The two-time-level semi-implicit semi-Lagrangian spectral transform dynamical core in CAM 3.0 evolved from the three-time-level CCM2 semi-Lagrangian version detailed in Williamson and Olson  hereafter referred to as W&O94. As a first approximation, to convert from a three-time-level scheme to a two-time-level scheme, the time level index n-1 becomes n, the time level index n becomes n+ , and becomes . Terms needed at n+ are extrapolated in time using time n and n-1 terms, except the Coriolis term which is implicit as the average of time n and n+1. This leads to a more complex semi-implicit equation to solve. Additional changes have been made in the scheme to incorporate advances in semi-Lagrangian methods developed since W&O94. In the following, reference is made to changes from the scheme developed in W&O94. The reader is referred to that paper for additional details of the derivation of basic aspects of the semi-Lagrangian approximations. Only the details of the two-time-level approximations are provided here.
The semi-Lagrangian dynamical core adopts the same hybrid vertical coordinate () as the Eulerian core defined by
In the system the hydrostatic equation is approximated in a general way by
The semi-implicit equations are linearized about a reference state with constant and . We choose
To ameliorate the mountain resonance problem,
Ritchie and Tanguay  introduce a perturbation
surface pressure prognostic variable
Variables needed at time ( ) are obtained by extrapolation
Lagrangian polynomial quasi-cubic interpolation is used in the prognostic equations for the dynamical core. Monotonic Hermite quasi-cubic interpolation is used for tracers. Details are provided in the Eulerian Dynamical Core description. The trajectory calculation uses tri-linear interpolation of the wind field.
The discrete semi-Lagrangian, semi-implicit continuity equation is obtained from (16) of W&O94 modified to be spatially uncentered by a fraction , and to predict
The surface pressure forecast equation is obtained by summing over all
levels and is related to (18) of W&O94 but is spatially uncentered
The corresponding equation for the semi-implicit development follows and is related to (19) of W&O94, again spatially uncentered and using .
This is not the actual equation used to determine in the code. The equation actually used in the code to calculate involves only the divergence at time () with eliminated.
The thermodynamic equation is obtained from (25) of W&O94 modified to be spatially uncentered and to use . In addition Hortal's modification  is included, in which
The calculation of follows that of the ECMWF (Research Manual 3, ECMWF Forecast Model, Adiabatic Part, ECMWF Research Department, 2nd edition, 1/88, pp 2.25-2.26) Consider a constant lapse rate atmosphere
The momentum equations follow from (3) of W&O94 modified to be spatially uncentered, to use , and with the Coriolis term implicit following Côté and Staniforth  and Temperton . The semi-implicit, semi-Lagrangian momentum equation at level (but with the level subscript suppressed) is
The gradient of the geopotential is more complex than in the system because the hydrostatic matrix depends on the local pressure:
The momentum equation can be written as
By combining terms, 3.294 can be written in general as
W&O94 followed Bates et al.  which ignored rotating the vector to remain parallel to the earth's surface during translation. We include that factor by keeping the length of the vector written in terms of the same as the length of the vector written in terms of . Thus, (10) of W&O94 becomes
Transform to spectral space as described in the description of the
Eulerian spectral transform dynamical core. Note, from (4.5b) and
(4.6) on page 177 of Machenhauer 
Define and so that
For each vertical mode, i.e. element of , and for each Fourier wavenumber we have a system of equations in to solve. In following we drop the Fourier index and the modal element index from the notation.
Substitute and into the equation.
At the end of the system, the boundary conditions are
For each and we have the general systems of equations
Assume solutions of the form
Divergence in physical space is obtained from the vertical mode coefficients by
The trajectory calculation follows Hortal  Let denote the position vector of the parcel,