Introduction
Micrometeorological measurements of vertical turbulent scalar fluxes in the
atmospheric boundary layer (ABL) are usually carried out at altitudes zM≥ 1.5 m due to technological limitations of the eddy covariance
method. The measurement results are often attributed to the exchange of heat,
moisture and gases at the surface. This procedure is not justified for
inhomogeneous surfaces because of a large area contributing to the flux, and
because of variability of the second moments with height. The relationship
between the surface flux Fs(x,y,0) and the flux Fs(xM,yM,zM), measured in point xM=(xM,yM,zM),
can be formalized via the footprint function fs:
Fs(xM,yM,zM)=∫-∞∞∫-∞∞fs(x,y,xM,yM,zM)Fs(x,y,0)dxdy.
Schematic representation of the footprint evaluation algorithm.
(a) Setup of the numerical experiment. (b) Example of two
trajectories (red and blue bold curves). Shifted trajectories are shown by
the dashed lines. The particle brings the impact into the value
fs(xS,yS,xM) if it intersects the test area δM
in the vicinity of the sensor position xM and the origin of the
modified trajectory belongs to the test area δS.
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Traditionally, footprint functions
fsd(xd,yd,xM)=fs(x,y,xM) are
expressed in a local coordinate system with the origin that coincides with
the sensor position (here, xd=xM-x is the positive upwind distance
from the sensor and yd=yM-y is the crosswind distance; see
Fig. a). In a horizontally homogenous case these functions do not
depend on xM and yM. In the ABL the surface area contributing to the
flux is elongated in the wind direction; therefore, the crosswind-integrated
footprint function fsy defined as
fsy(xd,zM)=∫-∞∞fsd(xd,yd,zM)dyd
is one of the
most required characteristics for the practical use.
The measurements of the scalar flux footprint functions in natural
environment are restricted e.g., due to the necessity to conduct the emission and
detection of artificial tracers. Besides, such measurements are not available
for the stably stratified ABL, where the area of the surface influencing the
point of measurements increases.
Modeling approaches used for footprint calculation include stochastic models,
such as single-particle first-order Lagrangian stochastic models based on the
generalized Langevin equation (LSMs) and zeroth-order stochastic models (also
known as the random displacement models, RDMs) (see the reviews listed in the
papers and the monograph
). Besides, one can use the analytical models (e.g.,
) and the parameterizations based on
the scaling approach . All of these models
should be calibrated against the data considered to be representative of real
processes. Their results depend on the choice of universal functions in the
ABL or in the surface layer (non-dimensional velocity and scalar gradients,
non-dimensional dissipation, dispersion of the velocity components, etc.).
Commonly, the applicability of the analytical models is limited by a
“constant flux layer” simplification, assuming that the measurement height
zM is much less than the thickness of the ABL zi. However, under the
strongly stable stratification the thickness zi may be several meters;
therefore, the vertical gradients of momentum and scalar fluxes near the
surface can be large. It can lead to incorrect functioning of the models
designed for and tested on the data gathered under different conditions.
Large-eddy simulation (LES), employing the Eulerian approach for the
transport of scalars, was applied for the first time for a footprint
calculation in . Modern computational technologies allow
one to combine Eulerian and Lagrangian methods for turbulence simulation and
particle transport e.g., and to perform detailed calculations of averaged
two-dimensional footprints under different types of stratifications in the
ABL and footprints fs(x,y,xM) over heterogeneous
surfaces (for example, urban surfaces and surfaces with alternating types of
vegetation). Some examples of such calculations are given in
and .
Lagrangian transport in LES is complicated by the problem of the description
of small-scale (unresolved) fluctuations of the particle velocity, which is
similar to the problem of subgrid modeling of Eulerian dynamics. A common
approach for Lagrangian subgrid modeling in LES is the application of subgrid
LSMs e.g.,.
This approach requires a number of additional calculations for each particle
(e.g., interpolations of subfilter stresses τij and subgrid
dissipation ϵ into particle position xp). Besides, it is
necessary to generate a three-component random noise for each particle, which
is a time-consuming computational operation. A numerically stable solution to
the generalized Langevin equation (see Sect. , Eq. )
in LES requires smaller time steps than the steps to solution of Eulerian
equations, because local Lagrangian decorrelation time TL(xp,t)
can be very small.
The statistics of simulated turbulence in LES may significantly differ from
the statistics of real turbulence. For example, the use of dissipative
numerical schemes or low-order finite-difference schemes usually results in a
suppression of fluctuations over almost the entire resolved spectral ranges
of discrete models see, e.g., Fig. 16 in.
Turbulent fluxes (in the Eulerian representation) associated with these
fluctuations are restored by subgrid closure. However, in terms of the
Lagrangian transport the effects of distortion of the small-scale part of the
spectrum are most often not considered.
Numerical simulations of Lagrangian transport in LES are also limited by the
low scalability of parallel algorithms. This is due to the impossibility of
uniform loading of processors in a joint solution to the Euler and Lagrangian
equations, a large number of interprocessor exchanges and an unstructured
distribution of characteristics required for Lagrangian advection in the
computer RAM memory.
Thus, all methods of numerical and analytical determination of the functions
fs have individual drawbacks. At the same time, due to the lack
of a sufficient amount of experimental data and due to their low accuracy,
there are no clear criteria for evaluation of different models.
According to the need for computational cost reduction, one of the objectives
of this study is to establish the role of stochastic subgrid modeling in the
correct description of the particle dispersion in LES. Is it possible to
simplify the calculation and to avoid the introduction of stochastic terms
without the loss of accuracy in some integral characteristics, such as the
footprints or the concentration of pollutants emitted from the point sources?
The role of subgrid fluctuations is reduced with an increase in spatial LES
resolution. Therefore, the independence of results from the mesh size is used
as a criterion for checking the quality of Lagrangian transport procedures in
LES. It will be demonstrated that the subgrid stochastic modeling in LES can
be omitted in most cases. Instead, we propose a “computationally cheap”
procedure of inverse filtering supplemented by divergent correction of
Eulerian velocity to replace the subgrid stochastic modeling in LES (see the
description below).
Subgrid transport is especially significant near the surface and/or under
stable stratification – all are the cases associated with small eddy size.
That is why the stable ABL was selected as the key test scenario in this
study. We slightly modified the setup of the GABLS numerical
experiment for this purpose.
LES results are used as the input data for the stochastic models (LSMs and
RDMs). These data are pre-adjusted using known universal dependencies and
taking into account an incomplete representation of turbulent energy in LES.
The comparison of results of different stochastic models and the results from
LES allows one to specify the parameters for the LSMs and permits one to
identify the differences between LSMs and RDMs under the conditions that have
not been tested previously.
The paper is organized as follows. Section contains the description
of some common features of approaches: the implemented numerical algorithm
for footprint estimation in the LES and LS models (Sect. ); LES
governing equations and the definitions of some terminology used for the
small-scale modeling description and for the testing of particle transport
(Sect. ); the definitions of stochastic models (LSMs and RDMs) and
pointing to some problems connected with uncertainty in the choice of
turbulent statistics for them (Sects. and ).
Section contains a short description of the numerical
algorithms, the turbulent closure for the LES model used in this study
(Sect. ) and the description of the different approaches for the
Lagrangian particle transport in the LES tested here (Sect. ).
Section is mainly devoted to the testing of the ability of the
LES model with rough spatial resolution to reproduce particle dispersion
correctly. To this end, we implemented a special setup of the numerical
experiment (see Sect. ), permitting one to compare Lagrangian and
Eulerian statistics (see Sect. ). The focus was placed on
the approaches with the limited use of subgrid stochastic modeling (see
Sect. , where the sensitivity of the computed footprints to
the spatial resolution was investigated). The footprints computed with the
LES model with simple subgrid LSMs and RDMs (traditional approach) are
presented in Sects. and . Two-dimensional
footprints are shown in Sect. . Due to high sensitivity of LSMs to
the turbulent statistics, we emphasize data preparation for them using LES
results, measurement data and similarity laws in Sect. .
Section contains the results of footprint modeling with the use
of the set of different RDMs and LSMs (specified in Sect. )
in comparison with LES results (see Sect. ).
Section summarizes the results.
In addition to the basic calculation, we carried out a series of tests (see
Supplement Sect. S1) under unstable stratification in the ABL with different
grid steps in the LES model. This allows us to compare the results presented
here with similar results obtained in previous studies
e.g., and to verify the performance of our
LES model in footprint evaluation. Furthermore, we demonstrate the results of
footprint calculations above the inhomogeneous surface (Supplement Sect. S2)
with a huge number of particles involved in calculations simultaneously.
Computational aspects of technology are discussed as well.
Modeling approaches
Numerical evaluation of footprints
Computational methods for determination of footprints often reduce to the
implementation of Lagrangian transport of marked particles. Each particle can
contain a number of attributes, including its initial coordinate
x0p and time t0p. Choose two small horizontal plates δS
and δM for averaging in the neighborhood of zero with the areas SS and SM, respectively. Define the time interval Tp=[t0,t2], during which new particles are ejected near the ground with the
intensity H (here H is the mathematical expectation of the new particle
number emitted per unit area per unit time) and the interval Ta=[t1,t2] (t1>t0), when particles are detected near the point of
measurement. If t1 is sufficiently large for the ensemble averaged flux to
attain constant value in time, and Ta is quite large for
statistically significant averaging, then the footprint fs can be
evaluated by the formula
fs(xS,yS,xM,yM,zM)≈1SM1Ta∑p=1nSM∫δSH(x0p+x′,y0p+y′,t0p)dx′dy′-1wp|wp|ISMp,
where nSM is the number of intersections of the plane z=zM by the
particle trajectories at horizontal coordinates x1p:(x1p-xM,y1p-yM)∈δM in time interval Ta; ISMp=1 if
the initial coordinates x0p of such particles satisfy the
condition ((x1p-x0p)-(xM-xS),(y1p-y0p)-(yM-yS))∈δS and ISMp=0 otherwise. Here, wp is the vertical
component of the particle velocity at the moment of crossing the plane z=zM. Schematic representation of the algorithm for the footprint function
determination in LES is shown in Fig. . In accordance with
Eq. () and the description above, the particle crossing the test
area δM brings the impact into the value
fs(xS,yS,xM) if the beginning of its trajectory belongs
to the test area δS after trajectory modification such that the point
x1p coincides with sensor position xM. For example (see
Fig. b), when the footprint value is calculated at the point
(xS,yS), only the red particle is counted, but not the blue particle.
Such an algorithm of averaging was selected because it permits one to refine
the footprint resolution in the vicinity of the sensor independently of the
area of δM using the assumption of some spatial homogeneity.
In the horizontally homogeneous case, one can calculate the footprint
fsd(xd,yd,zM) by performing averaging over statistically
equivalent coordinates of the sensor position. For this averaging in LES with
a periodic domain, one can prescribe the coordinates (xM,yM) to the
domain center and select the area δS to be equal to the whole domain
size. Analogical methods can be applied when using LSMs or RDMs, whereas in
the case of RDMs, particle displacement should be used in Eq. ()
instead of velocity.
The nonuniform Cartesian grid xijd=(xid,yjd) (where -20≤i≤160; -120≤j≤120), stretched with the distance
from the sensor position, was selected for the footprint function
accumulation in the following sections of this paper. The grid was prescribed
as (x0d,y0d)=(0,0); xid=Δx0γx|i|i/|i| and
yid=Δy0γy|j|j/|j| if i≠0 and j≠0;
Δx0=Δy0=2 m; and γx=γy=1.05. This
grid is independent of the LES model resolution and coincides with the
footprint grids selected for all runs with LSMs and RDMs.
Lagrangian particles embedded in LES
Lagrangian particle velocity up and the particle position xp can be computed in LES models as follows:
uip=u‾i(p)+u′′ip,dxip=uipdt.
Here u‾i(p) is the interpolation of the resolved Eulerian
velocity into the particle position; u′′ip are the small-scale
unresolved Lagrangian velocity fluctuations associated with Eulerian velocity
fluctuations belonging to “subgrid” and “subfilter” scales. Here and
later we shall use the designation “subfilter” to denote the fluctuations
that belong to the resolved spectral range of the discrete model, but are not
reproduced numerically, and the designation “subgrid” for the fluctuations,
which can not be represented on the grid due to smallness of the scales. LES
governing equations for filtered velocity u‾
are
∂u‾i∂t=-∂u‾iu‾j∂xj-∂τij∂xj-∂p‾∂xi+Fie‾,∂u‾i∂xi=0,
where Fie comprises Coriolis and buoyancy forces, p‾ is
normalized pressure and τij=uiuj‾-ui‾uj‾ denotes the modeled “subgrid/subfilter”
stress tensor. A system of equations () can be supplemented by
the Eulerian equations of scalar transport:
∂s‾∂t=-u‾i∂s‾∂xi-∂ϑis∂xi+Qs‾,
where Qs‾ denotes source intensity; ϑis=sui‾-ui‾s‾ are the parameterized
“subgrid/subfilter” fluxes. Usually, the fluctuations u′′p are
defined as dependent on some random function ξ, introduced in order to
provide the missing part of mixing. The particular approaches for computing
the unresolved part of particle velocity will be discussed and tested in the
following sections.
There is a great practical interest in the calculation of footprints, as well
as of spatial and temporal characteristics of pollution transport from
localized sources above heterogeneous surfaces and in the areas with complex
geometry (in the urban environment, over the surfaces with complex terrain or
over the alternating types of vegetation). LES of such flows becomes a
routine procedure with increasing performance of computers. However, the
calculation of statistical characteristics of Lagrangian trajectories is
complicated in this case by the need of transport of huge number of tracers
e.g.,. For example, it is necessary to calculate the
trajectories of about 109 particles (see Supplement Sect. S2) to obtain
the footprints above the inhomogeneous surface with the explicitly prescribed
obstacles (the task similar to that presented in
).
On the other hand, a large number of particles (see, e.g., Supplement
Fig. S2.1b) allows one to estimate the local instantaneous spatially filtered
concentration of the scalar:
sP(x,t)=∑p=1,NG(x-xp(t)),
where G is the function that coincides with the convolution kernel of the
LES filter operator and N is the total number of particles in the domain.
If the mathematical expectation Qp of a number of new particles
ejected in a unit volume during the unit time interval is proportional to the
Eulerian concentration source strength Qp(x,t)=CQ‾s(x,t), then sP(x,t)≈Cs‾(x,t). One can perform the same operations with the
“Lagrangian” concentration sP(x,t) as the operations with the
Eulerian scalar s‾. Below, we will compare the averaged values of
sP and s‾ and their spatial variability. Besides, we will use
the estimation of concentration sP(x,t) to correct the particle
velocities (see Sect. , Eqs. and ), in
order to approximate the effect of subgrid turbulence.
Single-particle first-order Lagrangian stochastic models (LSMs)
Another approach (more widespread due to a lower computational cost) is the
replacement of the entire turbulent component of velocity by a random process
(Lagrangian stochastic models (LSMs)):
uip=ui(p)+u′ip,dxip=uipdt.
Here ui(p) is the ensemble-averaged Eulerian velocity at
point xp. Note that LSMs are assumed to also be applicable under
the temporal evolution of turbulence statistics. In this paper we shall
consider the ABL as it approaches a quasi-steady state. Therefore, due to the
assumption of ergodicity, ensemble averaging can be replaced by averaging in
time and in the directions of spatial homogeneity: φ≈φx,y,t.
A single-particle first-order LSM is formulated as follows. Velocity
u′ip is described by the stochastic differential equation:
du′ip=ai(xp,up,t)dt+bij(xp,up,t)ξip,
where ξ stays for the delta-correlated (usually Gaussian) random noise
with the variance dt
ξip(t)ξjh(t+t′)=δijδphδ(t′)dt
and with the zero average ξip=0; ai and bij are
the functions depending on the Eulerian characteristics of turbulence and on
the Lagrangian velocity of the particle. Typically bij is calculated by
the formula
bij=δijC0ϵ,
where ϵ denotes the energy dissipation rate, averaged for a fixed
coordinate, and C0 is the Kolmogorov constant. This kind of random term
(arguments are given in and ) is
defined by a Lagrangian velocity structure function in the inertial range
see
Dij(t′)=(ui(t+t′)-ui(t))(uj(t+t′)-uj(t))=δijC0ϵt′
if τη≪t′≪TE (here, τη=(ν/ϵ)1/2 is
the Kolmogorov microscale, TE=E2/ϵ is the energy-containing
turbulent timescale and E is the turbulent kinetic energy).
The function ai (drift term) determines the behavior of particles at large
times t∼TL∼TE (here TL is the Lagrangian decorrelation
timescale). For spatially inhomogeneous and statistically non-stationary
turbulent flows, including the ABL, the choice of ai is usually made
according to the well-mixed condition WMC;. In general
WMC does not lead to a unique solution for ai. Different LSMs are
constructed by introducing the additional physical assumptions, and can lead
to inequivalent results.
Lagrangian models are very sensitive to the choice of universal functions
that define the normalized root mean square (RMS) of the vertical velocity
σ̃w=w′21/2/U* and non-dimensional
dissipation ϵ̃=ϵz/U*3 (here U* is the
friction velocity). Besides, the simulation results are affected by the
choice of value of C0. It can be shown
e.g., that for one-dimensional LSM, these
parameters determine the eddy diffusivity Ks for the scalar in
the diffusion limit (when t≫TL, i.e., at large distances from the
source):
Ks=2σw4C0ϵ=2σ̃w4C0ϵ̃U*z.
The data of measurements in the ABL demonstrate large variation. For example,
the values of σ̃w2 range from 1.0 to 3.1 see
Table 1 in . According to Eq. () it implies the
change in Ks by more than 9 times.
There is no consensus on the value of C0 either. Formally, C0 has the
meaning of a universal Kolmogorov constant in Eq. (11). The estimation of
this constant for an isotropic turbulence using the data of laboratory
measurements and DNS provides an interval C0=6.±0.5 (see
). However, the values C0∼3–4 are often used for the
LSM of particle transport in the ABL, independently of the type of
stratification. These values have been obtained by the different methods. For
instance, the value C0=3.1 for a one-dimensional LSM corresponds to a
calibration performed in according to observation data
. This calibration (see ) assumes that
the turbulent Schmidt number Sc=Km/Ks=0.64 is near the
surface (here Km is the eddy viscosity). It is known that determination of
the turbulent Prandtl number Pr=Km/Kh (Kh –
heat transfer eddy diffusivity) and the Schmidt number based on observation
data is complicated by large statistical errors associated with the problem
of self-correlation . Therefore, this method of
estimation of C0 can not be considered final, and should be confirmed by
future studies. In the values of C0 were determined using
the LES-based evaluations of the velocity structure functions and the
Lagrangian spectra in convective and neutrally stratified ABLs. In this study
the LES model had a relatively low resolution, which can be insufficient for
accurate determination of this constant in the inertial subrange (see the
discussion on the resolution requirements in ). Nevertheless,
the value C0∼3 in the paper by is relevant for LSMs
applied to the convective ABL; in that case, the value of C0 is also
responsible for the energy containing timescales that are well resolved in
LES. The detailed overview of the methods of determination of the constant
C0 can be found in , where the discussion on the
disagreements of the different approaches is also included. The results of
the LSMs are very sensitive to the choice for C0, as was shown earlier by
, , and many others. Below
we show that the commonly used value of C0∼3–4 can be greatly
underestimated for use as a parameter in LSMs applied to the stably
stratified ABL.
Zeroth-order Lagrangian stochastic models or random displacement models (RDMs)
The simplest approach for development of the models of particle dispersion
entails replacement of the Eulerian advection–diffusion equation
∂s∂t+ui∂s∂xi=∂∂xiKs∂s∂xi+Qs
by the stochastic equation for particle position (random displacement models
– RDMs):
dxip=uidt+∂Ks∂xidt+2Ksξip.
Probability density of particle position P is connected with scalar field
concentration s as follows:
s(x,t)=∫R3∫-∞tQs(x0,t0)P(x,t|x0,t0)d3x0dt0.
Using the Fokker–Planck equation, it can be shown that Eq. () is
equivalent to Eq. () from the point of view of concentration
transport when the time step dt tends to zero .
An RDM has some major disadvantages. First, it shares the limitation of
Eulerian eddy-diffusion treatment of turbulent dispersion, i.e.,
“K-theory”. Correspondingly, it is not able to describe the non-diffusive
near field of a source. Also, an RDM can not be applied for the convective
ABL, where the counter-gradient transport is observed. Besides, it requires
the exact values of diffusion coefficient Ks, which can not be
measured directly.
Details of the LES model used in this study
Numerical algorithms and turbulent closure
A system of equations (–) is discretized using an
explicit finite-difference scheme with the second-order temporal
approximation (Adams–Bashforth method) and fourth-order (fully conserved for
advective terms) spatial approximation of velocity and scalars on a staggered
grid .
A mixed model , expressed as the sum of the Smagorinsky
and scale-similarity models, is used for calculation of the turbulent stress
tensor:
τijmix=τijsmag+τijssm=-2(CsΔ‾)2|S‾|S‾ij+(u‾iu‾j‾-ui‾‾uj‾‾),
where S‾ij is the filtered strain rate tensor, and
Cs is the dynamically determined
dimensionless coefficient that depends on time and spatial coordinates. The a
priori tests using the data of laboratory measurements show that
scale-similarity models with Gaussian or box filters provide correlation
typically as high as 80 % between real and modeled stresses see the
overview in. The significant part of this correlation can be
attributed to non-ideality of the spatial filter and use of common
information for computing both the real and modeled stresses
. The discrete spatial filter used in this study has a
smooth transfer function in spectral space, so it can be supposed that the
scale-similarity part of Eq. () is mainly responsible for the
influence of velocity fluctuations belonging to “subfilter” scales.
The procedure of calculation of the coefficients X(x,t)=(CsΔ‾)2 reduces to minimization of the functional
Ψ(X)=∫Ωεij(x)εij(x)dx, where Ω is the model domain and
εij(x) is the residual of the overdefined system of
equations
XMijτ^-α2X(MijT)=Lij-Hij+εij,
obtained by substitution of the mixed model (Eq. ) into the Germano
identity as
Tij-τij^=ui‾uj‾^-ui‾^uj‾^.
Here Tij are subgrid/subfilter stresses for the smoothed velocity
u‾^, obtained by successive application of basic
FΔ‾ and test FΔ^ spatial filters;
α=Δ‾^/Δ‾ is the ratio of the
filter widths. Tensors MijT, Mijτ, Lij and Hij are
calculated as follows:
MijT=2S‾^S‾^ij,Mijτ=2S‾S‾ij,Lij=ui‾uj‾^-ui‾^uj‾^,Hij=ui‾^uj‾^‾^-ui‾^‾^uj‾^‾^-ui‾uj‾‾^-ui‾‾uj‾‾^.
The generalized solution to the discrete analog of Eq. () is
searched using the iterative conjugate gradients (CG) method with a diagonal
preconditioner. To do this, the problem is reduced to a linear system of
equations
AΔ*AΔXΔ=AΔ*RΔ,
where XΔ is the desired solution (a vector of dimension N=NxNyNz with the values defined in the center of the grid cells); AΔ
and RΔ=LΔ-HΔ are the discrete analogs of the
operator and the right-hand side of Eq. () correspondingly;
AΔ* is the transpose matrix. The diagonal preconditioner
PΔ for the CG method was selected as follows:
PΔ=α4MΔTMΔT*+μ(MΔτMΔτ*-2α2MΔTMΔτ*)-1,
where μ=const∼1 is the empirical coefficient independent on
time and spatial position. The solution XΔ contains negative values
(unconditional minimization of the functional is used), however, mixed model
(Eq. ) reduces their relative number compared with the dynamic
Smagorinsky model. In the algorithm, negative values are replaced by zeroes.
In fact, this dynamic procedure is close to approach proposed in
, with the difference that the mixed model was applied here
and iterative method was replaced by a faster CG method.
Eddy-diffusion models are used for subgrid heat and concentration transfer:
ϑis=-Khsubgr∂s‾∂xi;
here, Khsubgr=(1/Scsubgr)(CsΔ‾)2|S‾| is the eddy diffusivity, which is
independent of the type of scalar. Subgrid turbulent Schmidt and Prandtl
numbers are fixed: Scsubgr=Prsubgr=0.8.
A distinctive feature of this model is that the discrete spatial filter operator FΔ‾=FxFyFz is explicitly involved in calculation of stresses.
The following discrete basic filter is selected:
Fx(φ)i,j,k=18φi-1,j,k+34φi,j,k+18φi+1,j,k;
here, i,j,k denote a grid cell number. φ is any variable. Similar
filtering is applied along the coordinates y and z. It is reasonable to
expect that we get the velocity u‾, smoothed according to
the specified filtering operator as a solution to Eq. ()
supplemented by the mixed closure (Eqs. –). Since the
discrete filtering operator is invertible, we can find the following velocity
at any point and time:
ui*=FΔ‾-1u‾i,
which better reflects the small-scale spatial variability. The approximate
inverse filter is calculated as a series
FΔ‾-1≈Fn-1=∑k=0n(I-FΔ‾)k,
where I is a unity operator; in the calculations presented below we used
n=5. Spatial spectra of “defiltered” velocity u* under the
neutral, unstable and stable stratifications were obtained earlier
. It was found in
all cases that this procedure improves the small-scale parts of the spectra
according to dependence S∼k-5/3, provides better agreement of
spectra calculated with the different spatial resolution, and improves the
convergence of non-dimensional spectra if proper length scales are used for
normalization.
Methods for Lagrangian particle transport in LES
Subgrid and subfilter modeling
Below, the subgrid and subfilter modeling methods used for the simulations in
the current study are listed. These methods will also be used in combinations
as defined in Sect. .
(1) Improvement of Lagrangian transport using inverse filtering of Eulerian velocity field
First, we will use the recovering of “subfilter” fluctuations
(Eqs. and ) in order to transport Lagrangian particles
more precisely:
up=u*(p).
Note that for the use of such a procedure, LES models should exhibit the
properties of a model with an explicit filtering. A similar approach was
recently applied by in LES with an approximate
deconvolution subgrid model ADM; see, which can also be
considered as the model with explicit filtering. In most cases, the
suppression of small-scale fluctuations in LES (particularly in those that
use a low-order numerical scheme) occurs as a result of the combined effect
of approximation errors and the subgrid closure. Therefore, the shapes of
effective spatial filters of most models can only be determined by a
posteriori analysis of the calculation results.
(2) Lagrangian stochastic subgrid/subfilter model
Second, we will apply the subgrid stochastic model proposed in
:
duip=-∂p‾∂xi-1TL(uip-u‾i(p))dt+C0ϵξip.
The parameter C0 was specified to be equal to 6, because the stochastic
part of the model (Eq. ) is mainly responsible for spatial scales
and timescales in an isotropic inertial subrange of the turbulence. When
using the dynamic mixed model (Eqs. –), a value of ϵ is not calculated directly, and then it is assumed that the
dissipation is locally balanced by shear production and buoyancy production
or sink. In addition, since this model can produce a local generation of
kinetic energy, the averaging in a horizontal plane was performed to avoid
negative values of dissipation:
ϵ=-S‾ijτijxy+gΘ0ϑ3Θxy,
where ϑ3Θ is the vertical subgrid flux of potential
temperature and g/Θ0 is the buoyancy parameter. Timescale TL
was evaluated as
TL=(Esubgr+Esubf)/12+34C0ϵ.
Thus, the total unresolved kinetic energy was calculated as the sum of
“subfilter” energy
Esubf=12(ui*-u‾i)2xy
and “subgrid” energy:
Esubgr≈12∫kmini∞Si(ki)dki≈34CK′ϵ2/3∑i=1,3πΔgi-2/3.
To evaluate the value Esubgr it was supposed that “subgrid”
fluctuations belong to quite a wide inertial range with the component-wise
velocity spectra Si(ki)=CK′ϵ2/3ki-5/3, and that the
minimal wavenumbers for these fluctuations kmini=π/Δgi correspond to wavelengths in two grid steps. Here,
Δgi is the grid step in the appropriate direction and
CK′=1855CK=0.5 is the Kolmogorov constant (here, CK≈1.5 is the Kolmogorov constant associated with three-dimensional
wavenumbers).
All the values required for a application of this model were linearly
interpolated into the particle position everywhere except at heights
z<Δg/2, where we use the constant values
TL(Δg/2) and ϵ(Δg/2). This
procedure is rather arbitrary, but it does not have large impact on the
results due to the small decorrelation time TL(Δg/2).
Besides, there are no physically grounded reasons for the justification of
such interpolations in LES because the resolved velocity in the vicinity of
surface is greatly corrupted by the approximation errors. Such procedures
should be considered as an adjustments depending on the numerical scheme and
on the subgrid closure.
(3) Random displacement subgrid/subfilter model
Third, the RDM specified in Sect. will be adopted for the
Lagrangian particles subgrid dispersion. In this case we shall use the same
subgrid diffusivity Khsubgr both for the Eulerian
scalars (Eq. ) and for the particles displacement calculations:
dxip=u‾i(p)dt+∂Kssubgr(p)∂xidt+2Kssubgr(p)ξip.
This model does not contains the arbitrary specified parameters except those
which were already used in the Eulerian LES. The coefficient
Kssubgr was linearly interpolated into the particle
positions at heights z≥z0 with the assumption that
Kssubgr(x,y,0)=0. A constant value
Kssubgr(x,y,z)=Kssubgr(x,y,z0) was
used for z<z0.
(4) Divergent correction of the Eulerian velocity field
Finally, in order to find out whether the subgrid mixing is one of the key
processes in the dispersion of Lagrangian tracers, we introduced an
additional correction to the particle velocities:
ucor¯div(p)=u‾(p)+u‾div(p),
where u‾div is the deterministic divergent additive
to the velocity field u‾:
u‾div,i=ϑispsP,
with the imposed restriction u‾div,i=0 if sP=0.
Here, the “subgrid” flux ϑisp is calculated using the same
closure as the closure for Eulerian scalars s‾, with the only
difference that the concentration sP, estimated by the number of
particles in a grid cell, is used in Eq. (). The applicability of
this procedure justified because of the large number of particles involved in
simulation (in all the cases described below we have at least several dozens
of particles in each grid cell).
Correction given by Eqs. (), () does not provide
true small-scale mixing, but only introduces an additional “stretching” or
“compression” of the small volumes filled with particles and provides
concentration fluxes across the borders of grid cells close to “subgrid”
fluxes in Eulerian model. Using this correction, we are guaranteed to get a
high correlation between the “Eulerian” and “Lagrangian” concentrations
(in all our preliminary tests s‾′sp′xy/s‾′2s′p2≈0.9).
The idea of such a correction was based on the assumption that details of the
mechanism of subgrid mixing have a little influence on the statistics of
trajectories at sufficiently large distances from the source and at long
enough time t. It was assumed that the quick mixing on small spatial scales
can be implicitly substituted by the approximation errors arising in the
procedures of interpolation and by the errors of discrete solution to the
advection equation. Correction brings an additional systematic effect to
reduce incorrect particle transport by the large eddies.
Simplified velocity interpolation
In preliminary tests it became clear that trilinear interpolation of each
velocity component provides no advantages for footprint calculation in
comparison with the following simplified linear interpolation on a staggered
grid:
u(p)=u‾i-12,j,kxi+12,j,k-xpΔx+u‾i+12,j,kxp-xi-12,j,kΔx,v(p)=v‾i,j-12,kyi,j+12,k-ypΔy+v‾i,j+12,kyp-yi,j-12,kΔy,w(p)=w‾i,j,k-12zi,j,k+12-zpΔz+w‾i,j,k+12zp-zi,j,k-12Δz,
where position (i,j,k) is the center of a grid cell containing the
particle. Trilinear interpolation and interpolation given by
Eq. () provide nearly the same concentration fluxes across the
borders of a grid cell, but the latter does not result in additional
substantial smoothing of velocity. An exception was made for the grid layer
closest to the surface (zp<Δg) where the mean velocity
components were adjusted according to the Monin–Obukhov similarity theory
with the dimensionless functions taken from .
LES of stable ABL and footprint calculations
The setup of the numerical experiment
Stable boundary layer at the latitude 73∘ N in almost steady state
conditions was considered. The calculations were carried out according to the
GABLS scenario , with the difference that the geostrophic wind Ug has been rotated 35∘ clockwise such that the
wind direction near the surface approximately coincides with the axis x.
The duration of runs is 9 h. The initial wind velocity coincides with
geostrophic velocity |Ug|=8 m s-1. The initial
potential temperature Θ‾ is equal to the surface temperature
Θs|t=0=265 K up to the height 100 m and increases
linearly with the rate dΘ/dz=0.05 K m-1 if
z>100 m. During the calculations, the surface temperature decreases
linearly with time: dΘs/dt=-0.25 K h-1.
Dynamical and thermal roughness parameters z0 and z0Θ are set to
0.1 m. The calculations were performed at the equidistant grids with steps
Δg=2.0, 3.125, 6.25 and 12.5 m. The size of the
horizontally periodic computational domain was equal to 400×400×400 m3. The last hour of numerical experiments was used for
averaging the results and subsequent analysis.
This setup is based on the observation data (see ). As
was shown in , the LES results obtained under the same
conditions with the different models converged with the higher grid
resolutions. Later, this case was used for testing the LES models, e.g., in
and and many others, and for the
improvement of subgrid modeling, e.g., in , ,
and . The LES model presented here was tested earlier
under the non-modified setup of GABLS in , where the
turbulent statistics above a flat surface and above an urban-like surface
were investigated. In all of these studies, LES results were in agreement
with the known similarity relationships for the stable ABL. This allows one
to consider the LES data for GABLS as a reference case for testing of the
approaches utilizing the statistical averaging of the turbulence (e.g., see
, where the intercomparison of single-column models was
performed). Several of the non-dimensional relationships in the stable ABL
were collected and presented in . The considered
case is also included in the LES database for this study and fits well with
the different stability regimes after the appropriate normalization.
Therefore, the results obtained in this particular case can be generalized
for many cases due to the similarity of the stable ABLs. Besides, the
presented simulations are easily reproducible, and they can be repeated using
any LES model that contains the Lagrangian particle transport routines.
Mean wind velocity u (a) and
temperature Θ (b) in runs with different grid
steps (spatial step is pointed in legend). Gray dots are the data from other
LES models obtained in (wind velocity is rotated 35∘
clockwise). “Standard” wind profile for stable conditions in accordance
with is shown by the vertical dashes.
![]()
The mean wind velocity and the potential temperature, calculated with the
different spatial steps Δg, are shown in Fig. .
The model slightly overestimates the height of the boundary layer at coarse
grids; however, the wind velocity near the surface is approximately the same
in all runs. As one can see from Fig. 2, the results of the simulation are in
good agreement with the results from other LES presented in
(see http://gabls.metoffice.com for more information). The mean wind
profile computed in accordance with is shown in
Fig. by the vertical dashes; in the surface layer part of the
domain this “standard” profile for the stable conditions almost coincides
with the longitudinal velocity obtained in LES.
Passive Lagrangian tracers were transported simultaneously with the
calculations of dynamics. Each particle, when reaching a lateral boundary of
domain, is returned from the opposite boundary in accordance with periodic
conditions. The reflection condition is used at the ground. The particles are
ejected at the height z0=0.1 m (one particle per each grid cell
adjacent to surface) with regular time intervals Δtej=1 s. The
position of the new particle within a grid cell is set randomly with uniform
probability. The ejection of particles takes place continuously from the
seventh to the ninth hour of the experiment.
To limit the number of particles involved in the calculation, the absorption
condition is applied at the height of 100 m within the ABL. It was verified
previously that the upper boundary condition does not have a large impact on
the results of calculations of footprints for the heights zM up to 60 m
and for the distances x-xM considered in this paper (see Appendix A and
the test with the LSM shown by the orange curves in Fig. a, c,
e). This formulation of the numerical experiment allows direct comparison of
the concentration of particles sP, estimated by Eq. (), and the
scalar concentration s‾, calculated by the Eulerian approach
(Eq. ). For this purpose, an additional scalar s‾ is
calculated from the seventh till the ninth hour, with a constant surface flux
Fs=const=1, zero initial condition and the Dirichlet
condition s‾=0 at altitude 100 m.
In the last hour of simulation, the averaged number of particles in each cell
of the grid near the surface was approximately equal to 700–800, 350–400,
180–200 and 110–130 for grid steps Δg=12.5, 6.25, 3.125
and 2.0 m, respectively. Having such a number of particles, one can estimate
the concentration sp(xi,j,k,tm) at each time step, where
xi,j,k is the center of a grid cell. It was assumed that each
particle contributes to the concentration s̃P(xi,j,k) with
the weight ri,j,kp=(Vp⋂Vi,j,k)/Vi,j,k, where Vp is
the rectangular neighborhood of its position with the side
Δg, (Vp⋂Vi,j,k) is the volume of intersection
with a grid cell, and Vi,j,k is the cell volume. This averaging is close
to the filtering of an Eulerian scalar (Eq. ). The additional
normalization is performed as follows: sP=s̃PΔtej/Δz. The concentration sP corresponds to the number of
particles in one cubic meter under the condition that one particle per square
meter per second is ejected near the surface. Concentration sP is
numerically equal (excluding errors, determined by different methods of
transport) to the concentration of the scalar field s‾ if
scalar surface flux Fs=1.
Total Fstot=s‾w‾+ϑ3s (solid lines), resolved
Fsres=s‾w‾
(short-dashed lines) and “subgrid” Fssbg=ϑ3s (long-dashed lines with shading) scalar fluxes in the
runs with different grid steps Δg.
![]()
Figure shows the resolved and the parameterized components of
flux w′s′ in runs with different grid steps. It is seen that
the calculation time is not large enough to reach a steady state (the total
flux is not constant with the hight, so the average concentration continues
to grow during the last hour). However, it was checked that the flux
footprint close to the sensor is not affected by nonstationarity. Besides, we
can compare the values of s‾ and sP, because the boundary
and initial conditions are identical for them.
The unresolved fraction of the flux Fssbg=ϑ3s is an essential part of the total flux
Fstot=s‾w‾+ϑ3s. Accordingly, the vertical transport of Lagrangian
particles by resolved velocity u‾ may be significantly
underestimated. Thus, we have a “hard” enough test to verify Lagrangian
transport in LES with a poorly resolved velocity field.
Sensitivity of LES results on methods of particle transport and spatial resolution
Footprint calculation with limited application of subgrid stochastic modeling in LES
Crosswind-integrated scalar flux footprints fsy in
the stable ABL, computed by different methods and with different grid steps:
(a, c) sensor height zM=10 m; (b, d) zM=30 m. Grid
steps and methods are indicated in the legend: u – particles are
transported by a filtered LES velocity u‾; u* –
particles are transported by recovered velocity u*=F-1u‾; cor_div – the additional correction of velocity
(Eqs. and ); st_1l – the stochastic subgrid model
(Eq. ) is applied for the particles within the first computational
grid layer.
![]()
Figure shows the scalar flux footprints averaged in crosswind
direction fsy(xM-x,zM) computed by different methods and
with different grid steps. In all cases, we have avoided using the
subgrid-scale stochastic modeling, except for calculating the velocity of the
particles located within the first grid layer zp<Δg. For
the curves marked “st_1l”, the resultant velocity of the particles near
the surface was calculated as follows:
up=u(p)+r(zp)u′′p,
where the function r(zp) is defined as r(zp)=(1-zp/Δg) if zp<Δg and r(zp)=0
if zp≥Δg; u′′p is the random velocity
component, calculated using the stochastic subgrid model (Eq. ).
To take into account the memory effects in Langevin equation, the stochastic
model was implemented inside the layer zp<3Δg, so
(because of the smallness of scale TL) this procedure does not lead to
significant distortions in the random component of the velocity.
If the particles are advected by the filtered velocity u‾
without any correction then the vertical mixing is too weak and the maxima of
footprints fsy are strongly underestimated and shifted at the
large distances from the sensor position. Divergent correction of Eulerian
velocity (Eqs. , ) partially improves the results
(squares in Fig. a, b). For example, maximum of footprint
fsy for the sensor height zM=30 m (near the fifth
computational level) occurs to be close to the maxima of footprints, computed
at fine grids, but it is still shifted. Thus, the correction
(Eqs. , ) alone is not sufficient. Primarily this is
due to the weak mixing below the first computational level, where the
contribution of the subgrid velocity is crucial.
The inclusion of stochastics within the first layer improves the result
(dashed curves in Fig. a, b). However, it is not enough to
determine footprints at altitudes comparable to the grid spacing.
The advection of particles by the velocity u* leads to close
matching of functions fsy, calculated with different grid
steps (solid lines of different thickness in Fig. c, d). The
differences between these footprints are not significant from a practical
point of view, and can be equally explained by means of the incorrect
Lagrangian particles transport, as well as by means of the insufficiently
accurate solution to the Eulerian equations on the coarse grid.
Spatial variability of scalar concentration inferred by Eulerian and Lagrangian methods
While the particles were advected by the “defiltered” flow, we have also
used the correction (Eqs. and ). In this case the
subgrid diffusion coefficient was reduced twice: Kh*subgr=cKhsubgr, c=0.5 (coefficient c=0.5 was chosen
because about half of the subgrid flux can be restored using “defiltering”:
s‾w*-s‾w‾≈0.5ϑ3s). We note that when
the particles are advected by velocity u*(p), then the presence or
absence (crosses in Fig. c, d) of correction has no significant
effect on the function fsy. Nevertheless, this procedure may
be useful for the following reasons.
In the inertial range of three-dimensional turbulence along with the kinetic
energy the variance of a passive scalar concentration is transferred from
large scales to small scales with the formation of the spatial spectrum
Ss∼ϵsϵ-1/3k-5/3 (see
) (here ϵs is the dissipation rate of the
variance of concentration, caused by molecular diffusion). Lagrangian
transport of particles by a divergence-free velocity field u* with
the truncated small-scale spectrum is equivalent to Eulerian advection of
concentration s without any dissipation. The absence of a subgrid-scale
part of the velocity spectrum will lead to reduction of the forward cascade
and to the accumulation of variance σsp2 in the vicinity of the
smallest resolved scales.
(a) Variance σs2=s‾′2 of the concentration of Eulerian scalar (solid lines) and
variance σsp2=sP′2 of concentration sP,
determined by Lagrangian particles (symbols); grid steps and the methods of
calculations are shown in the legend, and symbolic notations are the same as
in Fig. ; stars – a stochastic model (LSM,
Eqs. –) is used throughout the domain. Open circles –
a subgrid RDM (Eq. ) is applied. (b) Correlation
corr(s‾,sP)=s‾′sP′xyt/(σspσs) between “Eulerian” and
“Lagrangian” concentrations. For remaining notations, see the caption of
Fig. .
![]()
Figure a shows the variances of “Eulerian” concentration
σs2(z)=s‾′2xyt computed at
different grids and the variances of “Lagrangian” concentration
σsp2(z)=sP′2xyt. One can see that if particles
are advected by the velocity u*(p) (crosses), variance
σsp2 is much larger than σs2. If the velocity
u*(p)+udiv(p) is used (filled circles), the
values of σsp2 and σs2 become closer to each
other. Besides, the correction (Eqs. and )
increases the correlation corr(s‾,sP)=s‾′sP′xyt/(σspσs) of two fields calculated
by means of “Eulerian” and “Lagrangian” approaches (see
Fig. b).
One can expect that in more complicated cases (e.g., the turbulent flow
around geometric objects and the formation of quasi-periodic eddies), the
accumulation of small-scale noise in the concentration field may lead to the
incorrect advection of concentration by the resolved eddies. This effect may
also be important for inertial particles when the nonphysical variance of
concentration can directly affect dynamics. In additional tests it was found
that the correction given by Eqs. () and () prevents
particle stagnation in zones with unresolved turbulence during the modeling
of urban-like environments. Thus, this correction is desirable for a number
of reasons as a practical replacement of subgrid stochastics, which requires
large computer resources.
Particle advection and footprint determination in LES with subgrid LSM
Crosswind-integrated scalar flux footprints fsy,
computed using the stochastic subgrid model (Eqs. –):
(a) sensor height zM=10 m; (b) zM=30 m. Grid steps
are given in the legend. Crosses denote footprints computed with the subgrid
LSM applied for the particles within the first grid layer only.
![]()
One can obtain footprints close to those presented in Fig. by
means of application of the stochastic subgrid model
(Eqs. –). The calculations for this model have been
carried out on the grids with steps 3.125, 6.25 and 12.5 m (solid lines in
Fig. a, b). One can note the shortcoming of this stochastic
subgrid modeling in LES, which can not be detected by study of the mean
characteristics. In the previous subsection, the recovered “subfilter” part
of velocity u′′=u*-u‾ and so the subfilter
Lagrangian velocity u′′(p) were highly correlated with the
resolved velocity u‾ in time and space. This is due to the
specifics of the spatial filter (Eq. ) used for the recovering given
by Eqs. () and (). This filter has a smooth transfer
function in spectral space. The analogous effects of non-ideal filters in LES
that lead to the high correlations between modeled and measured turbulent
stresses were obtained and discussed earlier in and
, where the laboratory data of turbulent flows were
studied. By contrast, additional mixing in the stochastic model
(Eqs. –) is due to random fluctuations, which are not
related to u‾ strictly. When one uses coarse grids, the
energy of these Lagrangian fluctuations should be large enough to restore
mixing in the vertical direction. This is accompanied by an excessive
suppression of the variability of concentration sP near the surface, where
the contribution of subgrid mixing is large (stars in Fig. a). The
correlation between “Eulerian” and “Lagrangian” concentrations is reduced
simultaneously (see Fig. b). Probably, this defect of the employed
Lagrangian stochastic model is connected to the horizontal averaging in the
evaluation of “subgrid” dissipation and energy. Nevertheless, this result
shows that in some cases the stochastic subgrid modeling can prevent correct
reproduction of the resolved spatial variability of particle concentrations
in LES along with improvement of the mean transport.
Footprints in LES with subgrid RDMs and the comparison of different methods
Crosswind-integrated scalar flux footprints fsy,
obtained in LES with Δg=6.25 m using different stochastic
Lagrangian subgrid models RDM (Eq. ) and LSM
(Eqs. –). The results obtained with these subgrid
models applied within the first computational grid layer in combination with
velocity recovering u*=F-1u‾ and correction of
velocity (Eqs. and ) are also shown. Black lines
are the footprints in LES with Δg=2.0 m.
![]()
In Fig. footprints obtained in LES with intermediate resolution
Δg=6.25 m are shown. We choose this resolution because
LES dynamics is still reproduced sufficiently well, but the effects from the
subgrid/subfilter Lagrangian parameterization are already clearly visible. In
addition to the approaches that were already discussed above, we applied the
subgrid RDM (Eq. ) and the subgrid RDM in combination with the
velocity recovering (Eqs. and ) and the correction
(Eqs. and ). In the former case we restricted the
activity of the subgrid RDM by the multiplying of the diffusivity coefficient
Khsubgr(p) in Eq. () on the following ramp
function: r(zp)=(1-zp/Δg) if zp≤Δg
and r(zp)=0 if zp>Δg.
Generally, results are in close agreement with the results of LES with the
fine grid, except for some details. One can see the intrinsic defect of the
RDM when it is applied to the dispersion of particles in a near field of a
source. That is, as the RDM is the approximation of the diffusion process
with the infinite speed of the signal prorogation, this model overestimates
values of fsy in the vicinity of the measurement point location
(see Fig. d, where this effect is highlighted in the logarithmic
scale). Nearly the same effect was obtained in (see
Figs. 1–3 in that paper, where the footprints from the RDM are also shifted
left in comparison with the other models). It was also observed that, along
with the overestimated vertical mixing, a subgrid RDM leads to the
propagation of some portion of the particles in the upwind direction (the
function fsy(xM-x,zM=10) has small but positive values if
xM-x<0). In LES with the intermediate resolution the mentioned
overestimated mixing exceeds the similar effect in RDM standing alone (see
Sect. ), because the coefficient Khsubgr is
highly variable in time and space, and it can achieve even larger local
values than the magnitude of the averaged turbulent diffusivity
Kh. At the higher levels of zM=30 m and zM=60 m, the
footprints are formed as a result of averaging of the turbulent motions over
the large spatial distances and over long temporal intervals, and the
diffusion approximation becomes acceptable. As will be shown in
Sect. , an RDM applied alone gives very close results to the results
of LSMs in this particular case of the stable ABL.
In contrast to the subgrid LSM and to the methods of velocity correction
proposed above, the advantage of the subgrid RDM consists in the absence of
the arbitrary prescribed parameters and in the absence of the need to involve
the additional suppositions. In terms of Eulerian statistics, this model is
identical to Eq. () (in the limit dt→0 and with the
precision defined by the spatial approximations). From this point of view,
subgrid RDM can be considered the “ideal” model, because it is determined
by the coefficients that are consistent with LES dynamics of the stratified
flow (the same subgrid diffusivity is used for the potential temperature that
defines the buoyancy and the interchanges between the kinetic and available
potential energy). One can see that the variance of “Lagrangian”
concentration computed with the use of a subgrid RDM (open circles in
Fig. a) is very close to the variance of the concentration
obtained by the Eulerian method. The correlation between “Eulerian” and
“Lagrangian” concentrations (open circles in Fig. b) is also
large, except for the first computational level; there, the Eulerian
non-monotonous numerical advection scheme produces significant numerical
noise. Thus, we have one more confirmation of the validity of the results,
except for the invariance with respect to the grid steps.
The impact from the subgrid RDM is reduced when it is applied within the
first grid layer only. In this case, the footprints are approximately the
same as the footprints computed using the other approaches.
Two-dimensional footprints
The trajectories of a large number of particles (∼1.8×108) were
simultaneously computed in LES with a grid step of 2.0 m. Accordingly, one
can get a statistically grounded estimation of two-dimensional footprint
functions fs(x-xM,y-yM,zM). These functions, computed for the
sensor heights zM=10 m and zM=30 m, are shown in Fig. a,
b. One can see that the area with the negative values of the footprint
exists. The negative values of the footprints are typical
e.g., of the convective boundary layer due
to fast upward advection by the narrow thermal plumes and slow downward
advection in the surroundings. Here, the negative values of the function
fs are connected to the Ekman spiral and to the mean transport of
the particles elevated to large altitudes in the direction perpendicular to
the near-surface wind. The negative values of the scalar flux footprint show
that the vertical turbulent transport of the scalar emitted in the relevant
area is basically directed from the upper levels down to the surface. For
example, the positive surface concentration flux in this area will lead to a
negative anomaly of the turbulent flux measured in the sensor position. This
does not contradict the diffusion approximation of the turbulent mixing,
because mean crosswind advection at the upper levels can produce the positive
vertical concentration gradient to the right of near-surface wind.
Two-dimensional footprints fs(x-xM,y-yM,zM) (×10-6 m-2) for sensor height zM=10 m (a) and
zM=30 m (b) and the corresponding crosswind-integrated
cumulative footprints F(xM-x) (c) and (d); long
dashed line – F+ (impact of the area with positive values of
fs); short dashed line – F- (impact of area with
negative values).
![]()
The contribution of the negative part of the flux to the “measured” flux is
significant, as shown in Fig. c, d, where cumulative footprints,
defined as
F(xd,zM)=∫-∞xdfsy(x′,zM)dx′,
are separated into positive and negative parts F(xM-x,zM)=F++F-.
Stochastic modeling and the comparison with LES
Preparation of turbulence data from LES for LSMs and RDMs
The LES results with grid step Δg=2.0 m were used for data
preparation. To apply an LSM (Eqs. and ), the following
Eulerian characteristics are required: the mean wind velocity components
u and v, the second moments ui′uj′ and the dissipation ϵ. Stochastic models are even more
sensitive to some of these characteristics than the advection of particles in
LES. For example, the underestimated values of the turbulent kinetic energy
in LES are the consequence of the suppression of small eddies. Nevertheless,
these eddies exert a relatively small influence on the mixing of scalars,
because the effective eddy diffusivity associated with them
(Khsmall∼Esmall1/2lsmall) is not
large due to the small spatial scale. However, the turbulent energy that is
substituted into the LSM affects results independently of the scale and has
to be evaluated with good accuracy.
Mean velocity
Mean wind velocity at the height z0<z≤Δg was computed using log-linear law:
ui=U*1κlnzz0+CmzL×ui|u|z=Δg/2,Cm=5,
and ui=0 at z<z0. Here, U* is the friction velocity,
κ=0.4 denotes the von Karman constant, L is the Obukhov length at
the surface
L=-U*3Θ0gQs,
where Qs is the kinematic potential temperature flux at the
surface, g=9.81 m s-2 is the acceleration of gravity and
Θ0=263.5 K is the reference potential temperature (as was prescribed
in presented simulations and in ). Note that the von Karman
constant is not included in the definition of the length L here and later
(this alternative definition of the Obukhov length is used along with the
traditional one; see, e.g., , Eq. 41). The linear
interpolation of velocity was used if z>Δg.
Momentum fluxes
The fluxes ui′uj′=u‾i′u‾j′+τijmix (i≠j) were interpolated linearly
and additionally smoothed everywhere in the domain. These fluxes are shown in
Fig. a.
(a) Total momentum fluxes obtained in LES with
Δg=2.0 m. (b) Normalized RMS of vertical velocity
σ̃w=σw/|τ|1/2 depending on a dimensionless
parameter z/Λ (solid red line - estimation using LES data σw=(w*2+2/3Esubgr)1/2; symbols – measurements
at different heights). (c) Variances of velocity
components (dashed line – resolved fluctuation; solid lines – the final
estimation for LSM; bold red lines – vertical component, green curves of
medium thickness – crosswind component, blue thin lines – longitudinal
component, circles – evaluation of σw2 by Eq. ).
(d) Vertical effective eddy diffusivity Ksww (red
solid line – coefficient calculated by the gradient and flux of scalar;
dashed line – estimation of coefficient using Eq. () with
C0=6); estimations of diffusion coefficients in crosswind direction
Ksvv (green dash-dot line) and coefficient in longitudinal
direction Ksuu (blue dash-dot-dot line).
![]()
Variances of velocity components
The variances of velocity components σi2=ui′2 were
estimated by the formula
σi2=(ui*′)2x,y,t+23Esubg,
where Esubg is the subgrid energy (Eq. ) and
(ui*′)2 are the variances of recovered velocity
components. The vertical velocity variance has the greatest impact on the
functions fsy. Figure b shows the comparison of
evaluated normalized RMS σ̃w=σw/|τ|1/2 (solid
line) with the SHEBA data (symbols; see description in
Fig. 15b; data kindly provided by Dr. A. Grachev).
The data are shown in dependence on non-dimensional stability parameter ξ=κz/Λ, where
Λ(z)=-|τ|3/2Θ0gQ
is the local Obukhov length, determined using values of fluxes of momentum
|τ| and temperature Q at the given height z (local scaling in the
stable ABL ). The measurements suggest that the mean value
of the normalized RMS σ̃w≈1.33 if the value ξ is
small. Figure b shows that our estimation of RMS is slightly less
than the measured values in the interval 0.03<ξ<0.2. Respectively, the
final values of vertical velocity variance designed for the substitution in
stochastic models were corrected as follows: σw2=1.332|τ| if
ξ<1. At the higher levels, the estimation (Eq. ) was applied.
The final estimations of the variances of velocity components are shown in
Fig. c by the solid lines. Dashed lines are the filtered resolved
velocity u‾i variances. The estimation of the variance
σw2 using Eq. () is shown by the circles. One can see
that significant parts of variances were not reproduced explicitly in LES and
were recovered using the above-mentioned assumptions.
Turbulent energy dissipation rate
Usual interpolation is not applicable to the calculation of dissipation
rate near the surface, where ϵ∼1/z. Besides, the values of
dissipation ϵΔk computed in LES at the levels zk=(k-1/2)Δg are approximately equal to the averaged values inside the
layers (k-1)Δg<z≤kΔg, but not to the
physical dissipation at given altitudes. Under the assumption that |τ|
is constant with height and neglecting the stratification inside first layer,
one can get the following corrected value of ϵ at the height
z=Δg/2:
ϵ|z=Δg/2≈2ϵΔ1/ln(Δg/z0).
Additional analysis showed that, if z<0.25zi, then the local balance of
turbulent kinetic energy (TKE) is well satisfied: ϵ≈S+B,
where S and B are shear and buoyancy production. Therefore, the
non-dimensional dissipation can be approximated by a formula
ϵ̃=ϵz|τ|3/2=ϕmzΛ-zΛ=1κ+(CmΛ-1)zΛ,
where
ϕm=∂u∂zz|τ|1/2=1κ+CmΛzΛ
is the non-dimensional
velocity gradient; CmΛ=5, according to the observation data
e.g., and LES results e.g.,.
Here, the assumption is used that the shear ∂u/∂z and the stress τ are collinear. Previous LES
studies of the stable ABL e.g., also give negligibly small
values of the transport terms in the TKE balance. The experimental
confirmation of the validity of Eq. () can be found in
, where the dissipation in the stable ABL was estimated
using the spectral analysis of longitudinal velocity in the inertial range.
In accordance with this paper, ϵ̃≈ϕm, which is
almost indistinguishable from Eq. () within the accuracy of the
experimental data and the ambiguity of the method of dissipation evaluation.
Discrete values of non-dimensional dissipation ϵΔkzk/|τ|3/2 are shown in Fig. a by circles. The dashed
straight line is the universal function (Eq. ). One can see that
the correction (Eq. ) makes the dissipation values closer to the
function (Eq. ). Finally, the profile of dissipation
ϵcf(z) for the LSM was corrected as follows (see
Fig. b). The dissipation was set to be constant below some height
ze, and was replaced by the universal function ϵ=ϵ̃|τ|3/2/z up to the level with z/Λ=1. The height
ze was chosen in such a way to equalize values of the dissipation averaged
in a layer 0≤z≤Δg and the dissipation
ϵΔ1. Figure b shows that the corrected
dissipation ϵcf (solid line) is very close to “discrete”
dissipation ϵΔk (circles), except for the first
computational level.
Diffusion coefficients
A random displacement model (Eq. ) requires the estimation of an
eddy-diffusion coefficient Ks. Note that, due to anisotropy, one
should use tensor diffusivity Ksij in a general case.
Neglecting this fact, let us assume that the principal axes of the tensor
Ksij are aligned with the coordinate axes. The corresponding
coefficients Ksww, Ksuu and Ksvv
(see Fig. d) can be calculated as follows:
Ksww=-w′s′/∂s‾∂z,
eKsuu=σu4σw4Ksww,Ksvv=σv4σw4Ksww.
The horizontal eddy diffusivities Ksuu and Ksvv
are estimated taking into account Eq. ().
(a) Discrete (LES) non-dimensional dissipation
ϵΔkκzk/|τ|3/2 (circles), corrected values
(solid line), universal function (Eq. ) (dashed straight line).
(b) Simulated discrete dissipation ϵΔk (circles)
and corrected dissipation ϵcf(z) for LSM (solid line). Dashed
horizontal line denotes the height ze, which was chosen in order to
equalize the integral values of the corrected dissipation and the discrete
dissipation.
![]()
One can see that the formula (Eq. ) provides a good
approximation for the coefficient Ksww if one sets the value
C0=6. We note that the data of LES were substantially corrected to get
this estimation. Very fine grid simulations are needed to verify and justify
the given value. There is no guarantee that this constant is actually
universal under different stratifications in the ABL.
Specification of LSMs and RDMs tested against LES
The following stochastic models were tested using the data prepared as
described above.
RDM0 is the random displacement model with uncorrelated components.
Particle position is computed by the formula similar to Eq. () but
with direction-dependent coefficients (see Eqs. and and
Fig.d). The components of the Gaussian random noise satisfy
Eq. ().
RDM1 differs from RDM0 by using the noise with
inter-component correlations:
ξip(t)ξjh(t+t′)=ui′uj′σiσjδphδ(t′)dt,
where σi=u′i21/2.
LSM0 is the Lagrangian
stochastic model without a WMC:
du′ip=-u′ipTLidt+C0ϵξip,TLi=2σi2C0ϵ.
LSM1 is based on the one-dimensional well-mixed model:
dwp=-wpTLw+12∂σw2∂z1+(wp)2σw2dt+C0ϵξ3p,TLw=2σw2C0ϵ,
supplemented by uncorrelated horizontal mixing similar to Eq. ()
with the appropriate variances σu2 and
σv2.
LSMT is a three-dimensional Lagrangian stochastic model satisfying a WMC,
which is proposed by . For the incompressible turbulent
fluid in a steady state and under the condition of zero mean vertical
velocity, this model formula 32 reads
aip=-12δijC0ϵ(τ-1)iku′kp+12∂τil∂xl+∂ui(p)∂xju′jp+12(τ-1)lj∂τil∂xku′jpu′kp,du′ip=aipdt+C0ϵξip,
where τ-1 is the tensor inverse to the stress tensor.
The setups of numerical experiments with RDMs and LSMs were close to particle
advection conditions in LES (absorbtion at altitude 100 m, ejection at
z0=0.1 m and reflection at z=0). The particles were generated
continuously within 2 h of modeling. The last hour was used for averaging.
Models LSM0 and LSM1 use the value C0=6. Three-dimensional model LSMT was
applied with C0=6, C0=8 and C0=4.
Modeling results
Crosswind-integrated footprints fsy (a, c, e) and cumulative footprints F (b, d, f) for sensor heights zM=10 m (a, b), zM=30 m (c, d) and zM=60 m (e, f). Solid lines – LES with grid steps
Δg=2.0 m. Blue triangles – LSMT with
C0=6 (absorption at z=100 m); open triangles – LSMT with C0=8; blue
dashed lines – LSMT with C0=4. Orange curves – LSMT with C0=6
(absorption at z=300 m). Short-dashed line – LSM0 (Lagranian stochastic
model without a well-mixed condition). Red circles – LSM1 (an LSM with a WMC
for vertical mixing). Open green circles – RDM0 (uncorrelated random
displacement model). Dash-dot green line – RDM1 (random displacement model
with correlation between the displacement components).
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Figure shows crosswind-integrated footprints fsy
and the corresponding cumulative footprints F, computed by LES (black bold
solid lines, Δg=2.0 m) and by stochastic models described
above. Footprints are shown for sensor heights zM= 10, 30 and 60 m.
Models RDM0, RDM1 and LSM1 provide very similar results. Faster mixing is
observed in stochastic models below altitude zM=10 m in comparison to
LES. These differences are not crucial and are compensated for in cumulative
footprints at the distances x-xM∼1000 m. The differences can be
explained either by insufficient subgrid mixing in LES or by an inexact
procedure of the data preparation for stochastic modeling. Very weak
sensitivity of the models with respect to correlations of particle velocity
components is observed as well. Thus, the results close to LES were obtained
in stochastic models having the “diffusion limit” with the same or similar
vertical diffusion coefficient. The significant advantages of LSMs compared
to RDMs were not observed in this particular flow.
The substantial disagreements with LES were obtained using the
three-dimensional Thomson model (Eq. ), with C0=4 and C0=6,
and the LSM0 model. The last one is designed for the isotropic turbulence and
does not satisfy a WMC under the conditions considered here. This model leads
to overestimated mixing, and such bias does not vanish at high altitudes.
LSMT (Eq. ) was proposed in as one of the
possible ways to satisfy a WMC in three dimensions. In our simulations the
error of LSMT with C0=6 is substantial and grows with sensor height. This
was shown by , who derived the diffusion limit of
Thomson's multi-dimensional model for Gaussian inhomogeneous turbulence and
showed that the implied effective eddy diffusivity for vertical dispersion is
Ks=2(σw4+u′w′2)C0ϵ.
Taking into account this expression and Eq. (), which is
valid for the one-dimensional LSM, one can estimate the appropriate value of
C0 for LSMT under the conditions considered here: C0≈6(1.334+1)/1.334≈8 (we assume that σw/|u′w′|1/2≈σw/|τ|1/2≈1.33). The
results of LSMT with C0=8 are in close agreement with the results of
other stochastic models and with the results of LES (open triangles in
Fig. a, c, e).
One can see that Thomson's multi-dimensional model with C0=4 produces a
very short footprint (blue dashed lines in Fig. ). Similar results
can be obtained using LSM1 with the values of C0<6 (not shown here).
Finally, it can be seen from Fig. that the top boundary condition
(absorbtion of particles at the height of 100 m) does not affect the
footprints considered here. See the orange curves, which are obtained in LSMT
(C0=6) with an absorption condition applied at the level above the
boundary layer height.
(a) Prandtl number Pr (dashed line) and Schmidt number
Sc (solid line), computed using Eulerian scalars. Symbols – Schmidt
numbers Sc, computed using the Lagrangian particles in LES, LSMs and RDMs.
(b) RMS of the crosswind position of particle Y′p=(yp-Yp)21/2 depending on the mean longitudinal position
Xp=xp. Dashed lines – RDM with
Ksuu=Ksyy=Ksww and one-dimensional
RDM Ksuu=Ksvv=0.
![]()
Turbulent Prandtl Pr and Schmidt Sc numbers computed using the Eulerian
approach are shown in Fig. a. These numbers coincide and are
approximately equal to 0.8 up to the altitude slightly less than 100 m,
where the boundary condition for a scalar is applied. Schmidt numbers Sc
were also calculated using the concentrations and the fluxes of Lagrangian
particles. Models RDM0 and LSM1 provide the values of Sc close to the
results of the Eulerian model. Calculations by LSMT (C0=6) result in Sc≈0.5–0.6, which is also the sign of the overestimated vertical
mixing.
Two-dimensional footprints fs(x-xM,y-yM,zM), computed by
models RDM0, RDM1 and LSM1 (figures are not shown here), were very close to
LES results presented in Fig. . In particular, this fact argues
for the mechanism of formation of the region with negative values of
fs having a simple nature, which can be easily reproduced in the
framework of the diffusion approximation.
The crosswind mixing can be characterized by an RMS of transversal
coordinates of the particles depending on the mean distance from the source:
Y′p(Xp)=(yp-Yp)21/2, where Xp=xp and
Yp=yp are the mathematical expectations of the particle
position. Functions Y′p(Xp) are shown in Fig. b. Models RDM0,
RDM1, LSM1 and LSMT (with C0=6) result in close horizontal dispersion. All
the stochastic models predict slightly less intensive mixing in comparison to
LES, which can be a consequence of the inaccurate data preparation algorithm
as well. If one neglects the anisotropy of eddy diffusivity, then this
dispersion would be substantially underestimated (see the short-dashed line
in Fig. b, computed by an RDM with coefficients
Ksuu=Ksvv=Ksww). One can see that
choice C0=8 in LSMT (open triangles) does not improve its overall
performance because the improved vertical mixing is accompanied by the
reduced dispersion of particles in the horizontal direction.
Wind direction rotation leads to widening of a concentration trace from the
point source (see the thin dashed line in Fig. b, computed with a
one-dimensional LSM). At larger distances from the source in the Ekman layer
the crosswind dispersion of pollution should be defined by the joint effect
of the wind rotation and vertical mixing, but not by the horizontal turbulent
mixing.
Conclusions
Scalar dispersion and flux footprint functions within the stable atmospheric
boundary layer were studied by means of LES and stochastic particle
dispersion modeling. It follows from LES results that the main impact on the
particle dispersion can be attributed to the advection of particles by
resolved and partially resolved “subfilter-scale” eddies. It ensures the
possibility of improving the results of particle
advection in discrete LES by the
use of recovering of small-scale partially resolved velocity fluctuations. If
one uses the LES model with the explicit filtering, then this recovering is
straightforward and consists of application of the known inverse filter
operator. Apparently, a similar method can be implemented for other LES when
the spatial filter is not specified in an explicit form. This would require,
however, the prior analysis of the modeled spectra to identify an effective
spatial resolution and the actual shape of the implicit filter. For
substantial improvement of particle transport statistics, it is enough to use
a subgrid Lagrangian stochastic model within the first computational layer
only, where the LES model becomes equivalent to the simplified RANS model.
When the particles are advected by a divergence-free turbulent velocity
field, then the variance of the particle concentration can be accumulated at
small spatial scales. In the considered case, it does not affect directly the
particle advection by the large eddies and has no significant influence on
the results of footprint calculations. In those cases, when the instantaneous
characteristics of the scalar field of a particle concentration are
important, additional correction to particle velocities may be required. It
can be done both through the introduction of stochastics, resulting in the
diffusion of concentration, and through the “computationally inexpensive”
divergent correction of the Eulerian velocity field.
Under the stable stratification, to calculate the flux footprint, it is
preferable to use stochastic models, which describe the particle dispersion
close to the process of scalar concentration diffusion with the effective
coefficient Ksww(z)=-w′s′/(ds/dz) in a vertical direction. RDM and
one-dimensional “well-mixed” LSM tested in this study are the examples of
such stochastic models. The optimal value for the parameter C0 for LSMs is
found to be close to 6 under the conditions considered here. This value
coincides with the estimation of Kolmogorov Lagrangian constant in isotropic
homogeneous turbulence. It provides additional justification for use of LSMs
in stable ABL, due extending their applicability over a wider range of scales
including the inertial subrange. Stochastic models that use smaller values
C0≈3–4 (this choice is widespread now) may produce extra mixing
and the shorter footprints, respectively. Note that the estimation C0=6 is
based on the LES results combined with the SHEBA data ,
where the non-dimensional vertical velocity RMS was evaluated as σ̃w≈1.33 (the exact estimation of this value in LES is restricted
by the resolution requirements). In the cases when LSMs utilize smaller
values of σ̃w the parameter C0 should be reduced accordingly
(for example, C0≈4.7 will be the best suited parameter for LSMs
with the widely used value σ̃w≈1.25 prescribed).
One-dimensional stochastic models can be supplemented by the horizontal
particle dispersion in a simple way. Introduction of the correlation between
particle displacement components in RDM does not improve or change results
substantially. However, the coefficients of horizontal diffusion
Ksuu and Ksvv for RDMs can be evaluated through
the vertical diffusion coefficient Ksww multiplied by the
square of velocity component variances ratio.
Model LSM1, constructed as a combination of independent stochastic models in
each direction (well mixed in the vertical direction only), gives reasonable
results, although this model does not satisfy a WMC in general. In contrast,
the three-dimensional Thomson model with a WMC and C0=6 provides
overestimated vertical mixing, which is manifested in too small Schmidt
number values and in reduced lengths of the footprints. The Thomson model
with C0=8 produces true mixing in the vertical direction, but
underestimates the mixing in the crosswind direction.
Accordingly, one can recommend another well-mixed stochastic model proposed
in . It was developed under the assumption that the vertical drift
term does not depend on the horizontal velocity components, and the vertical
component of this model coincides with LSM1. Prior to use, this model should
be modified in an appropriate way to take into account the variation of
momentum fluxes with height.
According to the presented LES, the source area and footprints in the stable
ABL can be substantially more extended than those predicted by the modern
LSMs and footprint parameterizations based on their results (e.g., the
parameterization by , which was calibrated with the use of
a stochastic model ). The following reasons were
identified in this study: (1) too small values of the parameter C0 are
used; and (2) the possible overestimated vertical mixing provided by some
stochastic models based on a well-mixed condition.
We emphasize that a very simple case of the moderately stratified stable ABL
in almost steady-state conditions was considered here. This setup of
numerical experiments permits the detailed intercomparison of different
approaches for the particle dispersion modeling, which utilize identical
simplifications. On the other hand, in a real environment the scalar flux
footprint functions can be greatly influenced by the meteorological
non-stationarity, the peculiarities of mixing inside the roughness layer,
internal radiative heating or cooling in the ABL, and so on. Also, a wider
investigation of different stability regimes from neutrality to strong
stratification must be undertaken in future studies to confirm the
universality of the findings.