Convection of Moist Saturated Air: Analytical Study. Robert Zakinyan, Arthur Zakinyan *, Roman Ryzhkov and Kristina Avanesyan

Transcription

1 atmosphere Article Convection of Moist Saturated Air: Analytical Study Robert Zakinyan, Arthur Zakinyan *, Roman Ryzhkov and Kristina Avanesyan Received: 9 October 015; Accepted: 30 December 015; Published: 5 January 016 Academic Editor: Robert W. Talbot Department of General and Theoretical Physics, Institute of Mamatics and Natural Sciences, North Caucasus Federal University, 1 Pushkin Street, Stavropol , Russia; zakinyan@mail.ru (R.Z.); romandesignllc@gmail.com (R.R.); awan.kristina@yandex.ru (K.A.) * Correspondence: zakinyan.a.r@mail.ru; Tel.: Abstract: In present work, steady-state stationary rmal convection of moist saturated air in a lower atmosphere has been studied oretically. Thermal convection was considered without accounting for Coriolis force, and with only temperature The analytical solution of geophysical fluid dynamics equations, which generalizes formulation of moist convection problem, is obtained in two-dimensional case. The stream function is derived in Boussinesq approximation with velocity divergence taken as zero. It has been shown that stream function is asymmetrical in direction contrary to dry and moist unsaturated air convection. It has been demonstrated that convection in moist atmosphere strongly depends on Keywords: rmal convection; Rayleigh Benard convection; moist air; stream function; analytical model 1. Introduction Atmospheric convection is involved in many of central problems in meteorology and climate science [1,]. Investigation and understanding of atmospheric convection is of great importance for development and improvement of global wear and climate prediction [3,4]. Convection interacts with larger-scale dynamics of planetary atmospheres in ways that remain poorly represented in global models [5,6]. The important role in convective instability is played by release of latent heat of condensation during phase transitions of water in moist saturated air [7]. Many experimental and oretical works, reviews and textbooks have been devoted to rmal convection [8 1]. Even so, we currently have not a complete understanding of this phenomenon. According to Alekseev and Gusev [13], free convection can be of two main types: The Rayleigh convection conditioned by hypercritical temperature gradient, and lateral convection resulting from temperature heterogeneity in horizontal plane. The interlatitude and monsoonal type air circulations belong to lateral convection. As for classic Rayleigh convection, examples are convection resulting in formation of clouds distributed in space in form of ular structures (Benard s). Here, convection will be studied. For analysis of a convective stability, linearized system of equations describing rmal convection in a plane has been solved by Rayleigh [14]. It is valid to say that Rayleigh ory is non-stationary two-dimensional linear analytical model of convection. The study of nonhydrostatic models commonly involves extensive numerical integrations. Such an approach was used by Ogura [15], who studied development of an axially symmetric moist convective toward steady state. The numerical studies have been furr developed by many authors (as, for example, in [16,17]). Much insight into problem can be gained by using highly truncated models. Saltzman [18] and Lorenz [19] studied dry convection by only taking a few spectral components of motion and temperature fields into Atmosphere 016, 7, 8; doi: /atmos

2 Atmosphere 016, 7, 8 of 10 account. They have developed Rayleigh ory by considering system of non-linear equations. However, y solved obtained system of equations numerically. These studies have been followed by many ors [0 ]. When moist processes are considered, formulation of a low-order model presents some special difficulties. These are due to complex expressions for latent heat release and asymmetric properties of condensation with respect to upward and downward motion. One simplified approach of physics of condensation has been proposed by Shirer and Dutton [3]. In ir model, condensation processes have effect of modifying critical Rayleigh number, i.e., critical temperature gradient needed for onset of convection. This model has been developed by Huang and Källén [4] by considering hysteretic effects. The effect of moisture is only introduced via condensational heating in [4]. Low-order models of atmosphere were also used to study specific problems such as blocking [5]. Several particular solutions for inviscid non-heat-conducting atmosphere have been analyzed in [6 8]. Currently, numerical methods of analysis of nonstationary two- and three-dimensional non-linear models of convection with use of high-performance computers are widely applicable in oretical studies. On or hand, because of ir relative simplicity, low-order models present considerably more qualitative insight into problem than complicated numerical computations. In spite of this, it is of interest to obtain analytical solution of stationary two-dimensional model of convection. It will allow us to understand better fundamental nature of atmospheric structures and dynamics. The obtained solution can be an effective tool at forecasting of convection parameters. Previously we have reported on two-dimensional analytical model of dry air rmal convection [9]. Here we present furr development of two-dimensional analytical convection model which takes into account atmospheric humidity. Hence, purpose of this article is to define conditions of moist saturated air free convection occurrence within framework of two-dimensional convection model. Here we extend classical parcel-based linear stability analysis to case with a finite displacement.. Basic Equations of Moist Saturated Air Convection Consider ideal fluid equation of motion, in x z plane, in Eulerian form, in inertial reference system, neglecting Earth rotation, in projections onto coordinates axes [30 3]: In equilibrium (statics): Bu Bt ` u Bu Bx ` w Bu Bz 1 ρ i Bw Bt ` u Bw Bx ` w Bw Bz 1 ρ i v 0, Bp Bx 0, 1 ρ e ˆBp Bx (1) ˆBp g () Bz ˆBp g 0 (3) Bz Here ρ i is moist air parcel density; ρ e air parcel surrounding atmosphere density; and g free fall acceleration. We accept parameters of surrounding atmosphere as an undisturbed state. Hence, pressure may be written as: p p ` p 1, here p 1 is pressure disturbance relative to statics state. In moist atmosphere air density equation in Boussinesq approximation will be [13] ρ i ρ e p1 α T β sq (4) where α = 1/T 0 is air rmal expansion coefficient; T 0 = 73 K; T(z) = T i (z) T e (z); T(z) overheat function; T i, T e air parcel internal and external temperatures; s water (also called specific humidity or moisture content); s=s i s e supersaturation function; β M d /M v 1 = 0.608; M d = 9 g/mol dry air molar ; M v = 18 g/mol water molar

3 Atmosphere 016, 7, 8 3 of 10. In or words, we assume that density depends on temperature and and not depends on pressure [33]. Here we do not consider water loading. The temperature change of surrounding atmosphere will be considered to follow law: T e pzq T ec γ pz z c q (5) where γ is surrounding air temperature gradient; and T ec surrounding air temperature at condensation level (z c ). The moist saturated air temperature gradient is determined by [34] γ ma dt i γ a ` L c p (6) where γ a is dry-adiabatic temperature gradient; L specific heat of condensation; c p specific heat capacity at constant pressure; s m saturated. The quantity / appearing in Equation (6) is a complicated function of temperature and pressure [34]; this is why Equation (6) is usually analyzed numerically. To obtain an analytical solution, we should propose an adequate parametrization for quantity /. We specify saturated change with altitude parametrically by expanding in a Taylor series: ˇ ` K pz z c q, K d s m ˇ (7) here K is some function of saturated at condensation level (s mc ). Then for moist-adiabatic temperature gradient we have parametrical form: «γ ma γ a ` L c p ff ˇ ` K pz z c q γ mac ` εpz z c q, γ mac γ a ` L c p ˇ (8) here γ mac is moist-adiabatic temperature gradient at condensation level [34] (this is known function of temperature and pressure). Note that Equation (8) presents a linear approximation to lapse rate. Analyzing and approximating table values of moist-adiabatic temperature gradient [34], for quantity ε we can get: ε = LK/c p « C/m. Then we have that temperature of adiabatically rising moist saturated air parcel follows law: T i pzq T ic γ mac pz z c q ε pz z cq (9) where T ic is rising air parcel temperature at condensation level. The function T i (z) graph presents state curve ; family of such curves is displayed on aerological diagrams. The table values of air temperature obtained from state curve demonstrates satisfactory agreement with values calculated by Equation (9) within convection layer of interest z w z c ď 5 km. Here z w is convection level. Taking into account Equations (5) and (9) overheat function will be: T pzq c T ` γ mac pz z c q ε pz z cq (10) where c T is overheat function at condensation level; γ mac = γ γ mac is difference of ambient air temperature gradient and rising air parcel temperature gradient at condensation level. The quantity γ mac determines angle between state curve and stratification curve at condensation level on aerological diagram.

4 Atmosphere 016, 7, 8 4 of 10 For adiabatically rising air parcel saturated follows law: s m pzq s mc ` ˇ pz z c q ` K pz z cq (11) Assume that decreases linearly with altitude in surrounding atmosphere: s e pzq s ec b pz z c q (1) where s ec is water in surrounding atmosphere at condensation level; b Then for supersaturation function we get s pzq c s ` ˇ ` b pz z c q ` K pz z cq (13) where c s = s mc s ec is supersaturation at condensation level. Taking into account Equations (10) and (13), Equation (4) for air parcel density will be written as: ρ i ρ e 1 α 0 α 1 pz z c q ` α pz z c q ı (14) where α 0 α c T ` β c s, α 1 α γ mac ` β ˇ ` b, α 1 pαε βkq (15) From here level of equalization of densities of rising air parcel and surrounding air can be found: b z ρ z c α 1 ` α 1 ` pαε βkq pα ct ` β c sq. (16) αε βk The level of equalization of temperatures ( level of neutral buoyancy) is determined by b z t z c γ mac ` p γ mac q ` ε c T. (17) ε The steady motion of air parcel is described by set of equations of moist saturated air convection: u Bu Bx ` w Bu Bz 1 Bp 1 (18) ρ e Bx u Bw Bx ` w Bw g pα T ` β sq Bz (19) T pzq c T ` γ mac pz z c q ε pz z cq (0) s pzq c s ` ˇ ` b In Equation (19) following assumption has been made: pz z c q ` K pz z cq (1) Bu Bx ` Bw Bz 0 () ˇ ˇ 1 Bp 1 ρ e Bz ˇ! g pα T ` β sq (3)

5 Atmosphere 016, 7, 8 5 of 10 To satisfy Equation (3), in present study we seek analytical solution for pressure disturbance in form: p 1 1 pxq p1 px, zqˇˇz z (4) where z is some level determined below. Substituting Equations (0) and (1) into Equation (19), we finally get set of equations of moist saturated air free convection in form: u Bu Bx ` w Bu Bz 1 Bp 1 ρ e Bx (5) u Bw Bx ` w Bw Bz g α 0 ` α 1 pz z c q α pz z c q ı (6) Bu Bx ` Bw Bz 0 (7) Furr we will find solution of obtained system of equations. 3. Solution of Moist Saturated Air Convection Equations It follows from continuity Equation (7) that for shallow convection condition stream function ψ may be introduced [30 3]: u Bψ Bz, w Bψ Bx (8) Substituting Equation (8) into Equations (5) and (6), we get: Bψ Bz Bψ B ψ Bz Bx ` Bψ Bx Equation (30) can be written as Bψ B ψ Bz Bx ` Bψ Bx B ψ BxBz Bψ Bx B ψ Bz 1 Bp 1 ρ e Bx (9) B ψ BzBx g α 0 ` α 1 pz z c q α pz z c q ı (30) B ψ BzBx N r mac rz ` pz z c q k 1 pz z c q ı (31) where rn mac g «α γ mac ` β ff ˇ ` b (3) is square of Brunt Väisälä frequency for moist saturated air; α c T ` β c s αε βk rz, k 1 «ff (33) α γ mac ` β ˇ ` b α γ mac ` β ˇ ` b We seek solution of resulting equations set in following form: Substituting Equation (34) into Equation (9), we get: ψ px, zq X pxq Z pzq (34) Z XX 1 1 ZZ 1 Bp 1 ρ e Bx (35)

6 Atmosphere 016, 7, 8 6 of 10 Substituting Equation (34) into Equation (31), we obtain: X ZZ 1 1 XX N r mac rz ` pz z c q k 1 pz z c q From Equation (36), separating variables, it follows that: ZZ 1 k r N mac rz ` pz z c q k 1 pz z c q (36) (37) Atmosphere 016, 7, 8 6 of 10 X 1 XX 1 k (38) ZZ = k N ( ) ( ) where k is constant requiring determination. mac ( z + z z The Equation c k 1 z z (37) c solution ) has form: (37) 1 Z kn r X XX mac d pz z c q rz ` pz = z j k cq k1 3 pz z cq (38) where k is constant requiring determination. The Equation (37) solution has form: It follows from here: ( z z ( ) c ) k 1 ( ) Z = kn mac z zc z + z z Z 1 kn r rz ` pz z c q k 1 pz z c q c (39) 3 mac d pz z c q rz ` pz z j (40) It follows from here: cq k1 3 pz z cq z + ( z zc) k1( z zc) Z = kn mac We seek Equation (38) solution with following ( z zc ) form: k1 (40) ( z zc) z + ( z zc) 3 X 1 cos kx (41) We seek Equation (38) solution with kfollowing form: Substituting Equation (41) directly into Equation 1 Х = cos (38), kx we see that solution is valid. Hence, for (41) stream function, we have expression: k Substituting Equation (41) directly N ψ ZX r d into Equation (38), we see that solution is valid. Hence, mac pz z c q rz ` pz z j for stream function, we have expression: cq k1 k 3 pz z cq cos kx (4) N ( ) mac z z ( ) c k 1 ( ) ψ = ZX = z zc z + z zc cos kx (4) 1 shows this function graph. k As is seen, stream 3 function is symmetrical in horizontal direction and asymmetrical 1 shows this in function graph. direction. As is seen, The level stream of maximal function is symmetrical velocities in horizontal is in upper part of direction convective and asymmetrical contrary in to direction. dry and moist The level unsaturated of maximal air convection velocities [9]. is in upper part of convective contrary to dry and moist unsaturated air convection [9]. (39) 1. Stream function at b = 10 6 m 1, ΔcT = 0.5 C, Δcs = 0, Δγmac = C/m, ε C/m, 1. Stream function at b = k = 10 3 m 1, d d 10 6 m 1 = m,. c T = 0.5 C, c s = 0, γ mac = 5 ˆ 10 4 C/m, ε = 3 ˆ 10 7 C/m, k = 10 3 m 1, { z zc 1.6 ˆ 10 6 m 1. From Equations (8) velocity projections can be written: z + ( z zc) k1( z zc) ( c ) 1 ( z z ) z + ( z z ) N mac u = ZX = cos kx k z z k c c 3 (43)

7 From here one can see that maximal velocities level is not in center of convective as it has been for dry air convection. The maximal velocities level for moist saturated air is at /3 of of convection. Substituting expression for zmax zc into Equation (44), we get for ascending moist Atmosphere 016, 7, 8 7 of 10 From Equations (8) velocity projections can be written: u Z 1 X rn mac rz ` pz z c q k 1 pz z c q d k pz z c q rz ` pz z jcos kx (43) Atmosphere 016, 7, 8 cq k1 3 pz z cq 7 of 10 ( ) z z ( ) w ZX 1 N r mac d pz z c q rz ` pz z c k cq 1 ( ) w= ZX = N j mac z zc z + k1 z z 3 pz c z sin kx 3 cq (44) sin kx (44) From here, using condition w = 0, we get expression for of convection From here, using condition w = 0, we get expression for of convection (or cloud ): (or cloud ): z w z c 3 c ` ` 16 c4k k zw z = 1 1 k z1rz 4k (45) shows shows dependence dependence of of convection convection on on water water As As is is seen, seen, convection convection rises rises monotonically monotonically with with.. Convection Convection vs. vs. water water The The atmospheric atmospheric parameters parameters are are same same as as in in Analysis and Discussion of Obtained Solution 4. Analysis and Discussion of Obtained Solution We find maximal velocities level from condition: w/ z 0. Taking derivative of We find maximal velocities level from condition: Bw/Bz = 0. Taking derivative of Equation (44) with respect to z, we get: Equation (44) with respect to z, we get: kz 1 zmax zc = (46) z max z c 1 ` a1 ` 4k k 1 rz 1 (46) k 1 If =0 ( overheat ΔcT and oversaturation Δcs are equal to zero at condensation level), If rz 0 ( overheat n for convection level c T and oversaturation and for maximal velocities c s are equal to zero at condensation level), level we have n for convection level and for maximal velocities level we have d s αδγ m mac + β ds + b 3 m z= zc zw zc = = 3 (47) z w z c 3 α γ mac ` β ˇ ` b k k 3 1 αε βk (47) 1 αε βk ds αδγ m mac + β + b 1 z= zc (48) zmax zc = = k1 αε βk

8 Atmosphere 016, 7, 8 8 of 10 z max z c 1 k 1 «α γ mac ` β αε βk ff ˇ ` b From here one can see that maximal velocities level is not in center of convective as it has been for dry air convection. The maximal velocities level for moist saturated air is at /3 of of convection. Substituting expression for z max z c into Equation (44), we get for ascending moist saturated air maximal velocity equation: w max r N mac k 1 g f e 1 ` a1 «` 4k 1 rz k 1 rz ` 1 ` a1 ` 4k 1 rz 4 (48) 1 1 ` a1 ff ` 4k 1 1 rz sin kx. (49) If rz 0, n we have w max r N mac? 3k1 sin kx (50) Let us find pressure spatial distribution. For this, integrating Equation (35), we get: p 1 px, zq 1 ρ ex Z 1 ZZ ` const (51) The constant of integration can be found from condition that at maximal velocities level pressure disturbance is equal to zero at points meeting condition cos kx = 1. From here we have Here we take into account that if rz 0, n const ρ e r N mac k (5) Z 1 3 kk 1N r k 1 pz z c q mac 1 j 3{, z max z c 1 (53) 3 k k 1 1 pz z c q From here, taking into account Equation (4) and assuming z z max, for pressure disturbance we get p 1 1 pxq p1 px, z max q ρ e r N mac k sin kx (54) It should be noted that analytical solution Equation (4) is obtained by integrating only one Equation (36); in so doing we neglected term 1 ρ e Bp 1 {B z in accordance with condition Equation (3). In such an approach to problem, Equation (18) plays a role of auxiliary equation for pressure disturbance determination. To make Equations (18) and (19) agree with condition Equation (3) we were compelled to consider Equation (18) at point z z max of maximal velocities where pressure disturbance is equal to zero. 5. Conclusions Hence, we have obtained new solution of equations of moist saturated air convection in atmospheric cloud layer. The solution is obtained within framework of stationary D analytical model. It has been shown that convection parameters in moist atmosphere strongly depend on water The expression for convective is obtained. The obtained solutions can be valuable in convection parameters forecasting practice. As it can be seen from obtained equations, important convection parameters, such as convection level,

9 Atmosphere 016, 7, 8 9 of 10 level of neutral buoyancy, maximal velocities level, and maximal velocity itself, are determined by characteristics which can be calculated from aerological diagrams using radiosounding data. These characteristics are angle between state curve and stratification curve at condensation level and water Acknowledgments: This work was supported by Ministry of Education and Science of Russian Federation in framework of base part of governmental ordering for scientific research works (order No 014/16, project 653). Author Contributions: Robert Zakinyan planned and supervised research, co-wrote paper; Arthur Zakinyan co-performed oretical analysis, co-wrote paper; Roman Ryzhkov and Kristina Avanesyan co-performed oretical analysis. Conflicts of Interest: The authors declare no conflict of interest. References 1. Emanuel, K.A. Atmospheric Convection; Oxford University Press: New York, NY, USA, Stevens, B. Atmospheric moist convection. Annu. Rev. Earth Planet. Sci. 005, 33, [CrossRef] 3. Sherwood, S.C.; Bony, S.; Dufresne, J.-L. Spread in model climate sensitivity traced to atmospheric convective mixing. Nature 014, 505, [CrossRef] [PubMed] 4. Bluestein, H.B. Severe Convective Storms and Tornadoes; Springer: Chichester, UK, Lambaerts, J.; Lapeyre, G.; Zeitlin, V. Moist versus dry barotropic instability in a shallow-water model of atmosphere with moist convection. J. Atmos. Sci. 011, 68, [CrossRef] 6. Yano, J.; Geleyn, J.-F.; Köhler, M.; Mironov, D.; Quaas, J.; Soares, P.M.M.; Phillips, V.T.J.; Plant, R.S.; Deluca, A.; Marquet, P.; et al. Basic concepts for convection parameterization in wear forecast and climate models: COST action ES0905 final report. Atmosphere 015, 6, [CrossRef] 7. Shmerlin, B.Y.; Kalashnik, M.V. Rayleigh convective instability in presence of phase transitions of water. The formation of large-scale eddies and cloud structures. Phys. Uspekhi 013, 56, [CrossRef] 8. Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability; Clarendon Press: Oxford, UK, Martynenko, O.G.; Khramtsov, P.P. Free-Convective Heat Transfer; Springer: Berlin, Germany, Ogura, Y. A review of numerical modeling research on small-scale convection in atmosphere. Meteorol. Monogr. 1963, 5, Arakawa, A.; Jung, J.-H. Multiscale modeling of moist-convective atmosphere A review. Atmos. Res. 011, 10, [CrossRef] 1. Derbyshire, S.H.; Beau, I.; Bechtold, P.; Grandpeix, J.-Y.; Piriou, J.-M.; Redelsperger, J.-L.; Soares, P.M.M. Sensitivity of moist convection to environmental humidity. Q. J. R. Meteorol. Soc. 004, 130, [CrossRef] 13. Alekseev, V.V.; Gusev, A.M. Free convection in geophysical processes. Sov. Phys. Uspekhi 1983, 6, [CrossRef] 14. Rayleigh, L. On convective currents in a horizontal layer of fluid when higher temperature is on underside. Philos. Mag. 1916, 3, [CrossRef] 15. Ogura, Y. The evolution of a moist convective element in a shallow, conditionally unstable atmosphere: A numerical calculation. J. Atmos. Sci. 1963, 0, [CrossRef] 16. Van Delden, A.; Oerlemans, J. Grouping of clouds in a numerical cumulus convection model. Contrib. Atmos. Phys 198, 56, Asai, T.; Nakasuji, I. A furr study of preferred mode of cumulus convection in a conditionally unstable atmosphere. J. Meteorol. Soc. Jpn. 198, 60, Saltzman, B. Finite amplitude free convection as an initial value problem I. J. Atmos. Sci. 196, 19, [CrossRef] 19. Lorenz, E.N. Deterministic nonperiodic flow. J. Atmos. Sci. 1963, 0, [CrossRef] 0. Van Delden, A. Scale selection in low-order spectral models of two-dimensional rmal convection. Tellus 1984, 36, [CrossRef] 1. Schlesinger, R.E.; Young, J.A. On transient behavior of Asai s model of moist convection. Mon. Wear Rev. 1970, 98, [CrossRef]

10 Atmosphere 016, 7, 8 10 of 10. Brerton, C.S. A ory for nonprecipitating moist convection between two parallel plates. Part I: Thermodynamics and linear solutions. J. Atmos. Sci. 1987, 44, [CrossRef] 3. Shirer, H.N.; Dutton, J.A. The branching hierarchy of multiple solutions in a model of moist convection. J. Amos. Sci. 1979, 36, [CrossRef] 4. Huang, X.-Y.; Källén, E. A low-order model for moist convection. Tellus 1986, 38, [CrossRef] 5. Charney, J.G.; Devore, J.G. Multiple flow equilibria in atmosphere and blocking. J. Atmos. Sci. 1979, 36, [CrossRef] 6. Kuo, H.L. Convection in a conditionally unstable atmosphere. Tellus 1961, 13, [CrossRef] 7. Lilly, D.K. On ory of disturbances in a conditionally unstable atmosphere. Mon. Wear Rev. 1960, 88, [CrossRef] 8. Gill, A.E. Spontaneously growing hurricane-like disturbances in a simple baroclinic model with latent: Heat release. In Intense Atmospheric Vortices; Springer: Berlin, Germany, Zakinyan, R.G.; Zakinyan, A.R.; Lukinov, A.A. Two-dimensional analytical model of dry air rmal convection. Meteorol. Atmos. Phys. 015, 17, [CrossRef] 30. Drazin, P.G. Introduction to Hydrodynamic Stability; Cambridge University Press: Cambridge, UK, Monin, A.S. Theoretical Geophysical Fluid Dynamics; Kluwer Academic Publishers: Dordrecht, The Nerlands, Zdunkowski, W.; Bott, A. Dynamics of Atmosphere: A Course in Theoretical Meteorology; Cambridge University Press: Cambridge, UK, Smith, R.K. The physics and parameterization of moist atmospheric convection. In Nato Science Series C; Kluwer Academic Publishers: Dordrecht, The Nerlands, 1997; Volume Matveev, L.T. Fundamentals of General Meteorology: Physics of Atmosphere; Israel Program for Scientific Translations: Jerusalem, Israel, by authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under terms and conditions of Creative Commons by Attribution (CC-BY) license (