The hurricanes (typhoons) have been extensively investigated during the last 60 years. Many of their features have been observed and experimentally studied using satellites, aircrafts, ships, and buoys. These observations created a detailed qualitative picture of hurricane structure, documented in several well-known texts by Dunn [1] , Anthes [2] , Hsu [3] , Cotton & Anthes [4] , Ogawa [5] , and Emanuel [6] .

Some idealized models [7] -[12] of several problems in hurricanes have also been developed. Complicated role of mesovortices in the hurricane eye was experimentally modeled in laboratory and discussed [13] . Lighthill developed a thermodynamic theory of ocean spray [14] , and its effect on the dynamics of near water air turbu- lence was revealed by Barenblatt et al. [15] . Detailed models of coupled interactions between the turbulent wind and oceanic waves near the air/sea interface have also been elaborated in text [16] (Ch. 3). Other conceptual ideas are mixed with numerical studies. Some works [17] -[20] modeled intriguing aspects of hurricane maturing. Many other papers developed turbulent baroclinic and barotropic numerical models (e.g. see paper [21] and ref- erences there). To forecast hurricane travel these models interact with the current synoptic and lower scale ob- servations (see recent extensive reviews in Refs. [22] [23] ).

Yet several fundamental problems in hurricane physics remain unresolved. These are the change in the direc- tions of hurricane rotation and radial wind in lower and upper parts of hurricane, radial increase in wind angular momentum in hurricane boundary layer, dramatic effect of ocean spray and its radial distribution, and a high in- crease in temperature at the upper region of boundary layer. The problems of hurricane genesis and maturing are also currently vaguely addressed.

Thus the main objective of this paper is to resolve the above problems by developing and analyzing some quantitative models, based on the author’s results [24] -[27] pre-published in Arxive. The models being rude enough still provide a consistent analytical description of the basic physical phenomena in hurricanes. The con- ceptual view of hurricanes as coherent structures, allows avoiding the common turbulent approximations except friction factors at the air/sea interface. The use in the model the aerodynamics of ideal gas requires implement- ing continuity for dynamic variables to avoid the Kelvin-Helmholtz (K-H) instability. Additionally, integral balance equations allow expressing all parameters in the distributions of field variables via only two external parameters—the sailing wind and temperature of warm air band the hurricane travels along.

The paper is organized as follows. The next Section briefly discusses the external forces causing horizontal travel of hurricanes, thermodynamics of air, dynamics of ideal liquids, and hurricane structure. Section 3 models the basic airflows in the upper layer of hurricane. Section 4 models the basic processes in the hurricane boun- dary layer. The last, Section 5 presents simple analytical models for hurricane genesis and maturing.

Neglecting the air band effects, air flows in upper layer of hurricanes can be modeled using the adiabatic ap- proximation. The structure and basic flows in the hurricane adiabatic layer is sketched in Figure 2, and the air- flows there are axi-symmetric. Here is a solid-like rotation of air in the eye region, and no vertical wind compo- nent exists in the outer region of hurricane. For convenience, we use in this Section the vertical axis z shifted upward by the height of boundary layer.

The following modeling equations are used below [25] :

In aeromechanical Equations (4), is the angular air velocity relative to the angular velocity of Earth rotation on -plane. Equations (5) represent the “jet approach” [29] for vertical mass balance and momentum in the eye wall averaged over radius. Introducing the stream function by common relations, , , yields the first integrals. Their linear forms, with numerical coefficients and allow an easy physical interpretation.

We now introduce two simplifying approximations:

(i); (ii) (6)

Here (6i) presents the “well mixing” assumption introduced by Deppermann [30] , and independency from air rotation introduced in (6ii) has been justified in Ref. [25] .

It is convenient to introduce the non-dimensional variables:

Here is the kinematic condition which relates the vertical and radial velocity at the outer jet radius. The common boundary conditions of continuity are employed for radial and rotational components of air field, with “frictional” kinks in distribution of angular momentum at the inner and outer walls of EW. Other na- tural conditions used in calculations are: when.

Tedious calculations of set (4) with approximations (6) yield the explicit expressions for radial non-dimen- sional distributions of dynamic variables:

,

;;

The approximation or was used in formulas (8). Here is the external radius of hurricane. In (8), parameter characterizes the effect of rotation on the EW jet cross-section, and parameter describes the frictional kink in the distribution of angular momentum at [25] . The struc- tural functions and are given by (A1) in Appendix.

Formulas (8) show that streamlines in hurricane are the circles in eye and outside EW, and ascending spirals in EW with kinks at,. Equations for jet radius and vertical velocity profile are then found upon substituting (8) into (5). It yields the equations for mass conservation, and evolution of the jet profile. These eq- uations are written in the non-dimensional form (7) as:

; (9)

Here, is the buoyancy parameter, and. The values and in (9) correspond to, the structural function is given by (A2) in Appendix, and.

There are simple asymptotic solutions of (9) in two limiting cases.

1): (10)

Here are some algebraic functions of and presented via formula (A2) [25] . The only physi- cally feasible case is where for stability, the heat supply from HBL to the hurricane jet should exceed the adiabatic cooling. Here the initial jet profile is convex down, with centripetal radial flow.

2):; (11)

When, the vertical component of air flows vanishes being converted to the radial one, with the jet ra- dius approaching to infinity.

It was found that the numerical solution of steady problem (9) exists only for physically feasible case. The following realistic parameters were accepted below in the demonstrative calculations:, , ,;, or 0.00363 (~3 or 1 m/s),.

The values of calculated parameters are:, , , , , , and.

Figures 3-9 illustrate the calculated radial distributions of basic variables, depending on altitude and initial value of vertical velocity. Figure 3 demonstrates the characteristic non-monotonic behavior of outer EW jet boundary, Figure 4 the increase in vertical velocity. Figure 5 shows that the angular velocity in EW jet de- creases with altitude, though the region when it is negative is not shown. Radial distributions of angular mo- mentum M and tangential velocity for two altitudes, presented in Figure 6 and Figure 7, demonstrate their increase in the eye and EW regions with two characteristic kinks, a plateau for M outside EW jet, and a sharp peak of at the outer boundary of EW. Figure 8 show characteristic distributions of radial velocity. It is

(a)

(b) (b)

Figure 6. Non-dimensional radial distributions of angular momentum at two non-dimensional altitudes: (a); (b).

(a)

(b) (b)

Figure 7. Non-dimensional radial distributions of tangential velocity at two non-dimensional altitudes: (a); (b).

negative (centripetal) at lower and positive at higher altitudes, with absolute maximum at the outer boun dary of EW jet. Finally, Figure 9 show two similar radial distributions of pressure which display a characteristic “depression” area at the center of hurricane. These results support a well-documented characteristic structure of hurricane sketched in Figure 10. Additionally, the radial distributions of tangential velocity and pressure at the bottom of adiabatic layer were found in [25] in a good agreement with these obtained using semi-empirical modeling and observation data by Deppermann [30] .

The structure and basic interactions in hurricane boundary layer (HBL) are sketched in Figure 11. Here the HBL is horizontally separated in the same three regions as in previous Section: the eye, HBL EW, and outer HBL region, the latter having generally a curvilinear upper surface. There are two major vertical sub-sections in HBL: upper aerodynamic and lover turbulent ones. Additionally, there is a condensation sub-layer located at the top of HBL EW and assumed to be very thin (~100 m). The height of HBL is restricted to the condensation level whose value is roughly evaluated using an empirical condition for the beginning condensation when the saturation point is achieved [4] . Here is the dew point temperature

(a)

(b) (b)

Figure 8. Non-dimensional radial distributions of radial velocity at two non-dimensional altitudes: (a); (b).

(a)

(b) (b)

Figure 9. Non-dimensional radial distributions of pressure at two non-dimensional altitudes: (a); (b).

Figure 10. Sketch of the total vertical distribution of EW jet.

Figure 11. Cross-sectional sketch of HBL and diagram of air/sea interactions.

depression at the sea surface. The common evaluation yields.

The emergence of hurricanes is still mysterious. Many observations of initial stages of hurricanes (e.g. see the text [2] ) found a threshold value of vorticity, exceeding which the hurricane is maturing. Analyses in papers by Ooyama [17] [18] and Emanuel [19] [20] have a mutual defect-adjustable diffusivity of angular momentum to fit the data. Also, most hurricanes in Atlantics are formed in near equator zone, indicating the importance of Co- riolis factor, which was not considered in the above papers.

Paper [27] proposed a two-steps scenario of hurricane’s genesis. In the first step, an emerged plume of warm and humid air formed in the near equator zone, moves upward (see the model of plume dynamics in [27] ). In the second step, the plume captures the rotation from a horizontally sheared wind, with restructuring of the plume and acquiring an initial value of angular momentum. If this plume is stable, the maturing stage begins. In this case the hurricane grows in the radial direction, presumably caused by the K-H instability with radial propaga- tion into ambient air under action of Earth rotation.

To describe the maturing stage of hurricanes we first consider the quasi-static relation for angular momentum extended to the external boundary of hurricane:

The absolute and relative tangential velocities are then defined as:

The slow evolution of and is now described by two heuristic equations:

The first equation in (35) describes propagation of the hurricane front due to the K-H instability with the rela- tive rotational velocity at the boundary. The second equation in (35) assumes that the radius change due

(a)

(b) (b)

Figure 13. Comparison of our adiabatic calculations and measurements [31] for radial distributions of tangential wind and surface pressure for hurricane Frederic for measurements on 09/11/1979: (a) Tangential wind; (b) Surface pressure; maximal wind speed, radius of eye jet Dashed and solid lines are measurements and calculations, re- spectively.

to the radial propagation of unstable boundary is the dominant contribution in the change of angular momentum.

The initial conditions are:

,. (36)

Here is the horizontal shear of wind initiated the plume rotation.

The solution of Equations (33)-(35) with conditions (36) is:

Formulas (37) show that depending on sign, the plume can rotate in cyclonic or anti-cyclonic directions.

1) In the cyclonic case, hurricane propagates outwards. It is the maturing case, when he functions, , and monotonically grow to their stationary values,

2) In the anti-cyclonic case, the disturbances propagate inwards, which cause the collapsing hurricane either in finite or in infinite time.

Thus the model selects as only stable, the cyclonic initial rotation, which naturally explains the observed cy- clonic rotation of matured hurricanes. However, the model does not describe the observed threshold in value of, seemingly because of the linear character of the first equation in (37).

To illustrate the model predictions we choose the following parameters . Calculations due to (37) and (38) yield:

i) Characteristic time of hurricane development:;

ii) Characteristic radius of developed hurricane:;

iii) Maximum speed of developed hurricane:;

iv) The grow of angular momentum: from to.

These results are consistent with observations in text [2] that the initial tropical cyclone is transformed into a hurricane during 5 - 6 days after the action of wind with vorticity.

The paper presents analytical two-layer hurricane model. The approach employed in the paper uses simplified aerodynamic equations for ideal humid gas with additional models for heat transfer, evaporation and condensa- tion. It mostly avoids the common turbulent approximations, except a thin near-water sub-layer.

Analysis of adiabatic aerodynamic modeling in the hurricane upper layer reveals a “hyperboloid” structure of eye wall (EW) jet. The radial and vertical distributions of basic variables have been theoretically calculated. It was found that upper layer of hurricane is stable when the thermal heat supplied into the layer exceeds the adia- batic cooling. The model also explains the change in the cyclonic/anti-cyclonic directions of hurricane rotation, as well as the directions of radial wind component in lower and upper parts of hurricane.

The model of hurricane boundary layer (HBL) employs aerodynamic approach only in its upper sub-layer and matches it with the turbulent approach in its lower sub-layer. The increase in the wind angular momentum in HBL is explained as an additional generation of wind by ocean waves propagating out of HBL EW. A dramatic effect of ocean spray and its radial distribution on evaporation has been modeled taking into account the ocean whitecaps generated by wind. A high increase in temperature in the upper sub-layer of HBL has been modeled by the condensation jump.

The balance relation applied to the HBL EW, presented the basic parameters governing the space distributions of field variables in hurricane via two external parameters-the sailing wind and horizontal temperature of a warm air band.

Additionally, a rude model for the hurricane genesis and maturing has also been developed. It explains the reason of cyclonic rotation of hurricanes.

All examples in the paper demonstrated a good correspondence with the existing observations when using common data for geometrical, fluid mechanical and thermodynamic parameters of hurricane.

It finally should be noted that developing the hurricane structure during hurricane genesis and maturing presents a very challenging numerical problem which by no means could be resolved by simplified analytical approaches.

The results of the paper could be used for easy tune-up of complicated numerical models, which take into ac- count real interaction of hurricane with environment.

The author thanks Dr. A. Benilov for extensive and highly productive discussions, as well as the participants of Physical Science Division Seminar at NOAA in Boulder, CO (July, 2012). A lot of thanks are also given to for- mer PhD Student, Dr. A. Gagov for help in calculations and graphics, and Dr. A. Voronovich for patiently read- ing the paper and making valuable suggestions.