If I'm alone in a survival situation and I have to cross a very cold river that's still flowing, how should I accomplish this safely?
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First off, weigh up whether it's worth crossing said river. I know this question is about if you "have" to cross it, but bear in mind that falling in is a real danger and if you do, hypothermia can onset very quickly and be deadly. It depends on the situation - if we're talking about a shallow, wade-able body of water that's not much more than a stream I'd be much less hesitant to give it the go-ahead than if we're talking about something 20ft deep and triple that in width! Sometimes your gut reaction can be good here, if you think it looks highly risky and you might not make it successfully, don't attempt it! And if you can't swim strongly and you're just relying on trying to balance on top of a few logs without falling in, don't even think about it.
If you decide you to cross it:
Look for a bridge - No, seriously. Some people really can miss the obvious.
Prepare - Be sure to prepare thoroughly first. Re-pack your rucksack so all the vitals you'll need are kept as dry as they can be, and accessible (you'll want things like chemical heating packs, stoves, fire fuel and dry, warm clothes to all be very handy when you reach the other side.)
Find a good point to cross - As a general rule avoid deep, fast flowing water. It's better to find somewhere that might be a bit wider, but shallower and easier overall. If the river narrows at a particular point don't be fooled, the water has to have gone somewhere and unless it's branched off to another stream, it's because it's gone down - i.e. it's deeper! Wider patches are generally shallower, slower flowing and therefore a better choice even if it seems like you've got more of a task ahead of you to cross. There are also other telltale signs you can look out for - reeds for instance tend to thrive better in shallower water.
Find a good time to cross - Leave yourself plenty of time before it gets dark - it may take longer than you think, and you don't want a lack of light providing an unnecessary complication. I'd advise crossing late morning if you can at all - that way you ensure you get plenty of light which crosses one of the list of things to worry about (and it's an easy one to cross off) and you also ensure that you get to the other side when the day, on average, should be warmest. If anyone's fallen in or got wet, you'll be glad of every little bit of heat you can get to warm you back up!
Stay out of the water if you can - See if there are any materials around which you could use to construct a raft or similar. The best thing to avoid getting frozen solid is to stay out of the water in the first place. Equally, try and see if you could build a bridge with logs or any other suitable material.
Rope yourself to a secure point on the bank..? - This is a contentious one. I initially recommended it because the plus side is that you won't get swept away in a current, the minus side is that it could get caught or pull you under. Judge the situation accordingly, but if you do use it, use a quick release karabiner to attach it to yourself (so you can remove it quickly and easily if you have to.)
As soon as you're across, get the heating supplies out - It may well be that adrenaline is keeping you going and you're colder than you think. Equally even if you feel ok, the next person of your team to cross may not be. Make the most of this by getting a fire going if you can, get warm clothes out, and keep an eye out for each other to make sure everyone's ok. Have some warm soup, run around - anything to get you warmed up again!
OK, this isn't a hypothetical question. You will have to ford numerous rivers fed directly from glaciers if you hike in the Swedish mountains. These are extremely cold, very rapid streams with rocks everywhere. Some basic advice is:
If the water is deeper and less rapid so you can / have to swim:
Synthetic garments are a saviour since you can get warm fast even if they get wet.
There is one more important technique you can use that I was taught in New Zealand, where you have to cross rivers all the time.
If you have a group of people (at least 3), you can greatly enhance safety by forming a chain in the following way:
A note on shoes: I was taught to leave them on. Riverbeds are slippery and can have sharp rocks. Your shoes protect you. And once out of the water, the small amount of water in your shoes warms up quickly and forms an insulating layer similar to a wetsuit. Or so they say :) I have done the above in relatively cold water (no ice floating on it, but it was of glacial origin) and didn't suffer frostbite.
EDIT: I forgot to say that, of course, the first and most important skill when it comes to river crossing is the skill to make the right decision. Rivers can swell quite quickly in rain, but once rain is over, those rivers that flood fast also drop fast. I was held up for a night by a waist-deep side-creek that flooded due to rain, and in the next morning it was only ankle deep.
If you are unsure whether you have a good reason to attempt a crossing or not, imagine that your crossing led to a fatality and you have to explain yourself to the grieving family. If you are in a survival situation, you might have a legitimate reason, but any sort of man-made deadline becomes irrelevant. (Wanting to be back in time for Super Bowl, or even wanting to be back in time for work, is not as important as not drowning).
The first thing is to think very hard about whether you really need to cross it, in most circumstances the answer will be NO so there needs to be a strong reason to do so.
DO NOT cross a river just because you are travelling in a straight line and the river is in the way. A river is a good source of water and, food and materials and they are an excellent guide to navigation as they will eventually lead you somewhere useful. They are also an obvious marker for anybody looking for you. There is also a good chance that if you follow a river you will eventually find a road or path crossing it.
If you have a tarp, bivi bag or heavy duty bin bag with you then use that to wrap your pack as well as you can this will both keep your kit dry and help your pack to float.
If you have water bottles empty and re close them, they will be useful flotation devices also inflate any plastic bags you have. Strap any logs etc you can find to your pack to help it float.
IF you have rope or cord attach it you an anchor on the 'home' side of the river and your pack this gives you an anchor if you slip and and you can retrieve your pack if you fall and need to drop it.