Atmospheric Sciences 101 Homework Assignment 5 Solutions and Discussion 1 The heavy black line in the figure above represents the temperature structure in the lower atmosphere the environmental lapse rate on a summer day Note that the environmental temperature decreases with height at a lapse rate of about 8 C km from the surface up to 2500 meters then there is a temperature inversion from 2500 3000 meters Suppose we have a parcel of air at the surface that gets lifted up through the atmosphere don t worry about how the parcel is lifted we will talk about this later Consider four different scenarios for this parcel Scenario A the parcel initially has a relative humidity of 100 When it rises condensation occurs immediately and the parcel rises at the moist adiabatic lapse rate through the entire layer from 0 3000 meters Scenario B the parcel initially has a relative humidity of 80 When it rises it first cools at the dry adiabatic lapse rate When it reaches 500 meters the air becomes saturated and as it rises further the parcel cools at the moist adiabatic lapse rate Scenario C the parcel initially has a relative humidity of 60 The parcel cools at the dry adiabatic lapse rate for the first 1250 meters then moist adiabatically as it rises above this point Scenario D the parcel initially has a relative humidity of 10 It does not become saturated as it is lifted through this layer up to 3000 m For each scenario above answer the following questions i What would be the altitude of the cloud base for this scenario if any ii What would be the altitude of the cloud top if any iii Where will the parcel be warmer than the surrounding environment Colder iv Based on iii where is the parcel less dense than the environment More dense v If a parcel is less dense than its surroundings it will be buoyant it can rise without the help of any lifting mechanism Based on iv where would you expect this parcel to be buoyant vi Would you call this scenario stable unstable or conditionally unstable Scenario A The parcel is saturated at the surface so any amount of lifting from the surface will cause the parcel to cool and thus the saturation mixing ratio in the parcel will decline Some of the water vapor in the parcel will condense immediately to form liquid water droplets Therefore the cloud base will be at the surface 0 km Note that as soon as the parcel rises a little bit it will be warmer than the environment because it will cool at the moist adiabatic lapse rate of 4 C km which is less than the environmental lapse rate near the surface Because the parcel is warmer it will also be less dense than the surrounding air and thus it will be buoyant immediately after it rises above the surface The buoyant parcel may rise freely without the aid of a lifting mechanism all the way through 3000 meters because it will be warmer than its surroundings the entire way up It cools at the moist adiabatic lapse rate the entire time because it remains saturated and as it rises and cools more of the water vapor condenses to form liquid water Thus the cloud will extend all the way from the surface through 3000 km and presumably even higher Because the parcel is buoyant all the way from the surface through 3000 meters we would call this scenario unstable i e as soon as the parcel is pushed up a little bit it takes off and would keep rising all the way through this layer Scenario B The parcel is not saturated at the surface in this scenario so as it is lifted it will cool at the dry adiabatic lapse rate of 10 C km When it reaches 500 meters the temperature of the parcel has decreased by 5 C and it becomes saturated the air in the parcel is now colder and can hold less water vapor than it could at the surface Note that the parcel is now colder and less dense than the surrounding air so if the lifting mechanism were turned off when the parcel reached 500 meters it would start to sink back to the surface quickly becoming unsaturated and warming dry adiabatically However if the lifting mechanism lifted the parcel above 500 meters liquid water would start to condense as the temperature and saturation mixing ratio decreased so the cloud base would be at 500 meters and the parcel will cool at the moist adiabatic lapse rate as it continues to rise Note that if the parcel is lifted up to 750 meters it will become warmer than its surroundings Thus it will also be less dense than its surroundings and it can rise freely even if we now turn off the lifting mechanism i e it is now buoyant This continues until we reach about 2800 meters at which point the parcel would again become colder and denser than the surrounding environmental air If we assume the lifting mechanism is still operating above 2800 meters then the parcel will keep rising cooling and condensing all the way through 3000 meters However if the lifting mechanism turned off somewhere below this point which is usually true in the real world then the parcel would not be buoyant above 2800 meters and we would expect the parcel to stop rising Thus the cloud top would be located at 2800 meters again assuming that the lifting mechanism is no longer operating Because the air parcel is buoyant only if we lift it up to 750 meters in this scenario we would call it conditionally unstable Scenario C Similar to scenario B except that now saturation doesn t occur until the parcel is lifted to 1250 meters and the parcel will be colder and denser than the surrounding environment until it is lifted up to about 1800 meters From 1800 meters to about 2600 meters the parcel would be warmer and less dense than the surrounding air and thus buoyant Above 2600 meters the parcel would be colder and denser than the environment Thus the cloud base would be 1250 meters the cloud top would occur around 2600 meters again assuming the lifting mechanism turned off sometime between 1800 and 2600 meters and we would again call this situation conditionally unstable Scenario D The parcel is so dry that it will remain unsaturated as we cool it dry adiabatically all the way through 3000 meters Note that the parcel will always be colder and less dense than its surroundings There will be no cloud formed by this parcel and it will never be buoyant This is a stable situation Note that in this exercise we only considered a parcel at the surface We could repeat the same exercise at any height and establish the stability of the atmosphere relative to a parcel originating at any level Clouds in the atmosphere don t always