1. Introduction
Land surface processes are a key component of climate models. Since Manabe (1969) designed the first land surface scheme for a climate model, many land surface models have been proposed and have become more sophisticated. A review paper by Pitman (2003) describes the history of land surface schemes in detail. A number of intercomparisons of land surface models were carried out in the 1990s, such as the Project for Intercomparison of Land-Surface Parameterization Schemes (PILPS) (Henderson-Sellers et al. 1996). Differences in the degree of land–atmosphere interaction among atmospheric general circulation models (AGCMs) were also investigated in detail by the Global Land–Atmosphere Coupling Experiment (GLACE; Koster et al. (2006)).
Despite the evolution of land surface schemes and atmospheric models, the role of the interface between the atmosphere and land basically remained unchanged. The land surface scheme provides the boundary conditions to the atmospheric model and vice versa; the atmospheric and land surface models exchange radiative energy, sensible heat, latent heat, and momentum with each other.
The state of the atmosphere–land interface is governed by the surface energy balance equation, which determines the surface temperature. As the energy flux exchange depends on the surface temperature, solution of the energy balance equation is one of the most important tasks in model simulation. Polcher et al. (1998) summarized the typical methods of coupling atmospheric and land surface models used in recent general circulation models (GCMs). They categorized coupling schemes into four methods, that is, implicit coupling, semi-implicit coupling, explicit coupling, and open explicit coupling. Among these methods, the last is not used in any GCM because of its insufficient numerical stability. Implicit coupling, which solves the atmospheric state and the surface state instantaneously at the same time level, is the most desirable method in terms of numerical stability. However, Polcher et al. noted that this method is now used in only a few models, as land surface models have become more complex. The semi-implicit coupling method, which solves the state of the atmosphere using the surface state at the previous time step, has problems associated with energy conservation, as reported by Schulz et al. (2001). Meanwhile, explicit coupling, which solves the surface state using the atmospheric state at the previous state, is a simple but promising method, although the numerical stability is reduced compared to the implicit method. Thus, each of these methods has advantages and disadvantages.
Besides the choice of methods, there is a more serious problem in solving of the surface energy balance equation; that is, in most models, the equation is not solved strictly but is instead solved approximately for reasons of numerical efficiency and/or technical considerations. In practice, the equation is linearized around the surface temperature at the previous time step, keeping the turbulent transfer coefficients at the ground constant. That is, the surface temperature is estimated too roughly, and therefore this conventional method sometimes shows oscillation of surface temperature from time step to time step. For avoiding this spurious oscillation, the predictor–corrector method seems to be useful (Schulz et al. 2001). However, it is shown in this paper that the method cannot avoid the problem completely.
In the practical sense, such oscillatory behavior may be a one-time phenomenon and may not be critical for stable simulations. In addition, as the spatial and temporal resolutions of climate models are not particularly high, it may be thought that this spurious behavior does not have such a large impact on the time-averaged climatology. However, it may be somewhat optimistic, and it is safe to remove such a nonphysical mode if possible. On the other hand, as model resolution becomes higher both in space and in time, diurnal variation is becoming an important issue. For example, the diurnal cycle over the Maritime Continent, such as Sumatera and Borneo Islands in the western Pacific Ocean, is a topic of interest in climate research (Ichikawa and Yasunari 2006; Sakurai et al. 2005; Hirose and Nakamura 2005; Mori et al. 2004). As such studies focus on precipitation, the exchange of heat and moisture between atmosphere and land plays an essential role. The surface radiative fluxes are also important factors because they affect the cloud features and vice versa. In this sense, an accurate solution of the surface energy balance equation at the interface between land and atmosphere is a crucial issue.
This paper discusses the numerically spurious behavior of the surface temperature. The goal is to overcome the difficulty of obtaining the true solution and to provide a new scheme that gives a fast solution to the surface energy equation. The next section, using a single-column experiment, shows the spurious mode of surface temperature solution, which sometimes appears in the conventional linearized method with the explicit coupling. The numerically spurious mode is discussed, including the reason why such a mode appears. In particular, the implicit treatment of the turbulent transfer coefficients is emphasized. In section 3, a new method based on a modified Newton–Raphson method that can be applied to any atmospheric and land surface situation is proposed. The single-column experiment demonstrated that the new method provided a better solution than the conventional method. Section 4 presents an evaluation of the new method, proposed in section 3, using an idealized test case and a realistic climate simulation. Finally, section 5 presents the conclusion.
2. Spurious mode in the surface energy balance
a. The surface energy balance equation
As described in section 1, Polcher et al. (1998) categorized four methods of coupling atmospheric and land surface models. In this paper, the “explicit coupling” method is applied to Eq. (2) for simplicity. This coupling method uses the atmospheric and ground states at the previous time step. Since these values are treated as constants while solving the energy balance equation by this method, Eq. (2) with Eqs. (3)–(7) depends only on the surface temperature Ts.
b. The conventional method
c. A single-column experiment
To evaluate the performance of schemes, an idealized single-column experiment is set up. In the atmospheric field, the temperature profile with a surface temperature of 300 K and lapse rate of 6.5 K km−1, no water vapor (completely dry), and very weak horizontal wind of 0.4 m s−1 are imposed as the initial conditions. The Mellor–Yamada level-2 scheme (Mellor and Yamada 1974) with the Nakanishi and Niino (2004) modification is used as the turbulent scheme. No cumulus parameterization and large-scale condensation are imposed. The radiation scheme developed by Sekiguchi and Nakajima (2006) is implemented, and the equinoctial condition is imposed. The turbulent transfer coefficient is calculated using the Louis (1979) scheme modified by Uno et al. (1995), who formulated the nonunity heat-to-momentum roughness ratio. The land surface scheme used is based on the Manabe bucket model (Manabe 1969). In the soil field, three layers for soil temperature and one layer for soil moisture are configured. The initial soil temperature is 300 K in every layer, and the soil moisture is 0.2. Thus, the soil is very wet, while the atmosphere is very dry. This situation seems to be somewhat extreme; however, it may be more probable as the horizontal resolution increases.
Figure 1 shows the surface temperature development during the first day for the conventional method with Δt = 15 min. Overall diurnal variation is captured, while oscillation of surface temperature appears around 0900 and 1600 local time (LT). This spurious oscillation occurs time step by time step. Figures 2a and 2b show the temporal variation of the turbulent transfer coefficient for water vapor (Ce |Va|) and the sensible/latent heat fluxes during the first day. Around 0900 and 1600 LT, the turbulent transfer coefficient Ce |Va| and the latent heat flux also oscillate in a manner similar to surface temperature. Moreover, the oscillation in the latent heat flux or the transfer coefficient is the inverse of that of surface temperature; that is, when the latent heat flux increases, the surface temperature decreases. This phenomenon implies that the overshoot or undershoot of surface temperature reflects the estimation of the turbulent transfer coefficient at the next time step. Therefore, we would expect that, if the time step is reduced, the overshoot or undershoot will be suppressed because the turbulent transfer coefficient at the previous step will give a good approximation of the current step. However, this expectation was not met. Figure 3 shows the surface temperature development with Δt = 5 and 1 min. Even if the time step Δt was configured as an extremely small value, such as 1 min, the oscillation behavior was not improved, as shown in Fig. 3b.
Schulz et al. (2001) noted that explicit calculation of the turbulent transfer coefficient sometimes leads to a similar problem. They argued that the predictor–corrector method would avoid this problem. In the predictor–corrector method, Eq. (8) is used twice: First, the temporary surface temperature is estimated by using Eq. (8). Then using this estimated temperature, the turbulent transfer coefficients are calculated again. Finally, a new surface temperature is estimated by using Eq. (8) with the updated turbulent transfer coefficients. Since the updated turbulent transfer coefficients are better approximation at time t + 1 than those estimated just by the values at time t, we can expect that the numerical oscillation is reduced. Figure 4 is the same as Fig. 1 except for use of the predictor–corrector method. However, as shown in Fig. 4, this method is still not satisfactory; some improvement was found around 0900 LT, but the oscillatory behavior appeared again around 1600 LT.
Thus, neither of these conventional methods can obtain a reasonable solution owing to the rough estimation of turbulent transfer coefficients. In addition, the latent heat flux is estimated by multiplying the turbulent transfer coefficient with the saturated water vapor mixing ratio, which increases exponentially with the surface temperature, as shown in Eq. (5). The exponential increase in saturated water vapor may also make the solution unstable.
d. The implicit treatment of all fluxes
The algorithm of the binary search tree method is briefly summarized below.
First, the initial interval
The midpoint is defined as
When the interval
In this case, we set T0 = 10 K and ε = 0.1 W m−2. Figure 5 shows the surface temperature development for the single column test using the binary search tree method with Δt = 15 min. No spurious oscillation appeared, as shown in this figure. Figures 6a and 6b present the temporal variation of the turbulent transfer coefficients of water vapor (Ce |Va|) and sensible/latent heat fluxes in the case of the binary search tree method. As shown in these figures, no oscillation was found around 0900 and 1600 LT. These developments were very smooth and reasonable.
Therefore, if we solve the surface energy balance equation in a completely implicit manner, the spurious oscillation can be suppressed completely. In particular, implicit treatment of the turbulent transfer coefficient for water vapor would be effective for reaching a stable solution. In terms of reliability in reaching a solution, the binary search tree method is robust. On the other hand, the method is much more time-consuming than the conventional method. This method shows only linear convergence. Even in this simple case, the averaged iteration number is 12. In the next section, we consider a more efficient method.
3. A new way to solve for the surface energy balance
a. Requirements for the Newton–Raphson method
In general, the most efficient method to solve a nonlinear equation is the Newton–Raphson method. When the guess point is sufficiently close to the true solution, this method has quadratic convergence. However, several requirements are imposed for the convergence.
Requirement 1: The residual function should be continuous and differentiable.
Requirement 2: The residual function should be monotonic in the interval within the guess point.
Requirement 3: An abrupt gradient change between two regions with relatively similar gradients does not exist.
The first requirement does not seem to be satisfied in Eq. (9) at the melting point of Ts = 273.15 K because the saturated water vapor is not differentiable at this point, and the latent heat Lh below the melting point includes the fusion heat. However, if two residual functions are defined separately in the cases of ice-covered ground and no ice-covered ground, each of residual functions is continuous and differentiable. In this section, keeping the second and third requirements in mind, a new method is constructed.
b. The Newton–Raphson method
Using Eqs. (10)–(12), the Newton–Raphson method is applied to the single-column experiment described in the previous section with Δt = 15 min. This method reached the solution until t = 0815 LT. However, at t = 0830 LT, the method could not converge to the solution. Figure 7 shows the residual function at 0830 LT and the intermediate values. The black square is the initial guess point that is the solution at the previous time 0815 LT. The intermediate value moves back and forth between the two points indicated by the white squares. The form of the residual function violates the third requirement; there is an abrupt, steep gradient between the two curved lines with relatively mild gradients.
Figure 8 explains the appearance of this abrupt, steep gradient. The figure shows each of the fluxes against the surface temperature. Obviously, the latent heat flux has an abrupt, steep gradient around Ts = 298.5 K, where the surface turbulent is rapidly enhanced. Thus, because of the rapid change in latent heat against surface temperature, the form of the residual function is not suitable for the Newton–Raphson method.
c. The modified Newton–Raphson method
d. The backward search
Although the technique described in the previous subsection overcomes the difficulty associated with the third requirement, we should consider the possibility of nonmonotonicity of the residual function, associated with the second requirement. Intuitively, the residual function is thought to be monotonic because the latent and sensible heat fluxes and the upward longwave radiation increase and ground heat flux decreases with increases in surface temperature. However, the monotonicity is sometimes broken, as shown in Fig. 11, which was detected in the realistic simulation, described in the following section. Figure 12 shows the contribution of each flux to the residual function in this case. Nonmonotonicity is again caused by the latent heat flux. As shown in Fig. 12, the latent heat flux has a negative minimum at Ts = 272 K.1 The fact that latent heat flux has a negative minimum is not unusual. In this case, the atmosphere in the lowest level is very wet. At the same time, the atmospheric temperature is higher than the surface temperature. Consequently, when the turbulent is enhanced at Ts = 272 K, the water vapor in the lowest atmosphere rimes on the ground, and the maximum downward latent heat flux enters into the ground through the surface.
Although this situation is physically valid, the modified Newton–Raphson method does not behave smoothly. In the worst case, the scheme tends to search for the solution around the white square shown in Fig. 11. As a result, no convergence solution can be obtained. Figure 13 shows a schematic figure for the nonmonotonic case. Let the initial point be the black circle. After one iteration, the initial point moves to the point indicated by the white square, which is in the region of the positive derivative. After one more iteration, the point indicated by the white square moves to the point indicated by the black triangle. Thus, we can no longer hope for convergence to the true solution. The scheme must search for the solution in the opposite direction at the point indicated by the white square. Then, let the reduced factor γ in Eq. (13) be −1. As a result, the intermediate point can escape the region of the positive derivative of the residual function. The factor γ is kept as −1 until the point escapes from that region. This backward search is effective for avoiding the difficulty associated with the second requirement.
e. Summary of the algorithm for γ control
If
otherwise, if |R(Tsn)| > | R(Tsn−1)|, then γn = aγn−1;
γn+1 = min(1.0, bγn).
The factors a and b are determined empirically. After some trial and error, the choice of (a, b) = (0.5, 1.1) was found to be stable and fast. From the single column experiment, the averaged iteration number by this method was 3.2, indicating that the Newton–Raphson method modified by the above γ control is markedly more efficient than the binary search tree method.
f. The possibility of multiple solutions
With regard to nonmonotonicity, the possibility of multiple solutions should be discussed. A schematic figure is shown in Fig. 14. On the solid line, there are three equilibrium solutions: A, B, and C. Although this seems to be a rare case, it is possible mathematically. Among the three solutions, solution B is not a physical solution. This is explained as follows. We will return to the original surface balance equation (1). Supposing that solution B is obtained at an arbitral time, let us consider the increase of incoming shortwave radiation with infinitesimal time advances. In Fig. 14, the solid line would be shifted slightly upward, as shown by the dashed line. As a result, the tendency of surface temperature at point B is positive from Eq. (1) so that the surface temperature increases only slightly. Nevertheless, the equilibrium point would also move to point B′ where the temperature is lower than that at point B. Owing to this contradiction, solution B is removed as a physical solution. Fortunately, the new scheme proposed in this section can avoid this nonphysical solution because the scheme does not search for the zero point within the positive derivative region.
On the other hand, solutions A and C are considered to be physical solutions because their equilibrium states move to higher temperatures. Which solution is selected depends on the solution at the previous time step. If the previous solution is near point A, the scheme makes that point converge. If the previous solution is near point C, then point C is selected as the next solution.
4. Test of the new scheme
So far, we have conducted a single column experiment. In this section, we apply the new scheme to a general circulation model. The model used here is the Nonhydrostatic Icosahedral Atmospheric Model (NICAM) (Satoh et al. 2008; Tomita and Satoh 2004). Although NICAM was originally developed for global cloud-resolving simulation (Tomita et al. 2005; Nasuno et al. 2007; Miura et al. 2007), the model is used as the conventional general circulation model owing to its relatively coarse resolution; that is, it uses the (Arakawa and Schubert 1974) cumulus parameterization and a large-scale condensation scheme (Le Treut and Li 1991).
a. Land planet experiment
The first case extends the single column experiment to a global simulation. The surface is covered by land, as in the single column experiment. The initial atmosphere and soil conditions are also the same as those in the single column experiment. The difference from the single column experiment is the inclusion of flow dynamics. The horizontal resolution is so-called glevel 5, which corresponds to a grid interval of approximately 240 km.2 The model has 54 vertical levels with the top level at 40 km. Integration time is one day.
Figures 15 and 16 show the results of surface temperature using the conventional method and the new method. If the conventional scheme is used, a noiselike distribution appears in the morning and evening, as shown in Fig. 15. These areas of the distribution are where the latent heat flux changes abruptly. On the other hand, no such noiselike distribution is seen in Fig. 16 for the new method.
b. The realistic experiment
Although the land planet experiment provided a good demonstration of the superiority of the new method over the conventional method, it is too idealized to test the performance under any atmosphere–land conditions. Therefore, a more realistic test is needed. For this purpose, we performed a 10-yr simulation with realistic topography and a seasonal cycle. The horizontal and vertical resolutions were the same as those in the land planet experiment. The monthly sea surface temperature and sea ice concentration, which were constructed by averaging the 40-yr European Centre for Medium-Range Weather Forecasts Reanalysis (ERA-40) data from 1979 to 1999, were prescribed. A time step of 20 min was used.
During such long-term simulation, the new method could find the solution everywhere and every time. All the climatology information obtained is not shown, because this is not essential for this paper. Instead, statistical results indicating the computational efficiency are shown in Fig. 17, which presents the histogram of the frequency ratio against the maximum number of iterations over the globe; in almost all situations, it is within 10 iterations. In approximately 80% of the cases, the maximum number is within five iterations. The largest maximum is 24 iterations, but this occurred only once during the simulation.
5. Conclusions
This paper focused on solving the surface energy balance equation. We performed a detailed analysis of the spurious oscillation behavior of the surface temperature between atmosphere and land surface models. Such a spurious mode appears during the abrupt enhancement and depression of the boundary layer turbulent when the surface energy balance equation is solved by the conventional method in which the surface energy balance equation is linearized around the surface temperature at the previous time. The turbulent transfer coefficients for water vapor and potential temperature on the ground surface depend on the stability and change abruptly when stability switches from stable to unstable, or vice versa. This leads to oscillation of latent and sensible heat fluxes. Numerical treatment of latent heat flux plays a key role in suppressing the spurious mode.
The results of the single column experiment indicated that the convectional method, which estimates the turbulent transfer coefficient explicitly, does not avoid this numerical oscillation. The predictor–corrector method also cannot sufficiently suppress this mode completely. To suppress the oscillation completely, the turbulent transfer coefficient must be estimated implicitly. The binary search tree method, which solves the surface energy balance equation fully implicitly against the surface temperature, obtains a completely smooth solution, even if the time step is relatively large.
The binary search tree method is a robust method to obtain the solution. Although this method guarantees the convergence of a solution, the convergence speed is very slow, that is, linear convergence. This problem becomes serious when the solution for surface temperature is high because the residual function is steeper at higher temperature due to the exponential increase in latent heat flux. This method is not suitable for numerical weather prediction and climate modeling in terms of computational efficiency. Therefore, a more efficient method is desirable. The Newton–Raphson method may be a unique candidate to quickly obtain the root of the nonlinear equation. However, the original Newton–Raphson method is not suitable because the form of the residual flux function sometimes does not allow us to use this method. This occurs when an abrupt gradient change between two regions with relatively mild gradients exists, as shown in Fig. 7. In this case, use of the Newton–Raphson method with a reduced factor can overcome this problem, although the convergence speed slows down. To accelerate the convergence after the intermediate solution nears the true solution, we make the factor increase gradually.
Another difficulty associated with the application of the Newton–Raphson method to the present problem is on the nonmonotonic form of the residual flux function, as shown in Fig. 11. As the true solution does not exist in the region of the positive derivative of the residual flux function, the backward search technique can be applied to escape this region.
Thus, a modified Newton–Raphson method was developed with reduction of updates, acceleration of convergence, and inclusion of backward searching. In this study, two tests—a land planet experiment and a long-term realistic experiment—were performed. The results showed that the physical performance of the modified method is comparable to that of the binary search tree method while reasonable computational performance is obtained.
Recently, the land surface schemes have become more sophisticated and complex, introducing the canopy layer (e.g., Takata et al. 2003). Consequently, equations for the surface and canopy temperatures must be solved simultaneously. If we can couple the two equations into a single equation, the new method proposed in this paper can be applied easily; a multidimensional Newton–Raphson method with an appropriate modification can be formulated; however, this is beyond the scope of this paper and will be addressed in future work.
Acknowledgments
Thanks to Dr. Seita Emori of the National Institute for Environmental Studies (NIES) for useful discussions regarding this paper and also to Dr. Michio Kawamiya of the Frontier Research Center for Global Change (FRCGC) of the Japan Agency for Marine-Earth Science and Technology (JAMSTEC), who suggested the possibility of multiple solutions. Part of the calculations in this paper was performed on the earth simulator at the Earth Simulator Center. This work was supported by the Innovation Program of Climate Change Projection for the 21st Century of the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
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Temporal variation of the surface temperature (K) by the conventional method with Δt = 15 min.
Citation: Journal of Hydrometeorology 10, 3; 10.1175/2008JHM1080.1
Temporal variation of the (a) turbulent transfer coefficient Ce|Va| and (b) sensible heat flux and latent heat flux (W m−2) by the conventional method with Δt = 15 min.
Citation: Journal of Hydrometeorology 10, 3; 10.1175/2008JHM1080.1
As in Fig. 1, but with (a) Δt = 5 min and (b) Δt = 1 min.
Citation: Journal of Hydrometeorology 10, 3; 10.1175/2008JHM1080.1
As in Fig. 1, but by the predictor–corrector method.
Citation: Journal of Hydrometeorology 10, 3; 10.1175/2008JHM1080.1
As in Fig. 1, but by the binary search tree method.
Citation: Journal of Hydrometeorology 10, 3; 10.1175/2008JHM1080.1
As in Fig. 2, but by the binary search tree method.
Citation: Journal of Hydrometeorology 10, 3; 10.1175/2008JHM1080.1
Residual flux (W m−2) at 0830 LT by the usual Newton–Raphson method.
Citation: Journal of Hydrometeorology 10, 3; 10.1175/2008JHM1080.1
Each of the fluxes (W m−2) against the surface temperature at 0830 LT.
Citation: Journal of Hydrometeorology 10, 3; 10.1175/2008JHM1080.1
As in Fig. 7, but by the modified Newton–Raphson method.
Citation: Journal of Hydrometeorology 10, 3; 10.1175/2008JHM1080.1
Temporal variation of the surface temperature (K) by the modified Newton–Raphson method with Δt = 15 min.
Citation: Journal of Hydrometeorology 10, 3; 10.1175/2008JHM1080.1
Nonmonotonic residual flux (W m−2) against the surface temperature.
Citation: Journal of Hydrometeorology 10, 3; 10.1175/2008JHM1080.1
Each flux (W m−2) as a function of the surface temperature.
Citation: Journal of Hydrometeorology 10, 3; 10.1175/2008JHM1080.1
Schematic figure of the backward Newton–Raphson method.
Citation: Journal of Hydrometeorology 10, 3; 10.1175/2008JHM1080.1
Schematic figure of multiple solutions.
Citation: Journal of Hydrometeorology 10, 3; 10.1175/2008JHM1080.1
The surface temperature (K) distribution by the conventional method.
Citation: Journal of Hydrometeorology 10, 3; 10.1175/2008JHM1080.1
As in Fig. 15, but by the modified Newton–Raphson method.
Citation: Journal of Hydrometeorology 10, 3; 10.1175/2008JHM1080.1
Frequency of the number of iterations of the modified Newton–Raphson method in the 10-yr run.
Citation: Journal of Hydrometeorology 10, 3; 10.1175/2008JHM1080.1
The resolution for an icosahedral grid has been defined by Tomita et al. (2001).
