The imbalance of the surface energy budget in eddy-covariance measurements is still an unsolved problem. A possible cause is the presence of land surface heterogeneity, which affects the boundary-layer turbulence. To investigate the impact of surface variables on the partitioning of the energy budget of flux measurements in the surface layer under convective conditions, we set up a systematic parameter study by means of large-eddy simulation. For the study we use a virtual control volume approach, which allows the determination of advection by the mean flow, flux-divergence and storage terms of the energy budget at the virtual measurement site, in addition to the standard turbulent flux. We focus on the heterogeneity of the surface fluxes and keep the topography flat. The surface fluxes vary locally in intensity and these patches have different length scales. Intensity and length scales can vary for the two horizontal dimensions but follow an idealized chessboard pattern. Our main focus lies on surface heterogeneity of the kilometer scale, and one order of magnitude smaller. For these two length scales, we investigate the average response of the fluxes at a number of virtual towers, when varying the heterogeneity length within the length scale and when varying the contrast between the different patches. For each simulation, virtual measurement towers were positioned at functionally different positions (e.g., downdraft region, updraft region, at border between domains, etc.). As the storage term is always small, the non-closure is given by the sum of the advection by the mean flow and the flux-divergence. Remarkably, the missing flux can be described by either the advection by the mean flow or the flux-divergence separately, because the latter two have a high correlation with each other. For kilometer scale heterogeneity, we notice a clear dependence of the updrafts and downdrafts on the surface heterogeneity and likewise we also see a dependence of the energy partitioning on the tower location. For the hectometer scale, we do not notice such a clear dependence. Finally, we seek correlators for the energy balance ratio in the simulations. The correlation with the friction velocity is less pronounced than previously found, but this is likely due to our concentration on effectively strongly to freely convective conditions.

The interpretation of the turbulent fluxes of latent and sensible heat at the
Earth's surface still suffers from the unresolved energy balance closure
problem of the eddy covariance (EC) measurement technique. That is, the
measured turbulent fluxes are not equal to the available energy at the earth's
surface

In fact, the studies by

The influence of heterogeneous landscapes on properties of the atmospheric
boundary-layer has already been investigated for a few decades with numerical
models, primarily large-eddy simulation. We will summarize a few results that
are relevant to the non-closure of the energy balance.

On the other hand,

In summary, the previously mentioned studies showed that landscape
heterogeneity can induce mesoscale motions in the boundary-layer, especially
for heterogeneity of length scales larger than the boundary-layer height. By
using a large-eddy simulation model coupled to a land-surface scheme,

Although the above findings indicate that surface heterogeneity at scales of
boundary-layer depth and larger can couple to the full boundary layer,
surface heterogeneity at scales considerably smaller than the boundary-layer
height appears to be blended, as observed by

Acknowledging the connection between the energy imbalance and
quasi-stationary flow on the one hand, and quasi-stationary flow and surface
heterogeneity on the other hand, we will investigate the effect of surface
heterogeneity on the energy balance closure problem in this work. To this
end, we will study a series of synthetic idealized landscapes that consist of
a chessboard pattern of surface fluxes with different amplitude and
different wavelengths in the

Let us stress again the research questions of this paper. The first aim is to investigate the average influence on virtual flux measurements of land surface heterogeneity in the form of a variable surface heat flux, for a given length scale of the heterogeneity. We focus on length scales of the order of kilometer, and also on length scales of the order of hectometers. The second aim is to correlate the simulated energy balance ratio to various observables that can be obtained from the simulation output and that are also measurable in a realistic setting.

For our simulations we have made use of the LES model PALM

Parameters of the LES configuration.

Relevant parameters of the simulation setup are summarized in
Table

Parameters of the simulations within the suite focusing on the landscape heterogeneity at kilometer scale.

We ran two suites of simulations, one suite with 144 simulated cases focusing
on surface heterogeneity of the kilometer scale (Table

Parameters of the simulations within the suite focusing on the landscape heterogeneity at hectometer scale.

Graphical representation of (

Fixed location of the virtual towers for the kilometer scale
heterogeneity. The surface heat flux pattern of this example corresponds to

The main aim of this parameter study is to find out the response of virtual
towers in heterogeneous terrain of a certain length scale with variable
surface parameters. For this reason we create two suites of simulations where
each simulated case has another combination of the surface parameters. The
surface parameters are the length scales

Within the domain, we have positioned nine virtual control volumes. These
control volumes are located at functionally different positions with respect
to the surface heterogeneity, as can be seen in Fig.

The
Gauss–Ostrogradski theorem or “divergence theorem” is a special case of
the Stokes–Cartan theorem in differential geometry. For our purposes, we also
restrict ourselves to three-dimensional space. We consider a compact volume

From a control volume point of view the net fluxes through the faces are what
balances the storage term inside the volume, and in this manner advection
effects are automatically included in the energy balance of the volume. Of
course, in analogy with measurements, the fluctuations at the top face yield
the “virtually measured” turbulent heat flux: first the temporal
correlations are calculated, then a spatial average over the upper face of
the volume is calculated. The latter average improves the statistical
significance of the virtual measurement. Although the subgrid fluxes become
small at the height of the control volume, we nevertheless include the
vertical component of the subgrid flux into the turbulent heat flux. In this
manner we can also capture the highest-frequency correlations. Real data from
measurement towers is usually sampled up to 10–50 Hz, whereas for
computational efficiency our simulation advances with a time step of one
second, i.e., our simulated data is obtained at 1 Hz. A higher sampling
frequency would not resolve the turbulence better, as the resolution of the
latter is limited by the grid spacing. The part of the total turbulent flux
that is not captured directly by the resolved turbulent flux by 1-Hertz
sampling is transported by the subgrid turbulent flux. For the advective
components we have made a distinction between advection due to the mean flow
versus advection due to the horizontal flux-divergence. In complex terrain we
do not know a well-defined choice of reference for the base temperature, in
contrast to the base temperature in homogeneous terrain that appeared in

We start our analysis with a discussion of the location of the updrafts and
downdrafts in heterogeneous terrain. For this purpose, we concentrate on a
few specific cases, more precisely

Analysis of the circulation patterns induced by the surface
heterogeneity by means of the vertical velocity (

We notice that for the heterogeneity lengths of

On the topic of circulations driven by a surface conditions that are by
design freely convective, we investigate how the domain average of

Control volume fluxes as a function of available energy (scaled by
the median value) for kilometer scale landscape heterogeneity. The fluxes are
normalized by the available energy at their respective location, in our setup
this means normalization by the surface flux. Please note that we have
plotted the non-closure (normalized energy balance residual) instead of the
energy balance ratio EBR (normalized turbulent flux). Panel

In Fig.

To analyze the tower response in more detail, we have separated the towers at
the centers (left panel) from those at the borders (central panel). We notice
that most towers show the typical underestimation of the energy balance
(i.e., positive energy balance residual), except for the tower located at the
warmest spot where there is an updraft. In fact, the closed energy balance
for the tower in the warm patch is similar to a result in

In the right panel, we show the data from four homogeneous control runs (with
data extraction window and data selection in the same manner as for the
heterogeneous runs). Each of these simulations has nine towers as well, but
now all towers have the same surface properties. The mean residual
(under-closure) of the homogeneous control runs is around 10 %, less than
for the heterogeneous cases but not negligible. There is significant spread
on the results, but the residual is mainly composed of advection and storage.
Compared to the towers at the edges (middle panel), which are locally
heterogeneous, the homogeneous case is clearly different. Compared to the
towers at the centers of the patches (left panel), the homogeneous case has a
different average but the difference is still within the spread. It is
remarkable that flux-divergence is very small in the homogeneous case, in
contrast to the heterogeneous terrain. The negligible flux-divergence for a
homogeneous site was also apparent in the desert site of

Correlation between flux-divergence and EBR for kilometer scale
heterogeneity

As the residual is formed by the sum of advection by the mean flow, storage
and flux-divergence, we now turn our attention to these flux components. It
turns out that primarily the advection by the mean flow determines the
different residuals, but that the flux-divergence has to be taken into
account as well for the full picture. In addition, the storage flux also
plays a role, but its signature is independent on the location of the tower,
and it is always small, which is understandable for our type of surface
conditions: there is only a storage flux due to the heating of the air inside
the control volumes. For different towers, the allocation of the residual to
advection by the mean flow versus flux-divergence varies. At first the
behavior of the flux-divergence appears irregular. Let us however take a
closer look in Fig.

Finally, we want to remark that due to computational constraints, the
virtual measurement height in our simulations lies at 50 m, which is an
order of magnitude larger than the typical tower height over short vegetation
with comparable surface roughness. This means that our findings for virtual
EC towers cannot be directly transferred to real eddy-covariance towers.
Other LES studies of the energy balance closure point towards a larger
imbalance at higher

In Fig.

Control volume fluxes as a function of available energy (scaled by the median value) for hectometer scale landscape heterogeneity. See Fig. 4 for the explanation of the captions and labels and the text for further discussion.

Correlation between flux-divergence and EBR for hectometer scale
heterogeneity

We investigate the possible connection between the energy balance ratio, the different flux contributions and variables such as friction velocity and boundary-layer height. We performed a linear correlation analysis between these variables and the energy balance ratio. We made one restriction on the data set, which is to limit the boundary-layer depth to values larger than 1 km, thereby excluding about 8 % of the data, in order to obtain a better representation of the boundary-layer depth (when the boundary-layer depths smaller than 1 km are included, the correlation deteriorates).

We found that friction velocity and boundary-layer depth cluster are
well-correlated with each other, but not with EBR. Although we might have
supposed that higher boundary-layer heights will arise if patches are present
with vigorous surface heating. However, we found that

The linear correlation analysis shows that the simulated EBR does not
linearly depend on easily measured characteristics. As we have learned from
Fig.

In this work, we have investigated the effect of idealized surface
heterogeneity on the components of the surface energy budget measured at
virtual measurement towers, by means of large-eddy simulation. By means of a
control volume approach, we have decomposed the modeled surface energy budget
to highlight its partitioning, and we have shown that the modeled energy
balance ratio exhibits values that are found in field experiments. In
addition, this approach allows us to investigate the energy balance closure
in two-dimensional complex terrain. We have found that for surface
heterogeneity with length scale of order kilometer, there is a clear relation
between the energy budget components and the location of the tower with
respect to the patches of surface heterogeneity. For surface heterogeneity of
hectometer scale, the response of the different towers appears to depend to a
lesser extent on their respective location. Towers located at the borders
between patches with different surface heat flux have worse closure than
towers located in the center of a patch. Although storage terms are not
negligible, the size of the residual depends mostly on the advection and
flux-divergence terms. Remarkably, flux-divergence and advection by the mean
flow separately correlate very well with the energy balance ratio, which
implies that the EBR can be explained by the advection or flux-divergence
only, as the latter two are well correlated among themselves. For the
kilometer scale heterogeneities, advection by the mean flow and
flux-divergence behave in opposite ways, while they are positively correlated
for hectometer scale heterogeneities. We did not find a high correlation
between the friction velocity and energy balance ratio but this could be due
to the limited range of

Please contact the authors directly for the data.

Even though the focus of this study is on virtual flux measurements, we can
look at an example of a real EC measurement site to make a qualitative
comparison of these virtual tower measurements with real tower measurements.
In a first approximation, the heterogeneity of the landscape around a
measurement site can be characterized by the dominant length scale of a
suitable surface variable. In

The authors declare that they have no conflict of interest.

We thank the anonymous reviewers for their detailed comments and insightful remarks, which significantly improved the quality of this work. This work was conducted within the Helmholtz Young Investigators Group “Capturing all relevant scales of biosphere-atmosphere exchange – the enigmatic energy balance closure problem”, which is funded by the Helmholtz-Association through the President's Initiative and Networking Fund, and by KIT. We thank the PALM group at Leibniz University Hanover for their open-source PALM code and their support.The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.Edited by: Heini Wernli Reviewed by: two anonymous referees