Zavisa Janjic 1 NMM Dynamic Core and HWRF Zavisa Janjic and Matt Pyle NOAA/NWS/NCEP/EMC, NCWCP,...
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Transcript of Zavisa Janjic 1 NMM Dynamic Core and HWRF Zavisa Janjic and Matt Pyle NOAA/NWS/NCEP/EMC, NCWCP,...
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
-2.5
0
0
0
00
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0
0
0
0
00
0
0
0
2.5
2.5
0
0
0
0
0
00
0
0
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0
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00
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0
00
0
0
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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285
300
300
300
300
315
315 315
315 315
330 330 330
330
345
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345
360 360
360
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375
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375
375
390
390
390
390
405 405 40540
5
405
420 420 420 420435
435 435 435450 450 450465 465 465480 480 480 480495 495 495 495510
510
510 510525
525 525540 54
0 540 540555 55
5 555 555570 570 570 570585 585 585 585
600 600
600 600615 615
615 615630 630 630 630645
645 645 645660 660 660675 675 675
690 690 690
705 705720
300
300
300300
315
315
315 315
330 330
330
330
345
345
345
360
360
360
375
375
375
375
390
390
390
405
405 405
420 420 420435 435 435450 450 450465 465480 480495 495 495510 510 510525 525 525540 540 540555 555570 570585 585600 600615 615630 630645 645660 660 660675 675 675690 705
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
Zavisa Janjic 29
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
Zavisa Janjic 30
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
Zavisa Janjic 31
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
Zavisa Janjic 32
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’
Zavisa Janjic 33
2D Divergence damping
Divergence damping damps both internal and external modes
ly
lx
DKtv
DKtu
1
1
Zavisa Janjic 34
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
Zavisa Janjic 36
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
2
3
4
5
6
7
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
7
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
Zavisa Janjic 37
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
Zavisa Janjic 38
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
Zavisa Janjic 39
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