The transport and mixing of pollution during the daytime evolution of a valley boundary layer is studied in an idealized way. The goal is to quantify horizontal and vertical tracer mass fluxes between four different valley volumes: the convective boundary layer, the slope wind layer, the stable core, and the atmosphere above the valley. For this purpose, large eddy simulations (LES) are conducted with the Weather Research and Forecasting (WRF) model for a quasi-two-dimensional valley. The valley geometry consists of two slopes with constant slope angle and is homogeneous in the along-valley direction. The surface sensible heat flux is horizontally homogeneous and prescribed by a sine function. The initial sounding is characterized by an atmosphere at rest and a constant Brunt–Väisälä frequency. Various experiments are conducted for different combinations of surface heating amplitudes and initial stability conditions. A passive tracer is released with an arbitrary but constant rate at the valley floor and resulting tracer mass fluxes are evaluated between the aforementioned volumes.

As a result of the surface heating, a convective boundary layer is
established in the lower part of the valley with a stable layer on top –
the so-called stable core. The height of the slope wind layer, as well as the wind
speed within, decreases with height due to the vertically increasing stability.
Hence, the mass flux within the slope wind layer decreases with height as
well. Due to mass continuity, this along-slope mass flux convergence leads to
a partial redirection of the flow from the slope wind layer towards the
valley centre and the formation of a horizontal intrusion above the
convective boundary layer. This intrusion is associated with a transport of
tracer mass from the slope wind layer towards the valley centre. A strong
static stability and/or weak forcing lead to large tracer mass fluxes
associated with this phenomenon. The total export of tracer mass out of the
valley atmosphere increases with decreasing stability and increasing forcing.
The effects of initial stability and forcing can be combined to a single
parameter, the breakup parameter

It is well known that slope winds provide a mechanism for the vertical
exchange of quantities like heat, moisture, and pollutants between a valley
atmosphere and the atmosphere aloft

In the early work of

In nature, rather rapid changes of stability or surface conditions with
height are the norm rather than the exception. For example, in an Alpine
valley, over an 18-day period in January, elevated inversions were found
during 84 % of the time

The intrusion of aerosol and other pollutants into the inversion layer has
implications for the definition and determination of the boundary layer
height over complex terrain. The CBL is usually defined over flat terrain as
a neutrally stratified layer bounded by a super-adiabatic surface layer and
topped by an inversion

Once the pollutants are exported out of the valley, they are partly injected
vertically into the free atmosphere above the capping inversion (mountain
venting,

Inside a valley, even without an elevated inversion or a cold-air pool
present at sunrise and with a homogeneous surface sensible heat flux, at
least one HI is likely to develop during the evolution of the valley boundary
layer. This is due to the development of a CBL with a capping inversion,
which leads to a sharp transition from a neutral to a stable stratification
and enables the development of an HI. For this reason a single HI formed in
many previous simulations of the valley boundary layer

However, tracer mass fluxes associated with HIs have never been quantified.
In this study, we aim to quantifying the horizontal and vertical fluxes of pollutants
between CBL, slope wind layer, stable core, and the atmosphere above an idealized
valley as well as the total export of tracer mass. This is done for a broad range
of atmospheric stability and amplitudes of surface sensible heat flux. The initial
atmospheric stability ranges from about one-half to 2 times the stability of the standard
atmosphere, and the reference forcing amplitude of 125 W m

This study is organized as follows: in Sect.

Valley topography with a schematic representation of volumes and tracer mass
fluxes and a typical profile of potential temperature at the centre of the valley
after a number of hours of model integration. The valley is homogeneous in

An overview of different sets of simulations. The reference simulation S1N10 is highlighted using bold typeset. See text for further explanation.

We use the Weather Research and Forecasting model (WRF-ARW), version 3.4.
This model has already been successfully used for simulations of thermally
driven winds in a previous study by

The model configuration is similar to the one in

The surface sensible heat flux is prescribed using a sine function with
amplitudes

The model is initialized with idealized soundings characterized by a constant
Brunt–Väisälä frequency

The passive tracer, which is used to measure the horizontal and vertical
exchange rates, is released at the lowest model level at every grid point
between

The procedures used to calculate vertical and horizontal fluxes follow the
averaging approach of

A variable

The averaging

The same decomposition can be applied to the total horizontal tracer mass
flux

As long as the valley atmosphere is characterized by at least one stably
stratified layer and anabatic slope winds exist, the valley volume can be
divided into three different sub-volumes (see Fig.

The first volume

The total tracer mass flux integrated spatially over the interface between
two volumes is called bulk flux and is denoted by

Vertical bulk tracer mass fluxes are calculated by integrating the total
tracer mass flux

In order to quantify tracer mass fluxes between the volumes described above,
working definitions of CBL height

In principle, various definitions of

Boundary layer heights and slope wind layer depths based on these three
definitions are shown for simulation S1N10 in Fig.

Vertical cross sections at 09:00 LT (left column) and 12:00 LT (right column) for
the simulation S1N10. Shown fields in

Since the

The general flow and tracer distribution are described in this section for the S1N10 simulation, which serves as a reference. The evolution of the other simulations is discussed briefly at the end of this section.

At the start of the simulation (06:00 LT), the whole atmosphere is stably
stratified and characterized by a constant Brunt–Väisälä frequency
(regime 0). With the onset of the surface-layer heating, a shallow CBL
begins to form in the centre of the valley and a slope wind layer develops
(regime 1, Fig.

Horizontal and vertical tracer mass fluxes for simulation S1N10 at 12:00 LT. Shown
are

Normalized bulk tracer mass flux for a set of S1 simulations with five different
values of initial static stability:

Tracer released at the valley bottom is distributed homogeneously within the
CBL and is also advected upwards by slope winds
(Fig.

As the boundary layer evolves, the slope wind layer grows in depth and the
wind speed within the HI increases (cf. Fig.

The tracer mass fluxes responsible for the redistribution of tracer mass are
shown in Fig.

Simulations with a stronger forcing or a weaker stability than the reference case S1N10 exhibit a similar evolution, but the inversion breakup (regime 2) is reached. In the case of a weaker forcing or a stronger stability than S1N10, the breakup does not occur. For example, the breakup is reached after about 4 h for S1N06, thus allowing for a larger amount of tracer export from the valley atmosphere, whereas for simulations S1N12 to S1N20 the breakup is never reached, which restricts vertical venting considerably. All simulations exhibit a similar pattern of tracer mass fluxes as long as the breakup is not reached. However, for a stronger stratification, the slope wind layer is shallower and the CBL grows more slowly. The tracer mass flux at the top of the slope wind layer is weaker and the recirculation within the valley is more pronounced. Only a single HI develops in all cases.

Figure

At about 08:30 LT the tracer reaches crest height and is exported out of the
valley. This is indicated by an increasing flux

In the afternoon, simulations differ greatly from each other depending on the
initial stability. While simulations with a weak stability reach the breakup
either before noon (S1N06, Fig.

For the case S1N10, the forcing is too weak to completely remove the
inversion which leads to a large maximum export of tracer mass but it grows
more slowly compared to simulations with a weaker initial stratification. Due
to this slower increase, more tracer mass accumulates before noon in

In the last half hour before sunset (18:00 LT),

The total tracer mass

The total mass of tracer passing each interface between two volumes

Once the tracer mass is in

The time of the inversion breakup and, consequently, whether it is reached at
all before sunset is an important parameter in describing the tracer mass
fluxes between the various valley volumes. This time is important since the
exchange with the atmosphere above the valley increases as the valley
atmosphere becomes neutral. Since the breakup time strongly depends on
initial stability and forcing amplitude, we study the impact of these
parameters here in more detail. The overall structure of the valley
atmosphere and the magnitudes of tracer mass fluxes between various volumes
of the reference simulation are very similar to those from runs with a weaker
or stronger forcing. Due to these similarities, we restrict the following
analysis to the fluxes at the top and the bottom of

Figure

Tracer mass flux

For the export of tracer mass out of

The dependence of the total tracer mass transport between

As in Fig.

In the case of the stronger stability, N18
(Fig.

The total export of tracer mass (

It is clear from the previous analysis, that the breakup time is an important
timescale for the evolution of the valley boundary layer, which depends on
both the initial stability and the forcing amplitude. Hence, the venting of a
valley strongly depends in a non-linear way on both parameters, but the total
export of tracer mass at crest height may be similar for certain combinations
of forcing and stability. A run with a strong forcing and a strong stability
exports about as much tracer as a run with a weaker forcing and a weaker
stability (Fig.

Total tracer mass exported out of the valley atmosphere at crest height normalized
by the total tracer mass released at the surface as a function of

After

Here,

Combining Eq. (

It is therefore useful to calculate the total energy provided until sunset

The ratio of required and provided energy,

In general, an initially stably stratified valley
atmosphere, which is heated from the surface can be divided into three
volumes: a convective boundary layer, a slope wind layer and a stable core.
These layers change in time and various flow regimes can be identified

The definition of these volumes are based on the crest height, CBL height and
slope wind layer depth

As in Fig.

The mechanisms responsible for tracer mass transportation between the
different valley volumes are up-slope winds, entrainment, HIs and
recirculation (cf. Fig.

The horizontal and vertical transport of tracer has been studied for a wide
range of stability conditions and forcing amplitudes, but only for a very
idealized framework. Soundings with a constant stability throughout the
atmosphere are used to initialize the model, which is not very realistic.
Multiple elevated inversions are common

The chosen quasi-two-dimensional valley geometry, which neither allows for the
development of along-valley winds nor for a plain-to-mountain circulation, is
a major simplification. The mountain-to-plain circulation provides a
mechanism to remove tracer from the vicinity of the mountain peaks

Due to the symmetry of the problem considered in this study, the two-slope
wind layers have been treated as a single volume. In reality, asymmetric flow
structures have been reported, e.g. in case of asymmetric solar forcing

We have shown that the export of tracer out of the valley is well described
by our breakup parameter

More specifically,

It is known that thermally driven winds can provide an effective venting
mechanism under favourable conditions. The number of turnovers of valley air
mass during daytime lies between zero and five, depending on the forcing
amplitude (see

An interesting question is whether the presented results for a passive tracer
can be applied to transport processes of heat and moisture as well. In
principle, both heat and moisture are transported by the slope flows; hence,
it is reasonable to assume similarities. However, heat is released over the
whole terrain surface area in contrast to the passive tracer, which is
released in our case only at the valley floor. An earlier onset of the export
of heat has therefore to be expected. At the same time, the exported heat
leads to an increase of the temperature above the valley, which affects the
breakup of the valley inversion

The evolution of a daytime valley boundary layer and
the transport of a passive tracer between three different valley sub-volumes
(convective boundary layer, slope wind layer, stable core) and the atmosphere
above the valley is investigated by means of large eddy simulations. The
model is initialized with a constant Brunt–Väisälä frequency and the
surface sensible heat flux is prescribed by a sine-shaped function. Numerous
simulations have been performed for initial Brunt–Väisälä frequencies

A horizontal intrusion forms above the convective boundary layer at the transition
zone between the slope wind layer and the stable core. This phenomenon is in
agreement with the circulation described by Vergeiner's idealized slope-flow
model

For a given heat flux amplitude, the efficiency of the vertical tracer mass transport
strongly depends on the static stability. For example, for a typical forcing
amplitude of 125 W m

The vertical transport of tracer mass from the convective boundary layer into the slope wind layer primarily depends on the forcing amplitude. A stronger forcing leads to an earlier onset of the associated tracer mass flux and a sharper increase in time. The horizontal flux from the slope wind layer into the stable core and the export at crest height depend on both forcing amplitude and initial stability. The export decreases drastically with increasing stability and decreasing forcing while the horizontal transport increases.

There is a similarity between simulations with different stability conditions and forcing amplitudes in the sense that a combination of weak forcing and weak stability leads roughly to the same export of tracer mass during the course of the day as a combination of strong forcing and strong stability.

A so-called breakup parameter

An early breakup of the valley inversion is required for effective venting of
pollutants. Half of the tracer mass released at the surface is exported for

Although this study is limited by the choice of idealized assumptions, i.e. quasi-two-dimensional topography, initially a vertically constant stratification and horizontally homogeneous surface heating, it provides an overview of the magnitudes of pollutant transport for a wide variety of stability conditions and forcing amplitudes. Also, we propose a single parameter, which describes the total export of tracer mass over the course of the day. In future studies, it would be desirable to test the dependence of pollutant venting on the breakup parameter for different valley topographies, including 3-D geometries, and to clarify whether the export of other quantities, such as heat and moisture, could be described by a similar relation. The impact of more realistic profiles of atmospheric stability on horizontal and vertical fluxes would be another important topic for a future study.

All data that are required to reproduce the numerical simulations presented in this article can be found in the Supplement. This includes changes to the WRF model code, namelist files, and input soundings as well as compiler information.

D. Leukauf designed the numerical experiments, carried them out, and prepared the manuscript. A. Gohm and M. W. Rotach provided suggestions for the design of the experiments, recommended relevant literature, discussed the results with the main author, and contributed to the manuscript by critical comments on text and figures, and proof reading.

This work was supported by the Austrian Science Fund (FWF) under grant P23918-N21, by the Austrian Federal Ministry of Science, Research and Economy (BMWFW) as part of the UniInfrastrukturprogramm of the Research Focal Point Scientific Computing at the University of Innsbruck, and by a PhD scholarship of the University of Innsbruck in the framework of the Nachwuchsförderung 2015. Computational results presented have been achieved in part using the Vienna Scientific Cluster (VSC). We thank two anonymous reviewers for their helpful comments. Edited by: B. Vogel Reviewed by: two anonymous referees