# Basic Concepts for Convection Parameterization in Weather Forecast and Climate Models: COST Action ES0905 Final Report

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## Abstract

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## 1. Introduction

#### 1.1. Overview

#### 1.2. A Key Achievement

**T1.1**. This is also considered a good example case for demonstrating the importance of theoretical guidance on the convection parameterization problem.

#### 1.3. Identified Pathways

**T2.4**below) of the QN formulation itself, but nothing to do with number of physical processes incorporated into QN.

#### 1.4. Model Comparisons and Process Studies

**Q1.2.1**,

**Q2.3.1**). This conclusion would have been certainly legitimate for guiding a direction for the further studies of the convective processes. However, Guichard et al. [7] even suggested this transformation process as a key missing element in parameterizations. The difference between these two statements must clearly be distinguished. A suggested focus on the transformation process does not give any key where to look within a parameterization itself: is it an issue of entrainment–detrainment or closure (cf. Section 2.1)?

**T1.1**, but rather in contrast to other earlier attempts. Here, one may argue that the transformation process is ultimately linked to the closure problem. However, how can we see this simply by many process experiments? In addition, how we identify a possible modification of the closure in this manner? It would be similar to asking to Orszag [6] to run many DNSs and figure out a problem in the skewness equation of QN. Note that even a skewness budget analysis of DNSs would not point out the problem in QN. We should clearly distinguish between process studies and the parameterization studies: the latter does not follow automatically from the former.

#### 1.5. Organization of the Action

- Mass-flux based approaches (Section 2.1)
- Non-Mass Flux based approaches (Section 2.2)
- High-Resolution Limit (Section 2.3)
- Physics and Observations (Section 2.4)

- Critical analysis of the strengths and weakness of the state-of-the-art convection parameterizations
- Development of conceptual models of atmospheric convection by exploiting methodologies from theoretical physics and applied mathematics
- Proposal of a generalized parameterization scheme applicable to all conceivable states of the atmosphere
- Defining suitable validation methods for convection parameterization against explicit modeling (CRM and LES) as well as against observations, especially satellite data

## 2. Tasks and Questions

#### 2.1. Mass-Flux Based Approaches

#### 2.1.1. Overview

**T2.4**below) and microphysical processes (including precipitation) can be neglected. The latter must either be drastically simplified in order to make it fit into the above standard formulation, or alternatively, an explicit treatment of convective vertical velocity is required (cf. [17]: see further

**Q2.1.2**below). The last is a hard task by itself under the mass-flux formulation, as reviewed in a book chapter [18].

_{B}(t)

**Q1.3.1**).

**T1.4**). Here, however, note a subtle point that strictly the mass–flux profile, η(z), also changes with time through the change of the entrainment and the detrainment rates by following the change of a large–scale state.

#### T1.1: Review of Current State-of-the-Art of Closure Hypothesis

^{∗}(in their own notation) for the range of 1–6 K against the reported choice of T

^{∗}= 1 K [34]. The reported model improvement does not depend on details of the entrainment-detrainment, either [35].

#### T1.2: Critical Review of the Concept of Convective Quasi-Equilibrium

**Q1.2.1**discusses this issue further, but in short, Arakawa and Schubert’s convective quasi–equilibrium is defined as a balanced state in the cloud–work function budget, which constitutes a part of the convective energy cycle. Importantly, the results from the energy–cycle investigations suggest that the concept of convective quasi–equilibrium could be more widely applicable than is usually supposed, with only a minor extension.

**Q1.7**could be a more serious issue.

#### Q1.2.1: How Can the Convective Quasi-Equilibrium Principle be Generalized to a System Subject to Time-Dependent Forcing? How Can a Memory Effect (e.g., from a Convection Event the Day before) Possibly be Incorporated into Quasi-Equilibrium Principle?

**T1.2**above. Though we are sure that there are lot of technical tests performed at an operational level, none of them is carefully reported in the literature.

_{ij}M

_{Bj}is a rate that the j–th convective type consumes the potential energy (or more precisely, cloud work function) for the i–th convective type, M

_{Bj}is the cloud–base mass-flux for the j–th convective type, and F

_{i}is the rate that large–scale processes produce the i–th convective–type potential energy. Here, N convection types are considered. The matrix elements, K

_{ij}, are expected to be positive, especially for deep convection due to its stabilization tendency associated with warming of the environment by environmental descent.

_{ij}, can be negative due to a destabilization tendency of shallow convection associated with the re–evaporation of detrained cloudy air.

_{i}is the cloud work function, K

_{i}the convective kinetic energy for the i–th convective type and the term D

_{K,i}represents the energy dissipation rate. Randall and Pan [44,45] proposed to take this pair of equations as the basis of a prognostic closure. The possibility is recently re-visited by Plant and Yano [9,41,42] under a slightly different adaptation.

_{i}and M

_{B,i}. Here, a key is to couple shallow and deep convection in this manner, which are typically treated independently in current schemes. Many prognostic formulations for closure have already been proposed in the literature in various forms, e.g., [46]. However, it is important to emphasize that the formulation based on the convective energy cycle presented herein is the most natural extension of Arakawa and Schubert’s convective quasi–equilibrium principle to a prognostic framework.

#### Q1.2.2: Are There Theoretical Formulation Available that could be Used to Directly Test Convective Quasi-Equilibrium (e.g., Based on Population Dynamics)?

**T2.4**, [48]). Thus, it is also natural to ask the question other way round: can we construct and test a closure hypothesis (e.g., convective quasi-equilibrium) based on a more general theory (e.g., population dynamics)?

**Q1.2.1**.

#### T1.3: Proposal for a General Framework of Parameterization Closure

**Q1.3.1**for an alternative possibility.)

_{i}, and convective response, D

_{c,i}, as in the case of the original Arakawa and Schubert’s quasi-equilibrium hypothesis, so that the closure condition can be written as

_{i}+ D

_{c,i}= 0

_{i,j}, in the discrete case, describing the interactions between different convective types, as in the Arakawa and Schubert’s original formulation based on the cloud-work function budget. As a result, the convective response is given by

**Q1.3.2**,

**T3.3**for further discussions (see also [51]).

#### Q1.3.1: Is It Feasible to Re-Formulate the Closure Problem as that of the Lower Boundary Condition of the System? Is It Desirable to Do So?

**T1.1**. See [19] for more.

#### Q1.3.2: How does the Fundamentally Chaotic and Turbulent Nature of Atmospheric Flows Affect the Closure of Parameterizations? Can the Quasi-Equilibrium still be Applied for These Flows?

**Q1.2.1**, is also the best approach for answering this question. In order to elucidate a chaotic behavior we have to take at least three convective modes. Studies have examined the one and two mode cases so far [9,42]. A finite departure from strict convective quasi-equilibrium may also be considered a stochastic process. Such a general framework, the method of homogenization, is outlined, for example, by Penland [55]. Specific examples of the applications include Melbourne and Stuart [56], and Gottwald and Melbourne [57].

#### T1.4: Review on Current State–of–the–Art of Entrainment-Detrainment Formulations

#### T1.5: Critical Review of Existing Methods for Estimating Entrainment and Detrainment Rates from CRM and LES

_{φc}, for an arbitrary physical variable, φ, which is defined by a vertical integral of

**Q1.8**. In principle, better estimates of entrainment–detrainment would be possible by systematically exploiting a model under SCA but without entrainment–detrainment hypothesis. However, such a possibility is still to be fully investigated (cf. [64]).

**T4.3**). Of course, the turbulence scheme must be developed in such a manner that it can give an eddy–transport value consistent with an LES–CRM diagnosis. This is another issue to be resolved (cf.

**T4.3**).

#### Q1.5.1: From a Critical Review of Existing Methods for Estimating Entrainment and Detrainment Rates from CRM and LES, What are the Advantages and Disadvantages of the Various Approaches?

**Figure 1.**A cross section of a thermal plume generated in a laboratory with use of a humidifier as a buoyancy source. Distribution of condensed water is shown by gray tone (courtesy: Anna Gorska and Szymon Malinowski).

**Q1.6.1**for related issues.

**T1.5**for further comparisons of the entrainment–detrainment evaluation methods.

#### T1.6: Proposal and Recommendation for the Entrainment-Detrainment Problem

**T1.5**, such methodologies have already been well established.

- (i)
- Critical fractional mixing ratio originally introduced in a context of a buoyancy sorting theory [70]: the critical fraction is defined as the mixing fraction between convective and environmental air that leads to neutral buoyancy. Mixing with less or more environmental air from this critical fraction leads to positive or negative buoyancy respectively. This division line is expected to play an important role in entrainment–detrainment processes.
- (ii)

**Q1.6.1**next.

#### Q1.6.1: What is the Precise Physical Meaning of Entrainment and Detrainment?

#### Q1.6.2: If They Provide Nothing Other Than Artificial Tuning Parameters, How could they be Replaced with More Physically-Based Quantities?

**T1.5**. In this very respect, entrainment and detrainment are far from artificial tuning parameters, but clearly physically given. On the other hand, as emphasized in

**Q1.6.1**, the basic physical mechanism driving these entrainment–detrainment processes is far from obvious. At least, the original idea of entrainment proposed by Morton et al. [77] for their laboratory convective plumes does not apply to atmospheric convection in any literal sense. Without such a theoretical basis, it may be rather easier to treat them purely as tuning parameters than anything physically based.

_{c}, and the fractional area, σ

_{c}, for convection,

_{c}w

_{c}

_{c}based on the moist–static energy budget (their Equation (11)). In this manner, the mass flux can be evaluated without knowing entrainment and detrainment rates. Note that once the mass flux is known, the entrainment rate may be diagnosed backwards under certain assumptions. (Entrainment would still be required to compute the vertical profile of in-convection variables φ

_{c}via Equation (8)).

**Q1.8**below.

#### Q1.7: How Strong and How Robust is the Observational Evidence for Self-Organized Criticality of Atmospheric Convection?

#### Q1.8: Can a General Unified Formulation of Convection Parameterization be Constructed on the Basis of Mass Fluxes?

- (i)
- entrainment–detrainment hypothesis (cf. Section 2.1.1,
**T1.5**,**Q1.5.1**,**T1.6**,**Q1.6.1**,**Q1.6.2**) - (ii)
- environment hypothesis: the hypothesis that all of the subgrid components (convection) are exclusively surrounded by a special component called the “environment”
- (iii)
- asymptotic limit of vanishing fractional areas for convection, such that the “environment” occupies almost the whole grid–box domain.

#### 2.2. Non-Mass Flux Based Approaches: New Theoretical Ideas

**T1.3**,

**Q1.3.2**,

**T3.3**).

#### Q2.0: Does the Hamiltonian Framework Help to Develop a General Theory for Statistical Cumulus Dynamics?

**Q2.3.5**for further.

#### T2.1: Review of Similarity Theories

#### Q2.1.1: What are the Key Non-Dimensional Parameters that Characterize the Microphysical Processes?

#### Q2.1.2: How can the Correlation be Determined between the Microphysical (e.g., Precipitation Rate) and Dynamical Variables (e.g., Plume Vertical Velocity)?

**Q1.8**).

**Q4.3.1**). The last point further leads us to a more general question: to what extent are microphysical details required for a given situation and a given purpose? Here, the microphysicists tend to emphasize strong local sensitivities to microphysical choices. On the other hand, the dynamicists tend to emphasize a final mean output. Such inclinations can point towards opposite conclusions for obvious reasons, and doubtless we need to find an appropriate intermediate position (cf.

**Q3.4.3**).

#### Q2.1.3: How should a Fully Consistent Energy Budget be Formulated in the Presence of Precipitation Processes?

**Q2.1.3**is typical of the issues that must be addressed when this pathway is pursued. Purely from a point of view of mechanics, this is rather a trivial question: one performs a formal energy integral for the vertical momentum equation. Although it is limited to a linear case, the clearest elucidation of this method is offered by [117]. A precipitation effect would simply be found as a water-loading effect in the buoyancy term. This contribution would be consistently carried over to a final energy-integral result. Furthermore, the water–loading effect can be reintegrated to the “classical buoyancy terms” under a consistent formulation [16,118].

**T2.1**) is based. The moment-based subgrid-scale description has been extensively developed in turbulence studies, with extensive applications in the dry turbulent boundary layer. This theoretical framework works well when the whole process is conservative. Constructing such a strictly conservative theory becomes difficult for the moist atmosphere, due to the existence of differential water flux [16,118,119], and once a precipitation process starts, the whole framework, unfortunately, breaks down even under standard approximations.

**T2.2**and the following questions) provides a more straightforward description for the evolution of the water distribution under precipitation processes so long as the processes are described purely in terms of a single macrophysical point. Note that the precipitation process itself would be more conveniently treated under a traditional moment-based description as a part of the eddy transport.

#### T2.2: Review of Probability-Density Based Approaches

#### Q2.2.1: How can Current Probability-Density Based Approaches be Generalized?

**Q2.2.5**. On the other hand, the transport part (eddy transport) is handled based on the moment-based description, invoking an assumed pdf approach (cf.

**Q2.2.3**).

#### Q2.2.2: How can Convective Processes be Incorporated into Probability-Based Cloud Parameterizations? Can Suitable Extensions of the Approach be Made Consistently?

**T2.4**). In general, it is not obvious how to describe convective evolution in terms of higher–order moments (e.g., skewness) regardless of whether the issues are handled in a self–contained manner or under a coupling with an independent convection scheme. The difficulty stems from a simple fact that a spatially–localized high water concentration associated with deep convective towers is not easily translated into a quantitative value of skewness.

#### Q2.2.3: Is the Moment Expansion a Good Approximation for Determining the Time-Evolution of the Probability Density? What is the Limit of This Approach?

#### Q2.2.4: Could the Fokker-Planck Equation Provide a Useful General Framework?

#### Q2.2.5: How can Microphysics be Included Properly into the Probability-Density Description?

**Q2.2.1**. Here, the assumed pdf approach becomes rather awkward, because it is hard to include microphysical processes (a process conditioned by a physical–space point) into moment equations.

#### T2.3: Assessments of Possibilities for Statistical Cumulus Dynamics

**Q1.7**). It is often argued that subgrid-scale parameterization is fundamentally “statistical” in nature (cf. [49]). However, little is known of the statistical dynamics for atmospheric convective ensembles. This is a domain that is clearly under–investigated. If we really wish to establish convection parameterization under a solid basis, this is definitely where much further work is needed. The present Action has initiated some preliminary investigations. Especially, we have identified renormalization group theory (RNG) as a potentially solid starting point [11]. We strongly emphasize the importance of more extensive efforts towards this direction.

#### Q2.3.1: How can a Standard, “Non-Interacting”, Statistical Description of Plumes be Generalized to Account for Plume Interactions?

**Q1.2.1**, this is a formulation that can be relatively easily implemented into operational models as well. Considering more direct interactions between convective elements is straightforward under the SCA framework (cf.

**Q1.8**). A key missing step is to develop a proper statistical theory under this framework.

#### Q2.3.2: How can Plume-Plume Interactions and Their Role in Convection Organization be Determined?

**Q2.3.5**).

#### Q2.3.3: How can the Transient, Life-Cycle Behavior of Plumes be Taken Into Account for the Statistical Plume Dynamics?

**T1.1**,

**Q1.2.1**for reservations for advancing towards this direction.

#### Q2.3.4: How can a Statistical Description be Formulated for the Two-Way Feedbacks between Convective Elements and Their “Large-Scale” Environment?

**Q1.2.1**would be the best candidate for this goal. Technically, it is straightforward to couple this convective energy-cycle system with simple models for large-scale tropical dynamics. The is an important next step to take.

#### Q2.3.5: How can Statistical Plume Dynamics Best be Described Within a Hamiltonian Framework?

#### T2.4: Proposal for a Consistent Subgrid-Scale Convection Formulation

- (1)
- Self-consistency
- (2)
- Consistency with physics

**Q2.2.2**provide a good example for further considering the issues of self–consistency of parameterization.

**Q2.2.2**. The first is to establish mutual consistency between the convective and the cloud schemes. “Consistency” here means a logical consistency by writing the same physical processes in two different ways within two different parameterizations. More precisely, in this case, an “equivalence” of the logic must be established. A classic example of such an equivalence of logic is found in quantum mechanics between the matrix–based formulation of Heisenberg and the wave–equation based formulation of Schrödinger. The equivalence of the formulations may be established by a mathematical transformation between the two. Such a robust equivalence is hardly established in parameterization literature.

**Q1.8**).

**Q3.4.2**for more).

#### 2.3. High-Resolution Limit

#### T3.1: Review of State-of-the-Art of High-Resolution Model Parameterization

**T.3.3**for further discussions.

#### T3.2: Analysis Based on Asymptotic Expansion Approach

**Q1.8**). For a parameter for the asymptotic expansion, we may take the fractional area, σ

_{c}, occupied by convection. This is a standard small parameter adopted in mass-flux convection parameterization, which is taken to be asymptotically small.

#### T3.3: Proposal and Recommendation for High-Resolution Model Parameterization

**T3.2**.

_{c}→ 0. This strategy, currently adopted at ECMWF [131], may be justified at the most fundamental level, by the fact that a good asymptotic expansion often works extremely well even when an expansion parameter is re-set to unity. Such a behavior can also be well anticipated for mass-flux convection parameterization. At the practical level, what the model can actually resolve (i.e., the effective resolution) is typically more than few times larger than a formal model resolution, as defined by a grid-box size, due to the fact that a spatial gradient must be evaluated numerically by taking over several grid points.

**T4.3**below, careful efforts are made in these studies to avoid any double counting in the interactions between the otherwise–competing computations of thermodynamic adjustment and convective latent heat release, as well as latent heat storage for downdrafts.

**Q2.1.3**). A hybrid approach combining the traditional moment-based approach and a subgrid-scale distribution is adopted for this purpose. When a relatively simple distribution is pre-assumed for a latter, a closed formulation was developed relatively easily at DWD.

**T2.4**).

#### 2.3.1. More General and Flexible Parameterization at Higher Resolutions

- (1)
- Start from the basic laws of physics (and chemistry: cf.
**T4.3**) - (2)
- Perform a systematic and logically consistent deduction from the above (cf.
**T2.4**) - (3)
- Sometimes it may be necessary to introduce certain approximations and hypotheses. These must be listed carefully so that you would know later where you introduced them and why.

**T4.3**).

**T2.1**). SCA introduced in

**Q1.8**may be considered a special application of mode decomposition.

_{t}) and thus dry-air content (q

_{d}= 1 − q

_{t}) are constant for a moving parcel in the absence of sources and sinks. Marquet [14] proposes a more general definition of specific moist-air entropy which can be computed directly from the local, basic properties of the fluid and which is valid for the general case of barycentric motions of open fluid parcels, where both q

_{t}and q

_{d}vary in space and in time. Computations are made by applying the third law of thermodynamics, because it is needed to determine absolute values of dry-air and water-vapor entropies independently of each other. The result is that moist-air entropy can be written as s = s

_{ref}+ c

_{pd}ln(θ

_{s}), where s

_{ref}and c

_{pd}are two constants. Therefore, θ

_{s}is a general measure of moist-air entropy. The important application shown by Marquet and Geleyn [14,15,16,118,129] is that values and changes in θ

_{s}are significantly different from those of θ

_{l}and θ

_{e}if q

_{t}and q

_{d}are not constant. This is especially observed in the upper part of marine stratocumulus and, more generally, at the boundaries of clouds.

_{s}is at the same time: (1) a Lagrangian tracer; and, (2) a state function of an atmospheric parcel. All previous proposals in this direction fulfilled only one of the two above properties. Furthermore, observational evidence for cases of entropy balance in marine stratocumulus shows a strong homogeneity of θ

_{s}, not only in the vertical, but also horizontally: i.e., between cloudy areas and clear air patches. It is expected that these two properties could also be valid for shallow convection, with asymptotic turbulent and mass-flux-type tendencies being in competition with diabatic heating rates.

**T3.3**(3MT and TKESV) are perfectly compatible with this new type of thinking.

#### Q3.4: High–Resolution Limit: Questions

#### Q3.4.1: Which Scales of Motion should be Parameterized and under Which Circumstances?

- (i)
- Any process in question cannot be characterized by a single scale (or wavenumber), but is more likely to consist of a continuous spectrum. In general, a method for extracting a particular process of concern is not trivial.
- (ii)
- Whether a process is well resolved or not cannot be simply decided by a given grid size. In order for a spatial scale to be adequately resolved, usually several grid points are required. As a corollary of this, and of point (i), the grid size required depends on both the type of process under consideration and the numerics.
- (iii)
- Thus the question of whether a process is resolved or not is not a simple dichotomic question.

**T3.3**).

**Q1.6.2**.

#### Q3.4.2: How can Convection Parameterization be Made Resolution-Independent in order to Avoid Double-Counting of Energy-Containing Scales of Motion or Loss of Particular Scales?

**T3.3**,

**Q3.4.1**.

**Q3.4.1**). In this respect, the best strategy for avoiding a double counting is to keep a consistency of a given parameterization with an original full system. For example, in order to avoid a double counting of energy–containing scales, a parameterization should contain a consistent energy cycle.

**T2.4**), as already emphasized, in order to avoid these operational difficulties.

#### Q3.4.3: What is the Degree of Complexity of Physics Required at a Given Horizontal Resolution?

**T4.2**).

#### 2.4. Physics and Observations

#### T4.1: Review of Subgrid-Scale Microphysical Parameterizations

**Q2.1.2**.

#### 2.4.1. Further Processes to be Incorporated into Convection Parameterizations

#### 2.4.1.1. Downdrafts

**Q1.8**).

#### 2.4.1.2. Cold Pools

**Q1.2.1**(see also [9,42]), deep convection is induced from shallow convection by the tendency of the latter continuously destabilizing its environment by evaporative cooling of non–precipitating water as it detrains. More precisely, this process increases both the available potential energy and the cloud work function for deep convection, and ultimately leads to an induction of deep convection, as manifested by a sudden increase of its kinetic energy. In a more realistic situation, the evaporative cooling may more directly induce convective downdrafts, which may immediately generate cold pools underneath. These transformations are clearly associated with an induction of kinetic energy from available potential energy. However, the existing literature does not tell us whether those pre-existing kinetic energy associated either with downdraft or cold pool is transformed into deep–convective kinetic energy, for example, by pressure force, or alternatively, an extra process is involved in order for a cold pool to trigger convection. The CRM can be used for diagnosing these energy–cycle processes more precisely. An outline for such a method is described by Figure 5 and associated discussions in Yano et al. [101]. However, unfortunately, this methodology has never been applied in detail even by the original authors.

**T1.1**,

**Q1.2.1**). As also emphasized in Section 2.1.1, in order to introduce a trigger process into a parameterization, a radical modification of its formulation is required.

#### 2.4.1.3. Topography

#### 2.4.2. Link to the Downscaling Problem

#### T4.2: Proposal and Recommendation on Observational Validations

**Q2.1.2**,

**Q3.4.3**). We especially refer to Jaynes [140] and Gregory [141] for the basics of the probability as an objective measure of uncertainties. From the point of view of probability theory, the goal of the model verification would be to reduce the model uncertainties by objectively examining the model errors. In order to make such a procedure useful and effective, forecast errors and model uncertainties must be linked together in a direct and quantitative manner. Unfortunately, many of the statistical methods found in general literature are not satisfactory for this purpose. The Bayesian principle (op. cit.) is rather an exception that can provide such a direct link so that from a given forecast error, an uncertainty associated with a particular parameter in parameterization, for example, can objectively and quantitatively estimated. The principle also tells us that ensemble, sample space, randomization, etc. as typically employed in statistical methods are not indispensable ingredients for uncertainty estimates, although they may be useful. Though there are already many applications of Bayesian principle to atmospheric science, the capacity of Bayesian for linking between the statistical errors and physical processes has not much been explored (cf. [179,180]). See also

**Q3.4.3**for a link to issues of required model complexity and uncertainties.

#### T4.3: Proposal and Recommendation for a Parameterization with Unified Physics

**T2.4**, Section 2.3.1, and [101] for further discussions.

**T2.4**. By relaxing SCA, it is straightforward to take into account of a certain distribution over a particular subgrid–scale component (segment); especially, over the environment. Such an idea (EDMF) is first introduced by Soares et al. [54], and an SCA procedure can derive such a formulation in a more self–consistent manner (cf.

**T1.5**).

**T1.5**). In order to maintain overall consistency, a fractional area occupied by a given subgrid–scale component must explicitly be added to the formulation. Note that this pre–factor is usually neglected in standard non–mass–flux parameterizations assuming that convection occupies only an asymptotically vanishing fraction. In the high resolution limit, this assumption becomes no longer true, as already discussed in

**T3.2**and

**T3.3**.

#### Q4.3.1: How can a Microphysical Formulation (Which is by Itself a Parameterization) be Made Resolution Dependent?

#### Q4.3.2: Can Detailed Microphysics with Its Sensitivity to Environmental Aerosols be Incorporated into a Mass-Flux Convection Parameterization? Are the Current Approaches Self-Consistent of Not? If Not, How can It be Achieved?

#### 2.4.3. How Can Observations Be Used for Convection Parameterization Studies?

## 3. Conclusions: Retrospective and Perspective

**T1.1**). MoU also provides a false anticipation (the 4th secondary objective: cf. Section 1.5) that extensive process studies by CRM and LES would by themselves automatically lead to improvements of parameterization. In the end of the present Action, we openly admit our misjudgment. Though the value of the process studies should hardly diminish in its own right, they do not serve for a purpose of parameterization improvements by themselves automatically unless we approach to the problem from a good understanding of the latter. This point is already extensively discussed in Sections 1.3, 1.4, 2.4.1,

**Q3.4.3**and re–iterated again below.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix: Derivation of the Result for T3.2

**T3.2**. For the basics of the mass–flux formulation, readers are advised to refer to, for example, [12]. For simplicity, we take a bulk case that the subgrid processes only consist of environment and convection with the subscripts e and c, respectively.

_{e}φ

_{e}+ σ

_{c}φ

_{c}

_{e}and σ

_{c}are the fractional areas occupied by environment and convection, respectively. Clearly,

_{e}+ σ

_{c}= 1

_{e}+ σ

_{c}(φ

_{c}− φ

_{e})

_{c}→ 0 is taken. As a result, the grid–point value may be approximated by the environmental value in this limit

_{e}

_{c}. For this purpose, we expand physical variables by σ

_{c}. For example, the grid–point value is expanded as

^{(0)}+ σ

_{c}φ

^{(1)}+ · · ·

^{(0)}= φ

_{e}

^{(1)}= φ

_{e}

^{(0)}− φ

_{e}

_{e}, can be most conveniently integrated directly in time without expanding in σ

_{c}by the equation:

_{e}and

**u**

_{e}are the environmental vertical and horizontal velocities, and F is the forcing term for a given physical variable. The time integration of Equation (A8) is essentially equivalent to that for the grid–point equation under the standard approximation (cf. Equation (7.8) in [12]), but with the grid–point value replaced by the environmental value.

_{c}. Here, we expect that the forcing, σ

_{c}F

_{c}, in convective scale is a leading–order quantity, thus we expand this term as

_{c}F

_{c}= F

_{c}

^{(0)}+ σ

_{c}F

_{c}

^{(1)}+ · · ·

_{c}, we obtain a correction equation for the above as

_{c}

^{(1)}. Furthermore, they are modified by a temporal tendency, ∂φ

_{c}

^{(0)}/∂t, already known from the leading order, given in the right hand side. It immediately transpires that this diagnostic procedure is nothing other than performing a time integral of the whole solution under a special “implicit” formula, thus the statement in

**T3.2**follows.

## References

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Yano, J.; Geleyn, J.-F.; Köhler, M.; Mironov, D.; Quaas, J.; Soares, P.M.M.; Phillips, V.T.J.; Plant, R.S.; Deluca, A.; Marquet, P.;
et al. Basic Concepts for Convection Parameterization in Weather Forecast and Climate Models: COST Action ES0905 Final Report. *Atmosphere* **2015**, *6*, 88-147.
https://doi.org/10.3390/atmos6010088

**AMA Style**

Yano J, Geleyn J-F, Köhler M, Mironov D, Quaas J, Soares PMM, Phillips VTJ, Plant RS, Deluca A, Marquet P,
et al. Basic Concepts for Convection Parameterization in Weather Forecast and Climate Models: COST Action ES0905 Final Report. *Atmosphere*. 2015; 6(1):88-147.
https://doi.org/10.3390/atmos6010088

**Chicago/Turabian Style**

Yano, Jun–Ichi, Jean-François Geleyn, Martin Köhler, Dmitrii Mironov, Johannes Quaas, Pedro M. M. Soares, Vaughan T. J. Phillips, Robert S. Plant, Anna Deluca, Pascal Marquet,
and et al. 2015. "Basic Concepts for Convection Parameterization in Weather Forecast and Climate Models: COST Action ES0905 Final Report" *Atmosphere* 6, no. 1: 88-147.
https://doi.org/10.3390/atmos6010088