Title Page A two-dimensional volatility basis set: 1

ACPD10 24091ndash24133 2010

2-D VBSthermodynamics

N M Donahue et al

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Abstract Introduction

Conclusions References

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Atmos Chem Phys Discuss 10 24091ndash24133 2010wwwatmos-chem-phys-discussnet10240912010doi105194acpd-10-24091-2010copy Author(s) 2010 CC Attribution 30 License

AtmosphericChemistry

and PhysicsDiscussions

This discussion paper ishas been under review for the journal Atmospheric Chemistryand Physics (ACP) Please refer to the corresponding final paper in ACP if available

A two-dimensional volatility basis set 1organic-aerosol mixing thermodynamicsN M Donahue S A Epstein S N Pandis and A L Robinson

Carnegie Mellon University Center for Atmospheric Particle Studies Pittsburgh USA

Received 23 September 2010 ndash Accepted 25 September 2010 ndash Published 15 October2010

Correspondence to N M Donahue (nmdandrewcmuedu)

Published by Copernicus Publications on behalf of the European Geosciences Union

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Abstract

We develop the thermodynamic underpinnings of a two-dimensional volatility basis set(2-D-VBS) employing saturation concentration (Co) and the oxygen content (OC) todescribe volatility mixing thermodynamics and chemical evolution of organic aerosolThis is an extension of our earlier one-dimensional approach employing Clowast only5

(Clowast = γCo where γ is an activity coefficient) We apply a mean-field approximationfor organic aerosol describing interactions of carbon and oxygen groups in individ-ual molecules (solutes) with carbon and oxygen groups in the organic-aerosol sol-vent In so doing we show that a linear structure activity relation (SAR) describingthe single-component Co of a molecule is directly tied to ideal solution (Raoultrsquos Law)10

behavior Conversely non-ideal solution behavior (activity coefficients) and a slightlynon-linear SAR emerge from off-diagonal (carbon-oxygen) interaction elements Fromthis foundation we can build a self-consistent description of OA mixing thermodynam-ics including predicted saturation concentrations and activity coefficients (and phaseseparation) for various solutions from just four free parameters the carbon number of15

a hydrocarbon with a 1microgmminus3 Co and the carbon-carbon oxygen-oxygen and non-ideal carbon-oxygen terms This treatment establishes the mean molecular formula fororganics within this 2-D space as well as activity coefficients for molecules within thisspace interacting with any bulk OA phase described by an average OC

1 Introduction20

It is well established that atmospheric organic aerosol (OA) exists as a complex mixtureof thousands of individual organic compounds (Hildemann et al 1991 Fraser et al1997 1998 Goldstein et al 2008) Furthermore it is clear that a large fraction of OAconsists of organic compounds that are difficult to separate in a gas chromatograph(Schauer et al 1996) and that very often a majority of OA consists of quite highly25

oxidized organic molecules (Zhang et al 2007 Aiken et al 2008 Ng et al 2009)

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As an example the most abundant identified organic molecule during the PittsburghAir Quality Study generally a carboxylic acid typically comprised about 1 of the totalorganic carbon mass (Robinson et al 2006 Shrivastava et al 2007) A typical 200nm diameter particle contains 5ndash10 million individual organic molecules with molarweights around 200 g moleminus1 The most abundant molecule appears approximately5

50 000 times in that single particle while hundreds to thousands of molecules appearof order 100ndash1000 times each in that same particle

A critical determinant of whether an organic compound is found in OA (as opposed tothe gas phase) must be saturation vapor pressure Here we shall use saturation massconcentration (Co and Clowast in microgmminus3) because mass measurements pervade aerosol10

science and we shall use the terms ldquovolatilityrdquo and ldquosaturation concentrationrdquo inter-changeably The difference between Co and Clowast is that Co is the pure (sub-cooled)liquid saturation compound of a substance while Clowast includes the activity coefficientin a mixture γ thus Clowast = γCo Furthermore because volatility spans an enormousrange we shall generally use a log axis employing the shorthand lCo = log10C

o and15

lClowast = log10Clowast

We have previously presented a framework describing OA mass ndash whether or notthe constituents are identified ndash with a series of volatility ldquobinsrdquo that are separatedby one decade (δlClowast = 1) which we shall refer to as the ldquoone-dimensional volatilitybasis setrdquo (1-D-VBS) (Donahue et al 2006) Here we shall extend the 1-D-VBS to20

a second dimension in order to improve our ability to predict the thermodynamicsincluding organic mixing and hydrophilicity and ultimately to coherently describe theoxidation chemistry

Because OA is a complex mixture the thermodynamics of mixtures are important(Pankow 1994) From both a modeling and a theoretical perspective it is desirable25

to be able to predict volatility under ambient conditions It is also important to knowwhether different organics will tend to form a single homogeneous phase or whetherthey will phase separate One reason is that phase separation can sharply reducethe predicted mass of the less abundant phase (Bowman and Melton 2004) while

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another is that separate phases will allow distinct OA types to remain externally mixed(for example with different sized modes) while a single OA phase will tend to drive OAcomposition toward a uniform internal mixture (Asa-Awuku et al 2009)

The challenge we seek to confront here is that the molecular composition of OA isunknown and very complex Consequently it will be useful to develop a thermody-5

namic treatment based on OA properties that we can directly observe and constrain tocomplement approaches based on surrogate molecules Each method contains intrin-sic errors ndash the simplified thermodynamics we shall present here cannot represent thefree-energy change of individual molecules upon solvation but the individual moleculesused in surrogate approaches may or may not adequately represent those of actual OA10

mixturesWith the surrogate approach one strives to carefully select a small set of surrogate

molecules and then to calculate the thermodynamics as accurately as possible Theuniversal functional activity coefficient (UNIFAC) method (Fredenslund et al 1975) hasbeen widely used to describe the activity of organic aerosol mixtures (Koo et al 200315

Bowman and Melton 2004 Pun et al 2006) UNIFAC is a group contribution methodthat gives the excess Gibbs free energy of each component in a specified mixturebased on the non-ideality arising from the size and shapes of the constituent moleculesand the non-ideality arising from their energetic interactions (Asher et al 2002 Chan-dramouli et al 2003) UNIFAC calculations require knowledge of the aerosol mixture20

composition and the abundance of each of the molecular subgroups (Bowman andMelton 2004)

UNIFAC calculations of some surrogate species also present challenges The mul-tifunctional and long-chain molecules which are typically found in the atmosphereare not accurately described by the UNIFAC method (Koo et al 2003 Bowman and25

Melton 2004) Also UNIFAC calculations can be computationally intensive (Bowmanand Melton 2004) Finally it is not obvious that surrogate molecules will accurately rep-resent complex ambient OA mixtures For instance Clegg et al (2008ab) have shownthat very similar seeming organics can have very different activities in model mixtures

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and thus that it is difficult a priori to choose ldquogoodrdquo surrogates for OA mixturesThe approach developed in this paper relies principally on data from in situ mass

spectrometers that reliably measure a large fraction of the OA mass and are able toconstrain the bulk atomic composition (C H O and sometimes N S) with goodprecision (Aiken et al 2008) The most precisely constrained measure is the oxy-5

gen to carbon ratio (OC) Here we shall show that we can develop a self-consistentthermodynamic framework for OA mixtures based on the total OA mass (COA) andOC Our key hypothesis will be that we can formulate the thermodynamics basedon a mean-field approximation in which individual molecules interact with aggregateaverage (bulk) properties of the OA solvent This is motivated by the example of com-10

position presented above ndash individual molecules in an OA particle are rarely if eversolvated by themselves but rather by a complex mixture of many molecules containingseveral different functional groups in many combinations While the individual solventmolecules can not be completely enumerated the necessary average properties cannow be measured routinely15

Both a mean structure activity relation (SAR) connecting measured Co and OC withcomposition (ie nCnH and nO for that Co and OC) and a generalized predictionof activity coefficients for individual organics interacting with a homogeneous OA mix-ture will fall out naturally and self-consistently from this formulation This rich behavioremerges from only four parameters a compound with a reference volatility (we use20

pentacosane) and three interaction terms expressing in turn carbon-carbon interac-tions (non-polar interactions) oxygen-oxygen interactions (polar interactions) and thenon-ideality of the carbon-oxygen cross interaction (the hydrophobicity) We shall ex-plore the implications of this both in the laboratory where different organics may ormay not serve as good ldquoseedsrdquo for other organics (Song et al 2007 Asa-Awuku et al25

2009 Vaden et al 2010) and in the atmosphereIt is very important to understand our objectives We do not seek to accurately predict

the phase partitioning of specific molecules (eg levoglucosan etc) Rather we seekto understand the bulk behavior of OA We will use this thermodynamic treatment within

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the VBS where the phase partitioning itself is known almost by definition We dowant to be able to invert the problem ndash to find nC nO nH etc for material with acertain Co and OC We also want to be able to understand the physical properties ofthe OA including hygroscopicity as well as the potential for different classes of OA tophase separate and we want to inform chemical mechanisms describing how these5

properties (including volatility and OC) evolve during photochemical oxidation

2 Theory

21 Saturation concentrations and vaporization enthalpies

Volatility is controlled principally by the vaporization enthalpy of a substance especiallyover liquids and other disordered condensed phases where the vaporization entropy10

is nearly constant (Prausnitz et al 1998) We recently showed that this holds for awide range of organics Specifically a pseudo-Arrhenius expression can describe thetemperature dependence of Co and there is a nearly linear relationship between lCo

300(log10C

o at 300 K) and the vaporization enthalpy (Epstein et al 2010) Here we shallassume that the relationship is precise15

Because we use Boltzmann terms extensively we shall scale energies by the gasconstant and report them in Kelvins θ = ER Thus the vaporization enthalpy be-comes ∆θvap A key value is the change in θ driving a one decade change in Co

at 300 K ∆θ10 = d∆θvapd lCo We also need to know the vaporization enthalpy forCo =1microgmminus3 ∆θvap

1microg So for a compound i 20

∆θvapi =∆θvap

1microgminus∆θ10times lCoi300 (1)

or

lCoi300 =

(∆θvap

1microgminus∆θvapi

) 1∆θ10

(2)

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This means that we need to determine or specify only one of ∆θvapi or lCo

i300 and weknow the other

Simple theoretical considerations suggest ∆θ10 minus690K (minus57kJmoleminus1) (Don-ahue et al 2006) ∆θ10 should cause a one-decade change in a Boltzmann termat 300 K On the other hand empirical correlations give a larger value of ∆θ10 5

1500K(minus13 kJ moleminus1) (Epstein et al 2010) Whatever the correct value of ∆θ10 animportant empirical finding is that it depends almost exclusively on lCo and it is inde-pendent of the degree of substitution (oxidation) of an organic molecule (Epstein et al2010) Consequently we shall make the assumption that the vaporization enthalpy issimply the energy required to remove a given organic molecule from an organic solvent10

22 Describing vaporization energies

It is useful to start with a very basic formulation far more detailed treatments can befound in any advanced thermodynamics text (eg Prausnitz et al 1998) The stepsinvolved in evaporating from a solvent are shown in Fig 1 Specifically the overallenergy change includes the energy required to remove a molecule i from a solvent s15

(θi s) but also the change in energy in the solvent when the interaction energy of themolecule with the solvent is replaced by the interaction energy of the solvent with itself(θss) What this boils down to (so to speak) is

θvapi s =θi s+

(θi sminusθss

)=2θi sminusθss (3)

Note that when the molecule is the solvent (a single component system) the term in20

parentheses vanishes and we are left with the simple and intuitive result of θvapi =θi i

This defines Co via Eq (2)We can also use Eq (2) to define Clowast when i 6= s meaning the excess entropy of i is

approximately zero (Prausnitz et al 1998) For an ideal solution γ = 1 and Clowast =Cotherefore ∆θvap

i s = θi i What we really want to know is the difference between the25

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vaporization energy of i in the solvent and the vaporization energy of pure i

δθi s =2θi sminusθssminusθi i (4)

For an ideal solution δθi s =0 and we obtain

θideali s =

12

(θss+θi i

)=θ (5)

This is a very general result for ideal solutions the cross term is the average of the5

corresponding diagonal terms We will therefore define this cross term in terms of thedeviation from the mean

θi s =θ+δθi s (6)

which is part of an interaction matrix of this form

θ=

(θi i θ+δθi s

θ+δθi s θss

)(7)10

All of this can be combined to obtain the energy of i in s

θvapi s =θi i +2δθi s (8)

We have already shown the exponential relationship between θ and both Co and Clowast

in Eq (2) so at an equivalent level of approximation the excess enthalpy (Prausnitzet al 1998) is just 2δθi s and the activity coefficient of i in s is given by a Boltzmann15

expression

γi s =exp(minus2δθi sT

)(9)

This is quite simply why activity coefficients are so difficult to predict or even param-eterize ndash an energy difference of only kT (about 25 kJ moleminus1 at 300 K) is enough to

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make the activity coefficient a factor of e and because γ is exponential it will grow (orshrink) very rapidly with increasing |δθ| Treatments attempting to predict activity coeffi-cients for individual molecules must carefully account for multiple thermodynamic terms(Jang et al 1997 Pankow and Chang 2008) that is appropriate for those specific sit-uations but the treatments require detailed knowledge of the solvent composition as5

well as the exact molecule of interest We have neither

23 Simple mixtures

So far the solvent has been of unspecified composition Now let us consider a moleculeA (i rarrA) in a solution s that is a two-component mixture of molecules A and B wherethe fraction of A in the solution is f sA and the fraction of B is f sB = (1minusf sA ) This means that10

from the perspective of a molecule A it will be interacting with a ldquosolventrdquo consisting ofan A-B mixture If the mixture is homogeneous and isotropic meaning that the localfractions (and orientations) of the molecules are the same throughout the mixture thenthe interaction energy an individual molecule A with the solvent is

θi s rarrθAs = f sAθAA+(1minus f sA

)θAB (10)15

From Fig 1 we also need to know the interaction energy of the solvent with itself Thisis a bit more complicated We shall assume that the solvent ldquomoleculerdquo replacing Awhen it evaporates is a composite containing fA of molecule A and (1minus fA) of moleculeB each of which will interact with the fractions of A and B in the solvent

θss = f sA2θAA+2f sA

(1minus f sA

)θAB+

(1minus f sA

)2θBB (11)20

We are now in a position to calculate δθAs the excess energy of A in s In a recurring

theme it can be shown that for an ideal mixture when θAB =12(θAA+θBB

)=θ then

δθAs =0 Consequently for non-ideal mixtures with θAB =θ+δθAB

δθAs =(1minus f sA

)2δθAB (12)24099

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In the end the activity coefficient of A in an A-B mixture is just what one might ex-pect ndash it depends on the non-ideality of the off-diagonal interaction element exactlyas in Eq (9) The total excess energy is now a function of the solution compositionincreasing quadratically as the solvent becomes increasingly dissimilar to A

In defining these mixtures as homogeneous and isotropic we have made our sec-5

ond major assumption Real molecules (for instance water) donrsquot interact randomlywith other molecules However given that there are few if any observables for OA tocontradict this assumption it is either robust or just necessary The very complexity ofambient OA may make it more homogeneous and isotropic than simpler mixtures Theextreme example is the tendency of mixtures to form liquid or amorphous liquid-like10

mixtures where the individual constituents would crystalize (Cappa et al 2008 Zobristet al 2008) but much more generally the complex mixtures found in atmospheric OAwill smooth out the rough edges of highly specific A-B interactions

231 Non-ideal activity

It is worth exploring the activity coefficient a bit further What we are really interested15

in is the activity of a compound αi defined relative to a pure reference state in vapor-liquid phase equilibrium and related to the activity coefficient and the fraction of A inthe solution fA This is simply

αA = fAγAs = fAexp(minus2δθAsT

)(13)

for the two-component mixture this becomes20

αA = fAexp(minus2(1minus f sA

)2δθABT)

(14)

The consequences of this are shown in Fig 2 The bottom line is that non-ideal be-havior emerges very quickly for quite modest values of δθ T and that this will inducephase separation However the real systems we are interested in consist of very com-plex mixtures Therefore we must ask whether it is possible to predict these small25

deviations from ideality with anything close to sufficient accuracy24100

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An ansatz here is the complex mixtures found in atmospheric OA will be sensitiveto mean-field properties and that we can indeed describe this mean-field behaviorusefully For any individual molecule the values we predict will not be accurate butwe hypothesize that on average they will be (with some semi-empirical calibration)However it is very important to appreciate the contrapositive of this claim predictions5

based on a few surrogate molecules are almost certain to be inaccurate for complexmixtures because the unique (and therefore unrepresentative) properties of those in-dividual surrogate molecules will likely not represent the average (bulk) properties ofthe mixture very well

3 Structure activity relations10

31 Data and mean trends

With the theoretical foundations of our approach developed we shall now make whatmay seem like an abrupt turn We would like to be able to describe lClowast in terms ofobservable average molecular properties of OA ndash specifically the average carbon hy-drogen and oxygen numbers (nCnHnO) Initially we will focus on lCo ndash the pure15

saturation concentrations There are a number of structure activity relations in the liter-ature one that is especially convenient in this context is SIMPOL (Pankow and Asher2008) in which (log10 of) the sub-cooled liquid vapor pressure of a pure substance canbe described as a simple sum of contributions from various functional groups

log10

(p0Li

)=b0+

sumk

νkibk (15)20

where bk is the effect of the kminus th ldquostructural elementrdquo and νki is the number of thoseelements in the molecule i That is tantalizingly close to what we want Here we shallbuild a SAR akin to SIMPOL with the minor difference of l Co for log10p

0Li focusing on

the carbon and oxygen numbers (nC and nO) of the organics Specifically our objective24101

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in this paper is to describe lCo = f (nCnO) neglecting for the time being other contribu-tions (for example N S) in favor of the ldquobedrockrdquo oxygenated organic compounds thatappear to comprise the bulk of OA

Data for a very wide range of compounds are plotted vs nC in Fig 3 These formthe basis for our SAR The data are from NIST as well as the literature (Koponen et al5

2007 Cappa et al 2007) The oxygenated compounds all have functionality identifiedin ambient OA (-OH =O -C(O)OH etc) We group molecules in classes defined byfunctionality (ie mono-alcohols with a single -OH) and plot each class with a differentsymbol The broad colored bands in the background are volatility classes we useelsewhere (Donahue et al 2009) here they serve to guide the eye through the various10

figures The important attribute for atmospheric partitioning is that compounds in thelight-green range will be semi-volatile in the atmosphere existing in significant fractionsin both the condensed and vapor phases Compounds in the light red and gray rangeswill be almost completely condensed under typical ambient conditions Note that thereare relatively few data in the semi-volatile (green) range and almost none below that15

This is because a saturation concentration of 1microgmminus3 corresponds to approximately10minus7 torr this is an exceedingly low value that is difficult to measure

There are several shortcomings with the available data The data in Fig 3 becomenoisy approaching the semi-volatile range and they do not include richly multifunctionalcompounds such as those that must comprise a large portion of the OOA pool Some20

of the noise at a low Co may be related to uncertainty in converting to sub-cooledliquid values and the rest is likely due to measurement uncertainties associated withvery low vapor pressures The diacid series does reach these low values partly be-cause the atmospheric community has devoted considerable effort to measuring theirvapor pressures (Tao and McMurry 1989 Bilde and Pandis 2001 Bilde et al 200325

Koponen et al 2007 Cappa et al 2007 2008)The most obvious feature of Fig 3 is the succession of parallel lines formed by the

different classes This means that there is a well-defined effect of increasing nC inde-pendent of molecule class We can thus define a robust relationship between nC and

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Co each class of molecule has an lCo vs nC slope of 0475 decades per carbon Thisdefines the carbon-number dependence of lCo for our SAR which is the slope of thetrend lines in Fig 3 We also need a reference point which we will define as the car-bon number nC at Co =1microgmminus3 for pure hydrocarbons From Fig 3 this is pentacosane(C25) indicated with the black square This reference nC may be uncertain by 2 car-5

bons (corresponding to a 10 uncertainty in the group contribution for carbon to Co)Some data for high-nC alkanes lie above the trend line while others fall on it Given theprecision of the relationship for lower-nC alkanes we believe this indicates the difficultyin measuring such low vapor pressures and that the lower envelope represented byour SAR represents the true values for the n-alkanes10

The next step in developing the SAR is to define the influence of oxygenation Achallenge is that we need to account for the influence of different functionalities In-creasing functionalization decreases volatility systematically with logarithmic factorsgiven by the vertical offsets of the various trend lines in Fig 3 from the pure hydro-carbon trend line For example each carbonyl group (=O blue and yellow symbols)15

decreases Co by 1 decade and each alcohol group (-OH green symbols) decreasesCo by 23 decades One organic-acid group decreases Co by about 35 decades or175 decades per oxygen (the carbon is not counted because it is included in nC)which is very close to a simple sum of (-OH) and (=O)

These effects are shown in a different way in Fig 4 which casts the effects of func-20

tionalization in a 2-D space with Co on the x-axis and OC on the y The different func-tional groups generate different slopes (shown with dashed red lines) The wedges inFig 4 describe the range of values one might expect as a carbon backbone is progres-sively functionalized These wedges thus represent the effect of oxidation chemistry onthe volatility and OC of a reduced precursor under the limiting condition that oxidation25

only functionalizes the backbone (ie nC remains unchanged)Despite the uncertainties in the underlying data there are enough data to suggest

that these single-substituent SAR effects on Co described by the trend lines in Figs 3and 4 are not grossly erroneous In the VBS we group compounds in bins separated by

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an order of magnitude in Co and overall we estimate the uncertainty in extrapolationin these simple SAR calculations to be somewhere between a factor of 10 and 100meaning plusmn1 bin to either side of the estimated volatility bin

If we are to construct a two-dimensional thermodynamic framework for bulk OA weneed to describe an average effect of added oxygen Multiple lines of evidence sug-5

gest that (-OH) (=O) and (-C(O)OH) functionalities are major contributors to ambientoxygenated OA known as OOA (Zhang et al 2007) First OOA is associated withlarge signals in the AMS at mz=18 (H2O+) and 44 (CO+

2 ) consistent with dehydra-tion and decarboxylation of organic acids (Aiken et al 2008) Second NMR analysesof solvent-extracted filter samples suggest high abundances of these functionalities10

(McFiggans et al 2005 Moretti et al 2008) Third FTIR analyses of ambient fil-ter samples show significant contributions of (-OH) (=O) and (-C(O)OH) functionalgroups (Hawkins et al 2010) Finally a van Krevlen analysis of ambient OA by Healdet al (2010) is consistent with either hydroxycarbonyl or organic-acid functionalizationdominating organic-aerosol oxidation on average (ie an average of one hydrogen is15

lost for every oxygen added to ambient OA)Based on this evidence we shall assume that on average OA consists of multi-

functional organics containing organic-acid terminal groups and an equal mixture ofalcohols and ketones along the backbone We can thus define the average effect ofeach added oxygen as decreasing Co by 17 decades shown in Fig 3 by thick dashed20

lines and in Fig 4 with black lines This is consistent with either acid or hydroxycarbonylfunctionality provided that these simple SAR values can be extrapolated to multifunc-tional compounds that presumably constitute OOA The end result of this analysis is asimple three-parameter structure activity relation

lCo =(n0

CminusnC

)bCminusnObO (16)25

with n0C =25 bC =0475 and b0 =17

Other functionalities are not used to establish the SAR but they fall within the rangealready defined Hydroperoxides (-OOH) are assumed to be important under low-NOx

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conditions (Logan et al 1981 Donahue and Prinn 1993) and we see evidence fortheir formation in 2-D-NMR samples of terpene ozonolysis (Maksymiuk et al 2009)The effect of -OOH on Co is similar to -OH (Pankow and Asher 2008)

Ethers (-O-) may also be significant in ambient OA They have a modest effect onvolatility slightly less than the =O effect of about 07 orders of magnitude per sub-5

stituent (Pankow and Asher 2008) Ethers may form through ring-closure reactionsduring the oxidation of larger organic compounds (Lim and Ziemann 2005) or as partof accretion reactions involving small monomers such as glyoxal (Kroll et al 2005Carlton et al 2006 Volkamer et al 2007 Galloway et al 2009) Again as their effectis not far outside of the =O line (even counterbalancing the -OOH effect) we do not10

believe that ethers will ruin the relation we proposeThe most significant omission in Fig 3 is the volatility of organic nitrates Organic

nitrates are certainly important in high-NOx environments (Lim and Ziemann 2005Presto and Donahue 2006 Zhang et al 2006) A nitrate group (-ONO2) typicallyreduces vapor pressure by about 25 orders of magnitude (Pankow and Asher 2008)15

so at a similar level of approximation we could introduce bN = 25 to a more completeSAR In any case nitrates and any other special functionality (organo sulfates aminesalkenes etc) will need to be treated separately if they are to be modeled

The diacids reveal the major shortcoming of this simple linear SAR While thereis significant uncertainty in their volatility the diacids (solid magenta) consistently fall20

about 1 order of magnitude below the 4-oxygen (dashed green) trend line of our simpleSAR which should be very close to the value for two acid groups This discrepancywill later provide an important constraint as we relax the ideal linear assumptions wehave used so far

32 Defining the 2-D volatility space25

The SAR described using Eq (16) allows us to determine the mean properties (specif-ically nC) in the 2-D space presented in Fig 4 The mean-property lines of 17 decadesper oxygen are shown in black The C20 lines are much shallower as OC increases

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much more slowly with each added O for larger molecules It is very important to re-alize that we are not using this SAR to predict lCo for a given nC and OC but ratherthe reverse In both laboratory experiments and field measurements we typically havea much better constraint on OC and lCo than nC and so we are using the SAR toinfer nC (and the average molecular formula) from those observables Likewise in the5

VBS modeling framework (Donahue et al 2006) we use lClowast explicitly and are thusnot directly sensitive to uncertainties in volatility

This SAR allows us to construct a family of contours (isopleths) for constant valuesof nC nH and nO with some uncertainty and spread in the lCo OC space This isshown in Fig 5 These contours interconnect predictive diagnostic and prognostic10

aspects of the 2-D space This paper addresses the predictive aspect focusing on therelationships between lCo and OC on the one hand and nC nH and nO on the otherIn subsequent publications we will explore the diagnostic and prognostic aspects Thediagnostic aspect allows us to evaluate the evolution of nC nH and nO in observational(laboratory and field) datasets where lClowast OC and total organic mass are constrained15

using the relationships established here The prognostic aspect allows us to predictthe evolution of lClowast OC and total organic mass in box and transport models withindividual reaction steps in part constrained by the SAR developed here

Figure 5 shows mean nC as diagonal black isopleths mean nO as green curvingisopleths and the sum of oxygen and carbon nM = nC+nO as cyan curving isopleths20

(this will be important later as nM is similar to the molar volume) The carbon isoplethsare labeled and for OC = 11 they intersect at nC = nO = 05nM (labeled in bold at 24 and 6) These isopleths define the average composition in this 2-D space A notablefeature is that the broad region of our 2-D space with 10minus5 ltCo lt 100microgmminus3 and 13lt OC lt 11 (more or less the OOA range) comprises compounds with 5ndash12 carbons25

and 4ndash8 oxygens Even though this is also the most uncertain region based almostcompletely on extrapolation the range in carbon and oxygen numbers in this region isnot so very large

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33 Solvation energies in the SAR

Though the ideal-solution and structure-activity treatments presented so far may seemdisconnected they are not Our next objective is to relate the SAR to interaction en-ergies of the functional groups in a molecule We will start with the linear SAR wejust defined in Eq (16) seeking to define the specific interaction energies between5

functional groups in a molecule with the functional groups in the solvent (the bulk OA)We shall demonstrate that the linearity of the SAR and ideal solution behavior givenby δθAB = 0 in Eq (12) are tightly linked Using this information we shall introducea single non-ideal interaction energy that simultaneously makes the SAR slightly nonideal (correcting the overestimation of diacid volatility) and the solution non-ideal intro-10

ducing realistic hydrophobic interactions leading to phase separation of non-polar andpolar organics in some cases

In keeping with the simple volatility SAR our central assumption is that we can calcu-late the solvation energy of an organic constituent as a linear combination of interactionenergies between different functional groups in the molecule and different functional15

groups in a solvent and that in a rich amorphous system these interactions will behomogeneous and isotropic In other words we will model OA as an amorphous col-lection of functional groups defined by the mean properties of the OA and without anyother regard to the molecular composition of the OA The central assumption of thiswork is that we can use a mean-field approximation to treat the interaction of individ-20

ual organic molecules with the complex organic solvent otherwise known as organicaerosol

In this paper we have made the simplifying assumption that we can reduce low-NOxOA to two ldquofunctional groupsrdquo carbon and oxygen The average composition of theOA solvent is given by the carbon and oxygen fractions f sC and f sO =

(1minus f sC

) We thus25

need only know two pairwise interaction energies θCC θOO and the cross term θCOThe situation is very much the same as with the simple solvation model presented inEqs (10) and (11) but now the solvated molecule i is in essence a polymer string of

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different functionalities describable by f iC and f iO =(

1minus f iC)

and the solvent is a mixture

of such molecules with a wide range of individual fC and fO values but a well-definedaverage given above We assume that it is the functionalities of the solvent and not thespecific molecules of the solvent that dominate the solute-solvent interactions

The situation is depicted in Fig 6 Applying this to organic aerosol for a given OA5

component i in the bulk OA solvent s

θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M

[f iC(f sCθCC+ (1minus f sC)θCO

)+

(1minus f iC)(f sCθCO+ (1minus f sC)θOO

)](17)

where we have used nC =nM middot fC and nO =nM middot fO the fractions of carbon and oxygen inthe molecule and solvent

When the component i is dissolved in itself we have a single-component system10

defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni

O(1minus f iC)θOO (18)

However there are two key assumptions in any linear SAR including Eq (16) Firstfunctional-group contributions are assumed to be independent of the specific molecularstructure (the location and proximity of the functional groups in a molecule) and sec-15

ond the contribution of a functional group to volatility is due only to its own abundance(and not to the abundance of other functional groups) In other words addition of an(-OH) functionality to an organic molecule will suppress the volatility by 23 decadesregardless of the rest of the molecular structure As we are assuming that volatilities

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and solvation energies have a one-to-one relationship the linear SAR is equivalent to

θi i =nCθCC+nOθOO (19)

with no apparent cross term θCO (as that would violate the second assumption juststated) Equations (18) and (19) can be equated which yields

θlinearCO

=12

(θCC+θOO

)=θ (20)5

This exactly mirrors Eq (5) to the point that we use the same symbol θ because theseterms will turn out to be identical For both a linear SAR and an ideal solution the crossinteraction term is simply the average of the diagonal terms This is fundamental Thereis a one-to-one correspondence between a linear SAR and ideal solution behaviorprovided that the ideal solution is defined consistently with the SAR ndash not in terms of10

the molarity of the solute but in terms of the molarity of the functionalities comprisingthe solute and the mean-field properties of the solvent

For compounds with similar densities in the OA system the appropriate fraction f iMis close to the volume fraction of the molecule i in the OA solvent s In practical termsthis means that a linear SAR corresponds to ideal solution behavior on a mass basis15

because the carbon and oxygen moieties all have very similar masses Formulatingthe thermodynamic interactions with different fractions (ie mole volume mass etc)simply requires transforming the x-axis in Fig 2 We are making no effort to exactlydescribe activity curves in that figure for individual molecules but rather seeking todevelop reasonably accurate values for bulk OA and various measurable fractions20

all with unknown molecular composition Given roughly constant density switchingbetween volume and mass fractions amounts to a slight skewing of the x-axis in Fig 2that will fall out with empirical calibration along with the other simplifying assumptionsmade so far This is consistent with the mass-fraction based solution formulation in the1-D VBS (Donahue et al 2006)25

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331 Non-linear SAR

If a linear SAR is equivalent to Eq (20) a non-linear SAR must be given by θCO 6= θThere are obviously many ways of introducing non-linearity but we shall show thatthis method introduces both in a self-consistent and useful way suitable for real OAmixtures Given the one-to-one relationship between lCo and θ already developed we5

can write a non-linear SAR in concise form as a matrix equation adding a non-linearcross term δθCO to the linear term introduced in Eq (20)

θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)

or for a single-component system the pure sub-cooled liquid solvation energy (andthus lCo SAR) is given by10

θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)

We still assume a linear relationship between the solvation energy of a pure substancegiven by Eq (22) and lCo

i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)

As we shall see the cross term is sufficient to provide interesting and realistic behaviorndash and we can constrain it

The non-linear SAR proposed here contains four free parameters24110

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1 The carbon-carbon interaction term (θCC =δθ10 middotbC)

2 The oxygen-oxygen interaction term (θOO =δθ10 middotbO)

3 The carbon-oxygen nonideality (δθCO =δθ10 middotbCO) and

4 The solvation energy of a 1microgmminus3 substance(θ0 =δθ10 middotbC middotn0

C)5

Our argument is that these four parameters underpin a reasonably accurate (andtestable) thermodynamic framework More information can be added ndash for examplesplitting the ldquooxygenrdquo moiety into =O and -OH would add another diagonal and twonew cross terms and adding a nitrate moiety will require a similar expansion We willcertainly need to add the nitrate term for high-NOx environments but it remains to be10

seen whether the oxygen term will need to be split for realistic atmospheric situationsWe can put some real numbers on the interaction matrix given the data presented

so far We will use δθ10 = 690 K as discussed earlier With this and the relationsfrom Fig 3 we have bC = 0475 rarr θCC = 328 K and we could also use bO = 17 togive θOO = 1172 K However we shall consider this together with the non-linear cross15

term The data shown in Fig 3 provide much better constraints on the volatility of ahomologous sequence of compounds (ie increasing nC with constant functionality)In addition most of the molecules constraining the solvation energy have many morecarbon than oxygen atoms Consequently we do not really constrain θOO as much aspartθpartnO generally with nO ltnC20

partθi i

partnOθOO+2δθCO =1172K (25)

Thus in the linear limit θ = 750 K this is equivalent to a linear θOO of 1172 K For thenon-linear case we add δθCO to the diagonal term instead of the off-diagonal term

θOO =1172Kminus2δθCO (26)

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This will keep the effect of oxygenated functionality consistent for relatively small nOwere the data constraints are robust resulting in the interaction matrix

θ=(

328 750750 1172minus2δθCO

)K (27)

The next question is what is a reasonable non-linearity for typical ambient OA con-stituents It is almost certain that δθCO lt 0 (the cross term is destabilizing) because5

this is what drives phase separation and we well know that oil and highly polar organicsdo not mix For a constraint we shall return to the volatility data presented in Fig 3noting two features Most importantly the diacids are less volatile than our simplelinear SAR suggests even though the same rules do a good job predicting mono-substituted organic acids In addition many of the sequences of oxygenated organics10

show a ldquohookrdquo at low nC with lower volatility in the data than predicted by the simplelinear trend lines in Fig 3 (for example the alcohols and mono-acids for nC lt 4) InFig 7 we show these data again but now employing the non-linear SAR described byEq (24) with δθCO =minus03times690=minus207K The consequence is an oxygen interactionterm bO = 23 but with bCO =minus03 generally compensating for nO nC (ie low OC)15

The new SAR again summarized with dashed lines gives better agreement with thedata Specifically the volatility of the diacids is now reproduced with good fidelity andthe SAR (the thick dashed lines) shows a low-nC hook

The revised SAR in our 2-D space is displayed in Fig 8 This is similar to Fig 5 butclose inspection reveals that the isopleths along the top line (OC = 11) have shifted20

toward lower volatility by about 1 decade For example a volatility of 100 microgmminus3 is ap-propriate for butanedioic (succinic) acid The carbon isopleths are very slightly curvedThis slight curvature toward lower volatility is crucial to the activity coefficients that drivephase separation between non-polar and polar organics This is important as it meansthat highly oxygenated organics of a given volatility will tend to have a somewhat lower25

carbon number than they would in a linear system Also by comparing Figs 7 and8 one can see that the region of this 2-D space that is reasonably constrained by

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data extends to roughly the nO = 4 green isopleth which curves from nC = nO = 4 atlog10C

o 2 downward and leftward This approximates the diacids The space to theupper left of this curve the SAR is an extrapolation and that is the very space occupiedby ambient oxygenated organic aerosol indicated by the dashed oval labeled ldquoOOArdquo(Jimenez et al 2009)5

While the nCnH and nO isopleths in the OOA region in Fig 8 are extrapolationsthey are not extreme extrapolations We consider this to be a reasonable projection oftypical values for ambient organic-aerosol compounds for all of the reasons discussedso far Again we emphasize that the primary quantities ndash log10C

lowast and OC can beconstrained by ambient observations and our primary application of the non-linear10

SAR is as a diagnostic to ascertain the typical values for nCnH and nO

4 OA activity coefficients

The SAR so far has focused on the pure solvation energies of molecules (really pseudomolecules) composed of carbon and oxygen groups When we relax our conditions andallow a molecule i to interact with an OA solvent s with a different average composi-15

tion we wish to calculate the effective saturation concentration including an activitycoefficient Clowast = γCo This will allow us to treat (model not just diagnose) realistic OAmixtures with atmospherically relevant composition Because in our volatility-basedspace we specify the saturation concentration lCo we need to know the activity coef-ficient As we showed in Eq (9) this depends on the non-ideal off-diagonal interaction20

term This is readily determined in our homogeneous solvent

δθi s =nM

[(f ic)2+ (f sc )2minus2f icf

sc

]δθCO (28)

This couples the non-ideal solution and non-linear SAR treatments we have devel-oped so far The non-linearity of the SAR (δθCO) leads directly to any non-idealityof a solute in the OA solution At this level of approximation all of the non-ideality25

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rests on differences between the carbon fraction (fC or OC) in the solvent and soluteThis is a substantial simplification however it amounts to asserting that polarity is themost important determinant of activity coefficients in organic solutions which is justi-fied (Prausnitz et al 1998 Pankow and Barsanti 2009) Furthermore the approximatemolar volume (nM) of the solute molecule is also a critical parameter5

402 Activity coefficients with a non-linear SAR

To illustrate the consequences of non-ideality we shall consider two fairly typical situa-tions typified by OA with two different bulk (single-phase) compositions First we shallconsider a highly aged background aerosol with OC = 0751 This is consistent withhighly oxidized OA characteristic of the remote atmosphere and generally called OOA10

(oxidized organic aerosol) (Jimenez et al 2009 Ng et al 2009) Second we shallconsider a much less oxidized aerosol with OC=0251 This is more consistent withvery fresh emissions possibly combining traffic emissions (known as HOA or hydro-carbon like organic aerosol to the AMS community) and either some biomass burningOA (BBOA) or slightly oxidized first-generation SOA (Lanz et al 2007 Jimenez et al15

2009)In Fig 9 we show organic activity coefficients γi s for the full range of organic solutes

dissolved in these two different bulk aerosol solvents This is calculated by using thenonlinear energy from Eq (28) in Eq (9) at 300 K using δθCO =minus207 K The OC ofthe bulk solvent is indicated on each panel The activity coefficients of organics are20

contoured in these panels for all Co OC combinations with contour intervals of 05ranging from 1 to 10 labeled at integer values from 1 to 5 For example material atlog10C

o =minus5 OC 0251 in Fig 9a has γ 10 resulting in log10Clowast minus4 This model

is highly simplified but it contains essential features that can be constrained by ambientobservations and that exhibit the fundamental characteristics of ambient OA25

In practical terms when γi s gt 5 or 10 it is likely that a constituent will phase sep-arate as the activity ai will be greater than 10 when fi gt 02 or 01 If there is anysignificant amount of a group of similar constituents they will form their own phase

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Once a class is less than 10 or 20 of the OA it is of secondary concern in any eventIn AMS jargon this implies that fresh HOA will not mix with aged OOA when OOA isin the majority (this is the ldquophase separation regionrdquo indicated along the lower edge ofFig 9a) Note again that this fraction is the volume fraction which is effectively themass fraction and we are judging ideality and modeling mixing on a volume-fraction5

basisWhat Fig 9a shows is that when the background aerosol is highly oxygenated (with

bulk average OC indicated by the dark green arrow) non-polar reduced primary emis-sions (POA with OC 0 and minus2le lCo le 2 for example the gray dot) will have a hardtime mixing into that background aerosol and will instead tend to phase separate (more10

accurately they will tend to remain in their original phase often in their own size modendash for example a traffic mode) The phase boundary will be somewhere near the γ =5 or10 contours depending on the exact composition distribution the uncontoured regionto the lower left of that boundary (low lCo low OC) will form a separate ldquoPOArdquo phaseThe transition is quite sharp and the location is not very sensitive to the exact value of15

the off diagonal term δθCO This is very consistent with much more sophisticated cal-culations for far simpler mixtures designed to simulate OA For example (Jang et al1997) showed that larger alkanes (exclusively) showed substantial activity coefficientsin model mixtures designed to simulate both woodsmoke and SOA

On the other hand when the bulk OA is not very oxidized as in Fig 9b almost all OA20

compounds can mix with the aerosol with only modest activity penalties (γ lt 3) Onlyfor semi-volatile compounds just at the volatility threshold (Clowast

i COA) will the smalldifference in volatility due to the non-ideality have any consequence These selectcompounds will shift toward the vapor phase because Clowast gtCo for γ gt 1 For the mostpart though when the bulk OA is relatively fresh with low OC we expect it to be25

characterized by a single condensed phase which will likely become progressivelymore oxidized as the mixture ages (Jimenez et al 2009) In AMS jargon OOA will mixwith HOA (or especially BBOA) when HOA is in the majority

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In general the consequence of incorporating activity coefficients into any represen-tation of OA will be to force some compounds into the vapor phase This may becomeeven more pronounced considering the dynamics of fresh mobile-source emissionswhich can be small enough for the Kelvin term to increase the volatility even more(Zhang et al 2004) This will be most dramatic when the systems phase separate5

and the region where a separate phase may have a relatively small mass fraction isthe lower left-hand portion of the 2-D space which comprises fresh primary emissions(POA or HOA in AMS jargon)

However it would be a gross oversimplification to conclude that incorporation ofactivity coefficients into a model coupling aerosol dynamics and chemistry would result10

in lowered OA levels due to the increased volatility The consequence of forcing moreHOA into the gas phase will almost certainly be rapid gas-phase oxidation (Shrivastavaet al 2008) generating products that will mix with the OOA phase with mass yields thatmay even exceed unity (Presto et al 2009 2010) The issue is thus not whether POAcan act as a seed for SOA but rather whether SOA (really OOA) is a good seed for15

POA The answer is no but the full answer is not that there will be less OA but ratherthat POA may for a time exist in a separate fresh size mode but that it will tend toevaporate be more rapidly oxidized in the gas phase and generate oxidation productsthat will mix into SOA Thus the major observable consequence of non-ideality may bekinetic ndash the transformation from POA to SOA or HOA to OOA will be more rapid that20

it would otherwise be were an ideal solution assumed

5 Conclusions

We have presented a self-consistent thermodynamic foundation for a two-dimensionalvolatility basis set relying on a minimum number of free parameters to describe real-istic and measurable properties of real-world organic-aerosol mixtures This treatment25

suggests that we can reasonably describe OA thermodynamics with a mean-field ap-proximation including a non-ideality term that simultaneously and self-consistently de-

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scribes non-linearity in the structure activity relation (SAR) as well as non-ideality inthe solution thermodynamics Because δ θCO lt 0 the consequences of all this will bethat more molecules will find themselves in the gas phase than in an ideal world

The formulation so far does not address the role of water (Pankow and Chang 2008Tong et al 2008 Barley et al 2009 Prisle et al 2010) but the strong correlation of5

OC and OA hygroscopicity reported in Jimenez et al (2009) suggests that we may beable to treat water in a similar fashion However because water can be a dominantcomponent and therefore its unique properties are more likely to challenge the mean-field assumption it is likely that explicit treatment of humidity effects will be required(Prisle et al 2010)10

The 2-D-VBS complements our well-established 1-D-VBS (Donahue et al 2006)by adding oxygen content (OC) as a second dimension in addition to saturation con-centration (Co) Using pairwise interaction energies we can formulate both a simplestructure activity relation to relate carbon and oxygen numbers to OC and Co Theformulation includes a non-linear coupling term that can be constrained by pure com-15

ponent vapor-pressure data that also leads to realistic activity coefficient predictionsThis formulation relies on measurable bulk properties of organic aerosol (most notablyOC) and uses a mean-field approximation for the molecule-solvent interactions ndash inessence we assume that a molecule interacts more with other functional groups in theorganic aerosol solution than with other molecules in that mixture Importantly the20

complex behaviors develop from a very simple formulation containing only four freeparameters all of which are constrained by vapor-pressure data of pure substances

This is a foundation for future work In several succeeding publications we shall ex-plore the uses of this 2-D-VBS for both diagnostic analyses and prognostic calculationsorganic aerosol properties ultimately extending the model to a fully coupled treatment25

of size-dependent organic-aerosol dynamics

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Acknowledgements This work was funded by the US EPA Science to Achieve Results (STAR)program

References

Aiken A C DeCarlo P F Kroll J H Worsnop D R Huffman J A Docherty K UlbrichI M Mohr C Kimmel J R Sueper D Zhang Q Sun Y Trimborn A Northway5

M Ziemann P J Canagaratna M R Alfarra R Prevot A S Dommen J DuplissyJ Metzger A Baltensperger U and Jimenez J L OC and OMOC Ratios of PrimarySecondary and Ambient Organic Aerosols with High Resolution Time-of-Flight Aerosol MassSpectrometry Env Sci Technol 42 4478ndash4485 2008 24092 24095 24104

Asa-Awuku A Miracolo M A Kroll J H Robinson A L and Donahue N M10

Mixing and phase partitioning of primary and secondary organic aerosols GeophysRes Lett 36 L15 827 doi1010292009GL039301 httpwwwaguorgjournalsglgl09152009GL039301 2009 24094 24095

Asher W Pankow J Erdakos G and Seinfeld J Estimating the vapor pressures of multi-functional oxygen-containing organic compounds using group contribution methods Atmos15

Environ 36 1483ndash1498 2002 24094Sensitivities of the absorptive partitioning model of secondary organic aerosol formation to the

inclusion of water Atmos Chem Phys 9 2919ndash2932 doi105194acp-9-2919-2009 200924117

Bilde M and Pandis S N Evaporation rates and vapor pressures of individual aerosol20

species formed in the atmospheric oxidation of alpha- and beta-pinene Environ Sci Tech-nol 35 3344ndash3349 2001 24102

Bilde M Svenningsson B Monster J and Rosenorn T Even-odd alternation of evaporationrates and vapor pressures of C3-C9 dicarboxylic acid aerosols Environ Sci Technol 371371ndash1378 doi101021es0201810 2003 2410225

Bowman F and Melton J Effect of activity coefficient models on predictions of secondaryorganic aerosol partitioning J Aerosol Sci 35 1415ndash1438 doi101016jjaerosci200407001 2004 24093 24094

Cappa C D Lovejoy E R and Ravishankara A R Determination of evaporation rates

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Discussion

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|D

iscussionP

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and vapor pressures of very low volatility compounds A study of the C-4-C-10 and C-12dicarboxylic acids J Phys Chem A 111 3099ndash3109 doi101021jp068686q 2007 24102

Cappa C D Lovejoy E R and Ravishankara A R Evidence for liquid-like and nonidealbehavior of a mixture of organic aerosol components P Natl Acad Sci USA 105 18 687ndash18 691 doi101073pnas0802144105 2008 24100 241025

Carlton A G Turpin B J Lim H-J Altieri K E and Seitzinger S Link between isopreneand secondary organic aerosol (SOA) Pyruvic acid oxidation yields low volatility organicacids in clouds Geophys Res Lett 33 L06 822 doi1010292005GL025374 2006 24105

Chandramouli B Jang M and Kamens R Gas-particle partitioning of semi-volatile or-ganics on organic aerosols using a predictive activity coefficient model analysis of the10

effects of parameter choices on model performance Atmos Environ 37 853ndash864 doi101016S1352-2310(02)00931-7 2003 24094

Clegg S L Kleeman M J Griffin R J and Seinfeld J H Effects of uncertainties in thethermodynamic properties of aerosol components in an air quality model Part 1 Treatmentof inorganic electrolytes and organic compounds in the condensed phase Atmos Chem15

Phys 8 1057ndash1085 doi105194acp-8-1057-2008 2008 24094Clegg S L Kleeman M J Griffin R J and Seinfeld J H Effects of uncertainties in the

thermodynamic properties of aerosol components in an air quality model - Part 2 Predic-tions of the vapour pressures of organic compounds Atmos Chem Phys 8 1087ndash1103doi105194acp-8-1087-2008 2008 2409420

Donahue N M and Prinn R G In-situ nonmethane hydrocarbon measurements on SAGA-3J Geophys Res-Atmos 98 16 915ndash16 932 1993 24105

Donahue N M Robinson A L Stanier C O and Pandis S N Coupled partitioningdilution and chemical aging of semivolatile organics Environ Sci Technol 40 2635ndash26432006 24093 24097 24106 24109 2411725

Donahue N M Robinson A L and Pandis S N Atmospheric Organic Particulate MatterFrom Smoke to Secondary Organic Aerosol Atmos Environ 43 94ndash106 doi101016jatmosenv200809055 httpdxdoiorg101016jatmosenv200809055 2009 24102

Epstein S A Riipinen I and Donahue N M A Semi-Empirical Correlation between En-thalpy of Vaporization and Saturation Concentration for Organic Aerosol Environ Sci Tech-30

nol 44 743ndash748 doi101021es902497z httppubsacsorgdoiabs101021es902497z2010 24096 24097

Fraser M P Cass G R Simoneit B R T and Rasmussen R A Air quality model eval-

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Discussion

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uation data for organics 4 C2-C36 non-aromatic hydrocarbons Environ Sci Technol 312356ndash2367 1997 24092

Fraser M P Cass G R Simoneit B R T and Rasmussen R A Air quality model evalua-tion data for organics 5 C6-C22 nonpolar and semipolar aromatic compounds Environ SciTechnol 32 1760ndash1770 1998 240925

Fredenslund A Jones R and Prausnitz J Group-contribution estimation of activity-coefficients in nonideal liquid-mixtures Aiche J 21 1086ndash1099 1975 24094

Galloway M M Chhabra P S Chan A W H Surratt J D Flagan R C Seinfeld J Hand Keutsch F N Glyoxal uptake on ammonium sulphate seed aerosol reaction productsand reversibility of uptake under dark and irradiated conditions Atmos Chem Phys 910

3331ndash3345 doi105194acp-9-3331-2009 2009 24105Goldstein A H Worton D R Williams B J Hering S V Kreisberg N M Panic O and

Gorecki T Thermal desorption comprehensive two-dimensional gas chromatography forin-situ measurements of organic aerosols J Chromatogr A 1186 340ndash347 doi101016jchroma200709094 2008 2409215

Hawkins L N Russell L M Covert D S Quinn P K and Bates T S Carboxylic acidssulfates and organosulfates in processed continental organic aerosol over the southeastPacific Ocean during VOCALSREx 2008 Geophys Res Lett 115 D13 201 doi1010292009JD013276 2010 24104

Heald C L Kroll J H Jimenez J L Docherty K S DeCarlo P F Aiken A C Chen20

Q Martin S T Farmer D K and Artaxo P A simplified description of the evolutionof organic aerosol composition in the atmosphere Geophys Res Lett 37 doi1010292010GL042737 2010 24104

Hildemann L M Mazurek M A Cass G R and Simoneit B R T Quantitative Charac-terization of Urban Sources of Organic Aerosol By High-Resolution Gas-Chromatography25

Environ Sci Technol 25 1311ndash1325 1991 24092Jang M Kamens R M and R S M A Thermodynamic Approach Using Group Contribu-

tion Methods to Model the Partitioning of Semivolatile Organic Compounds on AtmosphericParticulate Matter Environ Sci Technol 31 2805ndash2811 1997 24099 24115

Jimenez J L Canagaratna M R Donahue N M Prevot A S H Zhang Q Kroll J H30

DeCarlo P F Allan J Coe H Ng N L Aiken A C Docherty K D Ulbrich I MGrieshop A P Robinson A L Duplissy J Smith J D Wilson K R Lanz V A HueglinC Sun Y L Laaksonen A Raatikainen T Rautiainen J Vaattovaara P Ehn M

24120

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Kulmala M Tomlinson J M Collins D R Cubison M J Dunlea E J Huffman J AOnasch T B Alfarra M R Williams P I Bower K Kondo Y Schneider J DrewnickF Borrmann S Weimer S Demerjian K Salcedo D Cottrell L Griffin R Takami AMiyoshi T Hatakeyama S Shimono A Sun J Y Zhang Y M Dzepina K KimmelJ R Sueper D Jayne J T Herndon S C Trimborn A M Williams L R Wood5

E C Kolb C E Baltensperger U and Worsnop D R Evolution of Organic Aerosols inthe Atmosphere A New Framework Connecting Measurements to Models Science 3261525ndash1529 2009 24113 24114 24115 24117

Koo B Y Ansari A S and Pandis S N Integrated approaches to modeling the organic andinorganic atmospheric aerosol components Atmos Environ 37 4757ndash4768 2003 2409410

Koponen I K Riipinen I Hienola A Kulmala M and Bilde M Thermodynamic propertiesof malonic succinic and glutaric acids Evaporation rates and saturation vapor pressuresEnviron Sci Technol 41 3926ndash3933 doi101021es0611240 2007 24102

Kroll J Ng N Murphy S Varutbangkul V Flagan R and Seinfeld J Chamber studiesof secondary organic aerosol growth by reactive uptake of simple carbonyl compounds J15

Geophys Res-Atmos 110 doiARTND23207 2005 24105Lanz V A Alfarra M R Baltensperger U Buchmann B Hueglin C and Prevot A S

H Source apportionment of submicron organic aerosols at an urban site by factor analyticalmodelling of aerosol mass spectra Atmos Chem Phys 7 1503ndash1522 doi105194acp-7-1503-2007 2007 2411420

Lim Y B and Ziemann P J Products and Mechanism of Secondary Organic Aerosol For-mation from Reactions of n-Alkanes with OH Radicals in the Presence of NOx Environ SciTechnol 39 9229ndash9236 2005 24105

Logan J A Prather M J Wofsy S C and McElroy M B Tropospheric Chemistry A GlobalPerspective J Geophys Res-Atmos 86 7210ndash7254 1981 2410525

Maksymiuk C S Gayathri C Gil R R and Donahue N M Secondary Organic AerosolFormation from Multiphase Oxidation of Limonene by Ozone Mechanistic Constraintsvia Two-dimensional Heteronuclear NMR Spectroscopy Phys Chem Chem Phys 117810ndash7818 doi101039b820005j httppubsrscorgenContentArticleLanding2009CPB820005J 2009 2410530

McFiggans G Alfarra M R Allan J Bower K Coe H Cubison M Topping D WilliamsP Decesari S Facchini C and Fuzzi S Simplification of the representation of the organiccomponent of atmospheric particulates Faraday Discuss 130 341ndash362 httppubsrscorg

24121

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enContentArticleLanding2005FDB419435G 2005 24104Moretti F Tagliavini E Decesari S Facchini M C Rinaldi M and Fuzzi S NMR de-

termination of total carbonyls and carboxyls A tool for tracing the evolution of atmosphericoxidized organic aerosols Environ Sci Technol 42 4844ndash4849 doi101021es703166v2008 241045

Ng N L Canagaratna M R Zhang Q Jimenez J L Tian J Ulbrich I M Kroll J HDocherty K S Chhabra P S Bahreini R Murphy S M Seinfeld J H HildebrandtL DeCarlo P F Lanz V A Prevot A S H Dinar E Rudich Y and Worsnop D ROrganic aerosol components observed in worldwide datasets from aerosol mass spectrome-try Atmos Chem Phys Discuss 9 27745ndash27789 doi105194acpd-9-27745-2009 200910

24092 24114Pankow J F An absorption model of gasparticle partitioning of organic compounds in the

atmosphere Atmos Environ 28 185ndash188 1994 24093Pankow J F and Asher W E SIMPOL1 a simple group contribution method for predict-

ing vapor pressures and enthalpies of vaporization of multifunctional organic compounds15

Atmos Chem Phys 8 2773ndash2796 doi105194acp-8-2773-2008 2008 24101 24105Pankow J F and Barsanti K C The carbon number-polarity grid A means to manage the

complexity of the mix of organic compounds when modeling atmospheric organic particulatematter Atmos Environ 43 2829ndash2835 doi101016jatmosenv200812050 2009 24114

Pankow J F and Chang E I Variation in the sensitivity of predicted levels of atmo-20

spheric organic particulate matter (OPM) Environ Sci Technol 42 7321ndash7329 doi101021es8003377 2008 24099 24117

Prausnitz J M Lichtenthaler R N and Gomes de Azevedo E Molecular Thermodynamicsof Fluid-Phase Equilibria Prentice Hall Upper Saddle River New Jersey 3rd edn 199824096 24097 24098 2411425

Presto A A and Donahue N M Investigation of alpha-pinene plus ozone secondary organicaerosol formation at low total aerosol mass Environ Sci Technol 40 3536ndash3543 200624105

Presto A A Miracolo M A Donahue N M Robinson A L Kroll J H and Worsnop D RIntermediate-Volatility Organic Compounds A Potential Source of Ambient Oxidized Organic30

Aerosol Environ Sci Technol 43 4744ndash4749 doi101021es803219q httppubsacsorgdoiabs101021es803219q 2009 24116

Presto A A Miracolo M A Donahue N M and Robinson A L Secondary Organic

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Aerosol Formation from High-NOx Photo-Oxidation of Low Volatility Precursors n-AlkanesEnviron Sci Technol 44 2029ndash2034 doi101021es903712r httppubsacsorgdoiabs101021es903712r 2010 24116

Prisle N Engelhart G J Bilde M and Donahue N M Humidity Effects on Organic AerosolPartitioning Following the α-pinene + Ozone Reaction Geophys Res Lett 37 L01 8025

doi1010292009GL041402 httpwwwaguorgjournalsglgl10012009GL041402 201024117

Pun B K Seigneur C and Lohman K Modeling secondary organic aerosol formationvia multiphase partitioning with molecular data Environ Sci Technol 40 4722ndash4731 doi101021es0522736 2006 2409410

Robinson A L Donahue N M and Rogge W F Photochemical oxidation and changes inmolecular composition of organic aerosol in the regional context J Geophys Res-Atmos111 D03 302 httpwwwaguorgpubscrossref20062005JD006265shtml 2006 24093

Schauer J J Rogge W F Hildemann L M Mazurek M A and Cass G R Sourceapportionment of airborne particulate matter using organic compounds as tracers Atmos15

Environ 30 3837ndash3855 1996 24092Shrivastava M K Subramanian R Rogge W F and Robinson A L Sources of organic

aerosol Positive matrix factorization of molecular marker data and comparison of resultsfrom different source apportionment models Atmos Environ 41 9353ndash9369 doi101016jatmosenv200709016 2007 2409320

Shrivastava M K Lane T E Donahue N M Pandis S N and Robinson A L Effects ofGas-Particle Partitioning and Aging of Primary Organic Emissions using the Chemical Trans-port Model PMCAMx J Geophys Res A 113 D18 301 doidoi1010292007JD009735httpwwwaguorgpubscrossref20082007JD009735shtml 2008 24116

Song C Zaveri R A Alexander M L Thornton J A Madronich S Ortega J V Zelenyuk25

A Yu X-Y Laskin A and Maughan D A Effect of hydrophobic primary organic aerosolson secondary organic aerosol formation from ozonolysis of alpha-pinene Geophys ResLett 34 L20 803 doiARTNL20803 2007 24095

Tao Y and McMurry P Vapor-pressures and surface free-energies of C14-C18 monocar-boxylic acids and C5-dicarboxylic and C6-dicarboxylic acids Environ Sci Technol 2330

1519ndash1523 1989 24102Tong C Clegg S L and Seinfeld J H Comparison of activity coefficient models for atmo-

spheric aerosols containing mixtures of electrolytes organics and water Atmos Environ

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42 5459ndash5482 doi101016jatmosenv200802040 2008 24117Vaden T D Song C Zaveri R A Imre D and Zelenyuk A Morphology of mixed pri-

mary and secondary organic particles and the adsorption of spectator organic gases duringaerosol formation P Natl Acad Sci USA 107 6658ndash6663 doi101073pnas09112061072010 240955

Volkamer R Martini F S Molina L T Salcedo D Jimenez J L and Molina M J Amissing sink for gas-phase glyoxal in Mexico City Formation of secondary organic aerosolGeophys Res Lett 34 L19 807 doiARTNL19807 2007 24105

Zhang J Y Hartz K E H Pandis S N and Donahue N M Secondary organicaerosol formation from limonene ozonolysis Homogeneous and heterogeneous influences10

as a function of NOx J Phys Chem A 110 11053ndash11063 doi101021jp062836f httpdxdoiorg101021jp062836f 2006 24105

Zhang K Wexler A Zhu Y Hinds W and Sioutas C Evolution of particle number distribu-tion near roadways Part II the ldquoroad-to-ambientrdquo process Atmos Environ 38 6655ndash6665doiDOI101016jatmosenv200406044 2004 2411615

Zhang Q Jimenez J L Canagaratna M R Allan J D Coe H Ulbrich I Alfarra M RTakami A Middlebrook A M Sun Y L Dzepina K Dunlea E Docherty K De-Carlo P F Salcedo D Onasch T Jayne J T Miyoshi T Shimono A Hatakeyama STakegawa N Kondo Y Schneider J Drewnick F Borrmann S Weimer S DemerjianK Williams P Bower K Bahreini R Cottrell L Griffin R J Rautiainen J Sun J Y20

Zhang Y M and Worsnop D R Ubiquity and dominance of oxygenated species in organicaerosols in anthropogenically-influenced Northern Hemisphere midlatitudes Geophys ResLett 34 L13 801 doi1010292007GL029979 2007 24092 24104

Zobrist B Marcolli C Pedernera D A and Koop T Do atmospheric aerosols formglasses Atmos Chem Phys 8 5221ndash5244 doi105194acp-8-5221-2008 2008 2410025

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Organic AerosolfC fO= (1 - fC)

CCC

CO

O


CCC

CO

O

Fig 1 Solvation energy conceptualized A molecule i (teal circle) is disolved in a solvents To evaporate it must be removed from the solvent leaving a void and costing energy θi sto both the molecule and the solvent but the solvent will relax to fill the void releasing energyθss

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Fig 2 Activity of component A at 300 K for two-component (A-B) mixtures with non-idealinteraction energies δθAB of 200 300 400 and 500 K For δθ gt300 the system rapidly phaseseparates quickly achieving almost complete separation

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Fig 3 Structure-activity trends for singly and doubly functionalized oxygenated organicsVolatility (log10Co in microgmminus3) vs nC (carbon number) for different classes of organic compounds(indicated with colored symbols) forming the basis for the carbon and oxygen isopleths pre-sented in the top panel Each class contains the same oxygen functionality but a range of nCThe slopes of the class lines show the effect of increasing nC (minus0475 decades per C) whileoffsets from the hydrocarbon line show the effects of functionalization Carbonyls decrease Co

by about 1 decade per (=O) while alcohols decrease Co by about 23 decades per (-OH) For alinear approximation we adopt an average of minus17 decades per O indicated with dashed linesfor 0 1 2 and 4 oxygens later (Fig 7) we shall add a modest nonlinearity

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Fig 4 The effect of different functional groups on volatility (Co) and OC for a C6 and C20backbone HO- is highly polar decreasing lCo by 23 per group leading to the shallow linesO= is less polar decreasing lCo by 1 per group leading to steeper lines Acids (HO(O)-)decrease lCo by 175 per group The average effect of oxygentated functionality deduced inthis work (in black) leads to slopes similar to the structure-activity of organic acids

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Fig 5 Composition based on a simple linear structure activity relation described in the textdisplayed in a 2-D space with volatility (log10C

o) on the x-axis and oxygenation (OC) on the yaxis Isopleths are for of constant carbon number (nC labeled black lines) oxygen number (nOgreen curves) and total functional group number (nM =nC +nO cyan curves) The rough rangeof oxygenated organic aerosol (OOA) seen in the atmosphere is show with a dashed oval

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CCC

CO

O


CCC

CO

O

Fig 6 Solvation interactions in a simple Structure Activity Relation (SAR) for organics disolvedin an organic-aerosol solvent

24130

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Fig 7 Volatility (Co) predictions with modest nonlinearity (bCO = minus03) in a structure-activityrelation As in Fig 3 the average effect of oxygenated functionality is shown by thick dashedcurves The consequence of the nonlinearity is to increase the interaction energies of oxy-genated functional groups leading to lower volatility of diacids and a pronounced hook at lownC (and thus a high oxygen fraction) ndash these are both consistent with the data (eg alcoholsand acids for nC lt4)

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Fig 8 Organic composition with a small degree of non-linearity (minus03 decades per C-Ointeraction) in the SAR describing OA volatility Isopleths are for of constant carbon number(nC labeled black lines) oxygen number (nO green curves) and total functional group number(nM = nC +nO cyan curves) All isopleths are for even numbers only and the values for thegreen and cyan isopleths can be determined by the intersection points at OC = 11 (eg themeeting of nC = nO = 4 at lCo = 2 and OC = 1 with nM = 8) Compared with Figure 5 moreoxygenated organics are somewhat less volatile ndash the isopleths are shifted approximately 1decade to the left for OC=11

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Fig 9 Activity coefficients of organics in the 2-D space for two different mean OA compositionsIn the top panel the mean OC of the bulk OA is 0751 (OOA-like) indicated by the thick greenarrow while in the bottom panel the mean OC in the bulk OA is 0251 (POA-like) indicated bythe thick gray arrow The isopleths are for γi s the activity coefficient of organics with the lClowast

and OC indicated in each solution Contours are steps of 05 in γ starting at 1 and ending at10 Integer values are labeled up to γ = 5 Somewhere between 5ltγ lt 10 phase separationbecomes very likely For example the POA constituent indicated by the dark gray circle in in toppanel at log10C

o =minus2 OC = 00 has an activity coefficient in the bulk OOA 10 converselythe OOA constituent indicated by the green circle in in bottom panel at log10C

o =minus2 OC =0751 has an activity coefficient in the bulk OOA =25

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Abstract

We develop the thermodynamic underpinnings of a two-dimensional volatility basis set(2-D-VBS) employing saturation concentration (Co) and the oxygen content (OC) todescribe volatility mixing thermodynamics and chemical evolution of organic aerosolThis is an extension of our earlier one-dimensional approach employing Clowast only5

(Clowast = γCo where γ is an activity coefficient) We apply a mean-field approximationfor organic aerosol describing interactions of carbon and oxygen groups in individ-ual molecules (solutes) with carbon and oxygen groups in the organic-aerosol sol-vent In so doing we show that a linear structure activity relation (SAR) describingthe single-component Co of a molecule is directly tied to ideal solution (Raoultrsquos Law)10

behavior Conversely non-ideal solution behavior (activity coefficients) and a slightlynon-linear SAR emerge from off-diagonal (carbon-oxygen) interaction elements Fromthis foundation we can build a self-consistent description of OA mixing thermodynam-ics including predicted saturation concentrations and activity coefficients (and phaseseparation) for various solutions from just four free parameters the carbon number of15

a hydrocarbon with a 1microgmminus3 Co and the carbon-carbon oxygen-oxygen and non-ideal carbon-oxygen terms This treatment establishes the mean molecular formula fororganics within this 2-D space as well as activity coefficients for molecules within thisspace interacting with any bulk OA phase described by an average OC

1 Introduction20

It is well established that atmospheric organic aerosol (OA) exists as a complex mixtureof thousands of individual organic compounds (Hildemann et al 1991 Fraser et al1997 1998 Goldstein et al 2008) Furthermore it is clear that a large fraction of OAconsists of organic compounds that are difficult to separate in a gas chromatograph(Schauer et al 1996) and that very often a majority of OA consists of quite highly25

oxidized organic molecules (Zhang et al 2007 Aiken et al 2008 Ng et al 2009)

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o and15






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24095

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2 Theory




300(log10C





∆θvapi =∆θvap


or

lCoi300 =

(∆θvap


) 1∆θ10

(2)

24096

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θvapi s =θi s+

(θi sminusθss







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θideali s =

12

(θss+θi i

)=θ (5)





θ=

(θi i θ+δθi s

θ+δθi s θss

)(7)10





expression


)(9)


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23 Simple mixtures




)θAB (10)15



(1minus f sA

)θAB+

(1minus f sA

)2θBB (11)20



)=θ then



)2δθAB (12)24099

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)(13)



)2δθABT)

(14)



ACPD10 24091ndash24133 2010


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log10

(p0Li

)=b0+

sumk

νkibk (15)20


0Li focusing on


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lCo =(n0

CminusnC

)bCminusnObO (16)25

with n0C =25 bC =0475 and b0 =17


24104

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24106

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(1minus f sC

) We thus25


24107

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1minus f iC)





θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M


)+


)](17)



defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni




24108

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θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

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331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


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C)5





partθi i




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θ=(

328 750750 1172minus2δθCO

)K (27)








24112

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δθi s =nM


sc

]δθCO (28)


24113

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24114

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24115

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5 Conclusions



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24117

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References













24118

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24119

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24120

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24121

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24122

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24123

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24124

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CCC

CO

O


CCC

CO

O


24125

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24126

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24127

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24128

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24129

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CCC

CO

O


CCC

CO

O


24130

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24131

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24132

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24133

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o and15






24093

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24094

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24095

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2 Theory




300(log10C





∆θvapi =∆θvap


or

lCoi300 =

(∆θvap


) 1∆θ10

(2)

24096

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θvapi s =θi s+

(θi sminusθss







24097

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θideali s =

12

(θss+θi i

)=θ (5)





θ=

(θi i θ+δθi s

θ+δθi s θss

)(7)10





expression


)(9)


24098

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23 Simple mixtures




)θAB (10)15



(1minus f sA

)θAB+

(1minus f sA

)2θBB (11)20



)=θ then



)2δθAB (12)24099

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)(13)



)2δθABT)

(14)



ACPD10 24091ndash24133 2010


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log10

(p0Li

)=b0+

sumk

νkibk (15)20


0Li focusing on


ACPD10 24091ndash24133 2010


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24102

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24103

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lCo =(n0

CminusnC

)bCminusnObO (16)25

with n0C =25 bC =0475 and b0 =17


24104

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24105

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24106

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(1minus f sC

) We thus25


24107

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1minus f iC)





θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M


)+


)](17)



defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni




24108

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θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

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331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


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C)5





partθi i




24111

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θ=(

328 750750 1172minus2δθCO

)K (27)








24112

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δθi s =nM


sc

]δθCO (28)


24113

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24114

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24115

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5 Conclusions



24116

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24117

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References













24118

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24119

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24120

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24121

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24122

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24123

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24124

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CCC

CO

O


CCC

CO

O


24125

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24126

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24127

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24128

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24129

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CCC

CO

O


CCC

CO

O


24130

ACPD10 24091ndash24133 2010


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24131

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24132

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24133

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24094

ACPD10 24091ndash24133 2010


N M Donahue et al

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24095

ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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|D

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2 Theory




300(log10C





∆θvapi =∆θvap


or

lCoi300 =

(∆θvap


) 1∆θ10

(2)

24096

ACPD10 24091ndash24133 2010


N M Donahue et al

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|D

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Discussion

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|D

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θvapi s =θi s+

(θi sminusθss







24097

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N M Donahue et al

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|D

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Discussion

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θideali s =

12

(θss+θi i

)=θ (5)





θ=

(θi i θ+δθi s

θ+δθi s θss

)(7)10





expression


)(9)


24098

ACPD10 24091ndash24133 2010


N M Donahue et al

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|D

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Discussion

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|D

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23 Simple mixtures




)θAB (10)15



(1minus f sA

)θAB+

(1minus f sA

)2θBB (11)20



)=θ then



)2δθAB (12)24099

ACPD10 24091ndash24133 2010


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)(13)



)2δθABT)

(14)



ACPD10 24091ndash24133 2010


N M Donahue et al

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log10

(p0Li

)=b0+

sumk

νkibk (15)20


0Li focusing on


ACPD10 24091ndash24133 2010


N M Donahue et al

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|D

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24102

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|D

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24103

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|D

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lCo =(n0

CminusnC

)bCminusnObO (16)25

with n0C =25 bC =0475 and b0 =17


24104

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N M Donahue et al

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|D

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24105

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24106

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(1minus f sC

) We thus25


24107

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Discussion

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|D

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1minus f iC)





θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M


)+


)](17)



defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni




24108

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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|D

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θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

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331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


N M Donahue et al

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C)5





partθi i




24111

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θ=(

328 750750 1172minus2δθCO

)K (27)








24112

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δθi s =nM


sc

]δθCO (28)


24113

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24114

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24115

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5 Conclusions



24116

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24117

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References













24118

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24119

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24120

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24121

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24122

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24123

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24124

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CCC

CO

O


CCC

CO

O


24125

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24126

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24127

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24128

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24129

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|D

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CCC

CO

O


CCC

CO

O


24130

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24131

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24132

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24133

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24095

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2 Theory




300(log10C





∆θvapi =∆θvap


or

lCoi300 =

(∆θvap


) 1∆θ10

(2)

24096

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θvapi s =θi s+

(θi sminusθss







24097

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θideali s =

12

(θss+θi i

)=θ (5)





θ=

(θi i θ+δθi s

θ+δθi s θss

)(7)10





expression


)(9)


24098

ACPD10 24091ndash24133 2010


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23 Simple mixtures




)θAB (10)15



(1minus f sA

)θAB+

(1minus f sA

)2θBB (11)20



)=θ then



)2δθAB (12)24099

ACPD10 24091ndash24133 2010


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)(13)



)2δθABT)

(14)



ACPD10 24091ndash24133 2010


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log10

(p0Li

)=b0+

sumk

νkibk (15)20


0Li focusing on


ACPD10 24091ndash24133 2010


N M Donahue et al

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24102

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24103

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lCo =(n0

CminusnC

)bCminusnObO (16)25

with n0C =25 bC =0475 and b0 =17


24104

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N M Donahue et al

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24105

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24106

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(1minus f sC

) We thus25


24107

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1minus f iC)





θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M


)+


)](17)



defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni




24108

ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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|D

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θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

ACPD10 24091ndash24133 2010


N M Donahue et al

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|D

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Discussion

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|D

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331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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|D

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Discussion

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|D

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C)5





partθi i




24111

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N M Donahue et al

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|D

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θ=(

328 750750 1172minus2δθCO

)K (27)








24112

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δθi s =nM


sc

]δθCO (28)


24113

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24114

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24115

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5 Conclusions



24116

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24117

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References













24118

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24119

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24120

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24121

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24122

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24123

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24124

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|D

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CCC

CO

O


CCC

CO

O


24125

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24126

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24127

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24128

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24129

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|D

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CCC

CO

O


CCC

CO

O


24130

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24131

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24132

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24133

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Discussion

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|D

iscussionP

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2 Theory




300(log10C





∆θvapi =∆θvap


or

lCoi300 =

(∆θvap


) 1∆θ10

(2)

24096

ACPD10 24091ndash24133 2010


N M Donahue et al

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|D

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Discussion

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|D

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θvapi s =θi s+

(θi sminusθss







24097

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Discussion

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θideali s =

12

(θss+θi i

)=θ (5)





θ=

(θi i θ+δθi s

θ+δθi s θss

)(7)10





expression


)(9)


24098

ACPD10 24091ndash24133 2010


N M Donahue et al

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|D

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Discussion

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|D

iscussionP

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23 Simple mixtures




)θAB (10)15



(1minus f sA

)θAB+

(1minus f sA

)2θBB (11)20



)=θ then



)2δθAB (12)24099

ACPD10 24091ndash24133 2010


N M Donahue et al

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|D

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)(13)



)2δθABT)

(14)



ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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|D

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log10

(p0Li

)=b0+

sumk

νkibk (15)20


0Li focusing on


ACPD10 24091ndash24133 2010


N M Donahue et al

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|D

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Discussion

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|D

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24102

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|D

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24103

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|D

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lCo =(n0

CminusnC

)bCminusnObO (16)25

with n0C =25 bC =0475 and b0 =17


24104

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N M Donahue et al

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|D

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24105

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24106

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Discussion

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(1minus f sC

) We thus25


24107

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N M Donahue et al

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|D

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Discussion

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|D

iscussionP

aper|


1minus f iC)





θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M


)+


)](17)



defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni




24108

ACPD10 24091ndash24133 2010


N M Donahue et al

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θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

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331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


N M Donahue et al

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C)5





partθi i




24111

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θ=(

328 750750 1172minus2δθCO

)K (27)








24112

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δθi s =nM


sc

]δθCO (28)


24113

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24114

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24115

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5 Conclusions



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24117

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References













24118

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24119

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24120

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24121

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24122

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24123

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24124

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CCC

CO

O


CCC

CO

O


24125

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24126

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24127

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24128

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24129

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|D

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CCC

CO

O


CCC

CO

O


24130

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24131

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24132

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24133

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θvapi s =θi s+

(θi sminusθss







24097

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θideali s =

12

(θss+θi i

)=θ (5)





θ=

(θi i θ+δθi s

θ+δθi s θss

)(7)10





expression


)(9)


24098

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N M Donahue et al

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23 Simple mixtures




)θAB (10)15



(1minus f sA

)θAB+

(1minus f sA

)2θBB (11)20



)=θ then



)2δθAB (12)24099

ACPD10 24091ndash24133 2010


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)(13)



)2δθABT)

(14)



ACPD10 24091ndash24133 2010


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log10

(p0Li

)=b0+

sumk

νkibk (15)20


0Li focusing on


ACPD10 24091ndash24133 2010


N M Donahue et al

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|D

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|D

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24102

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|D

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24103

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lCo =(n0

CminusnC

)bCminusnObO (16)25

with n0C =25 bC =0475 and b0 =17


24104

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24105

ACPD10 24091ndash24133 2010


N M Donahue et al

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24106

ACPD10 24091ndash24133 2010


N M Donahue et al

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(1minus f sC

) We thus25


24107

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1minus f iC)





θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M


)+


)](17)



defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni




24108

ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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|D

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θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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J I

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Discussion

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|D

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Discussion

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|D

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331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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|D

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Discussion

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|D

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C)5





partθi i




24111

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θ=(

328 750750 1172minus2δθCO

)K (27)








24112

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δθi s =nM


sc

]δθCO (28)


24113

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24114

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24115

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5 Conclusions



24116

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24117

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References













24118

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24119

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24120

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24121

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24122

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24123

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24124

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|D

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CCC

CO

O


CCC

CO

O


24125

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24126

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24127

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24128

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N M Donahue et al

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24129

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|D

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CCC

CO

O


CCC

CO

O


24130

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N M Donahue et al

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24131

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24132

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N M Donahue et al

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|D

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24133

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N M Donahue et al

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Discussion

Paper

|D

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aper|




θideali s =

12

(θss+θi i

)=θ (5)





θ=

(θi i θ+δθi s

θ+δθi s θss

)(7)10





expression


)(9)


24098

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

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Discussion

Paper

|D

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aper|

Discussion

Paper

|D

iscussionP

aper|



23 Simple mixtures




)θAB (10)15



(1minus f sA

)θAB+

(1minus f sA

)2θBB (11)20



)=θ then



)2δθAB (12)24099

ACPD10 24091ndash24133 2010


N M Donahue et al

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|D

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Discussion

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|D

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)(13)



)2δθABT)

(14)



ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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|D

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Discussion

Paper

|D

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log10

(p0Li

)=b0+

sumk

νkibk (15)20


0Li focusing on


ACPD10 24091ndash24133 2010


N M Donahue et al

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Tables Figures

J I

J I

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Discussion

Paper

|D

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Discussion

Paper

|D

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24102

ACPD10 24091ndash24133 2010


N M Donahue et al

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|D

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|D

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24103

ACPD10 24091ndash24133 2010


N M Donahue et al

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|D

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Discussion

Paper

|D

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lCo =(n0

CminusnC

)bCminusnObO (16)25

with n0C =25 bC =0475 and b0 =17


24104

ACPD10 24091ndash24133 2010


N M Donahue et al

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|D

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Discussion

Paper

|D

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24105

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N M Donahue et al

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|D

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24106

ACPD10 24091ndash24133 2010


N M Donahue et al

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|D

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Discussion

Paper

|D

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aper|









(1minus f sC

) We thus25


24107

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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Paper

|D

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Discussion

Paper

|D

iscussionP

aper|


1minus f iC)





θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M


)+


)](17)



defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni




24108

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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Discussion

Paper

|D

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aper|

Discussion

Paper

|D

iscussionP

aper|




θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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Discussion

Paper

|D

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Discussion

Paper

|D

iscussionP

aper|

331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|





C)5





partθi i




24111

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θ=(

328 750750 1172minus2δθCO

)K (27)








24112

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δθi s =nM


sc

]δθCO (28)


24113

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24114

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24115

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5 Conclusions



24116

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24117

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References













24118

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24119

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24120

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24121

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24122

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24123

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24124

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CCC

CO

O


CCC

CO

O


24125

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24126

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24127

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24128

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24129

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CCC

CO

O


CCC

CO

O


24130

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24131

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24132

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24133

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23 Simple mixtures




)θAB (10)15



(1minus f sA

)θAB+

(1minus f sA

)2θBB (11)20



)=θ then



)2δθAB (12)24099

ACPD10 24091ndash24133 2010


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)(13)



)2δθABT)

(14)



ACPD10 24091ndash24133 2010


N M Donahue et al

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log10

(p0Li

)=b0+

sumk

νkibk (15)20


0Li focusing on


ACPD10 24091ndash24133 2010


N M Donahue et al

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24102

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24103

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lCo =(n0

CminusnC

)bCminusnObO (16)25

with n0C =25 bC =0475 and b0 =17


24104

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N M Donahue et al

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24105

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24106

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(1minus f sC

) We thus25


24107

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1minus f iC)





θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M


)+


)](17)



defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni




24108

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θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

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331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


N M Donahue et al

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C)5





partθi i




24111

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θ=(

328 750750 1172minus2δθCO

)K (27)








24112

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δθi s =nM


sc

]δθCO (28)


24113

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24114

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24115

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5 Conclusions



24116

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24117

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References













24118

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24119

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24120

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24121

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24122

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24123

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24124

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CCC

CO

O


CCC

CO

O


24125

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24126

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24127

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24128

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N M Donahue et al

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24129

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|D

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Discussion

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|D

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CCC

CO

O


CCC

CO

O


24130

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N M Donahue et al

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24131

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Discussion

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|D

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24132

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N M Donahue et al

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Discussion

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|D

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24133

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N M Donahue et al

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Discussion

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|D

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)(13)



)2δθABT)

(14)



ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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Discussion

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|D

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Discussion

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|D

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log10

(p0Li

)=b0+

sumk

νkibk (15)20


0Li focusing on


ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

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Discussion

Paper

|D

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Discussion

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|D

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24102

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N M Donahue et al

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24103

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N M Donahue et al

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|D

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Discussion

Paper

|D

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lCo =(n0

CminusnC

)bCminusnObO (16)25

with n0C =25 bC =0475 and b0 =17


24104

ACPD10 24091ndash24133 2010


N M Donahue et al

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24105

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N M Donahue et al

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24106

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N M Donahue et al

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(1minus f sC

) We thus25


24107

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N M Donahue et al

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|D

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Discussion

Paper

|D

iscussionP

aper|


1minus f iC)





θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M


)+


)](17)



defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni




24108

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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J I

J I

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Discussion

Paper

|D

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Discussion

Paper

|D

iscussionP

aper|




θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

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N M Donahue et al

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J I

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Discussion

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|D

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Discussion

Paper

|D

iscussionP

aper|

331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

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Discussion

Paper

|D

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aper|

Discussion

Paper

|D

iscussionP

aper|





C)5





partθi i




24111

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N M Donahue et al

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|D

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Discussion

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|D

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θ=(

328 750750 1172minus2δθCO

)K (27)








24112

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N M Donahue et al

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Discussion

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|D

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δθi s =nM


sc

]δθCO (28)


24113

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24114

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Discussion

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|D

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24115

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N M Donahue et al

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5 Conclusions



24116

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N M Donahue et al

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Discussion

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|D

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24117

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Discussion

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|D

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References













24118

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N M Donahue et al

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Discussion

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24119

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24120

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N M Donahue et al

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24121

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24122

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|D

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24123

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24124

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CCC

CO

O


CCC

CO

O


24125

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24126

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24127

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24128

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24129

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|D

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CCC

CO

O


CCC

CO

O


24130

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24131

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24132

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24133

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|D

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log10

(p0Li

)=b0+

sumk

νkibk (15)20


0Li focusing on


ACPD10 24091ndash24133 2010


N M Donahue et al

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24102

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24103

ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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|D

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lCo =(n0

CminusnC

)bCminusnObO (16)25

with n0C =25 bC =0475 and b0 =17


24104

ACPD10 24091ndash24133 2010


N M Donahue et al

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24105

ACPD10 24091ndash24133 2010


N M Donahue et al

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24106

ACPD10 24091ndash24133 2010


N M Donahue et al

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|D

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Discussion

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|D

iscussionP

aper|









(1minus f sC

) We thus25


24107

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N M Donahue et al

Title Page



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J I

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|D

iscussionP

aper|


1minus f iC)





θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M


)+


)](17)



defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni




24108

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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|D

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Discussion

Paper

|D

iscussionP

aper|




θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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Discussion

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|D

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aper|

331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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Discussion

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|D

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C)5





partθi i




24111

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N M Donahue et al

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Discussion

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|D

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aper|


θ=(

328 750750 1172minus2δθCO

)K (27)








24112

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N M Donahue et al

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Discussion

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δθi s =nM


sc

]δθCO (28)


24113

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N M Donahue et al

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24114

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24115

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N M Donahue et al

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5 Conclusions



24116

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24117

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References













24118

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N M Donahue et al

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24119

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24120

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N M Donahue et al

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Discussion

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|D

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24121

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N M Donahue et al

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Discussion

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|D

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24122

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N M Donahue et al

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Discussion

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|D

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24123

ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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|D

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24124

ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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|D

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Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24125

ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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Discussion

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|D

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24126

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N M Donahue et al

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|D

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24127

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N M Donahue et al

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Discussion

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|D

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24128

ACPD10 24091ndash24133 2010


N M Donahue et al

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|D

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Discussion

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|D

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24129

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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Discussion

Paper

|D

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aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24130

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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J I

J I

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Discussion

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|D

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Discussion

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|D

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24131

ACPD10 24091ndash24133 2010


N M Donahue et al

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|D

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Discussion

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|D

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24132

ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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|D

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Discussion

Paper

|D

iscussionP

aper|





24133

ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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|D

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aper|

Discussion

Paper

|D

iscussionP

aper|










24102

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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Discussion

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|D

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aper|

Discussion

Paper

|D

iscussionP

aper|









24103

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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J I

J I

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Discussion

Paper

|D

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aper|

Discussion

Paper

|D

iscussionP

aper|









lCo =(n0

CminusnC

)bCminusnObO (16)25

with n0C =25 bC =0475 and b0 =17


24104

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|











24105

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

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Discussion

Paper

|D

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aper|

Discussion

Paper

|D

iscussionP

aper|









24106

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|









(1minus f sC

) We thus25


24107

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


1minus f iC)





θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M


)+


)](17)



defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni




24108

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|




θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|

331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|





C)5





partθi i




24111

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


θ=(

328 750750 1172minus2δθCO

)K (27)








24112

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










δθi s =nM


sc

]δθCO (28)


24113

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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J I

J I

Back Close

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Discussion

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|D

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aper|

Discussion

Paper

|D

iscussionP

aper|











24114

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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J I

J I

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Discussion

Paper

|D

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aper|

Discussion

Paper

|D

iscussionP

aper|










24115

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

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Discussion

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|D

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Discussion

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|D

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aper|







5 Conclusions



24116

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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Discussion

Paper

|D

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Discussion

Paper

|D

iscussionP

aper|









24117

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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J I

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Discussion

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|D

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aper|

Discussion

Paper

|D

iscussionP

aper|


References













24118

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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Discussion

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|D

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aper|

Discussion

Paper

|D

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aper|















24119

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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J I

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Discussion

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|D

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Discussion

Paper

|D

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aper|















24120

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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Discussion

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|D

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Discussion

Paper

|D

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aper|












24121

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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Discussion

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|D

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Discussion

Paper

|D

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aper|
















24122

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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J I

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Discussion

Paper

|D

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aper|

Discussion

Paper

|D

iscussionP

aper|















24123

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

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Discussion

Paper

|D

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aper|

Discussion

Paper

|D

iscussionP

aper|










24124

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

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Discussion

Paper

|D

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aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24125

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

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Discussion

Paper

|D

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Discussion

Paper

|D

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24126

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N M Donahue et al

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|D

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Discussion

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|D

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24127

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N M Donahue et al

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Discussion

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|D

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Discussion

Paper

|D

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24128

ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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|D

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Discussion

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|D

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24129

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

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Discussion

Paper

|D

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aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24130

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

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Discussion

Paper

|D

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Discussion

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|D

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24131

ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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|D

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Discussion

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|D

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24132

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N M Donahue et al

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|D

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Discussion

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|D

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24133

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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Discussion

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|D

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Discussion

Paper

|D

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24103

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|









lCo =(n0

CminusnC

)bCminusnObO (16)25

with n0C =25 bC =0475 and b0 =17


24104

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|











24105

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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J I

J I

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Discussion

Paper

|D

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aper|

Discussion

Paper

|D

iscussionP

aper|









24106

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

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Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|









(1minus f sC

) We thus25


24107

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


1minus f iC)





θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M


)+


)](17)



defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni




24108

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|




θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|

331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|





C)5





partθi i




24111

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


θ=(

328 750750 1172minus2δθCO

)K (27)








24112

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










δθi s =nM


sc

]δθCO (28)


24113

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|











24114

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










24115

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|







5 Conclusions



24116

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|









24117

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


References













24118

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24119

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24120

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|












24121

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|
















24122

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24123

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










24124

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24125

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


24126

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|



24127

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


24128

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|



24129

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24130

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


24131

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


24132

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|





24133

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|









lCo =(n0

CminusnC

)bCminusnObO (16)25

with n0C =25 bC =0475 and b0 =17


24104

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|











24105

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|









24106

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|









(1minus f sC

) We thus25


24107

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


1minus f iC)





θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M


)+


)](17)



defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni




24108

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|




θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|

331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|





C)5





partθi i




24111

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


θ=(

328 750750 1172minus2δθCO

)K (27)








24112

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










δθi s =nM


sc

]δθCO (28)


24113

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|











24114

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










24115

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|







5 Conclusions



24116

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|









24117

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


References













24118

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24119

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24120

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|












24121

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|
















24122

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24123

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










24124

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24125

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


24126

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|



24127

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


24128

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|



24129

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24130

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


24131

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


24132

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|





24133

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|











24105

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|









24106

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|









(1minus f sC

) We thus25


24107

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


1minus f iC)





θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M


)+


)](17)



defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni




24108

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|




θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|

331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|





C)5





partθi i




24111

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


θ=(

328 750750 1172minus2δθCO

)K (27)








24112

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










δθi s =nM


sc

]δθCO (28)


24113

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|











24114

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










24115

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|







5 Conclusions



24116

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|









24117

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


References













24118

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24119

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24120

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|












24121

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|
















24122

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24123

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










24124

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24125

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


24126

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|



24127

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N M Donahue et al

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J I

J I

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Discussion

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|D

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Discussion

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|D

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24128

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

J I

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Full Screen Esc



Discussion

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|D

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Discussion

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|D

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24129

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24130

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

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|D

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Discussion

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|D

iscussionP

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24131

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

J I

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Discussion

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|D

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|D

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24132

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

J I

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Full Screen Esc



Discussion

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|D

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Discussion

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|D

iscussionP

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24133

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

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|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|









24106

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|









(1minus f sC

) We thus25


24107

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


1minus f iC)





θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M


)+


)](17)



defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni




24108

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|




θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|

331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|





C)5





partθi i




24111

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


θ=(

328 750750 1172minus2δθCO

)K (27)








24112

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










δθi s =nM


sc

]δθCO (28)


24113

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|











24114

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










24115

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|







5 Conclusions



24116

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|









24117

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


References













24118

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24119

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24120

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|












24121

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|
















24122

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24123

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










24124

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24125

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


24126

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|



24127

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


24128

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|



24129

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24130

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


24131

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


24132

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|





24133

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|









(1minus f sC

) We thus25


24107

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


1minus f iC)





θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M


)+


)](17)



defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni




24108

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|




θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|

331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|





C)5





partθi i




24111

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


θ=(

328 750750 1172minus2δθCO

)K (27)








24112

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










δθi s =nM


sc

]δθCO (28)


24113

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|











24114

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










24115

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|







5 Conclusions



24116

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|









24117

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


References













24118

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24119

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24120

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|












24121

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|
















24122

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24123

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










24124

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24125

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


24126

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|



24127

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


24128

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|



24129

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24130

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


24131

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


24132

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|





24133

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


1minus f iC)





θi s = niC

(f sCθCC+ f sOθCO

)+ni

O

(f sCθCO+ f sOθOO

)= ni

M


)+


)](17)



defined by

θi i = niC

(f iCθCC+ f iOθCO

)+ni

O

(f iCθCO+ f iOθOO

)= nC f

iCθCC+2

nCnO

nC+nOθCO+ni




24108

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|




θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|

331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|





C)5





partθi i




24111

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


θ=(

328 750750 1172minus2δθCO

)K (27)








24112

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










δθi s =nM


sc

]δθCO (28)


24113

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|











24114

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










24115

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|







5 Conclusions



24116

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|









24117

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


References













24118

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24119

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24120

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|












24121

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|
















24122

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|















24123

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










24124

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24125

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Discussion

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|D

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CCC

CO

O


CCC

CO

O


24130

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N M Donahue et al

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24133

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N M Donahue et al

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Discussion

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|D

iscussionP

aper|




θlinearCO

=12

(θCC+θOO

)=θ (20)5






24109

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

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Discussion

Paper

|D

iscussionP

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Discussion

Paper

|D

iscussionP

aper|

331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

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|D

iscussionP

aper|

Discussion

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|D

iscussionP

aper|





C)5





partθi i




24111

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


θ=(

328 750750 1172minus2δθCO

)K (27)








24112

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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J I

J I

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Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










δθi s =nM


sc

]δθCO (28)


24113

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

J I

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Discussion

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|D

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Discussion

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24114

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N M Donahue et al

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Discussion

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24115

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N M Donahue et al

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Discussion

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5 Conclusions



24116

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N M Donahue et al

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Discussion

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24117

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N M Donahue et al

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References













24118

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N M Donahue et al

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Discussion

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24119

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N M Donahue et al

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Discussion

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Discussion

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24120

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N M Donahue et al

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Discussion

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Discussion

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24121

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N M Donahue et al

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J I

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Discussion

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Discussion

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24122

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N M Donahue et al

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Discussion

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Discussion

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|D

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24123

ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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|D

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Discussion

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|D

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24124

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24125

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

J I

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Discussion

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24126

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N M Donahue et al

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24127

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N M Donahue et al

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Discussion

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|D

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24128

ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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|D

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24129

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

J I

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Full Screen Esc



Discussion

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|D

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aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24130

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

J I

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Discussion

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|D

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24131

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N M Donahue et al

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24132

ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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Discussion

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|D

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24133

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|

331 Non-linear SAR



θis =niM

(f iC f

iO

)[(θCC θ+δθCO

θ+δθCO θOO

)(f sCf sO

)](21)


θi i =niCθCC+ni

OθOO+2ni

CniO

niC+ni

O

δθCO (22)


i with

l Coi =(θ0minusθi

)δθ10 (23)

or equivalently15

lCoi =(n0Cminusni

C

)bCminusni

ObOminus2niCn

iO

niC+ni

O

bCO (24)



ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|





C)5





partθi i




24111

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


θ=(

328 750750 1172minus2δθCO

)K (27)








24112

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










δθi s =nM


sc

]δθCO (28)


24113

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

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|D

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aper|

Discussion

Paper

|D

iscussionP

aper|











24114

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



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J I

J I

Back Close

Full Screen Esc



Discussion

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|D

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aper|

Discussion

Paper

|D

iscussionP

aper|










24115

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

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|D

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Discussion

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|D

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5 Conclusions



24116

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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Full Screen Esc



Discussion

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|D

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Discussion

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24117

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N M Donahue et al

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J I

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Discussion

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Discussion

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|D

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References













24118

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N M Donahue et al

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J I

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Discussion

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Discussion

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|D

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24119

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N M Donahue et al

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Discussion

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Discussion

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24120

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N M Donahue et al

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Discussion

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Discussion

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24121

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N M Donahue et al

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Discussion

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Discussion

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24122

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N M Donahue et al

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Discussion

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Discussion

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|D

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24123

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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Discussion

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|D

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Discussion

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24124

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24125

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

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Full Screen Esc



Discussion

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Discussion

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24126

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N M Donahue et al

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Discussion

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Discussion

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24127

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N M Donahue et al

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Discussion

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Discussion

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|D

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24128

ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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24129

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

J I

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Discussion

Paper

|D

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aper|

Discussion

Paper

|D

iscussionP

aper|


CCC

CO

O


CCC

CO

O


24130

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

J I

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Full Screen Esc



Discussion

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|D

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Discussion

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|D

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24131

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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Discussion

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|D

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Discussion

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|D

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24132

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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Discussion

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|D

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Discussion

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|D

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24133

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

J I

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Discussion

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|D

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aper|

Discussion

Paper

|D

iscussionP

aper|





C)5





partθi i




24111

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

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aper|

Discussion

Paper

|D

iscussionP

aper|


θ=(

328 750750 1172minus2δθCO

)K (27)








24112

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










δθi s =nM


sc

]δθCO (28)


24113

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

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Discussion

Paper

|D

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aper|











24114

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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Discussion

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Discussion

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|D

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24115

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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Discussion

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Discussion

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5 Conclusions



24116

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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Discussion

Paper

|D

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Discussion

Paper

|D

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24117

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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Discussion

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|D

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Discussion

Paper

|D

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References













24118

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N M Donahue et al

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Discussion

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|D

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Discussion

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|D

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24119

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N M Donahue et al

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Discussion

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24120

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N M Donahue et al

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Discussion

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24121

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N M Donahue et al

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Discussion

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24122

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N M Donahue et al

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Discussion

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24123

ACPD10 24091ndash24133 2010


N M Donahue et al

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Discussion

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24124

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N M Donahue et al

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J I

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Discussion

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|D

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Discussion

Paper

|D

iscussionP

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CCC

CO

O


CCC

CO

O


24125

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

J I

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Discussion

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Discussion

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24126

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N M Donahue et al

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Discussion

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Discussion

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24127

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N M Donahue et al

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Discussion

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Discussion

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24128

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N M Donahue et al

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Discussion

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Discussion

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24129

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J I

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Discussion

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|D

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aper|

Discussion

Paper

|D

iscussionP

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CCC

CO

O


CCC

CO

O


24130

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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Discussion

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Discussion

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24131

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N M Donahue et al

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Discussion

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24132

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Discussion

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24133

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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Discussion

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|D

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Discussion

Paper

|D

iscussionP

aper|


θ=(

328 750750 1172minus2δθCO

)K (27)








24112

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



Discussion

Paper

|D

iscussionP

aper|

Discussion

Paper

|D

iscussionP

aper|










δθi s =nM


sc

]δθCO (28)


24113

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

J I

Back Close

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Discussion

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|D

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Discussion

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|D

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24114

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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Discussion

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Discussion

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24115

ACPD10 24091ndash24133 2010


N M Donahue et al

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J I

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Discussion

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Discussion

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5 Conclusions



24116

ACPD10 24091ndash24133 2010


N M Donahue et al

Title Page



Tables Figures

J I

J I

Back Close

Full Screen Esc



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24124

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24125

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24128

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24130

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24131

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24132

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24133

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24113

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24114

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24115

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24117

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24118

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24119

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24120

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24121

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24122

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24123

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24124

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24125

ACPD10 24091ndash24133 2010


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24126

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24127

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24128

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24129

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CCC

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O


24130

ACPD10 24091ndash24133 2010


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24131

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24132

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24133

ACPD10 24091ndash24133 2010


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24114

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24115

ACPD10 24091ndash24133 2010


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5 Conclusions



24116

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24117

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24118

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24119

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24120

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24121

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24122

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24123

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24124

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CCC

CO

O


24125

ACPD10 24091ndash24133 2010


N M Donahue et al

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24126

ACPD10 24091ndash24133 2010


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24127

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24128

ACPD10 24091ndash24133 2010


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24129

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CCC

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CCC

CO

O


24130

ACPD10 24091ndash24133 2010


N M Donahue et al

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24131

ACPD10 24091ndash24133 2010


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24132

ACPD10 24091ndash24133 2010


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24133

ACPD10 24091ndash24133 2010


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24115

ACPD10 24091ndash24133 2010


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5 Conclusions



24116

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24117

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References













24118

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24119

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24120

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24121

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24122

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24123

ACPD10 24091ndash24133 2010


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24124

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CCC

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CCC

CO

O


24125

ACPD10 24091ndash24133 2010


N M Donahue et al

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24126

ACPD10 24091ndash24133 2010


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24127

ACPD10 24091ndash24133 2010


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24128

ACPD10 24091ndash24133 2010


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24129

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CCC

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CCC

CO

O


24130

ACPD10 24091ndash24133 2010


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24131

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24132

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24133

ACPD10 24091ndash24133 2010


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5 Conclusions



24116

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24117

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References













24118

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24119

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24120

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24121

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24122

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24123

ACPD10 24091ndash24133 2010


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24124

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CCC

CO

O


24125

ACPD10 24091ndash24133 2010


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24126

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24127

ACPD10 24091ndash24133 2010


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24128

ACPD10 24091ndash24133 2010


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24129

ACPD10 24091ndash24133 2010


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Discussion

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CCC

CO

O


CCC

CO

O


24130

ACPD10 24091ndash24133 2010


N M Donahue et al

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Title Page A two-dimensional volatility basis set: 1