The Case for UsingHigh-Order Numerical Methods

in Mesoscale and Cloudscale NWP

Bill SkamarockNCAR/MMM

WRF (Weather Research and Forecast) Model

NCAR

NOAA - NCEP

NOAA - FSL

Air Force Weather Agency

Federal Aviation Administration

NRL, Universities and other labs

Collaborative developmenteffort by

Develop an advanced mesoscale forecast and assimilation system

Design for 1-10 km horizontal grids

Portable and efficient on parallel computers

Advanced data assimilation and model physics

Well suited for a broad range of applications

Community model with direct path to operations

The principal objective when developing anNWP model is to maximize efficiency

(1)Maximize forecast (solution) accuracy for a given computer resource, or

(2) minimize computer resource needed for a given forecast (solution) accuracy.

Consider …

Existing nonhydrostatic NWP modelsuse low-order accuracy numerics

(1st /2nd order time, 2nd order space)

MM5ARPS

COAMPSNMMGEMS

LMUKMO Unified Model

WRF dynamical core development projects

(1) Eulerian mass and height coordinate cores - 3rd order RK3 time int. - high order advection.(2) Semi-Lagrangian core - High order RK time int. - High order spatial operators for interp and gradient operators.

Question: Is the use of higher-order methods in the WRF cores justified?

Why are low-order numerics used in most mesoscale NWP models?

1. High-order methods examined in the early development of NWP models were generally not robust.

2. Traditional verification measures will not show increased accuracy of higher order methods when phenomena at small scale are inherently unpredictable at typical mesoscale NWP timescales (1-3 days).

3. There is a widely perceived need to conserve higher moments of the model solutions where possible. Conservation of this sort usually leads to the use of lower order methods.

Are these reasons (still) valid?

Are these reasons (still) valid? NO!

1. High-order methods examined in the early development of NWP models were generally not robust.

Robust higher-order methods have been developed.

2. Traditional verification measures will typically not show the increased accuracy of higher order methods when phenomena at small scale are inherently unpredictable at typical mesoscale NWP timescales (1-3 days).

Other error and verification measures should be used at mesoscale and cloudscale resolutions in addition to the traditional methods.

3. There is a widely perceived need to conserve higher moments of the model solutions where possible. Conservation of this sort usually leads to the use of lower order methods.

Conservation is not necessary or appropriate at small scales.

Outline for the remainder of this talk:

Describe the higher order numerical methodsused in the WRF model.

Present theoretical arguments for increased accuracyof these methods.

Present examples demonstrating this increased accuracyand efficiency.

Present some arguments suggesting that weshould expect increased efficiency using these methods.

Consider some arguments against the need forconservation of higher order moments in NWP modelsat small (and perhaps even large) scale.

Mention some issues that arise in global modeling that do not arise in limited-area modeling.

Conservation in numerical schemes

What quantities should we consider conserving? first-order quantities: mass, momentum, entropy. second-order quantities: energy, enstrophy?

Historically, energy conservation was used to analyze and prove stability (Keller and Lax, 1960’s) for nonlinear systems (esp. with recognition of nonlinear instability by Phillips, 1959).

Energy:Equations conserve energy (and mass, momentum, and entropy).

Enstrophy:Equations do not conserve enstrophy (barotropic vorticity equations do).Arakawa Jacobian formulation is unsuitable for flows exhibiting systematic downscale energy cascade.

No, but there is no theory supporting a yes or no answer.

Question: Does it make sense to conserve higher-order moments (energy) in numerical solutions when first order quantities are not conserved?

Question: Does it make sense to require conservation of energy in a numerical scheme when parameterized sources/sinks and boundary fluxes are orders of magnitude larger than the conservation errors in a non-conservative but more accurate scheme.

No, because this conservation would be meaningless.

Energy conservation:

Observation: Accurate solutions from non-conservative models should be very nearly conservative.

For short range high resolution NWP, conservation of higher order quantities is likely irrelevant to forecast skill. - frequent assimilation of new data - errors have little time to accumulate

For long range forecasts and climate applications, preceding arguments are still valid.

Our philosophy:First and foremost, we should conserve the first-order quantities in which the governing equations of the model are cast.

Global models must solve the equations of motion on a sphere: Pole problem: converging meridians (possibly severe stability/timestep restrictions). Solutions - spectral formulations (implicit), other implicit formulations, semi-Lagrangian formulations. No lateral boundary specification needed.

Dynamical solver issues for global and regional models

Regional models solve equations on some portion of the globe: Lateral boundary condition problem. No pole problem.

Thus, there is more latitude in the choice of numerical schemes for use in regional models than for use in global models.

What is in the WRF model?

The official core is the Eulerian mass coordinate core

ts

t

,

Hydrostatic pressure coordinate:

,,,, wWvVuU

Conserved state variables:

hydrostatic pressure

Non-conserved state variable: gz

gwdt

d

x

U

t

Qx

U

t

w

x

Uwpg

t

W

u

x

Uu

x

p

x

p

t

U

0

Inviscid, 2-Dequations without rotation:

Diagnosticrelations:

,,0p

Rp

WRF model flux-form mass coordinate equations

Height/Mass-Coordinate Model, Time Integration

3rd Order Runge-Kutta time integration

Rt

Rt

Rt

tt

t

tt

tt

1

1

2

3

advance

Amplification factor

241;;

41 tk

AAki nnt

Time-Split Leapfrog and Runge-Kutta Integration Schemes

Phase and amplitude errors for LF, RK3

Oscillation equationanalysis

ikt

Advection in the Height/Mass Coordinate Model

2nd, 3rd, 4th, 5th and 6th order centered and upwind-biased schemesare available in the WRF model.

Example: 5th order scheme

UFUF

xx

Uii

2

1

2

1

1

12132

32211

2

1

2

1

10560

1,1

60

1

15

2

60

37

iiiiii

iiiiii

iiUsign

UUF

where

For constant U, the 5th order flux divergence tendency becomes

TOHx

Cr

x

tU

x

Ut

x

Ut

iiiiiii

thth

..60

6152015660

1

6

6

321123

65

Advection in the Height/Mass Coordinate Model

The odd-ordered flux divergence schemes are equivalent to the next higher ordered (even) flux-divergence scheme plus a dissipation term of the higher even order with a coefficient proportional to the Courant number.

2nd Order 4th Order 6th Order

3rd Order 5th Order

Advection of Top-Hat Profile

xt U

Maximum Courant Number for Advection

(Wicker & Skamarock, 2002)

Time Scheme

3rd 4th 5th 6th

Leap Frog U 0.72 U 0.62

RK2 0.88 U 0.30 U

RK3 1.61 1.26 1.42 1.08

Spatial Order

xtUaC /

U = unstable

 

scheme convergencerotating cone test

-16

-14

-12

-10

-8

-6

-4

-2

0

0 1 2 3

ln(resolution)

ln(e

rror

)

leapfrog-4

RK3-4

RK3-5

RK3-6

leapfrog-6

(From Wicker & Skamarock, MWR 2002)

What can we take from theory?

(1) Increasing the order of the method: - increases the cost per timestep (e.g., in WRF, RK3 is twice the cost per timestep compared to leapfrog). - decrease in error is problem dependent (error reduction by small fraction to orders of magnitude are possible).

(2) Increasing the resolution for a given method: - cost scales as 1

1cost

n

h

where n is the number of spatial dimensions refined.Example: for a doubling of the horizontal resolution, the computational cost increases by a factor of 8.

Consider,

For multidimensional problems, higher-order methodsare usually more efficient than lower-order methods

Another example: On the same grid, if a high order dynamical core takes twice as long to produce a solution compared to a low order dynamical core, the high order model will run in the same time as the low order model on a grid with horizontal resolution

lh xx 3 2

km5.12 km,10e.g. hl xx

Practice (or the problem with theory)

In NWP models, the solutions are (1) not smooth, and (2) the solutions do not converge.

Why? Higher resolution leads to more fine-scale structure in the solution, because

So, we cannot rely solely on theory to guide usin choosing the most efficient methods for our models.

(1) model physics depend on the resolution, and (2) the resolution of the terrain, initial conditions and boundary conditions also increases.

(1) How do we define solution error in NWP? - verification measures? - can we associate errors with a model component, such as the dynamical core?

(2) What should we expect from our models as we increase resolution and resolve motions that are inherently unpredictable (e.g., convective cells for forecasts greater than O(hour)). - pointwise verification (implied determinacy) is not appropriate. - Need measures of resolution and spatial variability.

How do we evaluate the efficiency of a dynamical core?

5 min 10 min 15 min

Comparison of Gravity Current Simulations

HeightCoordinate

MassCoordinate

x = z = 100 m

2-D Mountain Wave Simulation

a = 1 km, dx = 200 m a = 100 km, dx = 20 km

Mass CoordinateHeight Coordinate

Comparison of Height and Mass Coordinates

Supercell Thunderstorm Simulation

Height coordinate model (dx = 2 km, dz = 500 m, dt = 12 s, 80 x 80 x 20 km domain )Surface temperature, surface winds and cloud field at 2 hours

Baroclinic Wave Simulation – Surface FieldsPressure (solid, c.i.= 4 mb), temperature (dashed, c.i.= 4K), cloud field (shaded)

Mass Coordinate, 4 days 12 h Height Coordinate, 4 days 6 h

Dx = 100 km, Dt= 10 min

From Takemi and Rotunno, 2002

WRF squall-line simulationsN-S periodic, W-E open b.c.

TKE tests

(t = 4h, dx = 1 km, gust front dashed)

From Takemi and Rotunno, 2002

36 h Forecast Valid 12Z 1 April 02, 24 h Precip

12 km ETA 22 km WRF 24 h RFC Analysis

Precipitation Threat Score and Bias

March 2002 April 2002

The problem with precip threat scores (and other pointwise verification schemes)truth forecast 1 forecast 2

Issue: the obviously poorer forecast has better skill scores

From Mike Baldwin NOAA/NSSL

Observations 22 km WRF10 km WRF

3 hour accumulated precip, forecasts, valid 18Z 4 June 200212Z runs, 15-18Z precip accumulation

4 km obs analysis (radar and gages)

0

4

8

12

20

50

precip(mm)

From Mike Baldwin and Matt Wandishin, NOAA/NSSL

8 km NMM 12 km opnl ETA

0

4

8

12

20

50

precip(mm)

3 hour accumulated precip, forecasts, valid 18Z 4 June 200212Z runs, 15-18Z precip accumulation

4 km obs analysis (radar and gages)

From Mike Baldwin and Matt Wandishin, NOAA/NSSL

Observations

Power spectra for precip (obs and forecasts)From Matt Wandishin and Mike Baldwin, NOAA/NSSL,

Spectra code from Ron Errico, NCAR

12Z forecasts,15-18 Z accum precip,valid 4 June 2002

Power spectra for precip (obs and forecasts)From Matt Wandishin and Mike Baldwin, NOAA/NSSL,

Spectra code from Ron Errico, NCAR

0Z and 12Z forecasts,3 hourly accum precip,averaged over June 2002

Conclusion: Given the high cost of large, multi-dimensional atmospheric simulations (i.e., NWP), the use of high-order numerical methods maximizes efficiency (accuracy for a given cost) in many (perhaps most?) error measures.

Power spectra for precip (obs and forecasts)

12Z forecasts,15-18 Z accum precip,valid 4 June 2002