Fig. 5: The application of mean-flow theory to probe flood hazard globally.

Flood hazards across cities worldwide are examined by (a) Subdividing cities into 1 km x 1 km tiles (shown in Vancouver, BC, Canada) and evaluating the topographic slope α, porosity ϕ, and order χ and (b) estimating the average chord length in the direction of descent to reveal (c) the porosity and chord length across 20 twenty cities; error bars show the margin of error with 90 percent confidence. Application of the mean-flow theory reveals: (d) the flow rate-normalized flood inundation \({H}^{*}=\langle \bar{h}\rangle /{h}_{ref}\) and non-dimensionalized flood intensity \({P}^{*}=\langle \bar{hu}\rangle /h{u}_{ref}\) for twenty cities with constant flow rate of Q = 1m³/s, (e) the theoretical flow rate-normalized flood inundation H^* and non-dimensionalized flood intensity P^* for recorded flood events. Finally, (f) The normalized monetary damages \({{{{{\mathcal{L}}}}}}\) are compared to the product of the inflow rate q and urban form factor \({{{{{\mathcal{F}}}}}}(\phi,\, {l}_{c},\, \alpha,\, D)\) among different flash flooding events. In panels (c-f), the markers' colors represent the average bottom slope of each city defined by the color bar to the right side. Building footprint data for Karachi and Mumbai may not be sufficient. Basemaps provided by OpenStreetMap (https://www.openstreetmap.org/copyright) used in conjunction with Microsoft’s Building Footprints and Google’s Open Buildings, data made available via an Open Database License (https://opendatacommons.org/licenses/odbl/).