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 e.g., and valley winds e.g.,, to synoptic-scale pressure systems that are responsible for modulations of the mean wind speeds e.g.,. In these examples, an oscillatory large-scale flow can periodically drive turbulence when the flow is at least temporarily hydrodynamically unstable. It is a well-established fact that large-scale flows driven by diurnal or synoptic-scale forcing exhibit time scales of hours to days, respectively, whereas turbulence occurs on the time scales of several minutes down to a few seconds . The scale separation of the forcing and the turbulence motivates the analysis of such temporally developing ABLs.

It has been recognized long ago that turbulent fluctuations govern the mean flow properties, but it is believed that small scales can be universally expressed (parameterized) based on physical relations to the large scales. Based on this assertion, developed a closed statistical description of atmospheric boundary layer turbulence which is known as the Monin–Obhukov similarity theory (MOST). MOST captures a number of mean field effects, in particular the mean temperature profile under near-neutral conditions, but it also fails in various respects, like an accurate account of the near-surface wind speed magnitude and direction e.g.,. Previous and ongoing research has been devoted to improving the MOST parameterization schemes. This has been accomplished with some success for distinguished flow regimes by an improved account of stability, wind shear, and turbulence phenomenology e.g.,. More recent approaches aim to develop parameterizations based on measured data by solving an inverse problem e.g.,. However, the dynamical variability of the ABL represented by the scatter in the observations e.g., can not yet be satisfactorily utilized for numerical simulations of ABLs which constitutes a fundamental lack in modeling.

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 , laboratory experiments , and theoretical analysis . These studies have revealed that linear and nonlinear dynamics are of equal importance for the boundary layer properties since the flow is periodically stabilizing and destabilizing. Numerical models of the ESBL require predictive capabilities and an account of variable flow properties. This challenge is addressed here by utilization of a reduced-order stochastic approach, known as the one-dimensional turbulence (ODT) model . ODT aims to capture the interplay of turbulent advection, Coriolis forces, and molecular diffusion by evolving momentary flow profiles in response to a prescribed boundary conditions or a bulk forcing mechanism. The model has been applied as standalone tool in order model and better understand momentary and asymptotic properties . Furthermore, ODT has already been utilized in LES as wall model and as a universal subgrid-scale model . These applications greatly profit from a comprehensive evaluation of the standalone model. Recently, demonstrated for a convective ABL how unrepresented variability in the vicinity of the surface can limit the predictive capabilities of a large-eddy simulation (LES). Results obtained by LES with a surface-layer parameterization scheme based on MOST suffered from a significantly delayed onset of the convective instability. The erroneous behavior was remedied by replacement of the surface paramerization with ODT.

The rest of this paper is organized as follows. In Sect.  the flow configuration is described with an account of the laminar solution and a brief outline of the simulated cases. In Sect.  an overview of the ODT modeling strategy is presented. In Sect.  ODT simulation results are shown and compared with reference data. Last, Sect.  closes with some concluding remarks.

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 e.g.,.

Figure a shows a sketch of the idealized planar boundary layer configuration considered in this study. A Newtonian fluid resides over a smooth planar surface, which is located at z=0. Both the fluid and the surface are placed in a rotating frame of reference that revolves around the vertical (wall-normal) z coordinate with uniform angular velocity Ω. A flow is driven relative to the rotating frame of reference by prescribed sinusoidal surface oscillations along the streamwise x direction with velocity amplitude U and angular frequency ω. Leading-order Coriolis effects induce flows in the spanwise y direction. The flow is resolved across the height H of the vertically oriented columnar computational domain (“ODT line”) as detailed below. H reaches up into the geostrophic free stream region that is taken at rest relative to the frame of reference. Here, H/δE=360, as defined below, has been selected. Note that the set-up is complementary to the pressure-driven flow considered by , which is a vertically mirrored version of the laboratory experiment investigated in , and the planar analog of the curved walls considered by .

Next, for the three ESBL cases mentioned above, the emerging multiscale properties are briefly discussed based on the laminar solution of the f-plane equations. This excludes turbulence effects but already hints at key properties of the solution, providing reference length scales for the boundary layer flow. The laminar solution is given by for a tidally driven flow, which is similar to who consider a periodically changing pressure gradient and who utilize wall oscillations as forcing mechanism. An extended solution for arbitrary wall oscillation frequencies and with an account of leading-order wall-curvature effects is given in . The latter formulation is adopted for the present study but applied to planar surface geometry only. The laminar solution is not repeated here. Instead, only the laminar boundary layer length scales are dicussed below.

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 D. (Experiments have provided evidence that this holds also for turbulent flow conditions for various forcing frequencies, .) The laminar solution yields four different boundary layer thicknesses, each of which can serve as reference scale D. The length scales are given by the laminar Stokes boundary layer (SBL) thickness δS=2ν/ω, the laminar Ekman boundary layer (EBL) thickness δE=2ν/f, and the two nested layer thicknesses δ±=2ν/|f±ω|, where f is the Coriolis parameter that is taken as f=2Ω for a notional polar boundary layer, and ν is the kinematic viscosity of the fluid. The relevance of the selected value depends on the forcing as it governs the flow regime.

Figure b shows the dependence of the laminar boundary layers on the normalized forcing frequency σ=ω/f. The Ekman property dominates for small forcing frequencies (σ≪1) so that D approaches the finite value δE from below. The Stokes property dominates for high forcing frequencies (σ≫1) due to which D→δS, which is asymptotically smaller than the Ekman layer thickness by the factor σ-1/2. This scaling suggests a vanishing boundary layer thickness, D≃δS→0, for asymptotically large forcing frequencies. This is more a hypothetical scenario since the wall oscillation aims to model slow processes on the large scales. The limit signalizes that molecular (viscous) surface fluxes are no longer able to respond to the forcing and the forcing only results in high energy dissipation in the vicinity of the surface.

Last, for forcing frequencies comparable to the Coriolis frequency (σ≃1) it can be observed that δ+ interpolates between δE and δS, whereas δ- exhibits an unbounded growth as σ→1. This signalizes a breakdown of the f-plane approximation at the Ekman layer resonance, which manifests itself by an “eruption” of the boundary layer thickness . Note that the diagram of length scales is further complicated under turbulent flow conditions due to additional turbulent inner and outer layer thicknesses that can likewise act as reference length scales . Nevertheless, also the turbulent simulation cases discussed in the following are denoted by E (forcing in the Ekman regime), NR (near-resonant forcing), and S (forcing in the Stokes regime), and are marked by gray lines in Fig. b. ODT cases NR and S correspond to the reference polar (PL) and mid-lattitude (ML) cases of , who did not investigate low forcing frequencies (E) as this is numerically too expensive.

The one-dimensional turbulence (ODT) model aims to resolve all relevant scales of a turbulent along a single physical coordinate (see for the original model description and for a recent summary). This is made feasible for high Reynolds number flows by modeling turbulent processes by a stochastically sampled sequence of mapping events that punctuate the deterministic advancement due to molecular diffusion, prescribed forcing terms, initial and boundary conditions. The model aims to resolve all scales of the flow along a physical coordinate (“ODT line” in Fig. ) by distinguishing the spatial redistribution of fluid elements due to turbulent advection from mixing of neighboring fluid elements by molecular diffusion.

Adopting the traditional boundary layer approximation on the f-plane , laminar Ekman flows can be described by a reduced version of the Navier–Stokes equations, except when the system is driven at the Coriolis frequency ω=f . noted that the traditional boundary layer approximation is not applicable when the flow becomes turbulent so that the full set of Navier–Stokes equations has to be solved. In order to formulate a feasible and self-contained dimensionally reduced turbulence model nevertheless, the laminar f-plane dynamics are retained whereas all remaining turbulence effects are modeled by a stochastic process . For the neutral ESBL, the ODT governing equations take the form 1∂u∂t+∑teEuδ̃t-te=∂∂zν∂u∂z+fv,2∂v∂t+∑teEvδ̃t-te=∂∂zν∂v∂z-fu,3∂w∂t+∑teEwδ̃t-te=∂∂zν∂w∂z, where t is time, te the stochastically sampled eddy occurrences, δ̃(t-te) the Dirac distribution function, representing a discrete “kick” of nominally infinite rate that becomes a finite “Heaviside jump” upon time integration across te, Ei with i=u,v,w the discrete stochastic eddy events that are described below, z the vertical coordinate, u the streamwise, v the spanwise, and w the vertical (wall-normal) velocity component. Oscillatory no-slip wall-boundary conditions with u=u(t) and v=w=0 are prescribed at z=0 as detailed in Fig. . Zero-flux (homogeneous Neumann) boundary conditions are prescribed for the upper domain boundary located at z=H far from the bottom wall. The deterministic terms in Eqs. ()–() are spatially discretized with a finite-volume method on an adaptive grid and an efficient explicit scheme is used for temporal integration .

Eddy events denoted by Ei model Navier–Stokes turbulence phenomenology along one-dimensional domain. This is accomplished by application of a map M(z) modeling fluid displacement due to turbulent advection and a kernel function K(z)=z-M(z) modeling the effect of pressure fluctuations, 4Ei:ui(z)→uiM(z)+ci(α)K(z), where the index i counts the velocity vector components (ui)=(u,v,w). The measure-preserving property of the selected triplet map addresses physical conservation principles . The coefficient ci in Eq. () controls the strength of the modeled pressure–velocity coupling by an exchange of kinetic energy upon eddy event implementation for details, see. The vector-valued coefficient ci is parameterized by a scalar model parameter 0≤α≤1, which controls the strength of the kinetic energy exchange relative to the maximum possible extractable value. Selecting α=0 means no exchange due to neglect of pressure-related effects, α=1 implies maximized exchange, and α=2/3 introduces the fastest tendency to local isotropy and is usually taken as default value unless there are specific reasons to adjust it . Note that the kinetic energy exchange is conservative so that no additional energy is introduced by turbulent eddy events. This distinguishes ODT from other stochastic approaches that tend to require additional damping terms balancing artificial energy sources e.g.,. This holds for model applications that resolve and do not resolve all flow scales. The latter situation is equivalent to a spatial filtering of the prognostic variables at the grid resolution, which yields additional subgrid-scale stresses . The main effect of these stresses is dissipation at the grid scale due to an extended turbulence cascade so that a turbulent eddy viscosity closure may be utilized in coarse-resolution ODT in complete analogy to LES . For unresolved surface roughness, a modification of the eddy sampling is also needed .

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 l and location z0 is probabilistically accepted with the momentary rate 5τ-1≃C2l-2(Ekin-ZEvp), where Ekin-ZEvp denotes the current available specific energy of the turbulent eddy event and τ the eddy turnover time. The eddy rate τ-1, the shear-extractable kinetic energy Ekin, and the viscous penalty energy Evp are detailed in and given here in the limit of constant-density flow. Very large eddy events that occur rarely in the sampling procedure can have a significant adverse effect on the boundary layer thickness leading to an unphysical “turbulent eruption”. The sampling of such unphysical eddy events has to be avoided. An additional rejection mechanism is needed for neutral boundary layers since potential energy is absent in Eq. () that would otherwise inhibit or drive turbulence due to stratification or onset of the convective instability, respectively . Here, the so-called “two-thirds mechanism” is used analogous to the temporally developing boundary layers investigated by and . Candidate eddy events that do not possess net extractable energy in at least two thirds of their size are rejected even if Eq. () evaluates to a real-valued τ. The guiding physical principle is turbulent scale locality that might otherwise be violated. The model parameters selected for this study are C=6, Z=200, and α=2/3 based on a calibration with statistically steady turbulent Ekman flow of low Reynolds number see.

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.

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 e.g.,.

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 e.g., that the f-plane boundary layer approximation should break down for near-resonant conditions, which nominally calls for a full account of Coriolis terms. In the present ODT formulation, however, only the traditional f-plane approximation is adopted for the deterministic (laminar) evolution. Nontraditional Coriolis effects are not explicitly implemented but presumably implicitly included to some extend in the kernel mechanism that models pressure–velocity couplings within the turbulent eddy event formulation. Either way, no artificial damping terms were used which hints at physical consistency and robustness of the modeling approach. This signalizes that the standalone formulation can be used as supplementary or advanced research tool wherever single-column modeling is applicable. Furthermore, the results obtained provide confidence for subgrid-scale ODT formulations by consolidating the recently reported improvement of the temporal evolution of a convective ABL within LES that utlizes ODT as a wall model .