Advanced Ocean Modelling: Using Open-Source Software

5.8 Exercise 24: The Abyssal Circulation 153Fig. 5.23 Illustration showing the grid configuration and float coordinate for the x-direction.Circles group grid points belonging to a certain grid indexdirection is slightly different from that of horizontal directions. Figure 5.23 showsthe grid configuration together with the float’s x-coordinate. The first step of theprocedure is to locate the grid cell containing a float with reference to the closestscalar grid point. This can be done with:( x∗)kpos = INTΔx + 0.5 + 1where x ∗ is the actual x-coordinate of the float, and “INT” truncates real numbersinto full (integer) numbers. The simplest scheme would then be to average surroundingvelocity grid points onto the scalar grid point, yielding a velocity of:〈u〉 = 0.5(u w + u e )where u w = u(kpos − 1) and u e = u(kpos). This velocity can be used to move thefloat around on the basis of the simple displacement equation:dx ∗dt= 〈u〉This scheme has been used in some of the previous exercises, noting that strandingof floats in dry grid cells can be avoided with use of additional conditions. Animproved scheme interpolates (rather than averages) the velocity from surroundinggrid points onto the location of the float. In the x-direction, for instance, this interpolationleads to:u(x ∗ ) = u w + δx ∗ u e − u wΔx(5.18)where δx ∗ is the distance from the float to the western cell face (Fig. 5.24), givenby:δx ∗ = x ∗ − Δx (kpos − 1) + 0.5Δx = x ∗ + (1.5 − kpos)Δx