Friday, February 22, 2013

Fixed

We're working up at 19,000'. I can't see the ground from my position in the cockpit because by the time the vector representing my angle of vision reaches the ground, it has dissipated into the mist. Mostly I see cirrus clouds around me, and they're kind of pinky orange, the sunset colour of a sun that won't set for a months, as this is a summer story and we're in the north. Between them in the distance I see blueish-green. If I were to put my head against the window so I could see in front of the wing root, down between the fuselage and the nacelle, I'd see little tiny trees. They're not just tiny because I'm way up in the sky. They're tiny because they have a short growing season and a long harsh winter. There are little cutlines and creeks and lakes, and probably rocks and bears, too. And mosquitoes.

I'm constantly doing fuel math, weather math, duty day math, and tweaking the heater controls to try and keep my feet warm. I judge it time to go home, and not just because my feet are cold. We're a decent distance from our intended airport of landing--indeed from any airport we can use--and I'm anticipating a headwind all the way south. I ask for down, both down south and down out of the cold and the worst of the headwind. The trick is to convert altitude to speed, descending rapidly enough in feet per minute to find more favourable winds quickly, but not so rapidly in airspeed that the benefit of the 'free' airspeed is lost to drag, which increases exponentially with airspeed. Someday I should set up the differential equations to determine how fast I should descend in this case. I think it's a solvable problem. Airbus computers probably do it. I just going by gut, though, and the desire to get down to where it's warmer. And today ATC makes that decision for me. They want me to cross a fix that is 30 DME from the destination at 5000'. Fix is aviation speak for any identifiable point at which you're supposed to report or turn, or do something. In this case it was identifiable only in as much as it was 30 nautical miles (DME is almost nautical miles. I'll explain that sometime) from the aerodrome. I'm asked to confirm I can do that.

I tell them I can make the crossing restriction. I didn't do any math for that either. I know how many nautical miles it is to the fix, how many thousand feet I have to descend and my current speed and rate of descent. That would be enough to calculate the altitude I would arrive at the fix, but to arrive there at 5000' I need to increase my rate of descent which will increase my speed (I don't plan to start powering back until I'm a lot closer to the airport than I am now). I used to know a rule of thumb for speed gain with fpm descent, but I think it may have been only applicable to Cessna 152s.

ATC amends my crossing restriction to 5000' at 20 DME, but I don't change my descent profile and reach it at at 30.8 DME, just to prove to myself that I was not incorrect in accepting the clearance. Actually 30.8 was a GPS distance, so it was a little more in DME.

From there I call the FSS and they give me traffic, nothing but a King Air taxiing out to head south. He'll be well out of the way by the time I'm on final. Land, fuel, park, secure for the night and wait for a cab. The airport manager is painting a railing at the front of the terminal. He tells us they frequently have to go out and get beavers off the runway here. We think this is hilarious. Why would they be on the runway? Maybe because it's warm. Maybe even beavers get tired of slogging through swamps all the time and like a nice paved road once in a while.

So 30 DME is a slant range?

Aviatrix said...

Yep, slant range. An old technology, but cool.

Jeremy P said...

I'm not sure I understand this part:

"And today ATC makes that decision for me. They want me to cross a fix that is 30 DME from the destination at 5000'"

"ATC amends my crossing restriction to 5000' at 20 DME, but I don't change my descent profile and reach it at at 20.8 nm from the fix, just to prove to myself that I was not incorrect in accepting the clearance"

If the original fix which had the crossing restriction was at 30 DME from the destination, and you reached 5000' at 20.8 nm from that location, are you saying that you were 50+ nm from the destination at that point - which was a full 30+ nm short of the revised crossing restriction? (I am not sure if the fix was in line with a direct track to the airport, so maybe there are some triangles involved here?)

Aviatrix said...

You're right. That makes no sense. I deleted my notes after I made them into this blog entry, but I must have meant 30.8 DME, as I have amended it.

Blogging isn't something I do at the high brainpower spot in my day.

LocalFlightEast said...

Could you briefly explain "slant range" for an aspiring PPL who's just about getting used to using an e6b.
I'm thinking triangles and hypotenuses or something?

Thanks!!

Frank Van Haste said...

To WMAP:

Exactly right. The line from the aircraft to the point of the fix on the ground is the hypotenuse of a triangle and DME provides its length. GPS, in contrast, provides distance from the fix to the spot on the ground under the aircraft. Hence GPS distance will always be a bit less than DME distance.

I suppose you could show that: (DME distance)^2 = ((GPS distance)^2 + (Altitude AGL in matching units)^2).

The difference between DME distance and GPS distance is normally inconsequential when the distances are large and hence the acute angle at the fix is small. When you get close in the difference (as a %'age ) will get to be quite a bit.

Obviously, the extreme case is if you fly directly over the fix at 6,000 feet AGL. The DME distance is 1.0 nm and the GPS distance is 0.0 nm.

Frank

Frank Van Haste said...

Dear Trix:

Wow, you have me thinking! My normal practice is to descend at the top of the green arc, thinking I'm getting back some of the time I lost while climbing to cruise altitude at Vy or some such. But of course, drag goes as the square of true airspeed and I suppose it's perfectly possible that some lower speed is actually more efficient.

Of course, the numbers will be different for descending 10,000 to 8,000 vs. descending 5,000 to 3,000 because 'rho' is different. I think...

I see some non-trivial math here.

Regards,

Frank

Sarah said...

Frank, the math got me too. It's your garden variety multivariate nonlinear optimization problem... just the thing to think about in the descent.

Maybe not so hard given high headwinds. If you're trying to optimize time to the goal, the most important factor is just getting down quickly out of the headwinds aloft. I agree, top of the green descent decaying to cruise at the target altitude is what you want. Now if you had a tailwind, it gets more interesting...

LocalFlightEast said...

Thanks Frank , that makes perfect sense and is something I can impress my instructor with , by knowing!

D.B. said...

I assume from the context that the Beavers on the runway were not the DeHaviland kind?

Anonymous said...

Without the complexity of an engine transforming fuel into speed, altitude, heat and noise, the optimal decend speed involves some happy birds flying the polar curve.