A baroclinic instability test case for dynamical cores of GCMs Christiane Jablonowski (University of Michigan / GFDL) David L. Williamson (NCAR) AMWG Meeting, 3/20/06

Overview Basic Idea & Design Goals Derivation of the test case & discussion of the initital conditions 4 dynamical cores: NCAR CAM3 & DWD’s GME Results of the test case –Steady-state conditions –Evolution of the baroclinic wave –Uncertainty of the ensemble of reference solutions Conclusions

The Idea & Design Goals Goal: development of a dynamical core test case (without physics, dry & prescribed orography) that –is easy to apply –is idealized but as realistic as possible –gives quick results –starts from an analytic initial state, suitable for all grids –triggers the evolution of a baroclinic wave Designed for primitive equation models with pressure-based vertical coordinates (hybrid, sigma or pure pressure coordinate)

Derivation of the Initial Conditions Initial data required: u, v, T, p s,  s Find a steady-state, balanced solution of the PE eqns: prescribe u, v and the surface pressure p s Plug prescribed variables into PE equations and derive the –Geopotential field  : based on the momentum equation for v (integrate), calculate surface geopotential  s –Temperature field: based on the hydrostatic equation

Initial Conditions v = 0 m/s p s = 1000 hPa u &  s

Initial Temperature Field

Perturbation Overlaid perturbation (at each model level) triggers the evolution of a baroclinic wave over 10 days Suggested: pertubation of the zonal wind field ‘u’ or the vorticity and divergence (for models in  -  form)

Characteristics of the Initial Conditions Instability mechanisms: –Baroclinic instability - vertical wind shear –Barotropic instability - horizontal wind shear But: –Statically stable –Inertially stable –Symmetrically stable

Characteristics Static stability

Characteristics Symmetric stability & Inertial stability

Test Strategy Initialize the dynamical core with the analytic initial conditions (balanced & steady state) Let the model run over 30 days (if possible without explicit diffusion) Does the model maintain the steady state? Perturb the initial conditions with a small, but well- resolved Gaussian hill perturbation 10-day simulation: Evolution of a baroclinic wave Step 1: Step 2:

Model Intercomparison Eulerian dynamical core (EUL), spectral Semi-Lagrangian (SLD), spectral Finite Volume (FV) dynamical core (NCAR/NASA/GFDL) Icosahedral model GME (2nd order finite difference approach) NCAR CAM3: German Weather Service (DWD):

Resolutions Default: 26 vertical levels (hybrid) with model top ≈ 3hPa. In addition: 18 and 49 vertical levels were tested. EUL & GME: varying horizontal diffusion coefficients K 4, no explicit diffusion in SLD and FV simulations. EUL / SLD (truncation) FV (lat x lon, degrees) GME (min. grid distance) T214 x 5≈ 440 km (ni = 16) T422 x 2.5≈ 220 km (ni = 32) T851 x 1.25≈ 110 km (ni = 64) T x 0.625≈ 55 km (ni =128) T x ≈ 26 km (ni = 256)

Steady-State Test Case Maintenance of the zonal-mean initial state (u wind) Wave number 5 effect Decentering parameter effect, with  =0 EUL is matched

Steady-State Test Case: GME GME shows a truncation error with wave number 5 Artifact of computational grid and low-order num. method

Steady-State Test Case as a Debugging Tool Discovery of a flaw in the SLD dynamical core (old CAM2 version): systematic decrease in the zonal wind speed over time Here: plotted for 30 days with an older version of the test case (u max = 45m/s)

Baroclinic Waves: 30-day Animation Movie: Courtesy of Francis X. Giraldo, NRL, Monterey Surface pressure Surface temperature North-polar stereographic projection

Evolution of the Baroclinic Wave Perturbation gets organized over the first 4 days and starts growing rapidly from day 6 onwards FV 0.5 x L26 dycore T (850 hPa)psps

Evolution of the Baroclinic Wave Explosive cyclogenesis after day 7 Baroclinic wave breaks after day 9 FV 0.5 x L26 dycore T (850 hPa)psps

Convergence with Resolution Surface pressure starts converging at 1 x 1.25 degrees FV L26 dycore, Day 9

Model Intercomparison at Day 9 Second highest resolutions, L26 p s fields visually very similar Spectral noise in EUL and SLD

Model Intercomparison at Day 9 p s fields visually almost identical Differences only at small scales

850 hPa Vorticity at Day 7 Differences in the vorticity fields grow faster than p s diff.

850 hPa Vorticity at Day 9 Small-scale differences easily influenced by diffusion Spectral noise in EUL and SLD (L26)

Impact of explicit diffusion EUL T85L26 with K 4 increased by a factor of 10 (1 x m 4 /s) No spectral noise, but severe damping of the circulation

Model Intercomparisons: Uncertainty Estimate of the uncertainty in the reference solutions across all four models using l 2 (p s )

Single-Model Convergence Single-model uncertainty stays well below the uncertainty across models Models converge within the uncertainty for the resolutions T85 (EUL & SLD), 1x1.25 (FV), GME (55km / ni=128)

Uncertainty in the Relative Vorticity Estimate of the uncertainty in the reference solutions using l 2 [  (850 hPa)] Errors grow faster, but conclusions are the same

Vertical Resolutions Model runs with 18 and 49 levels at mid-range horizontal resolutions are compared to the default 26-level runs Uncertainty stays well below the uncertainty across the models

Phase Errors Phase errors diminish with increasing resolutions Phase error at lower resolutions is substantial for GME, attributes to the relatively ‘late’ convergence of GME (55 km)

Energy Fixer: SLD Dynamical Core Baroclinic wave test revealed problem in the energy fixer of the SLD dynamical core (old CAM2 version) SLD problem with the energy fixer corrected

Conclusions Goal: Development of an easy to use baroclinic wave test case that serves as a debugging tool and and fosters model intercomparisons Test of the models as they are used operationally (no extra diffusion, no special tuning of parameters) Established an ensemble of reference solutions and their uncertainty Models converge within the uncertainty at the resolutions EUL & SLD T85, FV 1 x 1.25, GME (55km/ni=128) Convergence characteristics the same for T or  variable Accessibility: We make the ensemble of solutions (p s ) available to all interested modeling groups and offer to compute l 2 (p s ) norms

Publications Jablonowski, C. and D. L. Williamson, 2006a: A Baroclinic Wave Test Case for Dynamical Cores of General Circulation Models: Model Intercomparisons, NCAR Technical Note TN- 469+STR, 89 pp. (available online at /503/7631 (shortly: also available on NCAR’s CAM3 web page) /503/7631 Jablonowski, C. and D. L. Williamson, 2006b: A Baroclinic Instability Test Case for Atmospheric Model Dynamical Cores, Quart. J. Roy. Meteor. Soc., in review