Journal of Geophysical Research: Atmospheres

Analytical asymptotic solutions to determine interactions between the planetary boundary layer and the Earth's surface

Authors


Abstract

[1] This study derives an asymptotic analytical solution to calculate land skin temperature, planetary boundary layer (PBL) temperature, and turbulent heat fluxes over arid and wet ground surfaces. Applying the analytical solution to field data, the turbulent heat fluxes and the daytime canopy resistance (which are difficult to measure directly) can be easily determined on the basis of solar radiation and atmospheric radiation measurements, as well as other commonly available meteorological data. The results can be compared with a Lagrangian soil-plant-atmosphere model. Using the derived canopy resistance as input for the Lagrangian model, the simulated PBL temperature has a higher correlation coefficient with the real data than two other canopy schemes. The analytical solution proves that the mean skin temperature is slightly higher than the mean PBL temperature, and its amplitude is always larger than that of PBL temperature in general. An equation for determining the half-life of either PBL temperature or skin temperature is derived. The half-life is the time needed for the initial temperatures of the PBL or a land surface to decay to 50%. Note that the temperatures will alter in response to variations in the meteorological conditions such as changes in solar radiation. Over bare ground, vegetation, snow or ice, the half-lives are ∼8 hours. For a body of water, it is 1.4 months with a thermocline depth of 100 m. Although the algorithm of determining canopy resistance is verified in an urban area in Taiwan, it is likely to be valid worldwide.

1. Introduction

[2] The anomalies of sea surface temperatures over oceans (SSTs) [Gates, 1992] and canopy resistance over land [Walker and Rowntree, 1977; Shukla and Mintz, 1982] are important driving mechanisms for climate anomalies. The SSTs and canopy resistance are the parameters of the lower boundary conditions of the atmosphere. These two parameters are important because they partition the surface turbulent heat fluxes into the sensible heat flux and the latent heat flux, and their lifetimes are relatively long in comparison with a synoptic weather pattern. The lifetimes of SSTs and canopy resistance are of from a few months to as long as a few years, but that of a synoptic weather pattern is only of a few days [Avissar and Verstraete, 1990]. A proper description of these boundary conditions (such as SSTs) can be used by atmospheric models for the present climate simulations [Gates, 1992] such as a reproduction of the well-known El Nino phenomenon [Trenberth, 1997].

[3] The SSTs over oceans have been well monitored by satellites, ships, or buoys [Webster and Lukas, 1992], while the measurements with a global coverage of canopy resistance over land are still problematic. The major difficulties in determining the canopy resistance include the heterogeneity of land covers and the uncertainty of many parameters such as leaf area index, root depths, albedo, roughness, and soil moisture [Priestley and Taylor, 1972; Sellers and Dorman, 1987; Sellers et al., 1989; Parlange and Katul, 1992; Stannard, 1993; Smith et al., 1993; Liang et al., 1994; Franks et al., 1997; Franks and Beven, 1997a, 1997b; Bastidas et al., 1999; Gupta et al., 1999]. Besides, we are still not clear how to relate these parameters with canopy resistance [Sellers et al., 1989; Franks et al., 1997; Franks and Beven, 1997a, 1997b; Bastidas et al., 1999; Gupta et al., 1999; Sen et al., 2001; Tsuang and Tu, 2002]. Their relations can be varied with vegetation and soil types. Furthermore, human interference such as irrigation further complicates the problem [Tsuang and Tu, 2002].

[4] Unlike numerical models [Deardorff, 1978; Dickinson et al., 1986; Sellers et al., 1986, 1996; Tsuang and Wang, 1993; Hostetler et al., 1993; Deutsches Klimarechenzentrum (DKRZ), 1994; Ek and Mahrt, 1994; Segal et al., 1995; Avissar, 1996; Roeckner et al., 1996; Randall et al., 1996; Sakakibara, 1996; Jin et al., 1997; Tsuang and Tu, 2002], this study develops an analytical method to simulate the interactions between land and the atmosphere. Only a few studies have examined such interactions analytically. Carslaw and Jaeger [1959] derived an analytical solution for soil temperature at various depths under the boundary condition that the surface temperature is periodical in time. Otterman [1990] formulated a linear differential equation for extremely arid ground conditions to determine the column-mean potential air temperature of the planetary boundary layer (PBL temperature). This study derives a more general analytical equation set for calculating skin temperature, PBL temperature, and turbulent heat fluxes for both arid and wet ground conditions.

[5] For the sake of simplification, the following conditions are assumed and will be hereafter denoted as the hypothesis of “constant diurnal cycle”: (1) diurnal cycles of incoming solar radiation and atmospheric radiation are constant; (2) variables including aerodynamic resistance ra, canopy resistance rc, specific humidity qa, thermal capacity CA of a PBL, and thermal capacity CG of the ground are constant; (3) the advection term in the PBL is neglected; (4) latent heat released in the PBL is neglected; and (5) sensible heat is completely absorbed within the PBL. Note that Otterman [1990] made similar assumptions.

[6] Under the above-simplified meteorological conditions (constant diurnal cycle), an analytical solution can be obtained to compute skin temperature, PBL temperature, and turbulent heat fluxes. Since these properties become analytical, their relations with canopy resistance can be quantified explicitly, and hence canopy resistance can be determined explicitly if some of these temperatures and fluxes are available. The analytical asymptotic solution for skin/PBL temperatures involves the following procedures: (1) simplification of the closure problem by constructing a one-column PBL-and-land model as shown in section 2; (2) expressing the surface energy budget (SEB) and PBL energy budget (PEB) equations using the mathematical mean value theorem [e.g., Kreyszig, 1993] as shown in sections 3 and 4; (3) rewriting the SEB and PEB equations as an ordinary differential equation (ODE) set by introduction of timescale coefficients and truncation of the high-order terms of SEB and PEB equations; and (4) solving the ODE set analytically.

[7] This hypothesis of “constant diurnal cycle” works only if the amplitude of the daily fluctuation of ra, rc, qa, CA, and CG is small. Although the analytical solution is derived on the basis of the hypothesis, it is applied to a study site in an urban area of a subtropical island in Taichung, Taiwan, where the meteorological conditions are highly variable. To have an annual cycle, the daily means of surface air temperature and mixing height are used for calculating the sensible and the ground heat fluxes at the site. The daily daytime means of aerodynamic resistance and specific humidity are used for calculating the daytime canopy resistance. The errors involved in all the aforementioned simplifications are evaluated by comparison with the results simulated using the hourly meteorological forcings as inputs by a Lagrangian model, which has a more completed formulation of the interactions between the PBL and the land.

2. Model Formulation

[8] An analytical model for the study of the interactions between the planetary boundary layer (PBL) and a land surface is suggested. The model is based on the analysis of the heat balance of two reservoirs (Figure 1), namely the mixing height za of the PBL and the effective depth zg of soil ground (or the thermocline depth of a water body). The effective depth is determined as equation image (see Appendix A for details), where kg is the heat diffusivity of the soil, and ω is the Earth's angular velocity (2π/86400 s−1). In Figure 1, θa represents the potential air temperature of the PBL; the potential temperature is hereafter denoted as θ and is defined as θ ≡ T + (g/ca)z, with T being the temperature at height z, g the gravity acceleration on the Earth's surface, and ca the specific heat of air at constant pressure.

Figure 1.

Schematic of a two-reservoir PBL-and-land model where the shaded areas, namely, the mixing height za of the PBL and the effective depth zg of the ground, are the two energy reservoirs. The PBL-and-land model also can be described by time resistances of τs. Within the PBL the latent heat release is assumed to be negligible, the sensible heat is completely absorbed and there is cooling due to the net loss of longwave radiation ΔRl.

[9] No prescribed deep soil temperature is used to close the energy budget of the atmosphere-land system [DKRZ, 1994]. Assuming that the ground heat flux G is instantaneously and homogeneously distributed throughout the entire effective depth zg, the skin temperature θg is determined as

equation image

where t is time; CG is the heat capacity of the ground layer per unit area (≡ρgcgzg); ρg and cg are the density and the specific heat capacity of the ground, respectively. Arakawa and Mintz [1974] implemented the above equation in the UCLA general circulation model. Tsuang and Yuan [1994] showed that the formula could simulate the mean and the amplitude of the diurnal fluctuation of skin temperature at study sites in Taichung and Ilan, Taiwan.

[10] The ground heat flux G can be expressed through a heat balance equation [e.g., Brutsaert, 1982] as

equation image

where α is the albedo of the surface, Rs is the incoming solar radiation at the surface, Rld is the atmospheric downward longwave radiation, σ is the Stephan-Boltzmann constant, ε is the emissivity of the ground surface (ε ≈ 0.97 [e.g., Brutsaert, 1982]), LE is the latent heat flux, and H is the sensible heat flux. Note that Rld is a function of θa [Swinbank, 1963; Satterlund, 1979; Brutsaert, 1982; Tsuang et al., 1997]. Satterlund's, [1979] equation has been found to be valid for a wide range of air temperature [Brutsaert, 1982] and is used to calculate Rld.

[11] Latent heat flux, LE, and sensible heat flux, H, are parameterized [e.g., Brutsaert, 1982; Shuttleworth, 1989; Garratt, 1992; Sellers et al., 1996] as

equation image
equation image

where ρa is the air density, Lv is the latent heat of vaporization, q*(θg) is the saturated specific humidity at temperature θg, qa is the specific humidity of air at a reference height, ra is the aerodynamic resistance (s m−1), and rc is the canopy resistance (s m−1). q*(θg) is here expressed as in the work of Richards [1971].

[12] In the formulation of the heat balance for the PBL it can be assumed that the energy exchange between the mixed layer and the air above the mixed layer is very slow because the heat diffusivity at the interface above the mixed layer is very small. Moreover, the condensation process, which is accompanied by latent heat release, occurs only when the air temperature is equal to or lower than its dew point. For an unsaturated air parcel in vertical motion, condensation occurs at a height higher than the lifting condensation level (LCL) [e.g., Ahrens, 1994]. Thus if the LCL is higher than the PBL mixing height, the latent heat release can be neglected in the energy conservation equation for the PBL. Therefore the temporal variability of potential PBL temperature θa is determined by (1) neglecting the latent heat release in the mixed layer, (2) assuming that sensible heat is completely absorbed within the mixed layer, (3) neglecting the advection term in the PBL heat balance, (4) recognizing that the rate of temperature change is approximately equal to the rate of potential temperature change [Wallace and Hobbs, 1977], and (5) assuming that sensible heat flux is instantaneously and homogeneously distributed throughout the entire PBL; hence the heat balance of the mixing height can be written [Stull, 1973] as

equation image

where CA is the heat capacity per unit area of the PBL (≡ρacaza). The second term on the right hand side is a cooling term due to a net release of longwave radiation in the mixed layer, where τlw is the timescale of the radiation cooling (= 5.08 × 106 s = 1,411 hours), and θlw is the reference temperature of the cooling (= 167 K + (g/ca)z) [Paltridge and Platt, 1976]. This relation is equivalent to a temperature of 0°C yielding a cooling rate of 1.8 K day−1. Kuo [1979], using a more complete parameterization of longwave radiation, reported that for clear air with temperature and moisture profiles approximating that of the standard atmosphere [e.g., Ahrens, 1994] the infrared cooling rate is about 1.2 K day−1 [Pielke, 1984]. In what follows, an analytical solution of equations (1)(5) is provided, on the basis of the assumptions outlined in the introduction.

3. Analysis of the Energy Budget in the Ground Surface

[13] The surface energy budget (SEB) equation (1) can be decomposed into unsteady-state and steady-state equations by truncating the high-order terms of equations (B10) and (B12) as (see Appendix B for details)

equation image

and

equation image

where τgg is the life timescale (s) of skin temperature of a stand-alone land system and τag is the timescale (s) for energy transferring from a PBL reservoir to a land reservoir. The daily mean potential skin temperature and the daily mean potential PBL temperature are denoted as equation image and equation image, respectively, while their initial values are denoted as θg0 and θa0. In response to variations in the meteorological conditions (such as changes in solar radiation), equation image and equation image will change and eventually reach a steady state if the forcing is of a constant diurnal cycle. These steady-state values are denoted as equation image and equation image, respectively, that is, the superscript “=” represents the daily mean at steady state under the constant diurnal forcing. equation image is a temperature at which the mean theorem applies (i.e., equation image and equation image). Note that the half-life of θg0 can be determined as 0.693τgg if the PBL temperature θa is set at equation image by solving equation (6) analytically. Og and Oa are oscillation energies (W/m2) due to the diurnal oscillations of skin temperature and PBL temperature, respectively. equation image is the daily mean of equation image at steady state, and equation image. equation image and equation image are determined according to the mean value theorem for Gg, θa) as described in Appendix B. equation image and equation image are the standard deviations of θg and θa at steady state. Equation (7) describes the energy components of SEB at steady state, showing that at steady state the mean ground heat flux equation image approaches zero. According to the definitions of τgg, τag, Og, and Oa in equations (B14), (B13), (B8), and (B9), and inserting equations (2)(4) into them, they can be rewritten as

equation image
equation image
equation image
equation image

According to the definitions, τgg and τag are always positive and increase with CG. equation image and equation image can be determined as (see Appendix C for details)

equation image

where Sθ is the standard deviation of θ. n is the dominant exponent term of variable θ of function G. It is close to 17.9 for equation image because of the q*(θg) term and close to 4 for equation image because of the atmospheric radiation term of Rlda).

4. Analysis of the Energy Budget in the PBL

[14] The mean PBL energy budget equation (PEB) (equation (5)) under steady state conditions can be expressed as

equation image

If the thermal capacity CA of the PBL is constant, then the mean sensible heat flux at steady state can be written according to the above equation as

equation image

That is, the mean sensible heat flux is equal to the thermal radiation cooling flux in the PBL at steady state. The above equation yields an equation image of around 9 W m−2 when za = 300 m and equation image = 288 K. Substituting equation (4) into equation (14), the PEB equation (5) can be rewritten as

equation image

with

equation image

where τga is the timescale (s) for energy transferring from a land reservoir to a PBL reservoir and τaa is the life timescale (s) of PBL temperature of a stand-alone PBL system. Note that the half-life of θa0 can be determined as 0.693τaa if the skin temperature θg is set at equation image by solving the above equation (15) analytically. These timescales are defined as

equation image
equation image

Equation (16) shows that equation image is close to equation image as τga → 0 and equation image is close to θlw as τga → ∞. Putting equation (18) into equation (16) gives

equation image

This equation shows that the mean skin temperature is slightly higher than the mean PBL temperature at steady state since equation image is usually much larger than θlw (= 167 K) and τga and τlw are always positive.

5. Temperature Equations

[15] It is possible to use the Fourier series to analyze the solar radiation under a constant diurnal cycle in the frequency domain as

equation image

where equation image is the mean incoming solar radiation; rk is its amplitude of oscillation of 24/k-h period; trkm is the local time when the highest solar radiation occurs in that period; and th is local time. After truncating the high-order terms during the transient period, the heat conservation equations in the ground (equation (6)) and in the PBL (equation (15)) become

equation image

6. Analytical Solution

[16] Equation (21) is an ordinary differential equation set. Its analytical solution is

equation image
equation image

where θg0 and θa0 are the initial temperatures at local time th0. In addition, τs, τp, and τm are timescales (s), and a is a dimensionless value. They are defined as

equation image
equation image
equation image
equation image

The steady-state diurnal temperature functions θ*g(t) and θ*a(t) are determined to be

equation image
equation image

where equation image and equation image are amplitudes of skin and PBL temperature fluctuations of 24/k-h period at steady state, respectively; th0 is the local time at t = 0; tgkl and takl are the local times of day at which the highest skin and PBL temperatures of 24/k-h period occur, respectively; and tgc and tac are the terms for correcting the time lags inherent in the parameterization of equations (1) and (5). tgc has been determined to be 3 hours according to equation (A11), and the same value is suggested for tac. Note that according to the above equation the standard deviations of skin temperature and PBL temperature can be determined as

equation image

because of the orthogonal property of a cosine function [Kreyszig, 1993]. The analytical solutions of equation image, tgkl, equation image, and takl are found to be

equation image
equation image
equation image
equation image

where

equation image
equation image
equation image
equation image
equation image

Equations (22a) and (22b) include two exponential terms; since τm > τp, exp(−tm) ≫ exp (−tp). In addition, if t = 0.693τm, then the coefficients of θg0 and θa0 approximate 0.5. That is, the residues of θg0 and θa0 are only 50% after time 0.693τm Therefore we can define th ≡ 0.693τm as the half-life for both θg and θa of the coupled PBL-land system.

[17] Moreover, combining equations (25a) and (26a) yields

equation image

This proves that the amplitude of skin temperature is always higher than that of PBL temperature. The ratio increases with raza (or τga).

7. Typical Timescales

[18] Typical timescales over land under conditions of aerodynamic resistance (ra) = 25 s m−1, canopy resistance (rc) = 100 s m−1, equation image = 288 K, CG = 2.05 × 105 J m−2 K−1, za = 300 m, and ∂Rld/∂θa calculated from the Satterlund [1979] equation with a fractional cloud cover (mc) set at 30% are as follows: τag, 1.02 hours; τgg, 0.78 hours; τga, 2.08 hours; τaa, 2.08 hours; τs, 0.63 hours; τp, 0.60 hours; τm, 11.5 hours; and a, −0.5 according to equations (8)(9), (17)(18), and (23a)(23d). Note that ra and rc are typical values for a forest with wind speed of 3 m s−1 at 10 m height above the canopy [Brutsaert, 1982; Shuttleworth, 1989; Garratt, 1992]; CG is a typical value for a ground surface with soil moisture in the range between its wilting point and the field capacity of most soil types (Table 2 and Tsuang and Yuan [1994]). These timescales show that the half-life of a stand-alone land system over a soil surface is only 0.5 hours (= 0.693τgg), whereas that of a stand-alone PBL system is 1.4 hours (= 0.693τaa), but increases to 8.0 hours (= 0.693τm) for a coupled PBL-land system. Moreover, under the conditions, equation (19) shows that the mean skin temperature is slightly higher than the mean PBL temperature by about 0.2 K at steady state. Jin et al. [1997] also found that skin temperature is very close to surface air temperature after comparing a global data set of the two temperatures.

8. Hypothetical Case

[19] First, we verify the derived analytical asymptotic equations using “synthetic” data generated through the numerical solution of the model presented in the model formulation section, but with the mixing height being allowed to vary throughout the day and without using the time lag correction terms in equations (24a) and (24b) (i.e., setting tgc = tac = 0).

[20] A constant diurnal cycle of incoming solar radiation data Rs at the surface is generated on Julian date 265 (22 September) (autumn equinox) at a latitude of 45°N according to [Brutsaert, 1982]

equation image

where Rse is the solar radiation at the top of the atmosphere, and mc is the fraction of cloud cover (%). Rse is determined as [Oke, 1987]

equation image

where S is the solar constant (= 1367 W m−2) [Lean, 1989]. z is the solar zenith angle function, which varies with hour angle (ϕ), latitude (θ), and the Earth's declination angle (δ) as

equation image

where ϕ is zero at noon.

[21] Atmospheric longwave radiation data are generated as a function of emissivity, air temperature, and fraction of cloud cover [Satterlund, 1979]. Albedo is set at 0.2, while the fraction of cloud cover is set at 30%. Atmospheric pressure is set at the standard atmospheric pressure of 1013.23 mb. Specific humidity is fixed at 0.0062 kg kg−1. ra is set at 25 s m−1. rc is set at 100 s m−1. Note that the albedo is a typical value over land [Brutsaert, 1982].

[22] Mixing height according to Holzworth [1972] or Wallace and Hobbs [1977] can be determined as

equation image

where ∂θ/∂zinv and ∂T/∂zinv are the gradients of the potential temperature and air temperature above the mixed layer (K m−1), respectively. In addition, a minimum value of mixing height hmin is set and the above equation becomes

equation image

This case study sets hmin at 150 m, and fixes ∂T/∂zinv at 0.01 K m−1, a value measured in the early morning at 400 m height at O'Neill, Nebraska, on 25 August 1953 [Stull, 1973].

[23] Equations (1), (5), and (32) are ordinary differential equations (ODE). The Runge-Kutta method with time steps automatically chosen to meet an accuracy of six digits is used to solve these three equations simultaneously. In addition, a relatively high temperature of 27°C (300 K) is arbitrarily chosen as the initial temperature for both skin and PBL temperatures, and the initial mixing height is set to be hmin at local time 0 AM. Figure 2a shows the results of the numerical simulation for skin temperature, PBL temperature, latent heat flux, and sensible heat flux over a ground of various area heat capacities, CG.

Figure 2.

Skin temperature (θg), PBL temperature (θa), latent heat flux (LE) and sensible heat flux (H) simulated by the Runge-Kutta method (Figure 2a) and by the analytical equations (Figure 2b) over a ground with heat capacity per unit area CG (shown in the legend) in the range of 103–109 J m−2 K−1.

[24] The same results as Figure 2a are shown in Figure 2b, but they are calculated by the derived analytical equations using the following seven steps. First, we use the initial skin temperature, the initial PBL temperature and the initial mixing height to determine τaa, τga, τgg, and τag. Second, we use equations (25a) and (26a) to determine Δθgk and Δθak. Note that only the first two frequency components of solar radiation of 1 day−1 (r1) and 2 day−1 (r2) are used. Third, we use the Newton iteration method to solve Equations (7), (12), and (16) simultaneously to determine equation image, and equation image. Fourth, we again calculate τgg and τag. Fifth, we use equations (22a), (22b), and (33) to determine hourly θg, θa, and za for the next day. Sixth, we use the mean of za to calculate τga and τaa. Finally, we repeat steps 2 through 6 until the last day of simulation. The intensity of the frequency spectrum of solar radiation is determined by using the Fourier transform on equation (29), where trkm is determined to be hour 12 (noon) local time and rk of equation (20) is determined to be

equation image

There is a singularity in equation (34a) at k = 1. For this condition, r1 is determined to be

equation image

where ϕm is the hour angle at sunset, determined to be

equation image

Note ϕm = 0 occurs in the poles in their wintertime (perpetual night), and ϕm = π occurs in the poles in their summertime (perpetual day).

[25] A comparison between Figures 2a and 2b shows that their patterns are almost identical. Minor differences are shown in the first two days of simulation because of the truncation of high-order terms, and in the middle of night, when the analytical solution of the sensible heat flux is not smooth. This behavior is due to the limitation to only two frequency components of solar radiation in the analytical equations. More frequency components would smooth the plot. Nonetheless, part of the diurnal variations and the mean of temperatures are well matched both during the first two days and after the first two days in these two figures. Table 1 shows some statistics of the comparison. The bias of the mean skin temperature simulated by the Runge-Kutta (RK) method and by the analytical equations (Asy) is 0.14 K. The bias of the mean PBL temperature between the two methods is 0.06 K. The correlation coefficients between the two methods of skin temperature, PBL temperature, and latent heat flux are 1.00. The correlation coefficients of the mixing height, sensible heat flux and ground heat flux predictions are more than 0.93. This shows that the derived analytical equations can describe the behavior of the two-reservoir model.

Table 1. Comparison Between Skin Temperature θg, PBL Temperature θa, Latent Heat Flux LE, Sensible Heat Flux H, Ground Heat Flux G, and Mixing Height za Calculated by the Runge-Kutta Method and the Analytical Equationsa
 BiasRMSEr2
  • a

    Here, RK, Runge-Kutta method; Asy, analytical equations; RMSE, root-mean-square error; r2, correlation coefficient.

Asy θg − RK θg, K−0.140.681.00
Asy θa − RK θa, K−0.060.651.00
Asy LE − RK LE, W m−2−1.989.751.00
Asy H − RK H, W m−2−4.1829.240.93
Asy G − RK G, W m−26.6448.500.97
Asy za − RK za, m−5.5729.840.98

[26] Figure 2 also shows the sensitivities of skin temperatures and variations in PBL temperatures in response to changes in heat capacity of a ground CG, varying from 103 to 109 J m−2 K−1. No significant dependence on CG is found in the calculated skin temperatures or in the PBL temperatures when CG is <105 J m−2 K−1. Then, Δθg and Δθa decrease with CG when CG is larger than 105 J m−2 K−1. Note that the heat capacity CA of the PBL is equal to ρacaza, which approximates to 3 × 105 J m−2 K−1 when the mixing height za is 300 m. That is, when CG is <3 × 105 J m−2 K−1, CA dominates the magnitude of the diurnal fluctuations of both skin temperature and PBL temperature. Figure 3 shows τgg, τaa, and τm of the PBL-and-land system as functions of CG. It shows that τm is determined by τaa for CG < 106 J m−2 K−1 and is determined by τgg for CG larger than 106 J m−2 K−1.

Figure 3.

τgg, τaa, and τm as functions of the heat capacity per unit area of a ground CG.

[27] Table 2 shows CG of various ground types. As can be seen, for snow, soil and ice, the values of CG are <106 J m−2 K−1. However, for a water body with a mixing depth (the thickness of its thermocline) thicker than 0.24 m, the value of CG of the water body is larger than 106 J m−2 K−1. From this table we can conclude that for land surfaces (CG < 106 J m−2 K−1), the half-life is about 8 hours. The half-life then increases to 11 hours for a water body with a thermocline depth of 1 m, 4 days with the depth increasing to 10 m, 1.4 months with a depth of 100 m, and 1.1 years with a depth of 1000 m. Sensitivity analysis by varying meteorological variables is also listed in the table, showing that the half-life generally increases with ra, rc, mc, za, and equation image but decreases with equation image. The half-life is more sensitive to ra, rc, and za but is less sensitive to other meteorological variables.

Table 2. Properties and Sensitivity of Half-Lives tH of Various Land Typesa
 θ,b %ρgcg,c J m−3 K−1kg,d × 10−6 m2 s−1zg, mCG, J m−2 K−1Org2 × ra2 × rctH (h), mc = 02 × zaequation imageequation image
  • a

    Here, θ = water content; ρgcg = volumetric heat capacity; kg = heat diffusivity; zg = effective depth of ground (or thermocline depth of water body); CG = heat capacity per unit area; and tH = half-life of skin and PBL temperatures at 288 K. The default tH (org) is determined under conditions of aerodynamic resistance (ra) = 25 s m−1, equation image = 288 K, CG = 2.05 × 105 J m−2 K−1, za = 300 m, and ∂Rld/∂θa calculated from the Satterlund [1979] equation with a fraction of cloud cover (mc) set at 30%. The canopy resistance (rc) is set at 100 s m−1 over land and at 0 s m−1 for water.

  • b

    Dickinson et al. [1986].

  • c

    Hillel [1982].

  • d

    de Vries [1975].

Fresh Snow0.210.100.047.79 × 103691061265
Old Snow1.000.400.077.42 × 1047101171375
Dry Sand01.340.960.121.54 × 1058101281486
Sand at Wilting Point101.760.940.111.99 × 1058111381486
Sand at Field Capacity121.840.930.112.08 × 1058111381486
Wet Sand332.720.870.112.97 × 1059121591597
Dry Loam01.100.620.091.02 × 1057101171376
Loam at Wilting Point302.360.570.092.08 × 1058111381486
Loam at Field Capacity372.650.550.092.31 × 1058111481487
Wet Loam452.990.540.092.57 × 1059121491597
Dry Clay00.740.430.085.66 × 104791171375
Clay at Wilting Point522.920.360.072.05 × 1058111381486
Clay at Field Capacity573.130.350.072.17 × 1058111381486
Wet Clay633.380.340.072.32 × 1058111481487
Dry Peat00.680.400.075.03 × 104791171375
Peat at Wilting Point542.940.330.071.99 × 1058111381486
Peat at Field Capacity593.150.320.072.11 × 1058111381486
Wet Peat663.450.320.072.27 × 1058111481487
Ice1.931.200.132.48 × 1058111481497
Rock2.031.400.142.82 × 1059121591597
Concrete2.022.300.183.59 × 1051013161016107
Water4.1914.19 × 1061122111112118
Water4.19104.19 × 1079919699991009975
Water4.191004.19 × 1089791942979978979982742
Water4.1910004.19 × 10997841940297849774977598157412

9. Taichung Case Study

[28] This section applies the derived asymptotic equations to Taichung City, Taiwan, (24°09′N, 120°41′E) to calculate the skin temperature, turbulent heat fluxes and the daytime canopy resistance from 21 January 1996 to 19 October 1996. Conventionally, it has been very difficult to obtain canopy resistance because of high uncertainty in measuring latent heat flux and sensible heat flux [Winter, 1981]. This case study determines the surface energy fluxes by taking advantage of the derived asymptotic equations using solar radiation, atmospheric radiation and normal meteorological data as input. However, the asymptotic equations are derived under simplified conditions, so the errors involved in the approximations will be evaluated by comparison with a Lagrangian model, which does not have the simplifications.

[29] First, we can rewrite equation (2) to determine the daily latent heat flux as

equation image

where the daily sensible heat flux equation image and the daily ground heat flux equation image can be obtained by taking the means of equations (5) and (1), respectively, as

equation image
equation image

where CG is set at 2.05 × 105 J m−2 K−1, the typical value for a ground surface with soil moisture in the range between its wilting point and the field capacity of most soil types. In equation (37) assuming CA to be constant derives the right hand side, and CA is approximated by its daily mean. The above three equations only need net solar radiation, atmospheric radiation, PBL temperature, skin temperature, and mixing height as inputs.

[30] Next, we calculate skin temperature and mixing height as functions of PBL temperature. Note that PBL temperature is very close to surface air temperature, which is commonly measured in normal meteorological stations. For simplicity, the observed daily diurnal fluctuation of PBL temperature (surface air temperature) is approximated by a single frequency of 1/24 h−1 as

equation image

where tam is the local time when the highest PBL temperature occurs; equation image is daily mean PBL temperature. And Δθa is approximately equal to (θamax − θamin)/2, where θamax and θamin are the daily maximum and minimum PBL temperatures, respectively. Using the same approximation as the PBL temperature, the skin temperature can be also written as

equation image

where tgm is the local time when the highest skin temperature occurs. Assuming a steady state, equation image and equation image. In addition, equation image can be determined from equation image by equation (19), and equation image can be determined from equation image according to equation (28). Nonetheless, it is not suggested to use equations (25b) and (26b) for calculating tgm from tam because a time lag is inherent in the parameterization of equations (1) and (5). tgm is set at 1 PM according to observations [Tsuang and Yuan, 1994]. Although assuming a steady state, the half-lives of θg and θa over a land surface are only about 8 hours according to the previous hypothetical case study. Therefore the PBL and land system can reach a new steady state in 2–3 days with new meteorological conditions. Hence the derived skin temperature should be close to its true value except during the time when meteorological conditions are changing significantly.

[31] The daily mean mixing height can be determined by integrating the Holzworth equation (equation (32)), using equation (39) for θa as

equation image

As a result, both skin temperature and mixing height can be obtained if PBL temperature is measured. Consequently, the daily latent heat flux can be determined if the net solar radiation and atmospheric radiation are available as well.

[32] The derived daily latent heat flux can be used to determine its canopy resistance as follows. First, taking the daily mean of equation (3) it becomes

equation image

However, the canopy resistance in a day is not a constant, and its typical diurnal pattern is a “U” shape with the minimum value present for a considerable portion of the daylight hours [Blondin, 1988; Segal et al., 1995]. It approaches infinity during the night since the stomata of the leaves open during the day and close during the night. If we assume that those evapotranspiration processes only occur during the day, then the daytime canopy resistance, rcd, can be expressed by rewriting the above equation as

equation image

where rad is the daytime representative aerodynamic resistance; Φ is the ratio of daylight hours in a day; the brackets represent the daytime average; krc is a correction factor since the rcd derived involves many approximations. Later in this paper krc will be determined using a technique of an inverse approach. We refer to the canopy resistance derived from the above equation as “visible” daytime canopy resistance since it can be easily calculated using solar radiation, atmospheric radiation, PBL temperature, aerodynamic resistance and humidity without the need of a well instrumental latent heat flux measurement. Note that correct measurement of latent heat flux is very difficult [Winter, 1981].

[33] Hourly values of solar radiation Rs, atmospheric radiation Rld and terrestrial radiation Rlu were measured in the study area at National Chung-Hsing University of Taichung, Taiwan, by the author. Atmospheric pressure p, wind speed u and direction, humidity ea and air temperature θa and rainfall rate ppt were measured at a weather station 2 km from the study site, by the Central Weather Bureau, Taiwan (Figure 4). Two separated infrared radiometers (Eppley model PIR), one facing upward and the other facing downward, measured the atmospheric radiation and terrestrial radiation. The net solar radiation was measured by two pyranometers (Eppley model PSP), one of them measuring downward solar radiation and the other measuring the reflected solar radiation. The same instrumentation has been applied in other studies [Tsuang and Yuan, 1994].

Figure 4.

Daily incoming solar radiation (Rs), atmospheric radiation (Rld), atmospheric pressure (p), and surface air temperature (Ta) measured in the study area in Taichung, Taiwan, in 1996.

[34] Although the terrestrial longwave radiation and the albedo were measured, they are not used since they usually cannot represent the areal average of a heterogeneous terrain. Note that terrestrial longwave radiation and albedo are usually very site dependent. In this study, they were measured over a concrete roof surface in the university campus. In Taichung, there are many other land covers, such as trees, of which the albedo is very different from a concrete surface. Alternatively, the albedo in equation (36) is set at 0.2 since this value can represent many types of soils and vegetation [Brutsaert, 1982]. The daily longwave terrestrial radiation and skin temperature are determined from PBL temperature as presented previously since PBL temperature is less site dependent and is more commonly available from normal meteorological stations. Then, the daily longwave terrestrial radiation can be determined using the blackbody radiation equation with the emissivity of 0.97 (Rlu = εσθg4) [Brutsaert, 1982]. The lapse rate at the top of the mixing height is set at −0.0065 K m−1 (wet adiabatic lapse rate), a mean value measured at a nearby site [Chen et al., 2002]. Daytime aerodynamic resistances rad is calculated according to Businger et al. [1971] using daily daytime mean wind speeds under neutral stability conditions.

[35] Figure 5 shows the canopy resistance of the Taichung study site derived by equation (43) as krc = 1. Note that the data of rcd are discarded on the first day of a rain event and when the daily atmosphere pressure changed over 4 hPa. Under the above two criteria the meteorological conditions change significantly on the day and the previous day (Figure 4). Hence the assumptions of steady state for calculating skin temperature are not valid. This shows that there were sudden drops of canopy resistance from a harmonic mean of 129 s m−1 (arithmetic mean of 163 s m−1) during rainless periods to a harmonic mean of 85 s m−1 (arithmetic mean of 97 s m−1) during rainy periods, and in the wintertime, the canopy resistance was much larger than in the summertime. For example, in August the dates of decreases in the canopy resistance corresponded with periods of rain. During rainy periods, canopy resistance decreases because there is more water available for evaporation residing on skin reservoirs over soils, buildings or vegetation [e.g., Dunne and Leopold, 1978; Mintz and Walker, 1993; Tsuang and Tu, 2002]. In addition, there is more foliage during summer and less foliage during winter. Hence stomatal resistance decreases during the summer and increases during the winter. Other methods of determining canopy resistance are given by Sellers and Dorman [1987], Dorman and Sellers [1989], Sellers et al. [1989], Smith et al. [1993], and Liang et al. [1994].

Figure 5.

Observed precipitation rate (ppt) and daytime canopy resistance (rc) for Taichung in 1996 derived by equation (43) as krc = 1.

[36] It is difficult to verify the derived canopy resistance since no observation data are available. This study verifies the derived daily fluctuations of the canopy resistance by putting them into a Lagrangian column meteorological model to simulate air/soil temperatures and surface fluxes. The observed hourly solar radiation, atmospheric radiation, atmospheric pressure, wind speed and direction, humidity and rainfall rate are used as inputs. This Lagrangian model is a modified version of Tsuang and Tu [2002]. Its governing equations are almost identical to those listed in the model formulation section, except that it has 4 numerical levels in the soil and 16 levels in the atmosphere. Three canopy schemes are compared. The first one (denoted “visible”) sets the canopy resistance at that calculated by equation (43) for the daytime, where the missing values are filled with the harmonic means of the rainy or rainless periods depending on precipitation. During the night the canopy resistance is set at (a/b + c)/Lt, where a = 5000 J m−3, b = 10 W m−2, c = 114 s m−1, and Lt is leaf area index [Blondin, 1988; Tsuang and Tu, 2002]. The second canopy scheme (denoted “fixed”) is the same as the first, except that a fixed value is used for the daytime. The third scheme is simulated by the best combination of parameterizations suggested by Tsuang and Tu [2002] (denoted “optimum”). The “optimum” scheme uses Blondin's [1988] parameterization for the transpiration from vegetation, uses a parameterization after DKRZ [1994] for the evaporation from skin reservoirs after rain events, and uses a soil moisture-availability function by Holtan et al. [1975] further adjusted by a water table depth to determine the minimum effective water content during dry periods.

[37] Parameters of krc in the “visible” scheme in equation (43), the daytime canopy resistance rcd in the “fixed” scheme, and the initial (wini) and the minimum (wmin) effective water contents in the “optimum” scheme are identified using the Levenberg-Marquardt method by minimizing the root-mean square error (RMSE) between calculated and observed surface air temperature [Tsuang and Tu, 2002]. In the “visible” scheme, krc is identified to be 1.006, which is close to 1. Therefore the correction is not important. In the “fixed” scheme, rcd is identified to be 103 s m−1. In the “optimum” scheme, wmin is identified to be 88 mm and wini 106 mm where the maximum value (wmax) is set at 146 mm, a value identified by Tsuang and Tu [2002] for bare ground fraction. Note that 100% of land is classed as bare fraction by Hagemann et al. [1999] at the study site within a 5 km × 5 km square region.

[38] The results are shown in Table 3 and Figure 6. In addition, the results of the “visible” scheme without correction (i.e., krc = 1) are also shown in the table. As can be seen, using rcd derived by the “visible” schemes either with or without the correction provides the lowest RMSE among the three schemes, thus justifying the usage of equation (43) for determining canopy resistance. That equation decreases the RMSE by 0.44 K from the “fixed” scheme, and by 0.20 K from the “optimum” scheme. The “visible” scheme is significantly improved from the “fixed” scheme, and it is slightly better than the “optimum” scheme. The “visible” scheme has a better description in daily fluctuations. The “optimum” scheme can correlate these fluctuations with soil moisture availability, photosynthetically active radiation (PAR) and leaf area index. However, since the “visible” scheme has a better correlation than the “optimum” scheme, there might be a factor that was overlooked by the “optimum” scheme.

Figure 6.

Comparison of calculated daily surface air temperature among schemes of (1) uncorrected visible daytime canopy resistance (visible Rc) (top panel), (2) fixed daytime canopy resistance (fixed Rc) (middle panel), and (3) optimum canopy scheme (optimum Rc) (bottom panel) by a Lagrangian model.

Table 3. Statistics of Simulated Hourly Surface Air Temperature in Taichung From 21 January to 19 October 1996 by Four Canopy Resistance Schemes Using a Lagrangian Modela
SchemesObserved, KCalculated, KBias, KMAE, KRMSE, KCorrelation Coefficient (r2)Note
  • a

    Here, MAE, mean absolute error; RMSE, root-mean-square error.

1. Uncorrected Visible Rc297.35297.05−0.301.151.580.96krc = 1.
2. Corrected Visible Rc297.35297.07−0.291.141.570.96krc = 1.006
3. Fixed Rc297.35296.70−0.651.532.010.94rcd = 103.1 s m−1
4. Optimum Rc297.35297.15−0.211.281.770.95wini = 106 mm
wmin = 88 mm
wmax = 146 mm

[39] Figure 7 shows comparisons of the daily latent heat flux, the sensible heat flux, and the ground heat flux between those calculated by equations (36)(38) and those calculated by the Lagrangian model with the uncorrected visible canopy resistance scheme. The biases of these heat fluxes are all within 10 W m−2 and their RMSEs are within 20 W m−2. The latent heat flux has the highest correlation of 0.94, followed by the sensible heat flux of 0.80, and then the ground heat flux of 0.64.

Figure 7.

Comparison of the daily latent heat flux (LE), the sensible heat flux (H), and the ground heat flux (G) between those calculated by equations (36)(38) (denoted “asy”) and those by a Lagrangian model with the uncorrected visible Rc scheme (denoted “num”).

10. Conclusions

[40] Analytical equations (7), (16), (25a)(25b), and (26a)(26b) for calculating skin temperature, PBL temperature, and turbulent fluxes over various ground surfaces are derived. Using “synthetic” data generated under the assumptions for deriving the analytical equations, the solutions are found to be almost identical to those calculated by a numerical solution of the same equations. However, because of the simplifications required by the analytical approach, the error is evaluated by applying it to field data. It shows that the method is valid most of the time except on the first day of a rain event and when the daily atmosphere pressure changed over 4 hPa. Under the above two criteria the meteorological conditions change significantly on the day and the previous day. Hence the assumptions of constant aerodynamic resistance and specific humidity for calculating skin temperature are not valid. The analytical results show some interesting phenomena of the interactions between the PBL and the Earth's surface, which are difficult to prove numerically:

[41] 1. Typical values of half-lives of PBL/skin temperatures are listed in Table 2. Over bare ground, vegetation, snow or ice, the half-lives of skin/PBL temperatures are ∼8 hours, whereas the mean and the amplitude of the diurnal fluctuation of skin temperature are higher than those of PBL temperature, as shown by equation (19) and equation (28), respectively. It shows that τm is determined by τaa for CG < 106 J m−2 K−1 and is determined by τgg for CG > 106 J m−2 K−1. Over these land covers, the half-lives of the temperatures are determined by τaa.

[42] 2. Over a water body with a thermocline depth of 10 m, the half-life is 4 days, and the half-life increases to 1.4 months as the thermocline depth increases to 100 m. The half-lives are determined by τgg over a water body. Because of the long half-life, for a numerical weather forecast using a spinning-up time of one to three days, a prescribed skin temperature over the ocean is needed, but not over the land. For a climate run using a coupled ocean-and-atmosphere general circulation model, a spinning-up time of a few months is needed to adjust ocean temperature down to 100 m in depth.

[43] 3. At steady state the magnitude of sensible heat flux is approximately equal to the thermal radiation cooling flux in the PBL according to equation (14). Its daily flux can be determined by further accounting the heat storage in the PBL according to equation (37). The flux can be determined if data for air temperature and mixing height are available. In comparison with a multiple-layer Lagrangian model, the RMSE is 14 W m−2 with a correlation coefficient r2 of 0.80.

[44] 4. At steady state the magnitude of ground heat flux equals zero according to equation (7). The daily flux can be determined by further accounting the heat storage in the ground according to equation (38). In comparison with a multiple-layer Lagrangian model, the RMSE is 6 W m−2 with a correlation coefficient r2 of 0.64.

[45] 5. The daily latent heat flux can be determined according to equation (36). The flux can be determined if extra data of solar radiation and atmospheric radiation are available. In comparison with a multiple-layer Lagrangian model, the RMSE is 14 W m−2 with a correlation coefficient r2 of 0.94.

[46] 6. Daytime canopy resistance over land can be easily obtained according to equation (43) if extra data for specific humidity and aerodynamic resistance are available. Using the canopy resistance provides the highest correlation and the lowest RMSE in simulating surface air temperature by a multiple-layer Lagrangian model in comparison with two other methods of determining the canopy resistance. The RMSE of the hour surface air temperature is 1.6 K with a correlation coefficient r2 of 0.96.

[47] Although the algorithm of determining canopy resistance is verified in an urban area in Taiwan, it has a higher potential to be valid worldwide. If such a global canopy-resistance data set becomes available, the boundary conditions over land can be well described and atmospheric models should be able to conduct a better climate simulation.

Appendix A:: Effective Skin Layer Thickness

[48] A single soil layer of thickness zg is used for the simulation of temporal variability of soil skin temperature, θg, as shown in Figure 1. This appendix determines an expression of the effective heat thickness, zg, of the soil layer in order to capture the amplitude of the diurnal fluctuation of skin temperature.

[49] Considering an ideal surface where the heat diffusion coefficient of soil, kg, is constant, the heat transfer in the ground can be described as

equation image

where cg is specific heat of soil, ρg is soil density, G is heat flux in the ground. z and G are both positive upward. If the temperature at the ground surface can be described by a cosine function, with the highest temperature occurring at time tm as

equation image

where equation image is average skin temperature, Δθg is the amplitude of the diurnal fluctuation of skin temperature fluctuation and t is local time, the analytical solution of temperature profile case can be determined [Carslaw and Jaeger, 1959; Pielke, 1984] as

equation image

[50] Integrating equation (A1) in the region from the infinite depth, z = −∞, to the surface yields

equation image

where G0 is heat flux at the surface with positive sign in the downward direction. Note that the heat flux at the infinite depth is set at zero. The left hand side of equation (A4) can be rewritten by combining equation (A3) with equation (A4) as

equation image

Comparing equation (A5) with equation (A4), we obtain

equation image

In addition, using equation (A2) we obtain

equation image

Therefore ∂θg/∂t can be expressed by combining the above two equations and eliminating the Δθg term as

equation image

However, the above equation has a singularity at

equation image

and therefore it is difficult to implement in general circulation models [Arakawa and Mintz, 1974] for calculating θg using a finite difference method scheme with a finite time step of Δt. Neglect the phase-lag term

equation image

the above equation can be rewritten as

equation image

where the effective heat thickness zg of a soil layer is determined as

equation image

[51] Putting equation (A6) into equation (A9), θg can be determined as

equation image

A comparison with equation (A2) suggests that equation (A9) is able to provide values of skin temperature with the correct amplitude of diurnal fluctuation, but with a time lag of 3 hours.

Appendix B:: Skin Temperature Equation

[52] This section will describe methods of decomposing the SEB equation (1) into a steady-steady equation and an unsteady-state equation. These two equations can be used to determine the skin temperature at the two states. Both equations are derived to be functions of equation image and equation image on the basis of the assumption of a constant diurnal cycle. Using the Taylor's series expansion about equation image and equation image, the heat flux Gg, θa) can be determined as

equation image

where H.O.T. are higher-order terms. According to the mean value theorem, there is a skin temperature equation image and a PBL temperature equation image, which simplify the above equation as

equation image

Note that equation image and equation image are also functions of time. Using the Taylor's series expansion about equation image and equation image, where equation image and equation image are the daily mean temperatures of equation image and equation image under a constant diurnal cycle, ∂G/∂θg and ∂G/∂θa can be written as

equation image
equation image

In addition, θg, θa, equation image and equation image can be written as equation image, and equation image, where θ′g, θ′a, equation image and equation image are the deviation terms. Substitute the above equations (B2)(B4) into equation (1). Equation (1) becomes

equation image

Take the daily mean of the above equation. The daily mean temperature becomes

equation image

Note that equation image. Under a constant diurnal cycle, at steady state the daily mean temperature does not change with time, that is,

equation image

In addition, according to equation (C18) in Appendix C, equation image and equation image. Therefore at steady state the above equation can be rewritten as

equation image

with

equation image
equation image

where Og and Oa are oscillation energies due to the diurnal oscillations of skin temperature and PBL temperature, respectively. Note that the standard deviation image of θg is defined as equation image. Similarly, the standard deviation image θa is defined as equation image. The above equation (B7) tries to express the average daily flux equation image in terms of average daily temperatures equation image under steady-state conditions. Because of the nonlinearities these two fluxes are not the same and this is why this analysis introduces a correction (oscillations energies). equation image can be expressed as a function of equation image and equation image using equation (2). Then equation (B7) becomes

equation image

[53] Subtracting equation (B10) from equation (B5), we obtain

equation image

where the Rld, equation image, LE and H terms are eliminated in the above equation because these four terms in equation (B10) are canceled with those in equation (B5). Note that in equation (B5)equation image = (1 − α) + Rs + Rldequation image + equation imageLEequation imageHequation image. Reorganizing the above equation it becomes

equation image

where τgg and τag are defined as

equation image
equation image

Equations (B10) and (B12) can be applied in determining skin temperature in steady and unsteady-state conditions, respectively.

Appendix C:: Mean Theorem Value

[54] First we consider a function p as a monotonic function of a variable x, such as thermal radiation flux and saturated vapor pressure increasing with temperature, or kinetic energy increasing with wind speed; the general form of the function is

equation image

where a and b are constants, and n is a real number. During the averaging period L, we can always express the variable x in the time domain as

equation image

where equation image is the mean of x over L and x′(t) is a fluctuation term. For example, if L = 1 day, then equation image is the daily mean. Note that according to the mean value theorem [Kreyszig, 1993], for any continuous function p with a continuous first derivative, there is a value equation image ∈ [equation image, x] such that

equation image

In addition, since equation image(t) ∈ [equation image, x(t)], equation image(t) can be written as

equation image

where ξ(t) ∈ [0, 1]. Reorganizing the above equation, we can derive ξ(t) as

equation image

Now, let us determine ξ(t). Using the Taylor series expansion on both right-hand side and the left-hand side of equation (C3) on equation image, equation (C3) becomes

equation image

Moving the right-hand side of the equation to the left-hand side and using the notation ξ = (equation imageequation image)/x′ according to equation (C5), reorganizes the above equation as

equation image

Substituting equation (C1) into the above equation further reorganizes the above equation as

equation image

Dividing the above equation by aequation imagen−1, and using the notation of the dimensionless fluctuation term x′* ≡ x′/equation image, the above equation then becomes

equation image

Note that

equation image

for any variable y, where

equation image

Using the above equality (equation (C10)), equation (C9) can be further simplified as

equation image

The above equation can be reorganized as

equation image

Solving the above equation, we determine ξ as

equation image

Using the Taylor series expansion, for any n as ∣x′*∣ → 0, ξ is asymptotic to

equation image

where ξ = 0.5, while n = 2. The dimensionless mean theorem value equation image* can be determined by substituting equation (C15) into equation (C4) as

equation image

Therefore taking the mean of the above equation, equation image becomes

equation image

and

equation image

where v* is the variance of equation image, s*k is the skewness equation image, and k*u is the kurtosis equation image. Note that the second term of the first equality in equation (C17) is neglected because equation image.

[55] Finally, let's expand the mean of the nonlinear function p(x) according to the above derivations. Reorganizing equation (C3) yields

equation image

Substituting equations (C8) and (C17) into the above equation, and expanding ∂p/∂x on equation image by the Taylor series expansion, equation image becomes

equation image

where the third equality is derived because equation image is a constant. Note that since equation image is constant equation image is also a constant. The fourth equality is derived because equation image = 0.

[56] The above equation can be simplified in a form with the second order of accuracy. First, substituting equations (C17) and (C18) into equation image and equation image in the second term on the right-hand side of the last equality of the above equation, equation image and equation image can be rewritten as

equation image

and

equation image

[57] Then, substituting the above two equations into equation (C20) and neglecting the terms with errors higher than the third order, equation (C20) becomes

equation image

and

equation image

Hence equation (C23) has the second order of accuracy. It needs only the mean and the variance of the variable x of the nonlinear function p.

Acknowledgments

[58] The author would like to acknowledge the support for this work by the National Science Council, Taiwan, under grants 862621P005019 and 35110F. Thanks also to L. Bengtsson, K. Arpe, and L. Dumenil-Gates for hosting the author's visit to the Max Planck Institute for Meteorology, Hamburg, Germany, to a native English-speaking editor Terry Dean Sommers for proofreading, and to anonymous reviewers for their helpful and constructive comments for the presentation of this analytical concept.

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