parametrization of orographic processes in numerical weather processing andrew orr...
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Parametrization of orographic processes in numerical weather processing
Andrew [email protected]
Lecture 1: Effects of orographyLecture 2: Sub-grid scale orographic parameterization
History of orography parameterization
1. Pioneering of studies on linear 2d gravity waves (e.g. Queney, 1948)2. Gravity wave drag recognised as important sink of atmospheric momentum
(e.g. Eliassen and Palm, 1961)3. Observational and modelling studies of non-linear waves (e.g. Lilly, 1978)4. Modelling of 3d nonlinear waves 5. Development of envelope orography (not satisfactory technique for
representation of large-scale flow blocking)6. Alleviation of systematic westerly bias in numerical weather prediction
models through gravity wave drag (GWD) parameterization (Palmer et al. 1986)
7. High-resolution numerical modelling 8. Alleviation of inadequate representation of low-level drag through ‘blocked
flow’ drag parameterization (Lott and Miller 1997). This is the ECMWF orography parameterization scheme.
Alleviation of systematic westerly bias
Without GWD scheme
Analysis
With GWD scheme
Mean January sea level pressure (mb) for years 1984 to 1986 (from Palmer et al. 1986)
Icelandic/Aleutian lows are too deep
Flow too zonal
Azores anticyclone too far east
Siberian high too weak and too far south
Alleviation of systematic westerly bias
Analysis
Zonal mean cross-sections of zonal wind (ms-1) and temperature (K, dashed lines) for January 1984 and (a) without GWD scheme and (b) analysis (from Palmer et al 1986)
flow is too strong
temperature too low
Without GWD scheme
less impact in southern-hemisphere
Alleviation of systematic westerly bias
Zonal cross-sections of the differences in (a) zonal wind (ms-1) and (b) temperature (K)
slowing of winds in stratosphere and upper troposphere
0
fut
v
poleward induced meridional flow
descent over pole leads to warming
Parameterisation of gravity wave drag decelerated the predominately westerly flow
High-resolution numerical modelling
From Clark and Miller 1991
Sensitivity of pressure drag and momentum fluxes due to the Alps to horizontal resolution
No GWD scheme
large underestimation of drag
Specification of sub-grid orography
xh: topographic height above sea level
(from global 1km data set)
*
***
h: mean topographic height at each gridpoint-
From Baines and Palmer (1990)
At each gridpoint sub-grid orography represented by:
μ: standard deviation of h (amplitude of sub-grid orography) γ: anisotropy (measure of how elongated sub-grid orography is)θ: angle between x-axis and principal axis (i.e. direction of maximum slope) ψ: angle between low-level wind and principal axis of the topographyσ: mean slope (along principal axis)
2μ approximates the physical envelope of the peaks
Note source grid is filtered to remove small-scale orographic structures and scales resolved by model – otherwise parameterization may simulate unrelated effects
Specification of sub-grid orography
jiijxhxhH
Calculate topographic gradient correlation tensor
Direction of maximum mean-square gradient at an angle θ to the x-axis
Diagonalise
jiij x
h
x
hH
y
h
y
hH
y
h
x
hH
x
h
x
hH
221211 ,,
y
h
x
hM
y
h
x
hL
y
h
x
hK ,
2
1,
2
12222
)/arctan(5.0 LM
Specification of sub-grid orography
2
2
'
x
h
Change coordinates (orientated along principal axis)
Anisotropy defined as(1:circular; 0: ridge)
Slope (i.e. mean-square gradient along the principal axis)
If the low-level wind is directed at an angle φ to the x-axis, then the angle ψ is given by:
(ψ=0 flow normal to obstacle; ψ=π/2 flow parallel to obstacle)
sincos
sincos
xyy
yxx
2
2
2
'
'
xh
yh
x
xy
Resolution sensitivity of sub-grid fields
45°N 45°N
5°E
5°E 10°E
10°E 15°E
15°EERA40 mean orography / land sea mask
0
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45°N 45°N
5°E
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10°E 15°E
15°ET511 mean orography / land sea mask
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5°E
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10°E 15°E
15°ET799 mean orography / land sea mask
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5°E
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10°E 15°E
15°EERA40 slope
0
0.02
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0.08
0.1
0.12
0.14
45°N 45°N
5°E
5°E 10°E
10°E 15°E
15°ET511 slope
0
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0.04
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0.08
0.1
0.12
0.14
45°N 45°N
5°E
5°E 10°E
10°E 15°E
15°ET799 slope
0
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0.1
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0.14
45°N 45°N
5°E
5°E 10°E
10°E 15°E
15°EERA40 standard deviation
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45°N 45°N
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10°E 15°E
15°ET511 standard deviation
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45°N 45°N
5°E
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10°E 15°E
15°ET799 standard deviation
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ERA40~120kmT511~40kmT799~25km
Sub-grid scale orographic parameterisation
1. Compute surface pressure drag exerted on subgrid-scale orography2. Compute vertical distribution of wave stress accompanying the surface value
Gravity wave drag
Blocked flow drag1. Compute depth of blocked layer2. Compute drag at each model level for z < zblk
Scheme used for:ECMWF (Lott and Miller 1997),UK Met UM,HIRLAM, etc
zblk
hz/zblk
heff
h
blkeff zhh
Evaluation of blocking height
Characterise incident (low-level) flow passing over the mountain top by ρH, UH, NH (averaged between μ and 2μ)
Define non-dimensional mountain height Hn= hNH/UH
In ECMWF model assume h=3μ
ncrit
Zblk
HdzU
N
3
Blocking height zblk satisfies:
Where Hncrit≈1 tunes the depth of the blocked layer(uses wind speed Up calculated by resolving the wind U in the direction of UH)
Evaluation of blocked-flow dragAssume sub-grid scale orography has elliptical shape
For z<zblk flow streamlines divide around mountain. Drag exerted by the obstacle on the flow at these levels can be written as
l(z): horizontal width of the obstacle as seen by the flow at an upstream height z (assumes each layer below zblk is raised by a factor H/zblk, i.e. reduction of obstacle width)r: aspect ratio of the obstacle as seen by the incident flowCd (~1): form drag coefficient (proportional to ψ)B,C: constants Summing over number of consecutive ridges in a grid point gives the drag
This equation is applied quasi-implicitly level by level below zblk
See Lott and Miller 1997
2222 //1),(
byax
hyxh
1/ ba
2)()( 0
UUzlCzD dblk
2
||sincos
20,
12max)( 22
2/1UU
CBz
zZ
rCzD blkdblk
Evaluation of gravity wave surface stress
)cossin)(,sincos(4
222HHHHeffHHHs CBCBGhNU
Consider again an elliptical mountain
Gravity wave stress can be written as (Phillips 1984)
G (~1): constant (tunes amplitude of waves)
Typically L2/4ab ellipsoidal hills inside a grid point. Summing all forces we find the stress per unit area (using a=μ/σ)
)cossin)(,sincos( 222HHHHHHHs CBCBbGhNU
Evaluation of stress profile
Gravity wave breaking only active above zblk (i.e. λ=λs for 0<z< zblk)
Above zblk stress constant until waves break (i.e. convective overturning)
This occurs when the local Richardson number Rimin < Ricrit(=0.25), i.e. saturation hypothesis (Lindzen 1981)
zU
UhN
RiRi
NRi
/
/
1
12
22/1
2
min
:amplitude of wave
:mean Richardson numberRi
h
Values of the wave stress are defined progressively from the top of the blocked layer upwards
Evaluation of stress profile
2hNUk
2
22/1min
1
1
RiRiRi
Set λ=λs and Rimin=0.25 at model level representing top of blocked layer
Assume stress at any level
Uk-1,Tk-1
Uk-3,Tk-3
Uk-2,Tk-2
k-2
k-1
z=0; λ= λs
zk=zblk; λk= λs
Hei
ght
Calculate Ri at next level
Set λk-1=λk to estimate δhusing
Calculate Rimin
If Rimin>=Ricrit
estimate hset k-1= k
go to next level
If Rimin<Ricrit
set Rimin=Ricrit
estimate h=hsat
estimate = sat
go to next level
2hNUk
Repeat
Gravity wave stress profile
U Deceleration
Wave breaking
Wave breaking10km
Weak winds at low-level can result in low-level wave breaking.
Corresponding drag distributed linearly over a depth Δz (above the blocked flow)
zz
zk
kblk
blk
dzU
N
2
Note, trapped lee waves not represented in Lott and Miller scheme. However, accounted for in UK Met Office UM model (see Gregory et al. 1998)
Drag contributions
T213 forecasts: ECMWF model with mean orography and the subgrid scale orographic drag scheme. Explicit model pressure drag and parameterized mountain drag during PYREX.
From Lott and Miller 1997
Strong interaction/compensation between drag contributions
2.0m/s
60°S60°S
30°S 30°S
0°0°
30°N 30°N
60°N60°N
150°W
150°W 120°W
120°W 90°W
90°W 60°W
60°W 30°W
30°W 0°
0° 30°E
30°E 60°E
60°E 90°E
90°E 120°E
120°E 150°E
150°E
gravity wave + blocking stress (N/m2) 2004070412 T + 24 h
2.0m/s
60°S60°S
30°S 30°S
0°0°
30°N 30°N
60°N60°N
150°W
150°W 120°W
120°W 90°W
90°W 60°W
60°W 30°W
30°W 0°
0° 30°E
30°E 60°E
60°E 90°E
90°E 120°E
120°E 150°E
150°E
gravity wave + blocking stress (N/m2) 2005122512 T + 24 h
From ECMWF T511 operational model
Parameterized surface stresses
Sensitivity of resolved orographic drag to model resolution
From Smith et al. 2006
drag converging
parameterization still required at high-resolution
Weak flow: most drag produced by flow splitting
Strong flow: short-scale trapped lee waves produce significant fraction of drag (Georgelin and Lott, 2001
Orographic form drag due to scales <5000m
Effective roughness concept (Taylor et al. 1989)Enhancement of roughness length above its vegetative value in areas of orographyDisadvantages: Can reach 100’s of metersRoughness lengths for heat and moisture have to be reduced
New scheme: Directly parameterises TOFD and distributes it vertically (Beljaars et al. 2004)Vegetative roughness treated independentlyRequires filtering of orography field to have clear separation of horizontal scalesSpectrum of orography represented by piecewise empirical power lawIntegrates over the spectral orography to represent all relevant scalesWind forcing level of the drag scheme depends on horizontal scale of orography
6.0,005.0,1,12,1.0,00102.0,/2
,00035.0,003.0,000628.0,,)(,8.2
,9.1,for ,)(,for ,)(),/2,/2min(
with
,)()()(2/
10
111
10112
1212
112010101
/0
2
0
211
21
0
corrmdmHm
fltnnn
fltHflt
nnw
k
k
lz
wcorrmd
CCcmIzck
mkmkmkkaakIan
nkkkkakFkkkkakFkkl
dkekFl
kzUzUCC
zw
2
2
2
2
2
2
2
2
2
2
2
10
10
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10
1010
20
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2000
30°N 30°N
40°N40°N
50°N 50°N
120°W
120°W 110°W
110°W 100°W
100°W 90°W
90°W 80°W
80°Woper LSP+CP (mm/day) 20030706 12UTC + 30-36 h
-1
0
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30°N 30°N
40°N40°N
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120°W
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80°WT+36h IFS simulated GOES 8 First Infrared Band 2003070800
2000
30°N 30°N
40°N40°N
50°N 50°N
120°W
120°W 110°W
110°W 100°W
100°W 90°W
90°W 80°W
80°WGOES 8 First Infrared Band 2003070800
Enhancement of convection by orography: Simulation of mid-afternoon precipitation maximum
July 2003 mean operational T511 cross-sections of wind (m/s) and specific humidity (g/kg)
120 OW 116 OW 112OW 108OW 104OW 100OW 96OW 92OW
42.5N
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11
10.0m/smean operational July 2003 cross-section 12UTC + 30 h
-10123456789101112131415
120 OW 116OW 112OW 108OW 104OW 100OW 96OW 92OW
42.5N
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33
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10.0m/smean operational July 2003 cross-section 12UTC + 36 h
-10123456789101112131415
120 OW 116OW 112OW 108OW 104OW 100OW 96OW 92OW
42.5N
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10.0m/smean operational July 2003 cross-section 12UTC + 42 h
-10123456789101112131415
120 OW 116OW 112OW 108OW 104OW 100OW 96OW 92OW
42.5N
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10.0m/smean operational July 2003 cross-section 12UTC + 48 h
-10123456789101112131415
morning afternoon
evening night
References•Baines, P. G., and T. N. Palmer, 1990: Rationale for a new physically based parameterization of sub-grid scale orographic effects. Tech Memo. 169. European Centre for Medium-Range Weather Forecasts. •Beljaars, A. C. M., A. R. Brown, N. Wood, 2004: A new parameterization of turbulent orographic form drag. Quart. J. R. Met. Soc., 130, 1327-1347. •Clark, T. L., and M. J. Miller, 1991: Pressure drag and momentum fluxes due to the Alps. II: Representation in large scale models. Quart. J. R. Met. Soc., 117, 527-552.•Eliassen, A. and E., Palm, 1961: On the transfer of energy in stationary mountain waves, Geofys. Publ., 22, 1-23. •Georgelin, M. and F. Lott, 2001: On the transfer of momentum by trapped lee-waves. Case of the IOP3 of PYREX. J. Atmos. Sci., 58, 3563-3580.•Gregory, D., G. J. Shutts, and J. R. Mitchell, 1998: A new gravity-wave-drag scheme incorporating anisotropic orography and low-level wave breaking: Impact upon the climate of the UK Meteorological Office Unified Model. Quart. J. Roy. Met. Soc., 125, 463-493.•Lilly. D. K., 1978: A severe downslope windstorm and aircraft turbulence event induced by a mountain wave, J. Atmos. Sci., 35, 59-77. •Lindzen, R. S., 1981: Turbulence and stress due to gravity wave and tidal breakdown. J. Geophys. Res., 86, 9707-9714.•Lott, F. and M. J. Miller, 1997: A new subgrid-scale drag parameterization: Its formulation and testing, Quart. J. R. Met. Soc., 123, 101-127.•Queney, P., 1948: The problem of airflow over mountains. A summary of theoretical studies, Bull. Amer. Meteor. Soc., 29, 16-26.•Palmer, T. N., G. J. Shutts, and R. Swinbank, 1986: Alleviation of a systematic westerly bias in general circulation and numerical weather prediction models through an orographic gravity wave drag parameterization, Quart. J. R. Met. Soc., 112, 1001-1039.•Phillips, D. S., 1984: Analytical surface pressure and drag for linear hydrostatic flow over three-dimensional elliptical mountains. J. Atmos. Sci., 41, 1073-1084.•Smith, S., J. Doyle., A. Brown, and S. Webster, 2006: Sensitivity of resolved mountain drag to model resolution for MAP case studies. Submitted to Quart. J. R. Met. Soc..•Taylor, P. A., R. I. Sykes, and P. J. Mason, 1989: On the parameterization of drag over small scale topography in neutrally-stratified boundary-layer flow. Boundary layer Meteorol., 48, 408-422.