Concerning the term (ρ(u⨂u) + pI): If you are computing momentum for checking conservation of momentum and the wind is blowing North across the equator then there is thereby a quantity of Northward momentum moving North. If it is blowing South then there is Southward moving momentum moving south. Between the two cases we flipped the sign twice and ended with the same change in how much Northward momentum has moved North of the equator. Flipping the sign of u above does not change u⨂u.

If the wind is blowing North-west then there is some Westward momentum moving North and also some Northward momentum moving West. That is why we need the off diagonal terms.

If everything is standing still and the pressure is positive, then there is also Northward momentum moving North across the equator. Ditto South. That is why pI is added to the above term.

In tensor notation u⨂u would be uⁱu^j. Note its contribution to the Stress–energy tensor which is prominent in general relativity where T^μν = ρu^μu^ν + pg^μν.

I had wondered where this formalism originated.