  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.
