Numerical Weather Prediction

Advanced Synoptic M. D. Eastin

Numerical Weather Prediction


Dynamical Cores• Definitions• Grid architectures• Time stepping (hydrostatic vs. non-hydrostatic models)

Physical Parameterizations• Basic Concept• Planetary Boundary Layer (PBL)• Land-surface models• Grid-scale Microphysics• Sub-grid-scale Convection

Data Assimilation• Available observations• Assimilation techniques

Ensemble Forecasting• Basic Concept• Methods• Advantages and limitations

Numerical Weather Prediction


Dynamical CoreGrid “Cells” vs. Grid “Points”:

• Numerical models must provide a spatially-continuous representation of the full atmosphere

• “Points” represent small areas with large area between adjacent points

• “Cells” represents large areas with no area between adjacent cells

Example: Temperature – the single valuereported represents a spatial averageacross the total grid cell area

Cloud cover – the single value reported represents the fractionof the total grid cell area occupiedby cloud


Dynamical CoreGrid “Resolution” vs. Grid “Spacing”:

• The effective grid resolution is not the same as the grid cell spacing

• It takes several grid cells to truly resolve the spatial structure of a meteorological feature

Where is the feature’s maximum? Where are the feature’s edges?

• Model resolution is typically 5 times the grid cell spacing (at a minimum it is 3 times)


Dynamical CoreHorizontal Grid Architecture:

• Modern models use staggered grids • Mass (or thermodynamic) variables are computed for the grid cell center

• Wind (or kinematic) variables are computed along grid cell boundaries

• Such configuration improves the model’s computational efficiency and increases the effective resolution since the winds only advect mass across grid cell boundaries

• The NAM / WRF model and the RUC model use this staggered grid architecture



• Some models use spectral coordinates

• Represent global atmospheric structure as the sum of sine and cosine waves over a range of zonal wavenumbers (n)

• Both the GFS model and ECMWF model

are global spectral forecast systems

Advantage: Removes “truncation errors” that occur when strong gradients are present between adjacent grid cells

Limitation: Many processes cannot be represented with spectral techniques [Precipitation / Vertical advection] and must still be represented in grid cell space.

Operational models are really hybrid models – utilizing a variety of numerical techniques to integrate the governing equations

Observed

COS n=1

SIN n=1

SIN n=2



• Some models use spectral coordinates

• Represent atmospheric structure as the sum of sine and cosine waves over a wide range of zonal wavenumbers (n)

Example:

If we were to represent a given variable (temperature) using the first 30 waves in the zonal direction (n=1–30) then, the model’s effective grid spacing would be related to the zonal wavelength (λ) of the smallest resolved wave (n=30) via

λn=30 = 444 km

The current GFS model analyzes the first 574 waves (n = 574) for an effective grid cell spacing of ~23 km

Observed

COS n=1

SIN n=1

SIN n=2

nkmx 3/360111


Dynamical CoreVertical Grid Architecture:

• Some models use sigma coordinates

where: p = pressure at a given level

pT = pressure at the model toppS = pressure at the surface

σ = 1 at the surfaceσ = 0 at the model top

• The ECMWF model uses the sigma coordinate system

Advantage: Model is configured on pressure surfaces (like upper-air observations)

Limitation: Large numerical errors in computing the horizontal pressure gradient can occur in mountainous regions

TS

T

pp

pp


Dynamical CoreTime Stepping: CFL Condition

• In 1928, three mathematicians Courant-Fredrich-Lewy (CFL) determined that numerical models require a small time step (relative to the grid cell spacing) or numerical instabilities will occur and the model will generate very large errors (and “crash”)

Condition: where: c = fastest possible wave or wind Δx = grid cell spacing Δt = time step between forecasts

Time Stepping: “Advantages” of hydrostatic models

• The hydrostatic approximation eliminates small-scale pressure perturbations and pressure only varies over large (i.e., synoptic) horizontal scales• Recall from your dynamics course that sounds waves are essentially small-scale pressure fluctuations that move very quickly (the speed of sound in air is > 300 m/s)

• Hydrostatic models have less restrictive CFL conditions c = 100 m/s Δx = 10 km

and thus can complete a full forecast in less time Δt ≤ 100 s

• Non-hydrostatic models must account for sounds waves c = 400 m/s Δx = 10 km

with stricter CFL conditions, and thus require more time Δt ≤ 25 s to complete a full forecast

1

x

tc


Dynamical CoreTime Stepping: “Advantages” of hydrostatic models

• Since hydrostatic models run much faster, long-term prediction (all climate simulations and many weather forecast models) require the hydrostatic approximation in order to get results in a reasonable amount of “real” time

• The “most advanced” modern climate models running on the “fastest” supercomputers still require 1-2 months to complete 100-year simulations run in “hydrostatic mode”

• As a result – all vertical motions in climate models are diagnosed

• Only regional mesoscale models are non-hydrostatic

Hydrostatic models: GFS model (global weather/climate)ECMWF model (global weather/climate)CCM (global climate)

Non-hydrostatic models: NAM / WRF modelRUC modelMM5Hurricane WRF

What does the lack of non-hydrostatic vertical motions imply about ALL climate simulations?


Physical ParameterizationsBasic Concept:

• Regardless of model type (grid vs. spectral) or model resolution, there are always physical process that cannot be explicitly resolved by the model.• Any process that occurs on a spatial scale smaller than a grid cell length must be represented through analytic approximations

• Radiation (< 1 m)• Cloud Microphysics (< 1 m)• Planetary Boundary Layer (< 1 km)• Land-surface Processes (< 1 km)• Convection (< 1 km)

Parameterization: The means of expressing unknown orunresolved quantities in terms of otherexisting dependent variables (T, p, u, etc.)

Accounting for unresolved physicalprocesses without introducing additionaldependent variables

[ Remember: a numerical model consists ][ of a “closed set” of N equations with ][ N unknowns - or dependent variables ]


Physical ParameterizationsPlanetary Boundary Layer (PBL):

• The atmospheric PBL is a critical component of the “Earth System” since ALL heat, moisture, and momentum exchange between the atmosphere and the underlying surface occurs here – particularly in the surface and viscous sub-layers

• Then, small-scale 3-D turbulence must transfer the energy from the surface layer through the mixed layer and into the free atmosphere



• We have to represent sub-grid-scale 3-D turbulence in terms of grid-scale quantities without introducing any new predicted (or dependent) variables

Surface-Layer “Closure” Methods:

Bulk Aerodynamic:

Heat Flux

K-theory:

Heat Flux

Monin-Obukov Similarity Theory

see http://glossary.ametsoc.org/wiki/Monin-obukhov_similarity_theory

)( 1 SFCH TTVCTw

Note: Primes representthe

sub-grid-scale

[turbulent fluxes]

Overbars representthe

grid-scale

[large-scale means]

CH and KH are turbulent

transfer coefficients

determined through numerous field and lab

experiments

z

TKTw H

http://glossary.ametsoc.org/wiki/Monin-obukhov_similarity_theory



• We have to represent sub-grid-scale 3-D turbulence in terms of grid-scale quantities without introducing any new predicted (or dependent) variables

Mixed-Layer “Closure” Methods:

Local: Only mix turbulent quantitiesup/down to an adjacent modellevel through the PBL

Implies that all turbulent mixingis accomplished by eddies ofof the same small size

Non-local: Can mix turbulent quantitiesup/down through all modellevels in the PBL

Implies that turbulent mixingis accomplished by eddies ona broad range of scales

Most realistic (and more complicated)

Noah Land-Surface Model(used in the NAM and GFS models)


Physical ParameterizationsLand-Surface Processes

• We have to correctly represent the land surface type, vegetation, and soil properties in order to properly predict surface layer fluxes, PBL processes, convective initiation, and precipitation type/amount

Example: Differences in the PBL humiditydue to rapid evapotranspiration

from a corn field and relativelyslow evaporation from a nearbybare soil field may influencewhether storms develop

Predict: Soil temperature/moisture

Factors: Local albedoSoil typeVegetation typeSeasonal vegetation changeSnow cover


Physical ParameterizationsGrid-Scale Microphysics:

• We have to correctly account for the latent heat release / absorption from water phase changes of water during cloud and precipitation processes• We must also accurately predict the precipitation type / amount

Predicts: Degree of super-saturationLatent heat release / absorptionNumber concentrations of hydrometeor particles as a function of diameter [Six types: cloud water, cloud ice, rain, snow, graupel, and hail ] Fall velocities of each hydrometeor type

Lots ofsmall drops

Very fewlarge drops


Physical ParameterizationsGrid-Scale Microphysics:

• We have to correctly account for the latent heat release / absorption from water phase changes of water during cloud and precipitation processes• We must also accurately predict the precipitation type / amount

Two Types: Bin Method – actually predicts drop counts for each class Bulk Method – estimates drop counts using analytic formulas


Physical ParameterizationsSub-Grid-Scale Convection:

• Often times clouds are smaller than a grid cell – what do we do?• We need to accurately represent the radiation, latent heat, turbulent mixing, precipitation, and energy transfer associated with such sub-grid-scale clouds.

Convective Parameterizations (CPs):

• Required for models run with horizontal grid lengths > 2-3 km• Account for convection in a single grid column

• The “triggering” mechanism is unique to each CP scheme• Once “triggered”, all CP schemes adjust the temperature and humidity profile

through the column based of the fractional area covered by convection• Very few CP scheme adjust the momentum fields through the column

[ implies no updrafts or downdrafts – not realistic ]

• Numerous CP schemes exist – the most popular ones are:

• Betts-Miller-Janjic (BMJ) adjustment scheme• Arakawa-Schubert (AS) mass-flux scheme• Kain-Fritsch (KF) mass-flux scheme


Physical ParameterizationsBetts-Miller-Janjic (BMJ) Scheme:

• Used in the operational NAM / WRF regional model

Trigger: Checks grid column for non-zero CAPE extending > 200-mb in depth from the LFC• PBL must have sufficient deep moisture• Requires at most a weak capping inversion




Adjustment: If trigger criteria are met, then model adjusts the temperature and humidity profiles so a net warming (due to latent heat release) and a net drying (due to moisture removal via precipitation) are achieved through the CAPE layer.

No WarmingDrying

Some WarmingSome Drying

WarmingNo Drying




Result: Prolonged triggering of the scheme in a given grid column can be seen in model forecast soundings as very linear (i.e., unrealistic) profiles between the LFC and EL Caused by the lack of downdrafts in the scheme → No cooling in the PBL




Advantages:

• Low computational expense due to simplicity• Good performance in moist environments and with afternoon storms• Efficient drying and stabilization of the column

Disadvantages

• Neglects cooling due to downdrafts• Inability to trigger convection in dry environments• Difficulty handing convection in capped environments• Does not account well for shallow convection


Physical ParameterizationsArakawa-Schubert (AS) Scheme:

• Used in the operational GFS global model

Trigger: Checks grid column for non-zero CAPEChecks if column has been destabilizing (has increased CAPE) with time

• PBL warming due to advection or surface fluxes• PBL moistening due to advection or surface fluxes• Cold air advection aloft• Radiational cooling aloft

Result: Runs a 1-D cloud model for the cellGenerates an ensemble of clouds with different depths occupying some fraction of the grid cellReduces the instability in a manner proportion to its productionAdjusts temperature and moisture profiles accordinglyAccounts for downdraft cooling, entrainment / detrainment, and compensating subsidence


Physical ParameterizationsArakawa-Schubert (AS) Scheme:

• Used in the operational GFS global model

Advantages:

• Performs well in a variety of environments with realistic sounding adjustments• Represents downdrafts and handles capping inversions

Disadvantages:

• Computationally expensive• Performs better with larger grid lengths (> 40 km) in global models


Physical ParameterizationsKain-Fritsch (KF) Scheme:

• Not used in any operational models• Common choice in many mesoscale research models• Designed for smaller grid lengths (10-20 km)• Designed for midlatitude continental convection

Trigger: Checks grid column for non-zero CAPEChecks grid column for sufficient grid-scale vertical motion to lift parcels to LFC

Result: Produces clouds of single depth (only deep convection)

Accounts for downdraft cooling, entrainment / detrainment of both air and hydrometeors at

multiple levels, compensating subsidence, and storm outflow

Produces realistic adjustments to the thermodynamic profiles


Physical ParameterizationsKain-Fritsch (KF) Scheme:

• Not used in any operational models• Common choice in many mesoscale research models• Designed for smaller grid lengths (10-20 km)• Designed for midlatitude / continental convection

Advantages:

• Performs well in mesoscale numerical models• Produces the most realistic cold pools (compared to other CP schemes)• Involves the most realistic entrainment / detrainment processes• Can trigger realistic deep convection in capped environments

Disadvantages:

• Large computational expense• Tends to over-moisten the post-convective environment• Does not perform well in other regions (Tropics, over mid-latitude oceans)


Physical ParameterizationsExplicit Convection:

• When are convective parameterizations schemes no longer needed?• Current estimates suggest that CP is not needed with grid lengths less than 4 km

Why? Many precipitating clouds are greater than 4 km in diameterAll CP schemes were not designed to represent smaller clouds

• Nevertheless – great care must be taken to ensure precipitation is accurately represented (i.e., not “over-predicted”) when no CP scheme is used…

No CP Scheme - Explicit Convection BMJ Scheme


Physical ParameterizationsTwo “Flavors” of Numerical Model Precipitation:

1. Grid-Scale: Grid cell achieves saturation (or super-saturation) and precipitation is produced directly via the microphysics scheme

2. Sub-Grid-Scale: Grid cell does not achieve saturation but does reach the “trigger” criteria and the convective parameterization scheme produces precipitation

Sub-grid-scaleCumulonimbus

Grid-scaleStratonimbus

Contours = Total precipitationShading = CP precipitation


Physical ParameterizationsForecast Sensitivity to CP Choice:

• Model forecasts can change significantly due to ONLY choice of CP scheme!!!

• Shown are forecast fields valid at +30 h for two numerical simulations where the only difference was the CP scheme

SLP and Precipitation

KF

BMJ

1000-mb θe and winds

KF

BMJ


Data AssimilationA Not so Simple Requirement for a Good Forecast:

• As noted by Bjerknes (1904) - All good forecasts require a sufficiently accurate knowledge of the state of the atmosphere at the initial time…

• Observations serve a critical role in initializing all weather and climate model simulations – the observations must be accurate

Early Data Assimilation

• Generate regularly spaced grids from unevenly distributed observations• Objective analysis (inverse-distance-weighting schemes)• Smoothing (remove small scale “noise”)

Modern Data Assimilation

• Combining all available observations to construct the best possible estimate of the state of the atmosphere• Applied retrospectively to construct “re-analysis” datasets for climate studies• Applied in real-time to initialize weather prediction models• Use very sophisticated analysis techniques – 3DVAR and 4DVAR


Data AssimilationStep-1: Collect Available Observations BIG DATA – TeraBytes collected every hour

• In-situ surface observations (ASOS)• In-situ upper air observations (rawinsondes)• In-situ aircraft observations (commercial)• Satellite observations

• Imagers (VIS, IR, WV)• IR Sounders (T and RH profiles)• Microwave Sounders (liquid and ice)• Scatterometers (surface winds)• Cloud drift winds

• Radar observations (NEXRAD)• Lidar Systems• Unmanned drone aircraft• Neutrally-buoyant balloons


Data AssimilationStep-2: Interpolation of all available data onto an evenly spaced grid

• All observations are interpolated onto grids with the same resolution as the model

Data Source Weighting and Influence

• Some data types are more reliable than others (various error magnitudes)• Some data types are more representative than others (various observed resolutions)• All data types are assigned a unique “weight” before interpolating and merging with other data types• Weights are function of both error magnitude and the spatial distribution of the data source relative to the other sources

• Some data types have more influence on the the initial conditions than others

• These observed fields are NOT BALANCED so…

... SATSATRAWRAWfinal TWTWT


Data AssimilationStep-3: Creation of a “balanced initial” atmosphere (Analysis)

• The weighted / gridded observations are then compared to the “balanced” model field predicted by the previous forecast cycle but valid at the same data assimilation time

• This comparison provides a quantitative measure of the “distance” between the observed fields and the fields used to initialize the model (analysis fields)

• This distance is then reduced by applying “variational techniques” that repeatedly tweak the analysis fields while maintaining balance conditions (mass, hydrostatic, geostrophic, etc.) until an smaller more acceptable distance is found

• This final analysis is then used to initialize (or start) the numerical simulation

3DVAR: Observations within a large time window(±3h) are combined before the analysisfields are created by variational methods

4DVAR: Observations are combined into multiple smaller time windows (< 1h)


Ensemble ForecastingBasic Concept and Purpose:

• A prediction based not just a single (deterministic) forecast but on a suite of several individual forecasts

• All non-linear prediction systems suffer from “intrinsic chaos” or “the butterfly effect” whereby some seemingly miniscule differences in an early model state will amply until the large-scale forecasts at some later time are completely different• The realistic limit of deterministic prediction is about 2 weeks

• Ensemble forecasting is one method used to partially overcome such intrinsic chaos by quantifying the range (or spectrum) of possible atmospheric states

Sources of Intrinsic Chaos:

Initial Condition Errors: Instrument errorsErrors of representationErrors in the interpolation processSmall imbalances in the final analyses

Model Errors: Inappropriate physical parameterizationsInadequate vertical / horizontal resolutionInadequate representation of boundariesUnrepresented physical processes **


Ensemble ForecastingStrategies used to generate an ensemble of forecasts:

• The basic operational run of a model is called the control run• An ensemble of additional runs is generated by doing one or all of the following:

1. Introducing small variations into the initial conditions2. Perturbing the model physics (e.g., changing the CP scheme)3. Using a suite of different models (WRF, GFS, and ECMWF)

• The ensemble mean will (on average) represent the best forecast with the smallest error

• The range of forecasts from the ensemble can be used to determine forecast confidence


Ensemble ForecastingAdvantages:

• There are four primary advantages to ensemble prediction beyond what a single deterministic forecast can provide:

1. The ensemble mean (based on a simple average or a weightedaverage of the individual ensemble members) often exhibits moreskill than do the individual ensemble members

2. The ensemble provides a quantitative measure of forecast confidenceas a function of lead time and forecast location

3. A probabilistic forecast is immediately available from the ensemble

4. The ensemble system provides information regarding the optimallocations for additional targeted observations which can be used to improve the forecast (e.g., areas of large standard deviation**)


Ensemble ForecastingLimitation: NOT a “Silver Bullet”:

• In some situations the atmosphere can diverge outside the ensemble envelope (range)

• Large errors in the initial conditions• Model deficiencies • Unrepresented critical processes


Current Operational Forecast ModelsGFS Model

• Global• Hydrostatic• Spectral (27 km equivalent grid length)• Pressure-sigma (64 vertical levels)• Forecasts out to at least +16 days• 3DVAR (with 6-hr analyses)• 22-member ensemble forecast (MREF)• http://www.emc.ncep.noaa.gov/GFS

ECMWF Model

• Global• Hydrostatic• Spectral (25 km equivalent grid length)• Pressure-sigma (91 vertical levels)• Forecasts out to at least +10 days• 4DVAR (with 6-hr analyses)• 51-member ensemble forecast systems• http://www.ecmwf.int/

NAM / WRF Model

• Regional• Non-hydrostatic• Gridded (12 km grid cell length)• Pressure-sigma (35 vertical levels)• Boundary conditions from GFS• Forecasts out to at least +7 days• 3DVAR (with 3-hr analyses)• http://www.emc.ncep.noaa.gov/NAM

RUC / RAP Model

• Regional• Hydrostatic• Gridded (13 km grid cell length)• Isentropic-sigma (50 vertical levels)• Boundary conditions from NAM / WRF• Forecasts out to at least +24 hours• 3DVAR (with 1-hr analyses)• http://ruc.noaa.gov/

http://www.emc.ncep.noaa.gov/GFS

http://www.ecmwf.int/

http://www.emc.ncep.noaa.gov/NAM

http://ruc.noaa.gov/


Model Output Statistics (MOS)Extracting “Useful” Weather Forecast Information from Numerical Models

• Raw numerical forecast fields do not provide the information desired by the public• Useful MOS is obtained after combining (1) numerical model output parameters with (2) climatological information and (3) historical model errors to produce a new set of statistical forecasts that accounts for regional and seasonal differences through the use of multiple linear regression equations

CAUTION: Assumes the model is correct


ReferencesBarker, D. M., W. Huang, Y. R. Guo and Q. N. Xiao, 2004: A three-dimensional (3DVAR) data assimilation system for use

with MM5: Implementation and initial results. Mon. Wea. Rev., 83, 1-10.

Bjerknes, V., 1904: The problem of weather forecasting as a problem in mechanics and physics, Meteor. Z., 21, 1-7.

Bjerknes, V. 1914: Meteorology as an exact science. Mon. Wea. Rev., 42, 11-14.

Kalnay, E, 2003: Atmospheric Modeling, Data Assimilation, and Predictability. Cambridge University Press, 341 pp.

Lackmann, G. M., 2011: Winter Storms, Midlatitude Synoptic Meteorology - Dynamics, Analysis, and Forecasting, Amer. Meteor. Soc., Boston, 219-246.

Lorenz, E. N., 1965: A study of the predictability of a 28-variable atmospheric model. Tellus, 17, 321-333.

Molinari J. and M. Dudek, 1992: Cumulus parameterization in mesoscale numerical models: A critical review. Mon. Wea. Rev., 120, 326-344.

Ruth, D. P., B. Glahn, V. Dagostro, and K. Gilbert, 2009: The performance of MOS in the digital age . Weather andForecasting, 24, 504-519.

Strensrud, D. J., H. E. Brooks, J. Du, M. S. Tracton, and E. Rogers, 1999: Using ensembles for short-range forecasting.Mon. Wea. Rev., 127, 433-446.

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Numerical Weather Prediction