The short answer is that if 10,000,000 L/s of well mixed water passed over the wire, it’s temperature would increase by 0.75 degrees Celsius.
The long answer is that this is not quite the right way to view the question. There will be a temperature gradient around the cable - a zone where water is heated. The question is, how hot is this zone and how large is it. The cable will be buried, which makes the zone larger and less hot. However, this all depends on the type of soil in the ocean floor.
One thermal study estimated that, with typical operating temperatures and burying depths, the sea floor temperature (of the soil, not the water) would heat by 10 to 18 degrees Celsius. This is enough to potentially interfere with ocean life, according to the authors of the study. Because of limits in the operating temperature of cables, a project like this would make a larger zone like this, but the maximum temperature would stay the same.
It’s important to note that we already mess up aquatic life by discharging thermal energy into bodies of water. High temperature water discharge from power plants can make rivers more friendly towards invasive species and shift the balance of aquatic ecosystems. This project might be a net neutral for aquatic life by reducing high temperature discharge into rivers while creating hot spots in the ocean.
What about the magnetic field? I think distance is required between the two sides of the cable to prevent arc, which also means there will be a magnetic field.
Heat becomes an issue before magnets do. Magnets follow an inverse power law but thermal conduction regimes produce linear temperature gradients. The spacing required for cooling is greater than the spacing required for EM insulation except for very low power flows.
The long answer is that this is not quite the right way to view the question. There will be a temperature gradient around the cable - a zone where water is heated. The question is, how hot is this zone and how large is it. The cable will be buried, which makes the zone larger and less hot. However, this all depends on the type of soil in the ocean floor.
One thermal study estimated that, with typical operating temperatures and burying depths, the sea floor temperature (of the soil, not the water) would heat by 10 to 18 degrees Celsius. This is enough to potentially interfere with ocean life, according to the authors of the study. Because of limits in the operating temperature of cables, a project like this would make a larger zone like this, but the maximum temperature would stay the same.
It’s important to note that we already mess up aquatic life by discharging thermal energy into bodies of water. High temperature water discharge from power plants can make rivers more friendly towards invasive species and shift the balance of aquatic ecosystems. This project might be a net neutral for aquatic life by reducing high temperature discharge into rivers while creating hot spots in the ocean.
Thermal study: https://doi.org/10.1093/gji/ggw195