Introduction
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 τ(t)=h/w*, where h(t) is the CBL depth and
w*(t) is the convective velocity scale,
w*=gT〈wθ〉0h1/3,
the species mean and flux profiles may be significantly modified from
conserved species profiles. In Eq. (), g is gravity, T is the
mean CBL temperature, and 〈wθ〉0 is the surface virtual
potential temperature flux. Typical mid-day CBL values are h≈1 km
and w*≈1 m s-1; thus τ≈1000 s.
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 O3 in the CBL in
the presence of other reactive species such as NO and NO2. Another is
volatile organic compounds emitted by vegetation that react with OH and other
oxidants. The interactions among these species are affected by the turbulence
in the CBL so that, for example, their flux-gradient relationships are
different than for conserved species. Yet, regional air quality and global
climate models currently do not take into account these effects even though
they may affect the predicted species concentrations.
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 showed the potential
importance of chemical reactivity for the O3–NO–NO2 triad in the
surface layer of the CBL. This was followed by a more quantitative analysis
of this triad in the surface layer by and an
analytical study by . More detailed numerical studies in
the surface layer were carried out by , ,
, and . also used a
one-dimensional second-order closure model to study nitrogen oxide chemistry
in the nocturnal boundary layer.
pointed out that locally inhomogeneous mixing of species
involved in second-order reactions, as measured by the intensity of
segregation (the ratio of the species–species covariance to the product of
their means), can change (generally decrease) their reaction rates.
extended consideration of chemical reactivity effects for
two reacting species – one emitted at the surface and the other entrained
across the CBL top – to the entire CBL using large-eddy simulation (LES) as a
tool and quantified the relationship between the effective reaction rate and
intensity of segregation. used LES to further study the
effects of turbulent mixing on the effective reaction rate between two
species, and also compared LES results with a second-order turbulence model
using several closures for the triple correlation terms. used
LES with a more detailed chemical scheme that included OH, HO2, and a
generic hydrocarbon RH in addition to the O3–NO–NO2 triad and
obtained a significant reduction in the RH reaction rate in the CBL due to
segregation effects, and also showed that nonuniform surface fluxes of RH
further slowed its reaction rate. showed, via LES, that both
fair-weather cumulus and the concentration of NO + NO2 can further modify
the reaction rate of isoprene and the O3 concentration.
used LES to elicit more details of terms in the covariance budgets of
chemically reactive species and proposed a parameterization for the intensity
of segregation of reactive species.
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
, who developed a second-order closure model to
investigate reactive species in the CBL. This work by was subsequently used by as a basis for a
simple, one-dimensional second-order closure model to obtain continuous
equilibrium profiles of turbulent fluxes and mean concentrations of
non-conserved scalars (the O3–NO–NO2 triad) in a steady-state
convective boundary layer without shear. The development here combines a
simple mixed-layer model of the diurnally varying CBL
from which we obtain the depth h(t), the mean virtual potential temperature
Θ, and the virtual potential temperature difference across the assumed
infinitesimally thin CBL top ΔΘ with a second-order model of the
turbulence and mean CBL structure for both conserved and reactive species
with surface sources and sinks, and turbulent entrainment of FT air across
the top of the CBL. SOMCRUS differs from in that
it: (1) explicitly calculates h(t) rather than using a prescribed h(t),
and (2) does not include parameterized diagnostic equations for the
third moments that appear in the second-moment equations. We found that not
including the third-moment equations significantly simplified setting up and
running the model while not greatly impacting the results.
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 O3–NO–NO2 triad, and compare the results with LES.
Description of models
SOMCRUS
Basic equations
SOMCRUS is a further development of the model of who
carried out similar studies using a second-order closure model to calculate
profiles of mean and turbulence statistics, but they considered only
steady-state solutions (dh / dt=0), with the entrainment rate of FT air
into the CBL balanced by a mean subsidence velocity.
Here we extend the model of by considering a
diurnally varying h(t), which typically varies greatly throughout the day,
starting near the surface early in the morning and increasing to a typical
depth of a kilometer or more by mid-afternoon. We first solve for h(t), the
mean mixed-layer virtual potential temperature Θ(t), and the virtual
potential temperature across the inversion at the top of the CBL
ΔΘ(t)=Θh(t)-Θ(t) simultaneously using the
mixed-layer approach developed by ,
γdhdt-dΔΘdt+γ∂w∂zh=(1+A)〈wθ〉0h,
dhdt+∂W∂zh=A〈wθ〉0ΔΘ,
dΘdt=(1+A)〈wθ〉0h,
where γ=∂Θ/∂z is the FT lapse rate, θ
denotes fluctuations in virtual potential temperature, ∂W/∂z is the large-scale CBL subsidence, and
A=-〈wθ〉h〈wθ〉0
is the negative ratio of the virtual potential temperature flux at h to the
surface temperature flux. We use the computed h(t) as an input into
SOMCRUS.
SOMCRUS is a coupled second-order moment system for mean concentrations
Si(z,t), fluxes 〈wsi〉(z,t), species-temperature
covariances 〈θsi〉(z,t), and species–species
covariances 〈sisj〉(z,t) where angle brackets
〈⋯〉 indicate ensemble averaging, which here can be
interpreted as averaging over a large enough horizontal domain to obtain
stable statistics. The moment equations have the general form of time change
+ vertical transport + mixing = chemical reaction moments. The relevant
equations for this analysis follow and .
The first equation is the mass conservation equation for the concentration of
scalars s̃i(z,t), where s̃i(z,t) is decomposed
into a mean and fluctuation, s̃i(z,t)=Si(z,t)+si(z,t),
where for simplicity for single variables we use the notation Si=〈sĩ〉. The mean profiles Si(z,t) obey a system of
differential equations,
∂Si∂t+∂〈wsi〉∂z=Ri.
Similarly, R̃i(z,t), which is the rate of
concentration change due to reactions with all other species and to
photochemistry, is decomposed as
R̃i(z,t)=Ri(z,t)+ri(z,t),i=1,2,…,N,
where
Ri=R̃i.
The first- and second-order chemical reaction rates are given by bji and
kjmi, respectively, where the left side contains the reactants and the
right side the products:
sj→bjisi,sj+sm→kjmisi.
This notation can be extended to higher-order chemical reactions if needed. The reaction rates for a species i are then given by
Ri=∑j,mkjmiSjSm+〈sjsm〉+∑jbjiSj,ri=∑j,mkjmiSjsm+sjSm+∑jbjisj.
As described in detail by , Eq. () is combined
with the three second-moment equations for the flux, temperature–scalar
covariance, and scalar–scalar covariance,
∂∂t〈wsi〉+〈w2〉∂Si∂z+〈wsi〉τ1-(1-B)gT〈θsi〉=〈wri〉,
∂∂t〈θsi〉+〈wθ〉∂Si∂z+〈θsi〉τ4=〈riθ〉,
and
∂∂t〈sisj〉+〈wsi〉∂Sj∂z+〈wsj〉∂Si∂z+〈sisj〉τ3=〈rirj〉,
to obtain a set of equations that can be solved for the mean and second-order
moments. Here we have neglected moments higher than two since
found them to be relatively unimportant. Comparing the
two systems with and without parameterized third-order moment terms,
mathematically the latter is first-order in time and space variables while
the former contains second-order derivative terms and requires an additional
set of boundary conditions and empirically determined constants. We found,
however, that adding the third-moment diagnostic expressions given by
to the second-moment equations reduces the gradients in the
mean concentration profiles and improves somewhat the comparison with LES.
The chemical moments on the right-hand side of Eqs. ()–() are
〈wri〉=∑k,mkkmi(Sk〈wsm〉+Sm〈wsk〉)+∑kbki〈wsk〉〈θri〉=∑k,mkkmi(Sk〈θsm〉+Sm〈θsk〉)+∑kbki〈θsk〉
〈rirj〉=∑k,m(kkmi(Sk〈smsj〉+Sm〈sksj〉+〈sjsksm〉)+kkmj(Sk〈smsi〉+Sm〈sisk〉+〈sisksm〉))+∑k(bki〈sksj〉+bkj〈sksi〉).
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 assume that the mean virtual potential
temperature gradient term in Eq. () is negligible in the CBL. The
constants in Eqs. ()–() are obtained as follows. For the
pressure-scalar covariance term in Eq. () we follow ,
, , and and use the
parameterization
1ρsi∂p∂z=〈wsi〉τ1+BgT〈θsi〉,
where B≃0.4 is a dimensionless constant and τ1=τ1(z) the
“return to isotropy” timescale. This parameterization is based on
LES of the CBL, and is widely used in second-order models
of the CBL. Likewise, the viscous terms in Eqs. () and ()
have been parameterized by “return to isotropy” timescales τ4(z) and
τ3(z), respectively:
(νθ+νs)〈∇θ⋅∇si〉=〈θsi〉τ4(z)2νs〈∇si⋅∇sj〉=〈sisj〉τ3(z).
We also use the following parameterized second-order moments: (1)
the empirical formulation of for 〈w2〉
〈w2〉=1.8w*2z*2/31-0.8z*2,
where z*=z/h, and (2) the commonly accepted empirical formulation e.g., for 〈wθ〉(z),
〈wθ〉=〈wθ〉0(1-1.2z*).
These expressions result from a combination of both observations and laboratory experiments.
The time constants in Eqs. ()–() and Eqs. ()–() are parameterized as
τi=τTKE/ai=18aiκz(1-z*)〈w2〉1/2,i=1,3,4,
where ai are dimensionless constants, κ=0.4 is the
von Kármán constant, and τTKE is the turbulent kinetic energy
timescale in mid-CBL. This is similar to , except that we
use 18 instead of 10 as the constant in Eq. (). We do this so that
τTKE≈2.8h/w* in mid-CBL, as suggested by the LES results of
. This differs from , who assumed that
τTKE≈h/w*. Another difference from is that,
as pointed out by , the predicted free-convection
surface-layer relationship for the normalized eddy
diffusivity given by
Kθw*h=-1w*〈wsi〉0∂Si/∂z*=z*4/3,asz*→0,
leads to the relation
3a11.8+3a4=1.
In order to fulfill this condition, we modify the values of {a1,a4}={4.85,2.5} given by to {7.67,3.96} so as to both
maintain the same ratio a1/a4 as and fulfill Eq. (). The other two constants used here, {a3,B}={2.5,0.4},
are the same as in .
Description of LES model
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 and , and was
subsequently modified by , , , , and . Over the years, the NCAR LES has proven
its ability to simulate observed atmospheric statistics across a
wide variety of atmospheric situations and surface characteristics
e.g., and
has therefore become a close counterpart to field campaigns. Since
most of the LES code has been previously described,
we present here only a limited discussion of the current code.
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 's
() 1.5-order TKE model. Reactive scalars
are presumed to mix like passive scalars at scales smaller than the
filter width.
Horizontal derivatives are estimated using pseudospectral methods
, and vertical derivatives use a second-order
centered-in-space finite difference scheme for velocity fields and
's () method for all scalar
fields. A third-order Runge–Kutta scheme advances the solutions
in time .
The simulations use 256×256×256 grid points to resolve
a 5.12×5.12×2.56 km3 domain. Therefore, the grid
resolution is (20, 20, 10) m in the (x, y, z) directions,
respectively. Periodic boundary conditions are imposed in the
horizontal directions. 's ()
radiation boundary condition handles the upper boundary conditions.
No-slip conditions are enforced at the ground surface, where the
surface stress is calculated following Monin–Obukhov Similarity
Theory (MOST) from a prescribed surface roughness length and the
velocity or scalar mixing ratio at one-half grid point above the
surface, where no modification to MOST is imposed for reactive
scalars.
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 h is estimated using the LES fields as the height of
the minimum buoyancy flux.
Implementation of SOMCRUS
The SOMCRUS Eq. () and Eqs. ()–() contains 3n+n(n+1)/2 partial differential equations for the following variables: mean
concentrations, Si(z,t); vertical eddy fluxes, 〈wsi〉;
temperature-species covariances, 〈θsi〉; and
species–species variances and covariances, 〈sisj〉, where
1≤i≤j≤n, and n is the total number of species. The combined
PDE system is configured so that it can be solved in a space–time region
consisting of a full or partial diurnal cycle, t0<t<t1, where t0
is the initial time (e.g., sunrise, or earlier), and t1 is the final time
(e.g., sunset) with time-dependent spatial boundaries given by the CBL height,
0<z<h(t), using the mixed-layer Eqs. ()–().
We need to impose 3n+n(n+1)/2 boundary conditions (BCs). We impose an entrainment relationship for species fluxes
across the CBL top,
〈wsi〉h=-weSi(h+)-Si(h-),
where we is the entrainment velocity, Si(h+) is the concentration just
above the CBL top, and Si(h-) the concentration just below the top. We
also specify surface values for the temperature and species fluxes as well as
for the species variances and temperature–species covariances.
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 {0<z<h(t)} for the species concentrations and
second-order moments. There are, however, some serious mathematical and
numerical problems that can have significant impact on the CBL structure and
need to be addressed in the time-dependent CBL due to the singular nature of
the parameterized functions: namely, at the lower boundary (z*=0) the
parameterized moment 〈w2〉(z*), the timescales
τi(z*), and many coefficients (e.g., the eddy diffusivity) vanish. This
is a well-established feature of surface-layer dynamics
e.g., and has important implications for analysis and
solutions of CBL systems that attempt to simulate surface-layer structure,
namely: (1) proper choice and setup of BCs, (2) structure of the solutions,
and (3) mathematical and numerical techniques for solving such systems.
did not attempt to deal with this problem and thus did not
resolve surface-layer structure in a time-varying (diurnal) model as we do
here, which may have significant impact on the overlying CBL structure. In
the Appendix we lay out our technique for solving the set of Eqs. ()
to () in a way that allows us to resolve the surface-layer
structure and gives an efficient way to solve the moment equations throughout
the CBL.
Our boundary conditions (BCs) are similar to those used by .
We specify the surface species fluxes 〈wsi〉0(t); the surface
variances and covariances are specified based on relations obtained by
from observations in the free-convection regime:
〈θsi〉0=1.66〈wθ〉0〈wsi〉0w*2z*-2/3〈sisj〉0=1.66〈wsi〉0〈wsj〉0w*2z*-2/3.
At the lower boundary z=z0 (z0/h is set equal to 10-3 for
numerical calculations; note that z0 is not the roughness length but a
lower boundary condition for solving the differential equation set Eqs. ()–(), as we assume a free convection boundary
layer). Similarly, because of the discontinuity at the top boundary (z=h),
which causes numerical difficulties, we actually use z=0.993h in
SOMCRUS; henceforth for simplicity, we redefine h as the height used in
SOMCRUS.
We use Mathematica at all stages of the model
development, implementation, and simulations. The mixed-layer Eqs. ()–(), are first solved using the Mathematica
differential equation solver, and the calculated values for h(t) and
Θ(t) are used in SOMCRUS, Eqs. ()–().
SOMCRUS is designed to cleanly separate the turbulent mixing terms in the
moment equations from the chemical reaction terms in the system of Eqs. ()–(). Mathematica allows us to generate
the entire SOMCRUS system in two steps: (1) using symbolic algebra tools we
generate from the basic chemical suite of species and reactions the complete
moment chemistry; (2) parameterized CBL mixing along with the mixed-layer
solution for {h(t),Θ(t)} allows us to generate the turbulent
mixing part of the system in regularized form, Eqs. ()–().
The next step is to solve Eqs. ()–() with the given
boundary conditions. The Mathematica solver does this by a proper
spatial discretization scheme whose inputs (resolution, difference order,
etc.) can be controlled. Thereby a system of partial differential equations
is converted into a large (coupled) set of ordinary differential equations
solved by time-adaptive numeric codes. The output of the Mathematica
solver is a set of interpolating functions over a prescribed space–time
range. A single run for a conserved species with a spatial resolution of 100
points in x takes about 30 s of desktop computing time. A system of three
reactive species – i.e., the O3–NO–NO2 triad (15 equations) – at the same
resolution takes 100–200 s of desktop computing time, depending on the
spatial and temporal resolution used in solving the equations. The system
size increases with the number of reactive species; e.g., for 10 reactive
species, 85 equations must be solved.
SOMCRUS evaluation and results
Case description
In order to demonstrate the performance of SOMCRUS, we compare SOMCRUS
results with those from LES using the same meteorological case as
; namely, 15-day averaged observations from the Tropical
Forest and Fire Emission Experiment TROFFEE,. The initial
and boundary conditions in the numerical experiments are presented in Tables and . The geostrophic wind is 0 m s-1 (i.e.,
local free convective conditions). No large-scale forcings (i.e., no
horizontal heat and moisture advection, subsidence, nor radiative tendencies)
are prescribed. Turbulence is initiated in the LES by imposing a
divergence-free random perturbation field on the velocity and temperature
fields in the lowest 200 m. The LES results presented in Figs. – represent 1-hour averages centered at the depicted
times. The simulation begins at 05:00 local time (LT) and lasts 13 h
(sunrise is at 06:00 LT and sunset at 18:00 LT). The depth of the CBL
calculated by SOMCRUS and the surface temperature flux are shown in Fig. .
Conserved species means and moments
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 for the meteorological initial and boundary conditions of the
variables). Each scalar case (labeled “case A”, “case B”, and “case C”) has
different initial conditions and BCs as specified
in Table . We present these three conserved scalar cases to
demonstrate the ability of SOMCRUS to reproduce vertical mixing in the CBL
and the influence of surface or entrainment fluxes in the absence of
reactivity.
Diurnal cycles of virtual heat flux (blue) and boundary-layer height
(orange).
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Profiles for case A, which has a surface flux and an initial CBL
concentration, but zero concentration in the FT are compared in Fig. . This case illustrates the effects of both a surface source and
entrainment on the evolving CBL, but since the FT concentration is zero, the
total mass of species within the CBL (i.e., the area under the curve)
is not affected by entrainment and is the same for both SOMCRUS and LES. We
see that particularly at 10:00 LT the concentration distribution around the
CBL top is more spread out vertically in the LES than for SOMCRUS, which has
a step change in concentration at the CBL top. This smearing out is because
the LES resolves horizontal variations in the CBL structure – in particular,
horizontal variations in the CBL top. The LES also predicts a CBL depth about
150 m higher than SOMCRUS, which is consistent with the results of
, who used a similar mixed-layer model and made similar
comparisons of h with LES for the same case as here. These two features
result in a SOMCRUS CBL concentration that is larger than the LES
concentration. Furthermore, the LES predicts a smaller gradient throughout
the CBL, which increases the difference between the two concentration
profiles near the surface as compared to the upper part of the CBL. The
maximum difference of about 12 % occurs at 10:00 LT at z*≈0.06.
Later, at 12:00 and 14:00 LT these differences, although still present, are
less pronounced and thus the agreement between SOMCRUS and LES is improved.
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 ;
see also . All initial conditions are imposed at 05:00 LT, and t is
time in seconds. The subscripts ()0 and ()h refer to the surface and CBL top, respectively.
Property
Value
Initial CBL height, h (m)
200
Surface virtual potential temperature flux (K m s-1)
〈wθ〉0=0.19sinπ(t-8100)28 800
(from 07:25 to 15:25 LT)
SOMCRUS Ratio of entrainment to
〈wθv〉h/〈wθv〉0=-0.2
surface virtual temperature flux
Virtual potential temperature profile (K):
z<200.0 m
299.0
200.0 m <z< 212.5 m
300.0
z>212.5 m
300.0 + 6×10-3z
Surface moisture flux (g kg-1 m s-1)
〈wq〉0=0.13sinπ(t-3600)37 800
(from 06:00 to 16:50 LT)
Mixing ratio profile (gm kg-1):
z<200.0 m
15.0
200.0 m <z<212.5 m
15.0
z>212.5 m
10.0
Specifications for the conserved tracers and the O3–NO–NO2
triad in the numerical experiments with SOMCRUS and LES. The free-troposphere
(FT) concentration is constant in time; the convective boundary layer (CBL) concentration and the height h vary with time.
Scalar
Surface flux
FT concentration
CBL initial concentration
case A
1 unit m s-1
0
1 unit
case B
1 unit m s-1
6 units
0
case C
0
10 units
0
O3
-2.5×10-3 O3(5 m) ppbv m s-1
20 ppbv
2 ppbv
NO
5×10-4 ppbv m s-1
0
0.01 ppbv
NO2
0
0
0.1 ppbv
Comparing the vertical flux profiles in Fig. for case A at the
same three times, we see that the 10:00 LT LES flux is more spread out
vertically, analogous to the concentration, and extends to a higher level
than the SOMCRUS flux, with the difference increasing with height up to h.
This results in about a 12 % larger flux maximum for SOMCRUS than for the
LES. At later times, the LES and SOMCRUS fluxes are in very good agreement,
except near the top where the LES flux is again more spread out. The right
column of Fig. shows a comparison of SOMCRUS variances with LES
variances for case A. We see that the LES predicts the height of the variance
maximum near the CBL top to be about 150 m higher than SOMCRUS, consistent
with the predicted higher LES mixed-layer depth. The LES maximum variance is
slightly larger than SOMCRUS at 10:00 LT and subsequently decreases more
slowly than SOMCRUS so that by 14:00 LT the SOMCRUS variance is only about
17 % of the LES variance. This is likely occurring because the SOMCRUS
variance depends explicitly on the CBL growth rate and the jump in
concentration across the CBL top, while the LES variance, being a horizontal
average, also incorporates contributions from the horizontal variations in
CBL height, which are not included in the SOMCRUS results. The SOMCRUS
variance is also strongly dependent on the value of a3, but adjusting
a3 does not address the more rapid decrease in SOMCRUS variance with time
compared with LES; furthermore, decreasing a3 to obtain a better match to
the LES variance near the CBL top also increases the SOMCRUS variance near
the surface, which then worsens the comparison of SOMCRUS variance with the
LES variance.
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-1 initial surface flux, and zero
FT concentration (Case A) at 10:00, 12:00, and 14:00 LT.
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Comparison of SOMCRUS (blue curve) with the local free-convection
prediction of (green dashed curve) and with LES (red dots)
for conserved scalar case A at 10:00 LT. Each dot denotes a layer-averaged
LES value.
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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-1 surface
flux (Case B) at 10:00, 12:00, and 14:00 LT.
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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
O3.
![]()
Figure shows the variance of the same case A of Fig.
at 10:00 LT for the lowest 100 m of the CBL. Here we compare the variance with
both the LES and with the local free-convection prediction originally
presented by using dimensional analysis and observational
results for temperature variance; later found that this
relation, given below, also worked well for humidity variance observations:
〈s2〉s*2=1.8z*-2/3,
where s*=〈ws〉0/w*. Note that the dependency on h cancels out, and we have
〈s2〉=1.8〈ws〉02gT〈wθ〉0z-2/3.
We see that the SOMCRUS variance agrees well with the LES prediction to
within about 40 m of the surface, while the LES does not capture the
z-2/3 dependency close to the surface. We note that
have pointed out that it may be possible for the LES to reproduce this
additional near-surface scalar variance if an additional equation for
subfilter-scale scalar variance were incorporated akin to that used by
– a feature not yet implemented in the NCAR LES. The
SOMCRUS variance profile has a shape similar to that of the free-convection
prediction, but is systematically larger by about 0.2 units2.
Figure shows the same set of profiles for case B, which has no
initial CBL concentration, 6 units FT concentration, and 1 unit m s-1
surface flux. The results are very similar to case A; the combination of
surface flux and entrainment results in a CBL concentration remarkably close
to case A. Again at 10:00 LT the SOMCRUS concentration is larger than the LES
concentration throughout the CBL, with the difference decreasing towards the
CBL top, and the LES concentration exceeding the SOMCRUS concentration in the
entrainment region near the CBL top. At 12:00 and 14:00 LT, the
concentrations are in very good agreement, with the SOMCRUS concentrations
slightly exceeding the LES concentrations near the surface because of a
smaller vertical gradient in the LES concentrations.
Comparisons for nonreactive scalar case C at 10:00, 12:00, and 14:00 LT
are presented in Fig. . This case has no surface flux nor CBL
concentration, but an initial FT concentration of 10 units, so it illustrates
the effects solely of entrainment on the CBL vertical structure. Here we see
almost perfect agreement between the LES and SOMCRUS concentrations, except
near the top where the LES variables are again more spread out. The
comparison of SOMCRUS variances with LES variances shows that the variance
near the CBL top is similar to case A in that the SOMCRUS variance decreases
more rapidly with time than the LES variance. In the lowest 200 m of the CBL
the SOMCRUS variance becomes negligible since it depends on the surface flux,
while the LES variance, particularly at 10:00 LT, is still about 10 % of the
maximum variance near the CBL top. Thus, for the LES, variance generated by
the entrainment flux is transported all the way down to the surface.
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 free-convection prediction for
the z-2/3 dependency of scalar variance down to very near the surface.
O<inline-formula><mml:math display="inline"><mml:msub><mml:mi/><mml:mn mathvariant="normal">3</mml:mn></mml:msub></mml:math></inline-formula>–NO–NO<inline-formula><mml:math display="inline"><mml:msub><mml:mi/><mml:mn mathvariant="normal">2</mml:mn></mml:msub></mml:math></inline-formula> means and moments
We now consider the effects of chemical reactivity on the mean and moment
profiles for the O3–NO–NO2 triad. The reaction rates are given in
Table and the initial conditions in Table . These reactions
are fast enough (on the order of a hundred seconds around mid-day, increasing
at low sun angles) that the reaction time is comparable to the turbulence
timescale, h/w* early in the day. The LES surface O3 flux is specified
as a deposition velocity (0.0025 m s-1) times the resolved O3
concentration at the lowest grid level, which for scalars is 5 m above the
surface. It is not straightforward to apply this boundary condition directly
in SOMCRUS, although it can be done by extrapolating the 5 m O3 SOMCRUS
concentration down to the lowest level used in the SOMCRUS formulation
(z0/h=10-3). Therefore, to ensure as direct a comparison as possible
with the LES, we impose a boundary condition for O3 flux in SOMCRUS that
arises via a 30th-order polynomial fit to the time evolution of the
horizontally averaged O3 surface flux predicted by the LES, as shown in
Fig. .
The chemical reaction scheme used for the O3–NO–NO2 triad in
the numerical experiments with SOMCRUS and LES. χ is the zenith angle.
Number
Reaction
Reaction rate
R1 (bji)
NO2+hν→ NO + O3
1.67×10-2×exp[-0.575/cosχ]s-1
R2 (kjmi)
NO + O3→ NO2 + (O2)
3.00×10-12×exp[-1500/T(z,t)] cm3 molecule-1 s-1
The mean concentrations for all three species at 10:00, 12:00, and 14:00 LT are
shown in Fig. . We see that the agreement between SOMCRUS and LES
is very good for O3, again subject to the effects of a smaller CBL depth
h for SOMCRUS compared to that predicted by LES, but for NO + NO2 –
i.e., for the total odd nitrogen which is conserved – the LES predicts a higher
concentration than SOMCRUS. This is because the LES imposes a rough-wall
stability-corrected boundary condition that treats reactive scalars as
passive; that is, no reactivity is permitted between the surface and the
first grid point in the domain. As a result, for reactive species such as NO,
NO2, and O3 during daytime whose reactive timescale is of the order of
a minute or two, the LES domain produces a surface flux, in this case an NO
surface flux, that appears slightly larger than that imposed. The LES also
predicts a larger vertical gradient for NO than SOMCRUS for 12:00 and 14:00 LT.
This is somewhat puzzling since NO should be in approximate chemical
equilibrium throughout most of the mixed layer, but with positive surface and
entrainment fluxes.
Figure shows a comparison of SOMCRUS species flux profiles in the
CBL (blue lines) with LES predictions (red lines) for the O3–NO–NO2
triad. The SOMCRUS produces the non-linearity in the vertical flux profiles
resulting from the chemical reactions, similar to the LES. We also note the
effects of the greater vertical spread over which the entrainment processes
occur in the LES similar to what was observed for the conserved scalar cases.
Both models produce about the same curvature in the lower half of the CBL,
and because NO + NO2 is conserved, the sum of the NO and NO2 fluxes is
a straight line.
Comparison of SOMCRUS mean concentrations (blue lines) with LES
concentrations (red lines) of O3, NO, and NO2. Initial and boundary
conditions are given in Table . Top panel is at 10:00, the middle panel
at 12:00, and the bottom panel at 14:00 LT.
![]()
Comparison of SOMCRUS fluxes (blue lines) with LES concentrations
(red lines) of O3, NO, and NO2. Initial and boundary conditions are
given in Table . Top panel is at 10:00 LT, the middle panel at
12:00 LT, and the bottom panel at 14:00 LT.
![]()
Comparison of SOMCRUS θ-species covariances (blue lines) with
LES (red lines) of O3, NO, and NO2 at 12:00 LT. Initial and boundary
conditions are given in Table . Top panel covers the entire CBL, while
the bottom panel is up to 1 km to accentuate the region below the CBL top.
![]()
Comparison of SOMCRUS species variances (blue lines) with LES (red
lines) of O3, NO, and NO2 at 10:00, 12:00, and 14:00 LT. Initial and
boundary conditions are given in Table . Top panel is at 10:00 LT,
the middle panel at 12:00 LT, and the bottom panel at 14:00 LT.
![]()
Comparison of SOMCRUS species–species covariances (blue lines) with
LES (red lines) of O3, NO, and NO2. Initial and boundary conditions are
given in Table . Top panel is at 10:00 LT, the middle panel at
12:00 LT, and the bottom panel at 14:00 LT.
![]()
Intensities of segregation for the three combinations of O3 NO,
and NO2 at 10:00, 12:00, and 14:00 LT.
![]()
A comparison of the 〈θsi〉 covariance profiles at
12:00 LT in Fig. shows that near the surface, the LES and SOMCRUS
profiles are very similar. Since the surface flux of ozone is negative and
the temperature flux positive, 〈θsi〉 is negative; the
NO flux is positive at the surface and the NO2 flux is positive just above
the surface (due to chemical reaction), thus 〈θNO〉 and 〈θNO2〉 are both
positive near the surface. The SOMCRUS covariances decrease in magnitude
throughout the mixed layer and change sign near the CBL top, while the LES
covariances change sign about midway up, with a large positive 〈θO3〉 peak at the CBL top because of the positive jumps in
both Θ and O3 across the top, and large negative peaks in both
〈θNO〉 and 〈θNO2〉 because of the
negative jumps in NO and NO2 across the top. The SOMCRUS peaks behave
similarly, but with much smaller peak magnitudes. We note that in the
〈θsi〉 covariance equations, the generation term
〈wθ〉∂Si∂z
is a sink for 〈θO3〉 and a source for 〈θNO〉 and 〈θNO2〉 throughout most of the CBL.
On the other hand, the result of the SOMCRUS assumption of a zero gradient in
virtual potential temperature means that the term
〈wsi〉∂Θ∂z
is neglected in SOMCRUS, while in the LES, for ∂Θ/∂z>0, this is a source for 〈θO3〉, and a sink for
〈θNO〉 and 〈θNO2〉. Thus we
conclude that SOMCRUS may have some shortcomings in realistically modeling
this process compared to the LES; one possibility to address this may be to
incorporate a modeled virtual potential temperature gradient in SOMCRUS.
The species variances are compared in Fig. , and we see that the
LES variances are consistently larger than the SOMCRUS variances throughout
the CBL. Near the surface, the SOMCRUS species variances are negligible, as
in the conserved case C (Fig. ) with no surface flux, because the
surface flux for NO2 is zero, and the O3 and NO surface fluxes are not
large enough to generate variances comparable to those generated by
entrainment near the CBL top. On the other hand, the LES is able to transport
this entrainment-generated variance down to the surface, particularly at
10:00 LT.
A comparison of the 〈sisj〉 covariances in Fig.
shows that SOMCRUS generates generally smaller species peak covariances in
the entrainment region than the LES, and a more rapid decrease with time as
the entrainment rate decreases. As with the variance and the 〈θsi〉 covariances, throughout most of the CBL the SOMCRUS 〈sisj〉 covariances are considerably smaller than the LES. In the
entrainment region, SOMCRUS second moments are generated by the entrainment
flux and do not include contributions from the undulating capping inversion
that are present in the LES because of horizontal averaging. Covariances of
two species involved in a second-order chemical reaction can alter the
effective reaction rate since the rate is proportional to the concentration
of both species. For 〈O3NO〉, however, the covariance may
be significant near the surface, but is not large enough to significantly
impact the chemical reaction rate throughout the bulk of the mixed layer.
This is because the chemical reaction timescale (of order 100 s) is much
less than the mixing timescale h/w*; but for second-order reactions that
may occur on timescales comparable to h/w*, the covariances can
significantly affect the reaction rates throughout the CBL
e.g.,.
Intensity of Segregation
Intensity of segregation, defined as
Iij=〈sisj〉SiSj,
quantifies the change in effective reaction rate resulting from the
covariance of two species involved in a second-order chemical reaction.
Therefore, for the triad, the covariance 〈O3NO〉 can
change the effective reaction rate for these two species, according to the
relationship given by, e.g., ,
kkmi(effective)=kkmi(1+Ikmi).
Reaction (R2) in Table is first order, and therefore the other two
species–species covariances do not affect the reaction rates.
For the triad case modeled here, 〈O3NO〉 is relatively
small near the surface (Fig. ) because the surface fluxes of both
O3 and NO are relatively small. Therefore, the turbulence makes little
change to the reaction rate near the surface in both the SOMCRUS and LES
results, although for SOMCRUS the 〈O3NO〉 intensity of
segregation increases negatively very near the surface, as it should for
species with surface fluxes of opposite sign. Similarly, the 〈O3NO2〉 intensity of segregation also shows a negative increase
approaching the surface. This results from the negative O3 flux producing
negative fluctuations in NO2 via chemical reactivity. Similarly, the
positive NO flux produces positive NO2 flux, which produces positive
〈NONO2〉 intensity of segregation near the surface.
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. plots are cut off at the top of the
SOMCRUS-predicted h – i.e., about 150 m below the LES top – since above about
this level, the LES intensities of segregation become ill defined because the
mean concentrations of NO and NO2 are zero in the FT. For this case, at
10:00 LT 〈O3NO〉 reduces the reaction rate in both the
SOMCRUS and the LES results by as much as 5 % near the entrainment zone.
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
WRF-Chem,, which is used to simulate the emission,
transport, mixing, and chemical transformation of trace gases and aerosols
simultaneously with meteorology for investigation of regional-scale air
quality, field program analyses, and cloud-scale interactions between clouds
and chemistry; nor in the mixed-layer model described by which
examines the evolution of isoprene in the CBL. We also note that if we were
to use a more complete chemical mechanism such as Model for Ozone and Related
chemical Tracers, version 4 MOZART-4,, the influence of
the intensities of segregation might be enhanced/reduced as a result of in
situ species production via alternate chemical production.
Eddy diffusivity
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
K(z,t)=-〈ws〉/(∂S/∂z).
For a conserved scalar, using Eqs. () and () we have
∂∂t〈ws〉+〈w2〉∂S∂z+〈ws〉τ1-gT(1-B)〈θs〉=0
∂∂t〈θs〉+〈wθ〉∂S∂z+〈θs〉τ4=0.
For steady-state conditions, ∂∂t〈ws〉=∂∂t〈θs〉=0, and Eqs. () and () can be solved for 〈ws〉 and
〈θs〉:
〈ws〉=-τ1〈w2〉+gT(1-B)τ4〈wθ〉∂S∂z
〈θs〉=-τ4〈wθ〉∂S∂z.
Then the eddy diffusivity is
K=τ1〈w2〉+gT(1-B)τ4〈wθ〉.
considered the stationary case where the CBL depth did
not change with time because the buoyancy-driven entrainment rate was
balanced by the mean subsidence. In that case, Eqs. () and
() are exact. Here, however, the time changes are not zero, so
there is no reason to expect a priori that the stationary relation Eq. () correctly describes the dynamic case under consideration.
Interestingly, the “quasi-stationary” flux-gradient relation Eq. () holds consistently at all times t. To demonstrate this, we
use as an example a case with the same meteorological conditions as the
previous case, but with the following differences in the scalar variable: no
initial concentration and a surface flux of 〈ws〉0=0.05
units m s-1. We still use the same Mathematica implementation
scheme, including the changes in variables. Figure shows that
there is little difference between two sets of profiles.
We might expect, therefore, that we could use Eq. () to calculate
the S(z,t) profiles for the dynamic case considered here by solving the
eddy-diffusion equation
∂S∂t=∂∂zK(z,t)∂S∂z.
However, unlike SOMCRUS, whose solutions are almost completely independent of
z0, the eddy-diffusion approach is very sensitive to z0 because of the
singular surface boundary condition,
K(z,t)∂S∂zz0=〈ws〉0,
with K(z,t)∼O(z4/3). In Fig. we see that the eddy
diffusion approximation can capture the behavior of the concentration and
flux profiles for this test case, but it requires a high-resolution
calculation in Mathematica because this singular surface boundary
condition creates a large gradient in the concentration near the surface resulting in a sensitive dependence of computed profiles on surface flux and system discretization.
Figure shows that 100-point numerical resolution significantly
underestimates both the surface flux and concentration, but that both can be
adequately resolved with 1000 point resolution. SOMCRUS, however, is very
stable to boundary conditions at the surface because the flux and
concentration equations are separate and the flux equation is regular at
z=0, while in the explicit diffusivity formulation, the two equations are
linked. Another advantage of SOMCRUS, of course, is that it generates
second-order moments and intensity of segregation. Although it may seem more
straightforward to use an eddy diffusivity, we point out that this does not
save computational time compared to SOMCRUS.
A comparison of the flux-gradient profiles for the dynamic SOMCRUS
case considered here (red lines) versus the quasi-stationary diffusivity
K(z,t) derived from the SOMCRUS parameterizations (blue lines).
![]()
Conclusions
We have extended the model of to treat the behavior of
conserved and reactive species in the diurnally varying CBL by using: (1) the
mixed-layer model to calculate mixed-layer height, mean
virtual potential temperature, and virtual potential temperature jump across
the CBL top, and (2) a second-order moment closure model to calculate mean and
turbulence statistics of reactive species throughout the daytime. Comparing
SOMCRUS with a turbulence-resolving LES for a free-convection case, we note
that SOMCRUS has a discontinuous jump across the CBL top, while horizontal
averaging of the LES output smears out the variables across the top. We also
found: (1) generally good agreement for concentrations and fluxes of both
conserved and reactive species throughout most of the mixed layer, including
the curvature in the flux profiles throughout the CBL due to chemical
reactions; and (2) SOMCRUS mostly underpredicts the variances and covariances
compared to LES, indicating that the time constants used in the second-moment
equations in SOMCRUS for parameterizing the rates of dissipation and
return-to-isotropy terms may not be optimal. SOMCRUS is able to model the
rapid changes in concentrations, variances, and covariances in the surface
layer to within a few meters of the surface, as predicted by free-convection
similarity theory. We also show that using an eddy-diffusivity formulation
for vertical transport is problematical for a time-varying CBL because of the
inherent singularity as the diffusivity goes to zero approaching the surface,
which is not an issue for SOMCRUS because the flux and concentration
equations are separate and the flux equation is regular at z=0.
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 O3–NO–NO2 triad), this effect can be
significant for other reactive species in the CBL e.g.,.
A comparison of SOMCRUS profiles (solid lines) with profiles
obtained from the eddy-diffusion approximation Eq. () (dashed
lines) for concentration (left) and flux (right) of a conserved species for
three times: 10:00 LT (blue lines), 12:00 LT (orange lines), and 14:00 LT
(olive lines); and for two numerical resolutions: N=100 points (top) and
N=1000 points (bottom).
![]()
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
e.g., and large-eddy simulation models. SOMCRUS also
provides considerably more detail of the vertical variation of first- and
second-order species statistics than a mixed-layer model. Furthermore, it is
portable, requires little time to run on a PC or laptop using
Mathematica, and it is easy to change and to quickly make runs with
different scenarios.
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 Mathematica
code to Fortran and C++ (e.g., https://store.wolfram.com/view/app/mathcodef90)
and that the SOMCRUS code contains separate turbulent mixing and chemistry
modules that could in principle be independently incorporated into a
larger-scale numerical model. SOMCRUS can be obtained in the
currently reported scalar and O3–NO–NO2 triad Mathematica notebook
configuration by requesting a copy from lenschow@ucar.edu.