## Abstract

The large-scale diurnal variation of cloud cover is derived from diurnal variations of temperature, density, water content, and static stability in a linearized calculation. Forced by the diurnal cycle of solar heating, the calculated cloud distribution is broadly consistent with observed diurnal variations under maritime nonconvective, maritime convective, and continental convective conditions.

The calculated diurnal variation of low-cloud fraction follows primarily from the diurnal variation of temperature, which creates a diurnal variation of saturation vapor pressure. The calculated diurnal amplitude of low-cloud fraction is large under maritime nonconvective conditions, in which a well-mixed boundary layer promotes the transition between cloudy and clear conditions. The amplitude is further enhanced under continental conditions by the diurnal variation of vertical heat transport from the surface. The diurnal variation of high-cloud fraction under continental conditions follows primarily from the diurnal variation of low-level stability, which is large if the diurnal amplitude of surface temperature is large. The diurnal variation of high-cloud fraction under maritime convective conditions follows primarily from the diurnal variation of stability at cloud top, which controls the probability that convective cloud top exists in a height interval Δ*z.* The role of clouds in radiative heating is then important because high clouds concentrate shortwave heating in the upper troposphere, which enhances the diurnal variation of stability there.

## 1. Introduction

Clouds are an integral feature of the atmosphere, involving a wide range of time and space scales. Their interactions with shortwave (SW) and longwave (LW) radiation alter the distributions of surface and atmospheric heating, which in turn drives atmospheric motion. In addition, cloud variability is closely related to atmospheric convection, high-cloud cover serving as a proxy for precipitation and latent heat release (e.g., Richards and Arkin 1981; Fu et al. 1990; Hendon and Woodberry 1993).

Of the wide range of timescales exhibited by clouds, diurnal variations are particularly important. Diurnal variations of cloud cover (hereafter, “cloud diurnal variations”) are strong over continents and regions of maritime subsidence (Minnis and Harrison 1984; Rozendaal et al. 1995; Bergman and Salby 1996). They affect diurnal energetics of the atmosphere and surface (Harrison et al. 1988; Hartmann et al. 1991). Closely related are diurnal variations of latent heating in deep convective systems, which influence diurnal variability in the troposphere and above (Silva Dias et al. 1987; Hamilton 1981; Bergman and Salby 1994; Williams and Avery 1996). Cloud diurnal variations also contribute to time-mean energetics (Bergman and Salby 1997). As large as 20 W m^{−2}, that contribution follows from the nonlinear dependence of SW flux on cloud fractional coverage and solar zenith angle, both of which vary diurnally.

Cloud diurnal variations exhibit a characteristic phase and amplitude that depends on the time-mean or “climatological” conditions, for example, whether they occur over land or sea, in convective regions, or in nonconvective regions (e.g., Rozendaal et al. 1995; Minnis and Harrison 1984; Albright et al. 1985; Hartmann and Recker 1986; Hendon and Woodberry 1993; Bergman and Salby 1996). In fact, Bergman and Salby (1996) found that, in large part, the diurnal amplitude of cloud fraction can be explained in terms of local climatological conditions such as time-mean cloud cover, time-mean surface temperature, and the diurnal amplitude of surface temperature. Likewise, diurnal phase tends to fall into one of four categories, which typify high and low clouds over maritime and continental locations. The uniformity of diurnal phase across large regions makes cloud diurnal variations and their contributions to energetics spatially coherent (Salby et al. 1991; Bergman and Salby 1996). Cloud diurnal variations then accompany low-frequency variability in large-scale variations involved in climate, whereas other high-frequency variations operate on much smaller dimensions. The spatial coherence of cloud diurnal variations over scales much larger than individual clouds has two important implications: It suggests 1) a fundamental link between cloud diurnal variations and the large-scale atmospheric response to diurnal heating and 2) cloud diurnal variations might best be viewed statistically in terms of the spatially averaged probability of cloudiness.

This investigation utilizes these features to calculate the diurnal variation of cloud probability density, or “cloud fraction,” from diurnal variations of temperature, density, static stability, and water content. In the spirit of tidal investigations, a linearized calculation is used to derive the large-scale dynamical response to the diurnal variation of SW absorption in the atmosphere and at the surface. Diurnal variations of cloud fraction are then calculated as a function of height from that dynamical response. So, herein is tested the hypothesis that spatially coherent cloud diurnal variations are primarily functions of the diurnal variations of the large-scale dynamical and thermodynamical properties of the atmosphere, which, in turn, are driven by diurnal variations of SW absorption. In this context, the calculation also reveals the important mechanisms that link the diurnal variations of SW absorption and cloud cover.

The numerical framework of the calculation is discussed in section 2. Section 3 describes the cloud diurnal variations calculated under maritime nonconvective, maritime convective, and continental conditions. Forced only by the diurnal cycle of SW absorption in the atmosphere and at the surface, the calculated cloud diurnal variations are then compared to satellite observations of large-scale diurnal variability. Implications the results have for climate and climate simulation are discussed in section 4.

## 2. Numerical framework

Bergman and Salby (1996) identified four distinct cloud diurnal categories: high-cloud fraction *C*_{hi} (cloud-top pressure *p*_{c} < 440 mb) and low-cloud fraction *C*_{lo} ( *p*_{c} > 680 mb) over continental locations, high-cloud fraction over maritime convective locations (characterized by high-cloud fraction *C*_{hi} > 0.1), and low-cloud fraction *C*_{lo} over maritime nonconvective locations (*C*_{hi} < 0.1). Diurnal variations of cloud fraction are investigated here under each of those climatological conditions in terms of the specified time-mean component *C̄* and a calculated diurnal variation about that mean *C*′.

Figure 1 illustrates the calculation of cloud diurnal variability, which involves four stages. Time-mean climatological properties are first specified for one of the four cloud categories. Diurnal variations of diabatic heating are next calculated from radiative transfer in the time-mean atmosphere. The diurnal cycle of solar zenith angle provides the only diurnally varying input to this calculation. Since the time-mean conditions are specified to be horizontally uniform, only the migrating diurnal components (i.e., those components having the same zonal phase velocity as the solar diurnal cycle) contribute to the calculated response. Diurnal variations of temperature, density, static stability, and water content then follow from the diurnally varying heating in a linearized calculation of the atmospheric response, one in the spirit of tidal investigations,^{1} which successfully reproduce large-scale diurnal variations (e.g., Butler and Small 1963; Lindzen 1967). From that atmospheric diurnal variability, diurnal variations of cloud fraction are finally calculated in terms of the cloud probability density *C*(*z, t*) that cloud top is at height *z* and local time *t.*

### a. Time-mean conditions

Time-mean climatological properties are specified, in part, according to satellite observations from the International Satellite Cloud Climatology Project (ISCCP) C2 data archive, which provides monthly composite cloud diurnal statistics and monthly mean temperature and humidity statistics at 2.5° resolution (Rossow and Schiffer 1991; Rossow and Garder 1993), and from Global Cloud Imagery (GCI), which provides cloud-top IR brightness temperature at 0.5° spatial and 3-h temporal resolution (Salby et al. 1991). Observations are averaged over locations that contain each of the four cloud categories between 20°S and 20°N to obtain reliable climatological values. ISCCP C2 data is also averaged over the 7-yr period January 1984–December 1990. GCI is averaged over January 1984.

Each of the climatological categories has distinguishing climatological properties (Table 1) that are incorporated into the time-mean conditions specified for the calculation. Maritime nonconvective locations have very few high clouds, whereas maritime convective and continental locations have mean high-cloud fraction, *C̄*_{hi} > 0.2. Maritime convective locations are more humid than maritime nonconvective and continental locations. Continental locations have smaller low-cloud fraction and larger diurnal amplitude of surface temperature than locations in either of the maritime categories.

Time-mean properties specified in the radiative transfer calculation include: (1) surface albedo *A*_{s}; (2) the atmospheric temperature *T̄*(*z*) and relative humidity *RH*(*z*); and (3) the cloud fraction *C̄*(*z*), which represents the mean probability density that cloud top is at height *z,* cloud water density *ρ̄*_{c}(*z*), and the cloud liquid water path *LWP*(*z*), which, together with *ρ̄*_{c}, defines the cloud thickness.

The surface albedo is fixed at *A*_{s} = 0.1, which is typical of maritime or forested continental locations (e.g., Briegleb 1992). The time-mean temperature *T̄*(*z*) (Fig. 2a) is specified for all climatological conditions from the observed zonal-mean temperature at the equator (e.g., Holton 1979). The time-mean surface temperature is *T̄*_{s} = 300 K and the lapse rate is 6 K/km in the lower troposphere (*z* < 5 km), increasing to 8.5 K/km in the upper troposphere and then decreasing to 0 K/km by the tropopause at *z* = 16 km. The troposphere is capped by an isothermal stratosphere with *T̄* fixed at 196 K. The time-mean relative humidity in the upper troposphere (Fig. 2b) is specified to be larger under maritime convective conditions than under maritime nonconvective and continental conditions (cf. Table 1; Webster and Stephens 1980; Hartmann et al. 1992; Sun and Lindzen 1993).

Figure 3 displays the observed cloud fraction *C̄*_{obs}, as a function of temperature^{2} for (a) maritime nonconvective, (b) maritime convective, and (c) continental locations during January 1984 from GCI. Here, *C*_{obs} (*T*) is calculated from the fraction of observations within a climatological category that has cloud-top temperature *T.* So, it reflects the probability density that the observed cloud top occurs at *T*(*z*). Under all climatological conditions, *C̄*_{obs} increases slowly with temperature for *T̄* = 200–280 K. For larger *T̄,* there is a dramatic increase of *C̄*_{obs}. Those vertical distributions are represented in the calculation (Fig. 4) by a single cloud layer, with fractional coverage *C̄*_{lo} (Table 1), cloud-top temperature *T̄*_{c} = 290 K, and a linear distribution of probability density (i.e., cloud fraction) *C̄*(*z*) for *T̄* < 290 K. Time-mean cloud fraction is specified to be *C̄* = 0.1 km^{−1} at *T̄* = 290 K and *C̄* = 0 at *T̄* = 200 K under maritime convective (Fig. 4b) and continental (Fig. 4c) conditions. In contrast, maritime nonconvective conditions are specified to be cloud free for *T̄* < 290 K (Fig. 4a).

Under maritime conditions, the mean cloud water density (Fig. 5a) is fixed at *ρ̄*_{c} = 1.0 g m^{−3} for upper-level clouds (*T̄* < 250 K) and at *ρ̄*_{c} = 0.2 g m^{−3} for low clouds (*T̄* > 270 K). Under continental conditions, it is fixed at *ρ̄*_{c} = 1.0 g m^{−3} throughout. These values are in broad agreement with observed cloud properties in the Tropics (e.g., Stephens 1978; Bower et al. 1994). Cloud liquid water path *LWP*(*z*) (Fig. 5b) is specified under all climatological conditions at 30 g m^{−2} for low clouds (*T̄* > 270 K) and increases linearly with decreasing mean temperature for higher clouds. This behavior is in broad agreement with ISCCP C2 (asterisks in Fig. 5b) and with satellite observations of higher vertical resolution (Fu et al. 1990).

### b. Diurnal heating rate

The diurnally varying diabatic heating rate *q̇*′, which forces the calculated diurnal variations of the atmospheric state, is calculated with the radiative transfer model developed for the National Center for Atmospheric Research (NCAR) community climate model CCM2 (Hack et al. 1993). That calculation uses the time-mean vertical profile temperature, humidity, cloud fraction, and cloud water liquid water path specified in section 2a. The radiative transfer calculation is performed once for each hour of the day, with diurnal variations of radiative heating resulting from the diurnal variation of solar zenith angle.

### c. Dynamical response

Diurnal variations of dynamical properties, such as temperature, density, static stability, and total water content, are derived from the equations for hydrostatic motion on an *f* plane, which applies to large-scale atmospheric phenomena of narrow meridional extent:

is the Lagrangian time derivative, *r*_{w} is the total water mixing ratio, **F**_{h} represents frictional drag, *q̇* is diabatic heating rate, and *S*_{w} is a source of water into the atmosphere.

The total water mixing ratio can be expressed in terms of water vapor, cloud water, and cloud fraction *C*:

where *C* reflects the fraction of an air parcel containing cloud, *r*_{c} is the cloud water mixing ratio, *r*_{s} is the saturated vapor mixing ratio, and *r*_{υ} is the water vapor mixing ratio under cloud-free conditions.

Diurnal variability is spatially coherent, unlike other high-frequency variability, which operates on smaller dimensions. It is also of higher frequency than other variability operating on large dimensions (e.g., Salby et al. 1991) and, therefore, operates on a time–space scale separated from all other variability. For that reason, the governing equations (1) are averaged over a horizontal scale much larger than individual clouds, which separates diurnal variability from other high-frequency variability. The resulting large-scale state variables Ψ(*x, y, z, t*) are then composed of two components: a slowly varying or mean component Ψ̄(*x, y, z*) and the diurnal variation Ψ′(*x, y, z, t*) about it; that is,

Here Ψ̄ is treated as constant in time because of the wide separation of timescales between it and Ψ′. The diurnal variation is represented in terms of Fourier components with frequencies *σ* = ±1, ±2, ±3, . . . cycles per day.

Equations 1 are next linearized about a horizontally uniform and motionless mean state. The resulting system governing large-scale diurnal variability is then

Vertical transports of momentum, heat, and water by small-scale eddies, which act to dissipate large-scale motion, are represented in terms of linear friction, with rate *ν*; eddy diffusion of water, with coefficient *K*_{w}; and eddy diffusion of heat (i.e., of potential temperature *θ*), with coefficient *K*_{h}. Eddy correlation terms, for example, **u**′·**∇**Ψ′ for the generalized field variable Ψ, are small compared to the time derivative ∂Ψ′/∂*t* because Ψ′ has a large spatial scale and small temporal scale (e.g., analogous to treatments of tidal behavior), so they can be neglected. Specifying the diurnally varying diabatic heating rate *q̇*′ makes (4) a complete set of equations governing large-scale dynamical variations. If *K*_{h} is zero and boundary conditions specify *w*′ = 0 at the surface and finite kinetic energy *ρ̄***u**^{2} as *z* → ∞, then (4) can be solved analytically (Salby and Garcia 1987; Bergman 1996). That solution can be expanded to incorporate nonzero *K*_{h} using an iterative method (Bergman 1996), which includes two additional boundary conditions. 1) The diurnal variation of temperature at the surface *T*^{′}_{s} is specified from the diurnal variation of radiative flux *F*^{′}_{s} absorbed at the surface,

where *κ*_{s} is a prescribed constant that achieves a diurnal amplitude of surface temperature of 0.4 K under maritime conditions and 6 K under continental conditions (cf. Table 1). 2) The thermal energy *ρ**T* remains finite as *z* → ∞.

To account for dissipation of large-scale energy by small-scale eddies, which are strong inside the boundary layer, the frictional damping rate *ν*(*z*) in (4) is prescribed to decrease with height exponentially, with characteristic height 5 km (Fig. 6). Reflecting convective conditions, which prevail in the Tropics, damping rates of (4 day)^{−1} near the surface to (15 day)^{−1} at the tropopause are consistent with values in other numerical studies (e.g., Colton 1973; Salby et al. 1994; Bergman and Salby 1994). Eddy diffusivity also decreases exponentially with height, with *K*_{h} = 10 m^{2} s^{−1} at *z* = 0 (e.g., Kuo 1968) and *K*_{h} = 1 m^{2} s^{−1} at *z* → ∞ (e.g., Woodman and Rastogi 1984). For consistency, *K*_{w} equals *K*_{h} throughout.

### d. Cloud variability

Cloud fraction, along with other cloud properties, is not explicitly described by (4). The large-scale diurnal variation of cloud fraction is calculated here in terms of the diurnal variation of cloud probability density *C*′(*z, t*) that cloud top exists at height *z* and time *t.* It follows from the solution of (4) via parameterizations of the large-scale environment. Those parameterizations are validated a posteriori by comparing the numerical results with observed cloud diurnal variability.

To account for basic differences between high- and low-cloud formation, clouds are decomposed into two categories: “convective” (e.g., high cumuloform and anvil cirrus) and “nonconvective” (e.g., low stratiform and stratocumulus). Cloud fraction then follows from these two contributions as

where *C*_{c} is the convective cloud fraction and *C*_{nc} is the nonconvective cloud fraction.

Convective clouds are defined here to be upper-tropospheric clouds that derive their moisture from the lower troposphere through vertical motion. Their formation follows from atmospheric conditions both in the lower troposphere, where convection is initiated, and in the upper troposphere, where convective cloud tops exist. Convective cloud fraction *C*_{c}(*z, t*) then follows as the product of two probabilities:

where the probability density that convection occurs at time *t*: *P*_{c}(*t*) is related to conditions in the lower troposphere and the conditional probability density that, if convection occurs then, convective cloud top is at height *z*: *P*_{z}(*z, t*) is related to conditions at cloud top.

The diurnal variation of cloud fraction *C*′(*z, t*) follows from the linearization of (6) and (7) in terms of the “normalized diurnal variation”:

where

is the time-mean fraction of clouds at height *z* that are convective. That fraction (Fig. 7) is specified to be zero in the lower troposphere and 1.0 in the upper troposphere under all climatological conditions, in compliance with the definition of convective and nonconvective clouds. The probability densities *P*^{′}_{c}, *P*^{′}_{z}, and *C*^{′}_{nc} must be evaluated from the diurnal variations of temperature, density, stability, and moisture content, which follow from (4).

Diurnal variations of convection are closely related to diurnal variations of low-level stability, especially over continental locations (Wallace 1975; Gray and Jacobson 1977). Consequently, the first component of (8), the diurnal variation of convective activity, is dfined from the diurnal variation of static stability averaged over the boundary layer,

where *z̄*_{b} is the mean height of the boundary layer. The negative sign in (9) indicates that convection is strongest when the boundary layer stability is weakest, in accord with prevailing conceptual models of atmospheric convection (e.g., Emanuel 1994).

The second component of (8) involves the conditional probability that convective clouds exist *P*_{z}(*z, t*), given that convection occurs and reflects atmospheric conditions at cloud top. As shown in the appendix, an increase of *d**θ*/*dz* in the upper troposphere increases the density of isentropic surfaces, which, in turn, increases the probability *P*_{z} (*z, t*)Δ*z* that cloud top occurs in the height interval *z* to *z* + Δ*z.* The conditional probability then follows as

The third component of (8) involves *C*^{′}_{nc}(*z, t*), which is simply the diurnal variation of cloud fraction at height *z* if *C*_{c} = 0. To calculate *C*^{′}_{nc}, air is allowed to assume one of two states: cloudy or clear, each with distinct properties such as concentrations of water vapor and cloud water. Those two states are defined to be noninteracting, with fixed values for the mixing ratios of water vapor inside clear air *r*_{υ}, and cloud water *r*_{c}. Consequently,

The diurnal variation of nonconvective cloud fraction *C*^{′}_{nc} then follows from the linearization of (2):

Thus, *C*^{′}_{nc} is calculated from the diurnal variations of total water mixing ratio *r*^{′}_{w} and saturation vapor mixing ratio *r*^{′}_{s}. It is inversely proportional to the mean difference between the total water mixing ratios of cloudy air, *r̄*_{c} + *r̄*_{s}, and clear air *r̄*_{υ}. That difference can be small for maritime stratus clouds, which have small cloud water mixing ratios (e.g., Stephens 1978) and occur in well-mixed boundary layers, where the relative humidity is high and, therefore, *r*_{υ} ∼ *r*_{s} (e.g., Betts et al. 1995).

The cloud diurnal variation (12) follows from diurnal variations of atmospheric properties in (4), with *r*^{′}_{w}. Following directly from the solution to (4g). The saturation vapor mixing ratio

is a function of air density *ρ* and saturation vapor density *ρ*_{s}, which is a function of temperature. So its diurnal variation follows from (4) via *ρ*′ and *T*′.

## 3. Cloud response to the diurnal cycle of SW heating

### a. Low clouds under maritime nonconvective conditions

The troposphere under maritime nonconvective conditions is cloud free except for a low-cloud layer with cloud top at *T̄* = 290 K (Fig. 4a). The calculated large-scale diurnal heating rate *q̇*′ (Fig. 8) then follows primarily from the diurnal variation of SW absorption by atmospheric water vapor. That heating rate maximizes at 1200 local solar time (LST) and has its largest amplitude in the midtroposphere (*T̄* ∼ 250 K). The impact of the cloud layer on SW heating is evident in the lower troposphere, where enhanced SW absorption occurs just above the cloud layer (*T̄* ∼ 290 K).

The diurnal variation of temperature is closely related to the diurnal variation of saturation vapor mixing ratio, which, in turn, is important to the diurnal variation of nonconvective clouds in (12). Shown in Fig. 9 is the diurnal variation of temperature calculated from the first two harmonics (i.e., *σ* ± 1, ±2) of the diurnal heating rate in Fig. 8. A calculation using 12 diurnal harmonics (not shown) has a temperature variation that is virtually indistinguishable from Fig. 9, primarily because 98% of the solar diurnal variance is contained in the first two harmonics. The diurnal maximum of temperature occurs in the afternoon at all levels, reflecting the relationship

which is found by simplifying (4e). It decreases from a maximum near the surface, due to the vertical transport of heat from the surface, to a minimum at *T̄* = 290 K and then increases with height to a maximum at *T̄* = 200 K. A slight downward propagation of phase is evident over most of the troposphere. The same features are, in fact, evident in radiosonde observations over maritime locations (e.g., Harris et al. 1962; Frank 1980; Mapes and Zuidema 1996), as well as in numerical investigations of atmospheric tides (e.g., Lindzen 1967).

Figure 10 compares the normalized diurnal variation of low-cloud fraction *C*′/*C̄* from the calculation (solid line) to that of ISCCP C2 based on observations from IR only (short dashed line) and IR + visible (long dashed line). The agreement between the calculated and observed diurnal variations appears best for the IR + visible observations, although those observations are only available during daylight hours. The diurnal variation of cloud fraction is 12 h out of phase with that of temperature (Fig. 9), reflecting its relationship to the diurnal variation of saturation mixing ratio [cf. Eq. (12)]. This finding supports that of Brill and Albrecht (1982), who calculated diurnal variations of cloud fraction from diurnal variations of relative humidity in a one-dimensional boundary-layer simulation.

Table 2 lists the results of calculations that probe the sensitivity of the calculated diurnal variation of cloud fraction to changes of the specified model parameters. In those calculations, model parameters Ψ are changed from the “original” calculation by an amount, ΔΨ. Relative changes are denoted in Table 2 by percentages. For example, ΔΨ = 100% means that parameter Ψ was doubled in the “test” calculation. For some parameters, ΔΨ relates an absolute change. For example, the mean relative humidity RH in the test calculation is 0.1 larger, throughout the troposphere, than in the original calculation. The corresponding changes to the calculated diurnal variation of cloud fraction are shown via changes to the amplitude *D*_{1} and phase *ϕ*_{1} of the first harmonic in the Fourier spectrum,

Amplitude changes in Table 2 are related in terms of the relative change Δ*D*_{1}/*D*_{1} in percent. Here, Δ*D*_{1}/*D*_{1} = 0% means the diurnal amplitude of cloud fraction in the test calculation is unchanged from the original calculation, while Δ*D*_{1}/*D*_{1} = 100% means the diurnal amplitude in the test calculation is twice that in the original calculation. The absolute change in phase Δ*ϕ*_{1} is given in hours.

The phase of the calculated cloud diurnal variation (Table 2) is robust, changing by less than 1 h for a wide range of model parameters. The amplitude is more sensitive, principally to changes of relative humidity RH. The amplitude doubles if the relative humidity at the top of the boundary layer is increased from 0.85 to 0.95, which is typical of a well-mixed maritime boundary layer where convection is suppressed (e.g., Betts et al. 1995). This feature is consistent with satellite observations, which show that the diurnal amplitude of low-cloud fraction over maritime locations increases substantially as convective activity decreases (Bergman and Salby 1996).

In addition to testing the calculation for its sensitivity to the changes of the specified parameters, it was tested for its sensitivity to the impact of cloud diurnal variations on radiative heating. In the “original” calculation, radiative heating is calculated from time-mean cloud fields only, relying on the diurnal variation of solar angle for its diurnal variability. In the test calculation, the diurnal variations of radiative heating rates are recalculated incorporating the originally calculated diurnal variation of cloud fraction. Cloud diurnal variations are then recalculated from those heating rates. The differences between the the recalculated and the original diurnal amplitude and phase (Table 2; last line) measure the “cloud diurnal radiative feedback” (CDRF) on cloud diurnal variations. That feedback is small under maritime nonconvective conditions. The resulting change of diurnal amplitude is 9% and the change of diurnal phase is less than 1 h.

### b. High clouds under maritime convective conditions

High clouds over maritime locations exhibit large low-frequency variability (e.g., Salby et al. 1991). Those clouds are typically found in clusters and tend to be either concentrated over a specific location or scarce (e.g., Mapes and Houze 1993). This convective organization is demonstrated in Fig. 11, which displays the observed normalized daily deviation from the monthly mean cloud fraction in GCI, as a function of cloud-top temperature, over a 5° area in the west Pacific. During this time, the monthly mean high-cloud fraction *C̄*_{hi} = 0.20 (for cloud-top temperatures *T*_{c} < 250 K; Table 1) is explained entirely by low-frequency variations that produce nearly 100% high-cloud coverage 20% of the time. In contrast, low-frequency variations of low-cloud fraction (e.g., *T*_{c} > 270 K) are much weaker.

That low-frequency variability is important to diurnal variations of high clouds because it implies that variations of high clouds are restricted to extremely cloudy conditions, when the diurnal variation of SW absorption is strongly influenced by the presence of high clouds. For that reason, maritime convective conditions are specified to consist of two atmospheric states that are different only in their high-cloud fraction. In state 1, which occurs 20% of the time, the mean cloud fraction *C̄*_{1}(*z*) (Fig. 12a) is large for cloud tops at *T̄* < 250 K. In contrast, state 2, which occurs 80% of the time, has no cloud tops at *T̄* < 250 K (Fig. 12b). Both states have the same cloud fraction at *T̄* > 270 K, reflecting the uniformity of those clouds in Fig. 11. The time-mean cloud fraction follows from the weighted average over states 1 and 2 (Fig. 12c),

which reflects the time-mean cloud fraction observed over maritime convective locations (Fig. 3b).

The diurnal amplitudes of the heating rate (Fig. 13) calculated from those two states are quite different. In state 2, the diurnal amplitude of heating rate (short dashed line) is much like that under maritime nonconvective conditions, whereas, in state 1, diurnal heating (long dashed line) is more concentrated in the upper troposphere as a result of the influence of high clouds on solar heating.^{3} The time-mean heating rate (solid line) resembles that for state 2 because the atmosphere is in that state 80% of the time. The time-mean heating rate does not accurately represent the heating rate that occurs in the presence of high clouds and, therefore, underestimates the impact of SW heating in the upper troposphere on the diurnal variation of those clouds.

That impact is evident in the diurnal variation of temperature calculated under maritime convective conditions (Fig. 14). The temperature variation averaged over both states (Fig. 14a),

where *T*^{′}_{1} and *T*^{′}_{2} are the temperature variations calculated for states 1 and 2, respectively, resembles that under maritime nonconvective conditions (Fig. 8). However, the diurnal variation of temperature based on state 1 alone (Fig. 14b) shows a concentration of large diurnal amplitude high in the troposphere (*T̄* < 240 K), which results from the concentration of SW heating there (Fig. 13).

The vertical structure *C*′ is an important feature of cloud diurnal variability, one represented in the distribution of 11-*μ*m radiance observed from satellite (e.g., Albright et al. 1985; Hartmann and Recker 1986; Mapes and Houze 1993). The observed diurnal variation of cloud fraction is represented in terms of the number distribution of cloud-top temperature observations in the GCI *N*(*T, t*), where *T* is the brightness temperature of the observed radiance and *t* is local time. The normalized distribution

is a counterpart of the normalized cloud diurnal variation [*C*′(*T, t*)]/[*C̄*(*T*)]. Figure 15 displays *g*(*T, t*) obtained from maritime convective locations between 20°S and 20°N. The observed amplitude is typically 0.1–0.2, with maxima near *T̄* = 230 K and *T̄* = 200 K. Diurnal phase propagates systematically downward from a morning maximum (0300–0900 LST) at *T̄* = 200 K, to an evening maximum (1800 LST) near *T̄* = 250 K, and back to a morning maximum (0600 LST) at *T̄* = 290 K. These features are found in other investigations of diurnal variability of maritime convective clouds (e.g., Albright et al. 1985; Hartmann and Recker 1986; Mapes and Houze 1993).

The calculated diurnal variation of cloud fraction under maritime convective conditions (Fig. 16a) follows from the average over states 1 and 2,

where *C*^{′}_{1} and *C*^{′}_{2} are the diurnal variations of cloud fraction calculated for states 1 and 2, respectively. The calculated diurnal variation *C*′/*C̄* reproduces the dominant features of the observed diurnal variation in Fig. 15, particularly in the upper troposphere, where diurnal amplitude is typically 0.1–0.2. Maxima appear near *T̄* = 230 K and *T̄* = 200 K. Phase propagates downward from a morning maximum (0600 LST) at *T̄* = 200 K, to an evening maximum (1800 LST) at *T̄* = 230 K, and back to a morning maximum (0600 LST) at *T̄* = 290 K. However, the calculated phase is almost 3 h later than that observed in parts of the troposphere (Fig. 15), for example, at *T̄* = 200 and 280 K. In addition, the calculated vertical phase propagation occurs discretely in two phase shifts at *T̄* = 260 and 215 K, whereas the observed phase propagation is more continuous. That feature results from simplifying assumptions within the calculation. Nevertheless, results here indicate that the dominant features of the observed large-scale diurnal variation of cloud fraction are reproduced in this calculation, which contains only migrating diurnal components forced by the diurnal cycle of SW heating.

The specified vertical distribution of clouds is important to the calculated radiative heating and thus to *C*′. This sensitivity is illustrated by the diurnal variation of cloud fraction calculated from the time-mean heating alone (Fig. 16b), in which the impact of high clouds on radiative heating is underrepresented (Fig. 13). In that case, the calculated cloud diurnal variation is much too weak in the upper troposphere and the phase at *T̄* = 200 K is more than 6 h too late. High clouds concentrate radiative heating in the upper troposphere (Fig. 13), which enhances the diurnal variation of *d**θ*′/*dz* there (cf. packing together of contour lines in Fig. 14b), which in turn affects the diurnal variation of high-cloud fraction in (10). In addition, the diurnal amplitude of cloud fraction at *T̄* = 230 K is very sensitive to the specified cloud liquid water path (Table 3), which also alters radiative heating. Conversely, it is only slightly sensitive to changes of other climatological parameters. The role of high clouds in radiative heating was also found to be important to diurnal variations of maritime convection in studies with a general circulation model (Randall et al. 1991) and a cumulus ensemble model (Xu and Randall 1995).

Under convective conditions, the diurnal variation of latent heat release in precipitation could be important to cloud diurnal variations. To test that importance, the diurnal variation of latent heating rate is specified as

The vertical structure *Q*_{z} is prescribed according to the vertical structure of latent heating by mesoscale convective systems in the Tropics (e.g., Yanai et al. 1973; Houze 1989):

where *Q*_{l} is the amplitude of the heating rate, *z*_{0} is the base of the heating, and *z*_{t} is the top of the heating. The diurnal time dependence contains one harmonic,

with phase *ϕ*_{p}. For the sensitivity tests shown in Table 3, *Q*_{l} is specified based on a diurnal amplitude of precipitation rate of 2 mm day^{−1}, the base of the heating is at *T̄* = 270 K, the top of the heating is specified at *T̄* = 210 K, and the diurnal phase of the heating is specified at 0000, 0600, 1200, and 1800 LST. Even though the specified diurnal amplitude of precipitation rate is quite large for maritime conditions (Gray and Jacobson 1977), the corresponding diurnal variation of latent heating has little effect on the calculated diurnal variation of cloud fraction at *T̄* = 230 K. However, the vertical distribution of diurnal variations of latent heating could be very different from the time-mean distribution on which (21) is based. Further tests reveal that the impact of latent heating is highly sensitive to the assumed vertical profile of that heating. Furthermore, latent heating is likely to be more important at smaller horizontal scales. So results here regarding the importance of diurnal variations of latent heating to the diurnal variations of cloud fraction are inconclusive.

### c. Low clouds under continental conditions

Under continental conditions, turbulent heat transfer from the surface is important in the lower troposphere (e.g., Kuo 1968; Wallace 1975; Gray and Jacobson 1977). Its effect is found in the calculated diurnal variation of temperature under continental conditions (Fig. 17). In the lower troposphere (*T̄* > 280 K), amplitude decreases sharply with height, from a maximum at the surface, and phase propagates upward. The diurnal temperature variation in the upper troposphere (*T̄* < 260 K) is similar to that under maritime conditions (Figs. 8 and 13) but with a larger amplitude. The calculated diurnal variation of temperature is in broad agreement with radiosonde observations over continental locations (e.g., Harris 1959).

The calculated diurnal variation of low-cloud fraction under continental conditions (Fig. 18) captures the essential features of the observed continental low-cloud diurnal variation. A dominant maximum exists near noon, and cloud fraction is relatively uniform throughout the night. Like low clouds under maritime nonconvective conditions (section 2a), the diurnal variation of low clouds under continental conditions follows from the diurnal variation of temperature through the diurnal variation of saturation vapor density in (12). However, diurnal variability near the continental surface is dramatically altered by vertical heat transfer from that surface. The diurnal maximum of continental low-cloud fraction coincides with the diurnal minimum of temperature at *T̄* = 290 K (Fig. 17), which occurs at a later time and has a larger amplitude than under maritime nonconvective conditions because of eddy diffusion. There is a large uncertainty in the observed diurnal variation of low-cloud fraction as represented in the difference between the diurnal variations of observed cloud fraction from IR measurements alone (short dashed line) and those from both IR and visible measurements (long dashed line). That difference could result from systematic error introduced to IR cloud retrieval over continental locations by the diurnal variation of surface temperature, which introduces a diurnal variation to the temperature contrast between the surface and cloud top (Bergman and Salby 1996).

The calculated diurnal amplitude of continental low-cloud fraction is quite sensitive to changes of the model parameters (Table 4). In particular, it is very sensitive to the diurnal amplitude of surface temperature, which, when doubled, doubles the diurnal amplitude of cloud fraction. This relationship, which is also found in ISCCP C2 for low clouds over continental locations (Bergman and Salby 1996), follows from the impact of diurnal variations of vertical heat transport from the surface on atmospheric diurnal variability under continental conditions. In addition, the specified height of the cloud layer (as measured by cloud-top temperature *T̄*_{c}), relative humidity *RH*, vertical dependence of diffusivity (as measured by *H*_{ν}), and surface albedo *A*_{s} are important.

### d. High clouds under continental conditions

Over continental locations, the observed large-scale diurnal variation of cloud fraction (Fig. 19) is much stronger than over maritime locations (Fig. 14), with relative amplitude ranging from 0.2 at *T̄* = 270 K to almost 1.0 at *T̄* = 200 K. Downward phase propagation over continental locations is faster than at maritime convective locations. Cloud fraction has a diurnal maximum between 1500 and 1800 LST at *T̄* < 250 K, between 0000 and 0300 LST at *T̄* = 250–280 K, and near 1000 LST at *T̄* = 290 K. The evening-to-nighttime maximum for continental high clouds is consistent with other investigations (e.g., Minnis and Harrison 1984; Hartmann et al. 1991; Bergman and Salby 1996).

The calculated diurnal variation of cloud fraction under continental conditions (Fig. 20) mirrors its observational counterpart throughout. The diurnal maximum occurs at 1500–1800 LST for clouds at *T̄* < 250 K and occurs near midnight for clouds at *T̄* = 250–280 K. In addition, a strong midmorning maximum near *T̄* = 290 K is found in both the calculation and observations. Relative diurnal amplitudes are large, ranging from 0.2 at *T̄* = 270 K to 0.9 in the upper troposphere. However, the upper-tropospheric maximum occurs at *T̄* = 220 K in the calculation, whereas it is at *T̄* = 200 K in the observations.

The strong diurnal maximum at 1500–1800 LST for clouds at *T̄* < 250 K under continental conditions does not occur under maritime convective conditions (Fig. 16a). That feature results from the strong diurnal variation of low-level stability that occurs then under continental conditions, which determines the probability density *P*_{c} that convection occurs in (9). The importance of low-level stability to the diurnal variation of continental high clouds is supported by the model sensitivity (Table 5). The diurnal variation *C*′ at *T̄* = 230 K is strongly sensitive only to changes of the diurnal amplitude of surface temperature, which affects *d**θ*′/*dz* in the lower troposphere, and to the height of the specified boundary layer *z̄*_{b} over which *d**θ*/*dz* is averaged in (9) to calculate *P*^{′}_{c}. The good agreement between the calculated and observed cloud diurnal variations under continental conditions, therefore, follows from vertical heat transport from the surface by eddy diffusion. It lends supporting evidence for the hypothesis that diurnal variations of low-level stability govern the diurnal variation of convection over continental locations (e.g., Wallace 1975; Gray and Jacobson 1977).

## 4. Conclusions

The vertical distribution of large-scale cloud diurnal variability was calculated from the diurnal cycles of SW heating in the atmosphere and at the surface under horizontally uniform conditions. In so doing, clouds were viewed statistically in terms of the probability that cloud top is at a particular height. Thus, this investigation complements regional-scale studies (e.g., Xu and Randall 1995), which explicitly model the time evolution of individual clouds and cloud systems. The calculated diurnal variations are broadly consistent with satellite observations for four cloud categories, which characterize large-scale cloud diurnal variability (Bergman and Salby 1996): low clouds under maritime nonconvective conditions, high clouds under maritime convective conditions, low clouds under continental conditions, and high clouds under continental conditions.

This investigation was aided by simplifying approximations, particularly in the parameterization of cloud diurnal variability based on dynamic diurnal variability. Those simplifications render the calculation of cloud diurnal variations tractable and make the results straightforward to interpret. However, those same approximations introduce uncertainty into the calculations that should be assimilated into the interpretation of the results. That uncertainty is partially accounted for in sensitivity tests, which show the calculated cloud diurnal phase to be a robust feature but diurnal amplitude to be sensitive to the specified time-mean conditions. There is also uncertainty in the observed cloud diurnal variability to which the calculations were compared. So, although the mechanisms discussed here are found to be sufficient to explain the observed large-scale cloud diurnal variations, other mechanisms may also be important. In addition, this investigation addresses only large-scale cloud diurnal variations, which often do not typify diurnal variability at a specific location. For example, Bergman and Salby (1996) show that high clouds exhibit a wide range of diurnal phases at 2.5° horizontal scales but very little phase variability at 10° scales.

The calculated diurnal variations of cloud fraction are, in many respects, directly related to the diurnal variation of temperature forced by SW heating of the atmosphere and surface. The temperature variation in the upper troposphere is similar under all climatological conditions, which underscores the importance of global tidal oscillations there. However, differences in the diurnal variations of temperature in the lower troposphere (e.g., under maritime and continental conditions) lead to substantial differences in the corresponding cloud diurnal variations.

The diurnal variation of low-cloud fraction follows primarily from the diurnal variation of saturation vapor mixing ratio, which, in turn, follows from the diurnal variation of temperature. The low-cloud diurnal amplitude becomes very strong under maritime nonconvective conditions, in which the air at the height of the cloud layer is nearly saturated. Under those conditions, the transition from cloudy to clear conditions requires only a small perturbation of the moisture budget. Under continental conditions, in which the diurnal variation of surface temperature is large, the diurnal variation of low-cloud fraction is amplified by and its phase is advanced by the large diurnal variation of vertical heat transport from the surface. That diurnal mechanism is not necessarily restricted to continental locations, but may be important over some maritime locations, where diurnal amplitude of surface temperature as large as 1–2 K have been reported (e.g., Webster et al. 1996) and cumulus clouds are observed to have a midday maximum (e.g., Rozendaal et al. 1995).

The calculated diurnal variations of high-cloud fraction are related to diurnal variations of low-level static stability, which is related to the probability that convection occurs, and of static stability at cloud top, which determines the likelihood of convective cloud existing at a given level. Under convective conditions, the troposphere has near-neutral stability (i.e., *d**θ̄*/*dz* is small), so even small diurnal variations of potential temperature *θ*′/*θ̄* can introduce substantial diurnal variations of static stability. Under maritime convective conditions, in which the diurnal variation of low-level stability is small, diurnal variations of high clouds follow primarily from the diurnal variation of upper-level stability, which is enhanced by the concentration of radiative heating in the upper troposphere by high clouds. Under continental conditions, in which the diurnal variation of surface temperature is strong, the diurnal variation of low-level stability produces the strong diurnal variation of high clouds.

Large-scale cloud diurnal variations contribute substantially to time-mean energetics (Bergman and Salby 1997). They are, therefore, an important component of climate and should be accounted for in climate simulations. However, results here suggest that large-scale cloud diurnal variations and, therefore, their impact on energetics can be calculated directly from time-mean climatological properties such as temperature, humidity, cloud cover, surface heat capacity, and surface albedo without explicitly resolving atmospheric diurnal variability.

## Acknowledgments

This work owes its success, in part, to discussions with and contributing ideas by Judith Curry, Harry Hendon, Jeffrey Kiehl, Conway Leovy, Julius London, Brian Mapes, Murry Salby, Peter Webster, and anonymous reviewers.

## REFERENCES

### APPENDIX

#### Convective Cloud Probability

The convective cloud fraction *C*_{c}(*z, t*) reflects the vertical distribution of cloud tops associated with horizontally averaged behavior. The convective clouds discussed are clouds in the upper troposphere (e.g., *T̄* < 260 K) that have cloud tops far above the source of convective air parcels. Therefore, *C*_{c}(*z, t*) depends on atmospheric conditions in the lower troposphere, where convection is initiated, and in the upper troposphere, where cloud top exists. For that reason, the convective cloud fraction is expressed in (7) as the product of the probability density *P*_{c} that convection occurs and the conditional probability density *P*_{z}(*z, t*) that, if convection occurs, it produces cloud top at height *z.* The probability density *P*_{c} reflects atmospheric conditions in the lower troposphere, where convection is initiated, and is, therefore, interpreted as the “convective intensity.” The conditional probability density *P*_{z}(*z, t*) is a function of the atmospheric conditions at cloud top.

The diurnal variation of convective intensity alters the diurnal variation of cloud fraction in (10) solely in terms of the relative diurnal variation [*P*^{′}_{c}(*t*)]/*P̄*_{c}, so the time-mean convective intensity *P̄*_{c} can be set to unity without loss of generality. The conditional probability *P*_{z} then represents the convective cloud fraction if convective intensity is constant (i.e., if *P*^{′}_{c} = 0). That is,

The time-mean probability density *P̄*_{z}(*z*) then equals the time-mean convective cloud fraction *C̄*_{c}, which is available from observations (e.g., Fig. 3).

Convective cloud fraction can also be expressed in terms of the probability density *P*_{θ}(*θ*, *t*) that, if convection occurs, it produces cloud top at potential temperature *θ*(*z, t*). The probability densities *P*_{z} and *P*_{θ} are interrelated via the probability that convective cloud top lies in the height interval *z* to *z* + *dz*:

where

Equation (A2) constitutes a mapping between the cloud-top probability densities, one that is unique because, on large scales, *θ* increases monotonically with height. So *P*_{z} can be expressed in terms of *P*_{θ} as

Potential temperature at cloud-top *θ*(*z, t*) is a reflection of the buoyancy of low-level air inside the convection and, therefore, of convective intensity. For example, the energy required to lift an air parcel adiabatically is measured by the potential temperature difference through which it is lifted (e.g., Emanuel 1994). So, under conditions of constant convective intensity, the probability density *P*_{θ} depends on *θ* alone,

and is independent of time.

The diurnal variation of potential temperature is small compared to the time mean. Furthermore, *P** _{θ}* (Fig. A1) typically undergoes a relative change Δ

*P*

_{θ}/

*P*

_{θ}of less than 3% for a potential temperature change Δ

*θ*= 1 K, which is typical of diurnal variations. Therefore,

The relative diurnal variation of cloud-top probability density then reduces to

Figure A2 illustrates how convective cloud probability is altered by an increase of *d**θ*/*dz* in the upper troposphere. Figure A2a displays a hypothetical probability density *P*_{z}(*z*) (dashed line) for a given *d**θ*/*dz.* Here *P*_{z} goes to zero at height *z*_{t}, which marks the highest convective clouds. The probability that convective cloud top lies in the height interval *z*_{0} to *z*_{1} is represented by the stippled area. Figure A2b displays *θ*(*z*) (solid line) and *P*_{z}(*z*) + *θ*(*z*) (dashed line) under the same conditions as in Fig. A2a. Here *θ*_{t} is the potential temperature at *z*_{t} and *θ*_{1} is the potential temperature at *z*_{1}. In Fig. A2b, *P*_{z} is displaced by amount *θ* for illustrative purposes only and the area of the stippled region in Fig. A2b is the same as in Fig. A2a. Figure A2c displays *θ*(*z*) (solid line) and *P*_{z}(*z*) + *θ*(*z*) (dashed line) under conditions of larger *d**θ*/*dz,* with *P*_{θ} held constant. The following two effects are illustrated. 1) The increase of *d**θ*/*dz* increases the potential temperature at all heights greater than *z*_{0}, which changes *P*_{θ} at those heights. This affects the highest clouds most, which no longer reach the height *z*_{t}. Under actual upper-tropospheric conditions, this effect is small for reasons leading to (A5). 2) The increase of *d**θ*/*dz* increases the potential temperature change in a specified height interval, which, since *P** _{θ}* depends only on

*θ*, increases the probability that cloud top occurs in that interval. The lightly stippled region in Fig. A2c represents the same probability (i.e., cloud fraction) as it does in Figs. A2a and A2b because it corresponds to the same potential temperature interval. However, that region is now concentrated in a smaller height interval. The cloud fraction in the height interval between

*z*

_{0}and

*z*

_{1}is, therefore, increased by the amount represented in the area of the dark stippled region. Thus, an increase in

*d*

*θ*/

*dz*increases the probability that convective cloud top occurs in a specified height range.

## Footnotes

Current affiliation: Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, Colorado

*Corresponding author address:* Dr. John W. Bergman, CIRES, Campus Box 449, University of Colorado, Boulder, CO 80309-0449.

Email: jwb@cdc.noaa.gov

^{1}

However, unlike tidal investigations, which are global in extent, large-scale regional differences (e.g., continental vs maritime conditions) are explored here.

^{2}

To facilitate the comparison between the calculated cloud diurnal variations and those observed by satellite, mean temperature *T̄*(*z*) is used as a vertical coordinate.

^{3}

The deep vertical structure of the heating rate for state 1 in Fig. 13 results from the deep vertical structure of large-scale cloud fraction. In contrast, an individual cloud layer results in a relatively shallow vertical structure of the SW heating rate.