The Tropical Cyclone as a Divergent Source in a Background Flow

a. Observation, reanalysis, and full-physics model data

Infrared (IR; 10.7 μm) satellite observations from geostationary satellites are used. From the Geostationary Operational Environmental Satellite (GOES) line of satellites: GOES-13 and GOES-15; from the Himawari line: Himawari-8. The available temporal frequency of the scans is 30 min except for 2015 Joaquin, since we have access to rapid scan data and because it existed within the continental U.S. satellite scan area. Joaquin’s observations for its time period of interest range from 5 to 15 min. To augment the satellite data, we use the ERA5 (Hersbach et al. 2018) dataset whose horizontal grid spacing is 0.25° and is available every six hours. The wind fields of ERA5 are decomposed into divergent and rotational components using SPHEREPACK (Adams and Swarztrauber 1997).

Data from the idealized modeling simulations discussed in Part II and Part III are also used. Those data are from a slightly modified Cloud Model 1 (CM1; Bryan 2012), release 18. Full details of the simulations’ parameters can be found in Part II. A significant addition is the thermally balanced background specified for a prescribed environmental flow. Four simulations were run: three sheared and one nonsheared control. The model TCs are initialized using the standard Rotunno and Emanuel (1987) method, with a radial extent of 600 km, an initial RMW of 100 km, an initial surface maximum mean tangential wind of 20 m s⁻¹, and an initial maximum height of 10 km. The thermodynamic base state is the “moist tropical” sounding (Dunion 2011). The tropopause (N² = 2 × 10⁻⁴ s⁻²) is at 14 km. Sea surface temperatures (SSTs) are horizontally homogenous and invariant in time with a value of 27.5°C (Part I). An f plane at 20°N is used. Three background wind profiles are used: a Gaussian profile with a 13–1.5-km (proxy for 200–850 hPa) shear value of 7.5 m s⁻¹ (hence, G7.5), a cosine profile with a shear value also of 7.5 m s⁻¹ (C7.5), and an additional Gaussian profile with a shear value of 11.5 m s⁻¹ (G11.5). The shear is easterly and is initialized at the start of the simulation. The background environmental winds for the control are zero everywhere. For this paper, we focus on the G7.5 simulation and the control.

Multiple aspects of the simulations have already been discussed in Part II and in Part III, and we refer the reader to those articles for more details; however, we will provide a brief summary. The C7.5 and G11.5 TCs do not develop, the control TC develops, and the G7.5 TC develops. The G7.5 TC goes through two distinct intensification stages on its way to a Saffir–Simpson category-3-equivalent TC, as opposed to the control which continuously intensifies. The G7.5 TC exhibits tilt nutations of 9–10 h that exist on top of the larger, slower tilt precession. The magnitude of the nutations is approximately 15 km when analyzed using a center at 6-km height. After the initial 24 h during which the tilt grows, realignment occurs at 80 h into the G7.5 simulation, at which point the precession and the nutations both damp. Because of the outflow, vertical wind shear, calculated as the difference between winds at 13 and 1.5 km in a 200-km circle around the low-level center, reduces from 7.5 to 3.5 m s⁻¹. An abrupt rapid intensification follows at 90 h. The explicit intensity traces are shown in section 4.

b. Divergent shallow water model

The sink terms S in (2)–(4) are calculated to ensure that the specified vortex wind and height fields are exact solutions to the shallow water equations. This is critical to guarantee that any deviations we see from these fields is entirely a result of the background winds impinging on the vortex. We accomplish this task by simply brute-force numerically solving these equations for the sink terms after specifying the winds and the height of the vortex. It is important to note that we solve for the sink terms for the case in which U₀ = 0 so as to obtain the exact forcing needed to render the vortex an exact solution of the equations in the absence of the background winds.

The model is solved in time with the third-order Adams–Bashforth method. The spatial domain is solved spectrally using Fourier methods with the nonlinear terms evaluated using the nonlinear transform method of Orszag (1970). Because of the inherent periodicity of the spectral Fourier method, we employ a sponge boundary along all sidewalls to prevent reflections around the edge of the domain.

a. Observations and reanalysis review

Two fundamental findings from the original series were 1) the upshear movement and wobbling of the tilt as deduced from the cloud shield and as modeled explicitly in the CM1 and 2) the presence of a proximate upper-level anticyclone as the source of the shear. Four TCs are chosen to demonstrate these two aspects: 2015 northern Atlantic (NATL) Joaquin, 2016 NATL Matthew, 2017 northeastern Pacific (EPAC) Kenneth, and 2017 southern Indian (SIO) Ernie. While Joaquin (Part I; Part II; Part III) and Matthew (Part III) have been discussed previously, Kenneth and Ernie are shown to illustrate more examples of this process. Ernie also serves the purpose of demonstrating this phenomenon in a new basin (Courtney et al. 2019). Figure 1 illustrates the intensities and three shear calculations—two from SHIPS and one from the Cooperative Institute for Meteorological Satellite Studies (CIMSS)—for the four chosen TCs. All four of them endured sustained moderate shear (>5 m s⁻¹; Reasor et al. 2013; Rios-Berrios and Torn 2017) prior to their respective RI episodes.

View Full Size

Time series of best-track intensities (black line; left axis) and three shear calculations (red, orange, and brown; right axis) of (a) 2015 NATL Joaquin, (b) 2016 NATL Matthew, (c) 2017 EPAC Kenneth, and (d) 2017 SIO Ernie.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

If we assume that the coldest IR brightness temperatures are associated with the strongest convection that is in turn also associated with the tilt of the vortex (Part II), we can track the migration of the cloud shield upshear as a proxy for the movement of the tilt. We use a brightness temperature centroid (Ryglicki and Hart 2015) computed thusly:

$\langle x,y\rangle =\langle {\displaystyle \frac{{\displaystyle \iint \left(\overline{T_B}-T_B\right)r^2\hspace{0.17em}\cos \lambda \hspace{0.17em}dr\hspace{0.17em}d\lambda }}{{\displaystyle \iint \left(\overline{T_B}-T_B\right)r\hspace{0.17em}dr\hspace{0.17em}d\lambda }}},{\displaystyle \frac{{\displaystyle \iint \left(\overline{T_B}-T_B\right)r^2\hspace{0.17em}\sin \lambda \hspace{0.17em}dr\hspace{0.17em}d\lambda }}{{\displaystyle \iint \left(\overline{T_B}-T_B\right)r\hspace{0.17em}dr\hspace{0.17em}d\lambda }}}\rangle ,$(9)

where T_B is the IR brightness temperature in kelvin and the overbar indicates the average T_B over a 500-km circle surrounding the interpolated best-track position from HURDAT (Landsea and Franklin 2013) as per the methods from Part I. The centroid weighting area is a 150-km circle. The first guess at the initial time is the location of the coldest brightness temperature in the domain; all subsequent first guesses are the location of the centroid from the previous time. Figure 2 demonstrates that for each case, the centroid tracks the upshear migration of the clouds from 200 km away to left (or right) of the shear vector prior to the appearance of the eye. In the instances of Matthew (Fig. 2b), Kenneth (Fig. 2c) and Ernie (Fig. 2d), one can clearly identify manifestations of the nutations as the centroid nears the TC center in the left-of-shear or right-of-shear position, depending on the hemisphere.

View Full Size

Storm-relative tracks of the IR brightness temperature centroid, colored by day for each TC: (a) Joaquin, (b) Matthew, (c) Kenneth, and (d) Ernie. Angles are math convention: 0° is east, 90° is north, etc. Radial axes are distance in kilometers. Teal arrows are time-averaged shear direction from SHRD [defined as 850–200-hPa shear magnitude vs time (200–800 km)], SHDC (defined as being the same as SHRD but with vortex removed and averaged from 0 to 500 km relative to 850-hPa vortex center), and CIMSS methods for the time periods of the panels.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

A fundamental finding of Part I was that the TCs in this class are sheared by anticyclones. Usually, but not always,¹ this implies that the strongest environmental winds are confined to near the tropopause. This is the same layer as the outflow of the TC. This permits outflow blocking and shear reduction (Part III) that is then thought to permit the upshear movement of the tilt and subsequent realignment. Setting aside the expectation of capturing tilt nutations in reanalysis data, it is worth investigating how global reanalyses represent at least the outflow blocking portion of this process. Figure 3 is an illustration of the divergent and rotational winds at 200 hPa of the four TCs prior to their respective RIs. Per the rotational winds, upper-level anticyclones shear all four TCs. Generally, each TC exhibits divergent wind magnitudes upshear of at least 7 m s⁻¹ in coherent arc shapes, and the rotational winds are rerouted around the TCs to varying degrees.

View Full Size

Rotational (streamlines; right color key) and divergent (color shading; left color key) wind components (m s⁻¹) at 200 hPa surrounding each of the four TCs (yellow X) at a given date and time.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

b. Full-physics model review

As a summary of Part III, we present Fig. 4 which is a collection of three important features described in Part III from the G7.5 simulation. Figure 4a highlights the tilt-relative upper-level wind structure around the core of the TC during a TCA. In a tilt-relative frame, the stagnation point (where the winds reduce to zero) is located northeast (upshear right) of the core at 200 km radius. The peak winds exiting the TC are between 25 and 30 m s⁻¹. This has the consequence of creating a localized high pressure (Fig. 4b), the magnitude of which, approximately 45 Pa, is the same as the hydrostatic high pressure that is usually associated with the upper levels of TCs but is advected downshear in sheared cases (Wu and Emanuel 1993). There is also a second small lobe of low pressure located south-southeast (left of shear) of the TC on the downshear side of the block.

View Full Size

From the G7.5 simulation: (a) winds (m s⁻¹), smoothed with a 2-h running mean, around the center at 12.6-km height at 66 h 45 min; (b) pressure perturbation (Pa), smoothed with a 2-h running mean, at 12.6-km height; and (c) upshear–downshear divergent kinetic energy difference (green; positive is upshear), 13–1.5-km local vertical wind shear (red; see Part III); and tilt-relative upshear radial wind minimum (black) in the G7.5 simulation averaged vertically between 12.6 and 13.4 km. In (a), gray circles are 100-km intervals. In (b), the magenta X is the low-level center and the black line is the zero radial wind line.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

Figure 4c presents a time series of three important quantities: shear magnitude, where shear is defined as 13-km winds minus 1.5-km winds in a 200-km circle surrounding the low-level center; the upshear radial wind minimum location at 13-km height; and the difference between volume-integrated upshear and downshear divergent kinetic energy. For the difference, positive indicates upshear-oriented DKE is greater than downshear-oriented DKE and vice versa, and the bounds of integration are between 9 and 16 km and over the Cartesian domain. While the relationship among the three has been discussed at length in Part III, we instead focus on the two apparent regime shifts. For the first 48 h of the simulation, the outflow front is largely confined to within 150 km and the shear fluctuates between 5 and 8 m s⁻¹. At 48 h, the divergent kinetic energy difference shifts upshear, the outflow front expands to 250 km, and the shear drops to 3.5 m s⁻¹. Later in the simulation, at 114 h, the shear returns to 6 m s⁻¹ and intensification subsequently ceases. We will discuss this further in section 4d.

a. Helmholtz decomposition overview

In this section, we use a hierarchy of models to explore the features presented in section 3. The sequence is a simple 2D analytical expression, the nonlinear divergent shallow water simulations, and then finally the nonlinear full-physics simulation. Before we explore the analytical solutions, we will start with a basic analysis of the structure of the upper levels of the TC from the full-physics simulation by performing a Helmholtz decomposition of the flow to describe its basic characteristics using rotational streamfunction ψ and divergent velocity potential χ. Since the background flow is by definition initialized as neither irrotational nor nondivergent, it is completely filtered out by the Helmholtz decomposition.

Given the relative lack of idealized-TC-specific ψ and χ fields in the literature, we begin by presenting snapshots of the structures of the two fields in the G7.5 simulation and in the control at 13-km height at 85 h (Fig. 5). As a reminder, positive (negative) ψ is associated with anticyclonic (cyclonic) motion whereas positive (negative) χ is associated with convergence (divergence). The G7.5 ψ field (Fig. 5a) illustrates the downwind advection of the upper-level anticyclone, as previously demonstrated by Wu and Emanuel (1993, 1994). Two large, weak cyclonic gyres are created—one upshear right and one downshear. The control ψ field (Fig. 5c) is circular, as is expected for the upper levels of a TC in no background flow (Rappin et al. 2011). The χ fields for both simulations are similar in that they both expand radially from a small local area (Figs. 5b,d). In the G7.5 case, there is also weak convergence downshear. Figure 6 illustrates how the maximum ψ and minimum χ values change in time at 13.4-km height. This demonstrates that even though the shapes of the ψ field are noticeably different, the largest magnitude of the streamfunction is basically the same. For the χ field, the G7.5 TC is consistently more negative.

View Full Size

(top) Streamfunction ψ and (bottom) velocity potential χ of (a),(b) the sheared simulation and (c),(d) the control at 85 h into each simulation at 13.4-km height. The black arrow is the direction of the background flow.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

View Full Size

Time series of minimum χ and maximum ψ for each simulation at 13.4-km height.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

b. Analytical solutions

The simplest representation of the upper levels of the TC is an analytical representation of a point source collocated with a point vortex in two dimensions (Lamb 1932, 62–109). As noted in appendix A, one can prescribe a point source for both the anticyclonic rotational and divergent winds at the top of the TC in a background flow:

$u\left(x,y\right)=U_0+{\chi }_m\hspace{0.17em}{\displaystyle \frac{x}{x^2+y^2}}-{\psi }_m\hspace{0.17em}{\displaystyle \frac{y}{x^2+y^2}}\hspace{0.17em}\hspace{0.17em}\text{and}$(10)

$\upsilon \left(x,y\right)={\chi }_m\hspace{0.17em}{\displaystyle \frac{y}{x^2+y^2}}+{\psi }_m\hspace{0.17em}{\displaystyle \frac{x}{x^2+y^2}},$(11)

where x and y are coordinate locations from the origin and χ_m and ψ_m are magnitudes of the divergent and rotational sources, respectively. Effectively, the decay of the winds is r⁻¹. Figure 7 illustrates the simple examples here using 5 for χ_m and −5 for U₀ and ψ_m (note the signs). It can be shown that, for this series of equations, the stagnation point will be at the coordinate (−χ_m/U₀, ψ_m/U₀). In our example that uses both wind components (Fig. 7a), the stagnation point is at (1, 1). Comparing Fig. 7a with the full-physics results of Fig. 4a, the analytical solution reasonably represents the nonlinear numerical solution.

View Full Size

(a)–(c) Winds (m s⁻¹) and (d)–(f) pressure fields (Pa) of the analytical equations using different χ_m and ψ_m values. Streamlines are total winds. The green X in all plots is the location of the stagnation point, i.e., where the winds reduce to zero.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

In Figs. 7d–f, the maximum pressure perturbation is located between the stagnation point and the point source, while a lower pressure exists downwind. The individual pressure anomalies for the case with rotational and divergent vortex winds (Figs. 7a,d) are shown in Fig. 8. When the two pressure components of Fig. 8 are added together, the resultant pressure field is Fig. 7d. The high pressure is located between the vortex and the outflow front, mimicking the results of the CM1 numerical solution (Fig. 4b). This also explains the “drafting” downwind of the block and the localized lower pressure there found in the CM1 simulation south of the low-level center (Fig. 4b), as winds accelerate downwind of the block and lower the pressure. Fundamentally, the total pressure perturbation is the linear superposition of the two pressure anomalies from (12).

View Full Size

Pressure perturbations (Pa) associated with (a) the background and (b) the vortex. Black lines are total winds. The green X is the stagnation point. A combination of these two pressure fields yields Fig. 7d.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

c. Divergent shallow water results

Increasing complexity in the model hierarchy brings us to the DSW simulations. We base the initial conditions of the DSW model on the time-averaged ψ and χ fields from the control simulation at 12.6-km height (Fig. 9). These cross sections are essentially axially symmetric and do not show the marked asymmetry of the shear simulation. While we realize that the asymmetric structure of the χ field is essential for results in the real world and in the full-physics simulation, our demonstrations do not require this asymmetry. Initial conditions (appendix B) are thus

$\chi \left(r\right)={\chi }_m{e}^{-r^2/{\left(2{\sigma }_{\chi }\right)}^2}\hspace{0.17em}\text{and}$(13)

$\psi \left(r\right)={\displaystyle \frac{{\psi }_m}{\tilde{\psi }}}{\displaystyle \frac{{\left(\alpha r\right)}^{k-1}{e}^{-\alpha r/\theta }}{\mathrm{\Gamma }\left(k\right){\theta }^k}}.$(14)

Subsequently, we modify the inner cyclonic vortex region of the ψ field:

$\psi \left(r\right)=\psi \left(r\right)+\beta \left[{\psi }_m-\psi \left(r\right)\right]:r<\text{RM}\psi ;$(15)

ψ_m and χ_m control the peaks of the profiles, σ_χ controls the radial decay of the χ field, β controls the strength of the inner vortex, and RMψ is the radius of maximum ψ. Further explanations of these expressions can be found in appendix B. The initial height field h′ is

$h^{\prime }={\displaystyle \frac{{\text{Fr}}^2}{\text{Ro}}}\hspace{0.17em}\psi .$(16)

Figure 10 illustrates that our initial conditions are reasonable when compared with the results from the CM1 control simulation (Fig. 9). All fields are initialized at the origin.

View Full Size

Cross section of χ (red line) and ψ (orange line) at 12.6 km of the control simulation, time-averaged over the last 25 h of the simulation, as well as a cross section of χ (teal line) during a TCA in the Gaussian-7.5 simulation, averaged over 2.5 h.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

View Full Size

Initial conditions of the control experiment in the shallow water model: (a) the nondimensional ψ and χ fields, (b) the dimensionalized initial height field with the zero radial wind line in black, and (c) the dimensionalized perturbation wind field magnitude (color shading) and the total wind (streamlines).

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

Figure 11 shows the results of nine simulations of the DSW model, with details found in Table 1. All of these simulations are initialized with U₀ = −0.5, which is also part of the initial flow state. Our basic approaches are to modify the inner cyclonic vortex (Figs. 11a–c), initialize with only the ψ field (Figs. 11d–f), and initialize with only the χ field (Figs. 11g–i). The most important results are those of Figs. 11d–f without the divergent winds. The initialized ψ field is advected downstream (not shown), and no traces of the vortex remain at the origin. Conversely, in the χ-only simulations, the divergent field maintains coherence. This fully proves that the χ field and the divergent component of the outflow are primarily responsible for the outflow blocking phenomenon. In addition, in the DSW simulations that incorporate both ψ and χ, the strongest perturbation winds are “upshear” right, indicating that to match the complete structure of the full-physics simulation (Fig. 4a), both are required, but without the divergent component, no block occurs.

View Full Size

Results of the shallow water simulations after 130 h using nine different initial conditions (see Table 1). Shading is dimensionalized perturbation winds (m s⁻¹). Streamlines are total winds. The green line is the zero radial wind line.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

Table 1.

Initial condition variations among the divergent shallow water simulations. All unlisted variables are assumed to be the same as the control. Further information about constants of the analytical profiles is given in appendix B.

View Table

To continue to explore the evolution of the outflow front (the location at which the winds reduce to zero), we simply focus on the divergent component of the flow. Figure 12 highlights the location of the front as χ_m increases holding the background wind U₀ fixed (Fig. 12a) and its location as U₀ increases while holding χ_m fixed (Fig. 12b). The results are plotted as the ratio of the χ_m to U₀, both nondimensionalized, in order to relate the findings to the analytical solutions. For Fig. 12a, χ_m is increased by integer multiples over the initial value taken from the CM1, −1.25. For Fig. 12b, we take the value of χ_m of −6.25 (χ_m/U₀ ratio of 12.5) and increase the background wind magnitude incrementally by 2.5 m s⁻¹ (0.25 nondimensional wind units), spanning from −5.0 to −20.0 m s⁻¹ (from −0.5 to −2 nondimensional wind units). As χ_m increases linearly and the ratio increases, the outflow front distance does not increase linearly, indicating diminishing returns with regard to the increase of χ. If we apply the nondimensional scaling to the CM1 fields, the resulting ratio is approximately −3.5/−0.75, assuming Galilean invariance of the 13-km winds, or 4.66. According to Fig. 12, this should result in a dimensionalized outflow front location near 260 km. According to Fig. 4c, the average location of the outflow front between 48 and 90 h is approximately 250 km.

View Full Size

Steady-state locations of the outflow front location, plotted against the nondimensional ratio of the minimum value of the χ field to the background wind: (a) U₀ is held constant (0.5) while χ_m is increased and (b) χ_m is held constant (6.25) while U₀ is increased. In each simulation, there is no ψ field.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

There are two important caveats of the interpretation of these results. In addition to the magnitude, these results are also a function of the radial extinction as expressed by (13). As was shown in Fig. 11i, holding χ_m and U₀ fixed but increasing σ_χ does not result in an outflow front because the outflow is too weak as divergent winds are derived from the gradient of χ. The radial decay characteristics of the χ fields, such as r⁻¹ vice e^−r, will control the location of the stagnation point upwind. In addition, in the G7.5 simulation, the χ field is asymmetric. We noted that prior to the second abrupt RI, the DKE is skewed upshear during the nutations before shifting downshear (Fig. 4c). The downshear shift reduces the blocking effect. While obviously important, exploring the entirety of the upper level wind phase space is beyond the scope of this paper.

d. Full-physics divergent kinetic energy budget

One of the critical elements to the evolution at upper levels is the orientation of the terms relative to the flow. Figure 13 shows the spatial characteristics of these expressions at 13.4 km for the control (Figs. 13a–c), the G7.5 TC during a TCA (Figs. 13d–f), and the G7.5 TC later in the simulation after its intensification has ceased (Figs. 13g–i). The comparison of the three columns highlights the asymmetric nature of the upper-level structure in the G7.5 TC. In addition, Figs. 13f and 13i illustrate the important role of CrossKE on the upshear side of the TC, as it alters the total KE asymmetrically. For this reason, we include its time tendency explicitly in our budget calculation:

${\displaystyle \frac{\partial k_{\chi }}{\partial t}}+{\displaystyle \frac{\partial \left({\mathbf{V}}_{\psi }\cdot {\mathbf{V}}_{\chi }\right)}{\partial t}}=-c_p{\theta }_{\rho }\left({\mathbf{V}}_{\chi }\cdot \nabla \mathrm{\Pi }\right)+\left(\zeta +f\right)\left(u_{\chi }{\upsilon }_{\psi }-{\upsilon }_{\chi }{u}_{\psi }\right)+{\mathbf{V}}_{\psi }\cdot {\displaystyle \frac{\partial {\mathbf{V}}_{\chi }}{\partial t}}-ew{u}_{\chi }-{\mathbf{V}}_{\chi }\cdot \nabla k_{\chi }-{\mathbf{V}}_{\chi }\cdot \nabla k_{\psi }-{\mathbf{V}}_{\chi }\cdot \nabla \left({\mathbf{V}}_{\psi }\cdot {\mathbf{V}}_{\chi }\right)-w\left({\mathbf{V}}_{\chi }\cdot {\displaystyle \frac{\partial {\mathbf{V}}_{\psi }}{\partial z}}\right)-w\hspace{0.17em}{\displaystyle \frac{\partial k_{\chi }}{\partial z}}+R.$(18)

The derivation of (18) can be found in appendix C. From left to right, the terms on the right-hand side are pressure gradient or baroclinic generation (pgrad); divergent-rotational wind conversion (conv); rotational flux across local changes in divergent winds (chitt); meridional Coriolis (mercor); divergent fluxes of divergent kinetic energy (hadvchi), rotational kinetic energy (hadvpsi), and their cross term (hadvcross); divergent flux of the vertical flux of rotational winds (vadvpsi); and vertical advection of divergent kinetic energy (vadvkechi). We have grouped all sub-grid-scale effects into the residual R (Res). All terms of (18) are mass weighted during integration. For all following line plots, we smooth the data with a 2-h (nine point) running mean to aid in visualization and interpretation. An a posteriori analysis indicates that the three largest terms on the right-hand side ultimately become the divergent flux of divergent kinetic energy, the pressure gradient term, and the conversion term, but, as we shall show, there are important details in this evolution reflecting the transition shown in Fig. 4c at 48 h.

View Full Size

Spatial structure at 13 km of (top) rotational kinetic energy, (middle) divergent kinetic energy, and (bottom) cross-term energy for (a)–(c) the control at 84 h, (d)–(f) the G7.5 simulation at 65 h 30 min, and (g)–(i) the G7.5 simulation at 120 h. The black X is the low-level TC center. The black arrow in (d) is the direction of flow for all sheared images in (d)–(i).

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

We first illustrate the evolution of the three energy components for the two simulations in Fig. 14. The integration constraints are different from Part III: they are limited to 10–14 km in the vertical direction and a 500-km circle around the low-level center. The DKE in the G7.5 simulation is approximately a factor-of-2 larger than the DKE in the control simulation. The RKE in the control simulation is larger, but Fig. 5 illustrated that the ψ field of the Gaussian-7.5 simulation is advected downshear. Accordingly, the energy associated with that will also be advected downshear. CrossKE is comparatively smaller in the G7.5 simulation than the other two components, but it is practically zero in the control simulation.

View Full Size

The volume integral between 10- and 14-km heights and in a 500-km circle centered on the low-level TC center of the three kinetic energy components for each simulation. All terms are smoothed with a 2-h running mean.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

For our analysis of (18), we separate the time series into two overlapping time periods for each simulation: 6–60 h (Fig. 15), and then from 48 h to the end of the simulation at 144 h (Fig. 16). There is no significant DKE prior to 6 h. To aid in visualization, we have incorporated the three smallest terms into the residual: meridional Coriolis, rotational flux across local changes in divergent winds, and divergent flux of the vertical flux of rotational winds. The easiest way to interpret this evolution is by focusing on the G7.5 TC (Fig. 15b) from 24 to 30 h. The two largest positive terms are the pressure gradient and the horizontal divergent flux of RKE, while the largest negative term (aside from the residual) is the vertical advection of DKE. We can thus greatly simplify (18) for this time period:

${\displaystyle \frac{\partial k_{\chi }}{\partial t}}+{\displaystyle \frac{\partial \left({\mathbf{V}}_{\psi }\cdot {\mathbf{V}}_{\chi }\right)}{\partial t}}\approx -c_p{\theta }_{\rho }\left({\mathbf{V}}_{\chi }\cdot \nabla \mathrm{\Pi }\right)-{\mathbf{V}}_{\chi }\cdot \nabla k_{\psi }-w\hspace{0.17em}{\displaystyle \frac{\partial k_{\chi }}{\partial z}}+R.$(19)

A change occurs at 42 h. From this point until the end of the simulation (Fig. 16b), the horizontal divergent flux of DKE becomes the largest positive term as it, along with the conversion term, largely competes with the negative effects of the pressure gradient term. We can simplify (18) again for the second period:

${\displaystyle \frac{\partial k_{\chi }}{\partial t}}+{\displaystyle \frac{\partial \left({\mathbf{V}}_{\psi }\cdot {\mathbf{V}}_{\chi }\right)}{\partial t}}\approx -c_p{\theta }_{\rho }\left({\mathbf{V}}_{\chi }\cdot \nabla \mathrm{\Pi }\right)+\left(\zeta +f\right)\left(u_{\chi }{\upsilon }_{\psi }-{\upsilon }_{\chi }{u}_{\psi }\right)-{\mathbf{V}}_{\chi }\cdot \nabla k_{\chi }+R.$(20)

In recalling Fig. 4c, note this change occurs at the same time as the DKE shifts upshear, the outflow front gets pushed out to 250 km, and the shear drops to 3.5 m s⁻¹. A similar shift happens in the control simulation (Figs. 15a and 16a), albeit at smaller magnitudes. Simply stated, the DKE generation in the outflow can be divided into two phases: a pressure-driven phase and then a subsequent divergent-flux-driven phase. The second phase is largely what is responsible for the outflow’s pushing the environmental winds back. We also note that this change occurs after the precession of the tilt has reached left of shear (Part II).

View Full Size

Time series of the time-tendency terms of the divergent and cross-term kinetic energy budget equation for the (a) control simulation and (b) Gaussian-7.5 shear simulation from 6 to 60 h integrated between 10- and 14-km height and within a 200-km circle of the low-level center. All are smoothed with a 2-h running mean. The terms are defined in the text.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

View Full Size

As in Fig. 15, but for 48–144 h and for a change in the y axis.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

From a spatial perspective, Figs. 17 and 18 illustrate these changes relative to convection by analyzing the vertically integrated total condensed water of the column (TCWC), DKE, CrossKE, the pressure gradient term, the conversion term, and all of the horizontal flux terms grouped together at 28 h and then again at 64 h 45 min. From Fig. 17a, it is clearly seen that the convection is displaced downshear, while the pressure gradient generation is collocated with the convective maxima (Fig. 17d) with modest upwind advection (Fig. 17f). The CrossKE and conversion terms are small. At 64 h 45 min, during a TCA, Fig. 18a shows how convection is now left of shear. The DKE and positive CrossKE (Figs. 18b,c) are now in phase. While the pressure gradient is still locally positive over convection, it is weakly negative everywhere else (Fig. 18d) while the conversion term is negative over convection (18e). The flux terms are largely focused upshear and are predominantly positive (Fig. 18f), and this corroborates the time evolution seen in Fig. 16b.

View Full Size

Spatial characteristics of various quantities in the Gaussian-7.5 shear simulation at 28 h: (a) total condensed water of the column (first color bar on the right), (b) divergent kinetic energy (second color bar on the right), (c) cross-term kinetic energy (second color bar on the right), (d) pressure-work term, (e) rotational-divergent conversion term, and (f) sum of all horizontal energy flux terms. All are averaged over 2 h, and (b)–(f) are all at 13.4-km height. The black X is the low-level center. The black arrow is the direction of the environmental flow.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

View Full Size

As in Fig. 17, but for 64 h 45 min.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

Since the whole of this evolution is strongly tied to convection, we believe it would be worthwhile to investigate the evolution of convection and updraft distribution. Figure 19 is a collection of log-scale contour-frequency-by-altitude diagrams (CFADs; Yuter and Houze 1995) during a tilt nutation in the G7.5 simulation (Fig. 19a), when the tilt temporarily decreases to near zero in G7.5 (Fig. 19b), and a comparable time in the control (Fig. 19c). The CFADs indicate that there is substantially more coverage of updrafts above 20 m s⁻¹ when a TCA is present. If we use 20 m s⁻¹ as a threshold, Fig. 19d illustrates the evolution in time of the strongest updrafts. There are two clusters of strong updrafts in the G7.5 simulation prior to 42 h, and both of those can be related to increases in DKE via the pressure gradient generation term. The time period from 48 to 96 h indicates consistently strong updrafts, and this results in an increase of DKE aloft through upshear horizontal fluxes. Interestingly, after 100 h, the strong updrafts all but cease. This also corresponds to a leveling off of intensification and an increase of the shear from 3.5 to 6 m s⁻¹ (Fig. 4c). All of this would indicate that it is indeed the strongest convection that is driving the DKE generation and the blocking but in two different phases.

View Full Size

(a)–(c) Log₁₀ CFADs of vertical velocity (a) during a TCA in the Gaussian-7.5 simulation, (b) after a TCA after the tilt has reached a relative minimum, and (c) during the control at a time when the low-level intensity was similar. Values are smoothed with a 2-h running mean. (d) Total volume covered by updrafts greater than 20 m s⁻¹ for each simulation (bars; left axis) and the low-level maximum mean tangential wind of each simulation (lines; right axis).

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-20-0030.1

• Download Figure
• Download figure as PowerPoint slide

a. Summary

In this study, we have performed a more thorough, mathematical analysis of the outflow to complement the original series on atypical RI (Part I; Part II; Part III) using a hierarchy of analyses and models. We first reviewed the fundamental components of atypical RI: the return of the midlevel center upshear and outflow blocking. We showed how the movement of the midlevel center, based on findings from Part II, can be abstracted from infrared (IR) satellite imagery. The outflow blocking that permits this evolution (Part III) was then demonstrated structurally in ERA5. This highlighted that the divergent winds in ERA5, localized in an arc structure upwind, serve to block and reroute the environmental rotational winds. The environmental rotational winds are associated with upper-level anticyclones. We then revisited findings from Part III: the structure of the upper-level winds as the block occurs, the presence of an upshear Bernoulli high, and the time-varying shear and outflow front location.

Since the outflow is vertically confined to a layer approximately 2 km deep, we were able to reproduce facsimiles of the flow using an analytical 2D point-source solution and a forced divergent shallow water model. The simple analytical solution allowed us to demonstrate that the location of the Bernoulli high is a superposition of two pressure anomalies: one associated with the winds of the vortex, the other associated with the background winds. The shallow water model used initial conditions based on the time-averaged ψ and χ fields from the control simulation. The shallow water model simulations demonstrated that the existence of the χ field is critical to the maintenance of the block, as the ψ field by itself is advected downwind and offers little resistance to an environmental flow; however, we note that if we were using a two-layer model at minimum such as in Smith et al. (2000) then the vortex would exhibit more resistance to the flow through vertical coupling, and the single-layer shallow water model is in no way meant to model the full vertical structure of a TC. Nevertheless, based on our results of the simpler models, the steady-state outflow front location, while dependent upon the radial decay of χ, can be meaningfully assessed using the nondimensional ratio of the peak value of χ to the magnitude of the background wind: χ_m/U₀.

Given that the divergent wind is the most important part of the outflow for blocking, we derived a divergent kinetic energy budget equation to analyze the evolution of the divergent outflow of the CM1 simulation more carefully. We showed that the evolution of the divergent outflow can be divided into two regimes: a pressure-gradient-driven one and a divergent-flux-driven one. Both are strongly tied to convection. When the shift occurs at 48 h into the simulation, which is also the time the precession reaches left of shear (Part II), the outflow front is subsequently pushed out to 250 km. The divergent kinetic energy and the positive lobe of cross-term energy are in phase. Later in the simulation, after the abrupt rapid intensification at 90 h, the divergent kinetic energy maximum and the cross term are no longer in phase, the strongest vertical motions cease, the outflow front retreats to 150–200 km, the shear increases from 3.5 to 6 m s⁻¹, and the TC ceases its intensification.

b. Discussion

At the most basic level, what we have described herein is the Magnus effect (Magnus 1853) which is a specialized interpretation of the Bernoulli effect (Euler 1752). As a flow interacts with a rotating object, a high–low pressure couplet forms on either side of the object. This phenomenon has previously been used to explain the movement of tornadoes (Fujita 1965). The two differences here are that tornadoes are primarily rotational without a prominent divergent component and that the pressure anomaly caused by a modification of the tornado’s rotation was neglected. A TC possesses a large and significant divergent flow field, and this divergent flow is what is necessary to divert the environmental flow around the TC upwind. This creates an interesting paradox with regard to shear calculations used both operationally and academically. Many works have tried to remove the vortex in various ways to ascertain the environmental flow, but the original series plus this paper demonstrate that one cannot remove the divergent component of the flow at upper levels. This could be a significant explanation as to why operational shear calculations can be misleading, such as in 2016 Matthew (Stewart 2017; Part III). This also highlights the necessity of using shear calculations that incorporate the vertical structure of the environmental winds, as discussed by Finocchio et al. (2016), Onderlinde and Nolan (2016), Velden and Sears (2014), and Knaff et al. (2018), since outflow blocking is only effective between 150 and 250 hPa, and measures of vortex resilience for other wind profiles would fall to previously established research (Jones 1995; Reasor et al. 2004; Schecter 2015).

We also draw attention to the fact that, based on the time series of the χ minimum in the CM1 simulations, the only way to create a situation in which a weak TC can reduce oncoming moderate shear is by paradoxically shearing the TC, all else being equal thermodynamically. The minimum value of χ in the G7.5 simulation is 1.5–2 times as large as in the control when both are borderline hurricanes. In the control, the χ minimum is −1.0 × 10⁶ m² s⁻¹ while it is the equivalent of a category-1 hurricane. According to the nondimensionalized shallow water results in this case, to create an outflow front at a distance of 250 km the environmental shear can only be 2.6 m s⁻¹. We hypothesize that this stark contrast is due to the thermal anomaly caused by the tilt and the subsequent importance of buoyancy in modulating updraft characteristics (Part II) that in turn modulate the structure of χ, but we leave a more detailed investigation of this phenomenon for a future study.

An understated appeal of the simpler models—both shallow water and analytical—is that we can generalize our results beyond the realm of TCs. As noted in the introduction, various severe storm studies have noticed high pressures aloft upwind of severe weather and tried to attribute its existence to various mechanisms. Using our nondimensional studies as a framework, we have shown that the divergent component of any flow will act as a barrier to the oncoming winds; therefore, the high pressures and the blocking aloft noted by Mapes (1993), Schmidt and Cotton (1990), and Fritsch and Maddox (1981) are likely a result of the divergent outflow’s colliding with the environmental winds. The anticyclonic rotation of the winds around the system (Trier and Sharman 2009) is then associated with the rotational component of the outflow. Determining the extent of the blocking and the pressure perturbation is thus controlled by setting the nondimensionalization parameters such as length scale, wind scale, and so on to parameters that match the scale of whatever phenomenon one wishes to study.

More work still needs to be done. Throughout our simulations, we have neglected variations in radiative properties and SSTs, both of which would alter the thermodynamics of the situation considerably. This would then alter the structure and strength of the χ field through modulation of the updrafts. It has been shown that the two initial convective pulses in the G7.5 simulation are important to divergent kinetic energy generation prior to full pushback by the flow as the tilt reaches left of shear. We have limited our full-physics simulation to 27.5°C SSTs and a background shear magnitude of 7.5 m s⁻¹, as these were parameters motivated by observations. All else being equal, if SSTs play an important role in modulating the χ field through buoyant updrafts (Hendricks et al. 2004), then warmer SSTs would also enhance the outflow blocking mechanisms described throughout this paper and the original series. In addition, since convection is also favored diurnally at night (Kossin 2002), that will also modulate outflow associated with these features. Future work will address these details.

As a final note, we close by discussing the motivation behind this entire series. This study was motivated in large part by the Office of Naval Research Tropical Cyclone Intensification Defense Research Initiative (Doyle et al. 2017). One of the key science goals was to “understand the coupling of TC outflow with inner-core convection and its implications for intensity change.” We believe that we have firmly established a link between the two, albeit under specific circumstances and in an unexpected way. Convection in sheared TCs projects onto the divergent component of outflow, and this divergent flow serves to block the environmental flow, to lower the shear, and to permit realignment and subsequent vortex intensification if the TC is sheared by an anticyclone. These studies highlight the paramount importance of a new shear calculation: one that factors in both the vertical structure of the environmental winds and the ability of the TC to defend itself via its divergent outflow.