MOSAC-14

12-13 November 2009

Paper No. 14.5

Stochastic backscatter in the UnifiedModel

by

Glenn Shutts

1

1 Introduction

The growing popularity of stochastic parametrization (or ‘stochastic physics’) has been largelymotivated by deficiencies in medium-range ensemble prediction systems, seasonal forecastingand in climate modelling. A requirement for successful ensemble forecasting is that the spreadof member forecast states about the ensemble mean should match the mean forecast error.Spread, measured as a root-mean square difference, arises through the effects of initial condi-tion perturbations and model equation differences (stochastic or otherwise). A characteristicof all current operational ensemble prediction systems is that they have too little spread us-ing initial condition perturbations alone. The same problem been found at seasonal forecastingtimescales (Weisheimer et al, 2009). An under-spread ensemble prediction system (EPS) hasmember forecasts that are too similar and a significant possible meteorological event at day 5(e.g. European blocking) could be absent from all members. Palmer (2001) has speculated thatthe residual systematic error in climate models may result from a lack of stochasticity in physi-cal parametrization. Random perturbations with zero mean can, through nonlinear rectification,generate non-zero mean flow perturbations and it is possible that the absence of this stochasticitycould lead to systematic error.

This lack of spread in ensemble forecasts implies that the forecast model equations are in-sufficiently representing the chaotic nature of many atmosphere physical processes. The reasonsfor this remain unanswered though most of the current effort in stochastic parametrization hasbeen focussed on testing and tuning plausible representations of model error rather than at-tempting to understand the physical nature of the problem. The spread deficiency is greatest inthe tropics, consistent with the lack of low-frequency variability there (e.g. Madden-Julian os-cillation) and suggests that convection parametrization is mainly damping large-scale equatorialwave motion rather than exciting it. Another possibility is excessive energy dissipation throughnumerical diffusion (explicit or implicit) or parametrization which could suppress the growth ofsub-synoptic scale instabilities and remove energy that might have otherwise cascaded to largescales. Lastly, it is possible that the uncertainty associated with the representation of cloudsand radiation should be accounted for in modelling weather and climate. For instance, the well-known ‘forecaster’s nightmare’ in which stratocumulus cloud sheets may or may not break upin settled anticyclonic conditions. The existence of any such critical phenomena (e.g. triggeringof convective storms is another one) implies substantial sensitivity to small perturbations andtherefore model uncertainty.

Buizza et al (1999) addressed physical parametrization uncertainty by multiplying the ten-dencies of wind,temperature and water vapour mixing ratio by respective random number fields.The random numbers are spatially-correlated by assuming a piecewise constant representation(10 degree lat./lon. boxes) and are updated every 6 steps. The numbers were drawn from auniform probability distribution function (pdf) with range 0.5 to 1.5. The scheme has recentlybeen revised (Leutbecher, unpublished work) and now uses a spectral pattern generator drivenforward in time by a first-order autoregressive process amongst other modifications.

A different approach inspired by the stochastic backscatter scheme of Mason and Thomson(1992) considers model error to be due to unphysical energy sinks in the model and aims to injecta fraction of the energy dissipated back into the model (Shutts, 2005). Kinetic energy backscatterschemes are now used operationally at Environment Canada (Houtekamer et al, 2007) and theMet Office (Bowler et al, 2009) whilst at ECMWF a spectral backscatter scheme (SPBS) isbeing tested for implementation in the EPS (Berner et al, 2009). Differences in the approach atthese these centres relate to the spatial and temporal characteristics of the underlying forcingfield (typically through the rotational wind using a streamfunction forcing pattern) and thecalculation of a dissipation rate function that modulates the strength of the forcing.

2

In recognition of the uncertainty associated with the representation of mesoscale convectivesystems (MCSs) and their efficient upscale energy transfer, both the Met Office (MOGREPS) andUS Navy NOGAPS EPS include stochastic convection representations. The Met Office scheme(Stochastic Convective Vorticity; Bowler et al, 2008) randomly injects vertically-orientated po-tential vorticity dipoles (consistent with the known structure of MCSs) into regions of highConvective Available Potential Energy (CAPE). The US Navy scheme (Teixeira and Reynolds,2008) perturbs convective parametrization tendencies in a similar way to Buizza et al (1999)although does not impose horizontal spatial or temporal correlation scales.

As an alternative to the Buizza et al ‘perturbed tendency’ approach, the Met Office hasdeveloped a ‘Random Parameters’ (RP) scheme which targets important uncertain parametersin physical parametrization schemes. For instance the critical relative humidity parameter thatcontrols the functional dependence of cloud cover fraction on relative humidity is allowed tovary in time (as a first-order autoregressive process) between judiciously-chosen upper and lowerlimits.

The above techniques that account for model error are heuristic and essentially withoutrigorous physical basis. Neither is it entirely clear to what degree the stochastic perturbationsrepresent statistical fluctuations about an ensemble average or ‘knowledge uncertainty’ for whichthe value of some key parameter has large error bars. Frequently, parametrization schemes aredeveloped from idealized models whose key parameters cannot be readily related to complex, realsituations (e.g. the meaning of the mountain Froude number in complex terrain). In that caseit is the structural uncertainty of the parameterization scheme that is at issue not the numericaluncertainty of an essentially unmeasureable parameter. This makes it difficult to regard RandomParameters as more than a technical device.

2 Stochastic Kinetic Energy Backscatter scheme - SKEB2

2.1 Outline

The SKEB2 scheme is shortly due to replace the original stochastic backscatter scheme (SKEB1)developed by Alberto Arribas and implemented in MOGREPS in June 2006. With respect toSKEB1, SKEB2 has a substantially improved mathematical formulation with a spectral patterngenerator and energy dissipation rate calculation that includes a contribution from deep convec-tion. SKEB1 only addresses the numerical energy sink and uses the local kinetic energy as aconvenient proxy for its magnitude (thereby avoiding costly flow deformation rate calculationsand subsequent smoothing). SKEB2 derives from work originally done in the Met Office about10 years ago (Evans et al, 1998) and uses code written at that time. It parallels the developmentof the SPBS scheme at ECMWF but has several important differences in design and implemen-tation e.g. it includes velocity potential forcing; has no gravity wave/mountain drag component;has different formulations for both numerical and convective dissipation rates, and has a differentvertical structure algorithm.

SKEB2 computes a streamfunction forcing pattern field (Fψ) defined to provide a globally-uniform energy input rate of Btot (J.kg−1.s−1) with a spectral power distribution given by χ(n)where n is the degree of the associated Legendre function in a spectral expansion in terms ofspherical harmonics. Triangular truncation of the spherical harmonic expansion ensures that Fψis rotationally invariant and that the statistical properties of Fψ are homogeneous.

Fψ is calculated using:

3

F jψ =

N∑

m=−N

N∑

n=|m|

f jm,nP|m|n (µ) exp(imλ) (1)

where m is the zonal wavenumber, n is the degree of the associated Legendre function Pmn (µ),

f jm,n is the spectral coefficient satisfying the reality condition

f jm,n = f j∗−m,n

where the superscript ∗ denotes the complex conjugate , µ is the sine of the latitude, λ is thelongitude and j is the number of timesteps of length ∆t.

The spectral coefficients in eq. (1) are evolved according to the first-order autoregressiveprocess:

f j+1m,n = [1 − α(n)]f jm,n +

√

α(n) F0 χ(n)rjm,n (2)

where F0 is a constant that ensures an ensemble-average energy input rate of Btot and α(n)defines a wavenumber-dependent decorrelation time τ(n) through:

α(n) = 1 − exp(−∆t/τ(n)). (3)

If Dtot represents a certain specification of the model dissipation rate and bR is the fractionof this that should be backscattered (i.e. the backscatter ratio), then one can define a modulatedstreamfunction forcing Fψ by:

Fψ(λ, µ, z) =

(

bRDtot(λ, µ, z)

Btot

)1

2

Fψ(λ, µ, z). (4)

The modulated streamfunction forcing function Fψ provides a local energy input rate ofbRDtot so that areas with high dissipation will receive the largest streamfunction (or equivalently,vorticity) perturbations. Unlike its use in LES backscatter, the dissipation rate Dtot is meantto address those energy conversions that lead to the production of sub-filter scale kinetic energyand not thermal energy. For instance, boundary layer dissipation - the dominant sink of kineticenergy in the atmosphere - is excluded from Dtot as the characteristic eddy scale is orders ofmagnitude smaller than the filter scale of an NWP model.

Fψ is used to compute gridpoint u and v tendencies and these are multiplied by µ whichcrudely represents the latitudinal dependence of energy partitioning into rotational and diver-gent modes. Alongside a streamfunction forcing, SKEB2 also computes a velocity potentialforcing by similar reasoning. The divergent wind tendencies are correspondingly multiplied bycosine(latitude) so that their greatest impact is in the tropics..

2.2 Calculation of the total dissipation rate

Numerical dissipation, whether implicit or explicit, consumes disturbance energy in the filter-scale of an atmospheric model. The semi-Lagrangian advection algorithm for instance involvesinterpolation of fields to the departure point and this acts to smooth model fields and removeenergy (implicit dissipation). Also, horizontal diffusion may be implemented explicitly to smoothmodel fields. This numerical energy drain rate is unlikely to correspond to the real energydissipation rate given its artificial nature. The backscatter methodology supposes that the ‘drainrate’ should be matched by an upscale energy flow with the real dissipation rate being the smalldifference between them.

4

The numerical dissipation rate in SKEB2 is assumed to be equivalent to that implied bythe two-dimensional Smagorinsky horizontal diffusion scheme (Smagorinsky, 1963). This was apragmatic choice made possible by the existence of backscatter code used in the Met Office LargeEddy Model which uses the Smagorinsky-Lilly formulation of diffusion in a three-dimensionalcontext. The ECMWF SPBS scheme by contrast uses the mean-square vorticity gradient multi-plied by an effective bi-harmonic diffusion coefficient.

In SKEB2 then, the numerical dissipation rate (Dnum) is given by:

Dnum = (kH∆)2D3 (5)

where kH is treated as a tuning parameter, ∆ is the gridlength,

D =√

(D2S + D2

T ) (6)

with

DS =∂v

∂x+

∂u

∂y(7)

and

DT =∂u

∂x+

∂v

∂y. (8)

The convective dissipation rate Dcon represents a small-scale source of kinetic energy inthe filter scale. Kinetic energy generated by buoyancy forces in convective clouds can go intogenerating gravity waves and rotationally-balanced motions and the remainder will be dissipatedin turbulence. Convective parametrization schemes don’t explicitly address the fate of the kineticenergy released and so it must be considered to be dissipated. The fraction of energy retainedin balanced modes (e.g. meso-vortices) depends crucially on the amount of convective masstransfer in individual cloud systems and the magnitude of the background potential vorticity.Mesoscale convective systems and hurricanes represent extreme cases for which the efficiency ofbalanced flow production is very high. The dissipation rate contribution from deep convectionparametrization may be related to the rate of detrainment of convective kinetic energy ratherthan the rate of conversion of cloud kinetic energy to sensible heat energy.

The following expression for Dcon is heuristically motivated to associate regions of convectivemass flux entrainment/detrainment with the production of mesoscale kinetic energy:

Dcon = γ

∣

∣

∣

∣

∂Mc

∂z

∣

∣

∣

∣

Hc × CAPE (9)

where Mc is the convective mass transfer, Hc is the depth of the deep convection and γ is aparameter that depends on the mass flux profile.

With these definitions the total energy dissipation rate Dtot is given by :

Dtot = Dnum + Dcon (10)

and since it is a noisy field it is smoothed before being used in eq.(4).

5

2.3 Determination of the spectral power distribution function χ(n) -

the coarse-graining technique

The spectral power in the streamfunction forcing is governed by contributions from the patterngenerator Fψ and the smoothed total dissipation rate. The spectral function χ(n) is used tocontrol this power distribution and previous coarse-graining studies using a big-domain cloud-resolving model had suggested a power law dependence on n.

2.3.1 Methodology

Ideally, observational datasets from intensive field experiments would provide sufficient infor-mation about the statistical nature of physical processes in the atmosphere for improving theirrepresentation in NWP and climate modelling. For instance probability distribution functions(pdfs) for convective cloud population density (conditioned on CAPE, vertical wind shear, un-derlying land/sea, topography etc) could be determined and used to devise stochastic convectionparametrization schemes. Such datasets that do exist are generally insufficient for this purposeand it is necessary (and more convenient) to use numerical model simulation data. For instanceShutts and Palmer (2007) used a cloud-resolving model, configured to simulate deep convectionin a square domain of side length 7680 km, to compute pdfs of coarse-grained convective warmingrate. These pdfs were examined as a function of the strength of parametrized convective warm-ing based on the coarse-grained field values and used to validate the statistical assumption uponwhich the stochastic physics scheme is based i.e. that the standard deviation is proportional tomean parametrized tendency.

The coarse-graining procedure used for the cloud-resolving model involves averaging modelfields and their associated tendencies to a grid resolution typical of current NWP or climatemodels. By the Reynolds-averaging approach outlined below it is possible to compute the non-advective part of the total tendency that the corresponding coarse-grid model would see.

For the purposes of estimating a streamfunction forcing field the coarse-grained u momentumequation takes the form:

∂u

∂t+ V · ∇u − βyv +

∂

∂x

(

p′

ρ0

)

=[

V · ∇u −V · ∇u]

+ Fu = Fu (11)

where β is the meridional gradient of the Coriolis parameter at the equator; p′ is the perturba-tion pressure and ρ0(z) is a height-dependent reference density in this quasi-Boussinesq approxi-mation of the full momentum equation. Fu represents the combined effects of a Smagorinsky-Lillyturbulent diffusion scheme and a supplementary horizontal Laplacian diffusion. The first twoterms on the right-hand side represent the divergence of a Reynolds stress appropriate to thecoarse-graining scale. In this nomenclature the streamfunction forcing Fψ is defined as:

Fψ = ∇−2

(

∂Fv∂x

−∂Fu∂y

)

(12)

where the inverse Laplacian operator ∇−2 is applied in spectral space by expanding its operandas a double Fourier series.

2.3.2 Coarse-graining Large-Eddy Model simulations

The simulations to be described have been carried out with version 2.3 of the Met Office LEM(Large Eddy Model). The model uses an Arakawa C-grid staggering in horizontal planes and

6

Lorenz grid in the vertical with periodicity assumed in both horizontal directions . In all exper-iments, 50 vertical levels are deployed non-uniformly with height with resolution ranging from150 m in the boundary layer to 500 m in the free troposphere to 800 m in the stratosphere. TheCoriolis parameter is made a linear function of y with the equator at y = 0 thereby achievingan equatorial beta-plane. The domain is 10,000 km in the x direction and 5000 km in the ydirection with horizontal gridlength ∆x = ∆y = 2.44 km. The lower surface is treated as seawith a temperature SST (y) that varies parabolically in y according to:

SST (y) = 301.0− a(y/y0)2 (13)

where y0 is the meridional domain half-width (equal to 2500 km), a = 3.8125 and the units are inKelvin. Convection is forced by imposing a horizontally-uniform cooling function of −1.5 K/dayup to a height of about 11 km and then tailing off to zero by 15 km. This cooling function is usedin place of the model’s radiation scheme in order to reduce the considerable computational cost.The initial state is horizontally-stratified with a uniform geostrophic easterly wind of -5 ms−1.The imposed meridional pressure gradient that ensures initial balance is held fixed during theintegration and so acts as a kind of Trade Wind forcing function. Convection is initiated withsome small lower tropospheric temperature perturbations. The model configuration used here,together with some others that use anisotropic horizontal grids, is more fully described in Shutts(2006).

Convection develops in an equatorial band but after a few days splits into the familiar ‘double-ITCZ’ pattern with precipitation maxima at about 15 degrees from the equator (ITCZ is theInter-Tropical Convergence Zone). Figure 1 shows a Hovmoller plot of rain-rate averaged be-tween 10 degrees north and south. Rainfall rates increase up to day 3 but fall afterwards as itsdouble-ITCZ structure develops and moves outside of the Hovmoller averaging zone. Westwardpropagating rain cells dominate up to day 3 but thereafter, short zonal wavelength, eastwardpropagating convective clusters are evident with speeds of about 17 ms−1. As in the study ofShutts (2006), these are most likely to be driven by Kelvin waves. Beyond day 7, larger scaleconvective cloud clusters dominate with zonal wavelengths of 2000 to 4000 km and eastwardphase speeds of 7 to 11 ms−1. These systems have more in common with the cloud clustersassociated with the Madden-Julian oscillation - a desirable feature to have within the datasetsused for coarse-graining.

The initial easterly flow evolves into a pair of upper tropospheric westerly jets near themeridional limits of the domain and easterly jets near 8 degrees north and south (Figure 2).These easterly jets appear to result from the Coriolis torque acting on equatorward flowing airdriven by the outflow from the two ITCZs.

The streamfunction forcing, computed on a 40 km grid at a height of 10.5 km, is shown inFigure 3. The strongest variance in streamfunction forcing occurs in the jetstream regions andtypical gradients of about 20 m2s−2 per 1000 km imply flow accelerations of the order of a fewms−1 per day. Details of the spectral power distribution of the streamfunction forcing will begiven in the next section together with a similar estimate obtained from the IFS.

2.3.3 Coarse-graining ECMWF IFS forecasts

The approach taken here is to compute differences between forecasts made with the ECMWFIntegrated Forecast System (IFS) at two radically different horizontal resolutions. The highresolution run (here taken to be T1279 corresponding to a horizontal gridlength of ∼ 16 km) isregarded as ‘truth’ and its total (dynamical + physical) tendencies are spectrally-smoothed tothe resolution of the low resolution model (here T159 with an equivalent gridlength of ∼ 128 km). The difference between the coarse-grained tendency and the low-resolution model tendency

7

0

13.5

t

x (km) 10000

(days)

Figure 1: Hovmoller plot of the surface rain-rate averaged within a 20 degree band centred on theequator (Units: mm/hr). The dashed (dotted) line has a slope matching the rate of movementof a convective cluster moving eastwards and corresponds to a speed of 7.4 ms−1 (10.9 ms−1).At earlier times the solid line represents the movement of smaller zonal wavelength features thatare most likely to be Kelvin waves and corresponds to a phase speed of about 17 ms−1.

Figure 2: u at a height of 10.5 on day 7. Units: ms−1

8

Figure 3: The streamfunction forcing Fψ computed from the effective momentum forcing functionat a height of 10.5 on day 7 found when the momentum equation is coarse-grained to a 40 kmgrid. Units: m2s−2

can be considered to be a tendency error (see Hermanson et al, 2009) who carried out a relatedanalysis using the IFS). As with the cloud-resolving model, the aim will be to compute thestreamfunction forcing that is implied by the model coarse-grained tendency errors in u and v.

At a detailed level the procedure is as follows. Global gridpoint tendencies fields of u and vare transformed in a single spectral transform routine call to spectral (Spherical Harmonic (SH))coefficients of vorticity and divergence tendency. The vorticity tendency coefficients are dividedby −n(n + 1)/a2 (where n is the spherical harmonic degree and a is the Earth’s mean radius) togive streamfunction forcing SH coefficients. Subtracting the streamfunction forcing coefficientsfor T1279 and T159 (and neglecting those with n > 159 ) defines the streamfunction forcing‘error’ of the T159 forecast. Since the timestep of the T1279 run is 8 times smaller than the 1hour timestep of the T159 run, the T1279 streamfunction forcing is averaged over the single stepof the T159 run.

Streamfunction forcing power spectra have been computed from T+2 hour fields of IFSforecasts starting at 12Z August 17 2006 and from coarse-graining the momentum equationon day 7 of the LEM simulation (Figure 4) The cloud-resolving model curve has somewhat lesspower than that implied by the IFS coarse-graining but similar spectral slope. Considering thedifferences in the LEM and IFS simulations (global forecast versus idealized tropical simulation;parametrized versus explicit convection), it is hardly surprising to find some spectral powerdifferences in the computed streamfunction forcing. Both curves suggest power law behaviourbut particularly the IFS coarse-graining calculation. The red dashed line represents a spectralslope of −1.45 and this can be used to deduce the exponent in the power law assumed for χ(n).If µ is the exponent then the total spectral power goes as 2µ + 1 (= −1.45) implying thatµ = −1.225. Based on earlier, less realistic LEM simulations, a value of -1.27 is currently usedin SKEB2.

2.3.4 Conclusions

The coarse-graining methodology provides a way of designing and calibrating stochastic kineticenergy backscatter schemes.These preliminary results suggest that the current formulation ofstochastic backscatter is reasonably well supported by the coarse-graining analyses. Future workwill be directed at estimating the spectral power of Fψ in the wavenumber-frequency domain so

9

100

1000

10000

100000

1e+06

1e+07

1e+08

1e+09

1e−07 1e−06 1e−05 0.0001

a*p

ower

(n/a

) (m3 .s

−2)

wavenumber (n/a) (rad.m−1)

IFS T1270 − T159LEM 156x156 km

SKEB2

Figure 4: Power spectrum of the streamfunction forcing at level 53 (about 256 hPa) deducedfrom IFS (black line) and cloud-resolving model (blue line) coarse-graining analysis. The redline is the spectral power in the streamfunction forcing used in SKEB2 - also at about 250 hPa.The dashed black line corresponds to an approximate power law fit of the coarse-graining curvesand implies a power law with exponent equal to −1.43.

that the decorrelation time can be made a function of wavenumber. It will also be necessary tobe more careful in distinguishing between systematic and random contributions to the stream-function forcing. The coarse-graining methodology can be applied to temperature and humiditytendencies and it is planned to calibrate the revised stochastic physics scheme in this way.

3 Diagnostics and results using SKEB2

The streamfunction forcing is the product of an evolving isotropic pattern field and a modulatingfactor equal to the square root of the total dissipation rate. The dissipation rate in turn iscomposed of numerical and convective components whose 3-month (October-January) mean areshown in Figure 5. The numerical dissipation rate is largest outside of the tropics and, notunexpectedly, is largest in the storm track regions of the eastern sides of North America and Asia.The convective dissipation rate is by contrast, greatest in the tropics with maxima over Africa,the Indian Ocean, Maritime continent, ITCZ and South Pacific convergence zone. To get someappreciation of the spatial distribution of SKEB2 forcing, Figure 6 shows cross-section snapshotsof the wind speed increments (i.e. magnitude of the vector wind increment). The zonal-mean,latitude-height section shows the largest increments concentrated near the tropopause althoughsomewhat lower in the tropics where parametrized convective mass detrainment dominates. Largevalues in the tropical boundary layer are associated with the net source of convective mass fluxwhich is manifest in large positive values of ∂Mc/∂z. The global distribution of wind increment

10

at a height of 7.6 km is quite patchy with typical values of around 0.2 ms−2 which implies aforce per unit mass of 14.4 ms−1 per day. The longitude-height section at 15 S shows wind speedincrements associated with deep convection with lower and upper maxima associated with massentrainment and detrainment respectively.

Ensemble forecasts using SKEB2 have been made in the MOGREPS-G configuration with23 perturbed forecasts and one control forecast. Initial condition perturbations are generatedusing the Ensemble Transform Karman Filter (ETKF) method. 3-day forecasts in the operationalconfiguration (which use SKEB1 together with the random parameters scheme and the stochasticconvective vorticity scheme) are compared with forecasts using SKEB2 and random parametersbut no stochastic convective vorticity scheme. A total of 31 ensemble forecasts are made (eachday in May 2008) for SKEB2 and operational cases.

Figures 7- 9 show the improvements in ensemble forecast skill that result from SKEB2 (solidlines) relative to the current operational configuration (dashed lines). ‘Skill’ is measured simplyas the root-mean square (r.m.s) error of the ensemble mean and curves for the ‘spread’ aboutthe ensemble mean are included to show how increased spread is consistent with improved skill.The red curve in these plots shows the r.m.s. error of the unperturbed control forecast. At thistime we don’t have any probabilistic skill scores to show.

Results for temperature at 850 hPa in latitude ranges that roughly correspond to northern

hemisphere, tropics and southern hemisphere are shown in Figure 7. The spread, as representedby the green curves, is markedly smaller than the r.m.s. error showing that the ensemble systemis under-dispersive. The SKEB2 formulation however shows some useful improvement in spreadfor all three regions and this leads to some small reduction in the error in the ensemble mean.The error of the control is somewhat greater than the error of the ensemble mean in all cases.

Figure 8 shows the spread and skill for geopotential height at 500 hPa (omitting the tropics forwhich this field is too bland). Only small improvements in ensemble mean error can be detectedby day 3 although the spread is increased quite significantly in the northern hemisphere. In thesouthern hemisphere the spread at times less than 30 hours exceeds the error, otherwise the sameremarks apply.

The impact of SKEB2 on the east-west wind component (Figure 9) is similar to that of 850hPa temperature except that in the tropics the r.m.s error is significantly improved. On theother hand the very slow growth in spread and over-dispersion in the first day of the forecastare disappointing. The UM does not seem to be able to grow the perturbations injected into thetropical atmosphere.

4 Prospects for improved stochastic parametrization

A distinguishing feature of stochastic parametrization is the need to break away from the verticalcolumn representation that characterizes conventional ‘deterministic’ parametrization. Perturb-ing parametrization tendencies at the grid scale is ineffective in generating ensemble spreadbecause of their weak projection onto the balanced modes that underpin NWP. The requirementthat tendency perturbations be correlated in the horizontal brings with it technical problemsassociated with parallel computing and so efficient coding will be a major challenge.

Given the recent positive experience at ECMWF with their revised stochastic physics scheme(now renamed ‘Stochastically-Perturbed Physical Tendencies’ (SPPT) ) a similar scheme shouldbe developed for the UM to compare with the Random Parameters scheme. Although thisapproach remains controversial and not favoured by many who work in the field of physicalparametrization, the benefits in terms of improved spread and skill scores are hard to argueagainst. The Random Parameters scheme requires recalibration with every physics upgrade

11

Figure 5: Column-integrated numerical and convective dissipation rates (as defined in the SKEB2scheme) averaged from October 1989 to January 1990. Units: Wm−2

12

Figure 6: Magnitude of the vector wind increment plotted in sections : latitude-height sectionof the zonal average (left); global field at z = 7.6 km(upper right), and longitude-height sectionat 15 S (lower right). Units : ms−1

13

Figure 7: Spread and r.m.s. error of the temperature at 850 hPa for the regions 90 to 18.75degrees (upper); 18.75 to -18.75 degrees (middle) and -18.75 to -90 degrees. The blue linesrepresent the r.m.s. error of the ensemble mean (with respect to analyses) and the green linesrepresent the spread about the ensemble mean; solid lines (SKEB2), dashed lines (SKEB1). Allensemble forecasts use ETKF and include the RP scheme. The red line shows the r.m.s. errorof the deterministic forecast. The curves are the result of averaging 31 forecasts for each day inMay 2008. Units: K.

14

Figure 8: same as for Fig. 7 except for 500 hPa geopotential height and omitting the tropics.Units: m

15

Figure 9: same as for Fig. 7 except for u at 250 hPa. Units: ms−1

16

whilst SPPT is built on top of the deterministic parametrization formulations and requires fewerparameters to be adjusted. RP also contains some assumptions that cannot be justified physicallynamely that the parameters vary in time though not space. The Canadians are already using avariant of SPPT in their operational EPS alongside their version of SKEB.

Further development work is required for SKEB2 to bring it into line with the results of thecoarse-graining studies. In particular it will be necessary to improve the representation of thevertical structure of the streamfunction forcing. At present the forcing is too highly correlated inthe vertical - an undesirable feature and one that would be harmful for ensemble data assimilation.Another unexplored area is backscattering temperature and humidity variance. The CanadianEPS uses an implementation of temperature backscatter alongside kinetic energy backscatteralthough it’s impact is not thought to be great.

The stochastic backscatter approach has been tested in many configurations at EPS ECMWFand has proven to perform best at lower resolutions (e.g. T159 and T255) relative to SPPT.At T399 resolution SPPT tends to give better EPS probability skill scores than SPBS giventhe same amount of spread. Without using long decorrelation times (e.g. 30 days) both SPPTand SPBS have only weak impacts at the seasonal timescale. This finding is supported by earlyresults using SKEB2 in 10 year HADGEM3 runs which show only small impacts on systematicerrors. Stochastic forcing with decorrelation times of the order of one month will obviously leadto more spread in ensemble-based seasonal forecasting yet the justification for such an assumptionis lacking.

In view of the heuristic approach taken to stochastic parametrization in NWP so far thereis a pressing need to invest more effort into trying to understand the physical causes of the lackof spread seen in ensemble prediction. Evidence so far suggests that the error bars on somephysical parametrizations might be much larger than imagined. It should be incumbent uponthose working in physical process parametrization to quantify the uncertainty in their algorithmsand aim to express this in terms of probability distribution functions. For the purposes ofdeterministic forecasting, the mode of the distribution could be used to define parametrizationincrements whereas for ensemble prediction the distribution function could be sampled randomlyfor each ensemble member. A successful stochastic parametrization would complement a forecastmodel’s explicit grid-scale variability so that the resulting pdfs (e.g. of rain-rate ) comparefavourably with observation.

With a growing emphasis on ensemble forecasting at all time scales we need to bring greaterconsideration of the above issues to our research programme.

5 Acknowledgments

I thank Warren Tennant for supplying the results from the Unified Model EPS runs. Thiswork was carried out jointly with ECMWF and has greatly benefited from collaboration withstaff there. The use of their computing facilities is also gratefully acknowledged, particularly inrespect of the cloud-resolving model simulation.

References

[1] Berner, J., G. J. Shutts, M. Leutbecher and T. N. Palmer (2009) A spectral stochasticbackscatter scheme and its impact on flow-dependent predictability in the ECMWF ensembleprediction system. J. Atmos. Sci.,66, 603-626.

17

[2] Buizza, R., M. Miller and T. N. Palmer (1999) Stochastic representation of model uncertaintyin the ECMWF Ensemble Prediction System, Q. J. R. Meteorol. Soc., 125, 2887-2908.

[3] Bowler,N. E. A. Arribas, K. R. Mylne, K. B. Robertson, S. E. Beare (2008) The MOGREPSshort-range ensemble prediction system Q. J. R. Meteorol. Soc., 134, 703 - 722.

[4] Bowler, N. E., A. Arribas, S. E. Beare, K.R. Mylne and G. J. Shutts (2009) The local ETKFand SKEB: Upgrades to the MOGREPS short-range ensemble prediction system.Q. J. R.

Meteorol. Soc., 135,767-776.

[5] Evans, R. E., R. J. Graham, M. S. J. Harrsison and G. J. Shutts (1998) Preliminary inves-tigations into the effect of adding stochastic backscatter to the Unified Model. Met Office

Forecasting Research Technical Report No. 241, Met Office, Exeter, UK.

[6] Hermanson, L., B. J. Hoskins and T. N. Palmer (2009) A comparative method to evaluateand validate stochastic parametrizations. Q. J. R. Meteorol. Soc., 135, 1095-1103.

[7] Houtekamer, P. L., M. Charron, H. L. Mitchell and G. Pellerin (2007) Status of the global EPSat Environment Canada. Proc. ECMWF Workshop on Ensemble Prediction, 7-9 November,

ECMWF, Shinfield Park, Reading, Berkshire RG2 9AX, UK, 57-68.

[8] Mason, P. J. and D. J. Thomson (1992) Stochastic backscatter in large-eddy simulations ofboundary layers. J. Fluid Mech., 242, 51-78.

[9] Palmer, T. N. (2001) A nonlinear dynamical perspective on model error : a proposal fornon-local stochastic-dynamic parametrization in weather and climate prediction models. Q.

J. R. Meteorol. Soc., 127,279-304.

[10] Shutts, G. J. (2005) A kinetic energy backscatter algorithm for use in ensemble predictionsystems. Q. J. R. Meteorol. Soc.,131, 3079 - 3102.

[11] Shutts, G. J. (2006) Upscale effects in simulations of tropical convection on an equatorialbeta-plane. Dyn. Atmos. Oceans., 42, 30-58.

[12] Shutts, G. J. and T. N. Palmer (2007) Convective Forcing Fluctuations in a Cloud-ResolvingModel: Relevance to the Stochastic Parameterization Problem. J. Climate, 20, 187-202.

[13] Smagorinsky, J.(1963) General circulation experiments with the primitive equations.I. Thebasic experiment. Mon. Wea. Rev.,91, 99-164.

[14] Teixeira, J., and C.A. Reynolds (2008) Stochastic Nature of Physical Parameterizations inEnsemble Prediction: A Stochastic Convection Approach. Mon. Wea. Rev., 136, 483-496.

[15] Weisheimer, A., F.J. Doblas-Reyes, T.N. Palmer, A. Alessandri, M. Deque, N. Keenlyside,M. MacVean, A. Navarra and P. Rogel (2009) ENSEMBLES - a new multi-model ensemble forseasonal-to-annual predictions: Skill and progress beyond DEMETER in forecasting tropicalPacific SSTs.Geophys. Res. Lett. to appear.

18