The polar vortex is the pattern of swirling winds that trap the frigid air over the North and South Poles. A strong vortex keeps the cold air temperatures in the Arctic but a weakened one, allows the cold air to go to other regions. Weather patterns bring those arctic winds to lower regions or sometimes a section breaks off to travel to the lower regions, causing abnormally cold temperatures.
For northern areas well below the arctic circle, the vortex can bring actual temperatures to 36-below zero Fahrenheit or lower, with wind chills as low as 60-below zero. The vortex can bring below freezing temperatures to areas with subtropical climates, and the clash of frigid air with warmer air masses often creates winter storms.
Usually two or three times each year the cold arctic air affects small areas of Europe, North America or China but large masses of cold air covering half a continent or more are unusual and generally happen once in a decade.
The extreme cold of the polar vortex is a problem for a world where chaotic social breakdown has not occurred but add limited resources in a post apocalyptic world, you need your own plan for surviving the extreme cold for several days without relief. Both situations carry the danger at a minimum of frostbite and ultimately death due to such things as hypothermia, dehydration or carbon monoxide poisoning.
The ideal situation is when you can stay inside away from the extreme temperatures, for however many days the polar vortex is affecting your weather. If you have already reached your permanent bug out shelter, you should have already arranged for a heat source for cooking. A wood stove is ideal if you have plenty of hardwood to burn as it burns with minimal smoke and as you not only have it for heating but you can cook on it.
If you do not have a permanent structure or are faced with poorly burning fuel sources, Fired Up or EasyPrep EasyFire burn hot in any conditions and usually create enough heat to get the most reluctant fuel to burn. If you have access to propane, Emergency Essentials sells the Mr. Heater Little Buddy Propane Heater that heats 100 square feet, which also has a tip over, and low oxygen shut offs.
Your clothing during a deep freeze should keep your body dry and warm. It is better to wear multiple levels of lightweight clothes as the air space between layers provides insulation. The outer layer should be water repellent and mittens keep your hands warmer than gloves. You should have a bandana to cover your face and a poncho with a hood can double as a temporary shelter.
If you begin to sweat, you can remove your hat or hood as you lose heat quickly from exposing your head. It is critical to have a way to cool down without sweating, as the dampness lowers your body temperature and your damp clothing will not keep you warm.
Down sleeping bags are a valuable resource to keep you warm provided you can keep it dry. The ReadyStore sells the Kelty MIstral 0 Degree Sleeping Bag and although a polar vortex produces temperature below 0 degrees, if you are in a sheltered area and have a waterproof barrier on which to lay your bag, you should be fine.
Water is crucial to survival in extreme cold of a polar vortex and to keep it from freezing, you can keep a partially filled canteen or water bottle close to your body. The partially filled container allows the water to slosh around to prevent freezing. Be sure to melt the ice or snow before consuming as eating snow or ice will lower your body temperature.
If you have animals, you need to protect them from the extreme cold as well and be sure that they have water. You need to protect large animals by sheltering them in protected areas (wooden shelters can easily be bought and erected) - and you might have to bring smaller animals into your own living space.
When you have access to weather forecasts, you have some advance warning of an impending polar vortex coming to your area. After the apocalypse, it is likely you will realize the vortex has arrived when the extreme cold temperatures arrive and stay for a number of days. Prepping for the potential of a cold spell, especially in a normally subtropical area, will prepare you and your family in the event a blanket of artic air settles over your home or bug out location.