The ghyben herzberg closure relation above is

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Unformatted text preview: le 2 show some of the numerically computed moments of ΦSALT , assuming a linear trend <Φ>, and stationary fluctuations ϕ(x,y) around the linear trend: Φ (x, y ) = Φ (x ) ≈ ax σ Φ = ϕ (x )2 ϕ (x, y ) ≈ Φ (x, y ) − ax 1/ 2 (47) (48) ≈ constant Note: These relations hold only in a subdomain comprised between the sea boundary x = 0 (where φ = 0) and the tip of the salt wedge x ≈ L1+O(σ) (where φ ≈ 1+O(σ)). On the other hand, we demonstrate that the Φ-equation in the salt wedge zone is a stochastic PDE, analogous to the Boussinesq equation for vertically averaged groundwater flow with random K(x,y). Indeed, from eqs.(30), (31), (32), we have: ∂ ∂xi ⎛ ∂H ⎜ − K ( x1 , x2 ) ( H − Z SALT ) ⎜ ∂xi ⎝ ⎞ ⎟ (i=1,2) ⎟ ⎠ (49) The freshwater head H is given by the Ghyben-Herzberg relation Eq.(2.68): H = (1 + ε ) Z SEA − ε ΔZ − ε Z SALT Substituting H in Eq.(49), and using the Φ-transform, we obtain: ∂⎛ ∂φ ⎞ ⎟ = 0 (i=1,2) ⎜ − K ( x1, x2 ) ⎟ ⎜ ∂xi ⎝ ∂xi...
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