Generalized Adjoint for Physical Processes with Parameterized Discontinuities. Part II: Vector Formulations and Matching Conditions

Abstract

Generalized tangent linear and adjoint equations are derived for a vector equation that contains a parameterized source term with discontinuous on/off switches controlled by a threshold condition. As an extension of Part I, the key results here include a pair of interface matching conditions for coupled tangent linear and adjoint vectors across a switch point. Each matching condition can be expressed in either a forward form or a backward form that connects the vector values on the two sides of the switch point via a forward- or backward-matching matrix. The forward- and backward-matching matrices are mutually invertible. The backward/forward-matching matrix for the adjoint vector is the transpose of the forward/backward matrix for the tangent linear vector. By using the matching condition, the classic tangent linear (or adjoint) solution can be extended through a switch point, so a fundamental set of generalized tangent linear (or adjoint) solutions can be constructed, which leads to an explicit expression of the generalized tangent linear (or adjoint) resolvent?the inverse of the generalized tangent linear (or adjoint) operator. The generalized resolvents provide a complete description of the adjoint properties and yield an integral formulation for the gradient of the costfunction. When the parameterized process produces strong negative feedback, on?off oscillations can be produced numerically in vector forms due to essentially the same mechanism as previously illustrated by one-dimensional examples, and the oscillatory states yield to a marginal state in the limit of vanishing time steps in the numerical integration. Marginal states can impose multiple constraints on the tangent linear vector and thus cause multiple reductions in the effective dimension of the data-forcing vector in the backward integration of the adjoint equation. This extends the previous one-dimensional results.