Horizontal diffusivity

Trajectory models require spatiaUy and temporaUy resolved wind fields, mixing-height fields, deposition parameters, and data on the spatial distribution of emissions. Lagrangian trajectory models assume that vertical wind shear and horizontal diffusion are negligible. Other limitations of trajectory and Eulerian models have been discussed (30). 

In the application of gradient transfer methods, horizontal diffusion is frequently ignored, but the variation in vertical diffusivity must be approximated (11-14). 

Hints and Help Assume that as a first approximation horizontal turbulent diffusivity can be considered to be isotropic, that is, Ex = Ey= 2 x 104cm2s 1. Disregard the loss of HCB across the thermocline and to the atmosphere. In Chapter 22.3 we will see that because of horizontal water currents horizontal diffusion is, in fact, not isotropic. Nonetheless the above approximation yields reasonable results. Finally, note that in order to keep the total mass of HCB constant, CQ (t) must decrease as R(t) grows such that CaR2= constant. 

Consequently, the choice of the averaging time s determines which eddies appear in the mean advective transport term and which ones appear in the fluctuating part (and thus are interpreted as turbulence). The scale dependence of turbulent diffusivity is relevant mainly in the case of horizontal diffusion where eddies come in very different sizes, basically from the millimeter scale to the size of the ring structures related to ocean currents like the Gulf Stream, which exceed the hundred-kilometer scale. Horizontal diffusion will be further discussed in Section 22.3 here we first discuss vertical diffusivity where the scale problem is less relevant. 

If horizontal diffusivity were isotropic (Ex=Ey) and constant with time, then according... 

Thus, observing the growth of a tracer patch would allow us to calculate the horizontal diffusion coefficient. The above expression would still hold if the growing effect of random motion with patch size were considered as inferred by the picture of a spectmm of turbulent structures of different size. The turbulent diffusivities, Ex and Ey, would then depend on ax or oy. 

This empirical relation is based on data extending from Z, = 10 m to T = 1000 km, yielding horizontal diffusivities between h = 10 cmV and 108 cmV, respectively. 

The following considerations are based on an overview of horizontal diffusion theories by Peeters et al. (1996). Turbulence theory suggests a power law with unknown exponent m to describe the growth of cloud size a2 with elapsed time / ... 

The different exponents found in Eq. 22-43 indicate that the spreading of a tracer cloud is caused by two processes perpendicular to the flow direction, that is, along the minor principal axis, the spreading is compatible with normal Fickian diffusion with scale-independent horizontal diffusivity Eh (see Box 22.3, Eq. 6b). An additional effect is important along the axis of flow, the process of longitudinal dispersion. 

As pointed out by Peeters et al. (1996), based on their own experiments and on the reinterpretation of published field data, the adequate model to describe horizontal diffusion in lakes and oceans is the shear diffusion model by Carter and Okubo (1965). The model is described in Box 22.4. The most important consequence of this model is that the 4/3 law and the equivalent t3-power law for c2(t) expressed by Eq. 22-42 are replaced by an equation which corresponds to a continuous increase of the exponent m from 1 to 2 (Box 22.4, Eq.l) ... 

As shown by Peeters et al. (1996), horizontal diffusion experiments in lakes and oceans can be best described with the shear-diffusion model of Carter and Okubo (1965). The model yields the following relation between cloud size, ct2, and time, t ... 

The spreading results from a combination of two processes, (1) Fickian horizontal diffusion with scale-independent diffusivity Eb, and (2) dispersion by velocity shear in the direction of the mean flow. The process of dispersion is discussed in Section 22.4. It is related to the flow velocity difference (called velocity shear ) between adjacent streamlines. Since water parcels traveling on different streamlines (e.g., at different depth) have different velocities, a tracer cloud is elongated along the direction of the mean flow (see figure below). 

Horizontal diffusion. As the cloud expands horizontally, the particle size distribution may vary as a function of distance from the center if the diffusion coefficient is a function of particle size. This would cause the true correction factor to vary with horizontal position, and should be observable if observations are made for horizontal positions at the same altitude and time. Since the diffusion coefficient for clouds is known only within a factor of 10-100, no estimate of the variation with particle size is possible at this time. 

The ChemLeg program, in turn, focused on closer cooperation between policy makers, industry experts and national federations to facilitate the implementation of these standards. The main indicators of achievement were the enlargement of CEFIC with new full and associate CEFIC members representing nine of the ten Central and Eastern Europe states that subsequently joined the European Union the creation of an influential regional network of chemical industry and active involvement in European advocacy on the part of federations from Central and Eastern Europe (Doktor 2002). The political dynamics of trans-European industry mobilisation, organisation at the domestic and European level, and alliances with supranational and domestic policy elites and experts developed in full swing and in support of the horizontal diffusion of EU chemical safety policies and norms. 

Mean vertical advection is suppressed by the channel-like character of the marine layer, and horizontal diffusion is relatively unimportant because of a nearly uniform distribution of emission sources. Hence these terms do not appear in Equation 19. The chemical source term is calculated from the usual rate expressions. 

Because of its short half-life, the distribution of Ra can be sensitive indicator of small-scale horizontal mixing processes. Its distribution in Long Island Sound, a narrow embayment a few tens of kilometers wide, has been used to measure mixing rates of order 5-50 m s (Torgersen et al., 1996), a value compatible with the spatial scale (see Section 6.08.3, and LedweU et al., 1998). Moore (2000) has made measurements of all four isotopes in coastal waters of the Mid-Atlantic Bight. He assumed a steady-state horizontal diffusion-decay balance ... 

Further downwind on the continuum scale, as discussed in Section 2.4.2, the cloud has diffused vertically a significant distance so that its horizontal diffusion crzH > H. Its horizontal diffusion caused by the shear is such that the cloud streamwise dimensions crl above and below the canopy become approximately equal. Then the expressions (2.9)—(2.41) are no longer valid for the canopy cloud or plume. 

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