Statement 1: The ocean warming increases the frequency of tropical cyclone winds and rainfall, as well as an increase in extreme waves.

Ocean warming refers to the increase in the temperature of the Earth's oceans. As the oceans warm, it has several effects on the climate and weather patterns.

Effect on Tropical Cyclones:
Tropical cyclones, also known as hurricanes or typhoons, are large rotating storm systems that form over warm ocean waters. The energy for these storms comes from the heat and moisture in the ocean. When the ocean temperatures rise, there is more energy available for tropical cyclones to form and intensify. Therefore, ocean warming can increase the frequency of tropical cyclones.

Additionally, warmer oceans also lead to increased evaporation, which in turn increases the moisture content in the atmosphere. This increased moisture can result in heavier rainfall associated with tropical cyclones, leading to more intense and frequent rainfall events.

Furthermore, the warmer ocean temperatures can also contribute to the formation of more extreme waves during tropical cyclones. As the storms intensify, the stronger winds and increased energy can generate larger and more powerful waves, posing a greater risk to coastal areas.

Therefore, statement 1 is correct.

Statement 2: The productivity of the Oceans increases when the carbon emissions from the atmosphere are sunk by it.

The productivity of the oceans refers to the ability of marine ecosystems to support and sustain life. It is influenced by various factors, including nutrient availability, sunlight, and carbon dioxide (CO2) concentrations.

When carbon emissions, such as CO2, are absorbed by the oceans, a process called oceanic carbon sequestration, it can have both positive and negative effects on ocean productivity.

Positive Effects:
Some marine organisms, particularly phytoplankton, use CO2 for photosynthesis, the process by which they convert sunlight into energy. Increased CO2 availability can enhance the growth and productivity of these organisms, leading to increased primary production in the oceans. This can have cascading effects on the entire marine food web, benefiting higher trophic levels such as fish and marine mammals.

Negative Effects:
However, excessive absorption of CO2 by the oceans can also lead to ocean acidification. When CO2 dissolves in seawater, it reacts with water molecules to form carbonic acid, resulting in a decrease in pH. This acidification can have detrimental effects on marine organisms, particularly those that rely on calcium carbonate to build their shells or skeletons, such as coral reefs and shellfish.

Therefore, while the absorption of carbon emissions by the oceans can enhance certain aspects of ocean productivity, it also has negative consequences due to ocean acidification. Hence, statement 2 is incorrect.

In conclusion, statement 1 is correct as ocean warming can increase the frequency of tropical cyclone winds, rainfall, and extreme waves. However, statement 2 is incorrect as the absorption of carbon emissions by the oceans has both positive and negative effects on ocean productivity.