The ergodic hypothesis is a basic hypothesis typically invoked in atmospheric surface layer (ASL) experiments. The ergodic theorem of stationary random processes is introduced to analyse and verify the ergodicity of atmospheric turbulence measured using the eddy-covariance technique with two sets of field observational data. The results show that the ergodicity of atmospheric turbulence in atmospheric boundary layer (ABL) is relative not only to the atmospheric stratification but also to the eddy scale of atmospheric turbulence. The eddies of atmospheric turbulence, of which the scale is smaller than the scale of the ABL (i.e. the spatial scale is less than 1000 m and temporal scale is shorter than 10 min), effectively satisfy the ergodic theorems. Under these restrictions, a finite time average can be used as a substitute for the ensemble average of atmospheric turbulence, whereas eddies that are larger than ABL scale dissatisfy the mean ergodic theorem. Consequently, when a finite time average is used to substitute for the ensemble average, the eddy-covariance technique incurs large errors due to the loss of low-frequency information associated with larger eddies. A multi-station observation is compared with a single-station observation, and then the scope that satisfies the ergodic theorem is extended from scales smaller than the ABL, approximately 1000 m to scales greater than about 2000 m. Therefore, substituting the finite time average for the ensemble average of atmospheric turbulence is more faithfully approximate the actual values. Regardless of vertical velocity or temperature, the variance of eddies at different scales follows Monin–Obukhov similarity theory (MOST) better if the ergodic theorem can be satisfied; if not it deviates from MOST. The exploration of ergodicity in atmospheric turbulence is doubtlessly helpful in understanding the issues in atmospheric turbulent observations and provides a theoretical basis for overcoming related difficulties.
The basic principle of average of the turbulence measurements is based on
ensembles averaged over space, time and state. However, it is impossible to
make an actual turbulence measurement with enough observational instruments
in space for sufficient time to obtain all states of turbulent eddies to
achieve the goal of an ensemble average. Therefore, based on the ergodic
hypothesis, the time average of one spatial point, taken over a sufficiently
long observational time, is used as a substitute for the ensemble average for
temporally steady and spatially homogeneous surfaces (Stull, 1988; Wyngaard,
2010; Aubinet et al., 2012). The ergodic hypothesis is a basic assumption in
turbulence experiments in the atmospheric boundary layer (ABL) and
atmospheric surface layer (ASL). Stationarity, homogeneity, and ergodicity
are routinely used to link ensemble statistics (mean and higher-order
moments) of field experiments in the ABL. Many authors habitually refer to
the ergodicity assumption with descriptions such as “when satisfying ergodic
hypothesis
The ergodic hypothesis was first proposed by Boltzmann (Boltzmann, 1871; Uffink, 2004) in his study of the ensemble theory of statistical dynamics. He argued that a trajectory traverses all points on the energy hypersurface after a certain amount of time. At the beginning of 20th century, the Ehrenfest couple (Ehrenfest and Ehrenfest-Afanassjewa, 1912; Uffink, 2004) proposed a quasi-ergodic hypothesis and changed the term “traverses all points” in the aforesaid ergodic hypothesis to “passes arbitrarily close to every point”. The basic points of ergodic hypothesis or quasi-ergodic hypothesis recognize that the macroscopic property of a system in the equilibrium state is an average of microcosmic quantity in a sufficiently long time. Nevertheless, the ergodic hypothesis or quasi-ergodic hypothesis was never proven theoretically. The proof of the ergodic hypothesis in physics aroused the interest of mathematicians. Famous mathematician, Neumann et al. (1932) first theoretically proved the ergodic theorem in topological space (Birkhoff, 1931; Krengel, 1985). Afterward, a banausic ergodic theorem of stationary random processes was proven to provide a necessary and sufficient condition for the ergodicity of stationary random processes. Mattingly (2003) reviewed the research progress on ergodicity for stochastically forced Navier–Stokes equation, and that Galanti and Tsinober (2004) and Lennaert et al. (2006) solved the Navier–Stokes equation by numerical simulation to prove that turbulence that is temporally steady and spatially homogeneous is ergodic. However, Galanti and Tsinober (2004) also indicated that such partially turbulent flows acting as mixed layer, wake flow, jet flow, flow around the boundary layer may be non-ergodic.
Obviously, the advances of research on ergodicity in mathematics and physics have led the way for atmospheric sciences. We try first to introduce the ergodic theorem of stationary random processes to the atmospheric turbulence in this paper. The ergodicity of different scale eddies of atmospheric turbulence is directly analysed and verified quantitatively on the basis of field observation data obtained using eddy-covariance technique in the ASL.
Stationary random processes are processes which will not vary with time;
that is, for observed quantity
The mean
For the stationary random processes, the necessary and sufficient condition
satisfying the autocorrelation ergodicity is the autocorrelation ergodic
function Er(
The stationary random processes conform to the criterion, Eq. (4) or (5); then they satisfy the mean ergodic theorem or are intituled as the mean ergodicity. The stationary random processes conform to the criterion, Eq. (6a) or (7a); then they satisfy the autocorrelation ergodic theorem or are intituled as the autocorrelation ergodicity. If the stationary random processes are only of mean ergodicity, they are strictly ergodic or narrowly ergodic. If the stationary random processes are of both the mean ergodicity and autocorrelation ergodicity, they are namely wide ergodic stationary random processes. It is thus clear that the ergodic random processes are stationary, but the stationary processes may not be ergodic.
In the random process theory, calculating the mean or high-order moment
function requires a large number of repeated observations to acquire a sample
function
The scope of spatial and temporal scale of the atmospheric turbulence, which
is from the dissipation range, inertial subrange to the energy range, and
further the turbulent large eddy, is extremely broad (Stull, 1988). In such
wide spatial and temporal scope, the turbulent eddies include the isotropic
3-D eddy structure of high-frequency turbulence and orderly coherent
structure of low-frequency turbulence (Li et al., 2002). These eddies of
different scale are also different from each other in terms of their spatial
structure and physical properties, and even their transport characteristics
are not the same. It is thus reasonable that eddies with different
characteristics are separated, processed and studied using different methods
(Zuo et al., 2012). A major goal of our study is to understand what type of
eddy in the scale can satisfy the ergodic condition. Another goal is that the
time averaging of signals measured by a single station determines accurately
turbulent characteristic quantities. In order to study the ergodicity of
different scale eddies, Fourier transform is used as a band-pass filtering to
distinguish different scale eddies. That is to say, we aim to set the
Fourier transform coefficient of the part of frequencies, which it does not
need, as 0, and then we acquire the signals after filtering by means of
Fourier inverse transformation. The specific formulae are shown below:
The characteristics of the relations of Monin–Obukhov similarity (MOS) for the variance of different scale eddies are analysed and compared to test feasibility of the MOS relations for ergodic and non-ergodic turbulence. In order to provide an experimental basis for utilizing MOST and developing the turbulence theory of ABL under the condition of the complex underlying surfaces, the problems of eddy-covariance technique of the turbulence observation in ASL are further explored on the basis of studying the ergodicity and MOS relations of the variance of different scale eddies.
The MOS relations of turbulent variance can be regarded as an effective
instrumentality to verify whether or not the turbulent flow field is
steady and homogeneous (Foken et al., 2004). Under ideal conditions, the
local MOS relations of the variance of wind velocity, temperature and other
factors can be expressed as follows:
A large number of research results show that, in the case of unstable
stratification,
In this study two turbulence data sets are used for completely different purposes. The first turbulence data set is the data measured by the eddy-covariance technique under the homogeneous surface in Nagqu Station of Plateau Climate and Environment (NSPCE), Chinese Academy of Sciences (CAS). The data set in NSPCE/CAS includes the data that are measured by 3-D sonic anemometer and thermometer (CSAT3) with 10 Hz as well as infrared gas analyser (Li7500) in ASL from 23 July to 13 September 2011. In addition, the second turbulence data set of CASES-99 (Poulos et al., 2002; Chang and Huynh, 2002) is used to verify the ergodicity of turbulence observed by multiple stations. CASES-99 has seven observation sites, equivalent to seven observation stations. The data in the central tower of CASES-99 include those measured by sonic anemometer and thermometer (CSAT3) with 20 Hz and the infrared gas analyser (Li7500) at 10 m on a tower with 55 m height in ASL. The other six subsites of CASES-99 surrounding the central tower, sn1, sn2 and sn3 are located 100 m are away from the central tower, the subsite sn4 is 280 m away, and subsites sn5 and sn6 are located 300 m away. The data of subsites include those measured by 3-D sonic anemometer (ATI) and Li7500 at 10 m height on the towers. The analysed results with two data sets are compared to each other to test universality of the research results.
The geographic coordinate of NSPCE/CAS is 31.37
These data are used to study the ergodicity of turbulent eddies in ABL.
Firstly the inaccurate data caused by spike are deleted before data analyses.
Subsequently, the data are divided into continuous sections of 5 h, and the
signals of 1 h are obtained applying filtering of Eqs. (8) and (9) for each
5 h of data. In order to delete further the abnormal inaccurate data, the data
are divided once again into 12 continuous fragments of 5 min in 1 h. The
variances of velocity and temperature are calculated and compared to each other
for the fragments. The data with a deviation of less than
Applying the two data sets from NSPCE/CAS and CASES-99, the ergodicity of different temporal scale eddies is tested. Here as an example, we select representative data measured at a level of 3.08 m in NSPCE/CAS during three time frames, namely 03:00–04:00, 07:00–08:00 and 13:00–14:00 China standard time (CST) on 25 August under clear-sky conditions to test and demonstrate the ergodicity of different temporal scale eddies. These three time frames represent three situations, i.e. the nocturnal stable boundary layer, early neutral boundary layer and midday convective boundary layer.
Equations (8) and (9) are used to perform band-pass filtering from
The Monin–Obukhov stratification stability parameter
As a typical example, the eddy local stabilities,
Local stability parameter (
The Monin–Obukhov eddy local stability is not entirely the same as the Monin–Obukhov
stratification stability of ABL in the physical significance. The Monin–Obukhov
stratification stability of ABL indicates the overall effect of atmospheric
stratification in the ABL on the stability including all eddies in integral
boundary layer. The Monin–Obukhov stratification stability
Variation of mean ergodic function Ero(
Variation of mean ergodic function Ero(
Variation of mean ergodic function Ero(
Variation of mean ergodic function Ero(
In order to verify the mean ergodic theorem, we calculate the mean and
autocorrelation functions using Eqs. (2) and (3), then calculate the
variation of mean ergodic function Ero(
Comprehensive analyses of the characteristics of mean ergodicity of atmospheric turbulence as well as the relevant causes are discussed in the following sections.
According to the mean ergodic theorem, Eq. (4), the mean ergodic function
Ero(
As seen from Figs. 1–3, dimensionless mean ergodic function of the
vertical velocity is compared with respective function of the temperature and
humidity. It is 3–4 orders of magnitude less than those in the nocturnal stable
boundary layer; 1–2 orders of magnitude less than those in the early neutral boundary
layer; and about 2 orders of magnitude less than those in the midday convective
boundary layer. For example, at 15:00–16:00 (CST) during nighttime time
frame, the dimensionless mean ergodic function of vertical velocity is
10
For wind velocity of 1–2 ms
To facilitate comparison, Fig. 4 shows the variation of mean ergodic function
Ero(
Table 1 lists the corresponding relation of eddy local stabilities
Variation of the autocorrelation ergodic function of vertical velocity with relaxation time for different scale eddies.
In this section, Eqs. (7a) and (7b) are used to verify the autocorrelation
ergodic theorem. This is in accordance with Sect. 4.2 that the turbulent eddies
below 10 min at temporal scale satisfy the mean ergodic condition in the
various time frames; i.e. the turbulent eddies below 10 min at temporal
scale are at least strictly stationary random processes or narrow stationary
random processes – whether in the nocturnal stable boundary layer, in the
early neutral boundary layer, or midday convective boundary layer. Then we
analyse further the different scale eddies that satisfy the mean ergodic
condition and whether or not they also satisfy the autocorrelation ergodic condition,
so as to verify whether or not atmospheric turbulence is a narrow or wide stationary
random process. The autocorrelation ergodic function of turbulence variable
As an example of the vertical velocity, Fig. 5 shows the variation of
normalized autocorrelation ergodic function Er( After comparing Fig. 5a–c with Fig. 1a–c, i.e. comparing the
dimensionless mean ergodic function Ero( The above two basic characteristics imply that the autocorrelation
ergodic function Er( According to the autocorrelation ergodic function Eq. (7a), the
eddies of 30, 60 and below 10 min at the temporal scale, regardless of
whether they are in the stable boundary layer, neutral boundary layer or
convective boundary layer, all eddies satisfy the condition of
autocorrelation ergodic theorem. Therefore, in general ABL turbulence is the
stationary random process of autocorrelation ergodicity. The above results show that the eddies below 10 min at temporal scale
in the nocturnal stable boundary layer, early neutral boundary layer and
midday convective boundary layer satisfy not only the condition of mean
ergodic theorem but also the condition of autocorrelation
ergodic theorem. Therefore, eddies below 10 min at the temporal scale are
wide ergodic stationary random processes. Although the eddies of 30 and
60 min at temporal scale in the stable boundary layer, neutral boundary layer
and convective boundary layer satisfy the condition of autocorrelation
ergodic theorem, they dissatisfy the condition of mean ergodic theorem.
Therefore, eddies of 30 and 60 min at the temporal scale are neither narrow
ergodic stationary random processes nor wide ergodic stationary random
processes.
Variation of mean ergodic function
The basic principle of turbulence average is an ensemble average of the
space, time and state. Sections 4.2 and 4.3 verify the mean ergodic theorem
and autocorrelation ergodic theorem of atmospheric turbulence using field
observational data, so that the finite time average of a single station can
be used to substitute for the ensemble average for the ergodic turbulence.
This section examines the ergodicity of different scale eddies using the
observational data of a centre tower and six subsites of CASES-99, in all
seven sites, equivalent to seven stations. When the data are selected,
the following is considered: if the eddies are not evenly distributed at the seven
sites, then the observation results at the seven sites may have originated
from many eddies at a large scale. For this reason, the high-frequency
variance spectrum in excess of 0.1 Hz is compared firstly. Based on the
observational error, if the scatter of all high-frequency variances does not
exceed the average by
The results show ergodic characteristics of different scale eddies measured with multi-station observations as follows: Fig. 6a shows that the mean ergodic function of eddies below 30 min at temporal scale converges to 0 very well, except for the fact that the mean ergodic function of eddies of 60 min at temporal scale clearly deviates upward from 0. Figure 6b shows that autocorrelation ergodic function of all different scale eddies, including 60 min at temporal scale, gradually converges to 0. Therefore, eddies below 30 min at temporal scale measured with the multi-station observations satisfy the conditions of both the mean and autocorrelation ergodic theorem, while eddies of 60 min at temporal scale only satisfy the condition of autocorrelation ergodic theorem but dissatisfy the condition of mean ergodic theorem. These facts demonstrate that eddies below 30 min at temporal scale are the wide ergodic stationary random processes for time series of the above data sets composed by the seven stations. This signifies that, comparing data composed of the multi-station observations with data from a single station, the eddy temporal scale of wide ergodic stationary random processes is extended from below 10 to 30 min. As analysed above, if the eddies below 10 min at temporal scale are deemed the turbulent eddies in the ABL with height of about 1000 m, then the eddies of 30 min at the temporal scale, equivalent to the space scale greater than 2000 m, are deemed including eddy components of the local circulation in ABL. Therefore the multi-station observations can completely capture the local circulated eddies, which space scale is greater than 2000 m.
The atmospheric observations are impossible to repeat experiments exactly, must use the ergodic hypothesis and replace ensemble averages with time averages. The problem of how to determine the averaging time arises.
Variation of ogive functions of
The analyses on the ergodicity of different scale eddies in the above two
sections demonstrate that the eddies below 10 min at temporal scale as
relaxation time
Turbulent variance is a most basic characteristic quantity of the turbulence. Turbulence velocity variance, which represents turbulence intensity, and the variance of scalars, such as temperature and humidity, effectively describes the structural characteristics of turbulence. In order to test MOS relation of the different scale eddies with ergodicity, the vertical velocity and temperature data of NSPCE/CAS from 23 July to 13 September are used to determine the MOS relationship of variances of vertical velocity and temperature for the different scale eddies, and to analyse its relation to the ergodicity.
The MOS relation of vertical velocity variance is as follows:
Parameters of the fitting curve of MOS relation for vertical velocity variance.
MOS relation of vertical velocity variances of the different scale
eddies in NSPCE;
MOS relations of temperature variance of in different scale eddies
of NSPCE;
Figure 8 and Table 2 show that the parameters of fitting curve are greatly different, even if the fitting curve modality of MOS relation of the vertical velocity variance is the same for the eddies in different temporal scales. The correlation coefficients of MOS fitting curve of the vertical velocity variance under the unstable stratification are large, but the correlation coefficients under the stable stratification are small. Under unstable stratification, the correlation coefficient of eddies of 10 min in the temporal scale reaches 0.97, while the residual is only 0.16; under the stable stratification, the correlation coefficient reduces to 0.76, and the residual increases to 0.25. With the increase of eddy temporal scale from 10 (Fig. 8a) to 30 min (Fig. 8b) and 60 min (Fig. 8c), the correlation coefficients of MOS relation of the vertical velocity variance gradually reduce, and the residuals increase. The correlation coefficient in 60 min reaches a minimum; it is 0.83 under the unstable stratification, and only 0.30 under the stable stratification.
The temperature variance is shown in Fig. 9. MOS function to fit from eddies
of 10 min at the temporal scale under the unstable stratification is
as follows:
The above results show that the discreteness of fitting curve of MOS relation for the turbulence variance is increased with the increase of eddy temporal scale, whether it is the vertical velocity or temperature. The points of data during the stationary processes basically gather near the fitting curve of variance similarity relation, while all data points during the non-stationary processes deviate significantly from the fitting curve. However, the similarity of vertical velocity variance is superior to that of the temperature variance. These results are consistent with the conclusions of ergodicity test for the different scale eddies described in Sects. 2–4.4. The ergodicity of the small-scale eddies is superior to that of the larger-scale eddies, and eddies of 10 min at the temporal scale have the best variance similarity relations. These results also signify that when eddies in the stationary random processes satisfy the ergodic condition, both the vertical velocity variance and temperature variance of eddies in the different temporal scales comply with MOST very well – but, as for eddies with poor ergodicity during non-stationary random processes, the variances deviate from MOS relations.
Galanti and Tsinober (2004) proved that the turbulence, which is temporally steady and
spatially homogeneous, is ergodic, but “partially turbulent flows” such
as the mixed layer, wake flow, jet flow, flow around and boundary layer flow
may be non-ergodic turbulence. However, it has been proven through
atmospheric observational data that the turbulence ergodicity is related to
the scale of turbulent eddies. Since the large-scale eddies in ABL may be
strongly influenced by the boundary disturbance, they thus belong to “partial
turbulence”; however, since the small-scale eddies in atmospheric turbulence
may not be influenced by boundary disturbance, they may be temporally steady and
spatially homogeneous turbulence so that the mean ergodic theorem and
autocorrelation ergodic theorem are applicable for turbulence eddies at the
small scale in ABL, but the ergodic theorems are not applicable for the
large-scale eddies (i.e. the small-scale eddies in the ABL are ergodic and
the large-scale eddies exceeding the ABL scale are non-ergodic). The eddy-covariance technique for turbulence measurement is based on the ergodic assumption.
A lack of ergodicity related to the presence of large-scale eddy transport
can lead to a considerable error of the flux measurement. This has already been
pointed out by Mauder et al. (2007) or Foken et al. (2011). Therefore, we
realize from the above results that the large-scale eddies that exceed ABL
height may include components of non-ergodic random processes. The
eddy-covariance technique cannot capture the signals of large-scale eddies
exceeding ABL scale to result in the large error in the measurements of
atmospheric turbulent variance and covariance. MOST is developed under the
condition of steady time and a homogeneous surface. MOST conditions, steady
time and homogeneous underlying surface are in line with the ergodic
conditions. Therefore, the turbulence variances, even the turbulent fluxes of
eddies at different temporal scales, may comply with MOST very well, if the
ergodic conditions of stationary random processes are more effectively
satisfied. According to Kaimal and Wyngaard (1990), the atmospheric turbulence theory and observation method were feasible
and led to success under ideal conditions including a short period, steady state and homogeneous underlying surface,
and through observation in the 1950s–1970s, but these conditions are rare in reality. In the land surface processes and
ecosystem, the turbulent flux observations in ASL turn into a scientific issue. Commonly, there are interested researchers
in the fields of atmospheric sciences, ecology, geography sciences, etc. These observations must be implemented under
conditions such as complex terrain, heterogeneous surface, long period and unsteady state. It is necessary that
more neoteric observational tools and theories be applied with new perspectives in future research. The ergodic theorem of stationary random processes has successfully been introduced from mathematics into
atmospheric sciences. It undoubtedly provides a profitable tool for overcoming the challenges encountered in the modern
measurements of atmospheric turbulent flow. At least it offers a promising first step to diagnosticate directly the ergodic
hypotheses for ASL flows as a criterion. The necessary and sufficient condition of ergodic theorem can be used to
judge the applicable scope of eddy-covariance technique and MOST, as well as seek potential disable reasons for using them in the ABL. In the future, we shall continue our study of the ergodic problems for the atmospheric turbulence measurements under the
conditions of complex terrain, heterogeneous surface and unsteady, long
observational period, as well as seek effective schemes. The above results
indicate that the atmospheric turbulent eddies below the scale of ABL can be
captured by the eddy-covariance technique and comply with MOST very well.
Perhaps MOST can be a first-order approximation to deal with the
turbulence of eddies below ABL scale in order to satisfy the ergodic theorems, to
compensate for the effects of eddies dissatisfying the ergodic theorem, which may
be caused by the advection, local circulation, low-frequency effect, etc.
under the complex terrain, heterogeneous surface. For example, we developed a
turbulent theory of non-equilibrium thermodynamics (Hu, et al., 2007; Hu and Chen, 2009) to find the coupling
effects of vertical velocity, which is caused by the advection, local
circulation, and low frequency, on the vertical fluxes. The coupling effects
of vertical velocity may be a scheme to compensate for the effects of eddies
dissatisfying the ergodic theorems (Hu, 2003; Chen, et al., 2007, 2013). It is clear that such studies are preliminary, and many problems require further research. The attestation of
more field experiments is necessary.
From the above results, we can draw preliminary conclusions:
The turbulence in ABL is an eddy structure. When the temporal scale of turbulent eddies in ABL is about 2 min,
the corresponding spatial scale is about 120–240 m, equivalent to ASL
height; when the temporal scale of turbulent eddies in ABL is about 10 min,
the corresponding spatial scale is about 600–1200 m, equivalent to the
ABL height. For the eddies of larger temporal and spatial scale, such as
eddies of 30–60 min at the temporal scale, the corresponding spatial scale
is about 1800–3600 m to exceed the ABL height. The above results show that the ergodicity of atmospheric turbulence in ABL is relative not only to the atmospheric
stratification but also to the eddy scale of atmospheric turbulence. For the
atmospheric turbulent eddies below the ABL scale (i.e. the eddies below about
1000 m at the spatial scale and about 10 min at the temporal scale), the
mean ergodic function Ero( Due to above facts, when the stationary random process information of eddies below 10 min at the temporal scale
and below 1000 m of ABL height at the spatial scale can be captured, the atmospheric turbulence may satisfy the condition
of mean ergodic theorem. Therefore, an average of finite time can be used to substitute for the ensemble average to calculate
the mean of random variable as measuring atmospheric turbulence with the eddy-covariance technique. But for the turbulence of
eddies to be larger than 30 min at temporal scale (i.e. 2000 m at spatial scale magnitude), it dissatisfies the condition of mean
ergodic theorem so that the eddy-covariance technique cannot completely capture the information of non-stationary random processes.
This will inevitably cause a high level of error when the average of finite time is used to substitute for the ensemble average in
the experiments due to the loss of low-frequency component information associated with the large-scale eddies. Although the atmospheric temperature stratification has different effects on the stability of eddies in the different scales,
the ergodicity is mainly related to the eddy local stability, and its relation to the stratification stability of ABL is secondary. The data series composed from seven stations compare with the observational data from a single station. The results show that
the temporal and spatial scales of eddies belonging to the wide ergodic
stationary random processes are extended from 10 min to below 30 min and
from 1000 m to below 2000 m, respectively. This signifies that the ergodic
assumption is more likely to be satisfied well with multi-station
observations, and observational results produced by the eddy-covariance
technique are much closer to the true values when calculating the turbulence
averages, variances or fluxes. If the ergodic conditions of stationary random processes are more effectively satisfied, then the turbulence variances of
eddies in the different temporal scale can comply with MOST very well; however, the turbulence variances of the non-ergodic random processes deviate from MOS relations.
This study is supported by project grant nos. 91025011 and 91437103 of the National Natural Science Foundation of China and project grant no. 2010CB951701-2 of the National Program on Key Basic Research. This work was strongly supported by Heihe Upstream Watershed Ecology-Hydrology Experimental Research Station, Chinese Academy of Sciences. We would like to express our sincere gratitude for their support. We also thank Gordon Maclean at NCAR for providing the detailed data of CASES-99 used in this study as well as the referees and editor very much for their heartfelt comments, discussions and marked errors. Edited by: R. Sander