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 Advanced Synoptic M. D. Eastin

Dynamical Core

Grid “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 value reported represents a spatial average across the total grid cell area

Cloud cover

– the single value reported represents the fraction of the total grid cell area occupied by cloud Advanced Synoptic M. D. Eastin

Dynamical Core

Grid “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 spatial structure

the 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) Advanced Synoptic M. D. Eastin

Dynamical Core

Horizontal 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 Advanced Synoptic M. D. Eastin

Dynamical Core

Horizontal 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

COS n=1 SIN n=1 SIN n=2

Advanced Synoptic

Observed

M. D. Eastin

Dynamical Core

Horizontal Grid Architecture:

• 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 

x

  111

km

  360  /  λ n=30 = 444 km The

current GFS model

analyzes the first 574 waves (n = 574) for an effective grid cell spacing of

~23 km COS n=1 SIN n=1 SIN n=2

Advanced Synoptic

Observed

M. D. Eastin

Dynamical Core

Vertical Grid Architecture:

• Some models use

sigma coordinates

 

p p S

 

p T p T

where: p = pressure at a given level p T p S = pressure at the model top = pressure at the surface σ = 1 at the surface σ = 0 at the model top • The

ECMWF model

coordinate system uses the sigma 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 Advanced Synoptic M. D. Eastin

Dynamical Core

Time 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:  

c

 

x t

 1 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 and thus can complete a full forecast in less time c = 100 m/s Δx = 10 km

Δt ≤ 100 s

•

Non-hydrostatic models

must account for sounds waves with stricter CFL conditions, and thus require more time to complete a full forecast c = 400 m/s Δx = 10 km

Δt ≤ 25 s

Advanced Synoptic M. D. Eastin

Dynamical Core

Time Stepping: “Advantages” of hydrostatic models

• Since hydrostatic models run much faster, long-term prediction (

all climate simulations

and many

weather forecast models

)

require

the get results in a reasonable amount of “real” time

hydrostatic approximation

in order to • 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: Non-hydrostatic models: GFS model (global weather/climate) ECMWF model (global weather/climate) CCM (global climate) NAM / WRF model RUC model MM5 Hurricane WRF What does the

lack

of non-hydrostatic vertical motions imply about

ALL

climate simulations? Advanced Synoptic M. D. Eastin

Physical Parameterizations

Basic 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 or unresolved quantities in terms of other existing dependent variables (T, p, u, etc.) Accounting for unresolved physical processes without introducing additional dependent variables [ Remember: a numerical model consists ] [ of a “closed set” of N equations with ] [ N unknowns - or dependent variables ] Advanced Synoptic M. D. Eastin

Physical Parameterizations

Planetary 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

Advanced Synoptic M. D. Eastin

Physical Parameterizations

Planetary Boundary Layer (PBL):

• 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: K-theory:

w



T



w

  

C H V

 (

T

1

T

  

K H



T SFC

) 

T



z

Heat Flux Heat Flux

Note: Primes

represent the sub-grid-scale [turbulent fluxes]

Overbars

represent the grid-scale [large-scale means]

C H

and

K H

are turbulent transfer coefficients determined through numerous field and lab experiments

Monin-Obukov Similarity Theory

see http://glossary.ametsoc.org/wiki/Monin-obukhov_similarity_theory Advanced Synoptic M. D. Eastin

Physical Parameterizations

Planetary Boundary Layer (PBL):

• 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: Non-local:

Only mix turbulent quantities up/down to an adjacent model level through the PBL Implies that all turbulent mixing is accomplished by eddies of of the same small size Can mix turbulent quantities up/down through all model levels in the PBL Implies that turbulent mixing is accomplished by eddies on a broad range of scales Most realistic (and more complicated) Advanced Synoptic M. D. Eastin

Physical Parameterizations

Land-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 humidity due to rapid evapotranspiration from a corn field and relatively slow evaporation from a nearby bare soil field may influence whether storms develop

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

Predict: Soil temperature/moisture Factors: Local albedo Soil type Vegetation type Seasonal vegetation change Snow cover Advanced Synoptic M. D. Eastin

Physical Parameterizations

Grid-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-saturation Latent heat release / absorption Number 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 of small drops Very few large drops

Advanced Synoptic M. D. Eastin

Physical Parameterizations

Grid-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 Advanced Synoptic M. D. Eastin

Physical Parameterizations

Sub-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 Advanced Synoptic M. D. Eastin

Physical Parameterizations

Betts-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 Advanced Synoptic M. D. Eastin

Physical Parameterizations

Betts-Miller-Janjic (BMJ) Scheme:

• Used in the

operational NAM / WRF regional model

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.

Some Warming Some Drying No Warming Drying

Advanced Synoptic

Warming No Drying

M. D. Eastin

Physical Parameterizations

Betts-Miller-Janjic (BMJ) Scheme:

• Used in the

operational NAM / WRF regional model

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 Advanced Synoptic M. D. Eastin

Physical Parameterizations

Betts-Miller-Janjic (BMJ) Scheme:

• Used in the

operational NAM / WRF regional model

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 Advanced Synoptic M. D. Eastin

Physical Parameterizations

Arakawa-Schubert (AS) Scheme:

• Used in the

operational GFS global model

Trigger: Checks grid column for non-zero CAPE Checks 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 cell Generates an

ensemble of clouds

with different depths occupying some fraction of the grid cell Reduces the instability in a manner proportion to its production Adjusts temperature and moisture profiles accordingly Accounts for downdraft cooling, entrainment / detrainment, and compensating subsidence Advanced Synoptic M. D. Eastin

Physical Parameterizations

Arakawa-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 Advanced Synoptic M. D. Eastin

Physical Parameterizations

Kain-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 CAPE Checks 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 Advanced Synoptic M. D. Eastin

Physical Parameterizations

Kain-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) Advanced Synoptic M. D. Eastin

Physical Parameterizations

Explicit 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 diameter All 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

Advanced Synoptic M. D. Eastin

Physical Parameterizations

Two “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

Grid-scale Stratonimbus Sub-grid-scale Cumulonimbus

Advanced Synoptic

Contours = Total precipitation Shading = CP precipitation

M. D. Eastin

Physical Parameterizations

Forecast Sensitivity to CP Choice:

• Model forecasts can change significantly due to

ONLY

choice of CP scheme!!!

SLP and Precipitation 1000-mb θ e and winds

• Shown are forecast fields valid at +30 h for two numerical simulations where the

only difference

was the CP scheme

KF KF BMJ BMJ

M. D. Eastin Advanced Synoptic

Data Assimilation

A 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

Advanced Synoptic M. D. Eastin

Data Assimilation

Step-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 Advanced Synoptic M. D. Eastin

Data Assimilation

Step-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

T final



W RAW T RAW



W SAT T SAT

 ...

• Some data types have more influence on the the initial conditions than others • These observed fields are

NOT BALANCED

so… Advanced Synoptic M. D. Eastin

Data Assimilation

Step 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: 4DVAR

: Observations within a large time window ( ±3h) are combined before the analysis fields are created by variational methods Observations are combined into multiple smaller time windows (< 1h) Advanced Synoptic M. D. Eastin

Ensemble Forecasting

Basic 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: Model Errors: Instrument errors Errors of representation Errors in the interpolation process Small imbalances in the final analyses Inappropriate physical parameterizations Inadequate vertical / horizontal resolution Inadequate representation of boundaries Unrepresented physical processes

**

Advanced Synoptic M. D. Eastin

Ensemble Forecasting

Strategies 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 conditions 2. 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

Advanced Synoptic M. D. Eastin

Ensemble Forecasting

Advantages:

• 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 weighted average of the individual ensemble members) often exhibits more skill than do the individual ensemble members 2. The ensemble provides a quantitative measure of

forecast confidence

as 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 optimal locations for additional

targeted observations

which can be used to improve the forecast (e.g., areas of large standard deviation

**

) Advanced Synoptic M. D. Eastin

Ensemble Forecasting

Limitation: 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 Advanced Synoptic M. D. Eastin

Current Operational Forecast Models

GFS 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

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

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/

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/ Advanced Synoptic M. D. Eastin

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 Advanced Synoptic M. D. Eastin

References

Barker, 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 and Forecasting

,

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.

Advanced Synoptic M. D. Eastin