Andrei Nechayev PHYSICS of the ATMOSPHERIC VORTEX Hurricane Polar low Tornado The book contains the analysis of general properties, conditions of formation and possible intensifica- tion mechanisms of the natural atmospheric vortex: tropical cyclones, polar lows and tornadoes. For the first time the integrated physical approach to the theoretical description of the different scale vortex is undertaken. The universal overheat instability mechanism in an ascending moist air flow is proposed and analysed. It is shown that the intensification of wind speeds simultaneously with the temperature and pres- sure redistributions can be result of the reorganization of the dissipative structure which key parameters are the air lifting velocity and the air temperature. In the case of hurricane this reorganization can lead to eye formation. The transition of the dissipative structure to a new state may occur when the temperature lapse rate in the zone of air lifting reaches a certain critical value. An accordance of the observational data and the proposed theoretical description is shown. The book is of interest to the wide range of researchers specializing in Atmospheric Science and also to investigators of self-organization phenomena. 2 Foreword Hurricane is a huge atmospheric vortex well distinguishable from space thanks to an ex- tensive accumulation of clouds moving as a broad spirals around the center – a hurricane “eye”. The diameter of this cloudiness can reach hundreds of kilometers. The hurricane possesses a number of the properties obvious to any unsophisticated observer. The first one is a very strong (hurricane) wind with maximum speed exceeding 33 m/s (conven- tional magnitude). In the most powerful (“mature”) hurricanes wind velocity can reach 90- 100 m/s. Very high wind speeds are observed for hundreds kilometers from the hurricane center and at heights up to 10-13 km. The second property is an intensive and continuous precipitation, downpours and thunderstorms intensifying at the hurricane landfall. The third is that the hurricane center is warmed relatively to the surrounding atmosphere. This “warm core” is a cylindrical zone with a diameter of 20-50 km and a height up to 15 km with an overheat to the environment reaching 10-16 °C. Some properties are not so obvi- ous but rather characteristic. The wind in a hurricane always has a basic direction on a circle round the center, to be precise on a spiral converging to the center (in the bottom part of a hurricane) and diverging from the center in the top. The surface atmospheric pressure in the center of a hurricane is less than on periphery. Polar low as a natural phenomenon has been described for the first time in the mid-eight- ies of the XX-th century. Its surprising structural similarity to a tropical cyclone and its main difference – smaller spatial scale (<500 km) and essentially smaller time of generat- ing (within one day) – all indicates a resemblance of physical mechanisms of formation. Polar lows arise during winter time in high latitudes. For their formation some conditions are necessary: first of all is the presence of a wide area of open water and of a neighboring big air mass with a low temperature (may be an ice shield or a continental land). The tornado is an atmospheric vortex similar by structure to the hurricane but consider- ably smaller by scale. The diameter of the central part of a tornado as a rule makes tens or hundreds of meters. The maximum speeds exceed those of a hurricane. Destructive prop- erties of a tornado are known very well that one can not say about its physical properties: natural investigations of a tornado are complicated for clear reasons. The first idea which put a hurricane theory in motion belongs to the authors of [1]. They proposed and analysed the instability mechanism (Conditional Instability of Second Kind – CISK) which basic conception comes to the formula that “cumulus clouds and the large-scale circulation cooperate, rather than compete.” The theory of CISK has got an approval from some researchers and has been subjected to criticism from others [2-5]. The alternative theory [4-6] named WISHE (Wind Induced Surface Heat Exchange) was based on mechanic and thermodynamic conservation laws. The presence of deficiencies in basic assumptions of WISHE-theory was specified repeatedly in some works [7-8]. In present investigation an essentially new approach considering hurricane as an open nonequilibrium physical system having a number of steady states is proposed. Be- tween these states transitions are possible. The primary goal of the given work consists in search of the mechanism destabilizing a homogeneous state of the atmosphere and transforming it into a vortex with highly redistributed key parameters: wind velocity , pressure and air temperature. 3 1. Basic equations An atmospheric vortex observed in nature should be submitted to the classical physics laws. These laws are manifested in the following equations. Euler’s equation: ∂V 1 +∇()VV=− ∇+pF (1.1) ∂t ρ where V is a wind velocity vector, p is an atmospheric pressure, ρ is a dry air density, F is a resultant volume forces including gravity and inertia forces; The mass conservation law: ∂ρ +∇()ρV = 0 (1.2) ∂t The equation of ideal gas law: pR= ρ T (1.3) where T is an air temperature, R is the gas constant for air; R=287 J/kgK The equation of thermal balance: dT +− 2 (1.4) =+SQ++QQ++MT− ν ∇ dt Lr rpeT where S is a direct heat flow from ascending warm air; + – Qr , Qr are correspondingly radiating heating and radiating cooling, which depend on daytime, on height over the earth and on atmospheric transparency but do not dependent on a velocity field; Mp-e is an air cooling due to precipitation and evaporation; νT is a turbulent heat dissipation factor depending on velocity field; Q is the influx of latent L heat of condensation: Lc dqs QL =− B (1.5) cp dt . 6 where Lc is the latent heat of condensation (Lc = 2,5 10 J/kg); Cp is the specific heat of air at constant pressure (Cp= 1000 J/kgK ); B is a proportionality factor; qs is the saturated mixing ratio of water vapor: e q = 0,622 s (1.6) s p where es is the saturated vapor pressure [9]: 17,67t (0<t<60°C) es = 6,e105 xp (1.7) t + 243,5 [e]=1mb 4 The mathematical description of atmospheric vortex is carried out as a rule in cylindrical (r,φ,z) system of coordinates. In our case the dynamics equations can be written: ∂u ∂u v ∂u ∂u 1 ∂p v2 + u + + w =− ++fv (1.8a) ∂t ∂r r ∂ϕρ∂z ∂r r ∂v ∂v v ∂v ∂v 1 ∂p vu + u + + w =− −−fu (1.8b) ∂t ∂r r ∂ϕρ∂z ∂ϕ r ∂w ∂w v ∂w ∂w 1 ∂p + u + + w =− − g (1.8c) ∂t ∂r r ∂ϕρ∂z ∂z where u,v,w are radial, tangential (azimuthal) and vertical velocity, f is the Coriolis parameter, g is the gravitational acceleration. The forces of viscid and turbulent friction are not taken into consideration in the equations (1.8) as in the free air they are relatively small and become considerable near the surface only at heights less than 10 meters. The axisymmetry of stationary atmospheric vortex (the concentricity of hurricane’s isobar is well known) allow to consider all variables azimuthally independent. As long as tornadoes and hurricane central areas possess large Rossby number the Coriolis force in (1.8) can be neglected. Taking into account a relatively small value of vertical velocity compared with u and v the stationary equations can be written in a simplified form: ∂u v2 1 ∂p u −=− (1.9) ∂r r ρ ∂r ∂v vu u +=0 (1.10) ∂r r 1 ∂p − ++gb= 0 (1.11) ρ ∂z The corresponding mass conservation equation at ρ = const is given: ∂u u ∂w ++ = 0 (1.12) ∂r r ∂z Equation (1.9) is the basic equation for the stable atmospheric vortex. It is the equation of horizontal forces balance where the pressure force is compensated by the centrifugal one and by the expenditure of the radial flow energy. The equation (1.10) demonstrates the relation between the tangential and radial velocity under conditions of axisymmetry. In particular a zero vorticity (∂v/∂r+v/r=0) is required by this equation in zones where u≠0 (the periphery of the vortex). This relation is identically true at vr = const and signifies 5 mathematically the angular momentum conservation for the air parcel (the last is not cor- rect physically – see section 3) The equation (1.11) includes the hydrostatic one. Its accomplishment signifies the uni- form vertical motion or its absence. The meteorologists add into this equation the buoy- ancy value given as b=g(ρ-ρb)/ρb , where ρb is the density of some air parcel and ρ - the density of atmospheric environment. The buoyancy imitates the Archimedean force and can be both positive and negative. The mass conservation equation (1.12) plays one of the leading role in hydrodynamics. It governs the directions of air flows in the atmosphere. For instance, the updraft (w>0) is obligatory present near the earth surface when div u<0 (z=0, w=0). In a stationary vortex along the air jet (along the stream line) the Bernulli equation con- necting the absolute values of speeds with values of pressure in two given points should be carried out. Integrating Euler’s equation (1.1) along a stream line (from point 0 to point 1) and considering that p = ρRТ we will receive the Bernulli integral for barotropic atmo- sphere (V0 =0, T=const, z=const): V 2 1 1 ∂ρ V 2 ρρ− 1 + ∫ RTdl =+1 RT ln()10+ 10= (1.13) 2 0 ρ ∂l 2 ρ0 where V1 , ρ1 are velocity and density of air in the given point; ρ0 is the air density in the reference point (V0 = 0).