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

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

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)

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!

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

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

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

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

Reisner et al., 2005, MWR,

133, 1003-1022

Cloud field from an idealized smooth

hurricane simulation

Why the Rapid Intensification Phase?

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

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?

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

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)

Bulk Microphysical Model

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

⎛

⎝⎜⎞

⎠⎟

A multi-phase particle-based approachis being used to model spectraIn hurricanes

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

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

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

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!

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

Cloud-edge diffusion associated

with the movement of a 1-D cloud

Observations from DYCOMS-II,from Steven et al. (2005, MWR, 133, 1443-1462)

Almost all cloud modelsproduce too high of a cloud base

3-D Isosurface of Cloud Waterfrom

Smooth Cloud Model

3-D Isosurface of Cloud Waterfrom

Traditional Cloud Model

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

2-D Simulations: Moist Bubble Intercepting a Stratus Deck

Smooth Approach: Little Differences in Cloud Features withDifferent Time Step Sizes

Traditional Approach: Big Differences in Cloud Features withDifferent Time Step Sizes

Error in Cloud Waterfrom

Smooth Cloud Model

Error in Cloud Waterfrom

Traditional Cloud Model

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

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…

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

Biufurication

U Isosurfaces&

Wind VectorField

W Isosurfaces&

Wind VectorField

Rain Isosurfaces &Wind Vector

Field

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…?

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

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

Very Rapid Intensification!

Rain Isosurfaces &Wind Vector

Field

W Isosurfaces&

Wind VectorField

W Isosurfaces&

Wind VectorField

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