The core of the jet stream is found in the warm air, above the level of the cold air tropopause but below the warm air tropopause. If the temperature of the cold air tropopause is -40°C, the temperature in the warm air at that level would be -34°C. Above the tropopause in the cold air the temperature will be constant at -40°C but in the warm air the temperature will continue to decrease with altitude, at about 2°C/1,000ft, up to the warm air tropopause ten or twelve thousand feet higher. This means that the warm air tropopause temperature would be about -54°C.

 

Therefore, if you cross the jet stream from the cold sector to the warm sector below the core of the jet the temperature will increase (from -40°C to -34°C) but if you cross the jet stream from the cold sector to the warm sector above the core the temperature will decrease (from -40°C to -54°C).  The jet core (highest wind speed) is found at the level where there is maximum pressure gradient and zero temperature gradient (-40°C to -40°C).  If you stand with your back to the wind and have cold air on your left you are below the core of the jet and wind speed will increase as you climb.  If you stand with your back to the wind and colder air is on your right you are above the core of the jet and wind speed will decrease as you climb.