A Cold Core High is one in which the temperatures on the horizontal plane decrease toward the center.
The Pressure deceases rapidly with height.
Thicknesses are relatively low.
In the vertical, cold core highs slope toward the colder air aloft.
Because these highs are quiet shallow, it is common for an upper level low to exist above a cold core high.
If precipitation is caused by the upper level low, depending in the height of the freezing level associated with the high on the surface; a cold rain or snow can be expected.