^{*}

^{a}

An approximate method is developed for finding and analysing the main instability modes of a tropical cyclone whose basic state is obtained from a cloud resolving numerical simulation. The method is based on a linearised model of the perturbation dynamics that distinctly incorporates the overturning secondary circulation of the vortex, spatially inhomogeneous eddy diffusivities, and diabatic forcing associated with disturbances of moist convection. Although a general formula is provided for the latter, only parameterisations of diabatic forcing proportional to the local vertical velocity perturbation and modulated by local cloudiness of the basic state are implemented herein. The instability analysis is primarily illustrated for a mature tropical cyclone representative of a category 4 hurricane. For eddy diffusivities consistent with the fairly conventional configuration of the simulation that generates the basic state, perturbation growth is dominated by a low azimuthal wavenumber instability having greatest asymmetric kinetic energy density in the lower tropospheric region of the inner core of the vortex. The characteristics of the instability mode are inadequately explained by nondivergent 2D dynamics. Moreover, the growth rate and modal structure are sensitive to reasonable variations of the diabatic forcing. A second instability analysis is conducted for a mature tropical cyclone generated under conditions of much weaker horizontal diffusion. In this case, the linear model predicts a relatively fast high-wavenumber instability that is insensitive to the parameterisation of diabatic forcing. The prediction is in very good quantitative agreement with a previously published analysis of how the instability develops in a cloud resolving model on the way to creating mesovortices slightly inward of the central part of the eyewall.

Satellite and radar images of mature tropical cyclones commonly reveal deformed eyewalls and mesovortices along the periphery of the eye. There has been longstanding interest in understanding how such features develop and whether the process appreciably affects the temporal trend of vortex intensity. One plausible explanation for the emergence of prominent waves and mesovortices involves an instability of the local circular shear flow. Although such an explanation is prevalent in the literature, there has been limited progress in advancing an instability theory for realistically modelled tropical cyclones.

Basic insights have been gained through the study of idealised two-dimensional (2D) models. Such models show that a vorticity annulus similar to that on the inward edge of an eyewall is usually unstable. The onset of perturbation growth may involve the mutual amplification of counter-propagating vortex Rossby waves or a destabilising wave-critical layer interaction. Depending on specifics, the instability may generate robust arrays of mesovortices or engender transient turbulence that thoroughly redistributes inner core vorticity into a centralised monopole (Schubert et al.,

Additional insights have been gained from the study of three-dimensional (3D) stratified vortices whose basic states do not possess secondary circulations. The dominant modes of instability often resemble their 2D counterparts but differ in quantitative details [Nolan and Montgomery,

The final step toward a realistic perturbation theory is to generalise a 3D model to incorporate moisture and secondary circulation into the basic state of the vortex. The inclusion of cloud coverage alone has the effect of substantially reducing static stability (Durran and Klemp,

Needless to say, cloud coverage in a mature tropical cyclone is largely linked to the secondary circulation. Therefore, including one without the other in a model could yield misleading results. Naylor and Schecter (

NM02 contains the underpinnings of an appropriate linear model for investigating perturbation dynamics in a moist convective tropical cyclone. The NM02 model accommodates the incorporation of an adequately resolved boundary layer and the complete overturning secondary circulation of the basic state, but does not close the book on the thermodynamics. Proper parameterisation of the perturbation to diabatic forcing as a function of the prognostic fluid variables is necessary for a realistic instability analysis and remains an open issue. A separate challenge pertinent to analysing instabilities is to move beyond the conventional but questionable simplification of using constant eddy diffusivities.

The present study is based on a compressible nonhydrostatic model of a tropical cyclone. The equations of motion are expressed in a cylindrical coordinate system whose central axis corresponds to that of the vortex. The radial, azimuthal and vertical coordinates are respectively denoted by

The nonlinear equations of motion governing the tropical cyclone are given by
_{pd}_{d}_{vd}

A generic field

The azimuthal symmetry of the basic state facilitates an azimuthal Fourier decomposition of the reduced system. Letting _{B}_{B}

The feedback of an asymmetric linear perturbation on the mean vortex is essentially a second-order ‘eddy forcing’ of the symmetric (_{0}) is schematically given by
_{0} [see _{m}_{0} is initially subdominant to the asymmetric perturbation,

The definition of our chosen thermodynamic variable [_{v}_{v}_{t}

To devise a parameterisation for _{m}_{t}_{m}_{t}_{v}_{t}

In the preceding reversible moist-adiabatic vortex model, the perturbation to diabatic forcing can be written as follows:
_{b}_{b}

A more general linearised parameterisation of the perturbation to _{c}

One potential deficiency of the foregoing parameterisations [

The influence of small-scale turbulence on the velocity perturbation is parameterized with a linear eddy viscosity scheme that incorporates a modification of the oceanic surface drag. The velocity tendencies associated with turbulence can be expressed as follows:
^{−1}, ^{−1} and

Several remarks are in order regarding the preceding representation of turbulent transport in the velocity equations. To begin with,

Moving on to the thermodynamic equation, the effect of small-scale turbulence on

Several remaining formulas are required to complete the turbulence parameterisation in the linear system. To begin with, the momentum eddy diffusivities are given by

Perturbations to radiative transfer and dissipative heating are neglected in forthcoming sections of this paper. The potential impact of radiation on the development of instabilities has been examined to some extent by adding Newtonian relaxation of the form
_{r}

The perturbation to

The linear model employs a standard set of boundary conditions for a fluid in a rigid cylindrical enclosure. At _{nm}_{B}_{n}_{B}_{n}_{n}_{B}_{B}_{b}_{B}_{b}_{B}

Let _{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}

The _{t}^{i}_{n}_{R}_{R}

So as not to be lost in abstraction, it is worth remarking that the physical perturbation corresponding to a complex eigenmode is usually given by

Suppose that the system is initially perturbed with a single asymmetric (

Each fluid variable in the linearised model is discretized on a rectangular grid in the _{n}_{n}_{n}_{n}_{n}

The following simple formulas are normally used for finite differencing and linear interpolation of a generic field _{n}_{n}

The computation of the complete eigenbasis of a finely-structured tropical cyclone is usually too expensive to achieve with confidence of correct results. Although _{n}_{t}

The present study employs a less ambitious approach that begins by extracting the dominant eigenmode from a solution of the initial value problem. The discretized linear model [_{n}_{n}_{n}_{n}

Suppose that the eigenfrequencies _{n}_{n}

The primary basic state considered herein corresponds to a mature tropical cyclone simulated with Cloud Model 1 (CM1-r19.4) in an energy-conserving axisymmetric mode of operation [Bryan and Fritsch, ^{–3}. Radiative transfer is not explicitly calculated, but potential temperature (^{–1} in magnitude. The influence of subgrid-turbulence above the surface is represented by an anisotropic Smagorinsky-type scheme resembling that described in BR09. The nominal mixing lengths are given by CM1-formulas tailored for tropical cyclones in an axisymmetric framework or on grids that are deemed insufficiently dense for a standard large-eddy-simulation scheme. The horizontal mixing length increases from 100 m to 1 km as the underlying surface pressure decreases from 1015 to 900 hPa. The vertical mixing length increases asymptotically to 100 m with increasing

The computational domain extends radially to _{B}_{B}_{B}

The mature tropical cyclone is generated with a standard spinup procedure on the oceanic _{s}^{°}^{−1}. After approximately 7 days of intensification, the maximum azimuthal velocity of the tropical cyclone remains steady over an extended period of time. The basic state variables

_{bm}^{−1} is located 36 km from the centre of the vortex and nearly 1 km above the surface. The maximum azimuthal wind speed of 61.2 m s^{−1} at the lowest grid level (_{bm}

Selected fields associated with the basic state. (a) The azimuthal velocity _{b} (colour), density potential temperature _{b} (solid white contours). (b) A measure of gradient imbalance [_{bm} is located where the radius of the isoline is minimised.

_{bm}

_{zb}_{zb}

^{−1}) of the surface inflow is relatively strong but not much greater than typical observations pertaining to major hurricanes (Zhang et al., _{bm}

The secondary circulation of the basic state. (a) Magnitude (colour) and streamlines of the velocity field (_{b}, _{b}) in the ^{−1}. The dashed black curve is the principal AM isoline. (b) Magnified view of the secondary circulation in (a) in the lower troposphere.

_{bm}

The moist-thermodynamic structure of the basic state, illustrated primarily by the distributions of relative humidity (shading) and ^{−1} K^{–1}). The dotted blue curve is a selected contour of

The cloud structure of the basic state is important to the linear model insofar as it determines the proportionality between _{n}

Distributions of

As explained earlier, the eddy diffusivities used by the linear model are linked to those regulating the basic state. ^{2} s^{−1} and ^{2} s^{−1} are imposed on the distributions.

Horizontal (colour) and vertical (solid contours) momentum eddy diffusivities in the middle-to-lower tropospheric core of the simulated tropical cyclone. The dashed curve is the principal AM isoline.

The present section of this article examines the instability of the tropical cyclone described in

A few preliminary remarks are warranted. Henceforth, the meaning of _{bm}_{s}

Finally, although the physics parameterisations are varied, the domain size and peripheral sponge-layer of the linear model used to find the instability modes do not change from one calculation to the next. As in the CM1 simulation used to generate the basic state, the invariant domain of the linear model extends radially to

The dominant instability of the tropical cyclone under consideration is sensitive to the degree of diabatic forcing allowed in the linear model. The sensitivity is illustrated below by adjusting

The present method for computing the primary instability modes of the vortex follows the general procedure outlined in

Extensive computations reveal that the AMUM corresponds to _{R}_{I}_{R}

Complex eigenfrequency ^{−1} and ^{−1}) obtained for _{I}

(a)–(c) Basic inner-core structure of the

(a)–(c) Maximum values over

_{b}

(a) Intrinsic frequency

Moving outward to where

(a, b) Slices of the vertical velocity perturbation of the ^{−1}. Note that the units of the colorbar labels differ between (a) and (b). (c, d) As in (a, b) but for

_{p}

(a) The symmetric velocity perturbation that attends the growth of the _{0} whereas vectors depict (_{0}, _{0}). (b) The perturbation of kinetic energy density associated with the ^{−3}. (d)–(f) As in (a)–(c) but for _{bm}_{b}_{b}

_{0} regardless of whether

The contribution to _{zb}

Differences between the MUMs are also evident in various terms that formally contribute to the growth rate of _{b}

Domain integrals of the individual contributions to

The MUM associated with arbitrary

Variation of the complex eigenfrequency ^{−1} and ^{−1}) obtained for the MUM when

Consider first the group of linear systems that allow a medium degree of diabatic forcing (_{n}_{R}_{n}_{n}

(a) Spatial distribution of ^{−3}. (b) Domain integrals of the individual contributions to _{b}

Consider next the set of linear systems that allow relatively strong diabatic forcing (_{n}

It is worth remarking that decreasing the eddy diffusivity often magnifies the importance of higher wavenumber MUMs. For example, reducing _{n}^{−1}.

It is common practice to explain the instability of the primary circulation of a tropical cyclone in the context of a two-dimensional nondivergent barotropic model (see Appendix D). The foregoing analysis casts doubt on the general adequacy of such an approach. That is to say, the preceding results suggest that the three-dimensionality of the tropical cyclone under present consideration has a major impact on the prevailing mode of instability. The evidence includes MUMs with substantial horizontal vorticity and divergence. The evidence also includes major contributions from SC and/or the vertical shear component of PC to the volume integrated time-derivative of asymmetric kinetic energy (

Further insight is gained by directly comparing 2D and 3D instability theory. The 2D analysis requires reduction of the basic state to a circular shear-flow characterised by a 1 D vertical vorticity profile _{b}_{b}^{2}s^{–1}. The upper limit is roughly 1.4 times the

The nonmonotonic radial variation of _{b}^{2} s^{−1} diminishes the growth rate of each MUM with greater effect at larger

(a) Azimuthal wavenumber (^{−1} or less (at high

^{2}s^{–1}. Despite the reordering of growth rates, neither the low-frequency mode (

(a) Vertical vorticity (red and blue), streamlines (black) and corotation circles (dashed green) of the low-frequency mode of the 2D system with ^{2} s^{−1}. The streamline thickness is directly proportional to the local magnitude of the horizontal velocity perturbation

To some extent, the low-frequency mode of the 2D system that prevails under conditions of high viscosity resembles the lower tropospheric section of a typical 3D MUM that dominates under moderate diabatic forcing when turbulent transport is parameterized with

It is notable that (for ^{2}s^{–1}). Greater facilitation of diabatic forcing (

Reducing the general uncertainty of linear instability theory will require refinement of the physics parameterisations. Such refinement will require a comprehensive comparison of theory to state-of-the-art cloud resolving numerical simulations. While a comprehensive refinement effort is beyond the scope of this paper, a comparison of our linear model to the results of one of our earlier simulations is easy and worth reporting.

The simulation considered for illustrative purposes corresponds to the three-dimensional moist experiment of NS14 distinguished from others by the following ratio of surface-exchange coefficients: _{e}

For better compatibility with the model configuration of NS14, the physics parameterisations used presently in computing the linear instability modes differ somewhat from those used previously. Diabatic forcing is given by ^{2} s^{−1} where the pertinent instability modes are concentrated. The lower limits of the eddy diffusivities are given by ^{2} s^{−1} and ^{2} s^{−1}.

Comparison between the prevailing instability modes of a tropical cyclone simulated with CM1 and two theoretical predictions. (a) Growth rate versus azimuthal wavenumber

Scatter plot of the complex eigenfrequencies of the

Scatter plot of the complex eigenfrequencies of the MUMs of the tropical cyclone of Section 4 as determined by the 3D linear model with

The diamonds in _{R}_{I}_{R}_{I}

^{2}s^{–1}. As usual, the error bars extend from the minimum to maximum values of the data set for each

In this particular case study, the insensitivity of 3D linear instability theory to the degree of diabatic forcing allowed in the model is consistent with the concentration of modal wave activity inward of the eyewall cloud (NS14). Insensitivity to the reduction of ^{2}s^{–1} and

This paper has proposed a method to account for diabatic forcing and inhomogeneous eddy diffusivities in predicting and analysing the dominant instability modes of numerically simulated tropical cyclones. Excluding explicit moisture equations from the linearised model necessitated a partly intuitive parameterisation of the diabatic forcing

The instability analysis was illustrated for a mature tropical cyclone representative of a category 4 hurricane. The basic state was generated by an axisymmetric numerical simulation with two-moment cloud microphysics and typical settings for the parameterisation of subgrid turbulence. Initial consideration was given to linear systems having vertical and horizontal eddy diffusivities comparable to those regulating the basic state. With the diabatic forcing parameter

Reducing the eddy diffusivities with

Sensitivity of the foregoing analysis to the parameterisations of diabatic forcing and turbulent transport attests to the importance of details in predicting and understanding tropical cyclone instabilities. Improving the predictive skill of the linear model will require reducing the present degree of uncertainty in the aforementioned parameterisations. Refinements of

An initial test of our linear model produced encouraging results. The instability analysis showed very good quantitative agreement with the perturbation growth that leads to mesovortex formation slightly inward of the eyewall cloud in a previously conducted CM1 simulation with relatively low diffusivity (NS14). Such agreement helped validate the dynamical core of the linear model. On the other hand, questions regarding the parameterisation schemes were left unresolved. The instability was theoretically too fast for reasonable variants of turbulent transport to have an appreciable effect on its early development. Moreover, the perturbation seemed largely detached from moist processes (NS14). Accordingly, the instability predicted by the linear model showed little sensitivity to switching

The author thanks Dr. Konstantinos Menelaou and an anonymous reviewer for their comments on this study prior to publication. The author also expresses his gratitude to Dr. George Bryan for providing the cloud resolving model (CM1) used to generate the tropical cyclones whose instabilities were analysed herein. The NS14 study re-examined in

No potential conflict of interest was reported by the author.

The parameterisation of diabatic forcing given by _{pv}_{l}_{s}_{v}_{i}_{l}

As in reality, the simulated tropical cyclone providing the basic state of _{b}_{t}

The following analysis is restricted to the interior region of the eyewall updraft, defined to be where ^{−1}. In this saturated region of the tropical cyclone, _{b}_{s}_{b}_{0}. For simplicity, let us suppose that

Two estimates of Γ have been calculated.

It is common practice to use an anelastic approximation of the equations of motion when acoustic waves are deemed unimportant to the instabilities of interest (NM02; Hodyss and Nolan,

Recall from _{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}

As suggested earlier, acoustic filtering is expected to have minimal consequence on the primary inner core instability of a tropical cyclone. Consider the tropical cyclone of

In

The system under present consideration exhibits a somewhat complicated sensitivity to grid spacing. The _{I}_{R}_{R}

The equations of motion governing a nondivergent 2D vortex with kinematic viscosity

The asymmetric (_{b}_{B}_{B}

The Fourier transform of the horizontal velocity perturbation associated with a 3D instability mode can be decomposed into the following sum of irrotational (superscript-

The boundary conditions at the origin are regularity of the velocity potential _{n}_{B}_{n}