  The classic Lorenz equations were originally derived from the two-dimensional
Rayleigh-B\'enard convection system considering an idealised case with the
lowest order of harmonics. Although the low-order Lorenz equations have
traditionally served as a minimal model for chaotic and intermittent
atmospheric motions, even the dynamics of the two-dimensional Rayleigh-B\'enard
convection system is not fully represented by the Lorenz equations, and such
differences have yet to be clearly identified in a systematic manner. In this
paper, the convection problem is revisited through an investigation of various
dynamical behaviors exhibited by a two-dimensional direct numerical simulation
(DNS) and the generalized expansion of the Lorenz equations (GELE) derived by
considering additional higher-order harmonics in the spectral expansions of
periodic solutions. Notably, the GELE allows us to understand how nonlinear
interactions among high-order modes alter the dynamical features of the Lorenz
equations including fixed points, chaotic attractors, and periodic solutions.
It is verified that numerical solutions of the DNS can be recovered from the
solutions of GELE when we consider the system with sufficiently high-order
harmonics. At the lowest order, the classic Lorenz equations are recovered from
GELE. Unlike in the Lorenz equations, we observe limit tori, which are the
multi-dimensional analogue of limit cycles, in the solutions of the DNS and
GELE at high orders. Initial condition dependency in the DNS and Lorenz
equations is also discussed.
