1. Introduction

The accurate numerical prediction of weather and climate events requires the solution of the full, compressible Navier–Stokes equations within a narrow layer of atmosphere surrounding Earth, coupled with differential equations that govern physical processes such as the movement of heat energy and condensational phase changes of water vapor in the air. The number of degrees of freedom involved in this numerical prediction problem is immense, and it is unlikely that even the next generation of computer architecture will be able to simulate all the time and length scales of cumulus cloud behavior in this system accurately. To overcome this problem, temporally and spatially averaged forms of the governing equations are solved numerically on fixed spatial grids of more than one kilometer in the horizontal direction. While this makes the numerical solution to the system tractable, it means that weather and climate events occurring on scales below the resolution scales of the averaged system need to be added back into the system in an “averaged” sense via parameterizations of the subgrid-scale behavior.

An important subgrid-scale phenomenon that needs parameterizing in this way is moist cumulus convection (Arakawa 2004). This phenomenon plays a significant role in the vertical fluxes of entropy, air mass, moisture and air momentum in the atmosphere, each of which can have large magnitudes. This is particularly the case in the tropical regions of Earth’s atmosphere where these fluxes make an important contribution to the tropical circulation. This tropical circulation in turn helps to drive Earth’s climate. Currently it is only possible to directly represent convection in weather and climate simulations with grid blocks that are (at least) 1–2 km in horizontal length, and often much more than this. Today’s general circulation models (GCMs) are typically running with a horizontal spatial grid resolution of 10 km at most. Hence, we are a long way from the direct representation of convection in these models (the horizontal column footprints of approximately 100 km² need to be reduced to the order of 1 km²). One useful approach to modeling the small-scale effects of moist convection, and understanding its effect on the larger-scale dynamics of the system, is to consider one-dimensional single-column-atmosphere models developed (and investigated) by Yanai et al. (1973); see also Tiedtke (1989) and references therein.

Single-column-atmosphere models are simplified atmosphere models where physical processes such as moist convection or vertical heat transfer can be explored without directly simulating the large-scale horizontal dynamics [see Sobel and Bretherton (2000) and Ghan et al. (2000) and references therein for more details]. These single-column models have formed a significant part of many theoretical convection studies due to their ability to sensibly and accurately model convective processes while retaining a simple one-dimensional structure (Shutts 1994; Holt 1989; Lock and Norbury 2011; Cheng et al. 2017a,b). Such single-column models are readily used in GCMs, but due to “functional” errors in these models, which take the form of model biases and unwanted trade-offs (Bretherton 2007), their modeling of reality is still under some scrutiny. For example, they can lead to a weak or missing Madden–Julian oscillation (Madden and Julian 1971), and in particular, they are poor at predicting the diurnal transition between shallow and deep convection (Yang and Slingo 2001; Betts and Jakob 2002). Here we define shallow convection as nonprecipitating or weakly precipitating convection that is usually restricted to the lower half of the troposphere, while deep convection is heavily precipitating convection that typically approaches the tropopause. Many single-column convection schemes parameterize deep and shallow convection separately, but this is undesirable because in the physical atmosphere the transition from shallow to deep convection is a seamless process in both space and time. Hence it is desirable to have a unified scheme that incorporates both convection processes. There has been some progress in this direction over the last decade but this problem is by no means solved (Hohenegger and Bretherton 2011; Mapes and Neale 2011; Park 2014; D’Andrea et al. 2014). Here we examine a unified single-column model wherein deep and shallow convection events occur as different possible solutions to the same initial problem.

The particular parameterization scenario we consider is similar to that in Cheng et al. (2017a), which is a single column of compressible air parcels (an air parcel is a mass of air with homogeneous properties of gas density, heat energy and moisture density). Cheng et al. (2017a) devised a numerical algorithm that instantaneously adjusts the parcels from an initial buoyantly unstable configuration into a final statically stable (nowhere supersaturated) state. Such an unstable configuration can be triggered, for example, either (i) by a buoyantly stable column being lifted when passing over mountain topography, causing initially moist but unsaturated parcels to become supersaturated, and hence unstable; or (ii) by an unsaturated column becoming buoyantly unstable due to surface heating. The adjustment removes any instability from the column by moving parcels to a statically (buoyantly) stable arrangement that is nowhere supersaturated. Note, at no point does this approach solve the underlying three-dimensional (possibly turbulent) vertical dynamics, it solely models the physical laws of convection (i.e., the conserved thermodynamic processes) on unresolved short time scales via the transportation of air parcels.

In GCMs for weather and climate simulations, the vertical thermodynamic structure of the atmosphere generally needs to be “adjusted” at each time step so that the atmosphere is stable (or sufficiently stable) in the vertical direction so that the next time step can be applied. In current simulations the adjustment phase typically uses an algorithm based upon the moist available potential energy (MAPE) [or available potential energy (APE)], with the columnar atmosphere considered as a single-columnar plume (Randall and Wang 1992; Wang and Randall 1994). The MAPE of a single column (or, sometimes, a slightly more complex domain) gives an upper bound on the amount of kinetic energy produced through latent heating only, and is found by determining the global minimum total potential energy configuration of the air parcels. However, the single atmospheric plume construction is not always representative of the actual atmosphere, and in fact satellite observations show clouds forming in multiple layers, implying that the underlying process can be more complicated than a single plume. Various nonplume numerical weather and climate prediction schemes use an adjustment process based upon the convective available potential energy (CAPE). This approach calculates the available energy of a single (small) air parcel as it rises in the column (Emanuel 1994; Riemann-Campe et al. 2009). However, MAPE is usually preferred to CAPE because MAPE accounts for both ascending and descending air (Lorenz 1979; Randall and Wang 1992; Wang and Randall 1994). The algorithm presented by Cheng et al. (2017a) also deals with both upward- and downward-moving air parcels, and is based upon physical conservation laws, specifically those of air mass, thermal energy and moisture mass. For the convection process, in a column that does not lose or gain energy via radiation, Cheng et al. (2017a) conserve the quantities θ for unsaturated parcels and θ^M:= θ + Lq for saturated parcels. Here θ is the potential temperature, q is the specific humidity (of the water vapor in the parcel), and L is a physical constant that is large to denote that condensation of small amounts of water vapor leads to large increases in the potential temperature. These conserved quantities are suitable approximations (such as linearizations) of the exact conserved thermodynamic quantities when q is small, but in this paper we use more accurate forms of these thermodynamic conserved quantities in order to construct a more quantitatively accurate adjustment algorithm. The Cheng et al. (2017a) adjustment algorithm simultaneously rearranges all the parcels, with no defined order of parcel sorting; it assumes parcels remain saturated even when they may have been overtaken by other faster-moving parcels, and it neglects any effect of the parcels moving during the adjustment phase. As saturated parcels move upward, they conserve θ^M by condensing water vapor (decreases q) while simultaneously releasing latent heat (increases θ); L is the conversion factor for this process.

The algorithm of Cheng et al. (2017a) transforms some preadjusted state, comprising saturated, supersaturated and buoyantly unstable parcels, to a stable adjusted state, where no parcel is supersaturated. For certain initial data scenarios, the final stable state is a weak adjustment solution where the column is made up of vertical stacks of pancake-like regions interleaving moist and dry parcels, similar to appropriate relevant states observed in the real atmosphere, such as in lenticular clouds that form in laminar layers on the downside of mountains (Houze 1993; Whiteman 2000; Neiman and Shaw 2003). Formulations of the convection problem in columns of the atmosphere have the property that these new finescaled, vertically layered solutions may exist; these so-called weak solutions cannot be found using differential equation models, which average over these fine layers to get more continuously varying solutions. The usual numerical methods require a certain smoothness of solutions so that higher derivatives exist. Here the new solutions interleave dry and moist parcels such that the moisture field is not even continuous, let alone differentiable. In modern PDE jargon these are known as weak solutions; somewhat surprisingly these mathematically curious solutions can be seen in actual atmospheric scenarios, so it is a success of these column adjustment algorithms that they can computationally approximate these new solutions that may be of practical forecasting value. The results in Cheng et al. (2017a), including those with the interleaving weak solution, were shown to arise from the global maximum of an integral cost functional, which was confirmed by directly numerically maximizing the particular functional using the Munkres algorithm. The adjustment algorithm of Cheng et al. (2017a) is related to the simpler incompressible, Boussinesq, individual-parcel-sorting algorithm of Cheng et al. (2017b), which provides a proof of the well-posed convergence to weak solutions of the original PDE problem in the limit as the time step of the time-dependent problem Δt → 0. In the limit Δt → 0 the algorithms of Cheng et al. (2017a) and Cheng et al. (2017b) coincide.

The algorithm presented in Cheng et al. (2017a) behaves similarly to those algorithms assessed in Stansifer et al. (2017) and Harris and Tailleux (2018) in that it computes a parcel arrangement that globally maximizes (or minimizes) some integral functional. Here we propose a different adjustment algorithm (to that of Cheng et al. 2017a) that seeks to find local maxima of the same cost functional. Rather than sorting all parcels simultaneously, we propose an algorithm that sorts parcels by swapping neighboring parcels one by one in a clearly defined order. Such an algorithm allows us to consider other such local adjustments of the column, different to the global adjustment of Cheng et al. (2017a), from a given preadjusted state. In particular we demonstrate that the parcel-swapping algorithm delivers a potentially more appropriate (in terms of practical weather and climate forecasting) adjustment of the column for certain preadjusted states, and delivers the same adjustment for other initial states. It also allows for scenarios where an ascending parcels overtakes a saturated parcel, forcing it down the column and hence making it unsaturated. This effect is not present in Cheng et al. (2017a). This then poses the question, If more than one adjustment from the same initial data to different final adjusted states can be found via different algorithms (a shallow convection vs deep convection event perhaps), which one is more appropriate for a given time step in a numerical weather prediction simulation? As well as computing multiple adjusted solutions for the same preadjusted state, a feature of the proposed parcel-swapping algorithm is that it provides a sounder physical basis for implementing the convective inhibition criterion (CIC) (Colby 1984). This criterion states that an ascending parcel requires sufficient buoyancy in order to swap with its neighboring parcel directly above. In the Cheng et al. (2017a) algorithm this criterion was implemented only in an approximate way, because sorting all parcels simultaneously makes it difficult to identify the levels in the column where each ascending parcel moves above other stationary or descending parcels.

This paper is laid out as follows. In section 2 the moist convection problem is presented along with the formulation of the problem in terms of physical conservation laws for mass (of both air and moisture) and thermal energy as measured by the moist potential temperature. In section 3 we describe the new parcel-swapping algorithm used to find the adjusted, stable equilibrium state (or states) of an initially unstable atmosphere. Results of this algorithm showing the nonunique nature of the adjustment problem are presented in section 4, with a discussion on the implications of these results in section 5.

2. Formulation

Consider a single (vertical in height z) atmospheric column of moist air situated at a particular longitude and latitude on Earth’s surface and bounded between pressures p ∈ [p₀, p_top]. The column is assumed to satisfy the ideal gas in hydrostatic balance, thus ∂p/∂z = −ρ(T, p)g where the density ρ(T, p) satisfies the approximate ideal gas law for the pressure p = ρR(1 + 0.608q)T (see Gill 1982, p. 41). Here T is the absolute temperature, R = 287 J kg⁻¹ K⁻¹ is the dry-air gas constant, g = 9.81 m s⁻² is the gravitational constant and q = ρ_υ/ρ denotes the specific humidity for moist air with vapor density ρ_υ. For convenience we introduce the virtual temperature T_υ = (1 + 0.608q)T, which is the temperature at which dry air (q = 0) would have the same density as moist air at a given pressure, assuming ideal gas behavior.

Rather than using z as the vertical coordinate, it is more convenient to use pressure, as then equal sized pressure parcels of air contain the same mass of air, thus swapping parcels in our numerical scheme (automatically) conserves mass of air in the column. In the column itself, the p axis (or z axis) is orientated vertically, that is, perpendicular to Earth’s geoid surface, and the column contains two locally horizontally connected regions. One region (the pseudoadiabat) contains the upward motion in the column, and the other (the environment) contains the downward motion. The pseudoadiabat tends to have a much smaller relative volume within the column compared to the environment because the upward motion tends to be much faster than the downward motion, but typically the ratio of these volumes is unknown. Hence rather than model each region separately, we simultaneously model the respective upward and downward motions (Lock and Norbury 2011), and this provides the mechanism that allows parcels to pass through each other.

Other physical processes such as the net radiative heating/cooling of air parcels can be incorporated into the governing thermodynamic equations for practical applications, and in this case the governing quantities θ_υ and θ_e will no longer be simply conserved. Instead the evolution of these quantities will be governed by differential equations [see Lock and Norbury (2011) for a theoretical example for a Boussinesq atmosphere]. In this article our primary focus is solely on the latent heat release and we assume that the movement of parcels occurs on a “seconds” time scale much faster than effects such as daily column heating/cooling, which can hence be neglected for simplicity. The physical constant appearing in (2), L = L_υ/c_p = 2490 K, is the conversion factor that determines the increase in potential temperature within an air parcel when some fraction of the water vapor it contains is converted into liquid water at the same temperature. Here the constant L_υ is the latent heat of vaporization (condensation). In practical applications cloud generation with various forms of precipitation are also modeled, but here it is assumed that vapor goes directly to rain (which is then removed from the system) for simplicity. This removal of rain means that not only are the types and forms of melted/frozen precipitation not relevant, but that crucially we do not consider reevaporation to drier parcels here.

The origin of the (strong) nonlinearity of the problem comes from the fact that supersaturated/saturated (q ≥ Q_sat) air parcels have the potential to convert moisture into latent heat causing them to ascend the column releasing CAPE, while unsaturated air parcels do not. Therefore, while condition (3) is sufficient for the dry (nonsaturated) stability of the column, if any parcels are saturated or supersaturated there could potentially be alternative stable rearrangements of the air parcels in the column, determined by allowing parcels to condense moisture, take on the released heat energy, and rise. These alternative arrangements of parcels must also satisfy (3), with q ≤ Q_sat everywhere, in order for them to be stable. Here the function Q_sat(T, p) denotes the saturation specific humidity for condensation given by the Clausius–Clapeyron relation (Gill 1982), defined explicitly later in (12). Water vapor with q > Q_sat is assumed to quickly condense out as rain so that q = Q_sat; effectively q lowers to Q_sat, and then q = Q_sat holds during any rising of the parcel with appropriate condensation so that θ_e is constant.

In the physical atmosphere, complicated three-dimensional adjustment processes determine the transport of temperature and specific humidity, and to model this would require the time integration of the fully three-dimensional Navier–Stokes equations. In this work we replace this complex process by the vertical transport of individual air parcels, where the horizontal dynamics in the column is left unmodeled. Modeling this complicated 3D motion as an instantaneous parcel response means that we can allow parcels to move a finite distance in apparently zero time. While this appears unphysical, what it really means is the adjustment process is considered to occur on a faster time scale to the background dynamics (minutes to hours), which is consistent with the physical atmosphere in most practical (seconds to minutes) adjustment scenarios. This type of adjustment model has some similarities to mass-flux convection schemes such as in Tiedtke (1989) and Gregory and Rowntree (1990); however, in our adjustment model we do not model or solve for the ascending parcel dynamics, parcels are assumed to swap or jump to their new level instantaneously. Our model also neglects any mixing of the air parcels, or feedback from the condensing moisture, it makes no assumptions on entrainment rates between the environment and pseudoadiabat, and condensed moisture is assumed to rain out of the system immediately. The model only uses actual values of θ_υ, θ_e, and q, capturing the leading-order effect of moisture condensation in the atmosphere and its effect on the atmosphere column stability.

We consider the air column to be discretized into a vertical one-dimensional array of air parcels, and we consider the movement of the parcels in a Lagrangian-parcel-following sense (Emanuel 1994; Bokhove and Oliver 2006; Cheng et al. 2017a) via the conservation of the quantities θ_υ and θ_e in (1) and (2). Thus, we have the two scenarios:

• For unsaturated parcels (q < Q_sat),

  $\begin{array}{c}{q}_A=q\left[p_A\left(t\right),t\right]=q\left[p_A\left(0\right),0\right]\text{and}\\ {\theta }_{\upsilon A}={\theta }_{\upsilon }\left[p_A\left(t\right),t\right]={\theta }_{\upsilon }\left[p_A\left(0\right),0\right]\forall A,\end{array}$(5)

  at time t, where p = p_A(t) is the pressure location of the parcel with Lagrangian label A. The unsaturated parcels move adiabatically and tend to be forced down in a column as saturated parcels release moisture, gain latent heat, and rise above them. However, the parcels can adjust and ascend the column if (3) does not hold everywhere–if during their ascent they become saturated they then follow the process in the bullet point below.
• For saturated parcels (q = Q_sat), conservation of (2) means

  ${\theta }_A\hspace{0.17em}\exp \left({\displaystyle \frac{L{q}_A}{T_A}}\right)={\theta }_A\left[p_A\left(0\right),0\right]\hspace{0.17em}\exp \left\{{\displaystyle \frac{L{q}_A\left[p_A\left(0\right),0\right]}{T_A\left[p_A\left(0\right),0\right]}}\right\}={\theta }_{eA}\forall A,$(6)

  as the parcel ascends along the curve $T{\left(p_0/p\right)}^{R/c_p}\hspace{0.17em}\exp \left[L{Q}_{\text{sat}}\left(T,p\right)/T\right]={\theta }_{eA}$ in the (T, p) plane [denoted by C(A); this is the moist adiabat] for each parcel A. Therefore, condensed moisture is accompanied by a latent heat increase (with the conversion factor L) for the parcel as it ascends.

3. Numerical moist convection adjustment algorithm

If the atmospheric column is buoyantly unstable, that is, if (3) is violated anywhere in the column, or if the atmosphere is buoyantly stable, but contains parcels that are supersaturated, then the air parcels within the column adjust their positions (via the unresolved dynamics) to remove the instability. During this process the column is said to be convectively adjusted into a new stable state. For a column of unsaturated parcels [0 ≤ q < Q_sat (T, p) for all times], the final stable parcel arrangement within the column is unique, with θ_υ ordered monotonically as we ascend the column, with the hottest parcels at the top of the column (Shutts and Cullen 1987). However, for a column with saturated/supersaturated parcels there are potentially more than one stable rearrangement that satisfies (3) and has q ≤ Q_sat everywhere.

While the GMCAA of Cheng et al. (2017a) is fast and effective [O(N²) efficient, where N is the number of parcels in the column], it may be that adjusting the column toward a global maximum of (8) might not be the most meteorologically appropriate solution. It was speculated in Cheng et al. (2017a) that the GMCAA could be most appropriate when simulating deep convection events, due to the large amount of moisture removed from the system, but it might not be as suitable for shallow convection events in the atmosphere (convective instability), where an alternative arrangement of parcels that releases less moisture, and hence introduces less thermal energy to the parcels, might be more appropriate. In this article we consider two further algorithms based upon swapping neighboring parcels to identify whether other significant stable rearrangements of the same initial data are possible. As these algorithms are based on swapping neighboring parcels, we term them parcel-swapping moist convective adjustment algorithms (PSMCAA). While the GMCAA sorts all parcels simultaneously with no defined order of moving the parcels, the PSMCAA essentially assigns an order in which the parcels move so as to reach a stable configuration that simultaneously satisfies q ≤ Q_sat throughout the column. Potentially there are many different ways to determine the order of the parcel swaps leading to subtly different results. To highlight the different rearrangement possibilities, we consider two parcel swap algorithms. PSMCAA1 is a local approach based upon the amount of virtual potential temperature a parcel can gain by swapping with its neighbor. We swap parcels in the order based on the magnitude of this gain, moving the parcel gaining the most virtual potential temperature first. PSMCAA2 is a global approach based upon the largest gain in the functional (8), with θ replaced by θ_υ. We again move first the parcel that increases (8) the most.

In PSMCAA1 the superscript “0” denotes the initial preadjusted state, the superscript “1” denotes the final adjusted state, the superscript “Temp” denotes a temporary value during the switching phase, and the superscript “I” on $\tilde{\theta }$ denotes the potential temperature increase of swapping neighboring parcels. PSMCAA1 then proceeds as follows:

1. Convert the virtual potential temperatures to absolute temperatures ${\tilde{T}}_j^0=\tilde{\theta }{\left(p/p_0\right)}^{R/c_p}/\left(1+0.608{\tilde{q}}_j^0\right)$, and set the temporary variables equal to the preadjusted state, ${\tilde{T}}_j^{\text{Temp}}={\tilde{T}}_j^0$, ${\tilde{q}}_j^{\text{Temp}}={\tilde{q}}_j^0$, and $A_j^{\text{Temp}}=A_j^0$ for j = 1, …, N. If any parcels are initially supersaturated, then condense any excess moisture, and heat the parcel by solving

   ${\widehat{T}}_j{\left({\displaystyle \frac{p_0}{{\tilde{p}}_j}}\right)}^{R/c_p}\hspace{0.17em}\exp \left[{\displaystyle \frac{L{Q}_{\text{sat}}\left({\widehat{T}}_j,{\tilde{p}}_j\right)}{{\widehat{T}}_j}}\right]-{\displaystyle \frac{{\tilde{\theta }}_j^0}{1+0.608{\tilde{q}}_j^0}}\hspace{0.17em}\exp \left({\displaystyle \frac{L{\tilde{q}}_j^0}{{\tilde{T}}_j^0}}\right)=0,$

   for ${\widehat{T}}_j$ and set ${\tilde{T}}_j^{\text{Temp}}={\widehat{T}}_j$ and ${\tilde{q}}_j^{\text{Temp}}=Q_{\text{sat}}\left({\widehat{T}}_j,{\tilde{p}}_j\right)$. Convert absolute temperatures back to virtual potential temperatures ${\tilde{\theta }}_j^{\text{Temp}}$ using (1).

2. Calculate the virtual potential temperature increase of each parcel at ${\tilde{p}}_j$ if it swaps with its above neighbor at ${\tilde{p}}_{j+1}$, for j = 1, …, N − 1.

   1. If ${\tilde{q}}_j^{\text{Temp}}=Q_{\text{sat}}\left({\tilde{T}}_j^{\text{Temp}},{\tilde{p}}_j\right)$ (i.e., parcel is saturated) then excess moisture will condense during the swap, latent heat will be released, and its virtual potential temperature will change. Its new absolute temperature ${\widehat{T}}_j$ is found by solving the implicit relation

      ${\widehat{T}}_j{\left({\displaystyle \frac{p_0}{{\tilde{p}}_{j+1}}}\right)}^{R/c_p}\hspace{0.17em}\exp \left[{\displaystyle \frac{L{Q}_{\text{sat}}\left({\widehat{T}}_j,{\tilde{p}}_{j+1}\right)}{{\widehat{T}}_j}}\right]-{\displaystyle \frac{{\tilde{\theta }}_j^{\text{Temp}}}{1+0.608{\tilde{q}}_j^{\text{Temp}}}}\hspace{0.17em}\exp \left({\displaystyle \frac{L{\tilde{q}}_j^{\text{Temp}}}{{\tilde{T}}_j^{\text{Temp}}}}\right)=0,$

      and its moisture level is then

      ${\widehat{q}}_j=Q_{\text{sat}}\left({\widehat{T}}_j,{\tilde{p}}_{j+1}\right),$(10)

      with Lagrangian label ${\widehat{A}}_j=A_j^0$. The virtual potential temperature increase is defined as ${\tilde{\theta }}_j^I={\widehat{\theta }}_j-{\tilde{\theta }}_{j+1}^{\text{Temp}}$, for j = 1, …, N − 1, where ${\widehat{\theta }}_j={\widehat{T}}_j\left(1+0.608{\widehat{q}}_j\right){\left(p_0/{\tilde{p}}_{j+1}\right)}^{R/c_p}$.

   2. If ${\tilde{q}}_j^{\text{Temp}}<Q_{\text{sat}}\left({\tilde{T}}_j^{\text{Temp}},{\tilde{p}}_j\right)$ (i.e., the parcel is unsaturated) then there is no latent heat exchange and the parcel at ${\tilde{p}}_j$ rises adiabatically. If during this swap the parcel becomes supersaturated, then rain out excess moisture as in step 1. Then ${\widehat{\theta }}_j={\tilde{\theta }}_j^{\text{Temp}}$, ${\widehat{q}}_j=q_j^{\text{Temp}}$, and ${\widehat{A}}_j=A_j^{\text{Temp}}$ and the virtual potential temperature increase is ${\tilde{\theta }}_j^I={\tilde{\theta }}_j^{\text{Temp}}-{\tilde{\theta }}_{j+1}^{\text{Temp}}$.

3. The process in 2 above generates an array of virtual potential temperature increases ${\tilde{\theta }}_j^I$ for $j=1,\dots ,N-1$. We then determine $\max \left({\tilde{\theta }}_j^I\right)$, which occurs at $j=J$, say.

   1. If ${\tilde{\theta }}_J^I>0$ then the parcel at ${\tilde{p}}_J$ can gain enough virtual potential temperature to switch places with the parcel at ${\tilde{p}}_{J+1}$, hence we switch these two parcels and relabel

      $\begin{array}{l}{\tilde{\theta }}_J^{\text{Temp}}={\tilde{\theta }}_{J+1}^{\text{Temp}},{\tilde{q}}_J^{\text{Temp}}={\tilde{q}}_{J+1}^{\text{Temp}},A_J^{\text{Temp}}=A_{J+1}^{\text{Temp}},\\ {\tilde{\theta }}_{J+1}^{\text{Temp}}={\widehat{\theta }}_J,{\tilde{q}}_{J+1}^{\text{Temp}}={\widehat{q}}_J,A_{J+1}^{\text{Temp}}={\widehat{A}}_J.\end{array}$
   2. If ${\tilde{\theta }}_J^I\le 0$ then no parcels can gain enough virtual potential temperature to switch places with their neighbors, hence the algorithm stops and we define

      ${\tilde{\theta }}_j^1={\tilde{\theta }}_j^{\text{Temp}},{\tilde{q}}_j^1={\tilde{q}}_j^{\text{Temp}},A_j^1=A_j^{\text{Temp}}\text{for}j=1,\dots ,N.$

4. Steps 2 and 3(i) are repeated until case 3(ii) arises.

For PSMCAA2 the algorithm is essentially that of PSMCAA1 except in parts 2(i) and (ii) we also calculate $F_j^I=F_S-F_{PS}$ where F is the numerical value of (8) [with $\theta \mapsto {\theta }_{\upsilon }$, and α = 0.007 as in Cheng et al. (2017a)]. Here F_S is the value of the functional after the parcel swap and F_PS is the value of the functional prior to the swap. Thus in part 3 it is $F_J^I=\max \left(F_j^I\right)$ that determines the order of the swaps, rather than ${\tilde{\theta }}_J^I$. PSMCAA2 continues until $F_J^I\le 0$, and if $F_J^I>0$ this also means that ${\tilde{\theta }}_J^I>0$, hence the CIC is correctly implemented. Essentially PSMCAA2 places a larger weighting on parcel swaps that are higher up the column. This can be most simply seen by considering two parcel swaps that lead to the same value of ${\tilde{\theta }}_J^I$. In PSMCAA1 these are given equal weight and so we can choose which pair to swap first (our algorithm swaps the lower pair first), but in PSMCAA2 the exponential weighting factor in (8) prioritizes the higher pair. The two algorithms become identical if the exponential weighting function is replaced by a linear function of p.

For a time-dependent convection problem consisting of a rising air column as considered in Cheng et al. (2017b), it was shown that including the CIC in the calculation was essential for proving well posedness of the problem. Because the GMCAA sorts all parcels simultaneously, identifying the exact level at which two parcels overtake (or pass) each other is difficult, as such the GMCAA assumes that if a saturated parcel needs to “overtake” another saturated parcel as they both rise, then they meet at a height where this overtake can happen (Cheng et al. 2017a). However, the parcels may be moved around by other parcels rising above them, so the way we implement the CIC in a physically sound way is to define an order in which the parcels swap, as in PSMCAA1 and PSMCAA2.

Table 1.

Parameter values in (12) used for the numerical simulations of section 4.

View Table

4. Nonuniqueness results

In this section we present two idealized examples for the adjustment of an unstable column of moist air and compare the results of the algorithms discussed in section 3. We also examine the robustness of the algorithms to the form of the initial preadjusted state. For all the results in this section we use N = 1000 parcels, and p₀ =10⁵ Pa at Earth’s surface.

a. Example A

In example A we consider a column whose instability has been triggered by the inequality (3) being violated close to the planetary surface (i.e., the column is buoyantly unstable). We consider this example as one where the surface of Earth, in the tropics for example, has been heated leaving a warm but moist layer at the bottom of the column. The initial potential temperature and relative humidity distributions we use are

${\theta }^0\left(p\right)=f\left(p\right)\hspace{0.17em}\exp \left\{{\displaystyle \frac{3}{5}}\left[1-{\left({\displaystyle \frac{p}{p_0}}\right)}^{R/c_p}\right]\right\},$(13)

where

$f\left(p\right)=\{\begin{array}{cc}300-\mathrm{\Theta }\hspace{0.17em}\sin \left\{{\displaystyle \frac{35}{3}}\hspace{0.17em}\pi \left[1-{\left({\displaystyle \frac{p}{p_0}}\right)}^{R/c_p}\right]\right\}& \text{for}p>p_0{\left({\displaystyle \frac{67}{70}}\right)}^{c_p/R}\hspace{0.17em}\text{kPa}\\ 300-\mathrm{\Theta }& \text{for}p<p_0{\left({\displaystyle \frac{67}{70}}\right)}^{c_p/R}\hspace{0.17em}\text{kPa}\end{array}$(14)

and

$q^0\left(p\right)=\{\begin{array}{cc}{Q}_{\text{sat}}\left(T^0,p\right)& \text{if}p>85\hspace{0.17em}\text{kPa}\\ Q_{\text{sat}}\left(T^0,p\right)\hspace{0.17em}\exp \left({\displaystyle \frac{p-85\hspace{0.17em}000}{5000}}\right)& \text{if}p<85\hspace{0.17em}\text{kPa}\end{array},$(15)

respectively. Here T⁰ is the initial absolute temperature distribution found from (13). A plot of the initial preadjusted form of the virtual potential temperature ${\theta }_{\upsilon }^0=\left(1+0.608q^0\right){\theta }^0$, when Θ = 8 K, and q⁰ is given in Fig. 1. In (14) we leave Θ as a parameter so that we can use this same potential temperature distribution for example B in section 4b but with a different value of Θ.

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Plots of (a) the initial virtual potential temperature distribution ${\theta }_{\upsilon }^0\left(z\right)={\theta }^0\left(1+0.608q^0\right)$, with θ⁰ given by (13) with Θ = 8 K, and (b) the initial humidity distribution q⁰(z) given by (15). In (b) the dashed line gives the value of Q_sat.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0045.1

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In Fig. 2 we consider the forms of θ_υ(z) and q(z) postadjustment for (Figs. 2a,b) PSMCAA1, (Figs. 2c,d) PSMCAA2, and (Figs. 2e,f) the GMCAA. We observe that PSMCAA1 adjusts to essentially the same solution as the GMCAA. The structure of the solution is a weak solution consisting of saturated and dry parcels interleaving throughout the column. This type of weak solution, and its consequences for standard numerical techniques, was discussed in Cheng et al. (2017a). The adjustment of the column using PSMCAA2 has a slightly different structure. While there is still an interleaving of moist and dry parcel layers, this time the parcels are unsaturated, except at the very top of the column. There is also fractionally more latent heating at the bottom of the column. The reason for the difference between the two parcel-swapping algorithms is in the order that the parcels are swapped. In PSMCAA1 the parcels move from approximately the bottom of the column first, and these parcels contain more moisture, hence produce most latent heat when ascending. Therefore, these parcels ascend highest in the column. This can be seen in the initial and final Lagrangian levels of the parcels in Fig. 3. Then as parcels originally above the parcels that have already ascended begin to rise, they do not ascend as far in the column and thus remain saturated. However, as noted earlier the exponential weighting factor in the cost function puts more emphasis on the parcels at the top of the column, thus these parcels move first in PSMCAA2. Therefore, when a lower parcel ascends (and note it will ascend to a higher level) it overtakes the first parcel to move, forcing that first parcel back down the column, so that it becomes unsaturated, because its virtual potential temperature does not change but its height does. Note that it could become saturated again if reabsorption of precipitation was included in the model.

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Plots of (a),(c),(e) the virtual potential temperature postadjustment θ_υ(z) (solid line) and preadjustment ${\theta }_{\upsilon }^0\left(z\right)$ (dashed line) and (b),(d),(f) the moisture distribution q(z) postadjustment for the preadjusted state given in Fig. 1. Adjustments are via (a),(b) PSMCAA1, (c),(d) PSMCAA2, and (e),(f) GMCAA.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0045.1

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Plot of the final Eulerian position of each parcel compared to its initial level in the corresponding results of Fig. 2. The circles represent results of PSMCAA1, the squares represent PSMCAA2, and the crosses represent the GMCAA. We plot only every 10 points so as not to congest the figures.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0045.1

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Even in this simple example we observe that PSMCAA1 and PSMCAA2 produce different adjustments of the column, corresponding to different maxima of (8). Hence there is a nonuniqueness to the adjustment problem depending on the ordering of parcel swaps. This demonstrates that parcel rearrangement order matters. However, as PSMCAA1 and the GMCAA produce qualitatively very similar results, it can be argued that PSMCAA1 is the parcel-swapping algorithm that most accurately models the physics occurring in the atmosphere. In example B we demonstrate that, in contrast to the above, PSMCAA1 and GMCAA adjustments can lead to significantly different column adjustments for alternative initial datasets.

b. Example B

In example B we consider a column in which the instability is triggered by the lower layer of air near Earth’s surface becoming supersaturated but with an almost stable virtual potential temperature distribution [(3) is broken for only the bottom 200 m of the column]. This example can be considered as the case of a column of air being lifted as it passes over hilly topography, or as a front of cold air penetrates underneath (Cheng et al. 2017a). The initial potential temperature distribution is again given by (13) except we choose Θ = 4 K such that the profile almost satisfies (3) and the initial relative humidity distribution is

$q^0\left(p\right)=\{\begin{array}{cc}1.05Q_{\text{sat}}\left(T^0,p\right)& \text{if}p>85\hspace{0.17em}\text{kPa}\\ 1.05Q_{\text{sat}}\left(T^0,p\right)\hspace{0.17em}\exp \left({\displaystyle \frac{p-\mathrm{85\; 000}}{5000}}\right)& \text{if}p<85\hspace{0.17em}\text{kPa}\end{array}.$(16)

In this case the moist layer is a modest 5% supersaturated, and hence the column is marginally unstable, but this is sufficient to highlight nonuniqueness in the system. Plots of the initial ${\theta }_{\upsilon }^0$ and q⁰ profiles are given in Fig. 4.

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Plots of (a) the initial virtual potential temperature distribution ${\theta }_{\upsilon }^0\left(z\right)={\theta }^0\left(1+0.608q^0\right)$, with θ⁰ given by (13) with Θ = 4 K, and (b) the initial humidity distribution q⁰(z) given by (15). In (b) the dashed line gives the value of Q_sat.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0045.1

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In Fig. 5 we again compare the final adjustments of the three algorithms. The GMCAA in Figs. 5e and 5f rains out a large volume of moisture, producing a weak solution with interleaving saturated and unsaturated parcels through the column. Because of the large amount of condensation, a large amount of latent heating of the column has occurred. The adjustment via PSMCAA1 in Figs. 5a and 5b is significantly different. While this adjustment has a similar moist layer to the GMCAA for 6000 ≤ z ≤ 8000 m, the lower layer contains more moisture (less condensation has occurred), and hence this lower layer is cooler than the result in Fig. 5e. By considering the initial and final levels of the individual parcels in Fig. 6 we see that fewer parcels rise up the column for PSMCAA1 in this example compared to the GMCAA algorithm. The parcels in the lower moist layer are unsaturated, suggesting that these parcels ascended the column first, and then were overtaken by the parcels that ascended to the top moist layer. In this case, because the initial ${\theta }_{\upsilon }^0$ profile is almost stable, once the parcels at the bottom of the column are chosen to ascend, this choice forces all the saturated parcels above those chosen down the column, making them unsaturated and unable to rise. In the GMCAA, because all parcels are sorted simultaneously, and without a defined order, when a parcel at the bottom of the column is installed into its adjusted position, all the other saturated parcels are assumed to remain saturated and at their preadjusted positions until they are sorted. The results of the PSMCAA2 in Figs. 6c and 6d have a similar structure to the PSMCAA1 results except that the upper-layer parcels are not saturated throughout this upper layer, and both are at slightly higher levels in the column.

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Plots of (a),(c),(e) the virtual potential temperature postadjustment θ_υ(z) (solid line) and preadjustment ${\theta }_{\upsilon }^0\left(z\right)$ (dashed line) and (b),(d),(f) the moisture distribution q(z) postadjustment for the preadjusted state (13) and (16) with Θ = 4 K. Adjustments are via (a),(b) PSMCAA1, (c),(d) PSMCAA2, and (e),(f) GMCAA.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0045.1

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Plot of the final Eulerian position of each parcel compared to its initial level in the corresponding results of Fig. 5. The circles represent results of PSMCAA1, the squares represent PSMCAA2, and the crosses represent the GMCAA. We plot only every 10 points so as not to congest the figures.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0045.1

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The results in examples A and B show that the moist convective adjustment problem does not have a unique (stable) solution for every choice of initial data, and that the order in which parcels are swapped during the adjustment is significant to the final adjusted outcome. The scenarios in example A showed that for some initial data, PSMCAA1 and GMCAA adjust to the same result, the global maximum of (8), while in example B PSMCAA1 found a local maximum of (8) solution with less total latent heating of column parcels. In both examples PSMCAA2 demonstrates that the parcel rearrangement order matters, although given the agreement between PSMCAA1 and GMCAA in example A, it is likely that the algorithm for PSMCAA1 gives a better representation of the physics in an actual model atmosphere. Therefore, if two (or more) locally stable adjustment solutions exist for the same preadjusted state (i.e., initial data) then the sensible question to ask is, Which solution should my NWP model be adjusting to?

The column adjustment problems in these two examples can be considered as one time step in an inner loop of a time-dependent problem such as that of a rising column of atmosphere moving over a mountain range (see Cheng et al. 2017a), or of an NWP simulation. In either case, assuming that significantly different local and global adjustments exist, if the model continuously adjusts to the global solution then the NWP simulation will predict a large amount of moisture release, in a short amount of time, and hence a large amount of local latent heating of air parcels. This in turn will feed in to the background horizontal dynamics of the NWP model and could lead to an over prediction of precipitation, wind velocities and temperatures, etc. On the other hand, if the model always adjusts to the local solution at each time step then the amount of moisture released will be less, and potentially released more slowly, so there will be less thermal energy to feed back into the background dynamics. These alternatives may ultimately lead to very different global solutions. Hence care is needed when running such NWP and climate models to make sure the model is directed to the more appropriate solution.

c. Model sensitivity analysis

In this section we reconsider the column adjustment from example A, but here in Fig. 7 we compare the three results with Θ = 8 K, Θ = 6 K, and Θ = 4 K in order to demonstrate the robustness of the results, that is, that the results are not sensitive to the initial preadjusted state. Note that as Θ is reduced the overall stability of the column, as identified by (3), increases in the sense that for Θ = 4 K the column is only unstable for approximately the lowest 200 m, as noted in section 4b. In the results that follow we only plot the postadjusted moisture distribution as this is the clearest indicator of the structure of the resulting adjustment solution, and we only consider PSMCAA1 as, from the preceding results in sections 4a and 4b, this algorithm appears to give the best representation of the physics in a more realistic atmosphere.

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Plots of the moisture distribution q(z) postadjustment for the preadjusted state (13) and (16) with (a),(b) Θ = 8 K, (c),(d) Θ = 6 K, and (e),(f) Θ = 4 K. Adjustments are via (a),(c),(e) PSMCAA1 and (b),(d),(f) GMCAA.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0045.1

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The results in Fig. 7 show that the adjusted states for PSMCAA1 and GMCAA with Θ = 8 K and Θ = 6 K agree very well with each other; that is, the parcel-swapping algorithm adjusts to the same state as the global algorithm. Also, the Θ = 6 K solution has a qualitatively similar structure to that of the Θ = 8 K solution, showing that the adjustment algorithms are robust and not sensitive to the initial preadjusted state. The results for Θ = 4 K in Figs. 7e and 7f again show the nonuniqueness of the solution: compared to Fig. 5, the PSMCAA1 result here has no lower moist layer. These results show that there is a smooth transition from the results in Figs. 7c and 7d to those in Figs. 7e and 7f, but that the PSMCAA1 changes from the global to a local solution when the local forms, while the GMCAA continues to adjust to the global maximum of (8).

In Fig. 8 we consider one final test on the robustness of the solutions by considering the cases in Fig. 7 but with an additional layer of moist saturated parcels (q = Q_sat) placed in the preadjusted state for 48 ≤ p ≤ 55 kPa. This additional region of moist parcels is an example of an existing cloud layer in the midtroposphere, which has the potential to stop some rising parcels from passing through it, as well as acting as an additional region of convection. The results presented here are consistent with those in Fig. 7, with the Θ = 8- and 6-K results adjusting to the global solution while the Θ = 4-K result demonstrates the nonuniqueness in the problem. Again, the similarities of the Θ = 8- and 6-K results show that the adjustment results are not in general sensitive to the initial preadjusted state.

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Plots of the moisture distribution q(z) postadjustment for the preadjusted state (13) and (16) with an additional layer of saturated parcels between 55 and 48 kPa with (a),(b) Θ = 8 K, (c),(d) Θ = 6 K, and (e),(f) Θ = 4 K. Adjustments are via (a),(c),(e) PSMCAA1 and (b),(d),(f) GMCAA.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0045.1

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5. Conclusions and discussion

This article investigates moist convection of warm, moist air in a single-column atmosphere (which has no horizontal variation of any variable). The convection is modeled using conservation laws for the virtual potential temperature θ_υ and equivalent potential temperature θ_e (which measure the thermal energy of unsaturated and saturated air parcels, respectively; see Emanuel 1994), the air mass, and the moisture mass. We construct two numerical moist convective adjustment algorithms based upon swapping neighboring air parcels: each algorithm allows saturated air parcels to rise up the column to their neutral buoyancy level (following their moist adiabat). Each algorithm swaps parcels: the first via a local approach with the parcel choice being that with the largest gain in virtual potential temperature during swapping; and the second a global approach with the parcel swap making the largest gain in the cost functional (8) being actioned first. These algorithms are based on the physical conservation laws that underpin the usual differential equations that model weather evolution. The algorithms also include the convective inhibition criterion (CIC) (Colby 1984), where parcels only swap places if the lower parcel has sufficient buoyancy (arising from condensing water vapor and consequent latent heating of the air parcel) to vertically move it above the higher parcel.

The results of these parcel-swapping adjustment algorithms are compared directly with the results of the adjustment algorithm devised in Cheng et al. (2017a) in two examples A and B by choosing initial data that correspond to weather scenarios known as deep or shallow convection. The Cheng et al. (2017a) adjustment algorithm produces a large amount of latent heating of parcels that arises from the condensation of a large amount of water vapor: the algorithm produces a rearrangement of the air parcels that globally maximizes (8). In example A, the column is initially buoyantly unstable and our local parcel-swapping algorithm produces a near-identical adjustment of the column to the global-maximizing algorithm of Cheng et al. (2017a). However, in the marginally, or weakly, unstable case of the initial data of example B, the two algorithms give significantly different results. The global algorithm produces a result more similar to that in example A, where many parcels rise relatively large distances within the column giving a moisture profile with interleaving saturated and unsaturated parcel layers. By contrast the local parcel-swapping algorithm has fewer parcels rising from the bottom of the column, giving a much smaller total amount of condensation and hence a smaller total amount of latent heating of parcels within the column. The results in example B show that there are potentially multiple adjustments of a practical atmospheric column, where the initial data are chosen from realistic atmospheric forecasting situations. Therefore, if forecasting numerical codes (NWP) are always designed to adjust to the global maximum of (8) solution then it might turn out that this is not the desired solution, and which solution is more appropriate depends upon additional information, such as fine detail of the velocity, temperature and moisture distributions in the horizontal directions around the column. The consequence of this result for numerical weather prediction codes and climate models is that unstable vertical convection has to be numerically guided to the (meteorological or climate) desired solution depending upon both local and global features of the flow. As a bonus, we discover new types of finely layered, in the vertical, solutions, where drier layers are interleaved with moist layers, as shown in Figs. 2b, 2d and 2f, between the approximate heights of 3000–12 000 m.

The single-column model presented here gives insight into how a heated, moist column of compressible atmosphere adjusts when perturbed from a stable vertical arrangement to an unstable one. This model can be directly extended to time-dependent problems, such as the lifted column problem in Cheng et al. (2017a), where radiative cooling and sources of moisture and/or heat are significant. The main point of this work is to show how a modified parcel adjustment algorithm finds different stable solutions to those usually published in the literature. These different solutions mathematically are local maximizers, not the usual global maximizers, of an associated cost/energy functional. These new solutions generate less precipitation and hence can be thought of as shallow convection in the weather forecasting literature. Since shallow convection is widely present in both global weather forecasting and in climate prediction models, the message from the above examples is that careful thought is required when guiding numerical forecasting code in the presence of moist convection.