The shapes of stream channels Hydraulic geometry The shapes of stream channels

Continuity For cross-sectional flow area 𝐴 and mean flow velocity 𝑢, 𝑄=𝑢𝐴 For cross-sections that may be approximated by mean width 𝑤 and depth 𝑑, 𝑄≃𝑢𝑤𝑑

How big should a stream channel be?

Channel size Discharge varies in time and in space (downstream) Stable channels must transport the water and sediment supplied without requiring changes in channel cross-sectional geometry.

Regime theory Developed in early 20th century by British engineers designing irrigation canals in India and Pakistan Canals were designed to carry a particular discharge and maintain sufficient velocity to prevent siltation.

Regime equations (empirical) 𝑢=0.794 𝑄 1/6 𝑓 𝑠 1/3 𝑅=0.49 𝑄 1/3 𝑓 𝑠 −1/3 𝐴=1.26 𝑄 5/6 𝑓 𝑠 −1/3 𝑃=2.66 𝑄 1/2 𝑆=0.00053 𝑄 −1/6 𝑓 𝑠 5/3 Where silt factor 𝑓 𝑠 =1.59 𝑑 1/2 , 𝑑 is grain size in mm

Downstream hydraulic geometry How do 𝑢, 𝑑, 𝑤 vary as 𝑄 varies downstream? 𝑤=𝑎 𝑄 𝑏 𝑑=𝑐 𝑄 𝑓 𝑢=𝑘 𝑄 𝑚 Since 𝑄=𝑤𝑑𝑢, 𝑄=𝑎𝑐𝑘 𝑄 𝑏+𝑓+𝑚 so 𝑎𝑐𝑘=1 and 𝑏+𝑓+𝑚=1

At-a-station hydraulic geometry How do 𝑢, 𝑑, 𝑤 vary at a single cross-section as 𝑄 varies in time? (same math!) 𝑤=𝑎 𝑄 𝑏 𝑑=𝑐 𝑄 𝑓 𝑢=𝑘 𝑄 𝑚 Since 𝑄=𝑤𝑑𝑢, 𝑄=𝑎𝑐𝑘 𝑄 𝑏+𝑓+𝑚 so 𝑎𝑐𝑘=1 and 𝑏+𝑓+𝑚=1