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

Abstract

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.