Control of Gravity Waves
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Transcript of 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
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data2
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
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data3
500 hPa Geopotential height and windsApproximate mass-wind balance
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data4
MSL pressure and 10 metre windsApproximate mass-wind balance
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data5
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 Data assimilation and use of satellite data6
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
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data7
Mountain generated gravity waves should be retained
Rocky Mountains
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data8
Temperature cross-section over Norway Gravity waves in the ECMWF analysis
Acknowledgements to Agathe UntchNorway
Pre
ssu
re [
hP
a]
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data9
Analysis temperatures at 30 hPa
Acknowledgements to Agathe Untch
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data10
Equatorial Walker circulation
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data11
Divergent winds at 150hPa: ERA-40 average March 1989
Acknowledgements to Per Kållberg
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data12
Semi-diurnal tidal signal
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data13
Observed Mean Sea-Level pressure - Tropics
Semi-diurnal tidal signal for Seychelles (5N 56E)
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data14
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”
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data15
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 Data assimilation and use of satellite data16
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.
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data17
Non-dimensional wavenumber
Fre
qu
en
cy
Rossby modes and Gravity modes
The ‘critical frequency’ separating fast modes from slow.
Mixed Rossby-Gravity Wave
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data18
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
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data19
Non-linear NMI: USA Great PlanesSurface pressure evolution
Uninitialized field
Non-linear NMI
initialized field
Temperton and
Williamson (1981)
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data20
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:
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data21
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
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data22
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
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data23
Transfer function for Lanczos filter 6 hour window
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data24
Transfer function for Lanczos filter 12 hour window
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data25
Transfer function for Lanczos filter 6 and 12 hour window
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data26
Response to Lanczos filter with 6h cut-off
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data27
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)
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data28
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
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data29
Control of gravity waves within the variational assimilation
Primary control provided by Jb (mass/wind balance)
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data30
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.
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data31
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
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data32
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
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data33
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
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data34
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 Data assimilation and use of satellite data35
Minimization of cost function in 4D-VarValue of Jo, Jb and Jc terms
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data36
Minimization of cost function in 4D-VarValue of Jo, Jb and Jc terms – logarithmic scale
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data37
Himalaya grid point in 3D-Var - No Jc
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data38
Himalaya grid point in 3D-Var
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data39
Himalaya grid point in 4D-Var
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data40
Himalaya grid point in 4D-Var
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data41
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
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data42
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
Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data43
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
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