Lecture 1. Review of moist thermodynamics

Transcription

1 Tropical Meteorology Bretherton Lecture 1. Review of moist thermodynamics Moist thermodynamics for cumulus convection A good understanding of moist thermodynamics and of commonly used moistthermodynamic quantities is essential for the study of cumulus convection and reading its literature. Take-home points 1. Moist air can hold 7% more water vapor per degree it is warmer. 2. Water vapor makes air less dense; condensate makes it more dense; these effects are accounted for by using of virtual or density temperature to compute air parcel buoyancy. 3. Equivalent potential temperature θ e and moist static energy h are approximately conserved (except for latent heating by freezing/melting) and linearly mixing for typical conditions of tropical convection. For most purposes, θ e and moist static energy h/c p are functional equivalents, as are θ e and moist static energy h/c p. 4. The temperature excess of an adiabatically lifted boundary-layer air parcel (a measure of conditional instability that neglects virtual effects) can be assessed above its LCL by comparing its θ e or h with the environmental profiles of saturated! e * or h *. This type of plot is commonly used in tropical meteorology. We roughly follow the discussion in Sec of Cotton and Anthes, Storm and Cloud Dynamics (Academic Press, 1989). (a) Moist air as a mixture of components at thermodynamic equilibrium A rigorous treatment of moist thermodynamics starts by considering moist air as a thermodynamic system at a pressure p and temperature T composed of a density ρ d of dry air, ρ v of vapor, ρ l of liquid water and ρ i of ice. In C&A the subscript a is often used in place of the customary d for dry air. We define the water vapor mixing ratio q v = ρ v /ρ d and similarly for the mixing ratios q l, q i of liquid and ice. All of these mixing ratios are small (typical q v ~ 10-2 = 10 g kg -1, q l, q i ~ 10-3 = 1 g kg -1 ). The ideal gas law is assumed to hold for dry air and water vapor, with molecular weights m d = 29 daltons and m v = 18 daltons and hence gas constants R d = R u /m d = 287 J kg -1 (dry air) R v = R u /m v = 461 J kg -1 (water vapor), where R u is the universal gas constant and are the molecular weights of dry air and water. The gas constant and heat capacity of moist air (per unit of dry air) are R m = R d + q v R v, c pm = + q v c pv + q l c l + q i c i.

2 The isobaric heat capacities are: = 1005 J kg -1 K -1 (dry air), c l = 4190 J kg -1 K -1 (liquid) c pv = 1870 J kg -1 K -1 (water vapor) c i = 2100 J kg -1 K -1 (ice) For water, the latent heat of vaporization is L = L 0 + (c pv - c l )(T T 0 ), where T 0 = K is the freezing temperature and L 0 = J kg -1. Thus L decreases by ~0.1% for each 1 K increase of temperature. The latent heat of freezing L f = L f0 + (c l - c i )(T T 0 ), where L f0 = J kg -1. L f increases by ~0.7% for each 1 K increase of temperature. The saturation vapor pressure e*(t) satisfies the Clausius-Clapeyron relation d ln e * dt = L R v T 2! 0.07K "1 (doubles every 10 K) (1) Bolton (1980, JAS) provided the following useful approximate formula: e * " 17.67(T! ) % = hpa exp T! & '. (2) The saturation mixing ratio is q * = R d R v e * p! e * " R d R v e * p. (3) At 1000 hpa and 20 C, q * = 15 g kg -1 ; at 30 C, q * = 28 g kg -1 ; a typical 10 m q seen over the warm tropical oceans is about 20 g kg -1, corresponding to a dew point of 24 C. (b) Virtual and density temperature Since small differences in the density between moist air parcels are what drive density differences, it can be important to consider the contribution of water to the air density, including virtual effects due to vapor and loading effects due to condensate: p = p d + p v = ρ d R d T + ρ v R v T = ρ d R d T(1 + q v m d /m v ), ρ = ρ d + ρ v + ρ l + ρ i = ρ d (1 + q v + q l + q i ) = (p/r d T )(1 + q v + q l + q i )/ (1 + q v m d /m v ). Since m d /m v = 1.61 is larger than 1, more water vapor decreases the density, while the extra weight or loading of condensate (liquid + ice) adds to the density. This is often expressed in terms of a density temperature T ρ = p/ρr d that is defined to satisfy the ideal gas law for dry air: T ρ = T(1 + q v m d /m v )/(1 + q v + q l + q i ) T( q v - q l - q i ). (4) The density temperature is the temperature of dry air that has the same density as the moist air at the given pressure. The virtual temperature T v is like the density temperature, but without the condensate loading terms (many authors are sloppy about this distinction). Some scientists also use virtual potential temperature θ v and potential density temperature θ ρ, in which θ replaces T in (2.4).

3 A 1 g kg -1 water vapor mixing ratio increase raises the virtual or density temperature by 0.18 K at near-surface temperatures; the loading due to 1 g kg -1 of suspended liquid or ice would lower the density temperature by 0.3 K. (c) The heat equation The first law of thermodynamics for moist air is (C&A eq 2.34): c pm dt! dp " + L dq v! L f d i q i = Q (5) The symbol d i indicates an internal thermodynamic phase change (e.g. freezing of water droplets) but does not include external exchanges such as removal by precipitation. Q denotes diabatic heating. This is exact so far. If we divide (5) by T and use the ideal gas law, we get the equivalent form c pm d lnt! R m d ln p + L T dq v! L f T d iq i = Q T It is possible to go further without approximation, but it is cumbersome enough that only the most dedicated thermodynamicists do so. Instead, we usually approximate R m R d and c pm. Introducing potential temperature θ: d lnt! ( R d )d ln p! " + L T dq v! L f T d iq i = d ln" (d) Equivalent potential temperature (EPT) Q T The equivalent potential temperature θ e is the potential temperature achieved by adiabatically lifting moist air up to zero pressure without addition of heat Q, condensing water vapor above saturation. Traditionally, ice is not considered because in subfreezing clouds, ice and water are often far from bulk thermodynamic equilibrium (e.g. supercooled droplets can exist down to -40 C). Hence (6) implies p LCL L ln! e = ln! + " T d q *, (7) 0 A common further approximation is to take L L 0 (constant) and approximate T in the integrand by the saturation temperature T * at the lifted condensation level, which implies ln! e " ln! + Lq v Lq! T * e "! exp v ' & % T * ) ( (8) Eq. (8) can underestimate by θ e for warm moist air by up to 3 K. Bolton (1980) worked out a better approximation that I use (Matlab function theta_es.m on class web page): (6)! θ e = T " p ref p % & (1'0.28q v ) (! exp * q v ( q v ) ' 2.54 " T * % & - ),, (9)

4 where p ref = 1000 mb and the saturation temperature is estimated as follows: " 1 T * = T! 55! ln(q v q * )% 2840 ' &! K. (10) θ e is conserved in equilibrium vapor-liquid phase changes, and it is also approximately conserved during precipitation formation and evaporation. Thus it is an excellent tracer in moist air motions. However, freezing or sublimation can cause θ e changes up to a few K; from (2) we see that d ln! e " L f T d iq i + Q T If mass m 1 of an air parcel with EPT θ e1 is mixed with mass m 2 of an air parcel with EPT θ e2, the mixture EPT is to a very good approximation (but not exactly) linearly mixing, i. e. θ em = (m 1 θ e1 + m 2 θ e2 )/ (m 1 + m 2 ). The saturation equivalent potential temperature! e * is the θ e of air of a given T and p that is saturated with respect to liquid water, so (11)! e * (p,t ) =! e (p,t,q v = q * (p,t )) (12) At a given pressure, this is an increasing function of T. The 350 K moist pseudoadiabat is the temperature profile corresponding to! e * = 350 K (a typical near-surface θ e value in regions of deep tropical convection), and similarly for other temperatures. These are the moist adiabats usually shown on skew-t and other sounding plots. Because warmer air can hold more water, the temperature decreases less rapidly with p on a warmer moist pseudoadiabat. Comparing two moist pseudoadiabats, the temperature rise in the upper troposphere will be about twice as large as at 900 hpa (Slide 1.7, left). Some authors use the moist entropy s = c p log θ e /θ 0 instead of θ e. (e) Moist static energy Moist static energy (MSE) is a close relative to θ e. It is useful because even in the presence of moist convection, the atmosphere is nearly in hydrostatic balance, such that dp = -ρg dz. (13) Bernoulli s formula suggests that nonhydrostatic pressure perturbations p nh will be O(ρV 2 /2), where V is the convective velocity perturbation. Even in the strongest thunderstorms, V < 30 ms -1 and ρv 2 /2 < 5 hpa, which is small compared to the pressure changes felt by an air parcel ascending several km through a cumulus updraft. For most tropical convective drafts, the nonhydrostatic pressure perturbations are well under 1 hpa and the hydrostatic approximation is quite accurate. Applying (6) to the heat equation (5), we obtain c pm dt + gdz + Ldq v! L f d i q i = Q (14)

5 Approximating L = L 0 (constant) and c pm we can define moist static energy (MSE) h = T + gz + Lq v (15) and integrate (14) to obtain: dh! L f d i q i " Q For adiabatic displacements (Q = 0) of a moist air parcel, with no internal ice production or melting (d i q i = 0), we deduce that MSE is approximately conserved, even when there is liquid precipitation or evaporation. It is also linearly mixing. MSE can be interpreted as the sum of a dry thermodynamic energy T, gravitational potential energy gz, and latent energy of vaporization Lq v per unit mass. It is expressed in units of kj/kg (roughly equivalent to K). We can write h = s + Lq v, where s = T + gz is the dry static energy or DSE. Similarly, saturated MSE h * = T + gz + Lq * (14) is analogous to c p! e *. Saturated MSE is approximately constant along a moist adiabat. This provides a nice formula for the lapse rate along a moist adiabat: 0 = dh* dz "! dt % dz & ' ma = ( g + Lq* / H c p (1 + ) ), ) = L *q * c p *T + Lq* q * + (15) (1 c p T e 6 gkg Here H is the density scale height and T e = R v T 2 /L 15 K is the e-folding temperature for saturation mixing ratio. The term Lq * /gh is roughly 0.15γ. γ ranges from 3 near tropical cloud base down to 1 at the freezing level (5 km in the tropics) to near-zero in the upper troposphere. MSE is an energy, and θ e is an entropy proxy. MSE is roughly equal and functionally equivalent to c p θ e. Similarly, dry static energy is functionally equivalent to c p θ. Both are roughly conserved and linearly mixing, but each involves separate approximations to exact thermodynamics. For most purposes, one just should use the more convenient choice, MSE if working in height coordinates or θ e if working in pressure coordinates. The right panel of Slide 1.7 shows adiabats computed assuming constant saturated MSE. To compare with the left panel, height has been hydrostatically converted to pressure for each adiabat, assuming a surface pressure of 1000 hpa and computing density using the virtual temperature of saturated air. Dry adiabats based on constant dry static energy are plotted for comparison. The moist adiabats computed using saturated θ e and MSE are essentially identical, as are the dry adiabats computed using θ and DSE.

6 Vertical profiles of θ e and! e * or h and h * Plots showing vertical profiles of θ e or h and! e * or h * are commonly used in tropical meteorology, because these quantities nicely indicate conditional instability and don t vary too much across the tropical troposphere. The θ e of adiabatically lifted near-surface air tells us its! e * (i.e. its temperature) above the LCL. If the θ e of near-surface air exceeds the! e * in an extensive layer of the troposphere, that indicates the lifted parcel has a positive temperature excess, suggesting potential for surface-based moist convection in that layer. Similarly for h and h *. Slide 1.8 shows 52-day mean soundings from five locations near Kwajalein Island (9N 167W) in the western Pacific ocean, plotted using h and h *. The vertical line shows an adiabatically lifted boundary layer air parcel. It is well to the right of the profiles of h * throughout the free troposphere, showing substantial deep conditional instability.