What’s the math on finding your Freezing level?

Question

If the std temp lapse rate is 2c per 1000, and the freezing level is 0c.

If at the sfc my temp is 4c wouldn’t the freezing level be at 2,000?

If so I calculated this using a long method. Is there a formula that I can use to calculate this easily?

Answer 1

Here's how you can figure out the formula. You know that the temperature drops $2^\circ$ per thousand feet. You can represent this mathematically as:

$$ \text{Temperature} = \text{Surface Temperature} - 2^\circ\times\text{Altitude in 1000s of feet} $$

You want to find the altitude where the temperature is zero, so set $\text{Temperature}=0$:

$$ 0 = \text{Surface Temperature}- 2^\circ\times\text{Freezing Altitude in 1000s of feet} $$

You know the surface temperature and want to solve for the freezing altitude, so you want to get altitude on one side of the equation alone. This can be done in two steps:

• Add $2^\circ$ times the altitude to both sides of the equation: $$ 2^\circ\times\text{Freezing Altitude in 1000s of feet} = \text{Surface Temperature} $$
• Divide both sides by $2^\circ$:

$$ \text{Freezing Altitude in 1000s of feet} = \frac{\text{Surface Temperature}}{2^\circ} $$

It might be a good idea to read through an algebra book and work through some exercises, or take a course if possible. You don't need to know a lot of math to be a pilot, but you do need to know some.

Answer 2

According to the International Standard Atmosphere (ISA), temperature drops of 0.002°C per ft (0.0065°C per m) till a height of 36'000 ft (11'000 m). Knowing the temperature at SFC, the relevant math is quite simple:

$T_{@height} = T_{@sfc} - 0.002 \times height $

with temperatures in °C and height in ft.

If you want to find at which height a specific temperature is reached, you simply have to rearrange the terms of this formula to get:

$height = (T_{@sfc} - T_{@height}) / 0.002$

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You want to know for example at which height the temperature is 0°C? You just set $T_{@height} = 0$ and you get $height = T_{@sfc} / 0.002$. In your example where $T_{@sfc} = 4$ we get $height = 4 / 0.002 = 2'000$ ft.

You want to know at which height the temperature is 3°C? You just set $T_{@height} = 3$ and you get $height = (T_{@sfc} - 3) / 0.002$. In your example where $T_{@sfc} = 4$ we get $height = (4 - 3) / 0.002 = 1/0.002=500$ ft.

And so on.

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P.s.: sorry for being late but I had already prepared this answer and found no more time to post it.