American Institute of Mathematical Sciences

February  2022, 27(2): 769-797. doi: 10.3934/dcdsb.2021064

On the Lorenz '96 model and some generalizations

 Georgetown University, Washington, DC 20057, USA

* Corresponding author: Hans Engler

Received  November 2020 Revised  December 2020 Published  February 2022 Early access  February 2021

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.

Citation: John Kerin, Hans Engler. On the Lorenz '96 model and some generalizations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 769-797. doi: 10.3934/dcdsb.2021064
References:
 [1] R. V. Abramov and A. J. Majda, Quantifying uncertainty for non-Gaussian ensembles in complex systems, SIAM Journal on Scientific Computing, 26 (2004), 411-447.  doi: 10.1137/S1064827503426310. [2] R. V. Abramov and A. J. Majda, Blended response algorithms for linear fluctuation-dissipation for complex nonlinear dynamical systems, Nonlinearity, 20 (2007), 2793-2821.  doi: 10.1088/0951-7715/20/12/004. [3] R. V. Abramov and A. J. Majda, New approximations and tests of linear fluctuation-response for chaotic nonlinear forced-dissipative dynamical systems, Journal of Nonlinear Science, 18 (2008), 303-341.  doi: 10.1007/s00332-007-9011-9. [4] E. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, Springer Series in Computational Mathematics, 13. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-61257-2. [5] R. Blender, J. Wouters and V. Lucarini, Avalanches, breathers, and flow reversal in a continuous Lorenz-96 model, Physical Review E, 88 (2013), 013201, 5pp. doi: 10.1103/PhysRevE.88.013201. [6] J. G. Charney and J. G. DeVore, Multiple flow equilibria in the atmosphere and blocking, Journal of the Atmospheric Sciences, 36 (1979), 1205-1216.  doi: 10.1175/1520-0469(1979)036<1205:MFEITA>2.0.CO;2. [7] P. Chossat and R. Lauterbach, Methods in Equivariant Bifurcations and Dynamical Systems, World Scientific Publishing Co., Singapore, 2000. doi: 10.1142/4062. [8] J. A. Dutton, The nonlinear quasi-geostrophic equation. Part Ⅱ: Predictability, recurrence and limit properties of thermally-forced and unforced flows, Journal of the Atmospheric Sciences, 33 (1976), 1431-1453.  doi: 10.1175/1520-0469(1976)033<1431:TNQGEP>2.0.CO;2. [9] M. R. Frank, L. Mitchell, P. S. Dodds and C. M. Danforth, Standing swells surveyed showing surprisingly stable solutions for the Lorenz'96 model, International Journal of Bifurcation and Chaos, 24 (2014), 1430027, 14pp. doi: 10.1142/S0218127414300274. [10] G. Gallavotti and V. Lucarini, Equivalence of non-equilibrium ensembles and representation of friction in turbulent flows: the Lorenz 96 model, Journal of Statistical Physics, 156 (2014), 1027-1065.  doi: 10.1007/s10955-014-1051-6. [11] S. J. Jacobs, A note on multiple flow equilibria, Pure and Applied Geophysics, 130 (1989), 743-749.  doi: 10.1007/BF00881609. [12] D. L. van Kekem and A. E. Sterk, Wave propagation in the Lorenz-96 model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950008, 18 pp. doi: 10.1142/S0218127419500081. [13] D. L. van Kekem and A. E. Sterk, Travelling waves and their bifurcations in the Lorenz-96 model, Physica D: Nonlinear Phenomena, 367 (2018), 38-60.  doi: 10.1016/j.physd.2017.11.008. [14] D. L. van Kekem and A. E. Sterk, Symmetries in the Lorenz-96 model, International Journal of Bifurcation and Chaos, 29 (2019), 195008, 18pp. doi: 10.1142/S0218127419500081. [15] Y. A. Kuznetsov, Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODE's, SIAM Journal on Numerical Analysis, 36 (1999), 1104-1124.  doi: 10.1137/S0036142998335005. [16] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998. [17] E. N. Lorenz, Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. [18] E. N. Lorenz, Predictability: A problem partly solved,, in Predictability of Weather and Climate (eds. Tim Palmer and Renate Hagedorn), Cambridge University Press, (2006), 40–58. doi: 10.1017/CBO9780511617652.004. [19] E. N. Lorenz and K. A. Emanuel, Optimal sites for supplementary weather observations: Simulation with a small model, Journal of the Atmospheric Sciences, 55 (1998), 399-414.  doi: 10.1175/1520-0469(1998)055<0399:OSFSWO>2.0.CO;2. [20] E. N. Lorenz, Designing chaotic models, Journal of the Atmospheric Sciences, 62 (2005), 1574-1587.  doi: 10.1175/JAS3430.1. [21] V. Lucarini and S. Sarno, A statistical mechanical approach for the computation of the climatic response to general forcings, Nonlinear Processes in Geophysics, 18 (2011), 7-28.  doi: 10.5194/npg-18-7-2011. [22] S. A. Orszag and J. B. McLaughlin, Evidence that random behavior is generic for nonlinear differential equations, Physica D: Nonlinear Phenomena, 1 (1980), 68-79.  doi: 10.1016/0167-2789(80)90005-6. [23] L. R. Petzold and A. C. Hindmarsh, Lsoda, Computing and Mathematics Research Division, Lawrence Livermore National Laboratory, Livermore, CA, 1997. [24] K. Soetaert, T. Petzoldt and R. W. Setzer, Solving differential equations in R: Package deSolve, Journal of Statistical Software, 33 (2010), 1-25.  doi: 10.32614/RJ-2010-013. [25] A. E. Sterk and D. L. van Kekem, Predictability of extreme waves in the Lorenz-96 model near intermittency and quasi-periodicity, Complexity, 2017 (2017), Art. ID 9419024, 14 pp. doi: 10.1155/2017/9419024. [26] D. S. Wilks, Effects of stochastic parametrizations in the Lorenz'96 system, Quarterly Journal of the Royal Meteorological Society: A Journal of the Atmospheric Sciences, Applied Meteorology and Physical Oceanography, 131 (2005), 389-407.  doi: 10.1256/qj.04.03.

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References:
 [1] R. V. Abramov and A. J. Majda, Quantifying uncertainty for non-Gaussian ensembles in complex systems, SIAM Journal on Scientific Computing, 26 (2004), 411-447.  doi: 10.1137/S1064827503426310. [2] R. V. Abramov and A. J. Majda, Blended response algorithms for linear fluctuation-dissipation for complex nonlinear dynamical systems, Nonlinearity, 20 (2007), 2793-2821.  doi: 10.1088/0951-7715/20/12/004. [3] R. V. Abramov and A. J. Majda, New approximations and tests of linear fluctuation-response for chaotic nonlinear forced-dissipative dynamical systems, Journal of Nonlinear Science, 18 (2008), 303-341.  doi: 10.1007/s00332-007-9011-9. [4] E. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, Springer Series in Computational Mathematics, 13. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-61257-2. [5] R. Blender, J. Wouters and V. Lucarini, Avalanches, breathers, and flow reversal in a continuous Lorenz-96 model, Physical Review E, 88 (2013), 013201, 5pp. doi: 10.1103/PhysRevE.88.013201. [6] J. G. Charney and J. G. DeVore, Multiple flow equilibria in the atmosphere and blocking, Journal of the Atmospheric Sciences, 36 (1979), 1205-1216.  doi: 10.1175/1520-0469(1979)036<1205:MFEITA>2.0.CO;2. [7] P. Chossat and R. Lauterbach, Methods in Equivariant Bifurcations and Dynamical Systems, World Scientific Publishing Co., Singapore, 2000. doi: 10.1142/4062. [8] J. A. Dutton, The nonlinear quasi-geostrophic equation. Part Ⅱ: Predictability, recurrence and limit properties of thermally-forced and unforced flows, Journal of the Atmospheric Sciences, 33 (1976), 1431-1453.  doi: 10.1175/1520-0469(1976)033<1431:TNQGEP>2.0.CO;2. [9] M. R. Frank, L. Mitchell, P. S. Dodds and C. M. Danforth, Standing swells surveyed showing surprisingly stable solutions for the Lorenz'96 model, International Journal of Bifurcation and Chaos, 24 (2014), 1430027, 14pp. doi: 10.1142/S0218127414300274. [10] G. Gallavotti and V. Lucarini, Equivalence of non-equilibrium ensembles and representation of friction in turbulent flows: the Lorenz 96 model, Journal of Statistical Physics, 156 (2014), 1027-1065.  doi: 10.1007/s10955-014-1051-6. [11] S. J. Jacobs, A note on multiple flow equilibria, Pure and Applied Geophysics, 130 (1989), 743-749.  doi: 10.1007/BF00881609. [12] D. L. van Kekem and A. E. Sterk, Wave propagation in the Lorenz-96 model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950008, 18 pp. doi: 10.1142/S0218127419500081. [13] D. L. van Kekem and A. E. Sterk, Travelling waves and their bifurcations in the Lorenz-96 model, Physica D: Nonlinear Phenomena, 367 (2018), 38-60.  doi: 10.1016/j.physd.2017.11.008. [14] D. L. van Kekem and A. E. Sterk, Symmetries in the Lorenz-96 model, International Journal of Bifurcation and Chaos, 29 (2019), 195008, 18pp. doi: 10.1142/S0218127419500081. [15] Y. A. Kuznetsov, Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODE's, SIAM Journal on Numerical Analysis, 36 (1999), 1104-1124.  doi: 10.1137/S0036142998335005. [16] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998. [17] E. N. Lorenz, Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. [18] E. N. Lorenz, Predictability: A problem partly solved,, in Predictability of Weather and Climate (eds. Tim Palmer and Renate Hagedorn), Cambridge University Press, (2006), 40–58. doi: 10.1017/CBO9780511617652.004. [19] E. N. Lorenz and K. A. Emanuel, Optimal sites for supplementary weather observations: Simulation with a small model, Journal of the Atmospheric Sciences, 55 (1998), 399-414.  doi: 10.1175/1520-0469(1998)055<0399:OSFSWO>2.0.CO;2. [20] E. N. Lorenz, Designing chaotic models, Journal of the Atmospheric Sciences, 62 (2005), 1574-1587.  doi: 10.1175/JAS3430.1. [21] V. Lucarini and S. Sarno, A statistical mechanical approach for the computation of the climatic response to general forcings, Nonlinear Processes in Geophysics, 18 (2011), 7-28.  doi: 10.5194/npg-18-7-2011. [22] S. A. Orszag and J. B. McLaughlin, Evidence that random behavior is generic for nonlinear differential equations, Physica D: Nonlinear Phenomena, 1 (1980), 68-79.  doi: 10.1016/0167-2789(80)90005-6. [23] L. R. Petzold and A. C. Hindmarsh, Lsoda, Computing and Mathematics Research Division, Lawrence Livermore National Laboratory, Livermore, CA, 1997. [24] K. Soetaert, T. Petzoldt and R. W. Setzer, Solving differential equations in R: Package deSolve, Journal of Statistical Software, 33 (2010), 1-25.  doi: 10.32614/RJ-2010-013. [25] A. E. Sterk and D. L. van Kekem, Predictability of extreme waves in the Lorenz-96 model near intermittency and quasi-periodicity, Complexity, 2017 (2017), Art. ID 9419024, 14 pp. doi: 10.1155/2017/9419024. [26] D. S. Wilks, Effects of stochastic parametrizations in the Lorenz'96 system, Quarterly Journal of the Royal Meteorological Society: A Journal of the Atmospheric Sciences, Applied Meteorology and Physical Oceanography, 131 (2005), 389-407.  doi: 10.1256/qj.04.03.
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 $500 \leq t \leq 510$
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 $500 \leq t \leq 510$
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 $F \approx 8/9$
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 ($G_8$)
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 $F$
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 $(F_1, \, \alpha_0)$
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 $i \ge N/2)$
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
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 $E(0) = 400$
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
 $\mathcal{G}$-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
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 $>2$
 $N$ $F_1$ $m_1$ $F_2$ $m_2$ $F_3^\ast$ $\tilde F_3$ $\tilde F_4$ 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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