Parameterization of the planetary boundary layer and surface processes

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IACETH Institute for Atmospheric and Climate Science

Parameterization of the planetary boundary layer and surface processes

Figure: Trenberth, Climate System Modeling, 1992

Ulrike Lohmann (IACETH) PBL May 3, 2007 1 / 36

Introduction Cloud-free PBL Surface layer Cloud-topped PBL Breakup

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I Link to usefulscriptson parameterization of diabatic processes:

www.ecmwf.int/newsevents/training/lecture notes/LN PA.html

I Gibbs phenomenon(ringing artifacts):

Behavior of the Fourier series of a piecewise continuously differentiable periodic functionf at a jump discontinuity:

the nth partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself. The overshoot does not die out as the frequency increases, but approaches a finite limit.

(en.wikipedia.org/wiki/Gibbs phenomenon)

Typical flow chart in a GCM

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Introduction

I The interaction of the atmosphere with the Earth’s surface involves the exchange of heat, momentum, moisture and chemical species and hence is an important component of climate system modelling.

I Most GCMs can predict these surface fluxes reasonably well in the absence of complications due to terrain, canopy or precipitation.

I Outline for today:

I Cloud-free planetary boundary layer (PBL)

I Surface processes

I Cloud-topped PBL

I Because the PBL is the link between the Earth’s surface and the free atmosphere, it must be included in climate models.

I The PBL’s structure is dominated by surface processes that generate turbulent motions on a range of spatial scales.

I The surface layer (typically the lowest 100 to 200 meters) is defined by the condition of near constant fluxes of heat, momentum and moisture with height.

I Since the typical height of the PBL is 2 km or less, a large number of vertical layers are required to properly resolve the PBL. Because the computational cost of these additional layers is considered too large, PBL processes are parameterized in GCMs.

I The parameterizations should account for the dependence of the PBL height on surface conditions and surface types.

I It should also distinguish between an unstable, neutral or stable atmosphere; and it should couple in a consistent manner with the other parameterizations in the GCM.

www.mmm.ucar.edu/mm5/documents/MM5tut Web notes/MM5/mm5.htm

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Different wind speed profiles in the PBL

(Fig. 2 Beljaars, ECMWF lecture)

Daytime vs. nighttime PBL

Source: Wallace and Hobbs, Atmospheric Science, Acad. Press, 2006

I Start with the equation for the horizontal windu:

∂u

∂t +u∂u

∂x +v∂u

∂y +w∂u

∂z −fv =−1 ρ

∂p

∂x +ν∇²u (1) whereνis the kinematic viscosity.

I The spatial scales that are important in the PBL are much smaller than those present in the rest of the atmosphere.

I This means that the prognostic variables such asu,v,w,T,q are separated into mean and subgrid-scale terms (u=U+u⁰).

I After averaging and application of the Boussinesq approximation (retain density fluctuations in the buoyancy terms only), we obtain:

∂U

∂t +U∂U

∂x +V∂U

∂y +W∂U

∂z = fV − 1 ρ₀

∂P

∂x +ν∇²U (2)

− ∂

∂xu⁰u⁰− ∂

∂yu⁰v⁰− ∂

∂zu⁰w⁰ whereρ0is the density of the mean state.

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I The correlation terms likeu⁰w⁰are called eddy flux or Reynolds stresses

I Assume that we can neglect viscous effects for large Reynolds number (UL/ν >>1, whereLis a length scale) and thex,y scales are much larger than thez-scales (BL approximation):

∂U

∂t +U∂U

∂x +V∂U

∂y +W∂U

∂z −fV =−1 ρ0

∂P

∂x − ∂

∂zu⁰w⁰ (3)

I The simplest approach of obtaining these Reynolds stress terms is to assume they are described by an eddy diffusion process:

∂u⁰w⁰

∂z =−KM

∂U

∂z (4)

whereKM is the eddy diffusion coefficient for momentum.

I Although this is a straightforward model for the PBL, a constantKM coefficient is insufficient to model many observed states of the PBL.

Eddy diffusion process

(Wallace and Hobbs, 2006)

K

M

as a function of stability

I The Richardson numberRi is defined as the ratio of

mechanical generation of turbulence by wind shear vs. buoyant consumption by static stability. It is used as a dynamic stability measure to determine if turbulence will exist.

I Better: obtainKM as a function ofRi.

K_M =l²F_M

dU dz

(5) whereFM is a function ofRi andl is a turbulence length scale:

1 l = 1

κz +1

λ (6)

whereκis the von Karman constant (=0.4) andzis the height.

I It is confined by the asymptotic valueλsuch thatl is limited by the PBL height.

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Turbulent kinetic energy (TKE)

I Another approach is to obtainK_M from the turbulent kinetic energy (E):

K_M= 0.52l_M⁰ √

E (7)

where

E = 1

2(u⁰²+v⁰²+w⁰²) (8) andl_M⁰ is a modified mixing length.

I The prognostic equation for TKE is given as:

dE

dt =−u⁰w⁰∂U

∂z

| {z }

I

−u⁰w⁰∂V

∂z

| {z }

I

− g ρ₀w⁰ρ⁰

| {z }

II

+∂

∂z





 E⁰w⁰

| {z }

III

+p⁰w⁰ ρ

| {z }

IV







−

|{z}

V

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Turbulent kinetic energy (TKE)

I Terms I represent mechanical production of turbulence by wind shear and convert energy of the mean flow into turbulence

I Term II is the production of turbulence by buoyancy and converts potential energy of the atmosphere to turbulence or vice versa

I Terms III and IV are transport or turbulent diffusion terms because they equal zero when vertically integrated over the domain. They represent the vertical transport of turbulence energy by turbulent fluctuations and the pressure fluctuations respectively.

I Term V is the dissipation term which converts TKE into heat by molecular friction at very small scales.

TKE spectrum

(Fig. 9.5 Wallace&Hobbs, 2006)

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TKE budget

(Fig. 5.4 Stull, 1988)

www.mmm.ucar.edu/mm5/documents/MM5tut Web notes/MM5/mm5.htm

I The surface layer is defined by the near vertical constancy (variations less than 10%) of the heat, momentum and moisture fluxes.

I The turbulent flux of a variableχat the surface is obtained from the bulk transfer relation:

(w⁰χ⁰)S=−Cχ|UL|(χL−χS) (10) whereCχis the transfer coefficient. The subscriptsLandSrefer to values at the lowest model level (representing the top of the surface layer) and the surface, respectively, and|UL|is the horizontal wind vector at levelL.

I The transfer coefficients are obtained from Monin-Obukhov similarity theory by integrating the flux-profile relationships over the lowest model layer.

I The bulk transfer coefficients depend on the wind shear and static stability of the atmosphere above the surface, and the roughness of the surface. In general,C_χincreases in magnitude as the atmosphere becomes more unstable.

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Typical roughness lengths

Ground cover Roughness length (m)

Calm sea, paved areas, smooth desert 0.0002 Snow-covered fields, pack ice 0.005

Farm fields, tundra 0.03

Cultivated area with low crops 0.1

High crops, vineyards 0.25

Forest clumps, scattered buildings 0.5

Villages, mature forests 1

Centres of large cities, irregular forests ≥2

Surface temperatures over land

I Traditionally, AGCMs have calculated the land surfaceT_s from a surface energy balance condition:

C_L∂T_s

∂t =R_net+LE+H+G (11) whereC_Lis the heat capacity of the layer [J m⁻²K⁻¹],H= sensible heat flux,LE = latent heat flux,G = ground heat flux, andR_netis the net radiation:

Rnet= (1−αs)R_SW^↓ +R_LW^↓ −σT_s⁴ (12) whereα_s = surface albedo,R_SW^↓ = downwelling solar

radiation,R_LW^↓ = downwelling longwave radiation,= surface emissivity andσthe Stefan-Boltzmann constant.

I Recently, however, land surface models are being employed within AGCMs to evaluate the surface and subsurface T.

Background albedo in the ECHAM GCM

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Sea surface temperatures (SST)

I Over oceans, T_smust be either specified or calculated from an interactive ocean model.

I Many GCM studies of the atmosphere have employed fixed SSTs.

Daily SSTs are prescribed by linearly interpolating between mid-month observed SSTs and thus allow for seasonal variations.

Although this enables the AGCM to be run with great computational efficiency, it denies the atmosphere any realistic interaction over ocean regions. That is, surface fluxes are irrelevant to SSTs.

I It implies that the total surface-atmosphere system need not be in energy balance, i.e. the globally and annually averaged

top-of-the-atmosphere (TOA) net radiation balance need not to be 0!

I Typically, fixed SST models are “tuned” to guarantee this balance condition. But any perturbation to the “tuned” model, e.g. change in the model physics, will cause an imbalance.

I Over sea ice regions, Tsneeds to be evaluated from a sea ice model.

Surface hydrology

Easiest method: Budyko’s bucket model:

∂Ws

∂t =P_R−J_EV −R+M where:

Ws Soil water content P_R: Precipitation rate JEV: Evaporation rate R: Runoff

M: Melt water

I IfWs>Wsmax⇒Runoff Wsmax = Depth of ground water reservoir

Figure: www.icsu-scope.org/

downloadpubs/scope35/chapter05.html

Simulated vs. observed cloud cover

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Importance oceanic low clouds

I The cloud-topped boundary layer is a very widespread phenomenon with importance for global climate

I These clouds mainly form in regions of subtropical anticyclones or in response to cold air outbreaks

I A 4% increase in the area of the globe covered by low-level stratus clouds could offset the 2-3 K temperature increase due to doubling of CO2

I Strato cumulus (Sc) are important for the radiation budget because they reflect large amounts of shortwave radiation.

I They are difficult to predict in GCMs because they are thin and inversion is steep.

I Missing to capture them in coupled atmosphere-ocean (AO) GCMs is drastic as the ocean then absorbs too much radiation, which cause cause problems in simulating El Ninos.

Importance of low clouds for climate change

Figure: Model response to 2xCO₂and change in low cloud amount [Stephens, J.Climate, 2005]

Typical vertical structure of stratocumulus-topped BL

www.zamg.ac.at/docu/Manual/SatManu/CMs/ScSh/images/sscosk2k.jpg

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Temporal evolution of the cloud-topped BL

Source: Wallace and Hobbs, Atmospheric Science, 2006

I Envision the boundary layer to be characterized by a

population of buoyant plumes rising from the warm sea surface

I The boundary layer becomes well mixed as a result of the vigorous upward fluxes of sensible and latent heat

I TKE is generated both by buoyancy and by wind shear (cf.

equation 9):

I Mixing maintains a layer of constant mean Θ_ein the mixing layerh

I The equivalent potential temperature Θ_e is defined as:

Θ_e≡Θexp Lq_v

c_pT_s

(13) T_s(T,q_v,p) is the temperature at which an air parcel would become saturated by lowering its pressure dry-adiabatically.

I Θ_eis conserved when condensation/evaporation are the only diabatic effects

Schematic of the cloud-topped boundary layer

Figure: 5.10 Houze [1993]

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I his bounded by a layer of different Θ_e(usually higher) aloft

I Θ_eof layer above is maintained either by subsidence in the descending branch of the Hadley cell or in cold air mass behind a front

I What happens when air with higher Θ_e is entrained into the cloud elements?

I When cloud elements thicken to form a more continuous layer, radiative cooling becomes important

I Now ocean is shielded by the cloud layer and the turbulent heat flux in the subcloud layer diminishes

I TKE generation is concentrated in cloud layer caused by buoyancy and wind shear

I Stratus can breakup into Sc or cumulus (Cu) when boundary layer is slightly unstable or into Cu, cumulonimbus (Cb) else

Breakup of the cloud-topped boundary layer

Figure: 5.11 Houze [1993]; Randall, JAS [1980]

Cloud-top-entrainment instability (CTEI)

I Warm air entrained into top of Sc might cool and sink if it were initially dry enough to support considerable evaporative cooling of the neighboring cloud droplets.

I Evaporative cooling is sufficient to make some mixtures of clean and cloudy air negatively buoyant with respect to the cloud layer so that the parcel is accelerated downward.

Figure:Randall, JAS [1980]

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Role of entrainment

I The unstable environment is defined as (Lilly, 1968):

∆Θe= Θejust above CT −Θejust below CT<0 (14)

I This theory was further developed by Randall and Deardorff (1980) such that:

∆Θe<kL∆q_t cp

(15) wherek=0.23

I However, not all observations support this CTEI criterion. Some data showed nearly isothermal entrainment, i.e. no evaporative cooling

Role of entrainment

Figure: Cotton and Anthes: Storm and cloud dynamics, 1989

In 2/3 of the cases, breakup occurs when theory predicts it (shaded region).

Role of large-scale subsidence

I Large-scale subsidence establishes the pronounced capping inversion, which serves as an upper lid to the PBL and confines moisture and heat fluxes from ocean in shallow layer.

I Subsidence also dries the overlying air mass.

I Subsidence establishes environmental conditions favorable for maintaining a solid stratus deck, but too much subsidence may be responsible for breakup of Sc.