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  • View in gallery

    Conceptualization of (a) self-organized criticality and (b) homeostasis. The solid curves schematically show expected dependence of a system response P (e.g., precipitation rate), on an internal-state variable I (e.g., column-integrated water), under respective mechanisms.

  • View in gallery

    Snapshots showing time evolution of the simulation. (top) The precipitation rate (mm h−1) in logarithmic units and (bottom) column-integrated total water (mm) for days (a) 7, (b) 8, (c) 9, (d), 10, and (e) 16.

  • View in gallery

    Distribution of number occurrence as a function of column-integrated water (horizontal axis, mm) and the precipitation rate (vertical axis, mm h−1). Three options for column-integrated water are (a) the water vapor, (b) the total water, and (c) the condensed water. The bin size taken for the column water is 2.5 mm for (a) and (c), and 5 mm for (b), whereas the bin size for the precipitation rate is fixed at 10 mm h−1 in all cases. The solid curve in each panel shows the mean precipitation rate defined for each column-integrated water bin. Error bars are shown for the last few bins for the column-integrated water in order to indicate the uncertainties of the mean curve.

  • View in gallery

    The precipitation rate as in Fig. 3, but the results are by applying space averaging with an increment of a factor of 2 up to a 128-km scale (4, 8, 16, 32, 64, 128 km). The mean precipitation rate for a given column-integrated water is shown by marking a cycle with +, *, ○, × with increasing spatial scales (starting from the full resolution). The full-resolution case is shown with both the plus sign and the solid curve with a slight smoothing.

  • View in gallery

    As in Fig. 4, but for the number density.

  • View in gallery

    Time series of (a) domain-mean precipitation rate and (b) the domain-mean column-integrated water vapor (solid), total water (long dash), and saturated water vapor (short dash).

  • View in gallery

    Time evolution of the mean precipitation rate as a function of (a) column-integrated water vapor, (b) column-integrated total water, and (c) column-integrated condensed water. The mean precipitation rate is calculated as a function of the column-integrated water. Individual snapshot model outputs are shown every 2 days for the whole simulation period by a scatterplot. Four symbols (+, *, ○, ×) are used. Also shown are the curves for averages over 0–8 (solid), 10–18 (long dash), and 20–28 days (short dash).

  • View in gallery

    Time evolution of the number density distribution shown by scatterplots in 2-day intervals with a cyclic use of the symbols +, *, ○, and × for (a) Iυ, (b) It, and (c) Ic. Also shown are the curves averaged over 0–8 (solid), 10–18 (long dash), and 20–28 days (short dash). A chain-dash curve for average over 9–11 days is also added in (a).

  • View in gallery

    Analysis of time-averaged precipitation as a function of column-integrated water vapor with (a) 6- and (b) 12-hourly averages, and as a function of column-integrated total water with (c) 6- and (d) 12-hourly average. Number density is shown in the same format as in Fig. 3.

  • View in gallery

    Number of occurrences for a given precipitation rate (horizontal axis, mm h−1) and tendency (vertical axis, mm h−1) for the (a) column-integrated water vapor, (b) total water, and (c) condensed water. The number count is made over a bin of sizes 20 and 10 mm h−1 for the precipitation rate and the tendency, respectively. Counting (statistics) is limited to a high column-integrated water regime: (a) Iυ > 80 mm, (b) It > 100 mm, and (c) Ic > 30 mm.

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Self-Organized Criticality and Homeostasis in Atmospheric Convective Organization

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  • 1 GAME/CNRM, Météo-France and CNRS, Toulouse, France
  • 2 National Center for Atmospheric Research,* Boulder, Colorado
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Abstract

Atmospheric convection has a tendency to organize on a hierarchy of scales ranging from the mesoscale to the planetary scales, with the latter especially manifested by the Madden–Julian oscillation. The present paper examines two major competing mechanisms of self-organization in a cloud-resolving model (CRM) simulation from a phenomenological thermodynamic point of view.

The first mechanism is self-organized criticality. A saturation tendency of precipitation rate with increasing column-integrated water, reminiscent of critical phenomena, indicates self-organized criticality. The second is a self-regulation mechanism that is known as homeostasis in biology. A thermodynamic argument suggests that such self-regulation maintains the column-integrated water below a threshold by increasing the precipitation rate. Previous analyses of both observational data as well as CRM experiments give mixed results.

In this study, a CRM experiment over a large-scale domain with a constant sea surface temperature is analyzed. This analysis shows that the relation between the column-integrated total water and precipitation suggests self-organized criticality, whereas the one between the column-integrated water vapor and precipitation suggests homeostasis. The concurrent presence of these two mechanisms is further elaborated by detailed statistical and budget analyses. These statistics are scale invariant, reflecting a spatial scaling of precipitation processes.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Jun-Ichi Yano, GAME/CNRM, Météo-France, 42 Ave. Gaspard Coriolis, 31057 Toulouse CEDEX, France. E-mail: jun-ichi.yano@meteo.fr

Abstract

Atmospheric convection has a tendency to organize on a hierarchy of scales ranging from the mesoscale to the planetary scales, with the latter especially manifested by the Madden–Julian oscillation. The present paper examines two major competing mechanisms of self-organization in a cloud-resolving model (CRM) simulation from a phenomenological thermodynamic point of view.

The first mechanism is self-organized criticality. A saturation tendency of precipitation rate with increasing column-integrated water, reminiscent of critical phenomena, indicates self-organized criticality. The second is a self-regulation mechanism that is known as homeostasis in biology. A thermodynamic argument suggests that such self-regulation maintains the column-integrated water below a threshold by increasing the precipitation rate. Previous analyses of both observational data as well as CRM experiments give mixed results.

In this study, a CRM experiment over a large-scale domain with a constant sea surface temperature is analyzed. This analysis shows that the relation between the column-integrated total water and precipitation suggests self-organized criticality, whereas the one between the column-integrated water vapor and precipitation suggests homeostasis. The concurrent presence of these two mechanisms is further elaborated by detailed statistical and budget analyses. These statistics are scale invariant, reflecting a spatial scaling of precipitation processes.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Jun-Ichi Yano, GAME/CNRM, Météo-France, 42 Ave. Gaspard Coriolis, 31057 Toulouse CEDEX, France. E-mail: jun-ichi.yano@meteo.fr

1. Introduction

The present paper infers a basic mechanism for organized large-scale convection simulated by a cloud-resolving model (CRM). In general, self-organization is a consequence of strong interactions between many elements of a given complex system, with two of the mechanisms being criticality and homeostasis. Criticality spontaneously generates complex organized structures as invoked to explain self-organization (cf. Bak 1996; Sornette 2000), whereas homeostasis maintains the stability of some organized form as in physiology and biology (Cannon 1929, 1932).

To quote Wilson (1983, p. 583), “There are a number of problems in science which have, as a common characteristic, that complex microscopic behavior underlines macroscopic effects.” These problems are commonly called critical phenomena (Stanley 1971), associated with a wide range of interacting scales with a tendency for self-organization. It is tempting to consider atmospheric convection as an example of a critical phenomenon (Yano et al. 2012), considering the extensive roles of microscale cloud physics (Pruppacher and Klett 1997), and the tendency for convection to organize on scales up to planetary (Yano 1998; Moncrieff 2010).

Unlike purely thermodynamic problems, such as phase transition, these macroscale problems represent a further tendency to self-maintain at criticality instead of being simply controlled by an external parameter such as temperature. For this reason, the term self-organized criticality (cf. Bak et al. 1987) has been proposed to describe such macroscopic critical phenomena, such as organized structures associated with atmospheric convection.

One of the common quantitative features found with self-organized critical phenomena is the 1/f noise spectrum, where f is the frequency (Sornette 2000). Considering that 1/f noise variability is found in convective time series (Yano et al. 2001, 2004), atmospheric convection could be an example of self-organized criticality. This possibility is further supported by Peters and Neelin (2006) and Neelin et al. (2008) based on the analysis of the relationship between column-integrated water and precipitation rate.

In contrast, homeostasis has been proposed for the physiological–biological systems as a basic mechanism for maintaining internal stability. Thus, it is a self-regulating mechanism, or a coordinated response of their parts to external stimuli, that would disturb their normal function (Cannon 1929, 1932). More specifically, the mechanism is invoked for explaining the constancy of biological bodies, such as acidity, salinity, and the other compositions of the blood, body temperature, etc. The present paper is a generalization of this concept that adheres strictly to its etymology: the prefix homeo- means “similar” or “like” in Latin, whereas -stasis comes from Greek meaning “standstill.” Thus, the word can be translated as “quasi steady” or “quasi equilibrium.” In climate science recall that Arakawa and Schubert (1974) proposed convective quasi equilibrium as a principle for approximating convective clouds in climate models.

Raymond [2000; see also Raymond et al. (2009)] proposed a relationship between column-integrated water and precipitation rate based on a thermodynamic argument of how the column water should be self-regulated by precipitation. Assuming that tropical precipitation is self-regulating, this hypothesis suggests an analogy with biological homeostasis.

The present paper considers these two organization mechanisms—self-organized criticality and homeostasis—on an equal footing by suggesting a phenomenological interpretation based on a one-variable dynamical system (cf. Guckenheimer and Holmes 1983). By examining only a single variable at a “macroscopic” level, we elucidate how microscopic variables work integratively as in thermodynamic phase transition. These two competing mechanisms of self-organization are schematized by Fig. 1. Such a one-variable interpretation for self-organized criticality (cf. Fig. 1a) was presented by Peters and Neelin (2006). We further encompass Raymond’s thermodynamic control hypothesis (homeostasis: Fig. 1b) by extending Peter and Neelin’s interpretation of self-organized criticality.

Fig. 1.
Fig. 1.

Conceptualization of (a) self-organized criticality and (b) homeostasis. The solid curves schematically show expected dependence of a system response P (e.g., precipitation rate), on an internal-state variable I (e.g., column-integrated water), under respective mechanisms.

Citation: Journal of the Atmospheric Sciences 69, 12; 10.1175/JAS-D-12-069.1

A critical phenomenon occurs beyond a critical point, as indicated by Ic in Fig. 1a, for some internal-state variable I indicated by the horizontal axis. Beyond Ic, a system response such as precipitation rate P in the present study, increases with the increasing I. A critical point is typically associated with dynamical instability. Thus, once the system exceeds the critical point, dynamical instability drives a system away from the original equilibrium state, P = 0 (nonprecipitating state herein), with an increasing tendency for self-organization. An important qualitative aspect of the critical phenomenon is that the response, P is asymptotically bounded from above (Fig. 1a). Consequently, I continues to increase without bound when an external supply continues to increase, reflecting an inherent instability of the system under criticality. More quantitatively, the system response accords to a power law (cf. Stanley 1971),
e1
above the critical point with a constant α less than unity.
In contrast, homeostasis, depicted in Fig. 1b, maintains stability via negative feedback: I remains below some threshold Ic by an explosively increasing response P as I approaches the threshold from below. Raymond’s (2000) thermodynamic control theory hypothesizes such a relationship between the column-integrated water I and the precipitation rate P. More precisely, in its revised form (Raymond et al. 2007), this assumes a relation, defined as
e2
with Ic as a threshold value. Note that the precipitation rate diverges toward IIc. Thus, as an increasing external supply drives the system toward IIc, the precipitation rate keeps increasing so that the system never exceeds the threshold. The given singular negative feedback maintains the state below Ic, where the system can maintain self-organization in a stable manner.

The two relationships in Eqs. (1) and (2) suggest that qualitatively different physical processes contribute to the self-organization. We demonstrate how physical processes behind the self-organization can be elucidated by analyzing the IP relationship with these two contrasting physical mechanisms in mind. We interpret these two relationships in a qualitative manner. For example, in homeostasis, the tendency toward IIc should not be taken too literally, because any meteorological data would not strictly attain a singularity. However, the important aspect of homeostasis is that a system state I tends to be bounded from above due to a very steep increase of the response P with the increasing I. This tendency contrasts with that of self-organized criticality, in which P is asymptotically bounded from above. Thus, I may keep increasing without bound under a strong external supply. We should also keep in mind that the IP relationship does not prove a system is at self-organized criticality or homeostasis. For this reason, in the following we present some additional evidence in order to strengthen our arguments.

Previous analyses of both observational data as well as CRM experiments (which resolve mesoscale organization) do not definitely show whether the IP relationship is more consistent with self-organized criticality or homeostasis. As far as the authors are aware, no clear-cut reproduction of the original satellite data analysis by Peters and Neelin [2006; see also Neelin et al. (2008)] has been published. Conventional observational data (Bretherton et al. 2004), as well as some CRM experiments (Raymond et al. 2007, 2009), tend to support Raymond’s (2000) “homeostasis” hypothesis, as stated by Eq. (2).

More recent analyses give mixed conclusions. Holloway and Neelin (2010, p. 1095) investigated Atmospheric Radiation Measurement Program (ARM) data and concluded that it was “impossible to test the power-law relationship at high CWV [column water vapor].” A large-domain CRM experiment by Posselt et al. (2012) produced a scatterplot for precipitation that splits into two directions suggestive of both possibilities in Figs. 1a and 1b. Global model data analysis by Bechtold (2008) does show a flattening tendency of the precipitation rate as a function of column-integrated water for the largest values of column water (his Fig. 17), as expected from criticality. However, at lower values, this flattening is preceded by an exponential curve for precipitation rates varying over two orders of magnitude, consistent with a homeostasis tendency.

We analyze a CRM experiment with a constant sea surface temperature (SST) over a domain big enough to crudely emulate a planetary scale. A constant SST is especially important for testing these two theoretical mechanisms proposed for a homogeneous external state. Large-scale convective organization evolves over the first 10 days of the experiment. We present a more consistent picture of the relationship between the column-integrated water I and the precipitation rate P. By taking advantage of the completeness of a CRM dataset on the mesoscale, the three types of the column-integrated water are considered for this purpose: water vapor, total water, and condensed water.

The simulation is outlined in section 2. The following two sections are devoted to the analysis: Section 3 presents the relationship between the column-integrated water and the precipitation rate. Both space- and time-scale dependencies are examined. Section 4 examines the water budget in a search for physical interpretations. The paper is concluded by a discussion in section 5.

2. Simulation

The nonhydrostatic Eulerian/semi-Lagrangian anelastic model developed by Smolarkiewicz and Margolin (1997) is run over a channel domain with sizes 6000, 3200, and 25 km in longitude, latitude, and vertical directions, respectively. Grid spacings are 4 km and 500 m in the horizontal and vertical directions, respectively. The boundary conditions are periodic in longitude and rigid at the latitudinal boundaries. Rigid-lid and free-slip conditions are imposed at the top and the bottom of the domain. To damp unphysical wave reflections, 200-km-wide and 7-km-deep absorption layers are specified at the latitudinal and the upper boundaries, respectively. An equatorial β plane is assumed with the equator at the center of the latitudinal domain.

The model includes parameterization of mixed-phase bulk microphysics, first-order subgrid-scale turbulence for the boundary layer, and bulk aerodynamic-based surface flux. Constant and horizontally homogeneous radiative cooling of −1.5 K day−1 is imposed below the 12-km level, decreasing nonlinearly with height above this level, and set to zero above 18 km. A horizontally uniform SST of 302.5 K is assumed.

The initial condition is a state of rest with moisture and temperature profiles that are representative of the tropical western Pacific. The model is integrated for 28 days, and outputs of precipitation and column-integrated water (for the three types as defined below) every 6 h are analyzed. The 200-km-wide sponge layers at both latitudinal walls are excluded from this analysis. For each snapshot 700 × 1500 columns are analyzed, in total 1.176 × 108 column samples.

Figure 2 shows the evolution of rainfall (top row of each panel) and the column-integrated total water (bottom row of each panel). After formation of the initial cloud clusters at day 7 (Fig. 2a), convective systems rapidly organize upscale into large clusters in about 3 days (Figs. 2b–d), and within another few days into a single large organized system (Fig. 2e). This large-scale convective organization is maintained for the remainder of the simulation, propagating eastward at 8.5 m s−1, broadly similar to observed large-scale convective organization (Madden and Julian 1971; Nakazawa 1988; Yano 1998; Moncrieff 2010). An important feature to note in the coherent forms is a tendency for more extreme values for the precipitation rate and column-integrated total water. This feature is interpreted to be a signature of self-organized criticality, as shown in the following sections.

Fig. 2.
Fig. 2.

Snapshots showing time evolution of the simulation. (top) The precipitation rate (mm h−1) in logarithmic units and (bottom) column-integrated total water (mm) for days (a) 7, (b) 8, (c) 9, (d), 10, and (e) 16.

Citation: Journal of the Atmospheric Sciences 69, 12; 10.1175/JAS-D-12-069.1

3. Analysis: IP relationship

As discussed in the introduction, we take I as an internal-state variable for analysis. We examine how this variable controls the whole precipitation processes by examining the dependence of P, on I. In regard to I, we consider three options: (i) water vapor (only); (ii) total water, including the water condensate; and (iii) total condensate (i.e., the liquid and solid parts of water), distinguished by the subscripts, υ, t, and c. Note that by definition, It = Iυ + Ic.

The first option was taken by Peters and Neelin (2006). However, the second option is preferable for obtaining a closed budget of column-integrated water that has precipitation as a loss term (cf. section 4). The third option more closely examines the cloud–physical behavior of convective systems. Through the relation Ic = ItIυ, the budget of the condensed water complements the first two.

The first option may be normalized by using for the saturated atmospheric column with the same temperature profile. This quantity was called the “saturation fraction” by Bretherton et al. (2004). The basic idea of the saturation fraction is to provide a convenient normalization for the column-integrated water vapor such as relative humidity. The saturated column-integrated water vapor is expected to be a relatively stable normalization factor, because it varies weakly over time and space in the tropical atmosphere due to the weak horizontal temperature gradients (cf. Sobel et al. 2001; Yano and Bonazzola 2009). However, in our simulation, we found is not sufficiently time invariant. Note that the column-integrated water vapor Iυ is an unbounded variable, in the sense that the atmosphere can hold progressively more water vapor as it warms. In the present simulation, the warming tendency is not negligible (cf. Fig. 7b to follow): by artificially introducing an upper bound, the saturation fraction cannot detect an extremely warm and moist atmospheric column as a manifestation of self-organized criticality. For these reasons, we decided not to present the saturation fraction results, although we did include it in most of the subsequent analyses.

a. Time-mean statistics

We first present the relationship between the column-integrated water and precipitation rate for the entire simulation period. The self-organized criticality identified by Peters and Neelin (2006) consists of a tendency of the mean precipitation rate for a given column-integrated water to follow a critical curve, defined by Eq. (1). The issue for the present subsection is whether our CRM experiment reproduces that result.

With a slight modification of the analysis method of Peters and Neelin (2006), in Fig. 3 we show the distribution of number occurrence as a function of the column-integrated water (horizontal axis) and the precipitation rate (vertical axis). The mean curve for the precipitation rate defined for each column-integrated water bin is shown along with the error bars for the last few bins in extreme states (toward the right end of the axis). Here, the error bar is estimated by σP(I)/n(I)1/2 in terms of the standard deviation σP(I) of the precipitation rate for a given I bin and the sample number n(I) for the given I bin. Recall that the statistical uncertainty of a measurement scales by n−1/2.

Fig. 3.
Fig. 3.

Distribution of number occurrence as a function of column-integrated water (horizontal axis, mm) and the precipitation rate (vertical axis, mm h−1). Three options for column-integrated water are (a) the water vapor, (b) the total water, and (c) the condensed water. The bin size taken for the column water is 2.5 mm for (a) and (c), and 5 mm for (b), whereas the bin size for the precipitation rate is fixed at 10 mm h−1 in all cases. The solid curve in each panel shows the mean precipitation rate defined for each column-integrated water bin. Error bars are shown for the last few bins for the column-integrated water in order to indicate the uncertainties of the mean curve.

Citation: Journal of the Atmospheric Sciences 69, 12; 10.1175/JAS-D-12-069.1

Figure 3a corresponds to the case considered by Peters and Neelin (2006). Unlike their results, we do not see a clear tendency for the precipitation rate to saturate toward the high column-integrated water vapor state. Instead, the precipitation rate increases at a faster rate as the column water vapor increases. While the mean value for the last bin shows a slight dip, it is insignificant due to the small sample size; that is, the uncertainty of data for this bin is larger than the dip. Also, a saturation tendency is difficult to detect in the number density distribution as a whole. Therefore, we conclude that homeostasis (thermodynamic control)—rather than self-organized criticality—is salient for the column water vapor.

When the column total water is adopted as a controlling variable as in Fig. 3b, a different behavior is found. As in the case with the column water vapor, just above the critical point (ca. 70 mm for the total water), the precipitation rate rapidly increases as the column total water further increases. However, subsequent to this initial pick-up phase (up to ca. 130 mm), the rate of increase of precipitation slows as the column-integrated total water increases. A clear tendency for a saturation of precipitation rate occurs above 150 mm, and more so with the mean curve. Caution is warranted for extreme values of It because the sample size is relatively small for this range. There are only three occurrences each for the last two bins, and only 48 samples are found above 150 mm. Nevertheless, the saturation tendency is confirmed within the uncertainty of the data. Thus, the column-integrated total water It represents self-organized criticality in the same sense as demonstrated by Peters and Neelin (2006) for Iυ.

To further examine this self-organized critical tendency, we take the difference of the two types of column-integrated water, Ic = ItIυ, corresponding to the total amount of condensed water in a given atmospheric column. Its distribution against the precipitation rate, shown in Fig. 3c, displays an even clearer tendency for self-organized criticality.

Note that with the latter two variables (Figs. 3b and 3c), the distribution itself more closely follows the mean curve than the column water vapor (Fig. 3a). As a whole, the spread of precipitation against the given column-integrated value is more noticeable for water vapor than either total water or the condensate. This has an important implication as further discussed in section 4.

More statistics are presented in terms of space-scale dependence in the next subsection.

b. Spatial-scale dependence

An unexpected finding was that the above-mentioned statistics appear to be spatially invariant: the same overall distribution is recovered after spatial averaging, apart from the higher values being removed by averaging. The original data analysis by Peters and Neelin (2006) is presented with data of 0.25° horizontal resolution and approximately 25 km in longitude. Thus, it is natural to ask whether we can reproduce their result by degrading our data to that scale: an original motivation for our analysis.

Results are shown by Figs. 4 and 5 by plotting the mean precipitation rate and the number density, respectively, as functions of each of the three definitions of column-integrated water. Here, spatial averaging is applied with a sequential increment of a factor of 2 from 4 up to 128 km. The values are marked by +, *, ○, and × and in turn with increasing spatial scales (4, 8, 16, 32, 64, and 128 km). The first scatterplot with the plus sign corresponds to the original resolution (4 km). The results for the full resolution are also marked by the solid curve with a slight smoothing in order to elucidate the tendency. Unlike Fig. 3, the bin size is much smaller (1 mm) in order to more continuously depict the dependence on the column-integrated water. As a result, an explosive tendency of precipitation with increasing column water vapor is clearer in Fig. 4a than in Fig. 3.

Fig. 4.
Fig. 4.

The precipitation rate as in Fig. 3, but the results are by applying space averaging with an increment of a factor of 2 up to a 128-km scale (4, 8, 16, 32, 64, 128 km). The mean precipitation rate for a given column-integrated water is shown by marking a cycle with +, *, ○, × with increasing spatial scales (starting from the full resolution). The full-resolution case is shown with both the plus sign and the solid curve with a slight smoothing.

Citation: Journal of the Atmospheric Sciences 69, 12; 10.1175/JAS-D-12-069.1

Fig. 5.
Fig. 5.

As in Fig. 4, but for the number density.

Citation: Journal of the Atmospheric Sciences 69, 12; 10.1175/JAS-D-12-069.1

The main result from Fig. 4 is that the IP curve is itself scale invariant: the scatterplots for varying horizontal scales collapse into a single curve as obtained without spatial averaging, with the latter shown by the solid curve. A consequence of spatial averaging is the disappearance of high I values. A careful inspection of the scatterplot for the total water (Fig. 4b) would suggest that all the values above It = 150 mm are due to the original resolution (with plus sign). Just after applying the 8-km average, all of the points above It = 150 mm disappear; that is, no “saturation” of the precipitation rate with increasing It occurs above the 8-km scale. Thus, this is a major disagreement with Peters and Neelin (2006), who pointed out a criticality signature at 0.25° spatial resolution. A similar conclusion follows from the column condensed water (Fig. 4c), except for the wider saturation range. Consequently, smoothing up to 16 km is required to completely remove the saturation.

Distribution of the standard deviation (not shown) is also found to be scale invariant, though the tendency is substantially weaker than the mean value. High standard deviations found for high column water values erode with increasing scale averaging. Nevertheless, it is found that for all the variables, an ensemble of the standard deviation curves forms a single envelope.

The number density distribution of I (Fig. 5) is also scale invariant for all three options. With the water vapor (Fig. 5a), a quasi-Gaussian distribution slowly erodes with space averaging. With the total water (Fig. 5b), invariance is well established for It ≤ 80 mm. Above 80 mm, the long tail toward the larger value of It steepens with increasing averaging scales. The scatterplot suggests that the tail asymptotes to an exponential tendency below the 80-mm threshold as manifested by a straight line in the linear-log plot. A gradual erosion of distribution for the condensed water (Fig. 5c) with increasing scales may be understood in the same manner as a tail part of the total water distribution.

The scale invariance suggests some scaling law (fractal) for atmospheric convection (Lovejoy 1982; Yano and Takeuchi 1987). Furthermore, from the point of view of self-organized criticality theory, the invariance of the system under spatial averaging suggests an applicability of the renormalization group (RNG) approach (Wilson 1983; Fisher 1998; Amit and Martín-Mayer 2005). As a sequential averaging procedure applied to a complex statistical system satisfying a scaling behavior with many degrees of freedom, RNG provides a much simpler description of a system with reduced degrees of freedom.

Qualitative difference of the number density distribution between the column water vapor on the one hand and the column total water and condensate water on the other hand may also elucidate the overall tendency of the system with homeostasis and self-organized criticality. The column water vapor follows a Gaussian with a tendency for the number density to decrease more rapidly than an exponential rate for higher values. This reflects the fact that, under homeostasis, the system hardly exceeds Ic as indicated in Fig. 1b. However, the distribution of both the column total water and the condensed water decays more slowly than an exponential rate toward the higher values, reflecting the tendency of self-organized criticality to reach high values under strong external forcing, following the criticality curve shown in Fig. 1a. Consequently, the obtained number density distributions are consistent with homeostasis found for the column water vapor, and with self-organized criticality found for the column total water and condensed water by the (I, P) distributions in Figs. 2 and 3.

Finally, the peak at 55 mm in Figs. 5a and 5b represents the initial value. Different distribution shapes found both sides reflect the evolution of the different phases of the simulation as discussed in the next subsection: the lower side reflects the initial cloud-formation phase and the higher side the later mature stage.

c. Time evolution

There are two reasons for investigating the time evolution of the statistics: (i) to represent the organization of the system into large-scale coherence and (ii) to test the robustness of statistics by narrowing the analysis period.

To set the scene, we first present in Fig. 6 the time series of domain-averaged precipitation rate (Fig. 6a) and the column-integrated water (Fig. 6b). The development of convective organization is not exactly correlated with an increase of the precipitation rate. The precipitation quickly spins up in the first 2 days and remains relatively stable until day 8, when the convective organization begins. At this point, the precipitation rate suddenly increases and peaks at day 10, and then gradually decreases and exhibits an oscillatory behavior, as the system approaches an equilibrium state.

Fig. 6.
Fig. 6.

Time series of (a) domain-mean precipitation rate and (b) the domain-mean column-integrated water vapor (solid), total water (long dash), and saturated water vapor (short dash).

Citation: Journal of the Atmospheric Sciences 69, 12; 10.1175/JAS-D-12-069.1

The column-integrated water for total water (long dash), water vapor (solid), and saturated water vapor (short dash) in Fig. 6b evolve more smoothly than the precipitation rate and, interestingly, in concert. They monotonically decrease until day 10 and then steadily increase. Not surprisingly, the column total water is very close to the column water vapor. Thus, the contribution of the condensed water is small in the model domain average. It further suggests that the distribution of the condensed water is extremely inhomogeneous, considering the similar range of distribution between the water vapor and the condensed water in Figs. 5a and 5c. In contrast, the evolution of the saturated column water vapor reflects the density-weighted averaged column temperature. Thus, the system cools initially and then gradually warms, and maintains that tendency until the end of the simulation.

Time evolution of the precipitation dependency on column water and the number density distribution of the column water are shown in Figs. 7 and 8, respectively. Here, the mean precipitation rate, calculated as a function of the column-integrated water for individual snapshot model outputs, is shown every 2 days for the whole simulation period by a scatterplot for the four types of symbols (+, *, ○, ×) in turn.

Fig. 7.
Fig. 7.

Time evolution of the mean precipitation rate as a function of (a) column-integrated water vapor, (b) column-integrated total water, and (c) column-integrated condensed water. The mean precipitation rate is calculated as a function of the column-integrated water. Individual snapshot model outputs are shown every 2 days for the whole simulation period by a scatterplot. Four symbols (+, *, ○, ×) are used. Also shown are the curves for averages over 0–8 (solid), 10–18 (long dash), and 20–28 days (short dash).

Citation: Journal of the Atmospheric Sciences 69, 12; 10.1175/JAS-D-12-069.1

Fig. 8.
Fig. 8.

Time evolution of the number density distribution shown by scatterplots in 2-day intervals with a cyclic use of the symbols +, *, ○, and × for (a) Iυ, (b) It, and (c) Ic. Also shown are the curves averaged over 0–8 (solid), 10–18 (long dash), and 20–28 days (short dash). A chain-dash curve for average over 9–11 days is also added in (a).

Citation: Journal of the Atmospheric Sciences 69, 12; 10.1175/JAS-D-12-069.1

Figure 7a shows that the critical column water vapor marking the onset of precipitation increases with time, in particular the time-averaged curve. The tendency may be due to the gradual warming of the simulated atmosphere with time indicated by Fig. 6, consistent with Neelin et al. (2009). The scatter may well collapse were the data collocated by atmospheric-column-averaged temperature as presented by Neelin et al. This analysis is not attempted here in view of the relatively short simulation.

The homeostasis behavior does not fully develop until the last phase of the simulation. For the earlier transient phase, the precipitation rate tends to decrease with increasing column water in the high-value limit of the latter. Figure 7b shows that the relation between the precipitation rate and the column total water is relatively invariant with time, and independent of the phases of convective organization. As shown in Fig. 7c, the tendency with the column condensed water is at the middle point of the first two variables and represents a gradual tendency for a higher precipitation rate for a given column condensed water. Nevertheless, the evolution is much weaker than for the water vapor, and the curve is almost invariant with time.

Time evolution of the number density distribution for the column water vapor (Fig. 8a) is reflected in the time series of the domain-averaged column water vapor: the number density distribution changes little for the first 8 days (solid curve), until the convective organization begins. The whole distribution (not only the maximum) then suddenly shifts to lower values (the chain-dash curve for 9–11 days). After this short dry-out period, the column water vapor recovers gradually, associated with a widening of the distribution to the higher sides, as seen by the mean curve for 10–18 days (long dash). As the large-scale convective organization matures, the maximum of the distribution shifts to a higher value, as the mean curve for 20–28 days (short dash) shows.

Compared to the column water vapor, the number density distributions of the latter two variables are much more invariant with time. The total water (Fig. 8b) still shows a similar initial decrease and a later increase over time for the water vapor, but to a much lesser extent. The column condensed water distribution (Fig. 8c) remains stable for the whole simulation only after a short spinup during the first few days, associated with the initial formation of clouds. The distribution on day 2 (marked by a plus sign) is clearly distinguished from the others. However, the wider spread of instantaneous distributions than the mean curves suggests substantial temporal fluctuations in cloud distribution.

d. Statistics with time-averaged precipitation rate

Our analysis so far has been based on an instantaneous precipitation rate stored every 6 h. Observational measurements of rainfall normally use accumulated values over a fixed interval. For example, Peters and Neelin (2006) use the Tropical Rainfall Measuring Mission (TRMM) dataset averaged over 3 h. To more quantitatively compare our results with those observational analyses, the effects of time averaging on the precipitation rate are investigated in this subsection.

The precipitation rate (as an instantaneous value) was stored every 30 min (unlike the other variables) in the present simulation. The analysis of section 3a is repeated by taking a time mean of the 30-min precipitation data for 6 and 12 h. In contrast, the column-integrated water data are available only every 6 h. In combining with 6-hourly averaged precipitation data, the column-integrated water data averaged over two time points are matched to the middle point of the precipitation averaging period. In combining with 12-hourly averaged precipitation data, a 1–2–1 moving average is applied to the column-integrated water data in order to have data equivalent to the 12-hourly average. The results for the column water vapor and the column total water are shown in Fig. 9. Those for the condensed water are similar (not shown). The most emphatic aspect is that, unlike with spatial averaging, the system is not invariant with the temporal scale. Qualitatively, the same curve as for the instantaneous precipitation rate is obtained in a scaled-down manner.

Fig. 9.
Fig. 9.

Analysis of time-averaged precipitation as a function of column-integrated water vapor with (a) 6- and (b) 12-hourly averages, and as a function of column-integrated total water with (c) 6- and (d) 12-hourly average. Number density is shown in the same format as in Fig. 3.

Citation: Journal of the Atmospheric Sciences 69, 12; 10.1175/JAS-D-12-069.1

With 6- and 12-h averages, the precipitation rate scales down approximately by factors of 4 and 6, respectively. After this rescaling, the distributions of the number density are remarkably similar to the cases with instantaneous precipitation, especially for the column water vapor, as shown by a comparison of Figs. 9a and 9b with Fig. 3a. In contrast, for the column total water, we see more spread in precipitation rate than in instantaneous precipitation (cf. Figs. 9c and 9d with Fig. 3b). Behavior of the column condensed water (not shown) is similar to that of the column total water.

4. Water budget analysis

The analysis so far implies that precipitation works as homeostasis against the column-integrated water vapor, and that it behaves in a self-organized critical manner against the column-integrated total water. Why does the precipitation behave so differently against two different types of column-integrated quantities? In pursuing this question, we now examine the budget of the column-integrated water in the form
eq1
with the subscripts, υ, t, and c, to be added for both I and F. Here, F designates the rate of supply of water to a given column, against a loss due to the precipitation P. Strictly, the condensation rate is the true loss term for the column water vapor, not the precipitation rate. Since the latter is not available, we simply use the precipitation rate, which to a first approximation should not cause serious discrepancies.

The local tendency ∂I/∂t of the column-integrated water is evaluated over one time step by using the restart files, available every 24 h. The 28 outputs (snapshots), available for ∂I/∂t, are used for the following analysis. The number density distribution is plotted against the precipitation rate (horizontal axis) and the column-integrated water tendency (vertical axis) in Fig. 10 for the three states of water.

Fig. 10.
Fig. 10.

Number of occurrences for a given precipitation rate (horizontal axis, mm h−1) and tendency (vertical axis, mm h−1) for the (a) column-integrated water vapor, (b) total water, and (c) condensed water. The number count is made over a bin of sizes 20 and 10 mm h−1 for the precipitation rate and the tendency, respectively. Counting (statistics) is limited to a high column-integrated water regime: (a) Iυ > 80 mm, (b) It > 100 mm, and (c) Ic > 30 mm.

Citation: Journal of the Atmospheric Sciences 69, 12; 10.1175/JAS-D-12-069.1

Homeostasis with the column water vapor suggests that the precipitation works as a stabilization process against an external supply Fυ of the water vapor. To maintain such a stability, the precipitation (water condensation) must occur as soon as the water vapor is supplied to the column, leading to a quasi-equilibrium state as shown:
e3
and
eq2
This expectation is confirmed by Fig. 10a.

To maintain the quasi-equilibrium state defined by Eq. (3), the precipitation rate must be balanced by Fυ, rather than by the internal state Iυ—that is the main reason why we see a rather wide spread in distribution of the response (precipitation rate) for a given Iυ in Fig. 3a. In contrast, self-organized criticality with the column-integrated total water and condensed water suggests that the precipitation rate is more efficiently constrained by the internal state (column-integrated water), and that it does not increase as much with the increasing internal-state variables (column water values). Consequently, under strong external forcing (i.e., water supply), the precipitation process cannot cope, resulting in a rapid increase of the column-integrated water. The distributions in Figs. 10b and 10c clearly show that satisfies for It and Ic. The distribution is much more spread vertically than the case with the water vapor Iυ.

Asymmetry in distribution in ∂I/∂t against a fixed column-integrated water value is noted. The probability for having a high positive ∂I/∂t is much higher than for a high negative ∂I/∂t in both cases, with the tendency more pronounced for the column condensed water. This asymmetry suggests that the atmospheric column rapidly accumulates condensed water (as well as total water) but is relatively slow in consuming it. This storage tendency is an important signature of self-organized criticality manifested as an instability of the system.

Compared to Figs. 3 and 4, Fig. 10 shows a more robust contrasting tendency for homeostasis and criticality of atmospheric convection against the water vapor and the condensed water, respectively. There is an order of magnitude difference between ∂Iυ/∂t and ∂Ic/∂t for the entire range of the precipitation rate.

5. Discussion

Self-organized structures can be generated and maintained by two key mechanisms: self-organized criticality and homeostasis. A simple one-variable diagnosis identifies the working of these two major mechanisms. The method is applied to a CRM simulation that spontaneously generates multiscale convective organization.

Conclusions are drawn by examining the atmospheric-column water budget in different manners. When the precipitation rate is examined as a function of the column-integrated water vapor, the behavior of atmospheric convection is consistent with homeostasis: the system is self-regulated toward a state with a specific amount of column water vapor. There is an increasing tendency for precipitation with the increasing column water vapor (cf. Figs. 3a, 4a, 7a, 9a, and 9b). However, when the precipitation rate is examined as a function of the column-integrated total water, the system represents self-organized criticality: the precipitation rate reaches an asymptote with increasing total water of the column (cf. Figs. 3b, 4b, 7b, 9c, and 9d). The analysis with the column-integrated condensed water leads to the same conclusion (cf. Figs. 3c and 4c). The implication is that with increasing total water in the atmospheric column, the precipitation rate tends to be bounded regardless of a rate of increase in the atmospheric-column total water by an external supply. The higher the supply rate of water vapor, the more cloud water accumulates before it precipitates.

The column water budget supplements the above-mentioned two opposite interpretations based on a one-dimensional dynamical system (cf. Fig. 1). The column-integrated water vapor budget is characterized by a close balance between a loss by precipitation (used as a substitute for net condensation) and external supply (gain) of water vapor [Eq. (3)] as seen in Fig. 10a. The system is in quasi equilibrium broadly similar to that between the cloud work function and the large-scale forcing [cf. Fig. 13 in Arakawa and Schubert (1974)], commonly cited as an evidence for convective quasi equilibrium. By the same token, the column-integrated water vapor is interpreted to be in quasi equilibrium. In this respect, the two concepts of homeostasis and quasi equilibrium are related. The main function of homeostasis is to maintain a system in a state of stability, which implies equilibrium.

In contrast, when the column budget of either total water or condensed water is examined, the system is almost always away from equilibrium, as shown in Figs. 10b and 10c. In that case, the system is perpetually transient, because the precipitation rate is strongly constrained by an immediately available column total water independent of an instantaneous supply rate of water. This can furthermore be interpreted as a manifestation of inherent instabilities of the system. The system rapidly accumulates cloud water (total water) and more slowly consumes it, as suggested by the asymmetry in the distribution. The number density distribution of the column-integrated water (Fig. 5) also provides consistent interpretations for the three types of water.

Conceptual models have been proposed in order to explain both self-organized criticality (Muller et al. 2009; Stechmann and Neelin 2011) and homeostasis (Raymond 2000) of atmospheric convection. However, the contrasting behavior of the atmospheric convective system in terms of water vapor and condensed water presented herein has not previously presented, to the best of our knowledge.

To develop a more complete theory that encompasses both self-organized criticality and homeostasis associated with convection organization, complex convective processes could be formalized by an approach analogous to statistical mechanics (cf. Yeomans 1992). The renormalization group (RNG) theory (Wilson 1983; Fisher 1998; Amit and Martín-Mayer 2005) may be central, considering the observationally established scaling behavior of convection (Lovejoy 1982; Yano and Takeuchi 1987), as well as the scale independence of the IP statistics identified herein (cf. Fig. 4). To initially perform such a statistical analysis, a relatively simple but physically based model is required. An energy-cycle system investigated by Randall and Pan (1993), Pan and Randall (1998), and Yano and Plant (2012) are possible candidates.

Expecting it be difficult to develop a fully self-contained “realistic” statistical theory, large-domain CRM experiments are required to supplement the theories. For this reason, it is desirable to repeat the simulation at higher resolution and with a larger domain. The horizontal resolution of the present simulation is too coarse (4 km simulates mesoscale organization but not small-scale convection) and the domain is not global (6000 km in longitude). Longer simulations are also preferable, because self-organized criticality and homeostasis deal with the behavior of a system in association with rare extreme events. In particular, the scale invariance found in section 3b should be established for higher column water values with data from longer time series.

Both self-organized criticality and homeostasis are theoretically formulated under a homogeneous external condition, in the same sense as a homogeneous external temperature is assumed by classical thermodynamics. Thus, CRM experiments should continue to be run under homogeneous external conditions, with a homogeneous SST, in order to establish a robust correspondence with theory. A prescribed homogeneous radiative cooling could also focus attention on the self-organized criticality and homeostasis of convective dynamics in spite of the practical importance of cloud radiative feedback (cf. Stephens 2005).

Idealizations in our simulation may explain discrepancies with existing observational analysis. Ironically, inhomogeneity of SST over real oceans hinders a clear-cut interpretation of observations in terms of self-organized criticality (cf. Peters et al. 2009). For this reason, we expect that research based on idealized planetary-scale CRM experiments are required to further address the interplay between self-organized criticality and homeostasis introduced by our study. As emphasized in section 7 of Arakawa and Schubert (1974), a robust understanding of statistical cumulus dynamics would be an important prerequisite for the convection parameterization problem. Such a theory should first be established for homogeneous environments.

Acknowledgments

The present work is performed under a context of COST Action ES0905. JIY also acknowledged the support of NCAR’s Visitor Program.

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