## 1. Introduction

In this paper, we calculate exact, explicit analytical solutions to a mathematical model of radiative-convective equilibrium. We consider an analytical single-column model of a moist atmosphere of fixed height, which has no horizontal degrees of freedom. The solutions calculated represent a new type of partial differential equation (PDE) solution, in which ascending air jumps discontinuously in space and instantaneously in time, penetrating descending air parcels such that there is no net mass flux in the vertical. We use such analytical equilibrium solutions to test a numerical, Lagrangian-based, moist convective adjustment step, which acts as a convective parametrization. Our work suggests that it is fruitful to interpret column models in a Lagrangian sense.

In order to calculate analytical solutions, we consider a column atmosphere in which the number of physical processes is restricted. First, the role of clouds, and mixing and entrainment effects, are not included. Second, we assume simplified planetary-boundary-layer physics, in which the column atmosphere is subject to an infinite reservoir of thermal energy and moisture at sea level. We also make the Boussinesq approximation for simplicity. Given these simplifications, we focus on the latent heat effect of moisture in the atmosphere, where the condensation of water vapour to liquid water occurs according to Lorenz (1967). We allow for the phase change between water vapour and liquid water only; the ice phase of water is neglected.

In order to solve equations for the conservation of mass, thermal energy, and specific humidity, we adopt a Lagrangian view-point, and assume the column atmosphere to be a continuum of air parcels (Bokhove and Lynch, 2007); use a simple analytical formula for the saturation specific humidity, which provides a good approximation to that for Earth's atmosphere; assume a constant factor for converting the moisture released by condensation into the latent heat absorbed by the parcel during a pseudo-adiabatic process.

When the column atmosphere is in a dynamic, deep-convecting equilibrium, heat and moisture sources at sea level are balanced by the transport and release of heat and moisture in the body of the atmosphere, and by a (prescribed) net radiative cooling rate. In this equilibrium, air cools and descends at a finite rate until it reaches sea level. At sea level, the air becomes statically unstable due to an impulse of thermal heating at the boundary, and jumps to the top of the atmosphere in zero time, whereupon the air cools and descends, and the cycle repeats. Since such a solution is discontinuous, it is useful to reparametrize in terms of the vertical co-ordinate *z*, and instead to calculate a thermodynamic *solution path* followed by air during the course of a cycle. The solution path defines a vertical profile for potential temperature and specific humidity, and depends on whether or not air is ascending or descending. The actual path followed by the air is always vertical. The ascending branch of the solution path defines a pseudo-adiabat, while the descending branch defines the ‘environment’ for the column. Ascending air rises instantaneously along paths in the column atmosphere, which are uniformly distributed in the horizontal, but form a set of measure zero in a suitable sense.

We then use these analytical equilibrium solutions to test a Lagrangian-based moist convective adjustment (MCA) step, which acts as a convective parametrization. MCA has been commonly used as a convective parametrization within numerical models of the global atmosphere (e.g. Rennó *et al.*, 1994, p 14 430), and is still used in many general circulation models (GCMs). It is therefore of interest to test such an adjustment step in a robust manner, particularly given that many adjustment steps are implementation- or resolution-specific. This approach shares similarities with the benchmarking of dynamical cores of GCMs, initiated by Held and Suarez (1994), but applied to convective parametrizations.

We test our MCA step by incorporating it into a discretized version of our mathematical model, and allowing the system to reach equilibrium. We then compare the numerical and analytical equilibrium solutions. Numerical single-column models of the atmosphere have frequently been used in climate studies (e.g. Sobel and Bretherton, 2000, and references therein), and have also been used as a test-bed for physical parametrizations of subgrid-scale processes (Ghan *et** al.* 2000). Column models will remain useful tools for the immediate future because of their simplicity, and because, for the next decade, GCMs are likely to continue to have coarse meshes relative to moisture-related atmospheric processes (with horizontal spatial scales > 10 km, which will not resolve individual convective cells).

The structure of this paper is as follows. In section 2, we begin by formulating the mathematical model. We choose potential temperature *θ* and pseudo-height *z* (a monotonic function of pressure; Hoskins, 1971) as key variables, and not temperature and pressure. After introducing the model in standard Eulerian form, we subsequently rewrite the model in Lagrangian form to facilitate calculation of solutions.

In section 3, we introduce a formula for the saturation specific humidity, a function of both potential temperature and pseudo-height, which provides a good approximation to that of Earth's atmosphere in the ranges of interest. We use this formula to calculate an explicit expression for the ascending branch of the thermodynamic solution path for the column atmosphere in equilibrium.

In section 4, we first calculate an upper bound on the downward equilibrium speed. If this upper bound is exceeded, the ascending and descending branches of the solution path in the *θz*-plane attempt to intersect, allowing pockets of buoyancy to develop. This situation may occur initially, before equilibrium is established, but it cannot occur once the column atmosphere has reached a dynamic equilibrium. We then calculate the solution path followed by descending air. Finally, we consider an example where the prescribed radiative cooling is a piecewise-linear function of *z*.

In section 5, we show that our model supports equilibrium solutions in which an air parcel can release moisture as it descends. Such solutions exist provided the downward equilibrium speed is sufficiently slow, and provided the saturation specific humidity is a function of both potential temperature and pseudo-height, and not pseudo-height alone. For such solutions, the downward branch of the solution path (or ‘environment’) is always conditionally stable.

In section 6, we test a Lagrangian-based moist convective adjustment step which acts as a convective parametrization. We do so by incorporating the adjustment step into a numerical method which solves a discrete version of the time-dependent mathematical model. In the numerical case, we find that equilibrium is always attained when the downward equilibrium speed is less than its threshold value, and that the numerical equilibrium solutions converge to their analytical counterparts as the vertical mesh size decreases. We next use the numerical method to examine the behaviour of the column atmosphere when the upper bound on the downward equilibrium velocity is exceeded. In this case, equilibrium is not attained, and numerical evidence of radiative-convective disequilibrium (Randall *et** al.* 1994) is observed. Finally, we examine the time evolution of vertical profiles of potential temperature and specific humidity when an empirical formula for Earth's saturation specific humidity is used. We find that, for a suitably small downward velocity, the system evolves to equilibrium, and that, for suitably fast downward velocities, the system exhibits a periodic disequilibrium.

In section 7, we discuss the key elements of the model, and draw conclusions. Our main result is that a new type of weak, or modern non-differentiable, solution exists that captures the essential moist thermodynamics of traditional column models. Furthermore, we give an explicit formula for such a solution, and describe a numerical procedure that stably and accurately approximates it.