## 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 *r*_{a}, canopy resistance *r*_{c}, specific humidity *q*_{a}, thermal capacity *C*_{A} of a PBL, and thermal capacity *C*_{G} 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 *r*_{a}, *r*_{c}, *q*_{a}, *C*_{A}, and *C*_{G} 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.