2.1. Governing Equations
 We consider adiabatic, quasi-hydrostatic motions of a compressible atmosphere on an f-plane. Using the potential temperature θ as the vertical coordinate, the equations for the eastward velocity component u and the northward velocity component v are
where ζ = ∂v/∂x – ∂u/∂y is the isentropic relative vorticity, M = θΠ + Φ is the Montgomery potential, Π(p) = cp(p/p0)k is the Exner function, Φ is the geopotential, and the Coriolis parameter is chosen to be f = 5 × 10−5 s−1.
 The numerical model used here is based on the vorticity/divergence form of the primitive equations. The equations for the isentropic relative vorticity and the isentropic divergence δ = ∂u/∂x + ∂v/∂y are easily derived from (2.1) and (2.2), and can be written in the form
where u, v, ζ, δ are expressed in terms of the velocity potential χ and the streamfunction ψ as
 The hydrostatic and continuity equations are
where σ = −g−1(∂p/∂θ) is the pseudodensity. Equations (2.3)–(2.8), along with the definitions of σ and Π, constitute a closed system in the three prognostic variables ζ, δ, σ and the seven diagnostic variables χ, ψ, u, v, M, Π, p. Note that all variables are functions of x, y, θ, t, and all horizontal derivatives and time derivatives are taken on isentropic surfaces. Also note that a considerable simplification of the above equations has occurred because of the assumption = 0.
 The potential vorticity principle, obtained by eliminating the isentropic divergence between (2.3) and (2.8), is
where P = (f + ζ)/σ is the potential vorticity, and where D/Dt = (∂/∂t) + u(∂/∂x)+v(∂/∂y) is the material derivative. This material conservation relation indicates that the system (2.3)–(2.8) describes the PV mixing process in its purist form.
 The model is vertically discretized using the grid shown in Figure 2, with vorticity and divergence defined on 19 integer levels and pressure defined on the associated half-integer levels [Hsu and Arakawa, 1990]. The top boundary is assumed to be both an isentropic and isobaric surface, with θT = 360 K and pT = 106 hPa. The lower boundary was assumed to be the isentropic surface θB = 298 K, along which the pressure is variable in (x, y, t). The potential temperatures on the 19 integer levels are 298.5, 299.5, 300.5, 301.5, 303.0, 305.0, 307.0, 309.0, 311.0, 313.0, 315.0, 317.0, 319.5, 323.0, 327.5, 333.0, 339.5, 347.0, 355.5 K, which gives the finest vertical resolution in the lower troposphere where the most intense PV mixing occurs.
Figure 2. The vertical staggering of variables in the isentropic primitive equation model. The prognostic variables ζ, δ, σ are carried on the integer levels l = 1, 2, …, L, where we have chosen L = 19. Pressure is carried on the half-integer levels. The model top is assumed to be both an isentropic and an isobaric surface, with θT = 360 K and pT = 106 hPa. Pressure varies on the lower isentropic surface θB = 298 K.
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 The horizontal discretization is based on a double Fourier pseudospectral method having 384 × 384 equally spaced collocation points on a doubly periodic horizontal domain of size 600 km × 600 km, which results in a 1.56 km spacing between points. For quadratic nonlinearities, this grid gives 128 × 128 alias-free modes. The use of this double Fourier pseudospectral method makes it a simple matter to invert the relations ∇2ψ = ζ and ∇2χ = δ to obtain new values of the stream-function ψ and the velocity potential χ after new values of ζ and δ have been predicted from (2.3) and (2.4). During PV mixing there is a cascade of potential enstrophy to the highest resolved wavenumbers. To avoid spectral blocking, we have included the ordinary diffusion terms v∇2ζ, v∇2δ, and v∇2σ on the respective right hand sides of (2.3), (2.4), and (2.8), with the value of the diffusion coefficient chosen to be v = 100 m2 s−1. This value of the diffusion coefficient results in a 1/e damping time of 93 minutes for all modes having total wavenumber 128.
 A third order Adams-Bashforth explicit scheme [Durran, 1991] is used for the time discretization needed in (2.3), (2.4), and (2.8). Because this primitive equation model allows rapidly propagating Lamb waves and uses an explicit time differencing scheme, the CFL condition requires a rather small time step. The model simulations shown here used a time step of 1 s.
 Since inertia-gravity waves can be generated during PV mixing, a sponge layer was used near the lateral boundaries to minimize the false reappearance of these waves in the interior region of the model. It should be noted that the strict material conservation relation (2.9) is compromised by the use of a sponge layer near the domain edge and by the addition of ordinary diffusion terms to the right hand sides of (2.3) and (2.8). However, experience with the model indicates that these nonconservative effects are so weak that, for practical purposes, we can consider (2.9) to be satisfied. More information on the discretizations used in the model can be found in Hendricks .
2.3. Initial Conditions
 Because the model contains the three prognostic equations (2.3), (2.4), and (2.8), we must specify the three initial fields ζ(x, y, θ, 0), δ(x, y, θ, 0), and σ(x, y, θ, 0). To reduce the amount of inertia-gravity wave activity, we have chosen δ(x, y, θ, 0) = 0, so that χ(x, y, θ, 0) = 0 and the entire initial wind field is determined by ζ(x, y, θ, 0), or equivalently ψ(x, y, θ, 0). Two different initial conditions on ζ were chosen—one resulting in a thick PV tower and the other resulting in a thin PV tower. These initial conditions are three-dimensional generalizations of the two-dimensional vorticity rings used in Schubert et al. . The inner region is the low vorticity eye, the middle region is the high vorticity eyewall, and the outer region is the low vorticity environment. Hurricanes have often been observed to have such hollow profiles [Kossin and Eastin, 2001]. Using polar coordinates with the origin located at the center of the model domain, the initial condition for the isentropic relative vorticity has the separable form ζ(r, ϕ, θ, 0) = [ζsym(r) + ζpert(r, ϕ)]Θ(θ), where the radial structure of the axisymmetric part is given by
and the vertical structure is given by
 Here, S(s) = 1 − 3s2 + 2s3 is a cubic shape function that provides smooth transition zones, and θref = 302 K. For the thick hollow tower, ζ2 = 0.0021 s−1 and r1, r2, r3, r4 = 20, 24, 38, 42 km, respectively. For the thin hollow tower, ζ2 = 0.0045 s−1 and r1, r2, r3, r4 = 30, 34, 38, 42 km, respectively. The average vorticity over 0 ≤ r ≤ r4 is the same for both hollow towers, so that both vortices have a maximum initial velocity of approximately 30 m s−1. Because of the doubly periodic boundary conditions, the net circulation around the domain boundary vanishes on each isentropic surface, i.e., the horizontal average of the isentropic relative vorticity vanishes on each isentropic surface. Thus, after specification of r1, r2, r3, r4 and ζ2, the constant ζ3 is determined in such a way as to make the domain average of ζsym(r) vanish. This leads to a value of ζ3 that is negative, but its magnitude is small because the area outside r = r4 is approximately 64 times larger than the area inside r = r4.
 To initialize the σ field we first solve the isentropic coordinate version of the nonlinear balance equation as a two-dimensional Poisson equation for M, i.e.,
where ψ is the streamfunction associated with the initial axisymmetric part of the vorticity field, ζsym(r)Θ(θ). The solution of (2.12), obtained at each integer level in the vertical, gives the initial M field to within an additive function of θ. This additive function was determined so that the horizontal area average of M over the 600 km × 600 km domain resulted in a vertical thermodynamic profile in agreement with the Jordan mean sounding. With M (x, y, θ, 0) determined in this way, the σ(x, y, θ, 0) can be obtained from σ = −(p/kΠ)(∂2M/∂θ2), completing the initialization of the prognostic variables, except for the perturbation part of the vorticity, which was added to help initiate the instabilty process. This unbalanced, weak perturbation was added to the axisymmetric part of the vorticity at each isentropic level in the form
ζamp = 1.0 × 10−5 s−1, and ϕn is a random phase factor. This asymmetry in the initial vorticity, which mimics background noise associated with deep convection, is so weak that it is hardly detectable in the panels shown in Figures 3 and 6.
Figure 3. The initial condition for the thick PV hollow tower. Panels: azimuthal mean PV (top left), azimuthal mean tangential velocity with isosurfaces of normalized absolute angular momentum (top right), PV on the 303 K surface (bottom left), and PV on the 333 K surface (bottom right).
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