EurOtop Wave Runup R2p

EurOtop Wave Run-up (R2%)

Source: CHESS_Library/Structure_Reponses/Equations/Eurotop_r2p_q_Final.m (dispatched from compute_structure_response.m) Applies to: struc_type 1 (Levee), 3 (Rubblemound), 4 (Low-Crested Breakwater). Not computed for Type 2 (Floodwall) — calc_r2p is forced to 0 for floodwalls inside compute_structure_response.m. Enable flag: config.compute_r2p

What it models

The 2% run-up height (R2%) is the vertical distance, above the still water level (SWL), that is exceeded by the run-up of only 2% of the incident wave crests climbing a sloped structure face (embankment, levee, or rubblemound armor slope). It is used both as a stand-alone hazard (freeboard/crest design against wave run-up) and, in the LCS-CSR damage-progression module, as an input to the leeside overtopping-velocity calculation (see Van Gent Leeside Armor Stone Stability Dn50).

Conceptual model

• Waves climb the seaward face. Run-up grows with the wave height and with the breaker parameter ξ = tan α / √s (gentle, steep-fronted waves surge up steep slopes; steep waves break on gentle slopes and run up less). It is capped by a maximum that the run-up cannot exceed however surging the waves (Eq. 5.2 / 6.1).
• The sea state at the toe. Hm0 is the depth-limited wave height at the toe (Model Conventions) and the period is the spectral Tm-1,0 = Tp / 1.1.
• Surface and geometry reduce run-up through influence factors: roughness γf (grass, asphalt, rock), a berm γb (1 while toe berms are hidden), and wave obliquity γβ (1: perpendicular attack assumed).
• Rough, permeable slopes (rubble mound, low-crested breakwater) use γf_surge, which tends back to 1 for surging waves (1.8 < ξ < 10), because surging waves are less affected by roughness.
• Grass (roughness_ifactor = 2 in the input) is not a fixed factor: γf = 1.15·√Hm0 for Hm0 < 0.75 m, else 1.
• No waves, no run-up. With Hm0 = 0 (dry toe or a storm without waves) R2% is NaN.
• Outputs. R2p is the height above SWL; R2p_SWL = R2p + SWL is the run-up elevation, to compare with the crest.

Conceptual diagram

The equation

Source: EurOtop — Wave Overtopping of Sea Defences and Related Structures: Assessment Manual, August 2018 (cited directly in the function header, http://www.overtopping-manual.com/eurotop.pdf).

Breaker parameter (surf similarity parameter) based on the spectral mean period Tm-1,0:

L_m-1,0 = g · Tm-1,0² / (2π)
s_m-1,0 = Hm0 / L_m-1,0
ξ_m-1,0 = cot(α) / √(s_m-1,0)

Levee / embankment (struc_type = 1), gentle slopes (2 ≤ cot α ≤ 9.99) — EurOtop Eq. 5.1/5.2:

R2%_a   = Hm0 · 1.65 · γb · γf · γβ · ξ_m-1,0
R2%_max = Hm0 · 1.00 · γf · γβ · (4 − 1.5 / √(γb · ξ_m-1,0))
R2%     = min(R2%_a , R2%_max)         if R2%_max > 0
        = R2%_a                        otherwise

Levee / embankment, steep slopes (0.1 < cot α < 2) — EurOtop Eq. 5.6:

R2%_a = min( Hm0 · 0.80 / cot(α) + 1.6,  3 · Hm0 )
R2%   = max( 0,  max(R2%_a , 1.8 · Hm0) )

Rubblemound / LCBW (struc_type = 3, 4) — EurOtop Eq. 6.1, using a roughness factor γf_surge adjusted for the surf-similarity range 1.8 < ξ_m-1,0 < 10:

γf_surge = γf                                                    if ξ_m-1,0 ≤ 1.8
γf_surge = γf + (ξ_m-1,0 − 1.8) · (1 − γf) / 8.2                 if 1.8 < ξ_m-1,0 < 10

R2%_a   = Hm0 · 1.65 · γb · γf_surge · γβ · ξ_m-1,0
R2%_max = Hm0 · 1.00 · γf_surge · γβ · (4 − 1.5 / √(γb · ξ_m-1,0))
R2%     = min(R2%_a , R2%_max)          if R2%_max > 0
        = R2%_a                         otherwise

R2% is also reported as an elevation as, R2p_SWL = R2% + SWL, when SWL is referenced to a fixed datum.

Slope range. The levee formulas cover 0.1 < cot α ≤ 9.99. For a levee outside that range (near-vertical cot α ≤ 0.1, or gentler than 1:10) R2% is not computed and stays 0. For rubble mounds and low-crested breakwaters, cot α < 0.1 gives NaN. The in-code comment notes that 1:10 is outside the range of validity of the EurOtop equations.

Parameters

Symbol	Meaning	Units	Source
Hm0	Zero-moment significant wave height at the structure toe	m	Project forcing (project_forcing.Hm0, or LCS forcing column 6)
Tm-1,0 (T_m10)	Spectral mean wave period, computed in compute_structure_response.m as Tp / 1.1	s	Derived from project_forcing.Tp
SWL	Storm water level	m, datum	Project forcing (project_forcing.SWL, or LCS forcing column 5)
Rc	Freeboard = structure.crest_elevation − SWL	m	structure.crest_elevation (from config.crest_elevation) minus forcing SWL
cot(α) (slope)	Cotangent of the seaward structure slope	—	structure.seaside_slope (from config.seaside_slope)
γf	Influence factor for surface roughness	—	call_eurotop_ifactors.m, seeded by config.roughness_ifactor
γβ (gamma_beta_r2p)	Influence factor for oblique wave attack (run-up-specific)	—	call_eurotop_ifactors.m
γb	Influence factor for a berm (EurOtop Eq. 5.40): γb = 1 − rB·(1 − rdB), rB = B/Lberm, Lberm = B + 2·Hm0/tan(α_berm), bounded to [0.6, 1]	—	berm_influence_factor.m via call_eurotop_ifactors.m, from structure.berm_crest_width, berm_crest_elevation, berm_seaward_slope (cot α, used as tan α = 1/cot α). γb = 1 when config.add_toe_berm = 0, which is forced for levees, floodwalls and rubble mounds
g	Gravitational acceleration	m/s²	config.gravity_constant

Notes / limitations (from in-code documentation)

• Assumes no significant foreshore, non-impulsive breaking, and no composite (berm + wall) structures unless otherwise modeled through the influence factors.
• Does not model battered (sloped-face) walls.
• Wave characteristics are assumed to be measured/transformed to the structure toe.
• Thin run-up layers (below ~2 cm thickness) are not counted toward R2%.

References

• EurOtop. 2018. Wave Overtopping of Sea Defences and Related Structures: Assessment Manual. www.overtopping-manual.com/eurotop.pdf — cited exactly by this title, date, and URL in the function header of Eurotop_r2p_q_Final.m. (The manual's widely-known author team — Van der Meer, J.W., Allsop, N.W.H., Bruce, T., De Rouck, J., Kortenhaus, A., Pullen, T., Schüttrumpf, H., Troch, P., and Zanuttigh, B. — is not itself listed in the in-code citation; it is supplied here as commonly known bibliographic detail for the same document, not something transcribed from the source file.)