Math/environment question?

Question

A mass of rising air will cool along the dry adiabatic lapse rate (10 oC/km) unit it reaches the condensation level at which point it will continue to rise along the saturated adiabatic lapse rate (6oC/km)

A mass of air has a dew point temperature of 20oC and a temperature at the Earths surface of 30oC

At what altitude will the condensation level be reached

What will the temperature of the air mass be at 3100m above the earths surface

i need a mathematical way of representing these calculations and it cannot just be subtracting the different rates per km

any ideas

Answer 1

Effectively you are dealing with two different masses of air. That below the condensation altitude, and that above.

if

T0= initial temperature [deg C]         (KNOWN= 30)

A0=initial altitude [Km]                        (KNOWN =0)

Tc= temperature Condensation (Dew Point?) (KNOWN=20)

Ac =altitude of Condensation

Ah = altitude ABOVE Ac

R1= lapse rate before condensation

R2= lapse rate post condensation

first find Ac which you can get from rearranging

T0-Tc= Ac * R1

temperature DIFFERENCE between condensation altitude and an altitude higher than it got from rearranging:-

T_c- T_high= Ah *R2

(REMEMBER Ah is the difference between the higher altitude and the condensation altitude!)

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If you plot a graph of temperature against altitude. you get a "kinked" graph which kinks at the condensation altitude/temperature.  (It's altitude which controls the temp, so have altitude along the horizontal)

At altitudes lower than that of the condensation point the graphs gradient is 10 [Deg_C/km]. At higher altitudes the graphs gradient is 6[Deg_C/km]

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A graph is probably the simplest way of showing the predicted results.

If you've studied them then Differential equations are another way to detail the relationship between a number of variables/properties. I doubt it very much they the lapse rates are really linear!  (partial) Differential equations make it a lot easier to deal with non-linearity, so long as can write an equation which describes how the properties relate to each other,and you can find a solution. (sometimes you need to use complex numerical analysis to find a solution)

Answer 2

The mathematical way of representing this is shown by your knowledge of the two lapse rates.

1.  The mass of air will cool at 10 degrees per KM as it rises from the ground at a temperature of 30 degrees, until its temperature reaches the dew point temperature -- in this case, in 1 KM, it will reach the condensation level, and be at 20 degrees.  

2.  In 2100 more meters (2.1 more KM) above the ground, the temperature will have fallen 6 x 2.1 = 12.6 more degrees, to a temperature of 7.4 degrees, at 3100 meters altitude.

The math is simple... you could draw a graph of the temperatures, for example, to illustrate this.