Deep-Sea Research l, Vol 40, No 7, pp 1459-1473, 1993 0967-0637/93 $6 00 + 0 00 Printed m Great Britain Pergamon Press Ltd

Two-component model of sea particle size distribution

DUBRAVKO RISOVI~*

(Recetved 23 Aprd 1990; m revgsed form 8 September 1992; accepted 9 September 1992)

Abs t r ac t - -A general two-component model of particle size dls tnbuUon m sea water describes the components by a three pa ramete r generahzed g a m m a distribution. The values for two of these parameters , obtained from analysis of exper imental data, prove to be constants , umversal and valid for ohgotroph~c waters in all oceans The third pa ramete r is allowed to vary a round a mean value characteristic for each componen t In th~s way the componen t size d~stribuUons were reduced to a single parameter dis tnbuUon. The model was successfully applied to various measured size distributions. This model predicts measured data with few errors and provides a bet ter fit over a larger size range, 10-2-102 # m , than commonly-used size distribution models . The model is useful for hght sca t tenng and radiatwe transfer calculatmns, especmlly m oligotrophic waters

I N T R O D U C T I O N

THE size distributions of suspended particles in the sea are of interest to many ocean- ographers and physicists concerned with radiative transfer and light scattering in sea water. Apparent and inherent optical properties of all but the clearest natural waters are, to a large extent, determined by suspended particles. In turn, optical properties of oceanic particulates, especially those related to scattering, are strongly influenced by particle size distribution and composition.

Particulate matter in the sea consists of two basic types: biogenic and terrigenous. Abundances and relative concentrations of both vary considerably, and their dimensions can range from less than a micrometer to many micrometers (SHELDON et al . , 1972). Most terrigenous particles are small (HARRIS, 1977; BOGDANOV, 1974; LISITSYN and BOGDANOV, 1968; SHIFRIN, 1983), while submicron biogenic particles are mostly cyanobacteria and prochlorphytes (OLSON et al . , 1985; IrURRXAGA and MARRA, 1988; SIECEL et al . , 1990). Large particles are mostly biogenic.

Particle size distribution measurements indicate that the number of particles increases rapidly with decreasing size. Also the average concentration of a given particulate species decreases with species average size (SrIELDON and PARSONS, 1967; KIEFER, 1983).

Most authors have modeled particle size distributions by a one-parameter hyperbolic distribution (BADER, 1970), also known as Junge-distribution (JuNGE, 1963). This is the distribution of the Cr - k type, where C is a constant depending on particle concentration, r is particle radius, and parameter k has a value between 2.5 and 6, depending on particle size range and measurement site.

*"Ruder BogkoviC' Insu tu te , D ~ p a r t m e n t of Laser and Atomic Research and Deve lopment , 41001 Zagreb, P O B 1016, Croat ia

1459

1460 D Risovl~

Coulter Counter measurements (BRuN-COTrAN, 1971; GORDON and BROWN, 1972; CARDER and SCHLEMMER, 1973) and results obtained by optical methods and flow cytometry (ACKLESON et al., 1988; CARDER, 1970) show that concentrations of particles larger than 0.5/~m are m the range 102-105 cm -3. Values of the exponent k in the corresponding hyperbolic (Junge-type) distributions are 2.5--4.5. Measurements of par- ticles with dimensions less than 0.5/~m are relatively sparse. Particle concentrations between 0.02 and 2ktm are between 5 x 106 and 107 cm -3, and their size distribution may be described by hyperbolic distribution with an exponent 1.65 for the size range 0.02-2.0 #m, and 3.44 for a size range of 2-8 ~m (HARRIS, 1977).

Particle size distributions, especially in biologically active areas, tend to flatten out at diameters less than 5 ktm (WEIDEMANN et al., 1988; KITCHEN et al., 1978), and therefore the lower size limit of optically effective particles should be larger than predicted by the hyperbolic distribution. This has led a number of authors to conclude that size description of particles with hyperbolic distributions is inadequate, since it overestimates the number of small particles (which approaches infinity as the diameter approaches zero) and also underestimates the number of larger particles (DEL GROSSO, 1975; KITCHEN et al., 1982; KULLENBERG, 1974). Modeling the real particle size distribution over a wide size range (r = 0.01-50/~m), which is relevant for light scattering, is not possible using a single hyperbolic type distribution because errors would be too large. For adequate modeling it is necessary to employ several hyperbolic type distributions with various exponents and constants (e.g. BROWN and GORDON, 1974), SO that the simplicity and physical meaning of the model are lost. Also in the vicinity of the joints the fit is poor and the errors are usually high. Moreover one must be careful when extrapolating to smaller or larger size ranges. To overcome these difficulties I have attempted to derive a unique particle size distribution model that can describe particle size distribution over the entire relevant size range.

DERIVATION OF THE MODEL

In deriving the particle size distribution model the initial assumption was that a hyperbolic distribution cannot successfully describe particle size distribution within the entire size range between 10 -z and 102/~m. Hence, a different, more suitable type of the distribution has to be found.

Another assumption is that the global particle size distribution is obtained as a superposition of two size distributions, henceforth called the distribution components. These components are different, non-hyperbolic, dominant at different size ranges and in turn are the result of the superposition of various biogenic and terrigenous material size distributions.

We therefore first proceed with an analysis of size distributions for purely biogenic or terrigenous materials. Next, we find adequate distribution functions for the components and then determine the parameters in these distributions so as to obtain the best fit to the experimental data.

Biogenic material

Measurements of monocultures of phytoplankton show relatively narrow size distri- butions that are species-dependent (PAUL and JEFFREY, 1984; BRXCAUD and MOREL, 1986;

Model of sea pamc le size &strlbutlon 1461

1 20 o P l o t y m o n a s lue¢~¢o

S k e l e t o n e r n a ¢ ~ a t u m - - Gamma c h ~ n b u t l o n fit,

o0

~-o8o Z

~ 0 6 0 E~

E 7° O40

o \. 0 0 0 [ . . . . . . . . I . . . . . . . . . I . . . . . . . . . ~ . . . . . . . . I . . . . . . . . . ]

O0 2 O0 4 O0 6.00 8 O0 10 O0 D iamete r ( m i c r o m e t e r )

Fig 1 Normahzed size & s t n b u t m n of phytoplankton specms Platymonas sueclca and Skeleto- nema costatum (BmcAUD and MORZL, 1986) and corresponding g a m m a distributions fit. Para-

meters of the respectwe &s tnbu t ions are ~ = 20, b = 5.71 and ~ = 40, b = 7.279

JONASZ, 1986). These size distributions can be successfully modeled by the so-called gamma distribution (Fig. 1), a two-parameter distribution of the type:

0 r < O n(r) = (1)

Cr~, -br r _ > O ; / ~ > - l ; b > 0 ,

where n is a number of particles with a radius r, C is a constant related to concentration, and/~ and b are the parameters of distribution. Maximum distribution is at the mode radius r m =/Mb, and total number of particles is given by:

= c e -b' dr = cr( + +. 1) Ntot (2) 0

where F denotes the gamma function. This type of distribution was used by DEIRMENDIJAN (1964) and GREEN et al. (1971) for aerosol and fog particle distributions.

Each phytoplankton species can be described by gamma type size distribution, sugges- ting that biogenic material size distribution can be described by the sum of mono modal gamma distributions for various species:

N(r) = E, N~'~ (r), (3)

where N~, (r) represents i-th mono modal gamma distribution. It would be expected that there would be so many discrete contributions in (3), that the

overlap would eventually lead to a more or less smooth curve. This is indeed the case. In addition, the numerical concentration of certain species increases with decreasing average size of its members (SHELDON and PARSONS, 1967; SHELDON et al., 1972; KIEFER, 1983), SO that the slope of the total distribution (3) will be less steep than that of the gamma

1462 D RIsovl~

l°il lO

lO

~ 1 0 ~ '

E I O

n.- 10

2~ 7

10

10 -2

10

10

heterotrophlc bacteno

Cglroococcold eyonol~cterlo

monads

phytoplQrlktOn

I I [ l l l l l I I I I l l , I l l [ , , , l l l , [ I I I I H I I I

10 -2 10 °' 1 10 10 a RADIUS (m~crometer)

Fig. 2 Global s]ze distribution result ing from superposl t lon of certain b logemc specms described by a g a m m a size d]stributlon and corresponding generalized g a m m a dls tnbut]on fit ( p = 2, b = 17

and y = 0 280)

distribution. Therefore gamma distribution is not suitable for prediction of the global biogenic material distribution. For prediction of a global distribution, the so-called generalized gamma distribution is appropr ia te (Fig. 2). This is a th ree -paramete r distri- bution given by:

0 r ~ O n(r ) = (4)

C r ~' e -brr r > O.

For a specific/~ and y, the remaining constants can be uniquely determined by the mode radius, rm, where the distribution is at maximum. Thus, the constant C can be determined from the integral:

I0 Nto t = n( r ) dr = _ C b_(~,+l)/yr (5) Y

and constant b from the derivative, which can be written as

d n ( r ) = CH ' - 1 e - b ' ( p - (6) %br y)

which vanishes when r ~ = /a /by , or for r = r m = (/ .t/by)l/y, as well as for r = 0 and oo.

T e r r i g e n o u s m a t e r i a l

As measurements regarding the size distribution of purely terr igenous material in sea water are sparse, the size distributions of o ther terr igenous particles were considered:

Model of sea parucle size &stnbutlon 1463

1 0 ' A

10 3

E (9

V l 0 2 I l l LtJ m 10

Z 1

\

i0~ \ \ Hyperbolic d,str

~ \xLX "

',, \\ C ~ m m o t l l u r b a n

X 1 o -' ~ "~a,

0 Meosur~J dote \

1 0 -2 - - Ge~erollzed gommo dmtrlbutton ~)

I0 - 3 i i , iiiii I i i i iiiii~ i iiiii I i i i iiiii I

10 -~ 10 - ' 10 -' 1 10 RADIUS ( m , c r o m e t e r )

Fig. 3 Measured aerosol size dlstr ibuuon (JUNGE, 1963) and fit with certain size-&stnbutlon models Pararnetersofthed~stnbutlonsare/~ = l l . 5 , b = 65.21 forgamma&str ibut lonand/~ = 8.3,

b = 13 5 and 7 = 0 09 for generalized gamma distribution.

aerosol (GREEN et al., 1971; PATTERSON and GILLETTE, 1977), road dust (BADER, 1970), and kaolinite particles (PAK et al., 1971; JONASZ, 1986). These size distributions are rather broad and markedly skew with maxima at the side of very small particles. Moreover, long- term size distribution measurements of aerosol particles (RErrER et al., 1982) indicate that although the concentrations may vary by more than two orders of magnitude, the general shape of the distribution remains more or less similar.

As part of the terrigenous material in sea water is aeolian, size distributions similar to Fig. 3 might be expected, and in fact experimental results support this assumption (e.g. distribution of kaolinite (JONASZ, 1986) or red clay particles in the Northern Pacific nepheloid layer (PAr et al., 1971). These results, however, can not be described by a hyperbolic type distribution that increases monotonically nor by a gamma distribution. The appropriate distribution is the generalized gamma distribution (Fig. 3).

Particulates in the sea

As the generalized gamma distribution predicts the measured particle size distributions of some purely biogenic and purely terrigenous material, we can assume that each distribution component of a global particle size distribution in the sea can be predicted by a generalized gamma distribution.

To determine the parameters for the distribution components, we proceed as follows: First, we consider those size distributions that have unique (noticeable) components,

such as particle size distribution shown in Fig. 4. In this size distribution two components can be distinguished: one rather steep, narrow, and dominant at small sizes, and another much broader, less steep, and dominant at large sizes.

Second, we fit generalized gamma distributions to these experimental data and then determine the values of parameters #, b, 7 for the various size distribution components.

10 0 M e o s u r e d d o | o

__ e

10

~ ' 1 0 a I E 0

z l O

10

1464 D Rlsov]~

1 , , i i I f l , l ~ v i i i , l l , I

10 -' 10 Rad ius ( m i c r o m e t e r )

Fig. 4. Particle size distributmn from the Rarotonga area and the corresponding prediction obtained from 2C model with YB = 0.198 and YA = 0 154; C B = 1.136 × 109 cm -3/~m -3 , C A = 1.184

X 1025 cm-3,um -3

Once the values of these parameters have been determined, total particle size distribution can be obtained as:

dN(r) = dNA(r) + dNB(r) -- CAFA(r) dr + CBFB(r ) dr (7)

FA(r) = ~ A e-bAr~,; FB(r) = r"B e-bB ",

where dN(r) is the total number of particles per unit volume with radii between r and r + dr, and the indices A and B refer to the first and second components, respectively. The total number of particles per unit volume Nto t is:

Ntot = dN(r). (8) 0

Finally, although the particle size distributions in sea water vary considerably, they can be categorized into a few general types (SHELDON et al., 1972), suggesting that some of the parameters of distribution components could be held constant while others are permitted to vary, but providing adequate description of the real situation. Let us examine this more closely.

Particle size distributions vary with depth, and their shapes change in transit through water column (SHELDON et al., 1972; LAL and LERMAN, 1975). Moreover, variations in biogenic particulate production can b e extremely large, although sometimes limited to a few species, producing highly irregular distributions (JONASZ, 1978). However, consider- ing these changes in shape of a size distribution and neglecting monospecific blooms that produce multimodal distributions, we can see that the major change is in the steepness of the distribution. This is true especially for oligotrophic waters (SHELDON et al., 1972). The steepness of a generalized gamma distribution is primarily governed by parameter y. We therefore assume that for each size distribution component the average values of

M o d e l o f sea p a r u c l e size d i s t r i b u t i o n 1465

parameters/x and b, obtained through the fit, are general and globally vahd and that only the parameter ~ and the number (concentration) of component particles varies. Thus the component size distributions are reduced to one-parameter distributions.

In conclusion, we assume that relation (7) always holds and that the respective component parametersp and b are constants, while y is allowed to vary. CA and CB depend on particle concentration in the components, and their values are such that:

N A = C A F A ( r ) dr and N B = C B F B ( r ) dr (9) o o

N,o,=NA + NB,

where NA and NB are numerical concentrations of particles in respective distribution components.

However, the usual measured quantities are Ntot and N(Ar), but not NA or Na. Therefore, relations (9) should be modified to obtain CA and CB. Relations (9) show clearly that both components occupy the entire size range, but the first component ("A") has a narrower and steeper distribution than the "B" component. Since the number of particles in the "A" component decreases rapidly with increase in size, the contribution of this component to the large-size part of global particle size distribution is negligible. Hence most of the small particles in the global distribution belong to the "A" component, while large particles belong mostly to the "B" component. The mean particle radius for each component can be calculated from respective size distributions within a size range 0.01- 100 pm. These values differ within an order of magnitude.

Now, let us denote the number of particles measured in a size interval Ar (e.g. in one Coulter Counter channel) around some small radius by N< (Ar) and the number of particles measured in the interval Ar around some large radius by N> (Ar). Since the large particles are predominantly from the "B" component and the small particles predomi- nantly from the "A" component, we have:

I + Ar/2

N<(Ar) = CA Fm(r) dr (10) J -Ar /2

and

I + A r / 2

N>(Ar) ----- CB FB(r ) dr (11) J-- Ar/2

wherefrom we obtain C,4 and CB.

RESULTS AND DISCUSSION

Initial values of parameters for the distribution components were obtained by fitting a generalized gamma distribution to the particle size distribution measured in the Eastern Pacific (SHIFmN, 1974). This size distribution has two noticeable components character- ized by different concentration and shape (Fig. 4). The parameters for the "A" component were obtained by fit of a generalized gamma distribution to the measured size distribution of small particles (R < 1 btm), while the size distribution of large particles (R > 2 btm) was used to infer the parameters for the "B" component. Here we have assumed that, due to a

1466 D RIsovld

steep size distribution, contribution of component "A" particles to the large-size part of the measured distribution is negligible for particles w~th R > 2/zm. This size limit could be lower for biologically highly productive areas.

These initial values of the size distribution parameters were further used to predict other measured distributions. The parameters were slightly varied about the initial values to obtain the best fit of the considered data. As expected, the necessary variations in b and kt were smaller than variations in y. The optimal fixed values for the parameters b and/x that reasonably fit all considered experimental data were found to be:

PA = 2; b m = 52 Arm-l; (TA = 0.145 -- 0.195)

/~B = 2; bB = 17 ktm-1; (TB = 0.192 -- 0.322).

These results confirm our basic assumption that measured particle size distributions in the sea can be predicted by the two-component model in which particle size distribution is given by:

dU(r) = CAFA(r ) dr + CBFB(r) dr (12)

FA(r ) = r 2 e-52r~; Fo(r) = r 2 e -17r~,

where r is expressed in micrometers, and FA (r) and Fo(r) are size distribution components. The main difference between these two components is in the general shape (modal value, width and skewness) of the respective size distribution functions, although one also can argue about some other differences, like predominant composition, average index of refraction, etc. The parameter y varies slightly around mean values YA = 0.157 and Yo = 0.226. The constants CA and Co can be obtained from the relations (10) and (11), and the total number of particles is given by (9).

Validity of this model was confirmed by applying it to over 70 different measured particle size distributions. Partial results of this analysis, shown in Figs 4--8, represent some typical predictions obtained by this model. Data are from various sites in open ocean and coastal waters, and range in depth from surface to 3600 m.

Particle size distribution from the Rarotonga area in the Pacific (Fig. 4) shows data points representing total number of particles within a given size range (channel) at average size value for that channel (cf. Table 1). The predicted number of particles in a given size range was obtained as the sum of integrals of the distribution components over that range ( r 1 ~ r - - r2) :

J I: N(Ar) = NA(Ar) + NB(Ar) = CA r2 FA(r) dr + Co Fn(r) dr. (13) r l 1

Table 1 compares experimental data and predictions obtained from the two-component model (12) with CA = 1.184 × 1025 cm -3/2m -3 and CB = 1.136 × 109 cm -3/zm -3. The agreement is very good, and the difference in total particle number ANto t = 0.02%, while the mean difference per size interval AN = 9.5%. The mean difference per size interval is the average of the differences between the calculated and measured number of particles per size interval, taken regardless of sign, and is given by:

M 1

AN = ~ ~ bag, I, (14) I

t = l

Model of sea particle size dlstnbuuon 1467

Table i Measured and calculated parttcle stze dtstrtbutton

Model Radius (,urn) Measured N (cm -3) N o N A Nto t AN (%)

0 20-0.30 51,000 0 17.4 49,563 3 49,580 7 - 2 8 0 30-0 40 8600 0 14 0 9743 8 9757 8 13 5 0 40-0 50 2500 0 11 4 2726 6 2738 0 9 5 0 50--0 60 930 0 9 5 947.2 956.7 2 9 0 60-0 70 370 0 8 0 380 4 388 4 5 0 0 70-0 80 180.0 6 8 I70.1 176 8 -1 .7 0.80-0 90 98 0 5.8 82 5 98 4 - 9 8 0 90-1 00 56 0 5 0 42.8 47 8 -14 6 1.00-1 25 63 0 10 0 41 2 51.2 -18 8 1 25-2 50 41 0 23.5 17 5 41 0 0 0 2 50-5 00 10.0 11 7 0 3 12.0 19 8 5.00-12 5 5 2 4.4 0.0 4 4 -16.0

0 20-12.50 63,853 3 63,843 2 -0.02 la--NI -- 9 5

where AN, is percentage difference (error) for the i-th size range (channel), and M is the total number of channels.

Next, it should be pointed out that although CA and CB do relate to respective particle concentrations in the components, they do not represent the concentrations of "A" and "B" materials. Therefore, one should not try to infer relative abundances of "A" and "B" component materials from their ratio. The number of "A" or "B" particles per unit volume (concentration) is obtained through integration of the respective distribution over the considered size interval.

The second analysed data set was the measurements made by HARRIS (1977) in the Western Gulf of Mexico. Typical results are shown in Fig. 5. For these samples, taken from depths of 600 and 3600 m, the differences between the measured and predicted to..__tal number of particles are 0.01 and 0.06%, while mean differences per size interval are AN = 7.49 and 5.87%, respectively. The absolute average error for total particle numbers for the whole data set is 0.03%, and overall average error per measurement interval is 9.9%. To model these measurements with hyperbolic distributions, a combination of two distri- butions with different parameters was necessary (one for the size range <2/zm and one for sizes >2 k~m) (ibid.), producing much larger errors, namely ANto t = 8.9% and AN = 12.3%. The electron microscopic analysis of these samples showed that biogenic particle concentrations ranged between 2.4 x 103 cm -3 and 6.2 x 103 cm -3, representing less than 1% of total particle number (ibid.). Therefore these samples contain predominantly terrigenous material. The integrations of the "B" component size distributions corre- sponding to these measurements gave the total numbers of "B" component particles that represented 0.4-2.4% of the corresponding total particle numbers. If we take into account that identification process partially removed organic particles from the samples (ibid.), the total number of the "B" component particles obtained from the model agrees with the number of biogenic particles counted in the corresponding sample. Therefore it seems that the "B" component is predominantly biogenic while the "A" component contains both

1468 D Rlsovl~

iO

1o

I0

l

E u10

Z

10

10 2

*x

t a \ o M e a s u r e d dat(J (600m) ~ \ + M e a s u r e d da ta ( 3 6 0 0 m )

1 0 , , , , , , , , i , , , , , , , , i , , , , , , , , i 10 -2 10 -' I 10

R a d i u s ( m i c r o m e t e r )

Fig. 5. Particle size distributions from the Gulf of Mexico and the corresponding predictions obtained from the 2C model. YA = 0 145 and 7B = 0.301; C A = 2 547 x 1024 cm - 3 / t m -3, C B =

1.603 x 1011 cm-3/~m -3 (600 m) and 7A = 0 170 and 7B = 0.321; C A = 2.905 x 1023 cm -3/~m -3 , CB

= 1.497 x l0 ll cm-3 / tm -3 (3600 m)

biogenic and terrigenous materials in various proportions depending on measurement site. For this data set it seems that the "A" component is predominantly terrigenous.

Another analysed data set was the measurements made by CAROER et al. (1971) in the Eastern equatorial Pacific by means of a Coulter Counter. Typical results are shown in Fig. 6, where the data from two stations, one in the open ocean (YPT34) and one in the immediate vicinity of the Galapagos Islands (YPT41), are compared with the predictions obtained by our model. For these we obtain ANto t ~ 0.1% for both stations, while AN = 11.3 and 10.8% for YPT34 and YPT41, respectively. The corresponding (best fit) hyperbolic distributions produce higher average errors than our model, namely AN = 14 and 12% for YFI34 and YPT41, respectively. A two-segment hyperbolic distribution is needed to fit the data with similar, but somewhat greater errors than our model (cf. Table 2). Comparison of other measurements made by CARDER et al. (1971) and the predictions obtained from the two-component model give similar results.

Comparison of our calculations with some of the measurements performed by GROSSO (1978) in the Sargasso Sea is shown in Fig. 7. For sample no. 4 ANto t = 0 . 0 2 % , A,N = 7% and for sample no. 6 ANto t = 0.06% ; AN = 6%. Typical error in total particle number for other samples is less than 0.1% and AN _< 13.5%, which compares favorably with the predictions obtained using segmented hyperbolic distributions (cf. Table 2).

Applicability of the model to a broad particle size range with ruffle distribution is illustrated with the prediction for a surface sample from a site in the North Pacific (34°21'N, 150°00'W), which covers a size range from 0.5 to 80btm (SHELDON et al., 1972). In this c a s e ANto t = - 0 . 0 3 % , and the mean difference per size interval AN = 17.8% (Fig. 8). To obtain a similar fit with segmented hyperbolic distribution, at least three segments should be used.

To compare the performance of our model with other models, we first consider the

Model of sea parhcle size dlstrlbunon 1469

1 0 ~=

lO'

" ~ 1 0 3 I E £3

Z l o

10

o

Y P T 4 1 / 2 0 m

. . . . . . I() Rodtus (m~crometer)

Fig. 6. Pamcle slze &stnbutions from the Eastern Equatorial Pacific and the corresponding pre&cUons obtamed from the 2C model. YPT34: YA = 0 147 and YB = 0.192 for surface (CA = 4.390 x 1026 cm -3/xm -3, C B = 4.610 x 101° cm -3 #m-3) ; )'A = 0 190 and 7B = 0.230 for 100 m (C A

= 1.538 x 1026 cm-3#m -3, CB = 4.195 x 101° cm-a/zm-3); ~A = 0.190 and ~B = 0 192 for 1000 m (CA = 1.397 × 1026cm-a#m -3, C B = 9 453 x 109 c m - a # m -3) YPT41: }'A = 0 195 andyB = 0.230 for surface (CA = 2 170 X 1027 c m - a # m -3, C B = 3.150 x 1011 cm-3#m-3) ; y,~ = 0.195 and YB =

0 235 for 20 m (CA = 2.230 × 1027 cm-a/~m -3, CB = 8.123 x 1011 cm-a/zm-3).

1o

1o

~ ' 1 0 I E o

v

z 1o

10-

o M4msutetl dato t 4

_ 26 Mod l l

o

, , , i r l I

I0 I0 7

Rod,us (m,crometer)

Fig. 7. Particle size distributions from the Sargasso Sea and the corresponding predictions obtained from the 2C model with Y.4 = 0.177 and Ys -- 0.215 for sample no. 4 (C A = 8.330 x 1027 cm -3 #m -3, C B = 2.805 x 101° cm -3 #m-3) , Yx = 0.180 and )'B = 0 215 for sample no 6 (C A =

1 189 x 1027 c m - 3 # m -3, CB = 1.990 × 101°cm-a#m -3)

1o

10-

10

10 3-

~I0 2- r.~

I E u IO

Z

1

D Measured data

I 0 -2

10 -~ 10 -'

1470 D RlsovlC

' ' ' r''''I ' ' ' '''''I

1 I0 10 2

Rad ius ( m i c r o m e t e r )

Fig. 8. Particle size distribution from the North Pacific and the corresponding prediction obtained from the 2C model with Ya = 0.134 and YB = 0.216 (CA = 2.82 x 1026 cm -3 p m -3, CB = 2.20 x 10 l°

cm -3/~m-3) .

complexity of the model in question. Since our model is a two-parameter model (one parameter for each component) , we should compare it with other two-parameter models. Here we shall limit ourselves to the most frequently used model, the two-segment hyperbolic distribution of the following type:

N(r) = C1 C2 (15)

where k I and k 2 are parameters of the distribution. The first term is valid in the size range for which r -< r I (here C2 = 0) and the second term for the size range where r > r 1 (C1 = 0).

Next it is desirable to introduce some quantitative measure for the quality of the fit obtained by a certain model. We define a Q-number of the considered distribution model as the arithmetic mean of a mean difference per size interval and absolute value of the error in total particle number:

AN + I ANtotl Q-number - 2 (16)

Since the absolute error in total particle number [ANtotl is usually much smaller than the mean difference per size interval AN, the Q-number is roughly equal to a half of the mean difference (average error) per size interval. Hence a smaller Q-number corresponds to a bet ter fit. Thus we may consider the case for which Q-number <4 as an excellent fit, one with Q-number -<7 very good fit, etc.

Table 2 compares the predictions obtained from our two-component model (2C) and from one- and two-segment hyperbolic distributions (1SH and 2SH) by listing the corresponding Q-numbers. It is evident that the two-component model gives a much bet ter fit than the two-segment hyperbolic type distribution. Average Q-number for our model (for all presented samples) is Q2c = 5.72, which compares favorably with Q2sH =

Model of sea parucle size dlstnbuuon 1471

Table 2 Comparison of vartous models Excellent fit Q-number <4, poor fit" Q-number >15

Model Size range

Measurement r (am) 2C H1S H2S

Shlfnn/Rarotonga 0.2 -12.5 4 76 48 73 5.8 Harns/ 600 m 0.01- 4 0 3 75 22.30 2.99

3600 m 0.01- 4 0 2 97 23 75 5 64 Carder/YPT34/0 m 1 1 - 5.5 5 51 6.87 4 26

YPT34/40 m 1 1 - 5.5 5 10 8.72 7 08 YPT34/100 m 1 1 - 4.5 7 46 10.68 7 11 YPT34/1000 m 1.1 - 4.5 4.85 10 23 9 11 YPT41/0 m 1.1 - 5 5 4.67 8 71 5 29 YPT41/20 m 1 1 - 5 5 5.67 8.28 6 15 YPT41/100m 1.1 - 4 5 8.08 10.68 8.98 YPT41/1000 m 1.1 - 4.5 3 50 5.79 1.49

V.d. Grosso/#4 1.25-31.83 3.70 27.40 12 00 #5 1 25-25.26 14.49 50 84 19 88 #6 1 25-25 26 3 10 18.40 9.22

Sheldon/N. Pacific 0.5 -40.30 8.21 21 79 18.87

Average Q-number 5 7 18 9 8.3

2C = this (two component) model. HIS= hyperbolic 1-segment H2S = hyperbolic 2-segment.

8.26 o b t a i n e d for t w o - s e g m e n t h y p e r b o l i c m o d e l r e p r e s e n t a t i o n o f the s ame da ta . A o n e - s e g m e n t hype rbo l i c d i s t r ibu t ion gene ra l ly gives a p o o r fit ( e x c e p t p e r h a p s for a na r row range o f pa r t i c l e sizes). If a pa r t i c l e size r a n g e is b r o a d , e v e n the t w o - s e g m e n t hype rbo l i c d i s t r i bu t ion gives a p o o r fit (cf. T a b l e 2). T o ob t a in a fit c o m p a r a b l e to the one p r o d u c e d by the t w o - c o m p o n e n t mode l , one should use a m o d e l of m u c h g r e a t e r complex i t y . Even then the t w o - c o m p o n e n t m o d e l gives a b e t t e r fit in t r a n s i t i on z o n e b e t w e e n the p o w e r r e l a t ionsh ips .

G r e a t e r dev ia t ions b e t w e e n the p r ed i c t i ons and m e a s u r e m e n t s might be a resul t of i nc r ea sed p r o d u c t i o n of ce r ta in b io log ica l spec ies (cf. T a b l e 2, s a m p l e no . 5), bu t to some ex t en t t hey cou ld also be a resu l t o f a m e a s u r e m e n t e r r o r . T h e m a i n source of this e r ro r , a p a r t f r om ca l ib ra t i on e r r o r , is the spa t ia l i n h o m o g e n e i t y o f pa r t i c l e d i s t r i bu t ion in a given ( p r o c e s s e d ) v o l u m e o f sea wa te r . T h e e r r o r due to this i n h o m o g e n e i t y is smal l at the lower end o f the size r ange ( a b o u t 1%) , bu t cou ld be two o r d e r s o f m a g n i t u d e g r e a t e r at the u p p e r end of cumu la t i ve size d i s t r ibu t ion (JONASZ, 1983 a n d r e f e r e n c e there in ) . In add i t i on , the C o u l t e r C o u n t e r r e sponse can va ry up to 30% for a g iven pa r t i c l e vo lume due to the ef fec t o f pa r t i c l e shape and even m o r e due to t he w a y it c rosses the orifice (GOLIBERSUCH, 1973; KACHEL et a l . , 1970). This will i n t r o d u c e an a d d i t i o n a l r a n d o m er ro r , e spec ia l ly in the la rge-s ize pa r t of the d i s t r i bu t ion due to a g r e a t va r i e ty in shapes of b iogen ic pa r t i c l es . T a k i n g all these fac tors in to accoun t , t h e p r e d i c t i o n s o b t a i n e d f rom the t w o - c o m p o n e n t m o d e l for the cons ide r ed pa r t i c l e size d i s t r i b u t i o n s a re wel l within the l imits o f m e a s u r e m e n t e r ro r .

1472 D RlSOVI~

Acknowledgements--The author thanks Dr J C Kitchen and Prof J R V Zaneveld for helpful suggestions and critical reading of the manuscript, and Dr R W Sheldon for providing some of the data used in this work

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