# Radiative Convective Equilibrium and the Greenhouse Effect

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1 Radiative Convective Equilibrium and the Greenhouse Effect Weston Anderson September 19, 2016 Contents 1 Introduction 1 2 Basics of radiation 1 3 Emission temperature of earth Radiative equilibrium Greenhouse effect Radiative convective equilibrium Introduction In these notes we ll discuss the energy balance of earth. In general, energy can be transferred around a system (such as the earth) in three ways: 1.Radiation - Energy moving through space requiring neither a medium to travel through nor any exchange of mass. This is how the energy from the sun reaches the earth, traveling at the speed of light. 2. Convection - Transfer of energy through an exchange of mass. In the atmosphere parcels of air with different energy levels will change places and transfer energy around the system. 3. Conduction - No mass is exchanged, but energy is transferred through a medium by collisions between atoms or molecules. 2 Basics of radiation The sun emits a near constant amount of energy (solar luminosity) equal to L 0 = 3.9x10 2 6W. Because space is a vacuum, we can assume that the total amount of energy passing through a sphere of any size with the sun at its center 1

5 atmosphere entirely 2. Small ɛ - The atmosphere is optically thin. Emission to space is coming mainly from the surface. The diagram below illustrates the full solution between these two limits. As longwave opacitiy of the atmosphere increases, the outgoing flux increasingly comes from the atmosphere, where the surface emission is absorbed. At the surface, the emission (and temperature) rise with increasing atmospheric opacity to account for in increasing amount of radiation re-emitted to the surface from the atmosphere. This is the greenhouse effect. t Figure 4: Figure credit: notes of Ron Miller, NASA GISS 4.1 Radiative convective equilibrium In radiative equilibrium we assume that no convection takes place, and so that every vertical layer of the atmosphere is individually in equilibrium. In radiative convective equilibrium we begin to introduce concepts of atmospheric dynamics by allowing 1-D convection to take place within the vertical column of the atmosphere. To understand when this will occur, let us return to our radiative equilibrium model. In a simple 1-D model, the temperature difference between the atmosphere and the surface is a measure of the stability of the air column. If the surface is cooler than the atmosphere above it, the column is stable and no convection is necessary. If, however, the temperature of the surface is significantly warmer than the air above it, then the column is said to be unstable because warm air tends to rise. We can therefore introduce a measure of the stability of an air column as the difference in temperatures per unit height, or 5

6 a lapse rate : γ = dt dz. We can further estimate at what lapse rate the atmospheric column is stable by assuming that a parcel of air rises adiabatically that is, without transferring energy to its surroundings but that it cools as it rises. We begin with the first law of thermodynamics: Q = 0 = C p dt dp ρ where Q is the heating or cooling (Q=0 because the motions are adiabatic) and C p is the heat capacity of an ideal gas at constant pressure. We can then use the hydrostatic balance ( dp dz = ρg) to subsititute in for dp to get: or, equivalently Q = 0 = C p dt ρgdz ρ dt dz = g C p This dictates the rate at which an air parcel cools as it rises (or warms as it sinks) in a stable, dry, atmosphere (the dry adiabatic lapse rate). If in a column of air a parcel aloft is warmer than predicted by this rate, then the column is stable. If instead a parcel of air aloft is cooler than predicted by this relation, then the column is unstable and warmer air from below will rise as the cooler air sinks (i.e. convection will take place). We can now impose this lapse rate as a constraint in our simple model, which we will now refer to as a radiative convective equilibrium (RCE) model. That is, we are saying that if the temperature difference between the atmosphere aloft and at the surface is ever too great, the atmosphere in a column will mix vertically to relax the temperature profile to the dry adiabatic lapse rate, which is stable. We can now reconsider how atmospheric opacity affects the temperature profiles of the surface and the atmosphere, depicted below. We can infer from this figure that convection consistently transfers energy from the surface to further up in the atmosphere. That is, the radiative convective equilibrium solution always has a warmer T a and cooler T s compared to the radiative equilibrium solution. 6

7 t Figure 5: Figure credit: notes of Ron Miller, NASA GISS. The blue lines in the left panel indicate how the T s and T a profiles evolve once convection is introduced. The plateau of T s T a in the right panel is at the maximum allowable temperature difference before convection is triggered. The blue curve above the plateau is for the radiative equilibrium model only 7

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