Open boundary conditions were developed for atmospheric large-eddy simulation (LES) models and implemented into the Dutch Atmospheric Large-Eddy Simulation model. The implementation was tested in a “Big Brother”-like setup, in which the simulation with open boundary conditions was forced by an identical control simulation with periodic boundary conditions. The results show that the open boundary implementation has minimal influence on the solution. Both the mean state and the turbulent structures are close to the control simulation, and disturbances at the in- and outflow boundaries are negligible. To emulate a setup in which the LES is coupled to a coarser model, the influence of coarse boundary input was tested by smoothing the output of the periodic control simulation both temporally and spatially before feeding it as input to the simulation with open boundary conditions. When smoothing is applied over larger spatial and longer temporal scales, disturbances start to form at the inflow boundary and an area exists where turbulence needs to develop. Adding synthetic turbulence to the smoothed input reduces the size of this area and the magnitude of the disturbances.

Large-eddy simulation (LES) is a numerical simulation tool used to study turbulent motions in the atmospheric boundary layer (ABL). Employing resolutions ranging from 1–100

With the increase in computational power, the use of LES has shifted from idealized cases to more complex and realistic scenarios. Some examples are the simulation of urban areas

There is no consensus on the “best” implementation of open boundary conditions for anelastic turbulent flow. In 1991 two mini symposia were unsuccessfully dedicated to this topic, and the effort was summarized as a frustrating one

Summary of the open boundary implementations in the mentioned LES models.

The implementation of open LBCs make it possible to nest LES within both itself and mesoscale models

In this research we develop a set of open LBCs for anelastic LES and implement them in the Dutch Atmospheric Large-Eddy Simulation (DALES) model. The goal of the paper is threefold. First, we will give a clear and extensive description of the open LBCs developed in this research. Second, we will show the influence of the LBCs on the mean fields and turbulent characteristics. Third, we will see how, in an idealized setup, the results depend on the temporal and spatial resolution of the input data, as one would encounter when embedding the LES in a coarser, non-turbulence-resolving LAM. The LBCs are developed to minimize reflections and the area needed for turbulence to develop and to allow for potential future one-way nesting with coarser LAMs. To minimize reflections, the outflow boundary conditions will be based on the radiation boundary condition of

This section will describe the implementation of the open boundary conditions in the Dutch Atmospheric Large-Eddy Simulation (DALES) model

The boundary condition for the boundary-normal velocity components depends on whether the cell is an in- or outflow cell. An inflow cell for the boundary-normal velocity component is defined as

The outflow boundary condition is based on the Sommerfeld radiation boundary condition

For inflow cells, the boundary-normal velocity at the boundary

The use of radiation boundary conditions means that continuity is not guaranteed and a correction factor,

The input boundary-normal velocity components integrated over the lateral and top boundaries

The lateral and top boundaries are subdivided into patches

A 2D illustration of a nested setup in which the integration length scales are set to the grid size of the parent model. In this setup, the mass flux through a parent cell (blue) at the boundary of the child model (brown) is conserved, while the child model is free to generate turbulence on smaller scales.

The role of the correction term is to conserve mass integrated over the domain such that the pressure solver, which needs to find a solution that conserves mass locally, can find a solution. It is possible to implement the tendency from the correction factor as a non-homogeneous Neumann boundary condition for the modified pressure

At the moment, the vertical length scale of the integration patch is fixed to the vertical grid resolution. This allows for a straightforward implementation when using stretched vertical grids. We have also experimented with setting the vertical length scale of the integration patch to the domain height. This couples the boundary layer with the column above the inversion layer and gave unwanted results. In the future, the implementation can be extended to allow for a variable vertical integration length scale as well.

This section will discuss the boundary conditions for the cell-centred variables and the tangential velocity components. These variables are not computed at the boundary. Instead, ghost cells are used together with a second-order central discretization to determine the behaviour of the variable at the boundary. The implementation is different for in- and outflow boundaries. For the cell-centred variables and tangential velocity components, a boundary is defined as inflow if

For outflow cells, homogeneous Neumann conditions, Eq. (

For inflow boundaries, Dirichlet boundary conditions are a common choice

To derive the inflow boundary condition, we assume that advection is the only process taking place at the boundary.

To investigate the potential of synthetic turbulence in reducing the turbulence spinup area, the random flow generation (RFG) algorithm of

The test case setup used in this research is summarized in Fig.

Illustration of the simulation setup. The solid blue rectangles show the different simulations and the sections in which their results are analysed.

The simulation case used in the test setup is the development of a dry convective boundary layer. This case is well understood, and DALES is known to produce realistic results

Setup parameters for the reference case. From left to right; grid spacing, domain size, integration time step, surface heat flux, surface momentum flux and geostrophic wind forcing.

Evolution of the periodic reference case from initial profiles to the end of simulation (6 h). Left to right; slab-averaged potential temperature, slab-averaged east–west velocity, and slab-averaged resolved and total heat flux, slab-averaged east–west velocity variance.

The Big Brother-like experiment, as was first proposed by

Settings of the open boundary implementation for the sensitivity runs. The default settings are in bold.

In practice, the open boundary conditions will often be used to couple the LES to a coarser-resolution model, such as a mesoscale weather model. To study the impact of coarse-resolution (in space and time) boundary data, the periodic output is smoothed with a Gaussian filter before it is used to force the open boundary simulation. The simulation with open boundary conditions is repeated for different degrees of spatial and temporal smoothing. This setup emulates a one-way nesting setup and moves from the LES being nested in a turbulence-resolving model to a non-turbulence-resolving model. It also allows us to study the influence of resolution ratios between the parent and child model in a nested setup for both the spatial and the temporal resolutions. Since the smoothed fields come from the same model with the same model physics, resolution and subgrid parameterizations, any differences between the results of the simulation with the smoothed input and the reference (periodic) simulation must be caused by the boundary implementation and the smoothing. Comparison to the case without smoothing allows us to see the influence of smoothing, which relates to the resolution of and the turbulent scales present in the emulated parent model.

Different techniques exist to artificially add turbulence or increase the turbulent scales present in coarse data. To demonstrate the potential of one such technique, the synthetic turbulence algorithm of

This section will describe the results of the test case described in Sect.

In this section the results of the Big Brother experiment are shown. In this setup the periodic boundary output is input into the simulation with open boundary conditions at the same spatial and temporal resolution. This setup allows us to investigate the definition and implementation of the boundary conditions. Any disturbances present in the simulation with open boundary conditions must be a direct result of the boundary implementation, as the periodic simulation supplies “perfect” boundary fields. It is a first necessary test that needs to be passed. The challenging areas are mainly the outflow (east) boundary and the north and south boundaries. At the outflow boundary, fields should leave the domain unperturbed and the area affected by reflections upstream of the outflow boundary should be minimal. The north and south boundaries are both in- and outflow boundaries and will therefore challenge the capability of the boundary conditions to switch from in- to outflow in time and space. The results from the simulation with open boundary conditions are compared to the reference case with periodic boundary conditions. We would like the mean field and the turbulence properties such as the length scales and energy distribution to be unaffected by the numerics of the boundary condition implementation. The two simulations do not have to match from a deterministic point of view, as the chaotic nature of the system will result in different placement of eddies between both simulations.

To investigate the sensitivity of the solution to the parameters of the open boundary implementation, the simulation is repeated for different sets of parameters. Each of the parameters is individually perturbed around the default values. The parameters and their values are shown in Table

At and above the inversion height, the simulation with a larger timescale for the Robin inflow conditions,

For the integration length scale

Sensitivity analysis for the open boundary implementation parameters. Slab average profiles for simulations that have parameters perturbed around a default configuration (Table

Figures

Horizontal cross-section of the potential temperature perturbation with respect to the periodic slab average at a height of

Vertical cross-section of the potential temperature for the periodic simulation

A more quantitative comparison of the influence of the open boundary conditions on the magnitude of the turbulent perturbations is obtained by calculating

TKE profile derived from the cross-wind direction for the periodic

To further quantify the differences between the simulations, we vertically integrate the TKE over the boundary layer (Fig.

TKE integrated over the boundary layer. The dashed lines show the mean and the mean

A wavelet analysis of the potential temperature field is used to quantify the influence of the open boundary conditions on the power spectrum of the turbulence. Figure

Wavelet analysis of the potential temperature at a height of

From Figs.

This section will show and discuss the results of the smoothed-input simulations for different degrees of horizontal and temporal smoothing. This setup emulates the situation where the outer model provides boundary fields at a coarser spatial and/or temporal resolution than the LES. The panels in Figs.

Horizontal cross-section of the potential temperature perturbations with respect to the periodic simulation at a height of

Vertical cross-section of the potential temperature (similar to Fig.

TKE cross-section derived from the cross-wind direction (similar to Fig.

Figure

Once again, the results are quantified further by vertically integrating the TKE over the boundary layer along the cross-section for all simulations (Fig.

TKE integrated over the boundary layer. Each line represents one of the simulations from Fig.

The wavelet analysis for the smoothed-input simulations is shown in Fig.

Wavelet analysis of the potential temperature at a height of

Horizontal cross-section of the potential temperature perturbations with respect to the periodic simulation at a height of

Vertical cross-section of the potential temperature for different degrees of smoothing (similar to Fig.

TKE cross-section derived from the cross-wind direction for different degrees of smoothing (similar to Fig.

The results analysed in Figs.

The previous section has highlighted significant issues at the inflow boundary when the boundary values are smoothed in space and/or time, resulting in a more laminar flow near that boundary. A potential approach to reduce these issues

TKE integrated over the boundary layer. Each line represents one of the simulations from Fig.

Figure

A quantitative comparison once again using the vertical integral of TKE over the boundary layer (Fig.

Figure

Wavelet analysis of the potential temperature at a height of

The results analysed in Figs.

This paper introduced an open boundary implementation for atmospheric large-eddy simulation models that was implemented in the Dutch Atmospheric Large-Eddy Simulation (DALES) model. The goal of this research was to give a detailed description of the implementation, investigate its performance, and show the influence of open boundary conditions and boundary input on the solution.

Radiation boundary conditions were implemented as an outflow condition for the boundary-normal velocity components at the lateral and top boundaries. At the top boundary, buoyancy was also taken into account, which negated the need to add a sponge layer in the upper parts of the domain. Neumann conditions were used for the other variables at outflow boundaries. For inflow boundaries a Robin boundary condition was derived for the cell-centred variables and tangential velocity components to allow for a smooth transition between in- and outflow boundaries, and a nudging condition was implemented for the boundary-normal velocity components.

Using a Big Brother-like setup, where a simulation with open boundary conditions was forced by an identical control simulation with periodic boundary conditions on the same spatial and temporal resolution, it was shown that the influence of the boundary implementation on the solution was minimal. Slab-averaged profiles showed that the mean profiles are conserved. Furthermore, cross-sections of the potential temperature field showed that the turbulent input data were communicated well through the inflow boundary and that the turbulent fields left the domain without reflections or perturbations at the outflow boundary. Cross-wind turbulent kinetic energy cross-sections showed that the energy in the turbulent perturbations were the same in the simulation with open boundary conditions and the control simulation with periodic boundary conditions. The energy spectrum of the perturbations was also unchanged, which was shown with a wavelet analysis.

To investigate the influence of the spatial and temporal resolution of the input data, the output of the periodic simulation was smoothed before feeding it to the simulation with open boundary conditions. Different degrees of spatial and temporal smoothing showed that a mismatch between input turbulent scales and model scales results in the generation of wavelike disturbances downstream of the inflow boundary. The disturbances grow in size and magnitude when the ratio between input and model scales grows. The lack of turbulence in the input data also results in an area of reduced turbulent kinetic energy downstream of the inflow boundary, where there is no developed turbulence. This area grew as the smoothing increased. For large degrees of smoothing it was found that the turbulent energy overshoots before settling to values similar to the periodic control simulation. For these reasons, it is advised to be careful when coupling a large-eddy simulation model with open boundary conditions to a coarser model. Repeated nesting can be used and is currently being explored to step down in multiple steps from coarse data to the desired resolution. The results of this research indicate that the refinement factor when nesting should not exceed

The potential of adding synthetic turbulence to the LBCs was explored, and the results show that it can help to reduce the disturbances found in size and magnitude and to speed up the process of obtaining developed turbulence by artificially reducing the gap between the input turbulent scales and model scales. The strong wavelike character of the disturbances were removed, and the length of the inflow area required for turbulence to develop was reduced. The disturbances and development area also became less dependent on the degree of smoothing, and the development area is given by the ratio of the advective and convective velocity scales. However, if possible, we would still advise keeping the spatial and temporal ratios between the input data and the LES below the earlier-mentioned values.

In summary, the implementation of open BCs described in this study provides a suitable framework for further investigating the use of the DALES model in “nested” mode. This provides a major advance in its utility as a science tool, as it increases its applicability to problems for which periodic BCs have strong limitations, such as over heterogeneous terrain. Spatial and temporal averaging of the boundary values, as is typical of embedding an LES into coarser-resolution mesoscale models, causes the results to deteriorate. The smoothing effects are much larger than those from the implementation of the open BCs themselves. Some of the deterioration can be overcome by adding synthetic turbulence at the inflow boundaries.

The current version of DALES is available from the project website –

FLL conceptualized the paper, did the formal analysis and visualization, implemented the methodology and software, and wrote and edited the draft. CJ and APS supervised during the project, reviewed the draft and supported in the conceptualization. FJ supported the software implementation and reviewed the draft.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We acknowledge the use of ECMWF's computing and archive facilities; the funding provided by the Australian Research Council Centre of Excellence for Climate Extremes (CE170100023); and the support of the Ruisdael Observatory, scientific research infrastructure which is (partly) financed by the Dutch Research Council (NWO, grant no. 184.034.015)

This research is funded by the Australian Research Council Centre of Excellence for Climate Extremes (grant no. CE170100023) and by the Ruisdael Observatory scientific research infrastructure, which is (partly) financed by the Dutch Research Council (NWO, grant no. 184.034.015).

This paper was edited by Sylwester Arabas and reviewed by two anonymous referees.