Relatingthesubgridprocessestolarge-scalevariablesdependsonknowledgeofthefundamentalphysicsinvolvedintheprocess. Theonlymodelvariablesavailablefortheparameterizationsarethelarge-scaleﬁeldspredictedbythemodel. Anyphysicalprocess,suchasradiationorcumulusconv

(1)

IACETH Institute for Atmospheric and Climate Science

Parameterizations of physical processes

Ulrike Lohmann (IACETH) Parameterizations of physical processes April 26, 2007 1 / 35

Climate modeling Cloud cover Statistical schemes Prognostic schemes Microphysics Validation

Motivation

Figure:Courtesy Adrian Tompkins, ECMWF, 2005

IACETH stitute for Atmospheric and Climate Science

Parameterizations - general aspects

I Any physical process, such as radiation or cumulus convection, that occurs on scales smaller than the GCM resolution, must be parameterized.

I The only model variables available for the parameterizations are the large-scale fields predicted by the model.

I Relating the subgrid processes to large-scale variables depends on knowledge of the fundamental physics involved in the process.

(2)

Parameterized processes

I Radiation - usually 2 stream approach for shortwave (SW, solar) and longwave (LW, terrestrial) radiation

I Convection - usually mass flux approach

I Stratiform clouds:

I Usually prediction of cloud condensate (1 or 2 moment scheme) for water and ice

I Either relate cloud cover to relative humidity (RH), predict cloud cover (C) or use a statistical scheme

I Boundary layer and surface processes - usually predict turbulent kinetic energy

I Mechanical dissipation of kinetic energy (gravity wave drag)

Outline of the 3 parameterization lectures

I Today: Parameterization of large-scale clouds

I 3.5.: Parameterization of the cloud-topped boundary layer

I 24.5.: Parameterization of convection/radiation

I More details on parameterization of atmospheric radiation (see lecture “Radiation and climate change” by Wild and Blatter)

I More details on convection (see lecture “Cloud dynamics” by Lohmann and Spichtinger)

I Land-surface processes are subject of the lecture

“Land-atmosphere-climate interactions” by Seneviratne and Sch¨ar

Governing equations in climate models

∂~v

∂t = −~v· ∇~v−ω∂~v

∂p +f~k×~v=−∇Φ +D~M (1)

∂T

∂t = −∇ ·(~v T) +ω

κT

p −∂T

∂p

+Qrad

cp

+Qcon

cp

+D_H (2)

∂ω

∂p = −∇ ·~v (3)

∂q

∂t = −∇ ·(~v q) +ω∂q

∂p+E−C+Dq (4)

∂q_x

∂t = −∇ ·(~v q_x) +ω∂q_x

∂p +sources−sinks+D_q_x (5) whereD~M = (Dλ,Dφ) = dissipation terms for momentum

D_H, D_q= diffusion terms for heat and moisture E, C = rates of evaporation and condensation

q,qx = specific humidity and cloud condensate, x=cloud water, ice Φ = geopotential.

(3)

Boundary conditions

Boundary data Needed for the Dimensions

parameterization of

Sea surface (sfc) Radiation and lat, lon, time

temperature boundary layer

Surface type Sfc. temp. lat, lon

Surface roughness length Sfc. processes lat, lon

Land hydrology Sfc. processes lat, lon

Snow-free surface albedo Radiation lat, lon

Ozone mixing ratio Radiation lat, lon

Orography Dynamics lat, lon

Subgrid variance of orography Gravity wave drag lat, lon Aerosol emissions Aerosols, clouds, lat, lon, time

radiation

Parameterization of large-scale cloud cover

I Predicting cloud amount is very important for climate modelling.

I Clouds play a fundamental role in controlling the amount of shortwave and longwave radiation available to the climate system.

I Clouds range in size from hundreds of meters to hundreds of km in radius, which means that most clouds are smaller in area than the typical grid resolution of climate models.

I Unlike other climate variables (e.g., momentum, T, or q), there is no fundamental prognostic equation for cloud fraction (and cloud cover is not well defined).

I The cloud amount for a grid region must be related to the other predicted climate variables. However, the cloud fraction must be prescribed in association with the occurrence of condensation.

I Thus, when stable or convective condensation occurs a cloud amount is assigned to the same grid box.

Assumptions/definitions

I qs= saturation mixing ratio:

qs∼ e_s p−es

(6) where=R_d/R_v = ratio of the gas constant of dry air (R_d= 287 J kg⁻¹) to that of water vapor (R_v = 461.5 J kg⁻¹K⁻¹),e_s= saturation vapor pressure is obtained from the Clausius-Clapeyron equation:

e_s(T) =e_s0exp

L

Rv

1

To

− 1 T

(7) whereL= latent heat of vaporization, To= 273.2 K, es0= 611 Pa.

I Local criterion for formation of cloud: qt>qs, whereqt=qv+qc= total water (vapour+cloud) mixing ratio.

I Assumptions: 1) Condensation process is fast (compared to a GCM timestep), so thatqv=qsandqc=qt−qs. 2) No supersaturation is allowed to exist. Both assumptions are suspect in ice clouds

(4)

Partial cloud cover

I Partial coverage of a grid-box with clouds is only possible if there is an inhomogeneous distribution ofT and/orq

I Here concentrate on humidity fluctuations which are more important for water cloud formation

I Then, we can relate cloud amount to large-scale RH.

I The simplest approach is:

C=

( 1 RH≥RH_crit 0 RH<RHcrit

(8) whereRH_critis a critical relative humidity above which clouds can fill the grid box. It is a tunable constant (∼0.8).

I This parameterization would totally fill the grid box with clouds whenever RH exceedsRH_crit

I The value of 1 forRH≥RHcrit could also be viewed as a tunable constant, i.e., it can be set to<1.

Partial cloud cover

I More “complex” cloud fraction parameterization may assume the cloud fraction to a quadratic function of RH (Slingo, 1987):

C=

RH−a b

2

(9) whereaandbare viewed as “tunable” values.

I Or as square root functions of RH (Sundqvist et al., 1989):

C= 1− s

1−RH

1−RH_crit (10)

I Convective cloud cover is typically either assigned a small value (∼5%) or can be related to the convective upward mass flux Mu:

Cconv = 0.035ln(1.0 +Mu) (11)

Cloud blinking

I Some of these cloud cover schemes result in cloud “blinking”.

When condensation occurs a cloud is formed at that time step of the model.

I In the next time step RH may be less thanRH_critand the cloud fraction will be 0. Thus, a time series of cloud amount from this approach indicates high-frequency “on-off” cloud activity.

I Cloud blinking can be avoided by using:

I Statistical cloud schemes

I Prognostic cloud cover

(5)

Statistical cloud schemes

[Tompkins, ECMWF lecture, 2006]

Statistical cloud schemes

I Define a single cloud variables as the difference between the total waterqt within the grid volume and that which would exist if the air were just saturated (e.g. Smith, 1990):

s=a_l(qt−q_sl) (12)

I qsl is the saturation vapor mixing ratio with respect to the condensed water temperatureT_l =T−^L_c^v

Pq_l−^L_c^s

pq_i withc_p being the specific heat of dry air at constant pressure.

I The local deviation from the mean is:

s⁰=al(q_t⁰−αlT_l⁰) (13) where the coefficientsa_l andα_l are given by:

α_l = ∂qsl

∂T (14)

Statistical cloud schemes

I

a_l =

1 + L c_pα_l

−1

(15) The factoral accounts for latent heating.

I The fluctuations represented here are not necessarily due entirely to turbulence, but rather to all unresolved processes.

The grid-box mean ofs⁰is zero sinceq⁰_tandT_l⁰ are zero.

I The standard deviation ofs⁰ is:

σs=al

q

q_t⁰²+α²_lT_l⁰²−2αlq⁰_tT_l⁰ (16)

I The cloud water content is calculated on the assumption that sufficient cloud condensation nuclei are present to remove any supersaturation.

(6)

Statistical cloud schemes

I Partial cloudinessCand ensemble mean cloud water content qc, given a distribution functionG for the normalized variable t=s⁰/σ_s, can be expressed as:

C = Z ∞

−Q₁

G(t)dt (17)

q_c = 2σ_s Z∞

−Q1

(Q₁+t)G(t)dt (18)

whereQ1(=a_l[^q^t_σ^−q^s

s ]) is the normalized mean saturation deficit (condensation occurs ift>−Q1).

Choice of probability distribution function (PDF)

Figure:Tompkins, ECMWF lecture, 2006

Choice of PDF

(7)

Statistical cloud schemes

As an example, use a polynominal distribution function:

G(t) = 4/π

4 +t⁴ (19)

ThenC andq_care given by:

C = 1 2+ 1

4π

2[arctan(Q₁+ 1) +arctan(Q₁−1)] +ln

(Q1+ 1)²+ 1 (Q₁−1)²+ 1

q_c = 2σ_s b Q₁+1

2−arctan(¹₂Q₁²) π

!

(20)

Statistical vs. RH based cloud cover

Figure: Lohmann et al., JC, 1999

Prognostic cloud scheme

I Introduced by Tiedtke (1993):

∂qc

∂t = A(qc) +S_CV+S_BL+C−E−Gp−1 ρ

∂

∂z(ρw⁰q⁰_c)entr

∂C

∂t = A(C) +S(C)_CV+S(C)_BL+S(C)_C−D(C) (21)

I where A = transport, S - source by convection (CV), boundary layer turbulence (BL) and stratiform condensation (C), D - decrease of cloud area due to evaporation (E), G_p - generation of precipitation and the last term for condensation is the flux divergence due to cloud top entrainment of stratocumulus.

(8)

Prognostic statistical scheme

I Introduced by Tompkins (2002):

I Prognostic equations for the variance and skewness of theqt

PDF

Tompkins scheme

Processes affecting the total water PDF

Figure:Tompkins, JAS, 2002

I (a) Prior to cloud formation

I (b) Convective detrainment widens PDF

I (c) Precipitation generation narrows PDF

I (d) Mixing further reduces skewness

(9)

Example: Turbulence

I If a process is fast compared to a GCM timestep, an equilibrium can be assumed:

I

dq⁰_t²

dt =−2w⁰q_t⁰dq_t dz

| {z }

Source

− q_t⁰² τ

|{z}

Dissipation

(22)

I In equilibrium, the left hand side is zero, so that:

q_t⁰²=−2w⁰q_t⁰dqt

dzτ (23)

I Disadvantage: Can give good estimate in boundary layer, but above, other processes will determine variability, that evolve on slower timescales

Cloud water/ice mixing ratios

Current efforts in cloud parameterization focus on predicting the mass mixing rations of cloud waterq_l and iceq_i [e.g., Lohmann and Roeckner, Clim. Dyn., 1996]

dq_l

dt = Diff(q_l) +Adv(q_l) +Detr(q_l) +Q_cnd (24) +Qmlt−Qaut−Qacr−Qfrz−Qevp

dq_i

dt = Diff(q_i) +Adv(q_i) +Detr(q_i) +Q_dep (25)

−Qmlt−Qagg−Qacr+Qfrz−Qsub

where Diff=diffusion, Adv=advection, Detr=detrainment from convective outflow, cnd=condensation, mlt=melting,

aut=autoconversion, acr=accretion frz=freezing, evp=evaporation, dep=deposition, agg=aggregation, sub=sublimation

(10)

Autoconversion rates (rain formation in warm clouds) Derived from cloud-resolving models:

AUT = αq_l^4.7N_l^−3.3 Beheng(1994)

AUT = βq_l^2.5N_l^−1.8 Khairoutdinov&Kogan(2000)

Figure:Cloud droplet number concentration (Nl) = 40, 200, 1000 cm⁻³

Ways to obtainNl andNi: 1. Empirical relationships

Figure: a) marine clouds; b) stratiform continental clouds; c) convective continental clouds; d) all clouds [Boucher and Lohmann, 1995]

Ways to obtain N

l

and Ni : 2. Prognostic equations

[Lohmann, JAS, 2002]

dNl

dt = Diff(N_l) +Adv(N_l) +Detr(N_l) +Q_nucl (26) +Q_mlt^N −Q_aut^N −Q_self −Q_acr^N −Q_frz^N −Q_evp^N dN_i

dt = Diff(N_i) +Adv(N_i) +Detr(N_i) +Q_nuci (27)

−Q_mlt^N −Q_agg^N −Qacr+Q_frz^N −Q_sub^N where nucl=nucleation, self=self collection and^N refers to processes related to number, for example:

Q_frz^N =Q_frzN_l

q_l (28)

(11)

Cloud droplet nucleation

Derived from K¨ohler theory and observations:

N_l^t= 0.1

Na·w w+ 0.0023N_a

1.27

;Q_nucl=max N_l^t−N_l^t−1

∆t ,0

!

Figure: Green line:Nl= 375(1−exp[−0.00035Na])

Model tuning

I Basic criterion: Net radiation balance at the top of the atmosphere has to vanish (±1W m⁻²) in the long-term global mean

I Targets: SW≈LW≈235W m⁻²(best guess currently)

I Criteria for the choice of tuning parameters:

I Uncertainty

I Sensitivity

I Variation within “justifiable” limits

I Minimize the number of parameters

I For example tuneα’s in the autoconversion and aggregation rate in the prognostic equations forNl andNi above

Relatingthesubgridprocessestolarge-scalevariablesdependsonknowledgeofthefundamentalphysicsinvolvedintheprocess. Theonlymodelvariablesavailablefortheparameterizationsarethelarge-scaleﬁeldspredictedbythemodel. Anyphysicalprocess,suchasradiationorcumulusconv

Parameterizations of physical processes

Motivation

Parameterizations - general aspects

Parameterized processes

Outline of the 3 parameterization lectures

Governing equations in climate models

Boundary conditions

Parameterization of large-scale cloud cover

Assumptions/definitions

Partial cloud cover

Partial cloud cover

Cloud blinking

Statistical cloud schemes

Statistical cloud schemes

Statistical cloud schemes

Statistical cloud schemes

Choice of probability distribution function (PDF)

Choice of PDF

Statistical cloud schemes

Statistical vs. RH based cloud cover

Prognostic cloud scheme

Prognostic statistical scheme

Tompkins scheme

Processes affecting the total water PDF

Example: Turbulence

Cloud water/ice mixing ratios

Ways to obtain N

and Ni : 2. Prognostic equations

Cloud droplet nucleation

Model tuning

Model tuning

Evaluation of cloud parameterizations

Evaluation of cloud parameterizations