tropical meteorology by g.c. asnani - initialization and parameterization

7.1 Introduction

i) The subject of Meteorology is becominginter-disciplinary with important inputs fromdifferent fields like Oceanography, Agriculture,Atmospheric Pollution, Astronomy, Space Physics,Geophysics and Biophysics.

ii) As a science, Meteorology is becomingmore quantitative using Physics, Pure Mathematics,Applied Mathematics, Statistics, Chemistry,Computer Science, Satellite Sensing, etc.

iii) In respect of applications, Meteorology isfinding applications almost in all branches ofhuman activity in peace as well as in war.

iv) Meteorology is becoming more global inoutlook as well as in routine daily operations. It isa symbol of one-ness of this planet earth which isbecoming more like a "village".

v) Meteorology uses the most sophisticatedinstrumentat ion for observations,telecommunications and analysis of observations.Computers and satellites have become essentialtools in the meteorological world. Modelling of

7.1 Introduction :( Pages 7-1 to 7-10)

Present difficulties about meteorologicalobservation in the tropics; FGGE; conventionalmethods of forecasting; numerical weatherprediction in the tropics.

7.2 Initialization in the tropics :( Pages 7-10 to 7-32)

Necessity; model-consistent initialization;progress made in the field of initialization;inadequacy of observational network; accuracyof observations.

7.2.1 Objective analysis of a chart and interpolation atgrid-points; successive correction method;optimum interpolation of a single variable;multi-variate optimum interpolation.

7.2.2 Initialization for PE model; static initializationschemes; non-divergent balanced flow schemes;schemes with limited divergent flow; mass-windbalance by variational technique; dynamicinitialization schemes; forward and backwardtime-integration; normal mode initialization;Appendix 7.2(A) - Temperton’s(1988)scheme-normal modes; Appendix 7.2 (A’) - animproved version of the implicit non-linear NMIscheme; bounded derivative method.

7.2.3 Four-dimensional (4-D) data assimilation.Physical initialization introduced by T.N.Krishnamurti-what is physical initialization;physical initialization tested on track-forecastingof tropical cyclones; physical initialization testedagainst Climatology.

7.3 Parameterization of cumulus convection inthe tropics :

( Pages 7-32 to 7-49)Meso-scale model parameterizations;synoptic-scale models; moist convectiveadjustment; moisture convergence models.

7.3.1 Kuo’s parameterization schemes for deepcumulus convection; moist adiabatic process.

7.3.2 Arakawa-Schubert Scheme for CumulusParameterization1. Cloud Ensemble 2. Cloud sub-ensemble 3. Cloud-work function A (λ)4. Reduction of convective instability5. Influence of cloud-cloud interaction on A (λ)6. Influence of large-scale processes on A (λ)7. Quasi-stationary assumption about A (λ)8. Calculation of K (λ, λ′)9. Calculation of F (λ)10. Calculation of mB(λ′)11.Schematic diagram12.Some sub-problemsCombined updraft-downdraft model; furtherwork done on Arakawa-Schubert scheme ofcumulus parameterization; gravity waveparameterization in Arakawa-Schubert scheme.

7.4 Summary of Chapter 7 ( Pages 7-49 to 7-51)

CHAPTER 7Initialization and Parameterization

Contents

atmospheric phenomena with the help of computersis recognised as an essential tool in the developmentof science of meteorology. Satellites now constitutea regular system of observing the atmosphere, theland and the top layers of the oceans.

vi) The public and the Governments of theworld have become conscious of environmentalpollut ion and the consequent inadvertentmodification of the atmospheric constituents, withpossible and sometimes noticeable changes inclimate. In particular, depletion of stratosphericozone, increase of CO2 and consequent warming ofthe troposphere, melting of ice, rising of sea- leveland shifting of rain belts is causing concern at thehighest levels of science and administrationthroughout the world. As a result, Heads ofGovernments met at Rio de Janeiro (Brazil) in June1992, to decide on ways to protect theland-ocean-atmosphere system of the earth; themeeting is known as the ‘Earth Summit’.

vii) It is being realised that the anomalies ofsea-surface temperature (SST) in the tropical regionare capable of producing weather anomaliesthroughout the world. To study this problem,international scientific community has launched theprogramme TOGA (Tropical Ocean GlobalAtmosphere).

viii) Meteorologists throughout the worldare being urged to give weather forecasts for longerperiods of a few weeks to a few months, to help thenational and international administrators andplanners to take appropriate decisions and steps inadvance. At the moment, forecasting for about tendays in advance has become the immediate targetfor many meteorological Services. All of them havean eye on long-range weather forecasting a fewmonths in advance. At the moment, it is realised thatfor long-range forecasting statistical methods ratherthan the methods of time-integration as in GeneralCirculation models, are the most suitable methods.For medium-range weather forecasting, the GeneralCirculation models are the most suitable ones, butimprovements in these models are necessary,particularly in respect of parameterization ofsub-grid physical processes of boundary layerturbulence, cloud condensation and radiativeprocesses in the atmosphere. This is the mostchallenging scientific problem in NWP work.

ix) Observations and analysis of observationswill always have some errors. At what rate will the

errors amplify in course of time-integration bynumerical methods? What is the upper time limit towhich numerical time integration can proceed togive reasonable forecasts?

x) It has been realised that quantitativeweather forecasting is more difficult for the tropicsthan for the extra-tropics.

xi) Solar and wind energies are available inplenty. It is considered essential to developtechnology for using solar energy and wind energy.These forms of energy are to be used along withhydroelectric energy for which technology isalready available. These forms of energy areavailable in plenty in the tropics and in thesub-tropics. Some day, sub-tropics may become theworld’s best source for solar energy. However, forall these three forms of natural energy (sun, windand rainfall), meteorological observations areessential.

xii) Meteorologists have also attempted tomodify weather at various scales like fog-dispersal,artificial rain-making, artificial hail-suppression andtropical cyclone modification. Possibilities are seenof modifying even global climate by plannedinterference with soil surface over extensivemountains and by icebreaking at some selectedocean straits. After initial enthusiasm, the attitudetowards weather modification has somewhatchanged and it is now felt that we should go slowwith the operations in the field of weathermodification and instead concentrate more onresearch modelling and understanding of thephysical processes and anticipating consequences ofweather modification operations.

In the earlier Chapters, we have touchedupon different aspects of these meteorologicalproblems. Here in this Chapter, we shallconcentrate on problems connected withnumerical modelling of weather systems andalso on using observation obtained throughsatellites.Present difficulties about meteorologicalobservations in the tropics :

The world’s least developed countries lie inthe tropical region. The Governments of thesecountries are generally unable to invest their scarcefinancial resources in setting up the network ofsurface and upper air meteorological observatories.Already, more than two thirds of tropical region arecovered by the oceans with no regular observation

7-2 7.1 Introduction

points. The economy of the tropical land region ismostly agricultural. The Governments feel inclinedto invest in meteorological Services, provided thatthey can be convinced that the agricultural economyof their countr ies wil l benefit from themeteorological observations and Services. Thesemeteorological Services are in the form of :

i) weather forecasting for short range (upto 2days), medium range (about a week), and longrange (about a month or more)

ii) utilisation of rainfall water, reduction ofevaporational loss of water, etc. for betterwater-management

iii) improvement of agricultural production,and

iv) warnings against weather disasters.The conventional network of surface and

upper air meteorological observations is generallypoor in tropical countries. The development of thewhole field of meteorology in the tropics hasgenerally been at a low level. The methods ofweather forecasting are the old conventional ones.

It is also true that with the best network ofobservations, the old conventional methods offorecasting have reached their limit of success. Nofurther substantial improvement is possible evenwith the most experienced forecasters. Experiencehas provided the conventional forecasters withsome thumb rules of weather forecasting, but intheir case quantitative understanding is lacking.With these conventional methods offorecasting, the skill score of the mostexperienced meteorologists has reached anear- steady value like a plateau, with no hopeof further improvement.

FGGE (First Garp Global Experiment)Objectives :

The First Garp Global Experiment was thelatest international venture to improve the qualityand time period of weather predictions bydynamical models. The four specific objectives ofthe FGGE were :

i) To obtain a better diagnostic understandingof the large-scale dynamics of the globalatmosphere and of critical processes taking place init.

ii) To provide initial and verifying conditionsfor modelling experiments designed to extend therange of operational weather prediction towards its

ultimate limits.iii) To guide the design of an optimum

meteorological observation and prediction systemof operational weather prediction which will, on acontinuous basis, employ the technical andscientific knowledge developed during theexperiment; and

iv) Within the limitations of a one year periodof observation, to investigate the physicalmechanisms underlying fluctuations in climate.Duration :

After a year of preparatorybuild-up-activity, the Experiment started on 1stDecember 1978 and ended on 30th November 1979.During this one year of operations, the mostcomprehensive programme ever undertaken forobserving the earth’s atmosphere was successfullycarried out.Special Observing Periods (SOPs)

Within this one year of Experiment, therewere two special observing periods :

1) 5th January 1979 to 5th March 1979 and2) 1st May 1979 to 30th June 1979.

There were three regional experimentswithin the global experiment :

i) MONEX (Monsoon Experiment).i i ) WAMEX (West African Monsoon

Experiment).iii) POLEX (Polar Experiment).

In addition to the routine surface and upperair observations, there were the followingadditional observational aids :

i) About three hundred constant-levelballoons floating near 14 km above sea level.

ii) Drop-Wind-Sondes from six aircraftreleased each day, at 9 to 12 km level.

iii) 80 commercial jet aircraft taking in-flightobservations.

iv) Ocean vessels.v) Ocean buoys.

vi) 5 geo-stationary satellites and 2 polarorbiting satellites.Research data :

The data collected during the Experimentperiod were made available to research workersunder the following headings :

Level I data : Instrument readings convertedto standard physical units andreferred to earth co-ordinates.

7.1 Introduction 7-3

Level II data : a) Data collected through theGlobal Tele-communicationSystem within operationalcut-off time. (10-24 hrs afterobservation time).b) Same as (a) except for adelayed cut-off time.c) Data for climaticinvestigation collected in adelayed mode.

Level III data : Internally consistent data setsobtained from Level II data byapplying four-dimensionalassimilation techniques.

Level III-a data : Obtained from Level II-a data.Level III-b data : Obtained from Level II-b data.

The main FGGE III-b data sets wereproduced by the two institutions: European Centrefor Medium Range Weather Forecasts (ECMWF)and Geophysical Fluid Dynamics Laboratory(GFDL). Both these data sets are complete andcover the whole FGGE year. In addition, partialFGGE III-b data sets have also been produced bysome other institutions. In USA, four suchinstitutions were: Goddard Laboratory forAtmospheric Sciences (GLAS), NationalMeteorological Centre (NMC), Florida StateUniversity (FSU), and Naval EnvironmentalPrediction Research Facility (NEPRF).

The four-dimensional assimilationtechniques are in general highly complex and varyfrom one institution to another. Daley et al. (1985)presented a set of very useful tables summarisingthe parameters and processes of four - dimensionalassimilation models used by each of the six Centresmentioned above. The tables allow a reader to getnot only a good comparison but also a useful listingof some of the accepted techniques used at differentCentres. Tables 7.1 (1 to 4) taken from Daley et al.(1985) are presented for the same purpose.FGGE Reports :

Summary of operations during the FGGEyear was issued by WMO/ICSU in a series of elevenvolumes. Volumes I and II summarize theoperations on a system-by-system basis likeSatellite System, surface, upper air network,tropical wind observing ships, aircraft,dropwind-sonde system, tropical constant-levelballoon system, drifting buoy system and aircraftflight level data system. Volumes III, IV and V give

summary of data collected during the year. VolumeVI reports on the FGGE Data Management.Volumes VII, VIII and X summarize the operationsof the regional experiments of FGGE. Volume IXgives preliminary research results prepared by thescientists who participated in the field operations ofsummer MONEX. Volume XI summarizes theoceanographic operations undertaken in support ofFGGE.FGGE Research Results :

A few special seminars were held soon afterthe end of the FGGE year in November 1979. Thefirst preliminary assessment was done in Melbourne(Australia) in December 1979; the second sessionwas held in Budapest (Hungary) in June 1980; thethird in Bergen (Norway) in July 1980; the fourth atTallahassee (Florida, USA) in January 1981; andthe fifth at Denpasar (Bali, Indonesia) in October1981. Condensed papers and panel discussionsgiving the preliminary scientific results emergingfrom the FGGE data were circulated for the benefitof scientists. Subsequently, many research papersbased on FGGE data analyses have appeared in theleading scientific journals of the world.Conventional methods of forecasting :

A tropical forecasting office prepares chartsfor fixed standard hours of observation, 3 to 6 timesa day. Surface observations are available onone-hourly basis only from a few observatories; onthree-hourly, six-hourly and twelve-hourly basisfrom a progressively larger number of stations.Upper wind and radiosonde data are available on sixhourly basis from a few stations and ontwelve-hourly basis from a larger number ofstations.

A forecasting office prepares sea-levelcharts and upper air charts for standard isobariclevels 850, 700, 500, 300, 200, 150 and 100 mb(hPa). Some offices plot additional charts like24-hour pressure-change char ts, pressuredeparture-from-normal charts , maximum/minimum temperatures along with their 24-hourchanges and departures from normal, T − φ gramsare plotted in respect of limited number of stations.To detect easterly waves, forecasting offices alsoplot vertical-time section charts and x - t strips ofsatellite pictures. Whenever a tropical storm isdetected within the area of responsibility of aforecasting office, its past track is continuouslyupdated.


Isobars are drawn for sea level charts andstreamline-isotach analysis is done for the winddata. Geopotential height contours are also drawnfor various constant pressure charts. The flow

patterns and the weather patterns in the tropics areessentially seasonal with well-marked diurnaloscillations. A forecaster has got to be quite familiarwith the synoptic climatology of the region of his

Institution Horizontaldiscretization

Verticaldiscretization

Timediscretization

ModelVariables

aTopography Horizontal

dissipationECMWFb Second order

Arakawa Cgrid 1.875o

Sigma15 levels0.996-0.025

Semi-implicit∆ t = 15 min.filter = 0.05

T, u, v, ps

qModeratelysmoothed

k ∇4

k = 9 × 1015m4s− 1

GFDL SpectralR30


Semi-implicit8 min.≤ ∆ t ≤ 25 min

T, ζ, D,ps, q

Spectrallytruncated

k ∇2

k = 4.96 × 104m2s− 1

GLAS Fourth order

4o × 5oSigma9 levels0.945-0.065

Matsuno∆ t = 10 min.

T, u, v, ps

qSmoothedwith 8thorder digitalfilter

16th orderdigital filter(Shapiro)

NMC SpectralR30

Sigma12 layerswith midpoints0.962-0.021

Semi-implicit∆ t = 17 min.filter = 0.04

T, ζ, D,q, ps

SpectrallytruncatedR30

k ∇4

k = 6 × 1015m4s− 1

FSU Not applicableNEPRF Spectral

T40 andfourth order2.4o × 3o


Semi-implicit∆ t = 10 min.

T, ζ, D, ps, q −T, u, v, ps, q

Envelopeorography

NonlinearSmagorinsky

a T = temperature; ζ = vorticity; D = divergence; ps = surface pressure, q = mixing ratio; and u, v = wind components. b At 10, 20,30 mb the first guess was climatology from 1 December 78 to 10 January 79. Thereafter, persistent wind shear and thickness wereadded to model’s 50 mb ( Φ , v ) to obtain first guess.

TABLE 7.1(1) :Assimilating model−basic information. (Daley et al., 1985; Asnani, 1993).

Institution Boundarylayerflux

Sea surface

temperature

Land surface

temperature

C o n v e c t i v eparameterization

Radiation Vertical dissipation

ECMWF Monin-Obukhov

Specified fromclimatology

Predicted land surfacetemperature

Kuoscheme

Fully interactiveclouds, nodiurnal cycle

Mixing lengthfunction ofRia

GFDL Monin-Obukhov


Surface heatbalance,diurnal cycle

Moistconvectiveadjustment

Climatologicalclouds,diurnal cycle

Mixinglength

GLAS Bulkaerodynamic


Surface heatbalance,diurnal cycle

Arakawascheme

Fully interactiveclouds, diurnal cycle

Very weaklineardiffusion

NMC Bulkaerodynamic


No land surface heator moisture flux

Kuoscheme

None None

FSU Not applicableNEPRF Bulk

aerodynamicAnalyzed Surface heat

balanceArakawascheme

Fully interactiveclouds, diurnal cycle

Linear diffusion

aRichardson number.

TABLE 7.1(2) : Assimilating model−physical parameterization. (Daley et al., 1985; Asnani, 1993)).


responsibility.In terms of changes of weather from one day

to another day, a forecaster does not generally findoutstanding changes in the pressure or wind flowpatterns, except when there are well-markedorganised systems like depressions, storms orcyclones. When there are such well-markedpressure-wind systems, the task of a forecaster isrelatively easy; he goes by the knowledge gainedfrom past synoptic climatology of the region. Thedifficulty of the forecaster arises when the 24-hourchanges are very small in the pressure-wind system,but these are accompanied by large changes in therealised weather. For example, the rainfall mayincrease from 1⁄2 cm to 5 cm in 24 hours, but theremay be no easily detectable large change in thepressure-wind pattern.

The present network of observatories cancatch only synoptic-scale (wavelength ∼ 4,000 km),and larger systems. In the tropical region, aforecaster who is familiar with synopticclimatology of his region, and who analyses thecharts carefully and notes down the slow and small

changes which are taking place on the charts, soongets enough experience and confidence to be able toforecast large-scale variations in weather likely tooccur during the next 24 to 36 hours.

He can also issue a general outlook ofweather changes expected during the next 3 to 6days. For this purpose, he has to watch and infer thechanges in position and intensity of semi-permanentcentres of influence during the next few days. Thecentres of influence are :

i) ITCZ or near-equatorial trough. ii) Other seasonal troughs and ridges in the

lower, middle and upper troposphere. iii) Migratory middle-latitude cyclone waves

of the two hemispheres.The changes in these systems are generally

slow. Simple extrapolation in time along with theknowledge of synoptic climatology of the regioncan give fairly good idea of the changes likely tooccur in the near future.

The advantage of this system of forecastingis that a good and keen analyst can soon become agood forecaster for short-range (1 to 2 days) and

Institution Basictechnique

Coordinate system horizontal +vertical

Imposeddynamic

constraintson analysisincrements

a

Method ofdatainsertion

Number ofiterationsat eachinsertion

Time window

ECMWF 3-dimensionalmultivariateOIb of OMFDc

Local pressure

Non-divergence,geostrophy inextratropics

Discrete1

6 hours(correctedfor off time)

GFDL Horizontal andverticalunivariateOI of OMFD

Local pressure

None Continuous

1

2 hours

GLAS Horizontal univariateSCMd of OMFD

Local pressure

Geostrophic correction offirst guesswind field inextratropics

Discrete

3

6 hours

NMC 3-dimensionalmultivariateOI of OMFD

Local pressure


Discrete1

6 hours

FSUe

Local pressure

None Discrete1

6 hours(correctedfor off time)

NEPRF 3-dimensionalmultivariateOI of OMFD

Local pressure


Discrete1

6 hours

aAnalysis minus forecast differences. bOptimal interpolation. cObservation minus forecast differences. dSuccessive correctionmethod. eFirst guess obtained from ECMWF IIIb analyses and univariate SCM used on MONEX observations minus first guess.

TABLE 7.1(3) : Analysis technique (Daley et al., 1985; Asnani, 1993).


medium-range (∼1 week) periods. Thedisadvantage of this system is that it becomes an"art" rather than the "science" of weatherforecasting. The forecaster himself cannot explainthe basis of his forecasting the future position andintensity of the centres of influence. When he goeswrong, he cannot say why his prediction wentwrong and how he can avoid such errors in future.

Those who advocate dynamical methods offorecasting can, and do sometimes, ridicule this artof conventional forecasting. But the fact is that theadvocates of dynamical methods of forecastinghave yet to achieve a degree of success better thanthat of conventional forecasting in the tropics. Apoint in favour of dynamical forecasters is that on a

purely objective mathematical basis, they have beenable to simulate seasonal features of several weathersystems in the tropics. They have been able toun-cover the dynamical processes underlying theseweather systems which knowledge was otherwiseextremely difficult or impossible to achieve byfollowing the conventional methods of forecasting.The success of numerical weather predictionmodels in the middle latitudes and successfulsimulation of several weather situations in thetropics has created a general hope that in not toodistant future, the numerical weather predictionmethods will become operational in the tropics aswell.

Institution Functionaldependenceof analysisweightsa

Horizontaldimensions oflocalanalysis"bin"b

Vertical dimensionof "bin"

Maximumnumber ofobservationsper "bin" +max. searchradius

Interpolationof analysisback to modelcoordinates

Miscellaneous

ECMWF ABCD 660 × 660 km Whole columnunless morethan 192observations

192 obs≈ 2000 km

Interpolation of analysisvalues (full field)

Analysisincrements in each "bin"are averaged to produceglobal analysis

GFDL ABC 1 gridpoint Distancebetween 3consecutivepressurelevels

8 obs250 km

Interpolation ofanalysisincrements

No insertionabove σ = 0.052

GLAS AB 1 gridpoint Adjacentpressurelevels

no limit800 km


Search radiusis data - densitydependent

NMC ABCD 1 gridpoint 4 mandatorypressure levelsabove and belowanalysis level

20 obs1500 km

Interpolation of analysisincrements

No datainserted in top σ layer(midpoint = σ0.021)

FSU AB 1 gridpoint Procedure is strictly horizontal

> 100 obs2 gridintervals

Not applicable

NEPRF ABCD 1 gridpoint Whole columnunless morethan 200observations

200 obs900 km


aA = assumed data quality; B = distance between observation point and analysis point; C = distance to adjacentobservations and their quality; and D = assumed accuracy of forecast. bThe local three-dimensional volume in whichthe analysis is performed.

TABLE 7.1(4) : Analysis technique (Daley et al., 1985; Asnani, 1993).


Numerical weather prediction in the tropics :We cannot use quasi-geostrophic (Q.G.)

model in the neighbourhood of the equator. Wehave to use primitive equation (P.E.) model in thetropics, particularly when we are operating within5o of latitude from the equator.

The difficulties of dealing with P.E. modelare now well-known. After vertically-movingacoustic waves have been eliminated through theuse of quasi-static approximation, there remain thefollowing waves :

i) Horizontally-moving acoustic waves orLamb waves, with period of the order of a fewminutes to a few hours.

ii) Gravity waves with period of the order of afew minutes to a few hours.

iii) Inertio-gravity waves with period of theorder of half a day.

iv) Rossby waves with period of the order of afew days or more.

The speed of the propagation of the Lambwaves and the short-period gravity waves is of theorder of 100 ms−1; Rossby waves move with aspeed of the order of 10 ms−1; the speed ofinertio-gravity waves is intermediate between10 ms−1 and 100 ms−1.

The pressure amplitude of the short-periodacoustic waves is generally less than 1 mb (hPa),increases as we go towards longer-period waves andbecomes of the order of 5 mb (hPa) and more forpure Rossby waves in the tropics.

Rossby waves are of direct importance tometeorology. All synoptic-scale weather systemsbelong to the class of Rossby wave motion. Thefast-moving small-amplitude waves produce littleor no perceptible weather phenomena or theyproduce sub-synoptic-scale perturbations in theatmosphere. At any time, the energy content ofatmosphere in these very small waves is negligible.However, their presence in the model equations ofthe atmosphere creates a barrier in the timeintegration of the model. For example C.F.L.(Courant-Freidrich-Levy) condition for

computational stability is that c � t�x

� 1 where

c is the wave speed, ∆t is the length of time stepand ∆x is the space step or grid distance betweentwo neighbouring points in the numericalintegration of the model equations. For a given

value ∆x , the time step ∆t becomes smaller andsmaller for fast-moving waves. These fast-movingwaves reduce the speed of time-integration of themodel . Since their contribution to thesynoptic-scale weather processes is very small,these are regarded as ‘noise’ for numerical weatherprediction purposes.

This noise can be completely eliminated byuse of quasi-geostrophic approximation but thatwill be like paying a very high price becausegeostrophic approximation cannot be made in theneighbourhood of the equator. Hence we strike acompromise by allowing the Lamb waves and thegravity waves to remain in the model equations, butby suitable choice of numerical scheme fortime-integration, we achieve selective damping ofthe fast-moving waves. There are numericalschemes in which the faster-moving gravity wavessuffer greater damping than the slow-movingwaves. The analysis of these schemes has beenpresented by Kurihara (1965), Richtmyer andMorton (1967), Kreiss and Oliger (1973) andHaltiner and Williams (1980). Properties of some ofthe commonly used schemes are shown in Table7.1(5).

In P.E. models, we require reasonablyaccurate representation of both the pressurefield and the wind field at the initial time t = 0.This initialization of the field is a majorproblem in the integration of a P.E. model.Initial imbalances can lead to the generation ofsevere shock waves which can entail considerablewaste of forecasting time. We shall deal further withthis problem under the sub-heading "Initialization".Even before tackling this initialization, it must berecognised that the amount of data which pours intothe forecasting office and is to be used for the NWPmodel is so large that it becomes practicallyimpossible to analyse various charts manually andthence to pick up grid-point values of the variablesmanually. To be in line with the whole philosophyand concept of computer forecasting, the analysisof the weather charts has to be done througha computer. This subject will be further dealtwith under the heading "Objective Analysis".

To make maximum use of all availableinformation, particularly the one coming throughthe satellites, one has to find a way of feeding


asynoptic observations (those which do notbelong to the synoptic hour of other observations)while the time integration by the computer is inprogress. This problem is somewhat analogous tothe problem of initialization. We have to feed thenew information into the computer model withoutcreating serious problem of misfit of informationand consequent generation of shock waves. Thisquestion will be further dealt with under the heading"4-dimensional data assimilation". The usual3-dimensional space is now connected to the fourth

dimension of time.Conceptually, the system of meteorological

equations is closed; i.e. given inital distribution ofatmospheric parameters like wind, pressure, densityand temperature in the whole atmosphere and giventhe boundary conditions for these parameters for thewhole period of time-integration, it should bepossible to forecast the future values of thesemeteorological parameters in the wholeatmosphere. There is, however, an impliedcondit ion that we are able to represent

N u m b e rof Time

Computation- physical Mode Computa-tional Mode

Method Difference Equation Levels al Stability Amplitude Phase AmplitudeImplicit :(A)backward h r + 1 − h r = ∆ t F r + 1 2 Absolutely

stableHigh selectivedamping

Retardation None

(B)trapezoidal h r + 1 − h r = ∆ t (F r + 1 + F r) ⁄ 2 2 Absolutelystable

No change Little retardation

None

(C)partly

h r + 1 − h r

= ∆ t F 1r +

∆ t2

(F 2r + 1 + F 2

r )

2 Unstable formeteorologicalwave and onegravity wave

None

(D)partly h r + 1 − h r − 1

= 2 ∆ t F 1r + 2 ∆ t F 2r + 1

3 (very weak)unstable formeteorologicalwave

Damping ofgravity wave& weakamplifying ofmeteorological wave

Damping

Explicit :

(0)forward h r + 1 − h r = ∆ t F r 2 Unstable None

(1) leapfrog(centered)

h r + 1 − h r − 1 = 2 ∆ t F r 3 Conditionallystable (b < 1)

No change Moderateacceleration

No change

Iterative :(2)Eulerbackward

h ∗ − h r = ∆ t F r

h r + 1 − h r = ∆ t F ∗2 Conditionally

stable (b < 1)Moderatelyselectivedamping

Largeacceleration

None

( 3 ) m o d i f i edEuler-back-ward

h ∗ − h r = ∆ t2

F r

h ∗∗ − h r = ∆ t F ∗

h r + 1 − h r = ∆ t F ∗∗

2

Conditionally

stable (b < √⎯⎯ 2 )

Highlyselectivedamping

Moderate acceleration

None

(4)leapfrog-trapezoidal

h ∗ − h r − 1 = 2 ∆ t F r

h r + 1 − h r = ∆ t2

(F ∗ + F r ) 3Conditionally

stable (b < √⎯⎯ 2 )

Littledamping

Little error Veryeffectivedamping (inparticular ofmeteorological wave)

(5)leapfrog-backward

h ∗ − h r − 1 = 2 ∆ t F r

h r + 1 − h r = ∆ t F ∗3

Conditionallystable (b < 0.8)

Moderatelyselectivedamping

Moderate acceleration

Damping

F1 ≡ non-linear terms ; F2 ≡ linear terms ; F ≡ F1 + F2 ;µ ≡ 2πL

, where L ≡ wavelength in x-direction; b ≡ µ c ∆ t .

TABLE 7.1(5) : Properties of some commonly used numerical schemes (Kurihara, 1965; Asnani, 1993).


mathematically the distribution process for heatenergy, momentum and moisture inside theatmosphere. The atmosphere receives almost itsentire heat energy (sensible heat + latent heat)through the lower boundary. Proper mathematicalrepresentation of these lower boundary processesand also of internal diffusion and reflection isessential. This subject will be further treatedunder the title of "Parameterization ofphysical processes".

Even after the successful handling of theproblems of objective analysis, initialization,4-dimensional assimilation and parameterization ofphysical processes, we have to appreciate that thereis still a limit to the length of time for whichprediction is possible with the help of P.E. models.This limit arises out of two causes. Firstly, whateverbe the accuracy of observations and analysis, theassignment of a grid point value of a physicalparameter in terms of a limited-digit numberimplies generation of an error which is thedifference between the actual and the adopted valueof the meteorological parameter at a grid-point.Secondly, there are errors of observation andanalysis. The error field so generated is subjected tothe same set of model equations as the trueerror-free physical field. If there is an element ofdynamical instability in the physically real field, asmall error perturbation in the physical field willalso grow exponentially with time. This growingperturbation will also interact, in a non-linearmanner with the basic physical field. While this isas it should be in the case of physical field, wesimultaneously also have the error-field growingexponentially with time. These errors, howeversmall initially, grow to substantial size in course oftime, interact with and contaminate the physicalfield solution. The physical mode and the errormode get mixed up to such an extent that it becomesimpossible to separate the two. Consequently, thephysical field emerging in the real atmosphere losesits resemblance with the pattern evolving from thecomputation. Even if there be superficialresemblance, it is of little value for purposes ofscientific weather prediction. This leads us to theproblem of limit of predictability of theatmosphere by the conventional numerical weatherprediction methods. The current estimates suggestthat the conventional techniques of NWP can giveus meaningful weather predictions up to about 7 to

14 days. To achieve worthwhile prediction upto 7to 14 days by conventional NWP methods with P.E.models was the immediate objective of theinternational meteorological programme FGGE.

7.2 Initialization in the Tropics

Necessity : Before we fire a projectile, we must ensure

that the firing machine is at proper location and theprojectile has proper orientation in the horizontal aswell as in the vertical. Similarly, for correctscientific forecasting, the initial meteorologicalfield parameters should have not only reasonablycorrect instantaneous values at grid points butalso correct space gradients computable from theavailable grid-point values.Model-consistent Initialization :

The initial values of meteorologicalparameters at grid points and their space gradientsmust be consistent with the forecasting model. Forinstance, the forecasting model is designed to catchlow-frequency Rossby modes and to suppress thehigh-frequency gravity modes. Hence, the initialdata should have, as far as possible, not only theamplitudes of gravity modes equal to zero but alsotheir t ime-tendencies equal to zero. Thisrequirement should be fulfilled not only in respectof wind field but also in respect of all the physicalparameters of the model, like pressure, temperature,water vapour, diabatic heat ing, frictionaldissipation, and precipitation.

Different forecasting models have differentrequirements of model- consistent initialization.Progress made in the Field of Initialization :

The forecasting model of L.F. Richardson(1922) retained both gravity modes and Rossbymodes. The forecasting model of Charney (1947)removed the gravity modes throughquasi-geostrophic approximation. The first NWPmodel using electronic computers in late 1940s andearly 1950s initialized the grid-point data consistentwith the quasi-geostrophic forecasting model, or atmost Linear Balance forecasting model. Theoperational forecasting was then confined toextra-tropical regions; the weather processes intropical regions were considered to be only ofmarginal importance for the weather in theextra-tropics. Gradually, two complementaryperspectives emerged :

7-10 7.2 Initialization in the Tropics

a) The weather phenomena in the tropicsare of more than marginal importance forweather forecasting in the extra-tropics. Formedium and long-range forecasting in theextra-tropics, weather phenomena occurring in thetropics are of sufficient importance to merit moreattention.

b) We could embark on operational weatherforecasting in the tropics, with NWP models.

This type of perspective has encouragedgreater scientific interest and effort in the field ofinitialization for P.E. models in the tropics.

During initialization, attention has to begiven to all the meteorological parameters whichoccur in a forecasting model.

The f irst efforts in forecast ing andinitialization were confined to dry adiabatic models.Subsequently, moisture and diabatic heating havebeen incorporated in almost all operationalforecasting models. In a few specialised models,there is also provision for CO2, O3, other commonchemical constituents of the atmosphere and evenfor pollutant aerosols.

All initialization techniques imply some sortof smoothing which was first done through staticinitialization, then through dynamicinitialization and more recently through moresophisticated techniques like fitting of standardfunctions in the horizontal as well as in thevertical. In the horizontal , the smoothing is doneby fitting what are called "normal modes". Theseuse Fourier functions in x-direction and Legendrepolynomials or Hough functions in y-direction.Vertical smoothing is done through use of"equivalent depths" corresponding to thehorizontal normal modes.

Considerable progress has been made,during recent years, in normal-mode initializationof pressure field and horizontal wind field.However, corresponding initialization of othermeteorological fields like moisture, clouds,precipitation rate, diabatic heating, etc. are still inactive research and experimental stages. It is a"frontier" research problem.Inadequacy of observational network :

Tropics are known for inadequacy ofobservations. This peculiarity of the tropics isessentially due to two reasons:

i) Vast oceanic area where it is very costly toorganise regular systematic observations.

ii) Relatively low priority to meteorology dueto many demands on limited economic resources.

In order to catch a system reasonably well,we should have at least 10 to 12 grid pointobservations inside the system. Let us havesome estimate of the horizontal and verticaldimensions of the meteorological systems which wemust catch in our grid system designed forsynoptic-scale forecasting.

System Horizontal

dimensions(km)

Verticaldimensions

(km)Monsoon depressions andtropical oceanic depressions

2,000 9

Easterly waves in general 2,000 9African easterly waves 2,000 6Quasi-stationary planetarywaves

15,000 15

Upper tropospheric waves 10,000 -40,000

5 - 15

These systems can be caught reasonablywell if our horizontal grid length is of the order of2 degrees latitude and longitude. Tropical cyclonesneed special small grid lengths because theirimportant structural features like eye and cloud wallhave horizontal dimensions of the order of 10 km.Vertical resolution needs special consideration.Those systems which are actively fed by the tropicalPBL must have a fine resolution in the PBL andrelatively coarse resolution upstairs. In general,10-level resolution in the vertical is desirable.

It will be very long before we can achieve anet-work of radiosonde/rawin observatories whichwould enable us to assign grid-point values, withconfidence, for 2o latitude/longitude grid systemand at about 10 levels in the vertical.Accuracy of observations :

In the atmosphere, there is a mixture of agreat variety of scales of motion and temperature,ranging from a few millimetres to several thousandkilometres. These scales exist throughout theatmosphere and are not confined to any particularregion in tropics and extra-tropics. Even if ourobservational techniques were perfect to 100%acccuracy, which accuracy shall never be attained,the measurement includes the influence of all thosescales of motion which are always present in ouratmosphere. It is not possible and even worthwhileto deal with this totality of scales at one and thesame time. It is, therefore, essential to sift out, from

7.2 Initialization in the Tropics 7-11

the ideally "correct" measurement of observation,that component or those components in which weare interested at a particular instant.

For the time being, we are interested insynoptic-scale analysis of systems whosewavelength is of the order of 2000 km or more.Hence, for our purpose, atmospheric phenomenawith wavelengths of 200 km or less are justatmospheric noise which we would like to filter out,by suitable techniques, from our observations, evenif these were ideal and correct.

In our real observations, there is "noise"coming not only from these small-scaleatmospheric systems but also from "errors" ofobservation. These errors of observation arise fromthree sources :

i) Instrumental errors. ii) Computational errors, for example

computation of wind speed and direction fromobserved positions of a balloon.

iii) Rounding-off errors, for example retentionof a small number of decimal places.

At present, the random errors in variousmeasurements are estimated to be of the ordershown below in Table 7.2(1). Continuous effortsare in progress to reduce these errors which arisefrom a combination of small-scale phenomena andfrom errors of observations.7.2.1 Objective Analysis of a Chart andinterpolation at Grid-points :

For NWP work, we need machine methodsof analysis (Objective Analysis). The method ofanalysis has to be programmed so as to requireeither no human intervention at all or a minimalmachine-man mix. So far as speed is concerned,machine will generally come out to be faster thanman. It is also possible to programme "experience"of meteorologists for computer analysis, although,in general, it is difficult to get it from experiencedmeteorologists, in a form suitable for simplecomputer programming. Objective Analysis consists of four stages :

i) Quality Control of raw data received froman observation point, rejecting or modifying the rawdata.

ii) Method of interpolation to assign values atgrid points.

iii) Supply of "artificial" data (sometimescalled ‘bogus’ data, although this adjective soundstoo harsh) at grid points in a region where data are

totally absent or very sparse.iv) Final Check of analysis before the

grid-point values are fed into the computer formodel integration.

Each NWP analysis centre has its ownproblems and techniques of tackling points (i), (iii)and (iv). We shall concern ourselves here mainlywith (ii) in some detail. We shall assume that areasonable set of observations is available atobservation points in a reasonably dense networkand our problem is to interpolate at grid points.

For tropical regions, we need to analyse atleast the following three elements :

i) Pressure field, also referred to as mass fieldor contour field.

ii) Wind field.iii) Moisture field.

The following four methods are useful forobjective analysis in the tropics :

i) Successive Correction Method. ii)Variations of Successive Correction

Method using non-isotropic weighting functions.iii) Optimum interpolation of a single variable

(mass, wind, moisture).iv) Multi-variate optimum interpolation for

mass and wind fields jointly.Successive Correction Method :

It is an iterative process designed byBergthorssen and Doos (1955) with somemodifications by Cressman (1959). It can berepresented by

ψ gn+1 = ψ g

n + 1N

∑wi

i = 1

N

( ψ io − ψ i

n ) 7.2(1)

where ψ is a scalar parameter for which values arerequired at grid-points by objective analysis. Thesubscript g indicates grid-point while the subscripti indicates observation point.

ψ gn = Value of ψ at grid-point during scan

number n.

ψ gn+1 = Value of ψ at grid-point during scan

number n + 1.

ψ in = Value of ψ at observation point i

interpolated from grid point values during scannumber n.

ψ io = Observed value of ψ at observation

point i.


wi = weighting function which depends onthe distance of the observation point from thegrid-point. The form of wi changes in each scansuch that observations farther from the grid-pointreceive decreased weightage in successive scans.

N = number of observations used in thescan.

Several scans are made through the data, theinterpolations from the last scan being used to getimproved values during the next scan. In the firstscan, a first-guess value is used at the grid-point.Forecasted value or any other reasonable valuecould serve as a first-guess. This scheme has thefollowing advantages :

i) Programming is simple.ii) Convergence is fast.

iii) All the observations within a specifieddistance from the grid-point can be used to get thevalue at the grid-point.

Perhaps, one disadvantage is that twostations which are close together and which areproviding nearly the same information, getseparately the same weightage which an isolatedobservation gets in a different direction but at anequal distance from the grid-point. There can be toomuch weightage to a cluster of observations in oneand the same area. Another disadvantage is isotropyi.e. equal weightage in all directions, east-west aswell as north-south. This will be elaborated in thenext section.Variations of Successive Correction Methodusing non-isotropic Weighting Functions :

In the Successive Correction Methodoutlined in the previous section, the weightingfunction wi was isotropic; it depended only on thedistance of an observation point from a grid-pointbut not on its direction. In general, zonal windcomponent is stronger than meridional component.

Element Error

i) Surface pressure ± 0.2 mb (hPa)

ii) Surface temperature ± 1oC

iii) Surface wind speed ± 1 ms −1

iv) Surface relative humidity ± 5%

v) Radio-sonde temperature ± 1oC

vi) Radio- sonde relative humidity ± 5%

vii) Rawin wind speed ± 1 ms −1

viii)Commercial aircraft wind speeda) With inertial navigational systemb) Without inertial navigational system

± 2 ms −1

± 4 ms −1

ix) Satellite sea surface temperature ± 1oC to 2 oC

x) Satellite temperature soundingsa) 1000 - 800 mb (hPa)b) 800 - 300 mb (hPa)c) 300 - 100 mb (hPa)d) < 100 mb (hPa)

± 2.5 oC ± 2 o C± 2.5 oC ± 2 o C

xi) Satellite - measured relative humidity 20 to 30%

xii)Satellite - measured windsa) 900 mb (hPa) wind speedb) 500 mb (hPa) wind speedc) 250 mb (hPa) wind speed

± 3 ms −1

± 4 ms −1

± 4 ms −1

xiii)Constant level balloons at 200 mb (hPa)a) Temperatureb) Wind speed

± 0.5 oC ± 1.5 ms − 1

xiv)Drop-sondes from carrier balloonsa) Temperatureb) Wind speed

± 1 o C± 2 ms − 1

TABLE 7.2(1) : Errors of observing system for synoptic-scale analysis. (Source : GARP Pub. Ser. No. 20, 1978 and Bengtsson, 1975; Asnani, 1993).


Also, gradients of meteorological parameters aregenerally stronger in north- south direction than ineast-west direction.

To take account of these directionalvariations, the weighting function wi is made afunction of not only the distance of an observationpoint from a grid-point, but also of the direction ofits orientation relative to the grid-point, and eventhe direction of the wind. The weighting functionsare made non-isotropic.Optimum interpolation of a single variable :

It can be represented by

ψgA = ψg

F + 1N

∑wi

i = 1

N

( ψio − ψi

F ) 7.2(2)

where

ψ gA = Analysed value of ψ at grid-point.

ψ gF = First-guess value of ψ at grid-point.

ψ iF = First-guess value of ψ at observation

point i.

ψ io = Observed value of ψ at observation

point i, obtained by interpolation from grid-pointvalues.

N = Number of observations used in thescan.

The weighting function wi is obtained byminimising the error square E given by

E = ( ψgA − ψg )

2___________

where ψg is the true value at the grid-point and thebar indicates averaging over many past occasions;it involves statistical analysis of past observationsin the region. The formulat ion involvesco-variances between errors in the first guess field,observational errors and cross-covariance betweenthem.

Some of the advantages and disadvantagesof this scheme of interpolation (GARP Publ. series,1978) are given below:Advantages :

i) It allows for varying distribution ofobservations around the grid point.

ii) It allows for different levels of error indifferent types of observation.

iii) An estimate of the interpolation error ateach grid point is provided by this scheme.

iv) The interpolation may be made in thevertical as well as in the horizontal. This providesgreater consistency in multi-level objectiveanalysis.Disadvantages :

i) It needs much more computing time thanthe Successive Correction Method. It involvesinversion of an N × N matrix at each grid point,where N is the number of observations used.

ii) There may be ill-conditioning problems ininverting the matrix.

iii) A large number of statistics need to becalculated.Multi-variate Optimum Interpolation :

In this scheme, mass and wind fields may beanalysed simultaneously giving better consistencybetween the two interpolated fields. Verticalinterpolation is carried out first, at all observationpoints, interpolating height and wind separately.Then the horizontal interpolation is carried out.Equations for horizontal interpolation may bewritten in the form

⎛

⎜

⎝

⎜

⎜

ZgA

ugA

vgA

⎞

⎟

⎠

⎟

⎟ =

⎛

⎜

⎝

⎜

⎜

ZgF

ugF

vgF

⎞

⎟

⎠

⎟

⎟ +

1N

∑ i = 1

N

Ai

⎛

⎜

⎝

⎜

⎜

Zi0 − Zi

F

ui0 − ui

F

vi0 − vi

F

⎞

⎟

⎠

⎟

⎟ 7.2(3)

where Ai’s are (3 × 3) weighting matricesdetermined by simultaneously minimising theexpressions

( ZgA − Zg )

2___________

, ( ugA − ug )

2__________

, ( vgA − vg )

2__________

,

Z is the height of constant pressure surface and u, vare the wind components. Other symbols have thesame meanings as in the previous section.

While the basic principles of optimuminterpolation remain the same, several variations arebeing continually in troduced at differentoperational meteorological centres (e.g. Dey andMorone, 1985).7.2.2 Initialization for a P.E. Model :

In the tropics, for synoptic-scale systems, weare concerned essentially with a P.E. model. Evenafter objective analysis, we need to bring theindividual grid-point data into some sort of balancewith data at other neighbouring grid-points.Experience and theory have shown that if thisbalance is not provided initially, the computationswill lead to the development of ‘noise’ which may


become uncontrollable at later stages. Theoryshows that this noise comes from poorly-resolvedhigh-frequency waves. It is difficult to handle thesehigh-frequency waves in an NWP model. Theyobstruct the correct prediction of the moreimportant Rossby waves through non-linearinteractions. The problem is not so serious inquasi-geostrophic models because fast-movinghigh-frequency gravity waves are not present insuch models.

The initial state may be regarded asappropriate, if high-frequency oscillations are notpresent to any significant extent in the beginning,and do not have a chance of fast development, soonafter starting the time-integration.

During the sixties, this initialization wasdone mostly with schemes which may be termed as"static initialization schemes". Some sort of balancewas assumed between pressure and wind fields, thewind field being mainly rotational part, with orwithout some irrotational part.

In contrast, there have been subsequentschemes, still in the process of development, whichallow interaction to take place between thefirst-guess initial field and the P.E. forecastingmodel equations before the real-time integrationstarts. The result of initialization process is model- dependent. The initial state so obtained may not bea very accurate reproduction of the actual state ofthe atmosphere but it is model-consistent.Static Initialization Schemes :

Broadly speaking, there are three types ofstatic initialization :

i) Non-divergent balanced flow.ii) Non-divergent balanced flow plus limited

divergent flow. iii) Mass-wind balance by Variational

Technique.Upto mid-1970s, the balanced flows were

based on hierarchy of truncated models of vorticityand divergence equations, more popular modelsbeing known as non-linear balance model,linear-balance model and quasi-geostrophic model.These truncated models were energeticallyconsistent as per the analysis of Lorenz (1960).Subsequently , two different approaches,Normal-Mode initialization and BoundedDerivative method, have been developed toachieve the same purpose, i.e. to adjust the initialpressure and wind data in such a way that unwanted

meteorological noise does not grow tounmanageable intensity. Subsequent analysis hasalso revealed that these methods ofNormal-Mode ini t ial ization andBounded-Derivative method lead to a balancebetween pressure and wind fields which is notsignificantly different from the balance obtainedthrough truncated forms of vorticity and divergenceequations. However, in this section, these twomethods will be given separately after a briefdescription of "Dynamical InitializationSchemes".Non-divergent balanced flow Schemes :

Flow is represented by ψ field. Non-linearbalance equation is :

∇2(gz) = fo ∇2 ψ + β

∂ψ∂y

+ 2J⎛⎜⎝

∂ψ∂x

, ∂ψ∂y

⎞⎟⎠ 7.2(4)

This is also called reverse Balance equation.Geopotential is obtained from ψ-field. It becomeslinear Balance equation if the non-linear term

2 J ⎛⎜⎝

∂ψ∂x

, ∂ψ∂y

⎞⎟⎠ is omitted on the right-hand side of

the equation. If further, the term β ∂ψ∂x

is also

removed from the right-hand side, it becomesgeostrophic relationship with constant value of f. Inliterature, we come across experiments using eitherof the three balanced relationships between pressureand wind fields.Schemes with Limited Divergent Flow :

These schemes include non-divergent (ψ)component of flow as stated in the previous section.Additionally, they also include a limited componentof divergent flow (χ field). This comes fromω-equation. We have quasi-geostrophicω-equation corresponding to quasi-geostrophicmodel. We also have ω-equation correspondingto linear and non-linear balance models. Througheither of these forms of diagnostic ω-equation,we get ω-field corresponding to a given (ψ,z)field. Now,

∂ω∂p

= − ∇ 2 χ 7.2(5)

This enables us to determine χ-field. From this, weget Vχ, the divergent component of flow.


Vχ = � χ 7.2(6)

This procedure was first analysed by Phillips(1960).

A slight variation of the above procedure isto utilise observed fields of temperature T, thehorizontal motion V H and the surface pressure. The

rotational and irrotational components of the windare obtained through ω- equation as before. Fromsurface pressure and temperature in the vertical, onebuilds up the z-field. Krishnamurti (1969) andKanamitsu (1975) experimented with thesemethods.Mass-Wind Balance by Variational Technique :

This technique was developed by Sasaki(1970). It requires adjustment between mass andwind fields, using some diagnostic balancedrelat ionship between the two, subject tominimization of an integral like

I = ∫∫ [ α ( V − V∼ )2 + β ( z − z∼ )2

+ λ( relationship between z and V )2 ] dx dy

7.2(7)

The area integral is to be evaluated over theentire horizontal field under consideration. Here

V = finally adjusted velocity.

V∼ = velocity obtained from data analysis.

z = finally adjusted height.

z∼ = height obtained from data analysis.The constraint may be "strong" in which

case it is satisfied completely and the minimizationis also with respect to λ. Alternately, the constraintmay be "weak" in which case the extent of itsrealization depends on the value of λ relative toα and β. The relative values of α and β determinehow much the final fields depend on the analysedwinds or heights, and may be determined from theinterpolation errors given by an optimuminterpolation scheme.

The integral relationship may be extended tothree dimensions and the constraint may bespecified in terms of thermal wind and temperature(Lewis, 1972).

Dynamic Initialization Schemes :Necessity : The necessity of an alternate to static initializationschemes was appreciated after a few experimentswith the latter in tropical regions. These staticinitialization schemes suffer from the usual problemof pressure-wind adjustment in low latitudes.Krishnamurti (1969) found that it takes roughly anequivalent of 12 to 18 hours of integration before arealistic pressure-wind field emerges from staticallyinitialized pressure-wind field. In an experiment byMiyakoda et al. (1974), the first three days wereconsidered as initial adjustment period, duringwhich time, two tropical storms weakened veryrapidly.

Similar balancing procedures were also triedfor the spherical rotating earth (Houghton andWashington, 1969) by assuming that the pressurefield determines the wind field in extra-tropicallatitudes and that the wind field determines thepressure field in the tropics. The limited divergentcomponent of the wind consistent with thisbalancing could also be obtained over the sphere.However, this type of static initialization alsoresulted in unrealistic oscillations during the initialstages of the integration with NCAR GeneralCirculation model.

Similar large oscillations were alsoexperienced in integrations with a barotropic P.E.model in spectral form and also with a baroclinicP.E. model in spectral form (Bourke, 1974).

Experiences of this type have tended to turnattention to dynamic initialization schemes. Theobjective of dynamic initialization schemes is toestablish a dynamically adjusted pressure-windfield as the initial condition for a P.E. model ratherthan relying on its evolution from a crude initialstate during the course of integration and thenignoring the model forecast for the initial"adjustment" period.Forward and Backward time-integration :

Miyakoda and Moyer (1968) suggested aninitialization scheme in which a time-differencingscheme, having a property of selective damping(faster damping for high-frequency waves) wasused to perform a cyclic forward-and-backwardintegration repeatedly under some constraint on thevelocity potential. An iterative procedure suggestedby Nitta and Hovermale (1969) also did selectivedamping in cyclic forward-and-backward


integrations but without the constraint on thevelocity potential. Results of experiments show thatboth these methods yield a synoptic-scale verticalvelocity field which is close to the solution of thebalanced model ω-equation.

The shallow-water equations were treated byKiangi (1976) over Africa with equator roughlythrough the centre of the region. He started with agiven wind field and obtained the geopotential fieldvia the reverse Balance equation. This result ofstatic initialization became the starting point for thedynamic initialization. He then resorted toforward-and-backward integration for about 18iterations of one hour each. At this stage, theindividual disturbances became much more sharplydefined. He used this initialization procedure in aprediction model and showed that this iterativeinitialization produced a geopotential field whichwas roughly 10 metres different, in theroot-mean-square sense, from the balancedgeopotential field.

Kurihara and Tuleya (1978) proposed atwo-stage dynamic initialization scheme withemphasis on the dynamic adjustment in theboundary layer. The method assumes that at theanalysis stage before dynamical initialization starts,mass field is accurately known in the boundarylayer and the rotational wind field is accuratelyknown above the boundary layer. In the first stage,the dynamical effect of surface friction isincorporated in the boundary layer where the massfield is fixed as given at the analysis stage. This isdone by making a forward integration of the modelwhile anchoring the mass field everywhere in theboundary layer and the wind field at the top of theboundary layer. In the boundary layer with theprogress of time integration, the wind approaches aquasi-equilibrium state with friction included. Theboundary layer wind thus obtained yields frictionalmass convergence and divergence which can causeexternal gravity waves unless compensated for inthe free atmospheric layer. This compensation isartificially provided in the free atmospheric layer.

The second stage of dynamic initializationdeals with the free atmosphere. Here rotationalwind is taken as given by the analysis beforedynamic initialization starts. Given this wind field,a balanced mass field is computed by the simplifiedreverse Balance equation. Now the second stage ofdynamic initialization starts. It involves cyclic

forward-backward integration with restorativetechnique of Nitta and Hovermale (1969).

The primary function of this second stage isto suppress possible noise due to slightimbalances between the mass and wind fieldsat the end of the first stage. It is also anticipatedthat the divergent wind field which is required formaintaining a balance condition may evolve. Theboundary layer structure which was established atthe end of the first stage of initialization wouldremain practically unaffected by this second stageof initialization. The scheme was tested on a simplezonal flow and also on an isolated hurricane.Kurihara and Bender (1979) found out that thedynamical initialization scheme of Kurihara andTuleya (1978) could very well work for simplezonal flow and for isolated hurricane vortex but notfor the combination of the two. When a vortex wassuperimposed on a basic background flow, a largeacceleration of wind was observed in certain areasof the flow field during the first-stage dynamicinitialization of the boundary layer, which wouldtend not to assume a realistic quasi-equilibriumstate. This large acceleration was due to theadvection of the vortex by the basic backgroundcurrent away from the pressure low which (massfield) was not allowed to change during thisinitialization. It was realised that the boundary layerini t ial izat ion needed improvement. Thisimprovement is effected through an artificialforcing which is a function of wind speed at a pointsuch that the forcing is reduced as the wind speed isdecreased by the continuous action of the friction.Normal Mode Initialization :

A relatively new approach to initializationwas made through normal mode expansion of thedata. Normal modes are the solutions of dynamicprediction model equations without any diabaticheating or friction. They represent free oscillations.When the dynamical prediction equations arelinearised, then the normal modes are called"linear normal modes"; otherwise, the normalmodes are "non-linear normal modes". Thetheory of normal modes in classical hydrodynamicsis described in Lamb’s Hydrodynamics (1932, 6thedition, Section 168). As explained by Lamb,normal modes are also connected to normalco-ordinates and co-efficients of stability (see §6.10, Chapter 6).

The normal modes of a quasi-static P.E.


model correspond to gravity waves, Rossby waves,and waves which have mixed characteristics of bothgravity and Rossby waves. In general, gravity wavemodes have larger frequencies and smaller periodswhile Rossby wave modes have smaller frequenciesand longer periods.

In pract ice, i t has been found thatknowledge of linear normal modes is essentialfor construction of non-linear normal modes.As such, normal modes of the linear predictionequations form the basic modules for linear as wellas for non-linear normal mode initialization.

Barotropic non-divergent vorticity equationhas been the first amongst the prediction equationsin the field of NWP. Spherical harmonics are thenormal mode solutions of linearised barotropicnon-divergent vor t ici ty equation on theearth-sphere (Haurwitz, 1940; Craig, 1945;Neamtan, 1946). As such, spherical harmonics havebeen used as the basic functions for the spectralmodel of the global linear vorticity equation(Silberman, 1954; Platzman, 1960; Baer andPlatzman, 1961). Many research workers havesubsequently used spherical harmonics forexpressing single-level data in terms of sphericalharmonics for linear prediction model usingbarotropic non-divergent vorticity equation.

While spherical harmonics are the normalmode solutions for the linearised barotropicnon-divergent vorticity equation, they are not thenormal mode solutions for the linearised primitiveequation model over a sphere. We need a series ofspherical harmonics to construct normal modesolutions for linearised P.E. model, over a sphere.This is a very old problem connected with Laplace’slinearised tidal equation without externaltide-generating forces (see Lamb, 1932). Normalmodes of Laplace’s linear tidal equation includingthe method of obtaining the same, their asymptoticbehaviour and tables of their numerical values aregiven by Margules (1893), Hough (1898), Dikii(1965), Flattery (1967) and more completely byLonguett-Higgins (1968). In appreciation of theillustrious work done by Hough (1898), thesolutions corresponding to the normal modes oflinearised global shallow-water equations arereferred to as "Hough functions" or "Houghharmonics". For non-linear global shallow-waterprimitive equations, the normal modes are obtainedagain as a series sum of Hough harmonics.

Kasahara (1977, 1978) outlined the procedure forconstructing Hough harmonics and using the sameas basic functions in the solution of barotropic P.E.model equations on the earth-sphere.

Dickinson and Williamson (1972) suggestedthat one of the rational ways of filtering outunwanted modes (meteorological noise) during thetime-integration of a P.E. model, is to expand theinitial observed meteorological fields into series ofnormal modes, and selectively to filter out theunwanted normal gravity-wave modes from theinitial data fields. These authors outlined a generalprocedure for computation of normal modes in thecase of a linearised P.E. model. They also illustratedtheir general procedure for the actual calculation ofthe linear normal modes in the case of grid-pointrepresentation of a two-layer model ocean on therotating earth sphere. They also announced that theywere developing computer routines for expansionof the NCAR General Circulation Model (GCM)data into normal modes.

Williamson and Dickinson (1976) describedtheir procedure for expanding grid-point data intothe linear normal modes of the NCAR GCM. Theapproach assumed small-amplitude perturbationsabout a state of rest and involved separation ofvariables to give vertical and latitudinal structureequations for each longitudinal wave-number. Asan example of their procedure, they expanded theGCM model simulation data for 30 days, into thenormal modes. They showed that the computationalmodes, regarded as noise arising from numericalgrid-point representation of continuous fields, haveamplitudes at least an order of magnitude smallerthan the meteorologically important Rossby waves.They also showed that except for Kelvin waves, thegravity wave modes are also not significantly largerthan the computational modes.

Machenhauer (1977) pointed out thatputting the amplitudes of gravity wave modes equalto zero at initial time t = 0 does not prevent thegrowth of gravity-wave modes during the course oftime-integration of a P.E. model; the gravity-wavemodes do develop subsequently through non-linearinteractions among the initial Rossby-wave modes.He suggested that some horizontal velocitydivergence and hence some vertical motion (whichare otherwise characteristic of gravity-wave modes) should be retained initially at t = 0. Machenhauer(1977) also developed and described a procedure


which is cal led non-l inear normal-modeinitialization for a spectral model of shallow-waterequations on the spherical earth. This procedureinvolved the use of a non-linear balance equationand an iterative process to solve it.

Baer (1977) independently suggested the useof a more complete non-linear balancedrelationship. He tested it to a second orderapproximation for a mid-lat itude, β-plane,shallow-water model. The general theory ofnon-linear normal modes was further developed byBaer and Tribbia (1977) and applied by Tribbia(1979) to a simple equatorial model for whichseparation of gravitational modes is more difficultthan it is for mid-latitude region. The shallow-watermodel of Machenhauer (1977) was extended byDaley (1979) to multi-level spectral forecastingmodels and by Temperton and Williamson (1979)to multi-level grid-point models.

Studies have been undertaken to identifysimilarities and dis-similarities in this normal modeinit ia l ization and the earl ier methods ofinitialization. For example, Wiin-Nielsen (1979)found great similarity in this normal modeinitialization method and the earlier staticinitialization method which combined balancedmodel flow having limited amount of divergence.He concluded that the balance between the pressurefield and the wind field (divergent + non-divergentflow) is virtually identical to the quasi-geostrophicbalance except on the largest scales. Even for theselargest scales, the difference came entirely from thevariation of coriolis parameter on the sphericalearth. The Rossby-type normal modes would beidentical to those coming from quasi-geostrophicbalance theory if coriolis parameter is keptconstant. Wiin-Nielsen obtained this result fromlinearised shallow-water equations.

Leith (1980) examined in depth, the role ofsuccessive iterations adopted in the non-linearnormal-mode initialization procedures. He pointedout that application of these initializationprocedures, starting from a linearly balanced state,induces the secondary circulations needed for abalance in non-linear balanced states; that the firstiteration in the usual non-linear normal modeinitialization procedure for P.E. models is nearlyequivalent to quasi-geostrophic approximation; thatsubsequent iterations lead to higher orderapproximations including a 3-dimensional

non-linear Balance equation which can beconsidered as an extension of the classical2-dimensional non-linear Balance equationintroduced by Charney (1955) for initialization ofP.E. models.

In practice, normal mode initialization(NMI) method has not been without difficulties(Daley, 1981). There have been efforts to overcomethese difficulties. Convergence problems associatedwith Machenhauer’s iteration scheme have beenreduced through the work of Kitade (1983) andRasch (1985). Tribbia (1984) proposed a newapproach for obtaining higher-order corrections toMachenhauer’s scheme. In this method, theprojection of the non-linear forcing on thegravitational manifold is expanded in Taylor seriesaround the initial time. This gave a generalisediteration equation in which the first termcorresponds to the then accepted standard,non-linear, normal mode initialization by thismethod; more refined initial balance between windand pressure fields can be obtained to any order byincorporating higher and higher-order terms in theexpansion.

Normal mode initialization (NMI) is a"frontier" research problem at the time of writing.The research is progressing in the followingdirections :

i) Development of implicit initializationschemes which are equivalent to NMI schemes inthe final result but do not require explicitknowledge of normal modes.

ii) Development of schemes for limited areaforecasting.

iii) Development of schemes which areapplicable to forecasting models which contain notonly the usual three parameters (two windcomponents u and v and the heights of constantpressure surfaces) but also friction and diabaticheating.

iv) Incorporating vertical structure as a part ofNMI.

Temperton (1988) has given an implicitNMI scheme which can be implemented withoutexplicitly knowing the normal modes of ashallow-water model.

This implicit scheme is applicable tolimited-area models also. His presentation of thescheme is simple and clear. We shall present onlyan outline of Temperton’s scheme as Appendix 7.2


(A & A′).Appendix 7.2(A)Temperton’s (1988) SchemeModel :

Barotropic (shallow-water) model inCartesian coordinates on a polar stereographicprojection, with map scale factor m given by

m = 1 + sin θo

1 + sin θ

θo is the latitude where the map scale factor m is

unity, i.e. the standard latitude where the mapprojection is true. If u and v are the components ofthe wind vector along the axes of this coordinatesystem, then the wind images U and V are given by

U = um

, V = vm

The shallow-water equations are

∂U

∂t = f V −

∂φ∂x

+ QU (A-1)

∂V

∂t = − f U −

∂φ∂y

+ QV (A-2)

∂φ∂t

= − m2 Φ ⎛⎜⎝ ∂U

∂x +

∂V

∂y ⎞⎟⎠ + Qφ (A-3)

φ is the geopotential height expressed as aperturbation on the mean value Φ . The non-linearterms have been grouped together and put on theright-hand side of Eqs. (A-1, 2, 3) as QU,QV,Qφ.Setting of these non-linear terms equal to zero isequivalent to linearising the equations about a stateof rest with constant geopotential height Φ.

The model is bounded by a solid wall Γ inthe vicinity of the equator.

The horizontal wind is decomposed intorotational and irrotational components in terms ofthe usual functions ψ and χ :

U = − ∂ψ∂y

+ ∂χ∂x

; V = ∂ψ∂x

+ ∂χ∂y

(A-4)

ζ∼ = ∇2 ψ ; D∼

= ∇2 χ (A-5)

ζ∼ = ∂V

∂x −

∂U

∂y ; D

∼ =

∂U

∂x +

∂V

∂y (A-6)

Boundary conditions at the solid wall are that thereis no cross-boundary flow; i.e.

ψ = 0 on Γ (A-7)

n ⋅ � χ = 0 on Γ (A-8)

where n is the outward-pointing normal vector atthe wall.

The model Equations A-(1-3) can now bewritten as

∂ ζ∼

∂t = − f D

∼ + Qζ (A-9 )

∂D∼

∂t = f ζ∼ − ∇2φ + QD (A-10)

∂φ∂t

= − m2 Φ D∼

+ Qφ (A-11)

Here Qζ and QD contain the non-linear terms and

β-terms of the vorticity and divergence equationsrespectively. Qφ contains the non-linear terms of the

continuity equation.Normal Modes :

Normal modes of the model are the solutionsof Eqs. A-(9-11) when we set

Qζ = 0 = QD = Qφ ; i.e.

∂ζ∼

∂t = − f D

∼ (A-12)

∂D∼

∂t = f ζ∼ − ∇2 φ (A-13)

∂φ∂t

= − m2 Φ D∼

(A-14)

Some important Properties of the NormalModes:

f and m2 are not constant in space. Hence,Equations A-(12-14) are non-separable and thenormal modes cannot be found easily. However, wecan infer some useful properties of these normalmodes :

i) Time tendencies of slow (Rossby) modesare very small and hence can be regarded asnegligible terms in these Equations A-(12-14).Hence these Rossby modes are nearly


non-divergent (D∼

R = 0) and quasi-geostrophic.

∇2 φR = f ζ∼R (A-15)

ii) All (Rossby as well as gravity) modessatisfy Equations A-12 and A-14. These equationsgive

D∼

= − 1f ∂ ζ∼

∂t = −

1

m2Φ ∂φ∂t

∴ ∂∂t

⎛⎜⎝ ζ∼

f −

φm2Φ

⎞⎟⎠ = 0

i.e. ∂∂t

⎛⎝ m2 Φ ζ∼ − f φ ⎞⎠ = 0

We denote Z ≡ m2 Φ ζ∼ − f φ and call it"Potential Vorticity". We have

∂Z

∂t = 0 (A-16 )

iii) Associated with any mode, we can have ahorizontal field of Z whose time variation is givenby

Z ( x, y, t ) = Zo ( x, y ) eiυt

i.e. ∂Z

∂t ( x, y, t ) = i υ Zo (x, y) eiυt (A-17)

where υ is the frequency of the mode.

From Eq. (A-16), we have ∂Z

∂t ( x, y, t ) = 0. Hence

either υ = 0 or Zo ( x, y ) = 0.iv) For the fast (gravity) modes, υ ≠ 0; hence

Zo (x,y) = 0; i.e. gravity modes have zero potentialvorticity Z. Also

m2 Φ ζ∼G = f φG (A-18)

where subscript G denotes gravity mode.v) Entire potential vorticity is carried by

Rossby modes.Implicit Linear NMI :

Let the analysed model state be denoted by

Xo = ⎛⎝ ζ∼o , D∼

o , φo ⎞⎠T. Set the fast gravity modes

equal to zero and denote such gravity wave-free

initialized state by XI = ⎛⎝ ζ∼I , D∼

I , φI ⎞⎠T consisting

of only slow Rossby modes for which D∼

I = 0. Also

by (A-15)

∇2 φI = f ζ∼I (A-19)

If we add increments ∆ ζ∼ = − ζ∼G

and

∆φ = − φG to ζ∼o and φo fields, we shall get

gravity wave-free fields ζ∼I and φ I i.e.

∆ ζ∼ = − ζ∼G = ζ∼I − ζ∼o (A-20)

∆ φ = − φG = φI − φo (A-21)

These increments also define fast gravitymodes and hence by (A-18) they satisfy

∆ ζ∼ = f

m2 Φ ∆ φ (A-22)

Our immediate objective is to determine

∆ ζ∼ and ∆φ. Substituting A-(20-22) in (A-19) and

re-arranging, we get a Helmholtz equation for ∆ φ :

⎛⎜⎝∇2 −

f 2

m2 Φ

⎞⎟⎠ (∆ φ) = − ∇2 φ o + f ζ∼o (A -23)

R.H.S. is known from the initial analyzed field. By

solving (A-23), we get ∆φ. The corresponding ∆ ζ∼

can then be found immediately from (A-22).

By adding increments ∆φ and ∆ζ∼ to analyzed

fields φ o and ζ∼o we get fields φR and ζ∼R . Of

course D∼

R = 0.

Implicit non-linear NMI : Equations A-9 to A-11 can be written in the

form

∂∂t

⎡⎢⎣

⎢⎢ ζ∼

D∼

φ

⎤⎥⎦

⎥⎥ =

⎡⎢⎣

⎢⎢

0f0

− f 0

−m2Φ 0

−∇2

0

⎤⎥⎦

⎥⎥ ⎡⎢⎣

⎢⎢ ζ∼

D∼

φ

⎤⎥⎦

⎥⎥ +

⎡

⎢

⎣

⎢

⎢

QζQD

Qφ

⎤

⎥

⎦

⎥

⎥ (A-24)

As in the case of implicit linear NMI, we

determine increments ∆ζ∼ , ∆D∼

, ∆φ which whenadded to the analysed fields will take away gravity


modes. It is easy to see that the increments

∆ζ∼ , ∆D∼

, ∆φ will then satisfy the equation

⎡

⎢

⎣

⎢

⎢

( δt ζ∼ )G

( δt D∼ )G

( δt φ)G

⎤

⎥

⎦

⎥

⎥ =

⎡⎢⎣

⎢⎢

0− f 0

f 0

m2Φ 0

∇2

0

⎤⎥⎦

⎥⎥ ⎡⎢⎣

⎢⎢ ∆ ζ∼

∆ D∼

∆ φ

⎤⎥⎦

⎥⎥ (A-25)

The subscript G denotes the fast gravitymode component of the vector; the time -

tendencies δt ζ∼

, δt D∼

, δt φ written on the left

hand side of Eq. (A-25) are obtained by running the

model for one forward time-step; ∆ζ∼ , ∆D∼

, ∆φare the variables to be determined.

The second equation of (A-25) is

− f ∆ ζ∼ + ∇2(∆ φ) = (δt D∼ )G (A-26)

The slow Rossby modes are non - divergent;hence the observed divergence and itstime-tendency can be regarded as entirely due tofast gravity modes, i.e.

(δt D∼ )o = (δt D

∼ )G (A-27)

By virtue of (A-18), we also have

m2 Φ ∆ ζ∼ = f ∆ φ (A-28)

Substitution of (A-27) and (A-28) in (A-26)gives

⎛⎜⎝∇2 −

f 2

m2 Φ

⎞⎟⎠ ∆ φ = (δt D

∼ )o (A-29)

Eq. (A-29) is Helmholtz equation for therequired increment ∆ φ to the geopotential field φ;the right-hand side of (A-29) is simply the field of

tendency of D∼

obtained from one forward time stepof the model. Solve (A-29) for ∆ φ . Use this ∆ φ to

obtain ∆ ζ∼

from (A-28). Having obtained the

increment in ζ∼

and φ fields, now proceed to get

∆ D∼

, the increment in divergence field D∼

.The third equation of (A-25) is

m2 Φ ∆ D∼

= (δt φ)G (A-30)

To get ∆ D∼

, we now require (δt φ)G . It will

now be shown that this term (δt φ)G will be obtained

by solving another Helmholtz equation which

involves time tendencies of ζ∼ and φ. The argumentis as follows :

Eq.(A−18) gives m2Φ( δt ζ∼

)G = f ( δt φ )G

(A-31)

Eq. (A−15) gives f ( δt ζ∼

)R = ∇2 ( δt φ )R

(A-32)

Now f ( δt ζ∼

)R + f ( δt ζ∼

)G = f ( δt ζ∼

)o (A-33)

∴ f ( δt ζ∼

)G = f ( δt ζ∼

)o − ∇2( δt φ )R

= f ( δt ζ∼

)o − ∇2⎧⎨⎩( δt φ )o − ( δt φ )G

⎫⎬⎭ (A-34)

∴ ⎛⎜⎝ ∇2 −

f 2

m2Φ ⎞⎟⎠ ( δt φ )G

= ∇2( δt φ )o − f ( δt ζ

∼ )o

(A-35)

The RHS of A-35 is computed by runningthe model for one forward time-step. We have thus

obtained ∆ ζ∼

, ∆D∼

, ∆φ. We get corresponding

∆U , ∆V also. This completes one iteration. Its stepsare summarized below :

i) Run the model for one forward time-step to

obtain ( δt ζ∼

)o , ( δt D

∼ )o

, ( δt φ )o;

ii) Solve Helmholtz Equation A-29 for ∆ φ;iii) Solve Helmholtz Equation A-35 for

( δt φ )G;

iv) Obtain ∆ ζ∼

from ∆ φ through Eq. A-28;

v) Obtain ∆ D∼

from ( δt φ )G through Eq.

A-30;vi) Obtain ψ and χ functions from the vorticity

and divergence fields by solving Helmholtzequations

∇2 (∆ ψ) = ∆ ζ∼

∇2 (∆ χ) = ∆ D∼

vii) From ∆ψ and ∆χ functions, obtain windcomponents


∆ U = − ∂∂y

(∆ψ) + ∂∂x

(∆ χ)

∆ V = ∂∂x

(∆ ψ) + ∂∂y

(∆ χ)

viii) Add the increments ∆U , ∆V , ∆φ to themodel fields.

(Note : The Helmholtz equations mentionedin steps (ii), (iii) and (vi) above are to be solved byassigning suitable boundary conditions (A-7) and(A-8)).

This completes iteration number 1. Nowcheck if ∆U , ∆V , ∆φ are all less than somespecified small values. If not, go to iterationnumbers 2,3,....until ∆U , ∆V , ∆φ are all less thanthe specified small values; i.e. the iterative schemeconverges.

When the iterative scheme converges, theresulting model state satisfies

⎧⎨⎩

δt D∼

= 0

δt ( f ζ∼

− ∇2 φ ) = 0

⎫⎬⎭

(A-36)

Appendix 7.2� A� �An Improved Version of the Implicit Non-linearNMI Scheme :

Temperton (1988) further improved theimplicit non-linear NMI scheme given in Appendix7.2(A). It is believed that this scheme is better inhandling "Beta" terms, boundary conditions andmutual orthogonality of the derived Rossby andGravity modes. We shall now present an outline ofthis improved scheme retaining the same barotropyof the model in Cartesian coordinates on a polarstereographic projection, with the same map-scale

factor m given by m = 1 + sin θo

1 + sin θ and wind images

U and V given by U = um

, V = vm

.

The shallow-water equations are

∂∂t

⎛⎝∇2 ψ⎞⎠ = − Fχ + Bψ + Qψ (A′-1)

∂∂t

⎛⎝∇2χ⎞⎠ = F ψ + Bχ − ∇2φ + Qχ (A′-2)

∂φ∂t

= − m2 Φ ∇2χ + Qφ (A′-3)

ψ is the streamfunction, χ is the velocitypotential and φ is the geopotential height expressedas a perturbation on its mean value Φ. The operatorsF and B are defined by

F ≡ ∂∂x

⎛⎜⎝f

∂∂x

⎞⎟⎠ +

∂∂y

⎛⎜⎝f

∂∂y

⎞⎟⎠ (A′ -4 )

B ≡ ∂∂x

⎛⎜⎝f

∂∂y

⎞⎟⎠ −

∂∂y

⎛⎜⎝f

∂∂x

⎞⎟⎠ (A′ - 5)

The non-linear terms have all been groupedtogether and put on the RHS of Equations (A′-1),

(A′-2), (A′-3) as Qψ , Qχ , Qφ.

The model is bounded by a solid wall Γ inthe vicinity of the equator. Boundary conditions atthe solid wall are that there is no cross-boundaryflow; i.e. ψ = 0 and n⋅�χ = 0 on Γ; n is theoutward-pointing normal vector at the wall. Linear Normal Modes :

Linear normal modes of the model are thesolutions of the linear equations :

∂∂t

⎛⎝∇2 ψ⎞⎠ = − F χ (A′-6)

∂∂t

⎛⎝∇2 χ⎞⎠ = F ψ + Bχ − ∇2 φ (A′-7)

∂φ∂t

= − m2 Φ ∇2 χ (A′-8)

Some important properties of the linear NormalModes :

f and m2 are not constant in space. Hence,Eqs. (A′-6) to (A′-8) are non-separable and thenormal modes cannot be found easily. However, wecan infer some useful properties of these normalmodes :

i) Time-tendencies of slow (Rossby) modesdenoted by subscript R are very small and hence canbe regarded as zero in Eqs. (A′-1) to (A′-3); thesemodes are stationary. These modes are also nearlynon-divergent and satisfy the linear Balanceequation :

∇2 φR = F ψR (A′-9)

ii) Time-tendencies of fast (gravity) modes,denoted by subscript G, are comparable to otherterms in Eqs. (A′-1) to (A′-3) and hence cannot be


neglected as was done for Rossby modes. We know

the boundary condition ∂χ∂n

= 0 on Γ. We can then

invert Eq. (A′-8) and write

χ = − 1Φ

∇n−2

⎧⎨⎩

∂∂t

⎛⎜⎝

φm2

⎞⎟⎠

⎫⎬⎭

(A′-10)

where ∇n−2 is the linear operator which inverts the

Laplacian with homogeneous Neumann-typeboundary conditions. The consistency condition forthis inversion, namely that the domain integral of⎧⎨⎩

∂∂t

⎛⎜⎝

φm2

⎞⎟⎠

⎫⎬⎭ vanishes, is satisfied.

From Eqs. (A′-6) and (A′-10), we get thelinearised potential vorticity equation

∂Z′∂t

= 0

where

Z′ ≡ Φ ∇2 ψ − F ∇n −2 ⎛⎜⎝

φm2

⎞⎟⎠ (A′-11)

is the potential vorticity.iii) Associated with any mode, we can have a

horizontal field of Z′ with time variation given by

Z ′ (x , y , t) = Z ′o (x , y) ei υ t

∂Z′∂t

( x , y , t ) = i υ Z′o ( x , y ) ei υ t

where υ is the frequency of the mode. Eq. (A′-11) now gives that either υ = 0 or

Z′o = 0. The condition υ = 0 is associated withslow-moving Rossby modes and the conditionZ′o (x , y) = 0 is associated with fast-moving gravity

modes. Hence for gravity modes, we have

Φ ( ∇2 ψ )G = F ∇n −2 ⎛⎜⎝

φm2

⎞⎟⎠G

(A’-12)

iv) Slow-moving Rossby modes have zerodivergence but non-zero potential vorticity, withpressure-wind field satisfying linear Balanceequation; fast-moving gravity modes have non-zerodivergence but zero potential vorticity.

Implicit NMI :Arguing on the same lines as before, one

iteration of initialization scheme, using normalmodes based on Eqs. (A′-6) to (A′-8) is exactlyequivalent to

⎡

⎢

⎣

⎢

⎢

0

− F

0

F

− B

m2 Φ ∇2

0

∇2

0

⎤

⎥

⎦

⎥

⎥ ⎡⎢⎣

⎢⎢

∆ψ ∆χ ∆φ

⎤⎥⎦

⎥⎥ =

⎡

⎢

⎣

⎢

⎢

δt ( ∇2 ψ )

δt ( ∇2χ )

δt φ

⎤

⎥

⎦

⎥

⎥G

(A′-13)

where ∆ψ , ∆χ , ∆φ are the required changes to thevariables, and the tendencies on the RHS areobtained from one forward model time-step; thesubscript G denotes the fast-moving gravity modecomponent.

The third equation of (A′-13) is

m2 Φ ∇2 (∆ χ) = (δt φ)G (A′-14)

This equation will be used to find therequired changes to the χ-component of the windfield. To determine ( δt φ )G

, we use the same

argument as before.(A′-12) gives

F ∇n −2 ⎧⎨⎩δt

⎛⎜⎝

φm2

⎞⎟⎠

⎫⎬⎭G

= Φ ⎛⎝∇2 δt ψ⎞⎠G

(A′-15)

(A′-9) gives F ( δt ψ )R = ∇2 ( δt φ )R

(A′-16)

Total time tendency is the sum of its two componenttendencies

∴ F (δt ψ)G = F (δt ψ)o − F (δt ψ)R ( A′-17)

or F ( δt ψ )G = F ( δt ψ)o − ∇2 (δt φ)o + ∇2 (δt φ)G

(A′-18)

The boundary condition for ψ on the rigidboundary Γ is ψ = 0. With this homogeneousDirichlet-type boundary condition, Eq. (A′-15)

gives

Φ (δt ψ)G = ∇d −2 F ∇n

−2 ⎧⎨⎩δt

⎛⎜⎝

φm2

⎞⎟⎠ ⎫⎬⎭G

∴ F(δt ψ)G = 1Φ

F ∇d −2 F ∇n

−2 ⎧⎨⎩δt

⎛⎜⎝

φm2

⎞⎟⎠

⎫⎬⎭G


(A′-19)

∇d −2 denotes the linear operator which inverts the

Laplacian with homogeneous Dirichlet boundarycondition.

Combining (A′-18) and (A′-19), we get

∇2(δt φ)G − 1Φ

F ∇d−2 F ∇n

−2 ⎧⎨⎩δt

⎛⎜⎝

φm2

⎞⎟⎠

⎫⎬⎭G

= − F ( δt ψ)o + ∇2 ( δt φ )o

(A′-20)

Given the "observed" one forward time-steptendencies of φ and ψ , Eq. (A′- 20) helps us to

obtain the tendency ( δt φ )G. Eq. (A′-14) is then

solved for ∆χ, with boundary condition∂

∂n (∆ χ) = 0 on Γ. We now return to the second

equation of (A’-13).

− F ( ∆ ψ ) − B ( ∆χ ) + ∇2 ( ∆ φ ) = ⎧⎨⎩δt (∇

2 χ)⎫⎬⎭G

(A′-21)

As stated earlier, the slow Rossby modes arenon-divergent; i.e.

⎧⎨⎩δt (∇

2 χ )⎫⎬⎭G

= ⎧⎨⎩δt (∇

2 χ )⎫⎬⎭0

(A′-22)

From (A′-20) and (A′-14), we have already

obtained ∆χ. Now using (A′-22), Eq. (A′-21) gives

∇2(∆φ) − F(∆ψ) = ⎧⎨⎩δt(∇

2χ)⎫⎬⎭o

+ B(∆χ) (A′-23)

Since the changes ∆ φ and ∆ψ correspond tothe fast modes, having zero potential vorticity, wecan obtain, as in case of Eq. (A′-19),

F (∆ ψ) = 1Φ

F ∆ d−2 F ∆ n

−2 ⎛⎜⎝

∆φm2

⎞⎟⎠ (A′-24)

substituting (A′-24) in (A′-23), we get

∇2(∆ φ) − 1Φ

F ∇d−2 F ∇n

−2 ⎛⎜⎝

∆φm2

⎞⎟⎠ =

⎧⎨⎩δt (∇

2 χ)⎫⎬⎭o

+ B ( ∆ χ ) (A′-25)

This is the required equation for the change ∆φ tothe geopotential field φ.

To find the corresponding increments to therotational part of the wind field, we invoke theproperty that gravitational modes have zeropotential vorticity (Eq. (A′-12)).

i.e. ∇2 (∆ ψ) = 1Φ

F ∇ n−2

⎛⎜⎝

∆ φm2

⎞⎟⎠ (A′-26)

The whole algorithm for one iteration of theinitialization scheme is summarised below :

i) Run the model for one forward time-step toobtain the "observed" time tendencies

δt (∇2 ψ)o , δt (∇

2 χ)o , δt (φ)o.

ii) Solve (A′-20) for (δt φ)G and then (A′-14)

for ∆χ.iii) Solve (A′-25) for ∆φ.iv) Solve (A′-26) for ∆ψ.

v) From ∆ψ and ∆χ , obtain ∆U and ∆V .vi) Add the increments ∆U , ∆V , ∆φ to the

model fields.This is iteration No. 1. Now repeat the steps (i)

to (vi). Follow this iterative procedure until theincrements ∆U , ∆V , ∆φ are below some specifiedsmall values. If the iterative scheme converges, itfollows from (A′-20) and (A′-25) that the resultingmodel state satisfies

⎧⎨⎩

δt ⎛⎝∇2 χ⎞⎠ = 0

δt (Fψ − ∇2 φ) = 0

⎫⎬⎭ (A’-27)

Higher-order NMI Scheme : It is possible to build an hierarchy of higher

order normal mode initialization schemes, bothexplicit and implicit, along with increasingcomplexity. Temperton (1988) also gave an implicitversion of Tribbia’s (1984) higher-order explicitscheme. In going from explicit to implicit version,one has now to take two forward time-steps insteadof one forward time-step taken in the last scheme.For details of the corresponding iteration scheme,the reader is referred to the original paper ofTemperton (1988).BOUNDED DERIVATIVE METHODIntroduction :

The method of Bounded Derivative hasbeen in use for quite some time in fields like PlasmaPhysics. This method appears to have been firstbrought into Meteorology by Kreiss (1980).Browning, Kasahara and Kreiss (1980) formalised


the method systematically for initialization schemesin numerical weather prediction. In its final form,the bounded derivative method has muchresemblance with implicit non-linear normal modeinitialization.

Initialization problem in NWP work is tomodify the original analysed pressure and windfields in such a way that the amplitudes of thefast-moving gravitational modes are small and theirfirst and successive time derivatives are also small.As in NMI schemes, one can get an hierarchy ofbounded derivative (BD) schemes also. InFirst-Order approximation of pressure-windrelationship, BD scheme demands that the first timederivatives of u, v, φ at time t = 0 are negligible;using this approximate pressure-wind relationship,the Second-Order approximation demands that thesecond-order time-derivatives of u, v, φ are alsosmall; Higher-Order approximations demand thatfurther Higher-Order time-derivatives are alsosmall . The method keeps successivetime-derivatives of u, v, φ bounded; the upper limitsof the time-derivatives are derived from thecorresponding space-derivative terms occurring inthe model equations.

As is well-known, quasi-geostrophic modelgives the First-Order diagnostic relationshipbetween pressure and wind fields energeticallyconsistent with a prognostic form of the truncatedvorticity equation; Second-Order pressure-windrelationship is the diagnostic linear Balanceequation with some more terms in energeticallyconsistent vorticity equation. Non-linear diagnosticBalance equation between pressure and wind fieldswith corresponding additional terms in prognosticvort ic i ty equation gives the Third-Orderapproximation, and so on. These are all derivedfrom the consideration of energetic consistency(Lorenz, 1960).

Similarly, Bounded Derivative Methodalso gives an hierarchy of pressure-windrelationships based on the bounds oftime-derivatives. The reader may ask whetherthese refinements are worth the trouble. The answeris that at least conceptually, these exercises give adeeper understanding of and support for hierarchyof initialization schemes.Basic Steps :

Some basic steps in the understanding of

hierarchy of initialization schemes are outlinedbelow :

i) It is first accepted as a fact, based onobservations, that Rossby-wave modes haverelatively large horizontal-scales and relativelysmall frequencies and large periods in timecompared to gravity-wave modes.

ii) In Rossby-wave modes, rotationalcomponent of wind is larger than the irrotationalcomponent of wind.

iii) The pressure and wind fields in Rossbymodes are in near-geostrophic balance. The linearand non-linear balance relationships are only somerefinements of quasi-geostrophic balance and arenot drastically different from it.

iv) In Rossby-wave modes, Rossby NumberRo is much smaller than 1.

v) In terms of the above properties ofRossby-wave modes, the gravity-wave modes standquite in contrast; e.g. the horizontal scales ofgravity-wave modes are generally small. Even iftheir horizontal scales be large, yet their changes intime are fast; time-tendencies in gravity-modes arelarge. Their irrotational component of wind is muchlarger than the rotational component; flow is alsoessentially cross-isobaric.

vi) Keeping these facts in mind, one performsscale analysis of the primitive equations in terms ofcharacter ist ic length and time scales ofRossby-wave modes and then to find approximateforms of primitive equations which will beconsistent with Rossby-wave modes. For thecharacteristic length-scale chosen, a restriction isput on the values of the first time- derivatives in theprognostic terms of the primitive equations. Thisensures that the prognostic term is small comparedto space-gradient terms thereby ensuring smallvalue for the Rossby number Ro. In other words,time tendency can be neglected in comparison to

some large space-gradient terms, i.e. ∂∂t

∼ 0 in

prognostic model equations. This is the First-Orderapproximation.

vii) The next Second-Order approximation ismade by differentiating the prognostic primitiveequations with respect to time once more, and thendemanding that the Second-Order time-derivative issmall compared to the First-Order time derivativesof other space-gradient terms in the primitiveequation. The condition obtained during the


First-Order approximation is utilised in thisSecond-Order approximation.

viii) Third-Order approximation is made byfurther differentiating the primitive equation withrespect to time and then demanding that theThird-Order time derivative is small compared tothe Second-Order time derivatives of otherspace-gradient terms in the primitive equation. Theconditions obtained during the First-Order andSecond-Order approximations are utilised in thisThird-Order approximation, and so on.

Illustration :

We shall illustrate the method of BoundedDerivative Initialization (BDI) for the simple caseof a linearised shallow-water model in (x,t)co-ordinates given by Semazzi and Navon (1986).

Basic state consists of a uniform zonal current U−

geostrophically balanced with height h− . The stepsare as given below :

1) The linearised Equations are

⎧

⎨

⎩

⎪⎪⎪

⎪⎪⎪

∂u

∂t + U−

∂u

∂x − f v = − g

∂h

∂x

∂v

∂t + U−

∂v

∂x + f u = 0

∂h

∂t + U−

∂h

∂x + h−

∂u

∂x − f

U−

g v = 0

⎫

⎬

⎭

⎪⎪⎪

⎪⎪⎪

7.2(8)

2) Assume Perturbations of the form

⎛⎜⎝

u v h

⎞⎟⎠ =

⎛⎜⎝

u~

v~ h~

⎞⎟⎠ eikx 7.2(9)

where u∼

, v∼ , h

∼ are functions of time only.

3) Substituting 7.2(9) into 7.2(8), we get

⎧

⎨

⎩

⎪⎪⎪

⎪⎪⎪

∂u

∼

∂t + i k U− u

∼ + i k g h

∼ − f v

∼ = 0

∂v

∼

∂t + i k U− v

∼ + f u

∼ = 0

∂h

∼

∂t + i k U− h

∼ −

f U− v∼

g + i k h− u

∼ = 0

⎫

⎬

⎭

⎪⎪⎪

⎪⎪⎪

7.2(10)

4) For illustration, we shall adopt characteristicscales of a typical, middle-latitude synopticdisturbance.

L horizontal scale ≈ 106 m

7.2(11)

H∼ height perturbation ≈ 102 m

H− mean height ≈ 104 m

V∼ horizontal particle speed ≈ 10 m s−1

T characteristic time L ⁄ V∼ = 105 s

K characteristic wave number ≈ 10−6 m−1

G gravitational acceleration ≈ 10 m s−2

F Coriolis Parameter = 10−4 s−1

5) Non-dimensional quantities are all of the orderunity, except ∈ = O (10−1). To identify thesequantities, we show them, temporarily, as primedquantities.

u∼ ′ = u

∼ ⁄ V∼ , x′ = x / L

7.2(12)v∼

′ = v∼

⁄ V∼ , k′ = k / K

h∼′ = h

∼ / H

∼ , g′ = g / G

h−

′ = h−

/ H− ,

U−

′ = U−

/ V∼

t′ = t / T , f ′ = f / F

6) Substituting 7.2(12) into 7.2(10), we get a systemof non-dimensional equations. We illustrate this bysubstitution in the first of the three equations of7.2(10).

∂u∼

∂t →

∂u∼′

∂t′ × 10−4 ms−2

i k u∼

U−

→ i k′ u∼′ U− ′ × 10−4 ms−2

i k g h∼ → i k′ g′ h

∼ ′ × 10−3 ms−2

f v∼ → f ′ v

∼′ × 10−3 ms−2

or ∂u

∼′

∂t′ + ik′u

∼′ U− ′ + 10 ⎛⎝i k′g′h

∼′ − f ′ v

∼′ ⎞⎠ = 0

7) To simplify the notation, we now drop primes,though remembering all the time, that we aredealing with non-dimensional quantities each oforder unity in the equations except ∈= O(10−1).With this understanding, Eqs. 7.2(10) become

∈ ⎛⎜⎝

∂u∼

∂t + i k u

∼ U−

⎞⎟⎠ + ⎛⎝ i k g h

∼ − f v

∼ ⎞⎠ = 0


∈ ⎛⎜⎝

∂v∼

∂t + i k v

∼ U−

⎞⎟⎠ + f u

∼ = 0 7.2(13)

∈2 ⎛⎜⎝

∂h∼

∂t + i k h

∼ U

− − f

U−

g v∼

⎞⎟⎠ + i k h

_ u∼ = 0

8) To ensure that the First-Order time derivatives ofu, v, h correspond to Rossby modes, it has beenshown by Browning et al. (1980) that the First-Order approximation is given by

⎧

⎨

⎩

⎪

⎪

i k g h∼

− f v∼ = ∈ a ( x , y , t )

f u∼

= ∈ b ( x , y , t ) i k h

− u∼ = ∈2 c ( x , y , t )

⎫

⎬

⎭

⎪

⎪ 7.2(14)

where a,b,c are smooth functions.The simplest way to satisfy these conditions

is to assume that a = b = c = 0 at time t = 0. From7.2(14), we then get

i k g h∼ − f v

∼ = 0

f u∼

= 0 i k h−

u∼= 0

or u∼

h∼ = 0 ,

v∼

h∼ =

i k gf

7.2(15)

i.e. v∼

is in geostrophic balance while u∼

, which couldgive horizontal velocity divergence, is zero as aFirst-Order approximation (non-divergentgeostrophic balance model).9) To get Second-Order approximation at time t =0, we first differentiate 7.2(14) with respect to timeand get

⎧

⎨

⎩

⎪⎪⎪

⎪⎪⎪

i k g ∂h

∼

∂t − f

∂v∼

∂t = ∈

∂a

∂t

f ∂u

∼

∂t = ∈

∂b

∂t

i k h_ ∂u

∼

∂t = ∈ 2

∂c

∂t

⎫

⎬

⎭

⎪⎪⎪

⎪⎪⎪

7.2(16)

Now we put∂a

∂t = 0 =

∂b

∂t =

∂c

∂t ; i.e.

⎧

⎨

⎩

⎪⎪⎪

⎪⎪⎪

i k g ∂h

∼

∂t − f

∂v∼

∂t = 0

f ∂u

∼

∂t = 0

i k h_ ∂u

∼

∂t = 0

⎫

⎬

⎭

⎪⎪⎪

⎪⎪⎪

7.2(17)

In effect, these are two equations in 3 quantities

∂u∼

∂t ,

∂v∼

∂t and

∂h∼

∂t. Substituting for

∂u∼

∂t ,

∂v∼

∂t and

∂h∼

∂t.

in 7.2(13), we get

⎛⎜⎝ ∈−2 g k 2

h−

f + ∈−1 f

⎞⎟⎠ u∼

h∼ + 2 i k U−

v∼

h∼ = −

g k2 U−

f

i k U−

u∼

h∼ − ∈−1 f

v∼

h∼ = − i g k ∈−1 7.2(18)

or u∼

h∼ =

∈2 g k 2 U−

k2 g h−

+ ∈ f 2 − 2 ∈3 k 2 U− 2

v∼

h∼ =

i k gf

⎡⎢⎣ 1 +

∈3 k2 U− 2

k2 g h−

+ ∈ f 2 − 2 ∈3 k2 U−2

⎤⎥⎦

7.2(19)

If the terms of the order ∈3 in thedenominators of 7.2(19) are neglected, the balancerelationship between wind and pressure fieldbecomes identical with that given by Phillips(1960).7.2.3 Four-dimensional (4-D) Data Assimilation :

Under the topic of Objective Analysis, wedealt with 2-dimensional and 3-dimensionalinterpolation at grid points in 2-dimensional and3-dimensional space, when we are givenobservations at observation points, which are notlocated at our grid points. All the observations wereconsidered to be simultaneous or synoptic. Our nextquestion relates to the utilization of data which arenon-synoptic or asynoptic. This question has arisenparticularly during the last few decades whensatell ites give frequent observationsdistributed round the clock. In our NWP work,should we use only those satellite observationswhich are very close to the standard hours ofobservation? The same question may also arise in


respect of other observations which are taken afterthe initial time of NWP model and are receivedduring the course of the integration of the NWPmodel. There is a general desire to include all theseasynoptic observations in the model forecastscheme as far as possible. Naturally, this will haveto be done through some scheme of objectiveanalysis. When we add time dimension to objectiveanalysis in three dimensions of space, it becomes4-dimensional objective analysis.

4-dimensional objective analysis in thetropics presents some special difficulties. As statedearlier, even the objectively analysed two or threedimensional data at grid points have to be subjectedto special initialization processes (static, dynamic ora combination of the two) if we are to avoidunrealistic high-frequency oscillations of largeamplitudes. After all this sophistication at initialtime t = 0, it will not be advisable to feed into thecomputer model during the process of integration,raw data as they are received after time t = 0.Feeding of such raw data will create considerable"noise" inside the model and this may even ruin theforecast. The model cannot "assimilate" or "digest"the new data in raw form. We must subject the newraw data to some special process before feeding thesame into the computer model. Problems relating tothe feeding of these asynoptic data through4-dimensional objective analysis are referred to asthe problems of 4-dimensional data assimilation.This is a relatively new area of research althoughthe problems are basically the same as those ofstatic or dynamic initialization. The specialdifficulty in the tropics is the relative slownessof adjustment towards a balanced state. It hasto be ensured that gravity waves, which get easilyexcited at the slightest provocation, shall remainwithin tolerable limits before any new informationis introduced.

Miyakoda et al. (1976) subjected the globalsurface and upper air data for the entire GATEperiod (15th June to 24th Sept 1974) to theirtechnique of four-dimensional assimilation, usingGFDL general circulation model, with 9 verticallevels and approximately 220 km horizontalresolution on a modified Kurihara grid. For alimited period (4th to 17th Sept 1974), theyexperimented with three different versions of datainsert ion. Their scheme of damping thehigh-frequency oscillation is through repeated use

of Euler-Backward time differencing. The samedata at one grid point were inserted repeatedly everytime step for a 2-hour interval, beginning one hourbefore the valid forecast time. Their mainconclusions were as follows :

1) 4-dimensional assimilation approach isviable.

2) For the middle and higher latitudes, theassimilation results were better for the northernhemisphere than for the southern hemisphere,presumably due to relatively greater scarcity of datafor the southern hemisphere.

3) The results for the tropics were lesssatisfactory than for the extra-tropics.

4) The scheme somewhat over-smoothensmeteorologically significant systems also.

Ghil et al. (1979) experimented withinsertion of time-continuous satellite-soundingtemperatures obtained from polar-orbiting satellitesNOAA-4 and Nimbus-6 during the periodJanuary-March 1976. They used the GLAS(Goddard Laboratory for Atmospheric Sciences)General circulation model with 4o latitude × 5o

longitude horizontal resolution. They usedEuler-backward time differencing scheme, withstaggered distribution of variables in space grid andadditional smoothing at high latitudes. The schemestrongly suppresses high-frequency oscillations.

The vertical temperature profiles obtainedfrom satellite-based radiance measurements weregrouped at 10-minute time intervals, centered intime around a forecast model time step, which wasalso of 10 minutes. All the methods experimentedwith, carried out a correction of forecasttemperatures at grid points situated in theneighbourhood of a group or cluster ofsatellite-observed temperatures. This correctionwas applied to the forecast value at the model timestep closest to the observation time. It was based oninterpolation of (observed minus model forecast)temperature field. The difference between variousmethods consisted in the manner in which theinterpolation co-efficients were determined. Thisinterpolation was two-dimensional (on isobaricsurface). In principle, these methods are similar tothose used for objective analysis.

A number of experiments were performed tostudy the effects of using various amounts ofsatellite data and different methods of assimilation.These included the assimilation of data from the


NOAA -4 satellite only, from Nimbus-6 only, andof data from both satellites combined. The testsinvolved variations in the application of successivecorrection methods and optimum interpolationmethods. Intermittent feeding of the satellitesounding data was also tested, and its resultscompared with those of time-continuous feeding.

The satellite data were assimilated toproduce initial states for numerical forecasts. Foreach assimilation experiment, an evenly spacedsequence of initial states was selected, from which3-day forecasts were made.

The effects of satellite data feeding werejudged by the following three criteria :

i) Differences between the initial statesproduced with and without utilisation of satellitedata.

ii) Difference between numerical predictionsmade from these initial states.

iii) Differences in local city precipitationforecasts for 128 cities uniformly distributed overthe United States.

The experiments suggested the followingconclusions :

i) Feeding of satellite temperature data canhave a modest, but statistically significant positiveimpact on numerical weather forecasts, as verifiedover the continents of the northern hemisphere.

ii) This impact is highly sensitive to thequantity of satellite data fed, the more the better.The impact of two satellite systems was larger thanthat of one-satellite system by an amount roughlyproportional to the quantity of data fed.

iii) The method of data interpolation plays amajor role in the magnitude of the impact for thesame data. Direct Insertion Method had practicallyno impact. Successive Correction Method andStatistical Assimilation Method provided anappreciable impact.

iv) Time-continuous feeding of satellitederived temperatures was better than intermittentfeeding.

In actual practice, the method of 4-Dassimilation varies from one operational centre toanother, and also varies at one and the sameoperational centre from time to time. Bengtsson etal. (1982) described the method of 4-D assimilationused at the European Centre for Medium RangeWeather Forecasts, UK, for producing FGGE levelIII-b data set, out of the level II-b data collected at

different times during the FGGE period. Thismethod is an intermittent data assimilation system,using a multi-variate optimum interpolationanalysis, a non-linear normal mode initialization,and also a high resolution forecast that produces afirst estimate for the subsequent analysis. Data wereassimilated in 6-hour periods.

Hoffman (1986) suggested a 4-Dimensionalvariational procedure to minimize errorssimultaneously in three-dimensions of space andthe fourth dimension of time. Let T be time periodlarger than the adjustment time of the forecastmodel. To get a 4-Dimensionally consistent data setat time t = 0, one determines the field at t = − T suchthat the difference between the model-forecastedfield during time t = -T and t = 0 and all theobservations in this time interval satisfy thecondition of "best fit" in 4-Dimensions. Thisbecomes the initial data set at t = 0.

Lorenc (1988) presented an iterative schemeof 4-dimensional assimilation. It uses operationallyavailable observations, 3-dimensional objectiveanalysis and the forecast model equations. Itapproximates a variational algorithm and is anextension of 3-dimensional optimum interpolationscheme. The author performed a series ofexperiments and produced dynamically consistent4-dimensional analyses which fi t ted theobservations without too much of computer time.However, he admitted that the scheme still requiredconsiderable improvement before it could beadopted for operational work; that it could be usedon experimental basis for arr iving at anoperationally acceptable scheme. Indeed, thesubject of 4-dimensional assimilation needs muchtheoretical work and trial experiments forimprovement, even though some schemes are in useoperationally at the operational forecasting centres.Physical Initialization introduced by T.N.Krishnamurti :-What is Physical Initialization?

Krishnamurti et al . (1995, MWR,September, pp. 2771-2790) presented a review ofthe subject, quoting earlier work of Krishnamurti etal. (1984, J. Metero. Soc., Japan, 62, 613-649);Ramamurthy and Carr (1987); Kasahara et al.(1988); Donner (1988); Donner and Rasch (1989);Puri and Miller (1990); Puri and Davidson (1992)and others.


The main steps of Krishnamurti et al. (1995)are summarized below:

(i) Estimate Observed Rainfall Rate:Observed rainfall rate at time of satellite picture isestimated from regression equation involving a mixof surface-based systems (Like Rain gauge) andSpace-based systems (OLR, Microwave ... SatellitePictures SSM/I Special SensorMicrowave/Imager).

(ii) Reverse Cumulus Parameterization:Modify the vertical distribution of humidity so as togive the above "observed" rainfall through Kuo’sparameterization scheme modified by Krishnamurtiet al. (1983, Moistening, Heating and Rainfallrates). This is being called Reverse CumulusParameterization.

(iii) Physical Initialization : Modify otherparameters of the atmosphere so as to be consistentwith this rainfall rate, vertical distribution ofhumidity, thermodynamics, vorticity, convergence,etc. This initial field is now accepted for timeintegration. This is called "Physical initialization".

(iv) T.N. Krishnamurti et al. used globalspectral model T213.

(v) In one of their studies they got Mesoscalesystems with rainfall rate of the order of 4 cm / 3 hr(4 x δ = 32 cm / day).

(vi) Meso-scale systems present in satellitepictures get identified and structurally analysed;these are then incorporated in NWP models through"Physical initialization", with improved physicaland dynamical consistency. 24-hour prediction ofsuch systems appears possible.

(vii) Authors identified as many as 47meso-scale precipitation elements over the globaltropics (30o S - 30o N) at 1200 UTC on 22 August1992.

(viii) There is currently a line of thinkingthat within an easterly wave there exists apopulation of Mesoscale Convective Systems(MCSs), each containing its own relative vorticitymaximum.The prevailing lower- tropospheric flowadvects these MCSs. At favorable locations, thesegive rise to disturbances with wavelengths of theorder of 500-1000 km.

A few of them may develop into tropicalstorms and cyclones. If there is circular flow, thenthese MCSs are likely to converge and coalesce intoa spiral rain band. (Holland and Dietachmayer,1993; Lander and Holland, 1993; Ritchie and

Holland, 1993; QJRMS).Krishnamurti et al. (1998, Meteorology

and Atmospheric Physics, 65,171-181 ) havefurther shown that physical initialization gives auseful input for initial mapping of meso-scalesystems within the field of a tropical cyclone andthis improved initial input contributes towardsimproved prediction of future intensity of acyclone. The authors illustrated this improvementin the case study of hurricane OPAL over the Gulfof Mexico for the period Oct.2,1995 , 12 UTCthrough October 6, 1995 ,12 UTC. Physical initial izat ion tested onTrack-Forecasting of tropical cyclones :

Williford et al. (1998, MWR, May,pp.1332-1336) analysed the errors in the positionsof tropical cyclone centers in the Atlantic ocean, thenortheast Pacific Ocean and the northwest Pacificocean during the period 1989-1995. They comparedthe cumulative forecast errors of Americanoperational forecast centers (Tropical PredictionCenter / National Hurricane Center/ Joint TyphoonWarning Center) and the FSU global spectral modelforecast having physical initialization scheme.

The errors were calculated for 12-hour,24-hour, 36-hour, 48-hour, and 72-hour forecastpositions.

The errors are shown below :-Errors in position (km)

Forecast Model 12-hr 24-hr 36-hr 48-hr 72-hr

FSU Model 115 180 235 291 469

Operational Model 102 176 257 352 550

Physical init ial ization tested againstclimatology:Krishnamurti et al. (1999, Atomosfera, 12, pp.199-203) have shown the utility of physicalinitialization in preparing rainfall climatology andalso in improving the analysis of three-dimensionalinitial state of the atmosphere on a routine basis.Physical ini t ial izat ion leads to improvedforecasting for very short periods which may becalled "now-casting". They have shown that therainfall estimate through physical initialization in aforecasting model comes very close ( c.c ~ 0.9 ) tothe observed one, thus improving the climatologynot only of rainfall but also of other meteorologicalparameters l ike temperature, humidity,static-stability, etc.


To get this improved estimate of rainfall, theauthors follow what they call "Reverse CumulusParameterization Algorithm"; this algorithmrestructures the vertical distribution of specifichumidity such that the use of the forward algorithmin the forecast model produces nearly the samerain as was supplied to it. The specific humidity inthe constant flux layer near the surface is alsorestructured through reverse surface similaritytheory, such that the surface evaporation and theprescribed precipitation are in close balance.Specific humidity field in the upper troposphere isalso restructured to be consistent with the forecastmodel-based OLR and the satellite - observedOLR. This procedure provides a balance among thevertically integrated evaporation, precipitation andthe moisture sink in the vertical.

The high degree of accuracy is obtained byplacing the reverse algorithm adjacent to theforward forecast algorithm. This assures a high skillfor each short time-step of model forecast rainfall.

Accumulation of the model registers of eachsmall time step over a longer time period like amonth gives good climatology. The authorscompared FSU Model climatology of Oct.1991rainfall over the region 30o S to 30o N around thewhole latitude belt, against the rainfall climatologyof ECMWF and NCEP for the same period,October 1991. The comparison confirms the utilityof physical initialization for improved andmodel-consistent climatology.Physical initialization tested on low-levelstratus clouds

Bachiochi and Krishnamurti (2000,Monthly Weather Review, September, Vol. 128,pp. 3083-3103) introduced an improvedparameterization scheme for low-level stratusclouds in the FSU Coupled Ocean - Atmospheremodel and showed that this parameterizationscheme improves the simulation of low-levelstratus clouds along the west coast of the north andSouth American continents. In theirparameterization scheme, the PBL clouds dependon the PBL column thermal structure, low-levelstability, wind magnitude at the PBL top, relativehumidity and surface wetness. This scheme oflow-level stratus cloud parameterization appears togive better energy balance at the sea surface, in thePBL layer below the cloud base as well as inside thestratus cloud mass; land-sea thermal contrasts are

also better simulated; the scheme enhances coastalupwelling Ekman transport and low-level windcirculation.

7.3 Parameterization of Cumulus Convection inthe Tropics

Since early 1960s, there have been severalattempts at cumulus parameterization. Forsynoptic-scale models, meso-scale cumulusmotions are regarded as sub-grid motions. Formeso-scale models, individual cumulus motions areregarded as sub-grid motions. As such, scales ofparameterization have varied considerably. Themore of details we incorporate in these schemes, themore we wish we could incorporate. This way,there will be no end. Somewhere we have to stopand review the advantages and disadvantagesof incorporating more and more details.Principally, the cost and speed of computationdecide the limit for various investigators. As such,various schemes are being reported in literature.Also, slight variations are made by someinvestigators and they generally all report "good"results. It becomes pretty difficult to judge therelat ive meri ts of several cumulusparameterization schemes now in the field. Weshall briefly review some of the most widely usedschemes for meso-scale model parameterizationsand for synoptic-scale model parameterizations(Houze and Betts, 1981; Ooyama, 1982; Frank,1983b).Meso-scale Model Parameterizations :Kreitzberg and Perkey (1976, 1977) employed whatmay be called sequential plume model to simulatethe convection. The plumes were activatedwhenever grid-point conditional instabilityexceeded a critical value determined by clouddepth, and continued until the instability droppedbelow the threshold. The total mass flux at cloudbase was determined iteratively by requiring thatthe hydrostatic pressure in the plume becomesequal to the pressure in the subsiding environment.These authors simulated meso-scale rain bandssimilar to those found in extra-tropicalcyclones.

Brown (1979) used a one-dimensionalupdraft plume model to simulate convection in histwo-dimensional meso-scale model. He includedsubsidence between plumes and somewhat detailed

7-32 7.3 Parameterization of Cumulus Convection in the Tropics

specification of the mass fluxes and transformationof condensation products. He assumed that the totalcloud mass flux at 900 mb (hPa) was directlypropotional to the large-scale mass flux at that level.The constant of proportionality was empiricallydetermined. He modelled evaporation-drivenmeso-scale downdrafts as well as meso-scale anvilupdrafts occurring in associat ion withcumulonimbus convection.

Fritsch and Chappel (1980) included bothconvective updrafts and downdrafts in their model.Both were one-dimensional entraining plumes.Convective mass flux was determined iteratively.Their formulation is based on the hypothesis thatthe buoyant energy available to a parcel, incombination with prescribed period of time for theconvection to remove that energy, can be used toregulate the convection in a grid-point element ofthe meso-scale model. The following were the mainassumptions and constraints of theparameterization:

i) Moist convection occurs only when air isforced to its level of free convection by low-levelconvergence, air mass overturning, or when low-level heating and mixing remove any stable layerssuppressing moist convection, i.e. when potentialbuoyant energy becomes available.

ii) Mass transports by moist convection areclosely approximated by model cloud ensemblewhich treats deep convection as the dominant cloudform.

iii) Precipitation efficiency of the convectiveclouds is related to the vertical wind shear acrossthe cloud depth.

iv) There is a prescribed period of time for theconvection to remove available buoyant energy(ABE) from a grid-point element and to stabilize it.The stabilization rate (removal of available buoyantenergy) is numerical ly given by themodel-generated ABE divided by the estimatedtime for the convective cells to move across thegrid-element.

v) The changes in the temperature and mixingratio at a grid-point are the net effects ofcompensating subsidence in the environment andarea-weighted cloud updrafts, downdrafts andenvironmental advections.

vi) Momentum is vertically exchanged throughbulk-mixing processes in the cloud updrafts anddowndrafts and compensating environmental

vertical motions.All the above mentioned parameterization

schemes have considerable flexibility in respect ofspecifications for vertical eddy heating and eddymoistening. It is also possible to incorporateadditional features like eddy transports ofmomentum and vorticity. However, the followingremarks are relevant here :

a) One has to be careful in respect of time andcost of computation. Some parameterizationschemes can prove too time-consuming and alsotoo costly.

b) Even if time and cost of computaion be ofsecondary importance for research purposes insome institutions, reasonable justification must begiven for introducing complicated time-consumingparameterization schemes. The justification can bein terms of theoretical reasoning, observationalevidence or a mixture of both.

c) There might be some instances ofimprovement in the performance of the modelsthrough introduction of complicatedparameterization schemes, but verification andvalidation of the scheme must be carried out oversufficiently large number of occasions. This is noteasy to achieve in practice.Synoptic-scale Models

We have several parameterization schemesfor moist convection in synoptic-scale systems. It isnow believed that moist convection helps thegrowth of synoptic-scale systems in both tropicsand extra-tropics. Intense moist convection canproduce layer-type (thick cirrostratus, thickaltostratus or thick nimbo-stratus) or toweringcumulus and cumulonimbus-type clouds. Ingeneral, both of these clouds occur in combinationin all latitudes. However, it is presently believedthat towering cumulonimbus clouds dominate thetropical regions and layer-type clouds dominate theextra-tropical regions. Hence, for tropical regions,the parameterization schemes are generally forcumulus-type convection.

In literature, one comes across a widevariety of tropical cumulus convection typeparameterization schemes; almost each author hashis own scheme. Each author produces a verylimited number of synoptic-scale situations wherehis scheme shows some skill score. It is difficult atpresent to objectively evaluate the relative merits ofthese numerous schemes. However, these schemes

7.3 Parameterization of Cumulus Convection in the Tropics 7-33

generally fall into three broad categories which weshall briefly describe in the following paragraphs.These three broad categories are :

i) Moist convective adjustment. ii) Moisture convergence models.

iii) Cumulus cloud models.All these models focus their attention on

arriving at reasonably correct synoptic-scaleconditions in respect of temperature, humidity andrainfall. These schemes either give no attention orgive very little and inadequate attention to the roleof cumulus convection in influencing themomentum and vorticity budgets of large-scalesystems. The importance of cumulus convection inmomentum and vorticity budgets of the large-scalesystems has been acknowledged only recently. It isexpected that cumulus parameterization schemestaking proper note of cumulus contributions tolarge-scale momentum and vorticity budgets willsoon appear on the scene.Moist Convective Adjustment

The technique consists of first predicting thelarge-scale temperature and moisture for a time step∆ t without looking at super-saturation, if any. It isthen examined whether super-saturation has beenreached at a grid point. If super-saturation is notreached, no moist convective adjustment is made.If super-saturation has been reached, then theexcess moisture is taken out as rain from the volumeof air represented by that gird point. The excessmoisture is taken out by condensation isobarically;the latent heat of condensation is used to warm upthe air, consistent with the first and second laws ofthermodynamics. Care is also taken to avoidunstable lapse rates in the vertical in respect of drybulb temperature as well as equivalent potentialtemperature. In this adjustment, lapse rate of asaturated conditionally unstable layer is adjusted toneutral lapse rate in a specified time interval whichis usually taken to be less than one hour.

Krishnamurti et al. (1980) tested twovariations of this scheme called "Hard" and"Soft" convective adjustments. In the hardconvective adjustment, adjustment was effectedfor the whole volume represented by a grid point;in the soft adjustment, the adjustment was assumedto occur over a small fraction (of the order of 4%)of the total volume represented by the grid point.For the remaining volume of about 96%, it wasassumed that the vertical profiles of temperature

and equivalent potential temperature remainunchanged during that time-step. The value of thefraction ( 4%) was obtained by Krishnamurti et al.(1980) by ‘trial and error’ method to get the best fit,in the root-mean-square sense, between thecomputed and the observed rainfall over GATEarea, during the period between 1 September and 18September 1974. The soft adjustment method isbased mainly on the schemes proposed by Manabeet al. (1965), Miyakoda et al. (1969) and Kurihara(1973). Hard adjustment method over-estimatesrainfall by about one order of magnitude; hence itis superceded by soft adjustment method.Moisture Convergence Models

This method introduces statisticallyaveraged influence of cumulus clouds on thetemperature and moisture distribution in the verticalcolumn in terms of moisture convergence in thevertical column. In their classical paper, Charneyand Eliassen (1964) used total moistureconvergence in a vertical column to estimatecondensational diabatic heating. The verticaldistribution of this diabatic heating was assumed tobe proportional to the heat released from a parcel ofsaturated air ascending moist adiabatically afterreaching its level of condensation.

This classical paper has been the basis ofseveral subsequent parameterization schemes ofthis category (Kuo, 1965, 1974; Ooyama, 1969;Anthes, 1977; Krishnamurti et al., 1979, 1980; andothers). Of all these schemes, Kuo’s (1965, 1974)scheme has been most widely used along with slightvariations. Hence, for illustration, we shall outlineKuo’s parameterization scheme.7.3.1 Kuo’s Parameterization Scheme for DeepCumulus Convection :

Kuo (1965) gave parameterization schemefor deep wet cumulus convection.

Later, he (Kuo, 1974) gave parameterizationschemes for both deep wet cumulus convection aswell as shallow dry convection; he also gave whathe called "a more rigorous derivation" of hisparameterization scheme for deep wet convection.Without going into his rigorous derivation, we shallhere highlight the main ideas of theparameterization scheme. Further, we shall not dealwith shallow dry convection but only with deep wetconvection. Krishnamurti et al. (1983) and severalother authors have introduced slight modificationsinto Kuo’s main scheme; but among these


modifications, we shall confine ourselves to the onegiven by Krishnamurti et al. (1983). The main ideasof Kuo’s scheme (1965, 1974) including slightmodification given by Krishnamurti et al. (1983)are given below :

1. Many of the large-scale disturbances in thetropical atmosphere are driven by the release oflatent heat in deep cumulus towers and CB clouds.

2. The horizontal scale of cumulus cloud ismany orders of magnitude smaller than thehorizontal grid scale used in large-scale numericalmodels.

The time-scale of cumulus cloud is alsomany orders of magnitude smaller than thetime-scale of the large-scale motion.

The exact location and the exact time ofoccurrence of the individual clouds are consideredas unknown from the large-scale point of view. It isassumed that within the large-scale grid area(∆x.∆y) and the time step ∆τ used in a numericalmodel, the individual clouds are randomlydistributed, such that we can handle the cloudsinside the area (∆x.∆y) during the timeinterval ∆τcollectively in a statistical way. Correspondingly,in the prognostic and diagnostic equations of themodel, one will have to consider non-linear eddyterms. These eddy terms will be expressed interms of the mean values of the meteorologicalparameters applicable to the grid area (∆x.∆y) andthe time-interval ∆ τ .

3. Deep convection takes place in that regionof the atmosphere which satisfies the twoconditoins :

i) Atmosphere is convectively unstable.ii) The flow has low-level horizontal

moisture-flux convergence.These two conditions ensure that there is

large-scale vertical upward motion which lifts thesurface air to its condensation level and then totrigger the release of convective instability. Let pBdenote the lifting condensation level for the surfaceair; this is taken as the base of the deep convectivecloud. Let pT be the pressure where the moistadiabat from the condensation level cuts theenvironmental temperature curve on athermodynamic (T-Φ) diagram. This moist adiabatis given by

∂∂p

⎛⎝ Cp Ts+g z+L qs ⎞⎠

=0

where the subscript s denotes saturation (seeAppendix). pT is taken as the top of the cloud (infact, by theoretical reasoning, the cloud top wouldbe higher than the level pT).

The cloud is considered to have the shape ofa vertical cylinder.

iii) In this large-scale environment favourablefor deep cumulus convection, there is fractionalarea "a" in which new clouds get producedcontinuously. This "a" varies in time and from onegrid area to another grid area. We take "a" as therate of production of new cloud area per unit time,per unit area of horizontal cross-section of thelarge-scale environment.

4. This cloud does not stay for long as cloudin the environment; it exists only in concept and thattoo momentarily. This momentary existence isfollowed by immediate dissolution. This concept ispartly justified on the ground that the life-period ofan individual cloud is small compared to thelife-period of the synoptic scale system.

5. In the vertical column between levels pBand pT there is the large-scale supply of moisture Igiven by

I = ∫ p

B

pT

⎧⎨⎩�⋅(q V)+ ∂

∂p(q ω) ⎫

⎬⎭

dpg

= − 1g

∫ pT

pB

⎧⎨⎩�⋅(q V)+ ∂

∂p(q ω) ⎫⎬

⎭ dp 7.3(1a)

In addition to this supply through large-scaleconvergence of horizontal and vertical moistureflux, there is also supply of moisture throughconvergence of moisture flux by small-scale eddies,which supply is not directly derivable fromlarge-scale flow parameters. Krishnamurti et al.(1983) denote this supply by ηI ; this η is to beassigned a plausible value through consideration ofits effect on large-scale environment as deducedfrom analysis of observations. Hence, the totalsupply of moisture through large-scale motion andsmall-scale eddies is (1+η)I.

6. This total supply of moisture per unithorizontal area, per unit time, is partitioned into twocomponents :

i) (1+η)Iq; this part of moisture is imagined tobe used in moistening the environment throughdissolution of the cloud immediately after itsproduction.


ii) (1+η) Iθ; this part of moisture is supposed

to be used in increasing the temperature of theenvironment through dissolution of the cloudimmediately after its production.

I=Iq+Iθ

(1+η)I=(1+η)Iq+(1+η)Iθ 7.3(1b)

The moist-adiabatic thermodynamic processby which, part of the moisture goes for moisteningand part for warming the environment, is explainedin Appendix to this section.

In this parameterization scheme, verticaldistributions of moistening and of warming at anylevel are supposed to be proportional to

qs−q

∆ τ and θs−θ∆ τ respectively where ∆τ is

cloud time-scale parameter. The total supply ofmoisture and heat to the vertical cloud column isdistributed at different levels, in proportion to thevalues of these vertical structure functions.

7. The moisture supplied to the verticalcolumn is assumed to affect potential temperatureθ and specific humidity q of the environment asshown in the following two equations :

∂θ∂t

+ V⋅� θ+ω ∂θ∂p

=aθ ⎛⎜⎝

θs−θ∆ τ + ω

∂θ∂p

⎞⎟⎠ 7.3(2a)

∂q∂t

+V⋅� q = aq ⎛⎜⎝

qs−q

∆ τ⎞⎟⎠ 7.3(2b)

where aθ and aq are proportionality factors assumed

to be independent of pressure; these may vary intime.

We define the two functions Qθ and Qq by

Qθ = Cp

g L ∫

pT

pB

Tθ

⎛⎜⎝

θs−θ∆ τ + ω

∂θ∂p

⎞⎟⎠ dp 7.3(3a)

Qq = 1g

∫pT

pB

qs−q

∆ τ dp 7.3(3b)

Then moisture supply aθ Qθ goes to warm

the vertical column and aqQq goes to moisten thevertical column. Rainfall rate R is given by

R = Cp

g L∫pT

pB

aθ Tθ

⎛⎜⎝

θs−θ∆ τ +ω

∂θ∂p

⎞⎟⎠dp 7.3(3c)

8. Now the total moisture supply (1+η)I is

partitioned in the ratio b : (1-b) for moistening andwarming of the environment, i.e.

aq Qq = (1+η) I b 7.3(4a)

aθ Qθ = (1+η) I (1−b) 7.3(4b)

Rainfall rate R is given by that part of themoisture supply which warms the environment andfalls down as precipitation.

∴R=(1+η)I(1−b) 7.3(4c)

η and b are the two parameters which need to bedetermined. At present, there is no theoreticalformulation to give the values of η and b.Krishnamurti e t al . (1983) adopted anempirical-cum-statistical method to assignreasonable values to η and b.

Ooyama-Esbensen-Chu (1977, personalcommunication to Krishnamurti et al., 1983) hadanalysed GATE B-scale data sets for 18 day-period1-18 September 1974, 4 times a day. Thus, therewere 72 map times of data sets. The heating rates,moistening rates and rainfall rates were obtainedfrom these data sets.

It was further assumed by Krishnamurti etal. (1983) that the moistening rate and the warmingrate, through their representat ion by(1+η) b and (1+η)(1−b) respectively, may beconsidered as related to two large-scale flowparameters :

i) Vertically averaged vertical velocity ω__

, and

ii) 700 mb(hPa) relative vorticity ζ.They assumed a multi-linear regression

relationship between η and b on one side and ζand ω

__ on the other side in the form

(1+η) b=a1 ζ+b1 ω__

+c1

and(1+η) (1− b)=a2 ζ+b2 ω__

+c2

i.e.η=⎧⎨⎩(a1+a2) ζ+(b1+b2) ω

__+(c1+c2)

⎫⎬⎭−1

7.3(5a)

and b = a1 ζ+b1 ω

__+c1

(a1+a2) ζ+(b1+b2) ω__

+(c1+c2)

7.3(5b)

The regression coefficients were determinedby statistical least-square fit method, using theempirical data of Ooyama-Esbensen-Chu (1977),


and got the values :

a1 = 0.158 × 105s a2 = 0.107 × 105s

b1 = 0.304 × 103 mb-1s b2 = 0.107 × 103 mb-1s

c1 = 0.476 c2=0.870(dimensionless) (dimensionless)

7.3(5c)

From the experiments of Krishnamurti et al.(1983), it appears reasonable, for the present atleast, to adopt these values of regressioncoefficients in the modified Kuo’s scheme of deepcumulus parameterization, in other tropical regionsalso, including India (Keshavamurty and Sawant,1989, personal communication).

9. The steps for calculating, from large-scaleparameters, the rates of heating, moistening andrainfall due to deep moist convection are givenbelow :

i) Determine Qθ from Eq. 7.3(3a) and Qq from

Eq. 7.3(3b).ii) Assuming the values of regression

coefficients given by Eq. 7.3(5c), calculate the values of η and b

given by Eq. 7.3(5a and 5b). Use these values of ηand b and also already calculated values of Qq andQθ to determine aq and aθ from Eqs. 7.3(4a, 4b and

1a).iii) Warming rate for environment is given by

Eq. 7.3(2a). Moistening rate for environment isgiven by Eq. 7.3(2b). Rainfall rate R is given by Eq.7.3(3c).

10. From the experiments performed byKrishnamurti et al. (1983), they got the values in theneighbourhood of η = 0.4 and b = 0.3. Hence, (1+η)(1-b) = 0.98.

As stated earlier, Krishnamurti et al. (1983)had extended the parameterization scheme of Kuo(1974). The latter would be a special case of theextended scheme of Krishnamurti et al. (1983) ifwe put η = 0 and b = 0 hence (1+η) (1-b ) = 1.0.

It would be appreciated that the warming of

the environment and the rainfall rate are bothproportional to (1+η) (1-b). In both these schemesof Kuo (1974) and Krishnamurti et al. (1983), (1+η)(1-b) is having approximately the same value, 1.0.Hence, the warming rate and rainfall rate by boththe schemes would be approximately the same. Butthe moistening of the environment by the scheme ofKuo (1974) would be less than the moistening bythe scheme of Krishnamurti et al. (1983), exceptthrough strong vertical diffusion of moisture.Experiments with Kuo (1974) scheme and withKrishnamurti et al. (1983) scheme suggest that theatmosphere shows results closer to those given bythe scheme of Krishnamurti et al. (1983).AppendixMoist Adiabatic Process

Let a saturated air parcel at pressure p+∆p,height z, temperature (T+∆T) and specific humidity(q+∆q) rise up to level p, (height z+ ∆z) moistadiabatically. The parcel remains saturated at levelp, height ( z+ ∆ z ), has temperature less than(T+ ∆ T ) and specific humidity which is less than(q+ ∆ q ) but is yet saturated. The parcel has lostmoisture quantity ∆q. This is the amount of watervapour which has condensed and precipitated out,down towards the ground. Latent heat releasedduring condensation is L∆q. This process can beimagined to consist of three parts :

i) In moving up through differential ∆ p, theparcel moved dry adiabatically through height∆ z=⎪∆ p⎪ / gρ. It cooled dry adiabatically by

g

Cp∆ z= g

Cp

⎪∆ p⎪g ρ =⎪∆ p⎪

ρ Cp

ii) Water vapour ∆ q condensed and releasedlatent heat L ∆ q. This latent heat warmed up the air

parcel by temperature L ∆ q

Cp so that the temperature

at the level z +∆z becomes

⎧⎨⎩(T+∆ T)− g

Cp ∆ z+L ∆ q

Cp

⎫⎬⎭.

Level ofparcel

Cp× temperature g × heightL × Sp.hum.

MoistStatic Energy

New ( z + ∆z)Cp

⎧⎨⎩(T + ∆ T) −

gCp

∆ z + L ∆ q

Cp

⎫⎬⎭ g(z + ∆z) Lq

Cp (T+∆T ) + gz +L(q +∆q)

Old ( z ) Cp ( T + ∆ T ) gz L(q + ∆q)Cp (T+∆T ) + gz + L ( q

+ ∆q )


iii) Rainfall amount is ∆q.In going through this process, the moist

static energy components of the parcel at theoriginal level and the new level are as shown intable below.

The moist static energy of the parcelremains constant; in fact, this property gives theequation of moist adiabat.

When the cloud amount a dissolves andmixes with the environment, then the amount ofmoisture added to the environment is proportionalto a(qs-q). This quantity depends on the humidityof the environment rather than on the amount ofrainfall which has gone down towards the ground.We can thus treat the amount of moisture whichgoes into moistening the environment as almostindependent of the amount of moisture which aftercondensation falls as rain to the ground. However,it is to be borne in mind that the amount of diabaticheating which is released in the atmosphere isentirely due to the amount of rainfall.

Temperature T and specific humidity q ofthe large-scale environment increase due to cloudformation and cloud dissolution. This can happenonly so long as the whole environment is not filledwith cumulus cloud (a < 1). When the environmentis totally filled with cloud (a=1), then T and q havereached their maximum values of Ts and qs givenby the moist adiabat. No further increase oftemperature and humidity can result fromadditional accession of moisture and condensationunless there is change of condensation level to anew condensation level having higher values of Tsand qs even though latent heat of condensation isstill being released; rainfall will continue to becopious without change of T and q at any level.7.3.2 Arakawa-Schubert Scheme for CumulusParameterization :

At the time of writing, this scheme isconsidered to be the best in terms of Physics andDynamics of the cloud system in relation tolarge-scale synoptic system. At the same time, it isregarded to be somewhat too complicated andtime-consuming in NWP modelling of thesynoptic-scale systems.

The scheme has appeared in literature infour parts in J. Atmos. Sciences :

Part I, Arakawa & Schubert ( April 1974)Part II, Lord & Arakawa ( December1980)

Part III, Lord ( January 1982)Part IV, Lord et al. ( January 1982)Part I gives the core theory of the scheme

explaining several approximations and also giving200 equations ( 158 in the main paper and 42 in theAppendix).

Part II can be divided into 3 sections : i) Summary of core theory in the form of 10

equations and 3 schematic-diagrams.ii) Calculation of Cloud Work Function from

observations at tropical and some sub-tropicallocations.

iii) Cloud Work Function is shown to be anearly-universal function of cloud depth.

Part III is important in 2 respects :a) Varification of the model is given against

the observed data from phase III of GATE, using asemi-prognost ic approach.Actual andmodel-calculated rates of precipitation are shown toagree pretty well.

b) Sensitivity experiments were performed toshow little effect of changing the seeminglyarbitrary values of some of the parameters andassumptions occurring in the scheme.

Part IV gives numerical analogue of thewhole scheme of parameterizat ion forincorporation into an NWP model.

We shall briefly outline the basic conceptsand formulations in Arakawa-Schubert schemewhich have been subsequently elaborated orverified by Lord and others.

Towards the end of this Section 7.3, we alsoindicate extension of this Arakawa-SchubertScheme by Cheng and Arakawa (1990, 1992). Thisextension emphasizes the role of downdrafts andpresents a combined updraft-downdraft model,which can be incorporated into theArakawa-Schubert Scheme given in parts I to IVabove.1. Cloud Ensemble

It is visualised that at a level between thecloud base and the cloud top, there is an ensembleof clouds, each cloud having its own entrainmentrate and its own flux of mass at the base. All theclouds in the ensemble are imagined to have thesame cloud base but different cloud tops. The areaoccupied by this cloud ensemble is sufficientlylarge so as to include all clouds in various stages ofgrowth and life-cycle; but the area occupied by thecloud ensemble is considered to be small compared


to the area of the large-scale system for which theeffet of cumulus convection is being parameterized.2. Cloud Sub-ensemble

The cloud ensemble consists ofsub-ensembles. It is assumed that the members of acloud sub-ensemble are at random phases in theirlife-cycle; the summation of a cloud property overall members of the sub-ensemble is proprotional tothe property of a single cloud averaged over itsentire life-time, the constant of proportionalitybeing the number of clouds.

Let λ be a parameter which characterises acloud type. Let λ be such that the entire ensembleis covered when λ takes positive values betweenzero and λmax. Then the interval (λ , λ + d λ)denotes a cloud sub-ensemble.

Let mz denote the mass flux across a cloudsub-ensemble at level z. Then fract ional

entrainment rate is given by 1mz

∂mz

∂z . Arakawa and

Schubert defined

λ= 1mz

∂mz

∂z 7.3(6)

Larger values of λ denote larger rates ofdilution of the air inside the cloud sub-ensembleand hence smaller depths and lower cloud tops forthe sub-ensemble.

It is assumed that for a given sub-ensemble,λ is constant with height z. A sub-ensemble isdistinguished from other sub-ensembles by thevalue of its fractional entrainment rate λ.

Since λ is constant with height, the verticaldistribution of mz (λ) is immediately given by,

mz (λ)=mB (λ) eλ (z − zB)≡mB (λ) η (z , λ) 7.3(7)

η( z , λ )≡eλ (z − zB) 7.3(8)

Here mB (λ) denotes the mass flux at the

base of the cloud sub-ensemble; η ( z , λ ) givesvertical distribution of mz.

Total mass flux MB into the cloud ensembleat the cloud base level zB is given by,

MB=∫λ = 0

λmax

mB ( λ )d λ 7.3(9)

Total mass flux Mz across level z inside thecloud ensemble is given by

MZ=∫λ = 0

λmax

mZ ( λ )d λ 7.3(10)

It is assumed that for any giventhermodynamic structure of the environment,fractional entrainment rate λ determines all theproperties of the cloud sub-ensemble representedby λi in the interval (λ , λ + d λ). The properties are:

i) Level of cloud top.ii) Vertical mass flux at any level including the

level of the cloud base zBiii) Buoyancy of air inside the cloud.iv) Speed of updraft inside the cloud.v) Work done by buoyancy force.

vi) Rate of precipitation. vii) Total mass of cloud air detrained at its top.viii) Rate at which cloud processes destroy

convective instability of the environment.Hence, every thermodynamic structure of

the environment has its own cloud ensemble withdeterminate cloud properties including the rate ofprecipitation and also the rate at which cloudprocesses reduce the convective instability of theenvironment.3. Cloud-Work Function A ( λ )

Cloud-Work Function is the work done bybuoyancy force, between the level of the cloud baseand the level of zero buoyancy,per unit mass flux atthe cloud base, per unit time. It is denoted by

A (λ) = ∫z

B

zt

η (z , λ) gT__

(z) ⎧⎨⎩Tvc (z , λ) − Tv

__ (z) ⎫

⎬⎭ dz

7.3(11)zt is the level of zero buoyancy; it is

sometimes called as cloud top. Tvc is the virtual

temperature of the cloud air; Tv

__ is the virtual

temperature of the environment. A (λ) is animportant parameter in the Arakawa-Schubertscheme of the cloud parameterization. In particular:

i) It is a property of the environment for aspecified value of λ

ii) It is the rate of generation of kinetic energyof vertical motion.

iii) Positive values of A (λ) indicate presenceof moist convective instability in the environment. 4. Reduction of Convective Instability

Cumulus convection tends to destroyconvective instability of the environment through


subsidence outside the cloud in the following way :i) Subsidence warms the environment and

hence reduces the buoyancy of the cloud updraft.ii) Subsidence dries the environment in the

lower and the middle levels of the layer in whichcloud exists.

iii) Subsidence outside the cloud pushes downthe top of the Planetary Boundary Layer (PBL).

iv) Subsidence outside the cloud reduces theinflux of saturated air into the cloud base.

Just as buoyancy is counted as the propertyof convectively unstable environment, so also,subsidence outside the cloud can be counted as aproperty of the cloud, reducing the convectiveinstability of the environment. It also reduces thebuoyancy and kinetic energy of the cloud updraft.Its function is opposite to that of buoyancy. Like A(λ), one can define D (λ) as the rate of destructionof convective instability, in a form analogous to A(λ), but performing opposite function.5. Influence of Cloud-Cloud Interaction on A (�)

By tending to destroy the instability of thecloud-free environment, each cloud is reducing thebuoyancy of itself and also of other clouds.

Let K ( λ , λ′) represent the rate ofincrease of kinetic energy of vertical motion of thecloud - type λ due to the influence of the othercloud-type λ′. This will be negative. mB (λ′) whichis the rate of mass flux at the base of the cloud-typeλ′ is +ve . Hence, ⎧

⎨⎩K (λ , λ′)mB (λ′)⎫

⎬⎭ is -ve.

Arakawa-Schubert Scheme puts

⎡⎢⎣ ddt

A (λ) ⎤⎥⎦C

=∫0

λmax

K ( λ , λ′ ) mB ( λ′ )d λ′

7.3(12)The effect of cloud-cloud interaction is to

reduce the convective instability of the environmentand hence to reduce A (λ) of the environment.K ( λ , λ′) is called the kernel.6. Influence of Large-scale Processes on A (λ)

Large-Scale processes other than cloudsinfluence the convective instability of theatmosphere through diabatic heating includingradiation, through horizontal advection, large-scaleupward motion, etc. Some of these processes mayincrease the convective instability of theatmosphere and some may decrease it. The

large-scale upward vertical motion is believed to bethe most important element affecting theenvironment and increasing convective instability,increasing A (λ), increasing the depth of PBL anddecreasing the static stability of the atmosphere. InArakawa-Schubert Scheme, the total contributionof large-scale process is to increase A (λ) and thiseffect is represented through the equation

⎡⎢⎣

dd t

A ( λ ) ⎤⎥⎦L S

=F ( λ ) 7.3(13)

7. Quasi-stationary Assumption about A (λ)Cloud-Cloud interaction represented by

∫0

λmax

K (λ , λ′) mB (λ′) d λ′ reduces the value of A

(λ). Large-scale processes represented by F(λ)increase the value of A(λ). Now, Arakawa-Schubert Scheme makes an important assumptionand uses it as the mathematical basis of the scheme.It assumes that the negative contribution ofcloud-cloud interaction exactly balances thepositive contribution of F (λ).

In other words,

∫0

λmax

K (λ , λ′) mB (λ′) d λ′+F (λ)=0 7.3(14)

This equation helps to make a closed scheme ofparameterization such that the mutual interactionbetween the environment and cloud ensemble istaken care of, in terms of the original prognostic setof equations for the large-scale environment. In thissense, Equation 7.3(14) is also called the closurecondition.

Justificat ion for this assumption ofquasi-balance between opposing influences on theenvironment in terms of A (λ) is given on thefollowing lines :

The period of a synoptic-scale system forwhich the parameterization scheme is devised is ofthe order of a few days. The interval of time forwhich this assumption of quasi-balance is made issmall, of the order of half an hour. We can assumequasi-equilibrium state of the synoptic-scale systemfor such a small interval of time. Assumingquasi-equilibrium state of the large-scaleenvironment for a small interval of time, wecalculate the change which will occur in thelarge-scale environment during this first smallinterval of time; incorporate this change in thelarge-scale environment at the end of the first small


time interval, get the new parameters for thechanged large-scale environment; again work outthe change which will occur in this environmentover the second small interval of time, introduce thechange in the environment and go over the thirdsmall interval of time, and so on. It is like using aforward time-differencing scheme.8. Calculation of K � � , ��

i) ⎡⎢⎣

dd t

A (λ) ⎤⎥⎦C

=∫0

λmax

K (λ , λ′) mB (λ′) d λ′

i.e.⎧⎨⎩A (i)t + ∆t−A (i)t

⎫⎬⎭=∑K

j = i

j = imax

(λ , λ′) mB (λ′) ∆ t

7.3(15)ii) Take small arbitrary values for mB(j) and ∆t ; forconvenience of numerical calculation, let mB(j)∆t= Constant . By using the set of equations given inthe original text (Arakawa and Schubert,1974), findthe value of

K (i , j)=⎧⎨⎩A (i)t + ∆ t − A (i)t

⎫⎬⎭/ mB(j) ∆ t 7.3(16)

for each value of j i.e. the change in cloud workfunction of the i-th sub-ensemble due tomodification of the large-scale environment by j-thsub-ensemble, per unit mass flux mB(j), per unittime.9. Calculation of F(λ)

i) ⎡⎢⎣

dd t

A (λ)⎤⎥⎦LS

=F (λ)

F (i)=⎧⎨⎩ A (i)t + ∆ t−A (i)t

⎫⎬⎭/∆ t 7.3(17)

ii) By using the set of equations given in theoriginal text (Arakawa and Schubert, 1974),calculate F(i) , the effect of large-scale processes onlarge-scale environment, in particular, in changingA (i); ∆t 30 minutes is acceptable.10. Calculation of mB (λ′)

In integral form, we have to solveFredholm’s Integral Equation of first kind

∫λ′ = 0

λmax

K (λ , λ′)mB (λ′) d λ′+F (λ)=0

under the condition that mB(λ′) is positive. Inpassing, it may be mentioned that there are twokinds of Fredholm’s Integral euqations :

First Kind :

∫a

b

K (x , y)f (y)d y=g (x) 7.3(18)

Second Kind :

∫a

b

K (x , y)f (y)d y=µ g (x)+f (x) 7.3(19)

K is known and is called Kernel. f isunknown and is to be found; µ and g are known.11. Schematic Diagram

Fig. 7.3 (1) is a schematic diagram of a partof the parameterization Scheme. T

__ and q

_ denote

the values of temperature and specific humidity ofthe synoptic-scale environment at time t. Fromthese parameters, one can get λ and A (λ). i ) Effect of Large-scale Processes onEnvironment :

Choose a small time interval ∆ t′ andcalculate changes in temperature and specifichumidity and get T

__ and q

_ at time t + ∆ t". From

these, calculate λ and A (λ) at time t +∆ t′ . The timerate of change of A (λ) gives F (λ) , i.e. F(i).ii) Effect of Cloud-Cloud Interaction onEnvironment :

Similarly, calculate changes intemperature and specific humidity in small timeinterval ∆ t ′′ due to cloud-cloud interaction. Forthe changed values of T

__ and q

_ , calculate λ and A

(λ) at time t +∆ t ′′ . From the time rate change ofA (λ) due to cloud-cloud interaction, findK (λ , λ′),i.e.K (i , j) .iii) For these given values of F ( i ) and K(i, j ), solve Fredholm’s integral equation :

∫λ′

λmax

K (λ , λ′)mB (λ′) d λ′+F (λ)=0

or∑⎡⎣K (i , j)mB(j)⎤

⎦j = 1

imax

+F (i) = 0 7.3(20)

In other words, for a particular i, given F(i)and K ( i , j ) for j = 1,2,3,......,imax, find the set ofvalues mB ( j ) for j = 1,2,3,...., imax 12. Some Sub-problems

We shall now outline some of thesub-problems of the Arakawa-Schubert Scheme,also using some numbers in place of symbols i andj when useful for fixing the ideas.


Sub-problem I : Given distribution of T

__ and q

_ , how to

calculate λi ?Ans.

i) Specify sub-ensemble λi by pressure levelof zero buoyancy, p̂i. This can be calculated from

given distribution of T__

and q_ . We may call this as

pressure of cloud-top of sub-ensemble λi. Equatingthe level of zero buoyancy to cloud top levelimplies that detrainment takes place in a thin layeraround this level, because level of zero buyoyancyis the level of maximum upward velocity.

ii) To get the correct value of λi for a givenp̂i, adopt the following iterative process :

a) Assume a guess value of λi

b) Using this guess value of λi and the givenvertical distribution of T

__ and q

_, calculate the

level of zero buoyancy. Let it come as p̂̂ i

c) If p̂̂ i

< p̂i , increase λi by a small amount.

If p̂̂ i

> p̂i , decrease λi by a small amount.

(Larger value of λ will give smaller height andlarger pressure value of cloud-top).

d) Go to step (b) and calculate new value of

p̂̂ i

for new value of λi . Go to step (c).

e) Use this iterative process till you get proper

λi such that ⎪ p̂̂ i

= p̂i⎪ for all practical purposes

i.e. ⎪ p̂i − p̂̂ i

⎪ is smaller than some specified

FIG. 7.3(1) : Schematic diagram for determining K ( λ , λ′ ), F(λ) &mB(λ′ ) (Asnani, 1993).


small quantity. This gives mutually compatible

values of λi and p̂ i for any given vertical

distribution of T−

and q_

.Sub-problem II :

Given T−

, q_

, λ and p how to get A (λ)?

Ans.

A (λ)=∫zB

zt

η (z , λ) g

T−( z) ⎧

⎨⎩Tvc(z , λ) − T

−v (z)⎫

⎬⎭ d z

Lord and Arakawa (1980) chose cloud-toppressure to denote a cloud sub-ensemble. For eachvalue of cloud-top, they calculated correspondingvalue of λ. Field observations directly givecloud-tops and vertical distribution of and q barand not λ . For this type of data set, it is possible tocalculate λ corresponding to each cloud -top. Lordand Arakawa (1980) chose the following 17 valuesfor cloud-top pressure (mb, hPa) to identify 17cloud sub-ensembles :

100, 150,200,250, 300, 350, 400, 450, 500,550, 600, 650, 700, 750, 800, 850, 912.5 mb (hPa).For all these cloud-tops, the base of the cloud wastaken to be 950mb. Corresponding to these 17values of cloud-tops, they calculated λ and A(λ) for

different, vertical distributions of T−

and q_. For

observations of p̂ ,T−

and q_ , they took 7 different

locations and synoptic conditions in the tropicsand sub-tropics. The 7 locations were :

i) The Marshall Islands data set from 15April to 22 July 1956 (Yanai et al. 1973, 1976).

ii) VIMHEX data set over north-centralVenezuela from 22 May to 6 Sep 1972 (Betts andMiller, 1975).

iii) GATE data set from 31 August to 18September 1974 (Thompson et al., 1979).

iv) AMTEX data set from 14 to 28 February1974 and 14 to 28 February 1975 (Nitta,1976).

v) Mean West Indies Tropical Sounding forhurricane season July to October for 10-year period1946-1955 (Jordan,1958).

vi) Composited Northwest Pacific Typhoondata set from mean soundings for 10-year period1961-70, given by Frank(1977).

vii) Composited West Indies Hurricane data setbased on observations at coastal and island stations

in and near the Gulf of Mexico and the CaribbeanSea (Nunez and Gray, 1977) for the 14-year period1961-74.

For each of these data sets, Lord andArakawa drew the graph of A(λ) versus cloud-toppressure. All the graphs showed great similarity,suggesting something like a universal relationshipbetween A(λ) and λ, the latter being represented bycloud-top pressure p̂̂. This type of universalrelat ionship gives a good support toArakawa-Schubert Scheme of parameterization.We may here remind ourselves that while A(λ) iscalculated from formula 7.3 (11) from the vertical

distribution of T−

and q_

of the environment, theArakawa-Schubert Scheme of Parameterizationfurther assumes quasi-stationarity of A(λ) oversmall intervals of time of the order of 30 minutes.This additional property of universal relationshipbetween λ and A(λ) in situations widely-separatedboth in space and time, lend some additionalsupport for taking A(λ) as a basic parameter in theparameterization scheme.

Fig. 7.3 (2) shows the relationship betweenfractional entrainment rate λ and A(λ) for theGATE dataset. For the other 6 data sets analysed byLord and Arakawa (1980), the distributions weresimilar to this in many respects. Fig 7.3 (3) showsthe relationship between fractional entrainment rateand cloud-top pressure for the GATE data. Thisdiagram enables the transformation of x-axis fromcloud-top pressure to fractional entrainment rateand vice-versa. Fig. 7.3 (4) shows the distributionof cloud-work function A(λ) versus cloud-toppressure for the 4 data sets of Marshall Islands,VIMHEX,GATE and AMTEX, taken from Lordand Arakawa(1980).Sub-problem III :

Given T−

, q_ , λ , p̂ (λ), and A(λ) ; how to

get K (λ , λ′ )andF (λ) ?

Ans.

i) d

d t A (λ)=∫

λ′ = 0

λmax

K (λ , λ′ ) mB (λ′) d λ′+F (λ)

ii) Specify all possible discrete values of λ .As already stated, Lord and Arakawa (1980) chose17 values for p̂ . Corresponding to these 17 valuesof p̂, they calculated 17 values of λ and A(λ) for


given vertical distribution of T__

and q_

. The set of 17values completely defines the cloud-ensemble.

Sub-ensemble λ≡ i , sub-ensemble λ′==j ;i,j=1,2,3,........,17.

iii) Choose a particular value of i, say i = 3.

dd t

A (3)=∑Kj = 1

j = 17

(3 , j)mB (j)+F (3)

iv) For the moment, consider changes inA(3) due to cloud-cloud interaction only.

a) Choose a small time interval ∆ t ′′.

A (3)t + ∆ t′′−A (3)t=∑Kj = 1

j = 17

(3 , j)mB (j)∆ t ′′

b) Take j =1 and an arbitrarily chosen smallamount of mass flux at cloud base mB′′ (1), of cloud

sub-ensemble 1.Calculate ⎧

⎨⎩ A (3)t + ∆t ′′−A (3)t

⎫⎬⎭ due to mB′′

(1). Then

K (3 , 1)=⎧⎨⎩A (3)t + ∆t ′′−A (3)t

⎫⎬⎭/mB′′ (1) ∆ t ′′ 7.3(21)

c) Similarly, take j = 2,3,4,...,17 and calculateK(3,2),K(3,3),....,K(3,17).

v) Now consider change in A(3) due tolarge-scale processes only. Choose a small timeinterval ∆ t ′.

F (3)=⎧⎨⎩A (3)t + ∆ t ′− A (3)t

⎫⎬⎭/∆ t ′ 7.3(22)

Sub-problem IV :

Given T−

, q_ , λ , p̂ (3), A(3), F(3), K(3,1),

K(3,2), K(3,3),......, K(3,17); how to get mB(1),mB(2), mB(3),.......,mB(17)?Ans.

i) Using the assumption of quasi-stationarityof A (3) over small interval of time ( 30 minutes),we have to solve the equation

∑Kj = 1

j = 17

( 3 , j )mB(j)+F (3)=0 7.3(23)

or

∫λ′ = 0

λmax

K (λ , λ′ ) mB (λ′) d λ′+F (λ)=0

Under the condition that mB(λ′) is positive.

K(3,1) mB(1) + K(3,2) mB (2) +K(3,3) mB(3)

FIG. 7.3(2) : Relationship between fractional entrainmentrate λ (10−2 km −1) and cloud work function A(λ) (J kg−1)for GATE data set (Lord & Arakawa, 1980; Asnani,1993).

FIG. 7.3(3) : Relationship between fractional entrainmentrate λ (10−2 km−1) and cloud top pressure p̂ (mb) forGATE data set (Lord & Arakawa, 1980; Asnani, 1993).

FIG. 7.3(4) : Relationship between cloud top pressure p̂ (mb) and cloud work function A (λ) (J kg−1) for 4 datasets. Mean values and one standard deviation from meanvalue are shown on the two sides of the mean value. (Lordand Arakawa, 1980; Asnani, 1993).


+......+ K(3,17) mB (17) + F(3) = 0When mB(λ′) is zero or negative, that cloud

sub-ensemble is supposed not to exist.ii) In this equation, K(3,1), K(3,2), K(3,3),....,

K(3,17) and F(3) are known. But the 17 quantitiesmB(1), mB(2), mB(3),....,mB(17) are unknown. Toget these 17 unknowns, we write 17 linearinhomogeneous equations in these unknowns :K(1,1) mB(1) + K (1,2) mB(2) + K(1,3) mB(3) +..............+ K(1,17) mB(17) + F(1) = 0K (2,1) mB (1) + K(2,2) mB (2) + K (2,3) mB(3) +..............+K(2,17) mB(17) + F(2) = 0....

K(17,1) mB(1) + K(17,2) mB(2) + K(17,3) mB(3)+...............+ K(17,17) mB(17) + F(17) =0 7.3(24).

iii) These 17 linear equations in 17 unknownsare solved by standard methods.

iv) Lord (1982) suggestted simplex linearprogramming algorithm for solving these 17 linearinhomogeneous equations subject to the conditionthat mB(j) are non-negative. This algorithm isfurther explained in the paper by Lord et al. (1982),which forms part IV of the Arakawa-SchubertScheme.Sub-problem V :

Given T__

, q_ , λ , p̂ ( λ ) , A ( λ ) , K(λ, λ′) , F (λ)

and mB (λ′) ; how to get the rate of precipitation and

the rates of change of T__

and q_ ?

Ans.i) Obtaining of mB(λ′) elements is a major

step which has been explained in Sub-problem IV.After mB(λ′) elements are determined, the

calculations for the rate of precipitation and therates of change of T

__ and q

_ of the environment are

relatively simple and straightforward. The steps forthese calculations are explained by Lord (1982) andLord et al. (1982). Readers may refer to theseoriginal papers for details of calculation. In thesepapers, the authors have also compared their modelresults with observations during Phase III of GATE,1-18 Sep ’74.

ii) The aim of presenting Arakawa-SchubertScheme here has been to make the outline of theirapproach clear to the reader, as far as possible. Theexplanation of the scheme is otherwise spread over

4 parts between 1974 and 1982.iii) Quantitative precipitation forecasting

(QPF) is major problem in tropical forecasting. Theresults of QPF for GATE area presented by Lord(1982) are quite impressive. The techniquedeserved to be tested for more occasions in differenttropical locations. In the process of testing, somesimplifications may also suggest themselves foradoption.

For Indian region, calculations of Cloud WorkFunction had been presented by Rama Varma Raja(1994, 1996, 1999).Combined Updraft-Downdraft Model :

Arakawa-Schubert model described abovedoes not include convective downdrafts, which areimportant components of tropical convective cloudsystems; convective updrafts are, however,included explicitly.

Strength of updrafts in these convectivecloud systems is intimately related to their tilt in thevertical, the height of cloud tops and the amount ofrainwater in the clouds. Cheng (1989 a,b)emphasized the role of downdrafts and presented acombined updraft-downdraft spectral cumulusensemble model, which can be incorporated into theArakawa-Schubert cumulus parameterizationscheme given above.

Cheng and Yanai (1989) further utilizedthis combined updraft-downdraft model to studythe effects of meso-scale convective system on theheat and moisture budgets of larger-scale tropicalcloud clusters, using the GATE Phase III data. Theyconcluded that the inclusion of convectivedowndrafts resulted in warming and drying in theupper troposphere, and cooling and moistening inthe lower troposphere.

Cheng and Arakawa (1990) described indetail, the incorporation of Cheng’s (1989 a,b)model into Arakawa-Schubert’s original model.This incorporation necessitated slight modificationin the definition of cloud work function originallygiven by Arakawa and Schubert (1974).

However, they found that this revision madeno significant difference in the normalised cloudwork function (Cheng and Arakawa, 1992).

Cheng and Arakawa (1992) also madesemi-prognostic tests (one-step predictions) withupdraft-only model and updraft-downdraft modelin the UCLA General Circulation model, using dataset of GATE Phase III.


They came to the following conclusions :i) Both the models predict cumulus heating

profiles which agree amongst themselves and alsowith the actual atmospheric conditions.

ii ) The updraft-only model tends toover-estimate the cumulus drying rates throughoutthe entire cloud layer. On the other hand, theupdraft-downdraft model predicts the cumulusdrying effects reasonably well.

iii) With their computer code in use, theinclusion of the downdrafts slowed down the entireGeneral Circulation computation by a factor of 2.8

Further work done on Arakawa-SchubertScheme of Cumulus Parameterization

In a series of three papers published in J.Atmos. Sci., 1 June 1989, Cheng and Yanai haveemphasized that a parameterization scheme shouldinvolve not only thermodynamic features as done inthe Arakawa-Schubert scheme, but also dynamicfeatures like Vertical Wind Shear. The three papersare :

(i) Effects of downdrafts and Mesoscaleconvective organization on the heat and moisturebudgets of tropical cloud clusters. Part I: Adiagnostic cumulus ensemble model (M.D. Cheng,pp. 1517-1538).

(ii) Effects of downdrafts and Mesoscaleconvective organization on the heat and moisturebudgets of tropical cloud clusters. Part II: Effectsof convective-scale downdrafts (M.D. Cheng, pp.1540-1564).

(iii) Effects of downdrafts and Mesoscaleconvective organization on the heat and moisturebudgets of tropical cloud clusters. Part III: Effectsof Mesoscale convective organization (M.D. Chengand M. Yanai, pp. 1566-1588).

They have presented this revisedArakawa-Schubert model and cal led i tUpdraft-downdraft Model. The main featuresof this model are summarized below:

(i) It is a diagnostic model.(ii) It gives updraft model and downdraft

model.(iii) Gives the formula for tilting angle of the

updraft in terms of horizontal and vertical velocitycomponents of the updraft.

uc = horizontal component of velocity ofupdraft air relative to the cloud.

vc = 0

wc = vertical component of updraft velocityrelative to the cloud.

Then, tan θ = uc

Wc

where θ is the angle between the updraftand the vertical direction.

(iv) Horizontal and vertical components ofvelocity of rain drops ur and wr are given by

ur = ucwr = wc-Vtwhere Vt is the mean terminal fall velocity

of raindrop given by Soong and Ogura (1973, J.Atmos. Sci., 30, 879-893):

Vt = 36.34 (ρ̂ qr)0.1364

⎛⎜⎝

ρo

ρ̂

⎞⎟⎠

1⁄2

ms−1

q r = rainwater-mixing ratio

ρ̂ = density of updraft air

ρo = density of air at ground level

(v) The updraft tilting angle can beinterpreted as the angle required to maintain updraftbuoyancy against loading effect of rainwater.

For each sub-ensemble, it is assumed thatthere is statistically steady updraft. Tilting angle isconsidered to be a constant for each sub-ensemble.

FIG. 7.3(5) : The up draft tilting angles of various types ofclouds obtained by Scheme A (solid), Scheme (B) (dashed)and Scheme C (long dashed). (Ming - Dean Cheng., 1989).


The updraft parameters uc and wc are obtained fromArakawa-Schubert Scheme of heat and moisturebudgets.

(vi) The tilting updraft model is testedagainst the data of GARP (Phase III). Thehorizontal distribution of θ is nearly uniform whenthere is scattered convection; however, when thereis organized cumulus convection, the updraft-tiltingangle shows local maximum.

(vii) Fig. 7.3(5) gives the updraft-tiltingangle θ for various types of clouds. The tiltingangle increases with the depth of the cloud havingthe same cloud base near 960 mb (hPa). The tiltingangle is less than 3o for clouds having tops at 700mb (hPa). For clouds having tops near 200 mb(hPa), the tilting angle at the top is of the order of30o.

For details of schemes A, B, and Cmentioned in Fig. 7.3(5), the reader may refer toPart I of the three-paper series in J. Atmos. Sci.,1989, pp. 1517-1538. However, it is sufficient forour purpose to note that the schemes A, B, and Care simplified versions with a constant tilting anglefor each ensemble updraft in the vertical. Theresults for the three schemes are not very different.

(viii) Downdraft model is also presented.(ix) In Part II, Cheng presents the effect of

convective-scale downdrafts on heat and moisturebudget of tropical cloud cluster.

(x) In Part III, Cheng and Yanai present theeffect of meso-scale convective organization onheat and moisture budgets of tropical cloud clusters.

(xi) The three papers together quantifyrelationship between updraft tilting angle,thermodynamic properties of the air mass and windshear.

(xii) Anvil effects are : Warming and dryingin the upper troposphere (of secondary importance,but not negligible). Correspondingly, there iscooling and moistening in the lower troposphere.

(xiii) Local maxima of the tilting angleappear well before the organized precipitationpatterns, associated with squall clusters, and can bedetected by radar; this implies the existence ofthermodynamically preferred regions for theformation of cloud clusters.

(xiv) Larger tilting angle generally giveslarger downdraft mass flux relative to the updraftmass flux.

(xv) Degree of cloud organization is

related to the vertical wind shear. When verticalwind shear is large, then squall-clusters are likely tooccur; when vertical wind shear is moderate,"non-squall clusters" are more likely.

(xvi) Large vertical wind shear favors larget i l t ing angle and deep cumulus inthermodynamically preferred regions. This provescoupling between the wind field and thethermodynamic field.

(xvii) The tilting angle and the cloud workfunction are negatively correlated in time; thetilting angle usually increases during the periodswhen the mass flux of the deep clouds associatedwith squall clusters is diagnosed.

(xviii) Occurrence of cloud clustersgenerally follows a long-term build-up of thecloud work function and the vertical windshear. Short-term fluctuations are interpreted asthe result of development and decay of organizedcumulus convection.

(xix) The meso-scale organization ofcumulus convection is a consequence of interactionbetween cumulus clouds and the environment underthe influence of vertical wind shear. Dynamicparameters such as low-level wind shear should betaken into consideration in future cumulusparameterization schemes.Inclusion of Updraft-Downdraft Phenomena inArakawa-Schubert Scheme of CumulusParameterization

In an International Symposium held inIndian Institute of Tropical Meteorology, Pune,India during 1992, Cheng and Arakawa presentedthe results of numerical experiments in GeneralCirculation Models in which Cheng and Yanai’s(1989, J. Atmos. Sci., 1 June, pp. 1517-1588)three-ser ies papers were included inArakawa-Schubert Scheme of cumulusparameterization. They brought out the followingpoints:

(i) In the original Arakawa-Schubertscheme, the cumulus ensemble model did notinclude convective downdrafts, which areimportant components in tropical convectivesystems.

(i i ) Downdrafts which invar iablyaccompany the updrafts inside the cumulus clouds,tend to decrease the cumulus heating and dryingabove the cloud base through a reduction of thesubsidence between cumulus clouds.


(iii) The outflow from downdrafts below thecloud base may also significantly modify thethermodynamic properties of the sub-cloud layer.

(iv) Incorporation of the downdraft effects isan obvious improvement in the originalArakawa-Schubert scheme of cumulusparameterization.

(v) Cheng’s (1989 Part I, J. Atmos. Sci., pp.1517-1538) model of combined updraft-downdraftspectral cumulus ensemble can be incorporated intothe original Arakawa-Schubert scheme as outlinedbelow:

(a) The rainwater generated in the updraftis assumed to fall partly inside and partlyoutside of updraft.(b) The mean tilting angle determines thispartitioning of the falling rain water.(c) This mean tilting angle is estimated byconsidering stable statistically-steadystates with random perturbations on thecloud-scale horizontal velocity.(d) The vertical velocity and thethermodynamical properties of theassociated downdraft are then calculatedconsidering the effects of rainwaterloading and evaporation.

(vi) The updraft-downdraft model of Cheng(1989, Parts I & II, 1 June, pp. 1517-1564) wasincorporated in the original Arakawa-Schubertscheme and tested diagnostically using a dataset forGATE Phase III. The results of this testing exerciseare summarized below:

(a) The updraft-downdraft model predictsthe cumulus drying rates reasonably well.(b) The updraft-downdraft model performsbetter than the updraft-only model.(c) These are the results of One-stepPrediction. The time-step was 10 minutes.(d) The computer time requirement for thiscombined Arakawa-Schubert-Chengmodel was about twice the time requiredfor Arakawa-Schubert model. Moreexperiments are required to reduce thecomputer time requirement. There is nodoubt that the introduction of dynamics ofdowndrafts in the thermodynamic model ofArakawa-Schubert scheme is essential.

Xu, Arakawa, and Krueger (1992, J.Atmos. Sci., 2402-2420) used two-dimensionalUCLA cumulus ensemble model (CEM), covering

a large horizontal area with a sufficiently smallhorizontal grid size. They performed a number ofsimulation experiments to study the macroscopicbehavior of cumulus convection under a variety ofdifferent large-scale and underlying surfacecondit ions. They came to the fol lowingconclusions:

(i) In all simulations, cumulus activity israther strongly modulated by large-scale processessuch as large-scale advection and basic wind shear.When the basic wind shear is strong, there arisemeso-scale organizations which process createssome phase delays in the modulation.

(ii) The budget of eddy kinetic energy(EKE) shows that the net EKE generation rate isnearly zero for a wide range of cumulus ensembles.

(iii) Horizontal resolution of the large-scalemodel influences the results of Arakawa-Schubertscheme in as much as quasi-balance of Cloud WorkFunction (CWF) by destructive influences of cloudson CWF and generative influence of large-scaleprocesses is better achieved when the modelresolution is finer so as to catch meso-scaleprocesses.Gravity wave parameterization inArakawa-Schubert Scheme

Kim and Arakawa (1995, J. Atmos. Sci.,1st June, 1875-1902) examined the influence ofintroducing gravity wave drag inArakawa-Schubert scheme of cumulusparameterization. Their work is summarized below:

(i) Firstly, they presented a very useful Tableshowing intercomparison of several importantschemes of parameterizing sub-grid-scaleorographic gravity wave drag for the stratosphereand the troposphere, including the work of Boer etal. (1984), Palmer et al. (1986), Stern et al. (1987),Surgi (1989), and Hayashi et al. (1992).

(ii) Kim and Arakawa (1995) devised ascheme in which the numerical model explicitlyresolves gravity waves. They used a meso-scale2-dimensional non-linear anelastic, non-hydrostaticmodel to numerically simulate gravity waves for avariety of orographic conditions. They tested thethen-existing schemes of gravity wave dragparameterization and showed that a large number ofthese schemes do not properly treat theenhancement of the drag due to low-level breakingthrough resonant amplification of non-hydrostatic


waves. (iii) The revised parameterization scheme

proposed by Kim and Arakawa (1995) seems toovercome the above-mentioned difficulty of earlierschemes by including additional statisticalinformation on sub-grid-scale orography inside theparameterization scheme.

7.4 Summary

1. IntroductionTrends are on the following lines :i) Subject of Meteorology is becoming

inter-disciplinary with inputs from Physics,Mathematics, Oceanography, Agriculture,Atmospheric Chemistry, Statistics, ComputerScience, Space Science, Remote Sensing, etc.Meteorology is also using most sophisticatedcomputers and satel l i tes in col lect ion,communication, and analysis of data and also inusing the data for automatic weather forecasting.

ii) Public and Government agenciesthroughout the world, have become consciousabout the importance of Meteorology in the studyand protection of environment on global scale, forsafety of human and other forms of life on the earth.

iii) It is being realised that weather andclimate in the tropical region have significantinfluence on the global weather and climate.

Ocean is regarded as an importantconst ituent of cl imate; hence, combinedocean-atmosphere dynamical models are beingdeveloped, for forecasting weather and climate, inplace of the earlier models, which involvedatmosphere alone or introduced the ocean, at mostas static lower boundary of the atmosphere.

iv) For weather forecast ing,parameterization of sub-grid physical processes,though difficult, is considered to be the mostimportant and challenging component ofmodelling the atmosphere-ocean system.

v) Earlier enthusiasm of the 1950s-1970s forweather modification, has given place to caution;emphasis has shifted to theoretical model studies orphysical laboratory studies, before interfering withthe atmosphere.

vi) Extending the period of detailed weatherforecasts to ten days, and general weather forecaststo a few months, is the immediate objective of manymeteorological services of the world.

2. Initialization First, to convert the observatory-point data

into grid-point data, different methods of objectiveanalysis are presented: Successive correctionmethod, Optimum interpolation method for a singlevariable, and Multi-variate Optimum interpolationmethod.

Objectively interpolated grid-point variablesare then subjected to "initialization" processes.Static and dynamic initialization schemes arepresented. The static initialization schemespresented correspond to non-divergent balancemodel, balance model with limited divergence, andmass-wind balance model by variat ionaltechniques.

Dynamic initialization schemes presentedare forward-and-backward time integration,normal-mode initialization and bounded-derivativemethod.

Methods of Four-dimensional dataassimilation are also briefly given.

"Physical Ini t ial izat ion" has beensuccessful ly in troduced mainly by T.N.Krishnamurti and his collaborators. In essence, it is"reverse cumulus parameterization". You modifythe vertical distribution of humidity in your modelso as to give the "observed" rates of moistening,heating, and rainfall at the time of satellite picture.These so-called "observed" rates come from a mixof surface-based systems (like rain gauge) andspace-based systems (like OLR and SSM/I).

In this system of physical parameterization,mesoscale systems present in satellite pictures getidentified and structurally analyzed.

This scheme of physical parameterizationhas been tested and appears to give improvedanalysis and forecast, particularly for severeweather systems including tropical cyclones. 3. Parameterization of Cumulus Convection inthe Tropics :

For meso-scale models, individual cumulusmotions are regarded as sub-grid motions. Forsynoptic-scale models, meso-scale cumulusmotions are regarded as sub-grid motions.Meso-scale Models :

i) Kreitzberg and Perkey (1976, 1977)employed what may be called sequential plumemodel to simulate meso-scale convection. Theplumes were activated whenever grid-pointconditional instability exceeded a critical value and

7.4 Summary 7-49

continued until the instability dropped below thethreshold. They simulated meso-scale rain bands ofextra-tropical cyclones.

ii) Brown (1979) used a one-dimensionalupdraft plume model to simulate convection in histwo-dimensional meso-scale model. He employedhis model to simulate evaporation-drivenmeso-scale downdrafts as well as meso- scale anvilupdrafts occurring in association with Cb clouds.

iii) Fritsch and Chappel (1980) usedone-dimensional plumes for both updrafts anddowndrafts in a convective system.

All the above-mentioned schemes havein-built flexibility in respect of specifications forvertical eddy heating and moistening. It is alsopossible, through slight modifications in themodels, to incorporate additional features like eddytransports of momentum and vorticity.Synoptic-scale Models :

i) Moist Convective Adjustment Models: Inthese models, lapse rate of super-saturatedconditionally unstable layer is adjusted to neutralityafter every specified time-interval (usually less thanone hour).

ii) Moist Convergence Models : Large-scalesupply of moisture due to convergence ispartitioned as follows: The condensed part is usedfor warming the column and precipitating out; theuncondensed part is used for moistening theatmosphere inside the cloud and also for moisteningthe environment outside the cloud.

iii) Kuo’s parameterization scheme for deepcumulus convection: In this scheme, cumulus cloudperforms the following functions:

a) The cloud sends precipitation to theearth’s surface.

b) The cloud provides to the environment,the sensible heat which is released duringcondensation, through mixing with the environmentby dissolution immediately after its production.

c) Through dissolution, the cloud alsoprovides moisture to the environment.

The moisture which gets into a givenvolume of air gets partitioned into two components:that which condenses and immediately falls downto the earth’s surface as precipitation and that whichmixes with the environment immediately afterdissolution of the cloud. There has been someuncertainty and arbitrariness about the fraction ofmoisture which is supposed to fall down to the

surface as precipitation and that which mixes withthe environment. In this respect, there is somevariation between Kuo’s 1965 and 1974 schemes.Of course, in each of the schemes, there arespecified rates of precipitation, warming of theenvironment, and moistening of the environment.Observations at different places and on differentoccasions are to determine the most acceptablevalues of this fraction or partitioning factor.

Krishnamurti et al. (1983) suggested aplausible way of determining this partitioningfactor. They took Ooyama-Esbensen-Chu (1977)data sets as observations giving precipitation rates,warming rates and moistening rates in theatmosphere. Their objective was to find the valueof partitioning factor which should be incorporatedin Kuo’s scheme and which would be consistentwith these observations. In arriving at thispartitioning factor, they also used two additionalavailable parameters of the large-scaleenvironment; these are the mean vertical velocityω__

in the troposphere and the relative vorticity ζ inthe lower troposphere. It is understandable andlogical to postulate that the entire process ofprecipitation, warming and moistening of theatmosphere should be closely related to thelarge-scale vertical velocity and lower troposphericvorticity of the atmosphere. They obtained astatistical relationship (by linear multipleregression) between the partitioning factor and thelarge-scale environmental parameters ω

__ and ζ. If

the regression coefficients are accepted and keptconstant, then the partitioning factor varies withtime and place along with ω

__ and ζ. The rates of

precipitation, warming of environment andmoistening of the environment, of course, remainbound to this partitioning factor.

This scheme of Krishnamurti et al. (1983),derived from Ooyama-Esbensen-Chu (1977) datasets appears to be giving remarkable results inIndian region also (Keshavamurty and Sawant,1989, personal communication).Arakawa-Schubert Scheme :

This scheme is much more complex thanKuo’s scheme. Its outline is given below :

i) A functional relationship is suggestedbetween large-scale environment, and sub-gridscale convection; i.e. corresponding to eachlarge-scale environment, there is specific sub-gridscale convection, having specific time evolution

7-50 7.4 Summary

and interaction with environment.ii) Time-tendency of the meso-scale

cumulus convection at any instant of time is theresidual of two opposing forces: convectiveinstability of the synoptic-scale environmenttending to enhance convection, and the convectionitself tending to destroy the convective instability ofthe environment and thus tending to reduce its ownintensity.

iii) We visualise a series of small timeintervals, each with temporary quasi-equilibriumbetween the two opposing forces. The time-scaleof this quasi-equilibrium state is small (~ 30 min),compared to the characteristic time-scale ofsynoptic-scale environment (~ 1 day).

iv) Cumulus cloud ensemble is divided intoa number of sub-ensembles according to a spectralparameter l. Each cloud sub-ensemble has its ownarea of horizontal cross-section, cloud-top level,speed of updraft, buoyancy, rate of entrainment,rate of precipitation, rate at which it tends to destroythe convective instability of the environment, etc.

v) The main mathematical problem isreduced to solving an integral equation denoting thequasi-equilibrium condition

∫0

λmax

K (λ , λ′) mB (λ′) d λ′+F (λ)=0

where K (λ , λ′) mB (λ′) d λ′ represents the

rate of decrease of convective instability of theenvironment due to the cloud sub-ensemble andF(λ) represents the forcing from the large-scaleenvironment to enhance the intensity of theconvective cloud ensemble. Solution gives mB(λ′)for various values of λ’. One can then determineall other properties of the cloud ensemble.

vi) Spectral parameter can be any propertyof a cloud ensemble. Lord and Arakawa (1980)found it convenient to take cloud top level as auseful character ist ic property of a cloudsub-ensemble and divided the cloud ensemble into17 sub-ensembles. They took 7 different locationswith different synoptic conditions in the tropics andsub-tropics for which cloud-top data could beobtained. For each value of cloud top, they

calculated fractional entrainment rate λ and thecloud work function A(λ ) given by

A (λ)=∫zB

zt

η (z , λ) g

T−( z) ⎧

⎨⎩Tvc(z , λ) − T

−v (z)⎫

⎬⎭ d z

All the seven data sets gave something likea universal functional relationship between λ andA(λ).

vii) Inside the text, algorithms are presentedto show how from a given vertical distribution ofenvironmental temperature T and environmentalspecific humidity q , one can calculate the cloudensemble properties : fractional entrainment rate λ ,cloud work function A(λ), rate of increase ofkinetic energy of vertical motion of the cloud-typeλ due to the influence of other cloud-type λ ’ i.e K(λ ,λ′) > contribution of large-scale processestowards increasing A(λ) i.e. F(λ), total mass fluxMB(λ) at the cloud base level and then the rates ofprecipitation and changes of T and q.

A significant improvement in the classicalArakawa-Schubert scheme of cumulusparameterization is the introduction of moredynamics in the scheme through provision ofupdrafts and downdrafts which produce, and inturn are produced by thermodynamic structuregiven by Arakawa-Schubert scheme. Cheng andYanai (1989) in cooperation with Arakawa et al.(1992) have participated in this improvement of theArakawa-Schubert scheme. With this improvedscheme, one can handle slanting updrafts whichgive particularly heavy precipitation and othersevere elements of weather.

A further improvement inArakawa-Schubert scheme has been theintroduction of gravity wave drag in the originalscheme. This improvement is due to Kim andArakawa (1995), who used a meso-scaletwo-dimensional non-l inear anelastic,non-hydrostatic model to numerically simulategravity waves for a variety of orographic condition,including sub-grid-scale orography inside theparameterization scheme.

7.4 Summary 7-51

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References 7-57

tropical meteorology by g.c. asnani - initialization and parameterization

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