A physically motivated numerical technique for solving the advection
In the advective form, DQ
= 0, where D
is the total derivative
, and "mixing ratio" Q
is an invariant along a flow trajectory
. By tracing (along the flow trajectory) backward in time to the "departure point", the value at the "arrival point" can be obtained by an interpolation
or a remapping procedure (between the fixed Eulerian grid
and a time-dependent distorted Lagrangian grid
). Because of the discrete particle–like approach, total mass is generally not conserved. To ensure mass conservation, the semi-Lagrangian method can be formulated with the conservative flux form. The singular particle
discretization is replaced by a finite control-volume discretization
. Analogous to an Eulerian flux-form formulation, total flux
from the upstream
direction, computed in the Lagrangian fashion, is used for the prediction
of the volume-averaged quantity, which can be the density
or a density-weighted mixing ratio
–like quantity. Because the size of the time step is not limited by the CFL
condition, both the advective-form and the flux-form semi-Lagrangian methods are computationally efficient, particularly in spherical geometry.
Staniforth, A., and J. Cote 1991. Semi-Lagrangian integration schemes for atmospheric models – A review. Mon. Wea. Rev.. 119. 2206–2223.
Lin, S.-J., and R. B. Rood 1996. Multidimensional flux-form semi-Lagrangian transport schemes. Mon. Wea. Rev.. 124. 2046–2070.
Copyright 2022 American Meteorological Society (AMS). For permission to reuse any portion of this work, please contact firstname.lastname@example.org. Any use of material in this work that is determined to be “fair use” under Section 107 of the U.S. Copyright Act (17 U.S. Code § 107) or that satisfies the conditions specified in Section 108 of the U.S.Copyright Act (17 USC § 108) does not require AMS’s permission. Republication, systematic reproduction, posting in electronic form, such as on a website or in a searchable database, or other uses of this material, except as exempted by the above statement, require written permission or a license from AMS. Additional details are provided in the AMS Copyright Policy statement.