Scale‐Aware Evaluation of Complex Mountain Boundary Layer Flow From Observations and Simulations

Abstract

Boundary layers over complex, mountainous terrain are characterized by multi‐scale, complex flow structures, where the characterization of individual flow modes poses a fundamental challenge. We apply the novel multi‐resolution coherent spatio‐temporal scale separation (mrCOSTS) method to LIDAR observations and numerical data of the velocity components of complex mountain boundary‐layer flow. Using three distinct time scales (turbulent scales, mountain boundary layer, and diurnal scales) the underlying physical processes are explored. Furthermore, we identified the dominant flow patterns for each time scale, for example, down‐ and up‐valley flows, cross‐valley vortices, small‐scale turbulence, and large evening transition eddies. Applying mrCOSTS to simulated velocity components enables us to identify how coherent structures and the flow patterns are represented at various mesh sizes in the model. Using mrCOSTS we trivially retrieved complex dynamics that were previously difficult to resolve, enabling a direct, scale‐aware evaluation between the LIDAR observations and model results.

Introduction

The atmospheric boundary layer (ABL) is the lowest part of the atmosphere where the flow is turbulent, directly influenced by the Earth's surface, and responds to surface forcings on timescales of an hour or less (Stull, 1988). Generally speaking, the influence of complex, mountainous terrain on atmospheric processes is mediated through the ABL. However, the classical definition needs to be extended for the mountain boundary layer (MoBL) as “the lowest part of the troposphere that is directly influenced by the mountainous terrain, responds to surface and terrain forcing with timescales of about one to a few hours” (Lehner & Rotach, 2018; Rotach & Holtslag, 2025a), thereby adding “terrain forcing” to Stull (1988)'s “surface forcing.” The MoBL is subject to forcings acting on spatiotemporal scales spanning several orders of magnitude, resulting in multi-scale flows (Duine et al., 2017; Farina & Zardi, 2023; Rucker et al., 2008; Serafin et al., 2018; Weigel & Rotach, 2004; Zardi & Whiteman, 2013), while our understanding of these exchange processes is still incomplete (Rotach et al., 2014).

The majority of our theoretical and analytical tools are better suited for more homogeneous flow conditions. The limitations of theory in the MoBL are well-known (Farina & Zardi, 2023; Rotach et al., 2014; Serafin et al., 2018; Stiperski et al., 2019). A less well-recognized problem, but one that we argue is just as prominent, is the lack of robust tools for analyzing these conditions. Even simple characterizations, such as describing the type of boundary layer, are non-trivial (Babić et al., 2024; Wekker & Kossmann, 2015). Few methods perform well for these sort of multi-scale problems (Baj et al., 2015; Herrera-Meja & Hoyos, 2019; Lapo et al., 2025; Lareau et al., 2024; Pfister et al., 2024; Stawiarski et al., 2015; Träumner et al., 2015; Zeeman, 2021). Thus, objectively characterizing the coherent spatiotemporal processes present in multi-scale flows, such as those in the MoBL, is a fundamental research need.

In numerical models, increasing the horizontal grid spacing over complex terrain does improve the realism of MoBL simulation with improved topography representation (Wagner et al., 2014). However, increasing horizontal grid spacing in real-case simulations through the hectometric range toward the LES range does not necessarily improve model performance (Goger & Dipankar, 2024). A major challenge is representing the multi-scale interactions of the MoBL in high-resolution simulations (Schemann et al., 2020), that is, between the parameterized smaller-scale turbulence and resolved larger-scale mesoscale flows. Cuxart (2015) note that a successful simulation in complex terrain strongly depends on the scale of the phenomenon of interest. For modeling, although “misrepresented scale interactions” are often discussed as a challenge for simulations of boundary-layer flow over complex terrain (Calaf et al., 2023; Goger et al., 2022; Umek et al., 2021), a systematic model validation strategy to identify which scales are actually resolved in the models remains a substantially non-trivial task with effectively no guidance in the literature.

To overcome these challenges, we apply the new multi-resolution coherent spatio-temporal scale separation (mrCOSTS) method (Lapo et al., 2025) to complex boundary-layer flow in a major Alpine valley specifically for the purposes of describing coherent spatiotemporal modes across a range of scales from turbulent to meso-scales. We use LIDAR observations of the wind components (Adler et al., 2021) and numerical data at four horizontal grid spacings (Goger & Dipankar, 2024) for a day when boundary-layer processes dominate.

A scale aware analysis with mrCOSTS is performed to answer the following questions, (a) What are the dominant coherent structures in time and space for complex MoBL flow? (b) Which of the observed coherent structures are simulated by the model?

Data and Methods

Observations and Numerical Model

Our location of interest is the Inn Valley in the Austrian Alps at the i-Box study site (Rotach et al., 2017). In summer 2019, the CROSSINN campaign took place (Adler et al., 2021) to observe MoBL development (Babić et al., 2021, 2024). In this study, we will use data the vertically staring SL88 LIDAR, located at the valley floor (Figure S1 in Supporting Information S1). The vertical profiles of the three wind components ($$\mathbf{u}$$, $$\mathbf{v}$$, $$\mathbf{w}$$; bolded symbols denote a vector in space) were retrieved every 80 s (Adler et al., 2021; Gohm et al., 2021).

We perform numerical simulations with the Icosahedral Nonhydrostatic (ICON) model (Zängl et al., 2015) and employ the same full-physics (Hogan & Bozzo, 2018; Louis, 1979; Schulz & Vogel, 2020; Seifert, 2006), real-case set-up (European Environmental Agency, 2017; FAO/IIASA/ISSCAS/JRC, 2012; NASA/METI/AIST/Japan Spacesystems and U.S./Japan ASTER Science Team, 2009) as Goger and Dipankar (2024). Therefore, we only mention the most important aspects: In a nested set-up (Figure S1 in Supporting Information S1), we employ four domains at $${\Delta }x=1$$ km (spanning the Alps; DX1000), $${\Delta }x=500$$ m (DX500), $${\Delta }x=250$$ m (DX250), and $${\Delta }x=125$$ m (DX125) with 80 vertical levels (Leuenberger et al., 2010). In this work we focus on DX1000 and DX125, but results for all model domains can be found in the SI.

We employ the Smagorinsky closure (Dipankar et al., 2015; Smagorinsky, 1963) which treats turbulence as isotropic and is therefore fully three-dimensional (3D). We deem the fully 3D scheme as suitable for complex topography, because horizontal contributions to turbulence generation (e.g., horizontal shear production), are beneficial for the realistic simulation of the MoBL (Goger et al., 2018, 2019; Rohanizadegan et al., 2025). Still, it has to kept in mind that near-surface turbulence in nature is anisotropic (Stiperski et al., 2019), and anisotropy also has to be implemented in future sub-grid models (Efstathiou et al., 2024).

We choose to analyze 14 September, 2019, one of the case studies of Goger and Dipankar (2024), where thermally-induced flows dominate and model validation suggests satisfactory model performance. For our analysis, we use the vertical profiles from the meteogram output at the valley floor site CS-VF0 as in Omanovic et al. (2024). We receive a vertical profile of the simulated wind components at every time step ($${\Delta }t=10$$ s), which we homogenize to the LIDAR observation frequency to ensure consistent comparison. For analysis the horizontal wind components are rotated into the valley axis ($$u$$ is along valley, $$v$$ across valley), but as the valley axis is already approximately orientated east-west this has only a minor impact.

mrCOSTS

mrCOSTS is an unsuperivsed, data-driven method designed to diagnose multi-scale data (Lapo et al., 2025). A mrCOSTS fit describes the original data using coherent spatiotemporal structures describing narrow bands of time dynamics, $${\tilde{\mathbf{x}}}_{0},{\tilde{\mathbf{x}}}_{1},\text{\ldots },{\tilde{\mathbf{x}}}_{P}$$, with $${\tilde{\mathbf{x}}}_{p}$$ denoting the mrCOSTS approximation of the original data for frequency band $$p$$. Each $${\tilde{\mathbf{x}}}_{p}$$ is reconstructed using a collection of Dynamic Mode Decomposition (DMD) models fit over sliding windows of increasing size for each decomposition level such that 1 $${\tilde{\mathbf{x}}}_{p}(t)=\sum\limits _{(\ell ,j,k)\in p}{\boldsymbol{\phi }}_{k,j,\ell }\,\mathrm{exp}\left({\omega }_{k,{j}_{,}\ell }t\right){b}_{k,j,\ell }.$$ where the decomposition level is $$\ell $$, the time window is $$k$$, the rank is $$j$$, and $$t$$ denotes time. The DMD models collected in Equation 1 are described by an amplitude ($$b$$) scaling a complex vector representing a coherent spatial pattern ($$\boldsymbol{\phi }$$) which is governed by a single set of complex time dynamics ($$\omega $$). $$Im(\omega )$$ corresponds to an oscillation with a single frequency and $$Re(\omega )$$ corresponding to exponential growth or decay. mrCOSTS can be used to fit multi-variate data by finding the coherent structures and time dynamics shared among the variables.

mrCOSTS enables a robust description of data described by nonstationarity, processes occurring across multiple dimensions and scales simultaneously, and invariances such as translation, which are the characteristics of multi-scale data (Lapo et al., 2025). As such, mrCOSTS is well-suited for analyzing MoBL processes. Finally, $$\tilde{\mathbf{x}}$$ approximates the de-noised $$\mathbf{x}$$. See Lapo et al. (2025) and Text S1 in Supporting Information S1 for more details, hyperparameters, and fit evaluations.

Scale Separation and Reynolds Decomposition for Non-Ideal Conditions

Reynolds decomposition separates flows into a mean ($$\overline{x}$$, here taken as the time mean) and fluctuating component ($${x}^{\prime }$$). 2 $$x={x}^{\prime }+\overline{x}.$$ $${x}^{\prime }$$ approximately represents the turbulent fluctuations of a flow for which signals are non-stationary, occur over inhomogeneous conditions, and are corrupted by noise. The separation of the turbulent component of $${x}^{\prime }$$ from other processes is substantially non-trivial for real flows (Baj et al., 2015; Mahrt & Thomas, 2016; Wyngaard, 2010), especially in the MoBL (Falocchi et al., 2019; Stiperski & Rotach, 2016).

The typical way of formulating this scale separation problem is to assume the existence of a single scale at which distinct physical processes can be separated, for instance a spectral gap between the larger meso-scale motions and turbulence. However, the spectral gap is not guaranteed to exist, especially in the MoBL. Approaches better suited to the complexities of the MoBL take advantage of the universal return to isotropy of turbulence at small scales to provide guidance on separating the smallest scales of turbulence from all larger scale motions (Falocchi et al., 2019). Regardless, approaches conditioned for separating at a single scale do not provide guidance for analyzing multi-scale flows, especially in the MoBL where non-turbulent processes at a variety of scales contribute to exchange processes (Lehner & Rotach, 2018; Serafin et al., 2018).

For this reason, we reformulate Reynolds decomposition into a form that better represents non-ideal conditions using a generalization of the decomposition presented in Baj et al. (2015): 3 $$x={x}_{\mathit{turb}}+{x}_{\mathit{submeso}}+{x}_{\mathit{meso}}+{\cdots}+c+\eta .$$ where $${x}_{\mathit{turb}}$$ are the locally turbulent fluctuations in $${x}^{\prime }$$, $${x}_{\mathit{submeso}}+{x}_{\mathit{meso}}+{\cdots}={x}_{\mathit{slow}}$$ are the locally non-turbulent motions with time scales on the order of the decomposition window and are thus shared between $${x}^{\prime }$$ and $$\overline{x}$$, $$c$$ is a background component of the flow that evolves on time scales orders of magnitude slower than the decomposition window, and $$\eta $$ is white noise. Here turbulence should be understand as being turbulent relative to the time scale of the processes in question. This decomposition bears resemblance to triple decomposition (Reynolds & Hussain, 1972), which adds a third term to Reynolds decomposition for a linear wave with a well-defined phase. However, $${x}_{\mathit{slow}}$$ is not limited to such waves for real flows (Baj et al., 2015). Thus, Equation 3 should be understood as a generalized form of triple decomposition.

We can use mrCOSTS to approximate Equation 3 4 $$\mathbf{x}=\sum\limits _{p\in {P}_{\mathit{turb}}}{\tilde{\mathbf{x}}}_{p}+\sum\limits _{p\in {P}_{\mathit{submeso}}}{\tilde{\mathbf{x}}}_{p}+\sum\limits _{p\in {P}_{\mathit{meso}}}{\tilde{\mathbf{x}}}_{p}+{\cdots}+{\tilde{\mathbf{x}}}_{0}+\sum\limits _{0}^{P}{\boldsymbol{{\epsilon}}}_{p}.$$ Here $${\sum }_{p\in {P}_{\mathit{turb}}}{\tilde{\mathbf{x}}}_{p}={\tilde{\mathbf{x}}}_{\mathit{turb}}$$ are the mrCOSTS bands representing turbulent motions from the smallest time scales resolved by mrCOSTS to the largest turbulent scales, $${\sum }_{p\in {P}_{\mathit{submeso}}}{\tilde{\mathbf{x}}}_{p}+{\sum }_{p\in {P}_{\mathit{meso}}}{\tilde{\mathbf{x}}}_{p}+{\cdots}={\tilde{\mathbf{x}}}_{\mathit{slow}}$$ are the mrCOSTS bands representing spatially coherent motions from the smallest non-turbulent scales to the largest time scales resolved by mrCOSTS, $${\tilde{\mathbf{x}}}_{0}$$ is the poorly-resolved background mode, and $${\boldsymbol{{\epsilon}}}_{p}$$ is the error of each band. As long as $${\boldsymbol{{\epsilon}}}_{p}$$ is relatively small, which is the case for the mrCOSTS fits used here (see Supporting Information S1), the mrCOSTS Reynolds decomposition can be used to robustly approximate the coherent structures at an arbitrary scale, resulting in a decomposition that is not constrained by a single decomposition time. The reconstruction using each band is notated as for example, $${\tilde{\mathbf{x}}}_{\mathit{turb}}$$ for the turbulent bands.

A useful property of the mrCOSTS Reynolds decomposition is that each component can be calculated at any spatiotemporal resolution, including the original dimensionality. Thus, one can also derive terms such as variances, covariances, and turbulent kinetic energy ($$\boldsymbol{k}$$) at any desired spatiotemporal resolution, including the observation's or model's native resolution. For instance, one can calculate the contribution of coherent turbulent fluctuations to the variance as 5 $$\sigma (\mathbf{x})={\overline{{\tilde{\mathbf{x}}}_{\mathit{turb}}}}^{2}+{\overline{{\boldsymbol{{\epsilon}}}_{\mathit{turb}}}}^{2}$$ where the overline can refer to average over any time or space scale, including no averaging and $${{\epsilon}}_{\mathit{turb}}$$ is the mrCOSTS error for the turbulent bands. A similar operation can be used for $$\boldsymbol{k}$$ 6 $$\boldsymbol{k}=\frac{1}{2}\left({\overline{{\tilde{\mathbf{u}}}_{\mathit{turb}}}}^{2}+{\overline{{\tilde{\mathbf{v}}}_{\mathit{turb}}}}^{2}+{\overline{{\tilde{\mathbf{w}}}_{\mathit{turb}}}}^{2}+{\overline{{\boldsymbol{{\epsilon}}}_{\mathit{turb}}}}^{2}\right)$$

Defining Coherent Spatiotemporal Bands

Examining individual $${x}_{p}$$ made it apparent that distinct processes were described by a contiguous range of $${x}_{p}$$, motivating the application of Equations 3 and 4. Analyzing the individual or aggregated bands has superficial similarities to scale-separation techniques. However, mrCOSTS objectively finds $${x}_{p}$$ across all scales and thus it is not necessary to identify a specific scale for example, for filtering. Consequently, we explore all the found coherent spatiotemporal modes using the aggregated $${x}_{p}$$.

We define four aggregated bands that appear to describe ranges of $${x}_{p}$$ with similar behavior. These aggregated bands largely corresponded to the literature definitions of specific scales for different physical processes in the MoBL: turbulent scales ($${\tilde{\mathbf{x}}}_{\mathit{turb}}$$), MoBL scales ($${\tilde{\mathbf{x}}}_{\mathit{MoBL}}$$), diurnal scales ($${\tilde{\mathbf{x}}}_{\mathit{diur}}$$), and the background mode ($${\tilde{\mathbf{x}}}_{0}$$). $${\tilde{\mathbf{x}}}_{\mathit{turb}}$$ is the sum of bands with time scales from 100 s (the smallest resolved time scale by mrCOSTS) to 67 min corresponding to the definition of the ABL for flat terrain Stull (1988). $${\tilde{\mathbf{x}}}_{\mathit{MoBL}}$$ corresponds to the definition of the MoBL provided by Lehner and Rotach (2018) with time scales between 67 and 150 min. The largest scales, $${\tilde{\mathbf{x}}}_{\mathit{diur}}$$, the sum of bands with time scale from 150 min to 14.5 hr, and $${\tilde{\mathbf{x}}}_{0}$$, processes longer than 14.5 hr (See Text S1 in Supporting Information S1), clearly describe the thermally-forced valley-wind (VW) system. This choice of aggregation appears justified due to the clearly different processes described in each band as well as the clear correspondence to the known-time scales of interest for the MoBL (Lehner & Rotach, 2018; Rotach & Holtslag, 2025b; Wekker & Kossmann, 2015; Zardi & Whiteman, 2013), but is not strictly necessary as the individual bands could be analyzed as well. When calculating variances with $${\tilde{\mathbf{x}}}_{\mathit{turb}}$$ we rotate the coordinate system into the direction of the non-turbulent component of the flow ($${\tilde{\mathbf{x}}}_{\mathit{MoBL}}+{\tilde{\mathbf{x}}}_{\mathit{diur}}+{\tilde{\mathbf{x}}}_{0}$$) such that the covariance $${u}^{\prime }{w}^{\prime }$$ measures whether vertical transport is strengthening or weakening the mean flow.

Results and Discussion

Overview and Flow Phases

The observed processes are described by a convective boundary layer (CBL) phase from 09:00–14:00 UTC, a VW phase from 14:00–18:00 UTC, and an evening-transition (ET) from 18:00–21:00 UTC. During the CBL-phase, $$\mathbf{u}$$ transitions from being weakly down-valley to weakly up-valley (Figure 1e) and in the VW-phase there is a rapid strengthening of the up-valley wind. Finally, during the ET phase, $$\mathbf{u}$$ weakens. During the CBL-phase, $$\mathbf{w}$$ reveals a sequence of up- and downdrafts, typical for a CBL in the valley (Babić et al., 2024; Serafin & Zardi, 2010; Wekker & Kossmann, 2015) with subsidence aloft (above 250 m a.g.; Figure 1a). $$\mathbf{u}$$ (Figure 1b) remains small (less than 2 m s⁻¹). During the VW-phase, $$\mathbf{w}$$ increases with strong updrafts simultaneous with $$\mathbf{u}$$ strengthening to 10 m s⁻¹. This increase in wind speeds is related to the thermally-induced up-valley flow that occur with approximately a diurnal scale, a common feature at the valley floor at this particular location (Goger & Dipankar, 2024; Lehner et al., 2021). DX125 simulates the wind structure in a realistic way (Figures 1c and 1g), with general subsidence during the CBL-phase and a slightly delayed updraft after 17:00 UTC. The model correctly simulates subsidence (Figure 1d) but with much less intense turbulent structures, and a delayed updraft, placing it nearer the ET.

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Turbulent Scales

We define $${\tilde{\mathbf{w}}}_{\mathit{turb}}^{\mathit{obs}}$$ as the mrCOSTS diagnoses of the LIDAR-observed $$\mathbf{w}$$, indicating the band in the subscript and the data source in the superscript. During the CBL-phase, the structures in both $${\tilde{\mathbf{w}}}_{\mathit{turb}}^{\mathit{obs}}$$ and $${\tilde{\mathbf{u}}}_{\mathit{turb}}^{\mathit{obs}}$$ consist of subsequent eddy perturbations with a vertical extent of up to 400 m a.g.l. with most still being attached to the surface (Figure 1b). While, $${\tilde{\mathbf{u}}}_{\mathit{turb}}^{\mathit{obs}}$$ consists of surface-connected eddies of similar vertical extent as $${\tilde{\mathbf{w}}}_{\mathit{turb}}^{\mathit{obs}}$$, the perturbations were of a much stronger intensity (Figure 1f), with similar patterns in $${\tilde{\mathbf{v}}}_{\mathit{turb}}^{\mathit{obs}}$$ (Figure S8 in Supporting Information S1), indicative of strongly anisotropic eddies. Turbulence during this period is known to be driven by buoyancy (Goger et al., 2018).

The $${\tilde{\mathbf{w}}}_{\mathit{turb}}^{\mathit{DX125}}$$ perturbations did not exist near the surface except for one intense turbulent eddy at 10:30 (Figure 1c), also visible in the vertical velocity variance ($${\sigma }_{w}$$) profile (Figure 1i). Otherwise, the model does not simulate any significant $${\tilde{\mathbf{w}}}_{\mathit{turb}}^{\mathit{DX125}}$$ (Figure 1g), completely disagreeing with the observations. This lack of surface turbulence is also evident in $${\tilde{u}}_{\mathit{turb}}^{\mathit{DX125}}$$ for both the time time-space plots (Figures 1f and 1h) and the variance profiles, for example, the observed $${\sigma }_{u}$$ profiles before noon have a clear maximum corresponding to turbulent eddies and the model does not show any maximum at all (Figure 1j).

During the VW-phase, $${\tilde{\mathbf{w}}}_{\mathit{turb}}^{\mathit{obs}}$$ are largely still influenced by the surface while also becoming taller (up to 1,000 m a.g.l.; Figure 1b). The model simulates - in good agreement with the observations—the same increase in eddy size during the VW phase, which can be assessed both visually using $${\tilde{\mathbf{w}}}_{\mathit{turb}}^{\mathit{DX125}}$$ (Figure 1c) and the profiles of $${\sigma }_{w}$$ (Figure 1i). However, the simulated eddies are not attached to the surface, for instance as seen in the disagreement in $${\sigma }_{w}$$ below 400 m, suggesting that the perturbations aloft are not surface-driven, but are instead likely related to the larger-scale up-valley wind arriving in the valley. An overall similar situation exists for $${\tilde{\mathbf{u}}}_{\mathit{turb}}^{\mathit{DX125}}$$.

In contrast to the increase in eddy-scale suggested by $${\tilde{\mathbf{w}}}_{\mathit{turb}}^{\mathit{obs}}$$, the $${\boldsymbol{k}}_{\mathit{turb}}^{\mathit{obs}}$$ structure highlights that the height of the eddies in the CBL- and VW-phases remains effectively constant. Instead, during the VW-phase a distinct two layer structure emerges. The intensity of the surface-connected eddies increases (Figure 1k) while eddies in the horizontal direction remain vertically constrained, as shown by $${\tilde{\mathbf{u}}}_{\mathit{turb}}^{\mathit{obs}}$$ (Figure 1f). The elevated turbulence appears to be largely disconnected from the surface (Figure 1k), supporting the view that this elevated turbulence is driven by the up-valley wind.

Using covariance calculated from $${\tilde{\mathbf{u}}}_{\mathit{turb}}^{\mathit{obs}}{\tilde{\mathbf{w}}}_{\mathit{turb}}^{\mathit{obs}}$$ we can see directly that a small number of eddies contribute the majority of the momentum transport, meaning it is not necessary to capture every eddy in order to describe the transport of momentum, but rather just these most energetic eddies. However, the model does not capture these eddies.

Finally, the flow is strongly anisotropic at all turbulent scales resolved by mrCOSTS. $${\tilde{\mathbf{u}}}_{\mathit{turb}}^{\mathit{obs}}$$ and $${\tilde{\mathbf{v}}}_{\mathit{turb}}^{\mathit{obs}}$$ are much larger than $${\tilde{\mathbf{w}}}_{\mathit{turb}}^{\mathit{obs}}$$ (c.f. Figures S7f, S8f, and S9f in Supporting Information S1), likely as a result of the channeling effect of the valley, violating the isotropy assumption of the turbulence scheme, likely contributing to the difficulty of the model in representing the turbulent fluctuations.

The profiles of $${\sigma }_{u}$$ and $${\sigma }_{w}$$ show that both $${\tilde{\mathbf{u}}}_{\mathit{turb}}^{\mathit{obs}}$$ and $${\tilde{\mathbf{w}}}_{\mathit{turb}}^{\mathit{obs}}$$ captured the turbulent fluctuations, with strong agreement between $$\sigma $$ derived from the detrended observations and from $${\tilde{\mathbf{x}}}_{\mathit{turb}}$$ (Figures 1i and 1j). There is a slight underestimation of $$\sigma $$ from $${\tilde{\mathbf{w}}}_{\mathit{turb}}^{\mathit{obs}}$$ and $${\tilde{\mathbf{u}}}_{\mathit{turb}}^{\mathit{obs}}$$, however. We know that $${\tilde{\mathbf{x}}}_{\mathit{turb}}$$ is predisposed to underestimate $$\sigma $$ due to unfit processes while estimates of $${\mathbf{x}}^{\prime }$$ are predisposed to overestimate $$\sigma $$ due to the inclusion of non-turbulent processes and noise (Section 2.3). The main disagreements in $$\sigma $$ occurred when $$\mathbf{w}$$ and $$\mathbf{u}$$ exhibited strong non-stationary behavior and linear detrending would be insufficient for removing $${x}_{\mathit{slow}}$$ in Equation 3 (e.g., 16:00 UTC for $${\sigma }_{w}$$ and 17:00 UTC for $${\sigma }_{u}$$), a limitation not shared by $${\tilde{\mathbf{x}}}_{\mathit{turb}}^{\mathit{obs}}$$. As the two methods otherwise agree, we can therefore conclude that $${\tilde{\mathbf{x}}}_{\mathit{turb}}$$ can provide a more robust representation of turbulent fluctuations than the standard approach of estimating variances.

Mountain Boundary Layer (MoBL) Scales

A faint growth of MoBL eddies in $${\tilde{w}}_{\mathit{MoBL}}^{\mathit{obs}}$$ is apparent, although similar structures are not present in $${\tilde{u}}_{\mathit{MoBL}}^{\mathit{obs}}$$ or $${\tilde{v}}_{\mathit{MoBL}}^{\mathit{obs}}$$ (Figures 2a–2c).

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Other than the growth of MoBL-scale eddies in $${\tilde{\mathbf{w}}}_{\mathit{MoBL}}^{\mathit{obs}}$$ (Figure 2a), motions at this scale lacked a connection to the surface (Figures 2b and 2c), suggesting that they are not directly related to surface forcing. The exception are the dominant structures close to the surface with a vertical extent up to 300 m a.g. in $${\tilde{\mathbf{v}}}_{\mathit{MoBL}}^{\mathit{obs}}$$ from 15:00 UTC until 19:00 UTC (Figure 1c). These structures are related to the cross-valley vortex (CVV), a secondary circulation establishing from the along-valley flow due to the valley's curvature at this particular location (Babić et al., 2021). At a height of 800 m, the return circulation of the CVV is visible in $${\tilde{\mathbf{v}}}_{\mathit{MoBL}}^{\mathit{obs}}$$, suggesting that mrCOSTS is able to disentangle the secondary circulations from the mean flow.

The MoBL-scale eddies during the CBL-phase are not captured by the model at both horizontal grid spacings (DX125 and DX1000, Figures 2d and 2g). Although weak structures are visible in the decomposed vertical velocities of the DX125 run (Figure 2d), they are not coupled to the surface as the decomposition of the observations suggest (Figure 2a). In contrast, the MoBL structures with a vertical extent of up to 800 m a.g.l. during the VW-phase are present in the DX125 run both in size and strength, while the DX1000 simulation still completely misses the large up- and downdrafts during the VW's ET. The $${\tilde{\mathbf{u}}}_{\mathit{MoBL}}^{\mathit{DX125}}$$ and $${\tilde{\mathbf{v}}}_{\mathit{MoBL}}^{\mathit{DX125}}$$ reveal that the DX125 run simulates both the $${\tilde{\mathbf{u}}}_{\mathit{MoBL}}^{\mathit{obs}}$$ and $${\tilde{\mathbf{v}}}_{\mathit{MoBL}}^{\mathit{obs}}$$ well, including the CVV close to the surface after 14:00 UTC (Figure 2f). In the DX1000 simulation, the CVV is also present, while its structures are weaker and delayed.

Valley Winds at Diurnal Scales

Processes at these time scales are related to the diurnal cycle and build-up of the plain-to-mountain circulation. This process, also known as Alpine pumping, occurs due to differential heating between the mountain range and the surrounding foreland (Zardi & Whiteman, 2013), resulting in a pressure gradient and consequently flow toward the mountains (up-valley winds) in the daytime and outward flow toward the surrounding plain (down-valley winds) in the nighttime.

$${\tilde{\mathbf{w}}}_{\mathit{diur}}^{\mathit{obs}}$$ largely consists of subsidence during the CBL-phase (Figure 3a), while during the VW-phase one dominant large updraft spanning up to 800 m forms. Both $${\mathbf{u}}_{\mathit{diur}}^{\mathit{obs}}$$ and $${\mathbf{v}}_{\mathit{diur}}^{\mathit{obs}}$$ contain the expected VW dynamics, with a rapid transition to a strong up-valley component during the VW-phase (Figure 3). $${\mathbf{u}}_{\mathit{diur}}^{\mathit{obs}}$$ spans the entire observed column, while $${\mathbf{v}}_{\mathit{diur}}^{\mathit{obs}}$$ is of a smaller vertical extent. The model domains capture this general trend, but with a notably weaker up-valley component that has a limited vertical extent and a delayed ET, visible in all three wind components.

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To fully characterize the effect of the VW system we must consider the combined diurnal and background scales, as the background scales contain a relevant component of the VW system (Figure S8 in Supporting Information S1) that would require a longer study period to fully characterize (Lapo et al., 2025). The up-valley wind generates strong shear characterized by a positive peak near the surface and a secondary, negative peak aloft (Figure 3m). This double-peaked pattern of shear observed during the VW-phase likely drives the intensification of turbulence below 400 m that interacts with the surface while the secondary peak aloft likely contributes to the turbulence above 400 m that is not connected to the surface (Figure 1k). By underestimating the strength of the up-valley wind near the surface (Figure 3n), the models are underestimating the strong wind shear characterizing the VW-phase (Figure 3m).

Summary and Conclusions

We applied the novel mrCOSTS method to multi-scale observational and numerical data of the MoBL in a major Alpine Valley. We decomposed the multi-scale flow into bands of coherent spatiotemporal structures, which we aggregated to three time scales of boundary-layer development (Figure 4):

• Turbulent scales: During the classical CBL development before noon, anisotropic turbulent surface eddies form, while turbulence intensifies during the afternoon VW phase and an elevated, shear-induced turbulence layer forms.

• MoBL scales: MoBL-scale eddies continuously grow until 14:00, while the afternoon is dominated by the up-valley wind and the associated CVV.

• Diurnal scales: Before noon, the valley floor is mostly dominated by subsidence, while one large up- and downdraft dominates during the ET.

• The conceptual diagram of the morning CBL (Figure 4b) is consistent with existing conceptual understandings (Wekker & Kossmann, 2015), however, the more complex shear-driven MoBL lacks such a clear conceptual picture. Figure 4c summarizes how these conditions are characterized by complex, co-existing, and interacting flow features that are otherwise hard to disentangle without a method like mrCOSTS.

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To conclude, we posit that objectively analyzing multi-scale flow features, such as those found in the MoBL, is a major research challenge. In that regard, we demonstrated mrCOSTS’ use as a novel “scale-aware” evaluation method, both for diagnosing observations and for evaluating numerical models. We identified several “co-existing” boundary layers directly from a single data source (LIDAR observations), demonstrating how mrCOSTS can be used to decompose flows consisting of co-existing structures at a variety of scales which cannot easily be viewed as “separate” entities. This ability is in contrast to the more typical scale separation approaches which assume a single relevant scale of separation. Still, it has to be kept in mind that our analysis is representative for the valley floor during a case study. To explore the full potential of the mrCOSTS method for process understanding this analysis would need to be extended the MoBL more broadly. Such a broad-range dataset over complex terrain will be available after the TEAMx campaign (Rotach et al., 2022), and mrCOSTS is an excellent method to continue the scale-aware analysis of complex processes over mountainous terrain.

Acknowledgments

BG and AD are supported by the EXCLAIM project, funded by ETH Zurich. The computational results presented have been achieved using resources from the Swiss National Supercomputing Centre (CSCS) under project ID d121. KL was funded by the Austrian Science Fund (FWF) [10.55776/ESP214]. We thank the two anonymous reviewers for their comments that led to the improvement of the manuscript. Open access publishing facilitated by Eidgenossische Technische Hochschule Zurich, as part of the Wiley - Eidgenossische Technische Hochschule Zurich agreement via the Consortium Of Swiss Academic Libraries.

Data Availability Statement

The observational data were retrieved from Gohm et al. (2021). We employed the ICON model version 2.6.5; the open-source code can be retrieved at DWD et al. (2024). The mrCOSTS analysis was performed using the PyDMD package v2025.04.01 (Demo et al., 2018; Ichinaga et al., 2024). All processed data and analysis code are available at Lapo and Goger (2025) with the MIT license.

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