Transcript (Lott and Miller 1997). This is the ECMWF orography

Parametrization of orographic processes in
numerical weather processing
Andrew Orr
[email protected]
Lecture 1: Effects of orography
Lecture 2: Sub-grid scale orographic parameterization
History of orography parameterization
1.
2.
3.
4.
5.
6.
7.
8.
Pioneering of studies on linear 2d gravity waves (e.g. Queney, 1948)
Gravity wave drag recognised as important sink of atmospheric momentum
(e.g. Eliassen and Palm, 1961)
Observational and modelling studies of non-linear waves (e.g. Lilly, 1978)
Modelling of 3d nonlinear waves
Development of envelope orography (not satisfactory technique for
representation of large-scale flow blocking)
Alleviation of systematic westerly bias in numerical weather prediction models
through gravity wave drag (GWD) parameterization (Palmer et al. 1986)
High-resolution numerical modelling
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
Mean January sea level pressure (mb) for years 1984 to 1986 (from Palmer et al. 1986)
Icelandic/Aleutian lows
are too deep
Siberian high too weak
and too far south
Flow too zonal
Azores anticyclone
too far east
Without GWD scheme
Analysis
With GWD scheme
Alleviation of systematic westerly bias
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)
Without GWD scheme
temperature too low
flow is too strong
less impact in southern-hemisphere
Analysis
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
v
  fu  0
t
poleward induced meridional flow
descent over pole leads to warming
Parameterisation of gravity wave
drag decelerated the predominately
westerly flow
High-resolution numerical modelling
No GWD scheme
large underestimation of drag
Sensitivity of pressure drag and
momentum fluxes due to the Alps to
horizontal resolution
From Clark and Miller 1991
Specification of sub-grid orography
From Baines and Palmer (1990)
h: mean topographic height
- at each gridpoint
*
x
h: topographic height above sea level
(from global 1km data set)
*
*
*
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 topography
σ: 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
h h
Calculate topographic gradient correlation tensor H ij 
xi x j
h h
h h
h h
H11 
, H12 
, H 22 
x x
x y
y y
Diagonalise
2
2
2
2


 h h 
1  h   h  
1  h   h  
K       , L       , M  

2  x   y  
2  x   y  

x

y






Direction of maximum mean-square gradient at an angle θ to the x-axis
  0.5 arctan( M / L)
Specification of sub-grid orography
Change coordinates (orientated along principal axis)
x  x cos   y sin 
y  y cos   x sin 
y
x
2
 h 
 
Anisotropy defined as
 y ' 
2


(1:circular; 0: ridge)
2

h
 
 
 x' 
2
Slope (i.e. meansquare gradient along  2   h 
the principal axis)
 x' 
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)
x
Resolution sensitivity of sub-grid fields
5°E
ERA40 mean orography
/ land sea mask15°E
10°E
5°E
deviation
ERA40 standard
10°E
5°E
15°E
ERA40
slope
10°E
15°E
0.14
1000
900
2500
0.12
2250
800
2000
700
1750
600
1500
0.1
0.08
500
1250
0.06
400
1000
45°N
45°N
300
45°N 45°N
45°N
45°N0.04
750
200
500
0.02
100
250
5°E
5°E
15°E
5°E
T511 mean orography
/ land sea mask 15°E
10°E
5°E
10°E
0
0
0
10°E
deviation
T511 standard
10°E
5°E
15°E
15°E
5°E
10°E
T511
slope
10°E
15°E
15°E
0.14
1000
2500
900
2250
800
2000
700
0.12
1750
0.1
600
0.08
1500
500
1250
0.06
400
1000
45°N
300
45°N
45°N 45°N
45°N
750
45°N0.04
200
500
0.02
5°E
5°E
250
100
0
0
15°E
5°E
T799 mean orography
/ land sea mask 15°E
10°E
5°E
10°E
10°E
deviation
T799 standard
10°E
15°E
0
5°E
15°E
5°E
10°E
T799
slope
10°E
15°E
15°E
1000
2500
900
2250
800
2000
700
0.14
0.12
1750
0.1
600
0.08
1500
500
1250
0.06
400
1000
300
45°N
45°N
45°N 45°N
45°N
750
200
500
5°E
10°E
15°E
ERA40~120km
T511~40km
T799~25km
45°N0.04
250
100
0
0
5°E
10°E
15°E
0.02
0
5°E
10°E
15°E
Sub-grid scale orographic parameterisation
Gravity wave drag
1. Compute surface pressure drag exerted on subgrid-scale orography
2. Compute vertical distribution of wave stress accompanying the surface value
heff  h  zblk
hef
f
zblk
h
hz/zblk
Blocked flow drag
1. Compute depth of blocked layer
2. Compute drag at each model level for z < zblk
Scheme used for:
ECMWF
(Lott and Miller 1997),
UK Met UM,
HIRLAM, etc
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
3
Blocking height zblk satisfies:
N
Zblk U dz  H ncrit
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 drag
Assume sub-grid scale orography has elliptical shape
h( x, y ) 
h
1  x2 / a 2  y 2 / b2
See Lott and Miller 1997
  a /b 1
For z<zblk flow streamlines divide around mountain. Drag exerted by the
obstacle on the flow at these levels can be written as
Dblk ( z )    0Cd l ( z )
UU
2
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 flow
Cd (~1): form drag coefficient (proportional to ψ)
B,C: constants
Summing over number of consecutive ridges in a grid point gives the drag
1/ 2
1    Z blk  z 



Dblk ( z )   Cd max  2  ,0 
r  2  z   



U |U |
B cos   C sin 
2
2
This equation is applied quasi-implicitly level by level below zblk
2
Evaluation of gravity wave surface stress
Consider again an elliptical mountain
Gravity wave stress can be written as (Phillips 1984)
 s   HU H N H h2bG( B cos2  H  C sin 2  H , ( B  C) sin  H cos H )
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)
 s   HU H N H heff
2

G( B cos 2  H  C sin 2  H , ( B  C ) sin  H cos H )
4
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)

1

Ri min 
 Ri 
1/ 2

 1  Ri  2
  N h / U
N2



2


h :amplitude of wave
Ri
:mean Richardson number
  U / z
Values of the wave stress are defined progressively from the top of the
blocked layer upwards
Evaluation of stress profile
Set λ=λs and Rimin=0.25 at model level representing top of blocked layer
Assume stress at any level
  kNUh 2
Calculate Ri at next level
Uk-3,Tk-3
Set λk-1=λk to estimate δh
using   kNUh 2
k-2
Repeat
Uk-2,Tk-2
k-1
Calculate Rimin
Ri min
set Rimin=Ricrit
estimate
h=hsat
estimate =
sat
go to next
level

1

 Ri 
1/ 2
 1  Ri  2



2


If Rimin>=Ricrit
estimate h
set k-1= k
go to next
level
Uk-1,Tk-1
zk=zblk; λk= λs
Height
If Rimin<Ricrit
z=0; λ= λs
Gravity wave stress profile

U
Deceleration
Wave breaking
10km
Wave breaking
Weak winds at low-level can result in low-level wave breaking.
Corresponding drag distributed linearly over a depth Δz (above the blocked flow)

zblk  z
zblk
Nk

dz 
Uk
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
From Lott and Miller 1997
T213 forecasts: ECMWF model with mean orography and the subgrid scale orographic drag
scheme. Explicit model pressure drag and parameterized mountain drag during PYREX.
Strong interaction/compensation between drag contributions
Parameterized surface stresses
gravity wave + blocking stress (N/m2) 2005122512 T + 24 h
150°W
120°W
90°W
60°W
30°W
0°
30°E
60°E
90°E
120°E
2.0m/s
150°E
60°N
60°N
30°N
30°N
0°
From ECMWF T511
operational model
0°
30°S
30°S
60°S
60°S
150°W
120°W
90°W
60°W
30°W
0°
30°E
60°E
90°E
120°E
150°E
gravity wave + blocking stress (N/m2) 2004070412 T + 24 h
150°W
120°W
90°W
60°W
30°W
0°
30°E
60°E
90°E
120°E
2.0m/s
150°E
60°N
60°N
30°N
30°N
0°
0°
30°S
30°S
60°S
60°S
150°W
120°W
90°W
60°W
30°W
0°
30°E
60°E
90°E
120°E
150°E
Sensitivity of resolved orographic drag to model
resolution
parameterization still required at high-resolution
drag converging
Weak flow: most
drag produced by
flow splitting
From Smith et al. 2006
Strong flow: shortscale 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 orography
Disadvantages: Can reach 100’s of meters
Roughness 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 independently
Requires filtering of orography field to have clear separation of horizontal scales
Spectrum of orography represented by piecewise empirical power law
Integrates over the spectral orography to represent all relevant scales
Wind forcing level of the drag scheme depends on horizontal scale of orography
2
k k

 0 /   2Cmd Ccorr U ( z ) U ( z ) 
F0 (k )e  z / lw dk ,
k0 l
z
w
with
lw  min( 2 / k ,2 / k1 ), F0 (k )  a1k n1 , for k0  k  k1 , F0 (k )  a2 k n2 , for k1  k  k , n1  1.9,
n2  2.8, a1   2flt ( I H k nflt1 ) 1 , a2  a1k1n1  n2 , k0  0.000628m 1 , k1  0.003m 1 , k flt  0.00035m 1 ,
k  2cm / z0 , I H  0.00102m 1 , cm  0.1,   12,   1, Cmd  0.005, Ccorr  0.6
Enhancement of convection by orography:
Simulation of mid-afternoon precipitation maximum
GOES 8 First Infrared Band 2003070800
120°W
110°W
100°W
90°W
80°W
50°N
2000
50°N
40°N
40°N
30°N
30°N
120°W
110°W
100°W
90°W
80°W
T+36h IFS simulated GOES 8 First Infrared Band 2003070800
120°W
110°W
100°W
90°W
80°W
50°N
oper LSP+CP (mm/day) 20030706 12UTC + 30-36 h
50°N
120°W
110°W
100°W
90°W
2
80°W
50
1020
10
50°N
50°N
50
2
25
2000
20
40°N
2
10
20
50
40°N
2000
20
2
210
2
40°N
15
20
40°N
2
10
5
2
10
2
30°N
2
2
0
2
30°N
10
30°N
120°W
110°W
100°W
90°W
80°W
30°N
10
120°W
110°W
100°W
-1
90°W
80°W
July 2003 mean operational T511 cross-sections of wind
(m/s) and specific humidity (g/kg)
mean operational July 2003 cross-section 12UTC + 30 h
100
200
300
400
1
1
1
1
500
3
600
3
3
5
5
700
7
7
5
9
800
9
11
900
morning
1000
O
O
120 W
116 W
O
O
112 W
108 W
O
104 W
O
100 W
O
96 W
mean operational July 2003 cross-section 12UTC + 36 h
10.0m/s
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1
100
200
300
1
400
1
3
3
600
3
5
800
9
11
900
afternoon
1000
O
O
116 W
11
O
112 W
108 W
200
300
1
1
1
500
3
600
3
5
700
5
7
7
9
800
9
11
900
evening
1000
O
120 W
O
116 W
O
112 W
11
O
108 W
O
104 W
42.5N
O
104 W
O
100 W
O
96 W
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1
O
92 W
42.5N
100
3
7
9
O
mean operational July 2003 cross-section 12UTC + 42 h
1
5
7
700
42.5N
400
1
500
120 W
O
92 W
1
10.0m/s
O
100 W
O
96 W
O
92 W
mean operational July 2003 cross-section 12UTC + 48 h
10.0m/s
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1
100
200
300
400
1
1
1
1
500
3
3
3
600
3
5
700
5
7
7
800
5
900
9
9
night
1000
O
120 W
O
116 W
O
112 W
11
O
108 W
O
104 W
42.5N
O
100 W
O
96 W
O
92 W
10.0m/s
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1
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, 97079714.
•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.