temporal errors
DESCRIPTION
Towards Developing a “Predictive” Hurricane Model or the “Fine-Tuning” of Model Parameters via a Recursive Least Squares Procedure. Goal: Minimize numerical errors within a model to be able to accurately quantify the impacts of model parameters on the predicted fields - PowerPoint PPT PresentationTRANSCRIPT
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Towards Developing a “Predictive” Hurricane Model or the “Fine-Tuning” of Model Parameters
via a Recursive Least Squares Procedure
Goal: Minimize numerical errors within a model to be able to accurately quantify the impacts of model parameters on the predicted fields
Methodology: Use observational data, i.e., lighting, in combination with various data assimilation approaches to determine model parameters
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Temporal Errors Are typically the dominant errors Produced by time-splitting Grow rapidly in-time for time-steps above
the fastest time-scale; determined by the smallest grid spacing in the model, i.e., vertical sound wave propagation
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Time-Splitting
All terms, advection, diffusion, and various sourcesmust be at the same time-level; otherwise time-splitting
is the result…
∂ψ∂t
= Advection(ψ , t) + Diffusion(ψ , t) + Source(ψ , t)
= F(ψ , t)
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How to Avoid Time-Splitting Errors
First, use a time-stepping procedure that does not produce these errors, i.e., Runge-Kutta
Second, time-scales must be resolved with regard to the accuracy of the temporal integrator
Third, use Newton’s method to determine if time scales are being resolved!
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Jacobian-Free Newton Krylov (JFNK) Solution Procedure
Newton's method solves a system of nonlinear equations of the form,
F(x) =0By a sequence of steps,
xk+1 =xk +δxk
J kδxk=-F(xk ), usually inverted by a Krylov solver, where
J i,jk =
∂Fi
∂xj
(xk )
Krylov solver employs the following Matrix-free approximation,
Jδz=F(x+εδz)-F(x)
ε
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Newton’ Method
xk+1 =xk +δx
δx=−F(x)F '(x)
, scalar
δx=-F(x)J (x)
, vector
Current modelstypically produce an
order oneNewton error,
i.e., f(x)=x-sin(x)=1
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Mechanics of a Krylov Solver
res =J δx−F(x)
resk+1 =resk +ϕ J δr
βresk+1 < resk
Jδr =matrix-free approximation, expensive &memory intensive for more than 10 iterations
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Physics-Based Preconditioner For problems with a large separation in time scales,
convergence of a Krylov solver can be extremely slow A physics-based preconditioner is designed to remove fast
time scales in an efficient manner For a single phase Navier-Stokes equation set, the
preconditioner was designed to remove sound waves only With a physics-based preconditioner active, the JFNK
procedure is somewhat like a “predictor-corrector” type algorithm
Entire numerical approach has been used in the simulation of idealized hurricanes employing Navier-Stokes
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Reisner et al., 2005, MWR,
133, 1003-1022
Cloud field from an idealized smooth
hurricane simulation
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Why the Rapid Intensification Phase?
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An Example of a Physics-Based PreconditionerUsed Within the Hurricane Model
Reisner et al., 2005, MWR, 133, 1003-1022
Faster thanReal-time
20 times speedup
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5.0 s time step 2.5 s time step
1.0 s time step 60.0 s time step
Large Errors in Potential Temperature, As large as errors in physical models?
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Key Approximations in Idealized Hurricane Model
Microphysical model involved a simple conversion between water vapor and total cloud substance
Mesh Reynolds number in both horizontal and vertical directions where near 0.1 to resolve smoothly resolve cloud edges
Rex =κΔtΔx2 =0.1
Rez =κΔtΔz2 =0.1
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A More “Complex” Smooth Bulk Microphysical Model
Reworked the Reisner/Thompson et al. microphysical model so that it is smooth or “numerically differentiable” implying… An individual parameterization cannot take out more
cloud substance, i.e., rain, than exists in a given cell Sum of all parameterizations cannot take out more
than exists in a given cell Fastest time scale of an individual parameterization is
the sound wave time-scale Cloud quantities do not go to zero outside the cloud,
i.e., f(x)=x-sin(x)
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Bulk Microphysical Model
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Smooth Bulk Microphysical Model
Psacr =ϕqrqs(Vqr−Vqs
)tanhVqr
−Vqs
εnl
⎛
⎝⎜⎞
⎠⎟qr−smallqs−small
φ=qr−10qr−env0.01εnl
qr−small =0.5(1+ tanh(φ))
Psacr =0.1qs−sumtanhPsacr
0.1qs−sum
⎛
⎝⎜⎞
⎠⎟
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A multi-phase particle-based approachis being used to model spectraIn hurricanes
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Cloud-edge Problem Time-scales can be very fast near cloud
boundaries, i.e., boundary is not resolved Eulerian advection is the problem, including
positive definite schemes and flux-corrected transport (FCT) schemes
Diffusion is the answer…increases time & spatial scales
But, unlike the previous example, high levels of diffusion need not be added everywhere
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Advection canIntroduce a Small DynamicalTime Scale Near Cloud Edges
Advection (ADV) is typically of opposite sign to diffusion (DIFF) near cloud edges
By monitoring this time scale, accuracy and efficiency of a given numerical procedure is maximized
Can be tied to the convergence of Newton’s method Most cloud models use time steps that exceed this time scale, FCT enables
this…
ΨΔt
ΔΨ=
Ψ
ADV + DIFF + REACT
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Edge Problem (Con’t)
FCT Versus Cloud-Edge Diffusion FCT procedure implicitly adds diffusion near cloud edges,
but is not time accurate Cloud-edge diffusion explicitly adds diffusion near edges,
but may add too much diffusion… But, by knowing how much diffusion is being added,
evaporation can be limited Which approach is better? Depends on whether one cares
about resolving time and spatial scales during a simulation and also how implicit diffusion influences a given feature
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Evaporation combinedwith fast condensational growth can lead to sharp
cloud edges
Quadratic interpolation leads to oscillations nearthe edges, must either resolve the edges via diffusion or use
linear interpolation to minimize oscillations…the basis for flux-corrected advective transport (FCT)
Most cloud models employ FCT to keep cloud variables positive and free from oscillations!
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Cloud Edge Diffusionfrom Dimensional Analysis
kqc
x =φ(t)uΔxΔqc
qc
⎛
⎝⎜⎞
⎠⎟
2
Diffusion operator in x direction for cloud water field…
Adds a resolved spatial scale!
Two approaches for determining closure coefficient:• Constant, • Advective procedure
ϕ =qc* − (ui+1/2
* qci+1/2* − ui−1/2
* qci−1/2* )
φ(t) = 1− 0.5 1+ tanhφgradqcϕ
qc*
⎛
⎝⎜⎞
⎠⎟⎡
⎣⎢
⎤
⎦⎥
φ=0.01
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Cloud-edge diffusion associated
with the movement of a 1-D cloud
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Observations from DYCOMS-II,from Steven et al. (2005, MWR, 133, 1443-1462)
Almost all cloud modelsproduce too high of a cloud base
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3-D Isosurface of Cloud Waterfrom
Smooth Cloud Model
3-D Isosurface of Cloud Waterfrom
Traditional Cloud Model
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Time averaged X-Z Cross-Sections of Cloud Water from the
Smooth Cloud Model Using VariousTime Step Sizes
Time averaged X-Z Cross-Sections of Cloud Water from the
Traditional Cloud Model Using VariousTime Step Sizes
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2-D Simulations: Moist Bubble Intercepting a Stratus Deck
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Smooth Approach: Little Differences in Cloud Features withDifferent Time Step Sizes
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Traditional Approach: Big Differences in Cloud Features withDifferent Time Step Sizes
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Error in Cloud Waterfrom
Smooth Cloud Model
Error in Cloud Waterfrom
Traditional Cloud Model
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Idealized Hurricane Simulations-Next Iteration
Base equation set-Navier-Stokes+new smooth cloud model
Constant resolution in horizontal (10 km) and vertical (300 m)
Predictive fields were initialized using sounding data representative of the atmosphere during the rapid intensification phase of Rita
“Bogus-vortex” was used to help spin-up hurricanes
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Key Tuning Parameter Vertical heat and moisture transport are key to
rapid intensification Coarse model resolution implies that these
processes must be parameterized Hence, the key tuning parameter in the model is
related to how quickly these quantities are “diffused” in the vertical direction…helps force formation of hot towers
Parameter must be reasonably smooth…
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Key Tuning Parameter (Con’t)
κ qv
z =κ zSmag +κ extra
z
κ extraz = φ(x)Δz2 tanh(V / 30)
V = u2 + v2 + w2
φ − tuning coefficent
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Biufurication
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U Isosurfaces&
Wind VectorField
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W Isosurfaces&
Wind VectorField
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Rain Isosurfaces &Wind Vector
Field
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Rita Simulations Questions? For a “real” case, does rapid intensification
still occur? Does the model develop bands? How sensitive is the model to variations in
the model parameter…?
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Rita Setup Bogus vortex, Key West Nexrad radar data
(processed by Steve Guimond) for eyewall, LASA data for bands
DBZ radar data was used to initialize rain water and graupel via simple functional relationships
LASA data was used to initialize water vapor, I.e., where lighting was present within a column water vapor was added to force saturation
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Rita Setup (Con’t) 4 km horizontal resolution & 300 m vertical
resolution All fields were initialized from the same
sounding data used to initialize the idealized simulations
Currently investigating impact of diffusion parameter as well as band initialization on intensification
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Very Rapid Intensification!
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Rain Isosurfaces &Wind Vector
Field
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W Isosurfaces&
Wind VectorField
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W Isosurfaces&
Wind VectorField
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Conclusions & Future Work Intensification of a modeled hurricane at
coarse resolution is extremely sensitive to vertical diffusion
Time errors can be important during rapid intensification
Is rapid intensification predictable? Maybe, with reasonable observational data and advanced data assimilation approaches