Depth-averaged momentum equation for gravity currents with varying density: coefficient in pressure term

Gravity currents are often modeled by means of shallow water equations (SWEs). In these models, simplifications such as the consideration of a constant layer-averaged density are common. This note presents the complete and general derivation of 2D depth-averaged momentum equation for gravity currents with density and velocity varying in the bed-normal direction. Special attention is given to the pressure term which is evaluated for constant, linear and exponential density profile. The shape of the density profile has implications for the momentum balance: the assumption of constant density leads to an overestimation of the driving force due to pressure gradient by a factor of 33% for linear density profile and up to 50% for an exponential profile. It also leads to an overestimation of celerity in numerical models based on traditional SWEs by factor of 22% and around 40% for linear end exponential density profiles respectively.