Atmospheric boundary layers (ABLs) exhibit transient processes on various time scales that range from a few days down to seconds, with a scale separation of the large-scale forcing and the small-scale turbulent response. One of the standing challenges in modeling and simulation of ABLs is a physically based representation of complex multiscale boundary layer dynamics. In this study, an idealized time-dependent ABL, the so-called Ekman–Stokes boundary layer (ESBL), is considered as a simple model for the near-surface flow in the mid latitudes and polar regions. The ESBL is driven by a prescribed temporal modulation of the bulk–surface velocity difference. A stochastic one-dimensional turbulence (ODT) model is applied to the ESBL as standalone tool that aims to resolve all relevant scales of the flow along a representative vertical coordinate. It is demonstrated by comparison with reference data that ODT is able to capture relevant features of the time-dependent boundary layer flow. The model predicts a parametric enhancement of the bulk–surface coupling in the event of a boundary layer resonance when the flow is forced with the local Coriolis frequency. The latter reproduces leading order effects of the critical latitudes. The model results suggest that the bulk flow decouples from the surface for high forcing frequencies due to a relative increase in detached residual turbulence.

Temporal variability is vast feature of atmospheric boundary layers (ABLs) that manifests itself by the emergence of transient flows.
Prominent examples range from diurnally forced flows, such as sea breezes

It has been recognized long ago

Therefore, the present paper addresses the modeling of an idealized time-dependent ABL that is driven by an oscillatory large-scale forcing, the so-called Ekman–Stokes boundary layer (ESBL).
The ESBL has been investigated previously by numerical simulations

The rest of this paper is organized as follows.
In Sect.

In the following, the ESBL flow is introduced forming an idealized model for atmospheric dynamics in reaction to a periodic forcing.
Here, three idealized cases representative of diurnal forcing are investigated.
The first case is from the Ekman (E) regime that corresponds to a polar boundary layer flow.
The second case is from the Stokes (S) regime that corresponds to an ABL flow in the tropics or subtropics.
The third case is a near-resonant (NR) one that is representative of critical latitudes at which a boundary layer resonance occurs

Figure

Next, for the three ESBL cases mentioned above, the emerging multiscale properties are briefly discussed based on the laminar solution of the

The laminar Ekman–Stokes boundary layer is governed by linear dynamics such that a transient but fully periodic solution is obtained.
The repeated change of the sign of the wall-shear stress provides an alternating momentum source/sink situation that results in a constant effective boundary layer thickness

Figure

Last, for forcing frequencies comparable to the Coriolis frequency (

The one-dimensional turbulence (ODT) model aims to resolve all relevant scales of a turbulent along a single physical coordinate (see

Adopting the traditional boundary layer approximation

Eddy events denoted by

Eddy events are sampled from an unknown probability density function (PDF) that depends on the flow state.
The expensive construction of this PDF is avoided by utilizing a rejection sampling approach that is described next.
A turbulent eddy event of selected size

In this section, the applicability of the ODT model to the ESBL is assessed by statistical analysis that exploits the temporal periodicity of the forcing by application of a phase-average. First, phase-averaged velocity profiles are investigated in order to clarify if a turbulent boundary layer is established. After that, the temporal variability of ESBL turbulence is investigated based on surrogate eddy event statistics.

Figure

Simulated phase-averaged law-of-the-wall plots for the horizontal velocity.
The three ODT cases E, NR, and S (dashed and solid lines) as defined in Fig.

Long-time ODT simulations with statistical gathering over

The observed discrepancy between ODT and DNS can be attributed to an unphysical enhancement of the mixing in the near-surface region as a consequence of an overpredicted turbulent eddy event rate.
This is supported by the fact that the ESBL turbulence is currently decaying such that ODT provides a larger resistance against relaminarization than DNS.
Equation (

Despite these shortcomings of the standalone model formulation, it is remarkable that a simple one-dimensional model is able to capture salient dynamical features of the turbulent ESBL.
Present results demonstrate that the model exhibts dynamical complexity that emerges from its construction and provides predictive capabilities.
The demonstration of this previously assumed model property consolidates the transient LES-ODT predictions of

Figure

In order to separate turbulence in different phases in response to the forcing, a phase-conditioned mean eddy event rate

Figure

At a large forcing frequency (case S), the phase-conditioned eddy event rate has developed into a less uniform distribution than in the case NR.
This is due to less eddy event occurrences due to a reduction of the forcing period which leaves less time for turbulence evolution

The model results demonstrate that additional physical information can be extracted from ODT eddy event sequences as these are a simulation result.
In fact, the spatio-temporal density of eddy events can be viewed as a turbulence indicator that provides a simple (“bare-bone”) representation of turbulence activity.
Such information cannot be obtained from closure-based single-column models.
Only DNS or high-resolution LES can provide similar dynamical details, for instance, in terms of the Bradshaw number as used by

Oscillatory atmospheric flows occur due to large-scale forcings, such as time-dependent pressure gradients on the synoptic scale or diurnal forcings. An accurate representation of the inherently unsteady and nonuniversal boundary layer is important for numerical weather prediction since the near-surface flow governs the bulk–surface coupling. An idealized configuration that facilitates the investigation of fundamental questions in the modeling of such flows is provided by the Ekman–Stokes boundary layer (ESBL). Complex multiscale dynamics can occur that depend on the relative importance of the wind turning (Ekman) and unsteady momentum diffusion (Stokes) effects. Moreover, the ESBL can be globally intermittent, alternating between laminar and turbulent response. The present study demonstrates that a reduced-order stochastic modeling approach based on the one-dimensional turbulence (ODT) model, used as a wall model, is able to capture relevant features of such transient Ekman flows in a cost efficient way.

A standalone ODT formulation has been adopted for the present study of a planar ESBL configuration.
Phase-averaged horizontal velocity profiles and the phase dependence of the turbulent eddy event rate have been investigated.
Both quantities are model predictions that can be physically interpreted.
The results obtained suggest that ODT is able to capture relevant features of the periodic turbulence modulation and intermittent behavior of the ESBL.
This applies to all forcing frequencies investigated, ranging from low frequencies (Ekman regime) to high frequencies (Stokes regime) across the Ekman layer resonance when the flow is forced with the local Coriolis frequency.
A relative enhancement of detached residual turbulence is observed for forcing frequencies larger than the Coriolis parameter, which reduces the bulk–surface coupling in the Stokes regime.
The coupling saturates in the Ekman regime by a parametrically modulated turbulence intensity, but peaks at the boundary layer resonance.
The Stokes regime occurs for diurnally forced flows in the tropics, the Ekman regime in the polar region, and the boundary layer resonance in the mid latitudes at a critical latitude

It is worth to note that the model predictions have been obtained with fixed model parameters that were previously calibrated for a stable Ekman boundary layer.
The latter is remarkable as it was criticized more than once

A “minimal” version of the fully adaptive ODT implementation used for this study is publicly available at the URL

MK wrote the paper, conceptualized the research, designed the numerical experiments, conducted the numerical simulations, and analyzed the data. HS supervised the research, contributed to the conceptualization, and reviewed and edited the manuscript.

The contact author has declared that neither of the authors has any competing interests.

The publicly available version of the fully adaptive ODT code (see above) is a “minimal” version and provided “as-is” without warranty.
The code version used for generation of present results is not yet publicly available, but it is a derivative of the public code that has been extended by Coriolis forces and a buoyant Boussinesq scalar (potential temperature) following

This article is part of the special issue “EMS Annual Meeting: European Conference for Applied Meteorology and Climatology 2022”. It is a result of the EMS Annual Meeting: European Conference for Applied Meteorology and Climatology 2022, Bonn, Germany, 4-9 September 2022. The corresponding presentation was part of session UP1.2: Atmospheric boundary-layer processes and turbulence.

This research is supported by the German Federal Government, the Federal Ministry of Education and Research and the State of Brandenburg within the framework of the joint project EIZ: Energy Innovation Center with funds from the Structural Development Act (Strukturstärkungsgesetz) for coal-mining regions.

This research has been supported by the Bundesministerium für Bildung und Forschung (grant nos. 85056897 and 03SF0693A) and the Brandenburgische Technische Universität Cottbus-Senftenberg (Graduate Research School (Conference Travel Grant)).

This paper was edited by Carlos Román-Cascón and reviewed by two anonymous referees.