Technical note: The horizontal scale dependence of the cloud overlap parameter α

The cloud overlap parameter α relates the combined cloud fraction between two altitude levels in a grid box to the cloud fraction as derived under the maximum and random overlap assumptions. In a number of published studies in this and other journals, it is found that α tends to increase with an increasing scale. In this Technical Note, we investigate this analytically by considering what happens to α when two grid boxes are merged to give a grid box with twice the area. Assuming that α depends only on scale, then between any two fixed altitudes, there will be a linear relationship between the values of α on the two scales. We illustrate this by finding the relationship when cloud cover fractions are assumed to be uniformly distributed, but with varying degrees of horizontal and vertical correlation. Based on this, we conclude that α increases with scale if its value is less than the vertical correlation coefficient in cloud fraction between the two altitude levels. This occurs when the clouds are deeper than would be expected at random (i.e. for exponentially distributed cloud depths).