Abstract
Päschke et al. (J Fluid Mech, 2012) studied the nonlinear dynamics of strongly tilted vortices subject to asymmetric diabatic heating by asymptotic methods. They found, inter alia, that an azimuthal Fourier mode 1 heating pattern can intensify or attenuate such a vortex depending on the relative orientation of the tilt and the heating asymmetries. The theory originally addressed the gradient wind regime which, asymptotically speaking, corresponds to vortex Rossby numbers of order unity in the limit. Formally, this restricts the applicability of the theory to rather weak vortices. It is shown below that said theory is, in contrast, uniformly valid for vanishing Coriolis parameter and thus applicable to vortices up to low hurricane strengths. An extended discussion of the asymptotics as regards their physical interpretation and their implications for the overall vortex dynamics is also provided in this context. The paper’s second contribution is a series of threedimensional numerical simulations examining the effect of different orientations of dipolar diabatic heating on idealized tropical cyclones. Comparisons with numerical solutions of the asymptotic equations yield evidence that supports the original theoretical predictions of Päschke et al. In addition, the influence of asymmetric diabatic heating on the time evolution of the vortex centerline is further analyzed, and a steering mechanism that depends on the orientation of the heating dipole is revealed. Finally, the steering mechanism is traced back to the correlation of dipolar perturbations of potential temperature, induced by the vortex tilt, and vertical velocity, for which diabatic heating not necessarily needs to be responsible, but which may have other origins.
1 Introduction
Atmospheric vortex intensification and attenuation, and the associated evolution of vortex structure remain a topic of intense investigations. As Smith and Montgomery [20] point out in their review article, intricate interactions of boundary layer processes, moist thermodynamics, multiscale stochastic deep convection, and the vortexscale fluid dynamics produce the observed, sometimes extremely rapid intensification of incipient hurricanes. They also emphasize that, despite the deep insights that have been gained in many studies of idealized axisymmetric flow models [6, 18, 19, 37], asymmetries of vortex structure, convection patterns, and boundary layer structure have been observed to be important for vortex intensification in reallife situations, see, e.g., recent work by Callaghan [2], RiosBerrios [34] and references therein.
This study focuses on the principal response mechanisms of Tropical cyclonelike (TC) atmospheric vortices to asymmetric diabatic heating following the theory of Päschke et al. [26]. Thus, neglecting boundary layer effects, we analyze both the structure and intensity of a tilted vortex under the influence of different configurations of explicitly prescribed asymmetric heating. In doing so, we adopt the point of view of, among others, Nolan and Montgomery [23], Nolan and Grasso [22], Nolan et al. [24] that latent heat release from condensation can be modeled, with limitations, by external diabatic heat sources in dry air. In the cited studies, nonaxisymmetric heating patterns were shown to have at most a small effect on vortex strength within the framework of linearizations about an axisymmetric upright vortex. These results of linear theory were corroborated in Nolan and Grasso [22], Nolan et al. [24] by comparison with fully nonlinear threedimensional simulations. In contrast, by both analytical and numerical examination, we demonstrate that the particular flow structure of a strongly tilted slender vortex, i.e., a vortex whose vorticity is confined to a narrow peak near the vortex center, does allow for leadingorder effects on vortex strength that are associated with asymmetric heating alone and that cannot be captured by linearized analyses.
In concentrating on the effects of purely asymmetric heating, we do not include any axisymmetric components of the heating pattern on purpose. As a consequence, the tests of the theory in this paper do not include secondary circulations (see explanation following Eq. (1) below). This also excludes phenomena such as the formation of an eye wall or of an anticyclonic upper level outflow which we hope to address in future studies. Investigating incipient hurricanes that develop in the tropical Atlantic, Marks et al. [17], Marks [16], and Dunkerton et al. [5] revealed, that such vortices can exhibit very strong tilt. Thus, for instance, the locations of the vortex center at heights equivalent to the 925 and \({200}\,{\hbox {h}\hbox {Pa}}\) pressure levels are located about \({200}\,{\hbox {k}\hbox {m}}\) apart, e.g., in ([16], figure 18) and ([5], figure 21). This amounts to an overall vortex tilt at a scale comparable to the vortex diameter, i.e., to a situation that without further assumptions does not allow for linearizations about an upright vortex. In fact, in characterizing the wind field of Hurricane Norbert (1984), Marks et al. [17] already utilized the concept of a height and generally also timedependent vortex center. Specifically, they considered barycenters of vorticity coinciding with a tilted centerline. Both, the interaction with a nonstationary environment and local selfinduction of the vortex lead to a generally timedependent vortex centerline. In a series of recent studies Chen and Gopalakrishnan [3] and Leighton et al. [13] examined the response of moist convection and vortex intensity to changes in the environmental large scale flow by means of highresolution idealized and nonidealized simulations. They found evidence of interactions between the asymmetric and tilted vortex core structure and the environment. The incorporation of such nonstationary strong tilts, described explicitly by a time dependent centerline, in a theory for vortex evolution is one of the key structural aspects in the analysis of Päschke et al. [26], which we revisit in the present work. Here we would also like to point to Davies et al. [4], who already provide evidence that asymmetric diabatic heating can directly influence vortex tilt.
Päschke et al. [26] analyzed the dynamics of such strongly tilted atmospheric vortices in the gradient wind regime by matched asymptotic expansions. They obtained a closed and coupled set of evolution equations for the primary circulation structure and the vortex centerline. Their theory describes, in effect, a nonlinear largetilt variant of the precessing quasimodes studied earlier in the context of small perturbations of an upright vortex by Reasor and Montgomery [30] (see also, e.g., [31], and references therein). Within their asymptotic framework, Päschke et al. [26] demonstrated that, in a strongly tilted vortex, purely asymmetric heating patterns with zero azimuthal mean can have a comparable impact on vortex intensity as axisymmetric ones.
As, by its very definition, the gradient wind regime is restricted to vortex Rossby numbers of order unity, this theory has been suspected to be applicable only to rather weak vortices with intensities relatively far from the interesting regime in which structural transitions from tropical storms to hurricanes may occur. To allow for vortices in this transition regime, we consider here in the first part of the paper the dynamics of mesoscale atmospheric vortices \(L_{\mathrm{mes}}\sim {100}\,{\hbox {k}\hbox {m}}\) that extend vertically across the depth of the troposphere \(h_{\mathrm{sc}}\sim {10}\,{\hbox {k}\hbox {m}}\) but feature large vortex Rossby number \(\mathrm{Ro}_{\mathrm{mes}}\gg 1\). We use the asymptotic techniques introduced by Päschke et al. [26] and recycle many of their technical steps. As indicated in Fig. 1, we assume vortices with nearly axisymmetric core structure at each horizontal level, and we allow for strong vortex tilt such that the vortex centers observed at different heights may be displaced horizontally relative to each other by distances comparable to the vortex core size \(L_{\mathrm{mes}}\).
One of the main findings of Päschke et al. [26] was the following evolution equation for the primary circulation described by the axisymmetric leadingorder circumferential velocity, \({u}_\theta \), valid for time scales large compared to the vortex turnover time scale,
Here (t, r, z) are the appropriately rescaled time, radial, and vertical coordinates, \(f_0\) is the Coriolis parameter, and \(w_0\) and \(u_{r,00}\) are the axisymmetric components of the vertical and radial velocities, i.e., the components of the secondary circulation, that are induced by the axisymmetric mean heating patterns. The apparent radial velocity \(u_{r,*}\) results from an interaction of the vortex tilt with the asymmetric first circumferential Fourier mode of the vertical velocity. In particular,
where \({\varvec{X}}(t,z)\) is the time dependent vortex centerline position at height z (see Fig. 1), w is the vertical velocity, and \({{\varvec{e}}}_r= {\varvec{i}}\cos (\theta ) + {\varvec{j}}\sin (\theta )\) is the radial unit vector of a horizontal polar coordinate system attached to the centerline.
\({\varvec{X}}(t,z)\) itself is governed by the centerline equation:
where \({\varvec{u}}_s(t,x,y,z)\) expresses the externally imposed background wind field, \(\delta \) is the small parameter giving rise to the asymptotic scaling, \({\varvec{k}}\) the vertical unit vector, \({\varvec{M}}_1\) is a weighted curvature measure of \({\varvec{X}}\) and \({\varvec{\varPsi }}\) evaluates Fourier1 modes of vertical velocity resulting from both, diabatic heating and adiabatic balances within a tilted vortex. In the adiabatic case and without vertical wind shear, equation (3) simplifies to a linear Schrödingerlike equation exhibiting undamped precession of eigenmodes as known from linear and nonlinear perturbation theories and from threedimensional simulations (see, e.g., [28, 30, 36, 40], and references therein). More details on the expressions for \({\varvec{M}}_1\) and \({\varvec{\varPsi }}\) follow in the further course of this article.
Note that for the rest of this paper we neglect the background wind \({\varvec{u}}_s\), because we are interested in highlighting a particular intrinsic interaction of asymmetric heating patterns (without an axisymmetric component) with the otherwise free dynamics of a tilted vortex.
The neglection of boundary layer effects and of the associated low level inward mass fluxes imply that the present analysis cannot describe the substantial secondary circulations, including those in the hurricane eye, observed in fully developed hurricanes. Therefore, we expect the theory to be applicable to incipient hurricanes and the early onset of a secondary circulation only. Especially for those comparably weak TCs, however, recent observational studies [8, 9, 36] have shown that phases of strong intensity changes are connected to localized and asymmetrically distributed convective events. To this end, in the present work we restrict to effects of purely asymmetric heating. Investigations of how these effects interact with external shear and a secondary circulation are deferred to forthcoming studies.
To summarize the findings of work that is to follow:

1.
The evolution equation for the core structure of a strongly tilted vortex from (1) is uniformly valid as \(f_0 \rightarrow 0\) so that it holds, in particular, also for \(\mathrm{Ro}_{\mathrm{mes}}\gg 1\), i.e., for vortices of moderate hurricane strength. In other words: For \(\mathrm{Ro}_{\mathrm{mes}}\ge 1\) the structure of the leadingorder equations does not change as \(f_0\rightarrow 0\).

2.
The mechanism of vortex spinup (spindown) by purely asymmetric heating of a tilted vortex is traced back analytically to an effective circumferential mean vertical mass flux divergence that arises when the first Fourier mode diabatic heating and the vortex tilt correlate negatively (positively).

3.
Asymmetric heating introduces a forcing of the vortex motion which can accelerate/decelerate the centerline precession and/or increase/decrease its tilt depending on the relative orientation of tilt and heating dipole.

4.
Equation (1) can be recast into a balance equation for kinetic energy, \(e_{\mathrm{k}}= \rho _0\frac{{u}_\theta ^2}{2}\),
(4)in line with the theory by Lorenz [14] for available potential energy (APE) generation and its conversion to mechanical energy. Here \({{\widetilde{p}}}\) is the relevant pressure perturbation, \({\widetilde{\Theta }}, Q_{\Theta }\) are the potential temperature perturbations and the diabatic heating, respectively, and \((\cdot )_0\) corresponds to the axisymmetric mean. In (4), it encodes the correlation of potential temperature perturbation and diabatic heating. N and \(\overline{\Theta }\) are the Brunt–Väisälä frequency and the background potential temperature stratification, respectively. Equation (4) states that, except for a conservative redistribution of kinetic energy due to advection and the work of the pressure perturbation, \({{\tilde{p}}}\), positive correlations of diabatic sources and potential temperature perturbations generate the potential energy available for increasing the kinetic energy of the vortex. Nolan et al. [24] study the effects of asymmetric diabatic heating on vortex strength in a linearized model. One of their conclusions is that “... purely asymmetric heating generally leads to vortex weakening, usually in terms of the symmetric energy, and always in terms of the lowlevel wind.” The present theory shows that this conclusion does not hold up in case of a strongly tilted vortex, but that in this case symmetric and suitably arranged asymmetric heating have vortex intensification/attenuation efficiencies of the same order of magnitude.

5.
The intensification (attenuation) mechanism due to asymmetric diabatic heating is traced back to geometrical properties of the vertical velocity, more precisely that part which positively (negatively) correlates with the potential temperature perturbation caused by the structure of the tilted vortex. Diabatic heating, in fact, turns out to cause those vertical motions at leading order following the weak temperature gradient approximation in the present setting. However, there is no need for diabatic heating in general as vortexscale averaged vertical motions can be (and most likely will be) caused by different driving forces, such as multiscale convection.

6.
The theory compares favorably with threedimensional numerical simulations based on the compressible Euler equations.
To arrive at these results, we first recount the governing equations and the principles of our analytical approach in Sect. 2, and then revisit the derivations by Päschke et al. [26]. A discussion of the scaling regime is given in Sect. 3 to investigate the influence of the Coriolis effect (item (1)). The asymptotic vortex core expansion is carried out in Sect. 4, and this analysis gives rise to a physical interpretation of the intensification/attenuation by asymmetric heating referred to in item (2). In Sect. 5 we establish the kinetic energy balance of item (4). Section 6 presents results of the theory in comparison with threedimensional computational simulations to corroborate items (3), (5), and (6). Section 7 provides conclusions and an outlook.
2 Dimensionless governing equations and distinguished limits
2.1 Governing equations
The dimensionless inviscid rotating compressible flow equations for an ideal gas with constant specific heat capacities in the beta plane approximation form the basis for the subsequent asymptotic analysis:
Here, \(p,\rho ,\Theta ,{\varvec{u}},w\) are rescaled pressure, density, potential temperature, and the horizontal and vertical velocities. \(\gamma \) is the specific heat ratio.
The threedimensional gradient is \({\varvec{\nabla }}= {\varvec{\nabla }_{\!\!\scriptstyle \parallel }}{} + {\varvec{k}}\,\partial /\partial z\) with the horizontal gradient \({\varvec{\nabla }_{\!\!\scriptstyle \parallel }}{} = {\varvec{i}}\, \partial /\partial x + {\varvec{j}}\, \partial /\partial y\), the zonal, meridional, and vertical coordinates (x, y, z), and the related unit vectors \(({\varvec{i}},{\varvec{j}},{\varvec{k}})\). Finally, t is the time variable and \(Q_{\Theta }\) is a diabatic source term.
Table 1 lists general characteristics of the neartropical atmosphere. Together with the combined values in Table 2 they form reference values for nondimensionalization. Let an asterisk denote dimensional quantities, then the unknowns and coordinates in (5) are
Note that \({u}_{\mathrm{ref}}/h_{\mathrm{sc}}\) is an estimate of the largescale thermal wind shear, and \({\varvec{x}}= {\varvec{i}}x + {\varvec{j}}y\) is the horizontal coordinate vector.
In deriving the dimensionless equations (5) using the quantities from Tables 1 and 2 the Mach, internal wave Froude, and Rossby numbers, and the \(\beta \)parameter
emerge naturally. These are replaced with functions of a single small expansion parameter \(\varepsilon \ll 1\) through the distinguished limits
in line with the multiscale asymptotic modeling framework of Klein [11]. Here \((N, f, \beta ) = \mathcal{O} ( 1 )\) as \(\varepsilon \rightarrow 0\), with concrete values
derived from (7) for \(\varepsilon = \mathrm{M}^{2/3} = 0.105\). Replacing the characteristic numbers according to (8) we obtain
whereas f and \(\beta \) appear explicitly in (10), N characterizes the background stratification of potential temperature and will be invoked below where we define the initial conditions for the vortex flow.
Equations (10) will form the basis for the subsequent asymptotic analysis for \(\varepsilon \ll 1\), although, following [26], much of the expansions will proceed in terms of the small parameter
3 Scaling regime for large vortex Rossby number and strong tilt
Here we set up the asymptotic scalings for our analysis in terms of typical orders of magnitude of several characteristics of tropical storms and hurricanes.
3.1 Vortex core size, intensity, and evolution time scale
Vortex core sizes, independent of what are the specific definitions used, are found in the range of \({50}\,{\hbox {k}\hbox {m}}\) to \({200}\,{\hbox {k}\hbox {m}}\), while wind speeds in the transition regime from storms to hurricanes are around \({30}\,{\hbox {m}\big /\hbox {s}}\) [7]. With \(\delta ^2 \equiv \varepsilon \sim 1/10\), \(h_{\mathrm{sc}}\sim {10}\,{\hbox {k}\hbox {m}}\), and \({u}_{\mathrm{ref}} \sim {10}\,{\hbox {m}\big /\hbox {s}}\), these data correspond well with
for a characteristic vortex core size \(L_{\mathrm{v}}\), a typical wind speed, and the associated depression in the vortex core, respectively. Note that the vortex core size itself is of subordinate interest since the effective Rossby number determines the scaling regime. Further, note that these scalings deviate from those adopted by Päschke et al. [26], who considered systematically larger radii of the order \(L_{\mathrm{v}}\sim h_{\mathrm{sc}}/\delta ^{3}\) needed for direct matching to a quasigeostrophic large scale outer flow. From their work we recall, however, that the vortex core structure and tilt develop on a time scale \(t_{\mathrm{v}}\) that is by \(1/\delta ^2\) longer than the vortex core turnover time scale \(t_{\mathrm{to}} = L_{\mathrm{v}}/u_{\mathrm{max}}\). Thus, in view of (12), we will follow the vortex core evolution on the time scale
The scalings in (12) and (13) include the regime of “rapid intensification,” defined by NOAA’s National Hurricane Center^{Footnote 1} to denote maximum wind accelerations of \(30\,\mathrm{kt}\) \((\sim \,{15}\,{\hbox {m}\big /\hbox {s}})\) in \(24\,\hbox {h}\).
Also, the adopted scalings describe a vortex in the cyclostrophic regime since
i.e., the Coriolis term is subordinate to the centripetal acceleration in the horizontal momentum balance in this regime. Accordingly, the vortex Rossby number is large,
3.2 Comoving coordinates for a strongly tilted vortex
Following Päschke et al. [26], we resolve the flow dynamics on the vortex precession and core evolution time scale \(t_{\mathrm{v}}\) from (13). The appropriate time coordinate is
For the core structure analysis we introduce vortex centered horizontal coordinates
where \({\varvec{X}}({{\widehat{t}}}, z)\) is the horizontal position of the vortex centerline at height z and \(\widehat{{\varvec{x}}}\) is the relative horizontal offset. With this scaling \(\widehat{{\varvec{x}}}\) resolves the core scale \(L_{\mathrm{v}}\) from (12) and the centerline covers comparable distances. This justifies the notion of “strong tilt.”
In the sequel we use polar coordinates in horizontal planes, i.e.,
with \({{\varvec{e}}}_r\) and \({{\varvec{e}}}_{\theta }\) the radial and circumferential unit vectors, respectively. The transformation rules for derivatives in these coordinates read
The horizontal velocity is decomposed into the vortex’ motion plus the relative velocity,
For later reference, here are the centerline, represented in the \(({{\varvec{e}}}_r, {{\varvec{e}}}_{\theta })\) basis,
and the Fourier expansion of functions of the circumferential angle, \(\theta \),
Note that we have exchanged the roles of \(F_{n1}\) and \(F_{n2}\) relative to their use in [26] as this will streamline the notation throughout the analysis of the orientation of a dipolar field characterized by \({\varvec{F}}_1 = (F_{11}, F_{12})^T\) relative to the tilt vector \(\partial {\varvec{X}} /\partial z\) below.
3.3 Vortex core expansion scheme
The circumferential velocity is expanded as
i.e., nonaxisymmetry relative to the centerline is allowed for scaling orders from \(\mathcal{O} ( \delta {u}_{\mathrm{ref}} )\) upwards. Across the core size length scale, \(L_{\mathrm{v}}\), such asymmetries induce horizontal divergences of order \({u}_r/ L_{\mathrm{v}}\sim \delta {u}_{\mathrm{ref}} / (h_{\mathrm{sc}}/ \delta ^2) = \delta ^3 {u}_{\mathrm{ref}}/h_{\mathrm{sc}}\), see (12). Since the flow field is anelastic to leading order as derived below, this implies the vertical velocity scaling,
Expansions for the thermodynamic variables are anticipated as follows,
[for plausibility arguments see [26], section 4.1.3]. The variables \(p_0(z)\), \(p_2(z)\), \(\rho _0(z)\), \(\rho _2(z)\), and \(\Theta _2(z)\) in (25) describe the stationary background (\(\Theta _0\) is a constant), \(\widehat{p}_{i}({{\widehat{t}}}, z)\), \(\widehat{\rho }_{i}({{\widehat{t}}}, z)\), and \(\widehat{\Theta }_{i}({{\widehat{t}}}, z)\), are higherorder horizontal means, and the quantities \(\widehat{p}^{(i)}({{\widehat{t}}}, \widehat{r}, \theta , z)\), \(\widehat{\rho }^{(i)}({{\widehat{t}}}, \widehat{r}, \theta , z)\), and \(\widehat{\Theta }^{(i)}({{\widehat{t}}}, \widehat{r}, \theta , z)\) are of prime interest with full spatiotemporal variation.
Note that, owing to the Fourier representation defined in (22), this notational convention “overloads” the subscript \((\cdot )_0\) with a doublemeaning, but the distinction should always be clear from the context.
The vortex centerline position is expanded as
4 Asymptotic analysis of the core structure evolution
This section revisits the analysis of Päschke et al. [26] for large vortex Rossby numbers focusing on the evolution equation for the primary circulation.
4.1 Asymptotic equation hierarchy for the vortex core
The governing equations transformed to the comoving coordinates are provided in Appendix A. Inserting the expansion scheme from the previous section we obtain
from the horizontal momentum balance at leading and first order, respectively. Each line in (27) displays the respective radial balance first and the circumferential balance as the second equation. We observe from the radial component in (27a) that the vortex is in cyclostrophic balance to leading order which implies large vortex Rossby number. The Coriolis effect enters only as a firstorder perturbation in the present regime as seen in the radial component of (27b). The pressure perturbations \(p^{(4)}, p^{(5)}\) inherit the assumed axisymmetry of \({u}_\theta ^{(0)}, {u}_\theta ^{(1)}\) thanks to the leading and firstorder circumferential momentum balances in (27a) and (27b), respectively.
The full secondorder horizontal momentum equations are listed in Appendix B, equations (69), but for the rest of the paper we only need the circumferential average of the circumferential component (69b). Letting \(\psi _0 \equiv \frac{1}{2\pi }\int _{\pi }^{\pi } \psi (\theta )\,{\mathrm{d}}\theta \) denote the circumferential average of some \(\theta \)dependent variable \(\psi \) in line with (22), we have
where
The flow is hydrostatic up to thirdorder and in the fourthorder horizontal mean, i.e., we have \(\frac{\partial p_{i}}{\partial z} = \rho _i\) \((i = 0, 2)\) and \(\frac{\partial {{\widehat{p}}}_{4}}{\partial z} = {{\widehat{\rho }}}_{4}\), whereas
The leading and firstorder velocities are horizontal and axisymmetric according to (23), (24) and thus divergence free. The secondorder velocity is subject to an anelastic divergence constraint obtained from the mass balance,
Similarly, the first nontrivial potential temperature transport equation reads
and the equation of state relates the thermodynamic perturbation variables through
4.2 Temporal evolution of the vortex structure
Päschke et al. [26] observed that with the aid of (27) and (29)–(33), and given the vortex tilt, \({\partial {\varvec{X}}^{(0)} / \partial z}\), as well as the diabatic source term, \(Q_\Theta ^{(0)}\), one may interpret (28) as a closed evolution equation for the leading circumferential velocity, \({u}_\theta ^{(0)}\).
To corroborate this, we use the Fourier decomposition, (22), for \(w^{(0)}\) and the of the centerline in polar coordinates from (21) to obtain
Expressions for \(w^{(0)}_0\) and \(w^{(0)}_{1k}\) for \(k = 1,2\) follow from the \(\Theta \)–transport equation in (32),
Since \(\widehat{p}^{(4)}\) is axisymmetric (see 27a), \(\widehat{p}^{(4)}_{1k} \equiv 0\) and the equation of state, (33), yields \(\widehat{\Theta }^{(4)}_{1k}/\Theta _0 =  \widehat{\rho }^{(4)}_{1k}/\rho _0\). With this information, the vertical momentum balance (30) yield
Using the cyclostrophic balance in (27a) to eliminate \({\partial \widehat{p}^{(4)} / \partial \widehat{r}}\), and going back to (35) we obtain expressions for the \(w^{(0)}_{1k}\) in terms of \({u}_\theta ^{(0)}\), \({\partial {\varvec{X}}^{(0)} / \partial z}\), and \(Q_\Theta ^{(0)}\),
where \(X^{(0)}_1 \equiv X^{(0)}\) and \(X^{(0)}_2 \equiv Y^{(0)}\). Upon insertion of this result in (34), the second term on the right cancels, so that
Here we have interpreted the cosine and sine Fourier1 components of \(Q_{\Theta }^{(0)}\) as the components of a heating dipole vector, \({\varvec{Q}}_{\Theta }\), in the horizontal plane.
To find a corresponding expression for \(u_{r,0}^{(2)}\) (see the third term in (28)), consider the circumferential average of mass continuity, (31). A brief calculation yields
or, equivalently,
with \(u_{r,*}^{(2)}\) defined in (34). Exploiting (37) in that definition and integrating in \(\widehat{r}\) requiring that \(u_{r,0}^{(2)}\) be finite at \(\widehat{r}=0\) we find
where
With (35) (first equation), (38), (41), and (42) we have now indeed expressed \(w^{(0)}_0, u_{r,0}^{(2)}\), and \(u_{r,*}^{(2)}\) in terms of \({u}_\theta ^{(0)}, {\partial {\varvec{X}}^{(0)} / \partial z}\), and \(Q_{\Theta }^{(0)}\) as announced. In the sequel, we may thus derive from (28) how vortex tilt and diabatic heating affect the evolution of the primary circulation.
Remark
(item 1): The results in this section match the corresponding result by Päschke et al. [26] with the Coriolis parameter \(f_0\) set to zero. This corroborates our statement (item 1) in the introduction that the vortex amplification/attenuation mechanism described in their work does not depend on the vortex Rossby number being at most of order unity.
5 Discussion of the asymmetric intensification/attenuation mechanism
5.1 The influence of asymmetric heating on the primary circulation
As elaborated in the previous section (28) describes the evolution of the primary circulation in response to external diabatic heating in the present vortex flow regime. Aiming to separate the influence of heating asymmetries from those of axisymmetric effects, we recall from (41) that the net circumferentially averaged radial velocity is entirely a response to diabatic effects, and that it consists of one part, \(u^{(2)}_{r,00}\), which, according to (42) is induced by axisymmetric heating, and a second part, \(u_{r,*}^{(2)}\), which, according to (38), arises from first Fourier mode asymmetric heating patterns. Using this decomposition in (28), we rewrite the equation as
which is the largeRossby (\(f_0 \rightarrow 0\)) version of Eq. (1) announced in the introduction. In this equation, the lefthand side captures the influence of the axisymmetric dynamics and axisymmetric diabatic heating, whereas the righthand side covers all effects due to the interaction of asymmetric heating and vortex tilt.
5.2 Mechanics of vortex intensification/attenuation by asymmetric heating of a tilted vortex
In the following section we analyze the leadingorder mass balance relations given in (39) and (40). We furthermore argue that \(u_{r,*}^{(2)}\), given in (38), plays a crucial role in explaining the spinup mechanism based on asymmetric diabatic heating. In this context we note that, according to (37), the firstorder Fourier modes of the vertical velocity involve a contribution from diabatic heating (first term) and one due to the adiabatic dynamics (second term). It is only the contribution by diabatic heating that has an impact on \(u_{r,*}^{(2)}\) as seen in (38).
The situation is sketched for the case of asymmetric heating antiparallel to the tilt in Fig. 2. From the anelastic constraint in (31), averaged in \(\theta \), we observe that a vertical velocity dipole with nonzero projection onto the tilt vector generates an averaged radial motion. From Fig. 2, this effect becomes physically transparent: the net vertical mass flow across the tilteddisc control volume through its radial boundary must be accompanied by a compensating radial motion to satisfy the anelastic constraint. In fact, considering mass continuity in centerlineattached coordinates in (39), we can identify the term in brackets as the axisymmetric mean of the vertical mass flux. Equation (40) reveals that this expression is equal to a horizontal mass flux governed by \(u_{r,*}^{(2)}\). We therefore conclude that the net vertical outflow in Fig. 2 is compensated in the present balanced vortex situation by a net horizontal inflow to preserve continuity.
This gives rise to a spinup mechanism that exploits the vertical (tilted) structure of the vortex to gain angular momentum by moving air parcels from larger radii to the center of the vortex. In contrast, the opposite orientation of diabatic Fourier1 modes leads to an attenuation of the vortex by transporting angular momentum away from the center. We therefore claim that by this mechanism it is possible to influence the overall strength of an atmospheric vortex as will be demonstrated in Sect. 6.
Remark
(item 2): This should support the second claim (item 2) in the introduction regarding the physical interpretation of the vortex spinup or spindown mechanism involving interactions of asymmetric diabatic heating with the vortex tilt.
5.3 Energy budget for the externally heated vortex
Here we elaborate on how the asymmetric diabatic heating is transferred to kinetic energy of the primary circulation in a tilted vortex. This will be particularly useful in assessing the derived equations within the framework of Available Potential Energy (APE, [14]).
To this end we multiply (43) by \(\rho _0 \widehat{r}\,{u}_\theta ^{(0)}\), use the \(\theta \)averaged leadingorder mass balance from (40) and recast the advective terms in conservation form to obtain,
Here we have dropped the \(^{(0)}\) superscript on \({u}_\theta ^{(0)}\) and \(w^{(0)}\) to simplify the notation, and we have used the cyclostrophic radial momentum balance from (27) to introduce the pressure gradient on the right.
This reveals the change of kinetic energy (lefthand side) to result from the work of the pressure force due to the mean radial motion (righthand side). Some straightforward but lengthy calculations, the details of which are given in Appendix C, yield a direct relation of the kinetic energy balance in (44) to the Lorenz’ theory of generation of available potential energy (APE) by diabatic heating,
where \(h_{\mathrm{k}}= e_{\mathrm{k}}+ p^{(4)}\), and \(\left( {\varvec{\Theta }},{\varvec{Q}}_\Theta \right) _{1} = \left( \Theta , Q_{\Theta }\right) _{11} {\varvec{i}}+ \left( \Theta , Q_{\Theta }\right) _{12} {\varvec{j}}\) are the dipole vectors spanned by the first circumferential Fourier components of the fourthorder potential temperature perturbation, \(\Theta ^{(4)}\), and of the diabatic heating function, \(Q_{\Theta }^{(0)}\), respectively.
Equation (45) poses the differential form of the kinetic energy balance. To end up with an integral form as presented in Lorenz [14] we make use of Gauss’s theorem which allows us to drop the radial and vertical derivatives assuming \(u_{r,00}^{(2)}\) and \(w_0^{(2)}\) vanish for sufficiently large \(\widehat{r}\) and z, respectively. Conditions under which this constraint for \(u_{r,00}^{(2)}\) is satisfied can be derived from (42). The equation shows that we do not only need \(Q_{\Theta ,0}\) such that the integral converges for large radii, but we need the integral to converge to zero. Thus, when assuming a concentrated pattern of heating with amplitude \(\mathscr {O}\left( 10^{4}\,\hbox {K}\big /\hbox {s}\right) \) close to the vortex center (in the eyewall) over a surface of \(\sim \) \(10^4\) km\({}^2\) it would need to be counteracted by contributions with opposite sign, i.e., cooling, but over a much larger surface of \(\sim \) \(10^6\) km\({}^2\). This simple scale approximation reveals cooling rates of \(\mathscr {O}\left( {{0.1}{K/d}}\right) \) which is by an order of magnitude smaller than observed radiative cooling rates [12, 34, 35, 38], and thus well within reach of the overall radiative balance on larger scales.
For the total (integrated) kinetic energy \(E_k\) we find
On the one hand, Lorenz [14] balanced the kinetic energy with the conversion rate from APE to kinetic energy (C) and the dissipation rate (D) whereas the latter is neglected in our case. In contrast, the analytical expression on the righthand side of the energy balance equation (46) coincides with the generation rate (G) of APE (see Appendix D for details) and not with the conversion rate. Therefore, we come to the conclusion that APE is directly converted into kinetic energy and does not accumulate (at leading order). This is the result of the timescale used in the asymptotic analysis as conversion between APE and kinetic energy is accomplished by the advective and pressurevelocity fluxes on faster timescales.
In line with Lorenz [14, 15] and as announced in (item 4) of the introduction, this result shows that positive correlations of temperature perturbation and diabatic heating lead to an increase in kinetic energy. The precise form of the righthand side of (4) as announced in the introduction (item 4) is obtained from (45) by realizing that \((1/\Theta _0) {\mathrm{d}}\Theta _2/{\mathrm{d}}z\) is the dimensionless representation of \(N^2\), the square of the Brunt–Väisälä frequency, and that the constant \(\Theta _0\) is the leadingorder dimensionless background potential temperature \(\overline{\Theta }= {T}_{\mathrm{ref}} (\Theta _0 + o\left( 1\right) )\).
Remark
(item 4): Nolan et al. [24], extending prior similar studies, investigate the influence of asymmetric diabatic heating on vortex intensification on the basis of a linearized anelastic model that includes a radially varying base state and baroclinic primary circulation. Their central conclusions are that (1) asymmetric heating patterns quite generally tend to attenuate a vortex, that (2) there are situations in which they can induce amplification, but in these cases their influence is (3) generally rather weak. In fact, they state in their section e: “... purely asymmetric heating generally leads to vortex weakening, usually in terms of the symmetric energy, and always in terms of the lowlevel wind.” Equation (45) shows, in contrast, that purely asymmetric heating in a tilted vortex can intensify or attenuate a vortex depending on the arrangement of the heating pattern relative to the tilt, and that the efficiencies of symmetric and asymmetric heating in generating kinetic energy are of the same order in the asymptotics as claimed in (item 4) of the introduction.
5.4 Diabatic forcing of the centerline motion
This section highlights aspects of the effects of asymmetric heating on the vortex centerline motion that have not been worked out in detail in our earlier publications. Examinations in Appendix E of the constituents of the centerline equation reveal that the term \({\varvec{k}} \times {\varvec{\varPsi }}\) splits into an adiabatic and a diabatic contribution due to the linear dependency of \({\varvec{\varPsi }}\) on the vertical velocity dipole \({\varvec{w}}_1\), where \({\varvec{w}}_1\) is composed of an adiabatic and a diabatic contribution (see (37)). In particular, the adiabatic expression is of the same analytical form as the diabatic one but evaluated with the adiabatic vertical velocity:
with
and the \({{\hat{R}}}_{\theta _0}\) indicating rotation by the angle \(\theta \). Inserting (47) into the explicit expressions for the field \({\varvec{\varPsi }}\) that arises in the centerline equation of motion in (3), one turns the terms describing the centerline evolution (see Eq. 90 in Appendix E) into a linear differential operator acting on \({\varvec{X}}\). The latter can be interpreted as the Hamiltonian of a (complexvalued) Schrödinger equation (54) that leads to a purely realvalued spectrum, i.e., to precession of the centerline in the complex (\(xy\)) plane.
The diabatic motion of the centerline on the other hand results from inserting some nontrivial \({\varvec{w}}_{1,\text {dia}}\) into \({\varvec{\varPsi }}\) which in our case, for purposes of setting up an elucidating test case for the subsequent comparison of asymptotics and 3D numerics, shall be a rotated version of (47):
Here \(\theta _0\) is the relative orientation of the diabatic vertical velocity dipole relative to the tilt. Clearly, \({\varvec{w}}_{1,\text {dia}}\) coincides with \({\varvec{w}}_{1,\text {ad}}\) for \(\theta _0=\pi /2\). Therefore, such a diabatic heating vertical velocity pattern will result in an additional contribution to the centerline motion of the same orientation and magnitude as the adiabatic contribution. By numerical experiments we observed that the adiabatic vertical velocity dipole, oriented \(\pi /2\) relative to the tilt leads to a centerline motion in the direction \(+\pi /2\).
In contrast, by varying \(\theta _0\) we expect a diabatic centerline forcing which is oriented \(\theta _0\) relative to the tilt. Therefore, for our formulation of diabatic heating the rotation angle \(\theta _0\) determines whether the diabatic forcing leads to acceleration (\(\theta _0=\pi /2\)) or deceleration (\(\theta _0=\pi /2\)) of the centerline precession or to increasing (\(\theta _0=\pi \)) or decreasing the tilt of the centerline. Note in this context that the resulting linear operator governing the diabatic centerline motion effectively is the rotated version of the adiabatic operator (multiplied with complex number on the unit circle), i.e., its spectrum is rotated in complex plane as \({\varvec{w}}_1\) is rotated. Real parts of the spectrum lead to precession while imaginary parts lead to growth/damping of the centerline amplitude.
In the further course of diabatic experiments we make use of our findings and specify heating patterns of the form
6 Comparison between asymptotic model and threedimensional simulations
In the course of this paper we presented the derivation of a system of partial differential equations (PDEs) governing the leadingorder dynamics of a tilted tropical cyclone in weak cyclostrophic regimes on time scales slow compared with the vortex turnover time (1 and 3).
The aim of this section is twofold: First, we intend to validate the reduced model equations against the threedimensional Euler equations, i.e., the relevant first principle equations (5), by solving both sets of equations in a suitable numerical framework. To this end, we follow the work of Papke [25] who presented the first results on this scenario. The second goal is to highlight principal mechanisms that are activated by purely asymmetric heating. Therefore, after analyzing the adiabatic dynamics of an initially tilted vortex in an environment without external wind shear (\({\varvec{u}}_s=0\)), we continue by constructing an idealized asymmetric diabatic heating pattern which will be imposed on the vortex under different angles relative to the tilt. We will refer to these experiments as adiabatic (reference simulation), stagnation, intensification and attenuation according to their influence on tilt and centerline geometry.
The quasitwodimensional Eqs. (1) and (3) are solved by a combination of appropriate numerical methods, details of which are presented in Appendix F. For the threedimensional simulations the general purpose atmospheric flow model EULAG (see, e.g., [29, 42]) provides efficient integration strategies for the compressible Euler equations (5), and EULAG’s compressible mode was used here. These two alternative representations of the tilted vortex flows will be referred to as asymptotic and threedimensional simulations.
Note that, although we worked out the scaling in the cyclostrophic regime (\(\mathrm{Ro}= \mathscr {O}\left( 1/\delta \right) \)) above, the vortex initial data adopted in the following sections are in the gradientwind regime (\(\mathrm{Ro}= \mathscr {O}\left( 1\right) \)) because we are principally interested in mechanisms active in the course of stormtohurricane transition. Therefore, in the numerical treatment terms involving the Coriolis parameter \({f}_{\mathrm{ref}}=7.3\,\times \,10^{5}\) 1/s, even if reasonably small, will not be neglected.
6.1 Numerical setting and initial data
According to the asymptotic analysis, a tilted vortex that is initially balanced w.r.t. acoustic and internal waves evolves on a slow time scale that is comparable to the synoptic time scale as discussed by Päschke et al. [26]. Higherorder dynamics occurs in the presence of initial perturbations (excitation of higherorder asymptotic modes) on faster time scales. However, we are interested in the leadingorder effects only, and thus we construct initial data to closely reproduce the leadingorder symmetries assumed in the asymptotic analysis of Sect. 3. This should allow us to obtain solutions following—qualitatively speaking—the evolution on the slow manifold of precessing quasimodes.
For both, the asymptotic model and EULAG, it is crucial to have a notion of a balanced background state. Prescribing the background potential temperature
we find the background density \({{\bar{\rho }}}\) and pressure \({{\bar{p}}}\) by integrating the hydrostatic equation with boundary value \({\bar{p}}(z=0) = {p}_{\mathrm{ref}}\) and making use of the dimensional form of (5e):
To motivate the construction of initial data, we are considering the case of an adiabatic vortex for which (1) is trivially stationary,
and the centerline equation (3) becomes a Schrödingertype equation,
with a Hamiltonian \({{\hat{H}}}\) depending on the tangential velocity \(u_\theta (r,z)\) and the zdependent background profiles \({{\bar{\rho }}}\), \({{\bar{p}}}\), \({{\bar{\Theta }}}\). Note, that we have introduced \(X := ({\varvec{X}} \cdot {\varvec{i}}) + i ({\varvec{X}} \cdot {\varvec{j}})\) as the complex representation of \({\varvec{X}}\) (see Appendix E). With suitable boundary conditions \({{\hat{H}}}\) takes the form of a Sturm–Liouville operator and therefore exhibits a realvalued, discrete spectrum [43], which sets the precession frequency of each eigenmode. The first nontrivial eigenvalue corresponds to the slowest varying solution and comes with a cosinelike eigenfunction [26] as the simplest time dependent solution with a nontrivial vortex tilt. Related details are omitted here but can easily be validated by numerical means.
As tilt is crucial for coupling asymmetric diabatic heating modes to the leadingorder vortex dynamics, we prescribe an initially barotropic tangential velocity profile
corresponding to a Gaussian vorticity profile
with \(q_m\) and \(\sigma \) chosen such that
and
where \(R_{\mathrm{mw}}=100\,\,\hbox {k}\hbox {m}\). Note that the radial velocity profile (55) results in a barotropic velocity distribution for an untilted vortex, only. The vertical displacement due to the centerline tilt renders the flow field genuinely baroclinic.
We take the first nontrivial eigenfunction of \({{\hat{H}}}\) to define the initial centerline geometry, scaled to a displacement of 160 km, as depicted by the bold(green) curve in Fig. 3 and arrange the tangential flow profile with respect to the centerline (see streamlines in Fig. 3).
For compliance with the rigid wall horizontal and vertical boundary conditions imposed by EULAG, and to avoid spurious wave reflections near the boundary of the computational domain, the velocity initial data for the threedimensional simulations are smoothly transitioned to zero at some finite radius by applying a mollifier:
Here \(r_1\) = 1 250 km and \(r_\infty \) = 1 750 km are the radii at which the mollifier starts and where it reaches fullsuppression, respectively. In addition, the profile within \(r_0\) = 100 km is adjusted to preserve differentiability at the origin. Although somewhat arbitrary, the choice of \(r_1\) and \(r_\infty \) is based on the sensitivity of tilt and tangential velocity toward these parameters. Tests (not presented here) yield that the solution remains essentially unchanged until decreasing both radii by a factor of \(\sim \)5. Thus, larger radii of the mollifier seem not to affect the vortex at all. Hence, we argue that the above choice is reasonable under the considerations of avoiding interactions with the domain boundary as well as essentially capturing the farfield behavior of the vortex. The resulting radial profile of initial tangential velocity is displayed in Fig. 4. (Anticipating the forthcoming discussion in Sect. 6.2.1, we note that \(r_1\) and \(r_2\) are both larger than the socalled critical radius [31]).
Section 4.1 yields higherorder diagnostic expressions \(u_r^{(2)}\), \(u_\theta ^{(2)}\), \(w^{(0)}\), \(\rho ^{(4)}\), \(p^{(4)}\), and \(\Theta ^{(4)}\), which we have used to achieve higherorder balance. An example is shown in Fig. 3 by colorizing streamlines with the potential temperature perturbation. The dipolar perturbation is negative in the tilt direction and positive uptilt.
In the threedimensional case the vortex is embedded into a domain of 4000 km extent in both horizontal directions, 10 km in the vertical, and a damping layer surrounds the domain near the horizontal boundaries to suppress gravity waves emerging from the inner core and to keep them from reflecting. The thickness of the layers is \({500}\,{\hbox {k}\hbox {m}}\), and the characteristic damping time scale is set to \(\tau = {10^4}{\hbox {s}}\), a setting that has proven to absorb internal waves resulting from small imbalances of the initial data.
In line with the structural properties of the tropical cyclone, i.e., an inner core with a smooth transition to the quasigeostrophic far field, we use the static mesh refinement capability of EULAG [29] and map equidistantly spaced coordinates onto a grid focused at the inner core. The actual mapping is accomplished by
where \(\alpha = 5\), \(c_{1,2} = 1/2\). \((x,y)_{p,c}\) are normalized coordinates on the domain \([1,1]^2\). Figure 5 demonstrates how the horizontal grid is focused toward the center of the computational domain.
The asymptotic equations are solved on a regular equidistant tilted polar grid on the domain \((r,z) \in [0, 1.12] \times [0, 12.5]\) in dimensionless units covering roughly a tilted cylinder of 1000 km around the centerline and the full vertical extent of \({10}\,{\hbox {k}\hbox {m}}\).
In the further course, we will compute the diabatic heating from Eq. (50) which involves the reconstruction of both, the current centerline position and circumferentially averaged tangential velocity. For the first, we borrow one of the concepts described by Nguyen et al. [21] and define the center of mass of the vorticity field at each horizontal level as
This formula is evaluated here utilizing a secondorder accurate discretization outlined by Papke [25]. The circumferential mean of \(u_\theta \) then is computed by the Biot–Savart integral
where \(B_r({\varvec{X}}) = \{{\varvec{x}}_{} \in {\mathbb {R}}^2  ({\varvec{x}}_{}  {\varvec{X}})^2 < r^2 \}\) denotes the circular domain centered at \({\varvec{X}}\) with radius r.
Remark
We have adopted here a Gaussian initial vorticity profile (cf. 56) which guarantees that the vorticity decays rapidly outside the vortex core in gradient wind balance. As a consequence, the vortices do not exhibit selfalignment due to critical damping of vortex Rossby waves as discussed in ([31, 40], and references therein), because the vorticity and its gradient are exponentially small in terms of the perturbation parameter \(\delta \) at the critical radius. The mollifier (59) does not change this circumstance. This choice allows us to focus in this work on changes in vortex tilt and intensity due to diabatic effects only. See also Sect. 7 for a discussion of critical damping in relation to the present theory.
6.2 Results and discussion
Here we present results of numerical simulations solving either the full threedimensional Euler equations (5) or the reduced asymptotic equation (1) for the primary circulation velocity \(u_\theta ^{(0)}\), and the centerline evolution (3). More information regarding the numerical methods used, including empirical convergence tests, is provided in Appendices E to G.
6.2.1 Adiabatic vortex
As a reference for the following experiments we first investigate the dynamics of a tilted adiabatic vortex. In Sect. 6.1 we constructed initial data to follow the first nontrivial eigenmode of the governing (adiabatic) Hamiltonian, and we found stationarity of the mean tangential wind according to (53). Hence, from the structure of (54) we expect undamped precession of this eigensolution.
Figure 6 compares the results of the threedimensional and asymptotic adiabatic simulations with initial data as discussed. Though exhibiting some smallscale oscillations and damping, the threedimensional simulation (Fig. 6, right panel) compares well with its asymptotic analogue. Time scales of one precession are 5.5 days for the asymptotic and 6.5 days for the threedimensional simulation. This difference leads to a deviation of the final positions of the centerline but given that the effective expansion parameter \(\delta = \sqrt{\varepsilon } \sim 1/3\), this is well within the error bounds of the leadingorder solution.
The asymptotic analysis revealed nontrivial leadingorder gradientwind (cyclostrophic) and hydrostatic balances for the vertical velocity w and for the thermodynamic quantities depending on the centerline tilt. In Fig. 7 both, w and \({{\tilde{\Theta }}}\) are visualized by a representative horizontal slice at 5000 m at \(t = 1.5\,\,\hbox {days}\) comparing asymptotic values (black contours) with the 3D numerical simulation results (shaded/colorcoded). The tilt vector \(\partial _z {\varvec{X}}\) is indicated by an arrow. Qualitative similarities are rather apparent, while deviations are again well within the asymptotic truncation order \(\mathscr {O}\left( \delta \right) \sim 1/3\). Both figures demonstrate the alignment of dipolar perturbations relative to the tilt as \({\varvec{w}}_1\) and \({\varvec{\Theta }}^{(4)}_1\) are rotated by \(90^{\circ }\) and \(180^{\circ }\), respectively. Jones [10] found similar patterns for vortices tilted by externally imposed shear.
We emphasize that the amplitude of the threedimensional representation of the vertical velocity dipole \({\varvec{w}}_1\) is weaker compared to the asymptotic one (0.37 m/s vs. 0.40 m/s in the present case of Fig. 7) which could, at least partially, explain the deviation of precession time scales found in Fig. 6. Reasons for that may be found in the discretization scheme of EULAG introducing special numerical viscosity [29].
Remarkably, due to the fact that the primary circulation velocity, \({u}_\theta ^{(0)}\), is time independent in the adiabatic case, the asymptotic problem reduces to linear Hamiltonian dynamics with time independent coefficients. The time evolution of the entire flow field is nevertheless nonlinear because of the alignment of the vortex within the solutiondependent tilted cylindrical coordinates. Whereas we can superimpose two centerline solutions of the homogeneous centerline equation to obtain a third one for given \({u}_\theta ^{(0)}\), we cannot not do so for the associated entire flow fields because the superimposed leadingorder velocity would not be axisymmetric with respect to the superimposed centerline.
Furthermore, readers familiar with earlier studies on vortex resilience against vertical shear (see, e.g., [31, 33]) may appreciate an assessment of the importance of resonance damping in the present case. This mechanism can stabilize a vortex against vertical shear and induces vertical selfalignment in an unsheared environment. It relies on a resonance between vortex precession and the propagation of vortex Rossby waves which arises at the critical radius. At this radius, the turnover time associated with the primary circulation matches the period of the vortex precession. The strength of the damping is a function of the radial gradient of (potential) vorticity at the critical radius. The present setup is based on the Gaussianshaped vorticity distribution from (55) with \(\sigma = 1/{94.5}\,{\hbox {k}\hbox {m}}\). The critical radius lies at \(r_{\mathrm{cr}} \approx {950}\,{\hbox {k}\hbox {m}}\), so that \(\exp (\sigma ^2 r_{\mathrm{cr}}^2) \approx e^{100}\), and with it the radial gradient of vorticity, is extremely small. Therefore, the resonance damping mechanism is not active in the present case.
6.2.2 Stagnation
The stagnation test follows the idea that the choice of \(\theta _0 = \pi /2\) for the diabatic heating pattern (50) leads to deceleration of the centerline precession by canceling the term \({\varvec{\varPsi }}\) in (3). Furthermore, such a heating structure has no impact on the leadingorder tangential velocity: With purely asymmetric heating, the terms involving the axisymmetric vertical and radial motions \(u_{r,00}\) and \(w_{0}\) vanish identically and the tangential velocity time evolution becomes
Considering that \(u_{r.*}\) is the projection of the diabatic heating onto the tilt vector,
we realize that the righthand side of (63) vanishes, too, due to orthogonality of \(\partial _z {\varvec{X}}\) and \(\hat{R}_{\pi /2} \partial _z {\varvec{X}}\).
By construction, inserting the heating \({\varvec{Q}}_{1, \pi /2}\) into (37) also satisfies \({\varvec{w}}_1 \equiv 0\) to leading order. Figure 8 shows the expected behavior for the threedimensional simulations. \({\varvec{Q}}_{1, \pi /2}\) attenuates the vertical velocity by a factor of \(1/5 \approx 0.66\, \delta \), i.e., it eliminates the leadingorder dipole of w visible in the left panel of Fig. 7. The residual depicted in the right panel of the present Fig. 8 is thus the result of higherorder asymptotic contributions and, possibly, additional numerical truncation errors. Our main point here is that with the cancellation of the circumferential Fourier mode one of the vertical velocity, both the influence of tilt and heating onto the vortex strength as represented by \(u_{r,*}^{(2)}\) in (34) and the selfinduced vortex motion due to \(\varPsi \) explained in Sect. 5.4, vanish identically. These conclusions are supported by the time independence of the primary circulation strength seen in Fig. 9 and by the substantial slowdown of the centerline precession in Fig. 10, respectively.
As discussed above, the leadingorder tangential velocity is not affected by asymmetric heating of the given orientation. Figure 9 presents both the time series of maximum tangential wind (dark/blue) and of maximum heating (light/red). Only small variations in the tangential velocity are apparent and, as we will see in the next subsection, this changes substantially when we alter the orientation of \({\varvec{Q}}_{1, \theta _0}\).
Experiments (not shown here) revealed instabilities caused by small perturbations due to discretization errors: The diabatic heating will exhibit large amplitudes where the local tilt \(\partial _z {\varvec{X}}\) (being result of the reconstruction (61)) is large. This affects the local velocity and as a consequence the spectral properties of the Hamiltonian of Eq. (54) (projecting onto higher frequency modes) increasing smallscale oscillatory features of the centerline \({\varvec{X}}\). Hence, we need make sure to maintain a certain regularity of \({\varvec{X}}\) to avoid obscuring the effect under consideration by triggering this feedback loop. We achieve this by restricting the heating to a concentrated pulse by applying a timedependent amplitude factor of the shape
For the current setting \(a=1\), \(b=1.5 \text{ days }\) and \(c=\frac{1 \text{ day }}{2\sqrt{2 \ln 300}}\).
Note that the proportionality of the heating strength to the vortex tilt mentioned above is an artifact of the present idealized choice of a heating pattern, which was not chosen to be realistic, but serves the purpose of testing our hypotheses. When latent heat release from convection will replace the present heating structure in subsequent studies, we expect a more robust response.
The heating distribution \({\varvec{Q}}_{1,\pi /2}\) constructed in such a way guarantees \(u_{r,*}\) to vanish (cf. 64) and by canceling \({\varvec{w}}_1\), the term involving the vector \({\varvec{\varPsi }}\) is cancelled from the centerline equation of motion (3). As discussed in the introduction of Sect. 6, we are initially in the regime of gradientwind balance. Hence, \({\varvec{M}}_1\) remains the only contribution to the centerline equation of motion. Numerical evaluations show that the amplitude of \({\varvec{M}}_1\) is about 1/6 of the amplitude of the \({\varvec{\varPsi }}\) in the adiabatic case which is why we observe a significant slowdown of the centerline precession in the asymptotic case in Fig. 10, left panel, during the heating time interval (cf. Fig 6 for reference). Although not as prominently as with the asymptotic simulation due, likely, to the aforementioned discretization errors, the centerline also slows down in the threedimensional simulation. It also becomes transparent that, due to the heating, the shape of the centerline acquires shortwavelength features that are excited due to a slight misalignment of the diabatic heating dipole.
6.2.3 Intensification
In Sect. 5.4 we constructed a dipolar heating pattern and, by aligning its orientation \(\theta _0=\pi /2\) relative to the centerline tilt, we found stagnation, i.e., deceleration of the centerline precession as well as suppression of the vortexinduced vertical velocity dipole. However, what is the influence of this heating dipole when oriented with \(\theta _0 = \pi \) relative to the tilt? \(\theta _0=\pi \) determines the sign of (64) to be negative and as a consequence the righthand side of (63) positive, i.e., leading to intensification. To examine the effect of the orientation of a dipolar heating pattern without altering its amplitude by the dependency of \({\varvec{Q}}_{1,\theta _0}\) on \(u_\theta \) and \(\partial _z {\varvec{X}}\) we fix \(u_\theta \) and \(\frac{\partial {\varvec{X}}}{\partial z}\) in (50) to values at \(t=1\,\hbox {day}\) but still keep track of the orientation of \(\frac{\partial {\varvec{X}}}{\partial z}\).
In contrast to Fig. 9, where we did not see any sizeable impact of diabatic heating on the mean circumferential velocity, Fig. 11 displays a clear increase for both the asymptotic and threedimensional equations. We notice that the intensification is more effective for the asymptotic simulation (\(\sim \) 3 m/s) than it is for the threedimensional simulation (\(\sim \) 1.5 m/s), even though the heating amplitude of the asymptotic simulation is tuned to meet the threedimensional simulation. This may be caused by insufficiencies of the centerline reconstruction method used for the threedimensional simulations ultimately leading to nonoptimal alignment of the heating dipole.
From the discussion in Sect. 5.4 we concluded that a heating pattern with orientation \(\theta _0 = \pi \) causes the centerline tilt to increase. This behavior is found valid in the current experiment as seen in Fig. 12. Both simulation approaches, asymptotic and threedimensional, exhibit stretching of the centerline during the phase of active heating. Since the increase in tangential velocity is not as efficient for the threedimensional simulation as it is for the asymptotic counterpart, after the period of heating, the centerline precesses with higher angular frequency in the asymptotic case. Again, this may be due to inaccuracies in the threedimensional simulation when computing and aligning the heating dipole.
In Sect. 5.2 we discussed the intensification/attenuation mechanism as result of the effective vertical velocity dipole resulting from the superposition of both adiabatic and diabatic contributions. In the current situation the diabatic vertical velocity dipole is oriented by \(\theta _0=\pi \) while the adiabatic dipole is oriented by \(\pi /2\). Both are of comparable amplitude, hence, we expect an angle of \(3\pi /4\) between tilt and resulting vertical velocity which is verified by results shown in the right panel of Fig. 13.
Although the intensification is rather weak, there is evidence that the orientation of the asymmetry of the diabatic heating matters for the evolution of the circumferential velocity. In fact, by allowing for stronger heating by increasing the duration of the heat pulse the effect on the tangential velocity is stronger (see Fig. 14), but it also affects the structure of the centerline in a more profound manner.
6.2.4 Attenuation
The final experiment of this work consists in switching the heating dipole pattern to a configuration where we expect attenuation of the vortex and vertical alignment of the centerline. Following Eq. (64) attenuation corresponds to \(\theta _0=0\), i.e., positive heating in the direction of the centerline tilt. Again, tilt amplitude and tangential velocity of Eq. (50) are set to values at \(t=1\) day to avoid nonlinear feedback and restricted by an amplitude factor to act over a short interval only.
Taking into account that initially, the velocity profile has uniform vertical structure in the tilted polar coordinate system (r, z), and that the heating dipole is constructed such that it lowers the tangential velocity mainly in regions at midaltitude, whereas regions above and below remain essentially unaffected, straightforward evaluation of the global maximum of \(u_\theta \) would not be insightful. To remedy this issue, in Fig. 15 we plot the quantity
as a proxy for the intensity change of the vortex. Both simulation approaches show the reduction in tangential wind speed due to asymmetric diabatic heating, now in remarkable congruence.
Finally, in Fig. 16 both, asymptotic and threedimensional simulations show how the centerline aligns when forced by the attenuating heating. In the same way as before, we see small deviations in the precession time scale and excitation of higherorder oscillatory modes but otherwise good agreement between the asymptotic and threedimensional experiments.
6.3 Energetics of asymmetric diabatic heating
In this section we want to analyze the energy budget of the vortex in some more detail following the theory on available potential energy (APE) by Edward Lorenz. To this end, we shall consider the seminal work of Lorenz [14] who includes a detailed account of why only part of the generation of potential energy by diabatic heating is available for conversion into kinetic energy. He provides explicit expressions for the total APE generation, G, and its conversion into mechanical energy, C, as a function of the current flow state and the diabatic heating rates. The relevant formulae are provided in Appendix D, Eqs. (76c) and (76d), respectively, and it is shown there how we use (85), (87e), and (88) to numerically evaluate these expressions in the present Cartesian coordinate system while Lorenz works with pressure coordinates.
We begin by analyzing the extended intensification experiment presented in Sect. 6.2.3. Figure 17 shows energy rates of change, in red with the scale on the right, and the corresponding time integrated amounts of energy, in blue with the scale on the left of the diagram. The solid black line reveals the increase in total kinetic energy, \(\varDelta E_{\text {kin}}\), whereas the dashdotted black line (both with the scale on the left) shows the accumulation of APE, labeled \(\varDelta \text {APE}\) in the figure. Both are defined as the difference between time t and some reference time, here \(t_0=1\,\hbox {day}\). The analysis demonstrates that the system’s kinetic energy increases as a consequence of the imposed asymmetric heating in line with our findings of Sect. 5.3. Furthermore, APE accumulation amounts to about one third of the total generation of APE which is seen in the dashdotted blue curve. This scaling is consistent with the asymptotic results which state that the accumulation of APE should be small of \(\mathscr {O}\left( \delta \right) \), with \(\delta = \sqrt{\varepsilon } \sim 1/3\) in the present case.
The dashdotted and dashed blue curves represent \(\int G\,{\mathrm{d}}t\), the total generation of APE, and \(\int C \,{\mathrm{d}}t\), the total conversion into kinetic energy. The corresponding rates, G and C, computed by the formulae (85) and (87e), are shown as the dashdotted and dashed red curves, respectively. Appendix D shows that, to leading order in the asymptotics, C obeys the same expression as G when one relates the vertical velocity to the diabatic heating rate via the weak temperature gradient approximation (cf. 85 and 88b). For a more detailed comparison, Fig. 17 also shows the leadingorder asymptotic expression for the conversion rate, \(C^{{\mathrm{LO}}}\) (dotted, red), which is evaluated on the basis of (88a) in Appendix D. Interestingly, one obtains this formula for the APE conversion rate from that for the total generation rate by replacing the heatingrelated term \(\frac{Q_\Theta }{\partial {\overline{\Theta }}/\partial z}\) in the expression for G with the vertical velocity w from the 3D simulation. This corresponds to replacing the leadingorder asymptotic result for w in (35) (see 119 for the general expression in physical dimensions) with the vertical velocity generated in the full 3D dynamics.
Figure 17 shows that \(C^{\mathrm{LO}}\) approximates C rather well throughout the entire \(6\,\,\hbox {days}\). In fact, it does so much better than one may expect from an \(\mathscr {O}\left( \delta \right) \) asymptotic truncation error. Furthermore, the time integrated change of kinetic energy, \(\varDelta E_{\text {kin}}\), and \(\int C^{\mathrm{LO}}\,{\mathrm{d}}t\) are practically equal as one may expect from Lorenz’ theory, i.e., \(\dot{E}_{\text {kin}} \simeq C^{LO}\). Finally, the difference between total APE generation, \(\int G \,{\mathrm{d}}t\), and kinetic energy change, \(\varDelta E_{\text {kin}}\), approximately equals the accumulated APE, \(\varDelta \text {APE}\), that is, \(\frac{\mathrm{d}}{\mathrm{d}t} \text {APE} \approx G  C\) as postulated by Lorenz, see (76a).
These findings primarily confirm, that the generation and conversion of APE are equal up to leading order (remember that \(\delta \approx 0.3\)). Next they reveal that APE does accumulate at first order and that the amount of APE accumulation is accounted for by the difference between G and \(C^{\mathrm{LO}}\). The only difference in the formulae used to evaluate these two quantities is that the vertical velocity w in (88a) is either evaluated by its leadingorder asymptotic approximation in (119) (for G) or taken directly from the 3D simulation (for \(C^{\mathrm{LO}}\)), respectively. The fact that only \(C^{\mathrm{LO}}\), using w from the simulation, closely reproduces the time derivative of kinetic energy suggests that the actual vertical motions are the central driver for the observed intensification rather than the heating itself. G on the other hand, assuming that \(Q_\Theta \) is translated into vertical motions via (119) introduces an offset which is accounted for by creating APE. Hence, the difference is caused by deviations from the weak temperature gradient approximation which is satisfied only at leading order. Further evidence is gained by considering the tail of C (or \(C^{\mathrm{LO}}\)) for \(t>4\,\,\hbox {days}\). Even without diabatic heating, conversion between APE and kinetic energy continues as the energy, intermediately stored in APE and essentially accounting for imbalances, now is getting converted (back and forth) to and from kinetic energy mediated by the vertical motions.
In addition, similar to the intensification experiment (cf. Fig. 17), Fig. 18 demonstrates the reduction in integrated kinetic energy due to the attenuating configuration of the heating dipole. In contrast to the previous experiment, we now find almost perfect agreement of all conversion rates and consequently the impact \(\varDelta \text {APE}\) is negligible. This once more reveals the stable nature of this configuration driving the vortex toward the solution of zero tilt and zero velocity.
Remark
(item 5): These considerations lead us to conclude that it is not the diabatic heat release which is primarily important for changes in kinetic energy but the resulting vertical motions and their correlation with temperature perturbations rooted in the structure of the tilted vortex. This settles our statement posed in the introduction in item 5.
6.4 Summary of tilt dynamics
As a final result, we present a quantitative analysis of the effect of asymmetric diabatic heating on the centerline tilt by considering the time series of tilt amplitude as measured by an \(L_2\)norm:
In Fig. 19 this quantity is plotted for all types of experiments performed in the course of this work (except the extended intensification experiment). The left graph shows the results from the evaluations of the asymptotic theory, and it confirms that the orientation of the diabatic heating dipole correlates with changes in the tilt amplitude. Aligning the heating dipole with the tilt (attenuation case) leads to decreasing tilt, i.e., the vortex tends to align vertically. This situation turns around if the heating dipole is rotated by \(180^{\circ }\) (intensification case) in that the tilt further increases. Although in the stagnation configuration (heating dipole is 90 degrees rotated relative to the tilt) neither the primary circulation nor the tilt magnitude are affected by the heating/cooling pattern, the centerline precession is slowed down during the period of active heating/cooling as predicted by the theory.
The graph on the right in Fig. 19 reveals that the threedimensional representation of the initial data does not project as well onto the first eigenmode of the centerline as in the asymptotic cases. Besides features which oscillate on a time scale of less than one day, all threedimensional simulations exhibit an additional systematic trend on the scale of roughly six days. However, in every case, asymptotic and threedimensional, the effects of activating an asymmetric diabatic heating dipole are superimposed onto the adiabatic reference. The stagnation simulation follows the adiabatic reference on both cases but exhibits slight distortions in the threedimensional case. As discussed before, this is the reason for restricting the diabatic heating to a short time period as this effect would increase for longer periods. However, for both experiments, intensification and attenuation, the tilt amplitude follows approximately the adiabatic reference curve albeit on increased and lowered levels, respectively, once the short time period heating activity has passed.
We argue that the response to asymmetric diabatic heating in the threedimensional simulations is not as direct as it is for the asymptotic analog simulations. This is probably due to imbalances excited through the diabatic heating, e.g., by misalignment of the heating dipole, and limits the efficacy of the heating in (50) in influencing tilt and circumferential velocity. Nonetheless, in all cases the intensification configuration increases the centerline tilt while the attenuation configuration decreases it.
Remark
(items 3 and 6): The analysis of this section corroborates our initial statement in (item 3) of the introduction which implies that the orientation of a purely dipolar diabatic heating pattern intensifies and shears an atmospheric vortex apart in the antiparallel orientation of heating dipole and tilt while it attenuates and aligns the vortex vertically in the parallel orientation. They also justify our claim in (item 6) that the predictions of the asymptotics are fully corroborated at a qualitative level by threedimensional simulations. As regards quantitative discrepancies, these are largely within the expected error of just a leadingorder asymptotic solution with values of the expansion parameter \(\delta \sim 1/3\).
7 Conclusions and outlook
With the present work we have extended the results of Päschke et al. [26] to vortex Rossby numbers larger than unity corresponding to wind speeds of \(\mathcal {O}(30\,\,\hbox {m}\big /\hbox {s})\) and showed that the principal structure of the reduced asymptotic model equations is retained in this limit. We found that the validity of the equations holds from the gradientwind regime up to (modest) cyclostrophic vortices. This corresponds to hurricanes of strength H1 on the SaffirSimpson scale.
Current models of tropical cyclone intensification rely on organized symmetric heating in an upright vortex [ [20], and references therein]. Observations show, however, that in incipient tropical storms the level of organization of convection is weak compared to that in fully developed hurricanes so that, in the former, patterns of convection can be strongly asymmetric relative to the nearly axisymmetric primary circulation. Our findings of leadingorder effects of asymmetric diabatic heating on both the strength of the primary circulation and on the vortex tilt may therefore add new insights into possible routes of acceleration from tropical storms to fully developed hurricanes.
Here, we have focused on highlighting the potential effect of purely asymmetric diabatic heating on the mean tangential velocity for an initially tilted vortex. This effect, we argued, can be of the same order of magnitude in the presence of strong vortex tilt as that of symmetric diabatic heating. Although the gain of horizontal wind speed was limited in our threedimensional simulations as we restricted to idealized heating configurations that maintain the overall flow structure which the asymptotic analysis was based on, we expect to see potentially stronger efficiencies in nature due to the selfregulation of moist convection in a sheared environment.
The effects of moist thermodynamics have been replaced here by artificial diabatic source and sink terms neglecting effects of water phase transitions. As we argued, the resulting asymmetric pattern of vertical velocity is the driver for the intensification/attenuation mechanism found in this study. In ongoing research, to be published elsewhere in the future, we find that mean vertical mass fluxes of an ensemble of convective towers can provide for similar effects. In fact, the heatinginduced vertical velocities from the present study would be replaced in this case by convective mass fluxes averaged over an ensemble of convective towers, while the rest of the theory remains largely unchanged. Thus, despite its somewhat artificial setup, the present study does reveal a potentially interesting physical mechanism.
In this work we did not discuss the interaction of environmental shear with the TC. However, observations of TCs just before rapid intensification often show a phase of relatively stationary configuration of background wind shear [39], vortex tilt, and asymmetries of convection [41]. In this situation, regions of strong convection develop downshear left from the vortex center [4, 32]. Taking into account that the shearinduced vortex tilt will lead to precession of the vortex in counter clockwise direction until shear and selfinduced precession tend to cancel each other [33, 44], we find that a downshear left position of convection is at the same time close to downtilt. Such a convection pattern will decompose into a mean axisymmetric and purely asymmetric components, the latter dominated by the first azimuthal Fourier mode. The symmetric part of the heating will, according to the present study (depending on the radial positioning of the heating center relative to the RMW), tend to accelerate the vortex, while it potentially aligns the centerline, i.e., reduces the tilt due to vertical advection (see second term on the right in Eq. 105). In contrast, the first Fourier asymmetric mode, according to the mechanism discussed in this paper, contributes an attenuating trend to the primary circulation while (also) working to reduce the tilt and thus to efficiently stabilize the vortex against the outer shear (see Sect. 6.2.4 and Fig. 16). In this scenario, which is in line with a situation described by Wang and Holland [44], a sudden breakdown of the outer shear will interrupt the attenuating effect of the asymmetric part of convection (that depends on the tilt) leaving the field to the symmetric and intensifying part. In addition to stabilization due to asymmetric diabatic heating, selfalignment due to critical damping [31, 40] can play an important role in that scenario, depending on the vortex’ vorticity distribution. The result is a potential pathway for rapid intensification, to be studied in detail in future work.
Critical damping of vortex Rossby waves has been identified as one mechanism responsible for the observed resilience of atmospheric vortices against tilting by background shear and for their capability of vertical (re)alignment (see [31, 40], and references therein). Critical damping arises when, at some radius, the rotation rate of a linear perturbation eigenmode of the vortex matches with the local angular velocity associated with its primary circulation. To address this phenomenon within the framework of the present theory, two regimes must be distinguished. In the first regime, the critical radius lies in the far field QG region, in the second within the gradient wind vortex core. In the first case, critical damping should be accessible within the present theoretical framework, and it will arise as part of the outer QG flow solution. To this end, the inner solution for the vortex core in gradient wind balance should be combined with the critical layer theory for nearly axisymmetric quasigeostrophic vortices of Reasor and Montgomery [30] as the outer solution. This should be interesting material for one future study. To address the second case, for which the critical radius lies within the gradient wind part of the vortex, a more sophisticated asymptotic approach is needed: In this case, the angular velocity of the primary circulation at the critical radius is comparable to the vortex turn over time scale, which is much faster than the time scale of the QG outer flow considered exclusively in the present work. To simultaneously capture the combined dynamics on both time scales, techniques of multiple scales asymptotics will have to be invoked. This could give rise to a second interesting future study.
Ongoing research aims to extend the asymptotic theory to include realistic moist process submodels and multiscale convection. In this context, the present observations regarding the net effects of vertical motions (see discussion of Fig. 15) will become important in that mean convective mass fluxes will take the role of the present heatinginduced vortexscale vertical velocities. Another interesting avenue for future research will be a thorough thermodynamic analysis of the heating/convection–tilt interactions in the spirit of, e.g., Pauluis and Zhang [27].
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Acknowledgements
T.D., A.P., and R.K.’s work has been funded by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 “Scaling Cascades in Complex Systems”, Project Number 235221301, Project (C06) “Multiscale structure of atmospheric vortices”, and by the Helmholtz Society of Research Institutions for funding through the “GeoSim” Graduate College. The authors also thank the European Centre for Medium Range Weather Forecast for supporting this work by R.K’s ECMWF research fellowship as well as hosting T.D., A.P., and R.K. for research stays. The authors acknowledge the NorthGerman Supercomputing Alliance (HLRN) as well as the German Climate Computing Center (DKRZ) for providing HPC resources that have contributed to the research results reported in this paper. The authors also gratefully thank Olivier Pauluis, Remi Taullieux, and Mike Montgomery for many fruitful discussions which have helped strengthen our interpretation of the asymptotic results. T.D. and R.K. further thank Sundararaman Gopalakrishnan, Frank Marks, Paul Reasor, and Dave Nolan for their hospitality and insightful discussions during a research stay at NOAA/HRD and University of Miami in the spring of 2019. NCAR is sponsored by the National Science Foundation. Finally, the authors express their gratitude to the two anonymous reviewers who helped to forge the final version of this paper through their challenging, yet fruitful questions and remarks.
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Tom Dörffel, Ariane Papke, and Rupert Klein are responsible for the study conception and design. The computational code for solving the asymptotic limit model was developed by Tom Dörffel, adaptation of the EULAG solver and its postprocessing engine to the test cases was pursued by Ariane Papke and Tom Dörffel supported at critical stages by Piotr Smolarkiewicz, data analysis and display was were performed and realized by Tom Dörffel and Natalia Ernst, and the interpretation and evaluation of the results was in the hands of Tom Dörffel, Rupert Klein, and Piotr Smolarkiewicz. The first draft of the manuscript was written by Tom Dörffel and all authors commented on intermediate versions of the manuscript. All authors read and approved the final manuscript.
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Appendices
A Governing equations in the comoving coordinates
Transforming (5) to the vortexcentered coordinates from Sect. 3.2 using (19) and defining \({{\varvec{U}}}\equiv \delta \,{\partial {\varvec{X}} / \partial t}\) and \({{\varvec{u}}}_{\mathrm{rel}}= u_r\, {{\varvec{e}}}_r+ u_{\theta }\, {{\varvec{e}}}_{\theta }\) we find
B Full secondorder horizontal momentum balances
C Derivation of the kinetic energy budget (45)
We start from (43), which is equivalent to (4.21) in [26] for \(f_0 = 0\). This is verified straightforwardly by using \(u_{r,0}^{(2)} = u_{r,00}^{(2)} + u_{r,*}^{(2)}\). The equation is multiplied by \(\widehat{r}\rho _0 {u}_\theta ^{(0)}\), and we use the mass conservation law in the form of Eq. (40), i.e.,
to generate the advective transport terms of kinetic energy in conservation form. We let
and obtain
Focusing on the righthand side of this equation, we rewrite the first term as
The square bracket vanishes according to (70), while we observe that by combining the axisymmetric part of the hydrostatic balance in (30) with the equation of state in (33) to replace \((p^{(4)}/\rho _0)_z\), and (35) to replace \(w_0^{(0)}\) one finds
Next we rewrite the second term on the right of (72) using the definition of \(u_{r,*}^{(2)}\) in (34), and the first Fourier modes of the vertical momentum balance in (36) to find
To obtain the second equality we have used the asymmetric WTG law from (37) and the fact that the second term in that equation contributes a component to \({\varvec{w}}_1\) that is orthogonal \(\partial {\varvec{X}}^{(0)}/\partial z\), and thus also orthogonal to \({\varvec{\Theta }}_1^{(4)}\).
Insertion of (73)–(75) generates the desired equation (45).
D Relation to Lorenz’ theory of available potential energy
We want to deepen the insight of the proposed asymmetric diabatic heating mechanism by finding connections to Lorenz’ concept of available potential energy (APE) [14]. As we will see, under certain assumptions the kinetic energy generation of Eq (45) is identical to Lorenz’ expressions of APE generation and conversion to kinetic energy. To this end, we start with Eqs. ( 16), (20), (17), (18) of Lorenz [14]:
Here, \(\overline{A}\) is the average available potential energy, \(\overline{K}\) the mean kinetic energy, C the conversion rate between APE and kinetic energy, G the generation rate of APE (all per unit area), \({p_{\text {s}}}\) the surface pressure, \(\omega = {\frac{\mathrm{d} p}{\mathrm{d} t}}\) the vertical velocity in pressure coordinates, \(\overline{\varGamma }= {\frac{\partial \overline{T}}{\partial z}}\) the lapse rate of mean temperature, \(\varGamma _d=g/c_{\mathrm {p}}\) the dryadiabatic lapse rate, \(Q = c_{\mathrm {p}}\pi Q_\Theta \) the diabatic source term with the Exner pressure \(\pi = \frac{T}{\Theta } = \left( \frac{p}{{p}_{\mathrm{ref}}} \right) ^{R/c_p}\), \(\overline{(\cdot )}\) the horizontal mean on constantpressure surfaces, and \({(\cdot )}'\) the deviation from the mean. In our case we neglect friction, hence Eq. (21) of Lorenz [14] is trivially zero.
First, we need to point out that (independently of our asymptotic theory) every quantity q(x, y, p, t) defined as field in pressure coordinates can be expressed as the sum of mean and perturbation,
where trivially \(\overline{{q}'} = 0\). By definition, there are no pressure perturbations on constantpressure surfaces \({p}' = {\pi }'=0\). This allows us to express Q conveniently as
Together with \({T}' / \overline{T} = {\Theta }'/ \overline{\Theta }\) and assuming the atmosphere to be hydrostatic (\({\frac{\partial p}{\partial z}} =  g \rho \)) we find \(\overline{\varGamma }= \overline{\pi {\frac{\partial \Theta }{\partial z}}} + \varGamma _d\). The generation rate G becomes
This expression involves the integration of horizontal means on surfaces of constant pressure. For comparing this expression with the findings of our asymptotic analysis, with means and perturbations defined on levels of constant height, we consider the following approximation. For some locally defined physical quantity q with representation q(x, y, p) in pressure coordinates, we introduce its representation in Cartesian space coordinates by
where \(p^c(x,y,z) = Z^{1}(z;x,y)\), Z(p; x, y) is the geometrical height as a function of pressure as given by the pressure coordinates, and \((\ )^{1}\) denotes the inversion of the \(z \leftrightarrow p\) relationship at frozen horizontal location (x, y). Let furthermore
denote the mean pressure on a surface of constant height z and its inversion, respectively. Then, taking into account that pressure fluctuations in horizontal planes are small, we use firstorder Taylor expansion in z to obtain
where \({Z}' = Z(p;x,y)  \overline{Z}(p)\) and, by the definition of Z(p; x, y) given above,
is the local pressure fluctuation in the Cartesian system. The findings of Sect. 4 reveal that we can neglect terms of order \(\mathscr {O}\left( p'\right) \) and higher at the cost of introducing an error of order \(\mathscr {O}\left( q \delta ^4\right) \), i.e., about \(1\%\) of the magnitude of q. This is a consequence of the fact that pressure is mainly vertically stratified and that the flow is essentially hydrostatic. These considerations justify approximating the average on constantpressure surfaces \(\overline{(\cdot )}\) by
where \(\varOmega \subset {\mathbb {R}}^2\) is the appropriate surface area to average over. With these results we approximate the total APE generation in (79) by the integrals in Cartesian coordinates, substituting \({\mathrm{d}}p =  g \rho \,{\mathrm{d}}z\) to obtain, up to higherorder terms (h.o.t.),
now with perturbations on surfaces of constant height. This expression approximates the original formulation (Eq. 76c) without further assumptions on the structure of \(Q_\Theta \).
Using the equation of state \(p = R T \rho \) in the definition of the APE conversion term C in (76d), we find
This integral gives the source term of (global) kinetic energy production (neglecting friction). \(\omega :=\frac{{\mathrm{d}}p}{{\mathrm{d}}t}\), however, turns out to be difficult in handling numerically if data is given on Cartesian coordinates as it requires tracing pressure changes along fluid trajectories in space and time. For achieving sufficient accuracy rather high temporal resolution would be necessary. We bypass this problem by the following derivation: Assuming a hydrostatic atmosphere this term reads in Cartesian coordinates as
Here we have used, in this sequence, the hydrostatic p–z transformation and the definition of the material derivative dp/dt (a \(\rightarrow \) b), the (vertically integrated) hydrostatic relation for pressure (b \(\rightarrow \) c), and mass continuity (c \(\rightarrow \) d). Then, the first term under the integral in (87d), owing to Gauß’ theorem, vanishes with the surfacetovolume ratio of \(\varOmega \) and is neglected in the sequel, while upon integration in \(z'\) the second term cancels the fourth, leading to the final result in (87e) which requires only spatial derivatives at constant time.
To relate (87e) to the asymptotic expression for the source term in the kinetic energy balance equation (45), we observe that the integrand in (87e) is the dimensional form of the r.h.s., \(\left( u^{(2)}_{r,00} +u_{r,*}\right) \partial p^{(4)}/\partial {{\widehat{r}}}\), of the preliminary version of the kinetic energy balance in (72). Repeating the derivations of the axisymmetric and Fourier mode 1 contributions to this term in (74) and (75), respectively, or using the general asymptotic expression for the vertical velocity in (119), we find
Note that \({Q}'_\Theta \) appears in the APE generation expression G in (85) as opposed to \(Q_\Theta \) in the expressions for APE conversion C in (88b). This difference implies an additional contribution to C relative to G which accounts for a coupling to the mean heating \(\overline{Q}_\Theta \). In the cases studied here, however, the diabatic heating has zero mean, so that (85) and (88b) match in these cases. We conclude that—at leading order in the asymptotics—there is no accumulation of APE since generated APE is immediately converted into kinetic energy. This establishes the desired link between the asymptotic kinetic energy budget in (45) and Lorenz’ APE theory.
E The centerline equation of motion
As preparation for Appendix F we give more details on the centerline equation of motion (3) and provide all the necessary information to close the system of equations (in conjunction with (1)).
We will present the missing terms necessary to close equation (3) in Sect. 7, provide a split into adiabatic and diabatic contributions in Sect. 7 and further deepen the analysis of this equation in Sect. 7 for being able to construct a stable and efficient numerical scheme to solve the asymptotic equations.
1.1 E.1 Formulation of Päschke et al. (2012)
In the course of this work we argued that the equations derived by Päschke et al. [26] are valid in the vanishingCoriolis case, i.e., \(f_0 \rightarrow 0\). The structure of the equations stays essentially the same, only terms proportional to \(f_0\) are to be dropped.
For completeness below we present the remaining expressions for \({\varvec{M}}_1\) and \({\varvec{\varPsi }}\) necessary to solve the system (1) and (3):
\({\varvec{\varPsi }}= L[{\varvec{{\varvec{{\mathscr {{{K}}}}}}}}] = {\hat{R}}_{\pi /2} L[{\varvec{{\mathscr {{{K}}}}}}_1]\) depends on the following expressions:
where \(L[\cdot ]\) is an integral operator,
and \({\hat{R}}_{\theta _0}\) denotes the matrix of twodimensional rotation by an angle \(\theta _0\).
Equations (90a–90g) involve the following expressions,
which are resolved in terms of \(u_\theta \), \({\varvec{X}}\), \(Q_0\) and \({\varvec{Q}}_1\). The expressions above give rise to the reformulated equation of centerline motion:
1.2 E.2 Split into diabatic and adiabatic contributions
To reveal more clearly the structure of the centerline evolution equation, we recall the formula (37) for the vertical velocity Fourier modes, which separates diabatic from adiabatic effects. Rewriting this formula in the dipole vector notation we have
and realize that the adiabatic part is a linear (differential) operation on \({\varvec{X}}\):
From the equations in the previous subsection we see that by linearity of the expressions in \({\varvec{w}}_1\) we can assemble \({\varvec{{\mathscr {{{K}}}}}}\) (and ultimately \({\varvec{\varPsi }}\)) by linear superposition of linear operations on \({\varvec{X}}\) (operators are symbolically denoted by \(\hat{~}\)) and in general nonlinear diabatic expressions.
\({\varvec{{\mathscr {{H}}}}}_1\) then becomes:
We first need to evaluate the expression for \({\varvec{W}}_1\),
to split \({\varvec{{\mathscr {I}}}}_1\) accordingly:
Together with \({\varvec{M}}_1\)
we get the following split for \(\tilde{{\varvec{{\mathscr {I}}}}}_1\):
Performing the split of \({\varvec{R}}_1\),
\({\varvec{\phi }}_1\) divides into
and therefore \({\varvec{{\mathscr {{J}}}}}_1\) into
For \({\varvec{Q}}_1\) we find the simple shorthand
Our final result is identifying three different contributions to \({\varvec{\varPsi }}\), a (generally nonlinear) diabatic term, an advective term, and a linear Sturm–Liouvilletype operator acting on \({\varvec{X}}\):
1.3 E.3 Characteristic structure of the vortex centerline equation
We continue rephrasing the original centerline tendency equation to further emphasize its structure. By trivially identifying \(\mathbb R^2\) with \({\mathbb {C}}\) we symbolically transform twodimensional vectors \({\varvec{a}} = (a_x, a_y) \in {\mathbb {R}}\) to \(a = a_x + i a_y \in {\mathbb {C}}\). Operations such as (\({\varvec{k}} \times \cdot )\) and \({\hat{R}}_{\pi /2}\) become multiplications with i. By that we can identify a substructure of the equation to be of Schrödingertype, and advective contribution and a source term, generally dependent on X, \(u_\theta \), and the coordinates (r, z, t) but without any further specification:
By identifying structural components, the centerline equation takes the form
Note that the lefthand sides of Eq. (107) resembles an advection operation while the righthand side involve linear and nonlinear source terms.
Päschke et al. ([26], section 6.2) pointed out that the adiabatic time evolution of the vortex centerline poses an eigenproblem. For that case the (homogeneous) centerline equation in the complex plane is written as
As the spectrum \(\omega _k\) of \({{\hat{H}}}\) is real, (108) can be interpreted as a Schrödingertype equation, hence eigenmodes \(X_k\) precess with the angular frequency \(\omega _k\) in the complex plane. For the adiabatic problem we therefore find, that
For the numerical experiments presented in Sect. 6 we use the first nontrivial eigenmode for initialization corresponding to the nonzero eigenvalue with the smallest magnitude.
F Numerical scheme for asymptotic equations
By providing a closure \((Q_0, {\varvec{Q}}_1) = F({\varvec{X}}, u_\theta , t)\) Eqs. (3) and (1) form a closed set of partial differential equations which in general cannot be solved analytically. Thus, we further analyze the structure of these equations seeking for an adapted numerical method to allow for efficient and stable time integration.
Weber [45] first presented a numerical scheme for solving the coupled system (3) and (1). While the author followed a methodoflines approach discretizing the spatial derivatives by fourthorder approximations to solve the resulting system of ordinary differential equations by generic integrators, it turned out that for certain settings it becomes stiff, and we therefore try to take advantage of the equations’ structure revealed in Appendix 7 to allow for a more efficient time integration by using adapted methods. We revisited all the equations presented by Päschke et al. [26] needed for closure and further performed the split of (3) into linear and nonlinear contributions which lead to a quasiHamiltonian substructure giving rise to the numerical scheme to be presented Sect. 7. In addition to the centerline equation (107) time evolution of the tangential velocity is rewritten as
to identify an advection term (in polar coordinates) on the lefthand side and source terms on the righthand side. The integration scheme of this equation is presented in Sect. 7.
By our analysis we learned that the system of equations (107) and (110) is assembled by prototypes of partial differential equations that are (1) advection equation, (2) Schrödinger equation, and (3) nonlinearly coupled ordinary differential equations (source terms).
Aiming at the scope of this work we restrict to asymmetric diabatic heating which allows to drop all terms referring to symmetric vertical and radial motions. We further neglect background wind shear. Hence, we drop all the advective terms from Eqs. (107) and (110) and set \(u_s=0\).
1.1 F.1 Integration of centerline
For the integration of the centerline position the general solution strategy is to integrate nonlinear source terms by the trapezoidal rule and the Sturm–Liouville operator (after appropriate spatial discretization) by the implicit midpoint rule. The choice of the latter is based on the principle of preserving unitarity during integration of the linear part of the equation. We also dropped contributions according to shear as they are not discussed in Sect. 6. The composition of the substeps reads:
\(\mathbbm {1}\) represents the identity operator. For the evaluation of \({{\hat{H}}}\) we need \(u^{n+1/2}\) which we obtain by the firstorder predictor \(u^{n+1/2} = u^*\). To maintain stability the timestep is completed by implicitly solving for \(X^{n+1}\) taking also \(u^{n+1}\) into account which is solved in the subsequent subsection.
1.2 F.2 Integration of tangential velocity
After canceling the advection part of Eq. (110), as \(u_{r,00}\) and \(w_0\) vanish, only the righthand side source term proportional to \(u_{r,*}\) remains which is treated numerically by applying the trapezoidal rule. The integration scheme for one timestep of size \(\varDelta t\) reads
For brevity we dropped the \({}_\theta \) subindex. Same as for Eq. (111c), it becomes obvious that the final step of the integration involves an implicit solution strategy as the term \(u^{n+1}_{r,*}\) depends on both X and u at time level \(n+1\).
1.3 F.3 Coupled integration
In both, explicit and implicit forcing, the equations are couples to each other. The latter requires information of X and u at time \(t^{n+1}\). A common procedure (c.f. [1, 42]) solving problems of this kind is to formulate it as a nonlinear fixedpoint problem and lagging \(X^{n+1}\) and \(u^{n+1}\) one iteration behind when evaluating Q and \(u_{r,*}\):
1.4 F.4 Details on the spatial discretization
The equations are discretized on an equidistant grid allowing for straightforward finitedifference approximations of the derivate operators. Boundary conditions are accommodated by extending the grid covering the physical domain plus a ghost layer of two cells. Solution values are stored in cellcenters while first derivate are computed typically on the corresponding faces.
Prototypical differential expressions such as \(\alpha \partial _z(\beta \partial _z\psi )\) are discretized as:
Integrals are computed via the trapezoidal rule where ghost cells are obtained by quadratic extrapolation. We further include an option to apply hyperviscosity to the centerline stabilizing time integration for activated diabatic heating. Further details may be taken from the source code available on demand by the corresponding author.
G Details on the numerical implementation
1.1 G.1 Dimensional variables
Though the derivation outlined above is carried out in terms of nondimensional variables, for the actual implementation into EULAG we used dimensional quantities of which some details will be presented in this section. In the spirit of asymptotic analysis, for reconstructing dimensional variables and to formulate leadingorder relations we use leadingorder or nexttoleadingorder modes. The following formulas are expressed for nonvanishing Coriolis parameters in the gradient wind regime. However, they are also valid in the cyclostrophic regime by setting \({f}_{\mathrm{ref}} = 0\).
Before presenting specific relation which arise from the asymptotic analysis, we want to relate the expansion modes with mean background values, denoted by bars \(\overline{(\cdot )}\) and perturbations, denoted by primes \({(\cdot )}'\).
Furthermore, we have
1.1.1 G.1.1 Pressure
Equation (27a) balances pressure gradient with radial forces. We find the dimensional version as
1.1.2 G.1.2 Potential temperature
For the deviation of potential temperature from its background mean value \(\overline{\Theta }\) we have for the first Fourier modes
1.1.3 G.1.3 Vertical velocity
In general, i.e., for arbitrary heating, the vertical velocity takes the following form in physical dimensions:
1.1.4 G.1.4 Diabatic heating
Finally, we defined a heating dipole aligned with an angle \(\theta _0\) relative to the tilt direction:
1.2 G.2 Convergence results
In addition to comparing both, simulation results of threedimensional and asymptotic equations, we check for convergence of each numerical schemes. For both simulations we check for selfconsistency by comparing results at increasing resolution with highresolved reference. Convergence of the EULAG simulations is displayed in Fig. 20 showing secondorder convergence.
The numerical scheme solving the asymptotic equation on the other hand is tested by evolving
for \(T=0.1\) (in asymptotic units) and comparing solutions of different resolutions against a reference solution with 1280 grid points. The results are plotted in Fig. 21 indicating secondorder convergence as expected.
H Antistagnation test
In sec. 6.2, we presented and discussed the results of an adiabatic reference as well as tests showing the results of asymmetric diabatic heat release under different angles \(\theta _0\) between tilt and heating dipole. We named these experiments stagnation test (\(\theta _0=\pi /2\)), intensification test (\(\theta _0=\pi \)) and attenuation test (\(\theta _0=0\)). Here we present the antistagnation test with \(\theta _0=\pi /2\) resulting in a heating dipole that is positively correlated with the tiltinduced adiabatic vertical velocity dipole. In analogy to Figs. 8, 10, and 9, we present vertical velocity, centerline time evolution and the time series of maximum tangential velocity in Figs. 22, 23, and 24.
During the phase of active heating, the vertical velocity essentially doubles in amplitude (cf. Fig. 7) and the centerline precession frequency is enhanced (cf. Fig. 6). The maximum of tangential velocity, however, remains approximately constant. We obtained the same result from the stagnation test since, as here, the heating dipole is orthogonal to the tilt vector yielding \(u_{r,*}=0\) and therefore no tendency in the tangential velocity.
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Dörffel, T., Papke, A., Klein, R. et al. Dynamics of tilted atmospheric vortices under asymmetric diabatic heating. Theor. Comput. Fluid Dyn. 35, 831–873 (2021). https://doi.org/10.1007/s0016202100591x
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DOI: https://doi.org/10.1007/s0016202100591x