Analytical Solutions for Coastal Currents
Authors: Pease Thomas E. 
Published: October 2008

Abstract
Several simple analytical expressions for wind- and slope-driven currents are critically examined. The solutions are found to be more widely applicable than is credited in reviews of them in published literature. Specifically, acceptable slopes of the bottom for "flat bottom" models, temporal variations in wind fields for "constant wind stress" models, and vertically varying friction coefficients for "constant coefficient" models are calculated. A method of computing analytical three-dimensional velocity fields is outlined using these concepts.

Introduction
This pager is a critical review of mathematicalexpressions for coastal currents. Emphasis is placed on the interpretation of the solutions and their implicit assumptions rather than on the wide variety of analytical expression that can be derived for coastal currents. The domain of interest is restricted to the coastal zone seaward of the surf zone, but in shallow water depths of less than several hundred feet. Such coastal waters on the continental shelves of the U.S. are characterized by density stratification during at least part of the year. Since the unstratified condition can be treated as a special case of generalized expressions for stratified currents, emphasis in this paper is placed on the stratified case.
Treatments of tidal currents are not included in this paper since prediction of these currents is relatively reliable based on observational data. The nonlinear interactions of tidal and nontidal coastal currents is a question worthy of detailed analysis, but this aspect has not been included in this paper.
Coastal circulation is one of the most complex phenomenon of oceanography. Three major factors contribute to this complexity:the numerous interacting forces,large ranges of characteristic time and space scales of motion andthe variable boundary and initial conditions.
The numerous forces controlling motions include wind stress, density gradients, and inertial forces.The time scale varies from hours for winds, to months for seasonal density changes. The spacial scales can range from a river outfall of 1 km width to storm surge slopes that extend over 200 km. Such diverse scales of motion compound the problem of selecting realistic boundary conditions and initial conditions. Such conditions must be functions of both time and position. Lateral boundary conditions depend on coastline topography and the adjacent deep ocean circulation. The dynamics of air-sea interaction determine the time-dependent surface boundary conditions. Due to these intricacies of coastal circulation a wide variety of solutions, using various combinations of assumptions and conditions, have been proposed in attempts to predict and explain the currents.
The earliest mathematical descriptions of coastal currents were based on analytical solutions to the simplified equations. Such results can be expressed as functional relationships between input variables and output circulation parameters. The primary deficiency of the approach has been that the assumptions required to obtain a solution are frequently not physically justifiable.
In order to reduce the number of assumptions needed to obtain solutions, recent researchers have turned almost exclusively from analytical solutions to numerical integration of the applicable equations.