Small-scale processes in atmospheric boundary layers are typically not resolved due to cost constraints but modeled based on physical relations with the resolved scales, neglecting expensive backscatter. This lack in modeling is addressed in the present study with the aid of the one-dimensional turbulence (ODT) model. ODT is applied as stand-alone column model to numerically investigate stratification effects in long-lived transient Ekman flows as canonical example of polar boundary layers by resolving turbulent winds and fluctuating temperature profiles on all relevant scales of the flow. We first calibrate the adjustable model parameters for neutral cases based on the surface drag law which yields slightly different optimal model set-ups for finite low and moderate Reynolds numbers. For the stably stratified cases, previously calibrated parameters are kept fixed and the model predictions are compared with various reference numerical simulations and also observations by an exploitation of boundary layer similarity. ODT reasonably captures the temporally developing flow for various prescribed stratification profiles, but fails to fully capture the near-surface laminarization by remaining longer in a fully developed turbulent state, which suggests preferential applicability to high-Reynolds-number flow regimes. Nevertheless, the model suggests that large near-surface turbulence scales are primarily affected by the developing stratification due to scale-selective buoyancy damping which agrees with the literature. The variability of the wind-turning angle represented by the ensemble of stratified cases simulated covers a wider range than reference reanalysis data. The present study suggests that the vertical-column ODT formulation that is highly resolved in space and time can help to accurately represent multi-physics boundary-layer and subgrid-scale processes, offering new opportunities for analysis of very stable polar boundary layer and atmospheric chemistry applications.

Atmospheric boundary layers are almost always turbulent so that they exhibit transient transport processes on a range of scales

A standing challenge, therefore, is the robust numerical representation and prediction of boundary-layer scaling properties that requires representation of transient and intermittent turbulent processes

The main goal of the present paper lies in the investigation of stratification-dependent turbulence modifications and long-time evolution of transient stable boundary layers.
Long-lived stable boundary layers occur in the polar regions, where measurements and fine-structure resolving simulations are sparse due to various technical difficulties.
In light of recent measurement efforts by means of the MOSAiC expedition

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

Stable stratification may entirely suppress turbulence.
In atmospheric flows, this effect is frequently observed for the nocturnal boundary layer.
Under dry conditions, strong near-surface stratification can develop as consequence of radiative surface cooling in combination with diffusive heat transfer to the air

Figure

Sketch of the stratified Ekman flow configuration investigated with application of the ODT model.
A warm (temperature

The initially neutral Ekman flow is fully developed and reaches an approximately statistically stationary state.
The initial condition exhibits thermal equilibrium

The steady laminar solution can be analytically obtained by solving the Navier-Stokes or ODT equations under absence of turbulence for the non-zero horizontal velocity components.
This solution is given by e.g.,

Based on the selected case set-up and the governing equations, stratified Ekman flow cases are described by the following nondimensional control parameters,

This section gives a brief overview of the stochastic ODT model and its application to stratified Ekman flow.
The general model set-up and details on the treatment of the Coriolis forces is given in

The one-dimensional turbulence (ODT) model

ODT has been applied to various types of flows ranging from free-shear flows and jets

For the flow configuration sketched in Fig.

The governing equations are the conservation equations of mass, momentum, and energy plus an equation of state.
Here we make use of the Oberbeck–Boussinesq approximation with a linear equation of state,

A sequence of ODT eddy events aims to reproduce the statistical properties and variability of the corresponding turbulent flow.
As a result of the oversampling and rejection algorithm summarized above, eddy events are implemented at a local time-varying rate governed by the frequency

Equation (

In the following, the adjustable model parameters

The ODT model parameters

For the stable atmospheric boundary layer,

In the present study, ODT is used in “DNS mode” in which all relevant flow scales, down to the viscous scales, are resolved along a 1-D physical coordinate.
Viscous suppression, hence, bounds the physically meaningful turbulence scales from below based on energetic considerations.
For

Eddy events are randomly sampled by oversampling of candidate eddies followed by a rejection step in which the eddy acceptance probability is evaluated using Eq. (

We note that

Parameterization coefficients for the turbulent drag over a smooth surface.

The surface drag law denotes the

Next, we present the case of neutral Ekman flow for the purpose of model calibration.
Figure

For a smooth surface, reference values of

To summarize, model predictions shown in Fig.

Details of the ODT simulation cases for neutral Ekman flow with comparison to reference DNS.

Inputs that characterize the ODT case set-up are given above the middle line.
Outputs that characterize the flow state are given below the middle line for ODT and corresponding reference DNS for ^{®} Xeon^{®} E5-2630 (

With the model calibrations at hand, we proceed by analyzing the boundary layer structure for neutral conditions. This is important because the fully developed neutral flow will be used below as initial condition for the stratified cases.

Figure

Two locations corresponding to layer heights (length scales) are marked by symbols and are given for orientation.
One is the turbulent boundary layer thickness

For completeness, but without detailed discussion, Figs.

We proceed by assuming statistically stationary Ekman flow and analyze the turbulent fluctuations resolved by the model.
Further insight into the dynamics is given by the turbulence kinetic energy (TKE) budget, which is discussed next.
Figure

Note that

Altogether, the ODT model is able to capture relevant features of the temporal-averaged mean state and the corresponding velocity fluctuations in turbulent Ekman boundary layers under neutral conditions. With the calibrated model set-ups, we move on by investigating how the flow interacts with a prescribed initial stratification profile.

In this section we investigate stratification effects with ODT as stand-alone flow model using ensemble-averaged and instantaneous property profiles for various

The case set-ups and characterizing bulk quantities are summarized in Table

Overview of the transient stratified cases simulated with ODT.

Case prefixes denote weak (W), intermediate (I), and strong (S) stratification regimes.
The asterisk (

Figures

Next, the profiles of the momentary geostrophic velocity component

An interesting dynamical feature is the persistent inertial oscillation that is exhibited by

Hovmöller diagrams of various observables showing the temporal evolution of the momentum and thermal boundary layer for a single ODT realization of the weakly stratified case W001 with

Same as Fig.

The inertial oscillations are oblique in the reference LES from

In the present ODT application, the forcing is entirely due to the uniform geostrophic bulk flow, but the origin of the inertial oscillation seems to correlate with the onset of the surface cooling at

Last, Figs.

Below we will come back to selected aspects of the presented time series that will serve to analyze the simulation results.

The sequence of ODT eddy events encodes physical information on turbulence spatial scales in the flow under consideration.
We analyze the participating length scales by the joint probability density function (JPDF) of ODT eddies using the stochastically sampled eddy size

Figures

Joint probability density function (JPDF) of eddy size and midpoint for

In the following, we focus on the ensemble-averaged property profiles of the temperature,

Figure

The latter is a numerically robust definition that will, for at least weakly turbulent cases and not too long simulations times, yield a value on the order of the height of the inversion.

Synopsis of various simulated vertical property profiles.
ODT results (solid and broken lines) at

The velocity profiles shown in Fig.

The inversion exhibited by the ensemble-averaged temperature profile and the “wiggly” nonmonotonic region of the instantaneous temperature profile are likewise reduced in height for stronger stratification.
We attribute this to a reduction in mixing efficiency, which reduces together with the range of turbulence scales in the flow.
In particular the stratification-induced suppression of large eddy events as shown by the eddy JPDFs in Fig.

In Fig.

Last, Fig.

Figure

Hodographs of the ensemble-averaged horizontal velocity

Next, Fig.

Probability density functions (PDFs) of the wind turning angle

The value range covered by the differently stratified ODT cases shown in Fig.

In order to backup the above statement on the model's resistance against leaving the fully developed turbulent state, Fig.

Velocity boundary layer after two inertial periods (

Finally, we assess the developing stratification itself as it affects also

Vertical profiles of the normalized buoyancy frequency

Figure

Note, however, that the buoyancy frequency recovers for

Turbulent Ekman flow over a smooth surface and its reaction to developing stratification due to a sudden surface cooling is investigated as canonical problem for polar and nocturnal boundary layers over land or ice.
Here, the stochastic one-dimensional turbulence (ODT) model has been used as numerical tool in order to explore stratification effects and their effects on turbulence scales, including flow statistics.
Various observables have been considered encompassing instantaneous and ensemble-averaged velocity and temperature profiles, bulk quantities, derived and surrogate variables.
The emphasis is on the wind turning effect and its variability which are both numerically challenging, requiring high-fidelity models.
Wind-turning effects are intimately related to cross-isobaric mass transport, which has been reported to be underestimated in numerical circulation models as demonstrated recently by reanalysis data of the mid latitudes

In the present study, the ODT model aims to resolve all relevant scales of the turbulent flow but only along a vertical coordinate for a columnar computational domain.
Molecular diffusion and Coriolis forces are directly resolved, whereas the effects of turbulent stirring motions are modeled by a stochastic sequence of discrete eddy events that punctuate the deterministic diffusive flow evolution.
Two adjustable model parameters were calibrated for neutral conditions utilizing reference data for the surface drag law.
Either the friction velocity or the surface wind-turning angle can be matched by two different sets of model parameters.
Model parameter adjustment is advisable when the flow regime changes from low

The model application to the stratified cases was performed with fixed model parameters.
Present results demonstrate that ODT is able to very reasonably capture the structure of the viscous and thermal boundary layer for weak and strong, but not for intermediate stratification.
We assert that this is by construction and the model is unable to accurately capture the laminar-turbulent transition except when changing from a fully laminar to a fully turbulent flow regime.
An analysis of the turbulence spatial scales based on model surrogate data and conventional flow statistics revealed that ODT prefers fully-developed turbulence already at low Reynolds numbers.
The stratification is initially localized at the surface and propagates upwards interacting with neutral Ekman flow turbulence.
The model suggests that the flow reacts by energetically cutting-off first the large-scale near-surface eddies responsible for outer layer turbulence due to which the inversion and near-surface jet lowers for stronger stratification.
By contrast, small-scale near-surface eddies in the well-mixed turbulent surface layer remain almost unaffected.
This is consistent with flow physics

Furthermore, we have shown that, albeit ODT is able to qualitatively capture surface wind-turning effects for various stratified conditions (

A “minimal” version of the fully adaptive ODT implementation used for this study is publicly available at the following 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 provided “as-is” without warranty. The code version used for generation of the results presented here is not yet publicly available. It is a derivative of the public code that has been extended by including Coriolis forces and a buoyant Boussinesq scalar (potential temperature) following previous studies as cited in the text. Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “21st EMS Annual Meeting – virtual: European Conference for Applied Meteorology and Climatology 2021”.

Marten Klein and Heiko Schmidt thank Roland E. Maier for assistance in the processing of numerical data.

This research has been supported by the BTU Graduate Research School (Conference Travel Grant).

This paper was edited by Gert-Jan Steeneveld and reviewed by Bert Holtslag and one anonymous referee.