1. Introduction
Closure is a key issue in the convection parameterization problem (cf. Yano et al. 2013). Convective quasi equilibrium, as originally proposed by Arakawa and Schubert [1974; see Yano and Plant (2012a) as a review], remains an important guiding principle for the convective closure even today (e.g., Zhang 2002, 2003; Donner and Phillips 2003; Bechtold et al. 2014), in spite of various criticisms (e.g., Houze and Betts 1981; Mapes 1997).
Equation (1.1) states that the convective response (first term) is always in balance with the large-scale forcing (second term). This closure is, intuitively speaking, physically sound, because the convective process is much faster than the large-scale processes. However, in spite of a series of subsequent efforts, this original form of the closure has never become fully operational, but only in variant forms (e.g., Moorthi and Suarez 1992). This study will explain why the formulation given by Eq. (1.1) is structurally difficult to implement as a closure from a mathematical point of view.
The original implementation (Lord and Arakawa 1980; Lord 1982; Lord et al. 1982) devoted much attention to maintaining positiveness of the convective mass fluxes, because only convective updrafts were considered. Unfortunately, in our opinion and as we will discuss below, a rather elaborate iteration procedure introduced for this purpose may have obscured some more basic issues with a strict convective quasi-equilibrium closure.
The present study focuses on the closure problem exactly as given by Eq. (1.1) without any further restrictions. This strategy may be partially justified by considering negative mass fluxes as detraining downdrafts (i.e., time-reversed updrafts). Importantly, regardless of whether this reinterpretation stands or not, this simplification enables us to elucidate more clearly and cleanly some basic problems with Arakawa and Schubert’s (1974) original convective quasi-equilibrium closure.
For the same reason, the original assumption of a spectrum of purely entraining plumes is maintained in the present study, because we believe it is important to establish a baseline. In the literature, the problems with the oversimplified entraining-plume hypothesis have been extensively discussed, and various alternative formulations have been proposed, as reviewed in, for example, de Rooy et al. (2013) and Yano (2014a). Analysis with a more elaborate plume model would be considered a future work.
A simple formulation for the terms in Eq. (1.1) is provided in the next section, and some basic demonstrations of the problems are made in section 3. The identified problems are investigated in section 4 by examining the completeness of the entraining-plume spectrum as well as the mathematical structure of the interaction (kernel) matrix.
The present paper focuses on a rather narrow question of mathematical difficulties with the original closure formulation by Arakawa and Schubert (1974). Various physical issues associated with this closure hypothesis as well as with the mass-flux formulation itself are extensively discussed in the literature. Some of these may be found in a review of quasi equilibrium by Yano and Plant (2012a), and more general issues associated with the mass-flux parameterization are covered by Plant and Yano (2015). In concluding, in section 5, the paper also turns to the physical implications from the present findings, also referring to background issues.
2. Formulation
a. Data
A tropical climatology based on the Jordan sounding (Jordan 1958) is adopted for specifying vertical profiles of temperature and moisture. The vertical resolution used for the profile data is 50 hPa from 1000 to 200 hPa, and with a surface value at 1015 hPa being separately given. Data are also available at the 175-, 150-, 125-, 100-, 80-, 60-, 50-, 40-, and 30-hPa levels.
The three types of the large-scale forcing profile considered: deep (solid and chain) dashed shallow (long dashed and double dotted–dashed), and very shallow (short dash and triple dotted–dashed). (a) The forcings are shown as a function of height for both the thermal (negative curves) and the moisture (positive curves) terms. (b) The forcings are shown in terms of the generation rate of cloud work function [as found in Eq. (1.1)] across the spectrum of fractional entrainment rates.
Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0165.1
The large-scale forcing on the cloud work function F from Eq. (1.1) is obtained by vertically integrating a linear combination of two large-scale forcings, as explicitly given by Eq. (B33) in Arakawa and Schubert (1974). The integration is defined with a weighting that is a function of the fractional entrainment rate ε (see next subsection) and the resulting integrated forcing is presented in Fig. 1b. We remark that the forcing has a relatively weak dependence on a microphysical parameter c0, which is defined by Eq. (2.5) below in section 2c. The vertical profile of the large-scale forcing as defined by Eqs. (2.1a), (2.1b), and (2.2) has a well-defined vertical scale but its projection onto the plume components in Fig. 1b, presents a very broad distribution of forcing as a function of the entrainment rate, despite the fact that the entrainment rate determines the vertical scale of each plume mode. Moreover, the main difference from changing the vertical scale of large-scale forcing is a change of the spectrum amplitude rather than a change of the spectrum shape.
We diagnose the convective quasi-equilibrium closure of Eq. (1.1) by closely following the mass-flux spectrum formulation introduced by Arakawa and Schubert (1974), and for formulation details we refer to the original paper. In the following two subsections, we describe two major assumptions for which some additional specifications are required: the entraining-plume spectrum (section 2b) and the precipitation formulation (section 2c).
b. Entraining-plume spectrum
For a larger fractional entrainment rate ε, the in-plume air is more diluted by the environmental air, and so becomes less buoyant. As a result, the plume top height zTi decreases with increasing ε. In essence, the fractional entrainment rate ε becomes a reverse measure of the convection depth zT. Some examples of vertical profiles of entraining plumes for the Jordan sounding are shown in Fig. 2. A full mass-flux profile for the ith plume is defined by Miηi(z), where Mi is the mass flux at the plume base for the plume type and is the ith component of the mass-flux vector M in Eq. (1.1).
Normalized mass-flux profiles, η = M(z)/M(zB), for selected entraining plumes under the microphysical formulation given by Eq. (2.5). In order from the deepest (solid) to the shallowest profiles (double–dotted chain), the plots are for values of ε = 1 × 10−5, 2 × 10−5, 4 × 10−5, 6 × 10−5, and 8 × 10−5 m−1.
Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0165.1
c. Precipitation efficiency
Dependence of the precipitation efficiency c0 on the fractional entrainment rate ε as defined by Eq. (2.5).
Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0165.1
3. Basic analyses
a. Interaction matrix
The interaction matrix (kernel)
The interaction matrix
Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0165.1
The evaporative cooling associated with detrainment leads to a further destabilization of the atmosphere, and thus
b. Response due to a single plume
Once a value of Mi, as a component of the mass-flux vector M, is specified [see also Eq. (3.2) below], the tendencies of temperature and moisture produced by each convective plume type, i, can be calculated, respectively, from Eqs. (3.6a) and (3.6b) of Yano (2015). Examples of the convective response from individual plume types are shown in Fig. 5. Here, we rather arbitrary assume Mi = 10−2 kg m−2 s−1. For the cases of ε = 6 × 10−5 and 8 × 10−5 m−1, the resulting plumes are relatively shallow, with relatively weak precipitation. This leads to strong cooling and moistening at the detrainment level associated with cloud evaporation. The effects are much less pronounced for the deep-plume example, because a high precipitation does not leave much cloud water for detrainment at the plume top. The values obtained for the strong cooling and moistening associated with the detrained-air reevaporation are shown in the appendix to be consistent with a simple scale analysis.
Profiles of the tendencies of (a) the temperature and (b) the moisture (mixing ratio) produced by convective plumes for given, selected entrainment rates: ε = 2 × 10−5 (solid), 4 × 10−5 (long dashed), 6 × 10−5 (short dashed), and 8 × 10−5 m−1 (chain dashed). Plotted in units of K day−1 and assuming the convective mass-flux amplitude of Mi = 10−2 kg m−2 s−1.
Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0165.1
The strongly peaked character of the thermodynamic tendencies from individual plume types raises potential issues for construction of the total convective response, obtained by taking a linear sum of these individual tendencies weighted by the convection-base mass-flux values, Mi. The total response is considered next.
c. Total convective response
The convective-base mass-flux vector, M = (Mi), is obtained from Eq. (1.1) by multiplying the inverted matrix
(a) The spectrum of convective-base mass flux as a function of the fractional entrainment rate, as obtained from inverting the matrix
Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0165.1
The most noticeable feature is a strong downdraft below the 4-km level, which is the lowest height achieved by plumes with largest fractional entrainment rates ε under the given mean thermal profile. Above this level, a substantial updraft reaches the 14-km level under deep large-scale forcing (solid curve), consistent with the depth of forcing in Fig. 1a. It is replaced by an updraft that decreases linearly with height between 4 and 14 km under shallow large-scale forcing (long dashed). This response is rather unintuitive considering the fact that shallow large-scale forcing only reaches the 5-km level (cf. Fig. 1). Only when very shallow large-scale forcing is considered does the convective response above the 4-km level becomes negligible (short dashed).
Figure 7 shows the corresponding convective tendency profiles for temperature (Fig. 7a) and moisture (Fig. 7b). Clearly these do not match well with the forcings in Fig. 1, even though the cloud work functions for each mode are in equilibrium by construction (cf. section 5a). The sudden increase of mass flux at the 4-km level (Fig. 6b) is associated with unrealistically strong heating and drying, with magnitudes circa 60 K day−1 for temperature and −120 K day−1 for moisture. The peaks are manifestations of those seen for individual plume types in Fig. 5, but with the signs reversed: entrainment (i.e., negative cloud-top detrainment) at the top of detraining-downdraft plumes causes this tendency. On the other hand, tendencies with more reasonable magnitudes are found at the other vertical levels.
Vertical profiles of the convective tendencies for (a) temperature and (b) moisture (the mixing ratio) for the three large-scale forcing profiles given in Fig. 1: deep (solid), shallow (long dashed), and very shallow (short dashed).
Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0165.1
4. Further analyses
a. Completeness of the spectrum of plumes
Figures 8a and 8b shows the plume spectrum {ηj(z)} and the plume matrix
(a) The vertical profiles for the plume spectrum {ηj(z)}, shown as a function of height (horizontal axis) and the plume index (vertical axis). (b) The plume matrix (i.e., the spectrum of plumes decomposed by the vertical-velocity normal modes)
Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0165.1
The eigenvalues λk for the plume matrix
Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0165.1
(a) The plume matrix
Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0165.1
In summary, the completeness analysis demonstrates the entraining-plume decomposition to be highly redundant, so that it does not directly permit a decomposition of any vertical mass-flux profile under the formulas (4.1) and (4.3) due to a singularity of the matrix
b. Eigenvalues and eigenvectors of the interaction matrix
The interaction-matrix eigenvalues κl are plotted in Fig. 11 in decreasing order of their absolute values. From Eq. (4.14), if the large-scale forcing were to contribute with the same order to all of the eigenmodes (cf. Fig. 1b), then the higher-order modes (say, l ≥ 14) would dominate the convective response.
The eigenvalues κl for the interaction matrix
Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0165.1
Considering the eigenmodes themselves, the spectra of the first eight right and left eigenvectors Ml and
The first eight right eigenvectors Ml (l = 1, …, 8) of the interaction matrix, as defined by Eq. (4.9a). (a) Real and (b) imaginary components. The first four vectors are shown by solid, long-dashed, short-dashed, and dot–dashed curves. They are followed by four other varying types of the curves. Note the change of scale in the horizontal axis.
Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0165.1
The first eight left eigenvectors
Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0165.1
These features have significant consequences in defining the response of convection M against a given large-scale forcing F. First, the expansion coefficients Fl for the large-scale forcing are defined by projecting the large-scale forcing F onto the left eigenvectors
c. Analysis of an idealized, highly truncated interaction matrix
Here, the order of the vector indices for M and
In this manner, the idealized matrix (4.15a) provides a very simple demonstration for the origin of the singular behaviors of the quasi-equilibrium closure that were seen in previous subsections.
d. Perturbation analysis
Thus, the perturbation analysis here more explicitly demonstrates how a strict application of the convective quasi-equilibrium condition tends to lead to an abnormally strong response of shallow convection to large-scale forcing.
5. Physical implications
The present paper has focused on a rather narrow question of mathematical difficulties with the original closure formulation of Arakawa and Schubert (1974). In concluding, we turn to the physical implications from the present findings, also referring to background issues.
a. Free-ride principle
The convective quasi-equilibrium closure of Eq. (1.1) is based on stationarity of the cloud work function, which is a vertically integrated quantity [cf. Eq. (133) of Arakawa and Schubert 1974]. Thus, the closure is also formulated in terms of vertically integrated quantities. However, we might intuitively expect that a certain quasi-equilibrium state (i.e., a balance condition) is achieved at each vertical level, at least to a good approximation, if a large enough number of convective modes is considered. The different modes provide different weighting functions and upper limits for the integrals in question.
Hence, we are led to ask whether, given enough plume modes in Eq. (1.1), we obtain a free-ride state corresponding to Eq. (5.2): Will this be actually accomplished in practice by the quasi-equilibrium closure?
b. Completeness of the plume spectrum
Equivalence between Eqs. (1.1) and (5.2) could be established if the mass-flux spectrum were able to represent any possible convective response that may be required to satisfy the free-ride state. Thus, a first consideration is whether the mass-flux spectrum is flexible enough to represent any possible vertical profile. This has been examined using normal-mode and singular-vector decompositions in section 4a. The entraining-plume decomposition is shown to be highly redundant, as expected from the individual plume profiles (cf. Fig. 2), and so a decomposition of the entraining plumes into normal modes does not provide well-defined expansion coefficients. However, this ill posedness of the decomposition can be resolved by removing all the singular vectors with almost-vanishing eigenvalues from the expansion. A reconstructed plume spectrum still remains fairly close to the original entraining-plume spectrum, but practically removing the redundancy.
Here, the mathematical question of the completeness of a plume spectrum addresses its capacity and flexibility to represent any physically feasible vertical structure of convection. As we have seen, the conclusion obtained is rather mixed, and further investigations from a more practical perspective could be warranted.
c. Convective response under the spectrum mass flux
The next consideration is how an individual plume mode modifies the large-scale thermodynamic state (i.e., convective response: section 3b). The effect of an individual entraining plume is composed of two main parts: (i) detrainment that cools and moistens the large scale by reevaporation of the detrained cloudy air and (ii) compensating environmental descent, in response to the convective updraft, that induces adiabatic heating and drying. A major difference between these two effects is that the detrainment effect is found only at a single level at the plume top, whereas the environmental descent is felt at all of the vertical levels spanned by the plume. As a result, the detrainment effect focused on a single vertical level tends to be abnormally strong, with cooling and moistening rates far exceeding 10 K day−1 and so strongly dominant at that level over the environmental-descent effect.
The consequence is that a straightforward inversion of the interaction matrix in the closure condition of Eq. (1.1) produces a full convective response against a given large-scale forcing that becomes very singular (section 3c). For idealized large-scale forcing profiles with a half-sine-shaped large-scale uplifting, we find that the convective response is dominated by singularly strong warming and cooling induced at the top of the detraining-plume downdrafts (i.e., entraining-plume updraft modes with a negative amplitude). Due to the tendency of entraining-plume modes to produce a singular response, the convective quasi-equilibrium closure condition does not achieve a thermodynamic state close to the free-ride balances. Thus, the mathematical analysis herein points out in an explicit manner how and why a physically unrealistic feature of the entraining-plume model causes a problem.
A very simple way of removing these singular cooling–moistening effects would be to neglect all of the detrainment effects from the interaction matrix
d. Interaction matrix analysis
Another important aspect of the convective response under the convective quasi-equilibrium closure is the dominance of shallow plumes regardless of the vertical extent of large-scale forcing. This is rather unintuitive. However, one must remember that as a matter of principle, large-scale forcing is projected to all the plume modes by design, as explicitly shown by Fig. 1a. The resulting spectrum of the convective response is rather nontrivial, mathematically taking the form of a matrix inversion. This character of the problem means that we need to pay attention to the mathematical behavior of the inversion calculation in order to better understand the structural issues involved.
First, a singular-vector decomposition is performed on the interaction matrix in section 4b. The left-eigenvector spectra are dominated by middle-height plume modes, with maximum heights of 8–10 km; thus, relatively deep components of large-scale forcing lead to a strong response by convection. On the other hand, the right-eigenvector spectra are dominated by shallow plume modes, and thus, relatively deep large-scale forcing modes are strongly projected onto shallow convective modes.
This rather strong asymmetry between the left and the right eigenvectors stems from a strong asymmetry in the interaction matrix itself. In turn, the asymmetry of the interaction matrix stems from the nature of the detrainment effect of a plume mode onto other plume modes: only the deeper plume modes are affected by detrainment from a given plume mode, and this gives rise to the triangular structure apparent within the interaction matrix (cf. Fig. 4b).
With increasing precipitating efficiency, the detrainment effect becomes weaker as less cloudy air is available to detrain at plume top. In a fully precipitating limit for all of the plume modes, then the asymmetry of the interaction matrix would disappear, and the singular response to the large-scale forcing would be removed. However, additional calculations (not shown) indicate that even a weak asymmetry of the interaction matrix can lead to a singular response. A relatively strong sensitivity of the convective response to the transition scale ε0 in precipitation efficiency [Eq. (2.5)] has also been found because this parameter controls the relative contribution of detrainment effects to the interaction matrix.
An idealized 3 × 3 interaction matrix (sections 4c and 4d) is able to reproduce the character of these results. A singular perturbation expansion is required for describing the convective quasi-equilibrium closure due the fact that the matrix elements related to shallow convection tend to be substantially smaller than those for the interactions between deep convection. As a result, shallow convection tends to respond to large-scale forcing in a singular manner.
e. Further physical implications
An important feature throughout the present analysis is the strong cooling and moistening induced by reevaporation of the detrained cloudy air. When this contribution is suppressed, the convective response under the quasi-equilibrium closure becomes much more reasonable. It is worth noting that some alternative formulations of mixing, beyond the simple entrainment formulation of pure Arakawa and Schubert (1974), may help to alleviate the problem (de Rooy et al. 2013; Yano 2015). Another legitimate way of suppressing this effect is to couple the convection parameterization with a stratiform cloud representation, and to transfer the detrained cloudy convective air to form part of a stratiform cloud rather than immediately reevaporating it into the environment. The importance of this procedure would probably be needless to emphasize, because such a coupling of convection with stratiform clouds is accomplished in most of the operational global models already. However, its significance, to the extent revealed here, appears to be not widely appreciated.
At the same time, completely suppressing the evaporative cooling of the detrained cloudy air would likely not be wise. Yano and Plant (2012b) suggest that the resulting destabilization tendency of shallow convection can be a key mechanism driving transformations from shallow to deep convection. Two solutions may be considered for this remedy. The first is to retain the tendency explicitly for shallow convection, rather than imposing a strict equilibrium constraint. In this case, a singular response of shallow convection to large-scale forcing associated with evaporative cooling must be tamed in a different manner. The second is to transfer the role of this destabilization tendency to the stratiform cloud scheme: the mechanism may be represented by the cloud-top entrainment process (cf. Deardorff 1980; Randall 1980) under this reformulation, which is also expected to lead to an equivalent destabilization.
The quasi-equilibrium closure has been justified based on an argument that an overall time scale for the response of convection to large-scale forcing is so short that we can drop the time tendency of the cloud work function on the left-hand side, which is expected to evolve by following a slow large-scale time scale.
However, more precisely, the response time scale is short only for deep convection, but not for shallow convection. As a result, Eq. (5.3) may be approximated by Eq. (1.1) for the deep convection part only. In other words, the full convective ensemble does not immediately respond to any slow large-scale forcing, as originally envisioned by Arakawa and Schubert (1974). Rather, a finite time scale for the convective response to large-scale forcing should explicitly be taken into account by retaining the temporal tendency of the cloud work function on the left-hand side of Eq. (5.3), so that the closure becomes fully prognostic. Suitable formulations are already in place (e.g., Pan and Randall 1998; Yano and Plant 2012c). Here, we point out a solid reason for moving toward this direction.
The issues appear to be further involved, because observational analyses by Zhang (2002, 2003), Donner and Phillips (2003) suggest that the boundary layer processes controlling the evolution of the convective available potential energy (CAPE), and thus also likely of the cloud work functions, are of a much shorter time scale than those found in the free atmosphere. Thus, boundary layer processes, neglected in the analysis herein, may further contribute to break down a strict application of convective quasi-equilibrium closure. Those implications warrant further investigations.
The present study further suggests needs for reconsidering the mass-flux convection parameterization formulation from more general perspectives. Such investigations are already under way (e.g., Yano et al. 2005b; Yano 2014b, 2016). These developments should more seriously be considered in operational contexts.
APPENDIX
Scale Analysis
The purpose of this appendix is to estimate the order of magnitude of cooling and moistening associated with reevaporation of the detrained cloudy air.
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