6. An overview of the effects of radiation and convectionCopyright 2013, David A. RandallBasics of moist convectionIn atmospheric science, the term “convection” refers to a buoyancy-driven circulation. In other fields this is sometimes called “natural convection.”Before discussing moist convection, it is useful to briefly investigate dry convection. Consider the equation of vertical motion and the statement of approximate conservation of dry static energy, linearized with respect to a resting, horizontally uniform basic state:ρ∂′w∂t= −∂′p∂z−′ρg,(1)∂′s∂t= −′w∂s∂z.(2)Here the overbars denote horizontal averages, and primes denote departures from those averages. The gravity term of (1) represents the effects of buoyancy. Eq. (2) describes dry adiabatic motion.We assume for simplicity that ∂s∂z is independent of height. Also for simplicity, we neglect the perturbation pressure term in (1), which usually acts to partially cancel the effects of buoyancy, and we use the approximation−′ρρ⎛⎝⎜⎞⎠⎟≅′ΤΤ=′scpΤ.(3)Then (1) reduces to! Revised Friday, January 30, 2009! 1An Introduction to the General Circulation of the Atmosphere∂′w∂t=g′scpΤ.(4)Eqs. (2) and (4) form a closed system. We look for solutions of the form′w t( )=′w 0( )Re eσt{ },′s t( )=′s 0( )Re eσt{ },(5)where σ may be either real or imaginary. Substituting, we find that for nontrivial solutionsσ2= −gcpΤ∂s∂z.(6)For ∂s∂z< 0, σ is real, and there is an exponentially growing solution with σ> 0; this is dry convective instability. We say that ∂s∂z< 0(7)is the criterion for dry convective instability. It can be seen from either (2) or (4) that in the exponentially growing solution, with σ> 0, ′w t( ) and ′s t( ) have the same sign for all time, so that ′w′s > 0, i.e., convection transports dry static energy upward. We will show in Chapter 7 that an upward temperature flux tends to lower the atmosphere’s center of gravity, i.e., it reduces the total potential energy of the atmospheric column. The reduction in potential energy coincides with a generation of convective kinetic energy through the work done by the buoyancy force, so that the total energy is conserved. The generation of convective kinetic energy through an upward flux of dry static energy can be seen directly by multiplying both sides of (4) by ′w. For ∂s∂z> 0, σ is imaginary, and the solutions are oscillatory; these are gravity waves. Their frequency, N, satisfies N2=gcpΤ∂s∂z; this is called the Brunt-Väisällä frequency. Using the analysis given above as a starting point, you should be able to show that ′w′s = 0 for a gravity wave, where the overbar represents an average over the period of the wave. This means that gravity waves do not transport dry static energy.! Revised Friday, January 30, 2009! 2An Introduction to the General Circulation of the AtmosphereThe analysis above shows that, in the absence of phase changes, convection and gravity waves are mutually exclusive; they cannot occur in the same place at the same time. We will see later that this conclusion does not necessarily apply when phase changes are allowed.Up to this point, we have considered dry adiabatic motion. To analyze moist convection, we will assume saturated moist adiabatic motion. As discussed in Chapter 4, the moist static energy, h, is approximately conserved under moist adiabatic processes. We replace (2) by∂′h∂t= −′w∂ h∂z.(8)For saturated motion, the moist static energy must be equal to the saturation moist static energy, h*, so we rewrite (6) as∂ h∗'∂t= −w '∂ h∗∂z.(9) Next, we have to relate the buoyancy term of (4) to h∗'. Recall that h∗≡ cpΤ+ gz + Lq*Τ, p( ),(10)where the saturation mixing ratio depends, as indicated, on temperature and pressure. Perturbations at fixed height, and at approximately fixed pressure, satisfyh*' ≅ s ' 1 +γ( ),(11)where, in the linearization,γ≡Lcp∂q*∂Τ⎛⎝⎜⎞⎠⎟p(12)is evaluated using the mean-state temperature and pressure. The nondimensional parameter γ is positive and of order one.We now write−′ρρ⎛⎝⎜⎞⎠⎟≅′ΤΤ=h*'cpΤ1 +γ( ).(13)! Revised Friday, January 30, 2009! 3An Introduction to the General Circulation of the AtmosphereThis is analogous to (3). Substitution of (13) into the equation of vertical motion gives∂′w∂t=gh*'cpΤ1 +γ( ).(14)We look for exponential solutions of the system (9) and (14), and find thatσ2= −gcpΤ1 +γ( )∂ h*∂z.(15)This shows that the criterion for moist convective instability of a saturated atmosphere is∂ h*∂z< 0.(16) Compare with (6).Before moving on, we need to do one more thing. The dry adiabatic lapse rate of temperature is given byΓd= −∂Τ∂z⎛⎝⎜⎞⎠⎟dry adiabatic=gcp.(17)This is the rate at which temperature decreases with height when the dry static energy is independent of height. We can rewrite (2) as∂′Τ∂t=′wΓ−Γd( )(18)and re-state the criterion for dry convective instability asΓ>Γd.(19)Similarly, we can express the criterion for moist convective instability in terms of the moist adiabatic lapse rate, which is given by ! Revised Friday, January 30, 2009! 4An Introduction to the General Circulation of the AtmosphereΓm= −∂Τ∂z⎛⎝⎜⎞⎠⎟moist adiabatic≅Γd1+Lq*Τ, p( )RdΤ1+L2q*Τ, p( )cpRvΤ2⎡⎣⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥..(20)Eq. (20) is derived the QuickStudy on the moist adiabatic lapse rate. The denominator of (20) is larger than the numerator, so Γm<Γd, although Γm→Γdat cold temperatures. For example, with a pressure of 1000 mb and a temperature of 288 K, we find that Γm= 4.67K km-1(see Fig. 6.1). As the temperature increases, the moist adiabatic lapse rate decreases. For saturated motion, we can rewrite (2) asFigure 6.1: A plot of the moist adiabatic lapse rate, Γm, in K km-1, as a function of temperature and pressure. p, mbT, K34567892002202402602803001000800600400200T, Kp, mb! Revised Friday, January 30, 2009! 5An Introduction to the General Circulation of the Atmosphere∂′Τ∂t=′wΓ−Γm( ).(21)Compare with (18). Eqs. (18) and (21) will be used later.Convective energy transportsRiehl and Malkus (1958; Fig. 6.2) argued from the