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Richard Forbes (With thanks to Adrian Tompkins and Christian Jakob ) forbes@ecmwf.int

Numerical Weather Prediction Parametrization of Subgrid Physical Processes Clouds (2) Sub-grid Cloud Cover (or “Sub-grid Moisture Variability”). Richard Forbes (With thanks to Adrian Tompkins and Christian Jakob ) forbes@ecmwf.int. GCM Grid cell 16-400km.

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Richard Forbes (With thanks to Adrian Tompkins and Christian Jakob ) forbes@ecmwf.int

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  1. Numerical Weather Prediction Parametrization of Subgrid Physical ProcessesClouds (2)Sub-grid Cloud Cover (or “Sub-grid Moisture Variability”) Richard Forbes (With thanks to Adrian Tompkins and Christian Jakob) forbes@ecmwf.int

  2. GCM Grid cell 16-400km 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)

  3. z ~500m ~100km x Macroscale Issues of Parameterization VERTICAL COVERAGE Most models assume that this is 1 This can be a poor assumption with coarse vertical grids (e.g. in climate models)

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

  5. z x Macroscale Issues of Parameterization Vertical overlap of cloud Important for radiation and microphysics interaction ~500m Random overlap Maximum overlap ~100km

  6. z x Macroscale Issues of Parameterization In-cloud inhomogeneity in terms of cloud water, particle size/number ~500m ~100km

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

  8. Csmall Clarge GCM grid box precipitation not equal in each case since autoconversion is nonlinear 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 condensate mass qland cloud fraction C The in-cloud mass mixing ratio isql/C Reminder: Autoconversion (Kessler, 1969) Complex microphysics perhaps a wasted effort if assessment of Cis poor

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

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

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

  12. cloudy qt RH=1 RH<1 x 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.

  13. qt RH=60% x 0 RH 60 80 100 Simple Diagnostic Cloud Schemes:Relative Humidity Schemes 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 1 C

  14. qt RH=80% x Simple Diagnostic Cloud Schemes:Relative Humidity Schemes 1 Add water vapour to the gridcell, the moistest part of the cell become saturated and cloud forms. The cloud cover is low. C 0 RH 60 80 100

  15. qt RH=90% x Simple Diagnostic Cloud Schemes:Relative Humidity Schemes 1 Further increases in RH increase the cloud cover C 0 RH 60 80 100

  16. RH=100% qt x 1 C 0 RH 60 80 100 Simple Diagnostic Cloud Schemes:Relative Humidity Schemes • The grid cell becomes overcast when RH=100%,due to lack of supersaturation • Diagnostic RH-based parametrization C =f(RH)

  17. 1 C 0 RH 60 80 100 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: Remember this for later! RHcrit = critical relative humidity at which cloud assumed to form (= function of height, typical value is 60-80%)

  18. Diagnostic Relative Humidity Schemes • Since these schemes form cloud when RH<100%, they implicitly assume subgrid-scale variability for total water, qt, (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 qt is such that, on average, we expect a cloud cover of Y%”.

  19. Diagnostic Relative Humidity Schemes • 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?

  20. Diagnostic Relative Humidity Schemes • 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? representative for all situations?) • Can we do better?

  21. Diagnostic Relative Humidity Schemes • Another example is the scheme of Slingo, operational at ECMWF until 1995. • 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?

  22. qt x G(qt) qt qs Statistical PDF Schemes • Statistical schemes explicitly specify the probability density function (PDF), G, for the total water qt (and sometimes also temperature) Cloud cover is integral under supersaturated part of PDF Sommeriaand Deardorff (1977), Mellor (1977)

  23. LIQUID WATER TEMPERATURE conserved during changes of state qs S qt T Statistical PDF Schemes • Others form variable ‘s’ that also takes temperature variability into account, which affects qs Cloud mass if T variation is neglected qs S is simply the ‘distance’ from the linearized saturation vapour pressure curve INCREASES COMPLEXITY OF IMPLEMENTATION

  24. Statistical PDF Schemes • Knowing the PDF has advantages: • Information concerning subgrid fluctuations of humidity and cloud condensate is available (for all parametrizations) , e.g. • More accurate calculation of radiative fluxes • Unbiased calculation of microphysical processes • Use of underlying PDF means cloud variables (condensate, cloud fraction) are always self-consistent. • Note, location of clouds within grid cell is not known.

  25. Cloud inhomogeneity in radiation scheme Can treat the inhomogeneity of in-cloud condensate and overlap in a consistent way between the cloud and radiation schemes Independent Column Approximation, e.g. MCICA Traditional approach (homogeneous)

  26. qs G(qt) qt cloud Cloud range precip generation Grid mean qL0 qL Cloud inhomogeneity in microphysics Most current microphysical schemes use the grid-mean or cloud fraction cloud mass (i.e: neglect in-cloud variability) Homogeneous Sub-grid PDF For example, Kessler autoconversion scheme: Result is not equal in the two cases since microphysical processes are non-linear In the homogeneous case the grid mean cloud is less than threshold and gives zero precipitation formation

  27. Building a statistical cloud schemeWhat do we observe? • Lack of observations to determine qt PDF • Aircraft data • limited coverage • Tethered balloon • boundary layer • Satellite • difficulties resolving in vertical • no qt observations • poor horizontal resolution • Ground-based radar/Raman Lidar • one location • Cloud Resolving models have also been used • realism of microphysical parametrization? Modis image from NASA website

  28. Aircraft Observed PDFs Wood and Field JAS 2000 Aircraft observations low clouds < 2km Height Heymsfieldand McFarquhar JAS 96 Aircraft IWC obs during CEPEX PDF(qt) qt

  29. Building a statistical cloud schemeObserved PDF of water vapour/RH Raman Lidar From Franz Berger

  30. Building a statistical cloud schemeObserved PDF example from aircraft PDF Data Example, aircraft data from Larson et al. 01/02 PDFs are mostly approximated by uni or bi-modal distributions, describable by a few parameters

  31. G(qt) qt Building a statistical cloud scheme • Need to represent with a functional form, specify the: (1) PDF shape (unimodal, bimodal, symmetrical, bounded?) (2) PDF moments (mean, variance, skewness?) (3) Diagnostic or prognostic (how many degrees of freedom?)

  32. PDF( qt) qt qt Uniform: Letreut and Li (91) Triangular: Smith QJRMS (90) qt qt Gaussian: Mellor JAS (77) s4 polynomial: Lohmann et al. J. Clim (99) Building a statistical cloud scheme(1) Specification of PDF shape Many function forms have been used symmetrical distributions: Bounded Unbounded: Can clip, but need additional parameters

  33. PDF( qt) qt qt qt Lognormal: Bony & Emanuel JAS (01) Gamma: Barker et al. JAS (96) Exponential: Sommeria and Deardorff JAS (77) qt qt Beta: Tompkins JAS (02) Building a statistical cloud scheme(1) Specification of PDF shape skewed distributions: Unbounded, always skewed Double Normal/Gaussian: Lewellen and Yoh JAS (93), Golaz et al. JAS 2002 Bounded, symmetrical or skewed

  34. Skewness Kurtosis saturation Moment 1=MEAN Moment 2=VARIANCE Moment 3=SKEWNESS Moment 4=KURTOSIS positive negative PDF(qt) cloud forms? negative positive qt e.g. HOW WIDE? Building a statistical cloud scheme(2) Specification of PDF moments Need also to determine the moments of the distribution: • Variance (Symmetrical PDFs) • Skewness (Higher order PDFs) • Kurtosis (4-parameter PDFs) Functional form – needs to fit data but be sufficiently simple

  35. (1-RHcrit)qs C G(qt) 1-C qt Building a statistical cloud scheme(3) Diagnostic or prognostic PDF moments • Some schemes fix the moments (diagnostic e.g. Smith 1990) based on critical RH at which clouds assumed to form. • Some schemes predict the moments (prognostic, e.g. Tompkins 2002). Need to specify sources and sinks. • If moments (variance, skewness) are fixed, then statistical schemes are identically equivalent to a RH formulation • e.g. uniform qt distribution = Sundqvistformulation Sundqvistformulation!!!

  36. convection turbulence microphysics dynamics Building a statistical cloud schemeProcesses that can affect PDF moments

  37. Example: Turbulence In presence of vertical gradient of total water, turbulent mixing can increase horizontal variability dry air moist air Rate of change of total water variance

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

  39. turbulence Building a statistical cloud schemePredicting change of qt variance due to turbulence If a process is fast compared to a GCM timestep, an equilibrium can be assumed, e.g. turbulence Local equilibrium Source Dissipation 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

  40. Building a statistical cloud schemeExample: Tompkins (2002) prognostic PDF • Tompkins (2002) prognostic statistical scheme (implemented in ECHAM5 climate GCM). • Prognostic equations are introduced for variables representing the mean, 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 qs

  41. Prognostic statistical scheme in action Evolution of stratocumulus cloud – Tompkins (2002) Minimum Maximum qsat

  42. Turbulence breaks up cloud Prognostic Statistical Scheme in action Minimum Maximum qsat

  43. Turbulence breaks up cloud Turbulence creates cloud Prognostic Statistical Scheme in action Minimum Maximum qsat

  44. Building a statistical cloud schemePredicting change of qt variance due to precipitation • Change in variance due to precipitation • However, the tractability depends on the PDF form for the subgrid fluctuations of qt, given by G. Where P is the precipitation generation rate, e.g:

  45. Some further 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 treat sedimentation ? • Is there a real advantage over existing cloud schemes ?

  46. Prognostic statistical PDF scheme:Which prognostic variables/equations? Take a 2 parameter distribution & partially cloudy conditions • (2) Can specify distribution with • Water vapour • Cloud water mass mixing ratio • (1) Can specify distribution with • Mean • Variance of total water qsat qt qsat Cloud Cloud ql qv Variance

  47. Prognostic statistical scheme:(1) Water vapour and cloud water ? qsat • Cloud water budget conserved. • Microphysical sources and sinks easier to parametrize. • Water vapour • Cloud water • mass mixing ratio ql qv But problems arise in... Overcast conditions (…convection + microphysics) qsat Clear sky conditions (turbulence) qv qv+ql

  48. Prognostic statistical scheme:(2) Total water mean and variance ? qsat • Mean • Variance • of total water • “Cleaner solution”. • But conservation of liquid water may be difficult (eg. advection) • Need to parametrize those tricky microphysics terms!

  49. Some further 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 treat sedimentation ? • Is there a real advantage over existing cloud schemes ?

  50. qs qcloud G(qt) y qt x Prognostic statistical PDF scheme:How do we treat ice (and mixed-phase) cloud? If supersaturation allowed, then the equation for cloud-ice no longer holds supersaturated clear region cloudy “activated” region sub-saturated region

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