- Shouldn't aircraft be able to withstand winds of up to 100 mph? When flying the wind is at a much higher speed than that.
- Shouldn't they be able to take-off through fog using instruments alone?
- Shouldn't they be able to take-off on a slippery runway, since the wheels are not powered.
Adding to what the others have mentioned, bad weather is often a combination of low visibility, low ceilings, precipitation, high winds and lightning.
Visibility and ceilings
As mentioned in @fooot's answer, ILS is available to land when in these conditions. Every airliner and crew (and just about every airport they'd go into) is trained and equipped for ILS operations. A standard ILS approach is good in weather down to 200 ft ceilings and 1800 or 2400 ft visibility along the runway (RVR). Further airplane and airport equipment and aircrew training can allow ILS category II and III operations, which reduce these numbers (cat II is 100 ft vertical visibility and RVR 1200, cat III goes down to essentially no visibility).
To takeoff, a 2 engined airliner needs 5000 ft RVR (1 mile vis) normally. Airport lighting and airline opspecs can reduce this as low as RVR 300.
High winds are fine, but can make the landing more challenging. When the winds are strong and particularly gusty, a lower flap setting can be used to land and a faster approach speed flown. This will, however, necessitate a long landing rollout. Exceptional crosswinds can also present problems aligning the aircraft with the runway, and if you can't line up, you can't land (note that some larger airplanes can land in a crab and do not need to align with the runway until the main gear are on the ground).
You mention the wheels are not powered, and this is true. However, the friction between the runway and the contact patch on the tire is the only thing keeping the airplane on the runway in strong crosswinds, so you need friction to land (or takeoff) if there is a crosswind component.
Anything reducing friction is problematic (rain, snow, ice, rubber, etc), With high water you risk hydroplaning and this (or snow/ice) can lead to departures from the runway sides or an overrun if braking action is insufficient due to lack of friction. Ice needs to be chemically treated and snow must either be treated or plowed at regular intervals (while it is snowing). Ground handlers at the airport must also have deice fluid and the equipment to apply it to aircraft available for operations in snow.
When lightning is in the vicinity, ground crews go inside as it is a deadly hazard to be on the ramp in those conditions.
All of these things contribute to lowered arrival rates into the airport and as conditions deteriorate you reach a point where you are spending more time dealing with keeping the airport open than departing airplanes and it makes sense to close the airport temporarily.
The problem with wind is that it can be from any direction, not necessarily the direction of the runway. When the wind is coming from the side instead of straight ahead, it is called a crosswind. This means that landing aircraft have to point into the wind to remain lined up with the runway. The slower they are flying, the further to the side they must point the nose. Airliners are designed to land in these conditions, but it requires more skill to do it well. The control inputs required to compensate for a crosswind affect aircraft on takeoff as well, as then must work harder to stay lined up with the runway and stay level through lifting off. High winds tend to come along with gusts, which makes it even harder to control an aircraft. Aircraft during flight are flying faster, so changes in wind are less problematic. Once they are flying slower, the wind is much larger compared to the aircraft's speed, and control is much more critical during landing. Aircraft have crosswind limits, and even below those limits, aircraft may need to go around from a landing attempt, which introduces delays.
Another issue is wind shear, which is when the direction and/or speed of the wind quickly changes. A change of 40 knots at higher speeds is turbulence. During landing or takeoff, it can cause the aircraft to stall and crash.
Aircraft can certainly use systems like ILS that allow them to take off or land in low visibility. However, there are many limitations with these systems. The crew must be trained to use them, the aircraft must be equipped to use them, and the airport must be equipped with them. However, when the pilots are relying on instruments, they no longer are able to see hazards like usual. Because of this, separation requirements are inreased in these conditions, which means an airport can't handle as much traffic.
It's true that aircraft wheels are not directly powered. However, traction is still very important during braking, since aircraft can hydroplane just like a car can. This means aircraft need more distance to stop, and can slide off the runway if they lose traction. Frozen water in the form of ice or snow makes a plane even more difficult to steer or stop.
Combine all of the above (a typical storm that would cause delays at an airport), and you can see why airports delay flights or close when conditions get worse.
The wind should be coming from the right direction and be reasonably steady. Gusting crosswind is a very dangerous thing.
Also it's not just the planes that need to be out in the cold wind but also the ground crew, if you force them to work in such condition you will get unions on your doorstep, unions.
On takeoff you need to maintain the center line, if you deviate a bit the plane will plough into the grass next to it and you will not be flying today.
- If the wind were exactly aligned with the runway this might be theoretically possible. Maybe. But, the wind never directly lines up with the runway, and a 100mph crosswind would blow the airplane right off the runway before it ever got off the ground, causing it to crash.
- They can, though it's better if you can at least see the runway lights. What is actually impossible is taxiing, there's no system that allows you to follow taxiways to the terminals with no visibility.
- Again, yes, they can use the runway so they can take off. But, when they land, they do use their brakes to stop. If their tires get no grip, the planes would generally slide off the runway (an no, engine braking is not powerful enough to stop a plane on their own.) So having no traction on the runway may not stop you from taking off, but it will stop you from landing.
These are some of several reasons weather can shut down an airport, though there are several more. You might considering asking about other conditions in separate questions though.
Shouldn't aircraft be able to withstand winds of up to 100 mph?
Can the airframe withstand it? Sure. They deal with twice that in the jet stream almost daily. Can they land or take off in it? Not so much, especially if the wind is not straight down the runway (which it almost never is.) Every aircraft has a maximum crosswind component in which it can complete a safe takeoff or landing roll. I'm not aware of any of them near the 100 mph figure. What happens when you try to take off with a crosswind component that is too high? Well, possibly something like Continental Airlines Flight 1404, which got blown off the side of the runway on takeoff in Denver after it had already accelerated above 100 knots. Everyone survived, but the Captain as well as another person sustain critical injuries and the aircraft was destroyed.
The worst part about winds from storms, though, is that they tend to be very gusty with rapid changes in both magnitude and direction. Say you're flying a final approach and you suddenly encounter a 50 knot tail wind. Your wings just stalled and, if you're low enough and/or the gust lasts long enough, you just crashed short of the runway. Also, sudden changes in crosswinds can cause runway excursions.
Thunderstorms are also capable of producing a nasty type of sudden downdraft called a microburst. These are particularly nasty to encounter at low altitude. As you approach a microburst, it causes you to have a sudden strong headwind (and, thus, higher airspeed,) which suddenly increases the lift of the wings, while also slowing down the ground speed of the aircraft. Pilots may naturally compensate for the increased airspeed by reducing power. This then becomes a huge problem seconds later when that headwind vanishes and the wind is instead pushing the aircraft straight downward, causing it to lose altitude quickly. To make a bad problem worse, as the aircraft flies out of the microburst, it now has a sudden tailwind, which quickly decreases the airspeed, causing the wings to produce significantly less lift and perhaps even stall. Encountering such a microburst on short final to DFW near a thunderstorm caused the crash of Delta Air Lines Flight 191 in 1985, one of the most deadly aviation accidents in U.S. history.
Shouldn't they be able to take off through fog using instruments alone?
Sure. They do it all the time (in the case of airliners, at least.) Visual flight rules are not allowed in fog, but all airliner flights use instrument flight rules anyway. Sometimes they execute visual approaches, but nearly every airport that handles airliners and all airliners and airline flight crews have the ability to perform an instrument landing. Many commercial airport runways (and most airliners) are equipped and certified for fully automated landings. Takeoff doesn't even need that.
General Aviation is another matter here, though. Pilots and/or aircraft not certified for instrument flight rules cannot operate in fog. While many GA pilots and aircraft are certified for IFR, many also aren't. Flights not using IFR must be operated under Visual Meteorological Conditions, which generally require that aircraft can only land or take off when the cloud ceiling is more than 1,000 feet above the ground level and horizontal visibility is at least 3 statute miles (at least in the U.S.)
Shouldn't they be able to take-off on a slippery runway, since the wheels are not powered?
Not really, no. The big problem here is not the lack of ability to accelerate with the wheels, but rather the lack of directional control and the inability to brake. On both takeoff and landing, the aircraft relies on the contact of the landing gear with the pavement to maintain directional control. Also, traction is also obviously required to use the wheel brakes (which provide the primary stopping force on airliners.) Ice is particularly nasty about causing loss of directional control on the runway. For example, Tower Air Flight 41, a 747 departing JFK for Miami, lost directional control on takeoff due to ice, resulting in a runway excursion. There were no fatalities, unless you count the 747 itself, which was damaged beyond repair.
If 100 miles an hour generates enough lift for the aircraft to fly, then a wind gusting 100 miles per hour might result in the aircraft taking off while the passengers are still climbing the steps.
Kites fly without any speed relative to the ground. When the wind blows strong enough, so do aircraft. In weather like this the aircraft are either secured to the ground with tie-downs or taken indoors.
(Not that tie-downs always work.)
If you look at historical behavior, you'll find that airports in the 1970s and 1980s generally didn't close for bad weather. Stapleton Airport in Denver for example, never closed for weather a single time until a major blizzard in 1982.
I believe the reason they do now is liability law. Though they could theoretically continue to operate in most bad weather with a 99% chance of safety, that 1% chance of having an accident and facing an expensive lawsuit from injured passengers or surviving families causes them to err on the side of caution.
A point to bear in mind with all of these, is that it doesn't take much for flight cancellations if your airport is already running very close to capacity with no time to makeup earlier delays.
This notably is the case with Heathrow where NATS (UK ATC) will be trialling time based separation over the current distance based separation between aircraft. Here's a good explanation from them about the impact of headwinds on landing rates and how measures like this can mitigate some of these impacts: http://www.nats.aero/tbs/
All of these questions answer from the point of view of the plane. Can it land, can it take off, can the pilots see, can they control the plane, etc. But there is another aspect not yet mentioned: ground staff.
It's incredibly common for long delays to be caused by thunderstorms because it's not safe for anyone to be outside during active lightning. This means the people who hold the sticks to show the planes where to park, the people who hook up the fuel trucks, the luggage handlers - dozens of people many of whom are handling dangerous equipment, and all of whom have to stay inside until the lightning stops. An airport may stop allowing landings and takeoffs even when the planes would have no trouble landing or taking off because it's not safe for ground crew to be out there.