ACPAtmospheric Chemistry and PhysicsACPAtmos. Chem. Phys.1680-7324Copernicus PublicationsGöttingen, Germany10.5194/acp-17-3507-2017Impact of vertical wind shear on roll structure in idealized hurricane
boundary layersWangShoupingshouping.wang@nrlmry.navy.milJiangQingfangNaval Research Laboratory, Monterey, CA 93943, USAShouping Wang (shouping.wang@nrlmry.navy.mil)14March20171753507352415September201626October20169February201719February2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://acp.copernicus.org/articles/17/3507/2017/acp-17-3507-2017.htmlThe full text article is available as a PDF file from https://acp.copernicus.org/articles/17/3507/2017/acp-17-3507-2017.pdf
Quasi-two-dimensional roll vortices are frequently observed in hurricane
boundary layers. It is believed that this highly coherent structure, likely
caused by the inflection-point instability, plays an important role in
organizing turbulent transport. Large-eddy simulations are conducted to
investigate the impact of wind shear characteristics, such as the shear
strength and inflection-point level, on the roll structure in terms of its
spectral characteristics and turbulence organization. A mean wind nudging
approach is used in the simulations to maintain the specified mean wind shear
without directly affecting turbulent motions. Enhancing the radial wind shear
expands the roll horizontal scale and strengthens the roll's kinetic energy.
Increasing the inflection-point level tends to produce a narrow and sharp
peak in the power spectrum at the wavelength consistent with the roll spacing
indicated by the instantaneous turbulent fields. The spectral tangential
momentum flux, in particular, reaches a strong peak value at the roll
wavelength. In contrast, the spectral radial momentum flux obtains its
maximum at the wavelength that is usually shorter than the roll's, suggesting
that the roll radial momentum transport is less efficient than the tangential
because of the quasi-two-dimensionality of the roll structure. The most
robust rolls are produced in a simulation with the highest inflection-point
level and relatively strong radial wind shear. Based on the spectral
analysis, the roll-scale contribution to the turbulent momentum flux can
reach 40 % in the middle of the boundary layer.
Introduction
The hurricane boundary layer (HBL) is well known for its critical role in
evolution of tropical cyclones (TCs) as the air–sea interaction represents
both the most important source and sink of the moist available energy and the
kinetic energy, respectively. One of the frequently occurring features in the
HBL is horizontal roll vortices, which have quasi-two-dimensional coherent
and banded structure extending from the surface to the top of the HBL. The
observed horizontal roll scale, i.e., the average distance between two
neighboring rolls, ranges from sub-kilometer to ∼ 10 km (Wurman and
Winslow, 1998; Lorsolo et al., 2008; Foster, 2013). Observational and
modeling studies suggest that these roll vortices make a significant
contribution to the vertical heat and momentum transport (Zhang et al., 2008;
Zhu, 2008) and thus provide a critical control of the wind, temperature, and
moisture profiles.
Previous studies have attributed the prevalence of the roll structure to the
existence of an inflection point in the mean HBL radial wind profile and
attempted to establish the link between the HBL environment and the roll
statistical characteristics (e.g., Foster, 2005; Nolan, 2005). These analyses
are generally consistent with observations: (1) the rolls are oriented at
0–10∘ to the left of the tangential wind; (2) the roll aspect ratio
(ratio of the horizontal scale to the vertical) ranges from 2 to 4; and
(3) the roll-generated momentum fluxes are non-local. A recent study by
Foster (2013) differentiates the standard boundary layer roll vortices, as
those highlighted above, from the observed large roll vortices from synthetic
aperture radar images, whose horizontal scale reaches up to 10–20 km. His
results from a two-dimensional nonlinear resonant triad interaction model
further suggest that the observed unusually large roll aspect ratio results
from the upscale energy transport through the nonlinear wave–wave
interaction. Gao and Ginis (2014, hereafter GG14) and Gao and
Ginis (2016) investigated the formation of
HBL rolls by solving a two-dimensional perturbation system driven by the mean
wind profiles that are the solutions of an axisymmetric HBL model. They
concluded that the mean wind shear intensity affects the roll growth rate and
the inflection-point level (IPL hereafter) impacts the roll wavelength. While
these two-dimensional quasi-analytical models have significantly advanced our
understanding of HBL roll dynamics, they cannot accurately represent
three-dimensional stochastic turbulent flows. This work and these conclusions
are worth revisiting using a large-eddy simulation (LES) approach.
There have been a few LES studies of HBL rolls. Zhu (2008) configured a
nested WRF (Weather Research Forecast) model to include an LES domain with a
horizontal resolution of 100 m and a vertical grid spacing varying from 5 to
65 m below 1.6 km. The WRF-LES was used to simulate a real case of
hurricane landfall. Organized large-eddy circulations with horizontal scales
ranging from 1 to 10 km were found to intensely enhance the vertical
momentum, heat, and moisture transport. He further proposed a framework of the
turbulent transport parameterization based on the conceptual model of
convective up- and down-draft representation for shallow cumulus convection.
While this mesoscale LES grid-nesting framework represents a realistic and
sophisticated numerical approach, it does not allow for sensitivity studies
to examine impact of various mean conditions, such as wind profiles, on the
roll structure. In an idealized study of HBL rolls, Nakanishi and
Niino (2012, hereafter NN12) adopted a traditional LES approach, which uses a
20 × 20 × 4 km3 domain with periodic lateral
boundary conditions. They concluded that the inflection-point instability in
the radial wind profile leads to the formation of the quasi-linear roll
structure with wavelengths between 1.5 and 2.4 km. The LES study by Green
and Zhang (2015) also confirmed many of these findings and further suggested
that the turbulence diffusivity varies considerably among different
simulations, an indication that the downgradient transfer model breaks down
for the momentum fluxes associated with HBL rolls.
Among these LES studies, only the WRF-LES nesting approach used by Zhu (2008)
explicitly simulates mesoscale circulations and thus their effects on the
roll structure. Others neglect the horizontal advection effects by assuming a
local balance among the turbulent mixing, gradient wind, Coriolis force, and
hurricane-induced centripetal force. Consequently, the wind profile based on
the local force balance may not represent the most relevant features with
respect to the roll development in the HBL in the LES studies. For example,
Morrison et al. (2005) provided both observed radial and tangential winds
from WSR-88D radar data, and the IPLs estimated
from these observations are about 300 to 800 m for the winds at the TC radius
of 29 to 122 km, respectively. These IPLs are generally
higher than those of the LES simulations by NN12 which are 100 and 300 m at
the radius of 40 and 100 km, respectively. Therefore, there is a need to use
more realistic wind profiles in the LES studies. The latest study of Bryan et
al. (2017) provided an improved HBL LES framework that accounts for the
influence of mesoscale advection on the wind profiles. The current work
introduces an empirical approach as discussed in the next section.
Boundary layer rolls have been a subject of many studies since 1960s, as
reviewed by Atkinson and Zhang (1996) and Young et al. (2002). Several
physical mechanisms have been proposed for different environments, including
combined surface shear–buoyancy instability (Moeng and Sullivan, 1994;
Glendening, 1996), the surface shear–cloud convection–radiation instability
(Chlond, 1992), parallel instability (Lilly, 1966), and inflection-point
instability (Brown, 1970; Brown, 1972; Foster, 2005). As discussed at the
beginning of the paper, the most relevant mechanism for the HBL rolls is the
inflection-point instability. This work aims to gain a new understanding of the
impact of the mean wind profile characteristics, that are directly associated
with the inflection-point instability, the radial wind shear, and IPL, on the
roll structure. We use a different LES approach, featuring a mean nudging
method which is applied to the momentum equations to strongly regulate the
mean wind profile. This approach enables us to conduct a systematic study of
the roll response, including the growth of the HBL, turbulence intensity, and
the spectral distribution, to changes in the mean wind profiles. The
remainder of the paper is organized as follows. Section 2 describes the LES
model and simulation setup. Sections 3 and 4 provide general description of
the simulation results and spectral analysis, respectively. Further
discussions on the wind shear are given in Sect. 5. Section 6 summarizes the
work.
ApproachCOAMPS-LES
The Naval Research Laboratory Coupled Ocean/Atmosphere Mesoscale Prediction
System large-eddy simulation (COAMPS-LES) is used in this study. The LES
model was first introduced by Golaz et al. (2005) for the study of boundary
layer cloud systems. It has been applied to investigate various types of
boundary layer turbulence, including topographic flows, and stratocumulus
dynamics (Golaz et al., 2009; Wang et al., 2012; Jiang and Wang, 2013).
Readers are referred to these papers for detailed descriptions as well as its
various applications. Briefly, the model applies the anelastic approximation
for efficient numerical computation and uses the Deardorff's prognostic
turbulence kinetic energy approach for the subgrid-scale model (Deardorff,
1980). The model coordinate is configured such that x is directed away from
the center of a TC in the radial direction, y is in the direction
90∘ counterclockwise from x, i.e., the azimuthal or tangential
direction, and z the vertical axis. Because our simulations are focused on
the dynamics and structure of the rolls, moisture is not included. The
predictive variables are radial wind u, tangential wind v, potential
temperature
θ, and subgrid-scale turbulence kinetic energy. The model
uses the horizontal resolution Δx=Δy= 50 m and a variable
vertical grid with Δz= 30 m below 3 km gradually increasing to
200 m. This grid covers a 25.2 × 25.2 × 4.9 km3
domain. The Rayleigh damping technique is applied near the model top to
reduce downward reflection of internal gravity waves. The surface momentum
flux is calculated using the roughness length (z0) formulation of
Donelan et al. (2004). That is, z0 increases with the 10 m wind speed
following the Charnock relationship for the wind speed less than
33 m s-1, above which z0 is set equal to 3.35 mm, which is
equivalent to a drag coefficient of 0.0025. Because the 10 m wind is usually
less than 33 m s-1 for all the simulations, this modification of
z0 on the Charnock relationship should not have major effects on the
results presented here. To accelerate the LES spinup process, a moderate
constant surface heat flux Fh= 20 W m-2 is applied.
Because of the strong near-surface winds (∼ 30 m s-1), the
application of the heat flux does not change the dominance of the shear
production of turbulence. For comparison purposes, all the simulations start
with the same initial conditions. The horizontal wind is specified as a
constant gradient wind speed (Vg) and the linear potential
temperature (θ) profile with a gradient of 0.00475 K m-1 and a
value of 298.5 K at the first vertical level. The gradient wind
Vg is fixed at 45.5 m s-1 as the value represents a
middle-to-high speed range in a hurricane environment (e.g., Willoughby,
1990). The model is integrated for 10 h with a time step of 0.5 s.
Mean wind nudging
As discussed in the introduction, the mean wind profiles from the LES
simulations that do not include the mesoscale circulations (e.g., HBL inflow)
may not adequately represent the wind characteristics in a hurricane
environment. It is highly desirable that observationally based wind profiles
be used and approximately maintained throughout the simulations. We adopt a
modeling approach that strongly regulates the mean wind profile according to
our specifications. A special relaxation term is added to each horizontal
momentum equation to nudge the mean wind toward a specified target wind
profile. A unique feature of these nudging terms is that they only nudge the
horizontally averaged wind. That is, at each time step, the horizontal mean
wind profile, which is dependent only on z, is calculated from the
predicted winds and used as the variable in the nudging term. Because the
target profile is only a function of z, the nudging tendency is exactly the
same at every horizontal grid point for the same level at each time step.
Consequently, the LES-simulated turbulent perturbations, which are defined as
deviations from a horizontal mean, are not directly affected by the nudging
terms. Both the turbulent perturbations and statistics are, of course,
regulated by the mean wind profiles. This nudging approach was used to spin
up LES simulations of stratocumulus clouds by Kazil et al. (2016).
The momentum equations with the nudging terms can be written as
∂u∂t=-v⋅∇u-1ρ0∂p∂x+SGS-Vg2-v2R-f(Vg-v)+UT(z)-u(z)τ
and
∂v∂t=-v⋅∇v-1ρ0∂p∂y+SGS-u⋅vR+fu+VT(z)-v(z)τ,
where R is the radius from the LES domain to the center of the TC,
UT and VT are the prescribed target radial and tangential wind
components, respectively, ρ0 is the air density of an atmospheric
reference state, Vg denotes the gradient wind, SGS represents
effects of subgrid-scale motions, 〈〉 is a horizontally
averaged variable at each time step, and
τ is a relaxation timescale. Other symbols in Eqs. (1) and (2) have their generally accepted meaning. These equations are the same as
those used by NN12 except the relaxation terms represented by the curly
bracket in each equation. The square bracket is the gradient wind imbalance
term associated with the centripetal force, the Coriolis force, and the
large-scale radial pressure gradient. This term represents a major forcing
that is responsible for the mean wind shear characteristics; it is designated
as the rotational term hereafter for simplicity. Sensitivity simulations have
been conducted to evaluate how well the mean wind profiles can be controlled
by the nudging term. We find that the mean wind profiles are better regulated
by the nudging if the rotational terms are removed and their removal has little
impact on the turbulence statistics. This is consistent with the previous
studies showing negligible effects from the curvature terms on the roll
structure as well as the turbulence generation in general (Foster, 2005;
NN12). Thus, the square bracket terms are set to zero in Eqs. (1) and (2) in this
paper unless specified otherwise. Therefore, the nudging terms are used to
represent all the major processes that control the mean wind profiles, except
turbulence mixing. It is noteworthy that this new approach has a number of
attractive advantages. Firstly, it maintains the mean wind profiles, which
are derived from observations or balanced dynamic models, and accordingly,
are more realistic. Secondly, it offers a convenient way to systematically
change the mean wind profile and therefore allows us to examine the roll's
response to these changes. Lastly, because the actual rotational terms are
not explicitly included in the momentum equations, the horizontal winds no
longer rotate with time and LES simulations may reach non-oscillatory
quasi-equilibrium solutions. A comparison of three test simulations is
presented in the Appendix.
Target wind profiles
We are interested in two sets of LES quasi-equilibrium solutions
corresponding to different mean wind characteristics with regard to both the
wind shear strength and IPL. These two parameters are chosen because,
according to previous studies, they are key parameters related to
inflection-point instability. The former is the main source of turbulence and
the latter is linked to the roll scales (e.g., Chlond, 1992; GG14). The
vertical shear of the radial wind above the surface layer is a main focus of
this study. The shear layer, where the inflection point is located, usually
extends from ∼ 100 m to the top of the HBL. To avoid confusion, we
use the term “surface wind shear” to describe the wind shear that is
concentrated in the lowest 100 m.
The target wind profiles are formulated based on the normalized typical
hurricane wind profiles obtained from a dynamical model of Foster (2005) and
from the observations by Morrison et al. (2005). The LES mean winds are
nudged toward the target profiles, which are formulated to represent various
wind shear conditions. This approach facilitates the study of the response of
roll formation and dynamics to wind profiles through sensitivity simulations.
We have experimented with dozens of LES simulations using a variety of target
wind profiles. The two groups of the target wind profiles (i.e., groups H and
L; see Fig. 1) are chosen from these additional trial simulations, and they
exhibit systematic variations in shear strength and infection-point levels.
The target radial wind UT of H2 and tangential wind VT of group
H generally follow those of Fig. 2 of Foster (2005) except for the HBL
height. In addition, the super-gradient wind shape is also included in
VT, in accordance with Fig. 3a of Morrison et al. (2005). The UT
profile of H2 is multiplied by 0.5 and 1.5 to provide UT for H1 and
H3, respectively. The target radial wind UT of L2 is obtained by
vertically suppressing UT of H2 and increasing the near-surface value
to 13 m s-1. Then, UT of L2 is multiplied by 0.5 and 1.5 to
give UT of L1 and L3, respectively. The target tangential wind profile
VT of group L is obtained by lowering the HBL height for VT of
group H.
Target wind profiles used in simulations of groups L and
H: (a) target radial wind profiles; (b) target tangential
wind profiles. Only one target tangential profile VT is used for each
group.
In summary, group L simulations are forced with the target radial wind
profiles (UT) that have three shear strengths with the IPLs
approximately located at 200 m (Fig. 1a). Similarly, group H simulations
also have three shear strengths with the IPLs between 400 and 500 m. The
target tangential wind profile (VT) is specified in Fig. 1b. The
VT profile with the shear occurring below 700 m (dash dotted) is used
for group L simulations, the other (solid) for group H. This paper is focused
on the radial wind shear because of its direct link to the inflection
instability (GG14). Therefore, only one target tangential wind is prescribed
for each simulation group, which has three target radial wind profiles as
discussed above. It is recognized that changes in the radial wind inevitably
affect the tangential wind. The sensitivity of the LES results to the
tangential winds is also explored. The simulations and relevant parameters
are listed in Table 1.
Simulation conditions and results with the following parameters:
individual experiments (Exp), maximum radial wind shear (RSHmax),
inflection-point level (IPL), target radial wind (UT), target
tangential wind (VT), wavelength at the peak of w′ power spectrum
(Lp), HBL height (zi), aspect ratio (Lp/zi), and the
ratio -zi/Lmo, where Lmo is the Monin–Obukhov
length.
While there is some quantitative difference between the target wind profiles
defined above and the ones derived from the basic HBL balance equations, such
as those of Foster (2005), they carry some essential features that are
similar to the model-derived or observed wind profiles, such as an inflection
point in the radial wind, the super-gradient wind in HBL, and the gradient
wind balance above the HBL. Given our objective of investigating the impact
of the wind shear (including both the shear strength and the inflection-point
level) on the roll structure, our choices of the target winds are
justified in the sense that they retain the basic HBL mean wind features and
provide a simple way to make a meaningful comparative study.
Overall turbulence structure
This section is centered on comparing instantaneous turbulence fields and
statistics between group L and H simulations (see Table 1). Special attention is given to the roll structure manifested by the coherent
and organized turbulent flow. All the profiles presented here are obtained
from ensemble averaging applied over the entire horizontal domain and
between 8 and 10 h with a sample interval of 30 s. A time series of an
average variable is constructed by taking the horizontal mean every minute.
Time evolution and mean state
To gain a general impression of the HBL development and differences among the
simulations, the time series of the HBL heights (zi) and mean profiles
are examined. As shown in Fig. 2, zi increases rapidly with time for
most simulations during the first 5 h, after which the growth slows down
considerably, implying a quasi-equilibrium state being reached. H2 appears to
be an exception; its HBL height grows slowly only after 8 h. There
is a clear tendency that stronger radial wind shear results in a higher
zi for each group (L or H). The simulations L1 and H1 have the lowest
zi in their group, in accordance with the weakest turbulence, likely due
to the weak radial wind shear for both cases. It is worth noting that H3
predicts the highest zi among all the simulations, suggesting that it
produces the strongest turbulence intensity even though it does not have the
strongest radial wind shear (Table 1). It will be shown in Sect. 4 that H3
produces the most vigorous roll structure, which likely contributes to the
highest zi through strong non-local mixing as discussed by GG14. In
addition, zi has a critical impact on the roll characteristics and their
coupling with internal waves (GG14), which will be discussed in later
sections. It also should be noted that the high zi may reflect the
fact that neither radial advection nor diabatic heating is included in the
heat balance. These processes may affect the growth of the mixed layer
(Kepert et al., 2016).
HBL height evolution and mean vertical profiles. (a) HBL height
zi; (b) mean radial wind u‾; (c) mean tangential wind
v‾; and (d) mean potential temperature θ‾. Black dots
in Fig. 2b denote the inflection-point locations.
For all the simulations, the parameter, -zi/Lmo, where
Lmo is the Monin–Obukhov length, is between 0.075 and 0.12
(Table 1). These values are considerably smaller than values of 0.5–0.65
that represent the shear–buoyancy regime transition found by Moeng and
Sullivan (1994) in their study of the shear- and buoyancy-driven boundary
layers, implying that the shear production of turbulence is dominant in all
the simulations. The maximum value of -zi/Lmo among
all the simulations is 0.13, which is considerably less than the lower
criterion -zi/Lmo= 1.5 for the formation of
buoyancy- and shear-driven roll structures (Glendening, 1996). Therefore, any roll
structure resulting from these simulations should not be explained by the
buoyancy–shear mechanism.
Plan view of w′ at 9 h at three different levels (i.e.,
z/zi= 0.2, 0.4, and 0.9, respectively) from group L and H
simulations.
Because the mean wind profiles are nudged toward the target winds, the last
hour average winds exhibit the characteristics that bear resemblance to the
target wind profiles (Figs. 1 and 2). For instance, the radial shear
increases with the radial wind speed within each group. Group L has stronger
radial wind shears and lower IPLs than group H. The mean tangential winds
are very similar within each group. The mean potential temperature
(θ‾) profiles show considerable variations because of different
entrainment rates primarily determined by the shear-generated turbulence as
well as the surface heat flux.
Roll visualization
Two major differences in the wind forcing among the simulations are
associated with the radial wind shear strength and the IPLs. How do these
differences affect the roll structure as well as turbulence in general? The
link between the wind shear profiles and flow pattern is evident in the
horizontal cross sections of w′ at three levels, z/zi∈ (0.2, 0.4, 0.9), from the two groups of simulations shown in Fig. 3. These
plan views demonstrate quasi-linear patterns defined by up and down motions
for all the simulations except H1 for which the pattern is not clearly
recognizable at z/zi= 0.4 and 0.9, although a narrowly
spaced and weak quasi-linear pattern is present at z/zi= 0.2.
The absence of the coherent structure from H1 is likely due to the
weakest wind shear associated with the inflection point, which fails to
generate strong turbulence to support the roll growth. The quasi-linear
structures from the other five simulations have strong vertical coherence shown
at three levels. Therefore, these flow patterns can be identified as “roll
structure”.
It is evident that the rolls appear stronger, in terms of the maximum |w′|, with the increases in the radial wind shear intensity within
each group, i.e., from L1 to L3 or H1 to H3. For example, |w′|max at z/zi= 0.2 from L1 is about
5 m s-1
compared with 7 m s-1 from L2 and 10 m s-1 from L3. The
increasing shear also leads to an increase in the roll horizontal scale
within each group. The scale can be roughly estimated based on the number of
the rolls. It is about 1, 2, and 2.5 km for L1, L2, and L3,
respectively, and 3 km and 3.6 km for H2 and H3, respectively. Different
IPLs in the radial wind profiles have a crucial impact on the roll structure. A
comparison of w′ between the simulations of these two groups (i.e., L2
vs. H2 or L3 vs. H3) in Fig. 3 indicates that the horizontal scales of the
rolls tend to be larger for group H (3 km for H2 and 3.6 km for H3) than
group L (2 km for L2 and 2.5 km for L3) due to the higher IPLs in the
former. It is noteworthy that H3, which has the highest IPL and moderately
strong wind shear (Table 1), is characterized by the vigorous rolls that have
the largest horizontal scale, implying the importance of the IPL in
regulating the roll intensity as well as the scale. The roll expansion from
these simulations is consistent with the general increase in the HBL heights,
the enhanced wind shear, and the rising IPLs (Fig. 2).
These simulations also show a strong signature of gravity waves. For example,
the linear roll patterns are well defined near the inversion base (i.e.,
z/zi= 0.9 in Fig. 2). These patterns even extend above
the HBL (not shown here). The wave amplitude is particularly robust in H3.
Strong evidence of gravity waves also comes from the turbulent statistics
discussed in the next section. It is likely that this roll-like patterns
within the inversion is connected to both the gravity waves and the roll
structure in the HBL. The fact that H3 produces the strong rolls as well as
the large gravity wave amplitude hints at the possibility of an interaction
between these two processes. This is consistent with previous studies, such
as GG14 and NN12, which found that internal gravity waves may be excited by
the roll motion in the HBL and they interact with the rolls to enhance the
associated turbulent transport.
Many of the above-discussed aspects of the roll structure are also evident in
the horizontal cross sections of other perturbation variables. Figure 4 shows
the wind component perturbations (u′ and v′) and their vertical
fluxes (w′u′ and w′v′) at z/zi= 0.2 from H3. The negative
v′ tends to correlate with positive w′along each narrow quasi-linear
band. These negative v′ bands are caused by the upward motion
transporting lower speed wind upward, directly resulting in a very similar
roll pattern in the w′v′ field (Fig. 4b). These patterns suggest that
the roll-scale tangential momentum flux is dominated by the downward
transport (i.e., negative momentum flux) driven by the vigorous upward
motion. The radial wind perturbations (u′) also show similar roll features,
with the black line indicating a convergence line with u′∼0,
corresponding to the positive w′ in H3 (Fig. 4c). It is interesting that
the roll patterns are barely distinguishable in w′u′ (Fig. 4d), in
contrast to w′v′, although they are evident in both w′ and u′
fields. The poor correlation between u′ and w′ near the surface is
likely due to the alignment of the roll axis, namely, nearly along the
tangential direction. This can be seen by assuming the rolls are strictly
two-dimensional and ignoring the small angle between the roll axis and the
tangential direction. The continuity equation reduces to ∂u′/∂z+∂w′/∂z=0 and w′ can be written as
w′=-∫0z∂u′/∂x× dz, which
implies that the vertical velocity near the surface is mainly driven by the
low-level convergence of the radial flow. The above expression also implies
that u′ and w′ are approximately 90∘ out of phase: when
u′ reaches a maximum or minimum, ∂u′/∂x∼0 and
therefore w′∼0. Similarly, when w′ reaches a maximum or minimum,
u′∼0.
Plan views of turbulent perturbations at 9 h from H3 at
z/zi= 0.2. The fields are (a)v′, (b)w′v′,
(c)u′, and (d)w′u′. An “eye-fit” black line is drawn
in (c) to show an example of convergence zone induced by the radial wind.
This argument is supported by further quantitative analysis. A coordinate
transformation is performed on the instantaneous fields so that the resultant
u velocity is perpendicular to the longitudinal roll alignment while v is
along the longitudinal direction. Then, all the turbulent perturbations can be
averaged over a distance (5 km in this case) along the roll direction to
provide a snapshot of mean roll circulations on an x–z cross section.
Figure 5 shows the roll velocity perturbations and the layer-averaged radial
convergence 1z∫0z∂u′/∂xdz at z= 90 m and z= 500 m. It is
evident that u′≅ 0 at 90 m coincides with the
strong convergence and positive w′ values near x∼ 6.4,
9.8 and 13.5 km indicated by the open circles, making positive
w′ correlate with both positive and negative u′. At 500 m, however,
the locations with u′≅ 0 move toward the rotation center
(i.e., toward the left), thus enabling a better correlation of positive
w′ with negative u′. The cross section of the roll circulation from
Fig. 6 shows that updrafts originate along the convergence slope where
u′≅0 and tend to coincide with negative u′above the slope,
leading to a downward (or negative) cross-roll momentum transport aloft. The
negative momentum flux (i.e., w′u′‾) in conjunction with
the positive wind shear represents energy production for roll circulations.
This result also agrees with those of Foster (2005) and GG14, which show that
the roll streamlines tend to tilt vertically to efficiently extract the
kinetic energy from the mean shear flow.
Phase differences between the along-roll-averaged
perturbation u′, w′, and the vertically integrated divergence,
div =1z∫0z∂u′/∂xdz at z= 90 m
(a) and z= 500 m (b) for H3. Open circles denote the
locations where u′=0 and ∂u′/∂x < 0. Note the
different vertical scales between the top and bottom panels.
Turbulence statistics
Turbulence statistics respond strongly to the different wind profiles as
demonstrated in Fig. 7. The negative radial momentum flux (w′u′‾)
is significantly enhanced with the increase in the radial shear
intensity for group L or H, particularly near the levels of the inflection
points, where the shear reaches its local maximum as pointed out by GG14. The
higher IPLs from group H enhance w′u′‾ in the upper portion
of the HBL because of the increased shear-layer depth (Fig. 2), which is also
discussed by GG14. The stronger turbulence aloft in group H simulations
further intensifies the entrainment across the inversion, leading to a deeper
HBL. Both H3 and L3 have similar w′u′‾ maxima in spite of
the large difference in their shear shown in Table 1. The tangential momentum
flux (w′v′‾) also strengthens from group L to H responding,
in part, due to the enhanced tangential wind shear above 300 m (Fig. 2). The
w′v′‾ of H3 increases the most since the turbulence (e.g.,
w′2‾) is considerably stronger in the upper part of the
HBL than in other simulations. One consequence of the w′v′‾ increase is to reduce the surface “friction” effect on the tangential wind
speed because the overall HBL flux gradient is decreased as a result of the
enhanced downward w′v′‾ in the middle and upper HBL. Recalling
the robust roll structure from H3 (Fig. 3), we interpret the strengthening of
turbulence as the result of the highly organized and effective roll
transport. This reasoning is supported by the spectral analysis presented in
Sect. 4.
Vertical cross section of along-roll-averaged
perturbations from H3. The cross-roll velocity u′ is shown by color
shading. The value of ∂u′/∂x=-5 × 10-3 s-1
is contoured by thick black lines. Flow vectors are also displayed.
Profiles of LES turbulence statistics. The variables are
(a)w′u′‾, (b)w′v′‾, (c)u′2‾, (d)v′2‾, (e)w′2‾,
(f)Cpρ0w′θ′‾, (g)θ′2‾, (h)w′3‾, and (i)Sw=w′3‾/(w′2‾)3/2.
The buoyancy flux (Cpρ0w′θv′‾)
decreases from the fixed value 20 W m-2 at the surface to the maximum
negative entrainment flux at the inversion base (Fig. 7f). It is well
documented that the ratio between the entrainment and surface heat flux is
-0.2 for free convection generated by the surface heat flux (Stull, 1976;
Conzemius and Fedorovich, 2006). Thus, the effect of wind shear on
Cpρ0w′θv′‾ is evident as the
magnitude of this ratio can be as large as ∼-1.5 for H3. Variance of
each wind component (i.e., u′2‾, v′2‾ or
w′2‾) increases with the shear strength for both groups H
and L (Fig. 7c–e). The simulation H3, which has the highest inflection point
and moderately intense shear, produces the strongest turbulence above 500 m.
Above the HBL, neither w′2‾ nor θ′2‾ is close to zero; they are in fact very large for L2, L3, H2, and H3. At
the same levels, w′θv′‾ is very small or close to
zero as shown in Fig. 7e–g. This strongly suggests the presence of internal
gravity waves above the HBL, which are presumably generated by mesoscale
perturbations associated with the HBL rolls. According to linear wave theory,
there is a 90∘ phase lag between the wave-induced vertical velocity
and potential temperature perturbations, and therefore the vertical heat flux
associated with wave-induced perturbation is zero, although the vertical
velocity and potential temperature variances can be large. The presence of
gravity waves above boundary layer rolls is consistent with results from many
studies including both LES (e.g., NN12) and 2-D model studies (e.g., GG14).
The skewness of vertical velocity, defined by Sw=w′3‾/w′2‾3/2
(Fig. 7i), represents the symmetry, or lack thereof,
in the turbulence structure. The fact that all the Sw values above
150 m are positive points to a positively skewed structure, that is, the
flow is characterized with narrower/stronger updrafts and broader/weaker
downdrafts (Zhu, 2008; Foster, 2005). In general, a high degree of the flow
asymmetry is reached in the upper portion of the HBL.
Some important features emerging from the above diagnosis are worthy of
emphasis: (1) all simulations except H1 produce well-defined roll structure
manifested by a quasi-linear pattern through the depth of the HBLs; (2) increasing
the vertical shear of the radial wind results in enhanced
turbulence, higher HBL height, and larger roll spatial scales; (3) rising IPL
also leads to a larger roll spatial scale in spite of the weakened radial
shear; (4) the vertical tilting (in the radial direction) of the low-level
convergence zone enhances the radial momentum flux associated with HBL roll
circulations, which is consistent with other studies (e.g., GG14); and (5) the
presence of internal gravity waves is strongly suggested by the
“roll-like” pattern above the HBL and the 90∘ lag between w′ and
θ′ implied by the turbulence statistics. Some of these features are
further confirmed by the spectrum analysis described in the next section.
Spectrum analysis
To understand how the turbulent flow at various scales respond to the
changes in the wind forcing and how effective rolls are in vertical momentum
transfer, we examine the 2-D power density spectra of the simulated w′
and its co-spectra with u′, v′, and w′2 at z/zi= 0.4
where the rolls are most robust. The focus on 2-D spectra instead
of 1-D is due to the fact that the former represents spectral peaks and
associated spatial information more reliably than the latter as discussed by
Kelly and Wyngaard (2006).
Turbulence spectra
All the spectra are calculated using the data collected between 8 and 10 h
with a sampling interval of 5 min. They are functions of the magnitude of the
horizontal wave number vector kh=|k|=kx2+ky2, where
kx and ky are the wave numbers in the radial and
tangential directions, respectively. Note that the subscripts “x” and
“y” represent the radial and tangential directions, respectively,
as defined in Sect. 2.1. Figures 8 and 9 compare various turbulence spectra
at z/zi= 0.4 among simulations within each group as well
as between the two groups. For each group, the power of w′ increases with
the enhancing wind shear at all wave numbers (Fig. 8a). This increase,
however, is more significant for the wave numbers less than 0.01 m-1,
i.e., the spatial scales larger than 600 m, which is particularly true for
group H. The changes in the spectral distribution from group L to H are more
complicated because the higher IPLs are associated with weaker wind shear
(Fig. 2). The major difference between the two groups occurs at the
wave numbers between 10-3 and 5 × 10-3 m-1. The H2
spectrum remains essentially the same as the L2 for the wave number
kh≥0.008 m-1, below which the H2 power becomes
lower than the L2 before it reaches the narrow peak at kh= 0.002 m-1.
For L3 and H3, their spectra are very close to each other
except that the latter (H3) exhibits a peak at a smaller wave number (i.e.,
kh= 0.0017 m-1, or wavelength ∼ 3.6 km) than
the former (L3) (i.e., kh= 0.0027 m-1, or the
wavelength ∼ 2.3 km). The spectral peak from H3 is the strongest and
its wave number is the smallest among all the simulations (Table 1). In
contrast to the relatively smooth shape of the group L spectra, the spectra
of both H2 and H3, which have higher IPLs than group L, exhibit a narrow peak
(Fig. 8a), indicating the presence of a highly energetic and single-mode
structure. This qualitative difference suggests that IPL plays a critical
role in determining the roll strength and the effectiveness of the turbulent
transport.
The 2-D power spectra of w′(a) and co-spectra of
w′-w′2(b) at z/zi=0.4.
Many of the essential features discussed for the w′ power spectrum are
also evident in the cospectrum of w′-w′2, w′-v′, and
w′-u′ in Figs. 8b and 9. Note that the covariance of w′
and w′2 gives w′3, which is related to the skewness (Sw). The
cospectrum of w′-w′2 from each of the simulations L2, L3, H2, and H3
is consistent with the corresponding w′ power spectrum in that both have
the same peak wavelength. The cospectrum peak from H3 is the most prominent
in that it is both large and narrow, implying that the roll vertical motion
is strongly and positively skewed.
A major feature of the cospectrum of w′ and v′ from both H2 and H3 of
Fig. 9 is its sharp negative maximum at the same peak wave number as that
from the w′ spectrum, suggesting significant roll contributions to the
longitudinal momentum flux. Compared with H2 and H3, the group L cospectra
show a much smaller maximum even though their peak wave numbers are the same
as those of the rolls derived from the w′ spectrum.
For w′-u′ cospectra, only H3 results in the same peak wave number as the
rolls defined by the w′ spectrum, while other simulations produce the peak
wave numbers that are larger than the corresponding rolls. Therefore, the
roll structure of H3 has the strongest spectral peaks, among all the
simulations, at the same roll wavelength in the w′ power spectrum and the
cospectra of w′-w′2, w′-v′, and w′-u′.
The 2-D co-spectra of w′-v′(a) and
w′-u′(b) at z/zi=0.4.
The following features associated with H3 are worth noting: (1) the highest
zi (Fig. 2); (2) the strongest turbulence intensity and momentum fluxes
above 500 m (Fig. 7); (3) the largest roll wavelength (Fig. 8 and Table 1);
and (4) the strongest peak at the roll wavelength of the turbulence power
spectra and co-spectra among all the simulations (Figs. 8 and 9). These
features suggest that H3 has produced the most robust roll structure because
of the highest IPL in the radial wind and associated relatively strong shear
(Fig. 2b).
It is also noteworthy that the presence of a significant narrow peak in the
momentum flux spectra is consistent with the observational analysis by Zhang
et al. (2008), which shows sharp peaks in all the cospectra of w′ and the
horizontal wind and temperature perturbations (their Fig. 9). A main
difference is that their observed peak occurs at 900 m with an aspect ratio
∼ 2 and our LES modeled is at 3.5 km with the ratio of 2.7.
Decomposition of turbulent fluxes for H3. Various spectral
components for turbulent flux profiles are presented in top panels;
fractional contributions from the components in bottom panels. Three spectral
groups are small scale (< 1 km), large eddy (1–2.5 km), and roll
(> 2.5 km), respectively. (a)w′2‾,
(b)w′v′‾, (c)w′u′‾, (d)w′3‾, (e) spectral fractional contribution to w′2‾, (f) contribution to w′v′‾, (g) contribution to
w′u′‾, and (h) contribution to w′3‾.
Spectral decomposition of turbulent fluxes
How significant are the HBL roll contributions to turbulent fluxes compared
to other turbulent eddies in the LES simulations? This issue has been
addressed previously with a decomposition method based on the roll coherence
feature. For example, the updraft–downdraft roll circulation can be defined
based on the quasi-linear longitudinal coherence of the roll structure
(Glendening, 1996); the roll-scale characteristics may also be represented
as conditional means of the turbulent flow based on the convection model
(Zhu, 2008). Because a key feature of the rolls is that turbulence is
organized in such a way that various flux spectral distributions reach their
maxima at the roll wavelength, a decomposition method based on spectral
analysis provides a more fundamental representation of roll characteristics.
This approach is also consistent with the observational analyses of HBL
rolls by Zhang et al. (2008).
To compute the contributions from different wave numbers, we integrate each
flux over three spectral bands to yield the subtotals at each model level.
The spectral bands are chosen, in principle, to represent turbulent fluxes
from the small scale, the large-eddy scale, and the roll scale based on the
H3 spectra (Figs. 8 and 9). The small scale ranges from 0.1 to 1 km; the
large-eddy 1 to 2.5 km; the roll 2.5 to 12 km. The calculation is
carried out from the surface to 2 km.
To emphasize the relative importance of the fluxes from the different
spectral groups, we calculate both the fluxes and the flux fractions defined
by the ratio of the specific group flux to the total, as shown in Fig. 10.
The small-scale contribution to w′2‾ dominates in most of
the HBL; the large roll variance increases significantly with height to the
top of the HBL above which it carries more than 70 % of the total
variance (Fig. 10a). This is consistent with the characteristics of the flux
profiles implying the presence of gravity waves above the HBL (Fig. 7e–g).
The longitudinal momentum fluxes (w′v′‾) from different
spectral groups exhibit different vertical distribution, with the small scale
reaching the maximum near the surface and the larger scale near the mid-HBL
(Fig. 10b). This difference reflects the different nature of the turbulence at
different scales. The small-scale turbulence is largely produced by the wind
shear near the surface; thus, the flux maximum is naturally close to the surface.
The roll circulation, caused by the inflection-point instability, generates
the momentum flux that depends on the wind shear in both the tangential and
radial directions in the mid-HBL. The momentum flux w′v′‾ obtains the largest roll fractional contribution of 43 % at the mid-HBL
among all fluxes (Fig. 10f). The combined roll and large-eddy fluxes account
for 65 % of the total. The roll contribution to w′u′‾ is
only 25 %; it is considerably weaker than the contribution to w′v′‾, a result in accordance with the previous discussion that the
radial flux has less roll coherence than the longitudinal one. The roll
contribution to w′3‾ reaches the maximum at 0.7 zi,
accounting for about 20 % of the total (Fig. 10d and h), while the
combined roll and large-eddy contribution is about 45 %.
Roll characteristics: correlation coefficients and skewness
We have argued that the correlation between the roll-scale w′ and u′
is weaker than that between w′ and v′ because the low-level
convergence is mainly driven by the radial wind component, thus leading to
the diminished u′ in the area where the roll w′ reaches the maxima.
This reasoning is based on both the instantaneous perturbation fields (Figs. 4–6)
and the momentum-related cospectra (Fig. 9). It is also supported by
quantifying correlations of w′-v′ and w′-u′ from the roll
contributions shown in Fig. 10. These correlation coefficients are shown in
Fig. 11. The absolute value of the coefficient Cwvr, defined by
Cwvr=w′v′‾r/w′2‾r×v′2‾r0.5, where the superscript r represents the
roll contribution, is around 0.47 from 30 to 500 m, then decreases to near
0 at 1 km. In contrast, Cwur increases from 0
near the surface to 0.3 at 200 m, keeps nearly constant up to 900 m, and then
gradually decreases to 0.2 at 1.5 km. The values of Cwur are smaller than those of Cwvr below
600 m, indicating a weaker roll correlation of w′ with u′ than
with v′. The increasing value of Cwur with
height in the lowest 300 m is also consistent with the understanding of the
tilted convergence zone allowing for more efficient radial momentum transfer
away from the surface (Fig. 6).
Vertical profiles of roll characteristics derived from
H3: w skewness (Swr); w′-v′ correlation coefficient
(Cwvr); and w′-u′correlation coefficient (Cwur).
The results of the roll contribution to the third moment w′3‾
may be used to characterize the roll structure such as the roll skewness
(Swr), which can be computed in the same fashion as the correlation
coefficients from the roll contributions to w′2‾
and w′3‾. Because the simulated skewness is likely
problematic near the surface due to the coarse resolution (Sullivan and
Pattern, 2011), only the profile above 200 m is plotted in Fig. 11. The
skewness Swr decreases from 1.6 at 200 m to 0.7 at 600 m and to 0
above the HBL, where gravity waves are likely present. This
decreasing-with-height tendency agrees with the calculation of Zhu (2008).
Therefore, the roll updraft fraction is generally less than 50 % and
increases with height. In addition, the roll-scale skewness is close to zero
above the HBL, indicating that the flow at these scales is symmetric. This
characteristic is consistent with the linear theory of internal gravity
waves. For a linear wave at a given level, the updrafts and downdrafts in a
horizontal domain with period boundary conditions applied along the side
walls should occupy approximately the same amount of the fractional coverage.
The spectral analysis in this section confirms that both the roll's
horizontal scale and intensity are highly dependent on the shear and IPL in
the radial wind profile. The stronger the radial wind shear is and the higher
the IPL is, the stronger and larger the rolls are. More importantly,
increasing IPL tends to produce a robust roll structure in the sense that a
narrow and sharp peak is present in the w′ power spectrum and its
wavelength is the same as the peak wavelengths from the co-spectra of
w′-w′2, w′-v′, and w′-u′. This is in contrast to the
weaker rolls (e.g., H2) for which the peak wavelength from the w′-u′
is shorter than the others
because of the weaker coherency between the roll-scale w′ and u′.
Momentum transfer coefficients
The momentum transfer coefficient, defined by the negative ratio of the
momentum flux to the mean wind shear according to the K theory
(Stull, 1988, p. 204), plays a central role in the representation of HBL. It
has been shown and argued that the roll-generated momentum flux cannot be
represented by the local transfer theory because of the large-scale
nature of the roll circulations in terms of its horizontal and vertical
scales as compared to zi (e.g., Foster, 2005; Zhu, 2008; and Gao and
Ginis, 2016). In this subsection, the issue of the transfer coefficient is
briefly discussed using the results from the spectral analysis. Because the
momentum fluxes have been decomposed into three spectral groups, it is
convenient to compute the transfer coefficient for each group. By definition,
the transfer coefficient for the radial momentum flux from each spectral
group Kui can be calculated by
Kui=-w′u′‾i∂u‾/∂z,
where superscript i∈s,l,r represents small scale
(< 1 km), large-eddy scale (1–2.5 km), and roll scale
(> 2.5 km), respectively. The transfer coefficient for the
tangential momentum flux Kvi is computed similarly. Because both the
momentum fluxes (except for w′u′‾r) and the vertical
gradient of the wind speed are very close to zero above 1400 m, all the
values of the computed transfer coefficients are removed for z≥1400 m.
These transfer coefficients of both wind components are shown in Fig. 12. The
values of Kui change little with height from 200 m to 1.1 km, above
which Kur increases significantly because of both the finite values
of w′u′‾r and near-zero gradient of u‾. The
non-zero w′u′‾r above HBL is likely caused by the internal
gravity waves which are connected to the roll structure and have the same
wavelength as the rolls as discussed previously (see also GG14). The transfer
coefficients Kui are ill defined around z= 200 m
because ∂u‾/∂z≈0.
Unlike the nearly constant Kui, the tangential transfer coefficients
Kvi increase with height from zero at the surface to ∼ 150 m2 s-1
at 850 m. They then sharply increase near the HBL top
where ∂v‾/∂z≈0,
which results in both very large positive and negative values of Kvi
because w′v′‾i is always negative while ∂v‾/∂z changes sign. This behavior
is contradictory to the downgradient transfer theory which assumes no
negative Kvi (e.g., Stull, 1988, p. 108). This is similar to the
result of the counter-gradient w′v′‾ for the same reason
from the two-dimensional roll model of Gao and Ginis (2016). The main
difference is that their counter-gradient feature occurs in the mid-HBL where
the momentum flux is significantly larger than that near the HBL top in our
simulation H3 (Fig. 10b). This difference is mainly caused by the different
mean tangential wind profiles obtained with different methods: the dynamic
model approach of Gao and Ginis (2016) and the mean nudging in this work.
Therefore, there is a need to apply the same mean wind profiles in both the
2-D roll and LES models for a more effective comparison. The subgrid-scale
parameterized flux is not included in either w′v′‾i or w′u′‾i. The inclusion of the SGS flux
would slightly change the small-scale transfer coefficient profiles
Kvs and Kus.
Momentum transfer coefficients for three spectral groups of H3 for
Ku(a) and Kv(b). The three spectral groups are
small scale (< 1 km), large-eddy (1–2.5 km), and roll
(> 2.5 km), respectively.
Overall, there are marked differences between Kui and Kvi in
the mid-HBL between 200 and 850 m. The values of either Kui or
Kvi do not vary greatly between the spectral groups even though the
differences are obvious. The counter-gradient feature occurs at the HBL top
where ∂v‾/∂z changes sign
and w′v′‾i remains negative. Its effect on the momentum
flux parameterization would be likely negligible in this case, because
w′v′‾ is very small near the HBL top.
Impact of tangential wind shear
We have so far emphasized the impact of the radial wind shear on both
turbulence intensity and spectral distribution. However, both the radial and
tangential winds may have significant shear above the surface layer (Fig. 2b
and c). What roles does the tangential wind shear play in regulating the
roll structure? This section attempts to address this issue by comparing the
simulations H3, L3, L3H, and H3L, which are forced with different radial and
tangential wind shear in the target profiles (Table 1). The simulation L3H
uses the same target radial wind profile as the L3, but the same target
tangential wind as the group H simulations (i.e., the profile H in Fig. 1).
Correspondingly, the H3L adopts the same target radial wind profile as the
H3, but the target tangential wind of the group L (i.e., the profile L in
Fig. 1). This target wind specification is designed to examine how the roll
structure responds to a change in one wind component while the other remains
the same.
The comparison of the turbulence statistics profiles from H3 and H3L with
those from L3 and L3H (Fig. 13) suggests that the radial wind plays a
dominant role in determining the turbulence intensity. The target radial
wind with a high IPL from H3 and H3L leads to both the stronger w′u′‾ and higher HBL tops than the wind profile with a low IPL from
L3 and L3H, regardless of the different target tangential wind used. The
tangential momentum flux w′v′‾ is, however, predominately
determined by the tangential wind shear (Fig. 13a). Both L3 and H3L result
in similar weak momentum fluxes (w′v′‾), which can be
attributed to the same target tangential wind profile L in Fig. 1b. The
stronger momentum fluxes are obtained from H3 and L3H, which have the same
extended higher-level tangential shear profile H in Fig. 1b.
Comparison among simulations H3, L3, H3L and L3H with
different wind shear. (a)w′v′‾; (b)w′u′‾; (c)w′2‾; and (d)θ‾.
The spectral response of the turbulence is displayed in Fig. 14. A dominant
feature is that there is a peak in the power spectrum of w′ as well as
the two co-spectra of w′-v′ and w′-u′ at the same wave number from
H3 and H3L, which have the same radial wind with the higher IPLs (Fig. 1a).
This is particularly true for the co-spectra of w′-u′. In contrast,
the peak values of the spectra from L3 and L3H are more broadly distributed
at higher wave numbers. It is worth noting that the peak in the
w′-v′ co-spectrum of H3L is considerably weaker than that of H3 because
of the tangential wind shear reduction at upper levels in the target wind
profile L (Fig. 1b).
The above results suggest that the radial wind shear plays a more dominant
role in determining the roll characteristics with regard to the scale
selection, while the tangential wind shear strongly influences the
tangential momentum flux w′v′‾. Consequently, the
tangential wind shear enhances the overall turbulence intensity,
e=12(u′2‾+v′2‾+w′2‾), through the shear production. It can also affect the kinetic
energy of roll circulations, (u′2‾+w′2‾),
through the return-to-isotropy terms in the respective variance budget
equations as shown in NN12. This result is largely consistent with the
analysis of GG14, which found that the radial wind shear and IPL defines the
roll characteristics regarding the mode selection and turbulence intensity.
The analysis, however, does not include contributions from the tangential
wind shear to the roll energetics because of the two-dimensional nature of
the dynamic model.
Comparison of the power spectra of w′(a),
co-spectra of w′-v′(b) and w′-u′(c) at z/zi=0.4
among L3, L3H, H3, and H3L.
Summary and conclusion
A series of LES simulations have been conducted to examine the response of
the roll structure to different mean wind shear conditions in terms of the
radial wind shear strength and the IPL in an idealized HBL. A unique feature
in our approach is that a mean wind nudging technique with specified target
wind profiles is used to maintain the horizontal-domain average wind profiles
without directly affecting turbulent perturbations. Two groups of simulations
(L and H) are conducted. Each group uses the same target tangential wind
profile, but three radial wind profiles with different shear. Group H is
designed to have higher IPLs (∼ 430 m) in the radial wind than group L
(∼ 200 m).
All simulations except H1, which has the weakest radial wind shear, produce
the rolls manifested by a quasi-linear structure with the horizontal scale
ranging from 1 to 3.6 km. The roll structure extends from the
near-surface level (z/zi∼ 0.1) to the HBL top
(z/zi∼ 0.9). Within each group of simulations,
increasing radial wind shear tends to enhance overall turbulence and increase
the HBL height. Both the w′ power spectral peak and its wavelength
increase with the enhanced radial wind shear, indicating that the shear
regulates both the rolls' intensity and horizontal scale. Increasing IPLs,
from group L to H, results in more vigorous rolls with distinctly narrow and
sharp peaks in the power spectra. The most robust rolls are produced in H3,
which is forced with the highest IPL and moderately strong shear in the
radial wind. A unique and important feature of this roll structure is that
the peak wavelength is the same among the power spectrum of w′ and the
co-spectra of w′-w′2, w′-v′, and w′-u′, implying that there is a
consistent large roll contribution to all the relevant turbulent fluxes. This
feature is in contrast to all other simulations in which the peak wavelength
from the w′-u′ is shorter than the others because of the weak coherence
between the roll-scale w′ and u′ due to the quasi-two-dimensionality of
the roll structure.
One of the important features regarding the roll contribution to the
vertical momentum flux is that the tangential wind is better correlated with
the vertical motion than the radial wind in the lower half of the HBL. It is
because the low-level convergence mainly comes from the radial wind, whose
roll-scale perturbation is close to zero where the upward motion is
maximized. The convergence zone is tilted with height toward the rotation
center to generate broader updrafts in the area of negative radial wind
perturbations. Consequently, the negative correlation of upward motion and
radial wind perturbation increases with height, which is supported by the
roll momentum correlation coefficients calculated based on the spectral
analysis.
Effects of tangential wind shear are also investigated. A sensitivity
simulation, in which the upper-level tangential wind shear is reduced, shows
that the basic roll structure is not significantly impacted in the sense
that both the power spectrum and the momentum flux co-spectra generally
maintain their distributions. The tangential momentum flux, however, changes
significantly with the tangential wind shear, which feeds back to the
turbulence generation and leads to some difference in the overall turbulence
intensity. This effect is also reflected in the w′ power spectrum and
tangential momentum flux cospectrum in which the peak values are reduced.
Therefore, the radial wind profile critically determines the roll's
presence, intensity, and scale, while the tangential wind shear has
considerable impact on the tangential momentum transport.
The results of the spectral analysis are used to compute the roll
contributions to various turbulent fluxes. The contribution from the
roll-scale (≥ 2.5 km) circulation accounts for 15 % of w′2‾, 40 % of w′v′‾, 20 % of w′u′‾, and 20 % of w′3‾, respectively, at the
mid-HBL. The corresponding large-eddy (1–25 km) contribution is 25 %
(w′2‾), 30 % (w′v′‾), 30 %
(w′u′‾), and 20 % (w′3‾),
respectively. These values are, in general, consistent with previous studies
(e.g., Zhu, 2008; Zhang et al., 2008). Because the magnitude of the negative
roll tangential flux increases from almost zero to the maximum near the
mid-HBL, the roll circulations tend to enhance the lower-level mean
tangential wind by upward transport of the weaker wind. Finally, the momentum
transfer coefficients derived from the three spectral groups show large
differences between the radial and tangential components. While the
counter-gradient behavior occurs at the HBL top where the tangential wind
maximum is reached, its effect is small as the momentum flux is almost
negligible there in the case of H3. This evaluation based on Eq. (3) is meant
to provide an example of the transfer coefficients. More in-depth analyses
are clearly needed to understand the nature of the turbulent transfer
organized by HBL rolls and develop new turbulence closure models for the HBL.
This study highlights the critical roles of the radial wind shear in
regulating the roll structure. As discussed in the introduction, the mean
wind shear should be a strong function of both the local rotational forcing
and the mesoscale tendencies. The mean nudging approach used in this work is
intended to bridge the gap between the commonly used LES configuration and
the need for including the mesoscale effects, and to facilitate sensitivity
simulations. Because of the strong nudging it is difficult to isolate the
impact of the rolls on the mean wind profile in this study. A more
comprehensive study of the roll structure requires incorporating effects of
the hurricane mesoscale environment such as radial wind advection. The LES
approach recently proposed by Bryan et al. (2017) and the nested LES in a
mesoscale model of Zhu (2008) provide attractive modeling frameworks that
can be used to address issues related to the feedback of the rolls to the
mean wind profiles in HBLs.
All data are available from Shouping Wang (shouping.wang@nrlmry.navy.mil).
Mean wind nudging
Comparison of the evolution of the boundary layer height
zi(a) and the radial wind component u‾ at the 60 m level (b) from three
tests RN1 (only rotation included), RN2 (both rotation and nudging included),
and RN3 (only nudging included).
The mean wind nudging method introduced in Sect. 2 is used to maintain LES-simulated
mean wind profiles and to make systematic changes in the mean wind
for sensitivity simulations; it has no direct influence on the resolved
turbulence. Three LES simulations are presented here to evaluate these
statements. The first simulation (RN1) uses the horizontal momentum equations
with the rotation terms (i.e., the square bracket terms with R= 44 km)
and without the nudging terms in Eqs. (1) and (2). The second
(RN2) keeps both the rotation and the nudging terms for which the target wind
profiles are the same as the 9–10 h averaged wind from RN1. The third (RN3)
removes the rotation term and keeps the nudging, and the target profiles are
the same as those from RN2 except the target radial wind is enhanced to
-16.5 m s-1 at 90 m as shown in Fig. A1a. The relaxation timescale
is 10 min.
In general, all the variables are in excellent agreement among the three
simulations, as shown in Figs. A1–A2. The simulations RN1 and RN2 have very
consistent zi after 4 h of simulation, while the RN3 predicts
zi that is 50 m lower than the others. The radial wind velocity at
60 m from RN1 oscillates around the mean value -9.5 m s-1 after 1 h,
which is consistent with that from RN2 and only 0.6 m s-1 stronger than
RN3 which excludes the rotation term. The significantly reduced oscillation in
RN2 is due to the strong nudging, and the absence of the oscillation in RN3
reflects the removal of the rotation term. Despite these differences, all the
mean and turbulence profiles compare well among these simulations. RN1 and
RN2 almost have identical results as seen from Fig. A2. RN3 predicts slightly
weaker turbulence in the upper HBL, being consistent with the weaker shear in
both u‾ and v‾ at these levels. These results confirm the
previous two-dimensional model simulations and LES analyses that the rotation
terms do not have major influence on the turbulence structure driven by the
wind shear, although these terms may play the dominant role in the case of
the parallel instability (Foster, 2005; NN12). They also demonstrate that the
mean wind nudging method can be used to examine the response of turbulence to
a specific mean wind profile that is strongly regulated by the nudging
process. All the simulations presented in Sects. 1–6 of this paper exclude
the rotation terms and keep the nudging terms in Eqs. (1) and (2) with the
relaxation timescale of 10 min.
Comparison of test simulations for the mean nudging
approach. All the profiles are averages between 9 and 10 h at a sampling
interval 30 s. Panel (a) shows the mean radial wind, (b) mean tangential wind,
(c)θ‾, (d)w′2‾, (e)u′2‾,
(f)v′2‾, (g)w′u′‾, and (h)w′v′‾.
The authors declare that they have no conflict of interest.
Acknowledgements
We thank James Doyle for discussions on the LES model setup and gravity
waves. The careful reviews and valuable comments by Kun Gao and Ralph Foster
greatly improved the clarity of the manuscript. This research was
funded by the Office of Naval Research (ONR) under program element (PE)
0602435N.
Edited by: H. Wernli
Reviewed by: R. Foster and K. Gao
References
Atkins, B. W. and Zhang, J. W.: Mesoscale shallow convection in the
atmosphere, Rev. Geophys., 34, 403–431, 1996.
Brown, R. A.: A secondary flow model for the planetary boundary layer, J.
Atmos. Sci., 27, 742–757, 1970.
Brown, R. A.: On the physical mechanism of the inflection point instability,
J. Atmos. Sci., 29, 984–986, 1972.Bryan, G. H., Worsnop, R. P., Lundquist, J. K., and Zhang, J. A.: A simple
method for simulating wind profiles in the boundary layer of
tropical-cyclone, Bound.-Lay. Meteorol., 162, 475–502,
10.1007/s10546-016-0207-0, 2017.
Chlond, A.: Three-dimensional simulation of cloud street development during
a cloud air outbreak, Bound.-Lay. Meteorol., 5, 161–200, 1992.
Conzemius, R. J. and Fedorovich, E.: Dynamics of sheared convective boundary
layer entrainment. Part I: Methodological background and large-eddy
simulations, J. Atmos. Sci., 63, 1151–1178, 2006.
Deardorff, J. W.: Stratocumulus-capped mixed layers derived from a three
dimensional model, Bound.-Lay. Meteorol., 18, 495–527, 1980.Donelan, M. A., Haus, B. K., Reul, B., Plant, W. J., Stiassnie, M., Graber,
H. C., Brown, O. B., and Saltzman, E. S.: On the limiting aerodynamic
roughness of the ocean in very strong winds, Geophys. Res. Lett., 31,
L18306, 10.1029/2004GL019460, 2004.
Foster, R. C.: Why rolls are prevalent in the hurricane boundary layer, J.
Atmos. Sci., 62, 2647–2661, 2005.Foster, R. C.: Signature of large aspect ratio roll vortices in synthetic
aperture radar images of tropical cyclones, Oceanography, 26, 58–67,
10.5670/oceanog.2013.31, 2013.Gao, K. and Ginis, I.: On the generation of roll vortices due to the inflection
point instability of the hurricane boundary layer flow, J. Atmos. Sci., 71, 4292–4307,
10.1175/JAS-D-13-0362.1, 2014.Gao, K. and Ginis, I.: On the equilibrium-state roll vortices and their
effects in the hurricane boundary layer, J. Atmos. Sci., 73, 1205–1222,
10.1175/JAS-D-15-0089.1, 2016.
Glendening, J. W.: Lineal eddy features under strong shear conditions, J.
Atmos. Sci., 53, 3430–3448, 1996.Golaz, J.-C., Wang, S., Doyle, J. D., and Schmidt, J. M.: COAMPS™ LES:
Model evaluation and analysis of second and third moment vertical velocity
budgets, Bound.-Lay. Meteorol., 116, 487–517, 2005.Golaz, J.-C., Doyle, J. D., and Wang, S.: One-way nested large-eddy
simulation over the Askervein Hill, J. Adv. Model. Earth Syst., 1, 1–6,
10.3894/JAMES.2009.1.6, 2009.Green, B. W. and Zhang, F.: Idealized large-eddy simulations of a tropical
cyclone-like boundary layer, J. Atmos. Sci., 72, 1743–1764,
10.1175/JAS-D-14-0244.1, 2015.Jiang, Q. and Wang, S.: Aerosol Replenishment and cloud morphology: A VOCALS
example, J. Atmos. Sci., 71, 300–311, 10.1175/JAS-D-13-0128.1,
2013.Kazil, J., Feingold, G., and Yamaguchi, T.: Wind speed response of marine
non-precipitating stratocumulus clouds over a diurnal cycle in cloud-system
resolving simulations, Atmos. Chem. Phys., 16, 5811–5839,
10.5194/acp-16-5811-2016, 2016.
Kelly, M. and Wyngaard, J. C.: Two-dimensional spectra in the atmospheric
boundary layer, J. Atmos. Sci., 63, 3066–3070, 2006.Kepert, J. D., Schwendike, J., and Ramsay, H.: Why is the tropical cyclone
boundary layer not “well mixed”, J. Atmos. Sci, 73, 957–973,
10.1175/JAS-D-15-0216.1, 2016.
Lilly, D. K.: On the instability of Ekman Boundary Flow, J. Atmos. Sci., 23,
481–494, 1966.Lorsolo, S., Schroeder, J. L., and Dodge, P., and Marks Jr., F.: An
observational study of hurricane boundary layer small-scale coherent
structures, Mon. Weather Rev., 136, 2871–2893, 10.1175/2008MWR2273.1,
2008.
Moeng, C.-H. and Sullivan, P. P.: A comparison of shear- and buoyancy-driven
planetary boundary layer flows, J. Atmos. Sci., 51, 999–1022, 1994.
Morrison, I., Bussinger, S., Marks, F., and Dodge, P.: An observational case
for the prevalence of roll vortices in the hurricane boundary layer, J.
Atmos. Sci., 62, 2662–2673, 2005.Nakanishi, M. and Niino, H.: Large-eddy simulation of roll vortices in a
hurricane boundary layer, J. Atmos. Sci., 69, 3558–3575,
10.1175/JAS-D-11-0237.1, 2012.Nolan, D. S.: Instabilities in hurricane-like boundary layers, Dynam. Atmos.
Oceans, 40, 209–236, 10.1016/j.dynatmoce.2005.03.002, 2005.
Stull, R. B.: The energetics of entrainment across a density interface, J.
Atmos. Sci., 33, 1260–1267, 1976.
Stull, R. B.: An introduction to boundary layer meteorology, Kluwer Academic
Publishers, 666 pp.,
1988.Sullivan P. P. and Patton, E. G.: The effect of mesh resolution on convective
boundary layer statistics and structures generated by large-eddy simulation,
J. Atmos. Sci., 68, 2395–2415, 10.1175/JAS-D-10-05010.1, 2011.Wang, S., Zheng, X., and Jiang, Q.: Strongly sheared stratocumulus
convection: an observationally based large-eddy simulation study, Atmos.
Chem. Phys., 12, 5223–5235, 10.5194/acp-12-5223-2012, 2012.
Willoughby, H. E.: Gradient balance in tropical cyclones, J. Atmos. Sci., 47,
265–274, 1990.
Wurman, J. and Winslow, J.: Intense sub-kilometre-scale boundary layer rolls
observed in Hurricane Fran, Science, 280, 555–557, 1998.
Young, G. S., Kristovich, D. A. R., Hjelmfelt, M. R., and Foster, R. C.:
Rolls, streets, waves, and more, B. Am. Meteorol. Soc., 83, 997–1001, 2002.Zhang, A. J., Kastsaros, K. B., Black, P. G., Lehner, S. French, J. R., and
Drennan, W. M.: Effects of roll vortices on turbulent fluxes in the hurricane
boundary layer, Bound.-Lay. Meteorol., 128, 173–189,
10.1007/s10546-008-9281-2, 2008.
Zhu, P.: Simulation and parameterization of the turbulent transport in the
hurricane boundary layer by large eddies, J. Geophys. Res., 113,
D17104, 10.1029/2007JD009643, 2008.