We present and validate a set of equations for representing the atmosphere's large-scale general circulation in an Earth system model of intermediate complexity (EMIC). These dynamical equations have been implemented in Aeolus 1.0, which is a statistical–dynamical atmosphere model (SDAM) and includes radiative transfer and cloud modules (Coumou et al., 2011; Eliseev et al., 2013). The statistical dynamical approach is computationally efficient and thus enables us to perform climate simulations at multimillennia timescales, which is a prime aim of our model development. Further, this computational efficiency enables us to scan large and high-dimensional parameter space to tune the model parameters, e.g., for sensitivity studies.

Here, we present novel equations for the large-scale zonal-mean wind as well as those for planetary waves. Together with synoptic parameterization (as presented by Coumou et al., 2011), these form the mathematical description of the dynamical core of Aeolus 1.0.

We optimize the dynamical core parameter values by tuning all relevant dynamical fields to ERA-Interim reanalysis data (1983–2009) forcing the dynamical core with prescribed surface temperature, surface humidity and cumulus cloud fraction. We test the model's performance in reproducing the seasonal cycle and the influence of the El Niño–Southern Oscillation (ENSO). We use a simulated annealing optimization algorithm, which approximates the global minimum of a high-dimensional function.

With non-tuned parameter values, the model performs reasonably in terms of its representation of zonal-mean circulation, planetary waves and storm tracks. The simulated annealing optimization improves in particular the model's representation of the Northern Hemisphere jet stream and storm tracks as well as the Hadley circulation.

The regions of high azonal wind velocities (planetary waves) are accurately captured for all validation experiments. The zonal-mean zonal wind and the integrated lower troposphere mass flux show good results in particular in the Northern Hemisphere. In the Southern Hemisphere, the model tends to produce too-weak zonal-mean zonal winds and a too-narrow Hadley circulation. We discuss possible reasons for these model biases as well as planned future model improvements and applications.

Numerical models of the Earth system play a key role in our understanding of
physical processes in Earth and atmosphere and can be used to simulate past
and future climate changes.
General circulation models (GCMs) are physically the most realistic tools
for studying and modelling climate variability and climate change in the
Earth system. However, due to their relatively high resolution, they are
costly in terms of CPU runtime, limiting their applicability to study
climate variability over long (

On the other hand, highly idealized and computational efficient models for the climate system are able to simulate long time periods, but those are often box or one- or two-dimensional models describing only a limited number of processes or feedbacks of the real world. Hence, their application is limited, but they have been applied to study paleoclimate (Berger et al., 1992; Harvey, 1989) and future global change (Xiao et al., 1997).

A third class of models are so-called Earth system models of intermediate complexity (EMICs) which form a compromise between the computationally expensive (but more realistic) GCMs and the highly simplified models (Claussen et al., 2002). The number of processes and feedbacks are comparable to GCMs; however, due to a reduction in resolution and/or complexity of some model components, it is possible to study climate simulations up to multimillennia timescales (Eliseev et al., 2014a, b; Ganopolski et al., 2001; Montoya et al., 2005). Other applications include determining quick assessments of climate change impacts or running thousands of parameter sensitivity experiments (Knutti et al., 2002; Schmittner and Stocker, 1999).

EMICs are thus particularly useful for understanding the roles of different Earth components on very long timescales (multimillennia and longer) and consequently form useful tools complementary to GCMs. Internal climate processes on such very long timescales are primarily driven by ocean and ice dynamics (Holland et al., 2001; Latif, 1998; Polyakov et al., 2003), with the atmosphere's role being likely limited to globally distributing any perturbations to the system. In GCMs, it is however often the atmosphere which takes most of the computational load due to the need to resolve synoptic weather systems, which requires a high-resolution discretization in space and time. For these reasons, a key step in the development of EMICs intended for studying ocean and ice dynamics on multimillennial timescales is the derivation and validation of statistical–dynamical equations which accurately represent atmosphere dynamics (Coumou et al., 2011).

EMICs have been used in many climate studies and several different types of
simplified atmospheric components that form part of an EMIC exist including
two-dimensional, zonally averaged atmosphere models, 2.5-D atmosphere
models (the vertical dimension is reconstructed using two-dimensional fields)
with simple energy balance or statistical–dynamical atmosphere models
(SDAMs) (Claussen et al., 2002; McGuffie and Henderson-Sellers, 2005). Most
EMIC studies focus on climate variability on very long timescales (e.g.,
glacial cycles), and therefore fast processes are normally parameterized. In
particular, SDAMs parameterize smaller-scale (and more short-lived) processes
like synoptic eddy activity in terms of the large-scale, long-term fields.
The assumption of those models is thus that atmospheric variables can be
expressed in separate terms of a large-scale, long-term component, with
characteristic spatial and temporal scales of

The essential difference compared to GCMs is the point of truncation in the frequency spectrum of atmospheric motion (Saltzman, 1978). GCMs solve all phenomena of frequencies lower than and including synoptic cyclones (and sometimes even mesoscale systems), whereas statistical–dynamical (SD) models parameterize all scales smaller than and equal to synoptic. Much of the difficulty in SD models is to define physically reasonable parameterizations occurring in the equations (Saltzman, 1978). For Aeolus, the synoptic parameterization has been described in detail in Coumou et al. (2011).

As written above, SD models are also spatially averaged since, for long-term climate simulations, we are typically interested in the large spatial aspects of the climate. It is further practical to split the large-scale, long-term field into two components: the zonally averaged mean field and the asymmetric departure of the field from the zonally averaged fields characterizing the east–west variations. The azonal variables can be, for example, resolved by one-dimensional Fourier components around latitude circles or into spherical harmonics (Saltzman, 1978).

Here, we present the Aeolus 1.0 dynamical core, developed at the Potsdam Institute for Climate Impact Research (PIK), a new SD model for the atmosphere. It uses some aspects of the atmosphere module of the EMIC CLIMBER-2 developed by Petoukhov et al. (2000). The dynamical core is completely new with novel equations for the large-scale meridional wind speed as well as quasi-stationary planetary waves. Together with the synoptic parameterizations presented in Coumou et al. (2011), these equations form the new dynamical core of Aeolus 1.0. The model is coupled with the cloud module consisting of a three-layer stratiform plus convective cloud scheme as presented and validated in Eliseev et al. (2013).

Further, we present the equations of the model and validate the dynamical core using a parameter optimization experiment. Aeolus 1.0 is forced with prescribed surface temperature, surface humidity and cumulus cloud fraction to test the model's performance. In particular, we examine the reproduction of the seasonal cycle and the influence of El Niño–Southern Oscillation (ENSO) and compare relevant dynamical fields of the model output against seasonal means of ERA-Interim reanalysis data (climatology 1983–2009). The effects of parameter tuning are evaluated to improve the performance of the model.

In Sect. 2, we present the novel equations of the Aeolus 1.0 dynamical core with the derivation of these equations presented in the Supplement (Sects. S1–S2). The dynamical core is coupled with a convective plus three-layer stratiform cloud scheme (which includes low-level, mid-level and upper-level stratiform clouds) developed by Eliseev et al. (2013). In Sect. 3, we describe the experiment setup and the used reanalysis data sets. In Sect. 4, we explain the model discretization, and in Sect. 5 we introduce our specific calibration method. For parameter optimization of the wind velocities, we use simulated annealing, which approximates the global minimum of a high-dimensional function (Flechsig et al., 2013). In Sect. 6, we present Aeolus' dynamical fields with pre-optimized and optimized parameters and compare them with observations and output from models of the Coupled Model Intercomparison Project phase 5 (CMIP5). We conclude by discussing performance and limitations of the model in Sect. 7.

Aeolus 1.0 is a 2.5-D SD model, with the vertical dimension being largely parameterized and only coarsely resolved, and it therefore belongs to the class of intermediate complexity atmosphere models. Water and energy conservation is achieved via a set of two-dimensional, vertically averaged prognostic equations for temperature and specific humidity (Petoukhov et al., 2000).

The three-dimensional structure is described by these two-dimensional surface fields with the vertical dimensions reconstructed using an equation for the lapse rate and assuming an exponential profile for specific humidity (Petoukhov et al., 2000).

For given temperature and specific humidity fields, the three-dimensional wind field is calculated using a set of diagnostic equations. These statistical–dynamical equations for the wind fields are coupled and thus need several time steps or iterations to equilibrate.

The equations of the dynamical core of Aeolus 1.0 are separated into
equations for the (1) synoptic-scale transient waves (or storm tracks),
(2) quasi-stationary planetary waves and (3) the zonal-mean wind. Thus,
following classical statistical–dynamical approaches (Dobrovolski, 2000;
Imkeller and von Storch, 2012), the key assumption is that the wind velocity
field

As derived in the Supplement in Sect. S2, the large-scale, zonal-mean meridional
wind velocity

Atmosphere model parameters.

The azonal component of the large-scale wind field describes
quasi-stationary planetary waves and depends on latitude, longitude and
height. At the equivalent barotropic level (EBL), azonal geostrophic
components of horizontal velocities are computed by employing the definition of
the stream function

The zeroth-order solution of the thermally induced waves of the barotropic
vorticity equation is given by (see Sect. S1.3 in the Supplement)

The standardized integrated heat content in Eq. (8)

To remove possible singularities near the poles, at high latitudes, the stream function is dampened by a fourth-order interpolation function. Planetary waves at other tropospheric levels are directly calculated from those at the EBL (see Sect. S1.1 in the Supplement).

Finally, the time-averaged kinetic energy of transient eddies

Here,

The terms for

This provides us with a coupled set of equations for

The simulations were forced by multiyear averages of monthly mean
climatological, El Niño and La Niña month data (surface temperature,
surface specific humidity, temperature at 500 mb, geopotential height at 500
and 1000 mb) using ERA-Interim reanalysis data (Dee et al., 2011) for
1983–2009, as our aim is to show that Aeolus captures year-to-year variability
associated with the ENSO cycle. We identified 87 El Niño (74 La Niña)
months using 3-month running means of Extended Reconstructed Sea
Surface Temperature (ERSST) v4 anomalies (Huang et al., 2016)
using the definition that at least five consecutive overlapping seasons of sea surface temperature
(SST) anomalies are greater than 0.5 K (less than

Multiyear averages of monthly mean, El Niño and La Niña month
cumulative cloud fractions are taken from ISSCP (Rossow and Schiffer, 1999).
The spatial resolution is

We chose this time period, because the cumulative cloud fraction data, which are needed to calculate the lapse rate, are only available for this time period.

To avoid strong temperature gradients in the specified boundary conditions
for the numerical experiments, we use the lapse rate equation to calculate
temperatures at 1000 mb from those at 500 mb. We first calculate the lapse
rate using the temperature field and specific humidity utilizing the equation as
given in Petoukhov et al. (2000) at 1000 mb. Then, we recalculate the
temperature field at 1000 mb by employing the temperature field at 500 mb and
the linear lapse rate equation. This way, we ensure that the temperature at
500 mb is close to observations, and at the same time we have a vertical
temperature realistic profile for a model like Aeolus. Since the ERA-Interim
500 mb temperatures contain an orographic component, we exclude

We optimized the parameters for the numerical solutions of the wind
velocities

Before use with Aeolus, all data sets are interpolated to

Aeolus operates on a reduced grid to overcome the restriction of small time steps near the poles due to the Courant–Friedrichs–Lewy (CFL) criteria (Jablonowski et al., 2009). In the grid generation, longitudinally adjacent cells are merged if their zonal width in meters would be less than half of the cell width at the Equator.

This way the reduced grid has the same resolution as a regular grid at the
Equator, but at nominal resolution (

Equations (1)–(14) are implemented in Aeolus and numerically solved on a

The calibration of the winds is divided into two parts. First, we optimize
the dynamical variables primarily driven by the thermal state of the
atmosphere: the azonal velocities in zonal and meridional directions

A common approach for parameter tuning is simulated annealing (Ingber, 1996; Kirkpatrick, 1984). It is one experiment type in the multirun simulation environment SimEnv for sensitivity and uncertainty analysis of model output (Flechsig et al., 2013) which we use for all calibration experiments. A schematic plot of the optimization process is shown in Sect. S3 in the Supplement.

For each model run, the thermal state of the atmosphere is kept constant (and initialized as described above) and the dynamical core is equilibrated to this thermal state. This typically requires only a few time steps. Since we tune only the parameters of the dynamical core, Aeolus first calculates the clouds using its cloud scheme (Eliseev et al., 2013) to determine the lapse rate and initialize the three-dimensional thermal state. After that, only the state of the dynamical core is updated each time step.

For a good starting point, the parameters are first tuned manually, providing “pre-optimized” values. Next, we define physically realistic parameter ranges for automatic tuning as listed in Table 2.

Pre-optimized and optimized parameter sets and parameter ranges for optimization step 1.

For the azonal wind velocities, we use a weighting function which excludes the
tropics (from 10

The non-excluded grid as well as the zonal-mean zonal wind are weighted by

The total skill score for the scheme in step 1 is calculated by multiplying
the individual skills for the azonal velocities in zonal and meridional
directions

Skill score functions for individual variables are computed as in Taylor
(2001):

Here, the variable

For tuning the zonal-mean meridional wind velocity

Total skill score for the scheme in step 2 is calculated by multiplying the
individual skills for the vertical integral of lower troposphere mass flux

The goal of the optimization procedure is again to maximize skill

The skill score function for the eddy kinetic energy is given by the Taylor skill score function (Eq. 15).

The skill score is then calculated by

Here,

The weights of the lower troposphere mass flux

In Table 3, the manually tuned (or pre-optimized) parameters and their ranges are listed.

Pre-optimized and optimized parameter sets and parameter ranges for optimization step 2.

We compared the numerical solutions using the optimized parameters for the
wind fields

The figures for azonal wind velocities are divided into six subplots. The left column shows observational data and the right column model data. The top row shows climatological monthly averages, the middle row multiyear averages of El Niño months and the bottom row multiyear averages of La Niña months.

Azonal large-scale zonal wind

Azonal large-scale zonal wind velocity

In Figs. 1 and 2, the azonal components of the zonal wind velocities (

Azonal large-scale meridional wind velocity

Azonal large-scale meridional wind velocity

Figures 3 and 4 show the same type of plots for the azonal meridional wind
velocity (

Zonal-mean large-scale zonal wind velocity

In Fig. 5, the zonal-mean zonal wind velocity

The optimized parameters are listed in Table 2. The

The last parameter is

We compared the numerical solutions using the optimized parameters for the
zonal-mean lower troposphere integrated mass flux

Zonal-mean large-scale mass flux

The plots in Fig. 6 show that in general the monthly mean zonal-mean mass
flux calculated with optimized parameters matches better with observational
data. Here, the gain of the parameter optimization is clearly better than we
saw with calibration step 1. The ENSO cycle is clearly visible. However, the
width of the Hadley cell (especially in August) is still too small compared
to the width of the Hadley cell obtained by reanalysis data. The figure shows
only plots with a latitudinal range from 60

Zonal-mean time-averaged eddy kinetic energy

Figure 7 shows the zonal-mean eddy kinetic energy

In Figs. 8 and 9, the eddy kinetic energies

Eddy kinetic energy

Eddy kinetic energy

Comparison to CMIP5 models. The orange line represents ERA-Interim data, the red line results from Aeolus and gray lines CMIP5 models (annual mean zonal-mean data).

The spatial position and the magnitude are well captured; seasons and the
ENSO cycles are also well resolved, with some discrepancies in the tropics
(i.e., the region over the Atlantic and Pacific oceans) and the Southern
Hemisphere. In February and August,

The optimized parameters are listed in Table 3. The parameters

The parameters

Figure 10 shows the comparison of February and August

The CMIP5 multimodel mean of

In this paper, we presented the atmosphere model Aeolus, which is a
statistical–dynamical atmosphere model and belongs to the class of
intermediate complexity models. The equations of Aeolus are time averaged and
the model has a spatial resolution of

We performed parameter optimization of the dynamical core consisting of a large multidimensional parameter space and can be searched due to its fast computation time. For this approach, we used the simulated annealing optimization algorithm, which approximates the global minimum of a high-dimensional function. We divided the calibration into two parts. First, the azonal velocities in zonal and meridional directions as well as the zonal-mean zonal wind velocity were optimized, because they are primarily driven by the thermal state of the atmosphere. In the second step, we optimized the zonal-mean synoptic kinetic energy and the lower troposphere integrated mass flux, and hence the zonal-mean meridional velocity, since those variables depend strongly on variables of step 1.

The results of the winds and eddy kinetic energy are in reasonable agreement with the reanalysis data and showed that our model is able to reproduce the dynamic response from the seasonal cycle as well as the ENSO cycle which is a prime goal of our model development efforts. Parameter optimization in particular improves representation of the Hadley cell in terms of strength and width.

In the Southern Hemisphere, the dynamical fields tend to be too weak. This model bias might be related to the missing Antarctic ice sheet, upper tropospheric ozone, the constant lapse rate assumption or fundamental limitations of the equations. These possibilities will be analyzed in future work using the Potsdam Earth Model (POEM) to which Aeolus has been coupled.

Compared to CMIP5 models, Aeolus captures reasonably well the dynamical state
of the atmosphere in the Northern Hemisphere, particularly for monthly mean
eddy kinetic energy

Code and data are stored in PIK's long-term archive and are made available to interested parties on request.

The supplement related to this article is available online at:

The authors declare that they have no conflict of interest.

We thank ECMWF for making the ERA-Interim data available.

The work was supported by the German Federal Ministry of Education and Research, grant no. 01LN1304A, (S.T., D.C.).

Alexey V. Eliseev's contribution was partly supported by supported by the Government of the Russian Federation (agreement no. 14.Z50.31.0033).

The authors gratefully acknowledge the European Regional Development Fund (ERDF), the German Federal Ministry of Education and Research and the Land Brandenburg for supporting this project by providing resources on the high-performance computer system at the Potsdam Institute for Climate Impact Research.Edited by: Didier Roche Reviewed by: two anonymous referees