ESCI 344 – Tropical Meteorology

Lesson 10 – Tropical Cyclones:  Formation and Structure

 

References:  A Global View of Tropical Cyclones, Elsberry (ed.)

The Hurricane, Pielke

Tropical Cyclones:  Their evolution, structure, and effects, Anthes

Forecasters’ Guide to Tropical Meteorology, Atkinsson

Forecasters Guide to Tropical Meteorology (updated), Ramage

Global Guide to Tropical Cyclone Forecasting, Holland (ed.), online at http://www.bom.gov.au/bmrc/pubs/tcguide/globa_guide_intro.htm

 

Reading:   A Global View of Tropical Cyclones, Chapter 3, Frank (e-reserve)

Tropical Cyclones:  Their Evolution, Structure, and Effects, Chapter 2 (e‑reserve)

Hurricane,  Chapter 2, Pielke (e-reserve)

Global Guide, Chapter 2 (online)

 

GENERAL

˜  Tropical cyclones are primarily driven by latent heating.

˜  As the air spirals in toward the center, it picks up latent heat and sensible heat through evaporation from the ocean.

¡  Once the air is saturated, it can still pick up some sensible heat, but latent heating is the dominant mechanism.

˜  As the air approaches the center of the vortex, it rises in convection either in the eye wall, or in the spiral convective bands.  As it rises, it cools and the water vapor condenses, giving off its latent heat.

˜  The warming of the air due to the latent heat release results in low-level height falls, and upper-level height rises, which helps maintain the low-level convergence of the warm-moist air.

˜  The air is exhausted out and away at the upper levels.

 

STRENGTH, SIZE, AND INTENSITY

˜  We will use the following definitions:

¡  Core intensity, based on maximum winds or minimum sea-level pressure.

¡  Size, based on the mean radius of the outermost closed isobar.

¡  Strength, based on the shape of the outer core wind profile.

˜  There is great variability in the size, intensity, and strength of tropical cyclones.

¡  Storms can be large and intense, small and intense, large and weak, etc.

¡  You can’t infer intensity based on size or strength.

 

DISTRIBUTION OF WIND AND ANGULAR MOMENTUM

˜  The diagram below shows the typical tangential wind structure in a tropical cyclone.

˜  The wind structure is often represented by a modified Rankin vortex,

where C, D, and a are empirically determined constants.

 

¡  a is usually 0.4 to 0.6.

¡  a = 1 would be a pure Rankin vortex, in which relative angular momentum is conserved.

˜  In cylindrical coordinates, the radial and tangential components of the momentum equation are

where u is the radial velocity, v is the tangential velocity, r is the distance from the center of the storm, 𝜽 is the angular measure, and Fr and Fq represent turbulent friction.

˜  If the vortex is steady, axisymmetric, and friction is ignored, then the tangential wind is

,                                                      (1)

so the vortex is in gradient balance.

˜  Dividing by fv, we get

.

˜  The first term is just the Rossby number, RO.

¡  In the core of the vortex, the Rossby number is large, so the Coriolis effects can be ignored.  Therefore, the core of the storm is in cyclostrophic balance.

¡  Outside the core, the Rossby number is of the order of unity, so gradient balance holds.

˜  In either case, the wind speed depends on the pressure gradient.

˜  Broadly speaking, the lower the central pressure, the faster the maximum wind will be.

¡  The relation between maximum winds and the central pressure is closely approximated by

where p¥ is the ambient sea-level pressure outside of the circulation, and pc is the minimum central pressure.

¡  The constant A is empirically determined.  A value of 6.3 is often used for Atlantic hurricanes.

˜  If no energy were added or subtracted from an air parcel as it spiraled toward the center of the vortex, then its absolute angular momentum would have to be conserved.

¡  The absolute angular momentum is the relative angular momentum plus the angular momentum due to the rotation of the Earth.

¡  A ring of air parcels surrounding the center point of the vortex at distance r and stationary with respect to the Earth  will have a specific angular momentum (angular momentum per unit mass) of

just due to the rotation of the Earth.

¡  If the parcels have a tangential velocity, v, then their specific relative angular momentum is

,

so the specific absolute angular momentum is

.                                                   (2)

¡  If the parcel started out at rest at a distance of 500 km, and spiraled in to a radius of 15 km, it would have attained a tangential velocity of over 600 m/s if its angular momentum were conserved.

˜  Obviously, parcels don’t conserve angular momentum as they spiral into the center of a tropical cyclone.

˜  In fact, the absolute angular momentum decreases toward the center of tropical cyclones, which means that air parcels must be losing angular momentum as they spiral inward.

¡  The parcels lose angular momentum through turbulent dissipation.

˜  The absolute vorticity of an axisymmetric vortex is

,                                                        (3)

where the second and third terms are just the shear and curvature terms.  From equation (2) we can show that the vorticity and the specific absolute angular momentum are related via

.                                                          (4)

 

INERTIAL STABILITY OF A VORTEX

˜  A fundamental parameter for assessing how a vortex interacts with its environment is the inertial stability, which is developed mathematically below.

˜  The radial momentum equation in an axisymmetric vortex without friction is

.

˜  In terms of specific absolute angular momentum this can be written as

or

.                                                   (5)

where

.                                                   (6)

¡  Notice that for an air parcel that is in gradient balance, G (r) = Ma(r).

¡  You can think of G(r) as being the angular momentum that an air parcel in gradient balance would have at radius r.

˜  Now, imagine that an air parcel at position r0  is initially in gradient balance with its surroundings.  In this case its absolute angular momentum, M0, will be

.                                                     (7)

¡  If the parcel is displaced radially a small distance, dr, its absolute angular momentum will be conserved and will remain equal to M0 as it is displaced.  However, G(r) can be represented by the first few terms of a Taylor series expansion around r as

.                                                  (8)

Using equations (7) and (8) in equation (5) we get that for the displaced air parcel

.                                 (9)

¡  We know that G(r) is the absolute angular momentum that an air parcel in gradient balance would have at radius r, so equation (9) becomes

.                                           (10)

˜  Solutions to equation (10) are oscillations with frequency of

.                                                     (11)

˜  If the frequency is real, then the parcel will just oscillate around its original radius, and the flow is inertially stable.  If the frequency is imaginary, then the parcel will accelerate away from its original radius, and the flow is inertially unstable.

˜  The conditions for inertial stability are

w 2 > 0

Inertially stable

w 2 = 0

Inertially neutral

w 2 < 0

Inertially unstable

˜  Since absolute angular momentum and vorticity are related, we can  write the oscillation frequency in terms of vorticity, as

.                                                     (12)

˜  The more inertially stable a vortex is, the less it will interact horizontally with its environment, since horizontal displacements are resisted.  We will apply this concept when discussing the structure of the core and outer regions of a tropical cyclone.

˜  Physically, inertial stability can be explained as follows:

¡  If the parcel is displaced outward, and in its new position find that the Coriolis force is stronger than the new pressure gradient force, then it will accelerate outward away from its initial position and the vortex is inertially unstable.

¡  If the parcel is displaced outward, and in its new position find that the Coriolis force is weaker than the new pressure gradient force, then it will accelerate inward toward its initial position and the vortex is inertially stable.

˜  In straight-line flow, or for weak vortexes where the curvature term in equation (12) can be ignored, equation (12) becomes

.                                                           (13)

¡  This is why you often hear it said that anytime absolute vorticity is negative  that the flow is inertially unstable.  However, keep in mind that for stronger vortexes, particularly at low latitudes, negative absolute vorticity doesn’t automatically imply inertial instability.  You must look at the radial gradient of absolute angular momentum to assess the inertial stability in these cases.

 

ROSSBY RADIUS OF DEFORMATION

˜  A fundamental horizontal length scale for a disturbance in a rotating fluid is the Rossby radius of deformation.

˜  The Rossby radius of deformation is the distance that a gravity wave (which are the means by which the fluid adjusts to equilibrium) will travel in one inertial period (w-1).

˜  The Rossby radius of deformation is therefore

                                                (14)

where c  is the horizontal group velocity a gravity wave, and the denominator is the inertial frequency from equation (12).

¡  Note:  For flows whose absolute vorticity is primarily due to planetary vorticity (i.e., flows where z</