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.
2. Mathematical model
2.1. Mathematical model formulation
We consider a moist single-column atmosphere, at a fixed latitude and longitude, of finite depth with a rigid-lid tropopause at z = ztop. Neglecting horizontal advection terms, the Eulerian equations for the conservation of thermal energy and moisture become
Making the Boussinesq approximation (constant density), conservation of mass is given by
We require the column atmosphere to be statically stable, i.e. we require
for 0 < z < ztop. There are no fluxes of mass, energy, or moisture through the boundary at sea-level z = 0, or through the rigid-lid height z = ztop. Equations (1a)–(1b) and (2) are closely related to standard tropical models of bulk cloud microphysics (Biello and Majda, 2010). (The vertical momentum equation is replaced by the stability condition on the assumed longer time-scales for which these models apply.)
In (1a)–(1b), the dependent variables are potential temperature θ, and specific humidity q. We recall that
where T is absolute temperature, p is pressure, p0 is a reference pressure, R is the gas constant for dry air, cp is the specific heat at constant pressure of dry air, ρv is the mass of water vapour per unit volume of moist air, and ρ is the density of moist air. Sθ and Sq are source terms for potential temperature and specific humidity, respectively. The independent variables are time t, and pseudo-height z (the vertical velocity w is defined in terms of z; see (18)). Pseudo-height, first introduced in Hoskins (1971), is a monotonic function of pressure defined by
where the pressure p satisfies hydrostatic balance. In (5),
provides an upper bound on z, where ρ0 is a reference density, and g is the constant acceleration due to gravity. For the typical values cp = 1004 J K−1kg−1, p0 = 105 Pa, R = 287 J K−1kg−1, ρ0 = 1.275 kg m−3, and g = 9.8 m s−2, we find za ≈ 28 km. We choose ztop< za.
For model clarity and analytical tractability, we consider simplified physical processes. First, at sea level z = 0, we set
where θPBL and qPBL are constants. We thus consider a thin planetary boundary layer (PBL) which acts as an infinite reservoir of thermal energy and water vapour. The potential temperature (and specific humidity) of the environmental air is not required to be continuous across z = 0. Thus, (7a) provides a mechanism for provoking convective instability. If the value of θPBL is greater than the potential temperature of the environmental air immediately above z = 0, free convection will occur in order to satisfy the static stability requirement (3). Physically, this mechanism of instability is more closely related to convection over continental regions than over oceans, where horizontal boundary convergence is typically required to provide uplift (Holton, 2004, p. 294). In (7b), we write
where 0 ≤ α ≤ 1 is a dimensionless fraction which determines how close to saturation air is at z = 0: when α = 0, the air is dry, and when α = 1, the air is saturated. The saturation specific humidity qsat(θ,z) is a known function of potential temperature and pseudo-height (e.g. (28)).
Second, in the body of the column 0 < z < ztop (which may be loosely interpreted as the free troposphere), we choose the following source terms:
Here, −r(z) is an imposed net radiative cooling term (where r(z) > 0), Ql represents latent heating associated with the release of moisture due to condensation of water vapour, and
is a constant conversion factor. In (10), we have used the value L = 2.5 × 106 J K−1 for the latent heat of condensation of water. We use the relationship
for converting the moisture released to latent heat, during the condensation of water vapour to liquid water. We write
where the one-dimensional material derivative of a (continuously differentiable) scalar quantity is
and where the switch parameter δ is defined as
This nonlinear ‘switch’, adapted from Lorenz (1967), p. 16, determines whether or not a parcel releases moisture and gains latent heat. It is essentially a parcel, or Lagrangian, calculation. Equation (14) models vapour condensation simply, and does not allow feedback from the cloud/rain so formed. In this way, we focus on the ‘leading order’ effect of moisture in the atmosphere by means of moisture release and latent heat absorption. We thus ignore: clouds, entrainment and detrainment effects, the ice phase of water, and the interactive effect of water vapour on the radiative cooling rate.
Consider a column of constant cross-section. Let I, the space of measurable subsets of the column volume (taking unit density so that mass is equivalent to volume), denote the label space for a continuum of air parcels in the column, and let D denote physical space (the vertical column). Let χ, the measure-preserving mapping from I to D, for each value of time , denote the flow by the map
(Here, measure-preserving means that the volume of the image set equals the volume of the mapped, or domain, set.) If we write χ = (χ1,χ2,χ3), where χ1 and χ2 are uniformly distributed in the horizontal, then, for a distinguished parcel labelled A, we can write its vertical position z, at time t, as
For such a parcel, we write a scalar quantity ψ as
If χ3(a,t), for a ∈ I, is differentiable in t, we write the Lagrangian (vertical) velocity of the parcel labelled A as
If, on the other hand, χ3(a,t) jumps a finite distance in zero time, then ‘w = ∞’, and χ3 is no longer a conventional function.
We thus rewrite (1a)–(1b) and (2), together with (9a)–(9b), as
with mass conservation given by
In the column cross-section, the measures implied by χ1, χ2 are uniformly distributed in the horizontal, that is, ‘filaments’ of upward-moving air uniformly penetrate the downward-moving air ‘mass’, which we call the ‘environment’. The static stability condition (3) is now given by the requirement that potential temperature, along the path followed by a parcel, increases monotonically with pseudo-height.
In sections 3–5, we solve (19a)–(19b) for an atmosphere at equilibrium, by calculating thermodynamic solution paths, which provide vertical profiles of potential temperature and specific humidity. In Section 6, we use these solution paths to test a Lagrangian-based, numerical, moist convective adjustment step.
For a column atmosphere in radiative-convective equilibrium, the net input of thermal energy and moisture to a parcel over the course of a cycle must be zero. We write this as
where A denotes the parcel label, and where the path integrals are evaluated in the θz- and qz-planes, respectively.
In our model, with a thin PBL at z = 0, (21a)–(21b) describe the following deep-convective equilibrium. Air at z = 0 receives an impulse of heat and moisture according to (7a)–(7b), and then rises instantaneously to z = ztop in order to satisfy the stability condition (penetrating all downward-moving parcels as it rises). Once at z = ztop, the air cools and descends in finite time to z = 0, whereupon the cycle repeats. When a descending air parcel that reaches z = 0 is heated and becomes statically unstable, it is no longer appropriate to assign a single value for the height of that parcel's potential temperature and specific humidity at that instant (that is, χ3 in (16) is no longer single-valued), as it jumps a finite distance in zero time. It is instead more useful to reparametrize solutions in terms of pseudo-height z, and not time t. In doing so, we describe vertical profiles for potential temperature and specific humidity of ascending and descending air. We refer to the collection of these vertical profiles for a column atmosphere at equilibrium as the solution path.
Let θ↑(z), q↑(z) and θ↓(z), q↓(z), for 0 ≤ z ≤ ztop, denote the ascending and descending branches of the solution path, respectively. At z = 0,
by (7a)–(7b), and we define
At z = ztop,
In order for equilibrium to be stable, we require
for 0 ≤ z < ztop, so that the ascending branch has greater buoyancy than the descending branch. Equation (25) differs from a typical environmental sounding (in which the atmosphere is not at equilibrium) plotted on a pseudo-adiabatic chart, and in which regions of convective available potential energy are often apparent (e.g. Shutts, 1990).
Let w↑ and w↓ denote the upward and downward equilibrium speeds, respectively. For air in a column at equilibrium, when the lowermost parcel receives an impulse of thermal energy, it jumps from z = 0 to z = ztop in zero time. All other parcels will be statically stable, and so will descend at a finite speed. We thus have
where w0 > 0 is a constant. (In (26b), w0 is a constant in order to satisfy (2), which is a consequence of the Boussinesq approximation, and of no entrainment or detrainment.) Since ascent occurs in zero time, ascending air is unaffected by the net radiative cooling rate −r(z). From (19a), we see that when δ = 0, θ↑(z) defines an adiabat (potential temperature remains constant); q↑(z) describes unsaturated air at a constant specific humidity. When δ = 1, θ↑(z) defines a pseudo-adiabat, along which saturated air continuously releases moisture as it ascends, and q↑(z) = qsat(θ↑(z),z) describes air that is saturated up to z = ztop. The downward branches θ↓(z), q↓(z) describe the ‘environment’ of the column atmosphere. We note that both θ↑(z) and θ↓(z) increase monotonically with pseudo-height: θ↑(z) must increase monotonically with z because ascent is either adiabatic or pseudo-adiabatic, and hence the potential temperature of ascending air can only increase with z (28); and θ↓(z) must increase monotonically with z in order for the environment to be statically stable.
One interpretation of the infinite upward equilibrium speed in (26a) is given by considering the limit in which the finite cross-sectional area of narrow ‘filaments’ of ascending air tends to zero. Figure 1 shows this interpretation when the total cross-sectional area of the ascending air filaments (which are uniformly distributed in the horizontal) is combined into a single narrow ‘pipe’ of ascending air. In this case, we view the column as consisting of separate regions of upward and downward motion which do not interact (e.g. Shutts, 1990). We require net zero mass flux in the vertical at each horizontal cross-section of the column since w = 0 at z = 0,ztop. Mass flux is the product of density and vertical speed. Thus, for a column with unit circular cross-sectional area, and for a narrow circular pipe of ascending air of radius ε, we require
Here, wu denotes the (finite) upward speed, and wd denotes the downward speed. Then, as ε → 0, wu → ∞, wd → −w0 and and this gives our new solution: the upward-moving air occurs on filaments with zero volume measure in a column cross-section, but have finite flux because w↑ = ‘∞ ’. This upward flux balances the downward flux of w↓.
We calculate the ascending branches of the solution path using an analytical formula for the saturation specific humidity which approximates that for Earth's atmosphere.
3.1. A formula for qsat(θ,z)
Figure 2 plots contours of , an empirical formula for the saturation specific humidity of Earth's atmosphere (Appendix). In much of this paper, we shall approximate using the formula
where , β, and γ are positive constants.
Equation (28) provides a simple analytical formula that captures the main features of : both formulae decrease monotonically with pseudo-height, and increase monotonically with potential temperature; and Figure 3 shows, using the parameter values from Table I, that (28) provides a reasonable approximation to . The greatest difference between Figures 2 and 3 occurs when assumes its largest values and is least likely physically.
Table I. Parameters for qsat(θ,z) for use in (28).
θ at z = 0
qPBL for α = 1
25 g kg−1
0.012 K m−1
3.2. Solution path – ascending branch
In order to calculate the ascending branches θ↑(z) and q↑(z) of the solution path, with qsat(θ,z) defined by (28), it is first convenient to consider the more general formula
We thus consider a formula for qsat(θ,z) with straight-line contours in the θz-plane. Equation (28) is recovered by choosing
We begin by considering the lifting condensation level (LCL) of air in the ascending branch of the solution path. We remark that we only consider the LCL of a single parcel, with potential temperature θPBL at sea level z = 0, and not the LCL of parcels lying along the ‘environment’ curve θ↓(z), as is done when considering the stability of the atmosphere implied by an atmospheric sounding.
The LCL zLCL, of air at z = 0 with potential temperature θ↑(0) = θPBL and specific humidity q↑(0) = qPBL (22a)–(22b), is determined by
If 0 < α < 1, the air does not become saturated after receiving an impulse of moisture at z = 0. For a column at equilibrium, the air must jump to z = ztop in zero time in order to satisfy the static stability requirement on the column. For unsaturated air, δ = 0 and so, by (19a), potential temperature is materially conserved until the air reaches z = zLCL (in zero time), the height at which it first becomes saturated. Similarly, by (19b), the specific humidity of the unsaturated ascending air remains constant until z = zLCL. In this way, air ascends along an adiabat from z = 0 to z = zLCL. Once at z = zLCL, δ = 1 and so, by (19a), air ascends along a pseudo-adiabat from z = zLCL to z = ztop. Along a pseudo-adiabat, the air continuously releases moisture to adjust its specific humidity to its saturation value, while simultaneously absorbing the associated latent heat released. By (19b), air remains at its saturation value (which decreases with pseudo-height) between z = zLCL and z = ztop.
We now calculate θ↑(z), q↑(z) for qsat(θ,z) given by the general formula (29). Reparametrizing (19a)–(19b) in terms of z, and considering ascending air, yields
where we have used (18) and (26a). In (33a), the net radiative cooling term −r(z) is absent because ascent occurs in zero time. Using (29), we rewrite (33a)–(33b) as
where, by (30),
and where zLCL satisfies (32).
We next rewrite (34a) as
Integration, and resubstitution of (35), then yield the solution
where we have used
In theory, we may write q↑(z) in terms of θ↑(z). Using (14) and (19b) we have
In practice, since θ↑(z) will usually be defined implicitly, it may not be possible to write q↑(z) explicitly.
In the general case, (37) provides an implicit expression for θ↑(z). However, in the case where the formula for qsat(θ,z) is given by (28), substitution of (31) into (37), and rearrangement of terms allows us to solve for z(θ↑) explicitly. We find
Here, since qsat(θ,z) is defined by (28), (32) yields
where we assume α is chosen such that zLCL< ztop.
We note that
for 0 ≤ z ≤ ztop, and hence the argument of the logarithm in (40) is always positive. The upper bound is the equivalent potential temperature θe of a parcel with potential temperature θPBL and specific humidity , at z = 0. (Given (11), θe = θPBL + ΘLαqsat(θPBL,0); also (57).) The upper bound determines the maximum amount of latent heat that can be released by the parcel, and is only approached as z → ∞. The lower bound in (42) is attained at z = 0.
In Figure 4, we plot θ↑(z) from (40), using the parameter values from Table I, with ztop = 13.5 km (that is, p = 100 mb; (5)). (It is due to an artefact of the plotting in Figure 4 that θ↑(z) is not exactly flat in the region of adiabatic ascent.) In the region of pseudo-adiabatic ascent (above ∼ 1 km), we see two broad types of behaviour. For heights below ∼ 6 km, the amount of (moisture and) latent heat released increases approximately linearly with height. Above this height, the curve flattens out (that is, ascent becomes adiabatic) due to the large negative argument in the exponential term in (28), and relatively little additional (moisture and) latent heat is released. Because the majority of the latent heat released occurs at low altitudes, the equivalent potential temperature of a parcel is sensitive to the fraction α. This sensitivity is related to the formula used for qsat(θ,z). For example, if qsat(θ,z) were a linear function of θ and z, θ↑(z) would be linear in the region of pseudo-adiabatic ascent.
We calculate the descending branches of the solution path. We first calculate an upper bound on the equilibrium downward speed, necessary for equilibrium to be maintained. Throughout this section, we assume that a descending parcel in a column at equilibrium does not release moisture. In section 5, we relax this assumption.
4.1. Upper bound on w0
By adding (19a) to (19b) multiplied by ΘL, we can remove the nonlinear effect of the switch parameter. If we do so, for a column atmosphere at equilibrium, and reparametrize using z by substituting w↑ or w↓ = dzA/dt for the ascending and descending branches, respectively, we find
Equation (43a) is consistent with (33a)–(33b). If no moisture is released during descent then dq↓/dz = 0, and (43b) becomes
Thus, the environmental lapse rate dθ↓/dz for a column at equilibrium is completely determined by the formula for the imposed net radiative cooling rate −r(z) and by w0.
Now, if we write (25), the condition for equilibrium to occur, as
substitution of (43a) and (44) into (45) yields
where we have also used (24). Thus, for equilibrium to occur, we require
which provides an upper bound on w0. In (47), the numerator is positive because r(z) > 0, and, given a formula for qsat(θ,z) that decreases with z and increases with θ, the denominator of (47) is also positive, with
The upper bound wmax is completely determined by the prescribed net radiative cooling rate r(z) and by the formula for the saturation specific humidity qsat(θ,z). If w0 were greater than wmax, the ascending and descending branches θ↑(z) and θ↓(z) could intersect in the θz-plane, leading to buoyancy instability, and preventing equilibrium from being established; this is examined numerically in section 6.
4.2. Solution path – descending branch
We calculate θ↓(z) by integrating (44). Doing so yields the general expression
where we have used (24). In (49), we assume w0< wmax, and, as in section 3, θ↑(ztop) is determined by the formula chosen for qsat(θ,z). We consider an example below where the prescribed radiative cooling rate is a piecewise-linear function of z, and where the formula for the saturation specific humidity is defined by (28).
Let r(z) be defined by
where ra, rb, and zc are constants such that 0 < ra ≤ rb, and zc> zLCL, where zLCL is given by (41). Equation (50) provides a crude approximation to a typical expression for the tropical-mean radiative cooling rate (e.g. Folkins and Randall, 2005).
We calculate θ↓(z) by substituting (50) into (49). Figure 5 plots θ↓(z) in the case where ra = 0.1 K d−1, rb = 2 K d−1, zc = 9 km, w0 = 5 mm s−1, and zLCL ≈ 925 m (using (41), since qsat(θ,z) is given by (28), with α = 1/e ≈ 0.37). The upward segment of the dashed curve shows θ↑(z) from Figure 4. Air at z = 0 receives an impulse of heat and moisture (horizontal dashed line), and ascends (pseudo-) adiabatically in zero time (dashed curve). Once at z = ztop, air descends along θ↓(z) (bold curve), at constant speed w0. Since δ = 0 during descent, specific humidity is constant, and q↓(z) is given by
Once the air reaches z = 0, it receives an impulse of heat and moisture, and the cycle repeats itself. The inputs of heat and moisture at z = 0 exactly balance their losses during the cycle.
5. Descent with moisture release
Thus far, we have assumed that moisture is not released during descent. We now show that, in our model, moisture may be released if the downward equilibrium speed is sufficiently slow. In this case, the descending branch θ↓(z) of the solution path is conditionally stable.
5.1. Upper bound on w0
According to (14), for an air parcel to release moisture it must remain saturated and follow a path in the θz-plane such that qsat(θ,z) decreases as the parcel moves. We examine this condition for a descending parcel.
Consider the case where the saturation specific humidity qsat(θ,z) is a known function, but is not specified. Then, for a saturated parcel, labelled A, we have
For a descending parcel in a column at equilibrium, dzA/dt = w↓ = −w0. If we assume dqsat(θA,zA)/dt < 0, then δ = 1, and substitution of (19a) into (52) yields
which provides a condition for a descending parcel to release moisture in a column atmosphere at equilibrium.
If we assume that
in agreement with observations (Figure 2), then we can write (53) as follows. In order for moisture to be released somewhere in the column during descent, we must have
Equation (55) provides an upper bound on the downward equilibrium speed. (Implicit in (55) is the assumption that w0 is always sufficiently large such that θ↓(0) > 0 so that the potential temperature θ↓(z) is positive throughout the column.)
Equation (55) shows that if the formula for qsat is a function of z only then w∗ = 0, that is, descent with moisture release is not possible. In the case where qsat(θ,z) = f(θ − βz), and where r(z) = r0, a constant, (55) gives
In this case, when w0< w∗, the radiative cooling is sufficiently strong to draw the path of a descending parcel in the θz-plane across the (straight-line) contours of u = θ − βz such that qsat decreases as the parcel descends.
If qsat(θ,z) = 0 in a region of the θz-plane, a descending parcel may ‘rain out’ all of its water vapour (provided the solution path enters this region). From this point onwards, the downward branch of the solution path is determined by (44). For qsat(θ,z) defined by (28), a parcel cannot release all of its moisture because is everywhere positive.
5.2. Conditional stability
We show that, for a column atmosphere at equilibrium in which descending air releases moisture, the environment is conditionally stable.
We first calculate an expression for the saturated equivalent potential temperature , that is, the potential temperature a saturated parcel would have if all its moisture were condensed and the resultant latent heat used to warm the parcel. We assume a saturated atmosphere, and integrate (11) from a state with potential temperature θ and specific humidity qsat(θ,z), to one at a sufficiently great height such that all of the water vapour is condensed out. We thus find
This expression differs from the more standard formula θ exp(ΘLqsat/T) (e.g. Simpson, 1978; Bolton, 1980) because (11) is a linear expression for relating the latent heat associated with the condensation of water vapour.
An environmental sounding in the atmosphere is conditionally stable if . In our model, the ‘environment’ is given by the descending branch θ↓(z) of the solution path. We consider (57) along the environmental path θ↓(z), and write
We consider two cases: (i) , in which moisture is released during descent; and (ii) , in which no moisture is released during descent, and in which the specific humidity remains constant.
In case (i), w0< w∗, and . By (43b), (58) becomes
since r(z) > 0 and w0 > 0. Hence, the environmental profile – determined by moisture release during descent – is always conditionally stable.
In case (ii), w∗< w0< wmax, dθ↓/dz is given by (44), and (58) becomes
where dqsat/dz < 0, and the atmosphere may or may not be conditionally stable.
Figure 6 plots contours of using (57), where . We see that, in the case where no moisture is released during descent, conditional instability is most likely near sea level, in agreement with observations of Earth's atmosphere (Holton, 2004, p. 294), that is, the descending branch θ↓(z) is most likely to cut contours of such that the contours are increasing, close to sea level.
6. Numerical model
We introduce and test a Lagrangian-based, numerical, moist convective adjustment step, which acts as a convective parametrization. We test the adjustment step by incorporating it into a numerical method for solving the time-dependent equations introduced in section 2. We first show that the system evolves in time towards equilibrium when the downward equilibrium speed is less than its threshold value, and that these numerical equilibrium solutions converge to the analytical equilibrium solutions as the vertical mesh size decreases. We then use the numerical method to examine the behaviour of the system when the downward equilibrium speed exceeds its threshold value.
6.1. Moist convective adjustment step
We introduce a moist convective adjustment (MCA) step which provides a convective parametrization of the latent heat processes in the mathematical model of section 2, and which acts to ensure that the column atmosphere is statically stable and nowhere supersaturated.
We begin by dividing the column atmosphere with z ∈ [0,ztop] into N segments of equal vertical extent Δz = ztop/N, with centre
where i = 1,…,N. We refer to these segments as air parcels. The value of potential temperature and specific humidity of each parcel, (θi,qi), is assumed uniform over the parcel.
We then define the MCA step as follows.
1.If the values of θi, i = 1,…,N, are not monotonically increasing, perform a sorting operation so that they are. The values of qi ‘travel with’ the sorted air parcels.
2.If qi> qsat(θi,zi) is now true for any of the swapped air parcels, convert the excess water vapour to latent heat according to
and heat the air parcel by dθ. All excess moisture is assumed to precipitate immediately, and not to re-evaporate. If more than one parcel has become supersaturated, each parcel gains latent heat simultaneously.
3.Repeat steps 1 and 2 until the column is statically stable and nowhere supersaturated.
Parcels can thus jump vertically to ensure the static stability of the column. If, after reaching a new vertical position, a parcel has become supersaturated, it releases its excess moisture, and absorbs the associated latent heat of condensation. This additional heat may cause the parcel to become unstable again, and this process – of testing for static stability, parcel swapping, moisture release, and latent heat absorption – is repeated until the column atmosphere is stable and nowhere supersaturated. In this way, the MCA step acts to transport heat and moisture upwards, and allows precipitation to occur. Mass conservation and no flow through the boundary are automatically enforced because we consider N air parcels of equal vertical extent which lie in the fixed interval [0,ztop] with no overlap.
This MCA step provides a convective parametrization of the moisture and latent heat processes in our mathematical model, that is, of terms involving the switch parameter δ in (19a)–(19b). In essence, it is a Lagrangian-based hard convective adjustment step (Renno et al., 1994, p. 14 431), as it assumes that free convection is strong enough to maintain the constant equivalent potential temperature of an ascending parcel.
The MCA step only provides one method of parametrizing moisture and latent heat processes. For instance, we have chosen to sort all of the air parcels at once before considering supersaturation of the parcels. Several other methods are possible. For example, parcels could be sorted pairwise, or by any method that seeks to reduce the potential energy of the column. Further, there are many ways for implementing supersaturation of parcels, for example, dividing the release into smaller parts to avoid supersaturation values from being unrealistically large. It is therefore important to test the behaviour of the MCA step. More generally, such testing is important because many similar adjustment steps are often implementation- or resolution-dependent.
We test the MCA step by incorporating it into a numerical method, or algorithm, for solving the time-dependent equations (19a)–(19b). We restrict our attention to the case where the radiative cooling rate is constant, assume that no moisture is released by a descending air parcel, and set α = 1 in (8). We state the algorithm below.
1.Initialize the values of (θi,qi), for i = 1,…,N.
2.Perform the following operations:
(a)apply constant net radiative cooling to each air parcel, that is,
for i = 1,…,N;
(b)impart impulses of heat and moisture to the lowermost parcel, at z = z1, such that
where qPBL = qsat(θPBL,z1);
(c)perform the MCA step.
3.Increment the time by the time step Δt, and repeat the operations of the previous step.
In this case, it is possible to calculate explicit expressions for the vertical profiles of potential temperature and specific humidity when the system is at equilibrium. In order to test the MCA step, we allow the algorithm to evolve the system to equilibrium, and compare the equilibrium profiles with the expected numerical equilibrium solutions.
We calculate explicit expressions for the expected numerical equilibrium solutions θeqm(zi) and qeqm(zi), where i = 1,…,N, as follows (θeqm(zi) and qeqm(zi) are the discrete analogues of θ↓(z) and q↓(z), respectively.). Since we consider constant net radiative cooling −r0 throughout the column, and assume that no moisture is released during descent, we have the following relationship
for i = 2,…,N. Here, the downward equilibrium speed w0 is determined by the time step, from the relationship
Equation (63) simply states that, for constant net radiative cooling, the vertical potential temperature profile is linear.
Next, given a general formula for the saturation specific humidity, we have
for i = 1,…,N, where Δθ denotes the amount of latent heat absorbed by a parcel during its ascent from z = z1 to zN. Equation (65) states that, for a column at equilibrium, the total gain in potential temperature by an air parcel (left-hand side) is equal to the total loss in potential temperature (right-hand side) (also (21a)).
In the case where , and for typical atmospheric values of θPBL, we have
In this case, if we let N → ∞, then Δz → 0, z1 → 0, and zN → ztop, and (66a)–(66b) converge to their analytical equilibrium solution counterparts (49) and (51), for r(z) = r0.
We remark that we have assumed that no moisture is released by a parcel during descent. In order for this to be true, we require
for i = 2,…,N, that is, the saturation specific humidity of a parcel at z = zi, for a column at equilibrium, should not exceed that of a parcel which has been cooled by an amount r0Δt, and which has descended by a distance Δz. If qsat(θi,zi) is given by , this inequality becomes
which is consistent with (56).
Given an initial state, we use the numerical method to consider the time evolution of vertical profiles of potential temperature and specific humidity θ(zi,t), q(zi,t), for i = 1,…,N, where t = nΔt, for .
We consider a column atmosphere with an initial potential temperature profile that is linear and statically stable, for an atmosphere that is initially dry. We choose
for i = 1,…,N, where θ0 = 290 K, θtop = 350 K, and ztop = 13.5 km. Figures 7–9 show simulations where the number of air parcels is N = 64, the magnitude of the constant net radiative cooling rate is r0 = 2 K d−1, θPBL = 300 K, qPBL = αqsat(θPBL,z1), with α = 1, and where qsat(θi,zi) is given by (28) using the parameter values from Table I.
Figure 7 examines the time evolution of θ(zi,t) by plotting as a function of time, where the time step is Δt = 13 h. Here, θeqm(zi) is given by (66a) using (68), and, for , the Euclidean norm | · |2 is defined by
We see that the potential temperature profile θ(zi,t) evolves to the expected steady-state equilibrium θeqm(zi) after approximately 800 h or about 60 time steps. Once the column is at equilibrium, a parcel takes four jumps within each time step (not shown) to jump from z = z1 to z = zN. Since the maximum number of jumps within a time step is given by N − 1 (here, N = 64), four jumps per time step suggests that w0 is not close to its upper bound wmax (47). We note that the time taken to reach equilibrium depends on the initial conditions. For example, an air parcel with an initial value of potential temperature greater than the equivalent potential temperature of a parcel with potential temperature θPBL at z = 0 will require an initial ‘transient’ phase to be cooled by the action of the radiative cooling.
In all of the simulations performed given the initial state (71a)–(71b), and with , it was found that equilibrium was achieved provided w0< wmax, regardless of the resolution (number of air parcels) chosen. In cases where different (and not necessarily linear) initial states were chosen, and for different formulae for qsat(θ,z), equilibrium was always attained for w0 suitably small. This suggests that the MCA step works correctly and is an appropriate implementation for parametrizing the moisture and latent heat processes of our mathematical model.
We next use the numerical method to examine the time-dependent behaviour of the vertical profiles when the upper bound on w0 is exceeded.
If Δt is such that w0> wmax, the downward branch θ↓(zi) attempts to overlap the upward branch θeqm(zi), and equilibrium cannot be attained.
Figure 8 examines the time evolution of θ(zi,t) given the same initial conditions as Figure 7, but with w0 increased by reducing the time step from Δt = 13 h to Δt = 11 h. In doing so, w0 now exceeds the upper bound on the downward equilibrium speed, and equilibrium can no longer be achieved. Instead, we see that periodic behaviour results. This may provide numerical evidence of radiative-convective disequilibrium (Randall et al. 1994). The initial (transient) phase of the time evolution (approximately the first 650 h, which corresponds to a similar number of time steps as in Figure 7) is again determined by the initial conditions. After this phase, the column continually attempts to reach equilibrium, but parcels descend too quickly for a ‘true’, or steady-state, equilibrium to be established. Figure 9 shows a snapshot of θ(zi,t) from Figure 8, at t = 500 time steps (5500 h), which illustrates this. The figure also shows the initial potential temperature profile (71a) (‘initial θ’) and the downward equilibrium profile that would be attained if w0 were less than its upper bound.
We next computed with the saturation specific humidity given by the physically realistic empirical formula (Appendix). We examined the time-dependent behaviour of the potential temperature profile θ(zi,t), for various time steps. For w0 sufficiently small, we observed similar convergence to that shown in Figure 7. Here, in Figure 10, we only show the case in which equilibrium is not achieved. We see that the solution does not reach a steady equilibrium, and exhibits periodic behaviour. We note that, since equilibrium is not reached, we have used the equilibrium profile θeqm(zi) from Figure 7 in the norm . This example shows the periodic disequilibrium when w0 exceeds its threshold value.
In this article, we have calculated exact, explicit solutions for a mathematical model of radiative convective equilibrium. These solutions capture the leading-order moist thermodynamics of a single-column atmosphere, with no horizontal degrees of freedom, in a dynamic deep-convecting equilibrium. These solutions were used to test a Lagrangian-based moist convective adjustment convective parametrization scheme.
In our model, statically unstable air rises instantaneously, jumping a finite distance in zero time. Parcel descent, on the other hand, occurs at constant speed, a direct consequence of the Boussinesq approximation. Although upward velocities in Earth's atmosphere are necessarily finite, the ratio of the upward to downward velocity in moist tropical convection may be three orders of magnitude or greater (May and Rajopadhyaya, 1999). Our (new) Lagrangian solution allows a (constant), non-zero downward velocity for the environmental air which still satisfies the no-flux boundary conditions at z = 0 and z = ztop. Zero net vertical flux is obtained by the balance of the upward and downward fluxes.
We calculate the solution path of a column atmosphere, that is, we calculate vertical profiles of potential temperature and specific humidity for a column atmosphere at equilibrium. The ascending branch of the solution path defines a pseudo-adiabat, while the descending branch determines the environment of the column atmosphere. The thermodynamic properties of the solution path are completely determined by the formula chosen for the saturation specific humidity, by the prescribed net radiative cooling rate, and by the downward equilibrium speed. Key to obtaining analytical equilibrium solutions is our choice of a simple formula which provides a good approximation to the saturation specific humidity for Earth's atmosphere, and interpreting our column model from a Lagrangian viewpoint.
We introduced an MCA step for parametrizing the moisture and latent heat processes of a discrete version of our mathematical model. We used exact, explicit, analytical solutions to test a numerical procedure that moves air parcels discontinuously, and found that the numerical method robustly converged to the exact solutions as the vertical mesh size decreased. The proposed MCA procedure based on parcel stability correctly captures the PDE solutions.
Numerical simulations were also used to examine the behaviour of the column model when the upper bound on the downward equilibrium velocity is exceeded. Periodic time-dependent behaviour of the vertical profiles of potential temperature and specific humidity is observed, which may be related to the radiative-convective disequilibrium of Randall et al. (1994). Having tested and proved the numerical method, we also used the method to compute solutions for a physically realistic atmospheric formula for the saturation specific humidity.
Since our analytical and numerical solutions may only be sensibly interpreted from a Lagrangian viewpoint, our work suggests that column models may best be interpreted in a Lagrangian sense as weak solutions of the usual Eulerian partial differential equations.
The authors gratefully acknowledge the EPSRC for Anthony Lock's PhD Plus award. The authors are also grateful for the comments of the anonymous referees, and acknowledge helpful comments from Mike Cullen.
We determine the empirical formula for the saturation specific humidity, shown in Figure 2, as follows.
We begin by writing the specific humidity q in terms of the pressure of moist air p and the water vapour pressure e. We have (Gill, 1982, p. 41),
where ≈ 0.622 is the ratio of the molecular weights of water and dry air. Thus,
where esat(T) is the saturation vapour pressure of water.
We then use the following formula for the saturation vapour pressure of pure water over a plane water surface (Gill, 1982, p. 606),
Here, TC ≈ 273 K is the zero point of the Celsius scale. For (TC − 40) ≤ T ≤ (T + 40) K, (A.3) gives the value of the saturation vapour pressure (mb) in Table 94 of the Smithsonian Meteorological Tables, correct to 0.2%.
Finally, we define
Using the definitions of potential temperature (4a) and pseudo-height (5), we can write
Substituting (A.5a)–(A.5b) into (A.4) yields as a function of potential temperature and pseudo-height (Figure 2).