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 . 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 and . 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: 1dOxdt=F^(x1,x2,…,x55), where Ox is the VMR of the odd oxygen family containing O3, O and O(1D), and F^:R55→R. The Ox 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 Ox substitutes O3.

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. () 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: Cly=Cl+ClO+Cl2O2︸short-lived+ClONO2+HCl︸reservoirBry=Br+BrO+HBr+HOBr︸short-lived+BrONO2+BrCl︸reservoirNOy=N+NO+NO2+NO3︸short-lived+HNO3︸reservoirHOy=H+OH+HO2︸short-lived+H2O︸reservoir

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. ClOx or BrOx) 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 (NOy and HOy) 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 Cly and Bry the partitioning between the reservoir species needs to be considered. However, in most regions of the extrapolar stratosphere the lifetime of ClONO2 is shorter than the timescales of vertical or meridional transport so that ClONO2 also comes close to equilibrium state. The same can certainly be assumed for BrONO2, which has an even shorter lifetime than ClONO2.

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  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. ) sufficiently approximates the function in Eq. (), but reduces the dimensionality from 55 to 9. 2ΔOx=Fϕ,z,T,topO3,Cly,Bry,NOy,HOy,Ox, where ΔOx is the rate of change of ozone over 24 h and F:R9→R. After determining an approximation for F≈F̃ in Eq. () by SWIFT, the chemical change of the ozone VMR at each grid point in a GCM simulation can be calculated by Eq. (). Ox(t+24h)=Ox(t)+ΔOx(t)⋅24h≈Ox(t)+F̃(ϕ(t),z(t),T(t),topO3(t),Cly(t),3Bry(t),NOy(t),HOy(t),Ox(t))⋅24h

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. () and yields the rate of change of ozone over 24 h. The ΔOx function in Eq. () can be approximated by a polynomial function F̃: 4F(x1,…,x9)≈F̃(x1,…,x9)=∑j=1ncjfj(x1,…,x9), where x1,…,x9 represents the basic variables, fj represents polynomial terms (e.g. f1=x12x2) and cj 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 cj in Eq. () are determined such that the rate of change of Ox is calculated by the ATLAS CTM for m different values of the basic variables xi1,…,xi9, i=1,…,m, which are approximated with best accuracy: 5F(xi1,…,xi9)≈∑j=1ncjfj(xi1,…,xi9). 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. () in matrix notation we define an m×n matrix A with Aij=fj(xi1,…,xi9) and a vector F with Fi=F(xi1,…,xi9) , i=1,…,m. The polynomial coefficients cj are grouped into a vector c. Then, the linear equation system in Eq. () can be expressed as 6Ac=F. To determine c we employ the least-squares method, which is to minimize the Euclidian norm (‖‖) of the deviation between the approximation and F: 7‖Ac-F‖→min⁡. The minimization in Eq. () can be made more efficient and numerically stable by first transforming the matrix A into an orthogonal matrix. achieve this with successive Householder transformations which finally yield the QR decomposition of matrix A. and use the Gram–Schmidt process for orthogonalization. The literature suggests e.g. 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. (). 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. 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 ‖Ac-F‖ 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. () 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 8r=‖ATestc-FTest‖, where ATest is like the matrix A, only the rows of ATest correspond to the testing data points and the vector FTest contains the rate of change of ozone at the testing data points. The polynomial coefficients c are the ones determined via Eq. (), 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 Ox. 9τ=[Ox]R, where [Ox] is the concentration instead of VMR and R is the sum of the rates of all Ox-depleting catalytic cycles. In Fig.  the mean chemical lifetime of Ox taken from ATLAS data for January is displayed. The contour labels specify the lifetime in days. In the lower stratosphere the Ox lifetimes exceed 365 days. The longer lifetimes in the lower stratosphere are a consequence of the slower reaction rates of the catalytic Ox-loss cycles mostly due to fewer O atoms. The O atom is produced via the photolysis of O3, but its vertical distribution is mainly controlled by the three-body reaction O+O2+M→O3+M. The rate constant of this reaction increases with increasing pressure. The latitudinal (and seasonal) variation of the Ox 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 Ox is shorter than vertical and meridional transport timescales. In this quasi-chemical equilibrium state the Ox concentration is determined by the local meteorological conditions and the abundance of ODS. Consequently Ox can be calculated as a function Feq of the previously mentioned basic variables, but without the Ox VMR itself, so that Feq:R8→R. 10Ox=Feq(ϕ,z,T,topO3,Cly,Bry,NOy,HOy) Accordingly, in the upper stratosphere the Ox VMR at a point in time t+Δt is a function of eight basic variables at time t+Δt. The function Feq can also be approximated by a polynomial function F̃eq: 11Ox(t+Δt)≈F̃eq(x1(t+Δt),…,x8(t+Δt)), where x1,…,x8 corresponds to the eight variables in Eq. (). In SWIFT the polynomial functions calculate ΔOx and determine the Ox VMR of the next time step as in Eq. (). The Ox VMR at time t+Δt is a function of the nine basic variables x1(t),…,x9(t). 12Ox(t+Δt)=Ox(t)+F̃(x1(t),…,x9(t))⋅24h Both equations (Eqs.  and ) yield Ox(t+Δt). Setting them equal results in Ox(t)+F̃(x1(t),…,x9(t))⋅24h=F̃eq(x1(t+Δt),…,x8(t+Δt))F̃(x1(t),…,x9(t))⋅24h13=F̃eq(x1(t+Δt),…,x8(t+Δt))-Ox(t). This means that in the quasi-equilibrium region of Ox, F̃ is the result of the Ox polynomial function in equilibrium F̃eq minus the linear Ox term. However, the polynomial function F̃ contains various Ox terms of higher degree. These terms together with the higher Ox VMR in the upper stratosphere cause rather large errors. Consequently, the polynomial function F̃ is not suited to be used in the region where Ox is in quasi-chemical equilibrium. The altitude of 30 km roughly marks the transition between the equilibrium and non-equilibrium state of Ox. The lifetime is roughly 14 days in this altitude (see Fig. ). 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 Ox 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 Feq in Eq. () by fitting the Ox VMR directly instead of the rate of change of Ox.

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 ΔOx regime.

The lower boundary of the ΔOx 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. ). 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 (with θ0=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 . 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 . All partial species of the 4 ozone-depleting chemical families (Cly, Bry, NOy and HOy) are included in the 49 ATLAS species. The individual species are initialized from different sources. The VMR of H2O, N2O, HCl, O3, CO and HNO3 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 . The VMR of CH4 and NO2 (substitute for NOx) were taken from climatologies of the HALogen Occultation Experiment (HALOE) instrument . Initial values for Cly and Bry were derived from tracer–tracer correlations to CH4 and N2O measured during an aircraft and ballooning campaign described in . 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 . 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 .

For each month of the year, daily snapshot values of the basic variables at the current location of the trajectories and the corresponding ΔOx 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 105 throughout the lower and middle stratosphere. In order not to exceed the size of the computer's main memory, a random subsample of the 105 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 Cly, Bry and NOy. HOy is simply substituted by water vapour, since the H2O VMR is a factor of 1000 larger than the sum of all other HOy constituents. The ΔOx value is defined by the difference of the Ox VMR between two snapshots along the Lagrangian trajectory. A ΔOx value is associated with the beginning of a 24 h period: 14ΔOx(t)⋅24h=Ox(t+24h)-Ox(t).

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 ≈104 m vs. Bry VMR ≈10-11).

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.

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. . Future climate scenarios will shift the current probability density function (PDF) of the basic variables. The schematic in Fig.  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 ΔOx values. Therefore outliers need to be identified and the extrapolation must be prevented.

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

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.  the ΔOx in ATLAS and Extrapolar SWIFT is displayed as zonal averages. The ATLAS ΔOx is taken from the testing data sets and the SWIFT ΔOx 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 ΔOx 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 ΔOx 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 ΔOx, 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. ). 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.  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 ΔOx minus SWIFT ΔOx divided by the Ox VMR. 15Q=ΔOxSWIFT-ΔOxATLASOx⋅100 Q is given in units of percent per day [% day-1]. Q describes the positive or negative percentage drift of Ox VMR per day due to the error in SWIFT. The division by Ox makes the differences at small and high Ox VMR more comparable, instead of just interpreting the absolute deviation. Similar to the relative error, Q tends to have larger values for very small Ox 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 Ox 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 Ox 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. CO2) with a much higher concentration are much more important than the deviations at small Ox VMR. In Fig.  we discuss the distribution of Q. In Sect.  we use the absolute deviations between the SWIFT and the ATLAS simulation to discuss the error quantitatively.

Figure  shows the probability distribution of Q for the four representative months January, April, July and October. As in Fig.  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.  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 Ox 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.

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 Cly, Bry, NOy and HOy. We compiled monthly zonal climatologies to be distributed with the model if required. The H2O climatology (substituting the HOy family) is based on extensive observational data from Aura MLS. The Cly, Bry and NOy 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 NOx. The initialization of chemical species for the two ATLAS simulations was described in Sect. . For initialization and regions outside the ΔOx regime, an additional Ox 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 . 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 (ΔOx) by evaluating the polynomial function. Eventually the ΔOx is added to the Ox VMR from the last time step, according to Eq. (). In order to smooth the transition between two polynomial functions corresponding to consecutive months, we linearly interpolate between the ΔOx 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. .

Above the seasonally dependent upper boundary of the ΔOx regime, as introduced in Sect. , climatology values of Ox are used in the simulation. In a layer that extends over 2 km below this upper boundary the Ox VMRs are determined by computing an altitude-weighted average between values from the climatological Ox values and the ΔOx regime. Inside the polar vortex Ox 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 Cly, Bry, NOy, H2O and Ox, 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.  and 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 ΔOx regime, e.g. inside the polar vortex (white contour) or above the upper boundary (black dashed line), Ox values from the daily averaged Ox fields are used.

Figure  shows the entire annual cycle of 2005 (first simulation year) in bimonthly intervals. Figure  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 × 1012 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 Cly, Bry, NOy and H2O are the monthly climatologies described in Sect. . 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.  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.  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.  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. . 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. ). 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.  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. ). As already seen in the monthly means in Fig. , 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 ΔOx 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 104 times faster than in the CTM.