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Temporal Errors

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

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Temporal Errors

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  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,theJFNK 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, 1003-1022

  10. Why the Rapid Intensification Phase?

  11. An Example of a Physics-Based Preconditioner Used Within the Hurricane Model Faster than Real-time 20 times speed up Reisner et al., 2005, MWR, 133, 1003-1022

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

  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. 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 Diffusionfrom 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, 1443-1462) Almost all cloud models produce too high of a cloud base

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

  26. 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

  27. 2-D Simulations: Moist Bubble Intercepting a Stratus Deck

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

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

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

  31. 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

  32. 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…

  33. Key Tuning Parameter (Con’t)

  34. Biufurication

  35. U Isosurfaces & Wind Vector Field

  36. W Isosurfaces & Wind Vector Field

  37. Rain Isosurfaces & Wind Vector Field

  38. 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…?

  39. 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

  40. 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

  41. Very Rapid Intensification!

  42. Rain Isosurfaces & Wind Vector Field

  43. W Isosurfaces & Wind Vector Field

  44. W Isosurfaces & Wind Vector Field

  45. 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

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