- 3.2.1 Introduction
- 3.2.2 Vertical coordinate and hydrostatic equation
- 3.2.3 Semi-implicit reference state
- 3.2.4 Perturbation surface pressure prognostic variable
- 3.2.5 Extrapolated variables
- 3.2.6 Interpolants
- 3.2.7 Continuity Equation
- 3.2.8 Thermodynamic Equation
- 3.2.9 Momentum equations
- 3.2.10 Development of semi-implicit system equations
- 3.2.11 Trajectory Calculation
- 3.2.12 Mass and energy fixers and statistics calculations

3.2 Semi-Lagrangian Dynamical Core

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 [185] 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

where is pressure, is surface pressure, and is a specified constant reference pressure. The coefficients and specify the actual coordinate used. As mentioned by Simmons and Burridge [157] and implemented by Simmons and Strüfing [158] and Simmons and Strüfing [159], the coefficients and are defined only at the discrete model levels. This has implications in the continuity equation development which follows.

In the system the hydrostatic equation is approximated in a general way by

where k is the vertical grid index running from 1 at the top of the model to at the first model level above the surface, is the geopotential at level , is the surface geopotential, is the virtual temperature, and R is the gas constant. The matrix , referred to as the hydrostatic matrix, represents the discrete approximation to the hydrostatic integral and is left unspecified for now. It depends on pressure, which varies from horizontal point to point.

The semi-implicit equations are linearized about a reference state with constant and . We choose

(3.265) |

To ameliorate the mountain resonance problem,
Ritchie and Tanguay [148] introduce a perturbation
surface pressure prognostic variable

(3.266) | |||

(3.267) |

The perturbation surface pressure, , is never actually used as a grid point variable in the CAM 3.0 code. It is only used for the semi-implicit development and solution. The total is reclaimed in spectral space from the spectral coefficients of immediately after the semi-implicit equations are solved, and transformed back to spectral space along with its derivatives. This is in part because is needed for the horizontal diffusion correction to pressure surfaces. However the semi-Lagrangian CAM 3.0 default is to run with no horizontal diffusion.

Variables needed at time ( ) are obtained by extrapolation

(3.268) |

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

(3.269) | ||

where

(3.270) | ||

and
| ||

(3.271) |

denotes a vertical difference, denotes the vertical level, denotes the arrival point, the departure point from horizontal (two-dimensional) advection, and the midpoint of that trajectory.

The surface pressure forecast equation is obtained by summing over all
levels and is related to (18) of W&O94 but is spatially uncentered
and uses

(3.272) | |||

The corresponding equation for the semi-implicit development follows and is related to (19) of W&O94, again spatially uncentered and using .

(3.273) | ||

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.

(3.274) | ||

The combination is treated as a unit, and follows from (3.271).

The thermodynamic equation is obtained from (25) of W&O94 modified to be spatially uncentered and to use . In addition Hortal's modification [172] is included, in which

(3.275) |

is subtracted from both sides of the temperature equation. This is akin to horizontal diffusion which includes the first order term converting horizontal derivatives from eta to pressure coordinates, with replaced by , and taken as a global average so it is invariant with time and can commute with the differential operators.

(3.276) | |||

Note that represents the heating calculated to advance from time to time and is valid over the interval.

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

(3.277) | |||

(3.278) | |||

(3.279) | |||

(3.280) | |||

(3.281) | |||

(3.282) | |||

(3.283) | |||

(3.284) | |||

(3.285) | |||

(3.286) | |||

(3.287) | |||

(3.288) |

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 [43] and Temperton [171]. The semi-implicit, semi-Lagrangian momentum equation at level (but with the level subscript suppressed) is

(3.289) | |||

The gradient of the geopotential is more complex than in the system because the hydrostatic matrix depends on the local pressure:

(3.290) |

where is and is the gas constant for water vapor. The gradient of is calculated from the spectral representation and that of from a discrete cubic approximation that is consistent with the interpolation used in the semi-Lagrangian water vapor advection. In general, the elements of are functions of pressure at adjacent discrete model levels

(3.291) |

The gradient is then a function of pressure and the pressure gradient

(3.292) |

The pressure gradient is available from (3.263) and the surface pressure gradient calculated from the spectral representation

(3.293) |

The momentum equation can be written as

where contains known terms at times ( ) and ().

By combining terms, 3.294 can be written in general as

where and denote the spherical unit vectors in the longitudinal and latitudinal directions, respectively, at the points indicated by the subscripts, and and denote the appropriate combinations of terms in 3.294. Note that is distinct from the . Following Bates et al. [13], equations for the individual components are obtained by relating the unit vectors at the departure points ( , ) to those at the arrival points ( , ):

(3.296) | |||

(3.297) |

in which the vertical components ( ) are ignored. The dependence of 's and 's on the latitudes and longitudes of the arrival and departure points is given in the Appendix of Bates et al. [13].

W&O94 followed Bates et al. [13] 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

(3.298) |

where

(3.299) |

After the momentum equation is written in a common set of unit vectors

(3.300) |

Drop the from the notation, define

(3.301) |

and transform to vorticity and divergence

(3.302) | |||

(3.303) |

Note that

(3.304) | |||

(3.305) |

Then the vorticity and divergence equations become

(3.306) | |||

(3.307) |

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 [118]

(3.308) | |||

(3.309) |

and from (4.5a) on page 177 of Machenhauer [118]

(3.310) |

Then the equations for the spectral coefficients at time at each vertical level are

(3.311) | |||

(3.312) | |||

(3.313) | |||

(3.314) |

The underbar denotes a vector over vertical levels. Rewrite the vorticity and divergence equations in terms of vectors over vertical levels.

(3.315) | |||

(3.316) |

Define by

(3.317) | ||

and
| ||

(3.318) | ||

(3.319) | ||

(3.320) |

Then the vorticity and divergence equations are

(3.321) | |||

(3.322) |

Note that these equations are uncoupled in the vertical, i.e. each vertical level involves variables at that level only. The equation for however couples all levels.

(3.323) |

Define and so that

(3.324) |

Let denote the eigenvalues of with corresponding eigenvectors and is the matrix with columns

(3.325) |

and the diagonal matrix of corresponding eigenvalues

(3.326) | |||

(3.327) | |||

(3.328) |

Then transform

(3.329) | ||||

(3.330) | ||||

(3.331) |

(3.332) | |||

(3.333) | |||

(3.334) | |||

(3.335) |

Since is diagonal, all equations are now uncoupled in the vertical.

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.

(3.336) | |||

(3.337) | |||

(3.338) |

The modal index was included in the above equation on only as a reminder, but will also be dropped in the following.

Substitute and into the equation.

(3.339) | |||

which is just two tri-diagonal systems of equations, one for the even and one for the odd 's, and

At the end of the system, the boundary conditions are

(3.340) | ||||

the term is not present, and from the underlying truncation

(3.341) |

For each and we have the general systems of equations

(3.342) | |||

0 | (3.343) | ||

0 | (3.344) |

Assume solutions of the form

(3.345) |

then

(3.346) | |||

(3.347) |

(3.348) | |||

(3.349) |

(3.350) |

(3.351) |

Divergence in physical space is obtained from the vertical mode coefficients by

(3.352) |

The remaining variables are obtained in physical space by

(3.353) | |||

(3.354) | |||

(3.355) |

The trajectory calculation follows Hortal [77] Let denote the position vector of the parcel,

(3.356) |

which can be approximated in general by

(3.357) |

Hortal's method is based on a Taylor's series expansion

(3.358) |

or substituting for

(3.359) |

Approximate

(3.360) | ||

giving
| ||

(3.361) |

for the trajectory equation.