Editing Gravity current (section)

===Box models===
For a finite volume gravity current, perhaps the simplest modelling approach is via a box model where a "box" (rectangle for 2D problems, cylinder for 3D) is used to represent the current. The box does not rotate or shear, but changes in aspect ratio (i.e. stretches out) as the flow progresses.  Here, the dynamics of the problem are greatly simplified (i.e. the forces controlling the flow are not direct considered, only their effects) and typically reduce to a condition dictating the motion of the front via a [[Froude number]] and an equation stating the global conservation of mass, i.e. for a 2D problem
:<math>\begin{align}
\mathrm{Fr} &= \frac{u_\mathrm f}{\sqrt{ g' h }}\\
h l &= Q
\end{align}</math>

where {{math|Fr}} is the Froude number, {{math|''u''<sub>f</sub>}} is the speed at the front, {{math|''g''′}} is the [[reduced gravity#neutral buoyancy|reduced gravity]], {{math|''h''}} is the height of the box, {{math|''l''}} is the length of the box and {{math|''Q''}} is the volume per unit width. The model is not a good approximation in the early slumping stage of a gravity current, where {{math|''h''}} along the current is not at all constant, or the final viscous stage of a gravity current, where friction becomes important and changes {{math|Fr}}. The model is a good in the stage between these, where the Froude number at the front is constant and the shape of the current has a nearly constant height.

Additional equations can be specified for processes that would alter the density of the intruding fluid such as through sedimentation.  The front condition (Froude number) generally cannot be determined analytically but can instead be found from experiment or observation of natural phenomena.  The Froude number is not necessarily a constant, and may depend on the height of the flow in when this is comparable to the depth of overlying fluid.

The solution to this problem is found by noting that {{math|''u''<sub>f</sub> {{=}} {{sfrac|''dl''|''dt''}}}} and integrating for an initial length, {{math|''l''<sub>0</sub>}}.  In the case of a constant volume {{math|''Q''}} and Froude number {{math|Fr}}, this leads to
:<math>l^\frac32 = l_0^\frac32 + \tfrac32 \mathrm{Fr} \sqrt{g'Q}\, t \,.</math>