7th APEC - Tsukuba International Conference , 17 March 2013, Bunkyo School, Tsukuba University

Numerical Prediction of Mesoscale Weather
Kazuo Saito
ksaito@mri-jma.go.jp

Meteorological Research Institute (MRI)

1. History of numerical weather prediction
2. Fundamental equations for atmosphere
3. Importance of the initial condition
4. Ensemble prediction
5. The K-computer project1. History of Numerical Weather Prediction
Numerical Weather Prediction (NWP).. predicts future state of the atmosphere quantitatively by time-integratinglaws of physics.. regarded as one of the best application fields of computational physics fromthe earliest period

Bjerknes (1904) pointed out possibility of weather prediction based on
dynamics and physics
Richardson (1922) tried weather prediction by solving the equations of fluidwith hand calculation but failed by overwhelming of noiseso(10hPa)surface pressure

Horizontal and vertical grid taken by Richardson (1922)

time
145hPa/6hour
computation

Dawn of numerical weather prediction
1946 Pennsylvania Univ. Developed thefirst digital computer (ENIAC)
23 word memories, 18,800 tubes(300FLOPS)
Von Neumann of Institute for Advanced
Study, Princeton Univ. proposed application
to weather prediction
1950 First success of 24 hour forecast using
ENIAC by Charney et al.
Grid distance 736 km at 45N
Number of grid points 15×18
One level (500hPa)
2-dimensional barotropic model which predicts
the absolute vorticity preservation law
d/dt(f +ζ) = 0
35 days for two 24 hour computations

NWP in Japan
1959 Japan Meteorological Agency implemented IBM704
core Memory of 8 K words (36 bit)1960 Operation started with a northern hemispheric barotropic model(381 km, one level)

NWP models at JMA (1988-)

Operation of
JMA-NHM(10km)

(5km)

(2km)

2. Fundamental equations for atmosphere
Six variables which describe the state of dry atmosphere:
three velocity components, pressure, temperature and density
Prognostic equations• Momentum equation（three wind components: u, v and w ）• Continuity equation（pressure: p）• Thermodynamic equation (temperature: T)
Diagnostic equation• State equation (density: r)
In the case of moist atmosphere, preservation of water substances andthe phase change must be considered (cloud micro-physics).

Momentum equation• Momentum equation（three components）
du 1 p dif .u
dt r  x
dv 1 p dif .v
dt r  y
dw 1 p g  dif .w
dt r  z∂: partial derivative symbol

・Nwton’s law of motion: （Force）=（mass×acceleration）→
Navier-Sokes’ equation for fluid：（acceleration）=（pressure gradient force per unit mass）(+diffusion+gravity force for vertical direction ）Momentum equation• Momentum equation（three components）
du 1 p dif .u
dt r  x
dv 1 p dif .v
dt r  y
dw 1 p g  dif .w
dt r  z∂: partial derivative symbol

・Nwton’s law of motion: （Force）=（mass×acceleration）→
Navier-Sokes’ equation for fluid：（acceleration）=（pressure gradient force per unit mass）(+diffusion+gravity acceleration for vertical direction ）Coriolis’ force
Effect of Coriolis’ force is added on the earth

Northward motion shifts
eastward in Northernhemisphere due to the differenceof speeds of earth rotation.

North PoleHigh latitude
Earth rotation

Low latitude
Earth rotation

In vector formulation, Coriolis’ force is vector product ofangular velocity of the earth rotation  and wind vector V:dV1
 (2  V )  p  g  F ,dtrContinuity equation
Continuity equation（law of mass preservation）
r r u r v r w
0
t  x  y  z‥local time tendency of density=differences of massflux through surrounding boundaries

Mass flux (density x wind speed)

State equation for ideal gasses
Boyle-Charles’s combined law for ideal gas with a molecular weight of mp  r R *T  M R * T
mm V

R*：universal gas constant (=8.314J/mol/K)
In case of dry air (represented by subscript d),by Dalton’s law for partial pressure,
M

R*

i
Mi
R*
i
p d   pi TT r d RT
V
V
md
i
i mi

i is index to represent gas component such as nitrogen, oxygen, andargon, and md the weigh-average molecular of dry air (28.966 g/mol)
md 

M

i

i

Mi
i m
i

R (=R*/md)：gas constant for dry air (=287.05 J/Kg/K)

Forces on air and acceleration
contour of pressure

Coriolis’ force
low pressure

wind direction
acceleration

Frictional force

Pressure gradient force
Isobaric plain

Pressure gradient force

Low pressure

acceleration

Sum of pressure gradient
force, Coriolis’ force andfrictional force accelerates
the air mass toward the lowpressurehydrostatic equilibrium
1 dp
g

→
p dz
RT

→ p p0e



gz
RTm

Gravity force

Veridical pressure gradient
force usually balances withthe gravity force(hydrostatic equilibrium)

high pressure

1 dp
g 0
r dz

d
g
(log p) →dz
RT… well-known barometric height formula(pressure-height equation)

Thermodynamic equation
First law of the thermodynamics
dQ  dI  pd

‥heating to air mass is the sum of internal energy
increase and mechanical work by pressure. Here, Q is theadiabatic heating rate and  is specific volume (inverse ofdensity).
Qdt  Cv dT  pd  Cv dT  d ( p )  dp
 (Cv  R)dT  dp  C p dT  dp

Here Cv is the specific heat of air at constant volume(=5R/2), and Cp the specific heat of air at constant pressureFinite discretization
In the numerical model, differential equation is discretized by finite
difference, based on the Taylor series expansion:(x) 2f ( x  x)  f ( x)  xf ' ( x) 
f ' ' ( x)  ..
2 2
(x)
f ( x  x)  f ( x)  xf ' ( x) 
f ' ' ( x)  ..
2
Here, f’ (f’’) is the first (second) derivative of the function f.
The second order centered difference can be obtained as :ff ( x  x)  f ( x  x) O(x) 2x
2xor, in staggered grid,x
x
f (x  )  f (x  )f2
2  O( x ) 2

x
x
2
For advection term in momentum equations, higher order schemes are used e.g.,
x
x
3x
3x
f (x  )  f (x  )f (x 
)  f (x 
)
f 9
1
2
2 
2
2  O( x ) 4

x 8
x
8
3x
2

Pressure equation and implicit treatment
Pressure equation is obtained from the continuity equation and the state equation:

pr v r w
2 r u Cs ()  FPt
x y zHere, Cs is the sound wave speed (=(Cp/Cv×RT)1/2) and FP represents
forcing term such as the time tendency of (potential) temperature. Solutionsof above equations include sound waves due to the elasticity of air. In
atmospheric models for weather predication, pressure is treated implicitly inthe vertical direction. The following 1-dimensional elliptic (Helmholtz-type)pressure equation is obtained: 2 P 



(
h
P
)

e
'
P
 FP.HE2zz
A A    A 2, e'  {
}2 ,
t

(1   )Cs
2
FP 2
 ru  rv
rw
FP.HE  
 FW {()}  e' P .
2
(1   ) Cs  z(1   )
x
y
z

Gaussian sweep-out elimination method is used to solve above equation.

Physical processes in NWP modelSun

Turbulence

Cumulusconvection

evaporation
cloud
evaporation＊
Condensation
＊
＊
＊ ＊
＊ ＊
＊ ＊
＊ ＊
＊ Sensible heat,＊ latent heat,＊ momentum
Biosphere

Land

Turbulence

Boundary layer

Convective
boundary layer

Water
vapor

Water vapor

Condensation

Sensible heat, latent heat,
momentum

Ocean

＊ ＊ ＊
＊ ＊ ＊
＊ ＊ ＊
＊ ＊ ＊
＊ melting
＊ ＊
rain
＊ ＊ ＊
＊ ＊
＊ ＊
＊ ＊
evaporation＊ ＊ snow
＊ ＊
＊ ＊
＊ ＊
Sea iceCloud Microphysics• Typical prognostic variables– water vapor, cloud water, rain,
cloud ice, and graupel (Qv,
Qc, Qr, Qi, Qs, Qg)– number density of cloud ice,snow and graupel (Ni, Ns,
Ng)water vapor

cloud water

cloud icesnow

Tetens’ formula forsaturated vapor pressure [hPa]raines  6.11107.5t237.7  t

graupelBulk cloud microphysics
In bulk method, the size distribution function of water substance isexpressed by the inverse exponential function of the particle diameter D.

N ( D)  N0e D
Fall-out terminal velocity ofparticle is given as a powerfunction of D by the
Stokes’ law in the form ofV ( D)  aDb

Bulk cloud microphysics
The mass-weighted mean velocity is obtained by



V

6

r w D 3V ( D) N ( D)dD

 6 r D N ( D)dD3b  DDaDe dD
3  DD e dD3w

aG(4  b)6b

Here G(x) is the Gamma functionG( z )   t z 1e t dt
0

By Euler’s partial integration, the Gamma function has the following
characteristicG( z )   t
0

z 1









(e )'dt   t e ) 0  ( z  1)  t z  2e t dt

 ( z  1)G( z  1)

tz 1 t0

JMA nonhydrostatic model (Sep. 2004)
Mesoscale model of JMA which does not use approximation of thehydrostatic equilibrium•Fully compressible numerics developed by MRI and NPD•3 ice bulk cloud microphysics with the Kain-Fritsch convective parameterization scheme•Non-local boundary layer scheme

Cloud resolving model
Polar low simulation with JMA NHM
Horizontal resolution 2km
Initial time 15UTC 21 January 1997

GMS satellite IR image

Cloud resolving simulation of a polar
low (Yanase et al., 2002; GRL)

Cloud resoling simulation of winter monsoon clouds
over the Sea of Japan using Earth Simulator

Left) GMS satellite visible-Image at 15JST 14 January 2001
Right) Total water simulated by NHM with a horizontal resolution of 1 km

Eito et al. (2010; JMSJ)

3. Importance of the initial condition
Major difference between climate projection and NWP・the same point
predict future state of the atmosphere quantitatively by time-integrating laws ofphysics.
・major difference
Climate projection‥projects climate response to change of global environment such as CO2, SST⇒boundary condition and radiation-convection equilibrium are important
NWP‥predicts weather of a specific day in short time range⇒initial condition and time evolution are importantMaximum likelihood estimation indata assimilationｘ： analysis variableｘｂ： first guess of ｘｙｏ： observation dataｐ（・｜・）： conditional probabilistic density function (PDF)

Bayesian theorem（if ｘｂ and ｙｏ are independent,p(x | x b , y o ) 

pb (x b ) p o (y o ) p (x ) p (y )dxbbo

PDF ofobservation

PDF of first
guess

o

Baysian estimationxˆ  max[ pb (x b ) po (y o )]
x

xb

xa

yo

x

If PDFs of the first guess and observation are Gaussian normal
distribution as1

pb (x b ) 2 bpo ( y o ) 

2

1
2 o

2

exp{
exp{

( x  xb ) 2
2 b

2

}

( y  yo ) 2
2 o

2

}

The conditional PDF of x with background xb and observation yo is normal asp(x | x b , y o ) 


1
2 a

2

pb (x b ) po (y o ) pb (x b ) po (y o )dxexp{( x  xa ) 2
2 a

2

}

Maximum likelihood
estimation
PDF of firstguess

PDF ofobservation

Where analysis xa and its standard deviation σa are
xa


2

xb


2

yo

 a  b  o2
1
1
1


 a2  b2  o2

 o 2 xb   a 2 yo
→ xa  ( 2   2 )
a
o

xb
xa
yo
Analysis is weighted mean of first guess and observation, and analysis error
becomes smaller than the errors of first guess and observation.

xRelation between 1st guess, observation
and analysis
Probability

Analysis

Observation

Simple fitting
・Use observation data onlySpace
Probability

Analysis

Observation

Correction method
・Use observation data
・Use low quality background information

BackgroundSpace
Probability

Analysis

Forecast

Observation

Optimal interpolation／3D-VAR
・ Use observation data
・ Use forecast as the 1st guessbackground error is given by statistics

Space

4D-VAR
Observation ・ Use observation dataat actual time ・ Use forecast model and 1st guess
Analysis is obtained by time evolution ofSpace forecast/adjoint models

Analysis

Probability

Forecast

Time(continuous Assimilation)

4Dimensional –Variational method
Costfunction :J  Jb  Jo  Jc

Forecast model

















T
T
1
1
x0  x0b B1 x0  x0b  HMx 0  y o R 1 HMx 0  y o  J c
2
2
Penalty term
Gradient of  x 0 J   x 0 J b   x 0 J o   x 0 J c

costfunction: B1 x0  xb0   M T H T R 1 HMx 0  y o   x 0 J cx
Observation

Adjoint model

Jo

First guessxb
x0

First guessJoobservationJoJbobservation

Forecast
fromanalysisJoobservation

21UTC

Assimilation window

00UTC

Observation data can be assimilated at observation timetime

4Dimensional –Variational method
Costfunction :J  Jb  Jo  Jc

Forecast model













T
T
1
1
x0  x0b B1 x0  x0b  HMx 0  y o R 1 HMx 0  y o  J c
2
2Distant from the first guess
Penalty term
Gradient of  x 0 J   x 0 J b   x 0 J o   x 0 J c

costfunction:

Distant from observations B1 x0  xb0   M T H T R 1 HMx 0  y o   x 0 J cx
Observation

Adjoint model

Jo

First guessxb
x0

First guessJoobservationJoJbobservation

Forecast
fromanalysisJoobservation

21UTC

Assimilation window

00UTC

Observation data can be assimilated at observation timetimeMeso 4DVAR (Mar. 2002)(Koizumi et al., 2005； SOLA)The world first implementation of regional 4DVAR for operation.
LT and ADJ models based on MSM (a hydrostatic spectral model) of JMA.

RUC with PI

4D-Var3 hour accumulated rain for FT=18
Initial 12 UTC 9 September 2001

Observation

FT=15-18

Nonhydrostatic 4DVAR (Apr. 2009-)(JNoVA; Honda et al., 2009)
LT and ADJ models based on JMA-NHM
Radar-AMeDAS observation JNoVA (nonhydrostatic)

FT=24 from 2006 Aug 17 15UTC

Meso 4DVAR (hydrostatic)

Sawada and Honda (2009)

Threat score of MSM Mar.2001-Nov.2011 (FT=0-15)
Nonhydrostatic
4DVAR
10km to 5km
Meso4DVARRadar
GPS TPWV Reflectivity
Nonhydrostatic
modelPredictability of heavy rainfalls
Heavy
rains withorographic
forcing

Heavy rains
with synoptic
forcing‥relatively predictable in the current mesoscale NWP up to a point

Predictability of heavy rainfalls
Heavy
rains withorographic
forcing

Heavy rains
with synoptic
forcing‥relatively predictable in the current mesoscale NWP up to a point
Position is fixed by forcing
regardless the trivial errors
in initial condition

Example of orographic heavy rainfall

Radar-AMeDAS Observed rainfall
03-06 UTC, 19July 2011

MSM prediction FT=21
03-06 UTC, 19July 2011

A strong typhoon (T201106 Ma-on) hit western Japan and a record breaking 851mm
rainfall was observed in one day (19 July 2001).
MSM accurately predicted the orographically forced heavy rainfall.

Predictability of heavy rainfalls
Heavy
rains withorographic
forcing

Heavy rains
with synoptic
forcing‥relatively predictable in the current mesoscale NWP up to a point

Convective rains
without strong
synoptic/orographic
forcing‥difficult to predict due toPredictability of heavy rainfalls
Heavy
rains withorographic
forcing

Heavy rains
with synoptic
forcing‥relatively predictable in the current mesoscale NWP up to a point

Convective rains
without strong
synoptic/orographic
forcing‥difficult to predict due to• small horizontal/temporal scales• phenomena in unstable atmosphere
Result is very sensitive to small
perturbations in initial conditions

Approaches to predict local heavy rain
Forecast accuracy
Limit of deterministic forecast

Short range forecast for
precipitation

Current
Nowcasting

Current NWP model

Extrapolation
2h

6hr

1day

Forecast period

Approaches to predict local heavy rain (1)Forecast accuracy
Limit of deterministic forecast

Improved
nowcasting
Short range forecast for
precipitation

Improve the
nowcasting bydense observation
and advanced
techniques.

Current NWP model

Extrapolation
2h

6hr

1day

Forecast period

Approaches to predict local heavy rain (2)Forecast accuracy
Limit of deterministic forecast

Nowcasting

Cloud resolving modeland data assimilation
Short range forecast for
precipitation

Reduce the gap
between nowcasting
and NWP by highresolution data
assimilation

Current NWP model

Extrapolation
2h

6hr

1day

Forecast period

Local Heavy rainfall on September 2005 in Tokyo

Local heavy rainfall on 4 September 2005
100mm precipitation in 1 hour was observed inTokyo. No significant disturbances overTokyo metropolitan area.

Cloud resolving 4DVAR with cloud microphysics(Kawabata et al., 2011; Mon. Wea. Rev.)

Kessler warm rain process was implemented
in LT/ADJ models.
4DVAR assimilation of• Doppler Radar’s Radial Winds• Radar Reflectivity• GPS precipitable water vapor• Surface observations (wind, temperature)

4DVAR analysis

Precip. intensity (mm/h)

Assimilation of radarreflectivity 2030-2100JST

Observation(Kawabata et al., 2011; Mon. Wea. Rev.)

2130 JSTObs

4DVAR withassimilation ofradar reflectivity

4DVAR without
radar reflectivity

1st guess

Kawabata, T., T. Kuroda, H. Seko, and K. Saito, 2011: A cloudresolving 4D-Var assimilation experiment for a local heavy rainfall
event in the Tokyo metropolitan area, Mon. Wea. Rev. 139, 1911-1931.

4. Ensemble prediction
Initialcondition

Probabilistic density
function of analysis error

Forecast

deterministic
prediction

probabilistic
prediction

Time integration of the probabilistic density function is practically
impossible. In the ensemble forecast, finite members
approximate the features of probabilistic density function of
atmospheric states.

Approaches to predict local heavy rain
Forecast accuracy
Limit of deterministic forecast

Nowcasting

Cloud resolving modeland data assimilation
Short range forecast for
precipitation

Reduce the gap
between nowcasting
and NWP by highresolution data
assimilation

Current NWP model

Extrapolation
2h

6hr

1day

Forecast period

Approaches to predict local heavy rain (3)Predictability by probabilistic forecast

Forecast accuracy

Limit of deterministic forecast

Nowcasting

Cloud resolving modeland data assimilation
Short range forecast for
precipitation

Reduce the gap
between nowcasting
and NWP by highresolution data
assimilation

Current NWP model
Ensemble forecast

Extrapolation
2h

6hr

1day

Forecast period

2km ensemble prediction from JMA nonhydrostatic 4DVAR analysis for 2011 Niigata-Fukushima heavy rainfall

Observation
Control run
03-06 UTC, 29 July 2011

Ensemble spread

1mm/3h5mm/3h10mm/3h
Probability of precipitation at FT=18
Solid probability even for 50mm/3h

Ensemble mean

20mm/3h

50mm/3h

Saito et al. (2013)

Ensemble Kalman Filter
observation

analysis
R
Pa

T=t2
Pi 
f

Xif (Xif )T
m 1

Pf

Pa
analysis

T=t0

Ensemble mean



T=t1

NHM-LETKF
Miyoshi and Aranami (2006)~
X a  x f e  X f P a (HX )T R 1 ( y o  H ( x f ))e  m  1UD 1/ 2 UT~
 x f e  X f P a (HX )T R 1 ( y o  H ( x f ))e  X f T4. The K-Computer project“HPCI Strategic Programs for Innovative Research (SPIRE)” is being

carried out by RIKEN, with partners in industry, universities, national
institutes under an initiative by MEXTThe HPCI Strategic Programs for Innovative Research (2011.4-2016.3)
Field 3: Weather, Climate and Environment Prediction for disaster prevention
Sub theme ②：Super high performance mesoscale NWP
ａ．Cloud resolving 4DDA・feasibility of dynamical prediction of
local heavy rainfall in very sort range
forecast
MRI, JMA, DPRI/Kyoto Univ., NIED, ISM
ｂ．Cloud Resolving ensemble NWP and its verification

・quantitative prediction of the probabilityof local heavy fall with a lead time todisaster prevention
MRI, JMA, Tohoku Univ., DPRI/Kyoto Univ.
ｃ．High performance atmospheric model・Evaluation of model’s uncertainty throughsuper high resolution numerical experiments

JAMSTEC, MRI, Tokyo Univ., Nagoya Univ., etc.
Bin

LES

http://www.jamstec.go.jp/hpci-sp/kisyo/kisyo.en.html#kisyo_2

http://www.jamstec.go.jp/hpci-sp/kisyo/kisyo.html#kisyo_2

Summary• Prediction of weather is performed by numerical computation.
• Performance of sate of the art mesoscale NWP has beenremarkably improved, but still storm scale prediction ischallenging.
• High resolution data assimilation and ensemble prediction arenecessary. The K-computer will reduce compromise ofresolutions and ensemble members and show a prototype of the
future NWP system.

References

Thank you

Saito, K., T. Fujita, Y. Yamada, J. Ishida, Y. Kumagai, K. Aranami, S. Ohmori, R. Nagasawa, S. Kumagai, C. Muroi, T. Kato, H. Eito and Y.
Yamazaki, 2006: The operational JMA Nonhydrostatic Mesoscale Model. Mon. Wea. Rev., 134, 1266-1298.
Kawabata, T., H. Seko, K. Saito, T. Kuroda, K. Tamiya, T. Tsuyuki, Y. Honda and Y. Wakazuki, 2007: An Assimilation Experiment of the Nerima
Heavy Rainfall with a Cloud-Resolving Nonhydrostatic 4-Dimensional Variational Data Assimilation System. J. Meteor. Soc. Japan, 85, 255276.
Saito, K., J. Ishida, K. Aranami, T. Hara, T. Segawa, M. Narita and Y. Honda, 2007: Nonhydrostatic atmospheric models and operational
development at JMA. J. Meteor. Soc. Japan., 85B, 271-304.
Shoji, Y., M. Kunii and K. Saito, 2009: Assimilation of Nationwide and Global GPS PWV Data for a Heavy Rain Event on 28 July 2008 in
Hokuriku and Kinki, Japan. SOLA, 5, 45-48.
Kunii, M., K. Saito and H. Seko, 2010: Mesoscale Data Assimilation Experiment in the WWRP B08RDP. SOLA, 6, 33-36.
Saito, K., M. Kunii, M. Hara, H. Seko, T. Hara, M. Yamaguchi, T. Miyoshi and W. Wong, 2010: WWRP Beijing 2008 Olympics Forecast
Demonstration / Research and Development Project (B08FDP/RDP). Tech. Rep. MRI, 62, 210pp.
Seko, H., M. Kunii, Y. Shoji and K. Saito, 2010: Improvement of Rainfall Forecast by Assimilations of Ground-based GPS data and Radio
Occultation Data. SOLA. 6, 81-84.
Seko, H., T. Miyoshi, Y. Shoji and K. Saito, 2011: A data assimilation experiment of PWV using the LETKF system -Intense rainfall event on 28
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