1. 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

2. 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

3. Time-Splitting All terms, advection, diffusion, and various sources
must be at the same time-level; otherwise time-splitting
is the result…

4. 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!

5. Jacobian-Free Newton Krylov (JFNK) Solution Procedure

6. Newton’ Method Current models typically produce an order one
Newton error,
i.e., f(x)=x-sin(x)=1

7. Mechanics of a Krylov Solver

8. 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

9. Cloud field from an idealized smooth
hurricane simulation
Reisner et al., 2005, MWR, 133,

10. Why the Rapid Intensification Phase?

11. An Example of a Physics-Based Preconditioner
Used Within the Hurricane Model
Makes the hurricane problem doable…
Faster than
Real-time
20 times speed up
Reisner et al., 2005, MWR, 133,

12. Large Errors in Potential Temperature, As large as errors in physical models?
5.0 s time step
2.5 s time step
1.0 s time step
60.0 s time step

13. 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

14. 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)

15. Bulk Microphysical Model

16. Smooth Bulk Microphysical Model

17. A multi-phase particle-based approach
is being used to model spectra
In hurricanes

18. 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

19. Advection can Introduce a Small Dynamical Time 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…

20. 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

21. with fast condensational growth can lead to sharp cloud edges
Evaporation combined
with fast condensational growth can lead to sharp cloud edges
Quadratic interpolation leads to oscillations near
the 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!

22. Cloud Edge Diffusion from Dimensional Analysis
Diffusion operator in x direction
for cloud water field…
Adds a resolved spatial scale!
Two approaches for determining closure coefficient:
Constant,
Advective procedure

23. Cloud-edge diffusion associated with the movement of a 1-D cloud

24. Observations from DYCOMS-II,
from Steven et al. (2005, MWR, 133, )
Almost all cloud models
produce too high of a cloud base

25. 3-D Isosurface of Cloud Water from Smooth Cloud Model
Traditional Cloud Model

27. Time averaged X-Z Cross-Sections of Cloud Water from the
Smooth Cloud Model Using Various
Time Step Sizes
Time averaged X-Z Cross-Sections of
Cloud Water from the
Traditional Cloud Model Using Various
Time Step Sizes

28. 2-D Simulations: Moist Bubble Intercepting a Stratus Deck

29. Smooth Approach: Little Differences in Cloud Features with
Different Time Step Sizes

30. Traditional Approach: Big Differences in Cloud Features with
Different Time Step Sizes

31. Traditional Cloud Model
Error in Cloud Water
from
Smooth Cloud Model
Error in Cloud Water
from
Traditional Cloud Model

35. 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

36. 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…

37. Key Tuning Parameter (Con’t)

38. Biufurication

39. U Isosurfaces
&
Wind Vector
Field

40. W Isosurfaces
&
Wind Vector
Field

41. Rain
Isosurfaces &
Wind Vector
Field

42. 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…?

43. 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

44. 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

45. Very Rapid Intensification!

46. Rain
Isosurfaces &
Wind Vector
Field

47. W Isosurfaces
&
Wind Vector
Field

48. W Isosurfaces
&
Wind Vector
Field

49. 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