We examine the influence of the grid aspect ratio of horizontal to
vertical grid spacing on turbulence in the planetary boundary layer
(PBL) in a large-eddy simulation (LES). In order to clarify
and distinguish them from other artificial effects caused by
numerical schemes, we used a fully compressible meteorological LES
model with a fully explicit scheme of temporal integration. The
influences are investigated with a series of sensitivity tests with
parameter sweeps of spatial resolution and grid aspect ratio. We
confirmed that the mixing length of the eddy viscosity and diffusion
due to sub-grid-scale turbulence plays an essential role in
reproducing the theoretical

In meteorological simulations, the grid aspect ratio

Atmospheric phenomena have a wide range in terms of both time and space. One
of the initial targets of the studies involving meteorological numerical
simulations was the planetary wave. In this case, the horizontal scale, at

If we turn our attention to mesoscale phenomena, whose horizontal scales are

In the atmosphere, there are many smaller scale variabilities than
associated with cumulus clouds. For example, the spatial scale of
the dominant motion in the unstable planetary boundary layer (PBL)
is smaller than 1

LES is a vital tool to explicitly represent such small-scale variabilities.
It is designed to resolve turbulence at scales down to the inertial sub-range
by parameterizing sub-grid-scale (SGS) turbulence based on the theory of
energy cascades. LES for meteorological simulations has been used
since the 1970s

We present an overview of the model used in this study in
Sect.

The model used in this study is SCALE-LES
(Scalable Computing for Advanced Library and Environment-LES;

Detailed descriptions of the model, including the discretization method,
which are not directly related to the topic of this study, are given in the
appendices as follows: the governing equations are in
Appendix

Since the purpose of this study is to clarify the impact of the grid aspect
ratio on the turbulent aspects, the influences of the approximations to the
governing equations should be reduced as much as possible. For this reason,
we employ the set of fully compressible non-hydrostatic equations as the
governing equations. The three-dimensional momentum (

Notation of symbols.

The central difference schemes are used for spatial discretization; the second-order scheme is applied to the pressure gradient terms in the momentum equations and divergence terms of mass flux in the continuity equation, while the fourth-order scheme is applied to the advection terms in the momentum and thermodynamics equations. The reason the advection terms are discretized by the high-order scheme is based on the accuracy of the eddy viscosity and the diffusion terms representing the effect of SGS turbulence. The coefficients of the viscosity and diffusion terms are proportional to the square of the grid spacing, so that the magnitude of the terms would be comparable to the truncation error of the advection terms, in terms of order of accuracy, if the second-order scheme is employed. Additionally, the higher-order treatment for the advection terms is necessary from a different viewpoint as well. Since the advective term is a non-linear convolution, it requires a higher-order treatment to resolve the additional modes. The use of the lower-order scheme is justified by the scale separation of the fast modes (acoustic and fast gravity waves) and slow modes (advection). In the meteorological phenomena, the terms of the pressure gradient in the momentum equations and the divergence in the continuity equation are dominant for the fast modes, while the advection term is dominant for the slow modes. The interaction between the fast and slow modes is not significant generally. If we consider SGS mixing in a local field such as a several-grid scale, the fast waves would pass over this field soon before completing the SGS mixing. This means that fast waves do not participate much in the local mixing, compared with the mixing process itself.

The even-order schemes require the explicit numerical filter for numerical
stability. However, they have the advantage that it is easier to evaluate the
effect of the numerical diffusivity using the explicit numerical filter than
that introduced implicitly by an odd-order scheme. See
Appendices

A fully explicit scheme, i.e., HE–VE (horizontally explicit and vertically
explicit) scheme, is used for temporal integration in this study. This scheme
generally has less implicit numerical diffusion than implicit schemes such as
HI–VI (horizontally implicit and vertically implicit) and HE–VI (horizontally
explicit and vertically implicit) schemes. To focus on the influences of the
grid aspect ratio, an explicit scheme is more suitable than implicit schemes.
Additionally, we do not use a time-split scheme to avoid numerical damping of
the time splitting. See Appendix

In order to validate the dynamical core of this model, we performed a density
current experiment

The model for effect of SGS turbulence used in this model is
a Smagorinsky–Lilly-type model

The role of the sub-grid model is to parameterize the effect of SGS
turbulence based on the energy cascade theory of three-dimensional isotropic
turbulence. The eddy viscosity and diffusion model is employed as a sub-grid
model to represent the effect. For the determination of the amount of the
energy cascade, the mixing length of the eddy viscosity and diffusion is the
most important factor. The coefficient of the SGS eddy viscosity,

The mixing length depends on what type of spatial filter we employ on the
variables in the equations, and the length scale of the spatial filter is the
essential factor for the mixing length. The spatial filtering is inevitable
in the discretization of the equations. The spatial filter in a numerical
model is implicitly determined by the grid spacing and discretization
schemes. The artificial length characterizing the spatial filter owing to
discretization is defined as the filter length. Besides the filter length,
the shape of the grid should be also considered in the mixing length
determination. Its effect can be represented by the grid aspect ratio.

The effect of the static stability on the mixing length is introduced in this
model according to

There are two ambiguous factors in the configuration for determining the
mixing length in recent meteorological LES models. One problem is the
configuration of the filter length in Eq. (

Such rough treatment of the filter length and no consideration of the grid
aspect ratio lead to incorrect turbulent properties. A usual remedy for such
effects is a posteriori tuning of the Smagorinsky constant,

Here, we should note that each theory is based on its own basic concepts.
Although this means that different theoretical concepts lead to different
values for the constants, we should keep in mind that a certain constant is
uniquely determined by a particular model on a theoretical basis. The
Smagorinsky constant is derived by integration of the kinetic energy filtered
out by the spatial filter. If the cutoff filter is employed as the spatial
filter, the integration can be performed in the cubic

Let us return to the first problem, i.e., the determination of the spatial
filter length

The second problem is the treatment of the grid aspect ratio in LES in actual
grid systems. In the equilibrium condition with the universal Kolmogorov
spectrum, the energy flux cascaded into the SGS variability is equal to the
SGS dissipation. Since the dissipation does not depend on the artificial grid
configuration but on the physical configuration, energy cascaded to the SGS
turbulence should not depend on grid configuration, including the grid aspect
ratio. The energy cascade flux, which is equal to the dissipation, can be
written as a function of the strain tensor

We performed three PBL experiments to examine the influences of the
grid aspect ratio and filter length on simulated turbulence, which
are summarized in Table

In the three experiments above, a systematic parameter sweep of resolution
and grid aspect ratios was conducted. The spatial resolution of each run
performed in the control experiment is shown in
Table

Filter lengths and the effect of grid aspect ratio on mixing length in the three experiments performed in this study.

In all the runs, the domain size is 9.6

For the initial condition, we follow

The temporal integration is done for 4 hours for each run. In order to
focus on our purpose, we simplify the setting of the bottom boundary
condition so that a constant heat flux of 200

Different temporal intervals are used for the dynamical and physical
processes. We define the dynamical process as that related to fluid dynamics;
the advection, pressure gradient, and gravitational force terms in the
governing equations are treated as dynamical processes in this model. The
other processes are physical processes in this model. In this study, the
physical processes are only the surface flux for the momentum and the eddy
viscosity and diffusion for SGS turbulence. Note that we treat the eddy
viscosity and diffusion terms, which originated from the advection term, as
the physical process for correspondence with the sub-grid turbulence model in
RANS (Raynolds-averaged Navier-Stokes) mode in this model. The intervals for the dynamical process

Names of runs we performed in the control experiment. Columns and rows correspond to vertical and horizontal resolutions, respectively. The number following the character “AR” represents the grid aspect ratio.

We used an eighth-order numerical filter, i.e.,

Most sub-grid models are based on the idea of an energy cascade due to
three-dimensional isotropic turbulence so that the energy spectrum has the
slope of

Energy spectrum three-dimensional velocity at height of
500

On the other hand, the spectrum of horizontal kinetic energy

Same as Fig.

Horizontal cross section of the vertical velocity at

In most LES models, the filter length for the sub-grid model is set to be
equal to the grid spacing itself. In our model, the filter length is set to
be twice the grid spacing, as described in Sect.

The deviation from the theoretical

Dependency of the index of energy pile defined in the text on the
grid aspect ratio for runs in the

Next, we consider the influence of

In all the experiments, the magnitude of the SEP tends to be smaller for the larger-vertical resolution. This tendency is more apparent for the larger aspect ratio. It is possible that the amount of energy dissipation depends on the grid configuration, although it should be identical. The larger dissipation could result in smaller total energy and consequently a smaller SEP in the coarser resolution runs.

Vertical plots of horizontal mean

In this subsection, we show the influence of the grid configuration of the
runs on the turbulence statistics in the PBL in the control experiment.
Figure

Figure

Figure

Scatter diagrams between some quantities and grid configurations:

The variance of the vertical velocity is shown in
Fig.

The vertical profiles against

Theoretically, the variance of vertical velocity should converge to a certain
value with grid refinement for the following reason. The variance is
approximately equal to the sum of squares of each wave number component as

The profile of skewness shows an almost linear slope in the PBL except near
the surface, as shown in Fig.

Residuals from logarithmic linear regression of the skewness in the
horizontal resolution are relatively larger for coarser horizontal
resolution: e.g., the 5mAR20 (cyan square) and the 30mAR5 (green circle)
runs. Dependency of the skewness on the grid aspect ratio is one of the
reasons for the residuals. The dependency on the aspect ratio can be seen in
Fig.

We conducted a series of planetary boundary layer experiments to examine the influences of the aspect ratio of horizontal to vertical grid spacing on the atmospheric turbulence in a large-eddy simulation. In order to focus on the influences, we tried to avoid artificial effects as much as possible by employing the fully compressible governing equations. A fully explicit (i.e., HE–VE) temporal integration scheme and a high-order central difference scheme for spatial differentials are adopted to reduce implicitly introduced numerical viscosity and diffusion by discretization.

In the model used in this paper, we considered the effect of spatial filter
length and grid aspect ratio on the mixing length of the eddy viscosity and
diffusion, which is a parameterization of the energy cascade to SGS
variability. Explicit numerical hyperviscosity and diffusion is introduced to
reduce the two-grid-scale noise in this model. This can be considered
a spatial filter in LES. As a result, the filter length in this model is
double the grid spacing, while the grid spacing is used as the length in most
LESs. The effect of the grid aspect ratio on the mixing length proposed by

If we use a reasonable filter length and introduce the grid aspect ratio
effect, the energy spectra of all the runs show good correspondence with the
theoretical

The effect of the grid aspect ratio on the mixing length of the eddy viscosity and diffusion cannot be ignored. If the mixing length is not modified by the grid aspect ratio, spurious energy also piles at higher wave numbers because of an insufficient energy cascade to SGS turbulence at large grid aspect ratios. In previous studies, the Smagorinsky constant was often modified as a tuning parameter to obtain the expected energy spectrum. However, the constant should be determined theoretically and should not be tuned to make the energy cascade large instead of considering filter length and grid aspect ratio.

The horizontal resolution and grid aspect ratio also influence the turbulence statistics. The vertical profiles of several turbulence statistics depend on the grid configuration as follows. The depth of the unstable surface layer is larger for coarser horizontal resolution (but not vertical resolution). The variance of the resolved vertical velocity essentially depends on the horizontal resolution, mainly because of spatial averaging by discretization. The skewness of the vertical velocity shows dependency on grid aspect ratio as well as on horizontal resolution. In particular, the skewness is sensitive to the grid configuration around the top and bottom of the PBL. It becomes larger and smaller around the top and bottom, respectively, for coarser resolution and larger grid aspect ratio. Higher resolution and smaller grid aspect ratio are required in order to obtain accurate skewness. This is important, because the spurious strong upward wind near the PBL top, which is implied by larger skewness, would have a large effect on cloud microphysical processes. For example, the reproducibility of clouds around the PBL top, such as stratocumulus, would be sensitive to grid configuration.

We conclude that the grid aspect ratio influences the parameterization of the energy cascade to SGS variability and the reproducibility of skewness of turbulence in the PBL. Although there are many meteorological LESs in which the grid aspect ratio is large, such large grid aspect ratios have led to misinterpretation in some experiments in previous studies because of spurious energy piling at higher wave numbers and stronger vertical motion indicated by larger skewness. The aspect ratio should be taken into account properly in determining the mixing length of the eddy viscosity and diffusion as a sub-grid model for the reliability of simulations of boundary layer turbulence.

In this subsection, we introduce the governing equations for the prognostic
variables (

The continuity equations for each material can be described in flux
form:

The momentum equations for the gas, liquid, and solid materials are described
as

The equations of internal energies are given as

The sum of Eqs. (

Equations (

Equation (

The potential temperature for dry air, which is defined as

Using Eqs. (

From Eqs. (

The pressure expression is derived diagnostically as

Figure

Vertical profile of

Changing the prognostic variable for thermodynamics from the internal energy
to the newly defined potential temperature

The governing equations are summarized as follows:

The vertical boundary conditions are that the vertical velocities at the top
and bottom boundaries are zero. This causes the vertical flux at the top and
bottom boundaries for all the prognostic variables to be zero:

We conceptually separate the complete set of governing equations into
dynamical and physical parts:

According to this scheme, the dynamical processes can be written as

A Runge–Kutta (RK) scheme with three steps is used as the temporal
integration scheme for the dynamical processes. The RK scheme with three
steps used in this model is defined as

The advantage of this scheme compared with ordinary three-step schemes is the
reduction of memory load and storage, which is one of the most expensive
components of recent computers with low byte per flop (B

Time–height section of horizontal mean vertical velocity in runs
where

The acoustic wave is the fastest mode in the dynamical processes, and the temporal interval for dynamical processes must be less than the grid spacing divided by the speed of the acoustic wave, to satisfy the Courant–Friedrichs–Lewy (CFL) condition. However, the timescale of the physical processes is usually much longer than the interval, so the temporal interval for the physical processes can be much longer than for the dynamical processes. We use a larger temporal interval to calculate the tendencies of the physical processes than of the dynamical processes. We call the time step for the physical processes a large time step, and for dynamical processes, a small time step.

Traditionally, tendencies in physical processes are calculated with large time steps, and the prognostic variables are updated with the Euler scheme with the tendency for large time steps. This causes an artificial acoustic wave, as described below. In this model, the tendency in some physical processes, such as surface flux (sensible heat flux and latent heat flux) and eddy viscosity and diffusion, are calculated with large time steps in the same way, but they are added to the prognostic variables with the tendency of dynamical processes in small time steps.

Figure

In addition, such a wave radiating periodically from a fixed location can
cause a spurious stationary wave in the simulation output. This could lead to
misinterpretation of the simulation results, although this problem is not
directly based on physics or modeling. In most practical cases, the temporal
interval for historical output (

The artificial acoustic wave and resulting spurious stationary wave can be
avoided if the prognostic variables are updated with the tendency calculated
in the physical processes with a small time step, although the tendency is
calculated with a large time step. Figure

We employ the Arakawa-C staggered grid. Central difference schemes are used
for the spatial differential for the dynamical processes, because waves such
as gravity waves and acoustic waves generally cannot keep their spatial
symmetry with odd-order schemes. Based on a consideration of numerical
stability (see Appendix

Before the discretization of the differential equations, we diagnose several quantities from the prognostic variables.

Divergence in the continuity equation is calculated with the second-order
central difference scheme. The continuity equation is discretized as

The advection terms and pressure gradient term are calculated with the
fourth-
and second-order central difference schemes, respectively. The momentum
equation is discretized as

The thermodynamics equation is discretized as

The tracer advection process is done with the Euler scheme after the temporal
integration of the dynamical variables (

With the condition of no source

In order to satisfy the monotonicity of tracer advection, we employ the flux-corrected transport (FCT) scheme, which is a hybrid scheme with
a higher-order difference scheme and first-order upwind scheme

If the fourth-order central difference is applied,

Equation (

The tentative values are calculated using the low-order flux:

Allowable maximum and minimum values are calculated:

Several values for the flux limiter are calculated:

The flux limiters at the cell wall are calculated:

The boundary condition at the top and bottom boundaries is

For other prognostic variables, the vertical fluxes at the top and bottom boundaries are zero, except those from physical processes.

We impose an explicit numerical filter using the numerical viscosity and diffusion. Although the filter is necessary for numerical stability, too strong a filter could dampen down any physically meaningful variability. In this subsection, we describe the numerical filters used in this model, and discuss the strength of the filter.

In order to damp down the higher wave number component selectively, we adopt the
hyperviscosity and diffusion in the traditional way. The hyperviscosity and
diffusion of the

The Laplacian of

The hyperviscosity and diffusion can be discretized as

The coefficient,

For the numerical stability of the numerical filter itself, it should satisfy

The flux,

The numerical viscosity and diffusion in the

Vertical profiles of density, potential temperature, and water vapor usually
have significant (e.g., logarithmic) dependencies on height.
Equation (

Determination of the value of the non-dimensional coefficient

The

Odd-order advection schemes are generally more stable numerically than
even-order schemes. In general, odd-order schemes can be divided into
a central difference term and a filter term (and sometime other additional
terms) conceptually. This implicit numerical filter stabilizes the
calculations. We estimate the corresponding value of

The third-order scheme by

Although these schemes can be thought of as being similar to central
difference and explicit fourth-order numerical filters, the total coefficient
of the filter depends on the velocity

Currently, the following processes are implemented as physical processes in
SCALE-LES.

Cloud microphysics

A one-moment three-category bulk scheme

A one-moment six-category bulk scheme

A two-moment six-category bulk scheme

A bin method

Sub-grid turbulence

A Smagorinsky–Lilly-type scheme including stability effect developed by

A RANS (Reynolds-averaged Navier-Stokes) turbulence model

Radiation

A parallel plane radiation model

Surface flux

A Louis-type bulk model

A Beljaars-type bulk model

Urban canopy

A single-layer urban canopy model

In this model, we use the second-order central difference scheme for the spatial differential terms of the pressure gradient and the divergence of mass flux, and the fourth-order scheme for advection terms. In this section, we investigate the numerical instability of the terms and show why we use the second-order scheme for these terms.

For simplicity, we assume a case in which the initial potential temperature
is constant

The governing equations are

To investigate the stability, the equations are linearized. The density is
divided into the basic value and the deviation from the basic:

The time derivatives are written as

Now we consider the numerical stability of two-grid noise. An eigenanalysis
showed that the two-grid noise is the most unstable eigenmode in all the
cases we tested. The density with the noise can be written as

The necessary condition for the two-grid noise to decrease in time is

The accuracy of the pressure gradient term and divergence term could
especially affect the high-frequency modes of acoustic and gravity waves. The
high-frequency modes seem to be less significant meteorologically. We choose
the second-order spatial scheme for terms of the pressure gradient and
divergence terms to increase

Figure

Plot of potential temperature at

The authors would like to thank the editor and the two referees,
E. Goodfriend and an anonymous referee, for
fruitful discussions. This model was developed by an interdisciplinary team
at RIKEN Advanced Institute for Computational Science (AICS) called Team
SCALE, from the fields of meteorological science and computer science. Part
of the results in this paper was obtained using the K computer at the RIKEN
AICS. Figures in this paper were drawn using tools developed by GFD-Dennou
Club (