Control of Gravity Waves
Lars Isaksen
Room 308, Data Assimilation, ECMWF
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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 Isaksen, ECMWF, March 2006
1
Data assimilation and use of satellite data
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 extratropical 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
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
500 hPa Geopotential height and winds
Approximate mass-wind balance
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
MSL pressure and 10 metre winds
Approximate mass-wind balance
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
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.
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Ageostrophic motion – Jet stream related
An important unbalanced synoptic feature in the atmosphere
Wind and height fields at 250 hPa
Ageostrophic winds at 250 hPa
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Mountain generated gravity waves should be retained
Rocky Mountains
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Temperature cross-section over Norway
Gravity waves in the ECMWF analysis
Norway
Lars Isaksen, ECMWF, March 2006
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Acknowledgements to Agathe Untch
Data assimilation and use of satellite data
Analysis temperatures at 30 hPa
Acknowledgements to Agathe Untch
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Equatorial Walker circulation
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Divergent winds at 150hPa:
ERA-40 average March 1989
Acknowledgements to Per Kållberg
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Semi-diurnal tidal signal
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Observed Mean Sea-Level pressure - Tropics
Semi-diurnal tidal signal for Seychelles (5N 56E)
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Filtering the governing equations
Goal: Use filtered model equations that do not
allow high frequency solutions (“noise”) –
but still retain the “signal”
 Quasi-geostrophic equations/ omega equation
 Primitive equations with hydrostatic balance
 Primitive equations with damping time-step
like Eulerian backward
 Primitive equations with digital filter
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
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
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Normal-mode initialization
Linearize forecast model about a statically-stable state of rest:
dx
 iLx  N(x)  0
dt
L represents linear terms
where
N represents the nonlinear terms
and diabatic forcing
Diagonalize
L
by transforming to eigenvalue-mode - “Hough space”:
dx
 iEET x  N(x)  0
dt
where
Λ
is the diagonal eigenvalue matrix
Split eigenvalues into slow Rossby modes and fast Gravity modes.
dx R
 iER Λ R ETR x R  N R (x R , x G )  0
dt
dx G
 iEG Λ G ETG x G  NG (x R , x G )  0
dt
Lars Isaksen, ECMWF, March 2006
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x  x R  xG
Data assimilation and use of satellite data
Rossby modes and Gravity modes
Frequency
The ‘critical frequency’
separating fast
modes from slow.
Mixed Rossby-Gravity Wave
Non-dimensional wavenumber
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Non-linear Normal-Mode Initialization
The fast Gravity modes generally represent “noise” to be eliminated.
dx G
 iEG Λ G ETG x G  NG (x R , x G )  0
dt
dxk
 i k xk  N k  0 for one eigenvalue,  k
dt
If Nk is assumed constant (i.e. slowly varying compared to gravity waves):
x k ( t )  ( x k ( 0) 
dxk
0
At initial time set dt
N k ik t N k
)e

ik
ik
Nk
then xk (0)  
ik
so
Nk
xk ( t )  0 
ik
The high frequency component is removed and will NOT reappear.
Assumes that the slow Nk forcing balances the oscillations at initial time.
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Non-linear NMI: USA Great Planes
Surface pressure evolution
Non-linear NMI
initialized field
Temperton and
Williamson (1981)
Uninitialized field
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Optimal and approximate low-pass filter
Consider a infinite sequence of a ‘noisy’ function values: {x(i)}
We want to remove the high frequency ‘noise’.
One method: perform direct Fourier transform;
remove high-frequency Fourier components;
perform inverse Fourier transform.
This is identical to multiplying {x(i)} by a weighting function:
fk 

h x
n  
( nk )
n
sin n c t ( n  k )
 
x
n
n  

 c is the cut-off frequency
The finite approximation is:
fk 
N
 hn x
n  N
( n k )
sin n c t ( n k )
 
x
n
n  N
N
Lars Isaksen, ECMWF, March 2006
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f0 
N
(n)
h
x
 n 
n N
sin n c t ( n )
x

n
n N
N
Data assimilation and use of satellite data
Digital filter
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:
N
1
Init ( x )   hn x ( n )
h n N
( 0)
Lars Isaksen, ECMWF, March 2006
where
h
N
h
n N
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n
Data assimilation and use of satellite data
Fourier filter and Lanczos filter
c
Damping factor for waves
sin( n c t )
hn 
n
Gibbs Phenomenon for Fourier filter
sin( n c t ) sin[ n /( N  1)]
hn 
n
n /( N  1)
Broader cut-off for Lanczos filter
Wave frequency in hours
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Transfer function for Lanczos filter
6 hour window
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Transfer function for Lanczos filter
12 hour window
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Transfer function for Lanczos filter
6 and 12 hour window
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Response to Lanczos filter with 6h cut-off
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Diabatic non-linear normal mode initialization
Full-field initialization (ECMWF, 1982-1996)
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)
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Control of gravity waves within the
variational assimilation
Minimize: Jo + Jb + Jc
•
•
•
•
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
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Control of gravity waves within the
variational assimilation
Primary control provided by Jb (mass/wind balance)
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
NMI based Jc constraint
Still used at ECMWF in 3D-Var and until 2002 in 4D-Var
 dx   dx b 
Jc      

 dt G  dt G
2
I.
Project “analysis” and background tendencies onto gravity modes.
II.
Minimize the difference.
Noise is removed because background fields are balanced.
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
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 tangentlinear model back to initial time, t=0. Get J c (x 0 ) and  x J c (x 0 )
 Jc calculation is a virtually cost-free addition to Jo calculations
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Weak constraint Jc based on digital filter
Apply 12h digital filter to the departures from the reference trajectory
and obtain filtered values in the middle (6h): x N / 2 
N
N
 h x
n 0
n
n
Use tangent-linear model, R, to get: x N / 2   hn R (t0,tn )x 0
n 0
Define penalty term using energy norm, E:
J c (x0 ) 
1
x N / 2  xN / 2
2
2
E
The gradient of the penalty term is propagated by the adjoint, R*, of the
tangent-linear model back to time, t=0:
N
 x J c (x 0 )    n R (t0,t n )(x N / 2  x N / 2 )
*
n 0
N
  R * (t0,tn ) n E(x N / 2  x N / 2 )
  hn for n  N 2
n  
1  hn n  N 2
n 0
Jc calculation is a virtually cost-free addition to Jo calculations
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Hurricane Alma – impact of Jc formulation
Jc on divergence only with weight=100
versus
Jc on all prognostic fields with weight=10
MSL pressure and 850hPa wind analysis differences
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
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
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Minimization of cost function in 4D-Var
Value of Jo, Jb and Jc terms
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Minimization of cost function in 4D-Var
Value of Jo, Jb and Jc terms – logarithmic scale
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Himalaya grid point in 3D-Var - No Jc
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Himalaya grid point in 3D-Var
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Himalaya grid point in 4D-Var
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Himalaya grid point in 4D-Var
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Seychelles (5S 56E) MSL observations plus
3D-Var First Guess and Analysis
Observed value
First guess value
8 Feb 1997
14 Feb1997
Observed value
Analysis value
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
Seychelles (5S 56E) MSL observations plus
4D-Var First Guess and Analysis
Observations and first guess values
Observations and analysis values
4D-Var handles tidal signal very well !
Lars Isaksen, ECMWF, March 2006
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Data assimilation and use of satellite data
We discussed these topics today





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 Isaksen, ECMWF, March 2006
43
Data assimilation and use of satellite data