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    Numerical solution in the hurricane regime with αT = 10−1 (min)−1: (a) tangential wind u (m s−1) (solid and short dashes) and potential temperature θ (K) (long dashes; contours every 10 K) and (b) potential radius (km) Rp = 2m/f (solid) and streamfunction (kg s−1) ψ [dashed; contour interval (CI) is 0.5 × 109 kg s−1]. Isolines of Rp and ψ are not parallel to each other in the upper/outer portion of the plot, indicating that angular momentum is not materially conserved there; this is due to our “inertial adjustment” Xi (see section 2).

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    Illustration for the occurrence or nonoccurrence of an eye in two different stationary vortices for (a),(c) weak thermal relaxation αT = (300 min)−1 and (b),(d) strong thermal relaxation αT = (10 min)−1; (a),(b) vertical wind w (cm s−1) (negative values shaded) and (c),(d) vertically averaged temperature anomaly R (dashed), together with the vertically averaged equilibrium temperature anomaly eR (solid).

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    Stationary solution in the hurricane regime for a different profile of Te, specified according to (22).

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    (a) Eyeness (ε) and (b) normalized maximum surface wind for 100 different stationary vortices. In both panels, each cross represents the numerical solution for a specific pair of values of αT and cD, and the abscissa is log10F [with αT measured in (10 s)−1]. The normalization constant in (b) is 48 m s−1, which is the maximum wind of the TE solution for the specified Te.

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    Radial profiles of various variables (at the top of the boundary layer; i.e., z = 0) for the TE solution with different choices for Te: (left) tangential wind, (middle) absolute vorticity, and (right) vertical wind; (top row) standard profile Te according to (13); (middle row) Te modified so as to render continuous its second and third radial derivatives; and (bottom row) additional modification to Te so as to have radially increasing absolute vorticity in the inner core. The abscissa is scaled radius, = r/r1.

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    Streamfunction ψ (kg s−1; contours) and vertical wind (cm s−1; shading at steps 20%, 40%, 60%, and 80% of maxw) for four stages during the spindown integration at (a) t = 5 min, (b) t = 60 min, (c) t = 4 h, and (d) t = 96 h corresponding to the steady-state solution. The corresponding value of maxw is 57, 38, 57, and 67 cm s−1. The parameters are αT = (10 min)−1 and cD = 2 × 10−3.

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    As in Fig. 6 but for the spinup integration. The stages presented are (a) t = 5 min, (b) t = 3 h, (c) t = 10 h, and (d) t = 96 h for the steady-state solution; corresponding maxw is 530, 370, 170, and 67 cm s−1.

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    Eyeness ε as a function of time during the spinup integration for αT = (10 min)−1 and cD = 2 × 10−3.

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    Radial profiles of (a) the vertically averaged temperature anomaly (r) − R and (b) the surface wind u0(r) for the spinup integration in the hurricane regime. The solid line refers to the TE solution and the dashed lines refer to the actual vortex during the time integration at times t = 2.4 h, 4.5 h, 10 h, and 96 h.

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    As in Fig. 9 but with cD = 0.

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The Dynamics of Eye Formation and Maintenance in Axisymmetric Diabatic Vortices

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  • 1 University of Mainz, Mainz, Germany
  • | 2 NorthWest Research Associates, Redmond, Washington
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Abstract

This paper investigates the occurrence, formation, and maintenance of eyes in idealized axisymmetric balanced vortices with diabatic forcing. Two key elements of the model setup are temperature relaxation toward a specified equilibrium temperature Te and Ekman pumping from a turbulent boundary layer. Furthermore, the flow is assumed to be almost inviscid in the interior. The model does not attempt any closure for moist convection. Previous work by the authors has shown that there is a continuous transition from monsoonlike vortices to hurricane-like vortices. This transition is governed by the ratio F = αT /cD, where αT is the thermal relaxation rate and cD the surface drag coefficient.

An eye is defined in terms of the vertical wind with maximum upwelling occurring at some finite radius rather than at the origin. It is possible to obtain an eye even though Te maximizes at the origin, that is, even though Te does not directly predispose upwelling at some finite radius. The occurrence of an eye is controlled by F, and the transition between vortices without any eye and vortices with a clearly defined eye is rather sudden. These results are robust with respect to the amplitude of the forcing or the specific shape of Te. The key role of F is corroborated through a systematic nondimensionalization. In a steady-state hurricane-like vortex, mechanical forcing from Ekman pumping maximizes at some finite radius and is instrumental for the maintenance of an eyelike secondary circulation. On the other hand, eye formation during spinup is a purely inviscid process. The results imply that eye formation is a robust and general feature in vortices with strong diabatic forcing.

Corresponding author address: Dr. Volkmar Wirth, Institute for Atmospheric Physics, University of Mainz, Becherweg 21, 55099 Mainz, Germany. Email: vwirth@uni-mainz.de

Abstract

This paper investigates the occurrence, formation, and maintenance of eyes in idealized axisymmetric balanced vortices with diabatic forcing. Two key elements of the model setup are temperature relaxation toward a specified equilibrium temperature Te and Ekman pumping from a turbulent boundary layer. Furthermore, the flow is assumed to be almost inviscid in the interior. The model does not attempt any closure for moist convection. Previous work by the authors has shown that there is a continuous transition from monsoonlike vortices to hurricane-like vortices. This transition is governed by the ratio F = αT /cD, where αT is the thermal relaxation rate and cD the surface drag coefficient.

An eye is defined in terms of the vertical wind with maximum upwelling occurring at some finite radius rather than at the origin. It is possible to obtain an eye even though Te maximizes at the origin, that is, even though Te does not directly predispose upwelling at some finite radius. The occurrence of an eye is controlled by F, and the transition between vortices without any eye and vortices with a clearly defined eye is rather sudden. These results are robust with respect to the amplitude of the forcing or the specific shape of Te. The key role of F is corroborated through a systematic nondimensionalization. In a steady-state hurricane-like vortex, mechanical forcing from Ekman pumping maximizes at some finite radius and is instrumental for the maintenance of an eyelike secondary circulation. On the other hand, eye formation during spinup is a purely inviscid process. The results imply that eye formation is a robust and general feature in vortices with strong diabatic forcing.

Corresponding author address: Dr. Volkmar Wirth, Institute for Atmospheric Physics, University of Mainz, Becherweg 21, 55099 Mainz, Germany. Email: vwirth@uni-mainz.de

1. Introduction

Atmospheric flow is able to organize itself into various types of long-lived vortices with near-circular symmetry about a vertical axis. Some of these vortices have a pronounced structure called an eye, while others do not. A well-known example of a vortex with an eye is a mature hurricane (e.g., Anthes 1982). A hurricane eye is characterized by comparatively weak tangential wind, almost vanishing vertical wind or weak downwelling, often a lack of clouds except for low stratocumulus beneath the temperature inversion, and little or no precipitation. In stark contrast, surrounding the eye there is a stadium-like “eyewall” with outward sloping sides, which is characterized by strong tangential wind decreasing with height, strong upwelling, heavy deep convection, and torrential rainfall. Typically, there is a sharp gradient in tangential wind at the inner side of the eyewall (Malkus 1958). Close to the eyewall there is a ringlike structure of potential vorticity (PV), distinguishing a mature hurricane from a simple PV monopole or a Rankine vortex.

An eye seems to be common to all intense convective cyclones, although there may be important differences between tropical and extratropical conditions (Gray 1998). For tropical cyclones the Dvorak technique effectively uses the appearance in visual or infrared satellite images to estimate the storm’s intensity (Dvorak 1975). The utility of this technique indicates that storm intensity is closely related to storm structure.

For a definition of the term “eye,” it is useful to distinguish between the primary flow and the secondary circulation. The primary (swirling or azimuthal) flow is characterized by its tangential wind, whereas the secondary (toroidal) circulation is the flow in the meridional (radius–altitude) plane. We define an eye with reference to the secondary circulation. If the vertical wind maximizes right at the vortex center, the vortex has no eye. If, on the other hand, the vertical wind increases positively away from the vortex center and has a significant maximum at some finite radius, we consider this vortex to have an eye, and the location of the maximum upwelling is identified as the eyewall. For a well-developed eye the secondary circulation may have a two-cell structure with opposing sense of rotation inside and outside the eyewall and with downwelling at the center of the vortex.

There have been several efforts to explain the various aspects of an eye in a mature hurricane [for a review of early work, see Anthes (1982) and chapter 15 of Palmén and Newton (1969)]. A key element that needs to be understood is the location of the upwelling. One line of argument evokes Ekman pumping from the turbulent boundary layer, which for a mature hurricane favors maximum upwelling at some distance away from the center of the storm rather than at the center itself (Eliassen 1971; Emanuel 1986). Other studies have concentrated on the effect of convective heating in the free atmosphere, which in a mature hurricane has a ringlike structure. Using the Eliassen (1952) balanced vortex theory, Shapiro and Willoughby (1982) showed that this is associated with a reverse inner cell for the secondary circulation, producing many features in agreement with observations. As we will show, these two mechanisms (Ekman pumping and convective heating) are closely related. Regarding the genesis of hurricanes, one remaining key question is how the eye gets formed, that is, how, why, and when convection starts to concentrate into an annulus at some preferred radius.

Emanuel (1997) has shown that the eye dynamics is frontogenetic—producing the observed sharp gradients of tangential wind at the inside of the eyewall. Moreover, he argues that eyes play an active role for the temporal evolution during hurricane spinup, whereas they play a more passive role in the mature stage.

An important issue concerns the role of subsidence of air inside the eye in connection with the thermal structure of a mature hurricane (Jordan 1961). Eye subsidence is often used to explain why the maximum temperature, which is in the center of the storm, is not collocated with the maximum heating associated with deep convection, which is in the eyewall. On the other hand, this leaves open the question as to why there is eye subsidence at all since it implies a thermally indirect inner cell, while simple arguments based on buoyancy would suggest a thermally direct inner cell. It turns out that in a mature hurricane the eye secondary circulation must be mechanically forced (Emanuel 1997). Both Smith (1980) and Emanuel (1997) refer to turbulent exchange of angular momentum across the eyewall when talking about mechanical forcing. By contrast, the hurricane theory of Emanuel (1986) is effectively inviscid in the interior of the atmosphere, leaving Ekman pumping from the frictional boundary layer as the only possible source of mechanical forcing.

The above quoted theoretical approaches are not mutually exclusive. Rather, they describe different aspects and provide different perspectives by making different assumptions. Yet, most of them suffer from a criticism formulated very succinctly by Anthes (1982): “… the theories … are incomplete in that they require specification a priori of important structural aspects of the hurricane circulation (such as the tangential wind and horizontal pressure gradient profiles or the latent heating distribution). In reality the dynamic and thermodynamic variables are completely interacting, so that the specified structure of one variable may not be used in a cause–effect relation to explain the structure of another variable.”

In the present paper we refrain from dealing with the complexities of real hurricanes. Rather, we aim to investigate the dynamics of eye formation and eye maintenance in an idealized modeling framework similar to that of Wirth and Dunkerton (2006, hereafter WD06). Simplicity is considered to be an advantage rather than a deficiency. To be sure, we shall not evade Anthes’s criticism entirely, but we hope to shed some new light onto this problem.

Our focus will be on thermally forced vortices, also referred to as “diabatic vortices.” The model considers axisymmetric balanced flow of a stably stratified non-Boussinesq atmosphere on the f plane. The flow is forced by heating in the vortex center, which is implemented through relaxation of temperature T toward a specified equilibrium temperature Te. The flow is dissipated through surface friction, and it is assumed to be almost inviscid in the interior. As was shown in WD06, these constraints still allow vortices of rather different nature. Monsoon- and hurricane-like vortices were identified as two specific limits of this more general flow system, thus providing a unified perspective on the dynamics of such vortices. These two limits were referred to as “monsoon regime” and “hurricane regime,” respectively. Essential vortex properties in this model are controlled by the ratio
i1520-0469-66-12-3601-e1
where αT denotes the rate constant for thermal relaxation and cD is the dimensionless drag coefficient for surface friction (a nondimensional version of F is derived in the appendix). In other words, a specific combination of parameters allows one to characterize the primary vortex circulation and to quantify the transition between the monsoon and hurricane regime, respectively.

Depending on the location in parameter space, this system allows vortices either with or without an eye. As will be shown in this paper, variation of F mediates the transition from a more general vortex with a one-cell toroidal secondary circulation to a hurricane-like vortex with a two-cell toroidal secondary circulation. It is the goal of the present paper to understand these structural changes in the model, to relate the existence of an eye to the basic model parameters, to understand the eye maintenance in steady state, and to consider the eye formation during vortex spinup. In addition, our results will allow us to relate seemingly unconnected results from previous investigations.

The plan of the paper is as follows. First we give a short review of the model setup and the numerical solution technique in section 2. The occurrence of an eye despite no predisposition from the specified equilibrium temperature Te is investigated in section 3, and the controlling parameters for eye occurrence will be uncovered in section 4. These results will be set on a firm footing in section 5 through nondimensionalization of the governing equations. Mechanisms of eye maintenance in steady state are discussed in section 6, and eye formation during spinup is addressed in section 7. In section 8 we briefly talk about further aspects that are sometimes associated with an eye. Finally, a discussion of our results and a summary of our main conclusions are provided in section 9.

2. Model

The model setup and the numerical solution technique is basically the same as in WD06. Only the essential ingredients will be described here, while the details can be obtained from WD06.

We consider the primitive equations on a Northern Hemisphere f plane. As vertical coordinate we use log p altitude z = −H ln(p/p0), where p is pressure, H = 7000 m is a constant scale height, and p0 = 1000 hPa is a constant reference pressure. The flow is assumed to be rotationally symmetric about the vertical axis such that all fields only depend on radius r, altitude z, and time t. The domain extends from z = 0 to z = D in altitude and from r = 0 to r = R in radius. The flow is balanced in the sense that the equation for radial momentum can be replaced by the gradient wind equation. With these assumptions, the equations read
i1520-0469-66-12-3601-e2
i1520-0469-66-12-3601-e3
i1520-0469-66-12-3601-e4
i1520-0469-66-12-3601-e5
i1520-0469-66-12-3601-e6
Here u, υ, and w denote the tangential, radial, and vertical wind, respectively; Φ is the geopotential; T is the temperature; ρ0(z) = p(z)/(gH); f is the constant Coriolis parameter; g is the acceleration due to gravity; Ts = gH/Rd; Rd is the gas constant for dry air; in which cp is the specific heat at constant pressure; Q is the diabatic heating; and X represents material nonconservation of tangential momentum associated with subscale motions. Our notation regarding tangential and radial wind is the same as in WD06 and, thus, opposite to the convention used in the “hurricane community.” Combining (3) and (4) yields the thermal wind equation
i1520-0469-66-12-3601-e7
Writing the secondary circulation in terms of a mass streamfunction ψ,
i1520-0469-66-12-3601-e8
guarantees that (6) is satisfied automatically.
Equation (2) can be written in terms of angular momentum,
i1520-0469-66-12-3601-e9
as Dm/Dt = rX or
i1520-0469-66-12-3601-e10
where J(ψ, m) = (∂ψ/∂r)(∂m/∂z) − (∂ψ/∂z)(∂m/∂r). When the flow is inviscid in the interior (X = 0), one obtains material conservation of m. This will be referred to as the angular-momentum-conserving (AMC) limit. Note that the AMC limit still allows a frictional boundary layer (see below). According to (10), isolines of m and ψ are parallel for stationary flow in the AMC limit.
Diabatic heating Q is modeled as Newtonian relaxation toward a specified equilibrium temperature Te(r, z),
i1520-0469-66-12-3601-e11
where αT is the relaxation constant. Clearly, this is a highly simplified description of a set of complex and interacting processes including moist convection and radiation. For a given boundary layer moist entropy, Te can be viewed as the reversible moist adiabat emanating from the top of the subcloud layer (Xu and Emanuel 1989). The heat source according to (11) is internal and part of the solution (because T is part of the solution). Nevertheless, considering Te as given is one of the major assumptions in our approach. We do not try to obtain a closed theory by, for instance, trying to associate Te with boundary layer thermodynamics (see Emanuel et al. 1994). Rather, we consider Te as external and study the resulting dynamics. The relaxation coefficient αT is an effective parameter quantifying how quickly the atmosphere approaches Te. As detailed in WD06, we expect that αT is related to the fractional area of deep convection and therefore can vary by several orders of magnitude.
The equilibrium temperature is decomposed as
i1520-0469-66-12-3601-e12
with some reference temperature TR(z) depending on altitude only and with T ′ being small beyond a certain radius r0. The standard choice used throughout most of this paper is given by
i1520-0469-66-12-3601-e13
with r1 = 2R/15. It maximizes in the midtroposphere on the axis of symmetry and decays for larger radii like r−1 (for illustration, see Fig. 1 in WD06). Profiles of Te that maximize at some finite radius will be briefly discussed in section 8.
The nonconservative term X is given by
i1520-0469-66-12-3601-e14
where Xe is a superposition of horizontal and vertical eddy diffusion and Xi represents a relaxation toward inertial neutrality as a response to inertial instability (if it occurs). As discussed in WD06, both terms are small (but nonzero for numerical reasons). For the details we refer to WD06.
Surface stress TS is modeled through a drag law:
i1520-0469-66-12-3601-e15
where u0 is the tangential wind at z = 0, which is identified as the top of the boundary layer. The drag coefficient cd is assumed to depend on u0 according to
i1520-0469-66-12-3601-e16
with uS = 2.5 m s−1. In our numerical solutions we shall keep uS fixed and vary the strength of surface drag by varying the dimensionless parameter cD. In a separate study we varied the value of uS between 0 and 7.5 m s−1 and found that our results are practically insensitive to the precise choice of uS.
Given a balanced initial state, the equations are integrated forward in time as follows. At every time step the nonconservative terms Q and X can be derived from the knowledge of u and T. This allows one to compute the rhs of the Eliassen equation, which can symbolically be written as
i1520-0469-66-12-3601-e17
[Eliassen 1952; see (23) in WD06]. Here L is a linear elliptic differential operator and ψ is the streamfunction as defined in (8). The Eliassen equation is solved for ψ after suitable boundary conditions have been specified. The resulting secondary circulation (υ, w) is used to integrate (2) over one time increment, and the corresponding new temperature is obtained from (7).
The boundary layer is not treated explicitly, but its effect on the free atmosphere is accounted for through a simple parameterization. Following Schubert and Hack (1983) and Wirth (1995), the drag law (15) is translated to a lower boundary condition for ψ:
i1520-0469-66-12-3601-e18
where ζa denotes the vertical component of absolute vorticity. The vertical wind w associated with (18) represents the same effect as Ekman pumping. Although the approximations inherent in (18) prevent a precise quantitative representation of the boundary layer dynamics underneath a hurricane (e.g., Smith et al. 2008; Smith and Montgomery 2008), we believe that our parameterization captures the essential qualitative aspects, which is all we need in this study.

Numerical solutions are obtained through time stepping as described above. The computational domain was extended in the radial direction in order to prevent sensitivities associated with the outer boundary condition for ψ when solving (17). The domain is discretized by 67 grid points in the radial direction and 35 grid points in the vertical direction. To obtain the stationary solution, we typically started from a reference atmosphere at rest and integrated forward in time until stationarity was reached. Figure 1 illustrates the final result for the hurricane regime. It demonstrates that the AMC limit is well satisfied in the interior part of the vortex, where isolines of m and ψ are nearly parallel.

For the special case of no surface friction (cD = 0), our model allows a so-called thermal equilibrium (TE) solution. In this work, the latter is defined through
i1520-0469-66-12-3601-e19
where ue is obtained from Te by downward integration of the thermal wind equation (7) with ue(r, D) = −fr/2 [corresponding to ζa(r, D) = 0]. The corresponding surface wind is related to the temperature field via
i1520-0469-66-12-3601-e20
with Te substituted for T [the caret denotes a vertical average, cf. Eq. (38) in WD06]. As in WD06, we shall restrict our attention to supercritical forcing. This implies that the second term on the rhs exceeds fr/2 at intermediate radii and, hence, that one obtains cyclonic surface winds. It follows that the TE solution is possible only as long as cD = 0 since, otherwise, Ekman pumping (18) would force w ≠ 0, which according to (21) would prevent T = Te. The TE solution must not be confused with solutions from the “hurricane regime,” which are solutions for large αT /cD but cD ≠ 0.

In summary, the primary flow consists of tangential wind u and temperature T, which are diagnostically related to each other via (7). The secondary circulation (υ, w) is diagnostically related to Q and X via (17) and (8). The time evolution is given by (2), which is our “master equation.” The primary and the secondary circulation are tightly coupled since the evolution of the primary circulation depends on the secondary circulation [see (2)], whereas the secondary circulation is diagnostically related to the primary circulation [owing to (17) and the fact that Q and X depend on u and T].

We believe that—given Te—the above formulation represents the maximum conceptual simplification by explicitly building in all (approximate) balances that can be assumed. Geopotential (or pressure, if geometric altitude had been used) is eliminated because it is diagnostically related to other variables. Correspondingly, our arguments do not refer to geopotential or pressure, in contrast with earlier arguments (as, e.g., in Smith 1980). True, eventually vertical and radial accelerations are forced by small nonhydrostatic or nongradient imbalances, respectively (e.g., Bannon 1995). However, in a quasi-balanced vortex the assumption of hydrostaticity and the use of the Eliassen equation provide a better-posed formulation that avoids the notorious problem of small residuals from nearly canceling forces. To be sure, both approaches yield approximately the same results despite their conceptual differences. Care must be exercised to the extent that diagnostic relations per se prevent an interpretation in terms of cause and effect. For instance, the diagnostic Eliassen equation has often been used in the past to “explain” the secondary circulation in terms of a specified heating field Q (e.g., Shapiro and Willoughby 1982). On the other hand—as stated above—the heating field in our model is not specified but is part of the solution. In fact, in some situations it may be more meaningful to adopt the complementary perspective and view the heating field Q as forced by the secondary circulation (for reasons to be made clear below).

The key quantities that we shall vary in our numerical experiments are αT (quantifying the strength of the thermal forcing for a given TTe) and cD (quantifying the strength of boundary layer friction for a given surface wind). All other parameters are set to the following standard values: the domain size is D = 16 km and R = 450 km. The reference temperature TR(z) is specified in terms of potential temperature θR(z) = TR(z)eκz/H, which is set to θR = 300 K at z = 0 and increases linearly with altitude with a constant lapse rate R/dz = 4 K km−1. Furthermore, we use g = 9.81 m s−2, R = 287 J kg−1 K−1 (implying TS = 239 K), and f = 2Ω sinϕ0, with Ω = 7.292 × 10−5 s−1 and ϕ0 = 20° (i.e., f ≈ 0.5 × 10−4 s−1).

3. Eye occurrence despite no predisposition

We begin our analysis by considering the stationary solution for two different values of αT (Fig. 2). Apparently, for weak relaxation the upwelling maximizes right at the vortex center (Fig. 2a). On the other hand, for strong relaxation the maximum upwelling is located around r = 60 km, whereas at the vortex center there is almost vanishing w (Fig. 2b). Following the definition in the introduction, we associate the latter vortex with an eye, but not the former. In Fig. 2a the secondary circulation is thermally direct, whereas in Fig. 2b the part of the secondary circulation inside the eye (with weak subsidence in the upper troposphere) is thermally indirect.

How is it possible to obtain maximum upwelling at some finite distance from the center although our specified equilibrium temperature Te maximizes right at the center? Neglecting the small radial advection term, the stationary heat equation reads
i1520-0469-66-12-3601-e21
where the static stability S = ∂T/∂z + κT/H can effectively be taken as constant. In the monsoon limit (left column of Fig. 2) the spatial structure of the heating Q is essentially given by that of Te (see Fig. 2c). Maximum upwelling is right on the origin because maximum Te is right on the origin. In this sense, our thermal equilibrium Te has no predisposition for an eye. Note that our approach is in distinct contrast with studies like Shapiro and Willoughby (1982) or Schubert et al. (2007), who specify ringlike convective heating. Ringlike heating almost inevitably produces an eyelike secondary circulation, so in those studies the occurrence of an eye is built in by design.

The situation changes if we use a much larger αT while leaving Te unchanged (right column of Fig. 2). Now the structural similarity between Te and Q gets lost. The actual temperature T (dashed line in Fig. 2d), which is part of the solution, approaches Te and arranges such that the difference TeT (and, hence, w) maximizes around r = 60 km. So, in the hurricane limit we obtain an eye without direct predisposition from Te.

How robust is this result? We tested a number of different profiles for the equilibrium temperature Te(r, z). In all cases there was an eye as soon as the ratio αT /cD exceeded a certain threshold. For example, consider
i1520-0469-66-12-3601-e22
with T0 = 10 K and r1 = 0.4R. The radial dependence is a linear decrease out to r1 (solid line in Fig. 3). It is singular for r → 0 in the sense that the thermal wind relation prevents from approaching e (otherwise the angular velocity would be infinite at the origin). If anything, this profile seems to predispose upwelling at the origin even more than our standard profile (13). Nevertheless, for the stationary solution the fields of temperature and tangential wind are similar to that from the standard profile (see dashed line in Fig. 3, to be compared with the dashed line in Fig. 2d). As a consequence, TeT maximizes off the origin and the vortex has a clear eye.

4. What controls eye occurrence?

What determines whether the vortex has an eye or not? We define a variable ε to quantify “eyeness”—that is, the degree to which the vortex features an eye:
i1520-0469-66-12-3601-e23
If the mean vertical wind maximizes at r = 0, one obtains ε = 0. If ŵ maximizes at some finite radius with ŵ = 0 at the origin, one obtains ε = 1. Assuming maxrŵ > 0, a value ε > 1 indicates downwelling at the origin corresponding to a particularly strong eye structure. The radius at which ŵ maximizes is associated with the eyewall.

As in WD06, we computed 100 different steady-state solutions through varying both αT and cD by a wide margin, thus covering over three orders of magnitude for the scaling factor F. Figure 4a shows ε as a function of F for these 100 vortices. Apparently, for small values of F (corresponding to the monsoon regime) ε equals zero and the vortex does not show any hint of an eye. At some point ε starts to increase with increasing F, and for large values of F (corresponding to the hurricane regime) all vortices have a well-defined eye. Although there is some scatter, the points nearly collapse onto a single curve. Additional runs have shown that the remaining scatter is associated with the numerical diffusion, which had to be included in order to render the numerics stable. We conclude that F not only determines the essential features of the primary flow (see WD06) but is also the key parameter for the occurrence (or not) of an eye.

At first this result seems to conflict with the results of WD06. There it was shown that the vertical wind w does not scale with F. On the other hand, the eyeness ε, which is defined with reference to the vertical wind w, does scale with F. How is this possible? Note that the definition of ε contains the ratio of w at two radii. Equation (21) implies ŵαT (e); evaluating both sides at r1 and r2 and forming the ratio, the parameter αT drops out, leaving
i1520-0469-66-12-3601-e24
Since temperature does scale with F, this suggests that ε, too, should scale with F. The key point is that our parameter ε is sensitive to the shape of the radial profile of ŵ, but not to the magnitude of w.

Broadly speaking, the transition from “no eye” to “eye” is associated with the transition from weak to strong surface winds (Fig. 4b). However, the transition from ε ≈ 0 to ε ≳ 1 occurs over a rather small range of values of F, requiring F to increase only by a factor of about 4 (Fig. 4a). This is in distinct contrast with the transition in terms of the surface wind, which requires F to increase by a factor of about 22 (Fig. 4b), or the transition in terms of e, which requires F to increase by a factor of about 35 (not shown here; see Fig. 9 of WD06). In other words, the eye sets in rather suddenly, suggesting some nonlinear mechanism at work.

a. Dependence on forcing amplitude?

In our model setup the forcing amplitude T0 is fixed and specified. The parameter T0 represents the excess temperature of a moist adiabat in the inner part of the vortex against the environmental temperature. In reality, this moist adiabat and, hence, T0 depend on boundary layer moist entropy, which in turn is affected by the surface wind. This is the closure problem, which we intend to address in a future paper.

Here, we discuss briefly the sensitivity with respect to T0 in the context of the present model. The numerical computations were repeated with the forcing amplitude reduced by a factor of 2, that is, with T0 = 5 K instead of T0 = 10 K. The resulting scatterplots (not shown) look very similar to those in Fig. 4. In both cases the transition in terms of u0 is much more gradual than in terms of ε. For the smaller value of T0 the transition has shifted to a slightly larger value of F. We conclude that overall the behavior is only weakly dependent on the forcing amplitude.

5. Nondimensionalization

The results of the previous section can rigorously be understood through a nondimensionalization. We restrict our attention to stationary solutions in the AMC limit (i.e., X = 0) and write the governing equations in terms of the variables m, T ′ = TTR, and ψ. We also introduce the reference static stability:
i1520-0469-66-12-3601-e25
The dimensional scales for the different coordinates and variables are summarized in Table 1. The choices for the scales of u and m were motivated by considering the hurricane limit for which mru and the fu term is small in the thermal wind equation (7). Denoting the nondimensional coordinates and variables by an asterisk, the equation for angular momentum reads
i1520-0469-66-12-3601-e26
and the thermal wind balance becomes
i1520-0469-66-12-3601-e27
The upper boundary condition in the AMC limit is
i1520-0469-66-12-3601-e28
The nondimensional heat equation becomes
i1520-0469-66-12-3601-e29
with a nondimensional reference static stability
i1520-0469-66-12-3601-e30
and
i1520-0469-66-12-3601-e31
The lower boundary condition for ψ becomes
i1520-0469-66-12-3601-e32
where
i1520-0469-66-12-3601-e33
i1520-0469-66-12-3601-e34
Furthermore, the nondimensional vertical wind can be expressed as
i1520-0469-66-12-3601-e35
The complete set of nondimensional equations and boundary conditions is (26), (27), (28), (29), and (32). They contain four nondimensional parameters, A*, B*, C*, and S*.

We can now make rigorous statements about the dependence of the stationary model solution on αT and cD in the AMC limit (with cD ≠ 0). In the nondimensionalized equations, these two parameters only occur in A* and do so only in the form of a ratio; that is, A* ∝ F. This has several implications. For instance, u* can only depend on F, so does u (because U is independent of αT and cD). Accordingly, the nearly inviscid numerical solutions in Fig. 4b nearly collapse onto a single curve. By a similar argument, T can only depend on F, which is consistent with the numerical result of Fig. 9b in WD06. As far as eyeness is concerned, we note that ε is determined by the structure (not the magnitude) of ψ, that is, by ψ*. Thus, we have proven that ε can only be a function of F in the AMC limit, and this is again approximately borne out by the numerical solutions shown in Fig. 4a. On the other hand, the scale of the vertical wind is proportional to cD (but independent of αT), which together with (35) explains why w does not scale with F (see Fig. 10 of WD06).

The nondimensional version of (21) reads S*w* = A*(Te* − T*), which is equivalent to
i1520-0469-66-12-3601-e36
Here δz denotes the scale of the vertical (upward) penetration of ψe and F* is the nondimensional scaling factor (see appendix). The latter is defined such that F* ≈ 1 indicates the transition between the monsoon (F* ≪ 1) and hurricane (F* ≫ 1) regime. In the hurricane limit, which corresponds to (T*eT*) → 0 and F* → ∞, the rhs side of (36) is undetermined. However, in this limit the factors B* and C* in (32) can be neglected such that ψ* = O(1) and, hence, w* = O(1). On the other hand, in the monsoon limit, which corresponds to T*eT* ≈ 1 and F* → 0, the rhs of (36) tends to zero, indicating that w* → 0.

6. Eye maintenance in steady state

In the monsoon regime (i.e., for small values of F) the radial profile of temperature is rather flat in comparison with the equilibrium temperature (cf. the dashed and solid lines in Fig. 2c). It follows that the radial structure of e and, by virtue of (21), that of ŵ is essentially determined by Te. With our choice of Te maximizing at the origin, this prevents any eye from being formed.

The situation is very different in hurricane regime, that is, for vortices with large F. Figure 4 shows that these vortices have a well-defined eye. The question, which we shall address in this section, is: How is the eye maintained in steady state?

a. Linearization about the TE solution

The situation can be clarified by linearizing the problem about the TE solution (see section 2). For cD = 0, the TE solution solves the fully nonlinear equations. If cD differs from zero, the solution can be considered as a small perturbation to the TE solution as long as cD is small enough. Denoting the TE basic state by an overbar and the linearized variables by a prime, one obtains
i1520-0469-66-12-3601-e37
i1520-0469-66-12-3601-e38
i1520-0469-66-12-3601-e39
i1520-0469-66-12-3601-e40
i1520-0469-66-12-3601-e41
where S = ∂T/∂z + κT/H. Equation (41) is automatically satisfied if the meridional perturbation wind is defined through a perturbation streamfunction ψ′,
i1520-0469-66-12-3601-e42
Equations (38) and (39) can be combined into the linearized thermal wind equation,
i1520-0469-66-12-3601-e43
where (r, z) f + 2u/r. Equation (37) for the tangential wind can be written in terms of the basic state angular momentum as
i1520-0469-66-12-3601-e44
The linearized lower boundary condition reads
i1520-0469-66-12-3601-e45
The solution to the linearized equations can be obtained as follows. Apparently, the rhs of (45) is known, and the streamfunction ψ′ at z = 0 can readily be computed. Equation (44) can formally be written as an equation for ψ′(r, z) as follows:
i1520-0469-66-12-3601-e46
with V −∂m/∂z and W m/∂r. The characteristics of this equation are given in parametric form by
i1520-0469-66-12-3601-e47
that is, they are known since the basic state is known. The rate of change of ψ′ along the characteristics is −r2ρ0X′. The solution of this part of the problem is particularly simple if X′ = 0. In this case the value of ψ′(r, z) is given by the value of ψE(r), where the corresponding two points must be connected by a characteristic. If the characteristics emanating from the bottom boundary do not turn back to the bottom boundary but, instead, “vanish” at the outer boundary, the above is a simple recipe to compute ψ′(r, z) throughout the domain. One simple way of avoiding this “turning back” is to specify a Te for which the corresponding ue is inertially stable; that is, ∂me/∂r > 0, which is satisfied for our standard choice of Te. Next, one can compute T ′ from (40) since the lhs is now completely known. Finally, u′ can be obtained from the thermal wind equation (43) by assuming that u′ = 0 at z = D (i.e., insisting that the full solution u = u + u′ must satisfy ζa = 0 at z = 0).
From the foregoing it follows that
i1520-0469-66-12-3601-e48
throughout the entire domain. Using this in (40) and (43), one obtains
i1520-0469-66-12-3601-e49

The distinction between cause and effect is easier in the linearized system than with the fully nonlinear equations. The above approach to solving the linearized problem clarifies that both the secondary circulation and the deviation of the actual solution from the equilibrium solution are associated with boundary layer dynamics in the presence of nonzero surface wind. Ekman pumping dictates upwelling from the boundary layer, and for X = 0 this secondary circulation must flow along lines of constant m. The upwelling, in turn, produces a temperature anomaly T ′, which is associated with a u′ due to thermal wind balance. Finally, the temperature anomaly T ′ is also associated with a heating Q′ = −αTT ′. It transpires that the heating is driven by the secondary circulation, which emanates from the boundary layer. Given the surface wind, this is a form of “upward/outward control” analogous to the “downward control” of Haynes et al. (1991). Note that a superficial interpretation of the Eliassen equation (17) seems to suggest the opposite cause-and-effect relation that a given heating “drives” the secondary circulation. Of course, such an interpretation is misleading because the Eliassen equation is a diagnostic relation and cannot, per se, distinguish between cause and effect.

It follows that the secondary circulation of our steady-state vortices in hurricane regime can be viewed as mechanically forced through Ekman pumping (cf. “Ekman layer control” in Eliassen 1971). The upwelling w at the top of the boundary layer is governed by the radial profile of the tangential wind via (18). The surface wind, in turn, is related to T = Te via (20). Our Te is continuous at r = r1 with a continuous first radial derivative, but higher derivatives are discontinuous. It follows that ζa is discontinuous and the vertical wind w has a δ-like peak at r = r1 (first row in Fig. 5).

We slightly modified Te so as to obtain continuous second and third radial derivatives (second row in Fig. 5). Overall, this results in a smoother behavior, but the peak in w is still clearly visible. In yet another modification to Te we tried to make Te more hurricane-like with radially increasing ζa at small radii (third row in Fig. 5). This is associated with a concave or “U-shaped” profile of the tangential wind at small radii (see below section 8). However, there is a peak in w close to the radius of maximum wind in all three examples. This illustrates that maximum upwelling at some finite radius is a generic feature for our parameterization of Ekman pumping. It is common to all profiles of tangential wind because all of them increase with radius in the inner core, reach a maximum at some finite radius, and decay at larger radii. The peakedness of the upwelling is related to the amount of curvature of the u profile close to the radius of maximum wind [cf. Eliassen and Lystad (1977), who obtained much smoother maxima in w, associated with smoother u profiles. The forced upwelling at the peak propagates upward and outward following isolines of m and forms the eyewall. In other words, the specified equilibrium temperature Te indirectly predisposes an eye in the hurricane limit via the associated surface wind and through the mechanism of Ekman pumping. The role of Te is indirect because it is not the spatial structure of Te itself that matters but rather the corresponding balanced tangential wind and its effect on the turbulent boundary layer.

b. Spindown from the TE solution

It is instructive to compute the stationary solution in the hurricane regime through spindown from the TE solution (rather than spinup from rest as usually done). Figure 6 illustrates the behavior of the secondary circulation at four steps during the integration. As in WD06, the streamfunction ψ for the secondary circulation is partitioned into several components (the so-called Eliassen decomposition), two of which are quantitatively dominant: the component ψe owing to Ekman pumping and the component ψd owing to diabatic heating. The Ekman component ψe is obtained by solving (17) with F = 0 but using the correct lower boundary condition. The diabatic component ψd is obtained by solving (17) with X = 0 and the lower boundary condition replaced by ψ = 0. Initially T = Te such that Q = 0 and, hence, ψd = 0. Correspondingly, the secondary circulation is dominated by ψe early in the spindown integration (Fig. 6a). As argued above, close to the surface this results in maximum w at finite radius r1 = 60 km. Interestingly, in the middle and upper troposphere the maximum upwelling is located at the vortex center initially (Figs. 6a and 6b); only by t = 4 h has the maximum w moved to finite radius throughout the troposphere (Figs. 6c and 6d).

The upwelling associated with ψe in the lower troposphere cools the atmosphere; heating sets in as a consequence of (11), and some ψd adds to the ψe (the latter remains almost unchanged throughout the simulation). The effect of this ψd is quite dramatic, leading to a strong change of the streamfunction and associated w field (see Figs. 6b–d). As the final steady state settles out, there is a subtle balance with ψd dominating in the middle and upper troposphere; yet its existence is triggered by ψe. In a sense ψd can be viewed as a very strong feedback to a small ψe.

The last point is conceptually important. The Eliassen decomposition for the steady state (Fig. 3 in WD06) suggests that throughout most of the domain the diabatic contribution ψd is much larger than the Ekman contribution ψe. One may be tempted to conclude that the secondary circulation is primarily “driven” by diabatic heating. However, this spindown experiment, as well as the earlier argument from the linearized equations, indicates that the diabatic heating must be considered as a feedback and that the mechanical forcing from Ekman pumping is the “underlying cause” for the secondary circulation. Ekman pumping, in turn, is governed by the primary circulation, which in the hurricane limit is essentially given by Te.

c. Inside the eye

What about the weak downwelling in the upper troposphere inside the eye, which we get for our stationary hurricane-like vortices (e.g., Fig. 2b)? It can be viewed as an indirect consequence of Ekman pumping in the following sense. The contribution ψe is likely to yield upwelling throughout the inner part of the vortex (see Fig. 6a). On the other hand, ψd (which was identified as an indirect consequence of ψe) has a more complicated structure. Maximum we at finite radius implies maximum heating Q at finite radius. Since the rhs F[Q, X] of the Eliassen Eq. (17) contains ∂Q/∂r, one may expect a two-cell structure for ψd with a counterclockwise inner cell and a clockwise outer cell. The diabatic contribution ψd typically dominates the Ekman contribution ψe, thus explaining the eye subsidence as an indirect consequence of Ekman upwelling. Earlier arguments addressing eye subsidence, which are based on a specified heating distribution (Shapiro and Willoughby 1982; Schubert et al. 2007), remain valid and contribute important insights. However, care has to be exercised when interpreting such arguments in terms of cause and effect.

Interestingly, Ekman pumping is not the only mechanism in our model to yield downwelling inside the eye. This can be illuminated through a second spindown integration, for which we set cD = 0, that completely eliminates Ekman pumping. Nevertheless, the final steady state shows an eye with distinct downwelling inside the eye (not shown). It turns out that the Eliassen secondary circulation associated with interior friction X has—as before—its maximum upwelling at some finite radius away from the center. During the ensuing evolution, this causes T to deviate from Te and thus leads to nonzero ψd. The latter yields downwelling in the vortex center and dominates the secondary circulation in the final steady state. Note that in this experiment all vertical velocities are roughly an order of magnitude smaller than in the previous experiment with cD ≠ 0. This indicates that our “control experiment” is, indeed, almost inviscid in the interior (as promised) and that in the presence of a realistic drag coefficient cD Ekman pumping dominates over interior friction X regarding the secondary circulation.

In both cases downwelling inside the eye must be viewed as mechanically forced. In the control experiment the dominant mechanism is Ekman pumping, which is associated with turbulence inside the boundary layer. In the experiment without Ekman pumping (cD = 0), the downwelling inside the eye was traced back to interior friction X, which essentially parameterizes turbulent exchange of angular momentum in the radial direction.

The momentum balance inside the eye is dominated by two distinct contributions. On the one hand, there is a positive tendency of tangential wind owing to turbulent transfer of angular momentum across the eyewall as given by X. On the other hand, there is a negative tendency from outward advection. This balance is consistent with earlier findings (Malkus 1958; Emanuel 1997).

7. Eye formation during spinup

We now consider eye formation during the spinup integration from an initial state at rest in the hurricane regime. Figure 7 shows the vertical wind and the secondary circulation at four different stages. Initially (Fig. 7a) the surface wind is very small and the secondary streamfunction (contours) essentially consists of its diabatic contribution ψd. The corresponding upwelling (shading) maximizes right at the origin. Later, cyclonic surface winds give rise to a nonzero Ekman contribution ψe. At the same time, the maximum upwelling moves away from the origin toward some finite radius r ≈ 60 km, thus forming the eye.

The corresponding evolution of ε is shown in Fig. 8. During the first 2.5 h ε = 0, which is equivalent to maximum upwelling right at the origin. Then ε increases strongly, reaching a value of ε = 1.2 around t = 10 h and decays thereafter to reach the steady-state value of ε ≈ 1. Note, incidentally, that real hurricanes have been observed to have significant eye subsidence (i.e., ε > 1) during the formation stage, while the eye subsidence is weak and the air is essentially stagnant (ε ≈ 1) during the mature stage (Jordan 1961).

How is the eye formed in this spinup integration and what are the underlying processes? Figure 9 shows radial profiles of vertically averaged temperature and surface wind at four different stages during spinup. Early into the integration (t ≤ 2.4 h) the temperature increases with time at all radii in the core and near-core region (Fig. 9a) with e still maximizing at the vortex center. Correspondingly ε = 0, consistent with Fig. 8. At intermediate times the temperature at r = 0 increases faster than in the eyewall region (r ≈ 60 km), and for a while (0) even overshoots e(0) (see the curve t = 10 h).1 Finally the temperature settles into its steady-state profile (cf. Fig. 2d). During all times the heat equation can be approximated by (21), which means that warming is always a small residual between diabatic heating and compensating adiabatic cooling. This implies ŵe, and the overshoot in Fig. 9a corresponds to downwelling inside the eye and, hence, the section in Fig. 8 where ε > 1.

Owing to (21), the question “why does the vortex form an eye?” is equivalent to asking “why does T approach Te faster at the vortex center (r = 0) than in the eyewall region (at some finite radius)?”. So let us study the evolution of temperature during spinup. If temperature were governed by simple Newtonian relaxation ∂T/∂t = −αT (TTe), one would expect a “uniform” approach toward Te and no eye ever to form. However, the evolution of the T is also affected by the secondary circulation (υ, w), which in turn depends on both the equilibrium profile Te and the state of the primary vortex (T and u). The secondary circulation essentially redistributes the heating such that the warming is realized at some other location than where the heating is released (Ooyama 1969). Unfortunately, this whole argument is qualitative only and does not allow one to predict the precise development of temperature.

Owing to (20), the more rapid and closer approach to Te at the vortex center is associated with stronger than equilibrium values of tangential surface wind at small but finite radii, as shown in Fig. 9b. Points where the actual and equilibrium values of u0 intersect correspond to points in Fig. 9a where the actual and equilibrium values of T have the same slope. This result is arguably consistent with the fact that most of the overturning mass circulation consists of upward motion that forces a slight depression from Te at intermediate radii and a corresponding but much smaller elevation above Te at distant radii, in qualitative agreement with the equal-area construction of Held and Hou (1980).

In the hurricane regime the magnitude of the Ekman pumping for a given surface wind is so small in relation to the strength of the thermal forcing that it allows spinup of strong surface winds. This renders u0 even larger than that of the TE solution—that is, in some sense “overshooting” its truly inviscid steady-state value. Of course, this cannot go on forever because of the relaxational nature of the diabatic heating (13): as T approaches Te, the heating and the associated secondary circulation weaken. If cD were zero, the primary vortex would approach the equilibrium solution and no secondary circulation would be left. The role of Ekman pumping is to keep T slightly away from Te (and hence u slightly away from ue). In the final steady state the decelerating tendency from Ekman pumping is balanced by the accelerating tendency from the remaining small amount of the secondary circulation (which, in turn, owes its existence to the Ekman pumping).

This argument also sheds light on the nonexistence of an eye in the monsoon regime. There, the Ekman pumping for a given surface wind is much larger in comparison to the strength of the thermal forcing. This implies that the evolution of the fields shown in Fig. 7 terminates before the maximum upwelling moves away from the center because the stationary balance can be achieved with small or moderate surface winds.

The foregoing discussion suggests that in the hurricane regime Ekman pumping produces a significant effect only at the final stage, implying that at an early and intermediate stage the eye formation in our model is essentially an inviscid process. To test this prediction, we repeated the spinup experiment with cD = 0, that is, without any Ekman pumping. The result is displayed in Fig. 10. The initial development (t ≤ 10 h) is very similar to the standard run (Fig. 9), and there is a clear eye visible with maximum updraft at some finite radius and downwelling inside the eye. However, due to the lack of surface friction, T has no problem approaching Te, and the secondary circulation gradually decays as time increases. At later times (e.g., t = 48 h) the magnitude of the vertical wind is much smaller than for the previous run with cD ≠ 0.

We conclude that—even though the maintenance of an eye in steady state requires mechanical forcing (mostly from surface friction) as an essential ingredient—the eye formation during spinup in our integration is basically an inviscid process.

8. Further aspects

So far we have defined an eye through the secondary circulation, that is, through the radial profile of the vertical wind. As indicated in the introduction, there are additional features that can be used to characterize an eye. In this section we briefly comment on these aspects.

A hurricane eye has often been characterized by a concave (or U-shaped) radial profile of the tangential wind u. Indeed, we find a similar behavior in our model: vortices (and only those vortices) that have a fully developed eye in terms of secondary circulation are characterized by a concave profile u0(r) at small radii. This applies both to the evolution during spinup and to the parametric dependence of stationary vortices. It should be borne in mind that our axisymmetric model does not account for radial transports of angular momentum and moist entropy associated with eye mesovortices.

During the spinup integration, the u profile starts to turn concave several hours after the onset of the eye in terms of vertical wind. This sequence of events shows that the change in topology of the secondary circulation during eye formation preceeds—and, hence, causes in our model—the change of the shape of the u profile a few hours later. During the initial phase, the tangential wind inside the eye spins up through the inviscid mechanism of inward advection of angular momentum. Later, the toroidal secondary circulation develops a reverse inner cell; now the loss of angular momentum through outward advection opposes the gain of angular momentum from turbulent stresses into the eye that might attempt to restore solid body rotation when the u profile is initially concave. The latter is consistent with the results of Emanuel (1997).

Another important aspect is the observed fact that the temperature maximum is typically found inside the eye, even though the maximum heating is farther outside in the eyewall. It was shown by Emanuel (1997) that inviscid processes can cause enough eye subsidence to keep the eye temperature at the same temperature as the convective clouds in the eyewall, but no more. For the eye temperature to be greater than the temperature of the eyewall, one needs radial stresses.

Our simulated vortices in the hurricane regime are consistent with the observed structure in terms of the radial profile of temperature and heating (cf. section 3). However, this should not be overinterpreted. After all, our equilibrium temperature Te maximizes at the origin by design, from which it is almost inevitable to obtain a temperature maximum at the origin. Regarding eye formation in a hurricane, it would be more realistic to consider distributions of Te with an off-axis peak. This is because Te should be related to boundary-layer moist entropy, which increases more or less in proportion to the wind and so maximizes near the radius of maximum winds, at least at some time during the development. It turns out that such a ringlike equilibrium temperature requires a more refined model. We tried a scheme with two distinct values of the relaxation coefficient—a large one in convective regions and a much smaller one in nonconvective regions. Applying this refined model to a ringlike Te, we found that the actual temperature was rather flat in the eye, and there was very little horizontal flow in the eye with a front at the eyewall. It was only through a substantial increase in explicit horizontal eddy diffusion Xe that we obtained a temperature maximum at the origin. All this is entirely consistent with the predictions from Emanuel (1997) about the role of radial stresses for the temperature profile. Our main point in this paper, however, is a different one. Clearly it is not very surprising to obtain an eye when specifying a ringlike Te (cf. Shapiro and Willoughby 1982). By contrast, we have shown that, even with Te maximizing at the origin, one can obtain an eye, and we have quantified and discussed this within the framework of our model.

9. Summary and conclusions

We have investigated the occurrence, maintenance, and formation of eyes in axisymmetric diabatic vortices using the model setup of WD06. Key elements of this simple model are (i) relaxation toward a specified equilibrium temperature Te that maximizes at the vortex center and (ii) Ekman pumping from a turbulent boundary layer. The model does not attempt any closure for moist convection. An eye is defined in terms of the secondary circulation as maximum upwelling at some finite radius. These are our main results:

  1. It is possible to obtain a steady-state vortex with an eye even though the specified equilibrium temperature Te does not directly predispose any such structure. This means that maximum upwelling in the hurricane regime is off the vortex center, although the specified equilibrium temperature Te maximizes at r = 0.
  2. The occurrence of an eye is governed by the ratio of the strength of thermal forcing and the strength of surface friction, that is, by the scaling factor F = αT /cD. This result was proven rigorously for the AMC limit (i.e., inviscid in the interior) through nondimensionalization of the equations and is approximately borne out by our numerical solutions.
  3. The transition in eyeness as a function of F is rather sudden; that is, only a fairly small range of the parameter space is reserved for the transition between states without eye and states with a well-developed eye.
  4. The regime transition depends only weakly on the thermal forcing amplitude T0, which is fixed and specified in the current model. This implies that parameterizing T0 in terms of boundary layer moist entropy (rather than just specifying T0) would not change our conclusions regarding F as the key variable governing the rather sudden occurrence of an eye.
  5. If F exceeds a certain threshold, the steady-state vortex has an eye irrespective of the exact shape of the specified equilibrium temperature Te(r, z). In this sense, the occurrence of an eye is a robust feature. This may explain why it is relatively easy to obtain an eye in numerical simulations (Ooyama 1969; Kurihara and Bender 1982; Gray 1998).
  6. Although our equilibrium profile Te does not directly predispose upwelling off the origin in the hurricane regime, it does so indirectly via the associated surface wind and through the mechanism of Ekman pumping. In steady state, Ekman pumping keeps the solution from approaching the TE solution. Thus, for a given Te the secondary circulation, and hence the existence of the eye and the location of maximum upwelling, must be viewed as mechanically forced through boundary layer friction. The upwelling, in turn, gives rise to diabatic heating. The corresponding diabatic contribution to the secondary circulation from the Eliassen Eq. (17) dominates the direct Ekman contribution and is characterized by downwelling at the vortex center. This sheds new light on results from earlier investigations: some of them emphasize the role of boundary layer friction for the structure of the secondary circulation, while others emphasize the role of the diabatic heating (see our introduction). We have found that, given Te, the primary aspect is Ekman pumping, but that diabatic heating can be viewed as a strong feedback that determines to a large extent the structure and strength of the secondary circulation.
  7. Mechanical forcing from turbulent mixing across the eyewall can also lead to an eyelike secondary circulation, but this effect is much smaller than the effect of Ekman pumping as long as the interior is almost inviscid.
  8. In the hurricane regime, the formation of an eye is essentially due to an inviscid mechanism, in distinct contrast to the key role of the mechanical forcing for steady-state maintenance. The result was obtained by starting with an atmosphere at rest and analyzing the temporal evolution: in these spinup integrations the eye formation does not depend on the existence of an Ekman boundary layer. The model integration shows eye subsidence to be strongest during the intensification period and much weaker in steady state, consistent with the evolution of real hurricanes (Jordan 1961). Similarly, the fairly sudden onset of an eye during this spinup integration seems consistent with observations of tropical cyclogenesis (M. DeMaria 2007, personal communication).
  9. All our results are based on the Eliassen balanced vortex approach, which implicitly assumes that deviations from hydrostatic and gradient wind balance are small. This provides a novel perspective on eye occurrence and maintenance as essentially due to the diabatic forcing of a primary circulation and associated mechanical forcing of the secondary circulation.

What do these results imply for the formation of eyes in real hurricanes? Convective heating in mature hurricanes is believed to depend on the strength of the surface wind that enhances evaporation, further enhancing storm intensity (Emanuel 1991). Such wind-induced surface heat exchange (WISHE)-like feedback (Emanuel et al. 1994) is not allowed in the current version of our model, but it could be implemented by making equilibrium temperature Te a function of surface wind. Since the surface wind almost inevitably maximizes at some finite radius, this would yield maximum Te at some finite radius, with a strong chance to obtain ringlike heating. Owing to the diagnostic Eliassen relation (17), one then expects an eye—that is, a two-cell structure of the secondary circulation with maximum upwelling close to the radius of maximum wind. In fact, some earlier studies (e.g., Shapiro and Willoughby 1982; Schubert et al. 2007) prescribe convective heating with a ringlike structure, for which the eyelike secondary circulation is practically built in.

Even in our model the existence of an eye is tantamount to ringlike heating, but this is not obtained through thermodynamic feedbacks on Te from the boundary layer (because they are excluded in our approach). We conclude that eye occurrence is a robust and general property of any axisymmetric vortex with strong diabatic forcing. In particular, it does not depend on the details of convective closure. The simplicity of the model allows us to explicitly quantify what is meant by “strong diabatic heating.” In more refined models and in the real atmosphere, there are additional thermodynamic feedbacks from the boundary layer. However, as argued above, these are likely to increase the tendency for eye formation. What we have shown is that they are not a necessary prerequisite.

In a future publication we plan to implement closure for Te as indicated above. Preliminary experiments show that this, indeed, allows us to simulate the spinup of a hurricane-like vortex from a weak initial vortex. The results from this paper will be a useful starting point to interpret these fully coupled simulations.

Acknowledgments

We thank Thomas Frisius for helpful discussion that sparked off the derivation of our nondimensionalization. Furthermore we gratefully acknowledge Kerry Emanuel and two anonymous reviewers for their insightful comments, which led to substantial improvement of the presentation.

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APPENDIX

Nondimensional Scaling Factor for Regime Transition

This appendix provides the derivation of a nondimensional scaling factor, which characterizes the regime behavior. The point of departure is Eq. (51) of WD06, which is a condition for stationarity derived from the discussion of scales within the model. In current terminology it reads
i1520-0469-66-12-3601-ea1
where δz is the scale of the vertical (upward) penetration of ψe.
Assuming that the surface wind is not too small, we can make a number of simplifications. First, absolute vorticity at z = 0 is approximated as
i1520-0469-66-12-3601-ea2
Furthermore, we assume
i1520-0469-66-12-3601-ea3
and (20) turns into
i1520-0469-66-12-3601-ea4
In addition, we have θeθTeT = T0(1 − T*) and SR ≈ Δθ/H. Equation (A1) can then be rewritten as
i1520-0469-66-12-3601-ea5
The factor F* allows one to characterize the transition between the monsoon and hurricane regimes in terms of the model parameters. Note that δz may depend on the aspect ratio of the vortex; that is, for fixed D and H it may depend on L. Considering D, H, and Rd as fixed, (A5) shows that F* = F*(αT, cD, T0, SR, L). The monsoon regime, which can be defined through T* ≪ 1, corresponds to F* ≪ 1. The hurricane regime, which can defined through T* ≈ 1, corresponds to F* ≫ 1. Transition occurs around T* = ½; that is, F* ≈ 1. The ratio between αT and cD, for which F* = 1, is
i1520-0469-66-12-3601-ea6
Among others, this indicates a weak (inverse square root) dependence on T0, consistent with our numerical results from section 4a.

Fig. 1.
Fig. 1.

Numerical solution in the hurricane regime with αT = 10−1 (min)−1: (a) tangential wind u (m s−1) (solid and short dashes) and potential temperature θ (K) (long dashes; contours every 10 K) and (b) potential radius (km) Rp = 2m/f (solid) and streamfunction (kg s−1) ψ [dashed; contour interval (CI) is 0.5 × 109 kg s−1]. Isolines of Rp and ψ are not parallel to each other in the upper/outer portion of the plot, indicating that angular momentum is not materially conserved there; this is due to our “inertial adjustment” Xi (see section 2).

Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3031.1

Fig. 2.
Fig. 2.

Illustration for the occurrence or nonoccurrence of an eye in two different stationary vortices for (a),(c) weak thermal relaxation αT = (300 min)−1 and (b),(d) strong thermal relaxation αT = (10 min)−1; (a),(b) vertical wind w (cm s−1) (negative values shaded) and (c),(d) vertically averaged temperature anomaly R (dashed), together with the vertically averaged equilibrium temperature anomaly eR (solid).

Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3031.1

Fig. 3.
Fig. 3.

Stationary solution in the hurricane regime for a different profile of Te, specified according to (22).

Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3031.1

Fig. 4.
Fig. 4.

(a) Eyeness (ε) and (b) normalized maximum surface wind for 100 different stationary vortices. In both panels, each cross represents the numerical solution for a specific pair of values of αT and cD, and the abscissa is log10F [with αT measured in (10 s)−1]. The normalization constant in (b) is 48 m s−1, which is the maximum wind of the TE solution for the specified Te.

Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3031.1

Fig. 5.
Fig. 5.

Radial profiles of various variables (at the top of the boundary layer; i.e., z = 0) for the TE solution with different choices for Te: (left) tangential wind, (middle) absolute vorticity, and (right) vertical wind; (top row) standard profile Te according to (13); (middle row) Te modified so as to render continuous its second and third radial derivatives; and (bottom row) additional modification to Te so as to have radially increasing absolute vorticity in the inner core. The abscissa is scaled radius, = r/r1.

Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3031.1

Fig. 6.
Fig. 6.

Streamfunction ψ (kg s−1; contours) and vertical wind (cm s−1; shading at steps 20%, 40%, 60%, and 80% of maxw) for four stages during the spindown integration at (a) t = 5 min, (b) t = 60 min, (c) t = 4 h, and (d) t = 96 h corresponding to the steady-state solution. The corresponding value of maxw is 57, 38, 57, and 67 cm s−1. The parameters are αT = (10 min)−1 and cD = 2 × 10−3.

Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3031.1

Fig. 7.
Fig. 7.

As in Fig. 6 but for the spinup integration. The stages presented are (a) t = 5 min, (b) t = 3 h, (c) t = 10 h, and (d) t = 96 h for the steady-state solution; corresponding maxw is 530, 370, 170, and 67 cm s−1.

Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3031.1

Fig. 8.
Fig. 8.

Eyeness ε as a function of time during the spinup integration for αT = (10 min)−1 and cD = 2 × 10−3.

Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3031.1

Fig. 9.
Fig. 9.

Radial profiles of (a) the vertically averaged temperature anomaly (r) − R and (b) the surface wind u0(r) for the spinup integration in the hurricane regime. The solid line refers to the TE solution and the dashed lines refer to the actual vortex during the time integration at times t = 2.4 h, 4.5 h, 10 h, and 96 h.

Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3031.1

Fig. 10.
Fig. 10.

As in Fig. 9 but with cD = 0.

Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3031.1

Table 1.

Dimensional scales used to nondimensionalize the model.

Table 1.

1

The “overshoot” implies cooling inside the eye. This is qualitatively consistent with real hurricanes, where radiation leads to cooling inside the eye (Smith 2005), although our simple model does not by any means provide a realistic simulation of this process.

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