The use of adjoint equations to solve a variational adjustment problem with advective constraints

A methodology is developed to guarantee time continuity in a sequence of analyses. Coupling is accomplished by requiring the least squares minimization of adjustment to the analyses subject to dynamic constraints. In this paper, the analyses are assumed to be governed by advective constraints such as those used in vorticity conservation models; however, the method can easily be applied to other constraints. The results correspond to the application of a strong constraint as introduced by Sasaki, but the procedure used to accomplish the minimization is an alternative to the traditional methods for solution of the Euler-Lagrange equation. The method allows easier inclusion of more time levels in the analysis sequence as well as accommodation of more complicated constraints. The method is tested using both simulated and real data. The simulated data studies use a one-dimensional advection equation with progressively more complicated dynamics: constant advection velocity, spatially varying advection velocity, and nonlinear advection. The real data case study uses an advection of quasi-geostrophic potential vorticity constraint to examine the height adjustment process for three time periods on 6 March 1982. Separate studies are made for analyses derived from VAS and RAOB data. The results of this study indicate the method has excellent potential to reduce the random component of the analysis errors.