determine the temperature of a rising parcel of air at 1km intervals from 0 to 7km altitude

Question

this is about the environmental lapse rate for my meteorology class.

Answer 1

This question needs a bit more information. I will try to answer it the best I can though!

You are likely given a surface temperature and a surface dew point. When air rises in the atmosphere, it cools adiabatically (without latent heat transfer). If the air parcel is dry, meaning the temperature is NOT the same as the dew point, then you use the Dry Adiabatic Lapse Rate (DALR). If the temperature is the same as the dew point, you will use the Moist Adiabatic Lapse Rate (MALR). The MALR varies, but in most textbooks, it is about 5°C/km.

The DALR is about 10°C/km, so for every kilometer you go up in the atmosphere, your air will be 10º cooler if it remains dry. If you do not reach the dew point by cooling this way, then your answer will be 10°x7 cooler than your starting temperature.

Now, if you are including dew point and a dew point lapse rate, you will need to cool the dew point for every kilometer as well. Most textbooks will make the dew point lapse rate as about 2°C/km.

The key to answering this problem is to take your surface temperature and dew point and cool them by the appropriate lapse rates depending on if the air is saturated or dry. Each kilometer at a time, cool the surface temperature by the DALR and the Dew point by its own lapse rate. When these two become the same, cool the temperature and dew point by the MALR the rest of the way up.

Let's look at an example.

We will use Celsius here because it is the standard in meteorology (aside from Kelvin).

Let's say you have a surface temperature of 20°C. Your surface dew point is 4°C. You need to find the temperature in 1 km intervals all the way up to 7 km.

First, we need our lapse rates.

DALR (Dry rate): 10°C/km

MALR (Moist rate): 5°C/km

Dew Point Lapse Rate: 2°C/km

Let's do our first kilometer.

From 0-1 km, we take our temperature and drop it by the DALR of 10°C because the temperature is not the same as the dew point yet. Drop the dew point by 2°C as well. This leaves us with a temperature of 10°C and a dew point of 2°C at 1 km. Since our temperature and dew point are not the same, we continue with the dry lapse rate for the next kilometer up.

For the next kilometer, drop the temperature by the DALR of 10°C and the dew point by 2°C. That leaves us with a temperature of 0°C and a dew point of 0°C. Our temperature and dew point are the same now! We can now use the MALR!

The rest of the problem is simple. Since the temperature and dew point are the same, we can use the Moist Adiabatic Lapse Rate for the rest of the problem. The temperature AND dew point will fall at the Moist Adiabatic Lapse Rate (MALR) the rest of the way up (5°C/km)! The rest of the problem is shown below:

2-3 km: Temperature and dew point of -5°C.

3-4 km: Temperature and dew point of -10°C

4-5 km: Temperature and dew point of -15°C.

5-6 km: Temperature and dew point of -20°C

6-7 km: Temperature and dew point of -25°C.

I know it's been like 6 years since this question was asked, but maybe it can help others! Hope this helped!