It is well known that the representation of certain atmospheric conditions in climate and weather models can still suffer from the limited grid resolution that is facilitated by modern-day computer systems. Herein we study a simple one-dimensional analogy to those models by using a single-column model description of the atmosphere. The model employs an adaptive Cartesian mesh that applies a high-resolution mesh only when and where it is required. The so-called adaptive-grid model is described, and we report our findings obtained for tests to evaluate the representation of the atmospheric boundary layer, based on the first two GEWEX ABL Study (GABLS) inter-comparison cases. The analysis shows that the adaptive-grid algorithm is indeed able to dynamically coarsen and refine the numerical grid whilst maintaining an accurate solution. This is an interesting result as in reality, transitional dynamics (e.g. due to the diurnal cycle or due to changing synoptic conditions) are the rule rather than the exception.

Single-column models (SCMs) are often used as the building
blocks for global (or general) circulation models (GCMs). As such, many of
the lessons learned from SCM development can be inherited by GCMs and hence
the evaluations of SCMs receive considerable attention by the geoscientific
model development community

The discussion of limited grid resolution is present in many studies of SCMs
and GCMs. A prominent example is the nocturnal cumulus-cloud case

Over the years, the computational resources that are available to run
computer models have increased considerably

We test our model with the well-established cases defined for the first two
Global Energy and Water cycle EXchanges (GEWEX) ABL Study (GABLS)
inter-comparison projects for SCMs. As part of the GEWEX modelling and
prediction panel, GABLS was initiated in 2001 to improve our understanding of the
atmospheric boundary layer processes and their representation in models.
Based on observations during field campaigns, a variety of model cases has
been designed and studied using both LES and SCMs with a large set of models
using traditional static-grid structures. An overview of the results and
their interpretation for the first three inter-comparison cases are presented
in the work of

This paper is organized as follows. The present SCM is discussed in more detail in Sect.

As we focus on the merits of grid adaptivity in this study on SCMs and not on
the state-of-the-art closures for the vertical transport phenomena, we have
opted to employ simple and well-known descriptions for the turbulent
transport processes. More specifically, the present model uses a
stability-dependent, first-order, local

The authors of this work realize that there have been considerable
advancements in the representation of mixing under unstable conditions in the
past decades, e.g non-local mixing

For time integration, we recognize a reaction–diffusion-type equation
describing the evolution of the horizontal wind components and scalar fields
such as the virtual potential temperature and specific humidity (

The most prominent feature of the SCM presented in this work is that it
adaptively coarsens and refines the grid resolution based on the evolution of
the solution itself. As mentioned in the introduction, the associated
grid-adaptation algorithm is the same as described in

Apart from the imperfect representation of the physical aspects of a system
in numerical models, additional errors naturally arise due to the spatial and
temporal discretization. In general, a finer resolution corresponds to a more
accurate solution and a simulation result is considered to be “converged”
when the numerically obtained solution and the statistics of interest do not
crucially depend on the chosen resolution. The aim of the grid-adaptation
algorithm is to dynamically coarsen and refine the mesh so that the errors
due to the spatial discretization remain within limited bounds and are

For the cases in this work that focus on the ABL (i.e. in Sects.

The exact formulation of the methods are described at the online locations of the definition files for the different cases presented in this paper.

The wall-clock times are evaluated using a single core (processor model: Intel i7-6700 HQ).

Grid adaptation is carried out each time step. The tree-based
anisotropic-grid structure in Basilisk facilitates a convenient basis for the
multi-resolution analysis and the subsequent refinement and coarsening of
cells at integer levels of refinement. This entails that the spatial
resolution can vary by factors of 2

Apart from the Ekman-spiral case in Sect.

Before we focus our attention on cases that concern the ABL, this section
discusses the philosophy of the grid-adaptation strategy used based on the
analysis of a one-dimensional (1-D)

The locally evaluated error in the numerical solutions for

The locally evaluated error in the numerical solutions for

The scaling characteristics for the laminar Ekman-spiral case. Panel

The spatial-convergence properties for the equidistant-grid solver are
studied by iteratively decreasing the (equidistant) mesh-element sizes used (

Finally, the global convergence characteristics and the speed performance of
the two approaches are studied. The global error (

The following sections are devoted to testing the adaptive-grid approach in a
more applied SCM scenario, where the turbulent transport closures are applied
(see Sect.

Time-averaged profiles over the ninth hour of the run according to
the GABLS1 inter-comparison scenario. For

Evolution of

The first GABLS inter-comparison case focusses on the representation of a
stable boundary layer. Its scenario was inspired by the LES study of an ABL
over the Arctic Sea by

Comparison of the results obtained with the adaptive-grid SCM and
the participating models in the work of

Vertical profiles of the wind-speed magnitude

Evolution of

Due to the idealizations in the case set-up with respect to the reality of
the field observations, the model results were not compared to the
experimental data

Note that in general, results are of course sensitive to the closure chosen
to parameterize the turbulent transport, in our case given by
Eqs. (

The second GABLS model inter-comparison case was designed to study the model
representation of the ABL over the course of two consecutive diurnal cycles.
The case is set up after the observations that were collected on 23 and
24 October 1999 during the CASES-99 field experiment in Leon, Kansas,
USA

In this section we place our model output in the context of the results
presented in the work of

In this work we have
presented a one-dimensional (1-D) single-column model (SCM) that employs a
mesh whose resolution is varied adaptively based on the evolution of the
numerically obtained solution. This is an attractive feature because it is a
prerequisite to enable the computational effort required for the evaluation
of numerical solution to scale with the complexity of the studied physical
system. The adaptation algorithm based on limiting discretization errors
appears to function very well for a wide variety of geophysical applications,
e.g. 3-D atmospheric turbulence-resolving models

The present work does not include an in-depth assessment and discussion on
the performance of the presented methods in relation to the computational
speed. Even though this is an important motivation for the application of the
adaptive-grid strategy to GCMs, the authors argue that an SCM is not suitable
for speed-performance testing: the speed of single-column calculations is
virtually never a critical issue. Only in 3-D mode, when SCMs are “stitched
together” to enable the resolving of global circulations does the model's
computational efficiency become an issue. Furthermore, the performance of an SCM that employs a few tens of cells is not a good indicator for the
performance of a GCM that can employ billions of grid cells. For the latter,
parallel computation overhead and the so-called memory bottleneck are
important aspects. In contrast, for the SCM case, the complete instruction
set and solution data can typically be loaded onto the cache memory of a
single CPU's core. Nevertheless, for the readers' reference, the required run
times for the different SCM set-ups presented herein are listed in
Table

Following the turbulence-resolving study of

Basilisk is a freely available (GPLv3), multi-purpose
tool to solve partial differential equations and has its own website:

All authors contributed to the content of this paper, and it should be viewed as a fruit of the many discussions we have had. Furthermore, SP wrote the Basilisk code; he also designed and implemented the grid-adaptation algorithm. The numerical experiments were set up and performed by JAvH. The writing was led by JAvH and organized in an iterative procedure with BJHvdW.

The authors declare that they have no conflict of interest.

The authors gratefully acknowledge the funding by the ERC Consolidator grant
(648666). The LES ensemble results used for the GABLS1 inter-comparison are
kindly made available by Bob Beare (online via