Surface and Boundary Layer Parameterization Pathways of Information

Surface and boundary layer parameterization Pathways of Information

Clouds (CP/MP) Radiation

Surface The “atmosphere”

Boundary Layer

Surface Layer

Typical boundary layer evolution over land Parametrization of the planetary boundary layer (PBL) Martin Köhler & Anton Beljaars

• Introduction. Martin • Surface layer and surface fluxes. Anton • Outer layer. Martin • Stratocumulus. Martin • PBL evaluation. Maike • Exercises. Martin & Maike Los Angeles PBL

July 2001

Downtown LA

PBL top 10km Griffith Observatory

1000 to 10000 die annually in LA from heart disease resulting from SMOG. California stratocumulus and forest fires

Wolf Fire (6 June 2002)

Downtown LA

MODIS on Terra (res. 250m) visibleearth.nasa.gov Boundary layer: definition

The PBL is the layer close to the surface within which vertical transports by turbulence play dominant roles in the momentum, heat and moisture budgets.

Turbulent flows are characterized by fluctuating dynamical quantities in space and time in a “disordered” manner (Monin and Yaglon, 1973).

Why is PBL turbulent? -5 2 • high Reynolds numbers Re = UL/ν > 2000, ν ~ 10 m /s

g ∂θv

θv ∂z • low Richardson number Ri = 2 <1/ 4 ⎛ ∂u ⎞ ⎜ ⎟ ⎝ ∂z ⎠ Laboratory observations: transition to turbulence Laboratory observations: laminar and turbulent BL Space and time scales • Diffusive transport in the atmosphere is dominated by turbulence. • Time scale of turbulence varies from seconds to half hour. • Length scale varies from mm for dissipative eddies to 100 m for transporting eddies. • The largest eddies are the most efficient ones for transport.

cyclones microscale turbulence diurnal cycle

spectral gap

data: 1957 100 hours 1 hour 0.01 hour Power spectrum … which spectral gap?

10-2 Cabauw Data 1987 (10m) 24h diurnal harmonics 10-3 12h 30-80 days cyclones (t,radiative) 8h 10-4

10-5 cyclones t-5/3 -6 diurnal 10 cycle Power Spectrum of Wind / Period / Wind of Power Spectrum 10-7 spectral gap Brookhaven Data 1957 10-8 10000 1000 100 10 1 Period in Hours

10000 hours 100 hours 1 hour Spectrum from time series of wind (Stratus buoy)

Amplitude spectrum ( Power spectrum ) -5/6 (3D turbulence)

diurnal cycle

24 hours 2 hours Wave number spectra near tropopause

5000 km k-3 cyclones

500 km k-5/3

2 km shifted

GASP aircraft data near tropopause Nastrom and Gage (1985) Wave number spectra at z=150m below stratocumulus

U Spectrum

Reynolds Decomposition?

V Spectrum

W Spectrum 500m

Duynkerke 1998 T-tendencies due to turbulence scheme

[K/day]

Jan. 1999 T-tendencies due to convection scheme

[K/day]

Jan. 1999 U-Profile … Effects of Terrain

z0~1-10cm z0~50cm z0~1m

Ocean:! uz* Neutral: U = ln z0~0.1-1mm κ z0 Oke 1978 U-Profile … Effects of Stability

Neutral Stable Unstable Height

uz* Neutral: U = ln κ z0

ln(Height) surface layer

Oke 1978 Diurnal cycle of boundary layer height

Sunrise Sunset

(residual BL)

Local Time stable BL convective BL stable BL

Oke 1978 Diurnal cycle of profiles

convective BL

stable BL

Oke 1978 Conserved variables

For turbulent transport in the vertical, quantities are needed that are conserved for adiabatic ascent/descent.

R/c p For dry processes: θ = T( po / p) , pot. temperature dry static energy or s = cpT + gz.

Lθ For moist processes: θl =θ − ( )ql , liq. wat. pot. temperature cpT

or sl = cpT + gz − Lql , liq. water static energy

and qt = q + ql . total water Buoyancy parameter

unstable stable To determine static stability, move a fluid parcel adiabatically in the vertical and compare the density of the parcel with the density of the surrounding fluid.

dθ dθ v < 0 v > 0 Virtual potential temperature and dz dz virtual dry static energy are suitable parameters to describe stability:

{1 ( Rv 1)q q }, θv =θ + Rd − − l s c T{1 ( Rv 1)q q } gz, Rv 1 0.61 v = p + Rd − − l + Rd − ≈ Basic equations

∂u ∂u ∂u ∂u 1 ∂p 2 + u + v + w − fv = − +ν∇ u ∂t ∂x ∂y ∂z ρ ∂x mom. ∂v ∂v ∂v ∂v 1 ∂p 2 + u + v + w + fu = − +ν∇ v equ.’s ∂t ∂x ∂y ∂z ρ ∂y

∂w ∂w ∂w ∂w 1 ∂p 2 + u + v + w = − +ν∇ w − g ∂t ∂x ∂y ∂z ρ ∂z ∂u ∂v ∂w 1 dρ continuity + + = ∂x ∂y ∂z ρ dt Reynolds decomposition

u =U + u', v =V + v', w =W + w'

ρ = ρo + ρ', p = P + p'.

Substitute, apply averaging operator, Boussinesq approximation (density in buoyancy terms only) and hydrostatic approximation (vertical acceleration << buoyancy).

Averaging (overbar) is over grid box, i.e. sub-grid turbulent motion is averaged out.

Property of averaging operator: u ≡U + u' =U ≡U After Reynolds decomposition and averaging

∂U ∂U ∂U ∂U 1 ∂P 2 +U +V +W − fV = − +ν∇ U nd ∂t ∂x ∂y ∂z ρo ∂x 2 order ∂u'u' ∂u'v' ∂u'w' − − − ∂x ∂y ∂z

∂V ∂V ∂V ∂V 1 ∂P 2 +U +V +W + fU = − +ν∇ V nd ∂t ∂x ∂y ∂z ρo ∂y 2 order ∂u'v' ∂v'v' ∂v'w' − − − ∂x ∂y ∂z 1 ∂P 0 = − − g ρo ∂z ∂U ∂V ∂W + + = 0 ∂x ∂y ∂z The 2nd order correlations are unknown (closure problem) and need to be parametrized (i.e. expressed in terms of large scale variables). Reynolds equations

Boundary layer approximation ∂u'u' ∂u'w' << (horizontal scales >> vertical scales), e.g. : ∂x ∂z

High Reynolds number approximation ∂u'w' ν∇2U << (molecular diffusion << turbulent transports), e.g.: ∂z

∂∂∂∂UUUU1'' ∂∂ Puw +++U V W− fV =− − ∂∂∂txyz ∂ρo ∂∂ xz ∂∂∂∂VVVV1'' ∂∂ Pvw ++++=U V W fU − − ∂∂∂txyz ∂ρo ∂∂ yz

Reynolds Stress Simple closures

K-diffusion method: ∂U u'w' ≈ −K ∂z ∂∂∂uw'' U ∂2 analogy to ⎛⎞KKU ≈−⎜⎟ ≈−2 molecular diffusion ∂∂zz⎝⎠ ∂ z ∂ z

Mass-flux method:

up u'w' ≈ M (u −U ) mass flux (needs M closure)

∂ up up u = −ε(u −U ) entraining plume model ∂z Turbulent Kinetic Energy equation

local TKE: Euvw'1/2('≡22++ ' ') 2 mean TKE: E ≡1/ 2(u'2 + v'2 + w'2 ) Derive equation for E by combining equations of total velocity components and mean velocity components: Storage Mean flow TKE advection ∂∂∂∂EEEE +++UVW = Pressure ∂∂∂txyz ∂ correlation ∂∂∂∂UVgpw'' −−Ew ' ' uw ' ' −− vw ' 'ρε ' w ' +− ∂∂∂∂zzzzρρo Turbulent Shear production Buoyancy transport Dissipation Mixed layer turbulent kinetic energy budget

dry PBL

Stull 1988 normalized Literature General: Stull (1988): An introduction to boundary layer meteorology, Kluwer publishers. Oke(1978): Boundary layer climate, Halsted press.

Boundary layer in large scale atmospheric models: Holtslag and Duynkerke (eds., 1999): Clear and cloudy boundary layers, North Holland Press.

Surface fluxes: Brutsaert (1982): Evaporation into the atmosphere, Reidel publishers.

Sensitivity of ECMWF boundary layer scheme: Beljaars (1995): The impact of some aspects of the boundary layer scheme in the ECMWF model, ECMWF-seminar 1994. Parametrization of surface fluxes: Outline

• Surface layer (Monin Obukhov) similarity • Surface fluxes: Alternative formulations • Roughness length over land – Definition – Orographic contribution – Roughness lengths for heat and moisture • Ocean surface fluxes – Roughness lengths and transfer coefficients – Low wind speeds and the limit of free convection – Air-sea coupling at low wind speeds: Impact training course: boundary layer; surface layer Mixing across steep gradients

Stable BL Dry mixed layer Cloudy BL

θ θ θ

Surface flux parametrization is sensitive because of large gradients near the surface. training course: boundary layer; surface layer Boundary conditions for T and q have different character over land and ocean

Surface fluxes of heat and moisture are proportional to temperature and moisture differences: Lowest model level T1,q1 HcCUTT=ρ pH1 ()1 − s H E z1

ECUqq=ρ Es1 ()1 − Ts, qs Surface

Ocean boundary condition Land boundary condition

TT= s HEQG++=λ qq= () T cC U() T T C U { q q ()} T Q G sat s ρρλαpH1111− s+ E− sats +=

training course: boundary layer; surface layer Parametrization of surface fluxes: Outline

Considerations about the nature surface layer of the process: surf . • z/zo >> 1 0 τ o • distance to surface determines turbulence length scale • shear scales with surface Scaling parameters: friction rather than with zo z height or eddy size() m

u* friction velocity= τρo /(/) m s −u3 L Obukhov length= * () m g H κ o training θρc course: vp boundary layer; surface layer MO similarity for gradients

The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the ﬁle again. If the red x still appears, you may have to delete the image and then insert it again. ∂Uzκ z φm = is a universal function of ∂zu* L ∂Θ κ z φh = dimensionless shear Stability parameter∂z θ*

κφ(von Karman constant) is defined such thatm == 1 forzL / 0 ∂Θ κ z z φh = is a universal function of ∂z θ* L dimensionless Stability parameter z potential temperatureL gradient

−1 K 1 Note that with ⎛⎞∂U we obtain: m Kuwm =− ''⎜⎟ = ⎝⎠∂z κϕzu* m training course: boundary layer; surface layer MO gradient functions

Observations of φ m as a function of z/L, with κ = 0.4 Empirical gradient functions to describe these observations: −1/ 4 φm = (1−16z / L) for z / L < 0

φm =1+ 5z / L for z / L > 0

unstable stable

training course: boundary layer; surface layer Parametrization of surface fluxes: Outline

Dimensionless wind gradient (shear) or temperature gradient functions can be integrated to profile functions:

∂U u* u* ⎧ z ⎫ = φm ⇒ U = ⎨ln( ) − Ψm (z / L)⎬ ∂z κz κ ⎩ zom ⎭ with: zom integration constant (roughness length for momentum)

Ψm wind profile function, related to gradient function: ∂Ψ z φ =1−η , with η = m ∂η L Profile functions for temperature and moisture can be obtained intraining similar way. course: boundary layer; surface layer Integral profile functions: Momentum, heat and moisture

Profile functions for surface layer applied between surface and lowest model level provide link between fluxes and wind, temperature and moisture differences.

τ xo z1 + zom UzL11={ln( )−Ψm ( / )} ρκuz* om

τ yo z1 + zom VzL11={ln( )−Ψm ( / )} ρκu* zom

H z1 + zom θθ11−sh={ln( )−Ψ (zL / )} ρκcup* z oh

E z1 + zom q11−qzLsh={ln( )−Ψ ( / )} ρκuz* oq z0m Displacement height: Numerically necessary for large values of z0 training course: boundary layer; surface layer MO wind profile functions applied to observations

Stable Unstable

training course: boundary layer; surface layer Transfer coefficients Surface fluxes can be written explicitly as: CUU τρxM= 1 1 Lowest model level U1,V1,T1,q1 τρyM= CUV1 1 τ τ H E z1 x y HcCU() =ρθθpH1 1 − s 0, 0, Ts, qs Surface

ECUqq=ρ Es1 ()1 −

1/2 22 where UUV= 1 ( 11+ ) k 2 and Cφ = {ln(zz1111 /om )−−ψψ m ( zL / )}{ln( zz / oφφ ) ( zL / )}

M ⎧m ⎧m ⎧ ⎪ ⎪ ⎪ φ = h φ = h φ = ⎨H ⎨ ⎨ ⎪ ⎪ q ⎪ h training⎩ E ⎩ ⎩ course: boundary layer; surface layer Numerical procedure: The Richardson number

The expressions for surface fluxes are implicit i.e they contain the Obukhov length which depends on fluxes. The stability parameter z/L can be computed from the bulk Richardson number by solving the following relation:

gz1 θ1 −θs z1 {ln(z1 / zoh ) −ψ h (z1 / L)} Rib = 2 = 2 θ |U1 | L {ln(z1 / zom ) −ψ m (z1 / L)}

This relation can be solved: • Iteratively; • Approximated with empirical functions; • Tabulated.

training course: boundary layer; surface layer Louis scheme The older Louis formulation uses:

('')wCUFRizzzzφφφ=−φφnsbomo1 (111 − ) ( , / , / φ ) 2 With neutral transfer coefficient: κ Cφ n = {ln(zz11 /om )}{ln( zz / oφ )}

And empirical stability functions for Fφφ(,/,/) Ribomo z11 z z z

⎧uv() ⎪ Initially, the empirical stability functions, F , φθ= ⎨ φ ⎪ were not related to the (observed-based) ⎩ q Monin-Obukhov functions.

training course: boundary layer; surface layer Stability functions for surface layer

unstable stable

Land

Louis et al (dash) MO-functions (solid) Sea

training course: boundary layer; surface layer Surface fluxes: Summary • MO-similarity provides solid basis for parametrization of surface fluxes:

('')wCUφφφ=−φ 1 ( 1 − s ) k 2 Cφ = {ln(zz1111 /om )−−ψψ m ( zL / )}{ln( zz / oφφ ) ( zL / )}

zL111/(,/,/)= fRizzb om zz oφ

gz1 ()θθ1 − o Rib = 2 θ U1 • Different implementations are possible (z/L-functions, or Ri- functions) • Surface roughness lengths are crucial aspect of formulation. • Transfer coefficients are typically 0.001 over sea and 0.01 over land, mainly due to surface roughness. training course: boundary layer; surface layer Parametrization of surface fluxes: Outline

Example for wind: u z U = * ln( ) • Surface roughness length is defined on ln z κ zom the basis of logarithmic profile. 10 • For z/L small, profiles are logarithmic. • Roughness length is defined by 1 intersection with ordinate. 0.1 zom 0.01 Often displacement height is used U to obtain U=0 for z=0: u z + z U = * ln( om ) • Roughness lengths for momentum, κ zom heat and moisture are not the same. • Roughness lengths are surface properties. training course: boundary layer; surface layer Roughness length over land

Geographical fields based on land use tables: Ice surface 0.0001 m

Short grass 0.01 m

Long grass 0.05 m

Pasture 0.20 m

Suburban housing 0.6 m

Forest, cities 1-5 m

training course: boundary layer; surface layer Roughness length over land (orographic contribution)

• Small scale sub-grid orography contributes substantially to surface drag due to pressure forces on orographic features (form drag). • Effects are usually parametrized through orographic enhancement of surface roughness.

Drag is determined by U “silhouette area” per unit surface area. ΣA ΣA drag = C U 2 ΣA d S

Effective roughness length: S h + zoeff −2 h + zoveg −2 CD ΣA {ln( )} = {ln( )} + 2 trainingzoeff zoveg κ S course: boundary layer; surface layer Roughness length over land (orographic contribution)

Orographic form drag (simplified Wood and Mason, 1993):

τ os = 2αβ C θ 2 U 2 (h ) ρ m m α, β Shape parameters

Cm Drag coefficient θ Silhouette slope U Wind speed

hm Reference height

Vertical distribution (Wood et al, 2001): − z / hm τ o =τ os e

training course: boundary layer; surface layer Parametrization of flux divergence with continuous orographic spectrum:

∂τ o − 2ραβ Cm 2 2 −z / hm = θ U (hm )e ∂z hm Assume: hm ~1/ k ∞ θ 2 = ∫ k 2 F(k) dk ko

1000 m 100 m

Write flux divergence as: ∂τ ∞ o = −2ραβ C k 3F(k)U 2 (c / k)e−zk /cm dk ∂z m ∫ m training ko Beljaars, Brown and Wood, 2003 course: boundary layer; surface layer Roughness lengths for heat and moisture

• Roughness lengths for heat and moisture are different from the aerodynamic roughness length, because pressure transfer (form drag) does not exist for scalars. θθ01==(z 0) by extrapolation

θ* ⎛⎞z0m θθ0 −s =ln ⎜⎟ κ ⎝⎠z0h

θθ000==smh if zz • Vegetation: roughness lengths for heat and moisture are ~ 10x smaller than aerodynamic roughness.

training course: boundary layer; surface layer Parametrization of surface fluxes: Outline

Roughness lengths are determined by molecular diffusion and ocean wave interaction e.g.

2 u* ν zom = Cch + 0.11 , Cch is Charnock parameter g u* ν zoh = 0.40 u* ν zoq = 0.62 u* Current version of ECMWF model uses an ocean wave model to provide sea-state dependent Charnock parameter.

training course: boundary layer; surface layer Transfer coefficents for moisture (10 m reference level)

u2 ν ν z = 0.018 * + 0.11 , z = 0.62 om g u oq u 2 * * u* ν zom = 0.018 + 0.11 , zoq = zom g u*

• Using the same roughness length for momentum and moisture gives an overestimate of transfer coefficients at high wind speed

Neutral exchange evaporation coeff for

EN • The viscosity component increases the transfer at low wind speed

training course: boundary layer; surface layer Low wind speeds and the limit of free convection At zero wind speed, coupling with the surface disappears e.g. for evaporation: Lowest model level λE = ρ λCM U1 (q1 − qs ) U1,V1,T1,q1 H λE z1 H = ρ cp CH U1 (θ1 −θ s ) 0 qs Surface

Extension of MO similarity with free convection velocity: inversion

2 2 2 1/ 2 where U1 = (U1 +V1 + w* ) , h 1/ 3 and w* = ((g /T) (−H / ρcp ) h) Surface training course: boundary layer; surface layer Air-sea coupling at low winds

zz00mh≠

Revised scheme: Larger coupling at low wind speed (0-5 ms-1) training course: boundary layer; surface layer Air-sea coupling at low winds (control)

Precipitation, JJA; old formulation

training course: boundary layer; surface layer Air-sea coupling at low winds (revised scheme)

Precipitation, JJA; new formulation

training course: boundary layer; surface layer Air-sea coupling at low winds

Near surface Theta_e difference: New-Old

training course: boundary layer; surface layer Air-sea coupling at low winds

Theta and Theta_e profiles over warm pool with old an new formulation

new new

old old training course: boundary layer; surface layer Air-sea coupling at low winds

Zonal mean wind errors for DJF

Old

New

training course: boundary layer; surface layer IMET-stratus buoy / ECMWF (20 S 85 W)

Latent heat flux

training course: boundary layer; surface layer IMET-stratus buoy vs. ECMWF (20 S 85 W)

Sensible heat flux

training course: boundary layer; surface layer IMET-stratus buoy vs. ECMWF (20 S 85 W)

Horizontal wind speed

training course: boundary layer; surface layer IMET-stratus buoy vs. ECMWF (20 S 85 W)

Water/air q- difference

training course: boundary layer; surface layer IMET-stratus buoy vs. ECMWF (20 S 85 W)

Water/air T-difference

training course: boundary layer; surface layer BL budget considerations: IMET-stratus

Heat budget: ρcpwe (θi −θm ) − ρcpCH U (θm −θs ) − dL + dS + λP = 0 +40 +10 -80 +20 +10 W/m2 Moisture budget: we (qi − qm ) − CH U (qm − qs ) − P/ ρ = 0 -3 +3.3 -0.3 mm/day

θ qi i θ qm m

training q θs course: s boundary layer; surface layer