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    Idealized profile of a category 3 hurricane of (left) nondimensional velocity (solid; left axis) and thickness anomaly (dashed; right axis) and (right) relative vorticity . Different zones in the left panel are delimited by vertical dashed lines: core (zone 1), annulus (zone 2), and outer zone (zone 3).

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    (left) Pressure (colors) and velocity (arrows) distribution in the most unstable mode in the nondimensionalized xy domain. (center) Vorticity (colors) and location of the critical radii ( dashed lines). (right) IGW part of the unstable mode, as seen in the divergence field (enlarged domain).

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    Growth rates of unstable modes, as functions of (left) the width of the transition zone and (right) the width of the annulus . In the left panel Wr = 0.4 and in the right panel . Dashed lines account for the “Rossby compensated” growth rate, taking into account the variation of the global Rossby number due to changes in velocity distribution with d or . It is defined as the dimensional growth rate normalized by the (fixed) global Rossby number value at and in the left and right panels, respectively: .

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    Nonlinear dry saturation of the unstable mode: evolution of relative vorticity in the square domain in the xy plane with coordinates nondimensionalized by the RMW at times (a) , (b) , (c) , (d) , (e) , and (f) ().

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    Nonlinear saturation of the unstable mode: azimuthally averaged profiles of nondimensional (a) thickness , (b) azimuthal velocity , and (c) relative vorticity at times t = 0 (thick gray), 0.5 (black dashed), 1.5 (black dashed–dotted), and 4 (solid black).

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    Angular momentum budget contributions integrated in time between and in the dry case during nonlinear evolution of the (left) l = 3 unstable mode and (right) random initial perturbation. Lines represent the eddy term (blue), mean term (red), sum of the mean and eddy terms (black dotted), and actual difference of the radial distribution of mean angular momentum between and (gray dotted). Thin dark-green line in the right panel corresponds to the eddy term in the left panel, for comparison.

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    Evolution of the dissipation (black; left axis) and modulus of divergence of the wave field (gray; right axis) and of the amplitude of the modal perturbation with azimuthal wavenumber (blue; right axis) during the nonlinear saturation of the instability (dry case). Dissipation is calculated in an annulus encompassing the initial outer critical radius (see Fig. 2) and normalized by the total energy of the perturbation at . Divergence is averaged over an annulus around the vortex. Amplitude of the l = 3 perturbation is evaluated from the thickness. Both are normalized by their initial values. Perturbation amplitude shows good agreement with the prediction for the linear growth (dashed) at initial stages.

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    Nonlinear saturation of the moist precipitating instability. Colors represent the humidity anomaly (with respect to initial uniform value), with color shading scaled by a factor of 10 for positive values (see color bars). Contours indicate positive (black; with an interval of ) and negative (gray; with an interval of ) relative vorticity. Green contours show at an interval of 0.2. Simulation times are (a) , (b) , (c) , (d) , (e) , and (f) (). The square domain is in the nondimensionalized xy plane.

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    Azimuthally averaged (left) thickness perturbation, (center) azimuthal velocity perturbation, and (right) relative vorticity during the saturation of the barotropic instability in dry (black), moist precipitating (blue), and moist precipitating and evaporating (red) simulations at times (dashed) and (solid). Initial vortex profile is plotted in gray in the right panel.

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    As in Fig. 6, but in the (left) moist precipitating and (right) moist precipitating and evaporating models. The l = 3 unstable mode is used as the initial perturbation. Angular mean of the precipitation (magenta) is added. Thin dark-green line is the eddy term in the dry case (see Fig. 6, left).

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    Evolution of the relative vorticity during the moist destabilization (no evaporation) of the category 3 hurricane with random perturbation in the nondimensionalized xy domain at times (a) , (b) , (c) , (d) , (e) , and (f) . Green contours (interval of 0.2) indicate precipitation .

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    Evolution of the magnitude of the divergence averaged over an annulus around the vortex in different runs. Colors indicate dry (black), moist precipitating (blue), and moist precipitating and evaporating (red) cases. Initialization is with the unstable mode (solid) or random (dashed) perturbation. Thin gray line represents evolution of the amplitude of the unstable mode as predicted by the linear stability analysis. Sharp peak at the very beginning corresponds to the waves emitted during initial adjustment.

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    Snapshots of the divergence field at during the nonlinear saturation of the hurricane instability in the (left) dry and (right) moist precipitating and evaporating cases. Black contours are vorticity with an interval of , green contours (right panel only) are , with an interval of 0.2. Note that the color range is different in the two panels.

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    Angular means of the radial profiles of (left) pressure, (center) azimuthal velocity, and (right) vorticity at (dashed) and (solid) for dry (black) and moist precipitating and evaporating (red) simulations, with initial profiles (gray). Initialization with random initial perturbation (thick lines) and unstable mode (thin lines).

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Understanding Instabilities of Tropical Cyclones and Their Evolution with a Moist Convective Rotating Shallow-Water Model

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  • 1 Laboratoire de Météorologie Dynamique, CNRS-IPSL, ENS/UPMC, Paris, France
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Abstract

Instabilities of hurricane-like vortices are studied with the help of a rotating shallow-water model, including the effects of moist convection. Linear stability analysis demonstrates that dominant unstable modes are mixed Rossby–inertia–gravity waves. It is shown that, depending on fine details of the vorticity profile, a wavenumber selection of the instability may operate or not, leading in some cases to an unstable mode with a distinctively maximal growth rate and in other cases to an ensemble of unstable modes with close growth rates. Numerical simulations are performed in order to investigate nonlinear saturation of the instability and to understand the dynamical role of moisture. In agreement with previous studies, the authors confirm axisymmetrization of vorticity in the course of the development of the instability, which induces changes of intensity of the hurricane. In “dry” simulations, winds are intensified only inside the radius of maximum wind, while the maximum value of the wind decreases. “Moist precipitating” simulations (with and without evaporation) exhibit a net increase of winds, also at the radius of maximum wind, as compared to the dry simulations. Dynamical effects of moisture on the reorganization of the vortex and on the efficiency of inertia–gravity wave emission are quantified and shown to be considerable. Periodic bursts in the emission of waves related to the development of the unstable modes inside the vortex are evidenced, as well as the appearance of convectively coupled waves in the moist precipitating simulations with evaporation.

Corresponding author address: Noé Lahaye, Department of Mechanical and Aerospace Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411. E-mail: nlahaye@eng.ucsd.edu

Current affiliation: Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, California.

Abstract

Instabilities of hurricane-like vortices are studied with the help of a rotating shallow-water model, including the effects of moist convection. Linear stability analysis demonstrates that dominant unstable modes are mixed Rossby–inertia–gravity waves. It is shown that, depending on fine details of the vorticity profile, a wavenumber selection of the instability may operate or not, leading in some cases to an unstable mode with a distinctively maximal growth rate and in other cases to an ensemble of unstable modes with close growth rates. Numerical simulations are performed in order to investigate nonlinear saturation of the instability and to understand the dynamical role of moisture. In agreement with previous studies, the authors confirm axisymmetrization of vorticity in the course of the development of the instability, which induces changes of intensity of the hurricane. In “dry” simulations, winds are intensified only inside the radius of maximum wind, while the maximum value of the wind decreases. “Moist precipitating” simulations (with and without evaporation) exhibit a net increase of winds, also at the radius of maximum wind, as compared to the dry simulations. Dynamical effects of moisture on the reorganization of the vortex and on the efficiency of inertia–gravity wave emission are quantified and shown to be considerable. Periodic bursts in the emission of waves related to the development of the unstable modes inside the vortex are evidenced, as well as the appearance of convectively coupled waves in the moist precipitating simulations with evaporation.

Corresponding author address: Noé Lahaye, Department of Mechanical and Aerospace Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411. E-mail: nlahaye@eng.ucsd.edu

Current affiliation: Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, California.

1. Introduction

It is well known that mature tropical cyclones may undergo destabilization of their inner core. The instability is due to the annular character of the potential vorticity (PV) distribution (e.g., Schubert et al. 1999; Kossin and Schubert 2001; Hodyss and Nolan 2008; Hendricks et al. 2009, 2012, 2014; Menelaou et al. 2013a,b). This distribution of positive PV anomaly (PVA) results from the diabatic heating related to slantwise convection that takes place inside the radius of maximum wind (RMW), which is forced by the secondary circulation of the hurricane (e.g., Eliassen and Lystad 1977; Möller and Smith 1994; Hausman et al. 2006; Bui et al. 2009). Such profile of PV—with a reversal of the sign of its gradient—satisfies the classical criterion of barotropic instability. It allows for counterpropagating Rossby waves at each edge, which can phase lock and interact, inducing mutual growth (e.g., Montgomery and Shapiro 1995). The developing instability results in PV mixing and redistribution of angular momentum (e.g., Schubert et al. 1999; Kossin and Schubert 2001; Hendricks et al. 2009), thus considerably influencing the evolution of the hurricane (Emanuel 1997; Reasor et al. 2009). The diabatic effects are important, as they can modify the character of the instability (Rozoff et al. 2009; Lambaerts et al. 2011; Hendricks et al. 2012).

One of the greatest challenges in forecasting tropical cyclones is comprehension of the evolution of their intensity. Sudden changes in structure of tropical cyclones accompanied by intensification are commonly observed (Kossin and Eastin 2001; Reasor et al. 2009; Hendricks et al. 2012; Menelaou et al. 2013b) but remain far from being well understood (Wang and Wu 2004). Some of them were linked to the instability of the inner core (Hendricks et al. 2012; Menelaou et al. 2013b), thus urging an investigation of this aspect of the hurricane dynamics. During the last two decades, an increasing number of observations evidence the ubiquity of this instability in tropical cyclones (Kossin and Eastin 2001; Kossin et al. 2002; Kossin and Schubert 2004; Hendricks et al. 2012; Menelaou et al. 2013b) and indicate that it plays a major role in their life cycle.

The early studies of the instability of tropical cyclones were carried in the nondivergent barotropic (in fact, 2D Euler or Navier–Stokes) framework (Schubert et al. 1999; Kossin and Schubert 2001). These studies provided a link between the instability and the existence of mesovortices and polygonal eyewalls, well identified in observations (Lewis and Hawkins 1982; Muramatsu 1986). Numerical experiments showed that the PV mixing and momentum redistribution associated with the development of the instability diminish both the pressure at the center of the vortex and the maximum value of the wind. Yet, the overall impact of the inner-core instability on the intensity of the cyclone is not clearly understood: while the above-mentioned studies, as well as more recent ones using more sophisticated models (Yang et al. 2007; Hendricks et al. 2009), show a decrease of the intensity, it is often found that developing asymmetric disturbances of the inner core increase it (Chen and Yau 2001; Wang 2002; Menelaou et al. 2013a) or, at least, give a transient intensification, as found in observations (Hendricks et al. 2012) and in simulation of observed hurricanes (Menelaou et al. 2013b). In particular, diabatic effects may counteract the decrease of the maximal wind and restore a PV ring with sustained maximum around an eye with enhanced vorticity (Rozoff et al. 2009; Hendricks et al. 2014; Naylor and Schecter 2014).

Together with simplified 2D Navier–Stokes modeling of the inner-core instability, 3D instabilities of the hurricanes were also investigated in the literature (e.g., Nolan and Montgomery 2002; Hodyss and Nolan 2008). Their development was analyzed with the help of mesoscale atmospheric models (Kwon and Frank 2005), including the moist convection (Kwon and Frank 2008). In a recent study specially aiming at moisture effects (Naylor and Schecter 2014) the instabilities of hurricanes were analyzed with the help of a cloud-resolving model. However, “all inclusive” numerical models include a plethora of parameterized thermodynamical and dissipative processes, and the associated complexity is prohibitive for understanding the precise role of each kind of process in the developing instability.

Hurricane-like vortices are ageostrophic, and are therefore subject to inertia–gravity wave (IGW) emission (Ford 1994; Plougonven and Zeitlin 2002) by pure dynamical reasons. This so-called spontaneous IGW emission, which is important by itself as one of the sources of IGW in the atmosphere (e.g., Plougonven and Zhang 2014), is related to specific radiative instabilities (Le Dizès and Billant 2009) that result from the resonance between an outgoing IGW and a Rossby wave propagating across the PV gradient inside the vortex [these instabilities are thus relevant even for monotonic PV profiles (Ford 1994; Schecter and Montgomery 2004)]. The presence of critical levels is inherent for radiative instabilities (Le Dizès and Billant 2009), which, at least at large vertical wavenumbers, may be understood via the overreflection mechanism of the wave at the critical level [Billant and Le Dizès (2009) and references therein]. Yet, wave radiation may also result from the barotropic instability of the vortex. As an unstable vortex mode may couple with outgoing IGW (Hodyss and Nolan 2008), the barotropic instability is modified by compressibility. Convective activity, which is pronounced in hurricanes, is also one of the known sources of IGW emission (e.g., Spiga et al. 2008; Chane Ming et al. 2010). The impact of IGW on hurricane dynamics has been discussed in a number of papers without reaching a clear consensus (Chow and Chan 2003; Schecter and Montgomery 2004, 2006; Hodyss and Nolan 2008; Schecter 2008). The impact of moist convection on wave emission during development of barotropic or baroclinic instabilities, and on radiative instabilities, was not much discussed so far. Schecter and Montgomery (2007) found that cloud coverage and associated reduced buoyancy diminish the growth rate of linear perturbations by increasing the damping by critical layers. However, convergence associated with precipitation and related vertical motions are expected to increase the wave activity during nonlinear stage of the instability. Besides, development of the standard barotropic instability is not much sensitive to critical-layer damping (e.g., Lambaerts et al. 2011). The IGW emission in the course of nonlinear saturation of the barotropic instability is not sufficiently understood, especially in the presence of diabatic effects.

A reasonable compromise between oversimplified nondivergent 2D models and “all inclusive” models is suggested by rotating shallow-water (RSW) modeling, which induces horizontal compressibility, and thus allows us to incorporate IGW and their interaction with the vortex component of the flow. The RSW model retains the most pertinent dynamical features of the primitive equations but keeps the dynamics sufficiently simple to be accessible for detailed investigations. A considerable advantage of the model is that it allows for computationally cheap and efficient high-resolution numerical simulations. The horizontal divergence in the model provides a proxy for vertical velocity, which is related to moist convection and diabatic heating. Correspondingly, reasonable parameterizations of diabatic effects are available in the model, which was used recently in the context of hurricane dynamics by Hendricks et al. (2012) and with an emphasis on diabatic phenomena in Hendricks et al. (2014), where diabatic heating entered the model as a mass sink with time-dependent amplitude distributed over a ring inside the RMW. Yet, the RSW modeling can be pushed even further by including in a simple and reliable way moist convection, precipitation, and evaporation (Bouchut et al. 2009; Lambaerts et al. 2011), which leads to so-called moist convective RSW (mcRSW) model. A consistent numerical scheme has been developed for the model, giving a possibility to treat the combined effects of nonlinearity and precipitation, like precipitation fronts inherent to the moist convective dynamics (Frierson et al. 2004; Bouchut et al. 2009).

In the present paper we are using the mcRSW model in order to

  • understand and quantify the vortex-ring instability including effects of nonzero divergence at typical for hurricanes values of parameters,

  • understand nonlinear saturation of the instability and quantify the differences between dry and moist convective saturation,

  • quantify emission of IGW and convectively coupled IGW by the developing instability and understand the role of IGW in the saturation process.

The paper is organized as follows. In section 2 we sketch the mcRSW model and its properties and describe the vortex profile corresponding to an idealized category 3 hurricane. In section 3 we formulate and solve the linear stability problem and present a discussion of the sensitivity of the unstable modes to the variations of the profile of PV. Section 4 contains results of direct numerical simulations of nonlinear saturation of the instability in the “dry” model, where moisture is decoupled. We compare these results with moist precipitating and moist precipitating and evaporating saturations in section 5 and discuss the impact of the effects of moist convection upon the destabilization of the initial vortex. Section 6 contains a summary of the main results and a discussion. Dependence of the results on boundary conditions is discussed in the appendix.

2. The model and the vortex profile

a. The mcRSW model

The equations of the one-layer mcRSW model in polar coordinates (r, θ) read
e1
Here is velocity; h is thickness; Q is bulk humidity; P is condensation sink, with intensity governed by an adjustable constant β; E is the evaporation source in the moisture budget; g and f are gravity and Coriolis parameter, respectively; and the notation gives unit vectors along the axes. The Lagrangian derivative is . A relaxation parameterization is chosen for condensation (which will not be distinguished from precipitation below):
e2
where is a saturation value, τ is relaxation time, and H is Heaviside function. It should be emphasized that the relaxation time toward the equilibrium profile of humidity is small in the atmosphere. It is therefore useful to consider the limit of the system (1) and (2) without evaporation . Following (Gill 1982; Bouchut et al. 2009), one can show that in this limit,
e3
and precipitation and wind convergence are directly linked.
As to evaporation, we work with a frequently used parameterization (e.g., Goswami and Goswami 1991):
e4
where is an adjustable coefficient.

The model may be obtained by vertical averaging of the primitive equations in pseudoheight coordinates and the use of the conservation of moist enthalpy (Lambaerts et al. 2011). In the presence of condensation (but without evaporation), the mass is not conserved, but the moist enthalpy is, as well as the moist potential vorticity , where ζ is the relative vorticity. In the presence of evaporation, care should be taken to respect positivity of the moist enthalpy. If , the equation for moisture decouples. We will be referring to such case as “dry” in what follows, although moisture is still present but just passively advected. The case with , but will be referred to as “moist precipitating,” and the case with as “moist precipitating and evaporating.”

The main goal of the present paper is to understand the dynamical role of moisture during the saturation of the hurricane instabilities. It is therefore desirable to have some theoretical insight on the influence of precipitation upon the evolution of vorticity. The dry PV is not conserved anymore in the presence of precipitation, even in the absence of dissipation. Its evolution in the system (1) and (2) is given by
e5
This equation shows that PV will increase in precipitating regions. The reasoning can be pushed further in the quasigeostrophic limit (we will see below that, despite the essentially ageostrophic features of the hurricane’s instability, the unstable mode inside the vortex is close to balance). It can be shown (Lambaerts et al. 2011) that in the limit of immediate relaxation the geostrophic relative vorticity obeys the equation
e6
and thus increases in precipitating regions.

b. An idealized hurricane

The following scaling is used in order to nondimensionalize the equations. We choose as a velocity scale, the RMW as a spatial scale, as a time scale, and the nonperturbed thickness as a vertical scale, with , λ measuring the intensity of the pressure anomaly. The dynamical parameters are the global Rossby and Froude numbers:
e7
An axisymmetric vortex with azimuthal velocity and hydrostatic pressure in cyclo-geostrophic equilibrium,
e8
is an exact solution of the “dry’ system (1). Because typical Rossby numbers of tropical cyclones are large, we will suppose . We choose the basic vortex to have a smoothed piecewise-constant vorticity profile with two levels corresponding to the core and the annulus, respectively. This profile was used in previous studies (e.g., Schubert et al. 1999; Hodyss and Nolan 2008; Hendricks et al. 2014). Transitions between the different zones of the vortex are smoothed using a third order Hermite polynomial, which in nondimensional form is given by
eq1
We add a zone of very weak negative relative vorticity outside the annulus, with amplitude and radial extension defined in such a way that the circulation is zero far from the vortex, and the associated energy is finite. The corresponding vortex profile is shown in Fig. 1. Its main parameters are
  • the local Rossby number : the maximum value of the relative vorticity (inside the annulus) divided by f;

  • the width of the annulus, ; and

  • the steepness of the transition slopes of nondimensional widths d1 and d2 between the zones of constant vorticity.

The interior (core) relative vorticity is weak (), as we are interested in the instability of the annular vortex. The global Rossby number (Ro) depends mainly on the local Rossby number and the width of the annulus but also on the inner value of the relative vorticity and on and . Negative relative vorticity in the outer zone is very weak and the width of this zone is several times the typical vortex size. Its impact on the results of the linear stability analysis was verified and shown to be unimportant. Likewise, the Burger number is very high for realistic hurricane profiles, on the order of 104. We checked that it has no influence on the results below, if varied up to two orders of magnitude with respect to this value. The profile we investigate in this paper is typical for category 3 hurricanes (cf. Willoughby et al. 2006), with , , and . The associated global Rossby number and Froude number are 32 and 0.3, respectively.
Fig. 1.
Fig. 1.

Idealized profile of a category 3 hurricane of (left) nondimensional velocity (solid; left axis) and thickness anomaly (dashed; right axis) and (right) relative vorticity . Different zones in the left panel are delimited by vertical dashed lines: core (zone 1), annulus (zone 2), and outer zone (zone 3).

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0115.1

3. Results of the linear stability analysis: Unstable modes and their dependence on the vortex profile

Linearizing the dry equations of motion around the vortex of Fig. 1 and looking for solutions in the form of azimuthal modes
e9
gives an eigenproblem for eigenvalues ω and eigenvectors :
e10
where is the operator of differentiation in r, which becomes a differentiation matrix after discretization. Numerical solution of (10) is obtained with the help of pseudospectral collocation method and the use of orthogonal rational functions (Boyd 1987) as interpolating polynomials. The corresponding grid on the semi-infinite interval is refined toward the center while getting coarser for r, where precision is less important. The method is identical to the one used by Lahaye and Zeitlin (2015).

The most unstable mode is presented in Fig. 2. Typically, the range of azimuthal wavenumbers of unstable modes is . The growth rate associated with this mode is , which corresponds to the characteristic dimensional e-folding time of about 1 h. This is in fair agreement with previous findings, both in incompressible 2D (e.g., Schubert et al. 1999) and full 3D models, as well as with observations (Menelaou et al. 2013b). The structure of the unstable mode clearly reflects its link to barotropic instability: the velocity of the perturbation follows the isobars, and the patterns of vorticity exhibit two radial maxima, corresponding to two counterpropagating Rossby waves, each one running perpendicular to a radial gradient of vorticity associated with the corresponding edge of the vortex annulus. The magnitude of the divergence associated with the unstable mode is two orders smaller than that of vorticity, in agreement with its balanced character. The second most unstable mode has and . The properties of the unstable mode, as well as of all the unstable modes we found (not presented) are similar to those of the mode.

Fig. 2.
Fig. 2.

(left) Pressure (colors) and velocity (arrows) distribution in the most unstable mode in the nondimensionalized xy domain. (center) Vorticity (colors) and location of the critical radii ( dashed lines). (right) IGW part of the unstable mode, as seen in the divergence field (enlarged domain).

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0115.1

Note that the typical parameters of the flow—high Rossby number and Froude number below unity—match the conditions of the Lighthill radiation mechanism. We observe a distinct wave pattern appearing away from the balanced part of the unstable mode, and thus the instability is indeed of a mixed type, coupling a radiated IGW (or Poincaré) mode with the two Rossby modes of classical barotropic instability. We should emphasize that the structure of the mode here differs from the classical radiative unstable modes as described, for example, in Le Dizès and Billant (2009) and Schecter and Montgomery (2006). Those were studied within a monotonic vortex profile with vorticity localized inside the core and decreasing outwards and correspond to the mutual amplification of a single Rossby mode bound to the vorticity gradient and of a Poincaré mode. Typical growth rates of such unstable modes are usually weak (which we verified by repeating the stability analysis for a monotonic vortex profile) as compared to the high values associated with the mixed-type unstable modes we observe. The structure of our unstable modes is rather in a good agreement with the 3D results of Hodyss and Nolan (2008). The magnitude of the outgoing IGW is very weak. The neat modal structure of the unstable wave between the critical layers, without any sign of wave interference, seems to exclude the overreflection mechanism (cf. Le Dizès and Billant 2009) of instability. The influence of the position of critical layers of an unstable mode upon the value of the growth rate is not clear: for instance, the growth rates of the and modes in Fig. 3 (left) behave differently with a change of parameters, while their critical levels remain approximately the same.

Fig. 3.
Fig. 3.

Growth rates of unstable modes, as functions of (left) the width of the transition zone and (right) the width of the annulus . In the left panel Wr = 0.4 and in the right panel . Dashed lines account for the “Rossby compensated” growth rate, taking into account the variation of the global Rossby number due to changes in velocity distribution with d or . It is defined as the dimensional growth rate normalized by the (fixed) global Rossby number value at and in the left and right panels, respectively: .

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0115.1

The steepness of the vortex profile, and the associated magnitude of the radial gradient of PV, condition the celerity of the Rossby waves propagating at each edge of the vortex annulus. This, in turn, determines the efficiency of phase locking and instability. The changes in the edge width of the vortex profile necessarily change the distribution of PV gradients and, hence, the eigenmodes and eigenvalues of the linear stability problem. It is clear from (10) that the impact of the Rossby number is weak if its value is large ( is at least an order of magnitude less than unity for hurricanes). Results of the linear stability analysis indeed confirm that the value of the growth rate of the most unstable mode is not sensitive to the global Rossby number (and, hence, the dimensional growth rate scales as Rossby number, in our scaling).

In contrast, the evolution of the growth rates of unstable modes exhibits a strong dependence on the edge steepness parameters d1 and d2, as shown in Fig. 3 (left). In particular, a wavenumber selection occurs at small values of the edge width (i.e., strong PV gradients). As seen in the figure, the modes with wavenumbers l = 2, 4, and 5 become stable in this limit. The mode is very weakly unstable and the only mode with considerable growth rate is . (We should emphasize that convergence of the collocation method degrades for very small growth rates, where the singularities due to critical layers approach the real axis, and we did not seek to resolve them in detail). On the other hand, for larger values of d1 and d2, and hence weaker PV gradients, several wavenumbers (from 2 to 5 here) are unstable, with close growth rates, and may thus compete if the vortex is subject to a broad-spectrum perturbation.

The growth rate of almost every unstable mode decreases with increasing width of the annulus of vorticity, as seen in Fig. 3 (right). This is partially countered by the increasing global Rossby number of the vortex that enters in the scaling of the growth rate, so that dimensional growth rate may reach a maximum for a finite value of the annulus width, before vanishing for large annuli (see the dashed lines in Fig. 3). The sensitivity of the growth rates to the width of the annulus is known within divergenceless models (Schubert et al. 1999; Hendricks et al. 2009); here, we confirm it in a shallow-water model. The dependence of the growth rate of the shear instability on overlapping between Rossby modes (and, thereby, on the width of the jet) is well known and may be analyzed analytically for simple vorticity profiles (e.g., Heifetz et al. 1999). It appears that the width of the annulus also provides a wavenumber selection of the instability. Indeed, the increase of the growth rate with decreasing width of the annulus is more important for higher-wavenumber modes. Hence, several unstable modes with higher wavenumbers accumulate for decreasing thickness of the annular vortex (the wavenumber of the most unstable mode goes up to for the vortex considered), whereas it is unstable with respect to only one low-wavenumber () mode for thinner annuli. This fact may account for the irregularity of the observed developing instability patterns: while sometimes well-organized mesovortices emerge (Kossin and Schubert 2004), the flow often appears to be rather disorganized (Kossin et al. 2002). It is worth noting that a manifestation of the competition between different wavenumber modes in a hurricane has been reported in Menelaou et al. (2013b), where a change of wavenumber of the dominant asymmetric perturbation was observed.

4. Nonlinear saturation of the instability: Dry case

We now present the results of direct numerical simulations of destabilization of the vortex of Fig. 1, with the annulus size and the width of the transition zones . We use a finite-volume numerical code with grid cells, and the domain size times the deformation radius (Rd), which is 7.5 times the typical vortex diameter. The size of each grid cell is thus , which sets the resolution. Runs with different resolutions and domain sizes have been performed to check numerical convergence (with typically ranging within ). Neumann boundary conditions are used in order to evacuate the emitted IGW.

We will run three different types of simulations: the dry one with , where the moisture is decoupled, the moist precipitating one with , and finally the moist precipitating and evaporating one with . The last two types will be treated in the next section.

Two different types of initial conditions are used. The first one consists of superposition of the basic vortex profile and a perturbation corresponding to the most unstable mode with weak amplitude: the ratio of the maximum modulus of the velocity of the perturbation to that of the background vortex is 0.1 in the simulations presented below (other simulations with a smaller value of 0.02 have been performed as well). Such initial conditions enforce the flow to evolve through the selected instability pattern with the given azimuthal wavenumber, even in the presence of competing instabilities, and make the analysis of the dynamical processes easier, especially in what concerns the IGW emission. The second method consists of the superposition of the basic vortex profile and a noisy perturbation, which will be described in the next subsection.

a. Initialization with the most unstable mode

We first give a detailed analysis of the nonlinear saturation of the instability of the vortex with superimposed most unstable mode with wavenumber . The evolution of the relative vorticity is shown in Fig. 4. The initial perturbation of the vorticity amplifies and gives rise to a tripolar structure with well-defined maxima of PV—the hurricane mesovortices. Further evolution of these high-vorticity regions consists in filamentation of vorticity at the outer edge and merging, which in turn leads to formation of a monopolar, quasi-axisymmetric vortex structure. This scenario of evolution of the instability, as regards the redistribution of vorticity and PV (which looks very similar), shows a good agreement with previous findings in nondivergent barotropic models (e.g., Schubert et al. 1999). Such process has been identified as playing an important role in cyclone reintensification cycles, for it is associated with a redistribution of the angular momentum. Evolution of azimuthally averaged profiles of vorticity, tangential velocity, and pressure are given in Fig. 5. One sees that vorticity of the end state is monotonically decreasing with r (same for PV), so that the reorganized vortex profile is marginally stable with respect to the barotropic instability. In the absence of other instabilities for such a profile (the growth rate of radiative instabilities is negligible here, as follows from the linear stability analysis), the final monopolar structure is stable. The depression inside the core associated with the new vortex profile is amplified in course of the evolution of the instability, as well as the magnitude of the azimuthal velocity. In this simulation no clear change in the RMW was observed.

Fig. 4.
Fig. 4.

Nonlinear dry saturation of the unstable mode: evolution of relative vorticity in the square domain in the xy plane with coordinates nondimensionalized by the RMW at times (a) , (b) , (c) , (d) , (e) , and (f) ().

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0115.1

Fig. 5.
Fig. 5.

Nonlinear saturation of the unstable mode: azimuthally averaged profiles of nondimensional (a) thickness , (b) azimuthal velocity , and (c) relative vorticity at times t = 0 (thick gray), 0.5 (black dashed), 1.5 (black dashed–dotted), and 4 (solid black).

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0115.1

The process of intensification of the vortex can be understood by analyzing the angular momentum budget. Following Hendricks et al. (2014), the equation for the azimuthal mean (denoted by overbar) of the angular momentum is, in the absence of dissipation,
e11
Here is the mass-weighted average, and the asterisk denotes the perturbation with respect to this latter. The first term in the rhs of this equation corresponds to the processes involving the mean field and the second term corresponds to eddy processes. They will be referred below as the “mean flux” and the “eddy flux,” respectively.

The outputs of the numerical simulation corresponding to different terms in (11) are integrated in time from to using a trapezoidal integration with a time step of . The results are presented in Fig. 6 (left). There is no explicit dissipation in the model, only a numerical one, which we do not control. We can nevertheless estimate it from the difference between the time-integrated lhs and rhs (dashed black curves in the figure). As follows from the figure, the dissipation is mostly concentrated at the edges of the initial annulus of vorticity. The mean flux of angular momentum is negligible, because the angular mean of the radial wind remains very weak. Thus, the whole of the angular momentum change comes from the eddy flux and dissipation and consists of an increase inside the RMW and a decrease around it, which is consistent with Fig. 5.

Fig. 6.
Fig. 6.

Angular momentum budget contributions integrated in time between and in the dry case during nonlinear evolution of the (left) l = 3 unstable mode and (right) random initial perturbation. Lines represent the eddy term (blue), mean term (red), sum of the mean and eddy terms (black dotted), and actual difference of the radial distribution of mean angular momentum between and (gray dotted). Thin dark-green line in the right panel corresponds to the eddy term in the left panel, for comparison.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0115.1

Of particular interest is the evolution of the IGW field coupled to the instability. IGW radiation has been investigated in the context of hurricane-like vortices with monotonic profile, with the main goal to quantify its impact on the core dynamics. The magnitude of the wave flux was shown to be limited by absorption at critical layers (Chow and Chan 2003; Schecter and Montgomery 2004, 2006). The question of the impact of nonlinear effects on this phenomenon has been addressed by Schecter (2008), and it has been shown that critical layer stirring may overcome the coupling between the Rossby and the Poincaré modes of the standard radiative instability, thus rendering the impact of emitted waves negligible compared to other processes, such as Rayleigh damping and vorticity mixing. However, since the outgoing waves are coupled in our case to a strong barotropic instability, nonlinear evolution is, probably, different and the magnitude of the wave activity produced by a destabilizing vortex is to be investigated.

The magnitude of the radiated waves is measured by averaging the absolute value of the divergence over an annulus at some distance from the vortex. In Fig. 7, we show the time evolution of this quantity for an annulus located at and having a nondimensional width of 0.4. A peak in wave activity at the initial stage is visible and corresponds to an initial adjustment due to discretization errors and to the fact that the unstable mode superimposed on the vortex is not a solution of the nonlinear equations, especially when its amplitude is not negligible. The magnitude of the radiated waves follows fairly well the linear growth at initial stages and reaches a saturation roughly at the same time as the l = 3 mode of the perturbation of pressure and velocity. At subsequent times, the emission of waves from the destabilizing vortex exhibits curious periodic bursts, with the overall magnitude decreasing in time. The occurrence of these bursts coincides with the maxima of magnitude of the l = 3 component of the perturbation, which is superimposed. The lag between the two curves can be attributed to the wave propagation time to the radius where the divergence is evaluated. Generation of vorticity filaments is also observed at the outer edge of the vorticity annulus, with the same periodicity. These filaments, in turn, are associated with outgoing Rossby waves reaching a critical layer. The mechanism observed here may thus be similar to the one described by Schecter and Montgomery (2006) for vortices with a monotonic vorticity profile, which consists in a periodic reduction of critical-layer absorption due to nonlinear effects, implying back-and-forth energy exchanges between the critical layer and the bulk Rossby wave. Although such a mechanism is not strictly transposable to our case with a nonmonotonic vortex profile, this hypothesis is supported by the correlation between the l = 3-mode amplitude and dissipation in the zone comprising the outer initial critical layer (cf. Fig. 7). Indeed, the amplitude of the l = 3 mode follows with some lag the energy dissipation. Further investigations are needed to establish whether the mechanism described in Schecter and Montgomery (2006) is responsible for the observed bursts in the wave emission.

Fig. 7.
Fig. 7.

Evolution of the dissipation (black; left axis) and modulus of divergence of the wave field (gray; right axis) and of the amplitude of the modal perturbation with azimuthal wavenumber (blue; right axis) during the nonlinear saturation of the instability (dry case). Dissipation is calculated in an annulus encompassing the initial outer critical radius (see Fig. 2) and normalized by the total energy of the perturbation at . Divergence is averaged over an annulus around the vortex. Amplitude of the l = 3 perturbation is evaluated from the thickness. Both are normalized by their initial values. Perturbation amplitude shows good agreement with the prediction for the linear growth (dashed) at initial stages.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0115.1

b. Multimode random initialization

As mentioned in section 3, the annular vortex may be unstable with respect to several modes with different azimuthal wavenumbers and close growth rates. Thus, a competition between different unstable modes is possible. It is of interest and probably more realistic (as perturbations of real-life hurricanes, mainly through convection, are generally disorganized and do not possess a well-defined wavenumber) to investigate nonlinear saturation of the barotropic instability of a randomly perturbed annular vortex. The initial conditions used in the simulations described in this subsection consist of the category 3 hurricane with a superimposed perturbation that is radially confined in the annulus of vorticity and comprises azimuthal Fourier modes with random phases, close amplitudes, and wavenumbers in the range [1, 6]. The radial distribution of the perturbation is given by a Gaussian centered at the annulus and having slightly smaller width than the latter. Evolution of the relative vorticity for such initial conditions is shown later (see Fig. 11), where it is presented in the moist precipitating model (see the following section). The qualitative features of the vorticity field, however, are very similar in the dry case. We observe that mesovortices still emerge in course of the growth of the instability. However, their shape is less pronounced, they are more stirred, and their width is smaller. A wavenumber then emerges and neighboring mesovortices rapidly merge pairwise. The general evolution of the vortex, from the annular distribution of vorticity to a radially monotonic profile remains the same but occurs more rapidly than in the case of initialization with unstable mode. We see that the eddy flux of angular momentum is enhanced near the center of the vortex during the first half of the simulation (cf. Fig. 6, right panel). During this period, this term almost vanishes at the same location () in the simulation initialized with the l = 3 unstable mode. At later times (between and ; not shown) positive eddy flux appears around the center of the vortex in the simulation initialized with the most unstable mode, while it vanishes in the run with random initialization, as the vortex has almost reached a steady state already. Negative eddy flux slightly below the RMW in the simulation with random initial perturbation affects the difference in the radial distribution of the mean angular momentum [gray dotted lines in Fig. 6 (right)]. These differences in the eddy fluxes between the simulations with the two different initial conditions are in agreement with the shape of the final vortices at . The one with random initial perturbation possesses a slightly larger value of the tangential velocity inside the vortex core (for ) and a smaller one at RMW [cf. Fig. 14 (left), to be discussed in the next section].

The wave intensity [see Fig. 12 (black dashed)] does not exhibit regular bursts anymore, varying at a negligible rate compared to the runs with -mode initialization. Still, the mean amplitude of the wave field is comparable for the two initializations.

5. Nonlinear saturation: Moist precipitating cases with and without evaporation

Simulations including the effects of moist convection and evaporation use the same initial conditions for pressure and velocity as in the dry case. As for the initial humidity field, we impose a uniform distribution with a value slightly below saturation, so that precipitation is triggered already during the early stages of the evolution of the instability. Although not quite realistic, such distribution allows for a clear interpretation of the evolution at early stages, when humidity behaves as a passive tracer. The value of the humidity away from the zones of convergence is not of great interest, as it does not condensate, never reaching the saturation value.

a. Moist precipitating case without evaporation

For moist precipitating simulation, the value of saturation is taken to be and initial humidity is , following Lambaerts et al. (2011). We are close to the immediate relaxation limit, with relaxation time of several time steps of the code, and the parameter β is set to 1 (the dependence of the results on the values of parameters and β will be discussed below).

1) Initialization with the most unstable mode

The evolution of the humidity, relative vorticity, and precipitation fields is shown in Fig. 8. A clear pattern emerges in the anomaly of humidity generated by the growth of the instability (we recall that the background vortex state has no divergence and, thus, cannot change local values of a tracer). While humidity is decreasing inside the mesovortices and outward, a tongue of high values forms in between these mesovortices along the annulus and extends outward, turning in an anticyclonic sense. These positive anomalies are associated with precipitation, with the main event occurring before and the overall precipitation ending about . The outer precipitating regions are aligned with vorticity filaments (see Fig. 8 at and ) and correspond to outgoing outer-edge convectively coupled Rossby waves (e.g., Chen and Yau 2001; Wang 2002; Wang and Wu 2004) stretched at the critical level. The evolution of the perturbation of the relative vorticity in the moist case exhibits slightly larger values in the inner ring (i.e., at the inner edge of the basic vortex annulus; see Fig. 2) than in the dry case. On the contrary, perturbations of negative sign in the outer ring are weaker. The positive anomaly of the modulus of velocity is slightly larger. This results in a higher value of the azimuthal velocity inside the vortex core, associated with lower pressure and higher relative vorticity. Figure 9 highlights the impact of moisture on the saturation of the instability, as seen from the evolution of the azimuthally averaged dynamical fields.

Fig. 8.
Fig. 8.

Nonlinear saturation of the moist precipitating instability. Colors represent the humidity anomaly (with respect to initial uniform value), with color shading scaled by a factor of 10 for positive values (see color bars). Contours indicate positive (black; with an interval of ) and negative (gray; with an interval of ) relative vorticity. Green contours show at an interval of 0.2. Simulation times are (a) , (b) , (c) , (d) , (e) , and (f) (). The square domain is in the nondimensionalized xy plane.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0115.1

Fig. 9.
Fig. 9.

Azimuthally averaged (left) thickness perturbation, (center) azimuthal velocity perturbation, and (right) relative vorticity during the saturation of the barotropic instability in dry (black), moist precipitating (blue), and moist precipitating and evaporating (red) simulations at times (dashed) and (solid). Initial vortex profile is plotted in gray in the right panel.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0115.1

The impact of moist effects is also clearly seen in the angular momentum budget, which can be analyzed along the same lines as in the dry simulation above. Figure 10 (left) is the moist precipitating analog of Fig. 6 (left), where azimuthally averaged precipitation is also displayed. As is clear from the figure, precipitation changes both mean and eddy fluxes, while dissipation remains roughly the same as in the dry case. Appearance of a mean flux of angular momentum is caused by the inward mean radial wind due to the convergence associated with the precipitation, in a region where . The mean flux vanishes around , corresponding to . Enhancement of the eddy flux is due to larger values of in (11), which is directly linked to the precipitation; see (5). The net result of the moist effects is an increase of the angular momentum in the vortex, as compared to the dry case (cf. Fig. 9).

Fig. 10.
Fig. 10.

As in Fig. 6, but in the (left) moist precipitating and (right) moist precipitating and evaporating models. The l = 3 unstable mode is used as the initial perturbation. Angular mean of the precipitation (magenta) is added. Thin dark-green line is the eddy term in the dry case (see Fig. 6, left).

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0115.1

2) Multimode random initialization

The evolution of the relative vorticity in this case is given in Fig. 11. As far as the axisymmetrization process associated with the eddy flux of angular momentum is concerned, the same differences as in the dry case are observed between the runs initialized with random perturbation and the l = 3 unstable mode. Namely, axisymmetrization of the vortex occurs more rapidly, mainly associated with an enhanced eddy flux of angular momentum at the vortex center.

Fig. 11.
Fig. 11.

Evolution of the relative vorticity during the moist destabilization (no evaporation) of the category 3 hurricane with random perturbation in the nondimensionalized xy domain at times (a) , (b) , (c) , (d) , (e) , and (f) . Green contours (interval of 0.2) indicate precipitation .

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0115.1

Comparison of the precipitation patterns in Figs. 11 and 8 shows that precipitation appears less organized than in the simulation initialized with the most unstable mode. Nevertheless, precipitation occurs in the region of negative perturbation of vorticity, as in the previous case, and exhibits elongated filaments in the outer region as well. The time-integrated mean precipitation (as in Fig. 10) is qualitatively the same as in the simulation initialized with the most unstable mode, although precipitation is slightly larger in the vortex annulus (maximum value of 0.03 compared to 0.02) and slightly smaller in the outer region. As a result, the mean flux of angular momentum is larger for the run with random initialization (maximum value of 0.04 compared to 0.02 in Fig. 10).

b. Moist precipitating case with evaporation

We finally add evaporation to the moist precipitating simulations initialized with the most unstable mode. Evaporation parameter was taken to be . Saturation value for humidity is set to , so that initial value of the moist enthalpy is large enough to remain positive for sufficiently long times. All diagnostics, which follow the lines of the preceding subsections, exhibit a net increase of the impact of the moist convection due to evaporation that feeds and maintains precipitation. Indeed, in this case the maximum velocity further increases in the course of the destabilization and, correspondingly, the pressure and vorticity in the core are, respectively, lower and larger. Mean profiles of thickness, tangential velocity, and vorticity are given later [see thin red lines in Fig. 14, compared with the same profile in the dry case (thin black lines)]. The angular momentum budget is also affected, as the precipitation is prolonged in the evaporating case. Comparison of the left and right panels of Fig. 10 shows that the mean precipitation is larger in the evaporating case, which leads to slightly larger values of the eddy flux of angular momentum and, most importantly, to a net increase of the mean flux.

The effects of enhanced moist convection manifest themselves also in the wave generation, as shown in Fig. 12. The amplitude of the waves away from the vortex is augmented by precipitation inside the vortex, being forced by a strong convergence zone there. The amplitude of the waves is substantially greater over a much longer time as compared to the simulation without evaporation. Divergence and convergence of velocity associated with the waves and the increasing value of humidity (which grows as a result of the evaporation) trigger precipitation that couples with waves. The phenomenon is depicted in Fig. 13, where one can see that the pattern of the precipitation inside the vortex is very similar to the moist precipitating case without evaporation (cf. Fig. 8 at ) and exhibits additional features outside of the vortex. This leads to a strong nonlinear enhancement of the ageostrophic motions related to IGW: (i) inside the precipitation area, owing to the very high value of convergence; (ii) at the edges of these zones, associated with the formation of precipitation fronts; and, surprisingly, (iii) at the tip of the precipitating region of the convectively coupled wave. In addition, the precipitation zones coupled with the outgoing waves are situated in a region where the radial gradient of mean absolute angular momentum is negative and is associated with an inward mean radial velocity. Hence, precipitation induces a decrease of the mean angular momentum in this region, as follows from (11). The mean flux associated with convectively coupled IGW appears clearly in the outer region of the vortex in the right panel of Fig. 10. The mean flux divided by r is nearly constant in this region (between and ), and we use it as a proxy for the intensity of convectively coupled waves in the next section. Note that the angular mean of precipitation vanishes around , which is not visible as the figure is zoomed at the vortex center.

Fig. 12.
Fig. 12.

Evolution of the magnitude of the divergence averaged over an annulus around the vortex in different runs. Colors indicate dry (black), moist precipitating (blue), and moist precipitating and evaporating (red) cases. Initialization is with the unstable mode (solid) or random (dashed) perturbation. Thin gray line represents evolution of the amplitude of the unstable mode as predicted by the linear stability analysis. Sharp peak at the very beginning corresponds to the waves emitted during initial adjustment.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0115.1

Fig. 13.
Fig. 13.

Snapshots of the divergence field at during the nonlinear saturation of the hurricane instability in the (left) dry and (right) moist precipitating and evaporating cases. Black contours are vorticity with an interval of , green contours (right panel only) are , with an interval of 0.2. Note that the color range is different in the two panels.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0115.1

Evolution of the mean profile under random initialization is given in Fig. 14 (thick red lines). Regarding the impact of precipitation on the vortex evolution, the same conclusions as in the precipitating case without evaporation given in section 5a(2) are valid. In addition, convectively coupled IGWs still appear, with a dominant azimuthal wavenumber at the beginning of the simulation, and then , thus roughly following the mesovortices that form and merge pairwise, as described in section 4b. We found that their intensity and impact on the mean angular momentum are very similar to the previous case of the most unstable mode initialization. It is worth mentioning, however, that the value of the mean flux (integrated from to ) at is about 20% larger, associated with a larger value of the mean precipitation.

Fig. 14.
Fig. 14.

Angular means of the radial profiles of (left) pressure, (center) azimuthal velocity, and (right) vorticity at (dashed) and (solid) for dry (black) and moist precipitating and evaporating (red) simulations, with initial profiles (gray). Initialization with random initial perturbation (thick lines) and unstable mode (thin lines).

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0115.1

c. Dependence of the results on the values of parameters controlling moisture and precipitation

We further investigated the impact of the precipitation and evaporation in supplementary simulations by varying the values of the parameters characterizing the moist processes. One of the parameters , , and was changed at a time, while the others were kept at their reference values used in the previous subsections.

In the moist precipitating case without evaporation β was changed from 1 to ¾ in one run and was changed to 0.85 and 0.67 in two other ones. Note that changes of the parameters are constrained by the positiveness of the moist enthalpy. Results of these runs are very similar to what was observed in the reference moist precipitating run with and (cf. section 5a). Table 1 contains an ensemble of relevant dynamical quantities measured in each of these simulations. Quantitatively, we observe that the impact of the precipitation is reduced with the decrease of the values of β and , accordingly to (1) and (3). In the moist precipitating and evaporating case, three different values of have been used: αe = 0.05, 0.1 (reference value), and 0.2. Qualitative features of the evolution of the destabilizing cyclone are similar, yet precipitation is enhanced by a stronger value of evaporation—as expected—thus implying quantitative changes. Intercomparison of the simulations (cf. Table 1) shows that evaporation increases the value of wind at RMW at final stages and shifts its location toward the center. This is associated with higher value of the angular mean of the time-integrated precipitation and, hence, of eddy and mean fluxes of absolute angular momentum inside the vortex. Convectively coupled waves out of the vortex are also modified by a change of the evaporation intensity and enhanced, as follows from the value of the mean flux of angular momentum at (cf. Table 1, last column), which, as explained above, is a good proxy for the impact of convectively coupled waves. Evaporation is directly correlated with the loss of angular momentum in this region.

Table 1.

Impact of moist parameters: comparison of dry, moist precipitating (MP) and moist precipitating and evaporating (ME) simulations with l = 3 unstable-mode initialization. Maximum of angular mean of tangential wind and RMW at the final stage (), maximum of angular mean of the precipitation , maximum and minimum of the eddy flux of absolute angular momentum (EFT), maximum of mean flux of absolute angular momentum (MFT) and its value at r = 2, integrated in time between and (see Fig. 10 for comparison). Values in the last four columns are multiplied by 100.

Table 1.

6. Summary of the main results and discussion

Let us summarize the main results obtained in the present paper. Our linear stability analysis of the category 3 hurricane shows that the main instability is an ageostrophic barotropic instability. The corresponding unstable modes are mixed Rossby–inertia–gravity waves. The azimuthal wavenumber l of the most unstable mode is sensitive to the fine details of the initial vorticity profile, in particular to the overall width of the ring of vorticity, and to the widths of the transition zones. Sensitivity to the width of the vorticity ring was reported in studies with incompressible two-dimensional models (e.g., Schubert et al. 1999; Hendricks et al. 2009); here we confirm it in the shallow-water model. The sensitivity to the widths of the edges was not investigated previously. Both contribute to the wavenumber selection of the instability. Certain vortex profiles possess a range of unstable modes with different wavenumbers and close growth rates, while other profiles have a single distinctive maximum growth-rate mode.

In what concerns the dry saturation of the instability of the cyclone perturbed by the most unstable eigenmode, our numerical simulations confirm the axisymmetrization process resulting from the merging of the mesovortices produced by the primary instability. This mechanism induces wind enhancement inside the RMW. A new feature of the developing instability consists of bursts in the IGW emission correlated with the dynamics inside the vortex.

The moist precipitating counterparts of the dry simulations show amplification of the amplitude of the emitted IGW (as compared with the dry case) and an increase in the final magnitude of the wind. The simulations of the moist precipitating and evaporating saturation of the most unstable mode, with uniform initial moisture distribution and evaporation proportional to the intensity of the wind, show enhancement of the moisture effects consisting in a net increase of the intensification of the wind (up to RMW) and appearance of convectively coupled IGW with significant amplitudes far from the vortex.

If random perturbation is superimposed onto the vortex, the saturation of the instability is faster compared to the perturbation with a single mode. This, together with the wavenumber selection by the vortex profile, could give clues to the observed differences in the reorganization of hurricanes. Indeed, hurricanes with vortex profiles unstable with respect to several modes that grow all together, compete, and interact, may undergo a faster reorganization, and hence a faster axisymmetrization and intensity change, than hurricanes with a single unstable mode.

To recapitulate, the main differences between the dry and the moist precipitating saturation consist of

  • a net amplification of the wind intensification in the latter, especially in the presence of evaporation, including at the RMW,

  • an amplification of the IGW emission, and

  • emission of large-amplitude IGW coupled to precipitation fronts.

An important conclusion that follows from the present study is that the moist convective rotating shallow-water model is a valuable tool for understanding the evolution of tropical cyclones. The model reproduces the results of nondivergent models on instability of typical hurricane-like vortices and allows us to pinpoint the changes due to compressibility effects and inertia–gravity waves. A simple and consistent way to incorporate moist convection and precipitation allows us to obtain clear-cut conclusions on their dynamical influence and to show that moist effects do play an important role in intensification and reorganization of the hurricanes. While in recent studies with simplified models (Rozoff et al. 2009; Hendricks et al. 2014), the evolution of humidity was not considered and moist convection was mimicked by special forcing, our model includes moisture and precipitation in a self-consistent way and is unforced in the sense that it conserves the moist enthalpy.

We should emphasize that, in its baroclinic multilayer version (Lambaerts et al. 2012), the model allows us to introduce a vertical wind shear that is also an important ingredient in hurricane dynamics. This work is in progress.

Acknowledgments

The authors thank the referees for providing useful comments, which helped to improve the quality of the paper.

APPENDIX

Dependence of the Results on Boundary Conditions

As strict mass conservation is no longer ensured once Neumann boundary conditions are implemented in the numerical scheme, and because the velocity and pressure fields of the hurricane are relatively slowly decreasing, spurious mass and energy nonconservation could happen in the simulations. We therefore reran most of the simulations using periodic boundary conditions. Although the general evolution of the flow was mainly unchanged, small discrepancies were found in some cases and are worth discussion.

First, the amplitude of the emitted waves is weak, and their persistence in the periodic domain does not impact the vortical, nearly balanced part of the flow. Therefore, the dry simulations are very similar regardless of boundary conditions. For instance, the errors in the angular averages of the azimuthal velocity and relative vorticity at the end of the run (), as computed by the mean of the absolute value of their difference, are as low as approximately and (in units and f), respectively.

However, in moist simulations with Neumann boundary conditions, there is a weak overall positive anomaly of humidity that slightly enhances precipitation events. As a result, the relative vorticity in the core of the end-state monopolar vortex is weaker in the simulations with periodic conditions, although the shape of the final distribution is preserved. The value of the velocity at the RMW, as well as that of the radial gradient, is slightly weaker. This difference is stronger in the moist precipitating and evaporating simulations where, in addition, waves have larger amplitudes and, when reentering the domain, perturb the convectively coupled outgoing IGW. Relative differences at the center of the vortex are of about 10% in the moist precipitating and evaporating cases at . There is also a weak spurious mass nonconservation in simulations with Neumann boundary conditions, which is more or less uniform over the entire domain. Mean relative error in the angular averaged mass for the moist precipitating and evaporating simulation is 2.5%. For moist (without evaporation) simulations, corresponding errors are an order of magnitude smaller.

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