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    Latitude–height distribution of zonal-mean latent heating and its variance for JJA seasonal runs employing (a), (b) an MCA scheme, and (c), (d) the LSP of AGCM3 in the absence of the ZM scheme. The mean heating rate is displayed in units of 0.1 K day−1 with a contour interval of 10, and the variance of heating rate is displayed in units of (K day−1)2 with a contour interval of 10

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    Frequency spectra of ZM component of tropical precipitation (13°S–13°N) for a series of JJA seasonal runs in which the time scale τa was varied. The frequency spectra are displayed in three formats: (top) linear–linear, (middle) log–log, and (bottom) “energy preserving” log–linear. Positive frequencies correspond to eastward phase propagation while negative frequencies correspond to westward frequency propagation

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    Wavenumber–frequency spectra of upward E–P flux (13°S–13°N) through 57 hPa for the series of JJA seasonal runs (a) τa = 300, (b) 2400, (c) 7200 s, and (d) ∞ (LSP alone). Following Horinouchi et al. (2003), these are presented in energy preserving form. Positive and negative frequencies correspond to propagation as in Fig. 2

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    As in Fig. 2 except for LSP component of tropical precipitation. Note also, a different vertical scale is employed

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    Latitude–height distribution of zonal-mean latent heating variance of (a), (c), (e), (g) ZM and (b), (d), (f), (h) LSP for the τa sensitivity experiments presented in Figs. 2 and 4. Heating rates are displayed in units of 0.1 K day−1, with a contour interval of 2 K day−1 for ZM and 10 K day−1 for LSP

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    Latitude–height distribution of (a), (c), (e), (g) zonal-mean temperature and (b), (d), (f), (h) specific humidity anomaly away from the control run employing τa = 2400 s. Temperature anomalies are displayed in units of K while specific humidity anomalies are displayed in units of 10−4 kg m−3

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    The ZM component as in Fig. 2 except for the control, the control with μ = 0.2, the prognostic closure (4), and the prognostic closure (4) with μ = 0.2. The prognostic closure in these experiments used values of τd = 6 h and α = 108

  • View in gallery

    As in Fig. 7 except for LSP component of tropical precipitation. Note also, a different vertical scale is employed

  • View in gallery

    As in Fig. 5 except for the sensitivity experiments presented in Figs. 7 and 8. Heating rates are displayed in units of 0.1 K day−1 with a contour interval of 2 K day−1 for both ZM and LSP

  • View in gallery

    JJA seasonal (top) variance and (bottom) mean (13°S–13°N) for the two sets of sensitivity experiments employing the prognostic closure (4) in which (left) τd = 6 h, 2 × 106α ≤ 2 × 109 m4 kg−1 and (right) α = 2 × 108 m4 kg−1, 1200 s ≤ τd ≤ 12 h. Axes have been labeled with α̂ and τ̂d to facilitate a comparison with the closure of Pan and Randall (1998)

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    JJA seasonal-mean precipitation for (top) observations (Xie and Arkin 1996), (middle) 5-yr climatologies of the control, and (bottom) the control with μ = 0.2

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    Fig. B1. Assumed triangular distribution of subgrid relative humidity H about the grid-box mean value Hi. The width of the distribution, Δ, is taken to be Δ = 1 − Hc, where Hc is a tunable parameter referred to as the threshold relative humidity

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The Variability of Modeled Tropical Precipitation

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  • 1 Canadian Centre for Climate Modelling and Analysis, Meteorological Service of Canada, Victoria, British Columbia, Canada
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Abstract

This paper investigates the temporal properties of tropical precipitation in the Canadian Centre for Climate Modelling and Analysis (CCCma) third-generation atmospheric general circulation model (AGCM3). AGCM3 employs the penetrative mass-flux (PMF) scheme of Zhang and McFarlane (ZM) for the parameterization of deep cumulus convection. It is found that the temporal variability of the ZM scheme is sensitive to a number of its internal parameters as well to the use of a prognostic, rather than diagnostic, closure condition for the cloud-base mass flux. Sensitivity experiments suggest that the ZM scheme can produce realistic amounts of variability when compared to direct radar observations of deep cumulus convection in the Tropics.

A central finding of this study is that the resolved large-scale stratiform precipitation (LSP) in the model can participate in the modeling of deep latent heating and so compete with the ZM scheme in the Tropics. In modeling deep latent heating the LSP is found to mimic the behavior of a moist-convective adjustment scheme. In AGCM3 it is found that typical parameter settings of the ZM scheme place it in a regime in which the temporal variability of tropical precipitation is dominated by this behavior of the LSP, while the temporal mean is dominated by the ZM scheme. In such circumstances it is the LSP, and not the ZM scheme, that provides the primary source of resolved tropical Kelvin and mixed Rossby–gravity waves in the GCM. Such competition between LSP and the parameterization of deep convection appears to be active in other modeling studies. Consequently, it has the potential to complicate efforts to understand the nature of resolved tropical waves in GCMs and their role in the forcing of the quasi-biennial and semiannual oscillations.

Corresponding author address: Dr. John F. Scinocca, Canadian Centre for Climate Modelling and Analysis, 3694 Gordon Head Road, Victoria, BC V8N 3X3, Canada. Email: john.scinocca@ec.gc.ca

Abstract

This paper investigates the temporal properties of tropical precipitation in the Canadian Centre for Climate Modelling and Analysis (CCCma) third-generation atmospheric general circulation model (AGCM3). AGCM3 employs the penetrative mass-flux (PMF) scheme of Zhang and McFarlane (ZM) for the parameterization of deep cumulus convection. It is found that the temporal variability of the ZM scheme is sensitive to a number of its internal parameters as well to the use of a prognostic, rather than diagnostic, closure condition for the cloud-base mass flux. Sensitivity experiments suggest that the ZM scheme can produce realistic amounts of variability when compared to direct radar observations of deep cumulus convection in the Tropics.

A central finding of this study is that the resolved large-scale stratiform precipitation (LSP) in the model can participate in the modeling of deep latent heating and so compete with the ZM scheme in the Tropics. In modeling deep latent heating the LSP is found to mimic the behavior of a moist-convective adjustment scheme. In AGCM3 it is found that typical parameter settings of the ZM scheme place it in a regime in which the temporal variability of tropical precipitation is dominated by this behavior of the LSP, while the temporal mean is dominated by the ZM scheme. In such circumstances it is the LSP, and not the ZM scheme, that provides the primary source of resolved tropical Kelvin and mixed Rossby–gravity waves in the GCM. Such competition between LSP and the parameterization of deep convection appears to be active in other modeling studies. Consequently, it has the potential to complicate efforts to understand the nature of resolved tropical waves in GCMs and their role in the forcing of the quasi-biennial and semiannual oscillations.

Corresponding author address: Dr. John F. Scinocca, Canadian Centre for Climate Modelling and Analysis, 3694 Gordon Head Road, Victoria, BC V8N 3X3, Canada. Email: john.scinocca@ec.gc.ca

1. Introduction

The temporal characteristics of deep tropical convection is currently a topic of great interest (e.g., Horinouchi et al. 2003). There is a general feeling that current parameterizations of deep cumulus convection produce too little temporal variability in general circulation models (GCMs) employed for climate studies (Ricciardulli and Garcia 2000; Amodei et al. 2001; Lin and Neelin 2002). In particular, penetrative mass flux schemes, which tend to produce very reasonable seasonal-mean distributions of tropical precipitation, are thought to be associated with some of the weakest variability.

This is an important modeling issue since the variability of latent heating associated with deep convection in the Tropics is the primary source of resolved large-scale equatorially trapped waves such as Kelvin and mixed Rossby–gravity (RG) waves. In the tropical stratosphere the zonal-mean circulation is dominated by the quasi-biennial oscillation (QBO) which owes its existence to Kelvin, RG, and small-scale gravity waves (e.g., Dunkerton 1997). The structure of the semiannual oscillation (SAO) at the stratopause and mesopause is also in part driven by these waves and shaped by wave filtering induced by the QBO and SAO.

To properly model the tropical middle atmosphere with a GCM, one requires the correct “mix” of these waves and their effects. This turns out to be a notoriously difficult and subtle problem. It requires a realistic parameterization of unresolved small-scale gravity waves and a realistic representation of the resolved large-scale gravity, Kelvin, and mixed Rossby–gravity waves. It is becoming increasingly clear that the resolved large-scale component of this mix can be vastly different in any two GCMs.

Since equatorially trapped Kelvin and RG waves are global in scale there is the expectation that these waves should be well resolved in climate GCMs. Further, even in the absence of mean-flow driving from unresolved small-scale gravity waves, one might expect the mean-flow driving associated with these resolved waves to spontaneously produce a quasi-biennial-like oscillation (QBLO). Here, we shall use the term QBLO to refer to the generic mechanism of a wave-induced mean-flow oscillation consisting of descending shear zones (e.g., Plumb and McEwan 1978). The term QBO will be reserved for modeled QBLOs that possess 1) the observed spatial structure, 2) the observed period, and 3) are driven by the correct mix of waves.1 Until recently, however, GCMs have generally been unanimous in their inability to produce QBLOs (e.g., Boville and Randel 1992).

Recently, Takahashi (1996) and then Horinouchi and Yoden (1998) have shown that QBLOs can be driven by resolved waves in a GCM employing modest horizontal resolution, which is typical of current climate models. This result was found to depend on two key factors: increased vertical resolution in the stratosphere, and a parameterization of deep cumulus convection that produced latent heating with large temporal variability (e.g., moist-convective adjustment). Presumably, increased vertical resolution in the stratosphere is required to better model the nonlinear evolution of upwardly propagating waves as they approach their critical levels, while the requirement of large temporal variability in the latent heating of parameterized convection is needed to provide a source of resolved waves with sufficient momentum flux to drive a QBLO. This result has been validated and further explored in more conventional GCMs by Takahashi (1999) and Hamilton et al. (1999, 2001).

The requirement of large temporal variability in the latent heating associated with the parameterization of deep cumulus convection has focused attention on the temporal characteristics of current parameterizations of this process in climate GCMs. In a recent study, Ricciardulli and Garcia (2000, hereafter RG00) employed satellite observations of cloud-top brightness temperatures to derive the amplitude of wavenumber-frequency spectra of latent heating associated with observed tropical convection (following the method of Bergman and Salby 1994). This “observed” heating spectra was then compared to the heating spectra produced by two current parameterization schemes: the penetrative mass-flux (PMF) scheme of Zhang and McFarlane (1995, hereafter ZM), employed in the National Center for Atmospheric Research Community Climate Model CCM3, and the moist-convective adjustment (MCA) scheme of Hack (1994), employed in CCM2.

One of the central results of the RG00 study is that the ZM scheme, and by extension all PMF schemes (Horinouchi et al. 2003), severely underestimates the temporal variability of latent heating. The heating field of the MCA scheme of Hack (1994), on the other hand, was found to display an integrated temporal variance that was comparable to the satellite observations. This result was used to explain the ability of CCM2 and not CCM3 to produce a QBLO with increased vertical resolution in the stratosphere. Consequently, even though the PMF scheme of ZM produced more realistic monthly mean distributions of tropical precipitation, its weak temporal variability was seen to be a weakness in CCM3 (B. Boville 2001, personal communication).

There are a variety of other parameterizations of deep cumulus convection currently used in GCMs. In general, the temporal variability of heating in any of these schemes falls somewhere in the range between the PMF schemes and MCA schemes (Horinouchi et al. 2003). Given additional differences, such as numerical algorithms, spatial resolution, and other physical parameterizations, one expects resolved tropical waves to significantly vary between GCMs. Consequently, the mix of large-scale resolved waves and small-scale parameterized waves required in each GCM to produce a QBLO is constrained far more by these issues than by our understanding of the actual role that these waves play in the real system. This unsatisfactory state of affairs has lead to the general feeling that parameterizations of deep cumulus convection must be better constrained to give observed (i.e., RG00) levels of temporal variability (e.g., Amodei et al. 2001; Lin and Neelin 2002).

More recently, however, a study by Horinouchi (2002, hereafter H02) has suggested that RG00 significantly overestimated the variability of observed deep convective heating. In H02, an attempt is made to validate the method of Bergamn and Salby (1994) that RG00 used to estimate the amplitude of heating spectra. H02 used overlapping satellite and radar observations of tropical convection, obtained during the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE), to produce independent estimates of frequency spectra of latent heating.

From this analysis, H02 concluded that the satellite method used by RG00 produces estimates that, for typical climate GCM resolutions, are as much as an order of magnitude larger than those produced by direct radar observations. Clearly, definitive estimates of the heating variability associated with deep convection in the Tropics are required.

At the same time, we require a better understanding of resolved tropical waves and their forcing mechanisms in GCMs employing resolutions typically used for climate studies. In this study we perform a detailed investigation of the nature of modeled tropical precipitation and its role as a source of resolved tropical waves in the Canadian Centre for Climate Modelling and Analysis (CCCma) third-generation atmospheric general circulation model (AGCM3) and its upward extension, the Canadian Middle Atmosphere Model (CMAM). In addition, we consider the sensitivity of the temporal variability of the PMF scheme of ZM to several of its internal parameters and to the implementation of a simple prognostic, rather than diagnostic, closure condition for the cloud-base mass flux.

There are two central results. The first is that, contrary to current opinion, PMF parameterizations of deep cumulus convection such as the ZM scheme can produce tropical precipitation with observed levels of variability. The second, and perhaps more important result is that the resolved large-scale stratiform precipitation (LSP) scheme in the model can participate in the modeling of deep latent heating and so compete with the parameterization of tropical deep convection in the GCM. We find that typical parameter settings of the ZM scheme can allow this behavior of the LSP to dominate the variance of tropical deep latent heating and so provide the dominant source of resolved tropical waves in the CCCma GCM. While this study investigates the nature of tropical precipitation in a particular GCM, it is argued that this second result generalizes to other climate GCMs.

The outline of this paper is as follows. In section 2 we briefly review the PMF scheme of ZM and the large-scale stratiform precipitation scheme currently used in AGCM3. In section 3 we undertake a series of GCM experiments to investigate the sensitivity of tropical precipitation to several internal parameters of the ZM scheme as well as to the use of a prognostic, rather than diagnostic, closure condition for the cloud-base mass flux. A discussion of the results is provided in section 4 and we conclude with a brief summary in section 5.

2. Modeled precipitation and latent heat release

Precipitation and latent heat release occur as a consequence of two processes in AGCM3—resolved large-scale precipitation and parameterized deep cumulus convection. In this section we briefly review the properties of each of these processes.

a. Parameterization of deep cumulus convection

In AGCM3, the PMF scheme of ZM is used to parameterize the precipitation and latent heat release associated with deep cumulus convection. The scheme has already been discussed in detail by Zhang and McFarlane (1995). Here we shall briefly review its closure properties and its extension to a prognostic closure condition for the cloud-base mass flux.

The closure for the ZM scheme is based on a notional budget equation for convective available potential energy (CAPE). This may be written symbolically as
i1520-0469-61-16-1993-e1
where A represents CAPE, G represents the large-scale production of CAPE by resolved dynamics, and −MbF represents the subgrid depletion of CAPE by parameterized deep convection. Following ZM, Mb represents the cloud-base updraft mass flux, while F represents the rate at which cumulus clouds consume CAPE per unit cloud-base updraft mass flux. The quantity F is central to the parameterization problem. At any time its value will depend upon the current profiles of temperature and moisture as well as the assumed properties of typical updrafts and downdrafts in deep convective towers (see ZM).
Zhang and McFarlane (1995) employ the diagnostic closure condition:
i1520-0469-61-16-1993-e2
where τa is an adjustment time scale. Physically, this closure assumes that CAPE is consumed at an exponential rate (1/τa) by cumulus convection. This may be seen by combining (1) and (2) to obtain
i1520-0469-61-16-1993-e3
For example, if an amount of CAPE Ao exists at t = 0, in the absence of large-scale production (i.e., G = 0), we have the solution A = Ao exp(−t/τa) for t > 0.

An important parameter in this scheme is the adjustment time scale τa. The value used by RG00 in CCM3 for their study was τa = 7200 s. The value typically used in AGCM3 is τa = 2400 s. As we shall see in the next section, the choice of τa can significantly influence the amount of temporal variability produced by the ZM scheme.

In addition to the diagnostic closure condition (2) we will also consider a prognostic closure of the form
i1520-0469-61-16-1993-e4
which assumes that Mb increases proportionally with A and is dissipated with time scale τd. This closure is similar to that proposed by Randall and Pan (1993) and Pan and Randall (1998). A more complete description of these closures may be found in appendix A.

b. Large-scale precipitation

Supersaturation (i.e., relative humidities H that exceed a value of 1.0) will occur at times in the model as a consequence of a number of physical processes (e.g., radiative cooling, lateral convergence, adiabatic cooling associated with large-scale ascent, etc.). We shall refer to the condensation that ensues in these circumstances as large-scale or stratiform precipitation, since it arises from resolved processes in the model and since the latent heating and condensation occur coincidently.

Previous versions of the CCCma GCM (GCM2, McFarlane et al. 1992; and GCM1, Boer et al. 1984) employed a moist-convective adjustment (MCA) scheme (i.e., see Daley et al. 1976) for the parameterization of deep cumulus convection. The MCA scheme renders the basic state neutrally stable to moist (and therefore dry) static instability by ensuring that its lapse rate is no greater than that of a moist adiabat. Precipitation and latent heat release produced by the MCA scheme represents both deep convective and large-scale stratiform sources. Consequently, when MCA is used there is no explicit separation between these two components of the precipitation and heating.

In replacing the MCA scheme with the PMF scheme of ZM two issues arise. First, since the ZM scheme models only the deep convective component of latent heating and precipitation, an additional scheme is required to model the large-scale component of latent heating and precipitation. Second, in the absence of MCA, dry static instability is no longer treated by the physics package and may persist in the model.

In AGCM3, static instability is neutralized by employing a dry-convective adjustment (DCA) scheme. The DCA scheme is derived by a straightforward modification to the old MCA scheme. The maximum lapse rate is limited to a dry rather than moist adiabat. As in the MCA scheme, in neutralizing static instability the DCA scheme allows fluxes of heat and moisture between adjacent layers. However, in the DCA scheme, condensation is not permitted and the flux of moisture is limited so as not to produce supersaturated conditions in layer that is initially unsaturated.

For the large-scale component of the latent heating and precipitation, a threshold relative humidity Hc is assigned and used to determine the amount of liquid water and latent heating produced by condensation. The specific criterion and algorithm used for this purpose in AGCM3 is described in appendix B. We shall refer to this as the LSP scheme.

The order in which these schemes are called in the AGCM3 physics package is: DCA, LSP, and then ZM. This order is required since the ZM scheme formally assumes that input columns of temperature and specific humidity are everywhere statically stable and are not supersaturated.

3. GCM experiments

a. Model description

The GCM experiments described in this paper employ the Canadian Centre for Climate Modelling and Analysis third-generation atmospheric GCM (AGCM3; McFarlane et al. 2003, unpublished manuscript), which is similar in many respects to the second-generation model described in McFarlane et al. (1992). The model is spectral and, for the experiments discussed here, employs triangular truncation at total wavenumber of 47 (T47).

New or improved features of the parameterized physical processes in AGCM3 are: a new land surface scheme (CLASS, Verseghy et al. 1993), a new parameterization of cumulus convection (Zhang and McFarlane 1995), an improved treatment of solar radiation which employs four bands in the visible and near infrared, an “optimal” spectral representation of the earth's topography (Holzer 1996), a revised representation of turbulent transfer coefficients at the surface (Abdella and McFarlane 1996), and a new anisotropic orographic gravity wave drag parameterization (Scinocca and McFarlane 2000).

In addition to an analysis of the variability of tropical precipitation in AGCM3, we will also consider the effect of such variability on the forcing of resolved tropical waves (e.g., mixed RG and Kelvin waves). For this purpose we use a middle atmosphere configuration of the model. In the vertical, the model domain extends from the surface up to approximately 100 km and is spanned by 59 layers. The layer depths increase monotonically with height through the troposphere from approximately 100 m at the surface to approximately 3 km in the lower stratosphere and remain constant at this value above.

In this middle atmosphere configuration of the model we employ no parameterization of nonorographic gravity wave drag. Further, the levels in the troposphere are selected so as to exactly coincide with the levels used in the standard tropospheric configuration of AGCM3. Aside from the spatial resolution employed this middle atmosphere configuration is very similar to the Canadian middle atmosphere model (CMAM) initially reported by Beagley et al. (1997).

In this study we shall consider sensitivity experiments employing AGCM3 in three particular configurations. The first configuration is the normal AGCM3 setup in which LSP and the ZM parameterization of deep convection are active. This configuration will be used extensively here to investigate the properties of tropical precipitation and deep latent heating produced from each of the LSP and ZM schemes in the model and their sensitivity to several ZM parameters and the use of a prognostic rather than diagnostic closure condition for the ZM cloud-base mass flux. A second configuration, used for a sensitivity study conducted in this section, is one in which the ZM scheme is removed from AGCM3 and the LSP is allowed to act alone. The final configuration, also used for a sensitivity study conducted in this section, is one in which both the LSP and ZM schemes are removed from AGCM3 and the MCA used previously in AGCM2 is used in their place.

b. Precipitation in absence of parameterized deep convection

We first consider the properties of the LSP scheme in AGCM3 in the absence of the ZM parameterization of deep convection. As first demonstrated by Tiedtke (1984), one should expect the latitude–height distribution of zonal-mean seasonal-mean latent heating from the LSP scheme to be nearly identical to that produced by an MCA scheme. To test this result in AGCM3 the model was modified to use the MCA scheme employed previously in GCM2 (i.e., Daley et al. 1976). This MCA configuration of AGCM3 was then compared against a configuration of AGCM3 in which the deep cumulus convection scheme of ZM was removed and the LSP scheme acted on its own.

In Figs. 1a and 1c we present the latitude–height distribution of zonal-mean latent heating from a seasonal JJA run for each of these configurations of the model. The two patterns of seasonal-mean latent heating display strong similarities in structure and amplitude, corroborating the Tiedtke (1984) result in AGCM3. Furthermore, a comparison of the variance (Figs. 1b and 1d) between the MCA and the LSP scheme also reveals strong similarities in structure and amplitude. This would suggest that, in addition to the mean, the Tiedtke (1984) result extends to the variability of the latent heating as well.

The ability of the LSP scheme to mimic the behavior of an MCA scheme turns out to be a central issue in the present study. It is important, therefore, to be clear about the way in which this occurs. As discussed in section 2b, at every time step a MCA scheme will redistribute heat and moisture in the vertical so as to limit the temperature lapse rate to that of a moist adiabat. Clearly, the LSP scheme cannot effect a similar redistribution of heat and moisture at each time step. Such a redistribution by the LSP scheme must occur mainly through resolved processes over a number of time steps in the model. Even so, this difference appears to have little impact on the similarity of the seasonal mean and variance of LSP and MCA.

Their close similarity suggests that the MCA scheme redistributes heat and moisture over only a limited vertical extent at each time step. That is, rather than redistributing heat and moisture throughout the full depth of the troposphere, as occurs for full penetrative deep convective schemes (e.g., ZM; Arakawa and Schubert 1974), MCA schemes tend to induce more local adjustments to temperature and moisture to maintain an equilibrium of neutral moist stability. In this sense, one could question the classification of MCA schemes as parameterizations of deep penetrative cumulus convection.

c. Sensitivity of tropical precipitation to τa

As discussed in section 2, the implementation of the ZM scheme in CCM3 and AGCM3 differs in the value of τa, which sets the exponential rate at which CAPE is assumed to be depleted in the diagnostic closure (2). In CCM3, τa = 7200 s while in AGCM3 τa = 2400 s. To investigate the sensitivity of the variability of the ZM scheme to the value of τa a series of seasonal June– August (JJA) runs were performed for a number of different values of τa. Hereafter, we will refer to the τa = 2400 s experiment as the control.

In Fig. 2 we present the frequency spectra of tropical (13°S–13°N) deep convective precipitation produced from the ZM scheme for a number of seasonal JJA runs that employ different values of τa. These are produced from time series of hourly precipitation. The spectral density is displayed in three formats: linear–linear (top), log–log (middle), and “energy preserving” log–linear (bottom). These formats correspond to those employed by RG00. In Fig. 2, and all others which follow, positive frequencies correspond to eastward phase propagation while negative frequencies correspond to westward phase propagation.

Such precipitation spectra are a direct proxy for vertically integrated latent heating spectra. Following appendix B of RG00, assuming a uniform distribution of latent heating, one can relate the vertically integrated latent heating rate J/Cp (K day−1) to the precipitation rate P(mm day−1) through
i1520-0469-61-16-1993-e5
where L = 2.5 × 106 J kg−1 is the latent heat of condensation for water, Cp = 1004 J (kg K)−1, and ΔZ is the depth over which latent heat is released. A number of tests were performed in which corresponding latent heating spectra were produced (not shown). These indicated that, in this study, the relation (5) is roughly satisfied by ΔZ = 9 km for the deep convective precipitation and ΔZ = 10 km for the large-scale precipitation.

Figure 2 indicates that the variability of the deep convective precipitation produced by the ZM scheme is strongly sensitive to the value of τa used in the diagnostic closure condition (2). As the value of τa is reduced the spectral amplitude increases at all frequencies. In particular, the diurnal frequency and its higher harmonics significantly increase amplitude when τa is reduced. Further, there occurs a significant broadening of the frequency response about the diurnal cycle, particularly as τa is reduced below 2400 s.

It is anticipated that the dramatic increase in variability of the parameterized deep convective latent heating illustrated in Fig. 2 should be associated with an increase in the forcing of resolved tropical waves in the GCM. To investigate this issue, eastward and westward components of the resolved upward Eliassen–Palm (E–P) flux are calculated. Table 1 summarizes the contributions from the ZM and LSP schemes to the JJA integrated variance and mean of tropical precipitation as well as the integrated upward tropical E–P flux at 100 hPa, for each of the experiments illustrated in Fig. 2.

From Table 1 we see that as τa is reduced from 7200 to 300 s there occurs a seven fold increase in the total variance of precipitation (and latent heating) associated with the ZM scheme (first column). At the same time, however, the total upward E–P flux (rightmost column) displays little sensitivity to the value of τa over this range.

An extra experiment labeled τa = ∞ is also included in Table 1. In this experiment the ZM scheme is turned off and the LSP is allowed to act alone. This corresponds to the LSP-only simulation discussed in section 3b earlier. From Table 1 we see that the integrated upward E– P flux in the τa = ∞ experiment is a factor of 2–3 greater than any of the experiments that used the ZM scheme. In Fig. 3 we present the wavenumber–frequency spectra of upward E–P flux through 57 hPa from the τa = 300, 2400, 7200 s, and ∞ (LSP alone) experiments. For ease of comparison these have been presented in a form similar to Horinouchi et al. (2003).

The fact that there is more resolved tropical wave activity in the absence of the ZM parameterization of deep convection seems counterintuitive. However, as discussed in section 3b, in the absence of parameterized deep convection, the LSP acts essentially as a MCA scheme. Consequently, as documented by RG00, there occurs a dramatic increase in variability of precipitation and latent heating, as well as a dramatic increase in forced Kelvin and mixed Rossby–gravity waves.

The LSP and ZM precipitation display opposite sensitivity to the value of τa. This can be seen in Table 1 and in Fig. 4 where the frequency spectra associated with the LSP component of precipitation are displayed for each of the τa sensitivity experiments. As τa is increased the contribution from the ZM scheme decreases while the contribution from the LSP scheme increases. This compensating relation results in a total mean precipitation that is roughly conserved. For the variance, however, its total can vary dramatically with value of τa. Close inspection of the totals in Table 1 shows that the total variances generally exceed the sum of the ZM and LSP variances. This provides one measure of the degree to which the two are correlated.

The latitude–height distribution of zonal-mean seasonal-mean latent heating variance for both the ZM and LSP components are presented in Fig. 5 for each of the τa sensitivity experiments discussed earlier. The contour interval used for the LSP variance is made identical to that used in Fig. 1 while the contour interval used for the ZM variance is set to one-fifth this value. Figure 5 clearly illustrates the compensating relation between the ZM and LSP components of latent heating variance and that this relation is essentially constrained to the Tropics.

More importantly, as τa is made large (i.e., toward 7200 s) the LSP latent heating variance begins to take on structure indicative of MCA (cf. Fig. 1d with Fig. 5h). That is, as the ZM scheme is made less efficient the LSP acts more like a MCA scheme in the Tropics. The implication is that, in addition to stratiform precipitation, the LSP scheme may also act to model a significant portion of the deep latent heating in the Tropics. For example, the control setting of τa = 2400 s, typically used for AGCM3, places the model in a regime where most of the seasonal-mean tropical precipitation comes from the ZM scheme while most of its variance comes from the LSP scheme (i.e., see Table 1). Apparently then, the resolved tropical waves in the standard AGCM3 configuration (τa = 2400 s) are forced more by the LSP scheme than by the ZM scheme.

The extent to which either the LSP or ZM scheme dominates the modeling of deep latent heating in the Tropics has significant impact on the equilibrium profiles of temperature and moisture that are produced by the model. In Fig. 6 we present mean JJA anomaly plots of the temperature (left-hand column) and specific humidity (right-hand column) for the τa = 300 s, τa = 7200 s, τa = ∞ (LSP alone), and MCA experiments away from the control τa = 2400 s experiment. From Fig. 6 we can see that, relative to the control experiment, strong parameterized convection (Figs. 6a,b) implies a warmer wetter tropical troposphere above the boundary layer. Weaker parameterized convection, and stronger deep latent heating from LSP (Figs. 6c–f), implies a cooler dryer tropical troposphere above the boundary layer. This is consistent with the application of MCA in place of the LSP and ZM scheme in the model (Figs. 6g,h). Consequently, in addition to changing the mean and variability of vertically integrated deep latent heating, the compensating relation between LSP and parameterized convection also changes the basic-state profiles of temperature and moisture in the Tropics.

Ideally, we would like the parameterized deep convection to model all of the deep latent heating in the model. The sensitivity experiments described earlier show that for the ZM scheme this can be effected by a reduction of τa to values near 300 s. However, since τa corresponds roughly to the adjustment time scale for an ensemble of convective towers, such short time scales for τa are not realistic and, therefore, difficult to justify. In the next section we consider several additional sensitivity experiments where we attempt to similarly increase the ZM contribution to the modeled deep heating in AGCM3.

d. Downdraft sensitivity tests

In the previous section we found that the LSP participates in the modeling of deep tropical latent heating by acting like a MCA scheme when the ZM scheme acts weakly. Since the LSP is essentially resolved, its tendency to model deep heating can only be influenced indirectly through the properties of the parameterized deep convection (ZM). In this section we will consider another modification to the ZM scheme designed to reduce the participation of LSP in the modeling of deep latent heating.

The ZM scheme includes a parameterization of downdrafts in which part of the condensate generated in updrafts is evaporated. The amount of water evaporated is restricted to be at most a fraction μ of the total condensate. The strength of the downdrafts is controlled by this process. This is discussed in detail in appendix C.

The larger the value of μ, the larger the amount of moisture left in the environment, and the greater the potential for LSP. A value of μ = 1 has been used in all τa sensitivity experiments discussed in the previous section. Here we consider reducing the value of μ so that where deep convection is most vigorous (i.e., the Tropics) it will systematically leave less moisture in the environment and, therefore, tend to inhibit LSP.

In Figs. 7 and 8 we present the frequency spectra for the ZM and LSP components of JJA tropical precipitation, respectively, for the control and a new experiment in which μ has been reduced to a value of 0.2. The effect on the variance of LSP in the Tropics is dramatic. From Fig. 8 we can see that a reduction in μ from a value of 1 to a value of 0.2 has reduced the LSP variance by approximately an order of magnitude. At the same time the ZM scheme displays a more modest increase in variability (Fig. 7). The integrated contributions from the ZM and LSP schemes to the JJA variance and mean of tropical precipitation as well as the integrated upward tropical E–P flux at 100 hPa are presented in Table 2.

The latitude–height distribution of zonal-mean seasonal-mean latent heating variance for both the ZM and LSP components are presented in Fig. 9 for each of these experiments (Figs. 9a–d). (Unlike in Fig. 5, the contour interval used for the LSP component in Fig. 9 is identical to that used for the ZM component.) Figure 9 indicates that when μ is reduced in the control version of ZM, the dramatic reduction in LSP heating variance is essentially restricted to tropical latitudes (Fig. 9d). A comparison of these results with the results from the τa experiments discussed in the previous section indicates that a similar reduction in LSP variability may be realized by reducing μ or τa.

e. Closure sensitivity tests

In this section we consider two further modifications to the ZM scheme that involve the use of the prognostic closure condition (4) for the cloud-base mass flux. This different closure condition introduces the two new parameters α and τd. As we shall see later, each of these parameters provide additional ways to suppress (or enhance) the participation of LSP in the modeling of tropical deep convection.

Here we initially employ values of τd = 6 h and α = 108 m4 kg−1. As discussed in appendix A, this closure is similar to the closure scheme developed by Randall and Pan (1993) where analogous quantities α̂ and τ̂d are employed. An approximate relation between the two (see appendix A) is α̂ = α/2 and τ̂d = τd/2.

Employing these values of α and τd, two additional sensitivity experiments are performed—one with μ = 1 and the other with μ = 0.2. These have been added to Figs. 7–9 and to Table 2. From these diagnostics we can see that simply introducing a prognostic closure to the control ZM (μ = 1) increases its variance by roughly a factor of 3 and decreases the LSP variance by roughly a factor of 3.

The reduction of μ combined with the use of the prognostic closure in the ZM scheme results in some of the largest tropical variability obtained for the ZM scheme in the present study. The spectrum of tropical precipitation variance in Fig. 7 displays substantial broadening of the peak at the diurnal cycle and its higher harmonics. Table 2 indicates that this final experiment is also associated with a 25% increase in tropical upward E–P flux at 100 mb.

The sensitivity of these results to the value of α and τd is investigated by two additional sets of seasonal JJA sensitivity experiments. In the first set of experiments the value of τd was fixed to 6 h and the value of α was allowed to vary from 2 × 106 to 2 × 109 m4 kg−1. In the second set of experiments α was fixed to a value of 2 × 108 m4 kg−1 and τd was allowed to vary from 1200 s to 12 h.

In Fig. 10 we present the contributions from the ZM and LSP schemes to the JJA integrated variance and mean of tropical precipitation in these experiments. For the purpose of comparison with the convective parameterization of Randall and Pan (1993) and Pan and Randall (1998) (following appendix A, we shall collectively refer to these as RPC), we have labeled the axis with α̂ and τ̂d.

From Fig. 10 we can see that as the value of α is increased the contribution of the LSP to both the mean and variance of tropical precipitation increases while the contribution of the ZM scheme to the mean and variance of tropical precipitation decreases. When τ̂d = 3 h the crossover point, where the dominant contribution switches between the two, occurs in the range 108 < α̂ < 109 m4 kg−1 (left column). That is, as α is decreased the rate of change of Mb increases for a given amount of CAPE [see (4)] and the ZM scheme becomes more efficient at the expense of the LSP.

This behavior is consistent with the study of Lin et al. (2000) where the convective parameterization of RPC was employed to investigate the diurnal cycle of convective and stratiform precipitation over the Amazon Basin in the Colorado State University GCM. There it was found that stratiform precipitation in the model dominated the diurnal cycle of precipitation when α̂ = 109, while convective precipitation dominated the diurnal cycle of precipitation when α̂ = 108 (i.e., Lin et al. 2000, their Fig. 7). The fact that the crossover seems to occur over a similar range of α̂ for both the RPC and ZM schemes suggests that the two schemes are more similar than might have been expected given the differences outlined in appendix A.

Figure 10 indicates that a crossover point also occurs as the dissipation time scale τ̂d is varied. When α̂ = 108 m4 kg−1, the crossover point where the dominant contribution switches between the two occurs in the range 3000 s < τ̂d < 5000 s for the variance, and 1000 s < τ̂d < 2000 s for the mean (right column). This suggests that as the dissipation time scale τd becomes small it inhibits the ZM scheme thereby allowing the LSP to participate more in the modeling of deep latent heating as found in section 3c.

Taken together, the results of this and the previous two sections suggest four ways in which the dominant contribution to both the mean and variability of tropical precipitation can be switched between the LSP and ZM schemes. These are through adjustments of the ZM parameters μ, τa (diagnostic closure), and, α and τd (prognostic closure).

f. Seasonal-mean precipitation

From Tables 1 and 2 and Fig. 10 we can see that the changes to the ZM scheme discussed earlier also have an effect on the seasonal-mean tropical precipitation. In Fig. 11 we present the latitude–longitude distribution of seasonal-mean precipitation for JJA from observations (Xie and Arkin 1996) and from two 5-yr climatologies representing the control (i.e., diagnostic closure) and the control with μ = 0.2.

A number of tropical biases are immediately apparent in the control run. There is a large anomaly on the equator in the western Pacific with more than double the observed mean precipitation. Further, the control run displays a poorly structured Indian monsoon, which does not significantly extend below the equator. On the other hand, when μ is reduced to 0.2 much of these biases are alleviated. The extremum of precipitation on the western Pacific essentially vanishes and there is more of a tendency toward a split intertropical convergence zone. The structure of the Indian monsoon appears more realistic as well with a minimum now appearing over the southernmost tip of India. In addition, the precipitation pattern extends well below the equator as is found in the observations.

In an attempt to determine the source of these biases, the seasonal-mean precipitation for the control and μ = 0.2 experiments were decomposed into their LSP and ZM components (not shown). From this analysis it is found that the precipitation biases discussed earlier are caused primarily by the LSP scheme. As we have already seen, the reduction of μ from 1 to 0.2 significantly reduces the tendency of the LSP scheme to act like MCA in the Tropics. Consequently, ZM increases its contribution to the mean precipitation from 60% to more than 80% when μ is reduced in this way. For the same reason, a similar improvement is found for the low-τa experiments of section 3c and the low-α high-τd experiments of section 3e (not shown).

4. Discussion

a. Modeled versus observed variability

In the previous section we have seen that the latent heating produced by the ZM scheme can display a wide range of temporal variability. An obvious question is, does this range of variability encompasses that which occurs in the real system? If one employs the satellite-based estimates of RG00 for this comparison then the answer is clearly no. However, as discussed in the introduction, H02 has shown that those satellite-based measurements overestimate the variability of latent heating by as much as an order of magnitude when compared to overlapping direct radar measurements.

If we use the H02 radar estimates of tropical latent heating as representative of the real system, then one could reasonably argue that the ZM scheme can produce latent heating with the observed level of variability. For example, H02 show one-sided frequency spectra2 of radar derived deep convective precipitation coarse-grained to 0.5° and 2° spatial grids (Figs. 4 and 5 of H02). As the data is coarse-grained the power drops markedly. The physical grid size corresponding to the T47 spectral resolution employed for the experiments in section 3 is 3.75°. A further coarse-graining of the radar data is not possible (T. Horinouchi 2002, personal communication) and so we will compare the results of section 3 with H02's 2° spectra with the understanding that this is an overestimate of what should be expected from the model.

In section 3e it was found that the ZM scheme produced some of the largest variability when the prognostic closure (4) was used with μ = 0.2. Summing the eastward- and westward-directed energy preserving spectra for this experiment (Fig. 7, bottom), we see that the decade of frequencies centered on the diurnal cycle has an amplitude of roughly 20 mm2 day−2. This is roughly equivalent to the amplitude displayed for H02's 2° energy preserving radar spectra (Fig. 5 of H02). Given the limited amount of spatial and temporal data that went into the radar observations, it is difficult to derive any more information from such a comparison. However, if we take into account the fact that the H02 2° spectrum represents an overestimate for the experiments conducted here then one could also argue that the variability of the ZM scheme employing the prognostic closure with μ = 0.2 potentially exceeds the level of observed variability.

b. Participation of LSP in modeling deep latent heating

One of the central results of the present study is that the LSP, which is assumed to model only stratiform precipitation, has the tendency to act like a MCA scheme when the parameterization of deep convection acts too weakly or is inhibited. This was revealed by a compensating relationship in which the contribution of LSP to the mean and variance of tropical precipitation increased (decreased) if the contribution of parameterized deep convection was made to decrease (increase).

Here we have identified four methods whereby the interaction of the LSP and ZM schemes can be sufficiently altered to cause the mean and variance of tropical precipitation to be dominated by either of the two. Each of these four methods involves adjustments to the parameters of the ZM scheme. In principle, one should also be able probe this compensating relationship by a direct perturbation to the LSP. This is not possible here since the current model configuration does not offer a mechanism to directly perturb the LSP to substantially increase or decrease its contribution to tropical precipitation.

A recent study by K. von Salzen and N. A. McFarlane (2003, unpublished manuscript), however, offers some insight into the effect of inhibiting the LSP independent of the ZM scheme. In that study the authors consider the effects of introducing a parameterization of nonprecipitating shallow convection (K. von Salzen and N. A. McFarlane 2002, unpublished manuscript) into the CCCma fourth-generation GCM. There it is found that the shallow convection redistributes heat and moisture in such a way as to reduce the contribution of LSP in the Tropics. This reduction in LSP is found to be associated with an increase in the contribution of the ZM precipitation in the Tropics. From the perspective of the current study, this result may be viewed as a perturbation to the LSP scheme that is independent of the ZM scheme and so lends support for the robust nature of the compensating relationship between LSP and parameterized deep convection.

Furthermore, since the experiments of K. von Salzen and N. A. McFarlane (2003, unpublished manuscript) were conducted in a different configuration of the CCCma GCM (i.e., changes in the fourth-generation GCM include prognostic rather than diagnostic clouds, a new radiation scheme, and the addition of a sulfur cycle—for details see K. von Salzen and N. A. McFarlane 2003, unpublished manuscript), it suggests that the compensating relationship between LSP and parameterized deep convection in the Tropics is an effect that should generalized to other climate GCMs. Evidence of this compensating relationship in another climate model (Lin et al. 2000) has already been discussed in section 3e. Clearly, further investigation is required to determine the nature of this effect in other GCMs.

The tendency of LSP to participate in the explicit modeling of deep latent heating is related to the phenomenon of “gridpoint storms” (hereafter GPS) in the mesoscale modeling literature (e.g., Giorgi 1991). This is the tendency of high-resolution models to produce persistent grid-scale features during cumulus cloud events with intense vertical motion, latent heat release, and precipitation. In fact, one of the primary reasons cumulus parameterizations were introduced into mesoscale models was to reduce the occurrence of GPS (e.g., modified Kuo scheme of Anthes et al. 1987).

Even in global gridpoint models, at resolutions typically used for climate studies, there is potential for the development of GPS—particularly in the Tropics. To avoid this problem some models attempt to suppress the occurrence of LSP at tropical latitudes by, for example, preferentially enhancing the spatial diffusion applied to the moisture field. In some instances, such efforts can result in the complete absence of LSP in the Tropics (S. Webster 2002, personal communication). Clearly, in such circumstance one would not obtain the compensating relationship between parameterized deep convection and the tendency of LSP to act like MCA identified here.

However, the complete elimination of LSP at tropical latitudes is not a desirable solution either. There are physical mechanisms whereby deep convection and stratiform cloudiness interact. For example, as Pan and Randall (1998) discuss, stratiform clouds may be directly produced by cumulus detrainment. Artifically suppressing LSP at tropical latitudes would also inhibit this physical interaction between LSP and deep convection.

The tendency of LSP to participate in the explicit modeling of deep latent heating also has a potential bearing on efforts to increase the variability of parameterized deep convection by the introduction of triggering mechanisms. Such mechanisms are designed to inhibit the onset of deep convection until some criterion is reached, which is related to overcoming convective inhibition (i.e., the region of negative CAPE below the level of free convection). The idea is that this should make the temporal behavior of the deep scheme more sporadic.

However, here we have found that inhibiting the parameterized deep convection tends to cause the LSP to participate more in the modeling of deep convection in the GCM. The introduction of such a trigger mechanism into a deep convective scheme might, therefore, lead to the opposite behavior anticipated. That is, a decrease in the variability of the deep convective precipitation and an increase in the variability of the LSP. If the effect of such an approach were gauged by the change in the resolved waves alone, then one might find an increase in their upward E–P flux with the introduction of a trigger into the parameterization of deep convection. But, as we have found here, this increase may be due to enhanced variability of the LSP rather than the parameterized deep convection.

c. Tropical oscillations

One of the motivating factors for the present study was to understand the way in which tropical oscillations such as the QBO and SAO should be driven in GCMs employing resolutions typically used for climate studies. As discussed in the section 1, modeling the QBO requires 1) the observed spatial structure, 2) the observed period, and 3) driving by the correct mix of waves. Otherwise, we have referred to such modeled oscillations as a QBLOs.

It is now quite common for QBLOs to be driven in climate GCMs that satisfy conditions 1 and 2 (e.g., Takahshi 1999; Scaife et al. 2000; Giorgetta et al. 2002). The QBLO mechanism is so robust it does not matter that each of these efforts employ a widely varying mix of resolved versus parameterized waves to drive the oscillation. However, if one also requires such climate GCMs to, in addition, model realistic stratopause and mesopause semiannual oscillations (SAO) then the correct mix of these waves becomes much more important.

While it is not yet clear what the exact mix of resolved versus parameterized waves should be for GCMs employing spatial resolutions typical of climate studies, there are a number of issues that are becoming more certain. The careful study of H02 provides strong evidence that parameterizations of deep convection that produce realistic amounts of variability will be incapable of forcing resolved tropical waves with sufficient momentum flux to drive a QBLO in a climate GCM. This taken together with the present study also suggests that, contrary to current opinion, PMF parameterizations of deep cumulus convection, such as the ZM scheme, do not inherently possess weak variability.

This further underlines the importance of the parameterization problem for unresolved gravity waves in climate models. The need for the correct mix of waves applies to the parameterized waves as well. One could always drive a QBLO by filling in the required amount of momentum flux with such gravity wave drag parameterization schemes. However, the realistic modeling of the QBO and SAO in a climate model will almost certainly require a more sophisticated approach to the source problem in such parameterizations (i.e., temporal and spatial variation of the amplitudes, wavenumbers, and frequencies of the parameterized waves). This source problem is extremely difficult, given that many of the tropospheric processes that give rise to the waves are themselves parameterized in climate models (e.g., deep convection, boundary layer turbulence, etc.).

5. Summary

In this study we have investigated the temporal variability of the PMF scheme of ZM, which is used in the CCCma third-generation general circulation model (AGCM3), and the Canadian Middle Atmosphere Model (CMAM). Specifically, we have considered the sensitivity of the temporal variability of ZM to several of its internal parameters and to the implementation of a simple prognostic, rather than diagnostic, closure condition for the cloud-base mass flux. The analysis indicates that the ZM scheme can produce a wide range of temporal variability. Furthermore, a comparison of the temporal variability from these sensitivity experiments with the radar estimates of H02 suggests that the ZM scheme can produce a realistic amount of variability for spatial resolutions typically employed for climate GCMs. This result questions the motivation to append stochastic components to such parameterizations to directly enhance their variance (e.g., Lin and Neelin 2003).

One of the most interesting results of the present study is the discovery of a compensating relationship between the parameterized deep convection and the large-scale precipitation (LSP) in the Tropics. As the contribution to the mean and variance of tropical precipitation from either is made to increase, the contribution to the mean and variance from the other is found to decrease. This behavior is apparent in other studies (K. von Salzen and N. A. McFarlane 2003, unpublished manuscript) and other climate GCMs (Lin et al. 2000).

Here we have presented four different parameter adjustments to the ZM scheme that can each cause the mean and variance of tropical precipitation to be dominated by either the LSP or ZM scheme. In the present study the total mean precipitation in the Tropics was found to be roughly conserved, while the total variance in the Tropics displayed a wide range of values depending on which of the two schemes dominated. The largest total variance was found to occur when the LSP dominated in this relationship.

The source of this behavior was found to be due to the ability of the LSP to participate in the modeling of deep latent heating in the Tropics. In doing so, it was found that the LSP acted like a moist-convective adjustment (MCA) scheme. Since MCA schemes are much more variable than PMF schemes, such as ZM (e.g., RG00; Horinouchi et al. 2003), this explains the increased total variance of tropical precipitation when LSP dominates in the Tropics.

Ideally, it is preferred that the LSP primarily model stratiform precipitation and the ZM scheme model all deep convection in the Tropics. The sensitivity experiments of section 3 indicate that when this situation is approached, irrespective of the manner (e.g., small τa or small μ), there is an improvement in model biases. Seasonal-mean tropical precipitation patterns in JJA, for example, showed significant improvement.

Investigation of the source of resolved tropical waves in climate GCMs has focused primarily on the properties of parameterized deep convection in the Tropics (Ricciardulli and Garcia 2000; Amodei et al. 2001; Horinouchi et al. 2003; Lin and Neelin 2002). The ability of LSP to participate in the modeling of tropical deep latent heating in climate models means that it can provide an additional, possibly dominant, source of resolved waves. This has the potential to substantially complicate such studies. Further, the amount of parameterized gravity waves required in specific climate GCMs may implicitly depend on the extent to which LSP in each model acts as a source of resolved tropical waves.

Acknowledgments

The authors thank B. Boville, T. Horinouchi, K. von Salzen, and S. Webster for useful discussions during the course of this study and S. Kharin, V. Lorant, and K. von Salzen for their careful reading of the manuscript.

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APPENDIX A

Prognostic Closure Methods

The prognostic closure for cloud-base mass flux (4) used in the current study is similar to the prognostic closure derived by Randall and Pan (1993) and Pan and Randall (1998), hereafter collectively referred to as RPC. RPC applied their prognostic closure to a modified Arakawa and Schubert (1974, hereafter AS74) scheme. While the RPC approach is similar to that adopted here, there exist important differences and these are elaborated in the following.

The RPC closure is derived from a budget for the vertically integrated kinetic energy K(λ) of convective motions in a “subensemble” of cumulus clouds contained in a GCM grid cell. Each subensemble is characterized by a different fractional entrainment rate λ. In their derivation, RPC posit the relation
Kλα̂MBλ2
This relation between vertically integrated kinetic energy K(λ) and the cloud-base mass flux of each subensemble, MB(λ), is a concise statement that essentially defines the RPC closure. RPC further assume that K is dissipated at a rate K(λ)/τ̂d.
Following on (A1), RPC derive the following governing equations for their modified AS74 PMF scheme
i1520-0469-61-16-1993-ea2
where the term MB(λ) represents an integral over entrainment rates and is usually positive. In (A2) and (A3) Â(λ) is the “cloud work function” (AS74). The cloud work function Â(λ) is related to the CAPE, A, which was introduced in section 2a. This will be discussed further below. In the RPC scheme (as in AS74) each subensemble has a distinct cloud-base mass flux MB(λ) and entrainment rate λ. Consequently, each subensemble must be closed individually and accumulated to determine the ensemble cloud updraft mass flux Mu(z).

In order to relate the prognostic closure (4) for the ZM scheme to the RPC prognostic closure outlined above, it is useful to first relate the basic ZM scheme to the AS74 scheme. One important difference between the two is that the original AS74 scheme, and that modified by RPC, consider only updraft mass fluxes in their formulation. The ZM scheme on the other hand considers both updraft and downdraft mass fluxes in its formulation. Further information regarding the parameterization of the downdraft mass flux Md in the ZM scheme may be found in appendix C.

Another central difference is that the ZM scheme is a bulk cloud model, which is derived by the application of a number of simplifying assumptions to the AS74 scheme. One of the key assumptions in its derivation is that all subensembles have the same cloud-base mass flux, MB(λ) = Mb. In this way ZM derive a simple analytic expression for the ensemble cloud updraft mass flux Mu in terms of a continuous distribution of λ
i1520-0469-61-16-1993-ea4
where λD(z) is the fractional entrainment rate of the updraft plume that detrains at height z, λo is the maximum entrainment rate, which sets the height of the shallowest plume, and zb is the height of the cloud base. Zhang and McFarlane (1995) derive an approximate solution for the vertical profile λD(z) by requiring that the temperature of the plume that detrains at height z be equal to the environmental temperature. Then ZM define λo = λD(zo), where zo is taken to be the elevation at which the moist static energy has a minimum near midtroposphere. In this way the ensemble considered by ZM is limited to updrafts that detrain at or above zo ensuring that detrainment is confined to the conditionally stable region of the atmospheric column.

In deriving a prognostic closure for the ZM scheme, we could follow RPC and posit (A1) as our closure condition. In a similar manner we would then derive a cloud work function appropriate to the properties of the ZM scheme which would include the simplified expression (A4) for updraft mass flux and include the effects of downdraft mass fluxes (appendix C). In this way the cloud work function derived for the ZM scheme would differ in form from the cloud work function derived by RPC and AS74. Aside from this difference in work function the governing equations for such a prognostic ZM scheme would be identical to (A2) and (A3).

Rather than follow this approach, here we have chosen to employ a simpler strategy that is consistent with the bulk cloud formulation of ZM. That is, rather than posit (A1) for our closure we simply posit
i1520-0469-61-16-1993-eqa1
[i.e., (4)] with the constraint that Mb is positive and nonzero only when A > 0. This is isomorphic to (A3) and in this sense it is motivated by the RPC closure. Another motivating factor for the use of CAPE rather than the cloud work function  is that it eliminates the need for the solution of a complex integral equation for MB(λ) as a function of λ (e.g., Lord 1982).
It is of interest to note that the equilibrium solutions of the RPC governing equations, (A2) and (A3),
i1520-0469-61-16-1993-ea5
[i.e., Pan and Randall 1998, their Eqs. (26) and (27)], are essentially identical to the equilibrium solution of the ZM prognostic system (1) and (4)
i1520-0469-61-16-1993-ea7
A comparison of (A3) and (4) suggests that one should take
i1520-0469-61-16-1993-ea9
when relating the RPC scheme to the prognostic-closure form of the ZM scheme presented here. It is important to point out that, given the differences outlined above, (A9) on its own does not guarantee a correspondence between the two schemes. However, as is found in section 3e the two schemes display remarkable similarities in terms of their sensitivity to these parameters.

APPENDIX B

Large-Scale Precipitation

Here we provide a brief review of the stratiform, or large-scale, precipitation scheme used in AGCM3. The algorithm, which is used to determine condensation and latent heating based on grid-box mean quantities, has been used previously within the MCA parameterization employed in GCM2 (McFarlane et al. 1992) and GCM1 (Boer et al. 1984).

In the LSP scheme all liquid water produced as a result of condensation is assumed to fall out as precipitation. In each grid box there occurs an initial grid-box mean specific humidity qi and relative humidity Hi = qi/q*i, where q*i = q*i(Ti, Pi) is the saturation specific humidity and Ti and Pi are the initial grid-box mean temperature and pressure, respectively.

The change in specific humidity due to condensation, δq = qfqi produces a heating in the grid box. The associated change in temperature δT = TfTi is given by the Clausius–Clapeyron equation
CpδTLδq
where Cp is the specific heat at constant pressure, L is the latent heat of condensation, and qf and Tf are the final grid-box mean values of specific humidity and temperature, respectively.
In each grid box a simple triangular distribution fi(H) of relative humidity H is assumed about the grid-box mean value Hi. This is illustrated in Fig. B1. The width of the distribution about Hi is given by Δ ≡ 1 − Hc, where Hc is a tunable parameter referred to as the threshold relative humidity. A value of Hc = 0.95 is employed in AGCM3. The distribution fi(H) has the following properties:
i1520-0469-61-16-1993-eb2
The portion of fi with relative humidity exceeding unity has been shaded in Fig. 12. In the shaded region we have both vapor qυ|H≥1 and liquid ql|H≥1 due to condensation which total
i1520-0469-61-16-1993-eb4
The portion associated with liquid water ql|H≥1 is
i1520-0469-61-16-1993-eb5
while the portion associated with water vapor is
i1520-0469-61-16-1993-eb6

The quantity ql|H≥1 represents a first guess at the amount of condensed water in the grid box since it has ignored the associated release of latent heat. That is, ql|H≥1 has been determined using the initial mean grid-box mean temperature Ti in the definition of the saturation specific humidity q*i. The associated first guess for the final grid-box mean relative humidity may be written as Hiql|H≥1/q*i.

The strategy employed here is to assign the final grid-box mean relative humidity to be the first-guess value Hf = Hiql|H≥1/q*i and then determine the consistent final grid-box mean temperature Tf and associated liquid water ql. In doing so we implicitly assume a new, final distribution for water vapor ff(H) in which all of the remaining water vapor in the grid box exists at H ≤ 1. This is achieved by assuming that an amount of water vapor qυ|H≥1 occurs at H = 1 in the form of a δ function. This results in a ff(H) of the form
i1520-0469-61-16-1993-eb7
so that
i1520-0469-61-16-1993-eb8
We may now also write δq = qfqi, where
qfq*fTfPiHf
and where Tf, the final grid-box mean temperature, remains to be determined.
Using (B9) in (B1) we form an equation for the unknown Tf
i1520-0469-61-16-1993-eb10
This equation is nonlinear in Tf and one method of approximate solution has been outlined previously by Daley et al. (1976).
Given Hi, Hf, and the solution Tf, the actual amount of condensed liquid water that falls as large-scale precipitation in the model is then calculated to be
qlq*fTfPiHfHi

APPENDIX C

Evaporation in Downdrafts

As discussed in appendix A, ZM make the simplifying assumption that all subensembles have the same cloud-base mass flux Mb. Similarly, ZM assume an equal initial downward flux Md for an ensemble of downdraft plumes, which all originate at the bottom of the updraft detrainment layer zo (see appendix A for the definition of zo). In addition, ZM assume that the initial downward flux is proportional to the cloud-base mass flux Mb for the ensemble of updraft plumes.

As discussed in ZM, the downdraft mass flux may be written
i1520-0469-61-16-1993-ec1
where λm = 2λo is the maximum downdraft entrainment rate, and β is a proportionality factor referred to as the weight of the downdraft. The weight factor β is constrained by the availability of precipitation and the requirement that the net mass flux at cloud base zb be upward. The latter condition implies that 0 ≤ β ≤ 1.
The downdrafts are assumed to maintain a saturated state by the evaporation of rainwater. The maximum possible evaporation in the downdrafts may be written
i1520-0469-61-16-1993-ec2
where Ed is the downdraft entrainment rate, q is the environmental water vapor mixing ratio, and qd is the downdraft saturation water vapor mixing ratio.
The amount of rainwater evaporated into the downdrafts is β EVP. To ensure 0 ≤ β ≤ 1, this amount must not exceed the maximum EVP given by (C2). The current implementation of the ZM scheme constrains β EVP to be
i1520-0469-61-16-1993-ec3
where PC is the total precipitation produced in updrafts and μ is a tunable parameter. In addition to satisfying 0 ≤ β ≤ 1, (C3) also limits the evaporation to some fraction μ of the total precipitation PC. These conditions are satisfied by taking β to be
i1520-0469-61-16-1993-ec4
This expression differs slightly from that originally presented by ZM [i.e., Eq. (11)]. Currently a value of μ = 1 is used in AGCM3.

Fig. 1.
Fig. 1.

Latitude–height distribution of zonal-mean latent heating and its variance for JJA seasonal runs employing (a), (b) an MCA scheme, and (c), (d) the LSP of AGCM3 in the absence of the ZM scheme. The mean heating rate is displayed in units of 0.1 K day−1 with a contour interval of 10, and the variance of heating rate is displayed in units of (K day−1)2 with a contour interval of 10

Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<1993:TVOMTP>2.0.CO;2

Fig. 2.
Fig. 2.

Frequency spectra of ZM component of tropical precipitation (13°S–13°N) for a series of JJA seasonal runs in which the time scale τa was varied. The frequency spectra are displayed in three formats: (top) linear–linear, (middle) log–log, and (bottom) “energy preserving” log–linear. Positive frequencies correspond to eastward phase propagation while negative frequencies correspond to westward frequency propagation

Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<1993:TVOMTP>2.0.CO;2

Fig. 3.
Fig. 3.

Wavenumber–frequency spectra of upward E–P flux (13°S–13°N) through 57 hPa for the series of JJA seasonal runs (a) τa = 300, (b) 2400, (c) 7200 s, and (d) ∞ (LSP alone). Following Horinouchi et al. (2003), these are presented in energy preserving form. Positive and negative frequencies correspond to propagation as in Fig. 2

Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<1993:TVOMTP>2.0.CO;2

Fig. 4.
Fig. 4.

As in Fig. 2 except for LSP component of tropical precipitation. Note also, a different vertical scale is employed

Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<1993:TVOMTP>2.0.CO;2

Fig. 5.
Fig. 5.

Latitude–height distribution of zonal-mean latent heating variance of (a), (c), (e), (g) ZM and (b), (d), (f), (h) LSP for the τa sensitivity experiments presented in Figs. 2 and 4. Heating rates are displayed in units of 0.1 K day−1, with a contour interval of 2 K day−1 for ZM and 10 K day−1 for LSP

Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<1993:TVOMTP>2.0.CO;2

Fig. 6.
Fig. 6.

Latitude–height distribution of (a), (c), (e), (g) zonal-mean temperature and (b), (d), (f), (h) specific humidity anomaly away from the control run employing τa = 2400 s. Temperature anomalies are displayed in units of K while specific humidity anomalies are displayed in units of 10−4 kg m−3

Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<1993:TVOMTP>2.0.CO;2

Fig. 7.
Fig. 7.

The ZM component as in Fig. 2 except for the control, the control with μ = 0.2, the prognostic closure (4), and the prognostic closure (4) with μ = 0.2. The prognostic closure in these experiments used values of τd = 6 h and α = 108

Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<1993:TVOMTP>2.0.CO;2

Fig. 8.
Fig. 8.

As in Fig. 7 except for LSP component of tropical precipitation. Note also, a different vertical scale is employed

Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<1993:TVOMTP>2.0.CO;2

Fig. 9.
Fig. 9.

As in Fig. 5 except for the sensitivity experiments presented in Figs. 7 and 8. Heating rates are displayed in units of 0.1 K day−1 with a contour interval of 2 K day−1 for both ZM and LSP

Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<1993:TVOMTP>2.0.CO;2

Fig. 10.
Fig. 10.

JJA seasonal (top) variance and (bottom) mean (13°S–13°N) for the two sets of sensitivity experiments employing the prognostic closure (4) in which (left) τd = 6 h, 2 × 106α ≤ 2 × 109 m4 kg−1 and (right) α = 2 × 108 m4 kg−1, 1200 s ≤ τd ≤ 12 h. Axes have been labeled with α̂ and τ̂d to facilitate a comparison with the closure of Pan and Randall (1998)

Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<1993:TVOMTP>2.0.CO;2

Fig. 11.
Fig. 11.

JJA seasonal-mean precipitation for (top) observations (Xie and Arkin 1996), (middle) 5-yr climatologies of the control, and (bottom) the control with μ = 0.2

Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<1993:TVOMTP>2.0.CO;2

i1520-0469-61-16-1993-fB1

Fig. B1. Assumed triangular distribution of subgrid relative humidity H about the grid-box mean value Hi. The width of the distribution, Δ, is taken to be Δ = 1 − Hc, where Hc is a tunable parameter referred to as the threshold relative humidity

Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<1993:TVOMTP>2.0.CO;2

Table 1.

The ZM and LSP components of JJA seasonal mean and variance of tropical precipitation (13°S–13°N), and upward E–P flux at 100 hPa for the series of τ a sensitivity experiments in Figs. 2 and 4

Table 1.
Table 2.

The same as in Table 1 except for the sensitivity experiments presented in Figs. 7 and 8

Table 2.

1

The term QBLO used here, differs from the term QLO used by Takahshi (1999) and Hamilton et al. (2001), in that the QLO requires only conditions 1 and 2.

2

The sum of the eastward and westward spectra in Figs. 2 and 7.

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