Advection is defined as the transport of a conserved scalar quantity that is transported in a vector field.

In ecology, the scalar quantity is the mass of transported substance (or its amount per unit volume, called concentration C [ML-3]), while the vector field is the fluid flow field, identified by the three components of a velocity vector v=(«, v, w) [LT- ] defined at each point as a function of time. In the case of transport of heat, the discussion still holds, as long as the concentration of mass is substituted by the concentration of heat, which is normally expressed proportional to temperature. If the substance behaves like a solute, that is, has the same density as the medium or does not feel significant effect of its buoyant weight, each element of it (molecule or particle) will be displaced along the direction of the local velocity vector following the same path as if it were an element of the medium. This assumption allows the advective transport to be modeled in a relatively simple way. The mass flux F [ML^T-1] through a small area normal to direction n can simply be written as the product of the local concentration times the component of the local velocity vector along n (Figure 1):

In more general terms, the flux vector can be written as

where x = (x,y, z) is the position in Cartesian coordinates and t is time.

Deriving a mass balance for a control volume leads to the following differential equation:

where the symbol (V.) indicates the divergence applied to a vector quantity.

The continuity equation shown above can be simplified further if the flow field is solenoidal, that is, the divergence of the velocity is zero (V . v = 0). This condition applies to incompressible fluids, such as all liquid media found in ecological applications. Under this assumption, the continuity equation simplifies to qc(x,y,z,t) , ,qc(x,y,zt)

qt qx qc(x;y; Z; t) qc(x;y; Z; t) + v(x; Z; t)-;--f w(x;Z; t)-;-

It is simple to show (but omitted here for brevity) that for steady conditions, the solution of eqn  is a simple Figure 1 Schematic of the physical quantities used in eqn . The velocity field generates advective flux through an area only if it has a component in the direction orthogonal to the area, i.e., if v • n=0.

translation of the substance along the paths imposed by the flow field. Although this may not be good representation of the 'diffusive' reality as we experience it, the advective model finds important applications.

Pure advection does not exist alone in nature, as it is always associated at least to molecular diffusion, as discussed above. However, the significance of molecular diffusion in an advective process is negligible in most cases, due to the extremely low value of the diffusive flux. Advection is associated to only molecular diffusion when the flow field is slow, such as in laminar flows. One common application is the flow of water carrying substances in a porous medium, such as an aquifer or the hyporheic zone, as long as the process is modeled at the scale of the pores of the medium. Another useful property of the mass balance equation, valid also when diffusion is significant, is that the center of mass of the substance is subject to advective transport. This is often a sufficient tool to estimate the travel time of a mass of substance between two points of the domain, that is, two locations along a river. The substance disperses in the domain, but its average travel time is always determined by the advec-tive part of the process. Another application is the modeling of the transport of buoyant or heavy substances. These substances are affected by gravity, as their density is either smaller or larger than the density of the medium. Examples are the transport of colloids and suspended solids in water and in air. Particles are subject to the pulsating action of the flow and follow very irregular trajectories. Their behavior is often modeled by the addition of an advective effect, usually defined by a vertical velocity with an intensity dependent on the particle size, shape, and density.

Although the concept of advection and its mathematical description are quite simple, the solution of advective processes of ecological interest in complex flows poses a rather difficult task to modelers. The advec-tion equation is not simple to solve numerically, particularly when strong gradients of concentration (shocks) are treated. Complications arise because numerical methods are indeed curiously affected by a form of diffusion, in this case numerical diffusion rather than the physical one. 