ERA-40 Archive Plan

Annex I. Vertical integrals

The continuous, adiabatic, frictionless form of the model's primitive equations may be manipulated to give the following equations, in standard notation:

(A.1.1)

(A.1.2)

(A.1.3)

(A.1.4)

(A.1.5)

The notation is as in Simmons and Burridge(1981, Mon. Wea. Rev., 109, 758-766). The gas constant, , and specific heat at constant pressure, , vary with specific humidity, :

(A.1.6)

(A.1.7)

where subscripts and denote values for dry air and water vapour respectively.

It should be noted that should strictly be interpreted in terms of potential+internal energy only when vertically integrated:

(A.1.8)

Integrating the energy, mass, water vapour and ozone equations in the vertical, they may be written symbolically as:

(A.1.9)

(A.1.10)

(A.1.11)

(A.1.12)

(A.1.13)

The vertically integrated variables are:

(A.1.14)

(A.1.15)

(A.1.16)

(A.1.17)

(A.1.18)

(A.1.19)

The fluxes are:

(A.1.20)

(A.1.21)

(A.1.22)

(A.1.23)

(A.1.24)

(A.1.25)

(A.1.26)

(A.1.27)

(A.1.28)

and the energy conversions are:

(A.1.29)

(A.1.30)

The two energy equations can also be written:

(A.1.31)

(A.1.32)

where

(A.1.33)

The standard post-processing already produces (see Table 3.7) the net mass of water vapour (total column water vapour, above) and ozone (total column ozone, above) in kg/m2 for each grid point. Also available is surface geopotential, , enabling the component of to be simply computed if is known. Note that is a fixed field, so the calculation of can be done on monthly means if required.

In addition to these fields, analysis and selected forecast values of the following will be archived:

· , and

· , , , , and

·

These are most directly appropriate for use of the energy equation in the form given by equations (A.1.31) and (A.1.32). The flux can be simply calculated as , and the conversion term is given by the convergence of this flux. The vertical integral, , of a quantity that is defined by its analyzed or forecast values at the 60 full model levels is evaluated as .

If the alternative form of the energy equations given by (A.1.9) and (A.1.10) is preferred, the fluxes and can be computed simply by subtracting from and adding it to .The conversion term is given by:

(A.1.34)

All RHS terms in the budget equations (A.1.9)-(A.1.13) (or (A.1.31), (A.1.32), (A.1.11), (A.1.12) and (A.1.13)) can thus be computed in terms of the supplied integrals, applying a divergence operator and simple multiplications where needed. These operations can be carried out either on the instantaneous values or on their monthly means.