On an extension of the nonlinear feedback loop in a nine-dimensional Lorenz model Bo-Wen Shen
Department of Mathematics and Statistics San Diego State University
[email protected]
Nonlinear Feedback Loop in a 9D Lorenz Model
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CHAOS2016, University of London, UK, May 23-26, 2016
Outline 1. Introduction (3DLM, butterfly effect, nonlinear feedback loop) • The Three-Dimensional Lorenz Model (3DLM, Lorenz, 1963) -- the nonlinear feedback loop
• A Five-Dimensional Lorenz Model (5DLM; Shen 2014) -- an extension of nonlinear feedback loop -- analytical solutions of its critical points -- negative nonlinear feedback (stabilization)
• A Six-Dimensional Lorenz Model (6DLM; Shen 2015b) -- impact of additional heating term (destabilization)
2. Results • A Seven-Dimensional Lorenz Model (7DLM, Shen 2016a) -- hierarchical scale dependence
• A Nine-Dimensional Lorenz Model (9DLM V1, this study) -- a superset
3. Summary
rc: a critical value of Raleigh parameter for the onset of chaos
Nonlinear Feedback Loop in a 9D Lorenz Model
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CHAOS2016, University of London, UK, May 23-26, 2016
Is the Atmosphere More Predictable than We Assume?
6 May 2011
Nonlinear Feedback Loop in a 9D Lorenz Model
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Is TC Formation More Predictable Than We Assume? 1. Are the simulations of TC genesis (in-)consistent with Chaos theory? 2. Why can the high-resolution global model have skills?
Courtesy of Anthes (2011) and Wikipdeia Shen, B.-W., 2014a: Nonlinear Feedback in a Five-dimensional Lorenz Model. J. of Atmos. Sci., 71, 1701–1723. doi: http://dx.doi.org/10.1175/JAS-D-13-0223.1 • •
The butterfly effect of first kind: it means the sensitive dependence on initial conditions (Lorenz, 1963) The butterfly effect of second kind: it becomes a metaphor (or symbol ) for indicating that small perturbations can alter large-scale structure (Lorenz, 1972)
Nonlinear Feedback Loop in a 9D Lorenz Model
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Review of High-Dimensional LMs •
Curry (1978) reported that a larger r (i.e., Rayleigh number) is required for the onset of chaos in generalized LMs (e.g., a rc of approximately 43.5 in the generalized LM with 14 modes)
•
Curry et al. (1984) observed an irregular change in the degree of chaos as the resolution increased from a low resolution (i.e., three Fourier modes) and obtained a steady state solution with sufficiently high resolution.
•
Roy and Musielak (2007a) emphasized the importance in selecting modes that can conserve the system’s energy in the dissipationless limit.
•
Roy and Musielak (2007a,b,c) reported that some generalized LMs required a lager r (rc ∼ 40) for the onset of chaos, but others displayed a comparable (e.g., rc ∼ 24.74 in one of their LMs with 6 modes) or a smaller rc (e.g., rc ∼ 22 in their LM with 5 modes).
•
The aforementioned studies give an inclusive answer to the question of whether higher-dimensional LMs are more stable (predictable). è Under which conditions high-resolution LMs are more stable?
Nonlinear Feedback Loop in a 9D Lorenz Model
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The Nonlinear Feedback Loop in the
3-Dimenaional Lorenz Model (3DLM)
and
Its Extension in the
High-Dimensional Lorenz Models
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Nonlinearity and Forcing Terms Rayleigh-Benard Convection
By assuming 2D (x,z), incompressible and Boussinesq flow, the following equations were used in Lorenz (1963) This does not appear explicitly in the Lorenz model.
Non-linear terms
Boundary Forcing, which is represented by ‘r’.
• Navier-Stokes equation with constant viscosity • Heat transfer equation with constant thermal conductivity Nonlinear Feedback Loop in a 9D Lorenz Model
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3-Dimenaional Lorenz Model (3DLM) 1) r – Rayleigh number: (Ra/Rc) a dimensionless measure of temperature difference between top and bottom surfaces of liquid; proportional to effective force on fluid
M1 M2
2) σ – Prandtl number: (ν/κ) the ratio of the kinetic viscosity (κ, momentum diffusivity) to thermal diffusivity (ν)
M3
3) b – Physical proportion: (4/(1+a2)), b=8/3. 4) a – a=l/m, the ratio of the vertical height h of the fluid layer to the horizontal size of the convection rolls. b =8/3. • l=aπ/H and m=π/H,
Nonlinear Feedback Loop in a 9D Lorenz Model
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-XZ is associated with the J(M1, M3), indicating the impact of the M3 mode. With no –XZ, the above system is reduced to become a system with linear terms only, leading to an unstable solution as r>1. CHAOS2016, University of London, UK, May 23-26, 2016
Solutions by Three Fourier Modes These three modes were used in the original Lorenz model, referred to as a 3DLM.
Three additionalmodes modesare areincluded includedininthe thegeneralized generalizedLM, LM,referred referredtotoasasa a5DLM. 6DLM. Two additional Streamfunction
l=aπ/H and m=π/H, a=l/m, the ratio of vertical scale to its horizontal scale
Temperature
(X, Y, Z) represent the amplitudes of the three modes in the 3DLM. (X, Y, Z, Y X11, Z Y11), represent Z1) represent amplitudes of the modes in the 6DLM. the the amplitudes of the five six modes in the 5DLM. Nonlinear Feedback Loop in a 9D Lorenz Model
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The Nonlinear Feedback Loop in the 3DLM In the original 3D 5D Lorenz model, the Jacobian has only two terms:
≈
M2 M3
A loop appears as M2à M3 à (-M2) • • •
In the original 3D Lorenz model, a feedback loop forms with J(M1,M2) and J(M1,M3). In the 5DLM, the inclusion of M5 extends the feedback loop. Note that the feedback loop also indicates that any error growth of smallscale perturbations will be remained within the loop. Namely, they can not upscale further to alter large-scale (or environmental) flow.
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A Nonlinear Feedback Loop Downscaling
Upscaling
A loop appears as M2à M3 à (-M2) Nonlinear Feedback Loop in a 9D Lorenz Model
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CHAOS2016, University of London, UK, May 23-26, 2016
Five-Dimensional Lorenz Model (5DLM):
An Extension of the Nonlinear Feedback Loop
and
Negative Nonlinear Feedback Shen, B.-W., 2014a: Nonlinear Feedback in a Five-dimensional Lorenz Model. J. of Atmos. Sci., 71, 1701–1723. doi: http://dx.doi.org/10.1175/JAS-D-13-0223.1
Nonlinear Feedback Loop in a 9D Lorenz Model
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An Extension of the Nonlinear Feedback Loop 5DLM
neglected in the 3DLM
M3à M5 à M3
3DLM M3à M2 à M3
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Five-Dimensional Lorenz Model (5DLM) τ = κ (1 + a 2 )(π / H ) 2 t (dimensionless time) 1) r – Rayleigh number: (Ra/Rc) a dimensionless measure of temperature difference between top and bottom surfaces of liquid; proportional to effective force on fluid 2) σ – Prandtl number: (ν/κ) the ratio of the kinetic viscosity (κ, momentum diffusivity) to thermal diffusivity (ν) 3) b – Physical proportion: (4/(1+a2)) 4) d – (9+a2) /(1+a2) 5) a – the ratio of the vertical height h of the fluid layer to the horizontal size of the convection rolls. It turns out that for b =8/3, the convection begins for the smallest value of the Rayleigh number, that is, for the smallest value of the temperature difference. Nonlinear Feedback Loop in a 9D Lorenz Model
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Major negative feedback term
CHAOS2016, University of London, UK, May 23-26, 2016
Numerical Results with the 5DLM
Shen, B.-W., 2014a: Nonlinear Feedback in a Five-dimensional Lorenz Model. J. of Atmos. Sci., 71, 1701–1723. doi: http://dx.doi.org/10.1175/JAS-D-13-0223.1
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Phase Space Trajectories
in the 3DLM and 5DLM
5DLM with r=25
3DLM with r=25 strange attractors
stable critical points
5DLM with r=43.5 strange attractors
Shen, B.-W., 2014a: Nonlinear Feedback in a Five-dimensional Lorenz Model. J. of Atmos. Sci., 71, 1701–1723. doi: http://dx.doi.org/10.1175/JAS-D-13-0223.1 Nonlinear Feedback Loop in a 9D Lorenz Model
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CHAOS2016, University of London, UK, May 23-26, 2016
Stability Analysis 1. ensemble Lyapunov Exponent (eLE) 2. Linear Eigenvalue Analysis
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eLE
Ensemble LEs (eLEs) in the 5DLM
rc: 24.74 rc: 42.9
5DLM
3DLM
eLE >0 eLE 0 eLE 0 eLE
