Zavisa Janjic 1 NMM Dynamic Core and HWRF Zavisa Janjic and Matt Pyle NOAA/NWS/NCEP/EMC, NCWCP, College Park, MD.

Zavisa Janjic 1

NMM Dynamic Core and HWRF

Zavisa Janjic and Matt PyleNOAA/NWS/NCEP/EMC, NCWCP, College Park, MD

Zavisa Janjic 2

Basic Principles

Forecast accuracyFully compressible equationsDiscretization methods that minimize generation of computational noise and reduce or eliminate need for numerical filtersComputational efficiency, robustness

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Nonhydrostatic Mesoscale Model (NMM)

Built on NWP and regional climate experience by relaxing hydrostatic approximation (Janjic et al., 2001, MWR; Janjic, 2003, MAP, Janjic et al., 2010)Add-on nonhydrostatic module

Easy comparison of hydrostatic and nonhydrostatic solutionsReduced computational effort at lower resolutions

General terrain following vertical coordinate based on pressure (non-divergent flow remains on constant pressure surfaces)

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Inviscid Adiabatic Equations

TSfc Difference between hydrostatic pressures at surface and top

Hydrostatic pressurep Nonhydrostatic pressure

pRT Gas law

Hypsometric (not “hydrostatic”) Eq.

0

ss

ssst ss

v Hydrostatic continuity Eq.

Continued …

tyxsstsyx T ,,),,,( 21 Constant depth of hydrostatic pressure layer at the top

1 Zero at top and bottom of model atmosphere

2 Increases from 0 to 1 from top to bottom

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1p

Third Eq. of motion

),,(1

tyxWs

stgdt

dzw s

s

vIntegral of nonhydrostaticcontinuity Eq.

vkv fpdtd

ss )1( Momentum Eq.

p

ssp

tp

cT

ssT

tT

sp

s vv Thermodynamic Eq.

ws

swtw

gdtdw

g ss

v11

Vertical acceleration

Inviscid Adiabatic Equations, contd.

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Nonhydrostatic Dynamics Specifics

F, w, e are not independent, no independent prognostic equation for w!

More complex numerical algorithm, but no over-specification of w

e <<1 in meso and large scale atmospheric flows

Impact of nonhydrostatic dynamics becomes detectable at resolutions <10km, important at 1km.

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Horizontal Coordinate System, Rotated Lat-Lon

Rotates the Earth’s latitude and longitude so that the intersection of the Equator and the prime meridian is in the center of the domain

Minimized convergence of meridiansMore uniform grid spacing than on a regular lat-lon gridAllows longer time step than on a regular lat-lon grid

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Horizontal E grid

h v h v hv h v h vh v h v hv h v h vh v h v h

h v

v h

ddy

dx

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Vertical Coordinate

PDsgPDsgPp TOPT 21

TP

TOPT PDP

PDPDP TOPT

TOPPD

PD

rangepressure

rangesigma

0

10

2

1

sg

sg

10

1

2

1

sg

sg

Sangster, 1960; Arakawa & Lamb, 1977

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Vertical Staggering

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Conservation of important properties of the continuous system aka “mimetic” approach in Comp. Math. (Arakawa 1966, 1972, …; Jacobson 2001; Janjic 1977, 1984, …; Sadourny, 1968, … ; Tripoli, 1992 …)

Space Discretization Principles

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Nonlinear energy cascade controlled through energy and enstrophy conservationA number of properties of differential operators preservedQuadratic conservative finite differencingA number of first order (including momentum) and quadratic quantities conservedOmega-alpha term, transformations between KE and PEErrors associated with representation of orography minimizedMass conserving positive definite monotone Eulerian tracer advection

Space Discretization Principles

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Atmospheric Spectrum

Numerical models generally generate excessive small scale noise

False nonlinear energy cascade (Phillips, 1954; Arakawa, 1966 … ; Sadourny 1975; …)Other computational errors

Historically, problem controlled by:Removing spurious small scale energy by numerical filtering, dissipationPreventing excessive noise generation by enstrophy and energy conservation (Arakawa, 1966 … , Janjic, 1977, 1984; Janjic et al., 2010) by design

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NMM

* E grid FD schemes also reformulated for, and used in ESMF compliant B grid model being developed

Advection, divergence operators, each point talks to 8 neighbors

?

? ?

?Formal 4-th order

accurate

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Three sophisticated momentum advection schemes with identical linearized form, and therefore identical truncation errors and formal accuracy, but different nonlinear conservation properties. Janjic, 1984, MWR (blue); controlled energy cascade, but not enstrophy conserving, Arakawa, 1972, UCLA (red); energy and alternative enstrophy conserving, Janjic,1984,MWR (green)

Different nonlinear noise levels (green scheme) with identical formal accuracy and truncation error (Janjic et al., 2011, MWR)

In nonlinear systems conservation more important than formal accuracy

Second order schemes

Fourth order schemes

116 days

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-2.5

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Wind component developing due to the spurious pressure gradient force in the sigmacoordinate (left panel), and in the hybrid coordinate with the boundary between the pressure and sigma domains at about 400 hPa (right panel). Dashed lines representnegative values.

15 km

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435 435 435450 450 450465 465 465480 480 480 480495 495 495 495510

510

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525 525540 54

0 540 540555 55

5 555 555570 570 570 570585 585 585 585

600 600

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615 615630 630 630 630645

645 645 645660 660 660675 675 675

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10 km

sigma hybrid

minimum= .0000E+00 maximum= .3500E+04 interval= .2500E+03

Topography

13. 1.2005. 0 UTC + 00012

Potential temperature, January 13, 2005, 00Z12 hour forecasts, 3 deg contours

Example of nonphysical small scale energy source

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Lateral Boundary Conditions

Specified from the driving model along external boundaries4-point averaging along first internal row (Mesinger and Janjic, 1974; Miyakoda and Rosati, 1977; Mesinger, 1977)Upstream advection area next to the boundaries from 2nd internal row

Advection well posed along the boundaries, no computational boundary condition neededDissipative

HWRF internal nesting discussed elsewhere

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Lateral Boundary Conditionsh v h v h v h v h v h v h v h v h v h v h v h

v h v h v h v h v h v h v h v h v h v h v h v

h v h v h v h v h v h v h v h v h v h v h v h

v h v h v h v h v h v h v h v h v h v h v h v

h v h v h v h v h v h v h v h v h v h v h v h

v h v h v h v h v h v h v h v h v h v h v h v

h v h v h v h v h v h v h v h v h v h v h v h

v h v h v h v h v h v h v h v h v h v h v h v

h v h v h v h v h v h v h v h v h v h v h v h

v h v h v h v h v h v h v h v h v h v h v h v

h v h v h v h v h v h v h v h v h v h v h v h

Domain Interior

External BoundaryFirst Internal Row

Upstream Advection Area

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Time IntegrationExplicit where possible for accuracy, reduced communications on parallel computers

Horizontal advection of u, v, T, tracers (including q, cloud water, TKE …)Coriolis force

Implicit for fast processes that require a restrictively short time step for numerical stability, only in vertical columns, no impact on scalability

Vertical advection of u, v, T, tracers and vertically propagating sound waves

No time splitting and no iterative time stepping schemes in basic dynamics equations for accuracy and computational efficiency

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Horizontal advection and Coriolis Force

)(21

)(23

Δ1

1

yfyf

tyy

)(533.0)(533.1Δ

11

yfyft

yy

Non-iterative 2nd order Adams-Bashforth:

Weak linear instability (amplification), can be tolerated in practice with short time steps, or stabilized by a slight off-centering as in the WRF-NMM.

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0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

amplf1

amplf2

0.99

0.995

1

1.005

0 1 2 3

amplf1

amplf1

Amplification factors for the computational mode (red) and the

meteorological mode in the Adams-Bashforth scheme. Wave number is

shown along the abscissa.

Zoomed amplification factor near 1. The amplification factors of the

modified Adams-Bashforth scheme (green), and the original one (orange)

Cx

jjj

tmsC

x

uuC

t

u

31

,110

2

1

11

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Vertical Advection

Implicit Crank-Nicolson

Unconditionally computationally stableOff-centering option, dissipative

)]()([21

Δ1

1

yfyf

tyy

2)],()([21

Δ1

1

baybfyaf

tyy

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Advection of tracers

New Eulerian advection replaced the old Lagrangian

Improved conservation of advected species, and more consistent with remainder of the dynamicsReduces precipitation bias in warm season

Advects sqrt(quantity) to ensure positivityEnsures monotonicity a posteriori

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Advection of tracers

Twice longer time steps than for the basic dynamics

t1 t2 t3 t4 t5 t6time

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Gravity Wave Terms

Forward-Backward (Ames, 1968; Gadd, 1974; Janjic and Wiin-Nielsen, 1977; Janjic 1979)

Mass field computed from a forward time difference, while the velocity field comes from a backward time difference.

1D shallow water example

x

uH

t

h

x

hg

t

u

,

Mass field forcing to update wind from t+1 time

x

uHthh

Δ1

x

hgtuu

1

1 Δ

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Vertically Propagating Sound Waves

Actual computations hidden in a highly implicit algorithmIn case of linearized equations in a vertical column (Janjic et al., 2001; Janjic, 2003, 2011), reduces to

pressurechydrostatistatebasicfromdeviationp

z

pRT

c

c

t

ppp

v

p

'

,'''2'20

12

02

11

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Lateral Diffusion

Following Smagorinsky (1963) (Janjic, 1990, MWR; Janjic et al. 2010)

212222 ]'2)(4)(4)

cos(2)

cos(2[ TKEC

yw

xw

yu

xv

yv

xu

ndeformatio

sizegridscalelengthl

constantySmagorinskC

lCK

S

S

,

4.02.0,

)( 2

HWRF

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2D Divergence Damping

Dispersion of gravity-inertia waves alone can explain linear geostrophic adjustment on an infinite plain“In a finite domain, unless viscosity is introduced, gravity waves will forever ‘slosh’ without dissipating.” (e.g., Vallis, 1992, JAS)Numerical experiments by Farge and Sadourny (1989, J.Fluid.Mech.) strongly support the idea of dissipative geostrophic adjustment

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2D Divergence Damping

yxA

yxA

pressurechydrostati

xvyudv

xvyudu

A

dvduA

xvyuD

xyx

n

yyx

n

nynx

yy

xx

l

2'

4

''

''

'

)''()''(

3

2

)()(

3

1

'

'

''

x’y’

AA’

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2D Divergence damping

Divergence damping damps both internal and external modes

ly

lx

DKtv

DKtu

1

1

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2D Divergence Damping

External mode divergence

External mode divergence damping

top

bottomlext DD

exty

extx

DKtv

DKtu

2

2

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2D Divergence Damping

External mode damping combined with divergence damping, enhanced damping of the external mode

extyly

extxlx

DK

DK

tv

DK

DK

tu

21

21

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Dynamics formulation tested on various scales

0 0

0

0

0

0

0

0

0

Warm bubble

10

0

0

Cold bubble

Convection

Mountain waves

-5/3

Decaying 3D turbulence

Atmospheric spectra

0

1

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5

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8

-6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0

oceanphy3648

k^-3

k^-5/3

0

1

2

3

4

5

6

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8

-6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0

oceannop3648

k^-3

k^-5/3

No physics With Physics

-5/3

-3

-5/3

-3

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22nd Conference on Severe Local Storms, October 3-8, 2004, Hyannis, MA.

WRF-NMM

WRF-NMM

WRF-NMM

Eta Eta Eta

20044km resolution, no parameterized convection Breakthrough

that lead to application of “convection allowing” resolution for storm prediction

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Summary

Robust, reliable, fast

Extension of NWP methods developed and refined over a decades-long period into the nonhydrostatic realm

Utilized at NCEP in the HWRF, Hires Window and Short Range Ensemble Forecast (SREF)operational systems

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Horizontal Coordinate System, Rotated Lat-Lon

planeequatorialrotated

866.0)0cos(/)30cos(

3473.0)10cos(/)70cos(

)cos(Δ

7010

00

00

00

lonlatrotated

lonlat

x

NN

domain

ΔΔ

planeequatorial