6. Code aspects for ECMWF's LFULLIMP incremental form of the PC scheme

All grid point computations are perform with "non-incremental" (or full) variables X, the existence of "incremental" variables Y is totally limited to the spectral space. This requires the additional saving of the "background" variable in SP space in order to be able to compute the "increment" from the full variable and the "background" variable. The increase of memory of the incremental method come from this necessary eadditional storage.
During GP part of predictor step, the prognostic variables labelled "PxT9" refers to time-level "-" (however they are not used), and the prognostic variables labelled" PxT0" refers to time-level "0" (to be documented more carefully).
During GP part of corrector step, the prognostic variables labelled "PxT9" refers to time-level "0", and the prognostic variables labelled "PxT0" refers to time-level "*" (to be documented more carefully).

6.2 Data flow for incremental form

SUBROUTINE LATTEX :

PxNLT9 : is used for storage of old values for next iteration in PC (or next time-step in SI): - Predictor step : Contains ${\tt ZMOY1x} = (\Delta t / 2) {\bf M}X^0$; - Corrector step : not used (because the next time-step is 2TL non-extrapolating)
PxL9 : is used for High-Order interpolations at origin point: - Predictor step : Contains ${\tt PxT0}+{\tt ZMOY1x} = X^0 + (\Delta t / 2) {\bf M}X^0$; - Corrector step : Contains ${\tt PxT9}+{\tt PxNLT9} = X^0 + (\Delta t / 2) {\bf M}X^0$; (because PxT9 contains X⁰ in the corrector step,; and PxNLT9 contains $(\Delta t / 2) {\bf M}X^0$ in the corrector step)
PxT1 : is used for not interpolated quantities (at final point): - Predictor step : Contains ${\tt ZMOY1x} = (\Delta t / 2) {\bf M}X^0$; - Predictor step : Contains ${\tt ZMOY1x} = (\Delta t / 2) {\bf M}X^*$; (because ZMOY1x contains $(\Delta t / 2) {\bf M}X^*$ in the corrector step)
PxL0 : is used for Low-Order interpolated at origin point: - Not used in PC scheme (no Low-order interpolations)
PxSI : for Linear terms: - Not used in PC scheme and more generally in incremental form.

Hence, at the end of LATTEX, we have $X + {\bf M}(X)$ in PxL9 and ${\bf M}(X)$ in PxT1. We are thus ready to compute the RHS of ( 3) and (6 ) equations, except for the X_F term in the first RHS parenthesis of these equations.

SUBROUTINE LAPINEB :

In this subroutine, the RHS quantities of (3) and ( 6) computed at the final point and interpolated at the origin point are aggregated. Hence, they almost form the RHS of the linear equation inversion in its final form for the incremental variables Y^* and Y⁺ (as seen above, the X_F term is missing). Following the definition in the introduction, we use the notation tilde for the RHS of a linear inversion equation. Hence, after the LAPINEB aggregation we have $[\widetilde{Y^*} + X^0_F]$ for the predictor-step, and $[\widetilde{Y^+} + X^*_F]$ for the corrector-step.

Direct Spectral Transform

The method adopted to implement the incremental form is to transmit from GP to SP space only the agregated RHS $[\widetilde{Y^+} + X^*_F]$ of the linear model inversion equations. The "background" variables are needed in the spectral space to compute incremental variables from the full variables and vice-versa. The "background" variables are thus saved in the spectral space from one step or sub-step to the next one. With this method, no additional transforms are thus needed.

SUBROUTINE SPCSIX :

At the begining of spectral-space computations, we compute the definitive form of the linear inversion equation RHS from the provisional form and the "background" variable which has been saved at the previous step or sub-step (see below):

$\begin{eqnarray*}\widetilde{Y^*} & = & [\widetilde{Y^*}+ X^0_F] - X^0 \\ \widetilde{Y^+} & = & [\widetilde{Y^+}+ X^*_F] - X^* \end{eqnarray*}$

Then the linear equation is inversed, giving the new variables in incremental form:

$\begin{eqnarray*}Y* & = & {\bf B}^{-1}. \widetilde{Y^*} \\ Y+ & = & {\bf B}^{-1}. \widetilde{Y^+} \end{eqnarray*}$

Then the full variables are restored from the background variables:

$\begin{eqnarray*}X^* & = & Y^* + X^0 \\ X^+ & = & Y^+ + X^* \end{eqnarray*}$

The new full variables are stored for the next step or sub-step:

End of predictor step: X^* is saved for next sub-step.

End of corrector step: X⁺ is saved for next step.

These new full variables are spectrally-transformed back to the G.P space for next step or sub-step. The variable X^* becomes X⁰ in G.P space for the corrector step GP computations, and the variable X⁺ becomes X⁰ in G.P space for the next time-step GP computations.

6.3 Some details on Decentering

Provision is made in the code for more flexibility on the amount of decentering $\epsilon$ between the various terms to which it is applied.

In SUDYN : three values are defined:

-: BCOR_HORIZ : which corresponds to $\alpha_h$ in the formalism of Cullen QJ paper, or to $(1 + \epsilon_h)/2$ in the formalism of this note.
-: BCOR_VERT : which corresponds to $\alpha_v$ in the formalism of Cullen QJ paper, or to $(1 + \epsilon_v)/2$ in the formalism of this note.
-: BCOR = MAX(BCOR_HORIZ , BCOR_VERT) which will be noted $\epsilon$ in this section

In LATTEX, (computation of explicit model RHS) both $\epsilon_h$ and $\epsilon_v$ are used:

-: $\epsilon_h$ is applied to U and V equations
-: $\epsilon_v$ is applied to T and $\pi_s$ equations
-: the other variables are not decentered (q, q_i, q_l, O₃, ...)

In LAVENT , (trajectory research) only $\epsilon_v$ is used, but is applied only to the vertical component of the wind.

For the predictor step we thus have:

$\begin{displaymath}{\bf OF} = \frac{\Delta t}{2} \left[V^0_F + V^0_O + (1+\epsilon_v) w^0_F + (1-\epsilon_v)w^0_O \right] \end{displaymath}$

And for the corrector step we have:

$\begin{displaymath}{\bf O^*F} = \frac{\Delta t}{2} \left[V^*_F + V^0_{O^*} + (1+\epsilon_v) w^*_F + (1-\epsilon_v)w^0_{O^*} \right] \end{displaymath}$

where V is the horizontal wind vector, and w the vertical wind vector.

In SPCSIX , (inversion of linear model) only $\epsilon$ is used.

Next: About this document ... Up: Main page Previous: 5. Incremental form for

Pierre BENARD
2002-06-17

Home