Vortex Motion State of the Dry Atmosphere with Nonzero Velocity Divergence

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1 atmosphere Article Vortex Motion State of Dry Atmosphere with Nonzero Velocity Divergence Robert Zakinyan, Arthur Zakinyan * ID, Roman Ryzhkov Julia Semenova Department of General Theoretical Physics, Institute of Mamatics Natural Sciences, North Caucasus Federal University, 1 Pushkin Street, Stavropol , Russia; zakinyan@mail.ru R.Z.; romesignllc@gmail.com R.R.; brilliance_wave@mail.ru J.S. * Correspondence: zakinyan.a.r@mail.ru; Tel.: ceived: 5 December 017; Accepted: 8 February 018; Published: 10 February 018 Abstract: In present work, an analytical model of vortex motion basic state of dry atmosphere with nonzero air divergence is constructed. It is shown that air parcel moves along open curve trajectory of spiral geometry. It is found that for case of nonzero divergence, atmospheric basic state presents an unlimited sequence of vortex cells transiting from one to anor. On or h, at zero divergence, basic state presents a pair of connected vortices, trajectory is a closed curve. If in some cells air parcel moves upward, n in adjacent cells, it will move downward, vice versa. Upon reaching cell s middle height, parcel reverses direction of rotation. When parcel moves upward, motion is of anticyclonic type in lower part of vortex cell of cyclonic type in upper part. When parcel moves downward, motion is of anticyclonic type in upper part of vortex cell of cyclonic type in lower part. Keywords: atmosphere dynamics equation; vortex state of atmosphere; divergence; geostrophic wind; vortex motion 1. Introduction It is well known that geostrophic state is a two-dimensional basic state for large-scale atmospheric motion [1 3]. In geostrophic state, motion is nondivergent, thus, re is no vertical motion [4,5]. However, in reality, atmospheric motions are three-dimensional, motion is certainly upward or downward as a result of various reasons. Spiral structures of vortices in atmosphere in different scales form in se cases [6]. A typical three-dimensional basic state is cyclonic vorticity with rising motion due to surface convergence anticyclonic vorticity as well as sinking motion due to surface divergence. Exact solutions of Navier Stokes equations for a three-dimensional vortex have been discovered [7 11]; of particular interest is Sullivan s two-cell vortex solution, because flow not only spirals in toward axis out along it, but it also has a region of reverse flow near axis. It should be noted that re are a large number of works dedicated to oretical study of atmospheric vortices. However, most of se are concerned with numerical solution of atmospheric motion equations. Analytical models of steady-state vortexes in atmosphere have been examined in fewer studies e.g., [1,13]. In [14 16], three-dimensional atmospheric vortex motion is analyzed by considering primitive dynamic rmodynamic equations, including Coriolis, pressure gradient viscous forces. It should be noted that nondivergent motion has been studied in [14 16]. Furrmore, a strong assumption of model in [14 16] is that viscous friction force linearly depends on. In this paper, we furr develop oretical model of vortex motion state of atmosphere. The main goal of study is to refine model by taking into account Atmosphere 018, 9, 61; doi: /atmos

2 Atmosphere 018, 9, 61 of 10 nonzero divergence term in equations of motion. In addition, we use a more general expression for friction force in form of proportionality to aplacian.. Main Equations In local coordinate system, atmospheric state is described by set of Equations 1 5: u t + u u x + v u y + w u z = 1 p + ν u + ω 0z v ω 0y w 1 ρ x v t + u v x + v v y + w v z = 1 p ρ y p z w t + u w x + v w y + w w z = 1 ρ T t u x + v y + w z = αγ A γw + αu T x + ν v ω 0z u g + ν w + ω 0y u 3 + v, T = κ T 4 + αv T y + αw T z Here, T = T T; T T are air temperatures of disturbed undisturbed atmosphere correspondingly; u, v, w are projections of air on x, y, z axes; ρ is air density; p is air pressure; ω 0y, ω 0z are projections of earth s angular on y, z axes; ν is air kinematic viscosity; κ is air temperature conductivity coefficient; α = 1/T 0 is air rmal expansion coefficient, where T 0 = 73 K; γ is temperature vertical gradient; γ A = g/r d is autoconvective vertical gradient ; g is acceleration of gravity; R d is dry-air specific gas constant. The x-axis is tangential to parallel line in direction of earth s rotation, y-axis is tangential to meridian in direction from equator to North Pole, z-axis is perpendicular to earth s surface. Thus, ω 0x = 0. Here we neglect geoidal shape of earth assume it to be spherical. The air density has form ρ = ρ1 α T; static density ρ decreases with height by exponential law. Equation 5 has been obtained by substitution of density expression into continuity equation. The basic state of atmosphere is assumed to be in hydrostatic balance: p p = 0, x y = 0, 1 p g = 0 ρ z We let atmospheric parameters have form of perturbation of static state: p = pz + p x, y, z, t, T = Tz + θx, y, z, t, T = 0 T γ z, γ = γ a γ Here, γ a is dry-air adiabatic temperature gradient. It is convenient to convert equations into dimensionless form. The length scales for horizontal vertical directions are taken as H, scale in horizontal directions is U, time scale is /U, vertical scale is HU/, scale for pressure is ρu, temperature θ scale is z T = 0 T TH = γ H. At same time, we set δ = H/, = U/ν ynolds number, Ri = N H /U Richardson number, N Brunt Väisälä frequency, N = αg γ, Pr = ν/κ Prtl number, Ro = U/ω 0z Rossby number, = U/ω 0y H. If we drop advective terms in Equations 1 5 assume Oδ = 1 i.e., = H, n set of equations can be written as u t = p x + 1 v 1 w + 1 u Ro x + u y + u z 5 6

3 Atmosphere 018, 9, 61 3 of 10 v t = p y 1 u + 1 v Ro w t = p 1 + Ri T + z θ t = w + 1 Pr u + 1 x + v y + v z w x + w y + w z θ x + θ y + θ z u x + v y + w z = αγ A γ a Hw 10 Differentiating with respect to y x subtracting Equation 1 from Equation, we obtain Ω t = 1 D + 1 Ro Ω + 1 w y Here Ω = v x u y, D = u x + v y are vertical vorticity horizontal divergence, correspondingly. Analogously, differentiating summing Equations 6 7, we obtain Thus we have D t = p x + p y + Ω 1 w Ro x + 1 D Ω t + D t Ω = p Ro x + p y D = 1 Ro Ω + 1 w y + 1 D 1 w x 11 1 w t = p z + Ri T + 1 w + 1 u 13 θ t = w + 1 Pr θ 14 u x + v y + w z = αhγ A γ a w 15 We assume that three-dimensional atmospheric state is time independent stationary. Neglecting advective terms, Equations can be transformed: D = 1 Ro Ω + 1 w y Ω = p Ro x + p y + 1 D 1 w x = p z + Ri T + 1 w + 1 u 18 0 = w + 1 Pr θ 19 D + w z = αhγ A γ a w 0

4 Atmosphere 018, 9, 61 4 of 10 Excluding Ω, D, θ p, we obtain required partial differential equation: x + y [ Ra Pr w + w αhγ A γ a + Ta ] u + + w Ta z z = w z x Ro w z y 1 where Ra = Ri = N 3 Uν, Ta = = 4ω 0z 4 Ro ν Ra is Rayleigh number, Ta is Taylor number. 3. Solution of Main Equations Following previous studies [7 11,14 16], we assume that w has form w = XxYyWz, Xx = cos kx, Yy = cos ky, Wz = W 0 sinnπz + α 0 3 where k is wave number; α 0 is initial phase determining initial value of air vertical component at z = 0; n is mode number equal to number of wave lengths within characteristic length scale. In fact, n is number of vortex cell on vertical axis; hereafter we assume that n = 1. Substituting Equation 3 into Equation 1, we obtain k Ra Pr k + n π 3 n π Ta w + + x y u+ k + n π W 0 knπw 0 sin kx cos ky cosnπz + α 0 [ k αhγ A γ a + n π ] + Ta nπw 0 cos kx cos ky cosnπz + α 0 = Ro knπw 0 cos kx sin ky cosnπz + α 0 The functional form of zonal components differs from corresponding expressions in previous studies [7 11,14 16] because of fact that divergence is nonzero. Thus, we seek expression of u in form: u = W 0 A sin kx cos ky + B cos kx sin ky + C cos kx cos ky cosnπz + α 0 Substituting, we obtain [ k Ra Pr n π Ta k + n π 3 ] W 0 cos kx cos ky sinnπz + α 0 kka + nπ k + n π W 0 sin kx cos ky cosnπz + α 0 [ k k + n π ] B Ro nπ W 0 cos kx sin ky cosnπz + α 0 k k + n π [ k C αhγ A γ a nπ + n π ]} + Ta W 0 cos kx cos ky cosnπz + α 0 = 0 k { From here we have Ra cr = k +n π 3 +n π Ta k Pr C = αhγ A γ a nπ πn kk +n π, A = nπ k, B = Ro [ 1 + n π Ta + k k k +n π ]

5 Atmosphere 018, 9, 61 5 of 10 We also have N cr = νκ 4 k + n π 3 + n π ω 0z 4 ν k κ n π ν Assuming Pr = ν/κ = 1 k = π/, we obtain, for Brunt Väisälä frequency, s 1 or γ cr = κ n π ω 0z ν k αg 6 C 10 7 m From here it follows that critical value of vertical temperature gradient is close to value of dry-air adiabatic gradient. Finally u W 0 = nπ πn k sin kx cos ky + Ro kk +n π cos kx sin ky + C cos kx cos ky cosnπz + α 0 4 v = [ αhγa γ a k W 0 ] sinnπz + α 0 nπ k cosnπz + α 0 C sin ky Ro k ω 0z W 0 cos kx sin ky+ πn kk +n π cos ky sin kx cosnπz + α 0 5 w = W 0 cos kx cos ky sinnπz + α 0 6 Figure 1 shows field in accordance with Equations 4 6. In all calculations, we assume that α 0 = It is seen that air parcel motion has a three-dimensional vortex structure. Figure 1. Velocity field of vortex motion.

6 Atmosphere 018, 9, 61 6 of 10 Figure shows air parcel trajectory numerically calculated in accordance with Equations 4 6. As is seen from Figure, air parcel trajectory is an unclosed curve. In or words, parcel passes from one vortex cell to anor. It is also seen that while air parcel in lower level is spirally converged towards center, air is lifted up. After air parcel arrives at high level, air parcel is spirally diverged outwards. In same way, if air parcel is spirally converged towards center at upper level, n it moves downwards to lower level diverges. Thus, one can see that basic state of atmosphere in case of nonzero divergence is a sequence of connected vortex cells. On or h, at zero divergence, air parcel trajectory is a closed curve atmospheric state is a pair of connected vortices, as has been shown in previous studies. Atmosphere 018, 9, x FOR PEER REVIEW 7 of 11 Figure. The air parcel trajectory at nonzero divergence. Figure. The air parcel trajectory at nonzero divergence. It can be seen from Equation 0 that if divergence is equal to zero, n vertical It can be seen from 0 that if divergence is equal to zero, n vertical component has Equation following height dependence: component has following height dependence: H α γ γ z w z = w0e A a wz = w0 e HαγA γa z where w0 is initial value of at earth s surface at z = 0. From here it follows that at where w0 is initial value at upward earth s surface at z = 0.is From here it follows that at zero initial zeroofdivergence, air movement absent. etting zero initial zero divergence, upward air movement is absent. etting divergence be nonzero constant, n from Equation 0, we have divergence be nonzero constant, n from Equation 0, we have D H α γ γ z H α γ γ z w z = w0 0 e A a e A a 1 α γ γ H D A a wz = w0 0e HαγA γa z e HαγA γa z 1 Hαγ γ One can see that upward air movement in Alowera atmosphere is only possible at negative divergence Dsee < 0. One can that upward air movement in lower atmosphere is only possible at negative Equation 11 divergence D < 0. represents a vortex transport diffusion equation with a source of form 1 Ro Equation with aw source form followstransport that if Ddiffusion < Ro equation w y y < 0 of convergent D Ro11 Rorepresents H w y. aitvortex 1/Ro D Ro /Ro w/ y. It follows that if D < Ro /Ro w/ y w/ y < 0 convergent H H t > 0 Ω > 0. This means that horizontal air motion in lower atmosphere, D < 0, n Ω horizontal air motion in lower atmosphere, D < 0, n Ω/ t > 0 Ω > 0. This means that vortex has a cyclonic vorticity in lower atmosphere. In analogy, Equation 1 represents a vortex has a cyclonic vorticity in lower atmosphere. In analogy, Equation 1 represents a divergence transport diffusion equation with a source 0 of form p0 / y divergence transport diffusion equation with a source of form Ω/Ro p / x +. It iffollows from divergence Equation 17 that if negative Ω Ro w/ x. p Itx follows + p from y Equation 1 w x that 1/Ro 17 is constant H 0 0 D < 0, n p / x + negative p / y D = <1/Ro Ro.1 From Ω y = divergence is constant 0, n p /Ro x + H p w/ x Ro Ωhere Ro it follows Ro wthat x H. From here it follows that p x + p y > 0 if Ω > Ro w x. This means that pressure is low at center of cyclonic vortex. Equations relate three-dimensional flow pattern temperature field. At positive vertical, it follows from Equation 19 that θ < 0. This means that cyclonic vortex center is heated, that is, temperature maximum is at cyclonic vortex center.

7 Atmosphere 018, 9, 61 7 of 10 p / x + p / y > 0 if Ω > Ro / w/ x. This means that pressure is low at center of cyclonic vortex. Equations relate three-dimensional flow pattern temperature field. At positive vertical, it follows from Equation 19 that θ < 0. This means that cyclonic vortex center is heated, that is, temperature maximum is at cyclonic vortex center. The obtained solution describes basic state of atmosphere, which is vortex motion state. It can be demonstrated that traditional geostrophic state is a limiting case of vortex motion state. Indeed, at, that is, when viscosity is negligible, w = 0, from Equation 16, one can obtain that D 0. From Equation 18 it follows that 0 = p z + Ri θ + 1 u For stationary state, from Equations 6 10, one can obtain components of geostrophic : u g = Ro p y u g = p v g = Ro p x z Ri T u g x + v g y = 0 These expressions coincide with expressions for geostrophic state of atmosphere [1 3]. We return to system of Equations The expression for divergence can be obtained from Equation 0: D = αhγ A γ a w w z =[αhγ A γ a sinnπz + α 0 nπ cosnπz + α 0 ]W 0 cos kx cos ky From here it follows that horizontal divergence reverses sign at a height of z D = 1 { [ ] } nπ arctan α nπ αhγ A 0 γ a If within a layer 0 < z < z D horizontal divergence is negative convergence takes place, n within layer z D < z < z conv, horizontal divergence will be positive, vice versa. This is a manifestation of Dines s compensation principle. Thus, one can see that this principle takes place not only at zero total divergence but also at nonzero total divergence. The expression for temperature disturbance can be obtained from Equation 19: W 0 θ = Pr k + n π cos kx cos ky sinnπz + α 0 The expression for pressure disturbance can be obtained from Equation 18: 1 1 k +n π 3 +n π Ta p = k +n π 3 +n π Ta k Pr k 1 0 Tz γ z + nπk +n π W 0 cos kx cos ky cosnπz + α 0 + W 0 k sin kx cos ky + Ro 1 k cos kx sin ky + k +n π nπ C cos kx cos ky sinnπz + α 0

8 Atmosphere 018, 9, x FOR PEER REVIEW 9 of 11 3 k + n π + n πta k 1 + k + n π n π k nπ k + n π The expression for vorticity can be obtained from Equation 17: Ro Ω= Atmosphere 018, 9, 61 8 of 10 W0 cos kx cos ky cos nπz + α0 3 k +n π +n π Ta Ω= 1 + k + n π nπ W0 coskkx sin ky sin nπz + α0 + Wk cos + nkxπ cos ky cosnπz + α0 0 1 k k +n π W0 cos kx sin ky sinnπz + α0 + i h k Ro Ro αh k γ γ RoRo C k + n π cos kx cos ky sinnπz + α k + n π αha γ A a γ a RoH nπ WC0 W 0 cos kx cos ky sin nπz + α00 nπ Ro k 1 k nπk +n π Figures 3 4 show fields of vorticity, temperature, divergence, pressure at altitudes Figures 3 4 show fields of vorticity, temperature, divergence, pressure at z = 0.5 lower part of vortex cell z = 0.75 upper part of vortex cell, correspondingly. altitudes z = 0.5 lower part of vortex cell z = 0.75 upper part of vortex cell, As is seen, if motion field in lower part of cell has a cyclonic vorticity, n in upper part, correspondingly. As is seen, if motion field in lower part of cell has a cyclonic vorticity, it has an anticyclonic vorticity, vice versa. The divergence changes in sign at transition from n in upper part, it has an anticyclonic vorticity, vice versa. The divergence changes in sign one part to anor. The pressure field is in antiphase with vorticity field; that is, if pressure at transition from one part to anor. The pressure field is in antiphase with vorticity field; field in lower part is of cyclonic character, n vorticity field is of anticyclonic character. In that is, if pressure field in lower part is of cyclonic character, n vorticity field is of upper part, situation is opposite. anticyclonic character. In upper part, situation is opposite. a b c d Figure 3. The calculated fields of vorticity a, temperature b, divergence c, pressure d in Figure 3. The calculated fields of vorticity a, temperature b, divergence c, pressure d in lower part of vortex cell at z = 0.5. lower part of vortex cell at z = 0.5.

9 Atmosphere 018, 9, 61 9 of 10 Atmosphere 018, 9, x FOR PEER REVIEW 10 of 11 a b c d Figure 4. The calculated fields of vorticity a, temperature b, divergence c, pressure d in Figure 4. The calculated fields of vorticity a, temperature b, divergence c, pressure d in upperpart of vortex cell at z = upperpart of vortex cell at z = Conclusions 4. Conclusions In present study, mamatical model of stationary three-dimensional vortex state of In present study, divergence mamatical of stationary three-dimensional vortex state of atmosphere at nonzero has model been developed. The expressions for vortex atmosphere atbeen nonzero divergence has been developed. The expressions for vortex components have derived. It is shown that at zero divergence, basic state of atmosphere components have been derived. It is shown that at zero divergence, basic state of atmosphere presents a pair of connected vortices, air parcel trajectory is a closed curve. For case of presents a pair ofdivergence, connected vortices, air parcel closed curve. For of nonzero atmospheric basic statetrajectory presents is anaunlimited sequence of case vortex nonzero divergence, atmospheric basic state presents an unlimited sequence of vortex cells transiting from one to anor, air parcel trajectory is an unclosed curve. This result is in cells transiting from to anor, air parcel is an unclosed curve. Thisnot result is accordance with thatone observed in nature. Indeed, trajectory atmospheric vortices usually form as an in accordance Indeed, atmospheric vorticesitusually form notshown as an isolated couplewith but that as aobserved sequenceinofnature. alternating cyclones anticyclones. has also been isolated couple but as a sequence of alternating cyclones anticyclones. It has also been shown that that pressure field is in antiphase with vorticity field. pressure field is in antiphase with vorticity field. Author Contributions: Robert Zakinyan planned supervised research, co-performed oretical Author Contributions: Robert Zakinyan planned supervised research, co-performed oretical analysis, co-wrote paper; Arthur Zakinyan co-performed oretical analysis, co-wrote analysis, co-wrote paper; Arthur Zakinyan co-performed oretical analysis, co-wrote paper; paper; Ryzhkov Julia Semenova co-performed oretical analysis. RomanRoman Ryzhkov Julia Semenova co-performed oretical analysis. Conflicts of Interest: Interest: The authors declare declare no no conflict conflict of of interest. interest. Conflicts of ferences ferences Holton, Dynamic Meteorology; Elsevier: Amsterdam, The Nerls, Holton, J.R. J.R. An An Introduction Introduction totodynamic Meteorology; Elsevier: Amsterdam, Nerls, Gill, A.E. Atmosphere-Ocean Dynamics; Academic Press: San Diego, CA, USA, 198. Gill, A.E. Atmosphere-Ocean Dynamics; Academic Press: San Diego, CA, USA, 198. Pedlosky, J. Geophysical Geophysical Fluid Dynamics; Springer: New York, USA, Fluid Dynamics; Springer: New York, NY,NY, USA, illy, D.K.The Thedevelopment development maintenance of rotation in convective storms. In Intense Atmospheric illy, D.K. maintenance of rotation in convective storms. In Intense Atmospheric Vortices; Vortices; Bengtssoon,., ighthill, J., Eds.; Springer: Berlin, Germany, 198. Bengtssoon,., ighthill, J., Eds.; Springer: Berlin, Germany, 198. Kurihara, Y. Influence of environmental condition on genesis of tropic storm. In Intense Atmospheric Vortices; Bengtssoon,., ighthill, J., Eds.; Springer: Berlin, Germany, 198.

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