Data Assimilation Research Testbed Tutorial. Section 7: Some Additional Low-Order Models
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1 Data Assimilation Research Testbed Tutorial Section 7: Some Additional Low-Order Models Version.: September, 6 /home/jla/dart_tutorial/dart/tutorial/section7/tut_section7.fm 1 7/13/7
2 Low-order models in DART: Model Size Features lorenz_63 3 Chaotic, nearly integral attractor, bifurcations lorenz_84 3 More complex attractor, not as periodic 9var 9 Transient off-attractor dynamics lorenz_96 4 (variable) forced_ lorenz_96 lorenz_96 _scale 8 (variable) 44 (variable) lorenz_4 variable Higher dimensional system. Attractor dimension 13. Allows assimilation of model parameter (see Section ). Two primary interacting spatial/temporal scales. Multiscale dynamics. /home/jla/dart_tutorial/dart/tutorial/section7/tut_section7.fm 7/13/7
3 Lorenz 84 model: Attractor not sheet-like. Rare significant deviations. 1 1 Trajectories along deviations don t mesh back up with rest of attractor. 4 x This behavior can be challenging for certain filter variants. /home/jla/dart_tutorial/dart/tutorial/section7/tut_section7.fm 3 7/13/7
4 Lorenz 84 model: 3-variables: 1 dx = x dt x 3 ax 1 + af 1 dx = x dt 1 x bx 1 x 3 x + g dx = bx dt 1 x + x 1 x 3 x 3 4 x Parameters: a=.5, b = 4, f = 8, g = 1.5 can be set from model_nml. /home/jla/dart_tutorial/dart/tutorial/section7/tut_section7.fm 4 7/13/7
5 Lorenz 84 model: Run csh workshop_setup.csh in directory models/lorenz_84/work. Each state variable is observed every once every hour. Observational error variance is 1. Use matlab to examine the output. There s a new type of filter challenge represented here. Can you identify it? Can you propose ways to address it with techniques learned to date? /home/jla/dart_tutorial/dart/tutorial/section7/tut_section7.fm 5 7/13/7
6 9 Variable model:. Three groups of variables.1 Variables 1-3: Divergence Variables 4-6: Vorticity. Variables 7-9: Height In general, divergence is small. Height and pressure similar. Height and pressure have attractor similar to Lorenz_ /home/jla/dart_tutorial/dart/tutorial/section7/tut_section7.fm 6 7/13/7
7 9 Variable model: X i = U j U k + V j V k v a i X i + Y i + a i z i = U Y + Y V X v a Y j k j k i i i Y i ż i = U j ( z k h k ) + ( z j h j )V k g X i K a i z i + F i U i = b j x i + cy i (1) () (3) (4) V i = b k x i cy i (5) X i = a i x i (6) Y i = a i y i (7) X => Divergence, Y=>Vorticity, z=> height Parameters can be adjusted from model_nml. /home/jla/dart_tutorial/dart/tutorial/section7/tut_section7.fm 7 7/13/7
8 9 Variable model: When perturbed off the attractor, mimics gravity waves. Transient, high frequency oscillations dominate divergence variables. Can also appear in height and pressure variables. Run csh workshop_setup.csh in directory models/9var/work. Y1, Y, Y3 (the vorticity variables) are observed once every 6 hours Observational error variance is.4. Use matlab to examine the output. How do different filter kinds interact with gravity waves? /home/jla/dart_tutorial/dart/tutorial/section7/tut_section7.fm 8 7/13/7
9 Lorenz_96 (4-variable) model: One dimensional cyclic domain [., 1.]. Acts something like synoptic scale weather around mid-latitude circle Lorenz_96 state varnum 1 Ensemble Members of./prior_diag.nc True State Ensemble Mean Ensemble Members () model time (4 timesteps) Lorenz_96 state varnum Ensemble Members of./prior_diag.nc 1 True State Ensemble Mean Ensemble Members () The Attractor 1 3 True_State.nc Lorenz_96 true state model time (4 timesteps) Lorenz_96 state varnum 3 Ensemble Members of./prior_diag.nc 1 True State Ensemble Mean Ensemble Members () model time (4 timesteps) state variable # state variable # state variable # 1 /home/jla/dart_tutorial/dart/tutorial/section7/tut_section7.fm 9 7/13/7
10 Lorenz_96 (4-variable) model: Attractor dimension 13 by some measures. Start to explore model sizes closer to ensemble size. Can examine possible degeneracy issues with sample covariance. Naive application of small ensembles diverges in many cases. Run csh workshop_setup.csh in directory models/lorenz_96/work. 4 observations, randomly located in space, equally spaced in time. Observed once an hour; Observational error variance is 1.. Use matlab to examine the output. Need new techniques to fix this. /home/jla/dart_tutorial/dart/tutorial/section7/tut_section7.fm 1 7/13/7
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