## 1 Introduction

A wide range of problems in atmospheric transport and chemistry can be modeled, subject to suitable boundary conditions, by the linear equation

Here *c*(**x**,*t*) is the mass mixing ratio of the relevant trace gas, *s*(**x**,*t*) is its mass source, *ρ*(**x**,*t*) the density of air, and is the linear advection-diffusion-reaction-convection operator defined by

Here **u**(**x**,*t*) is the local mean wind speed, ** κ**(

**x**,

*t*) a symmetric eddy diffusivity tensor,

*l*(

**x**,

*t*) is the local loss rate, e.g., due to photolysis or reaction with a reservoir species, and is a linear operator modeling the nonlocal transport associated with unresolved convection. Note that (1) can easily be extended to multiple species by replacing

*c*and

*s*by vectors and

*l*by a matrix, which could be, for example, the tangent linear model to the chemistry scheme used by a chemistry transport model (CTM).

A typical objective in solving (1) is to evaluate an integral quantity , e.g., the (weighted) average of *c* over a given region and time period. The question of interest may then involve determining the sensitivity of to different configurations of the source distribution *s*. It is then well known [e.g., *Enting*, 2002] that rather than solve a large number of forward problems each with different *s*, it is more efficient to solve the adjoint or inverse equation to (1). Consequently, the theory, development, and numerical implementation of adjoints models for CTMs has been the subject of much research [e.g., *Vukicevic and Hess*, 2000; *Henze et al.*, 2007].

Many operational adjoint models are derived by (automated) differentiation of the discretized version of the forward model. Eulerian backtracking [*Hourdin and Talagrand*, 2006] provides an alternative based on deriving and then solving the *retro transport* equation, described in section 2 below, corresponding to (1). The advantages of Eulerian backtracking include

The conceptual framework for inverse problems corresponds closely to the “back trajectory” framework exploited in Lagrangian inverse problems [e.g.,

*Seibert and Frank*, 2004]. In contrast to other adjoint formulations, Eulerian backtracking inverse problems are therefore simpler to define, understand, and compare with Lagrangian results.The numerical transport scheme used to solve the retro transport equation is essentially the same as that utilized by the forward model. The qualitative behavior of numerical solutions is therefore well understood, and possible numerical stability problems [e.g.,

*Sirkes and Tziperman*, 1997] associated with adjoints are avoided.

Hitherto, the principal disadvantage has been that it is far from straightforward to obtain a high level of numerical accuracy in Eulerian backtracking calculations, e.g., *Hourdin et al.* [2006], comparing direct and adjoint sensitivities for a short-time test problem, report relative errors of order 10^{−2}. To address this, a new and accurate Eulerian-backtracking model (RETRO-TOM) [*Haines et al.*, 2014], based on the transport component of the CTM TOMCAT [*Chipperfield*, 2006] has been developed and tested by the authors. Relative accuracies of order 10^{−8} are reported for RETRO-TOM in comparable test problems.

In section 2, RETRO-TOM is shown to be an efficient alternative to, and a valuable numerical benchmark for, Lagrangian trajectory methods in a wide class of process studies. Problems in which the quantity of interest is either an integral (as above), or the flux of a species into a region or air mass, can each be addressed. In section 3, a key example problem addressing the question of how the stratospheric flux of a finite lifetime chemical species depends on the location(s) of its surface source(s), is revisited using RETRO-TOM. The question is of interest both in quantifying some fundamental climatological transport properties of the general circulation [c.f. *Berthet et al.*, 2007] and of practical value in evaluating the likely contributions of localized sources of very short lived halogenated species (VSLS hereafter) to the stratospheric loading of chlorine and bromine. Source location sensitivities for VSLS have to an extent been previously addressed in forward Eulerian [*Wuebbles et al.*, 2001; *Warwick et al.*, 2006a; *Levine et al.*, 2007; *Aschmann et al.*, 2009; *Holzer and Polvani*, 2013] and forward Lagrangian [*Levine et al.*, 2007; *Brioude et al.*, 2010; *Pisso et al.*, 2010] studies but not with the flexibility and resolution afforded by Eulerian backtracking. In section 4, conclusions are drawn.