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A two-layer theoretical model of hurricanes traveling (quasi-) steadily over open seas has been developed. The use of coherency concept allowed avoiding the common turbulent approximations, except a thin sub-layer near the air/sea interface. The model analytically describes 3D distributions of dynamic and thermodynamic variables in hurricanes and analyzes processes of evaporation and condensation. Using this modeling, the following fundamental problems were naturally resolved-change in the cyclonic/anti-cyclonic directions of hurricane rotation and the directions of radial wind in lower and upper parts of hurricane; increase in wind angular momentum in hurricane boundary layer; dramatic effect of ocean spray and its radial distribution; and a high increase in temperature at the upper region of boundary layer. Additionally, integral balances allowed expressing the governing parameters of field variables via two external parameters, the sailing wind and temperature of a warm air band, in which direction the hurricane travels. A rude model for the hurricane genesis and maturing has also been developed.

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 [

Some idealized models [

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 [

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.

Two factors affect the horizontal travel of hurricane: 1) stirring or “sailing” wind with velocity

We consider air motions in hurricane as axially symmetric flows of ideal compressible gas. The frame of refer- ence used below is a cylindrical coordinate system with vertical

Far away from hurricane, the atmosphere is assumed to be horizontally homogeneous with vertically distributed ambient density

Using adiabatic description of air,

and the static equation,

the vertical distributions of thermodynamic parameters are presented as:

Here

A typical structure of a mature hurricane traveling (quasi-) steadily over the open sea is sketched in

The vertical structure employed in the following models, includes the turbulent boundary sub-layer of thick- ness

In the HBL, the air-sea interaction directly affects the dynamics at the air/sea interface, generating oceanic waves which in turn interact with air flows in the outer part of HBL. There is also evaporation and the heat/mass exchange between the hurricane and environment. The moisture, sensible and latent heats are transported via HBL towards the EW jet. The height of HBL is limited by air moisture condensation, which causes the forma- tion of spiral rain bands, layered clouds and rainfall from them. Dynamic effects of rainfall can seemingly be neglected, though the rainfall can balance the evaporation from the oceanic surface. This results in a constant sa- linity level in the oceanic boundary layer.

Schematic structure of a hurricane

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

The following modeling equations are used below [

In aeromechanical Equations (4),

We now introduce two simplifying approximations:

(i)

Here (6i) presents the “well mixing” assumption introduced by Deppermann [

It is convenient to introduce the non-dimensional variables:

Sketch of adiabatic layer

Here

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

The approximation

Formulas (8) show that streamlines in hurricane are the circles in eye and outside EW, and ascending spirals in EW with kinks at

Here

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

1)

Here

2)

When

It was found that the numerical solution of steady problem (9) exists only for physically feasible case

The values of calculated parameters are:

Figures 3-9 illustrate the calculated radial distributions of basic variables, depending on altitude and initial value of vertical velocity

Non-dimensional altitude dependence of outer boundary of EW jet

Non-dimensional altitude dependence of vertical velocity

Vertical distribution of non-dimensional angu- lar velocity

(a)

(b) (b)

(a)

(b) (b)

negative (centripetal) at lower and positive at higher altitudes, with absolute maximum at the outer boun dary of EW jet. Finally,

The structure and basic interactions in hurricane boundary layer (HBL) are sketched in

(a)

(b) (b)

(a)

(b) (b)

depression at the sea surface. The common evaluation

They include coherent aerodynamic airflows in upper part of HBL, turbulent airflows in lower part of HBL, and dynamic interaction of oceanic waves with HBL airflows.

Models employ simplified equations of aerodynamics of ideal gas similar to Equations (4) with

Omitting the

The same assumptions as in the previous Section are employed here: the rigid-like airflow in HBL eye, the radial independence of vertical wind in HBL EW, and the same boundary conditions at the inner HBL EW in- terface. It is also assumed that the outer upper boundary

Introducing the stream function

Here

At the upper boundary HBL EW,

condition at

A huge air wind near the radius

waves propagating outside this region. The radial wind contribution can be neglected in this sub-layer because of very low variation assumed for

We use for friction factor

height

Evaluation of the roughness factor

The radial increase in angular momentum

Consider the oceanic waves initiated in the vicinity

;, , ,. (16)

Here the low indexes “e” and “r” denote the values of variables at the radii

. Values of roughness parameter

. Values of roughness parameter | . Values of roughness parameter | . Values of roughness parameter | . Values of roughness parameter |
---|---|---|---|

10 | 1 | 0.000511 | 270 |

10 | 2 | 0.00923 | 103 |

10 | 3 | 0.0333 | 70.0 |

10 | 4 | 0.0714 | 51.9 |

Evaporation from the oceanic surface and latent heat. Over calm oceanic water, the vertical air flux (per unit mass of vapor) caused by moisture evaporation can be approximated as

Here

Consider an example. Using (15) with

Using (15) and (17), the total mass flux of evaporation

Here

Transfer of sensible heat is considered in the steady model under condition of complete sea/air temperature balance

Condensation is assumed to happen in a relatively thin vertical layer whose height is less than hundred meters, where the over-saturated vapor comes into the upper layer of HBL. Neglecting the thickness of the layer, it is considered as a weak condensation jump, which is described by basic equations including the conservation of vertical fluxes of mass, momentum and energy [

Since the differences between velocities, pressures and densities over the jump are negligible,

The average temperature

In the following we use the plane Cartesian axes

Two fluxes of the “dry” air masses come from HBL to the EW jet via the bottom of adiabatic layer: 1) the flux from the radial airflow into the HBL and 2) fresh air coming because of horizontal travel of hurricane. Neglect- ing density variations, the balance is:

Here

In (22) the differences between heat capacities are neglected. The left-hand side of (22) describes the heat en- tering the hurricane condensation layer with unknown temperature

Here

Balance of the latent heat is presented by second formula in (18).

Assuming that the oceanic vapor is completely condensed in the condensation layer, the last formula in (19) along with (23) yields two useful chain equalities:

The values of

Entropy balance, detailed in [

Here entropy

Here numerical parameter

Affinity velocity of hurricane travel was determined in [

Thus, all the unknown parameters,

Recall that the non-dimensional temperatures

It is convenient to introduce non-dimensional wind components, scaled with the adiabatic speed

The above relations yield the five equations for parameters

Here

[

Substituting

1) External sensible heat supply is negligible—

One can see that due to (30)

2) Sailing wind is negligible—

Formulas (31) show that

In the limit cases,

Formulas (32) show that the steady, rotating hurricane can exist even without horizontal travel. Here the heat supply

1) Accepted and calculated parameters

Geometrical parameters known for the “standard” hurricane are:

Physical parameters are—

Note that values

Parameters calculated from (29) are shown in

2) Results of calculations. Using

In both the cases, the most striking result of calculations is a high increase in temperature

. Calculated non-dimensional parameters of standard hurricane.

Parameter | . Calculated non-dimensional parameters of standard hurricane. | . Calculated non-dimensional parameters of standard hurricane. | . Calculated non-dimensional parameters of standard hurricane. | . Calculated non-dimensional parameters of standard hurricane. | . Calculated non-dimensional parameters of standard hurricane. | . Calculated non-dimensional parameters of standard hurricane. | . Calculated non-dimensional parameters of standard hurricane. |
---|---|---|---|---|---|---|---|

Value | 0.0351 | 0.00341 | 1.64 | 0.727 | 0.00248 | 1.515 | 0.1 |

. Results of calculations of basic variables in hurricane travel under given values of sailing wind

. Results of calculations of basic variables in hurricane travel under given values of sailing wind | . Results of calculations of basic variables in hurricane travel under given values of sailing wind | . Results of calculations of basic variables in hurricane travel under given values of sailing wind | . Results of calculations of basic variables in hurricane travel under given values of sailing wind | . Results of calculations of basic variables in hurricane travel under given values of sailing wind | . Results of calculations of basic variables in hurricane travel under given values of sailing wind | . Results of calculations of basic variables in hurricane travel under given values of sailing wind | . Results of calculations of basic variables in hurricane travel under given values of sailing wind |
---|---|---|---|---|---|---|---|

0 | 49.5 | 1.74 | 20.0 | 0 | 0 | 0.097 | 8.31 |

5 | 50.4 | 1.77 | 18.4 | 0 | 0 | 0.097 | 8.31 |

10 | 51.3 | 1.80 | 16.8 | 0 | 0 | 0.097 | 8.31 |

15 | 52.2 | 1.83 | 15.2 | 0 | 0 | 0.097 | 8.31 |

. Results of calculations of basic variables in affine travel of hurricane under given values of

. Results of calculations of basic variables in affine travel of hurricane under given values of | . Results of calculations of basic variables in affine travel of hurricane under given values of | . Results of calculations of basic variables in affine travel of hurricane under given values of | . Results of calculations of basic variables in affine travel of hurricane under given values of | . Results of calculations of basic variables in affine travel of hurricane under given values of | . Results of calculations of basic variables in affine travel of hurricane under given values of | . Results of calculations of basic variables in affine travel of hurricane under given values of | . Results of calculations of basic variables in affine travel of hurricane under given values of | . Results of calculations of basic variables in affine travel of hurricane under given values of | . Results of calculations of basic variables in affine travel of hurricane under given values of |
---|---|---|---|---|---|---|---|---|---|

0 | 0 | 49.5 | 0 | 1.74 | 17.4 | 0 | 0 | 0.097 | 8.31 |

3 | 0.035 | 50.5 | 3.18 | 1.77 | 16.7 | 0.002 | 0.17 | 0.099 | 8.48 |

6 | 0.070 | 52.6 | 6.62 | 1.85 | 16.4 | 0.008 | 0.69 | 0.105 | 9.00 |

10 | 0.117 | 57.1 | 12 | 2.00 | 16.2 | 0.022 | 1.88 | 0.119 | 10.2 |

Calculations of the radial distributions of surface pressure and wind for hurricane Frederic, 1979, using the data according to paper [

The emergence of hurricanes is still mysterious. Many observations of initial stages of hurricanes (e.g. see the text [

Paper [

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

The absolute

The slow evolution of

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

(a)

(b) (b)

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

The initial conditions are:

Here

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

Formulas (37) show that depending on sign

1) In the cyclonic case

2) In the anti-cyclonic case

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

To illustrate the model predictions we choose the following parameters

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

These results are consistent with observations in text [

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.