Turbulence Dissipation Rate in the Atmospheric Boundary Layer: Observationsand WRF Mesoscale Modeling during the XPIA Field Campaign

DOMINGO MUÑOZ-ESPARZA AND ROBERT D. SHARMAN

National Center for Atmospheric Research, Boulder, Colorado

JULIE K. LUNDQUIST

Department of Atmospheric and Oceanic Sciences, University of Colorado Boulder, Boulder, Colorado

(Manuscript received 26 June 2017, in final form 17 September 2017)

ABSTRACT

A better understanding and prediction of turbulence dissipation rate « in the atmospheric boundary layer

(ABL) is important for many applications. Herein, sonic anemometer data from the Experimental Planetary

boundary layer Instrumentation Assessment (XPIA) field campaign (March–May 2015) are used to derive

energy dissipation rate (EDR; 5«1/3) within the first 300m above the ground employing second-order

structure functions. Turbulence dissipation rate is found to be strongly driven by the diurnal evolution of the

ABL, presenting a distinct statistical behavior between daytime and nighttime conditions that follows log–

Weibull and lognormal distributions, respectively. In addition, the vertical structure of EDR is characterized

by a decrease with height above the surface, with the largest gradients occurring within the surface layer (z,50m). Convection-permitting mesoscale simulations were carried out with all of the 1.5-order turbulent ki-

netic energy (TKE) closure planetary boundary layer (PBL) schemes available in the Weather Research and

Forecasting (WRF) Model. Overall, the three PBL schemes capture the observed diurnal evolution of EDR

as well as the statistical behavior and vertical structure. However, the Mellor–Yamada-type schemes un-

derestimate the large EDR levels during the bulk of daytime conditions, with the quasi-normal scale elimi-

nation (QNSE) scheme providing the best agreement with observations. During stably stratified nighttime

conditions, Mellor–Yamada–Janjic (MYJ) and QNSE tend to exhibit an artificial ‘‘clipping’’ to their

background TKE levels. A reduction in the model constant in the dissipation term for the Mellor–Yamada–

Nakanishi–Niino (MYNN) scheme did not have a noticeable impact on EDR estimates. In contrast, appli-

cation of a postprocessing statistical remapping technique reduced the systematic negative bias in theMYNN

results by ’75%.

1. Introduction

Turbulence dissipation rate « is a measure of the

strength of turbulence and is proportional to the energy

transferred from large to small turbulent eddies. While

most of the emphasis of the scientific community re-

garding turbulence has been on the structure of tur-

bulent kinetic energy and fluxes in the atmospheric

boundary layer (ABL), turbulence dissipation rates play

an important role in several applications of interest. In

the context of aviation applications, energy dissipation

rate (EDR; 5«1/3) is the International Civil Aviation

Organization (ICAO; ICAO 2001) standard to quantify

and report turbulence levels (Sharman and Lane 2016).

In that regard, extensive research has been carried out to

understand the behavior of « in the upper troposphere

and lower stratosphere (UTLS). In the UTLS region,

climatologies derived from in situ aircraft and pilot re-

ports (PIREPs) support the notion that EDR follows a

lognormal distribution (e.g., Nastrom and Gage 1985;

Frehlich and Sharman 2004; Sharman et al. 2014). In the

wind energy discipline, recent studies have found that

wind turbine wakes strongly depend on the background

turbulence dissipation rate (Smalikho et al. 2013), with

significant enhancement of turbulence dissipation

within the wind turbine wake (Lundquist and Bariteau

2015). Therefore, improving the understanding andCorresponding author: Dr. DomingoMuñoz-Esparza, domingom@

ucar.edu

Denotes content that is immediately available upon publica-

tion as open access.

JANUARY 2018 MUÑOZ - E S PARZA ET AL . 351

DOI: 10.1175/MWR-D-17-0186.1

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ability to predict the structure and characteristics of « in

the ABL are crucial.

In particular, only two studies have attempted to un-

derstand the statistical behavior of turbulence dissipa-

tion rates in the atmospheric boundary layer. Cho et al.

(2003) used aircraft data collected during the Transport

and Chemical Evolution over the Pacific (TRACE-P)

campaign (Jacob et al. 2003) to derive probability dis-

tributions of « in the atmospheric boundary layer and

free troposphere, using velocity field measurements at

20Hz. The authors found a bimodal distribution for the

boundary layer, clearly indicating a separation between

calm and turbulent conditions. The other study was

carried out by Frehlich et al. (2004). The authors ana-

lyzed the statistical behavior of « in the shear region of a

nocturnal low-level jet (LLJ) during the CASES-99 field

experiment using 200-Hz hot-wire measurements at

heights between 50 and 70m. In contrast, Frehlich et al.

(2004) identified a lognormal behavior of «. Given the

clear disparity in the research regarding the statistical

behavior of « in the ABL, further investigation is

therefore needed to elucidate the nature of dissipation

rates in the ABL.

From an atmospheric modeling perspective, turbu-

lence is typically fully parameterized in mesoscale

models using the so-called planetary boundary layer

(PBL) schemes. Some of these PBL schemes solve a

one-dimensional (1D) turbulent kinetic energy (TKE)

equation that is used to relate the subgrid turbulent

stresses to resolved vertical gradients of the particular

fluctuating magnitude. Such 1.5-order TKE closure

includes a dissipation term, often parameterized as-

suming equilibrium between production and dissipation

of turbulence based onKolmogorov’s second hypothesis

of similarity. In the vast majority of cases, research

studies focus on the intercomparison of PBL schemes

analyzing time evolutions and vertical profiles of wind

and thermodynamic variables (e.g., Shin andHong 2011;

Coniglio et al. 2013; Draxl et al. 2014; Cohen et al. 2015;

Mirocha et al. 2016), with no attention paid to the

evaluation of turbulent properties beyond surface

fluxes. In contrast, only a few studies have used turbu-

lent information to evaluate PBL schemes. Berg and

Zhong (2005) found that with some PBL schemes, TKE

can be underestimated in spite of overpredicting sensi-

ble and latent heat fluxes. Muñoz-Esparza et al. (2012)

examined the ability of PBL schemes to reproduce

vertical profiles of turbulent momentum flux and its link

to the ability to correctly represent vertical shear of the

wind speed profile as a function of ABL stability.

Foreman and Emeis (2012) proposed a modification to

the Mellor–Yamada–Janjic (MYJ) scheme closure

constants and an additional stability-dependent surface

length scale to increase TKE closer to the surface.While

not specifically targeting the evaluation of turbulence

quantities, recent studies by Yang et al. (2017) and Jahn

et al. (2017) found that the closure coefficient in the

dissipation term is one of the coefficients that has the

largest impact when forecasting turbine-height wind

speeds and wind ramp events with theMellor–Yamada–

Nakanishi–Niino (MYNN) scheme, respectively. These

results reinforce the need to evaluate this parameterized

turbulent quantity in PBL schemes.

Herein, we make use of the Experimental Planetary

boundary layer Instrumentation Assessment (XPIA)

campaign at the Boulder Atmospheric Observatory

(BAO), held 2 March–31 May 2015 (Lundquist et al.

2017), and use it to derive EDR and compare it to

Weather Research and Forecasting (WRF) mesoscale

simulations. The remainder of the paper is structured as

follows. The accuracy of second-order structure func-

tions and spectral energy distributions in estimating

dissipation rates is compared in section 2a, selecting the

most appropriate method to derive a 3-month-long

dataset of EDR within the lowest 300m of the ABL. In

section 2b, the statistical behavior of EDR as a function

of height and stability is identified. Comparison to

convection-permitting WRF Model simulations de-

scribed in section 3 is carried out in section 4, analyzing

the ability of the three 1.5-order TKE schemes in WRF

to reproduce the temporal evolution and vertical struc-

ture of EDR in the ABL. An attempt to improve the

performance of the MYNN scheme by modifying the

closure constant of the dissipation term is presented in

section 5. A postprocessing method based on statistical

remapping is applied in section 6 to the MYNN output

to remediate EDR bias. Finally, section 7 is devoted to

the conclusions and to describing current and future

research efforts.

2. Observations of turbulence dissipation rates inthe atmospheric boundary layer

To understand the behavior of « in the atmospheric

boundary layer, sonic anemometer data from the XPIA

campaign were employed (Lundquist et al. 2017), cor-

responding to the 3-month duration of the campaign,

March–May 2015. During the XPIA campaign, the

BAO tower was enhanced to have a total of 12 three-

dimensional sonic anemometers (Campbell CSAT3)

located every 50m from 50 to 300m above ground level

(AGL). Anemometers were mounted on booms point-

ing northwest (NW; 3348) and southeast (SE; 1548),providing high-frequency 20-Hz measurements of wind,

temperature, and relative humidity. Additionally, a 5-m

AGL tower located 200m southwest (SW) of the BAO

352 MONTHLY WEATHER REV IEW VOLUME 146

tower provided near-surface turbulent measurements

(boom pointed 3238). Continuous observations were

possible because of the presence of two anemometers

pointing at opposite directions at every height, ensuring

that one of them was always providing reliable data. To

remove artificially enhanced turbulence levels due to

wake effects from the tower on the sonic anemometers

readings, a sector encompassing 6308 around the total

wake angle was discarded, corresponding to (1248, 1848)for the NW pointing anemometers and (3048, 48) for theSE pointing anemometers. Detailed analysis comparing

sonic anemometer observations to profiling lidars dur-

ing the XPIA campaign revealed that wind speed defi-

cits up to 50% and an order of magnitude increase in

turbulent kinetic energy can be recorded if the wake

regions are considered (McCaffrey et al. 2017a). For

wind directions when neither of the two anemometers

was in the wake sector at a given height, «was calculated

independently and averaged over the NW and SE ane-

mometers for a more robust estimation.

a. Methods to estimate turbulence dissipation rate

Turbulence dissipation rate is often calculated based

on the so-called ‘‘inertial range’’ technique. Such an

approach is built upon Kolmogorov’s second hypothesis

of similarity (Kolmogorov 1941), which relates velocity

increments to energy dissipation in the inertial range of

three-dimensional isotropic turbulence:

Du(r)[ h[u(x1 r)2u(x)]2i5C

K«2/3r2/3. (1)

The previous relation is valid for spatial separations r

within the inertial range and in the direction of the flow,

CK ’ 2.0 is the Kolmogorov constant, and Du(r) is the

second-order structure function (e.g., Lindborg 1999;

Frehlich and Sharman 2004; Sharman et al. 2006). While

Kolmogorov’s second hypothesis was formulated in

terms of spatial increments, it is typical to invoke Tay-

lor’s hypothesis (Taylor 1938) in order to express « in

terms of temporal velocity increments:

«51

Ut

�D

u(t)

CK

�3/2, (2)

where Du(t) is the second-order structure function cal-

culated using the horizontal wind speed u over temporal

increments t. This is a convenient transformation when

single-point measurements are available, as is the case

for the sonic anemometers used herein. However, this

transformation assumes that the contribution to the

advection by turbulent motions is small, compared to

the mean flow advection, and only holds if u0/U � 1,

withu0 being the turbulent velocity fluctuation (u0 5 u2U).

Wecalculated the ratiou0/U for all the heights and found the

median and mean to be 0.12 and 0.16, respectively, which

validates the use of Taylor’s hypothesis and therefore en-

ables calculation of «.

In addition to the second-order structure function

method, it is quite popular within the atmospheric

boundary layer community to use the energy spectra as a

function of wavenumber (or frequency) and to fit it to

the Kolmogorov’s inertial range slope to calculate tur-

bulence dissipation rate (e.g., Champagne et al. 1977;

Oncley et al. 1996; Piper and Lundquist 2004; Lundquist

and Bariteau 2015; McCaffrey et al. 2017b). The

Kolmogorov law for the inertial range (or 2/3 law) can

be expressed in frequency domain terms (Obukhov

1962), which bear the well-known Kolmogorov’s 25/3

dependence

«52p

U

"Su( f )f 5/3

a

#3/2

, (3)

where Su( f ) is the power spectral density, f is the fre-

quency, and a is the inverse Kolmogorov’s constant

(5 C21K ). We assume Kolmogorov’s constant to be 0.5221,

based on measurements in the surface layer from

Wyngaard and Coté (1971). The selected value is con-

sistent with other estimates of Kolmogorov’s constant

where CK ’ 0.521 is typically found (Sreenivasan 1995).

Calculation of power spectral density involves the use of

the fast Fourier transform (FFT). To avoid spurious

noise arising when FFTs are performed over non-

periodic signals, a Hamming window was applied prior

to the FFT calculation w(t*)5 0:542 0:46 cos(2pt*),

with t* being the normalized time in the interval (0, 1).

Therefore, power spectral densities are consistently

underestimated because of the application of the win-

dow. To compensate for that effect (energy leak), Su( f )

is divided by the factorÐ 10w2(t*)dt* 5 0.3974, so the

original power spectral density of the signal is recovered.

Comparison of typical results from both methods is

presented in Fig. 1 for two instances: one where theABL

was stable [0500 UTC (2300 LT) 9March 2015], and one

where the ABL was convective [2100 UTC (1500 LT)

10March 2015]. These are typical examples of weak and

strong turbulence events in the ABL at the BAO tower

during theXPIA campaign.A 2-min windowwas chosen

for the calculation of « based on the inspection of the

data and the range of scales corresponding to the inertial

range. Larger intervals are not desired because in-

stantaneous large turbulence values will be washed out

through averaging over a large time period, and smaller

intervals provide noisy results. For the fitting to the

isotropic turbulent scales, the range 0.5–10Hz (2–0.1 s)

JANUARY 2018 MUÑOZ - E S PARZA ET AL . 353

was used, which properly displays the theoretical f25/3 or

t2/3 slopes for energy spectra and second-order structure

functions, respectively. For the sonic anemometer located

at 5m, the lowest frequency considered was 1Hz. This

restriction was applied to minimize the presence of energy

content from production scales because their characteristic

eddy size decreases as the surface is approached. Re-

alizations over the 2-min window using the two methods

reveal a much larger degree of variability for the energy

spectramethod (Figs. 1a,c). Such variability arises from the

truncation of the Fourier series for a finite number of

modes. Because the focus of this analysis is on the small

inertial range scales (0.5–10Hz), we investigated the pos-

sibility of subdividing the window into smaller windows

(60, 30, 12 and 6s) and applying subwindow overlapping to

reduce the noise. Figures 1a and 1c show that such an ap-

proach considerably reduces the noise, converging to a

much smoother spectral density distribution for the short-

est subwindow used (6 s). Further reduction of the

subwindow size is not possible because it would result in an

insufficient sampling of the lowest frequency of interest.

Even when using 6-s subwindows with 50% overlapping,

the structure function over a 2-minwindowdisplays amore

regular pattern that closely follows the expected t2/3 slope

(Figs. 1b,d). This good agreement with Kolmogorov’s

theory provides further evidence that the selected range of

time increments for the fittings to derive « is a good choice

where the assumption of local isotropy holds; otherwise,

there would be large differences as a function of t.

To quantify the uncertainty from the two methods in

estimating EDR, the spread of the fit to the Kolmogorov’s

slope over the range of time increments 0.1–2s was calcu-

lated. Following Piper (2001), this quantity can be related to

the uncertainty in the estimation of « using the following

expression:

s«5 1:5s

II21; (4)

FIG. 1. Energy spectra for different averaging subwindows valid at (a) 0500 UTC (2300 LT) 9 Mar and (c) 2100

UTC (1500 LT) 10 Mar at z 5 50m. (b),(d) As in (a),(c), but for second-order structure functions, respectively.

Dashed lines represent the theoretical Kolmogorov’s inertial range slope t2/3 ( f25/3) in time (frequency) domain.

354 MONTHLY WEATHER REV IEW VOLUME 146

where I is f 5/3Su( f ) or t22/3Du(t), and sI is the variance

of that quantity over every discrete frequency in the

sampling window. Probability distribution functions

(PDFs) of « uncertainty for the 3 months of sonic ane-

mometer observations are presented in Fig. 2a. All the

heights were combined for statistical robustness because

variability with height was negligible and did not follow

any pattern. When the spectral method without sub-

window averaging is used, as is often the case in many

studies, mean uncertainty of 151% in the estimation of

« is present due to the noise resulting from the FFT

calculations shown in Fig. 1. Splitting the 2-min window

into smaller intervals with 50% overlapping and aver-

aging the resulting FFTs progressively reduces the un-

certainty down to 34% when 6-s subwindows are used

(39 FFTs). In contrast, the structure function approach

further reduces the mean uncertainty to 11% and has a

symmetric distribution in logarithmic space, while all of

the energy spectrum–based approaches display a larger

right tail of the distribution (more probability of larger

uncertainties). Figure 2b reveals a good overall agree-

ment between the estimates from both methods, with a

tendency of the energy spectrum method to predict

larger « values than derived from structure function

calculations. In particular, such overprediction is found

to become more prominent for smaller values of «. We

attribute this effect to the noise becoming comparable to

« levels for small «. Moreover, the FFT noise is evenly

distributed around the mean in logarithmic scale

(Figs. 1a,c), meaning that for the same amplitude of

deviation, the contribution from above the mean values

is greater than deviations below the mean, contributing

to an artificial enhancement of « estimates. From this

analysis, it is concluded that second-order structure

functions provide a more robust approach to calculate

« and will therefore be used in the rest of this study.

b. Statistical behavior of turbulence dissipation rateduring the XPIA campaign

An algorithm to continuously derive turbulence dis-

sipation rate every 30 s was implemented and run for the

3 months of the XPIA campaign covering seven heights:

z 5 5, 50, 100, 150, 200, 250, and 300m. As a result, a

database of approximately 1.5 million « records was

created, after discarding times where data availability

was not sufficient. Because of the distinct behavior of

ABL turbulence as a function of atmospheric stability,

we classified the data into daytime/convective and

nighttime/stable conditions based on the sign of the

Obukhov length L52u3

*u/kghw0u0i, where u* is the

friction velocity, hw0u0i is the kinematic heat flux, u is

the potential temperature, k is the von Kármán con-

stant (5 0.4), and g is the acceleration due to gravity

(5 9.81m s22). When available, L at z 5 5m is used to

sort the date as a function of stability; otherwise, the

closest level aloft is used instead. Herein, we present the

dissipation rates in the form of EDR (5 «1/3; m2/3 s21),

because of this research being motivated by aviation

turbulence purposes. Nevertheless, « and EDR are in-

terchangeable, with the only difference being that EDR

produces larger values that span a broader range.

Figure 3 shows PDFs of EDR at the seven sonic an-

emometer heights for stable and convective conditions.

It is evident that the PDFs are remarkably different

depending upon ABL stability. During nighttime con-

ditions (Fig. 3a), PDFs at all the heights (5–300m)

display a well-defined lognormal distribution, charac-

terized by

FIG. 2. (a) PDFs of « uncertainty for the energy spectra method using different subwindow sizes and for the

structure functionmethod. (b) Correlation between the second-order structure function and energy spectra with 6-s

subwindows and 50% overlapping. Color scale indicates probability of occurrence (%), and the dashed back line

denotes perfect correlation.

JANUARY 2018 MUÑOZ - E S PARZA ET AL . 355

PDF(lnx)51ffiffiffiffiffiffiffiffiffiffiffi2ps2

p exp

"2

(lnx2m)2

2s2

#, (5)

with m representing the mean or expectation and s2

representing the variance. There is a progressive

shift toward smaller EDR values with height, with

the largest values present within the surface layer

(z 5 5m). Afterward, the decrease with height be-

comes more moderate. Our results agree well with the

specific observations by Frehlich et al. (2004) during a

2-h period in the shear region of an LLJ (50–70m),

demonstrating that stably stratified ABL bears a log-

normal distribution at all heights, with the mean of the

distribution presenting larger values within the sur-

face layer and decreasing with height. Our findings

regarding stable ABL EDR are consistent with those

at the UTLS, where lognormal EDR distributions

have been observed (Sharman et al. 2014). We attri-

bute this similarity to upper levels typically being

characterized by quasi-two-dimensional stratified

turbulence (e.g., Lindborg 1999), which appears to

share certain aspects with stably stratified turbulent

flows in the ABL.

During convective conditions (Fig. 3b), PDFs of

EDR exhibit an evident departure from a lognormal

distribution. Daytime distributions are skewed to-

ward larger EDR values, with a more progressive

decrease of probabilities in the lower-EDR portion

of the distribution. This distribution fits well to a log–

Weibull distribution, as will be shown later, and it is

defined as

PDF(lnx)5k

l

�lnx

l

�k21

exp 2

�lnx

l

�k#,

"(6)

where k and l are the shape and scale parameters of the

distribution, respectively. Again, we find the strongest

variability in the surface layer compared to the levels

higher up, which also presents the largest EDR values.

In contrast to nighttime conditions, the height de-

pendence tends to decrease the probability of the main

peak of the distribution instead of simply displacing it,

together with a progressive inclusion of the contribution

from smaller EDR values (,6 3 1022 m2/3 s21).

The PDF of EDR including all the heights and sta-

bilities is displayed in Fig. 3b as a dashed black line. The

combined PDF does not display a bimodal pattern as

found by Cho et al. (2003), who found a clear separation

between calm and turbulent conditions seeming to

occur at « 5 1025m2s23 (EDR5 2.153 1022 m2/3 s21). Ex-

amination of Fig. 4 in Cho et al. (2003) indicates that the

lower EDR peak of the bimodal distribution was likely

contaminated by measurements in the free troposphere

because the tropospheric peak of their distribution is

exactly at the same location, « ’ 1026m2 s23, therefore

artificially resulting in a bimodal distribution. The lower

« peak in their bimodal distribution exhibits values

that are even too low for what we find in the stable ABL,

and they are also much lower than those from Frehlich

et al. (2004) (« ’ 1023–1024m2 s23). Moreover, aircraft

measurements from the TRACE-P campaign used by

Cho et al. (2003) were taken during daytime conditions,

with flights departing between 0900 and 1100 LT and

FIG. 3. Probability distributions of EDR (m2/3 s21) at different heights during the 3months of theXPIA campaign

derived from 20-Hz sonic anemometer data using second-order structure functions. EDRs are calculated every 30 s

over 2-minwindows. (a) Stable conditions (748 960 samples) and (b) convective conditions (748 200 samples). Black

dashed line corresponds to the combined PDF including all the heights and stabilities. Top x axis displays mag-

nitude of « (m2 s23).

356 MONTHLY WEATHER REV IEW VOLUME 146

with a duration of 8–10h (Jacob et al. 2003). These flights

are likely to have mostly experienced convective ABL

turbulence conditions. However, observations report

frequent EDR ’ 1022 m2/3 s21 (« ’ 1026m2 s23), which

we have found to have probabilities of 0.1% during the

XPIA field campaign (two orders of magnitude smaller

than the characteristic convective and stable peaks of the

corresponding ABL distributions). Therefore, we con-

clude that EDR within the ABL does not follow a bi-

modal distribution even combining stable and convective

ABLs. Instead, it has a statistical behavior that differs

between convective and stable conditions, characterized

by log–Weibull and lognormal distributions, respectively,

with enhanced levels within the surface layer and pro-

gressive decrease with increasing height. The PDFs for

each of the individual months were in remarkably good

agreement, bearing the same structure and only display-

ing minor differences (not shown). Therefore, we expect

these findings to be valid at other times of the year, al-

though small variabilities in the mean and spread of the

probability distributions are anticipated. However, in the

presence of complex terrain or other heterogeneities, we

hypothesize that these PDFs will exhibit additional de-

parture from the results presented herein and likely de-

pend on the details of the local topography. Further

analysis including temporal series and vertical profiles of

EDR is included in the next section.

3. WRF mesoscale modeling setup

In this section, simulations with the Advanced Re-

search WRF Model (Skamarock et al. 2008; Skamarock

and Klemp 2008) version 3.8 are performed to evaluate

the ability of mesoscale parameterizations to reproduce

turbulence dissipation rate in the ABL. The model was

configured using two domains centered on the BAO

tower (see Fig. 4a) with horizontal grid resolutions of

9 and 3km that communicated via one-way nesting.

Physical parameterizations used include the longwave

and shortwave Rapid Radiative Transfer Model for

GCMs (RRTMG; Iacono et al. 2008), Kain–Fritsch cu-

mulus parameterization (only used in the outer 9-km

domain; Kain 2004), and the Thompson microphysics

scheme (Thompson et al. 2008). The Community Land

Model, version 4 (Lawrence et al. 2011), was used for the

land surface coupling, while the surface layer parame-

terization option was set according to the selected PBL

scheme, with all of them based on the Monin–Obukhov

similarity theory (Monin and Obukhov 1954). Physics

parameterizations were selected to resemble the oper-

ational WRF-based, state-of-the-art High-Resolution

Rapid Refresh (HRRR, http://ruc.noaa.gov/hrrr/;

Smith et al. 2008) configuration. Initial and boundary

conditions were specified from the North American

Regional Reanalysis, with a horizontal resolution of

32 km and available every 3 h (Mesinger et al. 2006).

Simulations were run for 48 h, with the initial 24 h

serving asmodel spinup and only the last 24 h considered

for analysis. To cover a wide range of atmospheric

conditions and boundary layer stabilities, the entire

month of March 2015 was simulated, and the model

output was saved every 10min.

The vertical coordinate was discretized using 73 grid

points in both domains (Fig. 4b), with the finest vertical

FIG. 4. (a) WRF domains with terrain elevation (m MSL). Domains have a horizontal resolution of 9 and 3 km,

with the dashed line indicating the lateral boundaries of the inner domain. The triangle denotes the location of the

BAO tower. (b) Vertical grid distribution, zoomed in over the first km in the upper-left subpanel.

JANUARY 2018 MUÑOZ - E S PARZA ET AL . 357

resolutions within the lowest 2 km, Dz ’ 10–142m (42

vertical levels), to ensure proper representation of gra-

dients in the ABL. Moreover, fine vertical grid spacing

has been shown to be required for PBL schemes to

avoid overestimating observed low-level jet heights

and overdiffusing the jet structure (Muñoz-Esparza2013). For z . 2 km, grid spacing was progressively

increased to span the remaining vertical domain ex-

tent with the model top placed at pt 5 100 hPa (z ’14 km). The three PBL schemes with 1.5-order tur-

bulent closure available in WRF v3.8 were exercised,

namely, MYNN (Nakanishi and Niino 2006), MYJ

(Janjic 1994), and Quasi-Normal Scale Elimination

(QNSE; Sukoriansky et al. 2005). These PBL schemes

solve a one-dimensional prognostic equation for tur-

bulent kinetic energy q in the vertical coordinate. The

1D TKE equation can be expressed in general form as

follows:

›q

›t52

›hq0w0i›z

2 hu0iw

0i›Ui

›z1

g

uhw0u0i2« . (7)

In Eq. (7),Ui is the zonal (i5 1) andmeridional (i5 2)

wind component, and u0i is the respective subgrid-scale

turbulent velocity fluctuations. Turbulent fluxes are

parameterized using a countergradient approach:

hu0iw

0i 5 2Km›Ui/›z, with the eddy diffusivity Km 5‘Smq

1/2. The same approach is used to parameterize

the turbulent heat flux hw0u0i and turbulent transport

hq0w0i, which accounts for the effect of pressure fluc-

tuations. The master length scale ‘ and the model

coefficients Sq, Sm, and Sh are specific to each PBL

scheme formulation. Herein, our focus is on the dis-

sipation term, which is parameterized in all cases

based on Kolmogorov’s hypothesis,

«5q3/2/b1‘ , (8)

where b1 is a model coefficient that has the value of

11.878 (MYJ), 24.0 (MYNN), and 6.0105 (QNSE).

Despite the selected grid size, the discretization of the

advection scheme acts as an implicit filter that reduces

the effective grid resolution (i.e., grid size/scale fully

resolved by the model). In the case of WRF, Skamarock

(2004) has shown that the typically used fifth-order ad-

vection scheme has an effective grid resolution of’7Dx.This implies that with Dx5 3 km, only scales of’21km

are fully resolved, and for smaller scales, the energy

content is damped out by the dissipative nature of the

odd-order advection scheme. Herein, we use the hybrid

centered upwind scheme recently proposed by Kosovic

et al. (2016), which considerably reduces the dissipative

errors in the numerical discretization of the advective

term. This hybrid scheme improves the effective grid

resolution to ’3–4Dx without increasing the computa-

tional cost and has been tested over a variety of high-

resolution large-eddy simulation (LES) cases. At this

point, we want to emphasize that while a finer horizontal

resolution could have been used, three-dimensional

underresolved convection can develop during daytime

conditions (e.g., Ching et al. 2014; Zhou et al. 2014;

Muñoz-Esparza et al. 2017; Mazzaro et al. 2017), which

invalidates the assumption of horizontal homogeneity

upon which 1D PBL formulations are based, and

therefore would have produced spurious results.

4. Temporal evolution and vertical structure ofEDR in the ABL

To judge the ability of the different PBL schemes to

reproduce temporal variability of EDR, Fig. 5 shows

comparison between EDR derived from sonic ane-

mometer data and mesoscale PBL schemes during

March 2015 at z 5 50m. Turbulence dissipation rate is

strongly modulated by atmospheric stability and there-

fore follows a clear diurnal pattern as the dominant

mode. In general, the three PBL schemes are able to

capture the diurnal evolution resulting in larger EDR

during daytime conditions, which progressively decay

during the evening transition to finally reach minimum

values during the nighttime. Also, the morning transi-

tion period is well captured by all the schemes, closely

matching the onset of the EDR enhancement at the

early morning hours as well as the rate of growth. The

two Mellor–Yamada schemes (MYJ and MYNN) tend

to underestimate EDR during daytime conditions, with

the QNSE scheme exhibiting the best agreement com-

pared to the sonic anemometer measurements. While

the evening transition is reasonably well represented, it

is worth noting that all the PBL schemes are more de-

ficient in reproducing EDR during stable ABL condi-

tions. The MYJ scheme (Fig. 5b) exhibits some sort of

‘‘flat’’ response during nighttime conditions, often clip-

ping to a constant value of EDR’ 6.4083 1022 m2/3 s21

in addition to overpredicting the lowest EDR events.

The MYNN scheme, in contrast, presents more realistic

trends during the nighttime but with a tendency to un-

derpredict EDR levels. Despite such underprediction,

the MYNN temporal evolution is the most realistic of

the three PBL schemes. In the case of QNSE, the per-

formance is similar to MYJ, with the difference being

that QNSE does not exhibit such a strong flat response

as observed in the MYJ scheme but still cannot re-

produce low EDR events during nighttime periods.

As mentioned in the introduction, we choose to ana-

lyze results in terms of EDR (5 «1/3) because of its

358 MONTHLY WEATHER REV IEW VOLUME 146

relevance for aviation turbulence. However, it is also

pertinent for the boundary layer community to present

the results using «. Therefore, we have incorporated

additional axes to allow the reader to simultaneously

visualize results in « and EDR terms, which will enable

an easier comparison with other studies. As seen in

Fig. 5, the differences in « between the PBL schemes and

the observations are larger than when EDR is used.

However, the behavior is equivalent, and therefore all

the discussion and conclusions regarding EDR are

equally applicable to «. For the sake of completeness,

the mean and standard deviation errors of the different

PBL schemes in predicting « are also included as an

appendix.

To gain a better understanding of the overall perfor-

mance of the three PBL schemes, scatterplots including

EDR for March 2015 at the seven anemometer heights

are included in Fig. 6 (left panels), comparing observa-

tions from sonic anemometer data to each of the three

PBL schemes. For a clearer representation, the data are

discretized in equally spaced EDR bins in logarithmic

scale, colored by probability of occurrence. The un-

derestimation for the larger EDR values observed dur-

ing daytime is consistent at all levels for MYNN and

MYJ, which accounts for the bulk of the probability

below the one-to-one correlation (dashed black line).

For QNSE, the scatterplot exhibits the best agree-

ment for moderate and large values (EDR . 7.0 31022 m2/3 s21), with a symmetric distribution around the

one-to-one correlation line, which is indicative of the

lack of a model bias in the prediction of EDR for this

particular scheme. The scatterplots are helpful in iden-

tifying the model flat response observed during night-

time conditions. In the case of MYJ, such effect results

in a horizontal line with large probabilities that is in-

sensitive to the variability in the observed EDR. We

found that these events are associated with instances

where the TKE production vanishes and is clipped to the

background value q0 5 0.01m2 s22, and the master

length scale ‘0 5 0.32m, resulting in EDR 5 6.4077 31022 m2/3 s21. It is worth mentioning that the default q0

for the MYJ scheme is 0.1m2 s22 and that we reduced it

FIG. 5. Time evolution of EDR at z 5 50m during the month of March 2015 at the BAO

tower comparing observations to (a) MYNN, (b) MYJ, and (c) QNSE PBL schemes. EDR

derived from sonic anemometer data using second-order structure functions are calculated

every 30 s over 2-min windows and averaged over 10-min intervals. Right y axis displays

magnitude of «.

JANUARY 2018 MUÑOZ - E S PARZA ET AL . 359

FIG. 6. Scatterplots of sonic anemometer vsmodel EDRs for each of the three PBL schemes duringMarch 2015 at

all heights (z 5 5, 50, 100, 150, 200, 250, 300 m). Right panels exclude cases where zsonic $ zi,WRF. The data are

discretized in equally spaced EDR bins in logarithmic scale, colored by probability of occurrence (%).

360 MONTHLY WEATHER REV IEW VOLUME 146

not to have an unrealistically large threshold during

stable conditions, as will be shown later. The MYJ

scheme exhibits the strongest flat response, followed by

QNSE with a tendency to frequently produce EDR ’3.0 3 1022 m2/3 s21 as revealed by the horizontal fringe

of large probabilities. For theMYNN scheme, this effect

is much less evident and also occurs at smaller values

(EDR , 1022 m2/3 s21).

Turbulence within the residual layer can play an im-

portant role during the nighttime at levels right above

the ABL height zi. In contrast, PBL schemes are de-

signed to parameterize the effect of turbulence strictly in

the ABL and do not typically account for the effect of

turbulence in the residual layer. Given that we have

continuous measurements during 1 month, some of the

levels at which the sonic anemometers are placed may

have fallen close to and above zi during nighttime con-

ditions. To understand the impact of the PBL scheme

formulation in predicting EDR at these heights, the

data were filtered to remove the model results when

zsonic $ zi,WRF for each scheme. The scatterplots after

removing these values are presented in Fig. 6 (right

panels), and clearly show that the unphysical clipping is

removed. This points out to a deficiency of PBL pa-

rameterizations within the residual layer. An accurate

prediction of zi is going to be critical in these conditions;

however, difficulties in evaluating boundary-layer

heights during nighttime periods (e.g., Steeneveld

et al. 2007) precluded a specific evaluation of this aspect

in the current study. Of the three TKE-based schemes

available in WRF v3.8, MYNN is the only one that al-

lows representation of low EDR values in a realistic

manner. Overall, QNSE is the most accurate scheme,

with amean error of 1.13 1022 m2/3 s21; themean errors

for MYJ and MYNN are twice as large. Mean absolute

errors from the three schemes are’5.03 1022 m2/3 s21,

slightly smaller in the case of the MYJ scheme. In gen-

eral, low EDR values present the largest scatter when

compared to sonic anemometer measurements, in-

dicative of the challenges in parameterizing stably

stratified ABLs in mesoscale models (e.g., Holtslag

et al. 2013).

Vertical profiles of average EDR for convective and

stable conditions are displayed in Fig. 7 for the cases

where zsonic # zi,WRF. EDR produced from all the

schemes exhibits a trend to decrease with height, con-

sistent with the nature of turbulence in the ABL. The

largest vertical gradients occur in the lowest 50m, where

the presence of the surface increases turbulence pro-

duction due to both shear and buoyancy in convective

conditions. During the daytime, observations show

EDR levels 60% larger on average than those found

during the nighttime, where the most significant vertical

gradients are limited to the lowest 50m. All the PBL

schemes reasonably capture the vertical distribution of

EDR. MYNN and MYJ have a bias of ’ 23.5 3 1022

m2/3 s21 (’270%) for 50 # z # 300m during daytime

conditions and of ’ 22.5 3 1022 m2/3 s21 (’270%)

during nighttime conditions. Interestingly, that ten-

dency is reversed at z5 5m for convective ABLs, where

both have a small positive bias of ’1.0 3 1022 m2/3 s21

(’10%). In contrast, average EDR profiles from the

QNSE scheme closely match the observations. Also, the

variability represented by error bars corresponding to

one standard deviation reveals a better behavior of the

QNSE scheme, with MYNN and MYN resulting in

narrower spreads. However, QNSE overestimates by

50% the average EDR at z 5 5m. We attribute this

effect to specific details of the surface layer parameter-

ization and the lower boundary condition of the TKE

equation for daytime conditions. In fact, it should be

mentioned that the theory upon which the QNSE

scheme is based is valid for stably stratified and weakly

unstable turbulence (Sukoriansky et al. 2005); therefore,

it may not be expected to perform well in more con-

vective regimes.

The diurnal evolution of TKE is well captured by all

the PBL schemes, which is expected, given the close

connection between EDR and TKE. The time evolution

of TKE at z 5 50m is presented in Fig. 8. Turbulent

kinetic energy was calculated from instantaneous tur-

bulent fluctuations over a 10-min running mean, in

contrast to the 2-min window used for the turbulence

dissipation. This is because « is related to inertial range

turbulence, associated with only a portion of the tur-

bulent spectrum, whereas the most important contri-

butions to TKE are made by production range eddies,

which are larger than the inertial range scales. If shorter

intervals are considered, these relevant scales will be

filtered out, while larger intervals will be contaminated

by nonstationary effects. However, there are some as-

pects that deserve specific mention. During nighttime

conditions, the three PBL schemes underestimate TKE,

typically by an order of magnitude. MYJ and QNSE

often default to their respective background TKE levels,

q0 5 0.01 and 0.005m2 s22. This indicates either a col-

lapse of turbulence production or TKE levels below the

background turbulence specified in the PBL scheme.

While MYNN produces a more natural evolution of

TKE, values after the evening transition decrease to

levels two orders of magnitude smaller. For daytime

conditions, it is found that while MYJ underestimates

TKE observations, consistent with the underestimation

of EDR shown in Fig. 5b, MYNN, in contrast, results in

the best agreement with TKE from the sonic anemom-

eters. The QNSE scheme tends to slightly underpredict

JANUARY 2018 MUÑOZ - E S PARZA ET AL . 361

TKEduring daytime; however, EDRduring nighttime is

often overestimated (Fig. 5a), while TKE is consistently

underpredicted. This contradictory behavior in terms of

parameterized turbulent quantities points to a modeling

deficiency in these 1.5-order TKE PBL schemes.

5. MYNN scheme sensitivity to b1 modelcoefficient

The MYNN model is often found to provide the most

accurate results in comparison to other PBL schemes in

WRF (e.g., Muñoz-Esparza et al. 2012; Huang et al.

2013; Coniglio et al. 2013; Cintineo et al. 2014;

Vanderwende et al. 2015), and it is the PBL scheme

upon which wind farm (Fitch et al. 2012) and swell wave

(Wu et al. 2017) parameterizations have recently been

developed and incorporated into theWRFModel. Also,

recent studies byYang et al. (2017) and Jahn et al. (2017)

have shown that b1 is one of the MYNN scheme model

parameters that has the largest impact when forecasting

turbine-height winds and wind ramp events. Moreover,

it is the PBL scheme used in theWRF-basedHRRR and

FIG. 7. Averaged vertical profiles of EDR from sonic anemometer observations (blue) and the three 1.5-order TKE PBL schemes in

WRF (red) during (a)–(c) daytime and (d)–(f) nighttime duringMarch 2015 for cases when zsonic # zi,WRF. Error bars denote one standard

deviation.

362 MONTHLY WEATHER REV IEW VOLUME 146

Rapid Refresh (RAP; Benjamin et al. 2016) forecast

products for the continental United States. Therefore, it

is worth investigating modifications to model parame-

ters in an attempt to improve the prediction of EDR.

From the top left panel in Fig. 6, the shift in EDR to

remove the negative bias was estimated as a multipli-

cative factor of 1.25, enhancing the turbulence dissipa-

tion. We modified the model constant in the dissipation

term to b1 5 1.2523 3 24.0 ’ 12.0 [Eq. (8)] and reran

WRF for the entire month of March with the MYNN

scheme using this modified b1 value. Figure 9 compares

EDR derived from sonic anemometer data and the

MYNN scheme for b1 5 24.0 and 12.0 at z 5 50m. In-

terestingly, lowering b1 by half did not have a noticeable

impact on EDR. The MYNN scheme continued to un-

derestimate the observed values throughout the entire

diurnal cycle, with the exception of the morning and

evening transitions (same as with the original b1).

To understand the reason for this behavior, scatter-

plots for some turbulent quantities for MYNN with

b1 5 24.0 and 12.0 are presented in Fig. 10. TKE com-

parison in Fig. 10a shows that the direct impact of

decreasing b1 (increase turbulence dissipation) is to

consistently decrease TKE. However, ‘ in Fig. 10b ex-

hibits no systematic changes apart from scatter around

the one-to-one correlation line due to changes in the

parameterized turbulent fluxes. These fluxes, in turn,

modify the meteorological variables that will feed back

into the turbulence parameterization. The budget term

for the turbulence dissipation, «dt/TKE, reveals a larger

dissipation of TKE due to the modification of b1 that

doubles the dissipation term (Fig. 10c). Then, when

combining a larger dissipation with a lower TKE, the

result is that «, and consequently, EDR, remain essen-

tially unchanged, as observed in Fig. 10d. This analysis

reveals that b1 controls the «/TKE ratio, likely arising

from the assumption of equilibrium in the PBL scheme,

but that does not affect the dissipation, which will re-

quire changes in the master length scale. The focus

herein resides on turbulent quantities, and further

FIG. 8. Time evolution of TKE at z 5 50m during the month of March 2015 at the BAO

tower comparing observations to (a) MYNN, (b) MYJ, and (c) QNSE PBL schemes. TKE

derived from sonic anemometer data are calculated and averaged over 10-min windows.

JANUARY 2018 MUÑOZ - E S PARZA ET AL . 363

analysis beyond the scope of the current work is needed

to quantify how these changes in the «/TKE ratio affect

wind, temperature, and moisture profiles in the ABL.

The question of why none of these 1D TKE PBL

schemes appear to have a consistent behavior between

TKE and « is rather difficult. In fact, several factors

may contribute to this issue, and they are hard to sep-

arate and/or evaluate when real-world conditions are

considered. One aspect is the assumption of equilib-

rium between turbulent production and dissipation

invoked by these PBL schemes, which would not be

expected to be satisfied in nonstationary conditions. In

other words, the dynamic ABL forcing can change in

time at a faster rate than the required time for turbu-

lence to reach instantaneous quasi equilibrium. In this

regard, we expect a two-equation closure formulation

solving an additional prognostic equation for « to

provide better results, instead of the diagnostic pa-

rameterization employed by 1.5-order TKE PBL

schemes. Also, the other assumption upon which 1D

PBL schemes are formulated, namely, the presence of

horizontal homogeneity, is likely to be violated in some

cases.Muñoz-Esparza et al. (2016) have recently shownthrough comparison to turbulence-resolving LESs that

these 1D PBL schemes fail to correctly capture

mountain waves because of the lack of a three-

dimensional turbulence parameterization. To com-

pensate for this deficiency, scale-aware (e.g., Shin and

Hong 2015; Ito et al. 2015) and 3D PBL parameteri-

zations (e.g., Jiménez and Kosovic 2016) should be

helpful; however, there is not yet a robust approach

that has been shown to improve this aspect and that has

been validated over a wide range of atmospheric con-

ditions. Another alternative is to couple mesoscale

to LES models, explicitly representing the most ener-

getic turbulence structures and therefore reducing the

impact of the parameterization. We are currently

leveraging recently developed multiscale capabilities

in WRF (Muñoz-Esparza et al. 2017) to better un-

derstand this issue, and that will be reported on in the

near future.

6. A posteriori correction with statisticalremapping

The analysis presented in section 4 reveals the

presence of systematic biases in the prediction of

EDR in the ABL. Attempts to correct these biases by

modifying the dissipation term coefficient as indicated

in section 5 have not proven to be successful. Another

alternative for improving the prediction of EDR from

the MYNN scheme is to apply a posteriori post-

processing technique. Sharman and Pearson (2017)

have recently proposed a method in the context of

turbulence forecasting for aviation purposes that

transforms a turbulence-related quantity (turbulent

index; D) derived from an NWP model into an EDR

value. While this method is conceived in a generic

sense to be able to transform any turbulence-related

metric of arbitrary units to the EDR space, it can also

be used to correct biases in EDR predictions. That

method requires the turbulent index to have the same

statistical distribution as the observed EDR and to be

correlated to it. If these two criteria are satisfied, D

can be then be statistically remapped into «1/3sr using a

simple linear transformation:

ln«1/3sr 5 a1 b lnD , (9)

where logarithms have been used because of the na-

ture of the observed EDR distributions as seen in

section 2b. The coefficients a and b in Eq. (9) are

calculated based on the expectation operator hi andthe standard deviation operator SD of the probability

FIG. 9. Time evolution of EDR at z 5 50m during the month of March 2015 at the BAO

tower comparing observations to the MYNN scheme for the default b1 5 24.0 and a reduced

value of the dissipation termmodel constant b1 5 12.0. EDR derived from sonic anemometer

data are calculated every 30 s over 2-min windows averaged over 10-min intervals. Right y

axis displays magnitude of «.

364 MONTHLY WEATHER REV IEW VOLUME 146

distributions of the observations and the quantity to

be remapped using the following expressions:

b5SD(ln«1/3obs)

SD(lnD), a5 hln«1/3obsi2bhlnDi , (10)

with D 5 «1/3 from the PBL scheme, and the subscript

‘‘obs’’ referring to the sonic anemometer observations.

For a more robust statistical description, April and May

2015 were additionally simulated with the MYNN

scheme. To construct the distributions, both the obser-

vations and the MYNN results were sampled every

10min, discarding instances where zsonic $ zi,WRF.

Figure 11 shows probability distributions of EDR from

sonic anemometer observations and the MYNN scheme

at two heights, z5 50 and 250m (the other levels behave

similarly). Daytime and nighttime PDFs were re-

spectively fitted to log–Weibull and lognormal distri-

butions using a nonlinear least squares minimization

(dashed lines). The PDFs corresponding to daytime

conditions (Figs. 11a,c) confirm that EDR in the con-

vective ABL bears a log–Weibull distribution (not to be

confused with the Gumbel distribution), with a good

agreement between the data and the fitted distribution

both for the observations and the WRF mesoscale re-

sults. The corresponding nighttime PDFs (Figs. 11b,d)

corroborate the lognormal behavior, also captured by

FIG. 10. Scatterplots of the default b1 5 24.0 vs the reduced b1 5 12.0 for theMYNN scheme duringMarch 2015 at

all heights within theABL (zsonic # zi,WRF). (a) TKE, (b)master length scale ‘, (c) dissipation budget term, «dt/TKE,

and (d) EDR. The figure is discretized in equally spaced bins in logarithmic scale, colored by probability of oc-

currence (%).

JANUARY 2018 MUÑOZ - E S PARZA ET AL . 365

the WRF simulations. MYNN results display the nega-

tive bias and, in general, tend to follow narrower dis-

tributions, which is consistent with the analysis of the

vertical structure of EDR presented in section 4 (Fig. 7).

Moreover, probabilities at the lower end of the EDR

distribution present some departure from the lognormal

distribution in the case of WRF results, attributed to the

tendency of the MYNN scheme to underpredict low

EDR values during the most stable stages of the night.

Scatterplots of sonic anemometer EDR observations

versus MYNN simulation results for March–May 2015

are displayed in Fig. 12, including the rawMYNNoutput

(Figs. 12a,c,e) and the results after performing the sta-

tistical remapping (Figs. 12b,d,f). For a more detailed

evaluation, daytime and nighttime conditions are also

plotted separately. The MYNN scheme exhibits a

negative bias during daytime conditions (Fig. 12a),

consistent with the observed pattern for March 2015.

The statistical remapping based on the log–Weibull

nature of convective ABL EDR removes the sys-

tematic negative bias of the model. This is clearly

observed in Fig. 12b and further confirmed by a sub-

stantial reduction of the mean error from 20.0215 to

0.0065 m2/3 s21 after the statistical remapping is applied.

A similar effect is found during nighttime (Figs. 12c,d) and

consequently in the combined scatterplot (Figs. 12e,f).

The original negative bias is reduced by ’75%, as in-

dicated by the mean error before and after the statistical

remapping, 20.0208 and 0.0055 m2/3 s21, respectively.

To complement this metric, the mean absolute error is

included, which gives an estimate of the spread around

the observations. While the systematic negative bias is

FIG. 11. Probability distributions of EDR (m2/3 s21) from samples taken every 10min during the 3 months of the

XPIA campaign (March–May 2015), derived from 20-Hz sonic anemometer data using second-order structure

functions and from WRF simulations using the MYNN scheme (zsonic # zi,WRF). Daytime conditions at (a) z 5 50

and (c) z5 250m. Nighttime conditions at (b) z5 50 and (d) z5 250m. Dashed lines corresponds to the best fit to

log–Weibull and lognormal distributions during daytime and nighttime conditions, respectively. Top x axis displays

magnitude of « (m2 s23).

366 MONTHLY WEATHER REV IEW VOLUME 146

removed, the statistical remapping slightly increases the

mean absolute error by 14%. This is because the PDFs

of EDR from the MYNN scheme tend to be narrower,

compared to the observations (see Fig. 11). Therefore,

the linear remapping increases the spread to match the

PDF from the observations, consequently increasing the

deviation from the observations and, in turn, the mean

absolute errors. This effect manifests in the increased

FIG. 12. Scatterplots of observed vs MYNN modeled EDR during March–May 2015 at all the sonic ane-

mometer heights (zsonic # zi,WRF). (a),(c),(e) Default MYNN scheme results compared to (b),(d),(f) the linear

statistical remapping. The figure is discretized in equally spaced bins in logarithmic scale, colored by probability

of occurrence (%).

JANUARY 2018 MUÑOZ - E S PARZA ET AL . 367

spread of the scatterplot after the statistical remapping

(cf. Fig. 12e and Fig. 12f). Nevertheless, this is a reduced

increase in the mean absolute error, which is desirable

to essentially remove the systematic negative bias of

MYNN EDR predictions.

7. Conclusions

In the present study, the structure of turbulence dis-

sipation rates in the atmospheric boundary layer has

been investigated. To that end, data continuously col-

lected during the XPIA field campaign held in March–

May 2015 at the BAO tower were employed, consisting

of high-frequency sonic anemometer measurements at

seven heights: z 5 5, 50, 100, 150, 200, 250, and 300m.

Two methods to derive « were compared, both based on

the fitting to the inertial range slope. The use of second-

order structure function results in a smaller uncertainty

of ’ 10%, quantified by the variability in the fitting to

the theoretical Kolmogorov’s inertial range slope. In

contrast, the use of fast Fourier transform reveals much

larger errors and a tendency to overestimate «, which

can be partially mitigated by applying subwindow av-

eraging. A database of’1.5 million EDR (5 «1/3) values

was generated and used to understand the statistical

behavior of EDR in theABL. EDR is strongly driven by

the diurnal evolution of the ABL, presenting a different

statistical behavior between daytime and nighttime

conditions, characterized by log–Weibull and lognormal

distributions, respectively. Our results do not support a

bimodal distribution, as found by Cho et al. (2003),

whose observations we suggest are likely influenced by

the inclusion of observations from the free troposphere.

In addition, the vertical structure of EDR is character-

ized by a decrease in height above the surface, and with

the largest gradients occurring within the surface layer

(z , 50m). Moreover, 60% larger EDR values are

consistently present, on average, during convective

conditions, with respect to stably stratified nighttime

turbulence.

Convection-permitting mesoscale simulations

were carried out with all of the 1.5-order TKE clo-

sure PBL schemes available in WRF v3.8 (MYJ,

MYNN, and QNSE) to understand and quantify the

ability of these schemes to predict turbulence dissi-

pation rate in the ABL. Overall, the three schemes

capture the observed diurnal evolution of EDR, with

the Mellor–Yamada-type schemes underestimating

large EDR levels during the bulk of daytime condi-

tions and QNSE resulting in the best agreement with

sonic anemometer observations. The simulated ver-

tical structure of EDR by all schemes is in good

agreement with observations, displaying the

expected decrease with height. Stably stratified

nighttime conditions are the most challenging sce-

nario, with PBL schemes tending to exhibit some sort

of artificial ‘‘clipping’’ to the background TKE values

specified in the MYJ and QNSE schemes. In contrast,

the MYNN scheme reveals a more realistic temporal

evolution but tends to underestimate EDR during the

strongest stratification events occurring in the middle

of the night. We find that most of these poor perfor-

mance cases for stable ABLs are related to instances

where the model-predicted boundary layer heights

are close to or below the sonic anemometer height,

therefore indicating the need for a residual layer

turbulence parameterization to be incorporated into

PBL schemes. Also, verification of the diagnosed ABL

heights by the PBL schemes would be required to further

understand the deficiencies.

We investigated the possibility of improving the

performance of the MYNN scheme by modifying the

constant in the dissipation term of the 1D TKE

equation b1 to correct for the negative bias of the

PBL scheme. However, it was found that such

modification (reduction of b1 from 24.0 to 12.0) did

not have a noticeable impact on EDR estimates,

which remained nearly unchanged in spite of the

remarkable decrease of TKE. This is explained by b1

affecting the «/TKE ratio, likely arising from the

assumption of equilibrium, but not the dissipation

itself, which will require changes in the master

length scale ‘. Moreover, an additional compari-

son of TKE exhibits no underestimation of the

MYNN results during daytime conditions, pointing

to a deficiency in the turbulence parameterization

approach.

A statistical remapping technique was used to

remove the systematic negative bias in the MYNN

scheme, which reduced the EDR mean error by

75%. The MYNN scheme [as well as MYJ and

QNSE (not shown)] reproduces the observed PDFs

both during daytime and nighttime conditions and is

well correlated to the observed EDR. A 14% in-

crease of the mean absolute (and root-mean-square)

error was found, related to the probability distri-

butions from the MYNN model typically being too

narrow. Nevertheless, this postprocessing approach

considerably reduces the systematic errors from the

PBL scheme parameterization in predicting EDR.

The results obtained herein regarding the

structure of EDR in the ABL can be applied in

turbulence-prediction algorithms. In particular, we

are currently working on extending the graphical

turbulence guidance (GTG) algorithm discussed

in Sharman and Pearson (2017) for low-level

368 MONTHLY WEATHER REV IEW VOLUME 146

turbulence to produce more realistic probability distri-

butions that are distinct for daytime and nighttime

conditions and that are representative of ABL turbu-

lence and structure. The current study was restricted to

the first 300m above ground that the BAO tower cov-

ered. However, other techniques, such as wind-profiling

radars (e.g., Jacoby-Koaly et al. 2002; Lundquist et al.

2017; McCaffrey et al. 2017b), can be used to extend

the current findings to higher heights, encompassing

deeper ABLs often occurring during convective day-

time conditions. Also, further research is needed to

understand how complex terrain features alter dissi-

pation rate in the ABL, as well as the ability of dif-

ferent modeling approaches to reproduce these

conditions.

Acknowledgments. This research was funded by

NASA Ames Research Center through Grant

NNX15AQ66G. JKL’s effort was supported by the

National Science Foundation (AGS-1554055) under

the CAREER program. All the simulations were

performed on the Yellowstone high-performance

computer (ark:/85065/d7wd3xhc) at the National

Center for Atmospheric Research (NCAR), pro-

vided by NCAR’s Computational and Information

Systems Laboratory, sponsored by the National Sci-

ence Foundation.

APPENDIX

Mean and Standard Deviation Errors of « Predictedby 1.5-Order TKE PBL Schemes

To quantify the performance of each PBL scheme in

predicting «, the relative errors of the mean and

standard deviation, lmean and lstd, respectively, are

provided in Table A1. These errors are calculated at

each of the sonic anemometer heights during daytime

and nighttime conditions separately, corresponding to

the month of March 2015, and are calculated as

follows:

lmean

51

n�n

i51

«wrf

2 «sonic

«sonic

, (A1)

lstd

5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�n

i51

(«wrf

2 h«wrf

i)2s

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�n

i51

(«sonic

2 h«sonic

i)2s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�n

i51

(«sonic

2 h«sonic

i)2s ,

(A2)

where hi is the ensemble average over the total number

n of 10-min samples, corresponding to the cases when

zsonic # zi,WRF.

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