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I am exchanging opinions with other pilots concerning climbing and descending with a wind gradient, and I'm facing strong opposition... In my opinion, a plane trimmed for S&L flight, if it encounters a headwind with a positive gradient (wind getting stronger with altitude) will gain altitude, since–because of the gradient–it will harvest some of the wind energy. In other words, part of the energy of the wind will be converted in potential energy of the airplane.

When descending in a tailwind with a positive gradient, things will take place in reverse: the plane will lose altitude, since a part of the potential energy of the plane will be transferred to the moving air, resulting in an increased KE of the wind...

Is my theory correct?

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You are correct.

Suppose the headwind just 10 meters above you is 10 knots stronger then where you are now. Climbing the 10 meters will cost you some kinetic energy which is transformed to potential energy. Suppose you were flying 200 knots airspeed initially, you will end up with 198.1 knots airspeed if the transformation from kinetic to potential energy is 100%.

I derived this from simple kinematic energy equations:

$\frac{1}{2}mV_0^2 + mgH_0 = \frac{1}{2}mV_1^2 + mgH_1$

$V_1 = \sqrt{V_0^2+2g(H_0-H_1)}$

But since your headwind is increased by 10 knots, your true airspeed will be 208.1 so it has increased by 8.1 knots. Free energy from the wind!

Note that your groundspeed will reduce during such a climb, but you will be able to generate a higher than usual climb rate at constant airspeed in a positive gradient headwind field.

Note that birds make use of this technique, for example the albatross can stay airborne for days with using only minimal energy for keeping in the air.

Albatross in windshear field (youtube)

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I would think you would loose 10 knots ground speed for an increase in headwind 10 knots and loose slightly less than 2 knots airspeed for the trade of 10 meters in altitude, per the rule of thumb 9feet per knot per 100knots. Interesting brain teaser.

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    $\begingroup$ I believe that the ground, as a reference, should be excluded from any analysis. You are flying, and the only valid reference is the mass of distant, immobile air... $\endgroup$ – xxavier Mar 15 '17 at 12:14
  • $\begingroup$ We can make a 'mental experiment': you descend in a tailwind with a positive gradient. You'll find a growing tailwind, and in order not to crash, you should give gas... By so doing, you supply energy to the wind through the propeller, while descending... Now, let's think in the inverse situation: climbing with headwind in a positive gradient. It seems reasonable that things will be precisely the opposite: The wind will now supply energy to the plane, while climbing... $\endgroup$ – xxavier Mar 15 '17 at 12:32

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