## Lagrangian Flocculation Model

This section summarizes the flocculation model developed by Winterwerp.8,9 We assume that the bulk properties of the suspended sediment, relevant for the present study, can be captured with one floc size only (denoted by Df in this chapter), which may vary with time and over space, however. We treat flocs of cohesive sediment as self-similar fractal entities, as proposed by Kranenburg15 and in line with Krone's1 hierarchical floc structure. The fractal dimension nf is obtained from the description of a growing object with linear size La, consisting of L seeds of linear size a, and volume V(La) (e.g. Vicsek16). Assuming that the linear size of the primary objects has unit dimension, V(La) a N(La), where N is the number of primary objects (seeds), the fractal dimension nf is defined as r ln(N(L))

From this approach, it follows that the differential (or excess) density Apf of mud flocs is given by (Kranenburg15)

Dp Df

where pf, pw, and ps are the densities of the mud flocs, the (interstitial) water, and the sediment (primary particles), and Df is a characteristic (mean) diameter of the flocs, and Dp is the diameter of the primary particles. From (12.2), it follows that the relation between mass (c) and volumetric concentration (f reads:

Df LDP

At the gelling point, defined as the condition that flocs form a space-filling network, the volumetric concentration of the flocs 0f becomes unity by definition. Hence, the gelling concentration cgel can be obtained from (12.3):

Such gelling conditions are found in fluid mud layers (e.g. Section 12.5); the gelling concentration is also known as the structural density in soil mechanics, as a measurable strength builds up in the network.

Measurements of the fractal dimension of flocs of cohesive sediment in the water column reveal values from about nf ^ 1.4 for very fragile flocs, like some marine snow to nf « 2.2 for strong estuarine flocs. Typical values within estuaries and coastal waters range from nf « 1.7 to 2.2, with an average value of nf « 2.0 (e.g. Kranenburg15).

This fractal approach implies that the relation between settling velocity ws of a single mud floc in still water, and floc size Df, which is based on a balance between gravitational and drag forces, does not follow Stokes' law, but (e.g. Winterwerp8)

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