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Non‐smoothness arises at cloud edge because, in moist thermodynamics, the thermodynamic properties of the atmosphere are different inside a cloud versus in clear air. In particular, inside a cloud, the vapor pressure of water is constrained by the saturation vapor pressure, which acts as a threshold. Due to this threshold, while the water vapor mixing ratio may vary continuously across cloud edge, its derivatives are not necessarily continuous at cloud edge. Similarly, non‐smoothness also arises for buoyancy and other variables. Consequently, this non‐smoothness in buoyancy and other variables can cause a degraded accuracy in computational simulations. Here we consider special treatment of numerical methods for the interface that arises from phase changes and cloud edges, in order to enhance the accuracy and potentially achieve second‐order accuracy. Numerical solutions are computed for the moist non‐precipitating Boussinesq equations as an idealized cloud‐resolving model with phase changes of water, that is, with cloud formation. Convergence tests, both spatial and temporal, are conducted to measure the numerical error as the grid spacing and time step are refined. While approximately second‐order accuracy is seen in root‐mean‐square (L2) error, the accuracy is degraded in the maximum (Linfinity) error. Discussion is also included on theoretical issues and potential implications for numerical simulations. 

Abstract The so-called traditional approximation, wherein the component of the Coriolis force proportional to the cosine of latitude is ignored, is frequently made in order to simplify the equations of atmospheric circulation. For velocity fields whose vertical component is comparable to their horizontal component (such as convective circulations), and in the tropics where the sine of latitude vanishes, the traditional approximation is not justified. We introduce a framework for studying the effect of diabatic heating on circulations in the presence of both traditional and nontraditional terms in the Coriolis force. The framework is intended to describe steady convective circulations on anfplane in the presence of radiation and momentum damping. We derive a single elliptic equation for the horizontal velocity potential, which is a generalization of the weak temperature gradient (WTG) approximation. The elliptic operator depends on latitude, radiative damping, and momentum damping coefficients. We show how all other dynamical fields can be diagnosed from this velocity potential; the horizontal velocity induced by the Coriolis force has a particularly simple expression in terms of the velocity potential. Limiting examples occur at the equator, where only the nontraditional terms are present, at the poles, where only the traditional terms appear, and in the absence of radiative damping where the WTG approximation is recovered. We discuss how the framework will be used to construct dynamical, nonlinear convective models, in order to diagnose their consequent upscale momentum and temperature fluxes. 

Abstract To define a conserved energy for an atmosphere with phase changes of water (such as vapor and liquid), motivation in the past has come from generalizations of dry energies—in particular, from gravitational potential energy ρgz. Here a new definition of moist energy is introduced, and it generalizes another form of dry potential energy, proportional to θ2, which is valuable since it is manifestly quadratic and positive definite. The moist potential energy here is piecewise quadratic and can be decomposed into three parts, proportional to bu2Hu, bs2Hs, and M2Hu, which represent, respectively, buoyant energies and a moist latent energy that is released upon a change of phase. The Heaviside functions Hu and Hs indicate the unsaturated and saturated phases, respectively. The M2 energy is also associated with an additional eigenmode that arises for a moist atmosphere but not a dry atmosphere. Both the Boussinesq and anelastic equations are examined, and similar energy decompositions are shown in both cases, although the anelastic energy is not quadratic. Extensions that include cloud microphysics are also discussed, such as the Kessler warm-rain scheme. As an application, empirical orthogonal function (EOF) analysis is considered, using a piecewise quadratic moist energy as a weighted energy in contrast to the standard L2 energy. By incorporating information about phase changes into the energy, the leading EOF modes become fundamentally different and capture the variability of the cloud layer rather than the dry subcloud layer.