12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

EQUILIBRIUM AND NON-EQUILIBRIUM SCALING RELATIONS IN
ATMOSPHERIC BOUNDARY LAYER.

Marta Wacławczyk
Institute of Geophysics, Faculty of Physics
University of Warsaw
Pasteura 5, 02-093 Warsaw
marta.waclawczyk@fuw.edu.pl

Jakub Nowak
Institute of Geophysics, Faculty of Physics
University of Warsaw
Pasteura 5, 02-093 Warsaw
marta.waclawczyk@fuw.edu.pl

Szymon P. Malinowski
Institute of Geophysics, Faculty of Physics
University of Warsaw
Pasteura 5, 02-093 Warsaw
marta.waclawczyk@fuw.edu.pl

ABSTRACTsider two distinct states of turbulence, ’equilibrium’ and ’nonequilibrium’. The ’equilibrium’ turbulence can be defined as astate where the classical Taylor’s dissipation law holds

In this work we aim to retrieve information about temporal changes of turbulence in the atmosphere and leadingorder terms in the budget of kinetic energy based on wind velocity measurements within the Atmospheric Boundary Layer(ABL). We focus on in-situ experiment performed by a research aircraft which flies horizontally and records current
state of turbulence along its track. We calculate turbulence
intensity and study its dependence on the integral length scale.
We also calculate two non-dimensional indicators, the dissipation factor and the integral-to-Taylor scale ratio and plot themas a function of the Taylor-based Reynolds number. This allows to divide flow regions into areas of equilibrium and nonequilibrium turbulence. In the equilibrium turbulence energy
spectra have a self-similar form and certain relations between
the length scale and turbulence intensity hold, which allow to
identify the leading-order terms in the energy balance. These
relations take a different form in non-equilibrium turbulence,
which is a sign of rapidly-changing external conditions.

ε = CεU3,L

Cε = const,(1)

This cornerstone relation is a basis of many turbulence models.
It also follows from Eq. (1) that
L
Cε
=
R ,
λ
15 λ

(2)

p
where λ = 15ν/εU is the Taylor’s length scale and Rλ =U λ /ν is the Taylor-based Reynolds number. Even thoughCε = const does not depend on the Reynolds number, Bos et al.
(2007) argued that in the decaying turbulence its value is typically about twice larger than in the stationary case where turbulence production balances the dissipation. This difference
follows from a finite time needed for the energy injected atlarge scales to reach the dissipative end of the energy cascade.
Moreover, if the equilibrium Taylor’s law (1) holds, the classification of turbulence states can be made by studying balanceof leading-order terms in the transport equation for turbulencekinetic energy. As it is discussed in the following, the balance
between shear production and dissipation implies L ∼ U , under the assumption of the constant shear. In the decaying turbulence, on the other hand, L ∼ 1/U a , with a > 0, (Sinhuber
et al., 2015).
A number of recent research works (Vassilicos, 2015;
Seoud & Vassilicos, 2007; Valente & Vassilicos, 2011; Bos& Rubinstein, 2018; Bos et al., 2007) questioned the validityof the scaling (1) and (2) in the presence of rapid time changesof the system. A different ’unusual’ dissipation scaling was
found in the laboratory experiments of Seoud & Vassilicos(2007) and Valente & Vassilicos (2011). Those authors argued
that Cε is not constant, but depends on the inlet conditions
and local Reynolds number. Bos & Rubinstein (2018) derived

INTRODUCTION
Turbulence plays an important role in the atmospheric
dynamics. It dissipates energy at small scales, but also contributes to the unpredictability of the weather (Wyngaard,2011). In spite of many measurement campaigns our knowledge on atmospheric turbulence is still far from sufficient due
to limited amount of data and measurement errors. Observations in the atmosphere can be divided into in situ and remotesensing. In situ measurements require that the instruments belocated directly at the point of interest, e.g. by installing themon a mast at certain, fixed heights. Another example are airborne measurements performed by research aircrafts, at altitudes higher than the heights of masts. Analysis of such measurement data is a subject of the current study.
Apart from calculating basic turbulence statistics, likethe turbulence intensity U , turbulence kinetic energy dissipation rate (EDR) ε and the integral length scale L along theflight track, we attempt to retrieve information on the temporal changes of turbulence. For this, we separately con-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
these non-equilibrium scaling laws for Cε and L /λ based ontheoretical arguments. They found

Cε
≈
Cε0



Rλ 0
Rλ

is the turbulent transport, p0 denotes fluctuations of the pressure, w0 = u03 is the fluctuating vertical component of velocity,15/14

P = −w0 u0i

(3)

∂ ui
∂z

is the production term, ε = 2si j si j , where si j = (∂ u0i /∂ x j +∂ u0j /∂ xi )/2 is the EDR andandL
Cε0 15/14≈Rλ
15 λ 0



1
Rλ

1/14
,

B = b0 w0(4)

denotes the buoyancy flux.
Balances of leading order terms in TKE equation were investigated e.g. in Yano & Wacławczyk (2022) to identify characteristic scales of atmospheric boundary-layer systems. We
will also consider here several simplified cases, assuming that
there are only two leading order terms in Eq. (5) which resultsin the balances between

where Cε0 and Rλ 0 are the ’equilibrium’ values of the nondimensional dissipation parameter and Reynolds number. If
turbulence decays Cε becomes larger than the equilibrium
value Cε0 . On the other hand, if turbulence production is locally larger than the dissipation Cε < Cε0 .
The non-equilibrium relations (3) and (4) are substantially different from their equilibrium counterparts (1) and(2). Hence, classification of turbulence into ’equilibrium’ and’non-equilibrium’ can be made on this basis. The purpose ofthis work is to use this procedure to analyse data recorded by aresearch aircraft in marine stratocumulus-topped ABL. In thiscase the flow conditions in the investigated region are unknowna priori and only limited information, that is 1D intersectionsof the flow field, along the flight track are available. We focus on the data from the recent ACORES campaign designed
for the observations of Aerosol, Cloud, Turbulence, and Radiation Properties at the Top of the Marine Boundary Layer overthe Eastern North Atlantic Ocean (Siebert et al., 2021; Nowak
et al., 2021). From this campaing, high-resolution (100 Hz)
data recorded by the ACTOS platform are available. We calculate turbulence statistics and the non-dimensional indicators
Cε and L /λ and plot them as a function of Rλ to identify
whether turbulence is in or out-of equilibrium. Moreover, byinvestigating values of Cε it is possible to divide the flow area
into regions where turbulence decays, develops or is in a stationary state. Additional criterion which we consider is the dependence of L on U , which allows to identify leading-orderterms in the turbulence energy budget. We believe our study
can deliver new, important information on the dynamics of turbulence in the atmosphere.

1. shear production and dissipation: P ∼ ε
2. buoyancy flux and dissipation B ∼ ε
3. turbulent transport and dissipation: T ∼ ε
4. time derivative and dissipation (decaying turbulence, no
forcing): ∂ k/∂t ∼ −ε
In case 1), using gradient diffusion hypothesis
w0 u0 = −νt

√ ∂u
∂u
∼L k
∂z
∂z

and assuming ∂ u/∂ z = S ≈ const we obtain
P∼L

√ 2
kS .

Using the equilibrium Taylor’s law (1), which predicts ε ∼k3/2 /L , for the balance P ∼ ε we obtainL√ 2 k3/2
,
kS ∼
L

hence
L 2 ∼ k,

or

L ∼U.

(6)

EQUILIBRIUM TURBULENCE
In the ’equilibrium’ turbulence the Taylor’s law (1) andEq. (2) are satisfied. The constant Cε in Eq. (1) may, however,be different in stationary and decaying or developing turbulence. As argued by Bos et al. (2007), it follows from the factthat there is a finite time T needed for the energy injected atlarge scales to reach the dissipative end of the energy cascade.
To study turbulence in the ABL we will consider turbulence kinetic energy (TKE) budget for horizontally homogeneous shear flow∂k
= T + P − ε + B,∂t

In the case of buoyancy-produced turbulence 2) we will assume thatB = w0 b0 ≈ const
and again make use of the equilibrium Taylor’s law (1). With
this for the balance B ∼ ε we obtainB∼(5)

hence L ∼ k3/2

orL ∼ U 3.

(7)

In case 3) where dissipation and advection are the leadingorder terms in the TKE budget, we will use the gradient diffusion hypothesis to express the transport term as

where k = u0i u0i /2, in the isotropic turbulence k = 3/2U 2 ,

T =−

k3/2
,
L

1 ∂ u0i u0i w0 1 ∂ p0 w0
−
2 ∂z
ρ ∂z

T =−

2

√ ∂k
√ k
1 ∂ u0i u0i w0
∂k
∼ νt
∼L k
∼L k .
2 ∂z
∂z
∂z
L

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
No conclusion can be drawn for the balance T ∼ ε. Possibly, the gradient diffusion relationship for the triple correlation
term should be appropriately modified in this case. We still
expect that the turbulence transport term will lead to decorrelation of L and U .
For zero forcing (case 4) we assume (9) still holds, i.e.
turbulence length scale is inversely proportional to U .

For the dissipation we will use the Taylor’s law:ε ∼ k3/2 /L .
Now, the balance T ∼ ε results in the identity relationship

L

√ k
k3/2
k
∼
,
L
L

(8)

SENSITIVITY STUDY
We analysed data of vertical wind velocity component w
in marine ABL recorded during the ACORES campaign. In
order to calculate turbulence statistics the instantaneous velocity should first be decomposed into the mean and fluctuatingparts. In practice, in the atmospheric turbulence, the presenceof large-scale convective motions and changes of atmosphericconditions along the flight track makes the choice of the averaging window difficult. In order to detrend the recorded signal we used the averaging window AWD = 50s, the same as
in Nowak et al. (2021) where the same ACORES data wereanalysed. Next, another window AWS was used to calculate
turbulence statistics from the detrended signal. We found thatAWS = 150s which corresponds to the window length larger
than 20L results in acceptable estimates of Cε , (with standard deviations below 0.1). This window was moved every 5s
along the signal and each time turbulence statistics werep es-hence, in this case no correlation between k (or U ) and L is
obtained.
Finally, in case 4) (decaying turbulence) we expect k andε to decrease and L to increase in time (Sinhuber et al., 2015)k ∼ t −n ,ε ∼ nt −(n+1) ,L ∼ t 1−n/2 ,where the coefficient n is related to the scaling of the energy
spectrum in low wavenumbers range. We assume
ka L = const,
where a is a constant. Differentiating above formula with respect to time
aka−1

dk
dL
L + ka= 0,
dt
dt

hence atimated. Turbulence intensity was calculated as U = w02 .
The integral length scale L was estimated by integrating thetransverse two-point correlation coefficient till the first zerocrossing and dividing the result by a factor 0.57, which, under additional assumptions should be equal to the longitudinallength scale (Wacławczyk et al., 2022). We also assume herethat the flow is locally isotropic, which is not always the casein marine stratocumulus-topped ABL (Akinlabi et al., 2019;
Nowak et al., 2021), especially close to the cloud top, where astably stratified layer is present. However, for data investigatedhere (from lower altitudes) our estimates should approximate
the integral turbulence length scale.
To find the energy dissipation rate ε we assume the following form of the transverse second-order structure function,in accordance with the Kolmogorov’s theory

1 dk
1 dL
=−
k dt
L dt

Because dk/dt < 0 and dL /dt > 0, hence we must have a > 0andL ∼ 1/kaor L ∼ 1/U 2a ,a > 0.

(9)

NON-EQUILIBRIUM TURBULENCE
The statistics of ’non-equilibrium’ turbulence follow theformulas (3) and (4). Also in this case Cε > Cε0 indicatesdecay of TKE whereas Cε < Cε0 would describe developing
turbulence, with production exceeding the dissipation (Bos &Rubinstein, 2018). The non-equilibrium scaling is found in thecase of rapid changes of the system (e.g. changes of forcing),such that the turbulence energy spectrum have not had enoughtime to relax to its self-similar state. The non-equilibrium relation (3) will modify the scalings (6) and (7). Instead of the
Taylor’s law, the dissipation will be proportional to

ε∼

k3/2
,
15/14 LR
1

S⊥ = (w0 (x + r) − w0 (x))2 = C⊥ ε 2/3 r2/3 ,where C⊥ ≈ 2.86. We estimate ε by fitting the calculated S⊥
in the inertial range to formula (13). Having U , ε and L we
calculated Cε as defined in Eq. (1) as well as L /λ and studied
its dependence on Rλ .

RESULTS(10)

Most of the data points considered here (coupled stratocumulus case) follow the equilibrium scaling (1) and (2)(Wacławczyk et al., 2022). Exception is for the data recordedat the lowest horizontal flight LEG of altitude 300 m, see Fig.
1. In this figure two parallel lines of a constant Cε0 = 0.45 andCε = 0.81 are plotted. The first one corresponds to the equilibrium case with the production equal, approximately dissipation. The second value corresponds to a free self-similar decayof turbulence with zero production. The mean value of Cε inthis region is equal, approximately Cε ≈ 0.25 and is lower thanCε0 , which suggests strong turbulence production and developing turbulence. Non-equilibrium scaling is also found for
L /λ , see Fig. (2).

λhence the leading-order balance between the production anddissipation gives
15/14

Rλ

L 2 ∼ k,

or

15/28

Rλ

L ∼U.

(11)

Instead of (7), for the balance B ∼ ε we obtain
15/14

L Rλ

∼ k3/2

15/14

or L Rλ∼ U 3.

(13)

(12)

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
10 4

1
2.3
2.25

0.8

2.2

L R 1/2 [m]

C

2.15

0.6

2.1
2.05
2

0.4

1.95
1.9

0.2
1.6

1.8

2

0.6

0.605

10 4

R

0.61

0.615

0.62

0.625

U[m/s]

Figure 3. Marine ABL, data for horizontal flight LEG at al√
titude 300m. L Rλ as a function of U follows the nonequilibrium scaling relation (11).

Figure 1. Marine ABL, data for horizontal flight LEG at altitude 300m. Cε as a function of Rλ . Solid black lines - equilibrium scalings Cε0 = 0.45 and Cε1 = 1.81Cε0 , dashed red line
- non-equilibrium scaling, Eq. (3), calculated statistics: symbols.

104

380
102

360
L[m]

100

L11 /

340
320

98

96

300

94

280

92
0.38

0.385

0.39

0.395

0.4

U[m/s]

260
1.6

1.8

R

2

Figure 4. Marine ABL, data for horizontal flight LEG at altitude 550m. L as a function of U follows the scaling relation
(9).

10 4

Figure 2. Marine ABL, data for horizontal flight LEG at altitude 300m. L /λ as a function of Rλ . Solid black line - equilibrium scaling (2), dashed red line - non-equilibrium scaling,
Eq. (4), calculated statistics: symbols.

equilibrium states. Dependence of L on U clearly follow
relation (7), which indicates dominant role of buoyancy anddissipation in the energy balance, see Fig. 5.

To additionally evaluate the data we studied the dependence of L on U and Rλ . The scaling seems to follow relation (11), where we assume 15/28 ≈ 1/2, see Fig. (3)Data recorded during horizontal flight at altitude 550m
follow the equilibrium scaling (1) and (2). In this region, below the cloud, turbulence production is much weaker and themean value of Cε ≈ 0.55 is larger than the equilibrium value
Cε0 . The integral length scale L becomes inversely proportional to U , which suggests that the scaling (9) holds, see Fig.
4. Here, scatter of the data is larger, which can indicate therole of turbulent transport which leads to decorrelation of L
and U .
The highest horizontal flight considered here was performed at altitude 800m, inside the stratocumulus cloud. Values of Cε changed along the flight track considerably, suggesting that the region of weaker turbulence alternate with areasof strong turbulence production. In spite of these variations
Cε and L /λ follow, in selected areas, the equilibrium scaling relations (1) and (2) which can be interpreted as a formof quasi-equilibrium, i.e. the function passes through series of

CONCLUSIONSWe presented estimates of non-dimensional indicators Cεand L /λ , as well as dependence of L on U from in-situ
measurements inside marine ABL, performed by a research
aircraft during ACORES campaign. We focused on the horizontal flight legs of the length of several kilometers. Stableconditions along the flight track allow to calculate statistics
such as the turbulence length scale L with a sufficient accuracy. We found that at the lowest flight leg at the altitude of
300m Cε < Cε0 which suggest that strong turbulence production takes place there. Statistics follow the non-equilibrium
scaling relations. Weaker turbulence is found at higher altitude of 550m, below the cloud, where Cε > Cε0 and L becomes inversely proportional to U . Inside the cloud Cε varies
considerably, however dependence of L on U suggest theleading order balance between the buoyancy production anddissipation in TKE budget. The analyses performed here canbe repeated for other measurement data in ABL’s. This can im-

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
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Phys. Fluids 19, 045101.
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2021 Coupled and decoupled stratocumulus-topped boundary layers: turbulence properties. Atmospheric Chemistry
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Siebert, Holger, Szodry, Kai-Erik, Egerer, Ulrike & et al. 2021
Observations of aerosol, cloud, turbulence, and radiation
properties at the top of the marine boundary layer over theeastern north atlantic ocean: The acores campaign. Bulletinof the American Meteorological Society 102 (1), E123 –E147.
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turbulence at high reynolds numbers. Phys. Rev. Lett. 114,
034501.
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generated by a class of multiscale grids. Journal of Fluid
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Rev. Fluid Mech. 47, 95–114.
Wacławczyk, M., Nowak, J. L., Siebert, H & Malinowski, S. P.
2022 Detecting non-equilibrium states in atmospheric turbulence. submitted to J. Atmos. Sci. .
Wyngaard, J. C. 2011 Turbulence in the atmosphere. Cambridge University Press.
Yano, J-I & Wacławczyk 2022 Nondimensionalization of theatmospheric boundary-layer system: Obukhov length andmonin–obukhov similarity theory. Boundary Layer Meteorology 182, 417–439.

180
160
140

L[m]

120

100

80

60
0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

U[m/s]

Figure 5. Marine ABL, data for horizontal flight LEG at altitude 800m. L as a function of U follows the scaling relation(7).

prove parametrizations of turbulence in the atmosphere, whichis crucially important for numerical weather predictions andclimate models.

ACKNOWLEDGEMENTS
The financial support of the National Science Centre,
Poland (Project No. 2020/37/B/ST10/03695) is gratefully acknowledged.

REFERENCES
Akinlabi, Emmanuel O., Wacławczyk, Marta, Mellado,
Juan Pedro & Malinowski, Szymon P. 2019 Estimating turbulence kinetic energy dissipation rates in the numericallysimulated stratocumulus cloud-top mixing layer: Evaluation of different methods. Journal of the Atmospheric Sciences 76 (5), 1471 – 1488.

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