Original Article Journal Section MOST, Monin-Obukhov similarity theory; ML,
Mixed Layer; TKE, Turbulence Kinetic Energy. [1]Equally contributing authors.
Ivana Stperski,
Department of Atmospheric and Cryospheric Sciences,
University of Innsbruck, Innrain 52f, Innsbruck, Austria
<EMAIL_ADDRESS>These results are part of a project that has
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (Grant agreement No.
101001691).

# Spectral scaling of unstably-stratified atmospheric flows: understanding the
low frequency spread of Kaimal model

Claudine Charrondière  Ivana Stiperski

###### Abstract

For more than five decades, a notable spread in the low frequencies of
velocity and temperature spectra, scaled using inertial subrange properties
and Monin-Obukhov similarity theory, has been observed in unstable
stratification. A large ensemble of 14 datasets, over relatively simple
terrain (from flat and homogeneous to gentle slope or valley floor) and very
complex terrain (steep slope, crater rim, mountain top), are used to assess
the reasons of this low-frequency behaviour.

Turbulence anisotropy is shown to be the primary factor influencing the
spectral energy at the largest scales and the spectral peak position of
streamwise and spanwise velocity spectra, with stability acting as a secondary
factor. On the contrary, the low-frequency behaviour of surface-normal and
temperature spectra is dominated by stability effects, while turbulence
anisotropy plays a secondary role. These observations are valid over both
simple and complex terrain, even though variability of the largest scales of
complex terrain datasets integrate other processes not included in turbulence
anisotropy or stability conditions.

Using a combination of non-dimensional turbulence parameters - temperature and
velocity variances as well as dissipation of turbulence kinetic energy and of
temperature variance - and of a semi-empirical model provided in the
literature, we are able to describe the behaviour of the velocity and
temperature spectra with only turbulence anisotropy and stability as input
parameters.

Finally, the paper provides some insights into the nature of anisotropic
eddies, as well as the relationship between the boundary layer height and the
spectral behaviour.

###### keywords:

Atmospheric Boundary Layer, Complex Terrain, Monin-Obukhov Similarity Theory,
Surface Layer, Turbulence Anisotropy

##  1 Introduction

Spatial structure of atmospheric boundary layer is tightly coupled to the
spectral characteristics of turbulence [59]. Turbulence spectra, thus, provide
invaluable information on the nature and structure of atmospheric turbulence,
in particular on how energy (or variance) is distributed among the different
scales of motions and how these scales interact with each others. In fact,
spectral analysis is widely used in studying atmospheric boundary layer flows,
especially in terms of understanding the process of turbulence cascade [e.g.,
26, 27, 54], the size and structure of most energetic eddies [e.g., 47, 53,
52, 57, 28], the departure of complex flows from canonical conditions [e.g.,
44, 63, 7, 3], the impact of large-scales atmospheric motions on the local
flow [10], in testing the numerical schemes of weather prediction models and
large eddy simulations [e.g., 56, 64, 12, 8], as well as in providing a
theoretical framework for widely used similarity scaling relations [e.g., 13,
22, 23, 24, 5, 31].

Surface-layer turbulence spectra are also one of the turbulence statistics
that follow the Monin-Obukhov similarity theory [MOST, 34]. The first and
still most widely used spectral scaling framework was proposed by [21], who
suggested scaling relations for the spectra of velocity and temperature, as
well as co-spectra of momentum and sensible heat fluxes. The spectral scaling
was based on the Kolmogorov’s hypothesis [27] such that the high-frequency
part of the scaled spectra collapse in the inertial subrange, allowing a study
of the low-frequency part of the spectra under MOST. Using data from the
Kansas field experiment over horizontally homogeneous and flat terrain [15],
[21] showed that under stable stratification the spectral energy at low
frequencies is driven by MOST stability parameter $\zeta=z/L$, where $z$ is
the height above the surface and $L$ the Obukhov length, defined as [55]:

$L=-\frac{\overline{\theta}}{\kappa g}\frac{u_{*}^{2}}{T_{*}}.$ (1)

Here, $\overline{\theta}$ is the mean potential temperature, $g=9.81$ m s-2 is
the effective gravity, $\kappa=0.4$ is the von Kármán constant,
$u_{*}=\left(\overline{u^{\prime}w^{\prime}}^{2}+\overline{v^{\prime}w^{\prime}}^{2}\right)^{1/4}$
is the friction velocity and
$T_{*}=-\overline{w^{\prime}\theta^{\prime}}/u_{*}$ is the temperature scale,
where $\overline{w^{\prime}\theta^{\prime}}$ is the kinematic sensible heat
flux, while $\overline{u^{\prime}w^{\prime}}$ and
$\overline{v^{\prime}w^{\prime}}$ are the momentum fluxes.

Still, in unstable conditions, [21] observed a large spread of the low-
frequency data that did not follow a particular scaling, concluding that the
spectra "tend to cluster in a random fashion". They only noted a small
separation of the surface-normal velocity spectra linked to $\zeta$ when
approaching neutral conditions, and offered no hypothesis to explain such a
spread. Later, [38] used an analytical model based on the same approach to
derive scaling laws for the velocity spectra both in the low-frequency and
high-frequency range, but also failed in predicting the behaviour of low-
frequency spectral energy for horizontal velocity spectra in unstable
conditions. Given that the integral of the spectrum yields the variance, the
non-scaling of the low-frequency part of the horizontal velocity spectra in
unstable stratification can be linked to the failure of MOST in explaining the
scaling of horizontal velocity variances, even over flat terrain [e.g., 65,
17].

Some refinements of MOST have been developed since. A mixed layer scaling
approach of [40] recognised that the horizontal velocity variances are driven
by inactive large-convective eddies, and thus that the mixed layer (ML) depth
$z_{i}$ and not the measurement height should be the relevant scaling height.
[33] extended this approach to spectra and co-spectra. For temperature spectra
and heat flux co-spectra, the scaling was then improved by a combination of
$z_{i}$ and the surface friction layer height [29] and explains the position
and spectral density of the energy peak. This means that the spectral shape
and density are related to both local [21] and bulk [29] information. Based on
the fact that the flow is influenced both by buoyancy and shear in the
majority of stability conditions (except highly convective/stable), [61]
developed the Multi-Point Monin-Obukhov theory, where the Obukhov length
serves not only as the vertical, but also as the horizontal length scale.
Through their approach, they could explain the commonly observed spectral
slopes.

Recently, [49], [51] and [50] showed that bulk turbulence statistics, such as
velocity variances, temperature variance and turbulence kinetic energy (TKE)
dissipation rate can be reconciled under MOST if turbulence anisotropy is
included as an additional scaling parameter. [35] extended the analysis to
scaled velocity and temperature gradients and obtained similar conclusion and
scaling relations. The novel scaling relations drastically improve scaling and
allow it to be extended to complex terrain, where the basic assumptions of
MOST (horizontally homogeneous and flat terrain, constant vertical flux
profile, no subsidence) are clearly violated. [50] suggested that, for
horizontal velocity variances, the scaling based on anisotropy outperforms the
ML scaling of [40], which implies that turbulence anisotropy incorporates the
majority of the information on the boundary conditions driving the flows
(including the ML height). Given the difficulties of obtaining ML height from
conventional turbulence measurements and the lack of routine boundary layer
height observations, the aforementioned scaling proposed in [33] or in [29]
are not easy to apply. A local parameter such as turbulence anisotropy is thus
preferred for understanding spectral behaviour. At the same time, spectral
analysis can help shed more light on the nature of turbulence anisotropy and
how it is impacted by terrain complexity.

Here we explore whether new results of [50] can be extended to two-point
statistics such as spectral scaling of [21], in order to explain the spread of
the low frequency range of velocity and temperature spectra in unstable
stratification. More specifically, we investigate the influence of turbulence
anisotropy on the spectral density and the location of the spectral peak for
datasets with varying degree of complexity. We first focus on canonical
conditions where the influence of terrain is expected to be small, and then
proceed to complex mountainous conditions to show how much our method can
account for the complexity of turbulence. Finally we explore turbulence
anisotropy in light of the newly developed spectral scaling.

Section 2 focuses on describing the theoretical framework on which the paper
is based. Section 3 provides an overview of the 14 datasets used, as well as a
description of the data processing. Section 4 presents the results with a
special focus on the influence of stability conditions and turbulence
anisotropy (Section 4.1 and 4.2), separation between turbulence anisotropy and
atmospheric stability effects (Section 4.3), analytical modeling (Section
4.4), and the influence of mountainous terrain (Section 4.5), followed by a
discussion on the contribution of the mixed layer height to spectral scaling
in Section 5 and conclusions in Section 6.

##  2 Theoretical framework

The spectral scaling proposed by [21] rests on two scaling frameworks: the
inertial subrange scaling as proposed by [26] and [27], and similarity
functions of MOST [34]. Given the observed non-constancy of fluxes with height
in non-canonical conditions [e.g., 36, 48], we use the local scaling approach
[37, 50] and scale the spectra at each measurement level with the respective
fluxes at that level. Note that the classic MOST is an asymptotic case of
local scaling in conditions when fluxes are constant with height.

According to Kolmogorov’s hypotheses [26, 27], the spectral density of the
velocity spectra $S_{u_{i}}$ (with $u_{1}=u$, $u_{2}=v$ and $u_{3}=w$ for
streamwise, spanwise and surface-normal velocity components respectively) and
temperature spectra $S_{\theta}$ in the inertial subrange are solely
determined by the dissipation rates of TKE $\varepsilon$ and of half the
temperature variance $N_{\theta}$ respectively, and can be written as [42,
66]:

$S_{u_{i}}(k_{x})=A_{i}\ \varepsilon^{2/3}\ k_{x}^{-5/3},$ (2)
$S_{\theta}(k_{x})=A_{\theta}\ N_{\theta}\ \varepsilon^{-1/3}\ k_{x}^{-5/3}.$
(3)

The wavenumber $k_{x}$ is parallel to the mean flow direction, $A_{i}$ is the
Kolmogorov constant, equal to $A_{1}\approx 0.55$ and
$A_{2}=A_{3}=\frac{4}{3}A_{1}$ for the three velocity components, and
$A_{\theta}\approx 0.8$ is the Obukhov-Corrsin constant for the temperature
variance. Following the Taylor’s frozen turbulence hypothesis [58], the
wavenumber spectrum can be converted from spatial to temporal domain,
$k_{x}S(k_{x})=fS(f)$, such that $k_{x}=2\pi f/\overline{u}$, where $f$ is the
frequency and $\overline{u}$ is the mean wind speed [66]. Finally, following
MOST, the spectra can be written in their scaled form as [21]:

$\frac{fS_{u_{i}}(f)}{u_{*}^{2}}=a_{i}\ \phi_{\varepsilon}^{2/3}\ n^{-2/3}\ ,$
(4) $\frac{fS_{\theta}(f)}{T_{*}^{2}}=a_{\theta}\ \phi_{N}\
\phi_{\varepsilon}^{-1/3}\ n^{-2/3}\ .$ (5)

Here the spectra are scaled based on the key surface-layer parameters: $z$,
$u_{*}$ and $T_{*}$. The scaled frequency $n=fz/\overline{u}$ and new values
for the constants are $a_{i}=A_{i}(2\pi\kappa)^{-2/3}$, which gives
$a_{1}\approx 0.3,a_{2}=a_{3}\approx 0.4$, and similarly $a_{\theta}\approx
0.43$. Finally, the MOST relations for TKE dissipation rate
$\phi_{\varepsilon}$ and for half the temperature variance $\phi_{N}$ are
defined as:

$\phi_{\varepsilon}=\dfrac{\kappa z\varepsilon}{u_{*}^{3}}\ ,$ (6)
$\phi_{N}=\dfrac{\kappa zN_{\theta}}{u_{*}T_{*}^{2}}\ .$ (7)

The scaled spectra based on Eqs. 4 and 5 collapse in the inertial subrange
(Fig. 1).

[21] observed that the low frequency ranges of both velocity and temperature
spectra vary according to the stability parameter for stable conditions
($\zeta>0$): the more stable the atmosphere, the less low frequency energy is
present in the scaled spectra, and the spectral peak is found at higher $n$,
such that energy containing eddies decrease in size and energy content as
stratification increases. However, for $\zeta<0$, no such trend could be
inferred. The only apparent separation between spectra according to $\zeta$ in
unstable conditions could be discerned for surface-normal velocity in close to
neutral stratification $-0.3\leq\zeta\leq 0$ (Fig. 1c).

Figure 1:  Scaled surface layer spectra for (a) streamwise, (b) spanwise, and
(c) surface-normal velocity, and (d) temperature, as a function of stability
parameter $\zeta$. Dark gray represents stable conditions, and light gray is
for unstable conditions. Figure adapted from [20].

To derive a universal model for the spectra in stable conditions, [21] studied
the link between the scaled frequency of the spectral peak $n_{m}$ and
$\zeta$. Using $n_{m}=f(\zeta)$, the spectra can then be described by a
universal shape for stable atmospheric conditions. However, for horizontal
velocity components in unstable atmospheric conditions, there is no clear
trend of $n_{m}$ according to $\zeta$. Some studies took a constant $n_{m}$
[e.g., 18, 6], but the low-frequency spectral density of unstable horizontal
velocity components was then not captured. This approach was employed later by
[38], who successfully reproduced spectra from the Kansas experiment in the
stable conditions using $n_{m}$ proportional to the scaled velocity gradient
$\phi_{m}$, taking the MOST relations for $\phi_{m}$ and $\phi_{\varepsilon}$.
In unstable conditions, [38] assumed that $n_{m}=f(\zeta)$ only for small
values of $\zeta$, and that it rapidly tends to a constant when stratification
approaches the free convective limit. This reasoning is able to explain the
behaviour shown in [21] for surface-normal velocity spectra.

According to [38], the scaled velocity spectra assume the following
expression:

$\frac{fS_{u_{i}}(f)}{u_{*}^{2}}=\frac{A_{u_{i}}n^{\gamma}}{(1+B_{u_{i}}n^{\alpha})^{\beta}},$
(8)

where parameters $A_{u_{i}}$ and $B_{u_{i}}$ determine the position of the
spectra on the ordinate, while $\alpha$, $\beta$ and $\gamma$ influence its
shape by providing asymptotic behaviour both at low and high frequencies.
Following Eq. 4, the inertial subrange asymptote $(n\gg 1)$ of Olesen’s model,
as described in Eq. 8, imposes two conditions on the free parameters:

$\gamma-\alpha\beta=-2/3\ ,$ (9)
$A_{u_{i}}=a_{i}\phi_{\varepsilon}^{2/3}B_{u_{i}}^{\beta}\ .$ (10)

Integrating Eq. 8 over the entire frequency range, [38] also derived an
expression for the scaled velocity variance $\phi_{u_{i}}$, which, when
combined with Eq. 10, provides an analytical solution for $B_{u_{i}}$:

$B_{u_{i}}=\left(C_{u_{i}}\dfrac{\phi_{u_{i}}^{2}}{\phi_{\varepsilon}^{2/3}}\right)^{P}\textnormal{
with
}C_{u_{i}}=\dfrac{\alpha}{a_{i}}\dfrac{\Gamma(\beta)}{\Gamma\left(\dfrac{\gamma}{\alpha}\right)\Gamma\left(\dfrac{2}{3\alpha}\right)}\textnormal{
and }P=\dfrac{\alpha}{\alpha\beta-1},$ (11)

where $\Gamma$ is the gamma function, and the scaled velocity variance is
defined as:

$\phi_{u_{i}}^{2}\ =\ \dfrac{\sigma_{u_{i}}^{2}}{u_{*}^{2}}\ .$ (12)

Equation 11, combined with Eq. 10, allowed to compute the parameters
$A_{u_{i}}$ and $B_{u_{i}}$, provided a set of values for $\alpha$, $\beta$
and $\gamma$. This approach has been used for example in [60], and has the
advantage of not requiring any information regarding the position of the
maximum spectral energy $n_{m}$. [60] described the mathematical shapes of
$A_{u_{i}}$ and $B_{u_{i}}$ for three sets of ($\alpha$, $\beta$, $\gamma$),
and showed analytically that the velocity spectra are a function of
$\phi_{u_{i}}$ only. Note that he used $\phi_{\varepsilon}=1$, which is a good
approximation only for near-neutral conditions. We thus do not expect this
solution to perform well in convective conditions.

Provided that $\phi_{\varepsilon}\neq 1$ in the range of stability considered
in the present study (down to $\zeta=-100$) we use the following adaptation of
Eq. 8:

$\frac{fS_{u_{i}}(f)}{u_{*}^{2}\phi_{\varepsilon}^{2/3}}=\frac{a_{i}B_{u_{i}}^{\beta}n^{\gamma}}{(1+B_{u_{i}}n^{\alpha})^{\beta}}$
(13)

One could adapt this semi-empirical model to temperature spectra consistently
with Eq. 5:

$\frac{fS_{\theta}(f)}{T_{*}^{2}\phi_{\varepsilon}^{-1/3}\phi_{N}}=\frac{a_{\theta}B_{\theta}^{\beta}n^{\gamma}}{(1+B_{\theta}n^{\alpha})^{\beta}},$
(14) $\textnormal{ where }\ \ B_{\theta}=\left(C_{\theta}\
\dfrac{\phi_{\theta}^{2}}{\phi_{\varepsilon}^{-1/3}\phi_{N}}\right)^{P},$ (15)

$C_{\theta}$ is defined as $C_{u_{i}}$ in Eq. 11, replacing $a_{i}$ by
$a_{\theta}$. The scaled temperature variance is defined as:

$\phi_{\theta}^{2}\ =\ \dfrac{\sigma_{\theta}^{2}}{T_{*}^{2}}\ .$ (16)

To include the information on turbulence anisotropy, we follow the work of
[50] who recently provided new empirical relations for $\phi_{u_{i}}$,
$\phi_{\theta}$ and $\phi_{\varepsilon}=f(y_{B},\zeta)$ that include the
degree of anisotropy $y_{B}$ as a second dimensionless parameter (Table 1).
The degree of anisotropy is quantified by the anisotropy invariant analysis of
the normalized anisotropy tensor $b_{ij}$ [32]:

$b_{ij}=\dfrac{\overline{u_{i}^{\prime}u_{j}^{\prime}}}{2e}-\dfrac{1}{3}\delta_{ij},$
(17)

where $\overline{u_{i}^{\prime}u_{j}^{\prime}}$ is the Reynolds stress tensor,
$e=0.5\overline{u_{l}^{\prime 2}}$ is the TKE, and $\delta_{ij}$ is the
Kronecker delta. The degree of turbulence anisotropy is then quantified
through the invariant $y_{B}$, determined from the smallest eigenvalue of
$b_{ij}$, namely $\lambda_{3}$, following [4]:

$y_{B}=(3\lambda_{3}+1)\frac{\sqrt{3}}{2}.$ (18)

where $y_{B}=\sqrt{3}/2$ corresponds to pure isotropy, while $y_{B}=0$ is the
most anisotropic state. For unstable stratification, the invariant $x_{B}$ was
shown not to be important [51]. More details on the anisotropy analysis are
provided in [49]. The new scaling relations for the $\phi$ functions thus make
it possible to include anisotropy into spectral scaling.

Table 1: Similarity scaling functions in the framework of MOST and according
to the new framework of [50]

| Monin-Obukhov similarity theory | Stiperski and Calaf (2023)
---|---|---
$\phi_{u}$ | $2.55\left[1-3\zeta\right]^{1/3}$ | $[0.784-2.582\log(y_{B})][1-3\zeta]^{1/3}$
$\phi_{v}$ | $2.05\left[1-3\zeta\right]^{1/3}$ | $[0.725-2.702\log(y_{B})][1-3\zeta]^{1/3}$
$\phi_{w}$ | $1.25\left[1-3\zeta\right]^{1/3}$ | $[1.119-0.019y_{B}-0.065y_{B}^{2}+0.028y_{B}^{3}][1-3\zeta]^{1/3}$
$\phi_{\theta}$ |  |  $0.99\left[0.063-\zeta\right]^{-1/3}$, when $\zeta<-0.05$
---
$0.015\left|\zeta\right|^{-1}+1.76$, when $\zeta>-0.05$
| $1.07\left[0.05+|\zeta|\right]^{-1/3}$
---
$+\left[-1.14+\left(0.017+0.217y_{B}\right)|\zeta|^{-9/10}\right]\left[1-\tanh{\left(10|\zeta|^{2/3}\right)}\right]$
$\phi_{\varepsilon}$ | $\left[1-3\zeta\right]^{-1}-\zeta$ |  | $a\left[1-3\zeta\right]^{-1}-\zeta b$
---
for $\varepsilon_{u}$: $a=0.371+1.811y_{B}$
$b=0.908+0.166y_{B}^{-1}+0.009y_{B}^{-2}$
for $\varepsilon_{w}$: $a=0.219+1.846y_{B}$
$b=0.635+0.052y_{B}^{-1}+0.006y_{B}^{-2}$
$\phi_{N}$ | $0.85\left[1-6.4\zeta\right]^{-2/3}/\phi_{\varepsilon}^{-1/3}$ |

##  3 Data

###  3.1 Datasets

The 14 datasets used in the present study are summarized in Table 2. These
datasets include: Advection Horizontal Array Turbulence Study [AHATS, 16];
Cabauw experimental site for Atmospheric Research [Cesar, 62]; Cooperative
Atmosphere Surface Exchange Study 1999 [CASES-99, 43]; five of the Innsbruck
Box [i-Box, 46] turbulence towers (CS-VF0, CS-SF1, CS-NF10, CS-NF27, CS-MT21)
together with the Im Hinteren Eis station; the NEAR and RIM towers from the
Second Meteor Crater Experiment [METCRAX II, stations MC-NEAR and MC-Rim, 30];
the F2 tower from the Mesoscale Alpine Programme MAP-Riviera experiment
[Riviera, 45] and the Central and West towers from Terrain-induced Rotor
Experiment [T-REX, stations T-RexC and T-RexW, 14].

The stations are separated into two categories. The first category groups
datasets that are closest to flat terrain (maximum slope is $3.25$°) : AHATS,
Cabauw, CASES-99, CS-VF0, CS-SF1, MC-Near, T-RexC and T-RexW. We will refer to
them as simple terrain thereafter. Note that CS-VF0, T-RexC and T-RexW are on
the valley floor, meaning that they are surrounded by mountains even though
they are on a relatively flat terrain. The second group is composed of
datasets acquired either on steep slopes (from $10$° to $40$°, CS-NF10, CS-
NF27 and Riviera), or in a highly complex configuration such on mountain tops
or ridges (CS-MT21, Im Hinteren Eis, MC-Rim). We will refer to them as complex
mountainous terrain.

Table 2: Information on turbulence datasets used in the study and site
characteristics

Experiment | Station name | Surface type |  | Terrain complexity
---
Slope angle
| Time range
---
available
| Measurement
---
heights (m)
| AHATS
---
California (USA)
AHATS | Fallow land | Flat | Jun-Aug, 2008 |  |  $1.55$, $3.3$, $4.24$,
---
$5.53$, $7.08$, $8.05$
| Cesar
---
Netherlands
Cabauw | Grass land | Flat | Jul-Oct, 2007 |  $60$, $100$
| CASES-99
---
Kansas (USA)
CASES-99 | Grass land | Flat | Oct, 1999 |  |  $5$, $10$, $20$, $30$,
---
$40$, $50$, $55$
| CS-VF0 | Mixed agricultural |  $0.5$° | Jan-Dec, 2015 |  $4$, $8.7$, $16.9$
| CS-SF1 | Alpine meadow |  $1$° | Jan-Dec, 2015 | $6.6$
| CS-NF10 | Alpine meadow |  $10$° | Jan-Dec, 2015 | $6.65$
| CS-NF27 | Alpine meadow |  $27$° | Oct-May, 2018 |  $1.3$, $6.06$
I-Box Tirol (Austria) | CS-MT21 | High alpine vegetation | Mountain top | Apr-Oct, 2015 | $4.67$
| Im Hinteren Eis
---
Tirol (Austria)
Im Hinteren Eis | Raw ground / snow | Mountain top | Jun-Oct, 2020 | $3.3$
| MC-NEAR | Desert |  $1$° | Oct, 2013 |  |  $10$, $15$, $20$, $25$, $30$,
---
$30$, $40$, $45$, $50$
METCRAX II Arizona (USA) | MC-Rim | Desert | Crater rim | Oct, 2013 |  |  $5$, $10$, $15$, $20$, $25$,
---
$30$, $35$, $40$
| Map-Riviera
---
Switzerland
Riviera | High alpine vegetation |  $40$° | Aug-Sept, 1999 |  $3.1$, $11$
| T-RexC | Desert |  $<0.5$° | Mar-Apr, 2006 |  $5$, $10$, $15$, $20$, $25$, $30$
T-REX California (USA) | T-RexW | Desert |  $3.25$° | Feb-Apr, 2006 |  $5$, $10$, $15$, $20$, $25$, $30$

###  3.2 Data processing

Turbulence data post-processing follows the procedure described in detail in
[51] and used in [50]. All datasets were double rotated [1] into the
streamline coordinate system where $u_{1}$ is the streamwise, $u_{2}$ the
spanwise, and $u_{3}$ the surface-normal velocity component. This coordinate
system is standard in complex terrain studies, for which surface-parallel
(streamwise and spanwise) direction, will diverge strongly from the
horizontal. Turbulence statistics were determined based on the 30-min Reynolds
averaging, with prior detrending, given that the present study focuses only on
unstable conditions [e.g., 2, 49]. Furthermore, the data were required to be
unstable at all measurement levels, and be acquired during daytime to avoid
including nighttime counter-gradient fluxes into the statistics.

The following quality control was additionally applied. We kept only
stationary data according to the stationarity test of [11], with a threshold
of $30$%. The stationarity test was applied to the mean wind speed, as well as
the sensible heat flux $\overline{w^{\prime}\theta^{\prime}}$ (except in near-
neutral conditions, $\zeta>-0.05$, where the non-stationarity could be solely
due to the small magnitude of the flux) and the turbulent momentum flux
$\overline{u^{\prime}w^{\prime}}$ (except in highly convective conditions,
$\zeta<-1$, where this flux has a small magnitude). Furthermore, we excluded
30-min segments for which the turbulent sensible heat flux
$\overline{w^{\prime}\theta^{\prime}}<0.01$ m K s-1 at the corresponding
measurement height, following [9] and [36]. Such a small flux can be a sign of
the (morning/evening) transition regime where
$\overline{w^{\prime}\theta^{\prime}}$ integrates contributions from both
unstable and stable regimes at different scales .

Spectral scaling of [21] requires the knowledge of the TKE dissipation rate
$\varepsilon$ and the dissipation rate of half the variance $N_{\theta}$ (Eqs.
4 and 5). These dissipation rates were determined from the second order
structure function for streamwise, spanwise and surface-normal velocity
variances [7] and for temperature variances [25], under the usual assumptions
of the existence of the inertial subrange [26]:

$\varepsilon_{u_{i}}=\left[\dfrac{D_{ii}(r)}{C_{2}^{u_{i}}}\right]^{3/2}r^{-1},$
(19)
$N_{\theta}=\left[\dfrac{D_{\theta\theta}(r)}{C_{2}^{\theta}}\right]\varepsilon_{u}^{1/3}r^{-2/3},$
(20)

where $r$ is the streamwise separation distance, and $C_{2}^{u}=1.97$,
$C_{2}^{v}=C_{2}^{w}=\frac{4}{3}C_{u}$, and $C_{2}^{\theta}=3.2$. To assure
that the second-order structure functions were evaluated within the inertial
subrange, only lags for which the slope of the structure functions fell within
the range of $2/3\pm 10\%$ were used.

Since some datasets cover a longer period or contain more measurement levels
than others (Table 2), subsampling with replacement was used to build an
ensemble of $8$ simple-terrain datasets, to ensure that each dataset is
equally represented in the analysis. Each dataset within the ensemble was
subsampled to have $500$ averaging periods within each bin of $\zeta$ or
$y_{B}$. Two ensembles were thus independently created for spectra binned
according to the logarithmically spaced $\zeta$ and linearly spaced $y_{B}$.
The complex terrain datasets were evaluated independently. In the following
figures, we present the ensemble medians and the interquartile ranges obtained
from the bins of the logarithmically spaced normalized frequency $n$. Each bin
of logarithmically spaced $n$ was required to have data from at least three
datasets, and more than ten 30-min segments.

##  4 Results

###  4.1 Effect of stability on the spectra

We first explore the classical scaling of [21] and bin the ensemble data
according to $\zeta$ (Fig. 2). In agreement with [18], we observe no
dependence of the surface-parallel velocity spectra ($S_{u}$ and $S_{v}$) on
$\zeta$, and only a small dependence of $S_{w}$. We, however, observe three
major differences between our and [18]’s results. The low-frequency part of
the spanwise spectra $S_{v}$ show reduced spectral density in the near-neutral
conditions ($\zeta>-0.1$) and a plateau-like shape, compared to more unstable
stratification. The existence of a plateau (observed also in individual
datasets), indicates a continuum of eddy sizes contributing significantly to
the total variance. In convective conditions the spectral peak of $S_{v}$ is
better defined, although still broader than in the streamwise velocity spectra
$S_{u}$. The spanwise velocity spectra $S_{v}$ also exhibits a slightly
concave shape between the frequency of the energy peak and lowest frequency
limit of the inertial subrange, especially visible for very anisotropic
conditions, as already pointed out in [19].

The small dependence of $S_{w}$ on $\zeta$ stems solely from the dependence of
the position of the spectral peak on $\zeta$. This peak is shifted from
approximately $n=0.1$ in very convective conditions to approximately $n=0.7$
when $|\zeta|<1$. This means that the surface-normal extent of the most
energetic eddies is largest in convective conditions, as expected. However, we
do not observe any impact of stability on the lowest frequencies.

The major difference between [18] results and ours arises for temperature
spectra, which show a very clear dependence on $\zeta$. An increase of
$|\zeta|$ is seen to correspond to a decrease of the low-frequency spectral
energy and a shift of the spectral peak towards smaller scales. [29] also
mentioned such a shift of the peak, with increasing instability during the
day.

Figure 2:  Ensemble of simple-terrain spectra of (a) streamwise, (b) spanwise
and (c) surface-normal velocity and (d) temperature binned according to the
logarithmically spaced stability parameter $\zeta$ (colour). Median of each
$\zeta$ bin is shown with a solid colored line, and the interquartile range in
shading. Solid black line corresponds to the spectral density expected in the
inertial subrange according to Eqs. 4 and 5.

###  4.2 Effect of anisotropy on the largest scales of turbulence

According to [50], velocity and temperature variances are functions of both
the stability parameter $\zeta$ and turbulence anisotropy $y_{B}$. Knowing
that turbulence anisotropy is scale-dependent, and that the dominant
anisotropy stems from the anisotropic forcing at energy-containing scales, the
question arises whether $y_{B}$ can explain the lack of scaling of surface-
parallel velocity spectra, as well as whether the low frequency contributions
among the different degrees of anisotropy show consistent behaviour.

To answer these questions, the velocity and temperature spectra are binned
according to the degree of turbulence anisotropy, irrespective of $\zeta$
(Fig. 3). Spectra of surface-parallel velocities, $S_{u}$ and $S_{v}$, as well
as temperature $S_{\theta}$, indicate a clear dependence on $y_{B}$ in the
range of frequencies that do not scale under the [21] scaling. This behaviour
is consistent with the results of [50] for the velocity variances and shows
that the degree of anisotropy is indeed the driving mechanism behind the non-
scaling of surface-parallel velocity spectra. On the other hand, the surface-
normal velocity spectra $S_{w}$ (Fig. 3c) show only a modest dependence on
$y_{B}$ in terms of the energy at peak frequency: isotropic turbulence
corresponds to a lower peak frequency which contains more energy than
anisotropic turbulence. This dependence, however, is not uniform as an
anisotropy threshold needs to be reached for $y_{B}$ to show effects on the
spectral peak.

Figure 3 also offers a deeper insight into the nature of turbulence eddies
associated with the different degrees of anisotropy. The results show that
quasi-isotropic turbulence (large $y_{B}$) is associated with smaller eddies
that carry less energy in the surface-parallel, but more in the surface-normal
direction than anisotropic turbulence, thus leading to an almost equal energy
distribution in all the three directions. Furthermore, the peaks in the
velocity spectra for isotropic turbulence occur at similar frequencies for all
three velocity components, indicating that isotropic turbulence is also quasi-
isotropic in eddy size. As the anisotropy increases ($y_{B}$ decreases), on
the other hand, the horizontal size and the energy content of the dominant
energy-containing eddies increases (the spectral peak is shifted towards lower
frequencies) compared to more isotropic turbulence. In fact, the surface-
normal velocity component shows strongest dependence on $y_{B}$ in quasi-
isotropic range, while the surface-parallel velocity spectra show largest
influence in anisotropic conditions.

Figure 3:  The same as Fig. 2 but binned according to the degree of anisotropy
$y_{B}$ (colour).

###  4.3 Joint effects of turbulence anisotropy and stability

Turbulence anisotropy, while defined only based on the Reynolds stress tensor
(Eqs. 17 and 18), is the result of multiple physical processes and their
interactions, including stability-related processes. To separate the effect of
$y_{B}$ from the effect of $\zeta$ in the scaled spectra, we use the spectral
peak frequency $n_{m}$ of $S_{u}$ as a marker. Figure 4 shows $n_{m}$ as a
function of $y_{B}$ for the four bins of $\zeta$ defined previously (cf. Fig.
2). The behaviour of $n_{m}$ differs significantly depending on whether
conditions are near-neutral ($|\zeta|<0.1$) or more convective
($|\zeta|>0.1$). For near-neutral conditions, the spectral peak frequency
increases linearly with $y_{B}$ while for convective conditions the growth is
exponential. For very anisotropic conditions ($y_{B}<0.35$) thus the increase
of $n_{m}$ with $y_{B}$ is weaker than in near-neutral conditions. In
addition, for a given value of $y_{B}$, the spectral peak occurs at lower
scaled frequencies $n_{m}$ than in near-neutral stratification. This feature
is lost when examining ensemble spectra binned according to $\zeta$ (Fig. 2),
since each stability bin contains different anisotropy types: very anisotropic
turbulence develops preferentially in near-neutral conditions, while the most
isotropic conditions are found only in convective conditions (Fig. 5a), as
already observed in [49] and [52]

Figure 4:  Spectral peak normalized frequency as a function of turbulence
anisotropy (abscissa) and stability (colored scale). Each point is computed
from the median spectra of a given bin of $y_{B}$ and $\zeta$.

Figure 5b-e shows the velocity and temperature spectra binned according to
turbulence anisotropy for two stability classes: near-neutral ($|\zeta|<0.1$)
and highly convective ($|\zeta>1|$). The low-frequency separation of the
streamwise and spanwise velocity spectra according to $y_{B}$ occurs in equal
measure for near-neutral and convective conditions (Fig. 5b-c), with however
several differences: (1) quasi-isotropic turbulence only occurs in convective
conditions, while more anisotropic turbulence is present in all stability
ranges as above-described; (2) for the most anisotropic data, the convective
spectra show a well defined peak at higher frequencies than in near-neutral
spectra, which instead show a plateau or a very broad peak with the maximum at
lower frequencies; (3) spectral energy is higher but also more variable at low
frequencies in convective conditions compared to near-neutral conditions.

The very small dependence of the surface-normal spectra on $y_{B}$ (Fig. 3c)
is related to the associated stability conditions (Figs. 2c and 5d). The most
striking result is the small but clear increase of energy in the surface-
normal direction with increasing $y_{B}$ in convective conditions. The
temperature spectra (Fig. 5e), on the other hand, show only a limited
variability according to $y_{B}$ for near-neutral stratification, but a clear
separation according to anisotropy in convective conditions. The clear low
frequencies binning of temperature spectra with $y_{B}$ in Fig. 3d is thus the
signature of the repartition of anisotropic states within the stability range
considered.

Figure 5 thus clearly highlights the different characteristics of anisotropic
eddies under the two stability regimes. Anisotropic eddies under near-neutral
stratification are predominantly driven by shear that injects energy into the
streamwise (and spanwise) velocity component. The streamwise and spanwise
spectra are thus characterised by larger surface-parallel size of the most
energetic eddies that occur over a wider range of scales (Fig. 5b, c), and
whose size scales with height ($n=1$) in the surface-normal direction (Fig.
5d). The anisotropic eddies that occur under convective conditions, on the
other hand, have surface-parallel spectra with a better defined peak that
corresponds to smaller eddies in the surface-parallel, but larger in the
surface-normal direction than in near-neutral conditions. The energy content
of the most energetic eddies, however, is larger than under near-neutral
stratification in all three directions, especially spanwise.

Figure 5:  (a) Distribution of anisotropy state within each bin of $\zeta$ in
the ensemble data. Spectra of (b) streamwise, (c) spanwise and (d) surface-
normal velocity and of (e) temperature, binned with respect to turbulence
anisotropy $y_{B}$ for convective (solid lines) and near-neutral (dotted
lines) conditions. The shading includes the interquartile range for the whole
stability conditions considered, regardless of turbulence anisotropy bins.

###  4.4 Analytical modeling of spectra

Following the semi-empirical model of [38] described in Sect. 2, the present
Section intends to describe the spectral shape, with a function requiring only
the knowledge of local parameters (namely $y_{B}$ and $\zeta$). To do so, we
use Eqs. 13 and 14 and take $\gamma=1$, as widely used in micrometeorology
[cf. 21, 38, 60]. As for $\alpha$ and $\beta$, there is no consensus on their
value in the literature. In stable conditions, [38] used $\beta=5/3$ for all
three velocity components. In near-neutral conditions, [60] differentiates
between ideal terrain (flat, smooth and uniform), where $\beta=1$ is the best
fit, and complex terrain (heterogeneous surface roughness, topography
conditions), where $\beta=5/3$ is more suitable. According to [19], a value of
$5/3$ is also the best fit for the surface-normal velocity spectra in unstable
conditions. In the present paper, we adapted $\beta$, without any a priori
assumptions, to the value that fits best to the spectra. We observe that
$\beta=5/3$ is a good fitting parameter only for $S_{u}$, while $S_{v}$ and
$S_{w}$ require respectively $7/3$ and $4/3$, and $S_{\theta}$ is best
described by $\beta=2$. The remaining exponent $\alpha$ is derived from Eq. 9,
knowing $\beta$ and $\gamma$. These coefficients, and the resulting
parameters, used to derive the solution are summarized in Table 3.

Table 3: List of coefficients used in the spectral model, Eqs 13 and 14 | Streamwise velocity
---
spectra $S_{u}$
| Spanwise velocity
---
spectra $S_{v}$
| Surface-normal
---
velocity spectra $S_{w}$
Temperature spectra $S_{\theta}$
$\alpha=1$ $\beta=5/3$ $\gamma=1$ |  | $C_{u}=2.24$
---
$P=1.5$
$\alpha=5/7$ $\beta=7/3$ $\gamma=1$ |  | $C_{v}=3.09$
---
$P=1.07$
$\alpha=5/4$ $\beta=4/3$ $\gamma=1$ |  | $C_{w}=1.94$
---
$P=1.875$
$\alpha=5/6$ $\beta=2$ $\gamma=1$ |  | $C_{\theta}=1.8$
---
$P=1.25$
Figure 6:  Scaled (a-c) streamwise, (d-f) spanwise, (g-i) surface-normal
velocity spectra and (j-l) temperature spectra. In each subplot, the solid
lines represent the median of a range of turbulence anisotropy from the
ensemble data of simple terrain dataset. The dashed lines are the solution of
Olesen’s model, computed for each individual spectrum before taking their
median. The shading is the interquartile range for each anisotropy bin (scale
on the right for the corresponding values of $y_{B}$) In the first row, the
solution is computed using the definition of $\phi$ (Eqs. 6, 7, 12 and 16),
the second row when using the $\phi$ functions according to MOST, and third
row when using $\phi$ functions according to [50] (Table 1). To facilitate the
reading, spectra in each subplot are shifted down compared to each other by
one decade, where the upper-most spectrum is in the correct position.

Figure 6 shows the velocity and temperature spectra along with the model
predictions. The model, given by Eq. 13 when $\phi_{u_{i}}$ and
$\phi_{\varepsilon}$ are determined directly from the data (Eq. 6 and 12), is
able to capture well the behaviour of velocity spectra, including the low
frequency spectral variation and the shift of the spectral peak with
turbulence anisotropy (Fig. 6a, 6d and 6g). For the streamwise velocity
spectra, the main divergence is that the model underestimates slightly the
energy peak. This is more pronounced for very anisotropic turbulence and tends
to disappear as $y_{B}$ increases. We also note an inability of the model -
constrained by its mathematical shape - to reproduce the concave region of
$S_{v}$ at high anisotropy. Except for these details, we thus consider that
the mathematical shape of the model is good enough to be used thereafter for
all velocity components.

Because of the inability of MOST to describe properly horizontal velocity
variances under unstable stratification, it was not possible to compute
correct spectral behaviour for horizontal velocity spectra using the
$\phi(\zeta)$ functions, when the model of [38] was published. Figures 6b and
6e, indeed show a huge discrepancy between the predictions of Eq. 13 when
using the classic MOST $\phi$ functions (left column in Table 1) and the data
both in terms of low frequency spectral energy and position of the spectral
peak. The discrepancy is particularly obvious for isotropic turbulence in
streamwise and all anisotropy types in spanwise velocity spectra. Sect. 4.2
showed that $S_{u}$ and $S_{v}$ are strongly dependant on $y_{B}$. Figures 6c
and 6f confirm that when using $\phi_{u}$, $\phi_{\varepsilon}=f(y_{B},\zeta)$
(right column in Table 1) as parameters of the model, we are now able to
better predict the shape of the spectra based only on stability $\zeta$ and
turbulence anisotropy $y_{B}$ as input parameters. Note that due to a lack of
scaling relation for $\phi_{\varepsilon_{v}}$ in [50] and under the commonly
applied assumption of horizontal isotropy, we used $\phi_{\varepsilon_{u}}$
also in the spanwise velocity spectra. For the surface-normal spectra, the
discrepancy between the data and the model based on MOST (Fig. 6h) is small,
because $\phi_{w}=f(\zeta)$ was already a good approximation in the MOST
framework (left column in Table 1). When using the $\phi_{w}$,
$\phi_{\varepsilon}=f(y_{B},\zeta)$ functions (right column in Table 1), as
parameters of the model, the spectra are, however, now even better represented
(Fig. 6i).

The model applied to temperature spectra (Eq. 14) using Eqs. 6, 7 and 16, is
shown in Fig. 6j. The solution based on MOST $\phi$ functions is relatively
good, with mostly a slight underestimation of low frequency spectral energy
for the most anisotropic bins (Fig. 6k). This is not a surprise provided that
both $\phi_{\theta}$, $\phi_{\varepsilon}$ and $\phi_{N}$ are already well
approximated by MOST. This confirms that the spread observed in Fig. 3d is
mostly linked to $\zeta$. Furthermore, [50] showed that $\phi_{\theta}$ does
not depend strongly on $y_{B}$ away from near-neutral stratification, while
$\phi_{\varepsilon}$ has a limited dependence on $y_{B}$ in the stability
range considered. If we now use $\phi_{\theta}$ and
$\phi_{\varepsilon}=f(y_{B},\zeta)$ and keep $\phi_{N}=f(\zeta)$ (cf. Table
1), the model improves for low $y_{B}$, while MOST scaling relations performs
best for $y_{B}>0.45$. The reason for using MOST function for $\phi_{N}$ stems
from no scaling relation for $\phi_{N}$ in [50], as well as no clear
dependence on $y_{B}$ is observed in the data (not shown).

###  4.5 Spectral scaling in mountainous terrain

Figure 7:  Scaled (a) streamwise, (b) spanwise, (c) surface-normal velocity
spectra and (d) temperature spectra. In each subplot, the black dotted lines
represent the median of a range of turbulence anisotropy from the ensemble
data of simple terrain dataset. The shading is the interquartile range for
each bin of anisotropy bin (scale on the right for the corresponding values of
$y_{B}$). The colored solid lines are the median of the spectra per bin of
$y_{B}$ for each of the 6 datasets included in the complex mountainous
category. To facilitate the reading, spectra in each subplot are shifted down
compared to each other by $0.3$, where the upper-most spectrum is in the
correct position.

For near-neutral stability conditions, [60] reported that the large scale
behaviour of $S_{u}$ and $S_{v}$ is highly related to the upwind topography,
causing differences in the spectra from one site to another, both in shape and
in spectral energy. Over mountainous terrain, this behaviour is expected to be
even more pronounced since the upwind topography is highly direction
dependent. The scaling results (Fig. 7) show that terrain indeed influences
the spectral distribution since the datasets show a large divergence amongst
themselves, but also that the effect of terrain is anisotropy dependent. For
the majority of the anisotropy range ($y_{B}>0.35$), the median spectra of six
mountainous datasets mostly fall within the spread of the simple terrain
datasets, despite the strong terrain complexity and the expected effect of
heterogeneity on the low frequency motions. For very anisotropic turbulence
($y_{B}<0.3$), however, spectra of all three velocity components over complex
terrain are found to always be above the simple terrain median, the spectral
peak is shifted towards larger scales and the low frequency energy increases
for $S_{u}$, $S_{v}$ and $S_{\theta}$ as $y_{B}$ decreases. The surface-normal
spectra $S_{w}$, on the other hand, show an increase of the spectral energy
around the frequency of the energy peak when approaching isotropy, as well as
a secondary peak for some datasets (cf. CS-MT21). It thus appears that terrain
complexity mostly impacts the surface-parallel components more for anisotropic
turbulence, and surface-normal component more for isotropic turbulence.

Figure 8:  Median scaled spectra of (a) streamwise, (b) spanwise and (c)
surface-normal velocity and (d) temperature for each individual dataset.
Datasets are classified on the scale from the simplest (blue) to the most
complex (red). The black dashed lines indicate spectral slopes of $1/2$ and
$1$.

According to [60], the scaled complex terrain spectra are characterized by a
low frequency energy higher than simple terrain spectra and a slope $\gamma$
lower than $1$ at these frequencies [although what they consider as complex
terrain is flat terrain with a ridge located $400$ m from the measurement
site, similar to T-REX which we characterize as simple terrain in our study,
41, 39]. Among the 14 datasets used in the present study, we also observe a
reduction of the low frequency slope of $S_{u}$ and $S_{v}$, when spectra are
plotted as the median over the whole dataset (Fig. 8). This slope decreases
from nearly $1$ for some of the simple terrain down to about $0.5$ for the
most complex terrain. Median temperature spectra $S_{\theta}$ show a low
frequency slope lower than $1$ for all of the datasets acquired over simple
and complex terrain. However, when binned according to $y_{B}$, low frequency
slopes of both velocity and temperature spectra are closer to $1$, hence the
use of $\gamma=1$ in the analytical model. Thus the slope decrease mentioned
by [60] and observed in Fig. 8 could be the result of the averaging of spectra
for data acquired over different conditions (stability, turbulence anisotropy,
wind direction etc.). Interestingly, low frequency slopes of $S_{w}$, whether
binned according to turbulence anisotropy conditions or not, are about $1$,
except for the two most complex terrain datasets (Im Hinteren Eis and CS-
MT21), for which $\gamma$ is lower. This feature is observed for all
anisotropy conditions (Fig. 7).

##  5 Discussion

###  5.1 Performance of the new spectral scaling

Figure 9:  Skill score of the improvement of the spectral model based on
$\phi=f(y_{B},\zeta)$ over the model using the original MOST $\phi$ functions,
for (a) the streamwise velocity spectra, (b) the spanwise velocity spectra,
(c) the surface-normal velocity spectra and (d) the temperature spectra for
each of the dataset used in the present study. The bins color corresponds to
turbulence anisotropy state. White and gray backgrounds are used for simple
and complex terrain respectively. The number of $30$-min segments of each
dataset is indicated at the top of each column.

Figure 9 quantifies the improvement of the model brought by using the scaling
relations of [50] over using MOST (Table 1) for each bin of $y_{B}$ in each of
the datasets, through the skill score $S_{s}$. The skill score is defined
based on the mean absolute deviation ($MAD$) of the spectra, as:

$S_{s}=1-\dfrac{MAD_{f(y_{B},\zeta)}}{MAD_{f(\zeta)}},$ (21)

with:

$MAD=\frac{1}{T}\sum_{t=1}^{T}\left(\frac{1}{N}\sum_{i=1}^{N}|\log(S_{\textnormal{model}}^{t}(n_{i}))-\log(S_{\textnormal{data}}^{t}(n_{i}))|\right),$
(22)

where $S_{\textnormal{model}}^{t}(n_{i})$ and
$S_{\textnormal{data}}^{t}(n_{i})$ are the spectra from the analytical model –
either $f(y_{B},\zeta)$ or $f(\zeta)$ – and from the data respectively, at the
normalized frequency $n_{i}$ ($N$ is the number of frequency points within
each spectra) and for the time segment $t$ ($T$ is the number of time segments
within each dataset). The skill score is computed for each individual spectrum
and the result is the average value per bin of $y_{B}$.

The skill score indicates a net improvement of the ability of the model to
reproduce the surface-parallel velocity spectra, by up to about $30\%$, when
using $f(y_{B},\zeta)$ compared to $f(\zeta)$, for both the simple and the
complex terrain (Fig. 9). The negative values for very anisotropic conditions
and the low values for intermediate anisotropic conditions in Fig. 9 are the
result of the shape constraint of the model, which does not allow to correctly
capture the energy peak for the most anisotropic bins (Sect. 4.4).
Qualitatively, looking at the individual datasets, the model is indeed still
better in its new version including $y_{B}$: the model using $f(\zeta)$
overestimates the energy peak and underestimate the lowest frequencies, while
the model using $f(y_{B},\zeta)$ only overestimates the energy peak but can
reproduce well the low frequencies. The surface-normal velocity spectra, that
were found not to be strongly influenced by anisotropy, are also improved, but
to a lesser degree (up to about $10\%$), for most of the simple terrain
datasets. There is no coherent pattern, however, over complex terrain.

Section 4.4 demonstrated that the prediction of temperature spectra for the
most anisotropic conditions is improved by the use of $y_{B}$ in the model
whereas it is worsened for the less anisotropic spectra for simple terrain.
Looking at each dataset separately, this behaviour is still globally valid for
simple terrain but the behaviour of the most anisotropic bins is not always
improved. This is also observed for the datasets acquired over sloping terrain
(CS-NF10, CS-NF27, Riviera) with a clear improvement of the prediction of very
anisotropic temperature spectra by up to about $15\%$. Mountain top stations
(CS-MT21, Im Hinteren Eis) show a relevant improvement for all anisotropic
conditions, as well.

Interestingly, spectra computed with data from MC-Rim station (Table 2),
located at the rim of a meteor crater, are not strongly affected by the
complexity of the surrounding terrain neither for velocity components nor for
temperature. This might be due to the fetch since only $25\%$ of the $30$-min
segments retained after quality control (Sect. 3.2) originates from the
direction of the crater while the rest originate from a more uniform terrain,
like MC-NEAR station.

###  5.2 Discussion on the mixing layer scaling of velocity spectra

The mixed layer scaling approach of [40] for horizontal velocity variances and
its expansion to velocity and temperature spectra [33] was developed in
response to the failure of MOST. ML scaling states that the dynamics of
horizontal velocity fluctuations is governed by the ML height $z_{i}$ as a
relevant length scale. [50] showed that mixed layer scaling does collapse the
surface-parallel velocity variances better than MOST, however, with a
lingering dependence on anisotropy. We therefore explore the ML spectral
scaling approach of [33] in light of anisotropy. For this we use the data from
the West tower during T-REX campaign (cf. Table 2), as lidar observations at
this site allowed retrieving the information on $z_{i}$.

Figure 10 shows scaled streamwise and spanwise velocity spectra for which
$z_{i}$, instead of the measurement height $z$, is taken to be the
representative length scale. The clear separation of the low frequency
spectral energy and the shift of the spectral peak position for surface-
parallel velocity spectra with $y_{B}$ observed in Fig. 3 nearly disappear
(Fig. 10). Only a slight shift of the spectral peak position for $S_{u}$ is
observed, by about one decade whereas it was of about three decades in Fig. 3.
These results confirm the importance of ML height in the surface-parallel
velocity dynamics.

As expected since $\phi_{w}$ and $S_{w}$ follow MOST, the mixed layer scaling
applied to $S_{w}$ shows the opposite behaviour (Fig. 11a), namely an evident
dependence on $y_{B}$ at low frequency, both in terms of spectral energy and
peak position: low frequency spectral energy increases and spectral peaks are
shifted towards larger scales as anisotropy decreases. This shift of the low
frequency spectral energy in $S_{w}$ is linked to the relative height of the
measurements within the mixed layer, and can be corrected by the factor
$(z/z_{i})^{-2}$ (Fig. 11b), derived from Fig. 5 of [33]. However, this causes
the spectra not to scale in the inertial subrange.

The comparison between Figs. 3 and 10 indicates that ML height does impose an
important scale on surface-parallel velocity spectra, but does not integrate
all of the processes affecting them. In fact, $S_{u}$ and $S_{v}$ scaled
according to ML approach, still exhibit significant scatter and a weak
dependence on $y_{B}$, charting the limitations of the ML approach.

Figure 10:  Scaled spectra of (a) streamwise and (b) spanwise velocity
component for the T-RexW dataset, scaled according to [33] with the ML height
$z_{i}$. Figure 11:  Surface-normal velocity spectra from the T-RexW dataset,
scaled according to [33] (a) with the ML height $z_{i}$, and (b) using a
combination of the the ML height $z_{i}$ and the measurement height $z$ as the
representative length scales.

##  6 Conclusion

Turbulence anisotropy has been shown to improve flux-variance and flux-
gradient scaling under Monin-Obukhov similarity theory (MOST) and allow it to
be extended to conditions where the assumptions of MOST are clearly violated
[50, 35]. Here we expended these findings to spectral scaling in the unstable
surface layer ($-100<\zeta<-0.01$) known not to obey MOST. We examined the
combined role of turbulence anisotropy $y_{B}$ and stability $\zeta$ in
explaining the dynamics of the largest turbulent scales, through the analysis
of velocity and temperature spectra scaled using inertial subrange properties
[26, 27] and MOST. Our analysis, based on an ensemble of 14 datasets covering
a wide range of stability and surface conditions, demonstrated that turbulence
anisotropy $y_{B}$ is able to explain the low frequency spread and the
position of the spectral peak in both the scaled streamwise and spanwise
velocity spectra, with a secondary effect of stability $\zeta$. On the other
hand, stability was shown to be the primary factor able to relate the spread
of the largest scales in temperature spectra and the peak position and
intensity for both temperature and surface-normal velocity spectra, while
$y_{B}$ was shown to have a secondary effect, most pronounced in convective
conditions. The combination of $\zeta$ and $y_{B}$ in a semi-empirical model
was therefore shown to be a powerful tool to describe the behaviour of the
three velocity components and temperature spectra.

The spectral scaling results shed light on the nature of turbulent eddies
under different states of anisotropy. Anisotropy was shown to govern the
spectral peak position, so that anisotropic turbulence is characterized by
eddies that peak at lower frequencies in the surface-parallel velocities and
temperature spectra, and to carry larger fraction of variance than isotropic
turbulence. On the other hand, isotropic turbulence, preferentially occurring
in convective conditions, was shown to be comprised of eddies that are smaller
in size both in spectra of streamwise and spanwise velocity and temperature,
and to carry a smaller fraction of energy than anisotropic turbulence. In
addition, isotropic turbulence was shown to be quasi-isotropic not only in
energy but also in size of the energy containing eddies. Finally, the results
highlighted that the spectral characteristics of anisotropic eddies depend on
stratification, so that anisotropic turbulence under near-neutral and very
unstable conditions differ.

The complementary analysis of six datasets acquired over highly complex
mountainous terrain is a first of its kind to systematically explore the
impact of highly complex terrain on turbulence spectra. We note a net
improvement of the ability to describe the surface-parallel velocity spectra
$S_{u}$ and $S_{v}$ in complex terrain when incorporating $y_{B}$ into the
analysis, compared to when using only $\zeta$. Complexity of turbulence in
velocity and temperature spectra are quite well predicted through the semi-
empirical model, including $y_{B}$ and $\zeta$ as parameters, for data
acquired on steep slopes. For mountain top locations, however, the spectral
model has only limited ability. The cause of this might stem from the use of
double rotation, which might incorrectly assign the velocity variances
compared to the gravity vector especially under conditions of flow separation.
In addition, other factors not taken into account might play a role, such as
for example streamline curvature.

In the end, we explored the contribution of the mixed layer scaling to the
understanding of the largest scales of turbulence for velocity spectra, and
found that, as previously observed, including the mixed layer height instead
of measurement height into scaling reduced the low frequency spread of $S_{u}$
and $S_{v}$. This approach, however, still exhibited large scatter and a weak
dependence on anisotropy, and therefore did not outperform the Kaimal scaling
with included anisotropy.

The present study highlights the advantage of examining a wide range of
stability and surface conditions in gaining insights into the complex surface-
layer dynamics. Thus, the clear dependence of temperature spectra on stability
$\zeta$, not observed by [21], is here shown to emerge out of the extensive
stability range probed. The results further highlight that turbulence
anisotropy, quantified through the invariant $y_{B}$, is not only a powerful
tool in expanding flux-variance, flux-gradient and spectral scaling to non-
canonical conditions, but through the modified spectral scaling offers a
deeper understanding of the nature and physical processes driving surface-
layer turbulence, both over canonical and complex surfaces.

## Acknowledgements

The providers of various data sets are acknowledged: the managers of the
database of the Cabauw Experimental Site for Atmospheric Research (CESAR),
UCAR/NCAR - Earth Observing Laboratory (EOL) for the access to the AHATS,
CASES-99, METCRAX II and T-REX data sets, the University of Basel and ETH for
the Riviera dataset, and all the collaborators of Department of Atmospheric
and Cryospheric Sciences (ACINN), who worked for many years to make the
reliable and continuous database of the i-Box study area, and Im Hinteren Eis
station available.

## Conflict of interest

The authors declare no conflict of interest.

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Figures/Graphical_abstract.pdfTurbulence anisotropy has been shown to improve
flux-variance and flux-gradient scaling, extending them to conditions where
the assumptions of Monin-Obukhov similarity theory (MOST) are violated. Our
work expends on these findings to spectral scaling in the unstable surface
layer. Turbulence anisotropy, together with the stability parameter, is able
to explain the low frequency spectral energy and the position of the spectral
peak for velocity and temperature spectra. The novel spectral scaling
elucidates on the nature of anisotropic eddies.