(A1)
The aerosol deposition model presented in chapter four is an implementation of the Slinn framework (Slinn, 1980) presented by Zhang et. al. (2002). In the interest of ease of repeatability, the major equations are reproduced here along with further definitions not presented in the above paper, but required in order to write the model. This is not intended as a review or critique of deposition models, but a record of the model used in chapter 4. Further details are available in the paper, or in other references used here. The general form of the model is presented first, followed by the required equations.
As explained in chapters two and four, total deposition velocity is the sum of gravitational and turbulent deposition velocities. Also, the turbulent deposition velocity is controlled by the aerodynamic resistance and the surface resistance to uptake. The total is represented as:
(A1)
The aerodynamic resistance has been approximated by a number of researchers. Here, either Monteith and Unsworth’s (1990) definition can be used, where:
(A2)
or Zhang et. al. (2002) suggest:
(A3)
where the only previously undefined quantity is the height integrated stability correction for heat (h; mentioned in chapter two) which is given by:
< 0 (A4a)
= 0 (A4b)
> 0 (A4c)
These are the stability corrections for heat and pollutant transport. For reference, those for momentum are also presented, although they are not used in this model.
< 0 (A5a)
= 0 (A5b)
> 0 (A5c)
vg (in equation A1) is the gravitational settling velocity, given by Stokes’ law as follows.
(A6)
where p is particle density, g is the acceleration due to gravity (9.81 m s-1), is the viscosity coefficient of air (1.71 10-5 m2 s-1) and Cc is the Cunningham slip correction, – a correction for the small non-zero component of perpendicular flow at the surface of small particles given by:
(A7)
where is the mean free path of air molecules (1 10-7 m). The final term in equation A1 is the surface resistance. This is the uncertain term, which gives rise to the variability in the predictions of different models, noted in chapter four. rs is defined as:
(A8)
here 0 is an empirical constant (set to 3 in the implementation used) and Rs is the size dependent sticking probability for aerosol (the probability that a particle will not bounce off the surface and be re-suspended). EB, EIM and EIN are, respectively, the collection efficiencies of Brownian diffusion, impaction and interception. There are several formulations of each of these parameters, and it is from these that the variability in rs stems. Rs is defined (Slinn, 1982) as:
(A9)
The efficiency of particle interception in this model is treated as a function of particle diameter and the radius of collectors in this model:
(A10)
with A being the radius of collectors. A value of 2 mm was used, to simulate blades of grass as the main collectors of aerosol. EB is a similarly empirical function, given by:
(A11)
where is another empirical constant, in this case a value of 0.66 was used, following the recommendation of Slinn (1982) for vegetated surfaces. Sc is the Schmidt number, the ratio of the kinematic viscosity of air (1.8 10-5 N m2 s-1) to the particle Brownian diffusivity. Brownian diffusivity is defined as:
(A12)
where K is the Boltzmann constant (1.38 10-23 J K-1), all other quantities as previously defined. Of the collection efficiencies, this leaves only EIM undefined. Many approximations have been suggested for EIM. Zhang et. al. (2002) present six of these, all functions of the Stokes number. Before presenting the EIM equation, the Stokes number is defined, for vegetated surfaces (Slinn, 1982) as:
(A13)
There is an alternative approximation for Stokes number over smooth surfaces (Giorgi, 1988), but the Slinn form was used in this case as it was designed for use with vegetated surfaces. Of the available forms for EIM, that presented by Giorgi (1988) was used, because it was derived for vegetated surfaces. It is similar to that reported by Peters and Eiden (1992) for forest, with slightly lower mean values, and significantly lower values for small particles.
(A14)
The equations presented in this appendix are sufficient to recreate the model used for comparison with measurements in chapter four of this thesis. The model was written in Visual Basic (in a Microsoft Excel spreadsheet), which is available on request from the author (currently j.dorsey@umist.ac.uk).
This is one possible implementation of the Slinn type deposition model framework. For more detail on this type of model and the advantages / disadvantages of various parameterisations of different terms, the reader is referred to Slinn (1980), Slinn (1982), Giorgi (1986), Zhang et. al. (2002) and especially to Ruijgrok et. al. (1997).
The Reynolds decomposition of time series outlined in chapter two is used widely throughout this thesis. It is not always convenient or desirable to use the full (over bar, prime) notation in every equation, and these have been omitted in certain cases, in the interest of removing unnecessary “clutter”, where it is clear which component should be used. In almost all cases the quantity has instead been capitalised, and the averaged component of the decomposition should be used. As a rule of thumb, if the equation in question is unrelated to eddy covariance calculation or time series analysis, the averaged quantity should be used. An exception to the capitalisation “rule” above is that time series associated with power and co-spectral analyses are denoted using capital subscripts “W”, and “T”.
Because of the rather large number of variables involved in a study of this type, several letters and symbols are occasionally re-used in different sections of the thesis to denote different quantities. Where this done, the text accompanying the equations always clarifies what variable the symbols refer to.
Finally, although units are listed with variables in appendix C, different units are used at different times for many variables. An example is the vapour mass flux, jv. In calculations jv must be treated in MKS, or at least in units consistent with the rest of the calculation. However when tabulated or plotted, jv appears in g m-3 hr-1 because this is a more suitable unit for its magnitude in general. Similarly Dp is always presented in either m or nm, but is always in m for calculation purposes. Wherever there could be any doubt, the required units are listed in the accompanying text with equations. However, care and common sense are required, as some equations taken from outside sources have mismatched units (e.g. equation 2.21, see kw).
# – Dimensionless
* – Variable units
Roman alphabet
A – Empirical constant (chapter six) – #
A – Aerosol collector radius (chapter four) – m
b – Empirical constant (chapter six) – #
C – General constant – *
CAB – Co-spectrum of time series A and B – *
Cc – Cunningham slip correction – #
cd – Surface drag coefficient – #
cp – Specific heat capacity of dry air at constant pressure – J kg-1 K-1
(= 1004.67)
cv – Specific heat capacity of dry air at constant volume – J kg-1 K-1
(= 717.02)
d – Zero plane displacement – m
D – Gas diffusion coefficient – s-1
D3 – Aerosol in range 11 nm < Dp < 3 m, typically – Range
Representative of 11 nm < Dp < 100 nm.
Dp – Particle diameter – m or nm
E – Water vapour loading – kg m-3
EB – Brownian diffusion efficiency – #
Ef – Total anthropogenis energy input – W m-2
Eff – Fossil fuel derived anthropogenic energy input – W m-2
EIM – Impaction efficiency – #
EIN – Interception efficiency – #
Enff – Non-fossil fuel anthropogenic energy input – W m-2
f – Nondimensional frequency – #
fmin – Minimum required logging frequency – Hz
fmA – Peak in power spectrum SA – *
fNy – Nyquist frequency – Hz
f – Pollutant (aerosol) flux – cm-2 s-1
g – Acceleration due to gravity – m s-2
(= 9.81)
H – Sensible heat flux – W m-2
hc – Canopy height – m
[i] – Concentration of species “i” – g m-3
jv – Vapour mass flux – g m-3 hr-1
k – von Karman constant – #
(= 0.4)
K – Bolzmann constant – J K-1
(= 1.38 10-23)
ke – Theoretical dissociation constant for NH4NO3 – hPa2
KE – Eddy diffusivity for water vapour – s-1
KH – Eddy diffusivity for heat – s-1
km – Measured vapour concentration product – hPa2
KM – Eddy diffusivity for momentum – s-1
Kn – Knudsen number – #
kspecies – Rate coeff. for second order reaction of “species” – #
kw – KH2O extinction coefficient – mV m3 g-1 cm-1
K – Eddy diffusivity for entrained pollutants – s-1
l – Mixing length – m
L – Obukhov length – m
Lv – Latent heat of vaporisation for water – J kg-1
mv – Molecular mass of condensing vapour – AMU
n – Natural frequency – Hz
Nair – Molarity of air – m-3
nh – Highest frequency in power spectrum (chapter four) – Hz
Ni – Molarity of species i – m-3
nl – Lowest frequency in power spectrum (chapter four) – Hz
nr – Highest properly resolved frequency in power – Hz
spectrum (chapter four)
nr – Radial frequency – rad s-1
N – Avogadro constant (chapter five) – #
(= 6 1023)
N – Number of samples in time series – #
Nmin – Minimum aerosol required for significant flux – #
Ntot – Total number of aerosol observed – #
p – Pressure – hPa
p0 – Reference pressure – hPa
(normally 1013.25)
p[i] – Partial pressure of species “i” – hPa
q – Water vapour loading (KH2O calculations) – kg m-3
Q – Measured (aerosol) flux (Schuepp model) – m-2 s-1
Q0 – (Aerosol) Flux normalisation (Schuepp model) – m-2 s-1
r – Particle radius – m
ra – Aerodynamic resistance – s m-1
rb – Laminar boundary layer resistance – s m-1
rc – Total canopy resistance – s m-1
rs – Canopy surface resistance to deposition – s m-1
rf – Sample flow rate – cm3 s-1
rg – Ground surface resistance – s m-1
Rg – Diameter dependent aerosol growth rate ratio – #
Ri – Gradient Richardson number – #
rL – Water vapour mass mixing ratio – #
RLu – Upward long wave radiation – W m-2
RLd – Downward long wave radiation – W m-2
Rn – Net radiation – W m-2
rp – Resistance to chemical uptake at surface – s m-1
Rs – Particle sticking efficiency – #
rsat – Saturation water vapour mass mixing ratio – #
rt – Total resistance to deposition – s m-1
rxy – Covariance of time series x and y – *
SA – Power spectrum of time series A – *
Sc – Schmidt number – #
St – Stokes number – #
St – Global short wave radiation – W m-2
T* – Surface layer scaling temperature – K
T – (Air) Temperature – K
Ta – Air temperature – K
Te – Arbitrary turbulence time scale – s
Ts – Surface temperature – K
u* – Friction velocity – m s-1
u – Instantaneous horizontal wind velocity – m s-1
– Mean
horizontal wind velocity – m s-1
u’ – Perturbation in horizontal wind velocity – ms-1
u0 – Rotated horizontal wind velocity – m s-1
v – Instantaneous transverse wind velocity – m s-1
–
Mean transverse wind velocity – m s-1
v’ – Perturbation in transverse wind velocity – ms-1
V – Voltage – V
v0 – Rotated transverse wind velocity – m s-1
vd – Deposition velocity – mm s-1
vdm – Measured deposition velocity – mm s-1
ve – Emission velocity – mm s-1
vg – Gravitational settling velocity – mm s-1
Vq – KH2O signal voltage – V
w – Instantaneous vertical wind velocity – m s-1
–
Mean vertical wind velocity – m s-1
w’ – Perturbation in vertical wind velocity – ms-1
w0 – Rotated vertical wind velocity – ms-1
x – KH20 optical path length – cm
x – Upwind distance (Schuepp model) – m
z – Height – m
z0 – Roughness length – m
z0’ – Sink level for ammonia vapour – m
zm – Measurement height – m
Greek alphabet
– Horizontal wind angle – Deg
– Vapour sticking coefficient – #
m – Transition regime vapour flux correction – #
– (Aerosol) Pollutant concentration – cm-3
c – Coincidence corrected aerosol concentration – cm-3
v – Condensable vapour concentration – g m-3
veff – Effective condensable vapour concentration – g m-3
– Proportional error – *
0 – Empirical constant (Zhang model) – #
(= 3)
h – Empirical stability correction for heat – #
m – Empirical stability correction for momentum – #
– Empirical stability correction for pollutants – #
h – Dissipation rate of temperature variance – #
– Dissipation rate of turbulent kinetic energy – #
– 
– #
– Viscosity coefficient of air – m2 s-1
(= 1.71 10-5)
– Latent heat of vaporisation for water – J kg-1
– Arbitrary turbulence length scale (chapter two) – m
– Mean free path of condensable vapour – m
(= 1 10-7)
E – Latent heat flux – W m-2
– Wind azimuth (vertical) angle – Deg
– Potential temperature – K
v – Virtual potential temperature – K
– (Air) Density – kg m-3
a – Density of dry air – kg m-3
(= 1.225)
p – Particle density – kg m-3
A – Standard deviation of time series “A” – *
– Reynolds stress – kg m-1 s-2
– Aerosol inlet lag time – s
m – Time period of measurement – s
h – Height integrated stability correction for heat – #
m – Height integrated stability correction for momentum – #
– Boundary layer stability parameter – #
First, for all my family, who were always supportive and seemed almost completely unfazed by my propensity for gibbering, howling and barking at the moon following long field projects. Thanks all of you for the beer, strange nights in Manchester, sympathy, and, belatedly, rescue missions down cliffs in the dead of night after I discovered that micrometeorology bites back.
At UMIST, thanks to Dr. Martin Gallagher, my supervisor, for putting up with me, – I hope your persistence has paid off at last! Also Prof. Tom Choularton. For assistance with fieldwork and sundry bouts of confusion, thanks to Dr. Mike Flynn, Dr. Karl Beswick, Dr. Keith Bower and especially Mr. Pete Kelly for building / assisting with several bits of field equipment. Many thanks to Dr. Paul Williams for the occasional sense of humour reboot…
Further afield, the staff at CEH (Bush) seem to have “borne the brunt” of my training. Dr. Eiko Nemitz is to be congratulated for never once taking out his frustration with me on animate objects (at least not in my presence). Thanks to Dr. Mark Sutton for lots of encouragement, and to Mark Theobald and Alan McDonald for beer in Edinburgh, and many scintillating conversations about anything but atmospheric science.
I acknowledge receipt of a UK Natural Environment Research Council (NERC) studentship during the course of these studies. Map of Edinburgh (chapter 6, figure 1) reproduced from 1999 Ordnance Survey map with the permission of the Controller of Her Majesty's Stationery Office, © Crown Copyright NC/01/516.