Bjerknes was the son of Vilhelm Bjerknes (1862–1951), a theoretical physicist who developed some of the first mathematical models of atmospheric and oceanic motions. Jacob followed the example of his father in studying meteorology. Born in the Swedish capital, Stockholm, he was educated at the University of Oslo, where he obtained his PhD in 1924, and worked at the Geophysical Institute at Bergen with his father from 1917, remaining there when Vilhelm moved to Oslo in 1926.

During World War I Bjerknes worked with his father in establishing a series of weather observation stations throughout Norway. From the data collected, and working with other notable meteorologists, including Tor Bergeron, they developed their theory of polar fronts, also known as the Bergen theory or the frontal theory. They had established that the atmosphere is composed of distinct air masses possessing different characteristics and applied the term ‘front’ to the boundary between two air masses. The polar front theory showed how cyclones (low-pressure centers) originated from atmospheric fronts over the Atlantic Ocean where a warm air mass met a cold air mass.

In 1939 Bjerknes moved to America and, unable to return to occupied Norway, became professor of meteorology at the University of California where he continued to study atmospheric circulation. In 1952 he became one of the first to use space techniques for meteorological research when he used photographs of cloud cover taken by research rockets for weather analysis.