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Representation of Convective Processes in NWP Models (part II)

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## Representation of Convective Processes in NWP Models (part II)

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**Representation of Convective Processes in NWP Models (part**II) George H. Bryan NCAR/MMM Presentation at ASP Colloquium, “The Challenge of Convective Forecasting” 13 July 2006**Outline**• Part I: What is a numerical model? • Part II: What resolution is needed to simulate convection in numerical models?**Part II: What resolution is needed to simulate convection**in numerical models? • An interesting question. • What do our commandments say?**Commandments (continued)**• Thou shalt use 1 km grid spacing to simulate convection explicitly • ….**Commandments (continued)**• Thou shalt use 1 km grid spacing to simulate explicitly convection • Honor thy elders • …. “There’s no need for grid spacing smaller than 2 km.”**Perspectives on resolution**• Historical Perspective • Theoretical Perspective • Pragmatic Perspective**The “1 km standard”**• Often quoted in journal articles, textbooks, at conferences, etc. • Clearly, there is some veracity to this “rule of thumb” • otherwise, it wouldn’t be so common • But, where did it come from?**The first cloud models**• Steiner (1973) • Perhaps first 3D simulation of convection • = 200 m • Cumulus congestus • Schlesinger (1975) • Perhaps first 3D simulation of deep convection • = 3.2 km • “a rather coarse mesh was used” • Schlesinger (1978) • = 1.8 km**The first cloud models (cont.)**• Klemp and Wilhelmson (1978) • A groundbreaking paper • The KW Model is the grand-daddy of the ARW Model • = 1 km • “… this resolution is admittedly rather coarse” • Tripoli and Cotton (1980) • = 750 m • Weisman and Klemp (1982) • = 2 km • “Finer resolution would be preferable …”**Commandments (continued)**• Thou shalt read the Old Testament • ….**The first cloud models (cont.)**• Klemp and Wilhelmson (1978) • A groundbreaking paper • The KW Model is the grand-daddy of the ARW Model • = 1 km • “… this resolution is admittedly rather coarse” • Tripoli and Cotton (1980) • = 750 m • Weisman and Klemp (1982) • = 2 km • “Finer resolution would be preferable …”**Summary of literature review**• of O(1 km) was there from the beginning • Many recognized/suggested that this was too coarse • In the decades that followed (80s and 90s), increasing computing power was utilized mainly for larger domains and longer integration times**Justification for 1 km**• Not a great deal of justification out there, other than: • The Sixth Commandment • “scientist A used this resolution; thus, I can, too.” • “It’s all I could afford.” • However …**Justification for 1 km**• Weisman et al. (1997) performed a large number of simulations, using from 12 km to 1 km • “… 4 km grid spacing may be sufficient to reproduce … midlatitude type convective systems” • They identified (correctly) that non-hydrostatic processes cannot be resolved unless 1 km**higher resolution**higher resolution Weisman, Skamarock, and Klemp, 1997: The Resolution Dependence of Explicitly Modeled Convective Systems (MWR, pg 527) ~4 km is sufficient to simulate mesoscale convective systems System-averaged rainwater mixing ratio (qr) weak shear strong shear “Clearly, the 1-km solution has not converged.” “…grid resolutions of 500 m or less may be needed to properly resolve the cellular-scale features …”**Looking beyond 1 km**• Only since the middle 90s have people looked below 1 km systematically • It’s expensive! • Need grids of O(1000 x 1000) • Small time steps • Droegemeier et al. (1994, 1996, 1997) • Found differences in simulations of supercells with 100 m • Turbulent details began to emerge**Supercell simulations: rainwater mixing ratio at z = 4 km,**t = 1 h from: Droegemeier et al. (1994)**Other recent studies**• Petch and Grey (2001) • Petch et al. (2002) • Adlerman and Droegemeier (2002) • Bryan et al. (2003) • All found that results were not converged with = 1 km • i.e., results are dependent on grid spacing • But why? • And what are consequences of coarse resolution?**Δx = Δz = 125 m:**θe, across-line cross sections with RKW “optimal” shear Δ x = 1000 m, Δz = 500 m:**Δx = Δz = 125 m:**θe, along-line cross sections with RKW “optimal” shear Δ x = 1000 m, Δz = 500 m:**along-line cross sections of θe: x=211 km, x=208 km, x=205**km**125 m:**Rainwater mixing ratio with “strong” shear 1000 m:**125 m:**θe, with “strong” shear 1000 m:**Perspectives on resolution**• Historical Perspective • Theoretical Perspective • Pragmatic Perspective**How big are convective clouds, anyway?**• Clouds are surprisingly small • Median updraft diameters are ~2-4 km • Updrafts of ~10 km are rare, and are usually found in supercells**Results of a thorough literature review**from: Bryan et al. (2006)**Some of my conclusions:**• Clouds are of O(1 km) • Grid spacing of O(1 km) should marginally resolve convective updrafts • I think the earliest cloud modelers knew this**The difference between resolution and grid spacing**• Grid spacing () is clear • The distance between grid cells • Resolution is nebulous • Recall that numerical techniques cannot properly handle features less than ~6**Analytic solution to the advection equation**• “E” = exact • “2” = 2nd order centered • “4” = 4th-order centered from: Durran (1999)**Analytic solution to the artificial diffusion terms**• “2” = 2 • “4” = 4 • “6” = 6 from: Durran (1999)**Effective Resolution**• This is a relatively new concept (to some) • The effective resolution of a numerical model is the minimum scale that is not affected by artificial aspects of the modeling system • In the ARW Model, this is ~6-8**Kinetic energy spectra from ARW simulations**from: Skamarock (2004)**Synthesis**• O(1 km) grid spacing is needed to resolve nonhydrostatic processes • Deep convective clouds are of O(1 km), and some supercells are of O(10 km) • The ARW Model needs ~6-8 to “resolve” a feature 1 km grid spacing is looking marginal**Scales in turbulent flows**• L is the scale of the large eddies • e.g., a Cu cloud • is the scale of the dissipative eddies • e.g., the cauliflower-like “puffiness”**Turbulence**• Small-scale turbulence cannot be resolved in numerical models • Theory is clear (Kolmogorov 1940) • To resolve all scales in clouds requires ~0.1 mm grid spacing (Corrsin 1961) • So, what should we do … ?**The filtered Navier-Stokes equations**Start with: Apply a filter, rearrange terms All sub-filter-scale flow is contained in the term (the subgrid turbulent flux) from Bryan et al. (2003)**Modeling subgrid turbulence**• We have a fairly good idea of how to parameterize for many flows • HOWEVER … a few rules apply**Scales in turbulent flows**• L is the scale of the large eddies • e.g., a Cu cloud • is the scale of the dissipative eddies • e.g., the cauliflower-like “puffiness”**1/η**1/L Turbulence Kinetic Energy Spectrum E(κ) κ**?**r s LES MM r s r s DNS r 1/Δ 1/Δ 1/Δ 1/Δ The Four Regimes of Numerical Modeling (Wyngaard, 2004) E(κ) κ**“Crude representation of average energy degradation**path” (A Roadmap!) Turbulent kinetic energy (small eddies) Turbulent kinetic energy (large eddies) Mean flow kinetic energy Internal energy of fluid (heat) Corrsin (1960)**Roadmap for LES**Turbulent kinetic energy (large eddies) Mean flow kinetic energy**Roadmap for LES**Turbulent kinetic energy (large eddies) Mean flow kinetic energy Transfer of kinetic energy to unresolved scales**LES subgrid model**• Works well if grid spacing () is 10-100 times smaller than the large eddies (L) • Recall: L~2-4 km • Suggests that needs to be ~20-200 m • If we want to use LES models … and we do … then of O(100 m) might be necessary**Early cloud modelers knew this**• Klemp and Wilhelmson (1978): • “. . . closure techniques for the subgrid equations are based on the existence of a grid scale within the inertial subrange and with present resolution [Δx = 1 km] this requirement is not satisfied.”**A problem: we want to do this ….**Turbulent kinetic energy (large eddies) Mean flow kinetic energy Transfer of kinetic energy to unresolved scales