Derivation of the 1D Exner equation
We begin with our original statement of mass conservation:
𝑑𝑆
𝑄𝑜𝑢𝑡 − 𝑄𝑖𝑛 = − ,
𝑑𝑡
where 𝑄 and 𝑆 are mass flux and storage (with dimensions of mass), respectively.
Considering a control area of soil surface with dimensions 𝛿𝑥𝛿𝑦, the change in mass storage
there is manifested as a change in the bulk sediment volume
𝑑𝑆 𝑑(𝜌𝑠 𝑉) 𝑑
𝜌𝑏 𝑑ℎ
=
=
𝜌𝑠 ℎ𝛿𝑥𝛿𝑦
=
𝛿𝑥𝛿𝑦 𝜌𝑠 1 − 𝑛 .
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝜌𝑠 𝑑𝑡
In the expression on the right, storage is now expressed as a change in the thickness ℎ of a
sediment layer with a porosity 𝑛 and particle mass density 𝜌𝑠 . This sediment thickness
changes in response to changes in the mass flux across the control area, which can be
expressed in terms of the flow per unit width 𝑢𝑥 𝛿𝑧 and sediment concentration 𝐶 in a
volume of overland flow passing through a control volume 𝛿𝑥𝛿𝑦𝛿𝑧:
𝑑𝑄
𝑑
𝑑(𝐶𝑢𝑥 )
𝑄𝑜𝑢𝑡 − 𝑄𝑖𝑛 =
𝛿𝑥 =
(𝜌𝑠 𝐶𝑢𝑥 𝛿𝑥𝛿𝑦𝛿𝑧) =
𝜌𝑠 𝛿𝑥𝛿𝑦𝛿𝑧.
𝑑𝑥
𝑑𝑥
𝑑𝑥
Substituting the new expressions into the original mass conservation expression, the
simplified expression is a form of the Exner equation:
𝑑(𝐶𝑢𝑥 𝛿𝑧)
𝑑ℎ
= −(1 − 𝑛)
𝑑𝑥
𝑑𝑡
Which is usually rearranged and stated by defining 𝑞𝑠 as a volumetric flux per unit width:
𝑑ℎ
1 𝑑𝑞𝑠
=−
𝑑𝑡
1 − 𝑛 𝑑𝑥