Adiabatic formulation of the ECMWF model 1
Adiabatic formulation of the ECMWF model
Agathe Untche-mail: [email protected]
(office 11)
Adiabatic formulation of the ECMWF model 2
Introduction
• Step by step guide through the decisions to be taken / choices to be made when designing the adiabatic formulation of a global Numerical Weather Prediction (NWP) model.
• In the process we are “constructing” the dynamical core of the ECMWF operational NWP Model.
Adiabatic formulation of the ECMWF model 3
Introduction (cont.)
• A numerical model has to be:– stable– accurate– efficient
• No compromise possible on stability!• The relative importance given to accuracy versus efficiency
depends on what the model is intended for. – For example:
• an operational NWP model has to be very efficient to allow the running of all applications (data-assimilation, forecasts, ensemble prediction system) in a tight daily schedule.
• a research model might not have to be so efficient but can’t compromise on accuracy.
Adiabatic formulation of the ECMWF model 4
Introduction (cont.)
• Essential to the performance of any NWP or climate-prediction model are a.) the form of the continuous governing equations
(approximated or full Euler equations?)
b.) boundary conditions imposed (conservation properties depend on these).
c.) the numerical schemes chosen to discretize and integrate the governing equations.
Adiabatic formulation of the ECMWF model 5
Euler Equations for a moist atmosphere on a rotating sphere
rFgpVDt
VD
1
2
00)(
VDt
DV
t
QDt
Dp
Dt
DqL
Dt
DTcv
RTp
3D momentum equation
Continuity equation
Thermodynamic equation
Equation of state
xii P
Dt
DX
qPDt
Dq Humidity equation
Transport equations ofvarious physical/chemicalspecies
Adiabatic formulation of the ECMWF model 6
Notations:
VtDt
D total time derivative
1
specific volume
q specific humidity
Xi mass mixing ratios of physical or chemical species (e.g. aerosols, ozone)
g gravity = gravitation g* + centrifugal force
L latent heat
Spherical geopotential approximation is made: neglect Earth’s oblateness (~0.3%). => spherical geometry assumed!
Adiabatic formulation of the ECMWF model 7
),,cos(),,(Dt
Dr
Dt
Dr
Dt
Drwvu
w
v
u
Fgr
pu
r
vu
Dt
Dw
Fp
ru
r
vw
r
u
Dt
Dv
Fp
rwv
r
uw
r
uv
Dt
Du
1cos2
1sin2
tan
cos
1cos2sin2
tan
22
2
rw
r
v
r
u
trr
tDt
D
cos
With
Euler Equations in spherical coordinates
x
),,( rMomentum equations in spherical coordinates :
Adiabatic formulation of the ECMWF model 8
Continuity equation in spherical coordinates
0coscos 322 Drr
Dt
D
r
r
r
v
r
u
Dt
Dr
rDt
D
Dt
DD
cos3With
Thermodynamic equation in spherical coordinates
QDt
Dp
Dt
DTcv
rw
r
v
r
u
trr
tDt
D
cos
With
Adiabatic formulation of the ECMWF model 9
Shallow atmosphere approximation
w
v
u
Fgr
pu
r
vu
Dt
Dw
Fp
ru
r
vw
r
u
Dt
Dv
Fp
rwv
r
uw
r
uv
Dt
Du
1cos2
1sin2
tan
cos
1cos2sin2
tan
22
2
a
a
a
z
r
z1. Replace r by the mean radius of the earth a and by
where z is height above mean sea level.,
2. Neglect vertical and horizontal variations in g.
tan3. Neglect all metric terms not involving
cos4. Neglect the Coriolis terms containing (resulting from
the horizontal component of ).
a
Adiabatic formulation of the ECMWF model 10
Euler Equations in shallow atmosphere approximation
RTp
wDt
D
c
Qw
c
RT
Dt
DT
nFz
pg
Dt
Dwn
FpkfDt
D
vv
rw
rz
0,v
,v
1
1v
v
h
h
vhh
For n=1 we refer to these equations asNon-hydrostatic equationsin Shallow atmosphere Approximation (NH-SA)
n is the tracer for thehydrostatic approximation:n=0 => vertical momentumequation = hydrostatic eq.
hv
is horizontal velocity
Adiabatic formulation of the ECMWF model 11
Choice of predicted variables
Combine continuity, thermodynamic & gas equation to obtain a prognostic equation for p.
RT
p
Tc
Qpw
c
pc
Dt
Dp
c
Q
Dt
Dp
cDt
DT
Fnz
pg
Dt
Dwn
FpkfDt
D
vh
v
p
pp
rw
rzhh
,v
1
1
1v
vv
RTp
wDt
D
c
Qw
c
RT
Dt
DT
nFz
pg
Dt
Dwn
FpkfDt
D
vv
rw
rz
0,v
,v
1
1v
v
h
h
vhh
=>
Predicted variables: ),,,,( Twvu ),,,,( pTwvuor ?
Form of NH-SA equations more commonly used in meteorology:
),,,,( Twvu ),,,,( pTwvu(Allows enforcement of mass conservation) (Thermodyn. Computations are simpler)
Adiabatic formulation of the ECMWF model 12
Hydrostatic Approximation (n=0)
• Benefits from hydrostatic approximation– Vertical momentum equation becomes a diagnostic
relation (=> one prognostic variable (w) less!)– Vertically propagating acoustic waves are eliminated
(these are the fastest waves in the atmosphere, causing the biggest stability problems in numerical integrations!)
• Drawbacks of hydrostatic approximation– Not valid for short horizontal scales (for mesoscale
phenomena)– Short gravity waves are distorted in the hydrostatic
pressure field.
• Operational version of the ECMWF model is a hydrostatic model. – operational horizontal resolution ~25km (T799), so
hydrostatic approximation is (still) OK.
Adiabatic formulation of the ECMWF model 13
Hydrostatic shallow atmosphere equations(Hydrostatic Primitive Equations (HPE))
RT
p
Tc
Qpw
c
pc
Dt
Dp
c
Q
Dt
Dp
cDt
DT
z
pg
FpkfDt
D
vh
v
p
pp
rzhh
,v
1
01
1v
vv
p is monotonic function of z and can be used as vertical coordinate. (Eliassen (1949))
Adiabatic formulation of the ECMWF model 14
Choice of vertical coordinate
Height above mean sea level z: - most natural vertical coordinate
Pressure p (isobaric coordinate): - has advantages for thermodynamic calculations - makes the continuity equation a diagnostic relation in the hydrostatic system - can be extended for use in non-hydrostatic models
Potential temperature (isentropic coordinate):pc
R
p
pT
0
- good coordinate where atmosphere is stably stratified (potential temperature increases monotonic with z).- adiabatic flow stays on isentropic surfaces (2D flow)- good coord. in stratosphere, not very good in troposphere
Most commonly used vertical coordinates:
Adiabatic formulation of the ECMWF model 15
Choice of vertical coordinate (cont.)
Generalized vertical coordinate s:
Any variable s which is a monotonic single-valued function of height z can be used as a vertical coordinate.
(Kasahara (1974), Staniforth & Wood (2003))
Coordinate transformation rules (from z to any vertical coordinate s):
ss
zz
ss
z
z
ssz
1
1
Adiabatic formulation of the ECMWF model 16
Pressure p as vertical coordinate in the hydrostatic system
Coordinate transformation rules (from z to p):
ss
zz
ss
z
z
ssz
1
1
ppp
zz
pg
pp
z
z
ppppz
1
1
=>
with geopotential
gz
p
Hydrostatic relation between p and z:
Adiabatic formulation of the ECMWF model 17
Hydrostatic Primitive Equations with pressure as vertical coordinate
pdzgd
RT
p
p
c
Q
cDt
DT
FkfDt
D
p
p
s
z
z
s
hp
pp
rphh
ss
1
0v
vv
v
Pressure gradient replaced by geopotential gradient (at constantpressure).
Continuity eq. is a diagnostic eq. in p-coordinates. Number of prognostic variables reduced to 3 (horizontal winds & T)!
Geopotential computed from hydrostatic equation.
Dt
Dp pressure vertical velocity
gz
p
Dt
Dp
ptDt
Dph
p
,v
Adiabatic formulation of the ECMWF model 18
Hydrostatic pressure as vertical coordinate for a non-hydrostatic shallow atmosphere model
Introduced by Laprise (1991)
In the hydrostatic system with pressure as vertical coordinate thecontinuity equation is a diagnostic equation.
The idea is to find a vertical coordinate for the NH system which makes the continuity equation a diagnostic equation.
Continuity equation in generalized vertical coordinate s: (Kasahara(1974))
0v
s
z
Dt
Ds
ss
z
s
z
t hs
s
For every s for which consts
z
=> continuity is diagnostic eq.!
Adiabatic formulation of the ECMWF model 19
Hydrostatic pressure as vertical coordinate for a non-hydrostatic shallow atmosphere model (cont.)
Choose 1 gconst and denote with the coordinate s for which
),,,(
),,,(),,,(
tz
zdtzgtz
TT
z
z
T
T
1
gs
z
gz
gz
1i.e.
For 0TT andz
is the weight of a column of air (of unit area) above a point at height z, i.e. hydrostatic pressure.
Adiabatic formulation of the ECMWF model 20
Hydrostatic pressure as vertical coordinate for a non-hydrostatic shallow atmosphere model (cont.)
wg
RT
p
Tc
Qp
c
pc
Dt
Dp
c
Q
Dt
Dp
cDt
DT
Fnp
gDt
Dwn
Fp
pkfDt
D
hh
h
vv
p
pp
rw
rhh
vvD
0v
D
1
1
1v
v
3
3
v
RT
p
Tc
Qpw
c
pc
Dt
Dp
c
Q
Dt
Dp
cDt
DT
Fnz
pg
Dt
Dwn
FpkfDt
D
vh
v
p
pp
rw
rzhh
,v
1
1
1v
vv
=>
D3
in Z in
Adiabatic formulation of the ECMWF model 21
Boundary conditions
Governing equations have to be solved subject to boundary conditions.
The lower boundary of the atmosphere (surface of the earth) is a material boundary (air parcel cannot cross it!) velocity component perpendicular to surface has to vanish (e.g. at a flat and rigid surface vertical velocity w = 0)
Unfortunately, the topography of the earth is far from flat, making it quite tricky to apply the lower boundary condition.
Solution: Use a terrain-following vertical coordinate.
sp
pFor example: traditional sigma-coordinate
(Phillips, 1957)
Adiabatic formulation of the ECMWF model 22
Terrain-following vertical coordinate
Therefore, we look to create a vertical coordinate which makes the upper and lower boundaries “flat”. That is, s is constant following the shape of the boundary (i.e. the boundary is a coordinate surface).
We are looking for a vertical coordinate “s” which makes it easy to apply the condition of zero velocity normal to the boundary, even for very complex boundaries like the earth’s topography.
The easiest case is a flat and rigid boundary where the boundary condition simply is: 0
Dt
Dss
1s 0s at the top at the bottom &e.g.
at the boundary.
Adiabatic formulation of the ECMWF model 23
Terrain-following vertical coordinate (cont.)
A simple function that fulfils these conditions is
Ts
T
ts
),,(
),( ss is a monotonic single-valued function of hydrostatic pressure and also depends on surface pressure in such a way that
s1),(0),( sssT sands
T is the pressure at the top boundary.)(Where
TTs tsts )),,((),,,(
For this is the traditional sigma-coordinate of Phillips (1957)0T
s
The ECMWF model uses a terrain-following vertical coordinate based on hydrostatic pressure. The principle will be explained based on hydrostatic pressure :
Adiabatic formulation of the ECMWF model 24
Sigma-coordinate(First introduced by Phillips (1957))
Drawback: Influence of topography is felt even in the upper levels far away from the surface.
Remedy: Use hybrid sigma-pressurecoordinates
),(),,( txtx s
),( txs
x
Adiabatic formulation of the ECMWF model 25
Hybrid vertical coordinate
),,()()(),,,( tBAt s
First introduced by Simmons and Burridge (1981).
1)1(,0)1( BA
The functions A and B can be quite general and allow to design a hybrid sigma-pressure coordinate where the coordinate surfaces are sigma surfaces near the ground, gradually become more horizontal with increasing distance from thesurface and turn into pure pressure surfaces in the stratosphere (B=0).
The difference to the sigma coordinate is in the way the monotonic relation between the new coordinate and hydrostatic pressure isdefined:
In order that the top and bottom boundaries are coordinate surfaces (=> easy application of boundary condition), A and B have to fulfil:
0)0(,)0( BA T
10
at the surface
at the top
Adiabatic formulation of the ECMWF model 26
Comparison of sigma-coordinates & hybrid η-coordinates
sigma-coordinate η-coordinate
Coordinate surfaces over a hill for
),()()(),,( txBAtx s ),(),,( txtx s
Adiabatic formulation of the ECMWF model 27
wg
RT
p
t
Tc
Qp
c
pc
Dt
Dp
c
Q
Dt
Dp
cDt
DT
Fnp
gDt
Dwn
Fp
pkfDt
D
hh
h
vv
p
pp
rw
rhh
11
3
3
1
v
1
vvD
0v
D
1
1
1v
v
Non-hydrostatic equations in hybrid vertical coordinate
wg
RT
p
Tc
Qp
c
pc
Dt
Dp
c
Q
Dt
Dp
cDt
DT
Fnp
gDt
Dwn
Fp
pkfDt
D
hh
h
vv
p
pp
rw
rhh
vvD
0v
D
1
1
1v
v
3
3
v
inin
prognostic continuity eq.
Adiabatic formulation of the ECMWF model 28
Hydrostatic Primitive Equations in hybrid η vertical coordinate
UUVaUUatU cos1cos12 UUvd KPpTRafV )(ln1
VVUaVVaVUatV 222 costancos1cos1
VVvd KPpTRafU )(lncos
dp
RT
p
ppp
t
c
Q
cDt
DT
FpkfDt
D
s
s
h
pp
rhh
1
0v
1v
vv
Continuity equation is prognosticagain because the (hydrostatic) pressure is not the vertical coordinate anymore.
In addition to the geopotential gradient term, a pressure gradient term again!
Dt
Dp
ptDt
Dph
p
,v
Adiabatic formulation of the ECMWF model 29
x
q
vds
h
v
vdhh
PDt
DX
PDt
Dq
dp
TR
ppp
t
KPpq
T
Dt
DT
KPpTRkfDt
D
ln
0v
)1(1
lnvv
1
TT
vv
Hydrostatic equations of the ECMWF operational model (incorporating moisture)
Adiabatic formulation of the ECMWF model 30
Notations:
Dt
Dp p-coordinate vertical velocity
q specific humidity
X mass mixing ratio of physical or chemical species (e.g. aerosols, ozone)
vT virtual temperature
q
R
RTT
d
vv 11
pdppdd cccR /,/ v dR gas constant of dry air, vR gas constant of water vapour
pdc specific heat of dry air at constant pressure
vpc specific heat of water vapour at constant pressure
contributions from physical parametrizationsxqT PPPP ,,,v
TKK ,v horizontal diffusion terms
Adiabatic formulation of the ECMWF model 31
0
pp
Vp
t h
0
pp
Vp
t h
From the continuity equation
pVdp
V hh
0
)(
dpV
d
dBp
td
dBp
Dt
Dshss )ln)(ln()(ln
1
0
dp
Vp
pt h
ss )(
1)(ln
1
0
dp
Vt
pph )(
0
1 0 0 atandatwith boundary conditions:
we can derive (by vertical integration) the following equations:
Needed for the energy-conversionterm in the thermodynamic equation
Needed for the semi-Lagrangianadvection
Prognostic equations for surfacepressure
spBAp )()( B from def. of vert. coord.
Adiabatic formulation of the ECMWF model 32
Prognostic equations of the ECMWF hydrostatic model
dp
TR
PDt
DX
PDt
Dq
dpd
dBp
td
dBp
Dt
D
KPpq
T
Dt
DT
KPpTRkfDt
D
vds
x
q
sss
v
vdhh
ln
)lnv)(ln()(ln
)1(1
lnvv
1
h
1
0
TT
vv
These equations are discretized and integrated in the ECMWF model.
Adiabatic formulation of the ECMWF model 33
Discretisation in the ECMWF Model
• Space discretisation– In the horizontal: spectral transform method– In the vertical: cubic finite-elements
• Time discretisation– Semi-implicit semi-Lagrangian two-time-level
scheme
Adiabatic formulation of the ECMWF model 34
Horizontal discretisation
• in grid-point space only (grid-point model)– finite-difference, finite volume methods
• (in spectral space only)• in both grid-point and spectral space and transform back and
forth between the two spaces (spectral transform method, spectral model)– Gives the best of both worlds:
• Non-local operations (e.g. derivatives) are computed in spectral space (analytically)
• Local operations (e.g. products of terms) are computed in grid-point space
– The price to pay is in the cost of the transformations between the two spaces
• in finite-element space (basis functions with finite support)
Options for discretisation are:
Adiabatic formulation of the ECMWF model 35
Horizontal discretisation (cont.)
ECMWF model uses the spectral transform method
Representation in spectral space in terms of spherical harmonics:
immn
mn ePY )(),( m: zonal wavenumber
n: total wavenumber
λ= longitudeμ= sin(θ) θ: latitudePn
m: Associated Legendre functions of the first kind
)()(
0;)1()1(!2
1
)!(
)!()12()( 22/2
mn
mn
nmn
mnm
nm
n
PP
md
d
nmn
mnnP
Ideally suited set of basis functions for spherical geometry (eigenfunctions of the Laplace operator).
Adiabatic formulation of the ECMWF model 36
The horizontal spectral representation
( , , , ) ( , ) ( , )N N
m mn n
m N n m
X t X t Y
1
1
2
0
)(),,,(4
1),(
ddePtXtX imm
nmn
FFT (fast Fourier transform)usingNF 2N+1points (linear grid)(3N+1 if quadratic grid)
Legendre transformby Gaussian quadratureusing NL (2N+1)/2“Gaussian” latitudes (linear grid)((3N+1)/2 if quadratic grid)No “fast” algorithm available
Triangular truncation(isotropic)
Spherical harmonics
associated Legendre polynomials
Fourier functions
m
Triangular truncation:n
N
m = -N m = N
Adiabatic formulation of the ECMWF model 37
Grid-points in longitude are equidistantly spaced (Fourier) points 2N+1 for linear grid 3N+1 for quadratic grid
Grid-points in latitude are the zeros of the Legendre polynomial of order NG 0)(0 m
NGP Gaussian latitudes
NG (2N+1)/2 for the linear grid.
Horizontal discretisation (cont.)
Representation in grid-point space is on the reduced Gaussian grid:
Gaussian grid: grid of Guassian quadrature points (to facilitate accurate numerical computation of the integrals involved in the Fourier and Legendre transforms) - Gauss-Legendre quadrature in latitude:
NG (3N+1)/2 for the quadratic grid.
- Gauss-Fourier quadrature in longitude:
Adiabatic formulation of the ECMWF model 38
The Gaussian grid
Full gridReduced grid
• Associated Legendre functions are very small near the poles for large m
About 30% reduction in number of points
Adiabatic formulation of the ECMWF model 39
T799 T1279
50°N 50°N
0°
0°
Orography at T1279
10
50
100
150
200
250
300
350
400
450
500
550
600
650
684.1
50°N 50°N
0°
0°
Orography at T799
10
50
100
150
200
250
300
350
400
450
500
550
600
634.0
25 km grid-spacing( 843,490 grid-points)Current operational resolution
16 km grid-spacing(2,140,704 grid-points)Future operationalresolution (from end 2009)
Adiabatic formulation of the ECMWF model 40
Spectral transform method
Grid-point space -semi-Lagrangian advection -physical parametrizations
Fourier Space
Spectral space -horizontal gradients -semi-implicit calculations -horizontal diffusion
FFT
LT
Inverse FFT
Inverse LT
Fourier Space
FFT: Fast Fourier Transform, LT: Legendre Transform
Adiabatic formulation of the ECMWF model 41
Horizontal discretisation (cont.)Advantages of the spectral representation:
a.) Horizontal derivatives are computed analytically => pressure-gradient terms are very accurate => no need to stagger variables on the grid b.) Spherical harmonics are eigenfunctions of the the Laplace operator => Solving the Helmholtz equation (arising from the semi-implicit method) is straightforward. => Applying high-order diffusion is very easy.
Disadvantage: Computational cost of the Legendre transforms is high and grows faster with increasing horizontal resolution than the cost of the rest of the model.
mn
mn Y
a
nnY
22 )1(
Adiabatic formulation of the ECMWF model 42
Comparison of cost profilesat different horizontal resolutions
0
5
10
15
20
25
30
35
40
45
50
no
rmal
ized
% c
ost
T511
T799
T1279
T2047
Adiabatic formulation of the ECMWF model 43
Cost of Legendre transforms
T511T799
T1279
T2047
Adiabatic formulation of the ECMWF model 44
Profile for T2047on IBM p690+ (768 CPUs)
Legendre Transforms~17% of total cost of model
Physics ~36% of total cost
Adiabatic formulation of the ECMWF model 45
L91L600.01
0.02
0.03
0.050.07
0.1
0.2
0.3
0.5
0.7
1
2
3
5
7
10
20
30
50
70
100
200
300
500
700
1000
Pres
sure
(hPa
)
60 levels 91 levels
1
2
3
4
5
6
7
8
9
10
12
14
16
18
20
25
30
35
40455055606570
91
Leve
l num
ber
Vertical resolution of the operational ECMWF model: 91 hybrid η-levels resolving the atmosphere up to 0.01hPa (~80km)(upper mesosphere)
Vertical discretisation
Variables are discretized onterrain-following pressurebased hybrid η-levels.
Adiabatic formulation of the ECMWF model 46
Vertical discretisation (cont.)
Choices: - finite difference methods - finite element methods
Operational version of the ECMWF model uses a cubic finite-element (FE) scheme based on cubic B-splines.
No staggering of variables, i.e. all variables are held on the samevertical levels. (Good for semi-Lagrangian advection scheme.)
Inspection of the governing equations shows that there are only vertical integrals (no derivatives) to be computed (if advection is done with semi-Lagrangian scheme).
) ( knownte and is l coordianhe verticaition of t the defincomes fromd
dB
Adiabatic formulation of the ECMWF model 47
Prognostic equations of the ECMWF hydrostatic model
dp
TR
PDt
DX
PDt
Dq
dpd
dBp
td
dBp
Dt
D
KPpq
T
Dt
DT
KPpTRkfDt
D
vds
x
q
sss
v
vdhh
ln
)lnv)(ln()(ln
)1(1
lnvv
1
h
1
0
TT
vv
These equations are discretized and integrated in the ECMWF model.
Reminder: slide 32Reminder: slide 32
Adiabatic formulation of the ECMWF model 48
0
pp
Vp
t h
0
pp
Vp
t h
From the continuity equation
pVdp
V hh
0
)(
dpV
d
dBp
td
dBp
Dt
Dshss )ln)(ln()(ln
1
0
dp
Vp
pt h
ss )(
1)(ln
1
0
dp
Vt
pph )(
0
1 0 0 atandatwith boundary conditions:
we can derive (by vertical integration) the following equations:
Needed for the energy-conversionterm in the thermodynamic equation
Needed for the semi-Lagrangianadvection
Prognostic equations for surfacepressure
spBAp )()( B from def. of vert. coord.
Reminder: slide 31Reminder: slide 31
Adiabatic formulation of the ECMWF model 49
Vertical integration in finite elements
0
( ) ( )F f x dx
can be approximated as
Applying the Galerkin method with test functions tj =>
-1 AC Bc C A Bc
2 2
1 1 0
( ) ( )
K M
i i i ii K i M
C d c e x dx
Basis sets
2 2
1 1
1 1
1 2
0 0 0
( ) ( ) ( ) ( ) for
xK M
i j i i j ii K i M
C t x d x dx c t x e y dy dx N j N
Aji Bji
Adiabatic formulation of the ECMWF model 50
Vertical integration in finite elements
1 -c S f
F PC
1 -1 -F P A B S f J f
Including the transformation from grid-point (GP) representation tofinite-element representation (FE)
and the projection of the result from FE to GP representation
one obtains
Matrix J depends only on the choice of the basis functions and the level spacing. It does not change during the integration of the model, so it needs to be computedonly once during the initialisation phase of the model and stored.
Adiabatic formulation of the ECMWF model 51
Cubic B-splines for regular spacing of levels (Prenter (1975))
0
)(
)(3)(3)(3
)(3)(3)(3
)(
4
1)(
32
31
211
23
31
211
23
32
3
i
iii
iii
i
i
hhh
hhh
hB
otherwise
for
for
for
for
ii
ii
ii
ii
21
1
1
12
Adiabatic formulation of the ECMWF model 52
Cubic B-splines as basis elements
Basis elementsfor the represen-tation of thefunction tobe integrated(integrand)
f
Basis elementsfor the representationof the integral
F
Adiabatic formulation of the ECMWF model 53
Benefits from using the finite-element scheme
in the vertical• High order accuracy (8th order for cubic elements)
• Very accurate computation of the pressure-gradient term in conjunction with the spectral computation of horizontal derivatives
• More accurate vertical velocity for the semi-Lagrangian trajectory computation
– Improved ozone conservation
• Reduced vertical noise in the stratosphere
• No staggering of variables required in the vertical: good for semi-Lagrangian scheme because winds and advected variables are represented on the same vertical levels.
Adiabatic formulation of the ECMWF model 54
Discretisation in time
Discretize on -three time-levels (e.g. leapfrog scheme) - produce a computational mode (time-filtering needed) -two time-levels - more efficient than three-time-level schemes - (less stable)
Decisions to be taken:
How to treat the advection: - in Eulerian way - in semi-Lagrangian way
How to discretize the right-hand sides of the equations in time: - explicitly - implicitly - semi-implicitly
Adiabatic formulation of the ECMWF model 55
Operational version of the ECMWF model uses
- Two-time-level scheme
- Semi-implicit treatment of the right-hand sides
- Semi-Lagrangian advection
Discretization in time (cont.)
Adiabatic formulation of the ECMWF model 56
Time discretisation of the right-hand sides
Discretisation of the right-hand side (RHS) of the equations:
- RHS taken at the centre of the time interval: explicit (second order) discretisation. Stability is subject to CFL-like criterion
- RHS average of its value at initial time and at final time: implicit discretisation (generally stable) => leads to a difficult system to solve (iterative solvers)
- treat only some linearized terms of RHS implicitly (semi-implicit discretisation)
Adiabatic formulation of the ECMWF model 57
Semi-implicit time integration
xRHSDt
DX
000 )(5.0)(5.0)( LLLRHSLLLRHSDt
DXxx
Notations:X : advected variableRHS: right-hand side of the equationL: part of RHS treated implicitlySuperscripts: “0” indicates value for explicit discretiz,
“-” indicates value at start of time step “+” indicates value at end of time step
0)(5.0 LLLLtt For compact notation define:
“implicit correction term”
L=RHS => implicit schemeL= part of RHS => semi-implicit
LRHSDt
DXttx 0=>
Benefit: slowing-down of the waves too fast for the explicit CFL cond.Drawback: overhead of having to solve an elliptic boundary-value prob.
Adiabatic formulation of the ECMWF model 58
Semi-implicit time integration (cont.)
Choice of which terms in RHS to treat implicitly is guided by the knowledge of which waves cause instability because they are too fast (violate the CFL condition) and need to be slowed down with an implicit treatment.
In a hydrostatic model, fastest waves are horizontally propagating external gravity waves (long surface gravity waves), Lamb waves (acoustic wave not filtered out by the hydrostatic approximation)and long internal gravity waves. => implicit treatment of the adjustment terms.
L= linearization of part of RHS (i.e. terms supporting the fast modes) => good chance of obtaining a system of equations in the variables at “+” that can be solved almost analytically in a spectral model.
Adiabatic formulation of the ECMWF model 59
Semi-implicit time integration (cont.)
1
0
h
TT
vv
)v(1
)(ln
)1(1
lnvv
dp
pp
t
KPpq
T
Dt
DT
KPpTRkfDt
D
ss
v
vdhh
DRHSpt
DRHSDt
DT
pTRTRHSDt
D
ttps
ttT
srdtth
)(ln
lnγv
v
semi-implicitequations
semi-implicit corrections
Adiabatic formulation of the ECMWF model 60
Semi-implicit time integration (cont.)
DRHSpt
DRHSDt
DT
pTRTRHSDt
D
ttps
ttT
srdtth
)(ln
lnγv
v
DRHSpt
DRHSDt
DT
pTRTRHSDt
D
ttps
ttT
srdtth
)(ln
lnγv
v
1
0
0
1
1
γ
dd
dpX
pX
dd
dpX
p
TX
dd
dp
p
XRX
r
sr
r
r
r
r
r
d
semi-implicitequations
Where:
Reference state for linearization:
rT ref. temperature
srp ref. surf. pressure
=> lin. geopotential for X=T
=> lin. energy conv. term for X=D
Adiabatic formulation of the ECMWF model 61
Semi-implicit time integration (cont.)
DRHSpt
DRHSDt
DT
pTRTRHSDt
D
ttps
ttT
srdtth
)(ln
lnγv
v
DRHSpt
DRHSDt
DT
pTRTRHSDt
D
ttps
ttT
srdtth
)(ln
lnγv
v
semi-implicitequations
Reference state for linearization:
rT ref. temperaturesrp ref. surf. pressure
By eliminating in above system all but one of the unknowns (D+) => DD
~I 2
rdTR γ operator acting only on the vertical
I unity operator
Adiabatic formulation of the ECMWF model 62
Semi-implicit time integration (cont.)
DD~
I 2
rdTR γ
Vertically coupled set of Helmholtz equations. Coupling through
Uncouple by transforming to the eigenspace of this matrix gamma(i.e. diagonalize gamma). Unity matrix “I” stays diagonal. =>
DDi
~1 2
One equation for each LevNi 1
In spectral space (spherical harmonics space):
mn
mni DD
a
nn ~)1(1
2
mn
mn Y
a
nnY
22 )1(
because
Once D+ has been computed, it is easy to compute the other variables at “+”.
Adiabatic formulation of the ECMWF model 63
Semi-Lagrangian advection
Semi-Lagrangian (SL) schemes are more efficient & more stablethan Eulerian advection schemes.
Coupling SL advection with semi-implicit treatment of the fastmodes results in a very stable scheme where the timestep canbe chosen on the basis of accuracy rather than for stability.
Disadvantage: Lack of conservation of mass and tracer concentrations. (More difficult to enforce conservation than in Eulerian schemes)
Adiabatic formulation of the ECMWF model 64
Semi-Lagrangian advection (cont)
x x x x
x x x x
x x x x
A
D *
xM( ) ( )
( )2
A DMt t t t
R tt
Centred second order accurate scheme
Three time-level scheme:
Ingredients of semi-Lagrangian advection are: 1.) Computation of the departure point (tajectory computation) 2.) Interpolation of the advected fields at the departure location
),( tRDt
D All equations are of this (Lagrangian) form:
(See slide 32)
Adiabatic formulation of the ECMWF model 65
Prognostic equations of the ECMWF hydrostatic model
dp
TR
PDt
DX
PDt
Dq
dpd
dBp
td
dBp
Dt
D
KPpq
T
Dt
DT
KPpTRkfDt
D
vds
x
q
sss
v
vdhh
ln
)lnv)(ln()(ln
)1(1
lnvv
1
h
1
0
TT
vv
These equations are discretized and integrated in the ECMWF model.
Reminder: slide 32Reminder: slide 32
Adiabatic formulation of the ECMWF model 66
Semi-Lagrangian advection (cont)
Unstable! => noisy forecasts
Two-time-level second order accurate schemes :
Forecast of temperatureat 200 hPa(from 1997/01/04)
( ) ( )( )
2
A DMt t t t
R tt
3 1( ) ( ) ( )
2 2 2
tR t R t R t twith
Extrapolation in time to middle of time interval
Adiabatic formulation of the ECMWF model 67
Stable extrapolating two-time-level semi-Lagrangian (SETTLS):
2 2
2
( )( ) ( )
2
DA D
t AV
d t dt t t t
dt dt
Forecast 200 hPa Tfrom 1997/01/04using SETTLS
( )
DD
t
dR t
dt
2
2
( ) ( )
A D
AVAV
R t R t td dR
dt dt t
With and
Taylor expansion to second order
( ) ( ) ( ( ) {2 ( ) ( )} )2
A D A Dt
t t t R t R t R t t
Adiabatic formulation of the ECMWF model 68
Interpolation in the semi-Lagrangian scheme
4
1
( ) ( )
i ii
x C x
4
4
)(
)(
)(
ikki
ikk
i
xx
xx
xCwith the weights
ECMWF model uses quasi-monotone quasi-cubic Lagrange interpolation
xx x xx
x x x x
x x x x
x xxx
x
x
x
x
y
x
Number of 1D cubic interpolationsin two dimensions is 5, in three dimensions 21!
To save on computations:cubic interpolation only for nearestneighbour rows, linear interpolation for rest => “quasi-cubic interpolation”=> 7 cubic + 10 linear in 3 dimensions.
Cubic Lagrange interpolation:
Adiabatic formulation of the ECMWF model 69
Interpolation in the semi-Lagrangian scheme (cont)
x: grid points
x: interpolation pointquasi-monotone procedure:
Quasi-monotone interpolation is used in the horizontal for all variables and also in the vertical for humidity and all “tracers” (e.g. ozone, aerosols).
xx
x
x
x
x
interpolated cubically
maxmin
Quasi-monotone interpolation:
Has a detrimental effect on conservation, but prevents unphysicalnegative concentrations.
Adiabatic formulation of the ECMWF model 70
shdd
s VTR
RHSDt
lD
TRl
1
][where *
=> Reduced mass loss/gain during a forecast.
Modified continuity & thermodynamic equations
][)()(ln * RHSllDt
Dp
Dt
Ds Continuity equation
Accuracy of cubic interpolation is much reduced when the fieldto be interpolated is rough (e.g. surface pressure over orography)
Idea by Ritchie & Tanguay (1996): Subtract a time-independentterm from the surface pressure which “contains” a large part of the orographic influence on surface pressure, advect the rest (smoother term)and treat the advection of the “rough term” with the right-hand side of the continuity equation [RHS].
Adiabatic formulation of the ECMWF model 71
Modified continuity & thermodynamic equations(cont)
b
bhTb T
TVRHSDt
TTD
)(][)(
TRp
T
p
ppT
d
s
refssb
with
Reduces noise levels over orography in all fields, but in particular in vertical velocity.
Thermodynamic equation
Similar idea for thermodynamic equation: (Hortal &Temperton (2001))
Approximation to the change of T with height in the standard atmosphere.
Adiabatic formulation of the ECMWF model 72
Trajectory calculation
M i
j
Ai
j
Di
j
Tangent plane projection
Semi-Lagrangian advection on the sphere
X
Y
Z
A
Vx
D
Momentum eq. is discretized in vector form (because a vector is continuous acrossthe poles, components are not!)
Trajectories are arcs of great circles if constant (angular) velocity is assumedfor the duration of a time step.
Interpolations at departure point are donefor components u & v of the velocity vec-tor relative to the system of reference localat D. Interpolated values are to be used at A, so the change of reference system from D to A needs to be taken into account.
Adiabatic formulation of the ECMWF model 73
Treatment of the Coriolis term
• Implicit treatment :0
00.5( )
h h
h h
V Vfk V V
t
• Advective treatment:
2
h
dRfk V
dt( 2
h
h h
dV dfk V V R)
dt dt
In three-time-level semi-Lagrangian:
In two-time-level semi-Lagrangian:
• treated explicitly with the rest of the RHS
Extrapolation in time to the middle of the trajectory leads to instability (Temperton (1997))Two stable options:
Helmholtz eqs partially coupled for individual spectral components => tri-diagonal system to be solved.
(Vector R hereis the position vector.)
Adiabatic formulation of the ECMWF model 74
Summary of the adiabatic formulation of the operational ECMWF atmospheric model
Hydrostatic shallow-atmosphere equations with pressure-based hybrid vertical coordinate
• Two-time-level semi-Lagrangian advection
– SETTLS (Stable Extrapolation Two-Time-Level Scheme)
– Quasi-monotone quasi-cubic Lagrange interpol. at departure point
– Linear interpolation for trajectory computations and RHS terms
– Modified continuity & thermodynamic equations to advect smoother fields (net of the orographic roughness)
• Semi-implicit treatment of linearized adjustment terms & Coriolis terms
• Cubic finite elements for the vertical integrals
• Spectral horizontal Helmholtz solver (and derivative computations)
• Uses the linear reduced Gaussian grid
Adiabatic formulation of the ECMWF model 75
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