Steady-State Dynamics of a Density Current in an f-Plane Nonlinear Shallow-Water Model

The authors study the nonlinear dynamics of a density current generated by a diabatic source in a rotating and a nonrotating system, both in the presence and in the absence of frictional losses, using a steady-state hydrostatic shallow-water model and producing solutions as a function of the Coriolis parameter and of the Rayleigh friction coefficient. Results are presented in the range of the parameter values that are relevant for shallow atmospheric flows as sea-land breezes and as cold pool outflows. In the shallow-water approximation, single-layer flows and two-layer flows with a lid have three degrees of freedom, and their steady-state dynamics are governed by three ordinary differential equations (ODEs), whereas two-layer flows bounded by a free surface have six degrees of freedom, and their dynamics are governed by six ODEs. It is shown that in the limit case of frictionless flow, the problem has an explicit analytical solution, and in the presence of friction, the system for a one-layer flow and for a two-layer flow bounded by a lid can be reduced to two algebraic equations, plus one second-order ordinary differential equation, which can be integrated numerically. Results show that the maximum runout length of the current occurs when the Rayleigh friction coefficient in the lower layer is on the order of the Coriolis parameter. This length is larger when the upper layer is deeper than the lower layer, but it shortens when the friction coefficient of the upper layer is smaller than that in the lower layer. In addition, the relative error of the solution to the linearized equations is computed. This error, which is enhanced when the width of the forcing is smaller than the Rossby radius, is sizable when the friction coefficient is smaller than the Coriolis parameter. In addition, by comparing the nonlinear solution with a lid (three degrees of freedom) to the nonlinear solution with a free surface as an upper boundary (six degrees of freedom), it is shown that the solution with the lid overestimates the geopotential for low values of the friction coefficient and it underestimates the geopotential for large values of this coefficient. The error, sizable when the two layers have a comparable depth, rapidly decreases when the upper layer becomes deeper than the lower layer; accordingly, a rigid lid can be safely adopted only when the depth of the upper layer is twice the depth of the lower layer, or deeper.