(Edit May 2012-- my thoughts on this matter have now changed significantly-- I'm still convinced that the earth "feels" a downward push from the wing of an aircraft in flight, equal in magnitude to the weight of the aircraft, which is also equal to the upward gravitational attraction that the aircraft exerts on the earth, but I no longer believe that this downward force need involve any specific amount of downward momentum of the air (downwash). For more, see posts 58, 61, and 72. End edit.)
Originally Posted by BMatthews
You're missing out on another option. The laws of thermodynamics.
You're suggesting that the downwash behind a plane connects to the surface of the planet. But what about the idea that the air is also a viscous gas? As such disturbances in it tend to damp out over time and the motion is converted to heat as it does so. There's lots of examples of how this works. So it's quite possible even on the vast sort of scale that you're considering that the reaction between airplane and air takes place within that mass of air only with the remaining energy being converted to heat instead of connecting the mass of the plane to the earth's mass.
You may be right, bear with me as I think this through out loud and maybe reach some conclusions at the end....
From an energy point of view the situation is complex-- we are burning fuel, dumping heat out the exhaust pipe, etc--
But from a conservation of momentum point of view-- and specifically looking at conservation of momentum in the up-down direction only-- I think the situation is much simpler. In order for the earth not to be pulled upwards by the plane's gravity (mg), it must experience a downward force equal to the plane's weight (mg). This must come from the downwash.
Can a downward pressure be transmitted to the earth's surface even with no downward velocity? As we increase the plane's altitude in a series of steps, we see the cross-sectional (horizontal slice) area of the downwash (as measured at the earth's surface) increase, and we see the velocity of the downwash decrease-- but if the earth is going to "feel" the plane's weight and not accelerate upwards, don't we expect the total downward momentum of the downwash to stay the same, no matter how far below the plane we are looking?
If the downwash is losing downward momentum the further below the plane it travels, doesn't that create a new conservation of momentum problem? What specifically would cause that loss of downward momentum?
It looks to me like the downward momentum of the downwash must be conserved, all the way to the earth's surface.
Can we think of the downwash as ultimately being converted to a pressure wave that presses down on the earth with no downward momentum? But wouldn't this pressure press upwards on the plane, too, complicating the picture further?
One way or another the earth must "feel" the plane's weight in the form of some sort of downward force, or else the earth will accelerate upwards due to the pull of the plane's gravity. I guess the question is, whether this downward force can be imparted by some sort of pressure that involves no downward momentum, or whether there must always be downward momentum, no matter how far below the airplane we are looking. I think there must always be downward momentum, even if it is mixed to the scale where it is a vanishingly small downward velocity spread out over a vanishingly large area.
We can pose a somewhat similar problem involving thrust forces-- what ultimately happens to the rearward momentum of the propwash or jetwash, if it is not conserved indefinitely-- but since there is no planet in existence where an aircraft can travel in one linear direction indefinitely, it's really not a problem. By the time the plane has flown all the way around the earth, it has blown the propwash through a full 360 degrees so there is no net momentum imparted to the atmosphere.
Maybe I'm assuming the molecules in the atmosphere, downwash, etc are perfectly elastic when they are really not? Like, if you have a row of marbles and hit a marble against the first marble in the row, the string of collisions doesn't go on forever? How is conservation of momentum satisfied in that simpler case?
So maybe the appropriate argument is that even if the momentum of the downwash is ultimately converted to heat, this still gets expressed as a localized pressure increase that acts (pushes downward) against the surface of the earth in a localized manner and counteracts the upward pull of the plane's gravity on the earth?
This discussion is starting to branch off into several branches:
1) The earth's surface must feel a localized downward force related to the creation of lift, whether that downward force is caused by the momentum of the downwash, or a high-pressure region due to air heated by the downwash, or whatever. Otherwise the earth would accelerate upward, due to the pull of the plane's gravity, and momentum would not be conserved. The earth must "feel" the plane's weight in some manner-- in some localized manner that is fundamentally different from the "globally dispersed" way that the earth feels the weight of a balloon. The earth must be prevented from accelerating upwards toward the plane, or momentum will not be conserved. That is the key point that is less important than the details of exactly how the downwash changes as it moves down through the atmosphere.
2) But looking in more detail, if there is a downwash at distance Y below the wing (ignoring horizontal distances, just looking at the vertical distances), with downward momentum Z, then at distance Y+1 below the wing, can the downward momentum of the downwash be less than Z? If so, where has the momentum gone? Is this loss of momentum (say through inelastic collisions) still consistent with the basic Newtonian laws? And bringing the problem back around to point #1, has the momentum gone into some form (e.g. pressure) that can still transmit the "weight" of the plane to the earth, acting in a localized manner at some point more or less below the plane, so that the earth is still prevented from accelerating upward toward the plane?
(When I say localized I mean-- it doesn't really matter if the action is spread out over an entire hemisphere-- as long as the net effect is to transmit to the earth a force equal to the plane's weight-- the direction of the force must be "downwards" as defined in our inertial reference frame--meaning that all the different vectors contributing to this force point not toward the center of the earth, but rather point parallel to the plane's weight vector, which at locations very far from the plane, will no longer be aligned with local gravity i.e. no longer point toward the center of the earth. It's hard to see how these forces could be created by increasing the pressure of parcels of the atmosphere that are very far from the plane-- say on the other side of the earth-- so the action must be somewhat localized...)