Statistical significance - meaningful or not

Statistical tests with large sample sizes can have large power. Power is the ability to detect an effect. Detection is indicated by a result which is statistically significant. A test with large power will detect a very small effect. This very small effect may not be meaningful in the context of the analysis being conducted. The courts have the perception that for an effect to be meaningful it is necessary for the effect to be statistically significant. However, statistical significance is not a sufficient condition for an effect to be meaningful. This can lead to a difficulty where testimony of no meaningful effect is interpreted by counsel as one of no statistically significant effect. Should the difference between a meaningful effect and a statistically significant effect be explained in reports and if so, how? Some possible answers are proposed.