5.8 Exercise 24: The Abyssal Circulation
153
Fig. 5.23 Illustration showing the grid configuration and float coordinate for the x-direction.
Circles group grid points belonging to a certain grid index
direction is slightly different from that of horizontal directions. Figure 5.23 shows
the grid configuration together with the float’s x-coordinate. The first step of the
procedure is to locate the grid cell containing a float with reference to the closest
scalar grid point. This can be done with:
kpos = INT
x
∗
Δx
+ 0.5
+ 1
where x
∗ is the actual x-coordinate of the float, and “INT” truncates real numbers
into full (integer) numbers. The simplest scheme would then be to average surrounding velocity 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 the
float 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 stranding of floats in dry grid cells can be avoided with use of additional conditions. An
improved scheme interpolates (rather than averages) the velocity from surrounding
grid points onto the location of the float. In the x-direction, for instance, this interpolation leads to:
u(x
∗ ) = u w + δx
∗ u e − u w
Δx
(5.18)
where δx
∗ is the distance from the float to the western cell face (Fig. 5.24), given
by:
δx
∗
= x
∗
− Δx (kpos − 1) + 0.5Δx = x
∗
+ (1.5 − kpos)Δx
153
Fig. 5.23 Illustration showing the grid configuration and float coordinate for the x-direction.
Circles group grid points belonging to a certain grid index
direction is slightly different from that of horizontal directions. Figure 5.23 shows
the grid configuration together with the float’s x-coordinate. The first step of the
procedure is to locate the grid cell containing a float with reference to the closest
scalar grid point. This can be done with:
kpos = INT
x
∗
Δx
+ 0.5
+ 1
where x
∗ is the actual x-coordinate of the float, and “INT” truncates real numbers
into full (integer) numbers. The simplest scheme would then be to average surrounding velocity 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 the
float 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 stranding of floats in dry grid cells can be avoided with use of additional conditions. An
improved scheme interpolates (rather than averages) the velocity from surrounding
grid points onto the location of the float. In the x-direction, for instance, this interpolation leads to:
u(x
∗ ) = u w + δx
∗ u e − u w
Δx
(5.18)
where δx
∗ is the distance from the float to the western cell face (Fig. 5.24), given
by:
δx
∗
= x
∗
− Δx (kpos − 1) + 0.5Δx = x
∗
+ (1.5 − kpos)Δx