control of gravity waves

Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data1

Control of Gravity Waves

Gravity waves and divergent flow in the atmosphere Two noise removal approaches: filtering and initialization Normal mode initialization Digital filter Control of gravity waves in the ECMWF assimilation system

Lars IsaksenRoom 308, Data Assimilation, ECMWF


Processes and waves in the atmosphere

Sound waves, synoptic scale waves, gravity waves, turbulence, Brownian motions ..

The atmospheric flow is quasi-geostrophic and largely rotational (non-divergent) – mass/wind balance at extra-tropical latitudes

The energy in the atmosphere is mainly associated with fairly slow moving large-scale and synoptic scale waves (Rossby waves)

Energy associated with gravity waves is quickly dissipated/dispersed to larger scale Rossby waves: the quasi-geostrophic balance is reinstated


500 hPa Geopotential height and windsApproximate mass-wind balance


MSL pressure and 10 metre windsApproximate mass-wind balance


Which atmospheric processes/waves are important in data assimilation and NWP?

Sound and gravity waves are generally NOT important, but can rather be considered a nuisance

Fast waves in the NWP system require unnecessary short time steps – inefficient use of computer time

Large amplitude gravity waves add high frequency noise to the assimilation system resulting in:

– rejection of correct observations– noisy forecasts with e.g. unrealistic precipitation

BUT certain gravity waves and divergent features should be retained in a realistic assimilation system. We will now present some examples.


Ageostrophic motion – Jet stream related An important unbalanced synoptic feature in the atmosphere

Ageostrophic winds at 250 hPa Wind and height fields at 250 hPa


Mountain generated gravity waves should be retained

Rocky Mountains


Temperature cross-section over Norway Gravity waves in the ECMWF analysis

Acknowledgements to Agathe UntchNorway

Pre

ssu

re [

hP

a]


Analysis temperatures at 30 hPa

Acknowledgements to Agathe Untch


Equatorial Walker circulation


Divergent winds at 150hPa: ERA-40 average March 1989

Acknowledgements to Per Kållberg


Semi-diurnal tidal signal


Observed Mean Sea-Level pressure - Tropics

Semi-diurnal tidal signal for Seychelles (5N 56E)


Filtering the governing equations

Quasi-geostrophic equations/ omega equation Primitive equations with hydrostatic balance Primitive equations with damping time-step like

Eulerian backward Primitive equations with digital filter

Goal: Use filtered model equations that do not allow high frequency solutions (“noise”) – but still retain the “signal”


Initialization

Goal: Remove the components of the initial field that are responsible for the “noise” – but retain the “signal”

• Make the initial fields satisfy a balance equation, e.g. quasi-geostrophic balance

or• Set tendencies of gravity waves to zero in initial fields

– Non-linear Normal Mode Initialization


Normal-mode initialization

0 N(x)Lxx

idt

d LLinearize forecast model about a statically-stable state of rest:

where N

represents linear terms

represents the nonlinear

terms and diabatic forcing

GR xxx

Diagonalize L by transforming to eigenvalue-mode - “Hough space”:

0, )x(xNxEΛEx

GRRRTRRR

R idt

d

0, )x(xNxEΛEx

GRGGTGGG

G idt

d

0 N(x)xEEx Ti

dt

dwhere Λ is the diagonal eigenvalue matrix

Split eigenvalues into slow Rossby modes and fast Gravity modes.


Non-dimensional wavenumber

Fre

qu

en

cy

Rossby modes and Gravity modes

The ‘critical frequency’ separating fast modes from slow.

Mixed Rossby-Gravity Wave


Non-linear Normal-Mode Initialization

0, )x(xNxEΛEx

GRGGTGGG

G idt

dThe fast Gravity modes generally represent “noise” to be eliminated.

for one eigenvalue,0 kkkk Nxi

dt

dx

k

kti

k

kkk i

Ne

i

Nxtx k

))0(()(

If Nk is assumed constant (i.e. slowly varying compared to gravity waves):

At initial time set0

dt

dxk

k

kk i

Ntx

0)(

k

kk i

Nx

)0(

The high frequency component is removed and will NOT reappear.

Assumes that the slow Nk forcing balances the oscillations at initial time.

k

then so


Non-linear NMI: USA Great PlanesSurface pressure evolution

Uninitialized field

Non-linear NMI

initialized field

Temperton and

Williamson (1981)


Optimal and approximate low-pass filter

)()( sin kn

n

ckn

nnk x

n

tnxhf

Consider a infinite sequence of a ‘noisy’ function values: {x(i)}

We want to remove the high frequency ‘noise’.

This is identical to multiplying {x(i)} by a weighting function:

c

One method: perform direct Fourier transform; remove high-frequency Fourier components;

perform inverse Fourier transform.

)()( sin knN

Nn

cknN

Nnnk x

n

tnxhf

is the cut-off frequency

)()(0

sin nN

Nn

cnN

Nnn x

n

tnxhf

The finite approximation is:


Digital filter

)()0( 1)( n

N

Nnn xh

hxInit

Consider a sequence of model values {x(i)} at consecutive

adiabatic time-steps starting from an uninitialized analysis

A digital filter adjusts values to remove high frequency ‘noise’

Adiabatic, non-recursive filter:

Perform forward adiabatic model integration {x(0),x(1),…,x(N)}

Perform backward adiabatic model integration {x(0),x(-1),…,x(-N)}

The filtered initial conditions are:

where

N

Nnnhh


Fourier filter and Lanczos filterD

am

pin

g f

ac

tor

for

wav

es

Wave frequency in hours

n

tnh c

n

)sin(

)1/(

)]1/(sin[)sin(

Nn

Nn

n

tnh c

n

c

Gibbs Phenomenon for Fourier filter

Broader cut-off for Lanczos filter


Transfer function for Lanczos filter 6 hour window


Transfer function for Lanczos filter 12 hour window


Transfer function for Lanczos filter 6 and 12 hour window


Response to Lanczos filter with 6h cut-off


Incremental initialization (ECMWF, 1996-1999)

Let xb denote background state, expected to be “noise free”

xU the uninitialized analysis

xI the initialized analysis and

Init(x) the result of an adiabatic NMI initialization. Then

xI = xb + Init(xU) – Init(xb)

Diabatic non-linear normal mode initializationFull-field initialization (ECMWF, 1982-1996)


Control of gravity waves within the variational assimilation

• Primary control provided by Jb (mass/wind balance)

• In 4D-Var Jo provides additional balance

• Digital filter or NMI based Jc contraint

• Diffusive properties of physics routines

Minimize: Jo + Jb + Jc


Control of gravity waves within the variational assimilation

Primary control provided by Jb (mass/wind balance)


NMI based Jc constraint

2

G

b

Gc dt

d

dt

dJ

xx

Still used at ECMWF in 3D-Var and until 2002 in 4D-Var

I. Project “analysis” and background tendencies onto gravity

modes.

II. Minimize the difference.

Noise is removed because background fields are balanced.


Weak constraint Jc based on digital filter

Implemented by Gustafsson (1992) in HIRLAM and Gauthier+Thépaut (2000) in ARPEGE/IFS at Meteo-France

Removes high frequency noise as part of 12h 4D-Var window integration

Apply 12h digital filter to the departures from the reference trajectory A spectral space energy norm is used to measure distance.

– At Meteo-France all prognostic variables are included in the norm

– At ECMWF only divergence is now included in the norm, with larger weight

Obtain filtered departures in the middle of the assimilation period (6h) Propagate filtered increments valid at t=6h by the adjoint of the tangent-

linear model back to initial time, t=0. Get and )( 0xcJ )( 0xcx J

Jc calculation is a virtually cost-free addition to Jo calculations


Weak constraint Jc based on digital filter

2

2/2/0 2

1)(

ENNc xxJ x

n

N

nnN h xx

0

2/

N

nNNnncx ttJ

02/2/,0

*0 ))(()( xxRx

N

nNNnntt

02/2/,0

* )()( xxER

Apply 12h digital filter to the departures from the reference trajectory

and obtain filtered values in the middle (6h):

N

nnnN tth

00,02/ )( xRx

Define penalty term using energy norm, E:

The gradient of the penalty term is propagated by the adjoint, R*, of the

tangent-linear model back to time, t=0:

Use tangent-linear model, R, to get:

Jc calculation is a virtually cost-free addition to Jo calculations

21

2

Nnh

Nnh

n

nn for


Hurricane Alma – impact of Jc formulationJc on divergence only with weight=100 versus Jc on all prognostic fields with weight=10

MSL pressure and 850hPa wind analysis differences


Impact of Jc formulation

Jc on divergence only with weight=100

versus Jc on all prognostic fields with weight=10

Impact near dynamic systems and near orography. Fit to wind data improved.

In general a small impact.

MSL pressure and 850hPa wind analysis differences


Minimization of cost function in 4D-VarValue of Jo, Jb and Jc terms


Minimization of cost function in 4D-VarValue of Jo, Jb and Jc terms – logarithmic scale


Himalaya grid point in 3D-Var - No Jc


Himalaya grid point in 3D-Var


Seychelles (5S 56E) MSL observations plus 3D-Var First Guess and Analysis

Observed value

Observed value

First guess value

Analysis value

8 Feb 1997 14 Feb1997


Observations and first guess values Observations and analysis values

4D-Var handles tidal signal very well !

Seychelles (5S 56E) MSL observations plus 4D-Var First Guess and Analysis


Gravity waves and divergent flow in the atmosphere Two noise removal approaches: filtering and initialization Normal mode initialization Digital filter Control of gravity waves in the ECMWF assimilation system

We discussed these topics today

control of gravity waves

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