Modeling and Numerical Simulation of the Recurrence of Ozone Depletion Events in the Arctic Spring

This paper presents a numerical study of the recurrences (or oscillations) of tropospheric ozone depletion events, ODEs, using the further developed one-dimensional chemistry transport model KINAL-T. Reactive bromine is the major contributor to the occurrence of ODEs. After the termination of an ODE, the reactive bromine in the air is deposited onto aerosols or on the snow surface, and the ozone may regenerate via NOx-catalyzed photochemistry or by turbulent transport from the free troposphere into the boundary layer. The replenished ozone then is available for the next cycle of autocatalytic bromine 5 release (bromine explosion) leading to another ODE. The recurrence periods are found to be as low as five days for the purely chemically NOx-driven oscillation and 30 days for a diffusion-driven recurrence. An important requirement for recurrences of ODEs to occur is found to be a sufficiently strong inversion layer. In a parameter study, the dependence of the recurrence period on the nitrogen oxides concentration, the inversion layer strength, the ambient temperature, the aerosol density, and the solar radiation is investigated. Parameters controlling the recurrence of ODEs are discussed. 10


Introduction
Oscillating chemical systems have been of scientific interest for well over a hundred years.One of the most simple, theoretical chemical oscillation was formulated by Lotka (1909), which are formulated in analogy to the predator-prey equations.Briggs and Rauscher (1973) found "an oscillating iodine clock", an oscillating reaction mechanism involving iodate, which could readily be reproduced in the laboratory.
Oscillations in tropospheric chemistry, involving the species NO x , HO x , CO, and O 3 with oscillation periods in the order of several weeks to centuries were found by several researchers (e.g.White and Dietz, 1984;Poppe and Lustfeld, 1996;Hess and Madronich, 1997;Tinsley and Field, 2001).Kalachev and Field (2001) investigated a system involving the species CO, O 3 , NO, NO 2 , HO, and HO 2 with a total of seven reactions and three emissions.They found an oscillation period of one month and managed to reduce the chemical system to four species.Moreover, low NO x , high HO x , high NO x , and low HO x regimes were identified.An oscillating chemical system can only occur, if the system comprises both non-linearities and feedback cycles.The chemistry of ozone depletion events (ODEs) consists of non-linearities and an auto-catalytic reaction cycle, suggesting the potential for an oscillating system.Tang and McConnell (1996) studied the ozone depletion events using a box model where the recurrence of an ODEs were found after about five days.Evans et al. (2003) found indications for chemical oscillations involving ODEs, where only photochemical recovery of O 3 was considered, and a recurrence period of approximately three days was found.This oscillation time scale is among the fastest found in a model of tropospheric chemistry.The chemical reaction mechanism consists of both gas-phase and aerosol-phase reactions.In addition, the oscillations are driven externally by emissions and depositions.In the present work, an extensive investigation of the oscillation potential of ODEs is conducted, and simulations with conditions similar to those described by Evans et al. (2003) are preformed in order to evaluate the present simulations, which, however, are preformed in a one-dimensional configuration considering a more advanced chemical reaction mechanism and a more sophisticated aerosol treatment.An overview of ODEs is given in the following paragraphs.
ODEs typically occur in the boundary layer in both, the Arctic and Antarctic during spring and sometimes also in fall.During a full ODE, ozone concentrations drop below 1 nmol mol −1 and for partial ODEs to levels of less than 10 nmol mol −1 (e.g.Oltmans, 1981;Bottenheim et al., 1986;Hausmann and Platt, 1994;Frieß et al., 2004;Wagner et al., 2007;Halfacre et al., 2014).Barrie et al. (1988) were the first to find an anti-correlation of the ozone and bromine concentrations during an ODE.Hausmann and Platt (1994) then found experimental evidence for the chemical reaction mechanism that is most likely responsible for the destruction of the ozone by Br atoms, which was suggested by Barrie et al. (1988): Br 2 + hν → 2Br, (R3) ODEs through a recycling of BrO since the reaction of ClO or IO with BrO is approximately one order of magnitude faster than the BrO self-reaction (R2), i.e. (Atkinson et al., 2007) BrO + XO → BrX + O 2 (R15) and with X = Cl or I compared to X = Br.The presence of chloride may also increase the speed of the bromine explosion: In the liquid phase, the reaction of deposited HOBr with chloride (Simpson et al., 2007) occurs at a much faster rate than the reaction with bromide due to the larger concentration of chloride and the higher reaction constant of (R17) compared to (R7).A large fraction of the BrCl can then react with bromide to ultimately produce Br 2 , which is then released into the gas phase.However, some of the deposited HOBr instead releases BrCl, effectively reducing the α described above.Whether the presence of chloride speeds up or slows down the bromine explosion depends on the reduction of α and the quicker release of Br 2 due to reaction (R17).A similar reaction involving HOCl also occurs although at a much smaller reaction rate.
As an alternative to the bromine explosion mechanisms, bromine may be released directly via a net heterogeneous reaction involving ozone (e.g.Oum et al., 1998;Artiglia et al., 2017) The underlying reaction mechanism may be an initial source for bromine, initiating the bromine explosion.The complete set of reactions can be found in Tables A3 and A4 in Appendix A. The release may need sunlight to occur efficiently (Pratt et al., 2013).
The meteorological conditions under which ODEs occur are also still under discussion.Often proposed are shallow, stable boundary layers (Wagner et al., 2001;Frieß et al., 2004;Lehrer et al., 2004;Koo et al., 2012).The inversion layer limits the loss of BrO from the boundary layer and also the replenishing of ozone from aloft.
ODEs occur predominantly at temperatures below −20 • C (Tarasick and Bottenheim, 2002), but could also be observed at temperatures of up to −6 • C (Bottenheim et al., 2009).Pöhler et al. (2010) found a nearly linear decrease of BrO concentrations with increasing temperature in the temperature range from−24 • C to −15 • C. Causes for the temperature dependence probably are a stronger surface-to-air flux of bromine, resulting from a stronger temperature gradient between the warm ice surface and cold air as well as the temperature-dependent reaction constants that may favor an ODE.
Frequently, successions of ODEs are measured at the same location over the year (e.g.Halfacre et al., 2014).To the authors knowledge, recurrences of ODES are hardly discussed in the literature.It is suggested (Bottenheim and Chan, 2006) that their cause is that air containing varying amounts of reactive Br and O 3 may be transported from different locations to the measurement site, leading to recurrence.
Alternatively, ozone in the polar boundary layer may also be replenished in-situ via two mechanisms: Ozone rich air is transported to the polar boundary layer from aloft by turbulent diffusion from the free troposphere.An inversion layer limits the rate of this replenishment.Ozone is also photochemically produced in-situ by the well-known NO x catalyzed O 3 -formation mechanism: NO 2 in turn is produced primarily by the reaction of NO and HO 2 where most of the HO 2 is produced by In the present study it is shown that the chemical system coupled with vertical turbulent diffusion shows periodicity even without horizontal transport.After a bromine explosion, the ozone concentration drops to a negligible level.As a consequence, the formation of BrO via (R5) drops to nearly zero, so that Br instead reacts with HO 2 , aldehydes, or alkenes to form HBr.
HBr then dissolves in the aerosols.Both gas-phase and dissolved HBr are chemically inert.Now that there is no more active bromine to destroy the ozone, the ozone concentration can regenerate by either downward mixing into the boundary layer from the free troposphere or via NO x -catalyzed photochemical O 3 formation.Together with the ozone, the active bromine species can also regenerate.However, due to the nonlinear nature of the bromine explosion, the reactivation speed of the inactive bromine in the aerosols scales with the amount of already active bromine.The reactivation of the inactive bromine thus starts out much slower than the ozone regeneration, allowing ozone to replenish before a new ODE occurs.
In the literature (e.g.Lotka, 1909;Tinsley and Field, 2001;Evans et al., 2003), reactions with periodic variations of some concentrations, such as the recurrences of ODEs, are called chemical oscillations.Since the term oscillation may suggest a constant recurrence period, the more general expression of recurrences is used in the remainder of this paper.
In the present study, the 1D Model KINAL-T (KInetic aNALysis of reaction mechanics with Transport) based on the work of Cao et al. (2016) is employed to calculate the recurrences of ODEs.Finding experimental evidence for recurrences is expected to be very difficult, since meteorological effects such as wind transport conceal the recurrent properties.It may be nearly impossible to disentangle the ozone regeneration via wind transport from the vertical diffusion or from the NO 2 photolysis.
Nevertheless, the present model provides important insight into the recurrences of ODEs.

Model and numerical solver
In the present study, the former model of Cao et al. (2016) is extended and optimized in order to account for the recurrences of ODEs.For simplicity, constant temperature, zero vertical velocity and prescribed turbulent diffusion coefficients (cf.section 2.1.1)are assumed.

The differential equations
The chemical reaction system is described by the temporal and spatial variations of the species concentrations c i,j , where i = 1, . . ., N is the species number and j = 1, . . ., M denotes the discretized grid number.Since the gas temperature is assumed to be constant, density changes of the gas phase are neglected.Using central differences for the discretization of the physical space, the governing equations for the species concentrations yield (Cao et al., 2016) dc i,j dt = P i,j − D i,j chemical production and consumption aerosol mass transfer . (2) The dry deposition term is assumed to be non-zero only in the lowest grid cell, j = 1.The diffusion flux is given by where the molecular diffusion coefficient D = 0.2 cm 2 s −1 and F i,1/2 = 0.In the one-dimensional grid under consideration, z j denotes the position of the center of grid cell j, and h j is the size of the grid cell j, h j = (z j+1 − z j−1 )/2.The turbulent diffusion coefficient at the interface of the grid cell j + 1/2 is denoted by k j+1/2 , cf.Eq. (3).
The evaluation of the turbulent diffusion coefficient needs special attention since its parameterization depends on the meteorological conditions, which will be given in the next subsection.Moreover, the gas-phase reactions and the aerosol treatment will be provided.

Turbulent diffusion coefficient
The height-dependent turbulent diffusion coefficient k(z) is chosen similar as by Cao et al. (2016), using the first-order parameterization of Pielke and Mahrer (1975) for neutrally stratified boundary layers using the following empirical polynomial equation: The discretized turbulent diffusion coefficients are determined by k j+1/2 = k (z j + h j /2).In equation ( 4), L is the boundary layer height up to the inversion layer.L 0 is the height of the surface layer which is assumed to be 10 % of the boundary layer height (Stull, 1988).k 0 = κu * L 0 is the turbulent diffusion coefficient at the top of the surface layer.κ = 0.41 is the von Karman constant and u * = κv/ ln(L 0 /z 0 ) the friction velocity, where v is the reference wind speed at the top of the surface layer, which is assumed to be v = 5 m s −1 .The surface roughness length for snow/ice, z 0 , is taken as z 0 = 10 −5 m (Huff andAbbatt, 2000, 2002).
A relation of L and the vertical potential temperature gradient is described by Neff et al. (2008) as: with the Brunt-Vaisala frequency The Coriolis parameter f = 1.458 × 10 −4 s −1 is calculated at the north pole.g = 9.81 m s −2 is the gravitational acceleration.
For the two different temperatures of T = 258 K and 238 K under consideration, a vertical potential temperature gradient of dΘ/dz = 6.4 × 10 −4 K m −1 and 5.9 × 10 −4 K m −1 , respectively, is considered both of which correspond to the boundary layer height of L = 200 m employed in this work.An inversion layer of thickness L inv is inserted at the top of the boundary layer, where the turbulent diffusion coefficient k t,inv is assumed to be constant and treated as a free parameter.
The turbulent diffusion coefficient k f in the free troposphere is assumed to be constant throughout the free troposphere.The reported values for turbulence in the free troposphere vary strongly between 0.01 and 100 m 2 s −1 (Wilson, 2004;Ueda et al., 2012).
An example of the resulting profile of the turbulent diffusion coefficient as defined through Eq. ( 4) is displayed in Fig. 1  are 10 cm 2 s −1 and 10 m 2 s −1 , respectively; these values refer to the base case discussed further below.The vertical diffusion between the boundary layer and the free troposphere is limited by a significantly reduced value of k(z).

Chemical reaction mechanism
The chemical reaction mechanism is based on the bromine/nitrogen/chlorine mechanism of Cao et al. (2014) with a few modifications to the gas phase mechanism and more complex aerosol modeling.Both modifications are described below.
The resulting mechanism encompasses 50 gas-phase species with 175 gas-phase reactions and 20 aerosol-phase species with 50 aerosol-phase reactions.The full reaction mechanism is described in Tables A1-A5 of Appendix A.
For simplicity, the gas-phase concentration of NO y is assumed to be a conserved quantity in the model.In reality, this is only partly true, since HNO 3 tends to dissolve quickly in aerosols and can become inert, acting as a strong sink.This sink may be compensated by the emissions of NO x from the snow, which was discussed in the introduction.The modeling of these processes would add more uncertainties since the emissions and depositions of the various reactive nitrogen species need to be parameterized.Also, in order to correctly model the deposition of HNO 3 , detailed aerosol chemistry is needed, which would increase the simulation time.Therefore, gas-phase NO y is assumed to be conserved in the present model, i.e. no emission and deposition of NO y and heterogeneous reactions involving NO y conserve gas-phase NO y .

Treatment of the aerosols
The aerosols are modeled as described by Sander (1999) and they are assumed to be liquid.A mono-disperse aerosol with a radius r = 1 µm is assumed.The pH value is fixed to 5. The aerosols are assumed not to undergo any dynamics except for turbulent diffusion, and the aerosol volume fraction in air is fixed at a value of φ = 10 −11 m 3 aq m −3 air .The aqueous reaction constants, acid/base equilibria, uptake coefficients, and Henry coefficients are taken from the box model CAABA/MECCA, version 3.8l (Sander et al., 2011), and they are summarized as follows.
The transfer rate for a gas species is given by with the species-and temperature-dependent non-dimensional Henry coefficients H i (T ), cf.Eq. ( 12).The gas and the aerosol concentrations c i,j,g and c i,j,a , respectively, are in molec cm −3 .The transfer coefficient k t is calculated as The diffusion limit for aerosol transfer k diff is where λ = 2.28 • 10 −5 T p Pa m K −1 (Pruppacher et al., 1998) is the mean free path with pressure p.In Eq. ( 9), use of has been made, including the assumption of a mono-disperse aerosol with radius r, the aerosol volume fraction φ, and aerosol surface area concentration A. The collision term k coll , cf.Eq. ( 8), for the aerosol transfer is Here, α i is the species-dependent uptake coefficient.
The temperature dependence of the Henry coefficient of species i is calculated by where T Hi = −∆ sol H/R is the enthalpy of dissolution divided by the universal gas constant R, and the uptake coefficients are obtained from where T 0 = 298.15K.The values for the Henry and the uptake coefficients are given in Table A3.The transfer rate for the corresponding aerosol species from the aerosol phase to the gas phase has the opposite sign.

The numerical grid
A sketch of the numerical grid is displayed in Fig. 2. The computational domain extends 1,000 m in vertical direction and the number of exemplary grid cells is M = 32.Different numerical grid resolutions were used to assure grid independence of the numerical solution of the equations, see discussion in the results' section.The lowest grid cell at the surface is z 1 = 10 −4 m.
The lowest M/2 + 1 grid cells are distributed logarithmically up to 100 m of the computational domain, cf.Fig. 3.In an intermediate regime, it is assured that at least one grid cell resides inside the inversion layer and that there are grid cells on the borders of the inversion layer to ensure a proper resolution the inversion layer.The remaining grid cells are distributed linearly up to the upper boundary at 1, 000 m.The numerical grid is displayed in Fig. 3 where the first 17 grid cells are logarithmically distributed, followed by five grid cells to resolve the inversion layer.The remaining ten grid cells are distributed linearly.A black vertical line marks transition from the logarithmic to the linear regime, and the grey area marks the inversion layer.
The choice of switching from the logarithmic to the linear grid at 100 m and at 200 m was tested, and the numerical results were not affected.The boundary layer height L is 200 m in the present simulations, cf.Tab. 1, so that the choice of 100 m for the switch of the numerical grid is chosen in order not to interfere with the height of the boundary layer.
Most simulations were conducted with 16 grid cells, simulations with 32, 48, and 64 grid cells were also performed to assure grid independence of the results.In subsection 3.1.1 it is shown that 16 grid cells are sufficient to calculate the recurrence periods with errors smaller than one percent.In total, several hundred simulations for 200 real-time days were conducted in order to study the model parameters (see Tab. 1), so that the small grid size is convenient to minimize the total runtime of the simulations.

The numerical solver
In order to study the recurrence of ODEs for different parameter settings, the typical realtime of about 20 days that was used by Cao et al. (2016) is extended to 200 days in the present study.Scanning the parameter space shown in Tab. 1 requires hundreds of simulations, so that at first, an optimization of the KINAL-T code (Cao et al., 2016) was conducted.Cao et al. (2016) decoupled the diffusion terms, the chemical reactions were solved in an implicit way using the Rosenbrock 4 solver (Gottwald and Wanner, 1981) This procedure has some disadvantages.Since the grid is logarithmic for z < 100 m, the cell size h j is h j ∝ z in that regime.
The diffusion time scale t d = h 2 j /k(z) ∝ z becomes very small for small z, which limits the time steps to the order of milliseconds if an explicit solver is chosen.Also, the heterogeneous reactions on the ice surface destroy all HOBr in the lowest cell in some tens to hundreds of microseconds, depending on the size of the grid cells, so that the heterogeneous reactions have to be solved as part of the diffusion equations in order to allow mixing of upper layers into the lowest cell during a single time step.Even then, however, time steps smaller than seconds are needed due to the strong coupling of the diffusion and the chemical reactions caused by the heterogeneous reactions, if the equations are solved completely decoupled.Thus, in the present code, the diffusion equations and the chemical reactions are solved fully coupled with the implicit, A-stable Rosenbrock 4 solver, resulting in a quite large Jacobian matrix of dimension n = N × M , i.e. the product of the number of species, N , and grid cells, M .The time steps are chosen adaptively, where most time steps are of the order of minutes.

Base parameters
The base parameter settings as well as the range in which they are varied are shown in Tab. 1.The values for the temperature T , pressure p, boundary layer height L, aerosol volume fraction φ, and solar zenith angle are chosen following Cao et al. (2016).
An inversion layer thickness of L inv = 50 m is chosen.Palo et al. (2017) found values ranging from 20 m to 1000 m, with a mean of 337 m.In the study of Neff et al. (2008), a shallow boundary layer with L ≈ 165 m and an inversion layer thickness of L inv ≈ 70 m were found.In the present study it was found that a larger inversion layer thickness L inv can be compensated (i.e.leads to asimilar behaviour in the boundary layer) by choosing a correspondingly larger turbulent diffusion coefficient k t,inv , with a nearly linear relationship k t,inv ∼ L 1.2 inv .The base parameter settings for the diffusion coefficient in the inversion layer has been determined by searching for the oscillating solutions of the numerical simulations.
The dependence of the recurrence period on [NO y ] was investigated for two different temperatures of 258 K and 238 K and The initial species concentrations are shown in Tab. 2. Initial concentrations of organic species are chosen to be consistent with the study of Hov et al. (1989).Nitrogen-contraining species concentrations are varied as shown in Tab. 2. Emissions of nitrogen from the snow are not considered, instead NO y is a conserved quantity in the model.Initial concentrations of bromine are zero in both the free troposphere and in the inversion layer.Starting with non-zero gas-phase bromine concentrations means that the initialization of the bromine explosion is not simulated, the simulation starts during the build-up stage of the bromine 10 explosion.

Boundary conditions and heterogeneous reactions at the ice surface
The upper boundary of the calculation domain at 1,000 m is a Dirichlet boundary where all species concentrations are set to the initial concentrations given in Tab. 2. The presumed large diffusion coefficient of 10 m 2 s −1 ensures that the free troposphere is nudged to the initial concentrations on a time scale of hours.
For the boundary at the ice surface, zero flux is assumed.The exchange with the snow/ice surface is modeled via the heterogeneous reactions listed in Appendix A in Tab.A5.An example of the general treatment of a representative heterogeneous reaction is which is represented as dry deposition reaction that occurs only in the lowest computational cell.The ice/snowpack itself is not modeled, instead, it is assumed that the salt content is infinite so that heterogeneous reactions on the ice surface are effectively treated as The first-order reaction constants are parameterized by where the thickness of the lowest layer is h 1 , cf.Eq. ( 3), and the dry deposition velocity is v d .The dry deposition velocity is modeled following the work of Seinfeld and Pandis (2006) as the inverse of the sum of three resistances R a , R b , and described as follows.First, the gas is transported from the center of the lowest grid cell z 1 to the top of the interfacial layer at the surface roughness length z 0 by turbulent diffusion, which leads to the aero-dynamic resistance Then, the gas must be transported through the interfacial layer via molecular diffusion resulting in the quasi-laminar resistance Finally, the surface resistance is estimated by with the thermal velocity v th = 8RT / (πM i ).M i is the molar mass of the gas species undergoing the heterogeneous reaction and R is the universal gas constant.For the range of γ in the present study (see Tab. A5), the aero-dynamic resistance is the

Results and discussion
In this section, the mechanism of recurrences of ozone depletion events as well as their possible termination are investigated.
First, the reasons for recurrence to occur is discussed and the recurrence period is defined.Second, a closed system with aerosols as the only surface for the recycling of bromine is investigated.Moreover, a comparison to an earlier study (Evans et al., 2003) is presented.Finally, parameter studies are performed on the base parameters presented in Tab. 1 in order to investigate the variation of the recurrence period.

Recurrence and termination of ODEs
This section concerns the study of recurrence and termination of ODEs.

Recurrence of ODEs
The recurrence period of an ODE is defined as the average time difference of two consecutive ozone maxima.An ozone maximum is only accepted, if the difference in mixing ratio to the preceding ozone minimum is at least 2 nmol mol −1 in order to avoid accepting smaller oscillations as a recurrence.The differences between the different grid sizes are small, the average recurrence period varies by less than 1%, and thus, 16 grid cells are sufficient to properly represent the major features of the ODEs and their recurrence.
Recurrence of ODEs may be explained as follows: After the occurrence of an ODE, there is not enough ozone left to sustain the BrO concentration through reaction (R1), causing bromine to be converted into HBr, which deposits onto the ice/snow surface or onto aerosols and then turns into bromide, cf.reaction (R12).The now inactive bromide needs to undergo another bromine explosion, see reaction (R9), in order to become reactive again, which does not occur in the absence of ozone.This allows the ozone in the boundary layer to replenish via the photolysis of NO 2 or by diffusion from the free troposphere through the inversion layer.Once the ozone mixing ratio is large enough, α, defined in equation ( 1), becomes larger than unity, allowing for another bromine explosion.Ozone and reactive bromine now replenish simultaneously where the formation of reactive bromine is becoming faster, increasing the ozone mixing ratio.Once the BrO mixing ratio has reached approximately 10 pmol mol −1 , the ozone destruction by bromine becomes larger than the ozone regeneration.Then, another ODE occurs and the cycle repeats.
Another sink of bromine in the model is the diffusion of part of the bromine species through the inversion layer, since bromine may leave the computational domain through the upper boundary, where the Dirichlet boundary conditions enforce the bromine concentrations to zero.
However, recurrences do not occur for all parameter settings as can be seen in Fig. 5 where k t,inv is increased from 10 to 50 cm 2 s −1 .For k t,inv = 50 cm 2 s −1 , after the initial ODE and an recurrence with a reduced value of [O 3 ], no further recurrence occurs.Instead, the reactive bromine and ozone establish a chemical equilibrium, and no further recurrence is observed, i.e. it terminates.The termination of recurrences will be further discussed next.

Termination of ODE Recurrences
In order to study the termination of ODE recurrences, the initial concentration of [NOy] is reduced to zero compared to the base case, cf.Tab. 1.Moreover, the turbulent diffusion coefficient in the inversion layer is increased from the base parameter of 10 cm 2 s −1 displayed in Fig. 6a to 20 cm 2 s −1 shown in Fig. 6b.
If the ozone recovery rate during an ODE is too large compared to the amount of reactive bromine left for α < 1 in Eq. ( 1), the remaining bromine is not sufficient to fully destroy the ozone, leading to shorter recurrence periods and lower levels of ozone peak concentrations as seen in Fig. 6b: Bromine levels drop until ozone can regenerate, the regeneration of ozone reactivates a part of the inactive bromine, which in turn depletes ozone until the ozone and the bromine concentrations achieve an equilibrium, and thus, only two recurrences occur.
The termination may occur directly after the initial ODE as shown in Fig. 5 or after a few recurrences as a dampened oscillation, see Fig. 6b.The initial ODE typically releases the largest amount of bromine because the initial ozone mixing ratio of 40 nmol mol −1 is much larger than the 10 nmol mol −1 mixing ratio of the recurrences.The total amount of bromine in boundary layer tends to drop after the initial ODE, mostly due to the dry deposition of HBr and to a lesser extent due to diffusion of bromine into the free troposphere.
This reduction in the bromine is the main dampening process.The smaller bromine mixing ratio may not be sufficient to destroy the remaining ozone once α < 1 and can thus result in a termination at a later recurrence instead of the termination after Both a large ozone regeneration rate or a higher Br release for each nmol mol −1 of ozone reduce the drop in bromine for the recurrences or can even result in an increase in the bromine concentration as shown in Fig. 7 where NO 2 is an ozone source leading to an increase of gaseous bromine after the first three recurrences..
Termination may not occur at all during 200 days.Typically, the recurrence period becomes constant after the first few recurrences.The first recurrences are affected by the initial value of 40 nmol mol −1 for the ozone concentration.The fate of most of the bromine after an ODE is stored in aerosols as bromide.While the initial bromine explosion is mostly driven by heterogeneous reactions on the ice/snow surface, the bromine explosions of the recurrences are driven by heterogeneous reactions on the aerosols, which now hold a significant amount of bromide.After a few recurrences, the bromine deposited on the ice surface or diffused to the free troposphere between each recurrence becomes equal to the bromine released from the ice surface during each recurrence, resulting in a constant recurrence period thereafter.
In order to observe fast recurrences, an O 3 recovery rate of about nmol mol −1 per day is required.However, as noted above, for an ODE to terminate properly, the O 3 recovery rate during the termination of an ODE may not be too large.

Initialization of an ODE with only aerosols
So far, the ODEs were initiated through the assumption of a fixed value of 0.3 pmol mol −1 Br x inside the boundary layer (cf. Tab. 2).In the present subsection, another mechanism for the initiation is studied, where aerosols are used to initiate the ODEs.
Five assumptions are changed with regard to the base case: -The concentration of Br − was set to 0.8 mol l −1 , corresponding to a mixing ratio of 160 pmol mol −1 in the gas phase, which is different from the base setting of 0.05 mol l −1 (equivalent to 10 pmol mol −1 ) shown in Tab. 2.
-The turbulent diffusion coefficient in the inversion layer is set to zero.
-The initial concentration of gas phase bromine is set to zero.
-All heterogeneous reactions on the ice/snow surface are turned off.
-The [NO y ] is increased from 50 to 100 pmol mol −1 .
The large concentration of Br − could be caused by a blowing snow event.In this simulation, the main source of the first reactive bromine is via the heterogeneous reaction of ozone, see reaction (R19).
As a result of the different settings, the total bromine concentration is conserved during the simulation.Furthermore, the boundary layer is a closed system for this simulation.The results are shown in Fig. 8.
After one hour, already 0.1 pmol mol −1 of reactive bromine is reactivated, which is sufficient for the bromine explosion on aerosols to become the dominant reactivation mechanism.Also, N 2 O 5 can activate the first bromine by producing BrNO 2 .
Reactive chlorine can also activate the first bromine, however more slowly.0.3 pmol mol −1 of reactive chlorine takes several days to produce an initial seed of 0.1 pmol mol −1 reactive bromine.Only in the first time steps, bromine is reactivated by the It is of particular interest, that oscillations occur without any external sources and sinks, such as dry depositions, emissions or heterogeneous reactions on the ice/snow surface.The density of each chemical element in the gas plus aerosol phase is conserved in this simulation, with hydrogen being the only exception due to the constant pH value of the aerosols.Due to the second law of thermodynamics, only reaction intermediates may oscillate.As an example, CO 2 is a permanent sink for other organic species in this simulation.It is expected for the oscillations to terminate after a sufficient amount of reactive organics are converted to non-reactive organics.

Comparison to studies in the literature
The most relevant research in the area of oscillating ODEs was performed by Evans et al. (2003) which is used to validate the present model.Similar to the aerosol-only simulation of the previous subsection, aerosols are the only source and sink for bromine in the system studied by Evans et al. (2003).
A comparison between the model of Evans et al. (2003) and the present study is displayed in Fig. 9, where Fig. 9a  -The aerosol transfer rates used by Evans et al. (2003) are approximately twenty times larger than these in the present work.Evans et al. (2003) employ the parameterization described by Michalowski et al. (2000).In that study, the diffusion-limit term k diff (Eq.( 9)), which appears in the gas-to-aerosol transfer constant k in (Eq.( 8)) of the present paper, is neglected and only k coll (Eq.( 11)) has been considered.However, the values used for k coll and for k in are also different.For the species HOBr, for instance, the base parameters in the present study are k coll = 8.9 × 10 −4 s −1 and k in = 1.2 × 10 −4 s −1 .Using the values for the oscillating result of Evans et al. (2003) and estimating the aerosol radius via Eq.( 10), which results in r = 0.3 µm, the corresponding values are k coll = 2.0×10 −3 s −1 and k in = 5.0×10 −4 s −1 .
These differences result not only from negligence of the diffusion limit but also from a larger accommodation coefficient (α = 0.8 compared to α = 0.5) as well as a larger aerosol surface area (A = 4.4 × 10 −7 cm 2 cm −3 versus A = 3 × 10 −7 cm 2 cm −3 ).
-NO x and HCHO are emitted from the snowpack as described by Evans et al. (2003).The emission rate is proportional to the photolysis rate of NO 2 with an average emission rate of 1.2 × 10 9 molec.cm −2 s −1 for NO x and 3.6 × 10 8 molec.cm −2 s −1 for HCHO.
-HNO 3 transfer to aerosols is considered and it acts as a sink for NO y .
-All heterogeneous reactions and depositions on the ice surface are neglected.However, following Evans et al. (2003), PAN and H 2 O 2 undergo dry depositions with velocities of v d = 0.004 cm s −1 and v d = 0.09 cm s −1 , respectively.
-The SZA is varied daily in the range of 65 -Initial concentrations and parameters are set to the values described by Evans et al. (2003).In particular, the initial mole fraction of bromide is set to 43 pmol mol −1 .
-Reactions involving the species BrNO 2 are neglected since the species BrNO 2 is not considered by Evans et al. (2003).
The results presented in Fig. 9 show that with an initial bromide mole fraction of 43 pmol mol −1 , the recurrence period is approximately five days.The chemical reaction mechanism used in the present KINAL-T code predicts larger HBr and HOBr mixing ratios compared to the model employed by Evans et al. (2003), resulting in smaller BrO mixing ratios for the same total bromine mole fraction and thus slower ODEs: Ozone is completely depleted approximately one day after the ODE has started, which is more than twice as long as predicted by Evans et al. (2003).Also notably, ozone replenishes to approximately 8.5 nmol mol −1 before the ODE starts, whereas Evans et al. (2003) predict only approximately 4.5 nmol mol −1 .This suggests that the bromine regeneration is slower or that less reactive bromine remains after an ODE in the present KINAL-T simulation.
After an ODE, reactive bromine mixing ratios drop to approximately 10 −4 pmol mol −1 in the present simulation.The bromine regeneration rate is approximately one order of magnitude per day in the present simulation.This means that if the bromine regeneration rate is the same but the reactive bromine mixing ratio obtained by Evans et al. (2003) drops to 10 −2 pmol mol −1 after an ODE, which might explain the difference.
Negligence of the BrNO 2 chemistry has been found to be of particular importance for finding recurrences of ODEs, since otherwise BrNO 2 acts as a sink for both, bromine and NOx.If BrNO 2 chemistry is considered, similar structures as seen in section 3.4.2for large NO y mixing ratios are found, where the large NO y concentrations cause a termination of the recurrences.
Another issue of importance is the larger aerosol transfer constants used by Evans et al. (2003) compared to the present study.
The gas-to-aerosol transfer constants used by Evans et al. (2003) are of the order of 10 −3 s −1 compared to 10 −4 s −1 for the base case.In the present study, the latter value has been adjusted to that used by Evans et al. (2003) in order to match their results.
These increased coefficients allow for a quick recycling of HOBr, HBr, and BrONO 2 .With smaller aerosol transfer constants, the bromine regeneration after an ODE slows down and, more importantly, a larger initial bromide mixing ratio (more than 100 pmol mol −1 ) is necessary to achieve BrO mixing ratios of at least 20 pmol mol −1 during an ODE.At an initial bromide mixing ratio of 43 pmol mol −1 , the ozone depletion occurs on a time scale of weeks with the slower aerosol transfer constant.
As discussed above, Michalowski et al. (2000) and Evans et al. (2003) ignored the diffusion-limit.Staebler et al. (1994) measured a maximum value of r = 0.1 µm in the aerosol size distribution at Alert, and therefore, Evans et al. (2003) assumed that the diffusion correction may be neglected for this small value of aerosol size.However, even at r = 0.1 µm, the HOBr transfer constants are calculated to decrease by a factor of two in the present study, which provides the motivation to consider its relevance in the present study.The aerosol transfer, however, is driven by the aerosol surface, which motivates the use of the aerosol surface distribution instead of the aerosol size distribution.This causes another shift towards larger aerosol sizes with an increased effect on the diffusion limit.
In order to reproduce the recurrence period of three days predicted by Evans et al. (2003), a second simulation with an increased initial bromide mixing ratio of 60 pmol mol −1 and increased NOx emissions by 35 % was conducted, cf.less than half a day and as a consequence, the recurrence period reduces by about half a day.The increased initial bromide mixing ration, however, barely affects the bromine regeneration speed, since it is limited by the low mixing ratio of reactive gas phase bromine (less than 10 −4 pmol mol −1 ) after the termination of the ODEs and not by the aerosol-phase bromide concentration.The increased NO x emissions affect both, the ozone regeneration and the bromine regeneration.The latter is not only increased by the larger ozone regeneration speed, but also by the bromine explosion mechanism involving BrONO 2 .
More BrO reacts to BrONO 2 , which quickly recycles bromide due to the large aerosol transfer coefficients.Consequently, the ODEs start at an ozone mixing ratio of approximately 6 nmol mol −1 compared to the 8.5 nmol mol −1 for the previously used emission rate.Thus, the increased emissions reduce the recurrence period by about one and a half days, resulting in the shorter recurrence period of ODEs found by Evans et al. (2003).
The differences between the numerical results by Evans et al. (2003) and the present study are most likely due to the different chemical reaction mechanisms.Even though Evans et al. (2003) used the reaction constants provided by the same group as the present study, the knowledge about chemical reaction constants has greatly improved in the last decade (Atkinson et al., 2007).
Moreover, it should be noted that Evans et al. (2003) used a box model whereas in the present study, the one-dimensional KINAL-T code with a more advanced model for the heterogenous reactions and the aerosol treatment is used.

Study of model parameters influencing the recurrence period
This subsection concerns the variation of some environmental parameters that affect the recurrence of ODEs: The strength of the inversion layer, the turbulent diffusion in the free troposphere, the NO y mixing ratio, the aerosol volume fraction as well as the solar zenith angle on the recurrence period of the ODEs, cf.Tab. 1.The variation of the NO y mixing ratio is investigated for T = 258 K (base setting) and T = 238 K, as well as simulations where the chlorine mechanism is used (base setting) or neglected.
In the following, three properties of the recurrences of ODEs will be considered: The average recurrence period, i.e. the time difference between two ozone maxima, the number of recurrences, and the average maximum of the ozone mixing ratio; these characteristics are evaluated for a real time of 200 days.

Strength of the inversion layer
Diffusion from aloft is one of the two mechanisms that replenishes the ozone in the model.Since the thickness of the inversion layer is fixed, see Tab. 1, the turbulent diffusion constant k t,inv is the most important parameter controlling the strength of this replenishment.The turbulent diffusion constant in the free troposphere, k f , also plays an important role.
In order to eliminate the influence of the ozone regeneration by NO 2 , the concentration of NO y is set to zero for evaluation purposes.In Fig. 10, the dependence of the recurrence characteristics on the variations of k t,inv and k f is shown, cf.The smallest recurrence period of approximately 20 days is found for the turbulent diffusion coefficient of k f = 10 5 cm 2 s −1 in the free troposphere and for k t,inv ≈ 40 cm 2 s −1 .For very small turbulent diffusion coefficients of less than k t,inv = 6 cm no recurrences occur since the ozone regeneration rate is too slow in the considered time of 200 days.
The recurrence period does not increase linearly with k t,inv since the ozone mixing ratio in the inversion layer changes with increased diffusion; three processes determine the ozone mixing ratio: -In the inversion layer, ozone is lost by diffusion into the boundary layer.
-Ozone is regenerated by its diffusion from the free troposphere into the inversion layer.-Bromine is mixed into the inversion layer and lost to the free troposphere, resulting in a partial ODE inside the inversion layer.
Inside the inversion layer, reactive bromine may survive due to the sustained ozone supply from the free troposphere.It turns out that larger diffusion coefficients inside the inversion layer result in increased ozone mixing ratios, converging to approximately 20 nmol mol −1 for k inv > 20 cm 2 s −1 , which is half of the value at the top boundary of the computational domain.
This is the reason for the sharp, nonlinear increase in the number of recurrences during 200 days.
For k inv < 14 cm 2 s −1 , the recurrence period decreases strongly, and for larger values of k inv , termination is initiated.The mixing ratios for O 3 and Br in the first regime, i.e. for k inv = 10 cm 2 s −1 , are displayed in Fig. 10d.After the first ODE, the ozone regeneration due to diffusion is not very much affected by an ongoing ODE since the ozone mixing ratio is only slightly varying inside the inversion layer, severely limiting the ozone regeneration rate without termination.
Since the standard value of k f = 10 5 cm 2 s −1 used in the present simulation corresponds to an almost perfectly mixed free troposphere, even larger values do not affect the simulation results.By neglecting horizontal mixing and transport, it is essentially assumed that the air mass in the boundary layer is confined.However, the upper troposphere will still have very different wind velocities, so it is reasonable that the air in the upper troposphere is exchanged quickly with fresh air even though the boundary layer is confined.A large turbulent diffusion coefficient in the upper troposphere ensures a quick exchange of the air with the upper simulation boundary.
The influence of a reduction of k f to values of 10 4 cm 2 s −1 and 10 3 cm 2 s −1 is presented in Figs.10a-10c.The value of k f = 10 4 cm 2 s −1 still corresponds to nearly perfect mixing inside the free troposphere as can be seen by the negligible differences in the mean recurrence period between k f = 10 5 cm 2 s −1 and k f = 10 4 cm 2 s −1 .All resulting profiles are very similar.
Reducing k f to 10 3 cm 2 s −1 , however, has a large impact, since bromine transported to the free troposphere will stay there for several weeks (as may be estimated from the diffusion time scale) before being transported to the upper boundary.Ozone is also transported much slower to the lower layers of the free troposphere, causing the ozone levels to drop to approximately 15 nmol mol −1 at 500 m for k t,inv > 25 cm 2 s −1 , decreasing of course with larger k t,inv and converging to 12 nmol mol −1 .
The ozone mixing ratio in the inversion layer drops to less than 10 nmol mol −1 , reducing the ozone regeneration in the boundary layer and also limiting the maximum ozone level that can be regenerated.
Recurrences occur only at larger turbulent diffusion coefficients of k t,inv > 14 cm 2 s −1 , and they terminate for k t,inv > 50 cm 2 s −1 , see Fig. 10b.In contrast to the larger values k f , the ozone mixing ratio in the inversion layer decreases with increasing k t,inv in the present case.Also, the time between two recurrences tends to increase with each further recurrence since a larger turbulent diffusion coefficient in the inversion layer causes as strong loss of bromine to the free troposphere, which in turn decreases the speed of the bromine explosion in the boundary layer.
If only two or three maxima occur, i.e. k inv > 16 cm 2 s −1 , due to the termination of the recurrences, the standard deviation of both the recurrence period and the ozone maxima increase sharply, since the first few recurrences are still affected by the first ODE, and the recurrences before the termination tend to have ozone maxima that are closer to the equilibrium mixing ratio of ozone.In the present model, NOy is treated as a conserved quantity.In this subsection, the NOy mixing ratio is varied as the major parameter influencing NOx-catalyzed photochemical O 3 formation, see reaction (R20) and its effect on the ODEs is investigated.
Figure 11 shows the variation of the recurrence period with NO y mixing ratios of up to 300 pmol mol −1 for two different down by the presence of chlorine.This is due to the fact that when chlorine is included, part of the heterogeneous reactions release BrCl instead of Br 2 , thus the amount of released bromine is reduced.Also, without chlorine, the slower ODE increases the duration of the bromine explosion, which also increases the bromine released, resulting overall in faster recurrences, since further recurrences contain more bromine in aerosols that can be reactivated.As a side effect, the ozone maximum value is slightly smaller without the chlorine chemistry.
For T = 238 K, the recurrence period is smaller compared to T = 258 K for small [NO y ], as can be seen in Fig. 11c.However, termination of recurrences starts already at around 70 pmol mol −1 of NO y instead of at around 200 pmol mol −1 for T = 258 K. Figure 11d shows a termination for T = 238 K after 80 days.In this temperature region, HNO 4 becomes very stable due to the decay of HNO 4 being nearly two orders of magnitude slower, see (R 56) in Tab.A1, and replaces PAN as the most abundant nitrogen species.The shift towards HNO 4 formation reduces the NO 2 concentration, retarding the ozone regeneration.
At the lower temperature, the ODE mechanism becomes more efficient, e.g. the reaction constant in the Br 2 -producing BrO self-reaction increases by 33 %, (R 5) in Tab.A1, while many of the HBr-producing reactions, e.g.(R 9) and (R 10) in Tab.A1, slow down by around around 20 %.The total amount of bromine released for the first ODE increases whereas the amount of bromine released for the recurrences decreases due to the slower ozone regeneration.
The main reason for the earlier termination at T = 238 K is a strong shift towards the increased HNO 4 formation.During an ODE, the HNO 4 can be destroyed to directly produce NO 2 by reacting with OH or by decaying through (R 56) and (R 59) in Tab.A1.PAN, however, is more stable during an ODE at 258 K due to a larger formation of CH 3 CO 3 via e.g.(R 34) in Tab.A1 caused by the larger OH formation during an ODE, consuming NO 2 (R 67) instead of producing it through (R 86) or through the photolysis of PAN.The shift from PAN as the most stable species towards HNO 4 for the lower temperature increases the ozone recovery during an ODE, resulting in earlier terminations of the ODEs.

The influence of the aerosol density
In order to study the influence of the aerosol characteristics, the standard value of the aerosol volume fraction of 10 −11 m 3 m −3 is varied between 10 −12 m 3 m −3 and 3×10 −10 m 3 m −3 , see Fig. 12.For small aerosol concentrations, the recycling of HBr is too weak for a full ODE to occur since only small bromine concentrations are released before the termination of an ODE.Only For larger aerosol mixing ratios, the faster bromine recycling reduces the recurrence period, however the ozone does not regenerate faster.Thus, the ozone maximum decreases so that the recurrences release less bromine, resulting in a larger net bromine loss per recurrence.Also, the reactivation strength of aerosol bromine increases relative to the activation on the ice surface, resulting in a lower bromine release from the ice, also increasing the net bromine loss for each recurrence, which ultimately leads to the termination of the recurrences for aerosol volume fractions larger than about 5.5×10 −11 m 3 m −3 .

Variation of the solar zenith angle
The mean recurrence period displayed in Fig. 13a hardly changes when the solar zenith angle SZA is varied from its standard value of 80 • (see Tab. 1) within the range of 70 • and 83 • .The variations stay within one standard deviation.For SZA > 83 • , the ODEs do no longer occur due to the slow photolysis frequencies.Surprisingly, the recurrence period does not monotonically decrease with increased SZA, instead, there is a minimum at SZA = 77 • .For a lower SZA, some or even all ODEs are only partial as Fig. 13b demonstrates for the value of SZA = 70 • .In particular, the minimum ozone mixing ratio for the six recurrences shown is approximately 10 nmol mol −1 and the ozone depletion restarts at an ozone mixing ratio of about 18 nmol mol −1 .For a SZA of 70 • , the NO 2 mixing ratio decreases to about 5 pmol mol −1 during the ODEs instead of to nearly zero at 80 • .Also, BrNO x and PAN are photolyzed faster, increasing the NO x formation.Only about 80 pmol mol −1 of bromine is released at SZA = 70 • during the first ODE, which is about two-thirds of the value at 80 • .Interestingly, gas-phase bromine does not drop to zero for the later recurrences, however, the BrO concentration drops to nearly zero.BrO mixing ratios do not exceed 10 pmol mol −1 , which is much lower than the typical mixing ratio of 30-40 pmol mol −1 of numerical simulations with a SZA of 80 • ; this is most likely a result of the increased formation of HO 2 .For SZA = 70 • , the fastest reaction of BrO is with NO, producing NO 2 and Br in the process.NO 2 is photolyzrefered to ozone, resulting in a net null cycle.For SZA = 80 • , the BrO self-reaction is stronger than its reaction with NO, favoring a full ODE.

Conclusions
In the present study, the one-dimensional model KINAL-T developed by Cao et al. (2016) was extended and optimized in order to study the potential of ODEs to recur.The extension concerns the chemical reaction mechanism as well as the treatment of aerosols and the improvement of the numerical solver.The model was employed to study both the recurrence and the termination of ODEs, and several parameters were varied to investigate their influence on the recurrence period, the maximum ozone mixing ratio, and the number of recurrences of the ODEs.After an ODE, ozone can be replenished by the diffusion of ozone from the free troposphere to the boundary layer and/or by the photolysis of NO 2 ; it is found that either of these two O 3 sources is sufficient to drive the recurrences.Another result of the present study is that the chemistry of ODEs coupled with the vertical diffusion alone can cause the recurrence of ODEs at the surface even without the existence of horizontal transport.
A strong inversion layer was found to be essential for the recurrence of ODEs since the steady mixing of the ozone back into the boundary layer may provide a sufficiently high ozone level to keep the reactive bromine in the boundary layer at a significant Without the presence of reactive nitrogen oxides, the system is a heterogeneous, diffusion-driven recurrence, the fastest periods of which were found to be approximately 20 days.Fast replenishment of the air in the free troposphere was found to lead to faster recurrences.It may be possible to find conditions leading to even shorter recurrence periods such as a slightly smaller SZA and moderately higher aerosol mixing ratios.
The replenishment of ozone via the photolysis of NO 2 is a chemical gas-phase process.Faster recurrence periods of approximately five days are found due to the destruction of NO x during an ODE.However, at sufficiently high nitrogen oxide levels, the amount of bromine released during the bromine explosion is not large enough to keep the NO 2 mixing ratio low, so that the recurrences can terminate due to the ozone regeneration, keeping the reactive bromine at a significant level.With high NO y mixing ratios, recurrences are possible even if the boundary layer does not interact at all with the free troposphere.Deactivation of the chlorine mechanism speeds up the bromine explosion, since the heterogeneous reactions of HOBr on aerosols and snow/ice surfaces always produce Br 2 instead of Br 2 and BrCl.The absence of chlorine thus results in faster recurrences.
More sunlight, for a SZA up to 77 • , and a higher aerosol volume fraction of up to 5.5×10 −11 m 3 /m 3 are beneficial for faster recurrences, at even higher values, the recurrence retards or terminates.Since bromine may be lost over time due to dry deposition and mixing into the upper troposphere, a strong release of bromine for each recurrence is important to enable the fast destruction of ozone so that no chemical equilibrium of bromine with the ozone may be established.
The present simulations were compared to results of an earlier study by Evans et al. (2003).Using the same initial bromide mixing ratio of 43 pmol mol −1 and the same NO x emissions, a shorter recurrence period of five days was found in comparison with three days predicted by Evans et al. (2003).The difference in the recurrence periods is caused by a slower reactive bromine regeneration after an ODE or a stronger bromine depletion during the termination of the ODEs in the present model.
By assuming an increased initial bromide mixing ratio of 60 pmol mol −1 and stronger NO x emissions by 35%, the recurrence period of three days found by Evans et al. (2003) could be reproduced.The differences may be attributable to different chemical reaction mechanisms, a more advanced treatment of the aerosol in the present study as well as to the use of a box model by Evans et al. (2003) versus a 1-D model in the present simulation.
An interesting extension of the present model could be the consideration of snow packs.A finite amount of sea salt that is consumed during the bromine explosion and redeposited after the bromine explosion may have an interesting effect on the recurrences.This may also allow for the modeling of NO x emissions from the snow, relaxing the present assumption of a conserved NO y mixing ratio.
Even though the present simulations are based on somewhat idealized assumptions, they demonstrate that there are additional reasons for the observed recurrences of ODEs that go beyond modified environmental conditions or advection of air masses with varying ozone and halogen content.Experimental validation of these simulations could be a challenge since these external causes of recurrences and intrinsic recurrences are likely to occur simultaneously.However, it is possible that the conditions simulated in the present paper can be found e.g. at high latitudes in the Arctic where day/night cycles do not play any role and recurrences may be observed.Thus, the present study provides valuable insight into parametric dependencies of the characteristics of the recurrences of ODEs and their termination.
2.9 × 10 −12 exp (500/T )      and Crutzen, 1996) for most species.Since the strongest resistance is the species-independent turbulent resistance, the deposition velocities for the different species vary only slightly around 21 cm s −1 .Deposition velocities in the present model are relatively large since the lowest grid cell is at 10 −4 m, reducing the turbulent resistance by a large factor compared to models using a linear grid.Also, the surface resistance is usually the largest resistance and widely calculated using parameterizations outlined by Wesely (1989), which, however, does not hold for ice/snow surfaces.Due to the large deposition velocity, the heterogeneous reactions are rate limited by the downward diffusion of the depositing species, replenishing in the lowest grid cell.

Figure 1 .
Figure 1.The turbulent diffusion coefficient k(z) as a function of altitude z for a boundary layer height of 200 m, a wind speed of 5 m s −1 , and the inversion layer thickness of 50 m . Phys.Discuss., https://doi.org/10.5194/acp-2018-1314Manuscript under review for journal Atmos.Chem.Phys.Discussion started: 13 February 2019 c Author(s) 2019.CC BY 4.0 License.

Figure 2 .
Figure 2. One-dimensional grid.Numbers are at the center of every numerical grid cell.Red grid cell resides inside the inversion layer.The grid cells at 200 m and 250 m are centered at the interface of the inversion layer

Figure 3 .
Figure 3. Numerical grid with M = 32 grid cells (cf.Fig. 2) plotted on a logarithmic scale (squares) and a linear scale (filled circles).The grey area shows the inversion layer , and diffusion was treated in an explicit way.The heterogeneous reactions were solved as 11 Atmos.Chem.Phys.Discuss., https://doi.org/10.5194/acp-2018-1314Manuscript under review for journal Atmos.Chem.Phys.Discussion started: 13 February 2019 c Author(s) 2019.CC BY 4.0 License.part of the chemistry equations.

Figure 4 .Figure 5 .
Figure 4. Evolution of O3 and total gaseous bromine mixing ratios for four different numbers of grid cells of M = 16, 32, 48, and 64

Figure 4
Figure4shows the recurrence of ODEs for the base setting of the present model, cf.Tab. 1, for 16, 32, 48, and 64 grid cells.
Figures 5 and 6 show a termination due to a sufficiently large k t,inv .The effect of a strong inversion layer as well as the ozone formation via nitrogen oxygen on ODEs will be studied in subsection 3.4.The next subsection concerns different ways of initialization of ODEs.

Figure 6 .
Figure 6.Recurrence and termination of ODEs for different values of kt,inv

Figure 7 .
Figure 7. Evolution of O3, NOx, and total gaseous and aerosol bromine mixing ratios for the base case with [NOy] = 150 pmol mol −1

Figure 8 .
Figure 8. Simulation neglecting the snow pack and the exchange between the boundary and the inversion layers

Figure 9 .
Figure9.Simulation of the recurrences of ODEs for the conditions ofEvans et al. (2003) Fig. 9b.The main effect of the increased initial bromide mixing ratio alone is a decrease of the duration of the ODEs from one day to somewhat Atmos.Chem.Phys.Discuss., https://doi.org/10.5194/acp-2018-1314Manuscript under review for journal Atmos.Chem.Phys.Discussion started: 13 February 2019 c Author(s) 2019.CC BY 4.0 License.
Figs. 10a-10c, and the variation of the mixing ratios of O 3 and Br at two different heights of 100 m and 225 m, Fig. 10d, for the base settings.
Figure 10.Dependence of the recurrence characteristics on the variations of kt,inv and k f , Figs. 10a-10c, and the variation of the mixing ratios of O3 and Br at two different heights of 100 m and 225 m for the base settings, Fig. 10d . Phys.Discuss., https://doi.org/10.5194/acp-2018-1314Manuscript under review for journal Atmos.Chem.Phys.Discussion started: 13 February 2019 c Author(s) 2019.CC BY 4.0 License.

5
Figure 11.Dependence of the recurrence characteristics on the variations of [NOy], Figs.11a-11c.Termination of an ODE: profiles of the mixing ratios of various species for the base settings and for 238 K and [NOy] = 100 pmol mol −1 , Fig. 11d Figure 12.Recurrence characteristics depending on the aerosol volume fraction after 200 days Figure 13.(a) Mean recurrence period and number of recurrences versus the solar zenith angle during 200 days.(b) Evolution of the mixing ratios of O3 and the bromine species for SZA = 70 • Atmos.Chem.Phys.Discuss., https://doi.org/10.5194/acp-2018-1314Manuscript under review for journal Atmos.Chem.Phys.Discussion started: 13 February 2019 c Author(s) 2019.CC BY 4.0 License.level, which then establishes chemical equilibrium with the remaining ozone.

Table 2 .
Initial trace-gas concentrations

Table A4 .
Reactions occurring in the liquid phase, forward reaction rate coefficients are shown as well as backward reaction rate constants if applicable.kin (X) denotes the gas-to-aerosol transfer rate for species X (Eq.

Table A5 .
Heterogeneous reactions and dry depositions occurring on the ice/snow surface