Boussinesq Navier Stokes to A Modification on the Lorenz-63 Model

We derive a five–dimensional Modified Lorenz System (MLS) from the incompressible Boussinesq Navier–Stokes equations with coupled potential temperature and moisture fields via systematic Galerkin projection onto a small set of physically motivated spatial modes. Beginning from
∂_t u+(u⋅∇)u=-∇p+ν∇^2 u+gα Θ z ̂, ∂_t Θ+(u⋅∇)Θ=κ_Θ ∇^2 Θ+S_Θ,
and a passive moisture equation, we nondimensionalize and project onto a divergence–free modal basis, leading to
q ̇_m=∑_n▒‍ L_mn q_n+∑_(i,j)▒‍ C_mij q_i q_j+F_m,
with closure hypotheses for unresolved modes. A judicious choice of five dominant amplitudes (T,H,P,W,R)^⊤ yields the MLS, which generalizes the classical Lorenz–63 model by incorporating moisture coupling and bounded nonlinear closures.
We provide a detailed analytical characterization of the MLS: (i) fixed points are identified and classified, (ii) linear stability is determined via the Jacobian eigenvalue spectrum, and (iii) Lyapunov exponents are defined through the tangent–linear system ξ ̇=J(X(t)) ξ, with the largest exponent λ_1 setting a predictability time scale T_L∼1/λ_1. Using energy estimates, we prove the existence of an absorbing set under admissible damping and closure conditions, ensuring global boundedness of solutions. The MLS thus serves as a mathematically well–posed reduced–order model capturing essential nonlinear dynamics of moist convective systems and providing a reproducible analytical testbed for future studies in predictability, ensemble design, and reduced modeling of complex atmospheric processes.

Modified Lorenz System, Galerkin Reduction, Lyapunov Exponents

1. Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20(2), 130–141. [Google Scholar] [Crossref]

2. Holmes, P., Lumley, J. L., & Berkooz, G. (1996). Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press. [Google Scholar] [Crossref]

3. Emanuel, K. A. (1994). Atmospheric convection. Oxford University Press. [Google Scholar] [Crossref]

4. Star, K., Stabile, G., Georgaka, S., Belloni, F., Rozza, G., & Degroote, J. (2019). POD-Galerkin reduced order model of the Boussinesq approximation for buoyancy-driven enclosed flows. In Proceedings of the International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2019) (pp. 2452–2461). [Google Scholar] [Crossref]

5. Rahman, S. M., Ahmed, S. E., & San, O. (2019). A dynamic closure modeling framework for model order reduction of geophysical flows. Physics of Fluids, 31(5), 051702. [Google Scholar] [Crossref]

6. Ahmed, S. E., San, O., Rasheed, A., & Pawar, S. (2021). On closures for reduced order models: a spectrum of first‑principle to machine‑learned avenues. Physics of Fluids, 33(9), 091301. :contentReference[oaicite:0]index=0 [Google Scholar] [Crossref]

7. Kim, H., & Kramer, B. (2025). Physically consistent predictive reduced‑order modeling by enhancing operator inference with state constraints. Journal of Computational Physics. :contentReference[oaicite:1]index=1 [Google Scholar] [Crossref]

8. Cavalieri, A. V. G., Maia, I., & Hwang, Y. (2025). State‑dependent convergence of Galerkin‑based reduced‑order models for Couette flow. Preprint/Research Article. [Google Scholar] [Crossref]

• Applied Partial Differential Equation Modeling of Grating-Based Devices for Enhanced Sensing and Monitoring in the Petroleum and Gas Industry
• Analysis of Critical Path Method (CPM) To Optimized Project Scheduling