Dynamical structure of vertically two-dimensional estuary in steady state

The dynamical structure of a two-dimensional (depth and axial directions) estuary is studied analytically. A set of governing equations describing the time-averaged velocity and salinity in the estuary is used, where all of the external parameters (depth, width, freshwater discharge and horizontal and vertical coefficients of eddy viscosity and eddy diffusivity) are assumed to be constant along the estuary.Two dynamical relations are taken into consideration in the theory. One of them is the dependence of the longitudinal scaleL _d on the balance of longitudinal salt transport, and the other is the relation between the vertical stratification of salinity and the the Prandtl numberP _r=A_v/K_v, whereA _v andK _v denote the coeffcients of the vertical eddy viscosity and diffusivity, respectively. The two relations result in an extension of the parameter range in which the linear balance of momentum holds.A linear state of motion (LM-state) is defined as the state where the momentum balance is linear. The LM-state comprises the so-called vertically homogeneous and the so-called partially mixed state. Perturbation analysis is introduced and dynamical theory is developed in the LM-state. Since the LM-state covers a fairly wide regime with respect to the balance of salt transport, the state is subdivided into three stages: the diffusive, intermediate and advective stages. In the diffusive stage the upstream salt transport is mainly attributed to the horizontal diffusion, whereas in the advective stage it is attributed to advection caused by gravitational circulation. The salinity balance is also linear in the diffusive and intermediate stages, whereas the balance is nonlinear in the advective stage. The advective stage of the LM-state is regarded as a stage bordering the salt wedge state.The longitudinal distribution of depth-mean salinity is found to take an exponential form in the diffusive stage, a nearly linear form in the advective stage and an intermediate form between them in the intermediate stage.