On the Lorenz '96 model and some generalizations

John Kerin; Hans Engler

doi:10.3934/dcdsb.2021064

Article Contents

2022, Volume 27, Issue 2: 769-797. Doi: 10.3934/dcdsb.2021064

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On the Lorenz '96 model and some generalizations

John Kerin and
Hans Engler^,

Georgetown University, Washington, DC 20057, USA

^* Corresponding author: Hans Engler
^* Corresponding author: Hans Engler

Received: October 31, 2020

Revised: November 30, 2020

Early access: February 2021

Published: February 2022

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Abstract

In 1996, Edward Lorenz introduced a system of ordinary differential equations that describes a scalar quantity evolving on a circular array of sites, undergoing forcing, dissipation, and rotation invariant advection. Lorenz constructed the system as a test problem for numerical weather prediction. Since then, the system has also found use as a test case in data assimilation. Mathematically, this is a dynamical system with a single bifurcation parameter (rescaled forcing) that undergoes multiple bifurcations and exhibits chaotic behavior for large forcing. In this paper, the main characteristics of the advection term in the model are identified and used to describe and classify possible generalizations of the system. A graphical method to study the bifurcation behavior of constant solutions is introduced, and it is shown how to use the rotation invariance to compute normal forms of the system analytically. Problems with site-dependent forcing, dissipation, or advection are considered and basic existence and stability results are proved for these extensions. We address some related topics in the appendices, wherein the Lorenz '96 system in Fourier space is considered, explicit solutions for some advection-only systems are found, and it is demonstrated how to use advection-only systems to assess numerical schemes.

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Mathematics Subject Classification: Primary: 34C23, 86A10; Secondary: 37C81, 37G05.

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Figure 1. A solution of the L96 system with $ F = 2 $ starting from random initial data. (a) All 36 sites at $ t = 500 $. (b) First site for $ 500 \leq t \leq 510 $. (c) Hovmoeller plot for $ 500 \leq t \leq 510 $

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Figure 2. A solution of the L96 system with $ F = 8 $ starting from random initial data. (a) All 36 sites at $ t = 500 $. (b) First site for $ 500 \leq t \leq 510 $. (c) Hovmoeller plot for $ 500 \leq t \leq 510 $

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Figure 3. Left: Eigenvalue curve of the linearization of the L96 system about the constant vector $ \mathbb{e} $ and eigenvalues on this curve for $ N = 36 $ sites. Right: Eigenvalue curves of $ FA - I $ for the L96 system. The curve in the left panel is stretched by $ F $ and shifted to the left by 1 unit. Black: $ F = -0.8 $. For even $ N $, a pitchfork bifurcation has occurred for $ F = -1/2 $. For odd $ N $, a Hopf bifurcation has occurred for some $ F < -1/2 $. Red: $ F = 0.5 $. The constant solution is stable. Blue: $ F = 1.1 $. A Hopf bifurcation has occurred for $ F \approx 8/9 $

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Figure 4. Images of the complex unit circle under the Laurent polynomials given in Table 1. (a): Ellipse ($ G_1, \, G_2, \, G_4 $). (b): Trefoil ($ G_3 $, L96 system). (c): Butterfly ($ G_5 $). (d): Kidney ($ G_6 $). (e): Vertical line ($ G_7 $). (f): Bee ($ G_8 $)

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Figure 5. Eigenvalue curves of advection terms in $ \mathcal{G}_2 $ for various values of $ F $. (a) $ G_1 $, no bifurcation occurs for positive $ F $. (b) $ G_5 $, a supercritical Hopf bifurcation occurs for positive $ F $. (c) $ -G_1 + \frac{1}{2} G_5 $, a supercritical Hopf bifurcation occurs for positive $ F $

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Figure 6. Left: Eigenvalues of $ F_1A - I $ (black and red) and of $ F_1A - I + \alpha_0 C_\ell $ (black and green), for $ N = 14 $. Right: Bifurcation diagram of system (38) in the $ (F,\alpha) $ plane. Blue line: Hopf bifurcation ($ \Re \, \lambda_k = 0 $). Blue stipples: A stable limit cycle exists. Green line: Hopf bifurcation ($ \Re \, \tilde \lambda_\ell = 0 $). Green stipples: A second periodic orbit exists (unstable). Red curves: Neimark-Sacker (N-S) bifurcation. Red stipples: Two stable limit cycles coexist. Magenta line: Linear approximation of N-S bifurcation curve. Red triangle: Hopf-Hopf bifurcation at $ (F_1, \, \alpha_0) $

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Figure 7. Stationary solutions of Eq. (41) with $ G = G_L, \, C = B = I $ and inhomogeneous forcing $ {\bf{F}} $ for $ N = 120 $ sites. Here $ F_i = 1 $ for $ 0 \le i < N/2 $ and $ F_i = M $ for $ i \ge N/2) $

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Figure 8. Hovmoeller plots showing inhomogeneous advection and dissipation as described in Eq. (46), for $ N = 100 $. Both panels use the same parameters $ (1,1,2) $ in the left half, but solutions have very different behavior. Left: Sites in the right half have parameters $ (0.5,1,1) $. Smaller advection in the right half leads to smaller spatial amplitudes. Perturbations are seen to travel to the right. Right: Sites in the right half have parameters $ (1,1.5,1) $. Larger dissipation in the right half leads to nearly constant solutions over a substantial range of sites

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Figure 9. Relative energy loss $ \Delta E(t)/E(0) $ for RK4 and scaled relative energy loss $ 10^3 \times \Delta E(t)/E(0) $ for lsoda, for $ N = 36 $, $ \Delta t = 0.05 $, and $ E(0) = 400 $

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Table 1. Description of the eigenvalue curves of the eight simplest 3-localized $ \mathcal{G} $-maps identified in Section 2.2. The two rightmost columns give the types of the first expected bifurcation for $ F>0 $ and $ F<0 $ as the magnitude of $ F $ increases. Asterisks indicate exceptions for certain site numbers

$ \mathcal{G} $-map	Laurent polynomial $ p_A(z) $	Shape of $ p_A(\mathbb{S}^1) $	$ F>0 $	$ F<0 $
$ G_1 $	$ - z^{-1} - 1 + 2z $	ellipse	none	pitchfork/Hopf
$ G_2 $	$ -z^{-2} - 1 + 2z^2 $	ellipse	none	pitchfork/Hopf $ ^\ast $
$ G_3 $	$ - z^{-2} + z $	trefoil	Hopf	pitchfork/Hopf
$ G_4 $	$ - z^{-3} -1 + 2 z^3 $	ellipse	none	pitchfork/Hopf $ ^{\ast} $
$ G_5 $	$ -z^{-2} - z + z^2 + z^3 $	butterfly	Hopf	Hopf
$ G_6 $	$ -z^{-1} + z - z^2 + z^3 $	kidney	Hopf	pitchfork/Hopf
$ G_7 $	$ -z^{-2} - z^{-1} + z + z^2 $	vertical line	none	none
$ G_8 $	$ -z^{-3} - z^{-1} + z^2 + z^3 $	bee	pitchfork/Hopf	Hopf

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Table 2. Multiple stable limit cycles are expected if approximately $ F \ge F_3^\ast $ and are found numerically for $ \tilde F_3 \le F \le \tilde F_4 $. Limit cycles may be characterized by their spatial periods $ m_1, \, m_2 $

$ N $	$ F_1 $	$ m_1 $	$ F_2 $	$ m_2 $	$ F_3^\ast $	$ \tilde F_3 $	$ \tilde F_4 $
12	1	4	1	6	1	1	$>2 $
14	.8901	7	1.1820	14	1.5206	not observed	not observed
18	.8982	9	1	6	1.1892	not observed	not observed
22	.9076	22	.9343	11	.9915	.996	$>4 $
28	.8901	14	.9457	28	1.0293	1.072	$>3 $
36	.8982	9	.9025	36	.9094	.904	$>2 $

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