Prof. S. K. Dash, IIT Delhi Basics of Numerical Weather...

Prof. S. K. Dash, IIT Delhi

Basics of Numerical Weather Prediction

The horizontal momentum equation,

Fpp

VkfdtVd

ss

rrr

+∇∂∂

+∇−=×+σφσφˆ

Continuity equation,

0)()(. =∂∂

+∂∂

+∇t

ppVp s

ss σσ

&r

Thermodynamic energy equation,

[ ] QpppC

RTdtd

R ssp

−=++⎥⎦⎤

⎢⎣⎡

∂∂

&& σσσφσ1

Prof. S. K. Dash, IIT DelhiHydrostatic equation,

σσφ RT

−=∂∂

Surface pressure tendency equation,

[ ] σdVpt

ps

s ∫∇−=∂∂ 1

0

.r

and the moisture equation,

[ ] [ ] [ ] SpqpVqpqpt ssss =

∂∂

+∇+∂∂ σ

σ&

r.

The above set of six equations can be solved in principle to getthe values of six unknowns viz., horizontal wind velocity V

r , surface pressure sp , temperature T , moisture q , geopotential φ and sigma velocity σ& . The meteorological parameters V

r , sp , T and q are time dependent while σ& and φ are called the diagnostic fields.

Prof. S. K. Dash, IIT DelhiThese equations constitute a closed system, which can be solved at all future times

From a given initial state With the prescribed boundary conditions

The governing equations can be written as

)()( XPXDtX

+=∂∂

Here, X (x, y, z, t) : any model variable e.g. wind, Temperature, Humidity, Surface pressure etc.

D : Dynamics e.g. advection, pressure forces etc.

P : Physics e.g. evaporation and Condensation of water, solar heating, infra-red cooling, Frictional

drag at the surface of the earth etc.


Richardson Factory (1922)

Picture Credit : A. Lannerback


Classification of Forecasts

Usually a weather forecast is for a given region and has a definite period of validity. According to the period of validity one can classify as: – Nowcasting : 0-6hrs– Short range : up to 3 days – Medium range : 3-10 days – Long/Extended range : more than 10 days

Monthly and seasonal means-- Climate : 30yrs or more


Important Surface Boundary Conditions

Sea Surface Temperature

Snow Cover and Snow Depth

Soil Moisture

Vegetation CoverInfluence of initial and surface boundary conditions at different time scales of model integrations:

Now-casting and

Short-range (1-3 days): Primarily initial values of atmospheric parameters

Medium-range (3-10 days): Both the initial values and surface conditions

Monthly and more : Mostly surface boundary conditions

Climate: Chemical composition of atmosphere


Weather observations and Forecasting

Values of meteorological parameters and weather conditions at a future time constitute weather forecasting:

• Values of Temperature, Wind, Humidity, amounts of Rainfall and Snowfall

• Cyclones, Clouds, Sunshine hours, Fog• Heat & cold wave conditions, Extremes


Important sources of weather/climate dataa) Surface

Surface observatoriesAutomatic Weather StationsShips

b) Upper air BalloonsAircraftsRadars

c) Satellite derived fields


Some landmarks in Weather Forecasting:–Invention of barometer (1643)

Torricelli–Invention of telegraph (1845)

Synoptic map displayed in London in 1851

–1850-1920 National Weather Organisations grew. Primary tool of forecast was the surface pressure map and the forecasts were done by empirical means.


- V. Bjerknes (1904) recognised forecasting as an initial value problem.He realised that the basic set of equations are very complicated with highly nonlinear terms and these partial differential equations do not possess closed solutions. However, there were inadequate data to define initial conditions at that time.

- L.F. Richardson (1922) sought to solve the system of equations by desk calculatorA failure – forecasted pressure changes were found one

order of magnitude greater than the actual.- 1930s invention of radiosonde for upper air data and

further empirical improvements in forecasting.


Synoptic conditions(mean sea level )


Synoptic conditions(850 hPa analysis)


Different Methods of ForecastingSynoptic, Numerical Weather Prediction (NWP) and Statistical MethodsSynoptic method is based on examination of synoptic charts and close watch of weather systems.NWP is based on basic equations and physical principles: Initial conditions and boundary conditions are important.

Forecasting techniques for medium range are basically different from short range, because the extrapolation technique decreases in accuracy as the period of extrapolation increases. For 10 days forecast one needs data from all over the globe.For longer days of forecasts, slowly varying boundary conditions and their variations become more important – Sea Surface Temperature (SST) anomalies, variations in surface albedo, snow cover and depth, soil moistureetc. In the climate time scale, atmospheric chemistry plays a dominant role.


• Statistical : Regression equation Simple y = a + bxy : predictantx : predictorMultiple linear regression equationy = a + b1x1+ b2x2 +….+ bnxn


Growth of NWP due to the factors:

1. Theoretical research by Rossby, Petterssen, Bjerkness, Charney, Eady, Eliassen etc. led to some direct application to practical problems: Radical departure from pure empiricism.

2. Large data collected by satellites, balloons and rockets.

3. High Performance Computers (HPC)


Accuracy of weather prediction by NWP limited by errors due to:

1) Truncations associated with the replacement of the fundamental atmospheric (partial differential) equations by numerical schemes.

2) Errors in the initial conditions of the atmospheric variables in terms of their measurements.

3) Approximations in physical processes in the parameterization schemes in numerical models..


Necessity of Global Models

• For short range forecasting Limited Area Models (LAM) can be used. It usually takes a couple of days for influences from the distant parts to reach the area.

• For medium and long-range forecasts: Impact of snow, sea surface temperature (SST), deserts and other surface conditions are important.– The reflection of sun’s energy back to space from

clouds.– Influence of large mountain ranges must also be

included.These forcings are of global scale and their effects are cumulative. Hence, for medium and long/extended range forecasts global models are necessary.

Prof. S. K. Dash, IIT Delhi Basics of Numerical Weather...

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