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- Author or Editor: John D. Wilson x

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## Abstract

To evaluate Reynolds-averaged Navier–Stokes (RANS) models of disturbed micrometeorological winds, steady-state computations using a second-order closure are compared with observations (see ) in which the surface layer wind was disturbed by a long, thin porous fence (height *h* = 1.25 m; thickness *d*
_{
x
} ≈ 1 mm). Starting with the case of neutral stratification and normal incidence, it is shown that low-resolution RANS simulations (streamwise grid interval Δ*x*/*h* = 1) produce reasonably good transects of mean wind speed, though with an ambiguity (or nonuniqueness) of at least 10%–15% of the amplitude of the relative wind curve, mainly arising from sensitivity to the choice of the solution mesh. For nearly perpendicular flows, the measured influence of stratification (stable or unstable) is to diminish the amplitude of the relative wind curve (i.e., windbreak is *less* effective), an effect that is replicated very well by the simulations. Obliquity of the incident wind, like stratification, also correlates with poorer shelter, but the computed response of the relative wind curve to obliquity is excessive. As for higher-order wind statistics, computed transects of velocity standard deviations compare poorly to those observed. Therefore, if this disturbed flow may be taken as representative, then caution must be recommended should it be thought that RANS-type models might be suitable (i.e., accurate, as well as convenient) for computing the disturbed wind statistics (typically mean velocity, shear stress tensor, and turbulent kinetic energy dissipation rate) that are needed to “drive” modern dispersion models in the complex wind regimes that must be confronted in such contexts as urban dispersion, or the wind migration of pollen.

## Abstract

To evaluate Reynolds-averaged Navier–Stokes (RANS) models of disturbed micrometeorological winds, steady-state computations using a second-order closure are compared with observations (see ) in which the surface layer wind was disturbed by a long, thin porous fence (height *h* = 1.25 m; thickness *d*
_{
x
} ≈ 1 mm). Starting with the case of neutral stratification and normal incidence, it is shown that low-resolution RANS simulations (streamwise grid interval Δ*x*/*h* = 1) produce reasonably good transects of mean wind speed, though with an ambiguity (or nonuniqueness) of at least 10%–15% of the amplitude of the relative wind curve, mainly arising from sensitivity to the choice of the solution mesh. For nearly perpendicular flows, the measured influence of stratification (stable or unstable) is to diminish the amplitude of the relative wind curve (i.e., windbreak is *less* effective), an effect that is replicated very well by the simulations. Obliquity of the incident wind, like stratification, also correlates with poorer shelter, but the computed response of the relative wind curve to obliquity is excessive. As for higher-order wind statistics, computed transects of velocity standard deviations compare poorly to those observed. Therefore, if this disturbed flow may be taken as representative, then caution must be recommended should it be thought that RANS-type models might be suitable (i.e., accurate, as well as convenient) for computing the disturbed wind statistics (typically mean velocity, shear stress tensor, and turbulent kinetic energy dissipation rate) that are needed to “drive” modern dispersion models in the complex wind regimes that must be confronted in such contexts as urban dispersion, or the wind migration of pollen.

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## Abstract

In numerical weather prediction models, drag on unresolved terrain is usually represented by augmenting the boundary drag on the model atmosphere, in terms of an effective surface roughness length. But as is shown here, if a terrain-following coordinate is defined relative to smoothed terrain, the residual unresolved terrain component implies a volumetric momentum sink, as recently implemented in the Canadian Climate Centre GCM, and as is implicit in the “orographic-stress profile” method. Thus treating drag on unresolved terrain by way of an internal (rather than enhanced surface) momentum sink is a better method in principle. While the skill of both methods hinges on limited fundamental knowledge of drag on terrain, a distributed momentum sink arguably offers greater flexibility to improve modeling of mountain winds, if necessary by tailoring the sink to achieve success, in specific regions, by trial and error.

A consequence of the new method is that unresolved terrain results in a ground-based (stress divergence) layer, that is somewhat analogous to a plant canopy layer, from the point of view of its momentum balance.

## Abstract

In numerical weather prediction models, drag on unresolved terrain is usually represented by augmenting the boundary drag on the model atmosphere, in terms of an effective surface roughness length. But as is shown here, if a terrain-following coordinate is defined relative to smoothed terrain, the residual unresolved terrain component implies a volumetric momentum sink, as recently implemented in the Canadian Climate Centre GCM, and as is implicit in the “orographic-stress profile” method. Thus treating drag on unresolved terrain by way of an internal (rather than enhanced surface) momentum sink is a better method in principle. While the skill of both methods hinges on limited fundamental knowledge of drag on terrain, a distributed momentum sink arguably offers greater flexibility to improve modeling of mountain winds, if necessary by tailoring the sink to achieve success, in specific regions, by trial and error.

A consequence of the new method is that unresolved terrain results in a ground-based (stress divergence) layer, that is somewhat analogous to a plant canopy layer, from the point of view of its momentum balance.

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## Abstract

Wind statistics were measured using cup and sonic anemometers, placed upwind and downwind from a porous plastic windbreak fence (height *h* = 1.25 m, length *Y* = 114 m, resistance coefficient *k*
_{
r0} = 2.4, and porosity *p* = 0.45) standing on otherwise uniform land (short grass with roughness length *z*
_{0} ∼ 1.9 cm). Intercomparison with collocated two-dimensional sonic anemometers suggested that, except in strongly stratified winds, cup anemometers (distance constant 1.5 m), subjected to a uniform overspeeding correction (here ∼10%), provide a reasonably accurate transect of the mean wind across the disturbed flow region. The measurements, binned with respect to mean wind direction and stratification, establish that the resistance coefficient of a windbreak of this type implies the maximum (or “potential”) mean wind reduction, a potential that is realized in neutral, perpendicular flow and for which a semiempirical formula is derived. Obliquity of the approaching wind *reduces* actual shelter effectiveness below the potential value, as was already known. However, a systematic influence of stratification could only be discriminated in winds that were not too far (say, within about ±30°) from perpendicular, under which conditions both stable and unstable stratification *reduced* shelter effectiveness. The “quiet zone,” in which velocity standard deviations (*σ*
_{
u
}, *σ*
_{
υ
}) are reduced relative to the approach flow, was found to extend farther downwind for the normal velocity component (*u*) than for the parallel component (*υ*).

## Abstract

Wind statistics were measured using cup and sonic anemometers, placed upwind and downwind from a porous plastic windbreak fence (height *h* = 1.25 m, length *Y* = 114 m, resistance coefficient *k*
_{
r0} = 2.4, and porosity *p* = 0.45) standing on otherwise uniform land (short grass with roughness length *z*
_{0} ∼ 1.9 cm). Intercomparison with collocated two-dimensional sonic anemometers suggested that, except in strongly stratified winds, cup anemometers (distance constant 1.5 m), subjected to a uniform overspeeding correction (here ∼10%), provide a reasonably accurate transect of the mean wind across the disturbed flow region. The measurements, binned with respect to mean wind direction and stratification, establish that the resistance coefficient of a windbreak of this type implies the maximum (or “potential”) mean wind reduction, a potential that is realized in neutral, perpendicular flow and for which a semiempirical formula is derived. Obliquity of the approaching wind *reduces* actual shelter effectiveness below the potential value, as was already known. However, a systematic influence of stratification could only be discriminated in winds that were not too far (say, within about ±30°) from perpendicular, under which conditions both stable and unstable stratification *reduced* shelter effectiveness. The “quiet zone,” in which velocity standard deviations (*σ*
_{
u
}, *σ*
_{
υ
}) are reduced relative to the approach flow, was found to extend farther downwind for the normal velocity component (*u*) than for the parallel component (*υ*).

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## Abstract

The simplest “random flight” models for the paths of heavy particles in turbulence have been tested against previous observations of the deposition of glass beads from an elevated source in the atmospheric surface layer. For the bead sizes examined (diameter 50–100 *μ*m), for which the ratio of particle inertial timescale to turbulence timescale *τ*
_{
p
}/Γ_{
L
} ≪ 1, it was found sufficient to adapt, as others earlier have done, a well-mixed first-order Lagrangian stochastic (“Langevin”) model of *fluid element* trajectories, simply by superposing a gravitational settling velocity *w*
_{
g
} and reducing the velocity autocorrelation timescale along the heavy particle trajectory (Γ_{
p
}) relative to the fluid-Lagrangian timescale (Γ_{
L
}). That is to say, unless details of the particle distribution very close to ground (where *τ*
_{
p
}/Γ_{
L
} is not small) are of interest, no advantage other than conceptual clarity can be found in the more faithful approach of explicitly modeling particle acceleration by means of the particle equation of motion.

With the timescale reduction parameter *β* ∼ 2, the Langevin model estimated the location and width of the bead deposit swath very well and fixed the peak deposit density to within (at worst) about 100% error (in very stable stratification), but more generally to within about 20%. In the case where trajectories intersected a tall crop canopy, uncertainties in the treatment of deposition proved more significant than nuances of the trajectory algorithm.

## Abstract

The simplest “random flight” models for the paths of heavy particles in turbulence have been tested against previous observations of the deposition of glass beads from an elevated source in the atmospheric surface layer. For the bead sizes examined (diameter 50–100 *μ*m), for which the ratio of particle inertial timescale to turbulence timescale *τ*
_{
p
}/Γ_{
L
} ≪ 1, it was found sufficient to adapt, as others earlier have done, a well-mixed first-order Lagrangian stochastic (“Langevin”) model of *fluid element* trajectories, simply by superposing a gravitational settling velocity *w*
_{
g
} and reducing the velocity autocorrelation timescale along the heavy particle trajectory (Γ_{
p
}) relative to the fluid-Lagrangian timescale (Γ_{
L
}). That is to say, unless details of the particle distribution very close to ground (where *τ*
_{
p
}/Γ_{
L
} is not small) are of interest, no advantage other than conceptual clarity can be found in the more faithful approach of explicitly modeling particle acceleration by means of the particle equation of motion.

With the timescale reduction parameter *β* ∼ 2, the Langevin model estimated the location and width of the bead deposit swath very well and fixed the peak deposit density to within (at worst) about 100% error (in very stable stratification), but more generally to within about 20%. In the case where trajectories intersected a tall crop canopy, uncertainties in the treatment of deposition proved more significant than nuances of the trajectory algorithm.

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## Abstract

Lagrangian stochastic (LS) dispersion models often use trajectory reflection to limit the domain accessible to a particle. It is shown how the well-mixed condition (Thomson) can he expressed in the Chapman-Kolmogorov equation for a discrete-time LS model to provide a test for the validity of a reflection algorithm. By that means it is shown that the usual algorithm (perfect reflection) is exactly consistent with the wmc when used to bound Gaussian homogeneous turbulence, but that no reflection scheme can satisfy the wmc when applied at a location where the probability distribution for the normal velocity is asymmetric, or locally inhomogeneous. Thus, there is no well-mixed reflection scheme for inhomogeneous or skew turbulence.

## Abstract

Lagrangian stochastic (LS) dispersion models often use trajectory reflection to limit the domain accessible to a particle. It is shown how the well-mixed condition (Thomson) can he expressed in the Chapman-Kolmogorov equation for a discrete-time LS model to provide a test for the validity of a reflection algorithm. By that means it is shown that the usual algorithm (perfect reflection) is exactly consistent with the wmc when used to bound Gaussian homogeneous turbulence, but that no reflection scheme can satisfy the wmc when applied at a location where the probability distribution for the normal velocity is asymmetric, or locally inhomogeneous. Thus, there is no well-mixed reflection scheme for inhomogeneous or skew turbulence.

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## Abstract

Wind velocity statistics from several points within a regular but sparse array of clumped corn plants are analyzed, with each clump consisting of 12 plants, having a mean height of 1.6 m and a collective leaf area of about 2.75 m^{2} and occupying approximately 0.8 m^{2} of ground area. The clumps defined the (484) nodes of a square lattice, with side length of 5.6 m, and this lattice covered an area of 120 m × 120 m within an otherwise uniform corn field. Forty-eight half-hour records of daytime mean “cup” wind speeds and turbulent kinetic energies *k,* from several points in the canopy, are displayed against a “reduced” mean wind direction that exploits the symmetries of the canopy. These data conform well with the corresponding fields from a three-dimensional, steady-state wind model (with eddy viscosity ∝ *λk*
^{1/2}, where *λ* is the length scale). Both the observations and the model confirm the importance of a set of special wind directions, some of which place a given point P in the shelter of a nearby clump (“blockage”) and others of which place P in a “corridor.”

## Abstract

Wind velocity statistics from several points within a regular but sparse array of clumped corn plants are analyzed, with each clump consisting of 12 plants, having a mean height of 1.6 m and a collective leaf area of about 2.75 m^{2} and occupying approximately 0.8 m^{2} of ground area. The clumps defined the (484) nodes of a square lattice, with side length of 5.6 m, and this lattice covered an area of 120 m × 120 m within an otherwise uniform corn field. Forty-eight half-hour records of daytime mean “cup” wind speeds and turbulent kinetic energies *k,* from several points in the canopy, are displayed against a “reduced” mean wind direction that exploits the symmetries of the canopy. These data conform well with the corresponding fields from a three-dimensional, steady-state wind model (with eddy viscosity ∝ *λk*
^{1/2}, where *λ* is the length scale). Both the observations and the model confirm the importance of a set of special wind directions, some of which place a given point P in the shelter of a nearby clump (“blockage”) and others of which place P in a “corridor.”

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## Abstract

From an analysis of scales in the cloud droplet collision problem, the authors infer that a trajectory model that is to be capable of predicting collisions between droplets of *all* possible sizes should be of second-order, that is should explicitly model particle acceleration. But for collisions between large droplets (radius about 50 µm or larger), which are still much smaller than raindroplets, a first-order model is appropriate.

The relative motion of large droplets are studied with a first-order, two particle trajectory model. Turbulence is found to be unimportant (relative to differential gravitational settling) if the (large) droplet sizes are sufficiently distinct. Zeroth-order two-particle models, of the type hitherto applied to be problem, deteriorate in accuracy as the influence of turbulence on the droplet separation increases, that is, for large σ_{
v
}/*v*′, where σ_{v} is the turbulent velocity scale and *v*′ is the droplet still-air terminal velocity. Under no circumstance is a single-particle model applicable.

## Abstract

From an analysis of scales in the cloud droplet collision problem, the authors infer that a trajectory model that is to be capable of predicting collisions between droplets of *all* possible sizes should be of second-order, that is should explicitly model particle acceleration. But for collisions between large droplets (radius about 50 µm or larger), which are still much smaller than raindroplets, a first-order model is appropriate.

The relative motion of large droplets are studied with a first-order, two particle trajectory model. Turbulence is found to be unimportant (relative to differential gravitational settling) if the (large) droplet sizes are sufficiently distinct. Zeroth-order two-particle models, of the type hitherto applied to be problem, deteriorate in accuracy as the influence of turbulence on the droplet separation increases, that is, for large σ_{
v
}/*v*′, where σ_{v} is the turbulent velocity scale and *v*′ is the droplet still-air terminal velocity. Under no circumstance is a single-particle model applicable.

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## Abstract

Three-dimensional empirical orthogonal functions (E0Fs), calculated from a large-eddy simulation of a weakly convective, planetary boundary layer (PBL), are used to decompose statistics for PBL turbulence into contributions from individual structures. The most energetic E0Fs, corresponding largely to boundary-layer-spanning eddies, together are responsible for about one-half of the turbulent kinetic energy (TKE) throughout the boundary layer, although they carry a substantial amount of the momentum and heat fluxes only near mid-PBL. Examination of the flux profiles also reveals coupling between large roll structures and inversion-borne gravity waves. By filtering the fields through the EOFs, skewness and intermittency (kurtosis) associated with the different vertical scales are determined. Positive skewness around mid-PBL is found to be attributable to the boundary-layer spanning eddies. lntermittency, however, cannot be attributed to either large- or small-scale structures: it results from interscale interactions. Finally, equations for the flux and energy budgets of individual structures are derived. The budget analyses show clearly that the main source of TKE for large roll structures is shearing stress, while the main loss mechanism is transfer to smaller scales. The inversion-borne gravity waves gain TKE from interscale transfers and buoyant acceleration and lose TKE to shearing effects.

## Abstract

Three-dimensional empirical orthogonal functions (E0Fs), calculated from a large-eddy simulation of a weakly convective, planetary boundary layer (PBL), are used to decompose statistics for PBL turbulence into contributions from individual structures. The most energetic E0Fs, corresponding largely to boundary-layer-spanning eddies, together are responsible for about one-half of the turbulent kinetic energy (TKE) throughout the boundary layer, although they carry a substantial amount of the momentum and heat fluxes only near mid-PBL. Examination of the flux profiles also reveals coupling between large roll structures and inversion-borne gravity waves. By filtering the fields through the EOFs, skewness and intermittency (kurtosis) associated with the different vertical scales are determined. Positive skewness around mid-PBL is found to be attributable to the boundary-layer spanning eddies. lntermittency, however, cannot be attributed to either large- or small-scale structures: it results from interscale interactions. Finally, equations for the flux and energy budgets of individual structures are derived. The budget analyses show clearly that the main source of TKE for large roll structures is shearing stress, while the main loss mechanism is transfer to smaller scales. The inversion-borne gravity waves gain TKE from interscale transfers and buoyant acceleration and lose TKE to shearing effects.

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## Abstract

Katul and Chang recently compared the performance of two second-order closure models with observations of wind and turbulence in the Duke Forest canopy, noting that such models “alleviate some of the theoretical objections to first-order closure.” This paper demonstrates that, notwithstanding those (valid) theoretical objections, Duke Forest wind simulations of comparable quality can be obtained using a first-order closure, namely, eddy viscosity *K* ∝ *λ*
*k*
*k* is the turbulent kinetic energy and *λ* is a turbulence length scale. It is concluded that, most often, uncertainty in the drag coefficient will limit the accuracy of modeled wind statistics, regardless of the turbulence closure chosen.

## Abstract

Katul and Chang recently compared the performance of two second-order closure models with observations of wind and turbulence in the Duke Forest canopy, noting that such models “alleviate some of the theoretical objections to first-order closure.” This paper demonstrates that, notwithstanding those (valid) theoretical objections, Duke Forest wind simulations of comparable quality can be obtained using a first-order closure, namely, eddy viscosity *K* ∝ *λ*
*k*
*k* is the turbulent kinetic energy and *λ* is a turbulence length scale. It is concluded that, most often, uncertainty in the drag coefficient will limit the accuracy of modeled wind statistics, regardless of the turbulence closure chosen.

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## Abstract

When a particle descends beneath the (nominal) lower boundary of the atmosphere, it may remain there for some time *τ* before it reemerges into the (resolved) flow. In particle trajectory models, *τ* is the random duration of unresolved trajectory segments, below the height *z*
_{
r
} at which an artificial reflection boundary condition is applied. By computing such paths, for realistic near-ground flows, it was found that the mean delay per reflection is *τ*
*z*
_{
r
}/*σ*
_{
w
} where *σ*
_{
w
} is the standard deviation of the vertical velocity at *z*
_{
r
}. The corresponding mean alongwind displacement per reflection, due to the mean horizontal wind *u*
*z*) below *z*
_{
r
}, is *δ*
*u*
*z*
_{
r
}〉 *τ*
*u*
*z*
_{
r
}〉 is the height average of *u*
*σ*
_{
δ
} ≈ 2*z*
_{
r
}. From simulations on the continental scale, with a lower boundary placed at *z*
_{
r
} ≈ 25 m, it was found that a typical particle suffered about 15 reflections per day, resulting in a net delay on the order of 30 min per day.

## Abstract

When a particle descends beneath the (nominal) lower boundary of the atmosphere, it may remain there for some time *τ* before it reemerges into the (resolved) flow. In particle trajectory models, *τ* is the random duration of unresolved trajectory segments, below the height *z*
_{
r
} at which an artificial reflection boundary condition is applied. By computing such paths, for realistic near-ground flows, it was found that the mean delay per reflection is *τ*
*z*
_{
r
}/*σ*
_{
w
} where *σ*
_{
w
} is the standard deviation of the vertical velocity at *z*
_{
r
}. The corresponding mean alongwind displacement per reflection, due to the mean horizontal wind *u*
*z*) below *z*
_{
r
}, is *δ*
*u*
*z*
_{
r
}〉 *τ*
*u*
*z*
_{
r
}〉 is the height average of *u*
*σ*
_{
δ
} ≈ 2*z*
_{
r
}. From simulations on the continental scale, with a lower boundary placed at *z*
_{
r
} ≈ 25 m, it was found that a typical particle suffered about 15 reflections per day, resulting in a net delay on the order of 30 min per day.