Supplement of Rapid formation of large

Supplement of Biogeosciences, 11, 4393–4406, 2014
http://www.biogeosciences.net/11/4393/2014/
doi:10.5194/bg-11-4393-2014-supplement
© Author(s) 2014. CC Attribution 3.0 License.
Supplement of
Rapid formation of large aggregates during the spring bloom of
Kerguelen Island: observations and model comparisons
M.-P. Jouandet et al.
Correspondence to: M.-P. Jouandet ([email protected])
Supplement S1: Model description
The biological model for phytoplankton growth is a modified form of that in Evans and Parslow (1985)
and Fasham et al (1990). In this case, there is only one nutrient, nitrate, and phytoplankton are lost to
coagulation as in Jackson (1995) and Jackson et al. (2005). There are no grazing losses. Values for
constants are given in Table 1.
Nitrate concentration
Change in nitrate concentration N:

!
= ! ! − 


(A1.)
where Kz=vertical mixing coefficient, G is the phytoplankton specific growth rate, and φ is the
phytoplankton concentration.
Phytoplankton growth
Phytoplankton growth rate at any given irradiance and nutrient concentration is given by
 = !"# min(! , ! )
(A2.)
where Gmaxis the maximum specific growth rate, rp and rn are the relative growth rates possible growth for
photosynthesis and nutrient limitation at I and N. They are calculated by
! =
! 
!
! !  ! + !"#
−

(A3.)
!"#
where r is the phytoplankton loss rate and α is the slope of the PI curve.
! =

! + 
(A4.)
!
! !"#
(A5.)
where KD is the half saturation constant for nitrate uptake.
Irradiance I given by
 = ! 
where I0 is the surface irradiance, k = kw+ φ kr, kw is the attenuation coefficient for water and 0.04 m-1 and
kr is the light absorption coefficient for plants. The value of kr was chosen so that k equaled the observed
attenuation at A3 (k=0.048 m-1 for P = 0.6 µg chl/L). Surface irradiance was calculated using the
relationships of Evans and Parslow (1985) for a latitude of 50°S and a starting time 120 d after winter
solstice.
A single phytoplankton cell was assume to have a diameter of 20 um and a density of 1.0637 g cm-3
(compared to a fluid density of 1.0275 g cm-3), for a resulting settling speed of 1.05 m d-1.
Describing particle size distributions
Standard coagulation theory describes particle size distributions using a number spectrum n(s), where s is
particle size, such as mass m or equivalent spherical diameter d. Number spectra in terms of m and d can
be related
  = 


(A6.)
The total number of particles in a small size interval dl < d ≤ d +Δd is approximately n Δd. For a
!
sufficiently small Δd, all particles have the same individual volume   =  ! . The total particle
!
volume in the interval is then V(d) n Δd. The total particle volume for any large particle range dl < d ≤ du
is
! = !!
(A7.)
    d
!!
which is proportional to the area under the curve when nV is plotted as a function of d. Plotting nV versus
the logarithm of d destroys the relationship between the area under the curve and the total particle
volume; plotting nVd as a function of log d restores it. We will discuss particle size distributions in terms
of the nVd form of the distribution, but note that it contains the same content as n.
How coagulation changes size distributions
Coagulation destroys small particles and creates new ones in a way that is mathematically expressed as
 , 
= +

!
!
!
 !   − !  ! ,  − !  −  
!
 !  , ! !
(A8.)
where β(m1,m2) is the coagulation kernel and describes the collision rate between particles of masses m1
and m2 and α is the stickiness, the probability of a collision destroying the two colliding particles and
forming a larger particle.
One of the techniques for solving this equation numerically is to break the size distribution n into
segments, called sections, each with a fixed shape but a variable magnitude, such that
 ,  =
! 
!!! 
(A9.)
within a range of ! >  ≥ !!! , where Qi(t) is the total particulate mass within the bounds (e.g.,
Jackson 1995) and the range represents the section boundaries. This approximation breaks n(m,t) into
separate time and mass varying parts. Gelbard et al. showed that doing so allows Eq. A8 to be
transformed from an integer-differential equation to a series of ordinary differential equations:
! 
=

2
!!! !!!
!!!
!
!!! !!!
!
!,!,! ! ! − !
!!!

!,! ! − !!,! !! −  !
2
!!"#
(A10.)
!
!!!!!
!,! ! where !!,!,! , !!,! , !!,! , and !!,! are sectional coefficients and lmax is the total number of sections. The
equations simplify if the mass of the upper boundary of a section is twice that of its lower boundary.
Jackson (1995) added a disaggregation term to Eq. A10 which moved mass from section i to the next
lower section i-1 at a rate λiQi, where λi is a size dependent disaggregation coefficient. We used the
values for λi from Jackson(1995).
The algae were assumed to be occupy the first section: φ=Q1. The equation describing their
concentration is
!
!
 ! !  !
= !  + !
− !
−
  ! −  !


 !
2 !,! !
!!"#
(A11.)
!
!,! ! + ! !
!!!
For particles in larger sections, the equation describing their concentration becomes
!
!
 ! !

= !
− !
+ !!! !!! − ! ! +
!



2
!!!
!
− !
!!!
!!! !!!
!
!!! !!!

!,! ! − !!,! !! −  !
2
where vl is the settling rate for particles in the lth section.
(A12.)
!,!,! ! !
!!"#
!
!,! ! !!!!!
Measures of aggregate size.
Because aggregate porosity increases with size, particle density decreases as particle size increases. The
relationship between the diameter assigned to a particle from analysis of an image da is frequently
!!
described using a fractal dimension Df:  ∝ ! . Increasing Df decreases the porosity of large particles
and results in smaller values of da for a given m. An apparent volume for a sphere with diameter da is
!
then ! = !! . A conserved volume Vc can be calculated from a particle’s mass and the density of the
!
single cell. Its diameter dc can be calculated assuming it is a sphere.
Numerical solution of equations
The equations were solved numerically for a 150 m mixed layer using a centered-difference scheme, noflux boundary conditions at the surface, fixed nitrate concentration= 30 µM and no diffusive particle flux
at the bottom boundary (150 m). Equations were solved at a vertical spacing of 2 m. Particle
concentrations were calculated in terms of the conserved volume.
Parameter values are given in Table 2.