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Corresponding author: Q. Yue, Joint Institute for Regional Earth System Science and Engineering, University of California—Los Angeles, Los Angeles, CA, USA. (Qing.Yue@jpl.nasa.gov)
 Cloud top entrainment instability (CTEI) is a hypothesized positive feedback between entrainment mixing and evaporative cooling near the cloud top. Previous theoretical and numerical modeling studies have shown that the persistence or breakup of marine boundary layer (MBL) clouds may be sensitive to the CTEI parameter. Collocated thermodynamic profile and cloud observations obtained from the Atmospheric Infrared Sounder (AIRS) and Moderate Resolution Imaging Spectroradiometer (MODIS) instruments are used to quantify the relationship between the CTEI parameter and the cloud-topped MBL transition from stratocumulus to trade cumulus in the northeastern Pacific Ocean. Results derived from AIRS and MODIS are compared with numerical results from the UCLA large eddy simulation (LES) model for both well-mixed and decoupled MBLs. The satellite and model results both demonstrate a clear correlation between the CTEI parameter and MBL cloud fraction. Despite fundamental differences between LES steady state results and the instantaneous snapshot type of observations from satellites, significant correlations for both the instantaneous pixel-scale observations and the long-term averaged spatial patterns between the CTEI parameter and MBL cloud fraction are found from the satellite observations and are consistent with LES results. This suggests the potential of using AIRS and MODIS to quantify global and temporal characteristics of the cloud-topped MBL transition.
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 Marine boundary layer (MBL) clouds over the subtropical oceans play an essential role in Earth's radiation budget and, consequently, the Earth's climate. Observational and numerical modeling studies have suggested that a moderate change in MBL cloud cover has the potential to offset or amplify global warming due to increased greenhouse gas concentrations in the atmosphere [Slingo, 1990; Klein and Hartmann, 1993; Zhu et al., 2007; Stephens, 2005; Bony and Dufresne, 2005; Bony et al., 2006].
 In the subtropical Pacific and Atlantic oceans, an important climatic feature is the transition from a stratocumulus to trade wind cumulus topped MBL. Over the past several decades, the characteristics of this transition have been extensively studied with satellite and in situ observations [e.g., Betts, 1990; Betts and Boers, 1990; Paluch and Lenschow, 1991; Minnis et al., 1992; de Roode and Duynkerke, 1997; Pincus et al., 1997; Norris, 1998; Wood and Bretherton, 2004; Sandu et al., 2010; Karlsson et al., 2010; Yue et al., 2011, Jones et al., 2011] and numerical simulations [e.g., Lilly, 1968; Krueger et al., 1995; Bretherton and Wyant, 1997; Stevens, 2000; Moeng, 2000; Lock, 2009; Sandu and Stevens, 2011; Chung and Teixeira, 2012; Chung et al., 2012]. Various physical processes have been shown to contribute to the occurrence of this transition. In a recent review paper by Wood , a summary is presented about the possible mechanisms of stratocumulus dissipation, including the so-called cloud top entrainment instability (CTEI). This mechanism was first defined by Lilly , in which CTEI can be understood as a positive feedback between entrainment mixing and evaporative cooling near the cloud top. However, in the Lilly  derivation of the criterion for this instability, the influence of liquid water on the buoyancy is neglected, and then a cloud layer would evaporate if the vertical gradient of equivalent potential temperature (θe) is negative across the inversion:
where Δ is the jump operator as the above cloud top value minus the in-cloud value. Randall  and Deardorff  refine Lilly's derivation by including the liquid water effect and obtain the Randall-Deardoff criterion for CTEI:
where sv is the virtual dry static energy and is often used as a measure of buoyancy, and (Δsv)crit is a positive critical value that, if exceeded, would result in the thermal stability being sustained. Many efforts have been made to derive the particular form for (Δsv)crit. As shown by Randall  and Deardorff , (Δsv)crit is proportional to the difference between saturated and actual water vapor mixing ratios above the cloud. Following the notation in Kuo and Schubert , the threshold for buoyancy reversal—thus CTEI occurrence—can be expressed as the following:
where θe is the equivalent potential temperature; qt is the total water mixing ratio, which includes both liquid and vapor; L is the latent heat of vaporization of water; cp is the specific heat capacity of dry air at constant pressure; and Δ is the jump operator as defined earlier. Upon inspection of equation (3), the parameter κ is affected by the thermodynamic properties of air both immediately above and below the inversion and represents the buoyancy of the mixture from cloudy air and the warm, dry, free atmosphere above the cloud. It is hypothesized that when κ is larger than a certain critical_value (equation (3)), CTEI will occur and cause the stratocumulus to break up into cumulus, and thus, a stratocumulus-topped MBL will transition into trade cumulus. Therefore, κ is termed the “CTEI parameter.”
 Two questions are key to this hypothesis: is CTEI efficient enough to prompt the MBL cloud transition, and is there such a critical_value of κ that marks the onset of stratocumulus transitioning into trade cumulus? Previous work provides contradictory evidence to these questions through laboratory and observational studies, theoretical analysis, and numerical models. For example, Lilly  presents three radiosonde observations from Oakland, California, during overcast stratus conditions and shows that all three cases have positive θe gradients across the cloud top to maintain the stability of the cloud layer against the penetration of above-cloud dry air. Using a low-resolution large eddy simulation (LES) model, Deardorff  finds that the entrainment rate increases as Δθe drops below a critical value, suggesting that CTEI may cause the cloud breakup. However, Hanson [1984a, 1984b] and Albrecht et al.  find no evidence of CTEI from aircraft observations. Based on both theoretical and mixed layer models, Hanson concludes that the Randall-Deardoff criterion is neither a necessary nor a sufficient condition for the stratocumulus breakup. Several more recent numerical modeling studies, such as Moeng  and Yamaguchi and Randall , use LES to show that evaporative cooling associated with CTEI cannot produce rapid entrainment, and thus the turbulent kinetic energy generated is too weak to maintain an effective positive feedback. As a result, the cloud dissipating effects of CTEI can be masked by other cloud-generating processes such as cloud top radiative cooling and surface moisture fluxes. Direct numerical simulations of cloud top mixing layers by Mellado  show that CTEI leads to the development of a turbulent convection layer within the cloud layer beneath the inversion; however, CTEI does not cause the cloud top breakup.
Kuo and Schubert  derive the critical_value for CTEI parameter, κ, which is a weak function of temperature near cloud top with a typical value of 0.23. After analyzing the available MBL cloud observations from in situ and field campaign measurements, they find that a large number of cases show persistent stratocumulus when the CTEI criterion is met. However, their simple model simulations indicate that initial conditions closer to the critical value lead to a longer stratocumulus lifetime. More strict criteria have been attempted in works such as those of Siems et al. , Duynkerke , and MacVean and Mason , the last of which suggested κ > 0.7 for CTEI to be sufficient to dominate other cloud-generating processes based on an energetics analysis during mixing and evaporation. These modified criteria are more consistent with the observations: when they are met, fewer cases with unbroken stratocumulus cloud are observed, or such clouds dissipate in a short time [e.g., de Roode and Duynkerke, 1997, MacVean and Mason, 1990].
 Despite the debate on the efficiency of CTEI to break up stratocumulus, Lewellen and Lewellen , Moeng , Lock , Xiao et al. , and Sandu and Stevens , using a wide range of simulations from LES, show that the CTEI parameter κ correlates to MBL cloud fraction, with κ < 0.2 corresponding to large cloud fraction and κ > 0.4–0.5 corresponding to universally small cloud fractions. Among these studies, Lock  and the simulations of well-mixed boundary layer by Xiao et al.  report a gradual decrease in cloud fraction when κ increases. A more abrupt decrease is noted by Sandu and Stevens , as is seen in the decoupled boundary layer simulation in Xiao et al. . Based on these results, Xiao et al.  and Wood  suggest that a higher possibility of buoyancy reversal at the cloud top in a decoupled boundary layer more likely causes a more rapid transition from stratocumulus-topped MBL to a cumulus-topped one, although this spontaneous entrainment can be compensated by cloud-generating processes and may not necessarily lead to cloud destruction [e.g., Yamaguchi and Randall, 2008]. The parameter κ is straightforward to measure in observations and calculate from model simulations, and it may prove useful to understand cloud cover changes in cloud-topped MBLs. Thus, κ has the potential to provide a valuable test to various parameterizations of MBL cloud and MBL transitions in models. In contrast, Sandu and Stevens  attributed the sharp cloud cover reduction at large κ to artifacts in the representation of cloud-top mixing in LES.
 More observations are required to understand how MBL cloud fraction and κ are interrelated. Currently, the limited observations also support contradictory conclusions. As mentioned earlier, Kuo and Schubert  describe in situ observations of stratocumulus cases in the midlatitudes and subtropics, and one trade cumulus case. They find that the Randall-Deardoff critical κ value of 0.23 does not mark the onset of the MBL transition. However, their data show κ ~0.7 for small cloud fractions, consistent with the κ value suggested by MacVean and Mason . Using the Lagrangian approach of following an air mass advected by the mean wind, de Roode and Duynkerke  noticed large κ values in a case when “cumulus clouds dominated the turbulence, penetrating into the thin and broken stratocumulus clouds above,” taken by aircraft during the Atlantic Stratocumulus Transition Experiment (ASTEX). More recently, from aircraft observations taken during the VOCALS Regional Experiment (VOCALS-Rex), Jones et al.  compute thermodynamic property jumps across the capping inversion as well as the parameter κ. From their data, although the general pattern of large (small) κ corresponding with small (large) cloud fraction somewhat holds, no transition from solid stratocumulus to cloud fraction less than 20% at κ ~0.4 can be seen as noted by Lock . Moreover, several cases showed near 100% cloud fraction at κ > 0.4.
 These field campaign and in situ observations provide detailed measurements on the fine structure of atmospheric thermodynamic profiles, radiation, surface fluxes, and cloud properties in the MBL, which enables the accurate estimation of various physical parameters. However, these observations are highly limited in space and time and only cover certain locations under very specific conditions. Satellite observations can overcome some of these shortcomings but are often constrained by coarser horizontal and vertical resolution, and sometimes by infrequent temporal sampling, so they cannot directly observe small-scale cloud processes. However, they provide useful information on the macro behavior of these small-scale processes. By observing this behavior, we can further our understanding of processes which are often difficult to monitor directly or indirectly, such as entrainment. This approach has been undertaken by several studies; for example, Minnis et al.  estimated boundary layer depths in the northeast Pacific from the Geostationary Operational Environmental Satellite data using a fixed lapse rate. Martins et al.  and Karlsson et al.  calculated MBL depths from temperature and water vapor profiles retrieved from AIRS observations. Yue et al.  used collocated NASA A-Train satellite observations to quantify the relationship between MBL cloud and lower tropospheric thermodynamic profiles. Using a combination of satellite observations and National Centers for Environmental Prediction (NCEP) reanalysis data, Wood and Bretherton  estimated the boundary layer depth, entrainment rate, and decoupling of the cloud-topped MBL over the northeastern and southeastern subtropical Pacific.
 In this study, we combine different satellite observations of cloud amount, cloud liquid water path, cloud top temperature and pressure, and atmospheric water vapor and temperature profiles with simulation results from an LES to estimate κ in order to examine the relationship between κ and MBL cloud fraction. The data and methodology is outlined in section 2. Results are presented in section 3, and a summary is presented in section 4.
2 Data and Methodology
2.1 Satellite Observations
 The Earth Observing System (EOS) Aqua satellite was the first of a constellation of satellites comprising the A-Train. Launched on 4 May 2002, Aqua carries six Earth-observing instruments collecting a variety of global data sets regarding radiation, cloud and aerosols, precipitation, atmospheric water vapor and temperature, and surface properties [Parkinson, 2003]. It has a near-polar low-Earth orbit with a period of 98.8 min and equatorial crossing times of 01:30 (descending) and 13:30 (ascending) LT. In this study, we use data from the Atmospheric Infrared Sounder (AIRS; Aumann et al. [2003b])/Advanced Microwave Sounding Unit (AMSU; Lambrigtsen and Lee ) sounding suite and the Moderate Resolution Imaging Spectroradiometer (MODIS; Barnes et al. ).
 The AIRS is an infrared grating spectrometer with a nadir spatial resolution of 13.5 km [Aumann et al., 2003b]. Groups of 3 × 3 AIRS footprints are coregistered to a single AMSU footprint [Lambrigtsen and Lee, 2003], and the combination of infrared and microwave radiances facilitates the retrieval of vertically resolved T and q profiles, along with additional information about clouds, the surface, and atmospheric trace gases. The AIRS/AMSU retrievals are obtained for an infrared effective cloud fraction (ECF), an emissivity-weighted cloud fraction, up to approximately 0.7 [Susskind et al., 2006]. The fraction of the highest-quality retrievals, obtained from infrared spectral information, diminishes with higher ECF [Fetzer et al., 2006; Yue et al., 2011].
 From AIRS/AMSU, we use temperature (T) and humidity (q) vertical profiles. Each day, the AIRS/AMSU sounding suite observes up to 324,000 vertical profiles of T and q. The Version 5 (V5) AIRS Level 2 (L2) Support product is used in this study. The L2 Support product has a horizontal resolution of ~45 km at nadir and is reported at 100 vertical levels with a nominal grid spacing of about 25 hPa in the MBL. This is about three times as many levels in the lower troposphere compared to the L2 Standard product. Fetzer et al. , and Martins et al.  show that AIRS is sensitive to the vertical gradients of the thermodynamic structure of the MBL. Yue et al.  quantified this sensitivity by combining the cloud observations from CloudSat and the AIRS data in the presence of shallow MBL clouds. Through comparisons with reanalysis data, they showed that although the broad averaging kernels (2–3 km width, [Maddy and Barnet, 2008]) of AIRS smooth the fine vertical structure in the boundary layer, AIRS contains quantitatively useful information about relative changes in vertical gradients of T and q within the MBL. AIRS/AMSU has a global yield of 60–70% for the highest-quality retrievals within the cloud-topped MBL, with larger yields of 80–90% over the extensive areas of the subtropical oceans and lower yields of 10–20% for overcast conditions in stratocumulus clouds. The high yield of AIRS/AMSU retrievals for partial low cloud cover makes this data set useful for describing large-scale T and q structure and the macro behavior of small-scale processes relevant to MBL clouds.
 The MODIS instrument is a spectrometer with 36 narrow band channels in the range of 0.4−14 µm. Its pixel size depends on the channel and varies from 0.25 to 1.0 km at nadir. At 1 km resolution, there are 1354 cross-track pixels resulting in a swath width of approximately 2330 km [Barnes et al., 1998]. In this work, we use Collection 5 L2 cloud products (MYD06_L2), which provide a cloud mask, effective radius, optical depth, cloud top temperature, and cloud top pressure. These cloud property measurements are collocated with T and q obtained from AIRS/AMSU using the method developed by Schreier et al. . In that method, approximately 200 1 km MODIS pixels are collocated with a given AIRS/AMSU field of view (FOV); only those within 45° of nadir are used.
 Rather than using the coarse resolution (5 km) MODIS cloud fraction data, pixels with the MODIS 1 km cloud mask reported as “confident cloud” or “probably cloud” are used to obtain the cloud fraction as the percentage of cloudy MODIS pixels in each AIRS/AMSU FOV as described by Platnick et al. . The 1 km data better characterize the subpixel cloud distributions in the AIRS/AMSU FOV. Previous studies have shown that cloud fraction values derived by this method suffer from small overestimation in MBL cloud scenes, and the uncertainty varies from scene to scene. The magnitude of this error is determined by several factors, such as instrument resolution, cloud detection algorithm, spatial extent and optical properties of cloud, and the effect of sunglint [Wielicki and Parker, 1992]. Zhao and Di Girolamo  and Kotarba  quantify the errors on daytime-only cloud fraction determined by MODIS cloud mask for MBL clouds by comparing with data from the Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER; 15 m resolution). They found that the MODIS cloud fraction error arises mostly from the sunglint scenes when the cloud fraction is smaller than 0.05, and average overestimation is less than 0.1. For sunglint-free scenes, MODIS overestimates cloud fraction by a few percent because of the competing tendency between overestimation by partially filled cloud pixels classified as cloud and underestimation by optically thin, partially filled cloud pixels classified as clear. To minimize these effects, FOVs containing cloud fraction less than 0.05 are excluded from the current study. Moreover, only daytime MODIS cloud mask data are used to avoid assessing consistency with the different retrieval methods used for nighttime cloud detection [Platnick et al., 2003].
 The subtropical northeastern Pacific region that is dominated by MBL low clouds is examined in our study, using two months of data (July and August 2009). Over one given AIRS/AMSU FOV, cloud parameters including cloud top pressure and temperature, optical depth, and effective particle size are represented as the mean of MODIS cloud retrievals. Cloud fraction is simply the percentage of MODIS cloudy pixels in an AIRS/AMSU field of view according to the MODIS cloud mask. MODIS MYD06_L2 cloud products use visible/near-infrared radiances [King et al., 2003] to derive optical depth, τ, and near cloud top effective radius, re,top, and these are available only under solar illumination. The mean τ and re,top for an AIRS FOV are then obtained as the average from all the cloudy MODIS pixels in the given FOV. We estimate the cloud liquid water path from cloud optical depth and effective radius using the equation below based on an adiabatic cloud assumption [Wood and Hartmann, 2006]:
where ρl is the liquid water density. Since our interest is focused on MBL clouds, we consider only the AIRS/AMSU pixels whose cloud top pressure from MODIS is greater than 700 hPa, and cloud fraction is larger than 5%. Some MBL clouds may be missed for four possible reasons: (1) because of the “top-down” view from satellites, middle and high clouds may obscure low clouds; (2) there is no sampling at nighttime; (3) in the presence of some T inversions, the MODIS cloud algorithm tends to place clouds above the inversion hence at cloud top pressures smaller than their true values [Menzel et al., 2008]; and (4) AIRS/AMSU only retrieves about 10–30% of the scenes covered by extensive Sc clouds (cloud fraction > 80%), resulting in a sampling bias toward the broken cloud scenes [Yue et al., 2011]. In this study, we are interested in the macro behavior of the cloud top buoyancy reversal and cloud fraction in the MBL as observed by Aqua. Therefore, obtaining a statistically significant sample size under various conditions is sufficient to fulfill our purpose. The uncertainty caused by the vertical T structure will be discussed later in section 3.
 Only the best AIRS retrievals are retained by requiring PBest = PsurfStd, which indicates that the highest-quality retrievals are obtained to the surface, as this study focuses on the MBL. Sea surface temperature (SST) is obtained from TSurf_forecast in the AIRS data. This is the surface temperature analysis from the Aviation (Global Forecast System) model (AVN), which, over the ocean, is essentially the RTG.SST (Real-Time-Global sea surface temperature analysis; Thiébaux et al. ). RTG.SST reports the bulk temperature measured by the floating buoys at about 1 m below the surface, while infrared sensors such as AIRS measure a thin surface layer. As a result, a 0.18 K adjustment is applied to the TSurf_forecast, which forms the SST used in our study [Donlon et al., 2002; Aumann et al., 2003a].
 Figure 1a shows the MBL low cloud fraction (CF) for the two summer months studied over the region of interest together with the Global Energy and Water Cycle Experiment cloud system study (GCSS)/Working Group for Numerical Experimentation Pacific Cross-Section Intercomparison (GPCI) transect at 1.5° × 1.5° resolution [Teixeira et al., 2011]. Over this track, ten 3° × 4° (latitude × longitude) boxes are selected from 35°N, 125°W, to 5°N, 165°W. The transition feature is apparent with solid stratocumulus cloud near the coast where the SST is lower and inversion strength is high, gradually changing to patchy and shallower trade cumulus where trade winds and warmer SST conspire to make a weaker inversion. The exception is a narrow strip of lower CF in proximity to the immediate coast [Betts et al., 1992; Bretherton and Wyant, 1997].
 The number of observations that meet the quality control standards in each grid box is shown in Figure 1b. As expected, fewer observations are obtained near the coast, where extensive stratocumulus sheets occur. White indicates regions with fewer than 100 observations. Except for near the coast, there are at least 250 observations in each 1.5° × 1.5° grid box, ensuring a statistically significant sample size.
2.2 LES Model and Experiment Design
 The LES model runs are carried out as described in Xiao et al. , so only a brief description on the model and experiment design is given here.
 We use the UCLA LES [Stevens and Moeng, 1999; Stevens et al., 2005; Stevens and Seifert, 2008]. Two types of experimental setups are constructed: the first is termed the “MIXED” series, and the second is the “DECOUPLED” series by Xiao et al. . They follow those used in the GCSS boundary layer working group intercomparison studies of (1) the second Dynamics and Chemistry of Marine Stratocumulus field study (DYCOMS-II) RF01 case [Stevens et al. 2003] and (2) the Atlantic Trade Wind Experiment (ATEX) [Stevens et al., 2001], respectively. The MIXED series show typical conditions of shallow, well-mixed stratocumulus-topped MBLs over the eastern subtropical oceans under a very strong inversion. The DECOUPLED series show features of the cumulus-transitioning-to-stratus regime, which has MBLs with high stratocumulus coverage but a substantially deeper and decoupled structure. There are 9 and 11 experiments carried out for MIXED and DECOUPLED series, respectively, by specifying different initial values of κ. To produce these κ values, the above cloud total water-mixing ratio, qt+, is varied with an interval of 0.5 g/kg for both series (ranges of qt+: 0.5 to 4.5 g/kg for MIXED, and 1.5 to 6.5 g/kg for DECOUPLED). Large-scale subsidence, radiative forcing, and surface fluxes are all specified in the same manner as Xiao et al. . All simulations are initialized with random perturbations added to spatially uniform temperature and moisture fields and are a total of 12 h long.
2.3 Calculating κ From Satellite Data and LES Results
 The CTEI parameter κ is proportional to the ratio of the jumps in equivalent potential temperature (Δθe) and total water (Δqt) across the inversion as in equation (3). Δqt = Δqv + Δql, where subscripts v and l stand for vapor and liquid, respectively. As the jump in liquid water across the inversion (Δql) is very difficult to assess in the observations, the present section will explore two different definitions of κ: the traditional definition as in equation (3) and a second definition based on the water vapor jump across the inversion (Δqv), which we term κv. These two definitions of κ are related as follows:
where . The profiles from in situ and field campaign observations [e.g., Jones et al., 2011; Carman et al., 2012] as well as from LES [e.g., Xiao et al., 2010] show that the contribution from Δql is much smaller than that from Δqv. Therefore, ignoring Δql, κv can faithfully represent the CTEI parameter κ (see Figure 3 and related discussions).
 The water vapor profile, qv, is taken from the AIRS data. While ql is not readily available from satellite observations, it is derived from LWP by assuming (1) an adiabatic cloud with cloud liquid water linearly distributed with height between zero at the cloud base and maximum near the cloud top and (2) that the cloud base is determined by the lifted condensation level (LCL). The LCL is computed using the expression LCL = 125(T − Td) and is determined from the average lapse rates of both T and dewpoint temperature (Td) [e.g., Lawrence, 2005] following Yue et al. . The forecast SST collocated with AIRS/AMSU FOVs is used to calculate LCL. q at the ocean surface is derived from SST together with the T and q at the first AIRS level above the surface assuming a conserved equivalent potential temperature. Assuming that ql above the cloud top is zero, we have
where ql,− is the ql at the cloud top, g is the gravitational constant [Li et al. 2002], and LWP is obtained from equation (4). Assumption (1) has been supported by various in situ observations of stratocumulus cloud [e.g., Duynkerke et al., 1995; Pawlowska and Brenguier, 2000], but breaks down in trade wind cumulus [e.g., Raga et al., 1990; Siebesma et al., 2003] or in MBL cloud with drizzle [e.g., Zuidema et al., 2005; Wood, 2005]. Due to its higher variability, cloud base is more difficult to determine in trade wind cumulus cloud and may be higher than the LCL [Bretherton and Wyant, 1997; Zuidema et al., 2012].
 When low clouds are present, the 11 µm infrared window brightness temperature data are used to estimate an opaque cloud-top temperature, and then pressure (or height) is inferred by comparison with a model analysis, working from the tropopause downward. In the case that low-level temperature inversions occur, the pressure level of the matching temperature above the inversion is selected, thus causing large uncertainties in the MODIS cloud top pressure [Menzel et al., 2008]. To minimize the effect of this uncertainty, we estimate the cloud top pressure from MODIS cloud top temperature and the AVN forecast SST collocated with AIRS/AMSU FOVs. Minnis et al.  used a fixed lapse rate of 7.1 K/km in the MBL to determine the cloud top height (MBL depth) over the northeastern Pacific Ocean. In this study, the parameterization developed in Wood and Bretherton  is used to calculate the cloud top height (MBL depth) zi:
where zi is in kilometers. Cloud top pressure is simply derived from zi using the hydrostatic equation. Temperature and humidity for layers just above and below the MODIS cloud top are used to calculate the property jump across the cloud top, Δθe and Δqv.
 As discussed in Yue et al. , the vertical gridding of AIRS is approximately 200 m below 700 hPa in the L2 Support product. Therefore, the layer depth over which the jumps are calculated from the satellite data is ~200 m, much coarser than the vertical resolution of typical LES, which is on the order of tens of meters. The limited vertical resolution and smoothed structure of the AIRS data does not capture the local fine vertical structure in the inversion-capped MBL [e.g., Martins et al., 2010]. However, as shown by Yue et al. , the relative change across the inversion can be quantified with the AIRS support product. As seen in equation (3), κ is defined as the ratio of the differences across a layer rather than the absolute differences; thus, the AIRS vertical profiles are applicable in this instance. We will elaborate further on this point later in the manuscript (see Figure 5 and associated discussion).
 In the LES results, cloud top is defined to be the level of the maximum gradient in liquid water potential temperature [Xiao et al., 2010]. All values are averaged over the entire model domain. The cloud fraction is defined as the fraction of cloud-water-containing grid columns in the domain, which is compatible with the MODIS cloud fraction defined as the fraction of cloudy pixels within one AIRS/AMSU FOV. The model levels above and below the inversion are defined as 100 m (50 m) above and below the horizontally averaged cloud top level for the DECOUPLED (MIXED) series in the standard calculation. To facilitate an “AIRS-like” calculation of κ from LES, two more sets of κ calculations are carried out in addition to the standard one. In the first set, the model levels to calculate the property jump are defined to be 200 m, comparable to the vertical grid of AIRS L2 Support profiles, while maintaining the original fine vertical resolution of the LES profile. In the second set, LES profiles are smoothed to the AIRS Support product vertical grid of ~200 m before calculating the property jump and κ.
 Note that these LES experiments are based on an ideal experimental setup and are entirely independent of satellite observations. As a result, a linear interpolation instead of an AIRS averaging kernel is applied to change the vertical resolution of LES profiles. Moreover, it is not our intent to reproduce satellite results with LES but rather to use these different depictions to quantify one aspect of the behavior of the cloud topped MBL: the relationship between cloud fraction and CTEI parameter κ during the transition between stratocumulus and trade wind cumulus.
2.4 Critical Value of κ
Kuo and Schubert  derived the following instability criterion:
where δ = 0.608 is a constant and γ = (L/cp) ∂ q∗/∂ θ. The saturated water vapor mixing ratio, q*, is expressed as
 Here Rv is the gas constant for water vapor, and is a known function of z and is given here as the height of inversion (z = zi). We calculate the saturated vapor pressure e* using the temperature of the layer just below the inversion. As shown by Randall  and Yamaguchi and Randall , κKS is not a universal value. Instead, it is weakly pressure dependent but strongly temperature dependent.
 Using an energetics analysis, MacVean and Mason  derived their instability criterion in the form of the following equation:
 Taking the same numerical values of θ0 and γ as in Kuo and Schubert , κMM is approximately 0.7. Similarly, κMM is also dependent on the MBL pressure and temperature.
 Instead of using constant values for cloud and MBL temperature and humidity, the instantaneous satellite retrievals are used to calculate κKS and κMM.
 Figure 2 shows the two-dimensional PDFs of κ versus MBL cloud fraction. Overlaid on the instantaneous AIRS and MODIS data are the steady state results for the MIXED and DECOUPLED series from the standard runs of the UCLA LES. The LES results are calculated from the domain average values of the parameters over the last 3 h of the simulation. Both satellite and LES show that the MBL cloud fraction decreases with κ. However, the transition from large to small CF is not as smooth as shown by Lock . Instead, a fairly sharp decrease from about 80% to 10% CF is seen in both the satellite and LES. In both the satellite data and the MIXED series simulations, the transition occurs for κ between 0.5 and 0.7, while in the DECOUPLED series simulations, the transition occurs for κ between 0.4 and 0.6.
Table 1. The Pearson Correlation Coefficients (r) at the 98% Significance Level, the Spearman Rank Correlation Coefficients (ρ), and the Significance of Its Deviation From Zero (R) Between the CF-κ Relationships From the Median Value Using AIRS/MODIS Observations and the Steady State Results From the UCLA-LESa
Correlation Coefficients With CF
aA small R indicates a statistically significant correlation.
Spearman Rank (ρ/R)
 We do not fully understand the reason for the larger κ during the MBL cloud fraction transition in the satellite data. The LES is initialized and carried out independently of the observations used in this study. As a result, some slight discrepancies in the numerical values are expected, but a similar pattern of the CF-κ relationship is found despite the discrepancies. While the LES has a high correlation, the satellite data has considerable scatter, especially at largest and smallest CF. This may be a result of comparing LES steady state results with instantaneous snapshots of satellite observations. It is likely that other physical processes are also regulating the MBL cloud fraction and are represented in the satellite observations (e.g., solar radiative heating of the cloud layer, gradual breakup of stratocumulus by increasing SST, light precipitation/drizzle, etc.), while in the LES experiments, the effect of CTEI has been considered in isolation from other processes. Therefore, one can only expect a general agreement between these satellite snapshots and LES results. However, as discussed later, the limited vertical resolution of AIRS and additional uncertainties in the satellite observations may also play a role.
 Previous studies show that, although decoupling does not always lead to stratocumulus cloud dissipation, it is a crucial first step in the MBL transition [e.g., Wyant et al., 1997, Bretherton and Wyant, 1997]. Xiao et al.  suggested that, in addition to strong buoyancy reversal at the cloud top, the MBL needs to be decoupled for stratocumulus clouds to break up into cumulus. Using MODIS cloud measurements and the atmospheric thermodynamic profiles from NCEP reanalysis, Wood and Bretherton  estimate the decoupling rate of northeastern and southeastern Pacific Ocean subtropical MBLs. Based on their result, most of these areas have a decoupled MBL with a general transition to a more decoupled structure moving away from the coast, consistent with the climatological trajectory and mixed-layer-model analysis by Bretherton and Wyant . Therefore, two groups are taken from the 10 boxes along the GPCI transect in order to make general comparisons with the two LES series. Shown in Figure 2b are data within boxes 1 to 4, which are for the northern part of the transect (32°–23°N) and closest to the coast. According to Wood and Bretherton , these regions have a decoupling rate of 0.0–0.3 and average CF ≥ 50% (Figure 1a). Observations from this area compare well with the steady state results from the LES MIXED series despite some scatter, and they both show a smoother transition from high CF to low CF than the satellite data along the entire transect. The minimum CF for the MIXED series (both LES and satellite) is around 60%. Figure 2c corresponds to boxes 7–10, that are for the transect between 14°N and 5°N, overlapped with the DECOUPLED series results, both of which show a sharp decrease in CF from nearly 100% to 5% as κ increases. Boxes 7–10 have decoupling rates > 0.4 according to Wood and Bretherton  and average CF < 50% (Figure 1a). By separating data in this manner, the instantaneous satellite observations are grouped according to the climatological characteristics of decoupling and transition behavior in the cloud-topped MBL. This does not suggest that every data point that is labeled under DECOUPLED or MIXED strictly has characteristics in MBL that are decoupled or well mixed. For both data sets in Figures 2b (GPCI: 23°N and 32°N, MIXED) and 2c (GPCI: 5°N and 14°N, DECOUPLED), strong correlations (r) are found between the median value of satellite data and the LES steady state results with r = 0.87 and 0.91, respectively, at a 98% significance level.
 To estimate the correlations between satellite observations and LES results on the CF-κ relationships, Pearson and Spearman rank correlation coefficients are calculated. The median values of κ are calculated from the satellite observations. Given in Table 1 are the Pearson correlation coefficients (r) at the 98% significance level, and the Spearman correlation coefficients (ρ) as well as the significance R. They all show strong correlations for the CF-κ relationships between the satellite data and LES. The small R values indicate that these correlations are significant.
 A significant number of data points have simultaneously large CF and κ values. Kuo and Schubert  and Yamaguchi and Randall  argue that a larger value of κ under persistent stratocumulus clouds indicates that the cloud would break up more rapidly and efficiently than in conditions with a smaller κ. However, this phenomenon is virtually impossible to quantify using existing polar-orbiting satellite observations since instantaneous “snapshot” observations without a rapid time evolution context cannot be obtained. However, the correlations obtained from these satellite observations can be used to constrain and diagnose the models; thus, it is crucial to develop the synergistic applications of observations and models for understanding physical processes in the atmosphere. For example, using a combination of satellite observations, meteorological reanalysis, and trajectory model results, Sandu et al.  investigated the cloud and environmental conditions in the MBL transition.
 Previously, the LES-derived κ values are based on jump calculations 100 m (50 m) over and under the cloud top level for the DECOUPLED (MIXED) series. The LES results are now used to quantify the effect of different layer thicknesses for jump calculations on κ, different vertical resolution/gridding of the T and q profiles, and the impact of neglecting the liquid water jump across the inversion. In Figure 3, the 30 min average κ (calculated with Δqv) and the difference between κv and κ (calculated with Δqt) are plotted as functions of CF for the MIXED series (upper row) and DECOUPLED series (lower row). First, the differences between κv and κ are very small in the LES results (less than 0.02), especially for small CF, supporting the assumption that Δqt is overwhelmingly dominated by water vapor contributions. The difference in κv and κ decreases to zero as CF decreases since the magnitude of Δql is smaller at lower CF (not shown).
 Next, we show the effect of the jump thickness on κ, which is tested as follows. For both the MIXED and DECOUPLED series, the layer thickness is increased to 200 m with the original fine vertical resolution of LES profiles unchanged. This thickness is similar to the AIRS Support product vertical gridding in the MBL. Results show that slightly larger values of κ are obtained from the standard calculations than those from the 200 m calculation. The difference is very small (0.033 and 0.012 for DECOUPLED and MIXED series, respectively) and does not show a dependence on CF. Furthermore, we interpolate LES T and q profiles to the vertical grid of the AIRS levels, and then κ is calculated using one layer above and below the domain averaged inversion top, similar to the calculation using satellite data. Compared to the results based on fine vertical resolution profiles, more scatter is present in the coarser resolution results, especially in the DECOUPLED series simulations. The MIXED series has additional scatter but only for CF near 100%. The κ calculated from coarser resolution profiles is larger than the standard κ in the MIXED series, and the magnitude of this difference decreases from 0.18 to 0.005 as CF decreases from 100% to 60%. However, in the DECOUPLED series, the κ difference between the coarser resolution and the standard calculation ranges from −0.14 to 0.4. The magnitude of the difference in the DECOUPLED series show little dependence on CF although more scatter shows up at higher CF. The above results suggest that increased variability in the satellite-based CF-κ correlations may be partially due to the inherent vertical resolution of AIRS T and q profiles. Moreover, AIRS and MODIS provide instantaneous observations, while comparing to the steady state results from LES model simulation, more scatter in the instantaneous satellite observations is expected.
 The spatial patterns of κ and boundary layer stability parameters such as lower tropospheric stability (LTS) and estimated inversion strength (EIS) are shown in Figure 4. LTS is defined as the potential temperature difference between 700 hPa and the surface. EIS is defined as , where is the moist adiabatic potential temperature lapse rate calculated from the mean of the surface and 700 hPa T and z700 is the altitude of the 700 hPa pressure level [e.g., Wood and Bretherton, 2004]. These two parameters are closely correlated with MBL CF on seasonal and annual time scales [e.g., Slingo, 1987; Klein and Hartmann, 1993; Wood and Bretherton, 2006; Yue et al., 2011]. Figure 4 shows averages of the instantaneous values over two months at 1.5° × 1.5° resolution over the northeastern Pacific. The CTEI parameter is calculated with (κ) and without (κv) the liquid water jump, and both are presented, where κ is computed based on the assumptions that clouds are adiabatic and that the calculated LCL is the cloud base. The two different κ parameters, as well as LTS and EIS, have spatial patterns that are consistent with the MBL CF in Figure 1. The correlation coefficients with CF (rCF) are −0.71, −0.52, 0.76, and 0.71 for κv, κ, LTS, and EIS, respectively. κ is somewhat noisier than κv and has the lowest rCF. One reason is that there are fewer observations in the calculation for Figure 4b compared to Figure 4a since successful MODIS retrievals of cloud optical depth and near-cloud-top effective radius are required. An additional reason is that the assumption of an adiabatic cloud and cloud base at the LCL break down as the clouds transition to trade cumulus.
 The mean LTS and EIS fields in Figure 4 are much smoother than κv, κ, and CF. Although the higher CFs generally (in the mean sense) correspond with larger values of EIS/LTS, as shown by Figure 5, correlations between EIS/LTS and CF have a much larger degree of scatter. These results suggest that LTS and EIS are more suitable as proxies for CF over larger scales and longer time periods [Klein and Hartmann, 1993], while κ is more strongly correlated to CF for instantaneous, point-by-point observations.
 Figure 6 illustrates the satellite data and LES steady state results in the (Δθe, Δqv) plane following the CTEI diagram in Kuo and Schubert [1988: Figure 1] and Yamaguchi and Randall [2008: Figure 8]. Figure 6a shows that the joint PDF of the observations and the straight lines are the linear fits of the satellite and LES data. Comparing with the in situ measurements shown in Kuo and Schubert , it is clear that the magnitudes of Δθe and Δqv from the satellite data are smaller. This is a result of the limited vertical resolution in AIRS retrievals, giving a smoothed boundary layer vertical structure and underestimating the strength of the inversion [e.g., Yue et al., 2011]. The LES experiments in this study samples the inversions within certain ranges of Δθe (0 to −12 K) and Δqv (−4 to −10 g/kg). However, the slopes of the lines are very similar between the satellite (0.75) and LES (0.90) data, suggesting that κ (a ratio between the property jumps) is much less affected by vertical resolution limitations. While κ estimated from AIRS data may not be as precise as that from finer-resolution in situ observations and LES calculations, realistic values of κ, especially in the mean sense, are obtained from AIRS.
 We also show linear fits for data over the GPCI transect between 23°N and 32°N (MIXED) and data over the GPCI transect between 5°N and 14°N (DECOUPLED), and the former has a smaller slope than the latter in both the satellite (23°N–32°N, slope = 0.71; 5°N–14°N, slope = 0.82) and the LES (MIXED, slope = 0.95; DECOUPLED, slope = 0.87). The larger difference between the satellite and LES results on MIXED series suggests that the effect of smoothed vertical structure has a larger impact on the scenes that are well mixed and contain stronger inversions with shallower MBLs.
 Figures 6b and 6c show the mean cloud fraction and LWP from the satellite data for the PDF shown in Figure 6a. The straight lines in the two figures represent stability lines given by the critical κ. Following the formula given by equations (8) and (10), the critical value of κ is calculated for each pixel using the MBL pressure and temperature observed by AIRS/MODIS. Results show that the range of variation is narrow and that the mean value is very close to the single value calculated from the reference MBL pressure and temperature. For the Kuo and Schubert  criteria, κKS = 0.25 ± 0.026; for the MacVean and Mason  criteria, κMM = 0.70 ± 0.02. The number of observations in Figure 6c is smaller than those in Figures 6a and 6b because of a reduced sampling with valid MODIS τ and re retrievals. Both cloud fraction and LWP decrease when approaching the κ = 1 line. Most of the large values of cloud fraction and LWP appear on the stable (right) side of the κ = 0.7 line; however, a number of bins with small values of cloud fraction and LWP are also located on the stable side. For LES, both the steady state (Figure 6, symbols) and transient state (not shown) results are located on the stable side of κ = 0.7 for all cloud fraction and LWP values. Yamaguchi and Randall  show with LES that the cloud evaporation time scale increases toward the stable side of stability lines. However, because of the inherent sampling limits of polar-orbiting satellite observations, such time scale dependencies cannot be derived from these observations and warrant further study with other observational approaches.
 Based on the previous analysis, κv is nearly identical to κ because the contribution of liquid water is small compared to water vapor. We further define
where Π is the Exner function and Δp is the pressure difference between the layers just above and below the inversion.
 Figures 7a–7c show the joint PDFs between κdry and the MBL CF based on instantaneous satellite data. Similar to the κv-CF PDF shown in Figure 2, the MBL CF is correlated with the parameter κdry, although κdry is always smaller. The correlation coefficient (r) between the instantaneous κv and κdry is 0.89. The two PDFs show similar degrees of scatter. Their difference (κv − κdry), as shown by Figure 8, varies between 0 and 1 and more scattered at low CF values than at high CF values, which is consistent with the fact that higher variability of inversion layer pressure is observed in the AIRS FOV when CF is small. The spatial patterns of κdry are also shown in Figure 7d based on the two-month average of satellite observations, which shows a high correlation with κv (r = 0.97) and the MBL CF (r = −0.76).
 The above analysis indicates that κdry is also a good indicator of CTEI parameter κ in the correlation of MBL CF.
 In this study, a combination of pixel-scale, collocated satellite data from the NASA Aqua Atmospheric Infrared Sounder (AIRS; Aumann et al. [2003b])/Advanced Microwave Sounding Unit (AMSU) sounding suite and the Moderate Resolution Imaging Spectroradiometer (MODIS; Barnes et al. ), and the simulations from the UCLA large eddy simulation (UCLA-LES) model to explore the relationship between the Cloud Top Entrainment Instability (CTEI) parameters and marine boundary layer (MBL) cloud fraction (CF) across the transition from the stratocumulus-topped to trade cumulus-topped MBL. The area of interest is over the subtropical northeastern Pacific Ocean. The satellite cloud observations are taken from MODIS, while the atmospheric temperature (T) and humidity (q) profiles are taken from the AIRS Level 2 Support product. These are used to estimate the CTEI parameter κ. The LES calculations are entirely independent of the satellite data used and are separated into MIXED and DECOUPLED series to represent a more well mixed MBL and a more decoupled MBL, respectively.
 Satellite observations and LES model simulations show that the MBL CF is closely correlated with κ. By separating satellite observations into two groups based on their distance from the coast and climatological patterns of the degree of MBL decoupling, we compare the observations with the two LES series. Observations in proximity to the coast compare well with the MIXED series simulations, with both presenting a smoother decrease in κ with CF from overcast to around 60%. The observations further away from the coast compare well with the DECOUPLED simulations, in which a sharp reduction of CF from 100% to 5% is present, although κ values derived from the satellite data are larger compared to LES. Our findings with AIRS and MODIS data support the theoretical and numerical results in Wood  and Xiao et al.  stating that larger κ in the decoupled MBL leads to higher possibilities of a transition to the trade cumulus-topped MBL. While it is difficult for polar-orbiting satellite observations to evaluate “cause and causality” in a temporally rapid physical process, the results presented here show that the collocated AIRS and MODIS data have the same large-scale behavior of MBL CF sorted by values of κ, consistent with LES simulations of a cloud-topped MBL.
 Compared with the LES, the satellite data have additional scatter around the mean values. We show that the coarse vertical resolution in the AIRS T and q retrievals is responsible for the scatter in κ-CF PDFs. However, it is also possibly due to other physical processes at play, which also regulate the MBL CF. For example, cloud top radiative cooling is a highly effective cloud generating process especially at large CF. Neglecting the liquid water jump across cloud top causes negligible changes in numerical values of κ. These findings suggest the need for improved vertical resolution in satellite soundings. Moreover, thermodynamic soundings within and under persistent stratocumulus cloud decks are necessary to improve our understanding of the MBL transition.
 The spatial patterns of CF, κ, and lower tropospheric stability parameters (LTS/EIS) are correlated with each other based on their mean fields in the satellite observations. However, a low correlation is found in the instantaneous satellite data between CF and LTS/EIS, suggesting that LTS/EIS correlations with CF reflect the large-scale and long-term effects on the MBL.
 Although we cannot rule out the possibility that CTEI may not be an effective processes and that the variations of the parameter κ and CF are due to other physical processes, our results provide observational support that domain-averaged values of the parameter κ may be useful in evaluating the parameterizations of MBL cloud in models [e.g., Lock, 2009; Xiao et al., 2010]. This work demonstrates that the combination of satellite observations and numerical models is a valuable approach for understanding the physical processes in the cloud-topped MBL.
 The research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. Q.Y. and E.J.F. were supported by NASA's Making Earth Science Data Records for Use in Research Environments (MEaSUREs) program. J.T. and K.S. acknowledge the support provided by the Office of Naval Research, Marine Meteorology Program under Awards N0001411IP20087 and N0001411IP20069, the NASA MAP Program, and the NOAA/CPO MAPP Program. Funding for H.X. is provided by NOAA Grant NA07OAR4310236 at UCLA and by U.S. DOE OBER grant KP/501021/58166 at PNNL. The Pacific Northwest National Laboratory is operated for DOE by Battelle Memorial Institute under Contract No. DE-AC05-76RL01830. Q.Y., E.J.F., J.T., M.M.S., and B.H.K. acknowledge the support of the AIRS Project at JPL. AIRS data were obtained through the Goddard Earth Services Data and Information Services Center (http://daac.gsfc.nasa.gov/). MODIS data were obtained through the Level-1 and Atmosphere Archive and Distribution System (LAADS; http://ladsweb.nascom.nasa.gov/). The authors would like to thank Jui-Lin Li and Terry Kubar for useful feedback in the preparation of this manuscript.