2. Primitive Equation Model (PE)
As illustrated in
Figure 1, all four models consider axisymmetric motions in three layers of an incompressible fluid on an
f-plane with subscripts denoting the layers from 0 for the lowest, 1 for the middle, and 2 for the top layer. The constant density in the lowest two layers is
, and that in the upper layer is
, where the constant
satisfies
. The lowest layer has constant thickness
, while the upper two layers have variable thicknesses
and
. The radial velocity components in the three layers are
,
, and
, and the corresponding radial mass fluxes are
,
, and
. The tangential velocity components in the three layers are
,
, and
.
The hydrostatic pressure at height
z, which depends on the weight of the overlying fluid, is given by
where the first line applies in layers 0 and 1 (the lower two layers), while the second line applies in layer 2 (the upper layer). Assume that, in the far-field, the upper two layer thicknesses approach the constant values
and
, so that the corresponding hydrostatic pressures in the far-field are
Defining the geopotential anomalies as
and
, it is easily shown from (
1) and (
2) that
so that the radial pressure gradient force in the lower two layers is
, while that in the upper layer is
.
The mass continuity equations are
where
w is the vertical velocity at the top of the boundary layer and
is the diabatic mass flux between the upper two layers with
being the upward diabatic mass flux (heating) from layer 1 to layer 2 and
being the downward diabatic mass flux (cooling) from layer 2 to layer 1. Adding the three continuity Equations (
4)–(
6), noting the cancellation of the terms on the right-hand side, and then integrating over the area and specifying the outer boundary condition to be
we conclude that
i.e., the total mass is conserved.
The radial momentum equations for the three layers are given by Equations (1.11)–(1.13) of
Table 1, where
f is the constant Coriolis parameter, the right-hand side terms are given by
, so
,
is the constant coefficient for the frictional stress between the layers,
is the drag coefficient,
is the boundary layer wind speed, and the three
terms are horizontal diffusive fluxes for the
u-equations as given by (
A2) in
Appendix A. Similarly, the tangential momentum equations for the three layers are given by Equations (1.14)–(1.16) of
Table 1, where the right-hand side terms are given by
where the three
terms are horizontal diffusive fluxes for the
m-equations as given by (
A4) in
Appendix A. Since the momentum Equations (1.11)–(1.16) may have a somewhat unfamiliar form, a derivation from first principles is given in
Appendix A.
Rather than predict
and
via (
5) and (
6), it is more convenient to predict
and
. The prognostic equations for
and
are easily obtained from (
3), (
5), and (
6), with the result given in Equations (1.17) and (1.18) of
Table 1. After
and
are predicted from (1.17) and (1.18), the layer depths can be recovered from (1.1) and (1.2), which follow directly from (
3). Note that
is a measure of static stability.
Now, consider the parameterization of the convective mass flux
. The equivalent potential temperature budget in the convection is
, where
is the entrainment parameter,
is the equivalent potential temperature in the convection in the upper layer,
is the equivalent potential temperature in the boundary layer, and
is the equivalent potential temperature in layer 1. We assume that the convection has neutral buoyancy in the upper layer, so that
, where
is the saturation equivalent potential temperature in the upper layer. Combining these last two relations, we obtain Equation (1.9) of
Table 1. Note that conditional instability occurs in a column when
, which means
since
. The equivalent potential temperature of the middle layer is assumed to be a specified parameter, while the boundary layer equivalent potential temperature,
(a prognostic variable), and the upper layer saturation equivalent potential temperature,
(a diagnostic variable), depend on radius and time. Finally,
is given by
, so it then follows that
as indicated by Equation (1.10) of
Table 1, where
is a specified parameter (typically zero).
Table 1.
Summary of the PE model. The upper half of the table lists the diagnostic equations, while the lower half lists the nine prognostic equations. The right-hand sides of (1.11)–(1.16) are defined in (
9) and (
10). Because the radial components
and
are predicted, the PE model allows propagating inertia-gravity waves. Because the boundary layer radial and tangential components
and
are predicted, and because large inflow often occurs in the boundary layer, the PE model can develop boundary layer shocks. The diabatic mass flux
depends on boundary layer convergence (
) and the entrainment parameter
, which in turn depends on the boundary layer equivalent potential temperature
, predicted by (1.19), and the upper tropospheric saturation equivalent potential temperature
, diagnosed from (1.4).
Table 1.
Summary of the PE model. The upper half of the table lists the diagnostic equations, while the lower half lists the nine prognostic equations. The right-hand sides of (1.11)–(1.16) are defined in (
9) and (
10). Because the radial components
and
are predicted, the PE model allows propagating inertia-gravity waves. Because the boundary layer radial and tangential components
and
are predicted, and because large inflow often occurs in the boundary layer, the PE model can develop boundary layer shocks. The diabatic mass flux
depends on boundary layer convergence (
) and the entrainment parameter
, which in turn depends on the boundary layer equivalent potential temperature
, predicted by (1.19), and the upper tropospheric saturation equivalent potential temperature
, diagnosed from (1.4).
Primitive Equation Model (PE) |
---|
Diagnostic Equations: |
Layer thicknesses: | Saturation equivalent potential temperatures: |
| | (1.1) | | | (1.3) |
| | (1.2) | | | (1.4) |
Air-sea interaction: | Ekman pumping: |
| | (1.5) | | | (1.7) |
| | (1.6) | | | (1.8) |
Diabatic mass flux: |
| | (1.9) | | with
| (1.10) |
Prognostic Equations: |
Radial wind components: |
| | (1.11) |
| | (1.12) |
| | (1.13) |
Tangential wind components: |
| | (1.14) |
| | (1.15) |
| | (1.16) |
Geopotential anomalies: |
| | (1.17) |
| | (1.18) |
Boundary layer equivalent potential temperature: |
| | (1.19) |
To include the effects of the warm core on convection, we need to interpret
in terms of potential temperature, even though potential temperature does not appear explicitly in the three-layer model. This can be accomplished by writing the hydrostatic equation for a compressible fluid in the differential form
, where
is the deviation of the geopotential
from its far-field value
,
is the deviation of the potential temperature
from its far-field value
,
hPa is the reference pressure, and
. Based on a discrete form of this hydrostatic equation, it is reasonable to define the middle-tropospheric potential temperature anomaly
, where
is the middle-tropospheric potential temperature in the far field, as
where we use 700 and 300 hPa as representative values of the pressure in the upper two layers. Thus, in the context of the three-layer model,
can be interpreted as proportional to the middle-tropospheric potential temperature anomaly. Using a Taylor series approximation (or simple inspection of the upper tropospheric spacing of
-lines and
-lines on a thermodynamic diagram), we can justify the approximation
, where
is the upper layer saturation equivalent potential temperature in the far field. Combining this approximation with (
11) yields Equation (1.4) of
Table 1. The spatial and temporal dependence of
can serve as an important negative feedback on tropical cyclone intensification, since an evolving warm core can reduce
and hence reduce
via Equation (1.9).
The equivalent potential temperature is predicted only in the lowest layer. In flux form, the boundary layer equivalent potential temperature equation is
where
is the exchange coefficient,
is the boundary layer wind speed, and
is the saturation equivalent potential temperature at the sea-surface temperature and pressure. When (
12) is converted to advective form, we obtain Equation (1.19) of
Table 1. Important controls on moist convection occur in the terms of (1.19). In the tropical cyclone inner region, where there is boundary layer pumping (
so that
), the term
vanishes. However, in the tropical cyclone outer region, where there is boundary layer suction (
so that
), the term
is positive and tends to produce a decrease of boundary layer equivalent potential temperature through the downward transport of dry middle-tropospheric air (
). Air–sea interaction plays a crucial role in maintaining high values of
through the surface flux term
. The factor
becomes large in regions of high surface winds, while the saturation equivalent potential temperature at the sea surface (
) becomes large in regions where the surface pressure is low, an effect that is incorporated through Equation (1.3) of
Table 1. Both effects enhance the surface flux
and can maintain
, and hence
slightly larger than unity, even though an upper tropospheric warm core is being produced and
is increasing through Equation (1.4).
Since
depends on both the sea surface temperature and pressure, and since the sea surface pressure depends on
, we should regard
as a dependent variable that increases as the sea surface pressure decreases. Using a Taylor series approximation (or simple inspection of the lower tropospheric spacing of
p-lines and
-lines on a thermodynamic diagram) the dependent variable
can be parameterized as given in Equation (1.3) of
Table 1. Thus, in the far field, where the geopotential anomaly
is very small, the value of
is
for the typical case examined in
Section 8. However, if
in the inner core is approximately
m
s
(i.e., a
hPa sea surface pressure anomaly), then the value of
is locally increased to
. This dependence of
on
plays an important role in enhancing the surface flux of
in the inner core of the vortex. Finally, we list in Equation (1.6) the assumed
and
formulas used by Ooyama for his standard case. The results of some very interesting experiments with other parameterizations for
and
are discussed by Ooyama ([
2], Section 14).
The nine prognostic variables of the primitive equation model are , , , , , , , , and , so all the other variables are diagnostic. The steps of the numerical integration procedure start with knowledge of these nine prognostic variables, either from the initial conditions or from the previous time step. The procedure for advancing one time level is as follows:
- 1.
Specify desired values for the various fixed parameters:
g,
,
f,
or
,
,
,
,
,
,
,
,
, and
K, where
K is the constant kinematic coefficient of eddy viscosity appearing in the equations for the horizontal diffusive fluxes given in (
A2) and (
A4) in
Appendix A.
- 2.
Using the known geopotential anomalies and , diagnose the layer depths and from (1.1) and (1.2), the sea-surface saturation equivalent potential temperature from (1.3), and the upper tropospheric saturation equivalent potential temperature from (1.4).
- 3.
Compute the boundary layer wind speed from (1.5), and then the exchange coefficient and the drag coefficient from (1.6).
- 4.
Using (1.7), diagnose the boundary layer vertical velocity from the known boundary layer radial velocity , then compute and from (1.8).
- 5.
Using the constant , the diagnosed , and the known , diagnose the entrainment parameter from (1.9); then, compute the diabatic mass flux from (1.10).
- 6.
Using the definitions of
and
given in (
9) and (
10), predict new values of the radial and tangential wind components from (1.11)–(1.16).
- 7.
Using the known radial velocity components and , predict the geopotential anomalies and via (1.17) and (1.18).
- 8.
Using the known boundary layer radial velocity , the diagnosed boundary layer suction , and the diagnosed , predict the boundary layer equivalent potential temperature via (1.19), and then return to step 2 for the beginning of the next time step.
The dynamical core of this model has no explicit thermodynamics. Since there is a discontinuity in the density at the interface between layers 1 and 2, we can regard this interface as containing many constant-density surfaces packed closely together. Through the analogy between density surfaces and potential temperature surfaces (see [
3]), we can also regard this interface as containing many closely packed
surfaces. The mass flux
can therefore be regarded as being due to latent heat release in the moist convective updrafts that carry cloudy air across
surfaces into the upper troposphere. The four models discussed here connect
to boundary layer convergence, boundary layer
, and upper tropospheric temperature in a very simple way. It should be noted that non-hydrostatic, full-physics models of tropical cyclones explicitly model the details of water and ice microphysics, the different categories of ice, the different fall velocities of liquid and ice, the role of aerosols in the size distributions of condensate, etc. The simplified three-layer models presented here neglect all such microphysical details, but do capture the most fundamental effect of precipitating moist convection, i.e., the convective mass flux across
surfaces from the lower troposphere to the upper troposphere. The dynamical adjustments to this mass flux are what produce the lower tropospheric cyclone and the upper tropospheric anticyclone.
Equations (1.1)–(1.19) constitute one of the simplest PE models capable of simulating tropical cyclone life cycles, including intensification, the formation of boundary layer shocks, and the development of an eyewall.
3. Hybrid Balanced Model 1 (HB1)
The equations for the HB1 model are listed in
Table 2. The most obvious difference between
Table 1 and
Table 2 is that for HB1 the number of diagnostic equations has increased from ten to fourteen, while the number of prognostic equations has decreased from nine to five. The ten diagnostic equations in
Table 1 carry over directly to
Table 2, and four additional diagnostic equations appear. In the HB1 model, the prognostic equations for
and
are discarded and replaced by the gradient wind relations (2.11) and (2.12). With gradient balance in the middle and upper layers, the predictions of
and
are redundant with the predictions of
and
. In other words, taking the local time derivative of the gradient wind equation yields
Using the tangential wind equation to eliminate
and the mass continuity principle to eliminate
, we obtain the diagnostic Equations (2.13) and (2.14) of
Table 2, where the local inertial stability
in layer
j is defined by
where
is the relative vorticity in layer
j and
is the potential vorticity in layer
j. Equations (2.13) and (2.14) constitute a coupled pair of second order diagnostic equations (i.e., the Eliassen transverse circulation equations given in reference [
4]) for the radial mass fluxes
and
. These radial mass fluxes are forced by the diabatic and frictional effects that appear on the right-hand sides of (2.13) and (2.14), while the spatial distribution of these fluxes is shaped by the inertial stability factors on the left-hand sides. Since (2.13) and (2.14) are diagnostic equations, an instantaneous change in the diabatic and frictional forcing produces an immediate response in the entire fluid. This “action at a distance” property does not occur in the PE model since fluctuations of the inner core forcing excite inertia-gravity waves, which require a finite time to spread information laterally and thereby adjust the radial mass fluxes in the surrounding fluid. The forcing terms on the right-hand side of the transverse circulation Equations (2.13) and (2.14) involve
,
,
, and
. For the idealized situation in which the absolute angular momentum is materially conserved in the upper two layers, we have
, so the forcing of the transverse circulation then involves only the radial derivatives of the boundary layer pumping
w and the diabatic mass flux
Q. If the diabatic mass flux
Q were to vanish and
w were to be largest at the center of the vortex, then
over a core region, which results in
and
, with the relative magnitudes of
and
determined by the relative magnitudes of the inertial stability factors
and
. Such radial outflow leads to the spin-down (at all radii) of the cyclonic vortex in layers 1 and 2. As discussed by Greenspan and Howard [
5] and Greenspan ([
6], Sections 2.3 and 2.4), this is what occurs in a laminar Ekman layer with a no-slip lower boundary condition. However, as discussed by Eliassen and Lystad [
7], the situation is different in a turbulent Ekman layer with a thin Prandtl surface layer and stress condition at the lower boundary. In that case, the Ekman pumping does not maximize at the vortex axis, but rather at some finite radius. This effect seems crucial in understanding the structure of the eye and eyewall. All four models discussed here can simulate this important effect.
Table 2.
Summary of the HB1 model. The upper half of the table lists the diagnostic equations, while the lower half lists the five prognostic equations. As indicated in (2.11) and (2.12), gradient balance is assumed in the upper two layers, but, as indicated in (2.15) and (2.16), the primitive forms of the momentum equations are retained in the boundary layer, thus allowing the formation of boundary layer shocks. Because the prognostic equations for and have been discarded and replaced by the coupled Eliassen Equations (2.13) and (2.14), propagating inertia-gravity waves are filtered in the HB1 model. A sufficient (but not necessary) condition for the uniqueness of the solutions of the Eliassen equations is that and . Numerical integrations of the Eliassen equations reveal that can become negative at some distance from the cyclone center, but no difficulty is encountered in the solution of the discretized Eliassen equations if a direct method of solution (such as Gaussian elimination) is used.
Table 2.
Summary of the HB1 model. The upper half of the table lists the diagnostic equations, while the lower half lists the five prognostic equations. As indicated in (2.11) and (2.12), gradient balance is assumed in the upper two layers, but, as indicated in (2.15) and (2.16), the primitive forms of the momentum equations are retained in the boundary layer, thus allowing the formation of boundary layer shocks. Because the prognostic equations for and have been discarded and replaced by the coupled Eliassen Equations (2.13) and (2.14), propagating inertia-gravity waves are filtered in the HB1 model. A sufficient (but not necessary) condition for the uniqueness of the solutions of the Eliassen equations is that and . Numerical integrations of the Eliassen equations reveal that can become negative at some distance from the cyclone center, but no difficulty is encountered in the solution of the discretized Eliassen equations if a direct method of solution (such as Gaussian elimination) is used.
Hybrid Balanced Model 1 (HB1) |
---|
Diagnostic Equations: |
Layer thicknesses: | Saturation equivalent potential temperatures: |
| | (2.1) | | | (2.3) |
| | (2.2) | | | (2.4) |
Air-sea interaction: | Ekman pumping: |
| | (2.5) | | | (2.7) |
| | (2.6) | | | (2.8) |
Diabatic mass flux: | Gradient balance equations: |
| | (2.9) | | | (2.11) |
| with
| (2.10) | | | (2.12) |
Transverse circulation: (where
and ) |
| | (2.13) |
| | (2.14) |
Prognostic Equations: |
Boundary layer radial and tangential wind components: |
| | (2.15) |
| | (2.16) |
Geopotential anomalies: |
| | (2.17) |
| | (2.18) |
Boundary layer equivalent potential temperature: |
| | (2.19) |
The five prognostic variables of the HB1 model are , , , , and , so all the other variables are diagnostic. HB1 is “hybrid” in the sense that gradient balance is used in the upper two layers but not in the boundary layer. The steps of the numerical integration procedure start with knowledge of these five prognostic variables, either from the initial conditions or from the previous time step. The procedure for advancing one time level is very similar to that outlined in the previous section for the PE model.
7. Potential Vorticity Aspects of the Models
Since it can lead to additional insights, it is interesting to interpret the four models in terms of potential vorticity (PV) dynamics (see [
14,
15,
16,
17]). Although the four models differ in their treatment of the boundary layer, they have identical continuity equations and angular momentum equations in the two main layers. Thus, all four models have the same PV dynamics in the main layers. Note that because
is a constant, there is no difference between vorticity and potential vorticity in the boundary layer. The PV equations for the two main layers are
where
and
for
. Since the HB1, HB2, and GB models use gradient wind balance, these three models possess an invertibility principle relating
. This invertibility principle is obtained by taking
of
to obtain
. Then, expressing
in terms of
, and making use of the gradient wind relation
, we obtain
This is a pair of coupled, second order equations for
and
, knowing the potential vorticity fields
and
predicted from (
17).
Neglecting the
and
terms in (
17) and considering the conditionally unstable vortex core region where
and
, the PV equations take the form
and
, so that
and
there. Note that, if the development of the upper tropospheric warm core reduces
to unity, this serves as a brake on the otherwise exponential growth of
. Very large values of
in the vortex core result not only in large cyclonic flow in layer 1 but also in substantial cyclonic flow in layer 2, a result that can be attributed to the coupling between the two equations in (
18).
Ooyama performed a linear stability analysis of the GB model ([
2], Section 8). In this simplified linear analysis, curvature effects are neglected, so axisymmetry is replaced by line-symmetry and gradient balance is replaced by geostrophic balance. The problem is reduced to the coupled equations for
and
listed in his entry (8.8). Although he does not interpret these equations as PV equations, they can easily be rearranged into the PV forms
where
is the basic state depth of each layer,
is the basic state entrainment parameter, and the constant
corresponds to
. The two quantities in large parentheses in (
19) are proportional to the perturbation PV in each layer, with
and
proportional to the relative vorticity in the two layers. The two terms on the right-hand sides of (
19), which are proportional to
, represent the effects of Ekman pumping. When coupled with convective mass fluxes and with
, these terms result in an inner core increase of PV in layer 1 and a decrease in layer 2. During such a
linear “CISK” process, both the primary and secondary circulations grow exponentially with time. However, in the
nonlinear context, described for layer 1 by
, the PV can grow exponentially even when the secondary circulation and
are nearly fixed in time. Since the assumptions of linear dynamics are violated even in tropical depressions (see the caption of
Table 5), it follows that it is the
factor in
that is most relevant to tropical cyclone rapid intensification, with the factor
serving as a convective limiter on the intensification process. Recently, Hendricks et al. [
18] have developed a shallow water axisymmetric model for intensification (SWAMI) and have shown how a PV limiter is important for the practical use of their model.
8. Concluding Remarks
A comparison of the four models is given in
Figure 2. The dependent variables of the four models are listed in the left column. The variables
,
,
,
,
U,
,
,
w,
,
, and
Q are diagnostic in all models, while the variables
,
, and
are prognostic in all models. The boundary layer wind components
and
are prognostic in the PE and HB1 models, but are diagnostic in the HB2 and GB models; this is related to the appearance of boundary layer shocks, as indicated by the inner arrow on the right. The radial and tangential wind components
,
,
, and
are prognostic in the PE model but are diagnostic in the HB1, HB2 and GB models; this is related to the occurrence of propagating inertia-gravity waves, as indicated by the outer arrow on the right.
Through a comparison of the GB model Equations (
Table 4) with the HB2 model Equations (
Table 3), it is apparent that the only difference in the model equations is the more general, nonlocal boundary layer dynamics of the HB2 model. However, this difference leads to significant changes to the tropical cyclone life cycles simulated by the two models. These result from different Ekman pumping distributions. To illustrate these differences, Ooyama made comparison runs [
1] using the constants listed in
Table 5, the initial condition
, and with the initial
in gradient balance with
where the constants
and
are also given in
Table 5. The initial condition on
is determined from
. Ooyama found in a numerical test that the values of
and
within 50 km from the vortex center are not influenced very much by their values outside 100 km. Solutions from HB2 are plotted as the solid curves in
Figure 3 below (adapted from [
1], Figure 2) and show three features found in later modeling studies [
9,
10,
11,
12,
19] and also in observations of Hurricane Hugo [
20]: (1) Subgradient boundary layer tangential flow outside the radius of maximum wind; (2) Supergradient boundary layer tangential flow near the radius of maximum wind; (3) Maximum Ekman pumping at a much smaller radius than in the GB model, resulting in a tighter vortex that intensifies more rapidly. As shown in
Figure 3, the GB model produces an Ekman pumping with a 2 m s
maximum at
km with a sharp edge on the outside of the eyewall, while the HB2 model produces an Ekman pumping that maximizes at
km with a sharp edge on the inside of the eyewall. This difference is due to the
term in the HB2 model, i.e., to the nonlocal character of the boundary layer dynamics in HB2. This term leads to a nearly discontinuous behavior of the boundary layer radial inflow near
km, and hence the maximum Ekman pumping there. This behavior of
in the HB2 model indicates the danger of formulating the nonlocal boundary layer dynamics without horizontal diffusion terms. If, without horizontal diffusion, the
term leads to the formation of a shock, the derivatives in the boundary layer equations can cease to exist, and the equations are no longer a useful description of the physics near the shock. This situation can be addressed by fitting in a shock condition or using horizontal diffusion to prevent multivalued radial inflow. The latter was the approach originally suggested by von Neumann and Richtmyer [
21].
Table 5.
Specified constants for the GB and HB2 model runs shown in
Figure 3. Note that the large value of the initial average vorticity inside
, given by
, makes linear CISK theory (see
Section 8 and references [
2,
22]) of limited use.
Table 5.
Specified constants for the GB and HB2 model runs shown in
Figure 3. Note that the large value of the initial average vorticity inside
, given by
, makes linear CISK theory (see
Section 8 and references [
2,
22]) of limited use.
Parameter | Value | Parameter | Value |
---|
|
0.9
| |
1000 m
|
|
0.1
| |
5000 m
|
g |
9.81
m s | |
5000 m
|
f | s | |
372 K
|
|
1004.5 J kg K | |
342 K
|
| m s | |
332 K
|
K |
1000 m s | |
10 m s |
|
0 m s | |
50 km
|
A particular limitation of the Ooyama (1969) cumulus parameterization scheme is the assumption that, if the column is conditionally unstable, the total cloud base vertical mass flux due to deep convection is equal to the vertical mass flux associated with the larger-scale horizontal convergence in the frictional boundary layer. Later studies showed that this assumption is not consistent with observations and with non-hydrostatic, full-physics models. For example, Yanai et al. [
23] performed a heat and moisture budget analysis using an array of five radiosonde stations in the Marshall Islands, which are located in the tropical Pacific ITCZ. From the apparent heat source
and the apparent moisture sink
, and using a simple bulk cloud model, Yanai et al. found that for cloud clusters in the ITCZ region, the total cloud base vertical mass flux is much larger than the vertical mass flux associated with the horizontal convergence in the frictional boundary layer. This conclusion about the vertical convective mass flux at the top of the boundary layer in weak tropical disturbances has been confirmed by several other observational studies, for example, [
24,
25,
26]. However, as found by Smith et al. [
27] using the full-physics CM1 model [
28], this may not be the case for hurricanes and typhoons in their mature and decaying phases.
In order to remove some of the limitations of the Ooyama model, Zhu et al. [
29] developed a three-layer,
-coordinate tropical cyclone model with the option of explicit moist physics or different types of parameterized moist physics. The parameterized moist physics is primarily based on generalizations of the parameterization scheme designed by Arakawa [
30] for the original 3-layer version of the UCLA general circulation model. An axisymmetric version of the Zhu et al. model was explored by Nguyen et al. [
31], with follow-up studies carried out by Zhu and Smith [
32,
33], and Zhu et al. [
34]. Together, these papers clarify the roles of shallow convection, precipitation-cooled downdrafts, and the vertical transport of momentum by deep convection on the dynamics of tropical cyclone intensification. Another family of simple hurricane models, not reviewed here, is the Maximum Potential Intensity (MPI) family first proposed by Emanuel [
35,
36,
37] to help understand the air–sea interaction aspects and the finite amplitude and steady-state aspects of tropical cyclones. An extensive review of the MPI models and their use in understanding the basic intensification process and longer-term evolution of tropical cyclones can be found in recent reviews by Smith and Montgomery [
38,
39].
It is interesting to note that the family of models defined in
Table 1,
Table 2,
Table 3 and
Table 4 and summarized in
Figure 2 has been organized in an unconventional manner. In the conventional organization (for example with community models such as WRF) all members of the family have the same dry dynamical core but have a variety of choices for the parameterized moist physics and air–sea interaction. In contrast, the four models in
Table 1,
Table 2,
Table 3 and
Table 4 and
Figure 2 all have the same parameterized moist physics and air–sea interaction but have four different choices for the dry dynamical core. These two approaches to the organization of model families seem complementary and useful for helping understand the physics of tropical cyclones.
Other extensions of the four models listed in
Table 1,
Table 2,
Table 3 and
Table 4 have been described by Bliss [
40] and DeMaria and Schubert [
41], who relaxed the axisymmetric and balanced assumptions and formulated the three-dimensional (spectral) problem on a middle latitude
-plane to simulate storm motion. As another extension, DeMaria and Schubert [
42,
43] have formulated a generalized version of the axisymmetric PE model (
Table 1) in order to apply the concepts of nonlinear normal mode analysis and initialization to the hurricane problem. The four models listed in
Table 1,
Table 2,
Table 3 and
Table 4 have no explicit thermodynamics, which at first sight seems problematic for a model of a phenomenon that involves latent heat release. However, DeMaria and Pickle [
3] have shown that a model with explicit thermodynamics, if formulated in terms of isentropic layers, is almost mathematically equivalent to the incompressible layer model. The most complete comparisons of the four models of
Table 1,
Table 2,
Table 3 and
Table 4, along with several additional refinements, were given by Frisius and Lee [
8], who performed numerical integrations that were, in the core region, twenty times the resolution (0.25 km vs. 5 km) used by Ooyama. Their results show additional fine structure and generally confirm the coarser resolution results shown in
Figure 3.
The PE model described in
Table 1 is not the only one of the four models that can be generalized to three dimensions. However, the generalizations of HB1, HB2, and GB to three dimensions can become somewhat complicated. The reason is that these involve generalizations of the classic
-equation of quasi-geostrophic theory, which is based on the selective use of the simplest relation between the wind and mass fields. This quasi-geostrophic
-equation is not valid for the inner core regions of intense tropical cyclones because of the large Rossby numbers found for the highly curved flows in the core. More general
-equations that are valid for the high Rossby number flows in tropical cyclones have been derived by Arakawa [
44], Thompson [
45,
46], and Shapiro and Montgomery [
47]. These arguments are based on generalizations of gradient wind balance, e.g., on the nonlinear balance equation in the derivations of Arakawa and Thompson and on asymmetric balance in the derivation of Shapiro and Montgomery. These generalized
-equations yield fully three-dimensional formulas for local inertial stability and local Rossby length. These second-order, elliptic,
-equations have the potential to provide diagnostic tools for a better understanding of such problems as the role of large-scale vertical wind shear on tropical cyclone structure and intensity change.
In closing, it is interesting to note that simple axisymmetric models have greatly expanded our understanding of tropical cyclone dynamics. For additional research that illustrates the rich history of these (and related models), see the studies by Emanuel [
48,
49], Shapiro [
50], Camp and Montgomery [
51], Smith and Vogl [
10], Smith and Montgomery [
52], Schecter and Dunkerton [
53], Schecter [
54,
55], and Frisius et al. [
56]. In many scientific fields, there is a tendency for the simplest models to experience a phase in which they yield new, remarkable physical insights (e.g., the Bohr model of the hydrogen atom, the Eady model of baroclinic instability, and the Ooyama model of tropical cyclones). Later, these simple models are used less in the research literature but are still very useful for introducing students to basic concepts. Although the simple family of tropical cyclone models described in
Table 1,
Table 2,
Table 3 and
Table 4 and
Figure 2 may be entering their pedagogic phase, they may still prove useful in providing an increased understanding of the remarkable inner-core, lower-tropospheric structures observed in intense storms such as Hurricanes Hugo [
20] and Patricia [
57].