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Numerical Weather Prediction Parametrization of Diabatic Processes Cloud Parametrization 2: Cloud Cover Richard Forbes and Adrian Tompkins

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2 Clouds in GCMs: What are the problems ? Many of the observed clouds and especially the processes within them are of subgrid-scale size (both horizontally and vertically) GCM Grid cell km

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3 ~500m ~100km Macroscale Issues of Parameterization VERTICAL COVERAGE Most models assume that this is 1 This can be a poor assumption with coarse vertical grids. Many climate models still use fewer than 30 vertical levels. x z

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4 ~500m ~100km Macroscale Issues of Parameterization HORIZONTAL COVERAGE, a Spatial arrangement ? x z

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5 ~500m ~100km Macroscale Issues of Parameterization Vertical Overlap of cloud Important for Radiation and Microphysics Interaction x z

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6 ~500m ~100km Macroscale Issues of Parameterization In-cloud inhomogeneity in terms of cloud particle size and number

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7 ~500m ~100km Macroscale Issues of Parameterization Just these issues can become very complex!!! x z

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8 Cloud Cover: Why Important? In addition to the influence on radiation, the cloud cover is important for the representation of microphysics Imagine a cloud with a liquid condensate mass q l The in-cloud mass mixing ratio is q l /a a large a small GCM grid box precipitation not equal in each case since autoconversion is nonlinear Reminder: Autoconversion (Kessler, 1969 ) Complex microphysics perhaps a wasted effort if assessment of a is poor

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9 q v = water vapour mixing ratio q c = cloud water (liquid/ice) mixing ratio q s = saturation mixing ratio = F(T,p) q t = total water (vapour+cloud) mixing ratio RH = relative humidity = q v / q s 1.Local criterion for formation of cloud: q t > q s This assumes that no supersaturation can exist 2.Condensation process is fast (cf. GCM timestep) q v = q s q c = q t – q s !!Both of these assumptions are suspect in ice clouds!! First: Some assumptions!

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10 Partial coverage of a grid-box with clouds is only possible if there is an inhomogeneous distribution of temperature and/or humidity. Homogeneous Distribution of water vapour and temperature: q x One Grid-cell Note in the second case the relative humidity=1 from our assumptions Partial cloud cover

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11 Another implication of the above is that clouds must exist before the grid-mean relative humidity reaches 1. q x cloudy= RH=1RH<1 Heterogeneous Distribution of T and q

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12 Heterogeneous Distribution of q only The interpretation does not change much if we only consider humidity variability Throughout this talk I will neglect temperature variability Analysis of observations and model data indicates humidity fluctuations are more important most of the time. qtqt x cloudy RH=1RH<1

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13 Take a grid cell with a certain (fixed) distribution of total water. At low mean RH, the cloud cover is zero, since even the moistest part of the grid cell is subsaturated qtqt x RH=60% RH C 1 Simple Diagnostic Cloud Schemes: Relative Humidity Schemes

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14 Add water vapour to the gridcell, the moistest part of the cell become saturated and cloud forms. The cloud cover is low. qtqt x RH=80% RH C 1 Simple Diagnostic Cloud Schemes: Relative Humidity Schemes

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15 Further increases in RH increase the cloud cover qtqt x RH=90% C 1 RH Simple Diagnostic Cloud Schemes: Relative Humidity Schemes

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16 The grid cell becomes overcast when RH=100%, due to lack of supersaturation qtqt x RH=100% C RH Simple Diagnostic Cloud Schemes: Relative Humidity Schemes

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17 Diagnostic Relative Humidity Schemes Many schemes, from the 1970s onwards, based cloud cover on the relative humidity (RH) e.g. Sundqvist et al. MWR 1989: RH crit = critical relative humidity at which cloud assumed to form (function of height, typical value is 60-80%) C RH

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18 Since these schemes form cloud when RH<100%, they implicitly assume subgrid-scale variability for total water, q t, (and/or temperature, T) However, the actual PDF (the shape) for these quantities and their variance (width) are often not known Given a RH of X% in nature, the mean distribution of q t is such that, on average, we expect a cloud cover of Y% Diagnostic Relative Humidity Schemes

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19 Advantages: –Better than homogeneous assumption, since clouds can form before grids reach saturation Disadvantages: –Cloud cover not well coupled to other processes –In reality, different cloud types with different coverage can exist with same relative humidity. This can not be represented Can we do better? Diagnostic Relative Humidity Schemes

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20 Could add further predictors E.g: Xu and Randall (1996) sampled cloud scenes from a 2D cloud resolving model to derive an empirical relationship with two predictors: More predictors, more degrees of freedom = flexible But still do not know the form of the PDF (is model valid?) Can we do better? Diagnostic Relative Humidity Schemes

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21 Diagnostic Relative Humidity Schemes Another example is the scheme of Slingo, operational at ECMWF until This scheme also adds dependence on vertical velocities use different empirical relations for different cloud types, e.g., middle level clouds: Relationships seem Ad-hoc? Can we do better?

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22 Statistical Schemes These explicitly specify the probability density function (PDF) for the total water q t (and sometimes also temperature) qtqt x qtqt PDF(q t ) qsqs Cloud cover is integral under supersaturated part of PDF

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23 Statistical Schemes Others form variable s that also takes temperature variability into account, which affects q s LIQUID WATER TEMPERATURE conserved during changes of state S is simply the distance from the linearized saturation vapour pressure curve qsqs T qtqt S INCREASES COMPLEXITY OF IMPLEMENTATION Cloud mass if T variation is neglected qsqs

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24 Statistical Schemes Knowing the PDF has advantages: –More accurate calculation of radiative fluxes –Unbiased calculation of microphysical processes However, location of clouds within gridcell unknown qtqt PDF(q t ) qsqs x y C e.g. microphysics bias

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25 Building a statistical cloud scheme Two tasks: Specification of the: (1) PDF shape (2) PDF moments Shape: Unimodal? bimodal? How many parameters? Moments: How do we set those parameters?

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26 Lack of observations to determine q t PDF –Aircraft data limited coverage –Tethered balloon boundary layer –Satellite difficulties resolving in vertical no q t observations poor horizontal resolution –Raman Lidar one location Cloud Resolving models have also been used realism of microphysical parameterisation? Modis image from NASA website Building a statistical cloud scheme TASK 1: Specification of PDF shape

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27 qtqt PDF(q t ) Height Aircraft Observed PDFs Wood and field JAS 2000 Aircraft observations low clouds < 2km Heymsfield and McFarquhar JAS 96 Aircraft IWC obs during CEPEX

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28 Conclusion: PDFs are mostly approximated by uni or bi- modal distributions, describable by a few parameters More examples from Larson et al. JAS 01/02 Note significant error that can occur if PDF is unimodal PDFData

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29 PDF of water vapour and RH from Raman Lidar From Franz Berger

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30 Building a statistical cloud scheme TASK 1: Specification of PDF shape Many function forms have been used symmetrical distributions: Triangular: Smith QJRMS (90) qtqt qtqt Gaussian: Mellor JAS (77) qtqt PDF( q t ) Uniform: Letreut and Li (91) qtqt s 4 polynomial: Lohmann et al. J. Clim (99)

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31 skewed distributions: q t PDF( q t ) Exponential: Sommeria and Deardorff JAS (77) Lognormal : Bony & Emanuel JAS (01) qtqt qtqt Gamma: Barker et al. JAS (96) qtqt Beta: Tompkins JAS (02) qtqt Double Normal/Gaussian: Lewellen and Yoh JAS (93), Golaz et al. JAS 2002 Building a statistical cloud scheme TASK 1: Specification of PDF shape

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32 Need also to determine the moments of the distribution: –Variance (Symmetrical PDFs) –Skewness (Higher order PDFs) –Kurtosis (4-parameter PDFs) qtqt PDF(q t ) e.g. HOW WIDE? saturation cloud forms? Moment 1=MEAN Moment 2=VARIANCE Moment 3=SKEWNESS Moment 4=KURTOSIS Skewness Kurtosis positive negative positive Building a statistical cloud scheme TASK 2: Specification of PDF moments

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33 Some schemes fix the moments (e.g. Smith 1990) based on critical RH at which clouds assumed to form If moments (variance, skewness) are fixed, then statistical schemes are identically equivalent to a RH formulation e.g. uniform q t distribution = Sundqvist form C 1-C (1-RH crit )q s qtqt G(q t ) Sundqvist formulation!!! where Building a statistical cloud scheme TASK 2: Specification of PDF moments

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34 Clouds in GCMs Processes that can affect distribution moments convection turbulence dynamics microphysics

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35 Example: Turbulence dry air moist air In presence of vertical gradient of total water, turbulent mixing can increase horizontal variability

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36 Example: Turbulence dry air moist air In presence of vertical gradient of total water, turbulent mixing can increase horizontal variability while mixing in the horizontal plane naturally reduces the horizontal variability

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37 Specification of PDF moments Example: Ricard and Royer, Ann Geophy, (93), Lohmann et al. J. Clim (99) Disadvantage: –Can give good estimate in boundary layer, but above, other processes will determine variability, that evolve on slower timescales turbulence Sourcedissipation local equilibrium If a process is fast compared to a GCM timestep, an equilibrium can be assumed, e.g. Turbulence

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38 Prognostic Statistical Scheme Tompkins (2002) prognostic statistical scheme (implemented in ECHAM5 climate GCM) Prognostic equations are introduced for the variance and skewness of the total water PDF Some of the sources and sinks are rather ad- hoc in their derivation! convective detrainment precipitation generation mixing qsqs

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39 Prognostic Statistical Scheme in action Minimum Maximum q sat

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40 Minimum Maximum q sat Turbulence breaks up cloud Prognostic Statistical Scheme in action

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41 Minimum Maximum q sat Turbulence breaks up cloud Turbulence creates cloud Prognostic Statistical Scheme in action

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42 Prognostic statistical scheme Production of variance from convection Klein et al. Change due to difference in variance Change due to difference in means Transport Also equivalent terms due to entrainment updraught We want this variance Detrained mass D = Convective Detrainment Rate

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43 Change in variance However, the tractability depends on the PDF form for the subgrid fluctuations of q, given by G: Where P is the precipitation generation rate, e.g: Prognostic statistical scheme Change in variance by precipitation

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44 Can quickly get untractable ! E.g: Semi-Lagrangian ice sedimentation Source of variance is far from simple, also depends on overlap assumptions In reality of course wish also to retain the sub-flux variability too Prognostic statistical scheme Complications - sedimentation

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45 Summary of statistical schemes Advantages –Information concerning subgrid fluctuations of humidity and cloud water is available –It is possible to link the sources and sinks explicitly to physical processes –Use of underlying PDF means cloud variables are always self-consistent Disadvantages –Deriving these sources and sinks rigorously is hard, especially for higher order moments needed for more complex PDFs! –If variance and skewness are used instead of cloud water and humidity, conservation of the latter is not ensured

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46 Issues for GCMs If we assume a 2-parameter PDF for total water, which prognostic variables should we use ? How do we treat the ice phase when supersaturation is allowed ? How do we link other processes with the total water PDF (microphysics, radiation, convection)? Is there a real advantage over existing cloud schemes ?

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47 Prognostic statistical scheme: Which prognostic equations? Take a 2 parameter distribution & partially cloudy conditions q sat Cloud Can specify distribution with (a)Mean (b)Variance of total water Can specify distribution with (a)Water vapour (b)Cloud water mass mixing ratio qvqv q l+i Cloud Variance qtqt

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48 q sat (a)Water vapour (b)Cloud water mass mixing ratio qvqv q l+i q sat qvqv q v +q l+i Cloud water budget conserved Avoids Detrainment term But problems arise in Clear sky conditions (turbulence) Overcast conditions (…convection + microphysics) (al la Tiedtke) Prognostic statistical scheme: Water vapour and cloud water ?

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49 q sat (a)Mean (b)Variance of total water Cleaner solution But conservation of liquid water compromised due to PDF Need to parametrize those tricky microphysics terms! Prognostic statistical scheme: Total water mean and variance ?

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50 Issues for GCMs If we assume a 2-parameter PDF for total water, which prognostic variables should we use ? How do we treat the ice phase when supersaturation is allowed ? How do we link other processes with the total water PDF (microphysics, radiation, convection)? Is there a real advantage over existing cloud schemes ?

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51 qtqt PDF(q t ) qsqs q cloud If supersaturation allowed, then the equation for cloud-ice no longer holds Ice cloud ? x y sub-saturated region supersaturated clear region cloudy activated region

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52 Issues for GCMs If we assume a 2-parameter PDF for total water, which prognostic variables should we use ? How do we treat the ice phase when supersaturation is allowed ? Is there a real advantage over existing cloud schemes ? How do we link other processes with the total water PDF (microphysics, radiation, convection)?

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53 Overheard recently in the ECMWF meteorological fruit bowl….… But a prognostic statistical scheme could provide consistent sub-grid information for all our physical parametrizations! We already have cloud variability in our models. Nearly all components of GCMs contain implicit/explicit assumptions concerning subgrid fluctuations, e.g: RH-based cloud cover, thresholds for precipitation evaporation, convective triggering, plane parallel bias corrections for radiative transfer… etc.

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54 Advantage of Statistical Scheme Statistical Cloud Scheme Radiation MicrophysicsConvection Scheme Can use information in other schemes Boundary Layer

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55 Variability in Clouds Models typically have many independent tunable parameters with a limited number of metrics for verification. M tuning parameters N Metrics e.g. Z500 scores, TOA radiation fluxes With large M, task of reducing error in N metrics becomes easier, but not necessarily for the right reasons Solution: Increase N, or reduce M Using a statistical cloud scheme with an underlying PDF of sub-grid variability would bring greater consistency between processes and reduce the number of independent tunable parameters M.

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56 Issues for GCMs If we assume a 2-parameter PDF for total water, which prognostic variables should we use ? How do we treat the ice phase when supersaturation is allowed ? Is there a real advantage over existing cloud schemes ? How do we link other processes with the total water PDF (microphysics, radiation, convection)?

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57 Use of Cloud PDF in Radiation Scheme Independent Column Approximation, e.g. MCICA Can treat the inhomogeneity of in-cloud condensate and overlap in a consistent way between the cloud and radiation schemes Traditional approach

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58 Result is not equal in the two cases since microphysical processes are non-linear Example on right: Autoconversion based on Kessler Grid mean cloud less than threshold and gives zero precipitation formation Cloud Inhomogeneity and microphysics biases qLqL q L0 cloud range mean precip generation Most current microphysical schemes use the grid-mean or cloud fraction cloud mass (i.e: neglect in-cloud variability) qtqt G(q t ) qsqs cloud Independent column approach? – computationally expensive!

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59 Summary GCMs need to have a representation of the sub-grid scale variability of the atmosphere (e.g. q, T) A prognostic statistical cloud scheme that can provide consistent sub-grid heterogeneity information across the model physics is an attractive concept (closer to the real world!) There are different implementations with different complexities and degrees of freedom. More degrees of freedom allow greater flexibility to represent the real atmosphere, but we need to have enough knowledge/information to understand and constrain the problem (form of pdf/sources/sinks)!

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60 Next time: The ECMWF Scheme ECMWF Scheme - Considers the physical processes and derives source or sink terms for cloud fraction (cover) and cloud condensate. HOW? To do this, assumptions about a distribution PDF and its moments are made for many of the terms Thus the ECMWF scheme is essentially another expression of a prognostic statistical scheme - the approach has advantages and disadvantages... convection turbulence dynamics microphysics

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