bred, singular, and lyapunov vectors in simple and ......bred vectors, singular vectors, and...

Bred Vectors, Singular Vectors, and Lyapunov Vectors in

Simple and Complex Models

Adrienne NorwoodAdvisor: Eugenia Kalnay

With special thanks to Drs. Kayo Ide, Brian Hunt, Shu-ChihYang, and Christopher Wolfe for their guidance and

suggestions.

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

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).

Computing Bred Vectors

• Integrate nonlinear model, M, to obtain control trajectory, 𝑥𝑐.

• Choose rescaling amplitude, 𝛿0, and integration window, IW, to target the desired mode of growth (Peña and Kalnay, 2004).

• Add a small perturbation

𝑥𝑝 𝑡𝑖 = 𝑥𝑐 𝑡𝑖 + δ0𝑝

| 𝑝 |,

where 𝑝

| 𝑝 |is the initial direction of the perturbation and δ0 is the amplitude.

• Integrate 𝑥𝑝 𝑡𝑖 forward using the nonlinear model.

• The bred vector is 𝑏𝑣 𝑡𝑖+𝐼𝑊 = 𝑥𝑝 𝑡𝑖+𝐼𝑊 − 𝑥𝑐 𝑡𝑖+𝐼𝑊 . Rescale to δ0, and repeat the process.

time

Initial random

perturbation

Bred Vectors

~LLVs

Control forcast

Forecast values

Computing Singular Vectors

• For nonlinear model, M, calculate the tangent linear model 𝐌𝐢𝐣 𝑡𝑖 , 𝑡𝑖+𝐼𝑊 =

𝜕𝑀𝑖

𝜕𝑥𝑗and the adjoint 𝐌𝐢𝐣 𝑡𝑖 , 𝑡𝑖+𝐼𝑊

∗.

• Initial singular vectors, 𝜉𝑗, are the columns of Ξ where𝐌∗𝐌𝚵 = 𝚵𝐒

where 𝐒 is diagonal with entries 𝜎𝑗2, the squares of the singular values

• The final singular vectors, 𝜂𝑗, are obtained by integrating the TLM forward starting from the initial singular vector.

𝐌 𝑡𝑖 , 𝑡𝑖+𝐼𝑊 𝜉𝑗 𝑡𝑖 = 𝜎𝑗𝜂𝑗 𝑡𝑖+𝐼𝑊𝐌

𝜉2

𝜉1

𝜎1𝜂1

𝜎2𝜂2

Fig. 6.7: Schematic of how all perturbations will converge towards the leading Local Lyapunov Vector

trajectory

random initial

perturbations

leading local

Lyapunov vector

Computing Lyapunov Vectors (Wolfe and Samelson, 2007)

• 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 𝜉𝑗and 𝜂𝑗, respectively.

• Every perturbation integrated forward in time aligns with the leading LV. Thus 𝜙1 =< 𝜙1, 𝜂𝑗 > 𝜂1.

• SVs are orthogonal so LV2 has a component in the direction of 𝜂2. Thus

𝜙2 =< 𝜙2, 𝜂1 > 𝜂1+< 𝜙2, 𝜂2 > 𝜂2, and so on.

• Every perturbation integrated backward in time aligns with the fastest decaying LV. Thus 𝜙𝑁 = < 𝜙𝑁, 𝜉𝑁 > 𝜉𝑁, where 𝑁 is the degrees of freedom of the system.

• Likewise

𝜙𝑁−1 =< 𝜙𝑁−1, 𝜉𝑁−1 > 𝜉𝑁−1+< 𝜙𝑁−1, 𝜉𝑁 > 𝜉𝑁,

and so on

Wolfe and Samelson (WS07) Cont’d

𝜙𝑛 = 𝑗=1𝑛 < 𝜂𝑗 , 𝜙𝑛 > 𝜂𝑗 (1)

𝜙𝑛 = 𝑗=𝑛𝑁 < 𝜉𝑗 , 𝜙𝑛 > 𝜉𝑗 (2)

• Use (1) to find < 𝜙𝑛, 𝜉𝑘 >, for 𝑘 ≥ 𝑛, and (2) to find< 𝜙𝑛, 𝜂𝑗 >, 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.

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, and 𝜌 = 28.

• The integration time step is .01.

Lorenz (1963) BVs, SVs, and LVs

• BVs were computed using 𝛿0 = .1 and 𝐼𝑊 = .02 units. These vectors are essentially equal to the LV1 and are labeled linear BVs.

• BVs were also computed using 𝛿0 = 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.

Results with Lorenz (1963)

• 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.

Linear BV Growth LV1 Growth Nonlinear BV Growth FSV1 Growth

Fast-Slow Coupled Model with “Extratropics” (Peña and Kalnay, 2004)

• The temporal scaling factor for the ocean, 𝜏𝑜 = .1 so the “ocean” is 10x slower than the other subsystems.

• The “extratropics” are like “weather noise”

when studying El Niño, which appears chaotically

every 2-7 years.

Fast “Extratropical Atmosphere”Slow “Ocean”Fast “Tropical Atmosphere”

1.08

“Ocean” Trajectory

Fast-Slow Coupled Model (FSCM; Peña and Kalnay, 2004) 𝑥𝑒 = 𝜏𝑒𝜎 𝑦𝑒 − 𝑥𝑒 − 𝑐𝑒 𝑆𝑥𝑡 + 𝑘1 𝑦𝑒 = 𝜏𝑒𝜌𝑥𝑒 − 𝜏𝑒𝑦𝑒 − 𝜏𝑒𝑥𝑒𝑧𝑒 + 𝑐𝑒 𝑆𝑦𝑡 + 𝑘1 𝑧𝑒 = 𝜏𝑒𝑥𝑒𝑦𝑒 − 𝜏𝑒𝛽𝑧𝑒 − 𝑐𝑡𝑧𝑡

𝑥𝑡 = 𝜎 𝑦𝑡 − 𝑥𝑡 − 𝑐 𝑆𝑋 + 𝑘2 − 𝑐𝑒 𝑆𝑥𝑒 + 𝑘1 𝑦𝑡 = 𝜌𝑥𝑡 − 𝑦𝑡 − 𝑥𝑡𝑧𝑡 + 𝑐 𝑆𝑌 + 𝑘2 + 𝑐𝑒 𝑆𝑦𝑒 + 𝑘1 𝑧𝑡 = 𝑥𝑡𝑦𝑡 − 𝛽𝑧𝑡 + 𝑐𝑧𝑍 + 𝑐𝑡𝑧𝑒

𝑋 = 𝜏𝑜𝜎 𝑌 − 𝑋 − 𝑐 𝑥𝑡 + 𝑘2 𝑌 = 𝜏𝑜𝜌𝑋 − 𝜏𝑜𝑌 − 𝜏𝑜𝑆𝑋𝑍 + 𝑐 𝑦𝑡 + 𝑘2 𝑍 = 𝜏𝑜𝑆𝑋𝑌 − 𝜏𝑜𝛽𝑍 − 𝑐𝑧𝑍

• 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.

LV Growth Approximately Corresponds to Particular Subsystems

Fast Extratropics

Fast Tropics

Slow Ocean

LV1 Growth

LV9 DecayLV6 GrowthLV5 Growth

FSV1 GrowthLV8 DecayLV4 Growth

Slow Mode BV GrowthLV7 DecayLV3 GrowthLV2 Growth

Fast-Slow Coupled Model with Convection

• We accelerate the “extratropical atmosphere” by setting 𝜏𝑒 = 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” Slow “Ocean”Fast “Tropical Atmosphere”

.08 1

LV and SV Growth Rates Approximately Correspond to Particular Subsystems

Fast Convection

Fast Tropics

Slow Ocean

LV1 Growth FSV1 Growth

Slow Mode BV GrowthFSV9 DecayFSV8 DecayLV7 Growth

LV6 GrowthLV3 Growth FSV7 Decay

LV2 Growth

A Quasi-Geostrophic Model

• Based upon Rotunno and Bao (1996), the nondimensional form is

𝑞𝑡 + 𝜓𝑥𝑞𝑦 − 𝜓𝑦𝑞𝑥 = 0

where potential vorticity 𝑞 = 𝜓𝑥𝑥 + 𝜓𝑦𝑦 +𝜓𝑧

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.

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.

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.

These two BVS collapse into a single vector, i.e. they converge to the leading LV.

Two different random initial BV perturbations on top of one anther.

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.

These two BVS collapse into a single vector, i.e. they converge to the leading LV.

Initial dimension ~ 5

Two different random initial BV perturbations on top of one anther.

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.

These two BVS collapse into a single vector, i.e. they converge to the leading LV.

Final dimension = 1

Initial dimension ~ 5

Two different random initial BV perturbations on top of one anther.

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 baroclinicinstabilities, 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.

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.

• 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.

SPEEDY Cont’d

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.

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!!

Thank you!!

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.

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!!

Definition of Growth Rates

• Bred vector (BV) growth rates are defined as 1

𝐼𝑊 ∗𝑑𝑡ln(

𝑏𝑣

𝛿0),

where 𝑑𝑡 is the integration time step.

• Singular vector (SV) growth rates are defined as ln 𝜎𝑗

𝑑𝑡 ∗𝐼𝑊.

• Lyapunov vector (LV) growth rates are defined as 1

𝜙𝑛

𝑑 𝜙𝑛

𝑑𝑡

Results with Lorenz (1963)

All of the leading vectors can be used to predict regime changes.

FSCM with Convection

• Weak coupling

• Weaker coupling

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.

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.

“Mixed” LVs and SVs

Slow Ocean

Fast Tropics