James Percival

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Last updated : 30th August 2018

The Exner equation - Part one

Lets derive the Exner equation from first principles. Consider two surfaces, \(z=\eta(x,y)\), denoting the boundary of a sediment bed and \(z=h(x,y)\), denoting the free surface of the water column, as well as a curve in the x-y plane, \(\mathcal{C}(x,y)\), bounding a cylinder, \(\Omega\times\mathbb{R}\), in space. The total amount of sediment contained in this cylinder in this viewpoint can be written as \[ S = \int_{\Omega} \left[\int_{-\infty}^{\eta} (1-\phi) dz+ \int_\eta^{\eta+\Delta} c_b^*\,dz +\int_{\eta+\Delta}^h c\,dz\right]dy\,dx.\] where \(\phi(x,y,z)\) is the bed porosity, \(c_b^*\) is the bedload sediment and \(c\) is the suspended sediment. Taking a time derivative, and applying the Leibnitz integral rule gives \[ \frac{d S}{dt} = \int_\Omega \left( (1-\phi)\frac{\partial \eta}{\partial} dy\,dx + \frac{d}{dt} \left[ \int_\eta^{\eta+\Delta} c_b^*\,dz +\int_{\eta+\Delta}^h c\,dz\right] \right) dy\,dx.\] Similarly we can consider the flux of material through the boundary, \(\mathcal{C}\), \[ \frac{d S}{dt} = \oint_\mathcal{C} \left[\int_\eta^{\eta+\Delta} \left( c_b^* \mathbf{u}_b -\kappa_b \nabla c_b^*\right)\cdot \mathbf{n} \,dz + \int_{\eta+\Delta}^{h} \left( c \mathbf{u} -\kappa\nabla c \right) \cdot\mathbf{n}\,dz\right]d\ell. \] Applying the divergence theorem and making the twin ansatzes that $\Delta$ does not change with time and that changes in \(c_b\) can be neglected, we obtain the relation \[(1-\phi)\frac{\partial \eta}{\partial t} + \nabla \cdot \mathbf{q}_b + \frac{d}{dt}\int_{\eta+\Delta}^h c\,dz +\nabla\cdot \int_{\eta+\Delta}^h \left( c\mathbf{u} -\kappa \nabla c\right)\,dz=0,\] where \(\mathbf{q}_b\) is the total flux in the bedload layer, \[ \mathbf{q}_b = \int_\eta^{\eta+\Delta} \left( c_b^* \mathbf{u}_b -\kappa_b \nabla c_b^* \right) \cdot \mathbf{n} \,dz.\] If we consider just the sediment in the water column, and assume that no sediment is lost through the free surface we see \[ \frac{d}{dt}\int_{\eta+\Delta}^h c\,dz +\nabla\cdot \int_{\eta+\Delta}^h \left( c\mathbf{u} -\kappa \nabla c \right) = F, dz \] where \(F\) is the inward flux of material at the bottom of the free water column.

Note that we have deliberately not written \(F\) as \(\left(c\mathbf{u}-\kappa\nabla\right)\cdot\mathbf{\hat{z}}\), since it might arise from a different modelling process than the turbulence modelling being applied inside the free water column. Substituting this into the total sediment conservation law gives the final version of the Exner equation, \[ (1-\phi)\frac{\partial \eta}{\partial t} + \nabla \cdot \mathbf{q}_b = -F. \]

Effectively the volume of sediment lost or gained by the bed must be balanced by a transfer of material to or from the suspended sediment, while the bedload tranport exists only to move, not hold material. It remains to define the bedload flux, \(\mathbf{q}_b\) and the flux into the water column, as well as better describing the physics within the water column itself.

tags: scour exner