# We focus on the numerical modelling of the saltwater

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Unformatted text preview: transform is now: φ (x, y ) = (1 − Z (x, y ))2 (38) with φ = 0 (exactly) on the sea boundary x = 0, and φ = 1 at some fixed distance L1, the characteristic length of penetration of the salt wedge. The latter is given, to order O(σ), by the analytical solution for a homogeneous aquifer: L1 = LSALT (σ ) ≈ LSALT (0 ) × (1 + O(σ )) (39) Thus, we may write the (approximate) boundary condition of the random case as: x = 0 : φ = 0; x = L1 : φ ≈ 1 + O(σ ) (40) The main idea, here, is that we prefer to solve for the Φ-field because it is more easily amenable to statistical analysis than the Z-field (more on this below). With this goal in mind, let us define the random fluctuations of φ and Z: (41) ϕ ( x, y ) = φ ( x, y ) − φ ( x) and z ( x, y ) = Z ( x, y ) − Z ( x) where the mean potential is given by: φ ( x) = 〈(1 − Z )²〉 . The brackets &lt;•&gt; represent either the shorewise spatial average (spatial mean of a single replicate along direction “y”), or the mathematical expectation E(•) over an ensemble of replicates : the two are equivalent...
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