We have developed a one-dimensional second-order closure numerical model to
study the vertical turbulent transport of trace reactive species in the
convective (daytime) planetary boundary layer (CBL), which we call the
Second-Order Model for Conserved and Reactive Unsteady Scalars (SOMCRUS). The
temporal variation of the CBL depth is calculated using a simple mixed-layer
model with a constant entrainment coefficient and zero-order discontinuity at
the CBL top. We then calculate time-varying continuous profiles of mean
concentrations and vertical turbulent fluxes, variances, and covariances of
both conserved and chemically reactive scalars in a diurnally varying CBL.
The set of reactive species is the O

The behavior of trace reactive species in the convective boundary layer (CBL)
is of considerable interest for determining the fate of substances emitted by
biogenic and anthropogenic sources or entrained into the CBL from the
overlying free troposphere (FT). These species may react photochemically or
with other species and may be aerosol precursors. If their reaction time
constants are between about 0.1 and 10 times the mixing time of the CBL,
which we estimate as

In order to model the behavior of reactive species correctly, it is important
to model both their vertical transport and effective reaction rates since the
coupling between the turbulence and the chemistry can have significant
impacts on the effective reaction rates and thus on the profiles of these
trace species and their products, many of which are important for air quality
and climate considerations. One example is the fate of O

The effects of chemical reactivity on mean and turbulence statistics of
species in the CBL have been investigated previously both with models and
observations. An early effort by

Here we report on continued development of a
second-order closure model of the CBL.
The immediate origins of the model – which we call the Second-Order Model for
Conserved and Reactive Unsteady Scalars (SOMCRUS) – go back to

Here we model a shear-free CBL and use free-convection surface-layer scaling,
but our scheme can easily be modified to run other parameterized boundary
layers (e.g., incorporating shear and canopy structure). We then apply SOMCRUS
first to a conserved species with differing surface and entrainment fluxes,
and second to the O

SOMCRUS is a further development of the model of

Here we extend the model of

SOMCRUS is a coupled second-order moment system for mean concentrations

The first equation is the mass conservation equation for the concentration of
scalars

The chemical moments on the right-hand side of Eqs. (

Equations (11)–(18) are formulated for first- and second-order chemical kinetics, but the moment chemistry scheme could be easily extended to other (higher-order) reactions.
Following

We also use the following parameterized second-order moments: (1)
the empirical formulation of

The time constants in Eqs. (

Due to the enormous complexities associated with real-world
observations, we turn to turbulence-resolving atmospheric
LES as a tool to evaluate the ability of SOMCRUS to simulate
the time evolution of passive and reactive scalars in the CBL. The
National Center for Atmospheric Research's (NCAR) LES was first
described in

The NCAR LES code integrates a set of three-dimensional,
wave-cutoff-filtered Boussinesq equations, where a Poisson equation
solves for the pressure. In the work described here, a thermodynamic energy
equation as well as a conservation equation for each of three passive
scalars and three reactive scalars are solved. Unresolved, or subfilter-scale
(SFS) processes, are accounted for by using

Horizontal derivatives are estimated using pseudospectral methods

The simulations use

Turbulent fluctuations from the LES are calculated as deviations from the
horizontal mean. Turbulence moments are then determined as
horizontally averaged fluctuation products which are then time-averaged using
a time-evolving vertical coordinate system according to the time-evolving CBL
depth. The CBL depth

The SOMCRUS Eq. (

We need to impose

In general, systems like SOMCRUS with top and bottom BCs are well-posed
mathematically, so we would expect a unique well-defined solution throughout
the domain

Our boundary conditions (BCs) are similar to those used by

We use

The next step is to solve Eqs. (

In order to demonstrate the performance of SOMCRUS, we compare SOMCRUS
results with those from LES using the same meteorological case as

We first compare the mean and moment profiles for three cases of a conserved
scalar using both SOMCRUS and LES at 10:00 LT, 12:00 LT, and 14:00 LT (see
Table

Diurnal cycles of virtual heat flux (blue) and boundary-layer height (orange).

Profiles for case A, which has a surface flux and an initial CBL
concentration, but zero concentration in the FT are compared in Fig.

Initial and prescribed values used for SOMCRUS and the LES numerical
experiments. The temperature and humidity surface fluxes, and mean profiles
are obtained from a
simple curve fit to observations from the Tropical Forest and Fire Emission
Experiment (TROFFEE), which is the same meteorological case used by

Specifications for the conserved tracers and the O

Comparing the vertical flux profiles in Fig.

Comparisons of concentration, flux, and variance between SOMCRUS
(blue curves) and LES (red curves) for a nonreactive scalar having 1 unit
initial CBL concentration, 1 unit m s

Comparison of SOMCRUS (blue curve) with the local free-convection
prediction of

Comparisons of concentration, flux, and variance between SOMCRUS
(blue curves) and LES (red curves) for a nonreactive scalar having no initial
CBL concentration, 6 units FT concentration, and 1 unit m s

Comparison of SOMCRUS concentrations (blue line) with large-eddy simulation (LES) (red line) of concentration, flux, and variance of a nonreactive scalar having zero initial CBL concentration and surface flux, and 10 ppbv FT concentration (Case C) at 10:00, 12:00, and 14:00 LT.

30th-order least squares polynomial fit to the LES surface flux of
O

Figure

Figure

Comparisons for nonreactive scalar case C at 10:00, 12:00, and 14:00 LT
are presented in Fig.

Overall we see from this comparison that the SOMCRUS and LES are in generally
good agreement for concentrations and fluxes, especially at the later times
when the differences in the entrainment process, which are most apparent at
10:00 LT, have less effect on the overall vertical structure because of the
increased CBL depth. However, SOMCRUS significantly underestimates the
variances near the CBL top – especially at later times. We also note that
SOMCRUS can reproduce the

We now consider the effects of chemical reactivity on the mean and moment
profiles for the O

The chemical reaction scheme used for the O

The mean concentrations for all three species at 10:00, 12:00, and 14:00 LT are
shown in Fig.

Figure

Comparison of SOMCRUS mean concentrations (blue lines) with LES
concentrations (red lines) of O

Comparison of SOMCRUS fluxes (blue lines) with LES concentrations
(red lines) of O

Comparison of SOMCRUS

Comparison of SOMCRUS species variances (blue lines) with LES (red
lines) of O

Comparison of SOMCRUS species–species covariances (blue lines) with
LES (red lines) of O

Intensities of segregation for the three combinations of O

A comparison of the

In order to maintain the convention of using capital letters for chemical species, we change the notation for mean/fluctuation of chemical species so that roman type represents a mean value and italic type represents a fluctuation.

andThe species variances are compared in Fig.

A comparison of the

Intensity of segregation, defined as

For the triad case modeled here,

The entrainment flux also generates species–species covariances that are
transported down to the surface, and here the covariances are relatively
large in magnitude so the intensity of segregation also becomes large in
magnitude. The Fig.

The effects of the intensity of segregation on the effective chemical
reaction rates are not included in, e.g., the boundary-layer parameterizations
of the Weather Research and Forecasting model coupled with Chemistry

The concept of an eddy diffusivity is often used in simplified models
involving diffusion in the CBL to parameterize turbulent mixing. We therefore
examine one obvious approach to this by applying the equations implemented in
SOMCRUS to derive an explicit formula for the eddy-diffusivity function

We might expect, therefore, that we could use Eq. (

A comparison of the flux-gradient profiles for the dynamic SOMCRUS
case considered here (red lines) versus the quasi-stationary diffusivity

We have extended the model of

Because SOMCRUS includes equations for species–species covariances, it can be
used to calculate intensities of segregation which can modify the reaction
rates for second-order chemical reactions. Although not very important
throughout most of the mixed layer for the case considered here (because of
the disparity between the turbulence mixing timescale and the chemical
reaction timescale for the O

A comparison of SOMCRUS profiles (solid lines) with profiles
obtained from the eddy-diffusion approximation Eq. (

We have shown that SOMCRUS provides a simple and robust tool for predicting
concentration, variance, and flux profiles of trace reactive species in the
CBL. SOMCRUS is intermediate in ease of use between simple mixed-layer models

SOMCRUS can easily be extended to include more complicated chemistry, such as
schemes involving isoprene and related reactions, and to incorporate
parameterizations for different surface boundary conditions and
meteorological regimes. Examples of this include a parameterized canopy layer
and surface stress. We believe that this tool has possibilities for use in
air quality models to more accurately simulate the behavior of reactive species
in the CBL. We note that software tools exist to convert

The standard technique for solving singular boundary-value problems known as
matched asymptotic expansions

Here we propose a regularization scheme for SOMCRUS that allows us to compute
solutions more efficiently than was the case for

We thank Jordi Vilà-Guerau de Arellano for his helpful discussions and comments, Leif Kristensen for paving the way with an antecedent version of SOMCRUS, and Mary Barth for her insightful review of the paper. We appreciate the encouragement of Alex Guenther and Thomas Karl in providing motivation for this work. The National Center for Atmospheric Research is sponsored by the National Science Foundation. Edited by: A. B. Guenther