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Gravity wave drag Parameterization of orographic related momentum

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Gravity wave drag

Parameterization of orographic related momentum

fluxes in a numerical weather processing model

Andrew Orr

Lecture 1: Atmospheric processes associated with orography

Lecture 2: Parameterization of subgrid-scale orography

(linearise around

mean state)

Drag effects acceleration through tendencies of the horizontal wind

Dyn: dynamics (resolved), e.g. advection, pressure gradient and Coriolis force

Param: physical parameterizations of sub-grid scale processes:

e.g. radiation, convection, boundary layer, orography

(Note a prime denotes sub-grid scale deviation and an overline denotes gridbox average)

Parameterized tendency related to the vertical divergence of sub-grid scale vertical momentum flux (N/m2) (also called stress)

Tendency’s due to sub-grid scale orography (SSO)

Equation for resolved part of the horizontal momentum

h: mean (resolved) topographic height at each gridpoint

*

*

*

*

x

From Baines and Palmer (1990)

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)

2μ approximates the physical envelope of the peaks

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)

Note source grid is filtered to remove small-scale orographic structures and scales resolved by model – otherwise parameterization may simulate unrelated effects

Additional filtering so that only forced at scales the model can represent well

Diagonalise

Direction of maximum mean-square gradient at an angle θ to the x-axis

Calculate topographic gradient correlation tensor

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:

Change coordinates (orientated along principal axis)

(ψ=0 flow normal to obstacle; ψ=π/2 flow parallel to obstacle)

Illustration of anisotropy and principle axis of sub-grid scale orography for a number of general-circulation model grid cells near the Tibetan plateau. Contours show the unresolved orography

From Scinocca and McFarlane 2000

mean orography

/ land sea mask

slope (σ)

standard deviation (μ)

Horizontal resolution: ERA40~120km; T511~40km; T799~25km

mean orography (UK Met Office Unified Model)

From Smith et al. (2006)

heff

zblk

h

hz/zblk

Contributions from (a) gravity wave drag and (b) low-level blocking drag

(a) Gravity wave drag

1. Compute gravity wave surface stress exerted on sub-grid scale orography

2. Compute vertical distribution of wave stress accompanying the surface value

(b) Blocked flow drag

See Lott and Miller (1997)

1. Compute depth of blocked layer

2. Compute drag at each model level for z < zblk

Compute gravity wave surface stress

- Determine where and how strongly waves are excited
- More, generally where is the amplitude of the displacement of the isentropic surface

Effects of anisotropy on gravity wave surface stress

Assume sub-grid scale orography has elliptical shape

Gravity wave stress can be written as (Phillips 1984)

G (~1): constant (tunes amplitude of waves)

Increasing this increases gravity wave surface stress

Typically L2/4ab ellipsoidal hills inside a grid point. Summing all forces we find the stress per unit area (using a=μ/σ)

See Lott and Miller 1997

Compute vertical distribution of stress

- Determine how strongly the waves are dissipated
- Strongest dissipation occurs in regions where the wave becomes unstable and breaks down into turbulence, referred to as wave breaking
- Convective instability: where the amplitude of the wave becomes so large that it causes relatively cold air to rise over less dense, warm air
- Kelvin-Helmholtz instability also important: associated with shear zones
- Lindzen’s saturation hypothesis: instability brings about turbulent dissipation of the wave such that the amplitude of the wave is reduced until it becomes stable again (Lindzen 1981), i.e. λequals the local saturated stress λsat
- i.e. dissipation is just sufficient to ensure that
- Which gives , i.e. strong dependence on U which shows that wave breaking is strong preferred in regions of weak flow. Similarly, the stronger/smaller N/ρ the more readily waves break (Palmer et al. 1986)

:amplitude of wave

:mean Richardson number

Saturation hypothesis in terms of Ri

- Can show that wave’s impact on static stability and vertical shear gives rise to a minimum Richardson number (Palmer et al. 1986)
- Instability occurs when Rimin < Ricrit (=0.25) (i.e. less than the critical value)
- Kelvin-Helmholtz: numerator becomes small
- Convective instability: denominator becomes large
- Following saturation hypothesis, when the wave is saturated Rimin=0.25 and δh=δhsat
- Wave breaking occurs more readily in this formulation than in a purely convective overturning scheme

Gravity wave breaking only active above zblk (i.e. λ=λs for 0<z< zblk)

Above zblk stress constant until waves break, i.e. saturation hypothesis (Lindzen 1981) when stress exceeds saturation stress

Excess stress dissipated → drag

This occurs when the local Richardson number Rimin < Ricrit(=0.25). Ricrit can be used as a tuning parameter. Increasing it makes it easier for waves to break, so drag momentum deposition occurs at a lower altitude.

zblk

λ

- Some rough figures based on a surface flux of 0.1 N m-2 (Palmer et al. 1986 This will saturate in the
- Boundary layer if U< 5 m/s (preferred region of wave breaking)
- Mid troposphere if U< 7 m/s (mid-troposphere winds generally like this)
- Lower stratosphere if U < 15 m/s (preferred region of wave breaking)

Assume stress at any level:

Calculate Ri at next level

λk-2

Set λk-1=λk to estimate δh

using

Uk-2,Tk-2

λk-1

Repeat

Uk-1,Tk-1

Calculate Rimin

zk=zblk; λk= λs

If Rimin<Ricrit

set Rimin=Ricrit

estimate h=hsat

Estimate λ= λsat

go to next level

Height

z=0; λ= λs

set λk-1= λk

go to next level

Values of the wave stress are defined progressively from zblk upwards

Set λ=λs and Rimin=0.25 at model level representing top of blocked layer

If Rimin>=Ricrit

Blocking height zblk satisfies:

(assume h=2μ)

Characterise incident flow passing over the mountain top by ρH, UH, NH (averaged between μ and 2μ)

Hncrit≈1 tunes the depth of the blocked layer.

If Hncrit is increased then Zblk decreases

Up calculated by resolving the wind U in the direction of UH

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)

Cd (~1): form drag coefficient (proportional to ψ)

Summing over number of consecutive ridges in a grid point gives the drag

r: aspect ratio of the obstacle as seen by the incident flow; B,C: constants

this equation is applied quasi-implicitly level by level below zblk

Consider again an elliptical mountain

From ECMWF T511 operational model

winter

summer

From Lott and Miller 1997

Large parameterized drag

(makes up difference between

resolved and measured drag)

resolved drag

measured drag

T213 forecasts: ECMWF model with mean orography and the subgrid scale orographic drag scheme. Explicit model pressure drag and parameterized mountain drag during PYREX.

Results from Canadian Climate model of northern hemisphere winter, showing westerly gravity wave drag (tendency in m/s/d)

Alleviation of systematic westerly bias

Icelandic/Aleutian lows are too deep

Siberian high too weak and too far south

Flow too zonal / westerly bias

Azores anticyclone too far east

Without GWD scheme

With GWD scheme

Mean January sea level pressure (mb) for years 1984 to 1986

alleviation of westerly bias

Analysis (best guess)

better agreement

From Palmer et al. 1986

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

Without GWD scheme

cold bias

westerly bias

less impact in southern-hemisphere

Analysis (best guess)

From Palmer et al. 1986

Flow-over case

Flow-blocking case

From Smith et al. 2006

Sub-grid scale contribution still significant

Strong flow: short-scale trapped lee waves produce significant fraction of drag (Georgelin and Lott, 2001

resolved drag

converging

parameterization still required at high-resolution

Relatively weak flow: flow blocking dominates

Zonal cross-sections of the differences in (a) zonal wind (ms-1) and (b) temperature (K)

With GWD scheme - control

slowing of winds in stratosphere and upper troposphere

parameterisation of gravity wave drag decelerated the predominately westerly flow

With GWD scheme - control

poleward induced meridional flow

descent over pole leads to warming / alleviation of cold bias

From Palmer et al. 1986

no rotation rotation

Nh/U=2.7 and Ro=∞Nh/U=2.7 and Ro=2.5

symmetrical flow

asymmetrical flow / diverted to left

Ro<1 for L>100km (with U=10 ms-1, f=10-4s-1)

Rossby number Ro=U/fL

f: Coriolis parameter

L: mountain width

i.e. important for large mountain ranges (e.g. Alps), but not individual peaks / sub-grid scale parameterization ??

From Olafsson and Bougeault (1997)

Drag normalised by that predicted analytically for linear 2d non-rotating frictionless flow

High-drag hydraulic state discussed previously

- For more realistic conditions, i.e. rotation and friction
- Constrains the drag to values predicted by linear theory
- Deviates by no more than 30% from the normalizing value
- Less dependence on Nh/U
- 3) Explains why observed pressure drags agree remarkably well with those predicted by linear 2d non-rotating frictionless flow

From Webster et al. (2003) (after Olafsson and Bougeault (1997))

Predicted drag is essentially an empirical fit to the idealized simulations of Olafsson and Bougeault (1997)

Uses simple 2d linear approximations to predict the surface pressure drag

Surface pressure drag partitioned into a ‘blocked-flow’ component and a ‘gravity wave’ component depending on the Nh/U of the flow

Gravity wave amplitude is assumed to be proportional to the amount of ‘flow-over’ , and the remainder of the surface pressure drag assumed to be due to flow-blocking

Deposition of assumes saturation process as in Palmer et al. 1986

Vertical deposition of assumed to be uniform throughout depth h

- 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.
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