A Hypothesis about Global Drying

I've been trying to understand how the PDSI seems to suggest that the land is on the whole drying out as a result of global warming (though there are regional exceptions), while in the paleo-climate, warmer periods were generally wetter, and colder periods drier (and in more modern times, for example, volcanically caused reductions in sunlight resulted in global droughts).  There would appear to be a contradiction: why won't a globally warmed world be wetter, not drier?

Here is a speculation about it.  Bear in mind the usual caveat that I'm not a climate scientist - just a non-specialist (albeit with a Physics PhD) trying to understand the implications for humanity.

The two temperature series shown above represent the global land temperature (on the left) and the global ocean temperature (on the right) - the figures come from the Wiki article on the instrumental temperature record.  To crudely summarize, the ocean has warmed about 0.7^oC since the pre-industrial era, while the land has warmed about 1.3^oC.  Thus the land has warmed 0.6^oC more than the ocean over the twentieth century.  The reason for this is that as we have thickened the CO₂ blanket over the planet, and thus warmed it, the oceans and land have responded differently.  The ocean, because it's a liquid which will allow convection, can transport heat into its depths much better than the land in which the heat must very slowly diffuse down through the layers of rock and soil.  Thus the land has warmed more, but only to a shallow depth, while the ocean has warmed less but deeper.  Since this fundamental issue will still apply as the planet gets further and further out of thermal equilibrium over the twenty-first century, we would expect the temperature gap between land and ocean to widen further and further until it is several degrees centigrade.

The global water cycle basically consists of water evaporating from the ocean, being transported to the land, falling as rain or dew, and then running off into rivers (I'm oversimplifying but that's the essence of it).  So the ocean is the source of the moisture in the atmosphere and the land is a moisture sink.

The amount of moisture that air can hold is a function of temperature and is governed by a thermodynamic equation known as the Clausius Clapeyron equation.  A decent approximation to this at normal temperatures and pressures is that the water holding capacity of the air increases by 7% for every  1^oC increase in temperature.  If we think the amount of water in the atmosphere is basically set by the global ocean temperature, but the land is warming more than the ocean, then we would expect relative humidity to drop (on average) near the land surface - there's more moisture in the air overall (as the oceans are somewhat warmer) but not enough to keep the land as wet as it was before the temperature increase since the land has warmed more than the ocean.  To a very rough approximation we might expect average relative humidity near the land to have dropped by about 7%x0.6 = 4% (percentage points) - the 0.6 being the average difference between the land temperature increase and ocean temperature increase.  This is not trivial and might be expected to result in more widespread drought.  If, late in the twenty-first century, the difference opened up to 3^oC then we'd expect average relative humidities near the land surface to drop by around 20% (again, percentage points - and this is 3x7 rounded since I don't pretend this is a precise calculation).  That is an enormous change that we'd expect to make land dramatically drier (on average).  That is broadly consistent with what the PDSI analysis of climate models seems to be saying: