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A Simple Model for Stratocumulus to Shallow Cumulus Cloud Transitions

D. Chung Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California

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J. Teixeira Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California

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Print Publication:
01 Apr 2012
DOI:
https://doi.org/10.1175/JCLI-D-11-00105.1
Page(s):
2547–2554
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Abstract

It is shown that essential features of stratocumulus to shallow cumulus cloud transitions can be represented by a simple stochastic model constructed from an ensemble of transitions, each of which depends on the amount of surface latent heat flux relative to initial cloud-top longwave net radiative flux. In its essence the simple model establishes a causal relation between the increase of sea surface temperature (SST) and the decrease in cloud fraction (CF) along the trade winds. The mean and variance of SST are taken from observations. Model predictions are compared with observations of CF along Lagrangian trajectories in four eastern subtropical ocean regions. The model reproduces well the decrease in mean CF and the peak in CF spread.

Corresponding author address: J. Teixeira, MS 233-300, Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109. E-mail: joao.teixeira@jpl.nasa.gov

Abstract

It is shown that essential features of stratocumulus to shallow cumulus cloud transitions can be represented by a simple stochastic model constructed from an ensemble of transitions, each of which depends on the amount of surface latent heat flux relative to initial cloud-top longwave net radiative flux. In its essence the simple model establishes a causal relation between the increase of sea surface temperature (SST) and the decrease in cloud fraction (CF) along the trade winds. The mean and variance of SST are taken from observations. Model predictions are compared with observations of CF along Lagrangian trajectories in four eastern subtropical ocean regions. The model reproduces well the decrease in mean CF and the peak in CF spread.

Corresponding author address: J. Teixeira, MS 233-300, Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109. E-mail: joao.teixeira@jpl.nasa.gov
Keywords: Cloud parameterizations

1. Introduction

It is now recognized that boundary layer clouds remain the largest source of uncertainty in current climate change projections (Cess et al. 1989; Bony and Dufresne 2005). Inadequate characterization of unresolved cloudiness severely affects climate predictions because of the vastly different radiative properties of stratocumulus versus shallow cumulus clouds. It is likely that the transition from stratocumulus to shallow cumulus regimes plays a key role in cloud–climate feedbacks (e.g., Teixeira et al. 2011) and it is thus crucial to understand the factors controlling this cloud transition. The stratocumulus to shallow cumulus transition has been investigated using a variety of approaches, ranging from large-scale statistical characterizations (often based on global observations) (Klein and Hartmann 1993; Wood and Hartmann 2006; Kawai and Teixeira 2010) to specific observational studies (Bretherton and Pincus 1995; Sandu et al. 2010; Mauger and Norris 2010) to mixed layer modeling studies (Bretherton and Wyant 1997) to high-resolution simulations (Krueger et al. 1995; Wyant et al. 1997).

For a better understanding of specific natural processes, it is often useful to have a hierarchy of models that allow insight into a particular problem at different levels (e.g., Held 2005). In the case of the stratocumulus to shallow cumulus transition, there seems to be a clear gap between the simplicity (and lack of causality) of observationally based statistical relations—for example, the linear relation between lower-tropospheric stability (LTS) and cloud fraction (CF) (e.g., Klein and Hartmann 1993)—and the complexity of high-resolution large-eddy simulation (LES) models. Even mixed layer models, which sit somewhere between LTS and LES models of the transition, suffer from particular problems: they can be quite complex—in a way that does not easily lead to a complete understanding of the model’s behavior—and yet they cannot even really be claimed as appropriate models of the transition, since the key variable that characterizes the transition, the cloud fraction, is not a variable in mixed layer models.

This is an unfortunate situation, and it is fair to say that simple models of the transition that do not involve the complexities of LES or mixed layer models but do include some measure of physical reasoning beyond empirical statistical correlations are clearly lacking. This is the main reason for us to propose the following simple stochastic model of the transition, where essentially we relate the decrease in cloud fraction during the transition to changes in the sea surface temperature (SST).

Our approach is to augment observed Lagrangian transition statistics with a simple model in an attempt to tease out some basic aspects of the transition. The simple model is constructed from simple physical (and dimensional) arguments that pertain to the changes experienced by the underlying physical processes across the transition. The Lagrangian data is taken from the analysis of Sandu et al. (2010), which details the statistics of CF along with the statistics of environmental factors that are often linked with the transition such as SST and large-scale divergence. They found that the transition, characterized by a decrease in CF during the first three days, is most correlated with an increase in SST; meanwhile, the large-scale divergence remains relatively constant during the same period.

2. A simple cloud transition model

As mentioned above, the development of this model is essentially driven by the following question: What is the simplest possible model that represents the key characteristics of the transition that still contains some basic physics and causality?

Let us assume that the essential fluxes that play an important role in the dynamics of a transitioning Lagrangian air column are surface latent heat flux (LHF) and cloud-top longwave net radiative flux (RAD). LHF drives the mixing in shallow cumulus boundary layers and clouds (e.g., Siebesma et al. 2003) while RAD drives the mixing in stratocumulus boundary layers and clouds (e.g., Lilly 1968). We then assume that the transition from stratocumulus to shallow cumulus clouds must depend in some way on the changes of these fluxes, ΔRAD and ΔLHF. Note that other key meteorological factors are related to these; changes in subsidence drying are largely compensated by ΔLHF, and similarly changes in subsidence warming are largely compensated by ΔRAD. We then choose ΔRAD and ΔLHF to be the only two independent flux changes, and let
e1
be a constant of the transition (which can also be argued from dimensional analysis). A somewhat similar criterion is proposed by Bretherton and Wyant (1997) to model boundary layer decoupling based on simplified mixed layer model dynamics.
We model RAD = F0 · CF, where F0 ≈ 50 W m−2, the initial cloud-top longwave net radiative cooling that drives stratocumulus convection, and CF is the cloud fraction. We define β ≡ LHF/F0. Then (1), on taking the limit, simplifies to −γ = dCF/dβ, which we integrate and supply with stratocumulus and shallow-cumulus boundary conditions, C1 ≈ 1 and C2 ≈ 0.1, to find
e2
The thresholds β1 and β2 mark the beginning and the end of the transition, related via −γ = (C2 − C1)/(β2 − β1).
We model LHF via the bulk aerodynamic formula
e3
where ρ0 ≈ 1.2 kg m−3, the density above the surface; L ≈ 2.5 × 106 J kg−1, the latent heat of vaporization; CTV(qS − q0) is the specific humidity flux; CT ≈ 0.015/7, the bulk aerodynamic coefficient; V ≈ 7 m s−1, the horizontal wind above the surface [Lilly (1968) used CTV ≈ 0.015 m s−1]; q0 is the specific humidity above the surface; qS is the saturation specific humidity at SST; and RH = q0/qS ≈ 0.9, the relative humidity above the surface, assuming that the difference between the SST and the temperature just above is negligible. We calculate qS from the Clausius–Clapeyron equation evaluated at SST and a pressure of 103 hPa.

3. Comparison with observations

a. Description of Lagrangian data

The present model relies on observed Lagrangian statistics of SST and CF: SST as model input and CF for comparison with model output. These are taken from the Lagrangian analysis by Sandu et al. (2010) of stratocumulus to shallow cumulus cloud transitions in four eastern subtropical ocean regions, namely the northeast and southeast Pacific and Atlantic Oceans (NEP, SEP, NEA, and SEA). In their analysis, SST is taken from the interim European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-Interim), with spatial resolution of 1.5° × 1.5° and temporal resolution of 6 h, while CF is from collection 5 of the Moderate Resolution Imaging Spectroradiometer (MODIS) level-3 products, with spatial resolution of 1° × 1°, observed twice per day at 1030 and 1330 UTC.

The data of Sandu et al. (2010), reproduced in Fig. 1, shows that the median SST rises during the first 3 days and then levels off thereafter. However, other environmental factors, such as the large-scale divergence and the temperature above the boundary layer, remain essentially constant during the same period (Sandu et al. 2010). A decrease in lower-tropospheric stability is also observed, but this is associated with the increase in SST. Figure 1 shows that the CF decreases on average but exhibits considerable spread during the first 3 days.

Fig. 1.
Fig. 1.

Lagrangian transition statistics of SST and CF from four eastern subtropical ocean regions taken from the analysis of Sandu et al. (2010).

Citation: Journal of Climate 25, 7; 10.1175/JCLI-D-11-00105.1

b. Simple model with SST fluctuations

For simplicity, we assume that SST obeys the normal distribution, N(μSST, σSST), with mean μSST and standard deviation σSST taken from the Lagrangian analysis of Sandu et al. (2010); for a normal distribution, the median is equal to the mean and the interquartile range is equal to 1.35 times the standard deviation.

At a given time t, the ensemble median and interquartile range of CF are respectively given by CF0.5(t) and CF0.75(t) − CF0.25(t), where CFp(t) is the pth percentile, defined according to
eq1
where X is the vector of random variables and f is its probability density function (PDF). For the present single-factor model,
eq2
where n is the PDF of a normal distribution. The integrals are computed using Monte Carlo integration, with 5.1 × 105 samples for each of the 41 instances of time distributed evenly over the duration of the transition.

Figure 2 shows that the main features of the transition are reproduced by the model. The mean CF decreases systematically and the peak in the CF spread is a robust feature of the model, although these results are somewhat sensitive to the sharpness of the transition, β2 − β1. The values of β1 and β2 are fitted to the NEP data by eye and no additional tuning of the model is performed for the other three regions. The fits (β1, β2) ≈ (1.4, 2.1), (1.5, 2.0), and (1.6, 1.9) correspond to γ ≈ 0.78, 0.56, and 0.33, respectively.

Fig. 2.
Fig. 2.

Comparison of observations with model predictions using different transition thresholds, β1 and β2. Increasing β2 − β1 simultaneously decreases the peak CF spread and the steepness of the median CF.

Citation: Journal of Climate 25, 7; 10.1175/JCLI-D-11-00105.1

c. Simple model with SST and wind fluctuations

To investigate the sensitivity of the results to the variability in wind speed, we extend the model to include the effects of wind fluctuations, V, appearing in the bulk aerodynamic transfer formula, (3). We introduce a stochastic component to the wind speed by assuming that V is independent of SST and V ~ N(7 m s−1, σV) for all time. Figure 3 shows that this simple stochastic model of the wind speed has a significant impact on the overall results: wind fluctuations increase the peak spread of CF without significantly changing the median CF. Except for the SEP region, the model shows remarkable good agreement with observations for both the CF median and interquartile range; the agreement occurs in three regions even though the key coefficients have been somewhat (loosely) optimized for just the NEP region. While we do not fully understand the discrepancy in the SEP region, we remark that the fit can be improved if F0 is lowered by 5%.

Fig. 3.
Fig. 3.

Effect of wind fluctuations on model predictions with fixed transition thresholds, (β1, β2) ≈ (1.4, 2.1). Increasing the wind fluctuations increases the peak CF spread uniformly but does not affect the median CF.

Citation: Journal of Climate 25, 7; 10.1175/JCLI-D-11-00105.1

d. CF distribution

Using the NEP data and (β1, β2) ≈ (1.4, 2.1) as an indication of transition time scale, we find that the median β roughly coincides with β1 and β2 on day 0 and day 6, respectively. Although this result appears to imply a fairly smooth transition, PDFs of β and CF over the course of 6 days in Fig. 4 paint a more complex picture of how the model behaves. At any time, there are pretransition, transitioning, and posttransition air masses, corresponding to β < β1, β1 < β < β2, and β2 < β, respectively. While the PDF of β is Gaussian, the PDF of CF has two dominant peaks at C1 ≈ 1 and C2 ≈ 0.1 at any time.

Fig. 4.
Fig. 4.

Snapshots of model (left) β and (right) CF PDFs of the NEP region produced with transition thresholds, (β1, β2) ≈ (1.4, 2.1) (vertical lines), and wind fluctuations, σV ≈ 1.0 m s−1.

Citation: Journal of Climate 25, 7; 10.1175/JCLI-D-11-00105.1

Comparing the global (for all time) model CF PDF from the NEP region with the observed CF PDF from the NEA region (Mauger and Norris 2010) (see our Fig. 5), we note that the present model captures essential features of the cloudiness distribution in transition regions: the probabilities are concentrated in two peaks with small, but nonzero, probabilities in the intermediate states. However, these are not completely equal comparisons since the Lagrangian study of Mauger and Norris (2010) tracks air columns that revert to their initial CF after 6 days, but the transitions of Sandu et al. (2010) are characterized by a monotonic decrease of CF over 6 days.

Fig. 5.
Fig. 5.

Global (for all time) model CF PDF from the NEP region. The two-peak PDF predicted by the present model is consistent with the Lagrangian analysis of Mauger and Norris (2010) in the NEA region.

Citation: Journal of Climate 25, 7; 10.1175/JCLI-D-11-00105.1

e. Two-peak model

To investigate assumptions on the CF PDF, independent of controlling factors, we consider the following two-peak model for conditional PDF of CF at time t:
e4
where p1 + p2 + α(C1 − C2) = 1. Assuming that p1, p2 < 0.5, or equivalently, that the median CF0.5 ∈ (C2, C1), the interquartile range can be written in terms of the median,
e5
independent of p1 or p2. Once the parameter α is fixed, we can calculate the model interquartile range from the observed median CF. As mentioned above, the purpose of this two-peak model is to test assumptions on the statistics of CF itself, independent of the controlling factors; presently, we are testing whether the conditional PDF of CF at time t can be modeled with two peaks at C1 and C2 given by (4). Fixing α = 0.9, C1 = 1, and C2 = 0.1, Fig. 6 shows fair agreement between model spread and observed spread at all times. This demonstrates that the piecewise form, (2), is not the reason for the discrepancy in the SEP region (Fig. 3).
Fig. 6.
Fig. 6.

Estimate of the interquartile range from the observed median, given by (5), using a two-peak distribution model for the conditional PDF at time t [(4)].

Citation: Journal of Climate 25, 7; 10.1175/JCLI-D-11-00105.1

4. Concluding remarks

A lack of simple yet physically based (even if minimally so) models that are capable of representing the key characteristics of the stratocumulus to cumulus transition is the main motivation for the development of the simple model described in this paper. The model is essentially based on an assumed linear change of cloud fraction, during the transition, as a function of the ratio between two key parameters controlling the transition: the surface latent heat flux (which controls cumulus mixing) and the stratocumulus cloud-top longwave radiative flux. The surface latent heat flux (LHF) is calculated using the bulk formula with a normal PDF of stochastic sea surface temperature (SST) based on observed mean and variance from Sandu et al. (2010). The wind speed can also be based on a normal PDF. This forms the basis of a simple stochastic model, which reproduces essential features of the transition, including the peak in CF spread and the gradual decrease in CF median. The model shows good agreement with observations provided that the variability of SST and V, which influences LHF, is taken into account. According to the model, increasing the V fluctuations increases the spread in CF uniformly but leaves the median CF virtually unchanged.

Although the median CF decreases gradually over 6 days, each realization of the transition could occur more rapidly over narrower SST ranges. Such processes generate PDFs with two peaks (pre- and posttransition states) joined by infrequent transitioning states, consistent with observations. To decouple CF statistics from the precise role of controlling factors, a simple two-peak PDF model is shown to successfully predict the unknown interquartile range given the median.

Acknowledgments

The authors acknowledge the comments from three reviewers and the support provided by the Office of Naval Research, Marine Meteorology Program under Award N0001408IP20064; the NASA MAP Program; and the NOAA MAPP/CPO Program. This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.

REFERENCES

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    • Search Google Scholar
    • Export Citation

  • Bretherton, C. S., and R. Pincus, 1995: Cloudiness and marine boundary layer dynamics in the ASTEX Lagrangian experiments. Part I: Synoptic setting and vertical structure. J. Atmos. Sci., 52, 2707–2723.

    • Search Google Scholar
    • Export Citation

  • Bretherton, C. S., and M. C. Wyant, 1997: Moisture transport, lower-tropospheric stability, and decoupling of cloud-topped boundary layers. J. Atmos. Sci., 54, 148–167.

    • Search Google Scholar
    • Export Citation

  • Cess, R. D., and Coauthors, 1989: Interpretation of cloud–climate feedback as produced by 14 atmospheric general circulation models. Science, 245, 513–516.

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    • Export Citation

  • Held, I. M., 2005: The gap between simulation and understanding in climate modeling. Bull. Amer. Meteor. Soc., 86, 1609–1614.

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    • Search Google Scholar
    • Export Citation

  • Klein, S. A., and D. L. Hartmann, 1993: The seasonal cycle of low stratiform clouds. J. Climate, 6, 1587–1606.

  • Krueger, S. K., G. T. McLean, and Q. Fu, 1995: Numerical simulation of the stratus-to-cumulus transition in the subtropical marine boundary layer. Part I: Boundary-layer structure. J. Atmos. Sci., 52, 2839–2850.

    • Search Google Scholar
    • Export Citation

  • Lilly, D. K., 1968: Models of cloud-topped mixed layers under a strong inversion. Quart. J. Roy. Meteor. Soc., 94, 292–309.

  • Mauger, G. S., and J. R. Norris, 2010: Assessing the impact of meteorological history on subtropical cloud fraction. J. Climate, 23, 2926–2940.

    • Search Google Scholar
    • Export Citation

  • Sandu, I., B. Stevens, and R. Pincus, 2010: On the transitions in marine boundary layer cloudiness. Atmos. Chem. Phys., 10, 2377–2391.

    • Search Google Scholar
    • Export Citation

  • Siebesma, P., and Coauthors, 2003: A large eddy simulation intercomparison study of shallow cumulus convection. J. Atmos. Sci., 60, 1201–1219.

    • Search Google Scholar
    • Export Citation

  • Teixeira, J., and Coauthors, 2011: Tropical and subtropical cloud transitions in weather and climate prediction models: The GCSS/WGNE Pacific Cross-Section Intercomparison (GPCI). J. Climate, 24, 5223–5256.

    • Search Google Scholar
    • Export Citation

  • Wood, R., and D. L. Hartmann, 2006: Spatial variability of liquid water path in marine low cloud: The importance of mesoscale cellular convection. J. Climate, 19, 1748–1764.

    • Search Google Scholar
    • Export Citation

  • Wyant, M. C., C. S. Bretherton, H. A. Rand, and D. E. Stevens, 1997: Numerical simulations and a conceptual model of the stratocumulus to trade cumulus transition. J. Atmos. Sci., 54, 168–192.

    • Search Google Scholar
    • Export Citation
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Journal of Climate

  • Bony, S., and J.-L. Dufresne, 2005: Marine boundary layer clouds at the heart of tropical cloud feedback uncertainties in climate models. Geophys. Res. Lett., 32, L20806, doi:10.1029/2005GL023851.

    • Search Google Scholar
    • Export Citation

  • Bretherton, C. S., and R. Pincus, 1995: Cloudiness and marine boundary layer dynamics in the ASTEX Lagrangian experiments. Part I: Synoptic setting and vertical structure. J. Atmos. Sci., 52, 2707–2723.

    • Search Google Scholar
    • Export Citation

  • Bretherton, C. S., and M. C. Wyant, 1997: Moisture transport, lower-tropospheric stability, and decoupling of cloud-topped boundary layers. J. Atmos. Sci., 54, 148–167.

    • Search Google Scholar
    • Export Citation

  • Cess, R. D., and Coauthors, 1989: Interpretation of cloud–climate feedback as produced by 14 atmospheric general circulation models. Science, 245, 513–516.

    • Search Google Scholar
    • Export Citation

  • Held, I. M., 2005: The gap between simulation and understanding in climate modeling. Bull. Amer. Meteor. Soc., 86, 1609–1614.

  • Kawai, H., and J. Teixeira, 2010: Probability density functions of liquid water path and cloud amount of marine boundary layer clouds: Geographical and seasonal variations and controlling meteorological factors. J. Climate, 23, 2079–2092.

    • Search Google Scholar
    • Export Citation

  • Klein, S. A., and D. L. Hartmann, 1993: The seasonal cycle of low stratiform clouds. J. Climate, 6, 1587–1606.

  • Krueger, S. K., G. T. McLean, and Q. Fu, 1995: Numerical simulation of the stratus-to-cumulus transition in the subtropical marine boundary layer. Part I: Boundary-layer structure. J. Atmos. Sci., 52, 2839–2850.

    • Search Google Scholar
    • Export Citation

  • Lilly, D. K., 1968: Models of cloud-topped mixed layers under a strong inversion. Quart. J. Roy. Meteor. Soc., 94, 292–309.

  • Mauger, G. S., and J. R. Norris, 2010: Assessing the impact of meteorological history on subtropical cloud fraction. J. Climate, 23, 2926–2940.

    • Search Google Scholar
    • Export Citation

  • Sandu, I., B. Stevens, and R. Pincus, 2010: On the transitions in marine boundary layer cloudiness. Atmos. Chem. Phys., 10, 2377–2391.

    • Search Google Scholar
    • Export Citation

  • Siebesma, P., and Coauthors, 2003: A large eddy simulation intercomparison study of shallow cumulus convection. J. Atmos. Sci., 60, 1201–1219.

    • Search Google Scholar
    • Export Citation

  • Teixeira, J., and Coauthors, 2011: Tropical and subtropical cloud transitions in weather and climate prediction models: The GCSS/WGNE Pacific Cross-Section Intercomparison (GPCI). J. Climate, 24, 5223–5256.

    • Search Google Scholar
    • Export Citation

  • Wood, R., and D. L. Hartmann, 2006: Spatial variability of liquid water path in marine low cloud: The importance of mesoscale cellular convection. J. Climate, 19, 1748–1764.

    • Search Google Scholar
    • Export Citation

  • Wyant, M. C., C. S. Bretherton, H. A. Rand, and D. E. Stevens, 1997: Numerical simulations and a conceptual model of the stratocumulus to trade cumulus transition. J. Atmos. Sci., 54, 168–192.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Lagrangian transition statistics of SST and CF from four eastern subtropical ocean regions taken from the analysis of Sandu et al. (2010).

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Comparison of observations with model predictions using different transition thresholds, β1 and β2. Increasing β2 − β1 simultaneously decreases the peak CF spread and the steepness of the median CF.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Effect of wind fluctuations on model predictions with fixed transition thresholds, (β1, β2) ≈ (1.4, 2.1). Increasing the wind fluctuations increases the peak CF spread uniformly but does not affect the median CF.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Snapshots of model (left) β and (right) CF PDFs of the NEP region produced with transition thresholds, (β1, β2) ≈ (1.4, 2.1) (vertical lines), and wind fluctuations, σV ≈ 1.0 m s−1.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Global (for all time) model CF PDF from the NEP region. The two-peak PDF predicted by the present model is consistent with the Lagrangian analysis of Mauger and Norris (2010) in the NEA region.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Estimate of the interquartile range from the observed median, given by (5), using a two-peak distribution model for the conditional PDF at time t [(4)].

Fig. 1.

Lagrangian transition statistics of SST and CF from four eastern subtropical ocean regions taken from the analysis of Sandu et al. (2010).
Fig. 2.

Comparison of observations with model predictions using different transition thresholds, β1 and β2. Increasing β2 − β1 simultaneously decreases the peak CF spread and the steepness of the median CF.
Fig. 3.

Effect of wind fluctuations on model predictions with fixed transition thresholds, (β1, β2) ≈ (1.4, 2.1). Increasing the wind fluctuations increases the peak CF spread uniformly but does not affect the median CF.
Fig. 4.

Snapshots of model (left) β and (right) CF PDFs of the NEP region produced with transition thresholds, (β1, β2) ≈ (1.4, 2.1) (vertical lines), and wind fluctuations, σV ≈ 1.0 m s−1.
Fig. 5.

Global (for all time) model CF PDF from the NEP region. The two-peak PDF predicted by the present model is consistent with the Lagrangian analysis of Mauger and Norris (2010) in the NEA region.
Fig. 6.

Estimate of the interquartile range from the observed median, given by (5), using a two-peak distribution model for the conditional PDF at time t [(4)].
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