About a third of the CO2 released into the atmosphere by humans each year is absorbed by the oceans. Normally this would be captured by diatoms and shellfish for making their shells. When they die, their shells become chalk, then limestone, thereby locking up CO2.
However, CO2 is building up in the seawater faster than the shellfish can remove it so the ocean are becoming acidic, and dissolving the shells, thereby preventing one of the main natural CCS processes. The pH has recently decreased 0.1 units. This means it is becoming more acid.
Let's look at the various proposals for storing CO2 in the ocean.
Adding iron to the oceans
Plankton that generate calcium or silicon carbonate skeletons, such as diatoms, coccolithophores and foraminifera, account for most direct carbon sequestration. When these organisms die, their carbonate skeletons sink relatively quickly and form a major component of the carbon-rich deep sea precipitation known as marine snow. Marine snow also includes fish fecal pellets and other organic detritus, and can be seen steadily falling thousands of meters below active plankton blooms.
Plankton growth is limited by a lack of iron and silicon in seawater. The idea is to fertilise the ocean and increase growth. An analysis of plankton show a molecular ratio of: 106 C: 16 N: 1 P: .001 Fe. So each atom of iron can fix 106,000 atoms of carbon. On a mass basis, each kilogram of iron can fix 83,000 kg of carbon dioxide. The 2004 EIFEX experiment reported a carbon dioxide to iron export ratio of nearly 3000 to 1.
However some researchers have found that the skeletons are dissolved and CO2 returns eventually to the ocean and atmosphere. There is optimism and pessimism on the topic, and much more work needs to be done.
The Azolla event occurred in the middle Eocene epoch, around 49 million years ago, when blooms of the freshwater fern Azolla are thought to have happened in the Arctic Ocean. As they sank to the stagnant sea floor, they were incorporated into the sediment; the resulting draw down of carbon dioxide has been speculated to have helped transform the planet from a "greenhouse Earth" state, hot enough for turtles and palm trees to prosper at the poles, to the icehouse Earth it has been since.
Since 1993, these 12 small-scale open ocean experiments (red dots) have shown that iron additions do indeed result in phytoplankton blooms, thereby drawing carbon dioxide out of the atmosphere and into the ocean. Source
Injecting at depth
If a pipeline from the shore or a ship injects CO2 at depth then it will dissolve but take hundreds of years to reach the surface.
Ocean bottom CO2 lakes
At greater depth CO2 is denser than water and will sink into enclosed basins forming a CO2 lake.
CO2 could be injected deep into ocean sediments forming a hybrid between ocean and geological storage.
One idea is to inject dry ice torpedoes that will hit the ocean floor deep enough to bury themselves.
Clathrate CO2-Water ice
In cold water CO2 can form a crystal with ice called a clathrate. Methane does this too, and there are proposals to push the methane out by displacing it with CO2.
Adding Lime to sea water
The idea is to convert limestone into lime and add it to the sea. This will decrease the acidity of the water and enable the it to absorb more CO2 from the air. The claim is that when you put lime into seawater it absorbs almost twice as much carbon dioxide as is produced by the breaking down of the limestone in the first place.
It would be a huge effort way out of all proportion to the result. You would need 83 T of limestone to be mined and converted to lime to collect 12 T of CO2. Then there is the energy required ......!
CaCO3 ---> CaO + CO2
The scale of the mining and processing of limestone would be huge.