Gravity was imposed on a single-species fluid using method (2). The initial density of the fluid was and the fluid was evolved for 10,000 time-steps with = 200.0 to ensure that the fluid reaches its equilibrium state. Figure 5-7
Figure 5-7: Density as a function of depth for a fluid with
, and g = 0.001 after 10,000 time-steps.
shows the density variation with depth when g=0.001. The density gradient is slowly increasing with depth. The shape of the curve shown in figure 5-7 is consistent with the expected exponential variation of the density with depth. The variation of the density gradient with depth is a function of the gravitational strength, when the gravitational strength is less than 0.005 the density gradient can be considered as being constant over the range of depths we are considering. Figure 5-8 shows the density variation with depth for a range of gravitational strengths and the best-fit straight line through the points. Even for the largest value of g = 0.0005 the linear approximation appears to fit well. The linear density gradient is shown in figure 5-9
Figure 5-9: The linear density gradient as a function of the
gravitational strength g for an initial density .
for a range of values of g less than 0.0005. Clearly the density gradient is proportional to the gravitational strength g. It is important to remember that these results show that the density gradient is approximately linear over the depths we are considering. The actual form of the density gradient is an exponential, as suggested by figure 5-7. This can be seen in figure 5-10
Figure 5-10: The linear density gradient as a function of the
gravitational strength g for an initial density .
which shown the linear density gradients plotted against g, as in figure
5-9, but for an initial uniform density of 4.0. The density gradients
shown in figure 5-10 are four times the gradients shown in figure
5-9.
We have seen that, provided the gravitational strength and the fluid depth are small, the density gradient is approximately linear and is proportional to g. For larger values of g, where the gradient can no-longer be considered linear, we would still expect the density gradient to be proportional to the gravitational strength.