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

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). If pressure deficit is small we have: ln(1+(ρ1-ρ0)/ ρ0)≈ (ρ1-ρ0)/ ρ0 and (1.13) gives us the classical Bernulli integral. In the most powerful hurricanes and tornadoes the horizontal pressure deficit usually doesn’t exceed 100-150 mb and therefore the Bernulli integral keeps its constancy with a known error.

6 2. Vertical warming-up and horisontal pressure drop. How does it occurs? The pressure in free atmosphere can be divided into two parts: the hydrostatic pressure (pressure of an atmospheric column) and the nonhydrostatic one. Everything that concerns the second category isn't defined by gravity. The nonhydrostatic part of pressure is notable for the big uncertainty. The hydrostatic pressure рh at some height z is calculated from the integrated equation ( 1.11): H = ρ (2.1) pzh () ∫ gdz z where H is the thickness of the troposphere. If the air density on the height z decreases by the value of Δρ(z) the total pressure of a column above z decreases accordingly:

H ∆=∆ρ pzh () ∫ gdz (2.2) z The change of air density Δρ at the given height is connected with changes of temperature and pressure by a relationship: ∆T ∆p ∆=ρρ()− + z (2.3) zzT p Correctness of (2.3) can be estimated from following speculations. The tropospheric warming-up in hurricane occurs at the release of latent heat of condensation. This process is isobaric so the correctness of the first term of the right side of (2.3) from here follows. But the density at each z level decreases also because the overlying column of air becomes lighter on the value Δρz defined by the formula (2.2) . Total change of the surface pressure can be evaluated having integrated (2.3) with the account of (2.2): HH∆T 1 H ∆T ∆=pg− ∫∫ρρdz − g ()∫ dz dz (2.4) 00T p z T It is clear that this bulky formula is the only first iteration in the calculation of the pres- sure of an atmospheric column in hurricane but nevertheless it gives an evaluation close to a reality. We will notice only that its applying is possible only in case of stationary and nearly static conditions which take place for example in a calm hurricane eye. The hydrostatic pressure drops owing to a warming-up of a tropospheric column repre- sent the basic but not unique part of the total pressure deficit between the hurricane center and its periphery. For exemple, the inertia forces bring in a contribution in the form of cen- trifugal forces. The solid-state rotation of some central zone with constant angular speed is characteristic of stationary atmospheric vortex (hurricanes, tornadoes, waterspouts, dust "devils"). This rotation can lead to additional pressure drop in the center. However just the hydrostatic pressure deficit Δp h provides the force which sets atmosphere in motion. This 7 active deficit will be named by "pressure forcing" or "forcing"to distinguish it from the concept "pressure" and "pressure drop". The primary warming-up of the troposphere at hurricane formation occurs in tropical clusters due to latent heat release in the rising moist air parcels. Air receives its high humidity from the ocean surface evaporation when water temperature exceeds by 1-2°C the temperature of air which is equal approximately to 26°C. This temperature difference doesn't allow air to rise upwards (tropical atmosphere in zones of hurricane genesis is con- sidered enough stable) however there are some poorly explored phenomena (for example, easterly waves or Ekman pumping [1]) which enable air to reach the condensation level. In rising parcels ("bubbles") there are two basic confronting processes: adiabatic cooling (-10°C on each kilometer of lifting) and an isobaric warming owing to the condensation process according to specific humidity of a parcel q. The rate of latent heat release of last process can be evaluated as:

Lc ∂qs ∂Tb (2.5) QL  w cp ∂Tb ∂z where qs is the saturated mixing ratio; Tbis a temperature of bubble; The most of this condensation heating goes on the compensation of adiabatic cooling with the lapse rate Γd= 10°C/km. Substantially smaller part dissipates in surrounding at- mosphere with corresponding warming-up. Some part remains in a bubble providing its further lifting. Process of warming-up of the atmosphere at the lifting of moist air isn't studied practically. It is possible to assume that the bubble air with Tb> T mixes up with environmental air partially. Thus losing a part of its volume and heat the bubble keeps "the core" with the greatest possible temperature and decreasing in mass moves to the top limit of the buoyancy. Nobody knows the temperature of a bubble. Modern dropwindsondes don't allow to measure it precisely as their time constants (at 20°C: 2,5s for temperature measuring and 0,1s for humidity [11]) and possibilities of the spatial resolution give obviously averaged values. There are objective data for functions qs(T), T(z) and w(z) but concerning the magnitude of ∂Tb/∂z in (2.5) it is possible to build assumptions only. Generally speak- ing, in the idealized case of bubble lifting its vertical temperature gradient must satisfy the equation:

∂TbcL ∂qsb∂T (2.6) =−()+ Γd ∂z cp ∂T ∂z

Takng into account that ∂qs/∂T ≈ qs /15 we can derive for bubble lapse rate:

∂TbdΓ Γb ≡− = (2.7) ∂zq16+ s / Practically (2.6) gives us the equation of moist adiabat according to the boundary condi- tion: Tb (z=W) = Tw At rather high altitudes when the specific humidityq is small enough ∂Tb/∂z should be close to dry adiabatic lapse rate. In the bottom layers ∂Tb/∂z should be between its mini- mum (the bubble keeps temperature) and the ϒ , the lapse rate in the surrounding atmo-

8 z km

Гb 5 w

4

zm

QL 3

q 2 s

W

0 6 12 18 qs g/kg 0 w0 2w0 3w0 w 0 3 6 9 Гb °C/km Fig. 1 Qualitative altitude dependences of qs , w , Гb and their product QL – the rate of latent heat release on the level of maximum warming-up zm ; W is the condensation level. . . sphere. Anyway the rate of latent heat release QL is proportional to ∂qs/∂Tb ∂Tb/∂z w(z) and therefore should have a layer or a zone of maximum values as ∂qs/∂Tb decreases with height and Γb and w(z) increase (Fig 1). Two conditions are necessary for simultaneous lifting and condensation of moist air: Tb > T and ∂Tb/∂z < 0. The time constant of relaxation of liquid phase in clouds at 0°C < t < 20°C makes approximately 1s [12]. At small w (in clusters w ≈ 10-2 m/s ) the pro- cess of condensation apparently must correlate to the lifting velocity (hardly the bubble moves jerky). Researchers believe [13,15] that at the beginning of updraft the bubble has a very small overheat (an order of 1°C) which increases in process of lifting in cold layers of atmosphere. If air with the moisture content q = 18 g/kg reaches 10 km of altitude (where all water vapor will be obviously condensed) it will be overheated concerning standard atmosphere by 3°C over all. If air has q = 21 g/kg (typical specific humidity in the hurricane center) the overheat will be 10°C. If the freezing of condensed water will be considered (latent heat of fusion is equal 3,3.10 J/kg) we will receive in addition 5-6°C and an overheat in 16°C which is characteristic of the top layers of mature hurricanes [16]. To understand complex physical processes some simple everyday analogies help. Let’s imagine a group of people standing in the middle of an enormous covered stadium and

9 z km

7

6

5

4

zm D

3

2

W

0 -20 0 20 t °C

Fig. 2 Qualitative temperature soundings illustrating the heating of the atmosphere by the ascending moist air. The temperature of "bubble"(heavy line), the temperature of atmosphere before (thin line) and after (dushed) the heating. Dush-dotted line is the dry adiabat. D is the level of maximum warming-up starting upwards the balls filled with warm air. The stadium has the flat roof ( the meteoro- logical analogy is tropopause)and balls rest against it and stop. Infinitely there are a lot of balls, they continuously rise collecting under the roof. Gradually they densely fill the first layer then the second, the third. Those balls that are in the center push aside that over them to the edges – so all the roof is filled. Filling a layer behind a layer balls actually replace the air under the roof by other warmer air. If you imagine as well that they can be slowly blown in the course of lifting they will warm up all atmosphere of the stadium. It is clear that the energy of hurricane comes from a temperature difference between ocean and atmosphere. Hurricane is an unclosed system whose energy initially increases and finally dissipates. The heat released from water vapor condensation is transferred to the surrounding air. Surface pressure decreases and the air from all directions is directed toward this place. It seems all should end with it. So it would if the air had no possibility to continue the movement upwards. But the moist air reaching the condensation level has such a possibility. Besides this air coming to the zone of low pressure loses its density which it had on periphery (according to the ideal gas law) and rising upwards it becomes

10 even lighter. Positive feedback is available. But it is sufficient to look at an usual cumulus cloud in which the same physical processes must go on and there are neither warming-up nor a hurricane-force wind. Thus the understanding vanishes. They say that hurricane is an enormous cumulus thunder cloud. But hurricane is not a cloud, it represents the system of clouds participating in a joint rotary motion round the center. Circulation is the main property of an atmospheric vortex. Presence of an axis of rotation gives the birth to special mechanical characteristic – the angular momentum. The law of angular momentum conservation is one of the strongest laws of physics. It is often applied to hurricane air parcels as they rotate around the center. However the angular mo- mentum is constant if vectors of all forces applied to a body or to its part pass through the rotation axis. The forces untwisting hurricane are pressure forces which accelerate air in the bottom part of hurricane in a boundary layer. These forces are not perpendicular to the rotation axis (there exist a so-called inflow angle) that is they have the radial component directed to the axis and tangential component creating linear acceleration. The points on a curve v (r) satisfying vr = const concern not the same air parcel (as should be at the real- ization of any conservation law) but different parcels and even different streams! The fact that the empirical v(r) is often close to inversely proportional function is connected likely with axial symmetry and its realization in both Euler's equations and Bernulli law that will be discussed later. Generally speaking, conservation laws in physics are a very strong remedy. And as any strong remedy it is necessary to apply it with care. One cannot forget that hurricane and tornado are open and nonequilibrium systems both in mechanical and in thermodynamical senses. Let's use an everyday analogy once more. Let's imagine a bathtub and the water flow- ing out through the drain hole. Water in the bathtub initially is quiet and motionless. Its angular momentum is equal to zero. You have opened a drain and have created pressure deficit. Streams have directed into the drain hole and a rotation of the water has started. What forces untwist it? Evidently the gravity of the water over the hole does. Where does the angular momentum come from? As the streams can't flow vertically downwards (the total flow is limited by the diameter of a hole). The rotating water has a non zero angular momentum. If you shut the drain hole the rotation quickly decreases due to water viscos- ity. Thus, in the system of water the angular momentum doesn't conserve: it appears and disappears. Probably, in the system of "water-bathtub-Earth globe" the angular momen- tum remains constant as this system is isolated and closed. But this fact is uninviting. In the hurricane all is essentially similar. Only the sink is above: the air flows upwards obeying a certain pressure gradient. The everyday model of hurricane is a flue with a chimney. Air in the flue is heated due to wood burning. It is much warmer than the sur- rounding air, and consequently the pressure in a chimney is less then atmospheric one by the magnitude Δp=ρghΔT/T where ΔT is an average temperature difference, h is the height of a flue with a chimney. If ΔT=30°С, Т=300 K, h=5m the pressure drop will be equal to 5 Pa. Calculating the wind velocity using the Bernulli law we will receive its magnitude close to 3 m/s. Not bad. It is clear why in the kindled stove an air hoots. Air stream in a flue don't rotate (though it is not obvious as the chimney isn't transparent) but it is turbulent. Similar processes go apparently in cumulus clouds and in «hot towers». For rotation the additional conditions are necessary.

11 3. Some properties of an atmospheric vortex

Vertical pressure gradient compensating gravity force is present in atmosphere constantly. The horizontal hydrostatic pressure deficit, or "forcing", always makes free air move. The character of this movement (rectilinear or vortical) depends on the form of surfaces of p/ρ potential. If this surface has a “gap” air streams will direct to its center and a circulation can start around there. Let's consider the simplified velocity distributions in a vortex with an axisymmetric pressure drop with a radius L (Fig. 3) applied to the boundary layer with thickness W where the main centripetal flows are observed. The structure of an idealized stationary vortex should satisfy the conditions: v = u = 0 at r = 0,L and w = 0 at z = 0. Pres- ence of wind extremums in functions v (r) and u (r) and existence of updraft area (w > 0) in a zone of negative divergence ( divu =u/r+∂u/∂r < 0 ) from here follows. Horizontal deficit of hydrostatic pressure is always maximum near the surface and the magnitude of pressure gradient ∂p/∂r changes weakly with height within the boundary layer (0 < z < W) as the main warming is characteristic of the top and middle layers of hurricane. So that the integral form of the mass conservation law gives:

2 (3.1) wr0ππmm=−2 rWum where um is the maximum radial velocity (the centripetal direction of flow is negative);w 0 is the vertical velocity at z = W; rm is the radius of maximum radial wind (we take it ap- proximately equal to the radius of updraft area which exactly is defined by the condition: divu < 0). Certainly um and w0 are averaged. We have:

2um (3.2) w0 =− W rm

Air acceleration takes place in a boundary layer z Rm is called as a zone of potential rotation, an air flow here moves by the ascending helical spiral tending to RMW. At heights where there is no surface friction the Bernulli law (within a stream) gives: 1 ∆=pp−=pvρρ()22++uw2 +−gz()z (3.3) 0 2 0 where р0 and z0 are pressure and height in some reference point r = L. Dependences v(r) and u(r) give the information of velocity field but don't reflect com- pletely the process of centripetal movement. Points on these curves concern as a rule dif- ferent streamlines and some sections belong to substantially different zones. For example, the area r < Rm in some vortex represents a zone of "solid-body" rotation (v = ωr, ω = vm/ Rm) in which we can use the equation of forces balance but the Bernulli law is incorrect. In a zone r > Rm (within a separate streamline) it is good to use both forces balance and 12 p

p0

pc

R r L v m m r

vm

vum

0 r Rm m L r -u

-um

0 rm L r w

w0

0 L r

Fig. 3 Pressure and velocity distributions in the idealized atmospheric vortex.

13 Bernulli law, the dependence v (r) here is close to inversely proportional. Really differen- tiating (1.13) on r (for a case of small w and z = const) and adding it to (1.9) we receive for v (r) and for vorticity ξ: v ∂v (3.4) + ≡=ξ 0 r ∂r –1 The equation (4.4) is satisfied at v ~ r . In a circle r < Rm vorticity ξ is constant and equal to the doubled angular speed: ξ = 2 vm/Rm. In the area r > Rm the vorticity may be equal to zero if v ~ r –1. In hydrodynamics there exists the theoretical vortex (the so- called Renkin vortex) satisfying conditions: v= vm r/Rm at r< Rm and v= vm Rm/r at r > Rm . The horizontal pressure deficit calculated for this vortex with the formula ∞ v2 ∆=p ρ∫ dr (3.5) 0 r

2 is equal ρvm and doesn't depend on Rm. In real atmospheric vortex v (r) usually differs from that of Renkin vortex but is close to it . The dependence u (r) at r > rm in the bound- ary layer of real hurricanes is also close to r –1 [17]. This fact allows an average radial flow 2πrρuW to remain invariable that corresponds to zero divergence and to absence of vertical motions of air (there is no cloudiness in a large area of hurricane air circulation). In tropical storms the air lifting occurs in total area r < rm with average velocity wa = 2|um|W/rm . In hurricanes it is basically concentrated inside the ring Rm < r < rm ( wa = 2 2 2|um|Wrm /(rm – Rm ) ). For steady state conditions it is obligatory that the stream lines of an atmospheric vortex had closed trajectories that is the horizontal flows must turn to the vertical direction. The moist air reaching the condensation level can continue its lifting only in the form of buyo- ant warm bubbles. It occurs if the bubble is surrounded by sufficient volume of the air with Т < Tb. In other words, if the flow I approaches condensation level only some part equal I/k where k > 1 can continue the upward motion. Actually 1/k is a part of the full area oc- cupied by bubbles. To understand better the structure of an atmospheric vortex let's make a mental experiment. Let's assume that in the layer 0 < z < W there is a horizontal pressure deficit Δp h creating a centripetal air inflow. The maximum possible magnitude of this flow 2 is: Imax = 2πrmρ|um|W , and its energy according to the Bernulli law (v=0) is equal: ρum /2 =Δph. This horizontal flow turns into the vertical one which for the reason specified above can't exceed Imax /k. To conserve the continuity of a flow and its energy (Δph=const) it is possible only having reduced um in k time and having established circulation with the speed vum satisfying the condition: u v22+=()m u2 (3.6) um k m As a result we come to the relationship (3.7) vu=−k 2 1 which doesn't depend neither on dimensions of a vortex nor on pressure drop magnitude

14 and is defined only by physics of the process of convection at the condensation level (factor k). The circulation direction can be obtained by means of Coriolis force or by a primary vortex . Due to numerous observations of hurricanes a surprising constancy of the inflow angle ( tgα =u/v) equal 22-25° in a wide area r > rm was discovered [17]. As tg22° = 0,4, so k=2,7. Hurricane is a huge "tube" providing air transportation across the troposphere from bottom to the top. The rough scenario is that: a pressure drop causes a centripetal flow in a boundary layer, the moist air accelerated by pressure forcing rises to the condensation level. Then "relay race" is picked up by a convection and it lifts the warming-up air to the greatest possible level. Probably the convection receives an aid from lateral pressure gra- dients “compressing” the central core from all sides. Then jets diverge all around forming characteristic spiral rainbands. The main link in this chain is the hydrostatic pressure drop. It is created by the warmed atmospheric column which is in the center of a vortex. There is a question: why does the warm core of hurricane exists so long time and its temperature anomaly isn't liquidate by horizontal gradients of pressure injecting into the warm center a cold air from periphery? The answer may be obvious. The centrifugal force of the rising and rotating air in the eyewall carries out this protective function! The basic warming-up of troposphere is observed at levels above 2 km. In mature hurricanes the overheat of 7-10 °C can be found at heights of 15 km[16,18]! Moist air in the eyewall rises to 10-13 km keeping its high speed of rotation and providing the vertical and radial advection of angu- lar momentum. The dry air which does not have buoyancy can't rise to these heights as the maximum height of its pure possible hidrostatic updraft can't exceed Δph/ρg that for ma- ture hurricanes is no more than 1200m. Hence, pledge of survivability of an atmospheric vortex is the constant lifting of rotating moist air above condensation level. The magnitude of the stationary pressure drop in hurricane (and in any other atmospher- ic vortex) depends not only on factors forming it (for example on tropospheric warming- up) but also on the value of outflow. If outflow isn't present or it is rather small the pres- sure drop is quickly liquidated owing to local increase in pressure. In case of low-level circulation the outflow from the surface can go only through the level of condensation. It should be provided by the additional nonhydrostatic pressure Δpnh and pressure force ∂pnh/∂z which appear from collision of centripetal streams in a boundary layer of a vor- tex. The magnitude of the outflow and of the ∂pnh/∂z should come to mutual accordance otherwise air flows will select a new way of motion (a new combination of u, v ,∂u/∂r). These nonhydrostatic forces counteract centripetal flows and turn air flows into the hori- zontal circulation and upwards. The magnitude of pressure gradient ∂p/∂r in the RMW area depends on these forces . The Coriolis force helps flows to move in mutual direction (counter-clockwise in Northern hemisphere). Probably it provides a vortex with primary circulation and with the primary total angular momentum which can't be changed sharply. For a rather weak atmospheric vortex (clusters, tropical storms) with small value of Δph and using the relationship rmw = –2umW the equations (3.1) and (3.2) lead to the formula for rm – the approximate radius of updraft area characterizing a certain effective size of a vortex:

12, ∆pW r = r ,(∆=pp− pr) (3.8) m w rm0

15 Taking Δpr = 1mb, we can receive for different atmospheric vortex.

w , m/s W , m rm , km Cluster 2 . 10–2 1000 600 Polar Low 2 . 10–1 1000 60 Mesocyclone 2 1000 6 Tornado 10 500 0,6

The zone of the horizontal pressure deficit in a hurricane boundary layer can be divided into three sections relatively. The first is a central area of the forced (solid-body) rotation 2 (a hurricane eye, r < Rm). The radial pressure drop over this section is close to ρvm /2. The second section is an area of free rotation (potential flow, r >rm). Here air goes in jet 2 2 and the pressure drop according to the Bernulli law is equal ρ(vum + um )/2 what may be 2 close to ρvm /2 if we assume that jets only change its direction at r < rm. The third section (Rm < r < rm) is intermediate between the first and the second. It is the area where flows begin to decelerate and change its direction untwisting an eye. The pressure drop here has a nonhydrostatic addition which is defined by Euler's equation and the mass conservation 2 law. It is possible to assume that the possible rest of pressure deficit equal to (Δp – ρvm ) is applied to this zone where ∂p/∂r has its maximum and the additional nonhydrostatic pressure rejects air flows upwards (Fig. 4,5). p v -u

pc Rm r Rm rm r Fig. 4 The hydrostatic (dashed) and total, contain- Fig. 5 Azimuthal (heavy line) and radial (thin line) ing the nonhydrostatic addition (solid), pressure dis- velocity profiles of a mature hurricane tributions

16 4. About the mechanism of hurricane formation. Dissipative structure

The hurricane history begins with cluster and tropical depression. Cluster is a gathering of cloudiness where a vertical lifting of moist air is observed within several days. The typical cross-section dimensions of clusters vary from 200 to 1000 kilometers. An atmo- spheric warming-up in clusters is very weak: no more than 1°C in the top layers however the earth surface has an appreciable pressure deficit (1-2 ГПа). There is an opinion that air lifting and pressure deficit are caused by the external reasons such as «easterly waves» or « Ekman pumping». We would like to present here physical circumstance capable to lead to a stable primary pressure drop in the cluster area. Indeed lifting and accumulation of water vapor above condensation level (z = W) is accompanied by replacement by this vapor of corresponding volumes of dry air. At heights z > W in a zone of cloudiness air pressure is equal to the sum of partial pressure of dry air and water vapor and it is equal to the pressure (at the same level) of dry air in a zone of clear sky. However total pressure of an atmospheric column in the cloudy zone will be less than that in the nearby areas of the "clear" sky because of the replacement of molecules of air by more light molecules of water. The maximum hydrostatic pressure deficit will be equal: H µµ− H ∆= ρ aw= ρ pgh ∫∫dz 06, gdw z (4.1) W µa W where ρw is the density of water vapor; µa , µw are molecular air and water weights; H is the the maximum height of water vapor lifting ; ρw = 1,2qs Assuming that at condensation level (z=500m) qs=18 g/kg and considering that in clusters until 5-6 km heights 80 % humidity remain [17], we will receive an evaluation Δph = 1,6 mb that corresponds to the data [19]. The dimensions of cluster (hundreds of kilometers) don't allow to low-level radial flows (which velocity is no more than 1m/s) to liquidate this pressure drop before the new portion of water vapor comes aloft. In the meteorological literature hurricane is compared to "Carno engine" converting the energy of heat in the mechanical energy [5]. Extending this analogy it is possible to com- pare hurricane with an internal combustion engine which can't start without ignition – a starter. In hurricane by such a starter the ascending flow of moist air and the primary pres- sure drop are. The water vapor serves as "gasoline" which is burnt down in a "fire cham- ber" of condensation. The hurricane engine starts to work steadily when due to "primary draft" the minimum pressure drop providing an uninterrupted delivery of water vapor to the condensation level is reached. The primary pressure drop should cause centripetal air flow which under corresponding conditions (Coriolis force or a certain vortex-germ) can lead to a stationary circulation. According to cluster observations [15,19,20] circulation especially in a boundary layer is the absolutely necessary condition for hurricane formation. Only it due to radial flow convergence is capable to support an air ascent in the vortex center and to provide here the maximum warming-up and the minimum heat dissipation as azimuthal speeds are

17 minimum. Just the circulation provides the positive feedback w → ΔТ → Δph → w. due to increasing of the maximum radial velociy um. The moist air ascent leads firstly to a warming-up of top layers, then middle and bottom layers become heated that increases the horizontal pressure drop Δp = р0– р which is applied to all boundary layer 0 < z < W. For considered air lifting area r < L there existsis a monotonously increasing function w0(Δph) where w0 is a vertical velocity at condensation level. For the hypothetical vortex described in previous section w0 may be expressed by the formula: 22()∆p 12/ (4.2) w ≈ h W 0 L Formula (4.2) expresses the essence of initial intensification of hurricane. In the presence of favorable external conditions (the main thing is the absence of vertical wind shear) necessity for the primary air lifting disappears. A tropical storm can support itself and can leave his "cradle" and set out on a big voyage. The further warming-up of troposphere lead to the direct increase of the forcing Δph and of all vortex velocities including um and w0 The warming of low and middle tro- pospheric layers goes according to the rate of a thermal flux which as appears from (2.5) should have a maximum at some height zm. Really the magnitude of ∂qs /∂T decreases with height (at levels from 1 to 4 km decreases in 4-5 times), and the magnitude of w increases (in clusters at the same heights increases in 3-4 times [19]). So there should be a layer where the warming-up goes faster than in other layers (Fig. 1,2). It leads to the reduction of the lapse rate ϒ in underlaying layers, as

∂T TTmW− ∂T (4.3)  ;ϒ ≡− ∂z zWm − ∂z where Тm is the temperature in a layer of the maximum warming-up, ТW is the tempera- ture at condensation level which practically doesn't change. Let's write down the equation of heat balance for an area rW having kept in (1.4) S, QL and turbulent dissipation .

dT TTb − Lc ∂qs ∂Tb 2 = − wTΦ +∇νT (4.4) dt τT cp ∂Tb ∂z The first term on the right-hand side of equation (6.2) is responsible for heat exchange of a bubble with the surrounding atmosphere going with the time constant τT (it is maxi- mum in the top layers of troposphere and it is minimum in bottom, where Тb ~ T); the second term – a rate of latent heat of condensation (it is minimum in the top layers and maximum at level zm); Φ is a heat transfer effectiveness ratio (Φ <1) depending on many factors and first of all on geometrical parameters of a bubble, on temperature difference (Тb –T) and degree of updraft turbulence ( actually on velocity field). The more intensive vapor condensation occurs, the more a bubble expands and the more intensively its mixing with atmosphere occurs: Φ increases. Boundary conditions for (4.4) correspond to axial symmetry : ∂Т/∂r = 0 (r=0, L)

18 Vertical velocity of bubble w should satisfy the equation (1.11). Let's exclude from it a hydrostatic part and add a standard Newtonian drag force: dw TT− = b gC−+dw22 ν ∇2w (4.5) dt T bb w where Тb is a bubble temperature, Т is the temperature of surrounding atmosphere, db is the diameter of a bubble, Cb is a proportionality factor. Thus, at the initial stage of a tropical storm there is a growth of Tm and wm at level of maximum warming-up zm together with growth of external forcing Δph=p0 – ph. We will introduce the quantity of a total air flowI through the vortex: L ∫ 2πρrwdr = It() (4.6) 0

At the initial stage the total air flow I grows with the increasing of Δph as the vertical velocity w0 increases according to (3.2). At some instant the reduction of the lapse rate in a layer of the maximum warming-up will decrease the bubble acceleration. Therefore wm will increase more slowly then will cease to increase and even can start to decrease (Fig. 6). However Тm and pressure forcing Δph will increase as the heating of total tro- pospheric column continues (w > 0). There the conflict known in the physics (basically in semiconductor electronics) between an air flow (current) and pressure drop (voltage) takes place. In electronics the presence of a "falling" segment of the current-voltage char- acteristic leads to the classical instability (for example the well-known effect of Gunn). Let's introduce the so-called "flow-forcing" characteristicI (Δph) as a certain analogue of current-voltage characteristic. It is imaginery characteristic but not fully virtual, theoreti- cally it can be measured and certainly it can help to understand, may be to explain the behaviour of atmospheric vortex subjected to intensification. In our case there should be a falling segment (a N-shape segment) of characteristics wm(Тm) and I (Δph) where ∂w/∂T <0 and ∂I/∂p<0 that gives a fundamental basis for redistribution of w and T leading the system to a transition to the new state (Fig 6,7). I C wm A A B

H G

TW Tcr Tm Δph

Fig. 6 Qualitative form of the dependence wm(Тm) Fig. 7 The flow-forcing characteristic I(Δph) of sup- in case of critical overheat (solid line) and with- posed dissipative structure. The transition ABC is out air flow decelerating (dashed line). Tcr cor- accompagnied by the redistribution of w(r) and the responds to critical lapse rate (when ∂w/∂T =0) by warming-up of a central column the formula: Tcr= (∂Т/∂z)cr (zm – W)+TW

19 In effect, the equations (4.4) and (4.5) represent classical (in the sense of Turing-Pr- igozhin theory) dissipative structure in which under certain conditions a horizontal strati- fication of two parameters (activatorw and inhibitor T )occurs. Stratification should begin in vicinities of the layer of maximum warming-up zm where the dependence w (T) can have a N-shaped appearance. The basic dissipative structure occupies the space 0

∂w TT− = b gC− dw22 (4.8) ∂t T bb Boundary conditions can be trivial: ∂T ∂w (4.9) ||=== = 0 ∂r rL00,,∂r rL

The dependence Ф(v) results from the fact that the latent heat release equal QLΔt is dis- tributed along an arch which a bubble follows. The length of this arch is equal v Δt . There- fore the heat equal QL/v at the unit of arch length is released. So the less is v in the given point the more this place is warmed-up. The equation (2.6 ) controling the bubble temperature must be added to the equations (4.7) , (4.8) with the equation of a full flowI through the structure: L πρ = ∫ 2 rw0 ()rdrI (4.10) 0 The equation (4.10) controls the horizontal redistribution of w(r). Boundary conditions for z= W are Т = TW and w=w0, where w0, is the initial vertical ve- locity of a bubble at the condensation level. It is defined by processes in a boundary layer and can't be set. The equation (4.7) defines a warming-up of structure and establishes a temperature field with zones of the minimum lapse rate The equation (4.8) defines ac- celerations and decelerations of bubbles and stimulates the warming-up by means of parameter w. The velocity w comes from the initial velocity w0, monotonously increasing with the growth of pressure drop (see (3.8)), and the velocity received from the equation (4.8 ) for corresponding level z. About possibility of sharp dependence w(T) it was told above. According to (4.8) at Tb →T the velocity w approaches zero. When it occurs the vertical flows must be redistributed. Certainly, the analysis of critical conditions even for such relatively simple dissipative structure is impossible without numerical simulation. We will try to evaluate these condi- tions proceeding from the simple reasons. In a zone of condensation the bubble tempera- ture apparently differs a little from the air temperature : no more than on 1-2 °C though no measurements do possible to confirm it. Bubble decelerating should come, when its minimum lapse rate, defined by the formula (2.7), becomes equal to the lapse rate of an

20 atmosphere. Hence, for the given height the critical state of the dissipative structure may require the condition: Γ d [qs]= 1mb ϒcr = (4.11) 16+ qS / where qs is the saturated mixed ratio of a bubble at the given height. For the supposed level of the maximum warming-up (650 mb) Т = 5 °C, q = 10g/kg. Ac- cording to the formula (4.11)ϒcr = 3,8 °C/km. We will compare this result to the data of observations published in [17].Table 1 gives the average temperature soundings for stan- dard tropical atmosphere (Jordan, 1958), clusters and weak hurricanes. Height Height Jordan Cluster Eye RMW 660 km mb km t °C t °C t °C t °C t °C sfc 0 26,3 26,1 – 24,5 25,7 900 1,0 19,8 20,7 22,1 21,1 20,2 800 2,1 14,6 15,4 18,7 16,7 15,5 700 3,3 8,6 9,4 14,3 11,3 10,2 600 4,7 1,4 2,0 10,0 5,0 3,3 500 6,3 -6,9 -5,7 2,7 -2,3 -4,7 400 8,3 -17,7 -15,8 -7,2 -11,2 -14,9

The minimum lapse rate (ϒ = 3 °C/km) can be seen in a hurricane eye in the layer between 600 and 700mb where the typical inversion of mature hurricanes is observed [16,21]. Ap- parently this level corresponds to the layer of maximum warming-up. In eyewall area the lapse rate is more than 4 °C/ km. Probably the magnitude of 3°C/ km exceeds the critical gradient as the dissipative structure was already reorganized. The maximum temperature of a bubble is defined by the equation (2.6) if the boundary condition Tb (z=W) and humidity q are known. If q=18g/kg the bubble temperature at the height of 3,7 km (650mb) will be equal to 11 °C. The eye temperature of a weak hurricane at this height (Table 1) is close to 12 °C; the typical initial temperature of the 650mb- inversion [16,21] is equal to 13 °C. Other words, the reorganization of the dissipative structure probably begins at the equality of a bubble and surrounding atmosphere lapse rate and close magnitudes of their temperatures. Unlike the bubble lifting to the level of free convection the full blocking of motion in our case doesn't occur, an ascending flow remains but the reorganization of its structure occurs. It is possible to use the concept of the critical overheat ΔTcr connected with the critical lapse rate by the relationship: (4.12) ∆ϒTTcr ≡−mcTz00=− rm()−+WTW −T The combined system of the simplified equations describing the behavior of the given dissipative structure is presented below. It is limited by area 0 < z < zm and 0 < r < rm where rm designate the outer border of updraft area.

21 ∂T ∂T TTb − Lc ∂qs ∂Tb 22 + u = + wvΦ()+∇νT T (4.13) ∂t ∂r τT cp ∂Tb ∂z

∂w ∂w 1 ∂p TT− + w =− nh + b gC−+dw22 ν ∇2w (4.14) ∂t ∂z ρ ∂z T bd w ∂ ∂ ∂ ∂ (4.15) Tb Lc qs Tb w0 u u =−()Γd − w =−()+ (4.16) ∂t cp ∂Tb ∂z W ∂r r

∂w ∂ρ ∂u u ∂ρ ρ + w =−ρ()+−u (4.17) ∂z ∂z ∂r r ∂r

∂u v2 1 ∂p H −=− h = ρ (4.19) (4.18) u ph ∫ gdz ∂r r ρ ∂r 0

22 2 (4.20) 2()pp0 −=h ρ()vu++wg+ z

(4.21) pp=+hn∆=pRh ρ T

rm W πρ = πρ (4.22) ∫∫22wr0 dr urm mdz 00

In the basic equations of dissipative structure (4.13) and (4.14) advective terms are add- ed, the equation (4.15) describes more correctly the bubble temperature rise; in the equa- tion (4.14) the nonhydrostatic pressure force for descriptive reasons is introduced. This pressure force appears in the boundary layer due to the air compression described by the equation (4.17) of mass conservation. This equation being reduced to the averaged form (4.16) can give us the vertical velocity at the condensation level w0(r) which serves as the bondary condition of the equation (4.14) replacing the unknown nonhydrostatic pressure force ∂pnh /∂z. The equation (4.19) gives an evaluation of the hydrostatic pressure drop depending on the tropospheric warming-up. This pressure drop controls air acceleration according to dynamic equations (4.18) and (4.20) which are correct for streamlines within a potential zone r > Rm. We must add to this system the integrated equation of mass conservation (4.22) in the boundary layer. Functioning of this dissipative structure is the following. In an area r < rm the lifting of moist air results in the tropospheric warming and the pressure deficit ( p0 – ph ) increases accordingly. The initial distributions of w(r),T (r) within this area are close to homogeneous. The average temperature grows slowly in all the layers so the pressure deficit increases, vertical speed w0 grows also. Streamlines for 22 this stage are represented in Fig 8a. As the center of the structure always has a weak overheat the lapse rate in the layer of maximum warming-up reaches its critical magni- tude near r = 0. As it was marked above, the dependence w(T) in a critical zone can be sharp enough: vertical velocity can reduce to zero at changes of Т less than one degree. The reduction of w0 near the center (due to decelerating of air lifting in the layer of the maximum warming-up) leads to corresponding increase of |divu| (due to narrowing of the zone of lifting) and to increase of w0 in areas remote from the center. The redistribution of initial vertical speed at condensation level w0 occurs: in the center it falls to a minimum, on periphery considerably increases (the total vertical flow is constant). The value of w0 serves as a boundary condition at z=W for the equation (4ю14). To provide the air updraft in a new place pressure forces must be redistributed too: according to N-shape section of the characteristic I(Δp) the domain of nonhydrostatic pressure forces must be formed. Really such domain exists in hurricanes in the zone of sharp pressure gradient within the area of eyewall. The centripetal forcing of rotating jets should create the zone of ring- shaped domain adjoining to the RMW from the inside. This domain (area of strong pres- sure forces) decelerates and reorients centripetal flows and also provides the basic updraft in the zone of inner ring of convection. Really the eye wall which has a width of 1-5 km always adjoins RMW (more subcentral only). Reorganization process can be avalanche. Decelerating of radial flows near the center increases the local nonhydrostatic pressure , air flows will search for a new way upward. The flows with u>0 will dissipate temperature z Fig. 8 Schematic depiction of secondary circulation of a tropical storm (a) and hurricane a (b). The ascending streamlines rounding the inversion form the outflow zone where divu>o and an air descent may occur W

r z

inversion b

W

r Rm

23 (the second term in the left side of the equation (4.13)), facilitating air lifting. In addition rising and rotating jets dissipate its own heat much less (in the equation (4.13) factor Ф reduces ). Thus on some distance from the center there will be created the zone of rotation updraft (an inner ring of convection) where the temperature gradient will be closer to the standart lapse rate of tropical atmosphere (Fig. 8b). The stabilization of vertical flows will be carried out by drag forces and turbulent viscosity (the fourth term in the right side of equation (4.14)). Strictly speaking, we must use in our speculations and formulas the nonhydrostatic forc- ing Δpnh instead of the hydrostatic one because just the nonhydrostatic vertical pressure gradient ∂pnh/∂z push the air upward. But it has too much of uncertainty. For our purposes it is quite enough that the Δpnh monotonnously depends on Δph, which can be measured. The physics of the process of redistribution of internal parameters of an atmospheric vortex is connected in our opinion with the reorganization of the inner structure of an air flow. Being accelerated near the surface and uniting in a jet air parcels lift toward a layer of maximum warming-up with mechanical energy and buoyancy sufficient for overcoming of forming inversion and for further ascent (Fig.8b). Highly rotating jets on the one hand disseminate layers which decelerate lifting, on the other hand they warm up surrounding atmosphere substantially less as their linear speed has considerably increased. As a result in RMW-zone the temperature lapse rate becomes greater than that in the central zone, it tends to a standard lapse rate of a tropical atmosphere [17], jets carry away its heat to the top of hurricanes forming an anticyclone here. Apparently here, in the bottom of RMW, the powerful hurricane jets are formed turn- ing then into spiral bands at top levels. One or two enormous jets are visible in hurricane space photos: in the bottom they, possibly, are rather narrow, but lifting upwards extend according to expansion of air forming them. We can suppose each standard cumulus cloud (and all the more a "hot tower") to be an independent dissipative structure in which a lifting of moist air and an accompanying warming-up of troposphere occurs. Whether it will come to a critical condition depends on many parameters including the geometrical parameters of structure: its cross-section dimensions, the height of air lifting and the moisture contents. We must notice that the basic equations of dissipative structure (4.13) - (4.14) include neither condition of vertical air lifting nor circulation. The hydrostatic pressure drop also is not obligatory. All these factors (together and separately) can strengthen the contradiction between pressure drop (a forcing), vertical flow and lapse rate which is introduced in basic equations due to nonlinear factors. If certain forcing δph is present in the low-levels (δph = lc∂ph/∂r , where lc is the radial size of cloud base) the dissipative structure of the cumulus cloud must have "a growing" segment of the characteristic I (δph) ). The overheat of some layers can result in decreas- ing of total I (Fig.9a,b ). The growth of δph (for example, in the zone of nonhydrostatic pressure gradient) may overcome the critical stage (Fig.9c) and result in chaotic streams redistributions but no regular reorganization occurs. If this cloud possess the circulation with an overheat in its center a mesocyclone or a tornado can be generated. Cumulus clouds of a hurricane providing the basic vertical air flow are summarized in a total dissipative structure. The characteristic I (Δph) of hurricane is the sum of "small" characteristics i (δph) of separate clouds. Some of them can already have the local N- shapeness of i(δph) (if the overheat becomes close to ΔTcr the ascending flow starts to be 24 decelerated), others still "grow" upwards (Fig. 9). When the large aggregate of clouds- cells will start to feel the weakening of an ascending flow (due to reduction of |∂T/∂z|), the total state of a tropical storm will approach the critical point: ∂I/∂p → 0. With the beginning of the reorganization the clouds in the central zone (future eye area) become "exhausted" (see observations of hurricane Gladys [31]) and the clouds on periphery (eyewall area where |∂pnh/∂r| is more) will be activated (amplified). If there is no large-scale circulation which accumulates a tropospheric warming-up in the central zone the clouds-cells which are in this zone won't have a chance to reach the necessary "collective" N-shapeness leading to the formation of a hurricane.

i i Fig. 9 Three types of cumulus cloud with different dissipative structures: a – normally grow- ing cloud without overheat a b (zone of maximum lapse rate); b – "exhausted" cloud with overheat and stable decrease of updraft (hurricane eye); c – cloud subjected to the flow redistribution (eyewall,"hot δph δph tower"). δp ∂p/∂r .l where i h= c lc – radial size of cloud

c

δph

25 5. Accordance with the empirical data 5.1 Hurricane

The description of hurricane Inez (1966) presented in [16] is one of the most informa- tive. This hurricane was surveyed within two days on August 27 and 28 when its force reached the maximum. The pressure drop in its center measured on August 28 was 927 mb, maximum azimutal velocity at level of 900 mb in 10 km from the center exceeded 70 m/s. By means of formula (2.4) we can evaluate the pressure drop in the hurricane center, using data of [16] on warming-up of a tropospheric column. The results of an approxi- mate calculation are tabulated.

3 Heigt km ΔT, K ρ, kg/m Ta , K Δpρ, mb Δpp , mb 1 – 3 5 1,1 300 3 10 3 – 10 10 0,7 250 20 6 10 – 15 15 0,3 210 10 3 total 33 19

Here ΔT is the average overheat, Δpρ is the primary pressure drop due to an isobaric warming-up; Δpp is the secondary pressure drop due to lightening of overlying layers. The total pressure drop is 52 mb. It is possible to assume that the solid-body rotation of the hurricane central core provides the remained 30 mb. Really centripetal jets converging to RMW area penetrate into the hurricane eye untwisting it at an angular speed of vm/Rm [23]. The air density in the eye center decreases under the action of centrifugal forces. 2 The corresponding pressure drop will be equal to ρvm /2 that in case of hurricane Inez makes just 30 mb. The presence of strongly pronounced inversion in a hurricane eye at level 650 mb can hardly be explained by adiabatic heating of air due to its descent. Addressing to the mecha- nism of overheat instability described above it is possible to explain the given "hump" as the increase of inversion at a layer of the maximum warming-up which has blocked the air lifting and led to sharp redistribution of ascending flows. Its lateral part was destroyed by turbulent dissipation of rotating jets but the central part has remained as the eye rotation goes with constant angular speed and is "laminar". It is possible to assume that 650 mb is exactly the level where according to (2.5) maximum warming-up occurs. The descent of the inversion in mature hurricanes to the level of 700-800 mb and air warming-up to 28-30°С not so considerably affects surface pressure drop: in Gloria [21] it added only 12 mb that confirm our calculations with formula (2.4): 9 mb on a warming-up and 3 mb on lightening of pressure of overlying layers. scent. The similar idea was stated by Shea and Grey in [13,14]. As it was already noted the circulation is the major condition of a hurricane formation. In clusters it is weak (v <4m/c), in tropical storm is stronger (v <18m/c). Dependences v (r) of hurricanes and tropical storm have characteristic differences. Storm's v(r) has wide "plateau" (Fig.10 ). If storm starts to gain the hurricane strength a specific"hump"

26 appears on the plateau. At first it is hardly noticeable and then become more and more expressed: the form of v(r) comes similar to v(r) of Renkin vortex. There is a zone of the maximum wind speed vm, a zone of solid-body rotation (r k>0,6). Similar dependences v (r) are observed practically at all heights. It is possible to assume that the basic jet having received an acceleration at r > rm, reaches the condensation level near RMW taking an azimuthal direction with v=vm. By means of dynamic friction the part of jet energy is transferred to the rotation of the hurricane center: the eye is untwisted while its angular speed becomes equal vm/Rm [23]. The free ascent of the rotating jets to the maximum lifting level occurs with invariable angular momentum as both operating forces – centrifugal and a pressure gradient – pass through a rotation axis. As ∂ph/∂r decreases with height, there is an outflow at the upper levels where the radial velocity u > 0. Jets move away from the center reducing its speed of rotation inversely proportional to the radius and forming the cumulus spiral rain bands typical of hurricanes . The prominent feature distinguishing hurricane from a tropical storm is the presence of one or several jet flows converging to RMW . Basically convection of moist air occurs along these streams and these lifting areas looking as arches or rings of cumulus and cir- rus clouds are well visible from space (see book cover). The eye and the inner ring of convection are formed apparently as a result of displacement of vertical air flows from the center to RMW and their further acceleration. According to the mechanism of over- heat instability it may occur when the absolute value of lapse rate ϒ in the central zone decreases to some critical size. The lifting area from round (r < rm ) will be transformed in a ring-shaped way with a higher (about two order) vertical speed. When the criterion (4.11) is satisfied and the redistribution of vertical flows starts the warming-up in the center ( especially in the top-levels)doesn't stop as w>0. The pressure continues to fall. This process can amplify itself as the rotating eyewall starts to play the role of a "chimney" isolating a warm core from cold surrounding atmosphere. In terms "flow - forcing" a critical condition of the storm-hurricane transition comes when the full lateral flow of air passing through hurricane boundary layer ceases to grow when the pres- sure drop (forcing) increases. On the characteristic I (Δph) it is a point A – the beginning of N-shape segment (Fig.7) . Transition A→B corresponds to a warming-up of the center of

v v

Rm r r

Fig 10 The typical radial profiles of azimuthal wind Fig. 11 The typical radial profiles of azimuthal ve- for hurricane (solid) and tropical storm (dushed) locity for hurricanes with two wind maximum: the upper is of mature hurricane passing its peak stage [18], the lower may belong to the hurricane in its deepening stage [47]

27 hurricane with formation of an inner ring of a convection – the area of high azimuthal and vertical speeds. The further pressure drop is accompanied by the contraction of the inner ring. Probably it occurs because the rotating jets pushed to the center by pressure gradient disseminate the inversion as though "eating" its edges. If hurricane loses its "fuel", the water vapor, for example during its landfall, protective force of the eyewall weakens, the cold air breaks into the center core and pressure deficit Δp decreases. The total air inflow I decreases too. On the characteristic I(Δp ) the segment ВG corresponds to hurricane weakening: there is the transition G→ H when the inner ring breaks up. In mature hurricanes and more often in the hurricanes which have reached 4th and 5th categories on a scale of Saffir-Simpson sometimes the so-called outer convective ring appears [26-29]. Dependence v (r) of such hurricanes has two expressed maxima (Fig.11 ) with corresponding peaks on w(r).The formation of the second (outer) ring can result from descending flows at r > rm, penetrating boundary layer and partially block- ing centripetal air flows. The area of descending flows – the so-called moat – can become a zone of superaccelerated radial speed when u ~ r-k, k> 1. In this case div u > 0 and descending flows of dry air mix up with the moist air arriving from periphery (Fig 12). The convection weakens resulting in the decrease of the updraft flow in the inner ring. Probably the decrease of vertical flow is compensated by a new convection in the outer ring which can be formed by the cut off of a part of a centripetal flow (at the top part of a boundary layer) which has found the way upward. The hurricane passes on to a new state – with two convective rings. Thus during the first moments the pressure drop doesn't change as center overheat remains unchanged. The total air inflow can increase due to convection in the outer ring. On the "flow-forcing" characteristic the transition C→D to a branch of two zones of lifting may occur.(Fig.13) Due to convection weakening in the inner ring and cooling of the hurricane warm core

Fig 12 Schematic depic- z tion of secondary circu- 3 lation of the mature hur- ricane with two rings of convection: 1 – ascending flow in the outer ring; 2 – centripetl flows discon- 1 necting the perifery; 3 – 4 weakening inner eyewall; 4 – descending flow of dry air. Thin dashed lines em- phasize the cold air flows coming into eye area.

inversion

W 2 r

28 the total pressure deficit decreases. "The point" on I (Δp) moves along the segment DE of a state with two rings: thus the outer ring contracts and the inner weakens. E→F is a definitive desintegration of the inner ring and transition in a state with one ring when the hurricane core again can starts to get warm. Under favorable conditions a hurricane can repeat all cycle BCDEFB. Some similar full cycles were observed and described in hur- ricane Allen (1980)[29]. After hurricane landfall the unlimited source of water vapor (warm ocean) disappears. Warming-up of a tropospheric column on all its height decreases. On the "flow-forcing" characteristic I(Δph ) it corresponds to the movement along the segment F → G. The states on this branch correspond probably to extratropical cyclone which differs both from a tropical storm (segment OA) and from mature hurricane. To get the state of extratropical cyclone it is possible only coming back along a branch BG thanks to a hysteresis of the characteristic I(Δph). Thus the external characteristic "flow-forcing" I(Δph ) (Fig. 13) reflects the basic set of states of hurricane and their mutual transitions. Branches ОА, BC, ED concern states of a tropical storm, of a hurricane with one convective ring and a hurricane with two rings ac- cordingly. Transitions between theses states occur at pressure drops changements resulting in a warming-up or cooling of the central core of hurricane.

Fig 13 Flow-forcing characteristic of the hurricane with two convective rings

I

D E

C A B Δph

H G F

O Δph

29 5.1 Polar low

Attempts to explain the origin of polar lows (and their less intense and more extensive ana- logs – "comma clouds"[35]) by means of traditional theories CISK and WISHE [33,36] have been accordingly undertaken. We will consider below the observational descriptions of some polar lows and we will try to explain them on the basis of the abovedescribed mechanism of the overheat instability. Polar low generation is favored by the layer of cold air extending over an open water surface and by the appearence of the 500mb-trough. In the case described in [33] there was a thick layer (over 1 km) with low temperature over the open area of the Barents sea. In another case [34] near Kadiak Island (Bering Sea) the cold air was advected from land and has cooled low layers on 5-6° C. This processes reduces atmospheric stability. As water temperature can't decrease below 4°C the surface air is appreciably overheated compared with surrounding atmosphere and its elevated humidity creates preconditions for a warming-up of an atmospheric column under condition of air lifting above the con- densation level. The humidity evaluation has shown [34] that the parcel which has risen on 500mb level will be overheated concerning atmosphere on 9°C. The pressure drop from 500mb-trough is transferred to the surface in the form of insignificant depression [34] which has started to play the role of “primary draft”. It provided both lifting of moist air to the condensation level and primary circulation increasing the potential vorticity in the bottom layers. Strong evaporation owing to the temperature difference between the open water and surrounding air gives a quite good source of water vapor. However the low moisture con- tent (q = 7-8 g/kg) and insufficiently large area of open water (in comparison with tropical cyclones) limits life time of the polar low by approximately 12 hours. The dissipative structure of polar low should be reorganized according to the formula (4.11) when the vertical temperature gradient in a layer of the maximum warming-up reaches its critical value. Let's consider theoretically the process of polar low formation using the observational data of [34] presented in Fig. 14 (the scale is changed for clearness). Qualitative curves show the temperature soundings obtained with the interval of 12 hours in the zone of for- mation of polar low over the Bering sea. The first sounding reports about initially stable atmosphere: the temperature drop on the first 2,5 km makes more than 5° C/km. In 12 hours after the intrusion of a cold air the bottom part of troposphere was cooled on the average of 5°C that resulted in lapse rate increase to a dry adiabatic. There was the convec- tion first sign – a condensation at the 850mb level (Fig 14b ) that prove the updraft of the air with low mixing ratio. In 12 hours the full influence of 500mb-trough became obvious: a strong updraft with the condensation in the layer about 2 km thickness occurs with an ad- ditional cooling of the middle layers due to the same trough. The atmosphere warming-up due to latent heat release and heat exchange between the superheated parcels of moist air and the atmosphere has started in a convection-condensation zone. However at this stage only insignificant general heating of all layers (till 500mb level) compared with the previ- ous sounding can be distinguished (Fig. 14c). Following sounding was made, apparently,

30 p mb p mb 600 Kodiak 600 Kodiak 21 Mar. 1975 22 Mar. 1975 1200 GMT 0000 GMT 700 700 [a] [b] 800 800

900 900

1000 t°C 1000 t°C –30 –20 –10 0 –30 –20 –10 0

p mb p mb Kodiak Kodiak 600 22 Mar. 1975 600 23 Mar. 1975 1200 GMT 0000 GMT

700 700 [c] [d] 800 800

900 900

1000 t°C 1000 t°C –30 –20 –10 0 –30 –20 –10 0

Fig.14 Temperature soundings taken in the time of polar low formation. Temperature and dew-point profiles ar eindicated by the heavy solid and dashed lines. Dry adiabats are shown as dash-dot lines, moist adiabat as thin dashed lines. Data are taken from [34]

when the reorganization of dissipative structure already was finished (Fig.14d ). At 850mb level (probably it was the level of the maximum warming-up zm) the air temperature has grown by 6°C [34] what resulted in the lapse rate decrease in the underlaying layers to 4°C/km that could be a critical magnitude for this case (just as in tropical cyclones, see Section 4) . The surface bubble lapse rate according to (2.5) was 3,8° C/ km. But critical stage becomes when the bubble and the surrounding air reach the identical magnitudes of its temperature and lapse rate at the same time. Near the surface the bubble had a rather overheat (Tb = 4° C, T = -10° C) and the rate of latent heat release was small due to practi- cal absence of condensation. That is why the 850 mb level could be the critical one and the maximum warming-up was observed here (Fig.14d). Though the magnitude of qs at this level was less than the surface one the vertical velocity of a bubble might be 2-3 times higher (due to mechanic acceleration) and the rate of heating here might have a maximum. After the reorganization of dissipative structure the further development of blocking in- version could proceed quite similar to the tropical cyclones eye formation with warming- up of low levels. 31 5.1 Tornado

The atmospheric vortex of a small spatial scale (tornadoes, waterspouts, dust "devils") are the phenomena wide-spread enough. The majority of them merely represent a beautiful show. But the minority which for some reasons develops the big strength bears destruc- tions and troubles sometimes heavier than hurricanes does. Structural similarity of hurricane and tornado has been known for a long time. But only recently it become possible to confirm it with exact measurements [37]. From the point of view of hydrodynamics the mechanisms of formation of hurricane and tornado shouldn't have basic distinctions. Stationary pressure deficit (the life time of a tornado is from sev- eral minutes to several hours) corresponds more to the assumption of its hydrostatic ori- gin. But direct acknowledgement of existence of a warm core in a tornado as it does in hurricane isn't present in the meantime. The presence of a mesocyclone or a supercell facilitates the tornadogenesis. Mesocyclone is a rotating cloud having the diameter of 10- 20 km with an ascending air flow in its center. But many tornadoes are connected with parent clouds with a substantially smaller diameter – less than 4 km. They were named "misocyclones". A hurricane, one of the greatest atmospheric vortex, can "spawn" the tornado itself [38,40]. Tornadoes arise under the hurricane cloudiness after its landfall. Observations of tornadoes [39] have shown that the leading role in the tornado forma- tion is played by a second low-level mesocyclone which is usually located in several kilometers from the center of the first mid-level mesocyclone. The interaction of these mesocyclones creates a known picture of tornado formation repeatedly documented in several works[37,39-41]. The low-level mesocyclone arises at the contact zone of main updraft and a rear-flank downdraft (RFD). Apparently it forms its own updraft due to the primary inflow jet which creates the negative divergence of radial speed near the vortex center. At the stage of tor- nadogenesis the upward motion in the low-level mesocyclone starts to be grasped by the main updraft of mid-level mesocyclone and results in a characteristic tilt of tornado core: the vertical flow of air from the low-level mesocyclone tends to an axis of the mid-level mesocyclone [39]. The next stage is an occurrence of the so-called occlusion dawndraft – a strong descending flow in the vicinity of the low-level mesocyclone. Just this moment the tornado is formed: the low-level circulation amplifies, the funnel becomes visible, an RMW area is forming with an eye strongly pronounced – a hole core zone in which the air moves downwards to the surface.[39-41]. These jets hitting the surface apparently create a characteristic dust cloud. Possibly the interaction of two mesocyclones causing the tilt of a tornado column pre- vents the formation of the strong tornado (tornadoes F0, F1, F2 categories were formed only [44]). If the tornado axis coincides with a mesocyclone axis (a vertical tornado) it is possible to expect substantial strengthening of intensity as in this case the hydrostatic and nonhydrostatic forces supplement each other. A «primary draft» apparently is obligatory for a tornado birth. As well as for hurricane there must be firstly an ascending air flow. Air lifting is characteristic of thunderstorms, of gust lines, of contact zones of warm and cold fronts. The speed of this lifting is rather

32 high (some m/s and more). A layer of moist air serves as source of water vapor (for exam- ple air from Gulf of Mexico in the USA). At temperatures 35-40 °C the saturated air can have specific humidity 40-45 g/kg. The maximum warming-up of a tropospheric column with such moist updraft can exceed the typical warming-up of mature hurricanes almost twice. The circulation which would accumulate a heating updraft in the vortex center is necessary for maintenance of positive feedback. It simultaneously preserved a warm core against cold centripetal streams. Such a kind of primary circulation can be provided by a misocyclone or a certain external vortex which can appear in the updraft low-level zone. Assuming that the mechanisms of tornado and waterspout formation should be simi- lar (if aren't identical), we want to pay attention on early and very informative work [45] where all stages of development of Florida Key Waterspout have been fixed in detail. "Primary draft" for a waterspout is provided, possibly, by a certain air flow which can form a turbulent vortex at the border of certain air mass . It is important that there exists a per- manent (rather weak) updraft of the surrounding air [46].At the initial stage the waterspout described in [45] had one-cell structure with the rotating air lifting in the vortex center. The column of ascending air becomes visible as small splashes from the water surface are grasped by ascending streams [45]. Probably at mid-levels there exists a mesocyclone which supports the primary draft of the waterspout. The ascending flow starts to warm up the surrounding atmosphere according to the equation (6.5). If there is no wind shear and the air lifting is vertical the primary pressure deficit (resulting from the air rotation) should have the hydrostatic addition. It shouldn't be considerable as the diameter of a col- umn is rather small and always has certain tilt. However, the pressure drop equal to 8 mb measured at height of 645m near the funnel basis [46] has possibly a hydrostatic origin. The reorganization of the dissipative structure comes when in some layer the critical value of the vertical lapse rate is reached. In the considered cases of hurricane and polar low the value of this critical overheat was 6°C (at 650 mb). Using formulas (2.5) it is pos- sible to evaluate the rate of a warming up (q = 10 g/kg, ∂Т/∂z = –10° C/km, w= 2 m/s, ). It is equal 1/30 °С/s. Hence, under favorable conditions the critical overheat can be reached for 3 minutes. This time quite corresponds to the average time of total life of the mature waterspouts which is equal 7,7 minutes [45]. The waterspout, apparently, can't reach a mature stage without the support of mesocy- clone (as in case of a tornado) which is described in [45] as a "collar cloud". Possibly, it provides the stability of an ascending flow and the attainment by the waterspout the stage when the supposed dissipative structure starts to reorganize. The waterspout in a mature stage has a characteristic two-cell structure. Its column looks like the hollow cylinder with internal and external walls between which the turbulent lifting of air occurs. So the waterspout has the parameters rm and Rm (in [45] they were equal 8m and 30m) typical for hurricane and tornado as well as for theoretical vortex which have undergone the reorga- nization of its dissipative structure. The most powerful and destructive tornadoes are born and go down to the surface from a misocyclone which is sometimes visible with the naked eye (Huge Manitoba Tornado, www.youtube.com ). All cloudy system resembles a wheel of a cart with a hub and an axis which is formed by a tornado funnel. A misocyclone has substantially three-dimensional structure in which the vertical and radial flows have identical spatial and time scales. The function of non hydrostatic pressure forces in misocyclone and tornado plays probably a defining role. 33 The intensification of a tornado occurs probably when in the layer of maximum warm- ing-up (the presumable height is 3-4 km) the lapse rate reaches its critical magnitude. When the criterion (4.11) is carried out , the dissipative structure starts to reorganize. The basic updraft is displaced from the center to the periferal area. The additional part of non- hydrostatic pressure (the domain of pressure forces) caused by the maximum centripetal and centrifugal accelerations provides the forcing of new circular updraft and air lifting to the condensation level. The local air compression in the RMW area results in a propor- tional increase of the partial pressure of water vapor and can cause its saturation and con- densation. So there is one of possible explanations of a tornado funnel appearence. Hence the visible part of a funnel is the zone of the maximum rotation speeds. The moment of funnel touchdown objectively corresponds to the tornado strengthening: the maximum speeds reach surface.

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