Statement 1: The ocean warming increases the frequency of tropical cyclone winds and rainfall.
Ocean warming refers to the increase in the temperature of the Earth's oceans. It is a consequence of climate change and is primarily caused by the accumulation of greenhouse gases in the atmosphere. The warming of the oceans has several effects, one of which is the impact on tropical cyclones.

Tropical cyclones, also known as hurricanes or typhoons depending on the region, are intense storms that form over warm ocean waters. They are fueled by the heat and moisture from the ocean's surface. As the ocean warms, it provides more energy and moisture to fuel the formation and intensification of tropical cyclones. This results in an increase in the frequency, intensity, and duration of these storms.

The warmer ocean waters also lead to increased evaporation, which in turn increases the amount of moisture in the atmosphere. This enhanced moisture content can result in heavier rainfall associated with tropical cyclones. Warmer oceans can also lead to an expansion of the area over which tropical cyclones can form, potentially increasing their geographic range.

Therefore, statement 1 is correct as ocean warming indeed increases the frequency of tropical cyclone winds and rainfall.

Statement 2: The productivity of the Oceans increases when the carbon emissions from the atmosphere are sunk by it.
Ocean productivity refers to the ability of the oceans to support and sustain marine life, including phytoplankton, which are the primary producers at the base of the marine food chain. Phytoplankton convert carbon dioxide and sunlight into organic matter through photosynthesis, thereby playing a crucial role in carbon cycling and the global carbon cycle.

When carbon emissions, primarily in the form of carbon dioxide, are absorbed by the oceans, they can have both positive and negative effects on ocean productivity. On one hand, increased carbon dioxide levels can stimulate the growth of phytoplankton, leading to enhanced productivity. This is because carbon dioxide is a key nutrient for phytoplankton, and higher concentrations can promote their growth.

On the other hand, the absorption of carbon dioxide by the oceans also leads to ocean acidification. As carbon dioxide dissolves in seawater, it reacts with water molecules to form carbonic acid, lowering the pH of the oceans. This can have detrimental effects on marine life, including phytoplankton, as it affects their ability to form shells and skeletons.

While increased carbon dioxide levels may initially enhance ocean productivity, the long-term implications of ocean acidification and its potential negative effects on marine ecosystems need to be considered. Therefore, statement 2 is not entirely correct as the increase in carbon emissions absorbed by the oceans can have both positive and negative impacts on ocean productivity.

In conclusion, statement 1 is correct as ocean warming increases the frequency of tropical cyclone winds and rainfall. However, statement 2 is not entirely correct as the impact of carbon emissions on ocean productivity is complex and depends on various factors, including ocean acidification.