Eulerian backtracking of atmospheric tracers. II: Numerical aspects

In Part I of this paper, a mathematical equivalence was established between Eulerian backtracking or retrotransport, on the one hand, and adjoint transport with respect to an air-mass-weighted scalar product, on the other. The time symmetry which lies at the basis of this mathematical equivalence can however be lost through discretization. That question is studied, and conditions are explicitly identified under which discretization schemes possess the property of time symmetry. Particular consideration is given to the case of the LMDZ model. The linear schemes used for turbulent diffusion and subgrid-scale convection are symmetric. For the Van Leer advection scheme used in LMDZ, which is nonlinear, the question of time symmetry does not even make sense. Those facts are illustrated by numerical simulations performed in the conditions of the European Transport EXperiment (ETEX). For a model that is not time-symmetric, the question arises as to whether it is preferable, in practical applications, to use the exact numerical adjoint, or the retro-transport model. Numerical results obtained in the context of one-dimensional advection show that the presence of slope limiters in the Van Leer advection scheme can produce in some circumstances unrealistic (in particular, negative) adjoint sensitivities. The retro-transport equation, on the other hand, generally produces robust and realistic results, and always preserves the positivity of sensitivities. Retro-transport may therefore be preferable in sensitivity computations, even in the context of variational assimilation.