The Extrapolar SWIFT model is a fast ozone chemistry scheme for interactive calculation of the extrapolar stratospheric ozone layer in coupled general circulation models (GCMs). In contrast to the widely used prescribed ozone, the SWIFT ozone layer interacts with the model dynamics and can respond to atmospheric variability or climatological trends.

The Extrapolar SWIFT model employs a repro-modelling approach, in which
algebraic functions are used to approximate the numerical output of a full
stratospheric chemistry and transport model (ATLAS). The full model solves a
coupled chemical differential equation system with 55 initial and boundary
conditions (mixing ratio of various chemical species and atmospheric
parameters). Hence the rate of change of ozone over 24 h is a function of 55
variables. Using covariances between these variables, we can find linear
combinations in order to reduce the parameter space to the following nine

For validation purposes, the Extrapolar SWIFT model has been integrated into
the ATLAS model, replacing the full stratospheric chemistry scheme.
Simulations with SWIFT in ATLAS have proven that the systematic error is
small and does not accumulate during the course of a simulation. In the
context of a 10-year simulation, the ozone layer simulated by SWIFT shows a
stable annual cycle, with inter-annual variations comparable to the ATLAS
model. The application of Extrapolar SWIFT requires the evaluation of
polynomial functions with 30–100 terms. Computers can currently calculate
such polynomial functions at thousands of model grid points in seconds. SWIFT
provides the desired numerical efficiency and computes the ozone layer

Modern climate models include an increasing number of climate
processes and run with ever higher model resolutions. Many processes that are
relevant for the climate system are already well understood, but they remain
computationally too demanding to be incorporated into climate models. One of
these processes is the stratospheric ozone chemistry. The feedbacks between
the ozone layer and the changing climate system have been investigated in
various studies

SWIFT is subdivided into a polar and an extrapolar module. The two
sub-modules follow separate approaches due to the differences in polar and
extrapolar ozone chemistry. The lack of sunlight and very low temperatures
during polar night extend the chemical lifetimes of various trace gases
relevant for ozone depletion. Under these conditions the individual species
within the chemical families Cl

In extrapolar conditions the diurnal average concentrations of the individual
species within the chemical families (partitioning) mentioned above are
sufficiently close to photochemical steady state because the photochemical
lifetimes of the involved species are sufficiently short compared to the
transport timescales. In a good approximation the chemically induced change
in ozone over 24 h is a function of the concentrations of the chemical
families, ozone itself and the physical boundary conditions. The Extrapolar
SWIFT model is based on the substitution of a comprehensive differential
equation system describing the ozone changes by algebraic functions. This
approach is also referred to as repro-modelling and has been successfully
applied to chemical models; see

Existing fast ozone schemes for climate models like the Cariolle scheme

In Sect. 2 of this paper the application of repro-modelling to the rate of
change of ozone is described. First we introduce the set-up of the
repro-model, containing polynomial coefficients as free parameters. Further,
the approximation algorithm determining these coefficients is described and
its modifications in comparison to previous studies are explained. Section 3
focuses on the domain of definition of the polynomial functions and how
outliers are handled in Extrapolar SWIFT. A validation and error estimation
of the polynomial functions are presented in Sect. 4. Eventually, two
different simulations with SWIFT are discussed in Sect. 5. A 2-year
simulation focuses on the error in the ozone field caused by the monthly
polynomial functions. A 10-year simulation mimics the set-up of SWIFT in a
GCM and demonstrates the stability of the model over a longer simulation
period. The development of Extrapolar SWIFT and the results of the
simulations are also discussed in

The Extrapolar SWIFT repro-model calculates the rate of change of ozone over
24 h by evaluating polynomial functions of fourth degree. Each polynomial
function is valid during 1 month of the year. To determine these polynomial
functions we use multivariate fitting of a representative data set which
comprises a wide range of stratospheric conditions, as suggested by

Nine

In order to set up a repro-model, we need to determine a set of

The stratospheric ozone depletion is driven by catalytic cycles involving the
short-lived species of the above-listed chemical families. Consequently, the
repro-model requires information on the concentration of the short-lived
compounds. This may be derived from the concentrations of the chemical
families. In the extrapolar regions the short-lived reactive species (e.g.
ClO

Apart from the VMR of the chemical constituents, the reaction rates depend on
temperature, air density and in the case of photolysis rates on the actinic
flux, particularly on the ultraviolet attenuation (UV attenuation). These
parameters must also be implicitly or explicitly included into the set of

The algebraic equation of the repro-model is a polynomial function of fourth
degree (i.e. the sum of the exponents of a term is

We start the fitting procedure with one polynomial term (

In our approach we are circumventing this problem by testing all polynomial terms individually as the next additional term. In other words, in each iteration each of the still available polynomial terms is temporarily added to the already selected terms and the fitting procedure is carried out. The term which reduces the residuum the most is permanently added to the polynomial function and removed from the pool of available terms. In the next iteration all remaining terms are fitted in combination with the previously accepted ones. By simply choosing the best fitting term we also avoid setting an arbitrary threshold for the minimum required reduction of the residuum. This polynomial term selection method makes the fitting procedure computationally much more extensive. However, the fitting procedure has to be carried out only once so that this additional computation time imposes no disadvantage during the application of SWIFT.

The more polynomial terms are added to the function, the better the
approximation will be; i.e. the residuum can be reduced further and further.
If as many polynomial terms (corresponding to columns of

It is important that the testing data set has the same probability
distribution of

In this section we discuss where in the stratosphere the Extrapolar SWIFT
model can be used, i.e. for which latitudes and altitudes the underlying
assumptions are valid. A key aspect for the definition of this
latitude–altitude region is the mean chemical lifetime

Zonal mean of O

Above roughly 30 km of altitude the mean lifetime of O

Since the lifetime of O

However, the upper stratosphere only contributes a few percent to the
stratospheric ozone column. The bulk of ozone dominating the total column
values is in the lower stratosphere below 30 km. This motivated our focus
on this part of the stratosphere which we will refer to as
the

The lower boundary of the

The regime boundaries between Extrapolar SWIFT and Polar SWIFT are defined by
the edge of the polar vortex. The horizontal extent of the polar vortex is
defined by 36 mPV units, where mPV is the modified potential vorticity
according to

The monthly training and testing data set for Extrapolar SWIFT are generated
with the stratospheric Lagrangian chemistry and transport model ATLAS

For each month of the
year, daily snapshot values of the

Individual chemical species in ATLAS are grouped into their respective
families and summed up to generate the mixing ratios of Cl

Before the fitting procedure, the

The Lagrangian trajectories in ATLAS are not distributed homogeneously. In general, higher trajectory densities can be found where there is strong horizontal and vertical wind shear, e.g. at the edge of the polar vortex. This is caused by the trajectory mixing algorithm in ATLAS, which initializes new or deletes existing trajectories based on their rate of divergence or convergence in a region of the model atmosphere. The regions of increased trajectory densities coincide with strong gradients of chemical constituents and meteorological parameters. Thus these gradients are well resolved in ATLAS, which is beneficial to Extrapolar SWIFT. The training and testing data sets simply contain the same unmodified sampling as in ATLAS and therefore also resolve the gradients well.

The extensive training data set derived from the ATLAS CTM fills a portion of the nine-dimensional hyperspace, which defines the domain of definition of the fitted polynomial functions. SWIFT is intended to be used in long-term climate simulations and it will certainly encounter inter-annual and decadal variability. Therefore we used data from ATLAS simulations covering a wide range of stratospheric variability. By taking the training and testing data from different decades we include maximum and minimum conditions of the solar cycle. The data also represent different QBO phases and the varying strengths and lifetimes of the Arctic and Antarctic polar vortices.

Climatological changes impacting the probability distribution of the

Apart from a PDF shift like the one illustrated in Fig.

Schematic of a shift in the probability density function of stratospheric temperature in a future climate.

When running a SWIFT simulation, the polynomial function should not be
evaluated outside the domain defined by the training data set. Polynomial
functions of higher degree tend to rapidly increase or decrease when
extrapolated. In order to determine if a data point lies outside or inside
the nine-dimensional domain of definition we need to be able to define its
boundaries. This could be achieved by enveloping the nine-dimensional cloud of
data points by a conjunction of nine-dimensional cells (cuboids) corresponding
to a nine-dimensional regular grid (look-up table). These grid cells are either
sampled by the training data set or not. A sampled grid cell is defined as
being inside the domain, and all the non-sampled grid cells are outside. Dealing
with a nine-dimensional grid with only a few nodes per dimension readily creates
a grid with millions of cells. However, the majority of these grid cells
represent combinations of

During the application of SWIFT within a GCM the following operations are
carried out at each spatial grid point. The domain polynomial is computed for
the values of the nine

As an initial validation step the rate of change of ozone in the testing data
set is compared to the rate of change of ozone calculated by the polynomial
functions. In Fig.

In general all four months show good agreement between ATLAS and SWIFT.
Especially in the tropics and mid-latitudes the amplitude of

Zonal and monthly mean of

To estimate the error of Extrapolar SWIFT, we examine the difference of ATLAS

Figure

As mentioned before, individual

Probability distribution of error quantity

The Extrapolar SWIFT module was coupled to the ATLAS CTM in order to perform
validation simulations. In this set-up the SWIFT scheme replaces the detailed
stratospheric chemistry model of ATLAS. Apart from the geographical and
meteorological variables provided by ATLAS, Extrapolar SWIFT requires the VMR
of the four ozone-depleting chemical families Cl

The SWIFT in ATLAS simulations are driven by ERA-Interim data

Above the seasonally dependent upper boundary of the

Initially, the Extrapolar SWIFT module coupled to ATLAS was used in a
simulation over a period of 2 years. With this short simulation we want to
compare the development of the ozone layer in SWIFT to a reference simulation
with ATLAS. The goal of the comparison is to investigate the error or drift
caused solely by the SWIFT polynomial functions. Therefore the simulation
conditions of both runs should be as similar as possible. To achieve this,
the SWIFT simulation does not use trace gas climatologies for Cl

The panels in Figs.

Figure

Further, it is unlikely for the monthly polynomial functions to produce the same deviations in exactly the same regions. If we compare the magnitude of the positive differences in January and March 2005 vs. January and March 2006 we see that the more positive deviations have switched from one month to the other. The variability of the magnitude can probably be attributed to the inter-annual stratospheric variability of the Northern Hemisphere, in particular the extent and lifetime of the polar vortex. In general the deviations of the year 2006 are not larger or more extensive than in 2005. Apparently no significant error is propagated from the preceding year to the following year.

The 2005 zonal and monthly mean stratospheric ozone concentrations
plotted in equivalent latitude vs. pressure altitude.

The 2006 zonal and monthly mean stratospheric ozone concentrations
plotted in equivalent latitude vs. pressure altitude. See also
Fig.

A SWIFT simulation over a period of 10 years demonstrates the stability of
the model. The set-up for this simulation mimics the coupling of SWIFT to a
GCM, although SWIFT is actually running in the ATLAS CTM. The trace gas
climatologies for Cl

Monthly mean values of the stratospheric ozone column (15–32 km)
over Potsdam (52.4

SWIFT vs. ATLAS scatter plot of daily averaged stratospheric ozone
columns (15–32 km) over Potsdam. The orange and green dots correspond to
the two training data periods in Fig.

SWIFT vs. Aura MLS scatter plot of daily averaged stratospheric
ozone columns (15–32 km) over Potsdam. The green dots correspond to the
second training data period, and the pink dots correspond to the pink period in
Fig.

Beginning in autumn 2004 observational data from the microwave limb sounder
Aura MLS are available and we additionally compare the SWIFT results with the
Aura MLS observations (black line in Fig.

The design of Extrapolar SWIFT enables full parallelization, since individual
model nodes can independently evaluate the polynomial functions. A function
consists of 30 to 100 polynomial terms, varying from month to month. Per
model node and time step, three polynomial functions have to be evaluated, one
domain polynomial and two

The version of Extrapolar SWIFT coupled to the ATLAS CTM is implemented in
MATLAB because the ATLAS model was written in MATLAB. SWIFT in ATLAS is not
optimized for speed and the evaluation of the polynomials is computed on a
single core. However, when comparing the full stratospheric chemistry scheme
of ATLAS vs. the evaluation of the SWIFT polynomial functions, the ozone
layer can be computed

The Extrapolar SWIFT model is a numerically efficient ozone
chemistry scheme for global climate models. Its primary goal is to enable the
interactions between the ozone layer, radiation and climate, while imposing a
low computational burden to the GCM it is coupled to. We accomplished this by
approximating the rate of change of ozone of the detailed chemistry model
ATLAS by using algebraic equations. Orthogonal polynomial functions of fourth
degree are used to approximate the rate of change of ozone over 24 h. An
automated and optimized procedure approximates one globally valid polynomial
function to a monthly training data set. In our repro-modelling approach we
reduce the dimensionality of the model through exploitation of the covariance
between variables. The polynomial functions are a function of only nine

Running the Extrapolar SWIFT model requires only the 12 monthly polynomial
functions and information about the nine

Simulations with the Extrapolar SWIFT model coupled to the ATLAS CTM have
shown good agreement to the reference model ATLAS. The stability of SWIFT has
been proven with a simulation over a 10-year period in which SWIFT was
validated against model and observational references. Errors did not
accumulate over the extended simulation period. Average deviations of the
integrated stratospheric ozone column (15–32 km) are

The source code of the Extrapolar SWIFT model (version 1.0)
and the Polar SWIFT model (version 2.0) is available via a publicly
accessible Zenodo repository at

The ATLAS CTM is available on the AWIForge repository
(

The authors declare that they have no conflict of interest.

This work was supported by the BMBF under the FAST-O3 project in the MiKliP
framework programme (FKZ 01LP1137A) and in the MiKliP II programme (FKZ
01LP1517E). This research has received funding from the European Community's
Seventh Framework Programme (FP7/2007–2013) under grant agreement no. 603557
(StratoClim). This study has been supported by the SFB/TR172 “Arctic
Amplification: Climate Relevant Atmospheric and Surface Processes, and
Feedback Mechanisms (AC)