Eulerian backtracking of atmospheric tracers. I: Adjoint derivation and parametrization of subgrid-scale transport

The problem of identification of sources of atmospheric tracers is most classically addressed through either Lagragian backtracking or adjoint integration. On the basis of physical considerations, the retro-transport equation, which is at the basis of Lagrangian backtracking, can be derived in a Eulerian framework as well. Because of a fundamental time symmetry of fluid transport, Lagrangian or Eulerian backtracking can be used for inverting measurements of the concentration of an atmospheric tracer. The retro-transport equation turns out to be the adjoint of the direct transport equation, with respect to the scalar product defined by integration with respect to air mass. In the present paper, the exact equivalence between the physically-derived retro-transport and adjoint equations is proved. The transformation from the direct to the retro-transport equation requires only simple transformations. The sign of terms describing explicit advection is changed. Terms describing linear sources or sinks of tracers are kept unchanged. Terms representing diffusion by unresolved time-symmetric motions of the transporting air are also unchanged. This is rigorously shown for turbulent eddy-diffusion or mixing length theory. The case of subgrid-scale vertical transport by non-time-symmetric motions of air is studied using the example of the Tiedtke mass-flux scheme for cumulus convection. The retro-transport equation is then obtained by simply inverting the roles of updraughts and downdraughts, as well as of entrainment and detrainment. Conservation of mass of the transporting air is critical for all those properties to hold.