Parametrization of orographic processes in numerical weather processing

Andrew Orrandrew.orr@ecmwf.int

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

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10°E 15°E

15°EERA40 mean orography / land sea mask

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15°ET511 mean orography / land sea mask

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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

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gravity wave + blocking stress (N/m2) 2004070412 T + 24 h

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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

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80°Woper LSP+CP (mm/day) 20030706 12UTC + 30-36 h

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80°WT+36h IFS simulated GOES 8 First Infrared Band 2003070800

2000

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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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10.0m/smean operational July 2003 cross-section 12UTC + 30 h

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10.0m/smean operational July 2003 cross-section 12UTC + 36 h

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10.0m/smean operational July 2003 cross-section 12UTC + 42 h

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10.0m/smean operational July 2003 cross-section 12UTC + 48 h

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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.