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Numerical simulations of nonhydrostatic atmospheric flow, based on linearly decoupled semi-implicit or fully-implicit techniques, usually solve linear systems by a pre-conditioned Krylov method without preserving the skew-symmetry of convective operators. We propose to perform atmospheric simulations in such a fully-implicit manner that the difference operators preserve both the skew-symmetry and the tightly nonlinear coupling of the differential operators. We demonstrate that a symmetry-preserving Jacobian-free Newton-Krylov~(JFNK) method mimics a balance between convective transport and turbulence dissipation. We present a wavelet method as an effective symmetry preserving discretization technique. The symmetry-preserving JFNK method for solving equations of nonhydrostatic atmospheric flows has been examined using two benchmark simulations of penetrative convection -- a) dry thermals rising in a neutrally stratified and stably stratified environment, and b) urban heat island circulations for effects of the surface heat flux $H_0$ varying in the range of $25 \le H_0 \le 930$Wm$^{-2}$.The results show that an eddy viscosity model provides the necessary dissipation of the subgrid-scale modes, while the symmetry-preserving JFNK method provides the conservation of mass and energy at a satisfactory level. Comparisons of the results from a laboratory experiment of heat island circulation and a field measurement of potential temperature also suggest the modelling accuracy of the present symmetry-preserving JFNK framework.