Bred Vectors, Singular Vectors, and Lyapunov Vectors in Simple and Complex Models

Transcription

1 Bred Vectors, Singular Vectors, and Lyapunov Vectors in Simple and Complex Models Adrienne Norwood Advisor: Eugenia Kalnay With special thanks to Drs. Kayo Ide, Brian Hunt, Shu-Chih Yang, and Christopher Wolfe for their guidance and suggestions.

2 Outline Introduction Computation of bred vectors (BVs), singular vectors (SVs), and Lyapunov vectors (LVs) Results Lorenz 1963 model which has 3 degrees of freedom A Fast-Slow coupled Lorenz model developed by Peña and Kalnay (2004) with 9 degrees of freedom A quasi-geostrophic model with 15,015 degrees of freedom, but a single type of instability. SPEEDY, a full atmospheric model with 135,240 degrees of freedom that contains several different types of instabilities. Summary

3 Bred, Singular, and Lyapunov Vectors Instabilities inherent in the atmosphere-ocean system would degrade forecasts even if models and observations were perfect. BVs, SVs, and LVs are three types of vectors frequently used to explore the instabilities of dynamical systems. BVs are by far the easiest (and cheapest) to compute while LVs are the most difficult (and expensive). All perturbations integrated with the tangent linear model (TLM) will, in time, align with the leading LV. Thus linear BVs, those with small amplitudes and short rescaling windows, align with LV1. However, when LV2 grows faster than LV1, BVs align with LV2 (Norwood et al., 2013). SVs are orthonormal and optimized for a particular integration window using a specific norm, but as the window length approaches infinity the final SVs converge to orthonormalized LVs (the basis of the Wolfe and Samelson 2007 method to obtain LVs).

4 Computing Bred Vectors Forecast values Initial random perturbation Bred Vectors ~LLVs Control forcast Integrate nonlinear model, M, to obtain control trajectory, x c. Choose rescaling amplitude, δ 0, and integration window, IW, to target the desired mode of growth (Peña and Kalnay, 2004). Add a small perturbation p x p t i = x c t i + δ 0, p where p is the initial direction of the perturbation and δ p 0 is the amplitude. Integrate x p t i forward using the nonlinear model. The bred vector is bv t i+iw = x p t i+iw x c t i+iw. Rescale to δ 0, and repeat the process. time

5 Computing Singular Vectors For nonlinear model, M, calculate the tangent linear model M ij t i, t i+iw = M i and the adjoint M x ij t i, t i+iw. j Initial singular vectors, ξ j, are the columns of Ξ where M MΞ = ΞS where S is diagonal with entries σ j 2, the squares of the singular values The final singular vectors, η j, are obtained by integrating the TLM forward starting from the initial singular vector. M t i, t i+iw ξ j t i = σ j η j t i+iw ξ 2 M σ 1 η 1 ξ 1 σ 2 η 2

6 Computing Lyapunov Vectors (Wolfe and Samelson, 2007) Fig. 6.7: Schematic of how all perturbations will converge towards the leading Local Lyapunov Vector Extend the integration window for the SVs forward and backward in time until the SVs converge. These asymptotic initial and final SVs shall be denoted as ξ j and η j, respectively. Every perturbation integrated forward in time aligns with the leading LV. Thus φ 1 = < φ 1, η j > η 1. SVs are orthogonal so LV2 has a component in the direction of η 2. Thus φ 2 =< φ 2, η 1 > η 1 + < φ 2, η 2 > η 2, and so on. Every perturbation integrated backward in time aligns with the fastest decaying LV. Thus φ N = < φ N, ξ N > ξ N, where N is the degrees of freedom of the system. Likewise φ N 1 = < φ N 1, ξ N 1 > ξ N 1 + < φ N 1, ξ N > ξ N, and so on random initial perturbations trajectory leading local Lyapunov vector

7 Wolfe and Samelson (WS07) Cont d n φ n = j=1 < η j, φ n > η j (1) N φ n = j=n < ξ j, φ n > ξ j (2) Use (1) to find < φ n, ξ k >, for k n, and (2) to find< φ n, η j >, for k n. Use substitution and patience to find the coefficients of (1) and (2). These Lyapunov vectors (LVs) will be invariant under the tangent linear model, i.e. they can be integrated with the TLM for intervals that are not too long.

8 Results: Lorenz (1963) Model This model has only 3 degrees of freedom. x = σ y x y = ρx y x z = xy βz using the standard parameters of σ = 10, β = 8 3 The integration time step is.01., and ρ = 28.

9 Lorenz (1963) BVs, SVs, and LVs BVs were computed using δ 0 =.1 and IW =.02 units. These vectors are essentially equal to the LV1 and are labeled linear BVs. BVs were also computed using δ 0 = 1 and IW =.08 units. These are less linear than the above and are labeled nonlinear. SVs were computed using integration windows of.02 units.

10 Results with Lorenz (1963) Linear BV Growth LV1 Growth Nonlinear BV Growth FSV1 Growth Red stars indicate periods of fastest growth Fast growth typically signals a regime change (Evans et al., 2004; Norwood et al., 2013) The linear BV is most similar to LV1, with an average correlation (cosine between the vectors) of.996. All of the leading vectors can be used to predict regime changes but FSV1 is the best predictor.

11 Fast-Slow Coupled Model with Extratropics (Peña and Kalnay, 2004) Fast Extratropical Atmosphere Fast Tropical Atmosphere Slow Ocean.08 1 The temporal scaling factor for the ocean, τ o =.1 so the ocean is 10x slower than the other subsystems. Ocean Trajectory The extratropics are like weather noise when studying El Niño, which appears chaotically every 2-7 years.

12 Fast-Slow Coupled Model (FSCM; Peña and Kalnay, 2004) x e = τ e σ y e x e c e Sx t + k 1 y e = τ e ρx e τ e y e τ e x e z e + c e Sy t + k 1 z e = τ e x e y e τ e βz e c t z t x t = σ y t x t c SX + k 2 c e Sx e + k 1 y t = ρx t y t x t z t + c SY + k 2 + c e Sy e + k 1 z t = x t y t βz t + c z Z + c t z e X = τ o σ Y X c x t + k 2 Y = τ o ρx τ o Y τ o SXZ + c y t + k 2 Z = τ o SXY τ o βz c z Z Lower case variables are the fast modes (extratropics and tropics), upper case are the slow modes (ocean). c s are coupling coefficients. k s are uncentering parameters τ and S are temporal and scaling factors, respectively.

13 LV Growth Approximately Corresponds to Particular Subsystems LV1 Growth LV4 Growth LV8 Decay FSV1 Growth Fast Extratropics LV5 Growth LV6 Growth LV9 Decay Fast Tropics LV2 Growth LV3 Growth LV7 Decay Slow Mode BV Growth Slow Ocean

14 Fast-Slow Coupled Model with Convection We accelerate the extratropical atmosphere by setting τ e = 10. This makes the weather noise more like convective noise. The coupling remains the same. The convective and tropical subsystems are weakly coupled and the tropical and ocean subsystems are strongly coupled. Fast Convection Fast Tropical Atmosphere Slow Ocean.08 1

15 LV and SV Growth Rates Approximately Correspond to Particular Subsystems LV1 Growth LV2 Growth FSV1 Growth Fast Convection LV3 Growth LV6 Growth FSV7 Decay Fast Tropics LV7 Growth FSV8 Decay FSV9 Decay Slow Mode BV Growth Slow Ocean

16 A Quasi-Geostrophic Model Based upon Rotunno and Bao (1996), the nondimensional form is q t + ψ x q y ψ y q x = 0 where potential vorticity q = ψ xx + ψ yy + ψ z S z z, ψ is the geopotential, and S z is the stratification parameter. The model has 7 levels on a 65 x 33 grid leading to 15,015 degrees of freedom. This model only exhibits baroclinic instability.

17 BVs for the QG Model Five different BVs with an amplitude of 1 and integration window of 24 hours were computed. Two different random initial BV perturbations on top of one anther.

18 BVs for the QG Model Five different BVs with an amplitude of 1 and integration window of 24 hours all converge to the leading LV. Two different random initial BV perturbations on top of one anther. These two BVS collapse into a single vector, i.e. they converge to the leading LV.

19 BVs for the QG Model Five different BVs with an amplitude of 1 and integration window of 24 hours all converge to the leading LV. Two different random initial BV perturbations on top of one anther. Initial dimension ~ 5 These two BVS collapse into a single vector, i.e. they converge to the leading LV.

20 BVs for the QG Model Five different BVs with an amplitude of 1 and integration window of 24 hours all converge to the leading LV. SVs computed using 4, 5, 6, and 7 day windows remain different. Consequently we cannot compute the LVs using the WS07 algorithm. Two different random initial BV perturbations on top of one anther. Initial dimension ~ 5 These two BVS collapse into a single vector, i.e. they converge to the leading LV. Final dimension = 1

21 What happens with SPEEDY? SPEEDY is a full atmospheric model with weather waves, convective instabilities, and even inertia gravity waves (i.e., Lamb waves) excited by tropical convection. There are 7 levels on a 96x48 grid with 6 variables (2 only at the surface) leading to 135,240 degrees of freedom. Five BVs were computed. With an amplitude of 1 m/s and integration window of 24 hours, this BV targets baroclinic instabilities, stronger in the winter hemisphere than in the summer hemisphere. They do NOT converge to a leading LV Two different BVs on top of one another.

22 SPEEDY Cont d We now reduce the amplitude of the wind to 1 cm/s and use a rescaling window of 6 hours to target convective instabilities. The BVs clearly remain distinct, although there are some unstable regions where they align to a local LV.

23 SPEEDY Cont d If we take a very small amplitude (1 mm/s) and a very short rescaling window (40 minutes), we obtain a leading LV corresponding to a global Lamb Wave, probably triggered by convection in the Warm Pool. There appears to be a leading LV for the SPEEDY model, but it probably would be useless for applications.

24 Summary BVs require the least amount of computational effort, time, and memory BVs can target instabilities through proper choice of perturbation size and integration window. BVs, SVs, and LVs can be used as predictors of regime changes in simple models. LVs can distinguish between various modes of growth, but not as cleanly as BVs, being influenced by more than one mode if there is strong coupling between the two. SVs can only differentiate between more than one mode of growth when the modes are very different in terms of frequency. SVs failed to converge for the QG model. Thus we were unable to use WS07 to compute the LVs for more complex models. BVs identified 3 types of instabilities: baroclinic waves, convection, and global Lamb waves. Contrary to what Toth and Kalnay (1997) hypothesized, there is a global leading LV for a weather model! However, the leading LV of complex systems with several types of instabilities will identify the fastest mode (such as global Lamb waves triggered by tropical convection), but this is not useful for weather forecasting applications.

25 Summary BVs require the least amount of computational effort, time, and memory BVs can target instabilities through proper choice of perturbation size and integration window. BVs, SVs, and LVs can be used as predictors of regime changes in simple models. LVs can distinguish between various modes of growth, but not as cleanly as BVs, being influenced by more than one mode if there is strong coupling between the two. SVs can only differentiate between more than one mode of growth when the modes are very different in terms of frequency. SVs failed to converge for the QG model. Thus we were unable to use WS07 to compute the LVs for more complex models. BVs identified 3 types of instabilities: baroclinic waves, convection, and global Lamb waves. Contrary to what Toth and Kalnay (1997) hypothesized, there is a global leading LV for a weather model! However, the leading LV of complex systems with several types of instabilities will identify the fastest mode (such as global Lamb waves triggered by tropical convection), but this is not useful for weather forecasting applications. Thank you!!

26 Thank you!!

27 Summary Cont d SVs failed to converge for the QG model. Thus we were unable to use WS07 to compute the LVs for more complex models. BVs identified 3 types of instabilities: baroclinic waves, convection, and global Lamb waves. The leading LV of complex systems with several types of instabilities will identify the fastest mode, which may not be useful for applications.

28 Summary Cont d SVs failed to converge for the QG model. Thus we were unable to use WS07 to compute the LVs for more complex models. BVs identified 3 types of instabilities: baroclinic waves, convection, and global Lamb waves. The leading LV of complex systems with several types of instabilities will identify the fastest mode, which may not be useful for applications. Thank you!!

29 Definition of Growth Rates Bred vector (BV) growth rates are defined as where dt is the integration time step. 1 IW dt ln( bv δ 0 ), Singular vector (SV) growth rates are defined as ln σ j dt IW. Lyapunov vector (LV) growth rates are defined as 1 φ n d φ n dt

30 Results with Lorenz (1963) All of the leading vectors can be used to predict regime changes.

31 FSCM with Convection Weak coupling Weaker coupling

32 Fast Mode Vectors The BVs can distinguish between the fastest and slowest modes of growth The FSVs are able to distinguish between the various subsystems and modes of growth, unlike with the previous setup The LVs correspond to particular subsystems.

33 Slow Mode Vectors Each vector is able to capture the slowest mode of growth, but the strong coupling between the tropical and ocean subsystems leads to the tropical subsystem influencing the growth of these vectors as well.

34 Mixed LVs and SVs Fast Tropics Slow Ocean