Aggregation Model of Marine Particles by Moments Method of Muilti Modes

Adrian Burd and Louis Yang Liu

July 11, 2008

Log-normal Distribution and Moments

By classical probability theory, the

k -th moment of the log-normal

distribution

N (x ) = √

1

e−

x ln log

(ln − µ)2 2 2σ

(1)

xlnσ ´∞ k ( k ) is dened as m = 0 N (x )x dx , where the lnµ is the mean and lnσ is the standard deviation of the logarithm of the variable. Note ln2 that the mean of x is µe 2 . σ

2π

Factors Determining the Sizes of Particles in Coagulation Process

For size distribution, we take the diameter of particles variable.

D as the

Factors Determining the Sizes of Particles in Coagulation Process

For size distribution, we take the diameter of particles variable.

D is determined by multiplicative factors, like:

D as the

Factors Determining the Sizes of Particles in Coagulation Process

For size distribution, we take the diameter of particles

D as the

variable.

D is determined by multiplicative factors, like: I The dynamic of marine ecological system (biological)

Factors Determining the Sizes of Particles in Coagulation Process

For size distribution, we take the diameter of particles

D as the

variable.

D is determined by multiplicative factors, like: I The dynamic of marine ecological system (biological) I Ocean currents (physical)

Factors Determining the Sizes of Particles in Coagulation Process

For size distribution, we take the diameter of particles

D as the

variable.

D is determined by multiplicative factors, like: I The dynamic of marine ecological system (biological) I Ocean currents (physical) I Aqua-chemical reactions (chemical)

Factors Determining the Sizes of Particles in Coagulation Process

For size distribution, we take the diameter of particles

D as the

variable.

D is determined by multiplicative factors, like: I The dynamic of marine ecological system (biological) I Ocean currents (physical) I Aqua-chemical reactions (chemical) Therefore, we can consider the distribution of particles with dierent sizes as log-normal distributions.

Pumps in Ocean Sphere

Figure:

Single Mode Let us consider a single-mode model based on Koziol-Leighton's model on aerosol dynamics, as the following ordinary dierential equation for this modeling

Single Mode Let us consider a single-mode model based on Koziol-Leighton's model on aerosol dynamics, as the following ordinary dierential equation for this modeling

dm(k ) (t ) = dt

´

´

D , D˜ )N (D )N (D˜ ) 3 k3 ˜3 ˜ ´ ∞ ´ ∞ (D + D ) dDd D k ˜ ˜ )D dDd D ˜, − 0 0 βBr (D , D )N (D )N (D 1 ∞ ∞ 2 0 0 βBr (

(2)

Single Mode Let us consider a single-mode model based on Koziol-Leighton's model on aerosol dynamics, as the following ordinary dierential equation for this modeling

dm(k ) (t ) = dt

where

´

´

D , D˜ )N (D )N (D˜ ) 3 k3 ˜3 ˜ ´ ∞ ´ ∞ (D + D ) dDd D k ˜ ˜ )D dDd D ˜, − 0 0 βBr (D , D )N (D )N (D 1 ∞ ∞ 2 0 0 βBr (

˜ ) = 4π × 10−9 × ( βBr (D , D

D + D˜ )(D + D ) 1

1

˜

is the Brownian kernel, a coagulation kernel of the interaction between two particles with diameters

D and D˜ due to Brownian

motion by related theory from particle physics.

Figure:

(2)

(3)

Multi-Modes

In most ocean environments or more realistic, we need to consider a modal dynamics for marine particles coagulation concentration, in other words, a composition of several dierent modes. In this situation, the distribution of marine particles

N (D , t ) = where

n X i =1

Ni (D ),

(0) ln D ln i 2 Ni (D ) = √mi (t ) e − 2ln2 i , 2π Dln σi (

for

i = 1, · · · n .

− µ ) σ

(4)

(5)

Coagulation Equations

Based on Koziol-Leighton's result on aerosol model, the dierential equation system of the

k -th moment is

´∞´∞ Pn dmi(k ) (t ) ˜ ˜ = j =1 cij 0 0 βBr (D , D )Ni (D )Nj (D ) dt k 3 3 ˜ + D ) 3 dDd D ˜ (D Pn ´ ∞ ´ ∞ ˜ − j =1 0 0 βBr (D , D ) Ni (D )Nj (D˜ )D k dDd D˜ ,

where

  0

cij =  12 1



dierent modes.

i < j, for i = j , for i > j

(6)

for

determined by the interactions between

Solving ODEs For the 0-th moment we dene

uij for

:=

´∞´∞ 0

i , j = 1, · · · , n.

0

mi(0) , which is the total number of particles, (ln

D −lnµi )2 − (ln D˜ −lnµj )2 2ln2 σi 2ln2 σj

˜ σi lnσj e D Dln −9 × ( 1 + 1 )(D + D ˜ )dDd D ˜. 4π × 10 D D˜

1

−

(7)

Solving ODEs For the 0-th moment we dene

uij for

:=

´∞´∞ 0

0

mi(0) , which is the total number of particles, (ln

D −lnµi )2 − (ln D˜ −lnµj )2 2ln2 σi 2ln2 σj

˜ σi lnσj e D Dln −9 × ( 1 + 1 )(D + D ˜ )dDd D ˜. 4π × 10 D D˜

1

−

(7)

i , j = 1, · · · , n.

Then the ODE system (6) becomes

n dmi(0) (t ) = X (0) (0) (cij − 1)uij mi (t )mj (t ), dt j =1 for

i = 1, · · · , n .

(8)

Numerical Experiments for the Model

Now let us consider the case of a combination of 3 modes which are algae, excretions of zooplankton, and aggregates.

Numerical Experiments for the Model

Now let us consider the case of a combination of 3 modes which are algae, excretions of zooplankton, and aggregates. We have

dm1(0) (t ) = − 1 u m(0) m(0) − u m(0) m(0) , 11 1 12 1 1 2 dt 2 dm2(0) (t ) = − 1 u m(0) m(0) − u m(0) m(0) , 22 2 23 2 2 3 dt 2

(9)

(10)

and

dm3(0) (t ) = − 1 u m(0) m(0) . 33 3 3 dt 2

(11)

Initial Data

Given:

I I I

µ1 = 6.5 × 10−6 µ2 = 9 × 10−6

meters,

meters,

µ3 = 2.5 × 10−5

σ1 = 2.1 × 10−6 ;

σ2 = 3.9 × 10−5 ;

meters,

σ3 = 1.0 × 10−5 .

Initial Data

Given:

I I I

µ1 = 6.5 × 10−6 µ2 = 9 × 10−6

meters,

meters,

µ3 = 2.5 × 10−5

σ1 = 2.1 × 10−6 ;

σ2 = 3.9 × 10−5 ;

meters,

σ3 = 1.0 × 10−5 .

The initial condition for each mode

m1(0) (0) = 1.0 × 107 ,

m2(0) (0) = 8.0 × 106 , m3(0) (0) = 3.0 × 107 .

We use MATLAB to compute numerically the integrals in (7) by quadrature approximation and choosing the domain of integral to be from 1

× 10−6

meters to 1

× 10−2

meters, the range of the sizes

we are interested in, to get an approximation for the singular integrals.

We use MATLAB to compute numerically the integrals in (7) by quadrature approximation and choosing the domain of integral to be from 1

× 10−6

meters to 1

× 10−2

meters, the range of the sizes

we are interested in, to get an approximation for the singular integrals. We mainly use the ODE solver ode45 which is based on Runge-Kutta Method to solve out

mi(0) (t ) for i = 1, 2, 3.

Output we plot the graphs of the size distributions. At

t=0

Figure:

Output At

t = 3600

Figure:

Output

Figure:

References

A. B. Burd, S. B. Moran, G. A. Jackson, A coupled adsorptionaggregation model of the POC/234Th ratio of marine particles, Deep-Sea Research Part I, 2000. A. S. Koziol, H. G. Leighton, The moments method for multi-modal multi-component aerosols as applied to the coagulation-type equation, Quarterly Journal of the Royal Meteorological Society, 2007. Park S. H., Lee K. W., Otto E., Fissan H., The log-normal size distribution theory of brownian aerosol coagulation for the entire particle size range - Part II: Analytical solution using Dahnekes coagulation kernel, Journal of Aerosol Science, 1999.

Sunrise or Sunset

Figure:

Sunrise or Sunset

Figure:

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