Can Existing Theory Predict the Response of Tropical Cyclone Intensity to Idealized Landfall?

Potential intensity is a theoretical upper-bound for the tropical cyclone intensity in a given thermodynamic environment. This theory is formulated by idealizing a mature tropical cyclone as a Carnot heat engine, where entropy fluxes from the ocean surface are used to maintain the circulation against surface frictional dissipation. Potential intensity $V_{p}$ is expressed as (Bister and Emanuel 1998),

where

$C_{k}$ and $C_{d}$ are bulk exchange coefficient for surface enthalpy and momentum, respectively; $\triangle{k}$ is the difference between the saturation enthalpy of the ocean surface and the enthalpy of the overlying near-surface air; $T_{ST}$ is the surface temperature, $T_{a}$ is the temperature of air overlying the surface; $T_{tpp}$ is the tropopause temperature; $L_{v}$ is the enthalpy of vaporization; $C_{p}$ is the specific heat capacity of air; $q^{*}$ is the saturation mixing ratio of the ocean surface at the local surface pressure; $q_{a}$ is the mixing ratio of air overlying the ocean surface; and $\epsilon$ is surface evaporative fraction, which is taken as $1$ over ocean surface. This formulation includes the effect of dissipative heating, which results in the tropopause temperature replacing the surface temperature in the denominator of the Carnot efficiency factor (Bister and Emanuel 1998).

For inland storms, where the underlying surface has a higher $C_{d}$ (i.e., a rougher surface) and smaller $\epsilon$ (i.e., a drier surface), Eq.(1) would predict a weaker maximum sustained wind speed compared to the ocean environment. Thus, we define the response of $V_{p}$ to any given surface forcing as the ratio of its post-forcing value to its initial pre-forcing value. Following CC20, the predicted $V_{p}$ response to surface roughening is given by

or surface drying given by

where $(C_{d})_{EXP}$ and $(\epsilon)_{EXP}$ are the modified value of each respective surface property parameter, representing various magnitudes of surface roughening or drying. $V_{p(CTRL)}$ is the potential intensity of a Control pre-landfall TC (defined in Section 3). As discussed in CC20, using normalized responses of $V_{p}$ can generalize the results to any mature storm intensity and also minimizes sensitivities associated with the precise definition of $V_{p}$. CC20 demonstrated that the final equilibrium intensity response in idealized simulations closely follows this prediction.

Here we expand this approach to simultaneous surface drying and roughening, i.e.

Mathematically, Eq.(6) indicates that the equilibrium intensity response to simultaneous forcing, $\tilde{V}_{p(C_{d}\epsilon)}$, is simply the product of the individual equilibrium responses, $\tilde{V}_{p(C_{d})}$ and $\tilde{V}_{p(\epsilon)}$, i.e.

The implication is that the response to any combination of surface roughening and drying may be predictable from the responses to each individual forcing. Inspired by Eq.(7), the complete time-dependent response of storm intensity $\tilde{v}_{m(C_{d}\epsilon)}(\tau)$ to simultaneous surface roughening and drying is hypothesized as the product of the individual transient responses, i.e.

The above outcome would have significant practical benefits for understanding the response to a wide range of surface properties as is found in nature.

E12 presents a theory for the time-dependent intensification of a TC, based on the role of outflow turbulence in setting the radial distribution of low-level entropy proposed in Emanuel and Rotunno (2011). The equation is given by

with initial condition $v_{m}=0$ at $\tau=0$, where $v_{max}$ is the final steady-state maximum wind speed and $h$ is a constant boundary layer depth scale within the eyewall. Ramsay et al. (2020) generalized this equation to predict intensification from a non-zero initial intensity. Here we further generalize Eq.(9) to represent the decay from an initial intensity $v_{m}(0)$ to a weaker equilibrium intensity $v_{f}$ (see Appendix). This can be written in a form normalized by the initial intensity given by

where $\tilde{v}_{m(th)}(\tau)=\frac{v_{m(th)}(\tau)}{v_{m}(0)}$ and $\tilde{v}_{f}=\frac{v_{f}}{v_{m}(0)}$. We use the subscript $th$ to denote the intensity predicted by theory. Examples of the theoretical prediction of Eq.(10) are presented in Fig.1a for a range of $v_{f}$, using the value of $h$ (5 km) that was applied in E12 and an initial intensity of $100\;ms^{-1}$.

When $v_{f}=0$, Eq.(10) reduces to

For this special case, the transient intensity response from a given $v_{m}(0)$ depends only on the boundary layer depth scale $h$. Fig.1b presents examples varying $h$ over a wide range of values, with using $v_{m}(0)=100\;ms^{-1}$ and $v_{f}=0$. Eq. (11) is the continuous limit of Eq.(10) as $v_{f}\rightarrow 0$ and is not singular. Note that setting $v_{f}=0$ produces a solution that does not actually reach zero intensity in finite time, but rather approaches zero only as $\tau\rightarrow\infty$; for realistic timescales, the solution still predicts a normalized intensity appreciably greater than zero ($\approx$15% after 5 days in Fig.1).

According to Eq.(10)-(11), the intrinsic time scale of intensity decay from the initial condition is determined by the boundary layer depth scale $h$ and the steady-state final intensity $v_{f}$, taking $C_{k}$ as constant, such that a smaller $h$ leads to a faster decay (Fig.1b). Effects from changes in the external environment, e.g., surface properties, are captured in the prediction of $v_{f}$ as discussed below. Notably, the precise definition of $h$ in E12 theory and its relation to the true boundary layer height, $H$, is uncertain both theoretically and practically. First and foremost, the TC boundary layer height $H$ is poorly understood even for storms over the ocean where the $H$ is approximated by its dynamical or thermodynamical characteristics (Kepert 2001; Emanuel 1997; Bryan and Rotunno 2009; Zhang et al. 2011; Seidel et al. 2010). Unfortunately, these estimates of boundary layer heights can vary substantially from one another (Zhang et al. 2011). Moreover, in E12 theory it is the boundary layer depth specifically within the deeply-convecting eyewall that is relevant, where air rapidly rising out of the boundary layer effectively blurs the distinction between boundary layer and free troposphere (Marks et al. 2008; Kepert 2010; Smith and Montgomery 2010); perhaps for this reason E12 found a value corresponding to an approximate half-depth of the troposphere (5km) to perform best. Finally, little is known about the TC boundary layer height during the landfall transition. Thus, when evaluating E12 theory, we simply test a range of values for $h$ and examine the extent to which variations in the best-fit values of $h$ across experiments align with variations in estimates of the boundary layer.

The pronounced spatiotemporal heterogeneity in surface properties from storm to storm in real-world landfalls requires sophisticated land-surface and boundary layer parameterizations (Cosby et al. 1984; Stull 1988; Davis et al. 2008; Nolan et al. 2009; Jin et al. 2010). However, experiments in axisymmetric geometry with a uniform environment and boundary forcing can reveal the fundamental responses of a mature TC to individual surface roughening or drying, as introduced in CC20. Thus, this work extends the individual forcing experiments of CC20 to simulations where the surface is simultaneously dried and roughened with varying magnitudes of each.

All experiments are performed using the Bryan Cloud Model (CM1v19.8) (Bryan and Fritsch 2002) in axisymmetric geometry with same setup as Chen and Chavas (2020); model parameters are summarized in Table 1. Dissipative heating is included. We first run a 200-day baseline experiment to allow a mature storm to reaches a statistical steady-state, from which we identify the most stable 15-day period. We then define the Control experiment (CTRL) as the ensemble-mean of five 10-day segments of the baseline experiment from this stable period whose start times are each one day apart. From each of the five CTRL ensemble member start times, we perform idealized landfall restart experiments by instantaneously modifying the surface wetness and/or roughness beneath the CTRL TC, which are then averaged into experimental ensembles analogous to the CTRL. Surface wetness is modified by decreasing the surface evaporative fraction $\epsilon$, which reduces the surface latent heat fluxes $F_{LH}$ through the decreased surface mixing ratio fluxes $F_{qv}$ in CM1 (sfcphys.F) as

where $C_{q}$ is the exchange coefficients for the surface moisture; $s_{10}$ is the 10-m wind speed; and $\Delta q$ is the moisture disequilibrium between the 10-m layer and the sea surface. Surface roughness is modified by increasing the drag coefficient $C_{d}$, which modulates the surface roughness length $z_{0}$ and in turn the friction velocity $u^{*}$ for the surface log-layer in CM1 as

where $\kappa$ is the von Kármán constant, $z=10m$ is the reference height, $s_{1}$ is the total wind speed on the lowest model, and $z_{a}$ is approximately equal to the lowest model level height. Readers are referred to CC20 for full details.

Our experiments are summarized in Fig.2. Surface roughening-only experiments ($2C_{d}$, $4C_{d}$, $6C_{d}$, $8C_{d}$, $10C_{d}$) and drying-only experiments ($0.7\epsilon$, $0.5\epsilon$, $0.3\epsilon$, $0.25\epsilon$, $0.1\epsilon$) are fully introduced in CC20. The combined experiments are simulated in the same manner but with the surface dried and roughened simultaneously. Our combined experiments are designed in a way where individual drying and roughening are systematically paired with each other; experiments are named by the corresponding modifications in $C_{d}$ and $\epsilon$. We focus on two specific subsets of experiments within this phase space. First, $0.7\epsilon 2C_{d}$, $0.7\epsilon 10C_{d}$, $0.1\epsilon 2C_{d}$, $0.1\epsilon 10C_{d}$ are chosen as the representatives for extreme combinations, where each forcing takes its highest or lowest non-zero magnitude. Second, $0.25\epsilon 4C_{d}$, $0.5\epsilon 2C_{d}$, $0.1\epsilon 8C_{d}$ are chosen to represent cases where the individual forcings are varied in ways that yield comparable equilibrium potential intensities; $\tilde{V}_{p(\epsilon)}$ and $\tilde{V}_{p(C_{d})}$ are labeled in Figure 2. Finally, we generate a special set of combined experiments, $0V_{p}XC_{d}$, in which $\tilde{V}_{p}$ is fully reduced to zero for a range of magnitudes of roughening. Experiments are modeled by setting both surface sensible and latent heat fluxes to zero while increasing the roughness by a factor of $X$. This set of experiments sharing the same $\tilde{V}_{p}=0$ and are applied to test the simplified form of E12 assuming $\tilde{v}_{f}=0$ (Eq.(11)).

In each experiment, the simulated storm intensity $v_{m}(\tau)$ is normalized by the time-dependent, quasi-stable CTRL value, where $\tau$ denotes the time since the start of a given forcing experiment. We primarily focus on the first 36-h evolution, during which $\tilde{v}_{m}$ decreases monotonically across all roughening, drying, and combined experiments. In addition, both the near-surface (50m) intensity response $\tilde{v}_{m(50m)}$ and above-BL (2km) intensity response $\tilde{v}_{m(2km)}$ are compared to theoretical predictions. The near-surface wind field is essential for predicting the inland TC hazards, while the above-BL wind field is generally used when formulating physically-based theories above the boundary layer. Over the ocean, $\tilde{v}_{m(50m)}$ and $\tilde{v}_{m(2km)}$ typically co-evolve closely (Powell et al. 2003). However, at and after landfall, the response of near-surface winds is expected to deviate from the above-BL winds, particularly in the case of roughening, as shown in CC20: there is a very rapid initial response of angular momentum to enhanced friction near the surface during the first 10 minutes (and especially the first 5-6 minutes) that subsequently propagates upward as the vortex decays. In contrast, the response to surface drying initially occurs aloft in the eyewall before propagating into the boundary layer, though this response is generally slower and smoother in time. Therefore, it is practically useful to test intensity theories against both near-surface and above-BL winds.

For E86 theory, we first compare the simulated equilibrium intensity against the equilibrium E86 prediction of Eq.(6). We then compare the full time-dependent simulated intensity response against that predicted by assuming the total response is the product of the individual responses, $\tilde{v}^{*}(\tau)$ (Eq.(8)). For E12 theory, we compare the simulated intensity evolution $\tilde{v}_{m}(\tau)$ against the E12 prediction $\tilde{v}_{m(th)}(\tau)$ of Eq.(10)-Eq.(11). In these solutions, the initial intensity $v_{m}(0)=100.4\,ms^{-1}$ for 2km wind field and $v_{m}(0)=90.38\,ms^{-1}$ for 50m wind field, respectively. The final intensity $v_{f}$ is set as the minimum value of $v_{m}$ in each simulation during the evolution. A range of $h$ will be tested in Eq.(10), which will be discussed in the following subsection. For real-world landfalls, we do not know the minimum intensity prior to the inland evolution. Thus, as a final step, we compare simulation results against the theory with $\tilde{v}_{f}$ predicted from $\tilde{V}_{p}$. This final step may be useful for potentially applying the theory to real-world landfalls.

As introduced in Section 2, $h$ is an uncertain boundary layer height scale both theoretically, as it is assumed to be constant, and in practice, because we do not have a simple means of defining the boundary layer depth particularly during the landfall transition. Thus, we do not aim to resolve this uncertainty in $h$ in the context of landfall, but rather we simply test what values of $h$ provide the best predictions and evaluate to what extent variations in $h$ across experiments align with variations in estimates of the boundary layer height $H$.

For each simulation, we test a range of constant values of $h$ from 1.0 to 6.0 km in 0.1 km increments in order to identify a best-fit boundary layer depth scale, $h_{BEST}$. We define $h_{BEST}$ as the value of $h$ that produces the smallest average error throughout the first 36-h evolution for each experiment. We then compare the systematic variation in $h_{BEST}$ against that of three typical estimates of boundary layer height $H$ calculated from each simulation.

Since the E12 solution applies within the convecting eyewall region, we estimate $H$ using three typical approximations and measure the value at the radius of maximum wind speed ($r_{max}$) at the lowest model level: 1. $H_{v_{m}}$ is the height of maximum tangential wind speed (Bryan and Rotunno 2009); 2. $H_{inflow}$ is the height where radial wind $u$ in the eyewall first decreases to 10% of its surface value (Zhang et al. 2011); and 3. $H_{\theta_{v}}$ is defined as the height where $\theta_{v}$ in the eyewall matches its value at the lowest model level (Seidel et al. 2010). The 36-h evolution of each estimate of $H$ across our simulations is shown in Supplementary Figure 1. Considering that $H$ varies in time during the experiment, the averaged value of $H$ during $\tau=0-6\;h$ and $\tau=30-36\;h$ are each compared to the $h_{BEST}$.

As discussed in Section 3b, the simulated intensity responses near the surface ($z=50m$) and above the boundary layer ($z=2km$) may differ, and thus we intend to test the theory against both. We begin by simply identifying important differences between the intensity responses at each level.

For drying-only experiments, $\tilde{v}_{m(2km)}$ responds to reduced $\epsilon$ before $\tilde{v}_{m(50m)}$ during the initial $\sim 5$ hours (Fig.3a), where stronger drying results in a larger deviation between the responses at each level. $\tilde{v}_{m(2km)}$ decreases slightly more rapidly than $\tilde{v}_{m(50m)}$ but they eventually converge to comparable equilibria. In contrast, for roughening $\tilde{v}_{m(50m)}$ responds nearly instantaneously across all roughening experiments, whereas $\tilde{v}_{m(2km)}$ responds more gradually. The magnitude of the deviation of $\tilde{v}_{m(50m)}$ from $\tilde{v}_{m(2km)}$ during the first 20 hours increases with increasing roughening (Fig.3b), though again both converge to comparable values after $\tau=20\;h$. The above behavior is consistent with the findings in CC20, where surface roughening has a significant and immediate impact on near-surface intensity while drying first impacts the eyewall aloft. As noted in CC20, $\tilde{V}_{p}$ provides a reasonable prediction for the long-term equilibrium intensity response to each individual forcing; there is a small overshoot in roughening experiments where $\tilde{v}_{m}$ reaches a minimum that is less than $\tilde{V}_{p}$ at approximately $\tau=36\;h$ before increasingly gradually towards $\tilde{V}_{p}$.

For our combined experiment sets (Fig.3c-d), both surface drying and roughening determine the total response of storm intensity. However, regardless of the relative strength of each forcing, $\tilde{v}_{m(50m)}$ always decreases more rapidly than $\tilde{v}_{m(2km)}$ due to the surface roughening. Similar to the roughening-only experiment, stronger roughening results in a larger deviation of $\tilde{v}_{m(50m)}$ from $\tilde{v}_{m(2km)}$ during the first 20 hours. Overall, the rapid initial response of near-surface intensity is controlled by the surface roughening regardless of the surface drying magnitudes. Moreover, similar to the individual forcing experiments, $\tilde{V}_{p}$ provides a reasonable prediction for the simulated minimum $\tilde{v}_{m}$ in combined experiments used to define the theoretical final intensity $\tilde{v}_{f}$.

Now we focus on the full time-dependent responses of the combined forcing experiments. As noted above, we hypothesized based on traditional potential intensity theory (Eq.(5)-(7)) that the transient response of storm intensity to simultaneous drying and roughening can be predicted as the product of their individual responses (Eq.(8)). Thus, we compare the $\tilde{v}^{*}(\tau)$ to the simulated $\tilde{v}_{m}(\tau)$ of near-surface winds for each combined experiment (Fig.4) for our experiment set with extreme cases of combined drying and roughening. Regardless of the magnitude of roughening and/or drying, $\tilde{v}^{*}(\tau)$ follows the simulated $\tilde{v}_{m}(\tau)$ closely through the initial rapid decay forced by roughening, the weakening stage through 36 hours, and the final equilibrium stage. There is a slight low bias in $\tilde{v}^{*}(\tau)$ relative to the $\tilde{v}_{m}(\tau)$ throughout the primary weakening stage, especially for strong drying (Fig.4c-d), indicating that there is a slight compensation in the response to roughening when strong drying is also applied. Overall, though, the full temporal evolution of the normalized responses to drying and roughening can indeed be combined multiplicatively. A very similar result is obtained for the above-BL intensity as well (not shown).

Although E86 theory is formulated for the equilibrium intensity, the above results indicate that the implication of its underlying physics also extends to the transient response to simultaneous surface drying and roughening. This behavior aligns with the notion that periods of intensity change represent a non-linear transition of the TC system between two equilibrium stable attractors given by the pre-forcing and post-forcing $V_{p}$ (Kieu and Moon 2016; Kieu and Wang 2017). Because the distance between attractors is multiplicative, evidently so too is the trajectory between them.

We next test the extent to which E12 theory (Eq.(10) and Eq.(11)) can predict the intensity evolution, taking as input $\tilde{v}_{f}$ and a best-fit value of $h_{BEST}$ from each simulation. The subsequent subsection compares the systematic variation in $h_{BEST}$ against that of the estimated $H$ in each set of experiments.

We begin by focusing on the above-BL intensity. The E12-based solution (Eq. (10)) can reasonably capture the overall transient intensity response across all roughening, drying, and combined experiments (Fig.5a-c). For drying-only experiments (Fig.5a), the theory initially ($\tau<5\;h$) underestimates $\tilde{v}_{m(2km)}$ (Fig.3a). $h_{BEST}$ in drying experiments takes a value around 5 km and shows little systematic variations with increased drying magnitude (Fig.5a). In contrast, $h_{BEST}$ decreases with increased surface roughening, from 4.3 km to 2.1 km (Fig.5b). In the following subsection, we compare this variation in $h_{BEST}$ to that of multiple estimates of $H$.

For near-surface winds, E12 theory can capture the transient response of $\tilde{v}_{m}$ in drying experiments (Fig.5d) using a slightly larger $h_{BEST}$. For roughening and combined experiments, though, it misses the very rapid initial decay due to enhanced $C_{d}$ (Fig.3b-d and Fig.5e-f), which was shown in Fig.1 of CC20 to occur within the first 10 minutes. The magnitude of this rapid initial decay depends on $C_{d}$, as surface roughening immediately acts to rapidly remove momentum near the surface first. To account for this initial response, we propose a simple model for this initial decay of $\tilde{v}_{m(50m)}$ that may be derived from the tangential momentum equation in a 1D slab boundary layer with a depth of $h^{*}$, given by

Taking $\left|\mathbf{v}\right|\approx v$, this equation may be integrated and then normalized by an initial intensity $v_{m}(0)$ to yield

Curiously, Eq.(17) is mathematically identical to the E12 solution with $v_{f}=0$ (Eq.(11)) except with $C_{k}$ replaced by $2C_{d}$, a topic we return to in the summary section.

We may combine Eq.(17) for the first 10-min evolution with Eq.(10) to model the complete near-surface intensity response to surface roughening:

where $\tilde{v}_{f}^{*}=\frac{v_{f}}{v_{m(th)}(10\;min)}$ since $v_{m(th)}$ now decreases from a new initial intensity $v_{m(th)}(10\;min)$ calculated from the first equation. For the $\tau\leq 10\;min$ solution, we find that the rapid decay by $\tau=10\;min$ across all roughening experiments can be captured by setting $h^{*}=1.78\;km$ constant, which yields a constant decay rate during the period $\tau=0-10$ min. In reality, the decay rate is very large in the first minute and monotonically decreases through the 10 minute period. This time-varying decay rate can be reproduced by allowing $h^{*}$ to increase with time from an initial very small value, which aligns with the physical response to roughening that may be thought of as the formation of a new internal boundary layer that begins at the surface and rapidly expands upward. Here though we employ a constant $h^{*}$ in order to retain a simple analytic solution; the 10-minute period is short enough that the difference is likely not of practical significance. Thereafter, we estimate $h_{BEST}$ for each simulation in the same manner as before.

The comparison of model against simulations is shown in Fig.5e-f. Eq.(18) can capture the simulated near-surface transient intensity response $\tilde{v}_{m}(\tau)$ across roughening and combined experiments. The values of $h_{BEST}$ for the prediction of $\tilde{v}_{m(50m)}$ are similar to those obtained for the above-BL case but slightly smaller in magnitude (Fig.5b,e and c,f).

Similar behavior associated with roughening is found in experiments for the special case $v_{f}=0$ (Fig.6). Thus, we again propose a two-stage model for $\tilde{v}_{m(50m)}$ given by

The comparison of model against simulations is shown in Fig.6, where $h_{BEST}$ exhibits a decreasing trend with enhanced surface roughening, similar to that found in the pure roughening experiments.

Note that multiplying both sides of (17) by $v$ yields a budget equation for kinetic energy given by $h\frac{d(KE)}{dt}=-C_{d}v^{3}$, where $KE=\frac{1}{2}v^{2}$ and the RHS is the expression for surface frictional dissipation of kinetic energy that is standard in TC theory (Bister and Emanuel 1998; Tang and Emanuel 2010; Chavas 2017)¹¹1Note: the KE budget equation may be expressed as $h\frac{d(KE)}{dt}=-2C_{d}|\textbf{v}|(KE)$. Hence, $2C_{d}$ represents the surface exchange coefficient of kinetic energy over a static surface.. Physically, then, the solution for the initial roughening response represents the intensity response to the dominant sink of kinetic energy in the absence of the dominant compensating thermodynamic source of kinetic energy from surface heat fluxes for a tropical cyclone. After this initial response, the solution follows a solution that accounts for both source and sink as encoded in E12 theory. The interpretation is that there exists a brief initial period where surface roughening directly modifies the near-surface air in a manner that is thermodynamically independent of the rest of the vortex. Thereafter, the vortex has adjusted and weakening proceeds according to processes governed by the full TC system, analogous to that of drying. In reality this transition is likely not instantaneous, though we have modeled it as so here for simplicity. The details of this adjustment process warrant more in-depth investigation that is left for future work.

We may combine all theoretical findings presented above to predict the near-surface intensity response to idealized landfalls for experiments combining drying and roughening. This represents the most complex of our experimental outcomes. We begin from the result of Eq.(8) and predict the intensity response to simultaneous surface drying and roughening as the product of their individual predicted intensity responses:

where $\tilde{v}_{th(\epsilon)}$ is generated by Eq.(10) and $\tilde{v}_{th(C_{d})}$ is generated by Eq.(18). When applying the transient solution for each individual forcing component, we use the previously-identified $h_{BEST}$ from Fig.7a-b and predict the $\tilde{v}_{f}$ using $\tilde{V}_{p}$ given by Eq.(4)-(5). In principle, an empirical model for $h_{BEST}$ for each individual forcing could be generated from our data since $h_{BEST}$ is approximately constant with enhanced drying and monotonically decreasing with enhanced roughening. But given the uncertainties in $h_{BEST}$ as a true physical parameter, we elect not to take such a step here.

The results are shown in Fig.8. Overall, our analytic theory performs well in capturing the first-order response across experiments, particularly given the relative simplicity of the method. There is a slight low bias in $\tilde{v}_{th(C_{d}\epsilon)}$ (i.e., too strong of a response) across all the predictions relative to the simulations, a reflection of the slight compensation found in the assumption that the combined response may be modeled as the product of the individual responses in Fig.4. Therefore, knowing how each surface forcing is changed after landfall provides a theoretical time-dependent intensity response prediction when given estimates of $\tilde{V}_{p}$ and $h_{BEST}$ for any combination of uniform surface forcing. This offers an avenue to link our theoretical understanding to real-world landfalls.

Finally, as a first step toward linking theory and real-world landfalls, we provide a simple comparison of our theoretical model with the prevailing empirical model for inland decay. The model was first introduced by DeMaria and Kaplan (1995) and most recently applied to historical observations by Jing and Lin (2019) as

where storm landfall intensity $V_{0}$ decays to a background intensity $V_{b}$ with a constant exponential decay rate $\alpha$. Jing and Lin (2019) estimated $V_{b}=18.82\;kt\;(9.7\;ms^{-1})$ and $\alpha=0.049\;h^{-1}$ using the historical Atlantic hurricane database.

Comparisons between theory and Eq.(21) predictions are shown in Fig.9, for $V_{0}$ from $100\;ms^{-1}$ to $23\;ms^{-1}$ similar to Jing and Lin (2019) (their Fig.3). Given that TCs in nature eventually dissipate (and typically do so rapidly), we choose the theoretical equation with $v_{f}=0$ (Eq.(11)); this is also by far the simplest choice as no final intensity information is required. We set $h=5km$ constant. Eq.(11) compares well against the empirical prediction for inland intensity decay, capturing the first-order structure of the characteristic response found in real-world storms. The comparison is poorer for weaker cases (Fig.9e-f), though uncertainties in intensity estimates are also likely highest for such weak storms. Note that the structure of the empirical model is constrained strongly by the assumption of an exponential model, so differences beyond the gross structure should not be overinterpreted. Ultimately, the consistency with the empirical model provides additional evidence that the physical model may indeed be applicable to the real world. Hence, it may help us to develop a physically-based understanding of the evolution of the TC after landfall.