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Home > Newsevents > Training > Rcourse_notes > DATA_ASSIMILATION > GRAV-WAVE_CONTROL >

The control of gravity waves in data assimilation
27 April 1999

By Adrian Simmons
European Centre for Medium-Range Weather Forecasts

Table of contents

1. Introduction

2. Non-linear normal-mode initialization

3. Control of gravity waves in the ECMWF varaitional assimilation

4. Digital filtering

Appendix A Definition of operators for the ECMWF vertical finite difference scheme

Appendix B The Lamb wave

Appendix C The non-recursive implementation of the recursive filter

References

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APPENDIX A . Definition of operators for the ECMWF vertical finite-difference scheme

Specification of the forms for the matrices

and

and the vectors

and

has been given by Simmons and Burridge(1981), although with a notation different to that used here. We present the general case of a reference temperature that varies with pressure, with values

at the "full" levels of the model, for

. We assume a hybrid vertical coordinate in which the pressure at the "half" levels,

, is defined as a function of the surface pressure,

, with

and

.

The forms are:

and

Here

and

All expressions involving a pressure are evaluated for the reference surface pressure

.

If the reference temperature profile is isothermal, with

, (A.2) reduces to

(A.7)

and noting that

(A.4) becomes simply

(A.8)

We illustrate results by comparing modes computed for the idealized temperature profile shown in Fig. 25 with modes computed for isothermal reference profiles with temperatures of 245K and 300K. The idealized profile is constructed to be linear in

in regions representative of the troposphere and stratosphere. This distribution of temperature has a mean value of approximately 245K

Figure 25 . An idealized vertical profile of temperature.

***Table 2 . Gravity-wave phase speeds for the first nine modes of a 50-level version of the ECMWF model for the reference temperature profile shown in*** ***Fig. 25 and for two uniform reference temperatures. The reference surface pressure is 1000hPa.***
Mode number	Phase speed (ms^-1) for reference temperature profile shown in Fig. 25	Phase speed (ms^-1) for 245K reference temperature	Phase speed (ms^-1) for 300K reference temperature
1	316	313	347
2	249	237	262
3	180	175	194
4	131	131	145
5	100	102	112
6	78	82	90
7	64	67	75
8	54	57	63
9	47	49	54

Gravity-wave phase speeds for the first nine modes are presented in Table 2 and mode structures for the profile shown in Fig. 25 and for the 245K reference temperature are shown in Fig. 26 . The corresponding mode structures for the 300K reference temperature can be seen in the right-hand panel of Fig. 10 . The gravity-wave phase speeds computed for the mean temperature of 245K are quite similar (within about 5%) to those computed for the reference state with varying tropospheric and stratospheric temperatures. Mode structures are qualitatively similar, but evidently reflect the differences in static stability in the troposphere and stratosphere. Phase speeds for the 245K reference temperature are smaller than those for the 300K reference by a factor equal to the square root of the ratio of the temperatures, and mode structures are identical for the two isothermal reference states. This is because the uniform reference temperature

appears only as a simple factor multiplying each element of the matrix

defined by equation (28)

Figure 26 . Structures of the first nine vertical modes of the 50-level model for the reference temperature profile shown in Fig. 25 and for a uniform reference temperature of 245K. The reference surface pressure is 1000hPa.

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07.06.2002

The control of gravity waves in data assimilation 27 April 1999 By Adrian Simmons European Centre for Medium-Range Weather Forecasts

The control of gravity waves in data assimilation
27 April 1999

By Adrian Simmons
European Centre for Medium-Range Weather Forecasts