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The atmosphere cannot not be seen isolated but is globally connected with the hydrosphere. Carbonate deposition removes carbon from this interconnected atmospheric hydrospheric system. To which extent equilibration processes transport carbon from the atmosphere to the hydrosphere or vice versa, is an interesting, complicated story, but has nothing to do with the importance of limestone deposition by coral and platform growth as a prominent mechanism which continuously removes carbon from the atmospheric hydrospheric system (when seeing this as a whole). Cases where the limestone lithosphere acts as a source for the atmospheric-hydrospheric system are much rarer (karstification, subduction, deep-water dissolution) and the enormous quantity of fossil carbonate platforms still preserved (e.g., Dolomites, the northern Calcareous Alps, the European Jurassic, the Rockies, the Devonian Classic Face of Australia, to name but a few) highlights the importance of this huge drawdown-process.
Some more details to these thoughts:
This means that the tropical/subtropical waters are constantly depleting from carbon dioxide as long as reefs and carbonate platforms grow. In a short-term view the atmosphere in the tropics actually becomes richer in carbon dioxide (these 0.6 mol. mentioned above). However, the same process also provides that at the same time (and in the long-term), the hydrosphere as a whole remains open as a carbon sink and hence maintains its abilities to buffer the climate. If reefs and carbonate platforms would not exist, an increase in carbon dioxide by other mechanisms (volcanism, sea level rise and associated warming of oceans due to enrichment of water vapor in the atmosphere, man-made increase by industry) might initially be counteracted by enhanced drawdown of carbon dioxide into the hydrosphere (due to increase in atmospheric pCO2). However, this physical sink would not remain open for long, because the upper hydrosphere would sooner or later become filled up with carbon dioxide, unless black shale events (biological pump) remove hydrospheric carbon in a large, catastrophic scale. Once the oceans capacity for drawing down atmospheric carbon is exhausted, the world could suddenly heat up at a rapid pace.
This convinces me that the growth of carbonate platforms and reefs (possibly besides the deposition of plantonic calcareous ooze) is an important continuous climatic buffer system which keeps the ocean reservoir open for necessary drawdown of carbondioxide from the atmosphere into the hydrosphere. In a very simplified way, for each calcium carbonate molecule precipitated, the atmosphere, at a global scale, might be cleared from 0,8 mol (i.e. 1,4 stored as limestone - 0,6 redistributed into the atmosphere) of carbon dioxide.
In geological terms, the start of a sea level rise might cause an initial (local?, regional?, global?) increase of atmospheric carbon dioxide by the onset of reef and platform development. Possibly, reefs even 'try' to create their preferred climate. This might have been an important process after the last ice age. However, sea level rise also largely increases accommodation space for reefs and carbonate platforms, so at in the medium and long term, a lot of carbon dioxide is drawn down as limestone to the lithosphere and the globe is stopped from boiling over, as long as carbonates are deposited. I believe that reefs and carbonate platforms (despite the ongoing discussion in as much living reefs are LOCAL sinks or sources) are important BULK sinks (just see all these shallow water limestones in Earth history!) which help keeping the ocean reservoirs open for eventual drawdown of atmospheric carbon dioxide and hence are one of the most important climatic buffer systems we have. We live in a world of rising sea level which should actually help coral reefs do their climate-buffering job; however, if we keep on destroying them, rising sea level will have positive feedback on the natural and man-made greenhouse effect and noone will help counteract this.