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**Model description paper**
01 Mar 2018

**Model description paper** | 01 Mar 2018

The Extrapolar SWIFT model (version 1.0): fast stratospheric ozone chemistry for global climate models

- Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research, Potsdam, Germany

- Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research, Potsdam, Germany

**Correspondence**: Daniel Kreyling (daniel.kreyling@awi.de)

**Correspondence**: Daniel Kreyling (daniel.kreyling@awi.de)

Abstract

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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
*basic* variables: latitude, pressure altitude, temperature, overhead
ozone column and the mixing ratio of ozone and of the ozone-depleting families
(Cl_{y}, Br_{y}, NO_{y} and HO_{y}). We will show that these nine variables are
sufficient to characterize the rate of change of ozone. An automated
procedure fits a polynomial function of fourth degree to the rate of change
of ozone obtained from several simulations with the ATLAS model. One
polynomial function is determined per month, which yields the rate of change
of ozone over 24 h. A key aspect for the robustness of the Extrapolar SWIFT
model is to include a wide range of stratospheric variability in the
numerical output of the ATLAS model, also covering atmospheric states that
will occur in a future climate (e.g. temperature and meridional circulation
changes or reduction of stratospheric chlorine loading).

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 10^{4}
times faster than the chemistry scheme in the ATLAS CTM.

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Kreyling, D., Wohltmann, I., Lehmann, R., and Rex, M.: The Extrapolar SWIFT model (version 1.0): fast stratospheric ozone chemistry for global climate models, Geosci. Model Dev., 11, 753–769, https://doi.org/10.5194/gmd-11-753-2018, 2018.

1 Introduction

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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 (e.g. Thompson and Solomon, 2002; Rex et al., 2006; Baldwin et al., 2007; Nowack et al., 2014; Calvo et al., 2015). All of them emphasize the importance of the interactions between climate change and the ozone layer. Climate simulations with a more accurate representation of the ozone layer lead to significant changes in tropospheric and surface variables. However, a frequently used approach to represent the ozone layer in general circulation models (GCMs) is the use of prescribed zonal mean ozone climatologies, as in many of the Coupled Model Intercomparison Project 5 (CMIP5) simulations (IPCC, 2014). By using prescribed ozone, the atmospheric dynamics cannot interact with the ozone field, the ozone hole is a static, zonally symmetric feature that does not interact with atmospheric waves and the ozone layer does not respond to climate change and vice versa. But this approach is computationally cheap and does not impede the GCM capacity for ensemble simulations. The incorporation of an interactive ozone layer instead of climatologies allows the ozone field to actually match the model dynamics and enables two-directional feedbacks. Chemistry climate models (CCMs) with a highly resolved stratosphere usually provide such an interactive ozone layer, but the computational cost of CCMs still limits their usefulness for long-term ensemble simulations (Eyring et al., 2010). In recent years different approaches were taken to efficiently incorporate interactive ozone in climate simulations (Eyring et al., 2013). One of these approaches is the development of stratospheric ozone chemistry schemes with a very low computational burden in comparison to the computation time of the GCM, for example the Cariolle scheme (Cariolle and Teyssèdre, 2007) or the Linoz scheme (Hsu and Prather, 2009). In this paper we introduce the extrapolar part of the numerically efficient and interactive stratospheric ozone chemistry scheme SWIFT. Its goal is to provide sufficient accuracy and efficiency to enable ensemble simulations with atmosphere–ocean coupled GCMs, while maintaining the physical and chemical properties of the processes that govern ozone chemistry in the stratosphere so that the SWIFT approach is valid for a wide range of climatic conditions, including future climate scenarios.

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_{y}, Br_{y}, NO_{y} and HO_{y} are too far
from chemical equilibrium so that their time evolution needs to be
calculated with differential equations. The Polar SWIFT model simulates the
time evolution of polar-vortex-averaged mixing ratios of ozone and four key
species during Arctic and Antarctic winters. A small coupled differential
equation system containing empirically determined fit parameters models the
most relevant processes of polar ozone depletion. The first Polar SWIFT
version was described by Rex et al. (2014) and the updated version
was published by Wohltmann et al. (2017).

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 Spivakovsky et al. (1990),
Turányi (1994) and Lowe and Tomlin (2000). As
in the previous studies we obtain the algebraic functions by fitting the
numerical solution of the chemical differential equation system with
orthonormal polynomial functions. Following the approach of
Turányi (1994) we use a wide range of input and output
values of a full chemical model to create a data set that is then used for
fitting the polynomial functions. However, a few modifications were
introduced, most prominently in the selection of the most suitable
polynomial terms. Moreover, we developed a termination criterion that does
not require the selection of arbitrary thresholds. It is important to note
that the repro-model is not a shortened subset of the full chemical system.
By approximating the output of the full system, we ensure that all physical
and chemical properties of the full chemical model are maintained in the
repro-model. In this application the rate of change of ozone in the lower and
middle stratosphere is parameterized by one polynomial function per month.
Each of these polynomials is a function of nine *basic* variables, which
are sufficient to parameterize the rate of change of ozone in the full
chemical system. The *basic* variables are latitude, pressure
altitude, temperature, the overhead ozone column, the volume mixing ratio
(VMR) of the ozone-depleting substances (ODSs) combined into four chemical
families and ozone itself. The calculation of the polynomial function values
instead of solving the chemical differential equation system drastically
reduces the computational cost and makes SWIFT a suitable candidate for
coupling to a GCM.

Existing fast ozone schemes for climate models like the Cariolle scheme (Cariolle and Teyssèdre, 2007) or the Linoz scheme (McLinden et al., 2000; Hsu and Prather, 2009) use a first-order Taylor-series expansion of the rate of change of ozone around mean atmospheric states of ozone mixing ratio, temperature and the overhead ozone column. In comparison to SWIFT, these schemes do not explicitly include the abundance of ODS as a variable in the model. Handling changes in stratospheric ODS abundance requires the repeated determination of production and loss rates and their derivatives. Including the ODS as additional degrees of freedom in the Extrapolar SWIFT model increases its resilience to ODS variability. Moreover, the linear Taylor-series functions tend to produce larger deviations when the rate of change of ozone is not linear with respect to the variability of the three variables. The Extrapolar SWIFT polynomial functions are continuous throughout the stratosphere and can cope with the non-linear parts of the rate of change of ozone.

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 Kreyling (2016).

2 Application of repro-modelling to stratospheric ozone chemistry

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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 Turányi (1994). As a source for the rate of change of ozone we use the comprehensive Lagrangian stratospheric chemistry and transport model ATLAS. The ATLAS model is described in detail in Wohltmann and Rex (2009) and Wohltmann et al. (2010). It contains 49 stratospheric trace gases interacting with each other in over 170 gas-phase and heterogeneous chemical reactions. Together with atmospheric and geographic initial and boundary conditions the differential equation system contains 55 variables and parameters. The rate of change of ozone may be represented as a function of 55 arguments:

$$\begin{array}{}\text{(1)}& {\displaystyle \frac{\mathrm{d}{\text{O}}_{\mathrm{x}}}{\mathrm{d}t}}=\widehat{F}({x}_{\mathrm{1}},\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{2}},\mathrm{\dots},\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{55}}),\end{array}$$

where O_{x} is the VMR of the odd oxygen family containing O_{3}, O and
O(^{1}D), and $\widehat{F}:{\mathbb{R}}^{\mathrm{55}}\to \mathbb{R}$. The
O_{x} family has a longer chemical lifetime than ozone, which is beneficial to
our approximation approach. Moreover, in the lower and middle stratosphere
odd oxygen almost entirely consists of ozone. Thus in Extrapolar SWIFT O_{x}
substitutes O_{3}.

In order to set up a repro-model, we need to determine a set of
*basic* variables which are sufficient for the parameterization of all
the physical and chemical processes in the full chemical system. The
determination of *basic* variables is a crucial aspect since their
number should be large enough so that the function in Eq. (1) is
approximated with sufficient accuracy. On the other hand, their number should
be as small as possible so that the repro-model is numerically efficient.
This is partly achieved by lumping the chemical species into chemical
families. The following four chemical families are relevant for ozone depletion
in the stratosphere and therefore constitute four of the *basic*
variables:

$$\begin{array}{ll}{\displaystyle}{\mathrm{Cl}}_{y}& {\displaystyle}=\underset{\mathrm{short}\text{-}\mathrm{lived}}{\underbrace{\mathrm{Cl}+\mathrm{ClO}+{\mathrm{Cl}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{2}}}}+\underset{\mathrm{reservoir}}{\underbrace{{\mathrm{ClONO}}_{\mathrm{2}}+\mathrm{HCl}}}\\ {\displaystyle}{\mathrm{Br}}_{y}& {\displaystyle}=\underset{\mathrm{short}\text{-}\mathrm{lived}}{\underbrace{\mathrm{Br}+\mathrm{BrO}+\mathrm{HBr}+\mathrm{HOBr}}}+\underset{\mathrm{reservoir}}{\underbrace{{\mathrm{BrONO}}_{\mathrm{2}}+\mathrm{BrCl}}}\\ {\displaystyle}{\mathrm{NO}}_{y}& {\displaystyle}=\underset{\mathrm{short}\text{-}\mathrm{lived}}{\underbrace{\mathrm{N}+\mathrm{NO}+{\mathrm{NO}}_{\mathrm{2}}+{\mathrm{NO}}_{\mathrm{3}}}}+\underset{\mathrm{reservoir}}{\underbrace{{\mathrm{HNO}}_{\mathrm{3}}}}\\ {\displaystyle}{\mathrm{HO}}_{y}& {\displaystyle}=\underset{\mathrm{short}\text{-}\mathrm{lived}}{\underbrace{\mathrm{H}+\mathrm{OH}+{\mathrm{HO}}_{\mathrm{2}}}}+\underset{\mathrm{reservoir}}{\underbrace{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}\end{array}$$

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_{x} or BrO_{x}) are sufficiently close to chemical equilibrium determined
by the local conditions (e.g. pressure, temperature, radiation and the
abundance of reaction partners). Consequently, in the chemical families
containing only one reservoir gas (NO_{y} and HO_{y}) the concentration of
the short-lived species is uniquely determined by the abundance of the total
family; i.e. we assume local chemical equilibrium between the short-lived and
reservoir species. For Cl_{y} and Br_{y} the partitioning between the
reservoir species needs to be considered. However, in most regions of the
extrapolar stratosphere the lifetime of ClONO_{2} is shorter than the timescales of vertical or meridional transport so that ClONO_{2} also comes
close to equilibrium state. The same can certainly be assumed for BrONO_{2},
which has an even shorter lifetime than ClONO_{2}.

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
*basic* variables. Table 1 summarizes the nine *basic*
variables we have identified. The column “Remarks” points out different
properties and processes parameterized by the variable. A function of these
nine
variables (Eq. 2) sufficiently approximates the function in
Eq. (1), but reduces the dimensionality from 55 to 9.

$$\begin{array}{}\text{(2)}& {\displaystyle}\mathrm{\Delta}{\mathrm{O}}_{\mathrm{x}}=F\left(\mathit{\varphi},z,T,\text{top}{\mathrm{O}}_{\mathrm{3}},{\mathrm{Cl}}_{y},{\mathrm{Br}}_{y},{\mathrm{NO}}_{y},{\mathrm{HO}}_{y},{\mathrm{O}}_{\mathrm{x}}\right),\end{array}$$

where ΔO_{x} is the rate of change of ozone over 24 h and
$F:{\mathbb{R}}^{\mathrm{9}}\to \mathbb{R}$. After determining an approximation
for $F\approx \stackrel{\mathrm{\u0303}}{F}$ in Eq. (2) by SWIFT, the chemical
change of the ozone VMR at each grid point in a GCM simulation can be
calculated by Eq. (3).

$$\begin{array}{ll}{\displaystyle}{\mathrm{O}}_{\mathrm{x}}(t+\mathrm{24}\phantom{\rule{0.125em}{0ex}}\mathrm{h})& {\displaystyle}={\mathrm{O}}_{\mathrm{x}}\left(t\right)+\mathrm{\Delta}{\mathrm{O}}_{\mathrm{x}}\left(t\right)\cdot \mathrm{24}\phantom{\rule{0.125em}{0ex}}\mathrm{h}\\ {\displaystyle}& {\displaystyle}\approx {\mathrm{O}}_{\mathrm{x}}\left(t\right)+\stackrel{\mathrm{\u0303}}{F}\left(\mathit{\varphi}\right(t),z(t),T(t),\text{top}{\mathrm{O}}_{\mathrm{3}}(t),{\mathrm{Cl}}_{y}(t),\\ \text{(3)}& {\displaystyle}& {\displaystyle}{\mathrm{Br}}_{y}\left(t\right),{\mathrm{NO}}_{y}\left(t\right),{\mathrm{HO}}_{y}\left(t\right),{\mathrm{O}}_{\mathrm{x}}\left(t\right))\cdot \mathrm{24}\phantom{\rule{0.125em}{0ex}}\mathrm{h}\end{array}$$

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 ≤ 4). The polynomial uses the same nine *basic*
variables as in Eq. (2) and yields the rate of change of ozone
over 24 h. The ΔO_{x} function in Eq. (2) can be
approximated by a polynomial function $\stackrel{\mathrm{\u0303}}{F}$:

$$\begin{array}{}\text{(4)}& {\displaystyle}F({x}_{\mathrm{1}},\mathrm{\dots},{x}_{\mathrm{9}})\approx \stackrel{\mathrm{\u0303}}{F}({x}_{\mathrm{1}},\mathrm{\dots},{x}_{\mathrm{9}})=\sum _{j=\mathrm{1}}^{n}{c}_{j}{f}_{j}({x}_{\mathrm{1}},\mathrm{\dots},{x}_{\mathrm{9}}),\end{array}$$

where ${x}_{\mathrm{1}},\mathrm{\dots},{x}_{\mathrm{9}}$ represents the *basic* variables, *f*_{j}
represents
polynomial terms (e.g. ${f}_{\mathrm{1}}={x}_{\mathrm{1}}^{\mathrm{2}}{x}_{\mathrm{2}}$) and *c*_{j} represents their coefficients for
*j*=1…*n*, where *n* is the number of polynomial terms. For a
polynomial function of fourth degree with nine variables, the maximum number of
terms is 715, including all mixed terms. The coefficients *c*_{j} in
Eq. (3) are determined such that the rate of change of O_{x}
is calculated by the ATLAS CTM for *m* different values of the *basic*
variables ${x}_{i\mathrm{1}},\mathrm{\dots},{x}_{i\mathrm{9}}$, $i=\mathrm{1},\mathrm{\dots},m$, which are approximated with
best accuracy:

$$\begin{array}{}\text{(5)}& {\displaystyle}F({x}_{i\mathrm{1}},\mathrm{\dots},{x}_{i\mathrm{9}})\approx \sum _{j=\mathrm{1}}^{n}{c}_{j}{f}_{j}({x}_{i\mathrm{1}},\mathrm{\dots},{x}_{i\mathrm{9}}).\end{array}$$

The *m* different values of the *basic* variables will be referred to
as training data points or the training data set. In order to write
Eq. (4) in matrix notation we define an *m*×*n* matrix
**A** with ${\mathbf{A}}_{ij}={f}_{j}({x}_{i\mathrm{1}},\mathrm{\dots},{x}_{i\mathrm{9}})$ and a
vector ** F** with ${F}_{i}=F({x}_{i\mathrm{1}},\mathrm{\dots},{x}_{i\mathrm{9}})$ , $i=\mathrm{1},\mathrm{\dots},m$. The polynomial coefficients

$$\begin{array}{}\text{(6)}& {\displaystyle}\mathbf{A}\mathit{c}=\mathit{F}.\end{array}$$

To determine ** c** we employ the least-squares method, which is to
minimize the Euclidian norm ($\Vert \Vert $) of the deviation between the
approximation and

$$\begin{array}{}\text{(7)}& {\displaystyle}\Vert \mathbf{A}\mathit{c}-\mathit{F}\Vert \to min.\end{array}$$

The minimization in Eq. (6) can be made more efficient and
numerically stable by first transforming the matrix **A** into an
orthogonal matrix. Spivakovsky et al. (1990) achieve this with
successive Householder transformations which finally yield the
**QR** decomposition of matrix **A**.
Turányi (1994) and Lowe and Tomlin (2000) use
the Gram–Schmidt process for orthogonalization. The literature suggests
(e.g. Golub and Van Loan, 1996) that the unmodified Gram–Schmidt process
has worse numerical properties which can impair the orthogonalization. In our
approach we are using a **QR** decomposition based on the Householder
transformation.

We start the fitting procedure with one polynomial term (*n*=1) on the right-hand side of Eq. (4). During the following iterations the
polynomial function is consecutively extended by one additional term. This
corresponds to an extension of the matrix **A** by one column.
Turányi (1994) started the approximation with the
constant term and continued with linear terms, then quadratic terms and so on
up to terms of maximum degree, also including all mixed terms. In each
iteration the residuum $\Vert \mathbf{A}\mathit{c}-\mathit{F}\Vert $ was calculated.
If the current residuum was reduced by a certain threshold relative to the
previous residuum, then the current term was accepted to be added to the
polynomial function. This method tested the terms in a given arbitrary order.
If the order of testing had been a different one, other polynomial terms
would be accepted and the overall quality of the polynomial function could
potentially be better.

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 **A**) are
fitted as there are training data points (corresponding to rows of
**A**) then the linear equation system in Eq. (5) is no
longer
overdetermined. In this case any small-scale structure originating
from the random distribution of training data points would have been fitted
and the polynomial function would contain an impractically large number of
terms. An overfitted polynomial function actually causes the residuum to be
higher when it is applied to an independent data set (i.e. not a subset of
the data the polynomial was fitted to). Consequently the fitting procedure
should be terminated before the random fluctuations in the training data set
are fitted. This termination criterion can be defined by applying the
selected polynomial terms and their coefficients to an independent data set
instead of the training data set. The independent data set is named the testing
data set here. The quality of the approximation is expressed by

$$\begin{array}{}\text{(8)}& {\displaystyle}r=\Vert {\mathbf{A}}^{\mathrm{Test}}\mathit{c}-{\mathit{F}}^{\mathrm{Test}}\Vert \phantom{\rule{0.125em}{0ex}},\end{array}$$

where **A**^{Test} is like the matrix **A**, only the
rows of **A**^{Test} correspond to the testing data points and
the vector *F*^{Test} contains the rate of change of ozone at
the testing data points. The polynomial coefficients *c* are the ones
determined via Eq. (6), and *r* is the residuum corresponding to
the polynomial function with one temporarily added term. At some iteration
during the fitting procedure, the residuum *r* will not be reduced by any of
the available additional terms. This defines the termination of the
approximation algorithm.

It is important that the testing data set has the same probability
distribution of *basic* variables as the training data set. We achieve
this by randomly separating the output of the ATLAS simulations into the
training and the testing data set, containing 2 ∕ 3 and 1 / 3 of the
total output, respectively.

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 *τ* of O_{x}.

$$\begin{array}{}\text{(9)}& {\displaystyle}\mathit{\tau}={\displaystyle \frac{\left[{\mathrm{O}}_{\mathrm{x}}\right]}{R}},\end{array}$$

where [O_{x}] is the concentration instead of VMR and *R* is the sum of the
rates of all O_{x}-depleting catalytic cycles. In Fig. 1
the mean chemical lifetime of O_{x} taken from ATLAS data for January is
displayed. The contour labels specify the lifetime in days. In the lower
stratosphere the O_{x} lifetimes exceed 365 days. The longer lifetimes in the
lower stratosphere are a consequence of the slower reaction rates of the
catalytic O_{x}-loss cycles mostly due to fewer O atoms. The O
atom is produced via the photolysis of O_{3}, but its vertical
distribution is mainly controlled by the three-body reaction $\mathrm{O}+{\mathrm{O}}_{\mathrm{2}}+\mathrm{M}\to {\mathrm{O}}_{\mathrm{3}}+\mathrm{M}$. The rate constant of
this reaction increases with increasing pressure. The latitudinal (and
seasonal) variation of the O_{x} lifetime reflects the varying length of the
day and the attenuation of solar radiation on its way through the atmosphere.

Above roughly 30 km of altitude the mean lifetime of O_{x} is shorter than
vertical and meridional transport timescales. In this quasi-chemical
equilibrium state the O_{x} concentration is determined by the local
meteorological conditions and the abundance of ODS. Consequently O_{x}
can be calculated as a function *F*_{eq} of the previously mentioned
*basic* variables, but without the O_{x} VMR itself, so that
${F}_{\mathrm{eq}}:{\mathbb{R}}^{\mathrm{8}}\to \mathbb{R}$.

$$\begin{array}{}\text{(10)}& {\displaystyle}{\mathrm{O}}_{\mathrm{x}}={F}_{\mathrm{eq}}(\mathit{\varphi},\mathrm{z},\mathrm{T},{\text{topO}}_{\mathrm{3}},{\text{Cl}}_{y},{\text{Br}}_{y},{\text{NO}}_{y},{\text{HO}}_{y})\end{array}$$

Accordingly, in the upper stratosphere the O_{x} VMR at a point in time *t*+Δ*t* is a function of eight *basic* variables at time *t*+Δ*t*.
The function *F*_{eq} can also be approximated by a polynomial
function ${\stackrel{\mathrm{\u0303}}{F}}_{\mathrm{eq}}$:

$$\begin{array}{}\text{(11)}& {\displaystyle}{\text{O}}_{\mathrm{x}}(t+\mathrm{\Delta}t)\approx {\stackrel{\mathrm{\u0303}}{F}}_{\mathrm{eq}}\left({x}_{\mathrm{1}}\right(t+\mathrm{\Delta}t),\mathrm{\dots},{x}_{\mathrm{8}}(t+\mathrm{\Delta}t\left)\right),\end{array}$$

where ${x}_{\mathrm{1}},\mathrm{\dots},{x}_{\mathrm{8}}$ corresponds to the eight variables in
Eq. (9). In SWIFT the polynomial functions calculate
ΔO_{x} and determine the O_{x} VMR of the next time step as in
Eq. (3). The O_{x} VMR at time *t*+Δ*t* is a function of
the nine *basic* variables ${x}_{\mathrm{1}}\left(t\right),\mathrm{\dots},{x}_{\mathrm{9}}\left(t\right)$.

$$\begin{array}{}\text{(12)}& {\displaystyle}{\text{O}}_{\mathrm{x}}(t+\mathrm{\Delta}t)={\text{O}}_{\mathrm{x}}\left(t\right)+\stackrel{\mathrm{\u0303}}{F}\left({x}_{\mathrm{1}}\right(t),\mathrm{\dots},{x}_{\mathrm{9}}(t\left)\right)\cdot \mathrm{24}\phantom{\rule{0.125em}{0ex}}\mathrm{h}\end{array}$$

Both equations (Eqs. 10 and 11) yield O_{x}(*t*+Δ*t*). Setting them equal results in

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\mathrm{O}}_{\mathrm{x}}\left(t\right)+\stackrel{\mathrm{\u0303}}{F}\left({x}_{\mathrm{1}}\right(t),\mathrm{\dots},{x}_{\mathrm{9}}(t\left)\right)\cdot \mathrm{24}\phantom{\rule{0.125em}{0ex}}\mathrm{h}\\ {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}={\stackrel{\mathrm{\u0303}}{F}}_{\mathrm{eq}}\left({x}_{\mathrm{1}}\right(t+\mathrm{\Delta}t),\mathrm{\dots},{x}_{\mathrm{8}}(t+\mathrm{\Delta}t\left)\right)\\ {\displaystyle}& {\displaystyle}\stackrel{\mathrm{\u0303}}{F}\left({x}_{\mathrm{1}}\right(t),\mathrm{\dots},{x}_{\mathrm{9}}(t\left)\right)\cdot \mathrm{24}\phantom{\rule{0.125em}{0ex}}\mathrm{h}\\ \text{(13)}& {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}={\stackrel{\mathrm{\u0303}}{F}}_{\mathrm{eq}}\left({x}_{\mathrm{1}}\right(t+\mathrm{\Delta}t),\mathrm{\dots},{x}_{\mathrm{8}}(t+\mathrm{\Delta}t\left)\right)-{\mathrm{O}}_{\mathrm{x}}\left(t\right).\end{array}$$

This means that in the quasi-equilibrium region of O_{x}, $\stackrel{\mathrm{\u0303}}{F}$ is the
result of the O_{x} polynomial function in equilibrium
${\stackrel{\mathrm{\u0303}}{F}}_{\mathrm{eq}}$ minus the linear O_{x} term. However, the polynomial
function $\stackrel{\mathrm{\u0303}}{F}$ contains various O_{x} terms of higher degree. These
terms together with the higher O_{x} VMR in the upper stratosphere cause
rather large errors. Consequently, the polynomial function $\stackrel{\mathrm{\u0303}}{F}$ is not
suited to be used in the region where O_{x} is in quasi-chemical equilibrium.
The altitude of 30 km roughly marks the transition between the equilibrium
and non-equilibrium state of O_{x}. The lifetime is roughly 14 days in this
altitude (see Fig. 1). We defined the 14-day contour to
be the upper boundary up to which the polynomial functions can be used or
rather up to which the training and testing data sets reach.

Since the lifetime of O_{x} is a function of the incoming solar radiation,
the altitude and tilt of the 14-day contour also depends on the season. In
the course of the year the tilt of the lifetime contours will shift. For each
monthly polynomial function we defined a separate upper boundary. In the
quasi-equilibrium region (upper stratosphere) the SWIFT simulations currently
require ozone values interpolated from stratospheric climatologies. For the
future a similar repro-modelling approach will be applied to function
*F*_{eq} in Eq. (9) by fitting the O_{x} VMR directly
instead of the rate of change of O_{x}.

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 ΔO_{x} regime.

The lower boundary of the ΔO_{x} regime is set to 15 km of pressure
altitude (roughly 120 hPa). In the tropics 15 km is approximately the
altitude of the tropical tropopause layer (TTL) defining the boundary between
tropospheric and stratospheric air. In the extratropical regions ozone-rich
air can also be found below 15 km, especially in the northern high
latitudes. However, at theses altitudes and latitudes the rate of change of
ozone is close to zero and the transport of ozone is much more relevant (see
also Fig. 1). When running Extrapolar SWIFT in a GCM,
treating ozone as a passive tracer below 15 km of pressure altitude is
recommended.

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 Lait (1994) (with *θ*_{0}=475 K). In the
vertical, the specified vertical extent of the Polar SWIFT domain goes from
roughly 18 to 27 km of pressure altitude. Above and below Polar SWIFT the
extrapolar module is used, although the rate of change of ozone is close to
zero during polar night.

The monthly training and testing data set for Extrapolar SWIFT are generated
with the stratospheric Lagrangian chemistry and transport model ATLAS
(Wohltmann and Rex, 2009; Wohltmann et al., 2010). The data used in this work
originated from two 2.5-year simulations, one from November 1998 to March
2001 and the second from November 2004 to March 2007. The chemistry module of
ATLAS contains a comprehensive set of gas-phase chemical reactions and a
heterogeneous chemistry scheme. Photolysis and reaction coefficients are
taken from the recent Jet Propulsion Laboratory (JPL) catalog
(Sander et al., 2011). All partial species of the 4 ozone-depleting
chemical families (Cl_{y}, Br_{y}, NO_{y} and
HO_{y}) are included in the 49 ATLAS species. The individual species
are initialized from different sources. The VMR of H_{2}O, N_{2}O,
HCl, O_{3}, CO and HNO_{3} were initialized from the
Aura Microwave Limb Sounder (MLS) climatologies for the 1998–2001
simulation. The 2004–2007 simulation used the measurements of Aura MLS
directly (Waters et al., 2006). The VMR of CH_{4} and NO_{2}
(substitute for NO_{x}) were taken from climatologies of the HALogen
Occultation Experiment (HALOE) instrument (Grooß and Russell III, 2005).
Initial values for Cl_{y} and Br_{y} were derived from
tracer–tracer correlations to CH_{4} and N_{2}O measured during an
aircraft and ballooning campaign described in Grooß et al. (2002).
The ATLAS trajectories are initialized in roughly 2 km thick pressure
altitude layers with a horizontal resolution of 200 km. On each trajectory
the chemistry is calculated like in a chemical box model. ATLAS solves a
coupled system of differential equations to obtain the rate of change of the
trace gases. The stiff numerical solver uses an automatic adaptive time step
and is based on the numerical differentiation formulas (Shampine and Reichelt, 1997).
After 24 h (mixing time step) the mixing algorithm merges or creates
trajectories and interpolates the chemical species accordingly. The ATLAS
trajectories are driven by ERA-Interim wind fields, temperatures and heating
rates (Dee et al., 2011).

For each month of the
year, daily snapshot values of the *basic* variables at the current
location of the trajectories and the corresponding ΔO_{x} at
a fixed time of day (00:00 UTC) are compiled into a data set which is later
split into a training and testing data set. The number of trajectories
computed in an average ATLAS run is roughly 10^{5} throughout the lower and
middle stratosphere. In order not to exceed the size of the computer's main
memory, a random subsample of the 10^{5} trajectories of each day of a month
was taken. This was done so that all monthly data sets contain the same
amount of data: 8 million and 4 million data points in the training and
testing data set, respectively. The monthly data sets are chosen such that
they also contain a fraction of data from the 10 days preceding and following
the current month. We do this in order to ensure a smoother transition of
polynomial functions from one month to the next.

Individual chemical species in ATLAS are grouped into their respective
families and summed up to generate the mixing ratios of Cl_{y}, Br_{y} and
NO_{y}. HO_{y} is simply substituted by water vapour, since the H_{2}O
VMR is a factor of 1000 larger than the sum of all other HO_{y}
constituents. The ΔO_{x} value is defined by the difference of the
O_{x} VMR between two snapshots along the Lagrangian trajectory. A
ΔO_{x} value is associated with the beginning of a 24 h
period:

$$\begin{array}{}\text{(14)}& {\displaystyle}\mathrm{\Delta}{\mathrm{O}}_{\mathrm{x}}\left(t\right)\cdot \mathrm{24}\phantom{\rule{0.125em}{0ex}}\mathrm{h}={\mathrm{O}}_{\mathrm{x}}(t+\mathrm{24}\phantom{\rule{0.125em}{0ex}}\mathrm{h})-{\mathrm{O}}_{\mathrm{x}}\left(t\right).\end{array}$$

Before the fitting procedure, the *basic* variables are normalized to
a range from 0 to 1. Otherwise the order of magnitude of the polynomial
coefficients would vary extremely due to the strongly varying magnitude of
the *basic* variables (e.g. pressure altitude ≈10^{4} m vs.
Br_{y} VMR $\approx {\mathrm{10}}^{-\mathrm{11}}$).

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.

3 Validity of repro-model in a changing climate

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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
*basic* variables can also be expected, e.g. changes in temperature
and meridional circulation. The resilience of SWIFT to such trends is
outlined in Fig. 2. Future climate scenarios will shift the
current probability density function (PDF) of the *basic* variables.
The schematic in Fig. 2 shows a shift of the temperature
PDF, assuming a normal distribution of the temperature, with the eight other
*basic* variables fixed. Most of the PDF in the training climate
(blue) and the future climate (orange) overlaps. The slightly colder
conditions of the future climate are thus mostly covered by the present
domain. Only at low temperatures when the probability is small do outliers
(red) occur. These outliers will force the polynomial functions to
extrapolate and likely produce erroneous ΔO_{x} values.
Therefore outliers need to be identified and the extrapolation must be
prevented.

Apart from a PDF shift like the one illustrated in Fig. 2,
there can be scenarios in which the shift of the PDF is too severe and the
repro-model cannot be applied. An example would be the reduction of
stratospheric chlorine by 50 %. The majority of the Cl_{y} PDF would be
outside the original PDF. In such a case the repro-model needs to be
refitted to an adjusted training climate, which can easily be done by
running the full ATLAS model for a few years driven by output from a climate
model or with modified levels of the ODS.

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 *basic* variables which do not occur in the
stratosphere (e.g. warm temperatures in the lowermost stratosphere).
Consequently less than 0.1 % of the grid cells are actually sampled by
the training data set. Using efficient ways to store and search this sparse
data set would be a feasible option for identifying outliers. However, in our
approach we make use of the regular grid but go one step further. Again we
employ a fitting procedure to determine a polynomial function that yields
positive values inside the sampled domain and negative values outside. This
polynomial function is hereafter called the domain polynomial. The regular grid
sampled by the training data set will be referred to as the training grid and is
used for fitting the domain polynomial. The domain polynomial is obtained
in the following way. First the cells of the training grid are assigned
either positive or negative values. The positive values (inside the domain)
are derived from the number of neighbouring cells also sampled as being inside
the domain. Outside the domain the cells are assigned negative values derived
from the cell's distance to the closest cell inside the domain. In order to
improve the quality of the fit at the domain boundary, some smoothing
operations were applied. By removing individual cells being isolated in the
opposing region the transition from positive to negative values becomes more
smooth. Additionally we removed outside cells which are adjacent to one cell
inside the domain, but not to any other. These cells are assigned values of
only −1 but are actually surrounded by outside cells with much lower
values. Finally the grid cells which were assigned values close to zero are
copied multiple times in the training grid to increase the weight of this
region during the fit.

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 *basic* variables in order to determine whether
these values reside inside or outside the domain of definition of the
original polynomial function. If inside, the ΔO_{x} is calculated as
usual. If the values of the nine *basic* variables prove to be outside,
we need to determine a close location inside the domain of definition, where
a ΔO_{x} can be calculated safely. Newton's method is applied to find
a nearby null of the domain polynomial, which defines the boundary of the
domain of definition. Within a certain margin of the null (±0.5) the
iteration of Newton's method is stopped and the ΔO_{x} value is
calculated at the current coordinates in the nine-dimensional space. An
advantage of using the domain polynomial is that its derivatives can be
computed easily and used in Newton's method.

4 Validation of polynomial functions

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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. 3 the ΔO_{x} in ATLAS and Extrapolar
SWIFT is displayed as zonal averages. The ATLAS ΔO_{x} is taken from
the testing data sets and the SWIFT ΔO_{x} from the polynomial
functions evaluated on the testing data set. The four months shown (January,
April, July and October) are selected as representative of each season. The
data are binned into equivalent latitude (5^{∘}) vs. pressure altitude
(1 km) bins and averaged. Grey shaded bins either mark areas outside the
ΔO_{x} regime (e.g. polar vortex, upper or lower regime boundary) or
indicate too few trajectories to yield a meaningful average. Since the
effective area of the zonal bands decreases towards the poles, the bins with
too few trajectories are found in high latitudes.

In general all four months show good agreement between ATLAS and SWIFT.
Especially in the tropics and mid-latitudes the amplitude of ΔO_{x}
and the extent of regions of production or loss compare very well. Even
detailed structures like the two local maxima in the tropical ozone
production region in January are visible in SWIFT. Steep gradients of
ΔO_{x}, e.g. around 25 km at mid-latitudes, are well reproduced by the
polynomial functions. Deviations between ATLAS and SWIFT occur at the upper
boundary of the summer hemisphere and in high latitudes at the beginning of
the winter season (e.g. Southern Hemisphere in April, Northern Hemisphere in
October). In the nine-dimensional hyperspace some boundary regions of the
training and testing data set are less densely populated with trajectories
than more central regions. This can have different causes, but the most
obvious one is the spatial difference of the trajectory density caused by the
mixing algorithm in ATLAS (see Sect. 2.4). Moreover the extreme
values of some of the nine *basic* variables occur less frequently if
they are approximately Gaussian distributed. Finally the selection criteria
for the trajectories described in Sect. 2.3 can cause sparsely
sampled regions; for example, due to the variability of the polar vortex in
1 year,
the January data set will include certain polar latitudes which will not be
included in the January of the next year. In regions with lower trajectory
density fewer squared errors need to be minimized during the least-squares
minimization. Consequently these regions have less weight in the
approximation than more densely sampled regions and the deviations will be
larger. However, we decided not to manipulate the trajectory density in the
training and testing data sets because we wanted to maintain the frequency
with which meteorological and chemical conditions occur in ATLAS. A sparsely
populated region in the nine-dimensional space implies infrequent and therefore
less relevant stratospheric conditions.

To estimate the error of Extrapolar SWIFT, we examine the difference of ATLAS
ΔO_{x} minus SWIFT ΔO_{x} divided by the O_{x} VMR.

$$\begin{array}{}\text{(15)}& {\displaystyle}Q={\displaystyle \frac{\mathrm{\Delta}{\mathrm{O}}_{{x}_{\mathrm{SWIFT}}}-\mathrm{\Delta}{\mathrm{O}}_{{x}_{\mathrm{ATLAS}}}}{{\mathrm{O}}_{\mathrm{x}}}}\cdot \phantom{\rule{0.125em}{0ex}}\mathrm{100}\end{array}$$

*Q* is given in units of percent per day [% day^{−1}]. *Q* describes
the positive or negative percentage drift of O_{x} VMR per day due to the
error in SWIFT. The division by O_{x} makes the differences at small and high
O_{x} VMR more comparable, instead of just interpreting the absolute
deviation. Similar to the relative error, *Q* tends to have larger values for
very small O_{x} VMR. These properties of *Q* need to be taken into account
when considering different regions with high or low ozone VMR. In the lower
tropical stratosphere where very small O_{x} VMR can be found, the absolute
errors of SWIFT are small in contrast to the *Q* values, which can exceed
±50 % day^{−1}. However, for the calculation of the total ozone
column the deviations at small O_{x} VMR are irrelevant. Also for the
computation of the atmospheric heating rates based on the SWIFT ozone field,
the absolute errors originating from other greenhouse gases (e.g.
CO_{2}) with a much higher concentration are much more important than
the deviations at small O_{x} VMR. In Fig. 4 we discuss the
distribution of *Q*. In Sect. 5 we use the absolute
deviations between the SWIFT and the ATLAS simulation to discuss the error
quantitatively.

Figure 4 shows the probability distribution of *Q* for the four
representative months January, April, July and October. As in
Fig. 3 the *Q* values of the roughly 4 million data points of
each monthly testing data set are discussed. The bin width of 1 bar in
Fig. 4 is 0.2 % day^{−1}. Thus over 20 % of
*Q* values reside within the interval of ±0.1 % day^{−1} in all
four months. The majority of *Q* values lie within the
±1 % day^{−1} interval. The mean (pink dashed line) and the median
(cyan dotted line) are close to zero. The strongest systematic biases (mean)
are −0.3 % day^{−1} in July and +0.25 % day^{−1} in
October; the median, however, is centred very close to 0.0 % day^{−1}
in both months. The grey shaded area shows the standard deviation (SD) around
the mean. The variability of the SD indicates that the quality of the
approximation actually varies significantly between the months. The errors of
the October polynomial function (SD of 3.5 % day^{−1}) are spread more
strongly than the errors of the April polynomial function (SD is roughly
0.6 % day^{−1}).

As mentioned before, individual *Q* values can surpass
±50 % day^{−1} when the O_{x} VMR is small, i.e. below 100 ppb.
But these extreme deviations are rare, which is demonstrated by the 5 and
95 % quantiles (black dotted lines); 90 % of the total *Q* values are
located between the two quantile lines.

5 Simulations with SWIFT

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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_{y}, Br_{y}, NO_{y} and HO_{y}.
We compiled monthly zonal climatologies to be distributed with the model if
required. The H_{2}O climatology (substituting the HO_{y} family) is
based on extensive observational data from Aura MLS. The Cl_{y}, Br_{y} and
NO_{y} climatologies are composed of the two ATLAS simulations used in the
training and testing data sets. All species in ATLAS contributing to one of
the chemical families are summed up and weighted according to their yield of
active chlorine, bromine or NO_{x}. The initialization of
chemical species for the two ATLAS simulations was described in
Sect. 2.4. For initialization and regions outside the
ΔO_{x} regime, an additional O_{x} climatology is required. This
climatology is also compiled from an extensive set of ozone measurements by
Aura MLS.

The SWIFT in ATLAS simulations are driven by ERA-Interim data
(Dee et al., 2011). Every 24 h the SWIFT module is called and the
rate of change of ozone is calculated based on the current conditions at the
beginning of each trajectory. The VMRs of the four ozone-depleting chemical
families are interpolated from the trace gas climatologies. Latitude,
pressure altitude and atmospheric temperature are defined by the trajectory
and the overhead ozone column is integrated from the ozone values of the
overhead trajectories. In combination with these eight parameters, the ozone VMR
of the last time step (24 h before) is used to calculate the rate of change
of ozone (ΔO_{x}) by evaluating the polynomial function. Eventually
the ΔO_{x} is added to the O_{x} VMR from the last time step,
according to Eq. (3). In order to smooth the transition between
two polynomial functions corresponding to consecutive months, we linearly
interpolate between the ΔO_{x} results of the two polynomial
functions. All other components of the ATLAS CTM, like the trajectory
transport or the mixing algorithm, remain unchanged. The SWIFT in ATLAS
simulations apply outlier handling as described in Sect. 3.2.

Above the seasonally dependent upper boundary of the ΔO_{x} regime, as
introduced in Sect. 2.3, climatology values of O_{x} are used
in the simulation. In a layer that extends over 2 km below this upper
boundary the O_{x} VMRs are determined by computing an altitude-weighted
average between values from the climatological O_{x} values and the
ΔO_{x} regime. Inside the polar vortex O_{x} climatology values are
used. The Polar SWIFT module is intentionally switched off to investigate
only the performance of the extrapolar module.

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_{y}, Br_{y},
NO_{y}, H_{2}O and O_{x}, but uses zonally and daily averaged trace gas VMRs
instead. These daily values are compiled from the reference ATLAS simulation.
Thus, apart from the averaging, the background trace gas fields are identical
in both simulations. Further, the simulation covers a 2-year time period
which coincides with the period from which half of the training data
originated (years 2005 and 2006). By selecting this simulation period we
ensure that the SWIFT polynomial functions were trained with the
stratospheric conditions of those years. In other words, the errors cannot be caused by
stratospheric variability unknown to SWIFT.

The panels in Figs. 5 and 6 show monthly
averaged ozone concentrations for the 2-year SWIFT simulation (middle
column). The reference ATLAS simulation is shown in the left column and the
difference between the two in the right column. Since it is the ozone
concentrations and total ozone columns that are crucial for the feedback of
ozone to the model radiation, we have transformed the mixing ratios produced
by SWIFT into ozone concentrations here. In the regions outside the
ΔO_{x} regime, e.g. inside the polar vortex (white contour)
or above the upper boundary (black dashed line), O_{x} values from
the daily averaged O_{x} fields are used.

Figure 5 shows the entire annual cycle of 2005 (first
simulation year) in bimonthly intervals. Figure 6 repeats
the sequence for the second simulation year 2006. Throughout both years SWIFT
shows excellent agreement with the ozone layer of the ATLAS simulation. The
seasonal cycle of the ozone layer is very well reproduced. The average
deviation oscillates at ±0.2 × 10^{12} per cm^{−3}. Over the
course of the year 2005 the positive differences in the lower stratosphere of
the Northern Hemisphere change sign to negative differences in the second
half of the year. This pattern can also be observed in the second simulation
year 2006. If the polynomial functions produce similar deviations in the same
month of different years, we can attribute the deviations to a suboptimal
approximation. However, the discussed deviations are in a region of strong
meridional transport where the residence time of air parcels is sufficiently
short so that no significant accumulation of errors occurs.

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.

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_{y}, Br_{y}, NO_{y} and H_{2}O are the monthly
climatologies described in Sect. 5.1. The simulation starts in
November 1998 and continues until December 2008. This period encompasses both
training data periods, the time between the two and a period after the last
training data period. The bright blue curve in Fig. 7
shows the seasonal and inter-annual variation of the stratospheric ozone
layer simulated by SWIFT. The depicted value is the integrated stratospheric
ozone column in Dobson units from 15 to 32 km of pressure altitude. In order to
observe a strong seasonal signal, we choose to display a location in the
Northern Hemispheric mid-latitudes (Potsdam at 52.4^{∘} N,
13.0^{∘} E). The orange and green shaded years in
Fig. 7 are the simulation periods of the training data
set. The red curve in both periods shows the values of the reference ATLAS
simulation. In both periods SWIFT reproduces the seasonal signal seen in
ATLAS quite well. Especially in the green shaded patch the agreement between
SWIFT and ATLAS seems to be as good as in the orange patch, although SWIFT
was running continuously for 4 years in between. To demonstrate this more
clearly, the scatter plot in Fig. 8 shows daily averaged
ozone columns of SWIFT on the *x* axis vs. the ones from ATLAS on the
*y* axis. The colouring of the dots corresponds to the two time periods in
Fig. 7. The scatter of data points from both periods
overlaps entirely and the magnitude and distribution of deviations from the
diagonal is identical. Clearly the errors of SWIFT did not accumulate over
the course of the previous 6 years.

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. 7). In autumn 2005 and 2006 ATLAS underestimates the ozone columns in comparison to the Aura MLS observations. Since SWIFT is trained with ATLAS data, SWIFT also reproduces this underestimation of about 30 DU and continues underestimating the autumn stratospheric ozone columns in the years 2007 and 2008 (pink shaded patch). During the first half of each year, however, SWIFT matches the Aura MLS columns quite well and even captures the inter-annual variability shown by the observations (compare spring maximum 2007 vs. 2008). The scatter plot in Fig. 9 shows daily averaged ozone columns from the green and pink shaded years. Some amount of deviation in this figure is also caused by the difference in geo-location between the MLS profile and the selected location in the SWIFT simulation (Potsdam). Days on which no MLS measurement was taken in a 200 km radius of Potsdam are excluded, which reduces the total amount of days by about 50 %. Again the colouring of the dots corresponds to the periods in the time series (Fig. 7). As already seen in the monthly means in Fig. 7, SWIFT underestimates the smaller ozone columns (autumn values below 200 DU). Otherwise, the spread of the dots agrees well in both periods, proving that the SWIFT simulation is not less accurate outside the training data period (pink) than under conditions which are part of the training data set (green).

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 ΔO_{x} polynomial functions for the
interpolation between two months. During the preparation of this paper
Extrapolar SWIFT was coupled to the climate model ECHAM6.3. The Fortran
SWIFT code is not fully optimized yet and the current estimates on the
computation time are preliminary. An initial estimate of the increase in
computation time caused by Extrapolar SWIFT is roughly 10 %. In
comparison to an ECHAM version employing full stratospheric chemistry (ECHAM
MESSy Atmospheric Chemistry model, or EMAC), the ECHAM + Extrapolar SWIFT
requires 6–8 times less computation time (only estimated).

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 10^{4} times faster than in the CTM.

6 Conclusions

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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
*basic* variables (latitude, pressure altitude, temperature, overhead
ozone column, total chlorine, total bromine, nitrogen oxide family, water
vapour and the ozone field). At the same time, all physical and chemical
processes contained in the full model output are parameterized in the
repro-model.

Running the Extrapolar SWIFT model requires only the 12 monthly polynomial
functions and information about the nine *basic* variables. The domain of
the polynomial function is defined by the nine-dimensional training data set. A
wide range of stratospheric variability needs to be included in the training
data set to increase the robustness of the polynomial functions. We have
shown that the SWIFT model can cope with a certain degree of unknown
variability induced, for example, by climate change. We estimate that
the polynomial functions can handle changes of up to a 10 % increase or
decrease in stratospheric chlorine loading without adjusting the current
training data set. More extreme changes, e.g. a 50 % reduction of
chlorine, requires an extension of the training data with values of disturbed
chemistry simulations. For handling occasional outliers, i.e. combinations of
the nine *basic* variables outside the domain of definition, Extrapolar
SWIFT includes a procedure to prevent extrapolation of the polynomial
functions.

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 ±15 DU between
ATLAS and SWIFT. The comparison to Aura MLS measurements showed an equally
good agreement with Extrapolar SWIFT, except for the periods of
underestimation of the stratospheric ozone column in autumn. This
underestimation, however, is a bias that originates from the source model
ATLAS. The computation of the solution of a polynomial function with up to
100 terms is significantly faster than solving a chemical differential
equation system. Extrapolar SWIFT requires 10^{4} times less computation time
than the chemistry scheme of the ATLAS CTM.

Code availability

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Code availability.

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 https://zenodo.org/record/1020048.

The ATLAS CTM is available on the AWIForge repository (https://swrepo1.awi.de/). Access to the repository is granted on request. Please contact Ingo.Wohltmann@awi.de. If required, the authors will give support for the implementation of SWIFT and ATLAS.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

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)^{3}” funded by the Deutsche Forschungsgemeinschaft
(DFG). We thank ECMWF for providing reanalysis data and the Aura MLS team for
observational data on stratospheric trace gas constituents.

The article processing charges for this open-access

publication were covered by a Research

Centre of the
Helmholtz Association.

Edited by: Fiona O'Connor

Reviewed by: two anonymous referees

References

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Short summary

The Extrapolar SWIFT model is a fast yet accurate stratospheric ozone chemistry module for global climate models. The importance of feedbacks between the climate system and the ozone layer has been demonstrated in previous studies. Therefore it is desirable to include an interactive ozone layer in climate simulations. However, ensemble simulations in particular have strict computational constraints. The Extrapolar SWIFT model provides an interactive ozone layer with small computational costs.

The Extrapolar SWIFT model is a fast yet accurate stratospheric ozone chemistry module for...

Geoscientific Model Development

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