Attenuation and lag of peak in routing

When a flood wave passes through a reach or reservoir, its shape changes. The hydrograph is flattened and delayed.

Concept

• Attenuation of peak: the peak outflow is lower than the peak inflow, $\Delta Q_p = I_{max} - O_{max}$. During the rising limb, part of the inflow fills the storage ($I > O$), so the outflow is smaller than the inflow. The stored water drains later, so the falling limb is stretched out.
• Lag of peak: the peak outflow occurs later than the peak inflow, $t_{lag} = t_{p,out} - t_{p,in}$.
• The peak outflow always occurs where the outflow curve crosses the inflow curve, because there $dS/dt = I - O = 0$ (storage is maximum).

Sketch

 Q
 |      inflow
 |       /\
 |      /  \      outflow
 |     /   /‾‾\
 |    /   /    \
 |   /   /      \__
 |  /___/__________\___
 +--------------------------> t
     t_pI  t_pO
     |<--lag-->|
 Attenuation = I_peak - O_peak

Why it occurs

• In a reservoir, the water surface is level, so storage is a function of outflow alone and the peak is cut by surcharge storage.
• In a river reach, the wedge and prism storage (Muskingum) delays the flood, and friction and floodplain storage flatten it. The wave peak decreases with distance travelled.

Attenuation and lag increase with the storage constant $K$ and are important in design of spillways, flood-control reservoirs, and in forecasting arrival times of a flood downstream.