged conditions and new (2014) cross linked polyolefin conditions formulations to account for submergence [email protected] H.IJ tubes) 𝐶? = 1.10 + Submerged ratio 𝑙K /ℎ. FG conditions - 𝑅𝑒 and 𝐾𝐶 considering vegetation I.NA diameter 2067.7 𝐶? = 0.11 + - 𝑅𝑒 range: 500 – 2300; 𝑅𝑒 𝑙K /ℎ LE.M 𝑅𝑒 𝑙K /ℎ LE.M E.NO range: 550 - 2650 744.2 - 𝐾𝐶 range: 25 – 110; 𝐾𝐶 𝑙K /ℎ LE.M 𝐶? = 0.97 + LE.M 𝐾𝐶 𝑙K /ℎ range: 30 - 130 Nature Based Solutions for Coastal Management Maria Maza, 2024  Paper Study Vegetation Formulation Considerations Hu et al. Lab Saltmarshes, 1.37 - Drag coefficient for wave and current 730 (2014) Mangroves 𝐶𝐷 = 1.04 + conditions 𝑅𝑒 (stiff wooden - 𝑅𝑒: Reynolds number considering rods) vegetation diameter and maximum velocity that is spatially averaged waves velocity for wave conditions and spatially averaged velocity field for waves and current conditions. - 𝑅𝑒 range: 300 - 4700 Losada et Lab Saltmarshes 50000 2.2 - Drag coefficient for wave and current 𝐶𝐷 = 0.08 + Regular waves al. (2016) (Spartina 𝑅𝑒 𝐷 conditions considering regular and irregular 75000 9 anglica and 𝐶𝐷 = 0.25 + 𝐷 Regular waves + current waves. 𝑅𝑒𝑤𝑐 Puccinellia - 𝑅𝑒 𝐷 ; 𝑅𝑒𝑤𝑐 𝐷 : Reynolds number obtained 50000 9 maritima) 𝐶𝐷 = 0.50 + 𝐷 Regular waves – current considering the deflected vegetation length 𝑅𝑒𝑤𝑐 22000 2.2 and maximum velocity at the beginning of the 𝐶𝐷 = 0.08 + Irregular waves vegetation field for wave conditions and 𝑅𝑒 𝐷 35000 9 waves and current conditions respectively. 𝐶𝐷 = 0.25 + Irregular waves + current 𝐷 𝑅𝑒𝑤𝑐 - 𝑅𝑒 𝐷 range: 50000 – 145000 (regular 27000 9 waves); 2000 – 55000 (irregular waves) 𝐶𝐷 = 0.50 + 𝐷 Irregular waves - current 𝐷 𝑅𝑒𝑤𝑐 - 𝑅𝑒𝑤𝑐 range: 135000 – 200000 (regular waves); 2500 – 55000 (irregular waves) López- Lab Mangroves 1.62 - Drag coefficient for waves 12.29 Arias et al. and (Rhizophora 𝐶𝐷 = 1 + - KC: Keulegan carpenter number 𝐾𝐶 (2024) Field sp.) considering root diameter - KC range: 10 - 220 Nature Based Solutions for Coastal Management Maria Maza, 2024  Wave energy attenuation Drag coefficient New approaches Nature Based Solutions for Coastal Management Maria Maza, 2024  New approaches need to go to the field, measure samples of stem,... The characterization of a vegetated ecosystem by measuring leaf traits, biomechanical properties and the number of individuals involves a lot of effort and is case-specific. CD is Problem unknown for many cases. Find a parameter easy to be quantified that allows estimating the coastal protection service provided by different ecosystems. Submerged Solid Volume Fraction and Standing Biomass Objective are explored. A new relationship that allows quantifying the coastal protection service provided by different species, avoiding the use of Benefits parameters that need to be calibrated. Nature Based Solutions for Coastal Management Maria Maza, 2024  New approaches IH Cantabria flume scale of 1:6 q Wave-Current-Tsunami Flume (COCOTSU) m 156 → m 26 Rhizophora mature tree 2 m → 12 m Nature Based Solutions for Coastal Management Maria Maza, 2024  New approaches Capacit y gauges Nature Based Solutions for Coastal Management Maria Maza, 2024  New approaches Nature Based Solutions for Coastal Management Maria Maza, 2024  New approaches Wave damping: 𝐻"#$ 1 q = 𝐻"#$,& 1 + 𝛽𝑋 3 different water depth Solid lines: wave height attenuation for the flume full of mangroves Dashed lines: wave height attenuation for the empty flume cases Nature Based Solutions for Coastal Management Maria Maza, 2024  New approaches How to measure the SVF? - define control volume? measure all roots q Submerged solid volumen fraction (SVF) frontal area distribution along the vertical and deviations due to rotating angle Frontal area analysis N: number of mangroves submerged volume fraction: per unit area, ratio between solid volum and water volume Ah: horizontal unit area Superposition of 8 pictures taken rotating the model to capture its 3-dimensional variability. vertical dotted lines display the value of the equivalent mean diameter (d*) for each water depth 𝐻:;< 𝛽 = 1.783 𝑆𝑉𝐹 ℎ Maza et al. 2019 relative wave height Nature Based Solutions for Coastal Management Maria Maza, 2024  New approaches ´ Numerical tool Mangrove forest in Costa Rica explain Lopez-Arias et al. 2024 increased control volume, in sparser meadown number of roots Nature Based Solutions for Coastal Management Maria Maza, 2024  New approaches ´ Numerical tool Implementación en el modelo SWAN de la nueva formulación: Nature Based Solutions for Coastal Management Maria Maza, 2024  New approaches Vegetation species Experimental set- Results Field campaign selection up standing biomass of ecosystem? = the weight of ecosystem per unit area? Halimione Salicornia Halimione Juncus Salicornia Spartina SHT Juncus Spartina MHT Upper saltmarsh Pioneer zone and lower saltmarsh can be observed and measured using remote sensing Nature Based Solutions for Coastal Management Maria Maza, 2024  New approaches 1 3 2 5 4 Nature Based Solutions for Coastal Management Maria Maza, 2024  New approaches Nature Based Solutions for Coastal Management Maria Maza, 2024  New approaches 𝐻 1 = 𝐻\ 1 + 𝛽𝑋 Nature Based Solutions for Coastal Management Maria Maza, 2024  New approaches L H h h hv hv Lv 𝐷𝑟𝑦 𝑊𝑒𝑖𝑔ℎ𝑡 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑆𝑡𝑎𝑛𝑑𝑖𝑛𝑔 𝐵𝑖𝑜𝑚𝑎𝑠𝑠 = ∗ 𝑚𝑖𝑛 ℎ] , ℎ ∗ 𝑆𝑅 𝑯𝒚𝒅𝒓𝒂𝒖𝒍𝒊𝒄 𝑺𝒕𝒂𝒏𝒅𝒊𝒏𝒈 𝑩𝒊𝒐𝒎𝒂𝒔𝒔 𝑯𝑺𝑩 = ℎ] = 𝑉𝑆𝐵 ∗ 𝐿] /𝐿 ∗ 𝐻⁄ℎ ℎ] 𝑆𝑅 = , 𝑤ℎ𝑒𝑟𝑒 𝑆𝑅 = 1 𝑖𝑓 ℎ] > ℎ ℎ multiplied by hydraulic conditions..? Nature Based Solutions for Coastal Management Maria Maza, 2024  New approaches Maza et al. 2022 Common linear relationship for 96 cases including 4 species, 2 meadow densities, 12 wave conditions. Nature Based Solutions for Coastal Management Maria Maza, 2024  New approaches Empirical Numerical Tool formulation Implementación en el modelo SWAN de la nueva formulación: Nature Based Solutions for Coastal Management Maria Maza, 2024  New approaches Validación de la herramienta: Scheldt estuary Scirpus maritimus Spartina anglica Bath: 228, 0𝑚 − 5𝑚 𝑏𝑖𝑜𝑚𝑎𝑠𝑎 (𝑔 ⁄𝑚 - ) = , 609, 5𝑚 − 15𝑚 1778, > 15𝑚 Heuner et al. (2015) Hellegat: and Schulze et al. (2019) 1102, 0 − 15𝑚 𝑏𝑖𝑜𝑚𝑎𝑠𝑎 (𝑔 ⁄𝑚 - ) = 8 1650, > 15𝑚 Vuik et al. 2018 Nature Based Solutions for Coastal Management Maria Maza, 2024  New approaches grey lines = applying drag formula from literature López-Arias et al. 2023 Nature Based Solutions for Coastal Management Maria Maza, 2024  Nature-based solutions for coastal management 𝐻 1 = 𝐻! 1 + β𝑥 4𝑎𝑁𝐻! 𝑘 𝑠𝑖𝑛ℎ" 𝑘𝑙 + 3𝑠𝑖𝑛ℎ𝑘𝑙 β= 𝐶# 9𝜋 𝑠𝑖𝑛ℎ2𝑘ℎ + 2𝑘ℎ 𝑠𝑖𝑛ℎ𝑘ℎ FLOW ENERGY ATTENUATION – ANALYTICAL FORMULATIONS Nature Based Solutions for Coastal Management Maria Maza, 2024