## 1. Introduction

The behavior of a tropical cyclone (TC) in the hours after it makes landfall is critical in determining its hazardous potential. Various modeling experiments and analyses have been used to gain insight into the factors affecting decay at landfall. A set of modeling experiments which test the response of an idealized TC to various levels of instantaneous surface roughening and drying is performed by Chen and Chavas (2020). However, the dependence on the initial state of the TC was not examined. Hlywiak and Nolan (2021) perform realistic landfall simulations and finds that surface wind decay sensitivity to surface roughness is dependent on the intensity and size of the TC as it decays although only one initial case is tested. Trends of reduced land-induced decay rates have also been reported for the United States (Li and Chakraborty 2020) and globally (Phillipson and Toumi 2021). Li and Chakraborty (2020) attribute this to warming SSTs via increased TC moisture content but determine that TC size does not affect the wind speed decay rate.

Batts et al. (1980) were among the first to examine the rate of decay of TCs after making landfall. They proposed a filling model for the central pressure, linear in time, and independent of the initial TC state. Among the first to examine the factors affecting the rate of pressure filling after landfall were Jarvinen et al. (1985), who found that the largest decreases in intensity occurred within the first six hours of the most intense storms making landfall and assumed a quadratic fill rate of central pressure. Ho et al. (1987) developed upon this showing that more intense TCs tend to fill faster while also noting that there is a dependence of size on intensity in their database of landfalling TCs. They do not attempt to separate the contributions from the two variables to the filling rate largely due to a scarcity of data. Vickery and Twisdale (1995) quantified the above observations by assuming an exponential decay model for central pressure deficit with a decay constant proportional to the initial deficit. Vickery (2005) extend this analysis to include TC size dependence and find that the decay rate is also inversely proportional to the radius of maximum wind in a study of many historical landfalling hurricanes. The intensity is diagnosed via the central pressure deficit, the difference between the environmental and central pressure, which is empirically assumed to decay exponentially. The TC size dependence of the decay rate is attributed to the rate at which a significant fraction of the storm area becomes affected by land as a storm moves over land. Kaplan and Demaria (1995) and DeMaria et al. (2006) identify translation speed and distance inland as key parameters controlling decay with later iterations considering the fraction of the circulation over land. The basis of these models is that the decay rate of Vmax is proportional to Vmax itself which leads to exponential decay of Vmax with a time constant. However, these empirical assumptions of exponential decay for both Vmax and pressure deficit and size dependence lack a physical framework. Chen and Chavas (2021) show that existing intensification theory may be adapted to predict the decay of surface winds in idealized landfall simulations and Phillipson and Toumi (2021) show that a physically based algebraic model for Vmax fits global observations of landfall decay. These theories do not consider the role of size.

Here we attempt to provide a physical basis for the pressure filling at landfall by proposing an analytic solution to the mass continuity equation in the tropical cyclone core. We demonstrate, through a set of numerical simulations, that the inner-core size and the radius of surface maximum wind speed are important physical variables which control the rate of the pressure deficit decay over land. We support this finding with a simple new analytical physically based framework which suggests the half-life of the central pressure deficit is proportional to the size of the inner core of a decaying TC.

## 2. Physical filling model

*r*is

*ρ*is the density at radius

*r*and height

*z*, and

*υ*is the radial wind at radius

_{r}*r*and height

*z*. The average surface pressure within the cylinder is

*r*and

*g*is the gravitational constant. We define

*r*,

*P*, at a given radius is given by

*r*as

*P*, is then

_{c}*r*tends to zero to avoid infinite central pressure tendency, and invoking L’Hôpitals’s rule,

_{0}is the radius of maximum wind speed at

*t*= 0 and

*χ*is the density-weighted column-mean radial wind speed at Rmax

_{0}[

*χ*is defined at the initial Rmax (Rmax

_{0}). We only require that

*r*out to Rmax

_{0}throughout the decay. It is also worth highlighting that this is the point at which the size of the TC core is introduced explicitly via Rmax

_{0}. The fundamental quantity controlling the central pressure tendency in Eq. (8) is the radial gradient of column speed but here we are effectively parameterizing this single quantity in terms of two independent terms, the column speed and the initial size of the core. This separation is motivated by the fact that during TC decay the column speed must tend to zero as the TC fills while the core size may remain finite. This technique is useful because we may then parameterize the decay of the core column speed in terms of the central pressure deficit [Eq. (12)], ultimately allowing us to express the central pressure tendency as a function of central pressure itself and other parameters [Eq. (13)]. Rmax

_{0}is simply a convenient scaling parameter we used to define the size of the TC core. The model is not sensitive to this choice as long as the above linearity assumption remains valid because

*χ*is defined as

_{0}and always appears in a ratio with Rmax

_{0}. This is also true of the precise definition of Rmax itself. We validate the model here using Rmax defined as the radius of maximum 10 m winds and show the linearity assumption is valid for this case. Alternative choices of Rmax, and indeed core size, may be valid but would require similar testing of the linearity assumption.

*P*is the environmental pressure. We then make a further assumption that

_{e}*χ*decays as

*k*is a positive constant to be determined,

*χ*

_{0}is the initial column speed. A relationship of this form is proposed as under decay conditions we expect the mass flux toward the center to be related to the frictional inflow in the boundary layer and the secondary circulation above the boundary layer. In both cases the pressure gradient force is the driver. We may expect that the secondary circulation driven by latent heating also includes upper level outflow and that this may decrease

*k*. This will be verified later. The precise nature of this relationship including the value of

*k*however is complex and related to the physics governing inflow and outflow speeds which we do not attempt to disentangle in this simple model. The assumed relationship in Eq. (12) and the value of

*k*is examined in section 2.

*k*) we can make the approximation

*α*is a decay rate (dimensions of inverse time) defined by

*k*= 2, Eq. (15) simplifies to an algebraic decay form and in the limit

*k*= 1, the decay is exponential:

*k*and the initial state parameters:

*k*= 1 is exponential, the initial condition

*k*will affect the magnitude of the half-life but not its dependence on the three TC state variables.

The above model makes two primary assumptions: (i) the column-integrated inflow speed is linear in *r* from the center until Rmax, and (ii) the core column-integrated speed *χ* decays as *k*. From this the decay half-life is derived in terms of the same parameters. Of the four model parameters only two are readily observable. Obtaining good estimates of _{0} through observations may be realistic for real-world landfalling TCs. The same cannot be said for *χ*_{0} which is not easily evaluated even in numerical simulations. This is largely because it is a small difference between large inflow and outflow terms. However, we show that the model may potentially be useful without direct estimates of *χ*_{0} per TC. It might be possible to parameterize *χ*_{0} in terms of other, more readily available physical quantities. Similarly, *k* is not a directly observable quantity, but we show the model may be useful regardless.

## 3. Methods

### a. Simulation setup

TC simulations are performed using the Weather Research and Forecasting (WRF) Model, version 4.0 (Skamarock et al. 2019). Three nested square domains with two-way interaction are used. The outer domain has a side of length 7200 km (600 × 600 grid cells, 12 km grid spacing) and contains the middle domain with side length 3208 km (802 × 802 grid cells, 4 km grid spacing) which in turn contains the inner domain with a side length of 1069 km (802 × 802 grid cells, 1.333 km grid spacing). The time steps for the domains are 60, 20, and 10 s, respectively, on an *f* plane at 30°N which is representative of a typical landfall latitude. Each domain has 41 vertical levels of which about 12 are below 2 km height. The initial environmental conditions and lateral boundary conditions are given by the Jordan mean tropical sounding during hurricane seasons (Jordan 1958). There is no background wind and a horizontal “sponge layer” with a width of 240 km absorbs noise along the outer domain boundaries by reducing the horizontal wind velocities to zero. A full suite of physics parameterizations are used and include the WRF single-moment 6-class microphysics scheme (Hong and Lim 2006), the Rapid Radiative Transfer Model for general circulation models (RRTMG) scheme (Iacono et al. 2008) for shortwave and longwave radiation, the Mellor–Yamada–Janjić (MYJ) planetary boundary layer scheme (Janjić 1994), and the Eta Similarity surface layer scheme (Janjić 2002). Given that the 4-km-grid-size middle domain is large enough to contain most of the structure of the simulated TCs no cumulus scheme is used in any domain. This idealized setup is also described in Wang and Toumi (2019).

### b. Experiment design

We first create a “Spinup” run which provides TCs with range of initial states from which we can simulate land-induced decay. The Spinup run is initialized with a bogus vortex, which is inserted in the center of the domains and has an analytic wind profile (Wang and Toumi 2016) with a near-surface maximum wind speed of 18 m s^{−1} and a radius of maximum wind speed of 100 km with a wind speed, which decreases linearly to zero from the surface to the WRF top at 20 km. The Spinup TC is allowed to develop and mature over 15 days with the entire domain over an ocean surface with fixed SST of 29°C.

We then create a set of land-induced TC decay experiments from the Spinup run by restarting the simulation with the ocean surface replaced by land over the entire domain. We perform these restart runs beginning two days into the Spinup run, then every day until 14 days, which gives 13 experiment cases which we refer to as the “Control” experiments. These are labeled according to the time in the Spinup run at which they were started. The land surface type in these experiments is designated “croplands” in the Modified IGBP MODIS 20-category land-use dataset (Friedl et al. 2010) and has a roughness length *z*_{0} of 0.15 m. Soil moisture and temperature was not fixed. An additional set of experiment cases labeled “Dry” was created by repeating the above method, but at the point of restart, as well as replacing ocean with land, we also remove all moisture from the atmosphere and set surface moisture fluxes to 0 for the duration of the run to keep the atmosphere dry (Li and Chakraborty 2020).

The Dry experiments serve multiple purposes. They provide a way to simplify the decay process by removing latent heating allowing us to focus on the effect of surface frictional forcing. They also allow us to estimate the contribution of latent heating to the maintenance of the TC during the decay. Finally, they offer a way of testing our new analytic decay framework across TCs decaying in extremely different environments.

All simulation variables in the decay experiments were output and stored at hourly intervals at *t* = 0, 1, 2, …, h. The density-weighted column-integrated radial wind speed, or “column speed” ^{−1}) and are the residual or large inflow and outflow speeds (∼10 m s^{−1}) so small fractional errors during the calculation lead to large fractional errors in the net flow. The coordinate system staggering in the numerical model, where the wind velocity components and pressure use coordinate systems offset from each other, is a further source of error especially in regions with large horizontal (especially radial) gradients of pressure and wind speed. We found that averaging the calculated values of *χ* are then available at times *t* = 0.5, 1.5, 2.5, …, h. All values of *χ* reported here are calculated using this method. To have TC state variables which are temporally aligned, the central pressure deficit fraction, *R*max are both evaluated at these times (*t* = 0.5, 1.5, 2.5, …, h) by linear interpolation from their on-the-hour values. We note that it may be possible to calculate a stable *χ*_{0} which then, via Eq. (15), provides the time dependent form of the central pressure deficit fraction. We cannot therefore consider the model fully “predictive” when

The parameter *k* is evaluated by regression from *χ* using Eq. (12). Since this cannot be observed or calculated from initial conditions it is also a nonpredictive parameter.

Given that the model depends on both observable (*χ*_{0}, *k*) parameters we test the performance in stages to investigate the sensitivity of its performance to the nonobservable quantities. We begin by estimating *χ*_{0} and *k* per experiment and evaluating the time dependent performance and half-life given by the model. Then we use a constant *k* equal to the mean *k* across the experiments calculated in the previous step to evaluate performance. Finally, we test the performance using a constant mean *k* and a constant mean *χ*_{0}. This final test is the fairest test of the model’s predictive skill since it uses only observable quantities.

## 4. Results

### a. Simulations overview

An overview of the Spinup run and the Control and Dry case experiments is shown in Fig. 1. The idealized “Spinup” TC initially contracts and intensifies during the first 4 days with the central pressure deficit fraction ^{−1} at a radius Rmax of 19 km. Following this Vmax decays relatively quickly over the next 3 days to 45 m s^{−1}, then slowly until the end of simulation at 15 days to 39 m s^{−1}.

We focus our analysis on the first 12 h of the Control and Dry sets of land-induced decay experiments as in most cases this is long enough for

Analysis of the decay of _{0}, and *χ*_{0}) refer to their values evaluated at 1.5 h after land is imposed in the numerical experiments. We note that the “shock” response may be shorter than 1 h, but allowing for 1 h was sufficient in all cases to ensure subsequent decay was monotonic and smooth. Initial shock responses in Vmax have been previously observed by Chen and Chavas (2020) in their surface roughening experiments. In our experiments Vmax always decreased over the first hour. Initial parameter values, both *t* = 0 and 1.5 h, for all experiments are provided in Table 1.

Exp is the experiments type: Control is normal; Dry is with all moisture removed from the atmosphere. *t _{lf}* is the time in days after the initialization of the Spinup run at which the land experiment began. Vmax is the maximum wind speed at 10 m (in m s

^{−1}), Rmax is the radius of maximum wind speed at 10 m (in km),

*χ*is the column radial wind speed (in m s

^{−1}) with subscript

*lf*denoting value at time land is imposed in simulation (

*t*= 0 h) and subscript 0 denoting “initial” values used in model taken at

*t*= 1.5 h.

*k*is the estimated decay exponent [Eq. (12)] and

*k*.

*t*

_{1/2}is the simulation half-life of the pressure deficit.

*k*.

*k*and

*χ*

_{0}.

### b. Control experiments

The half-life of the land-induced decay simulations is the time taken for *P*_{0} divided by initial radius of maximum wind speed Rmax_{0} for the Control experiments in Fig. 2. We compare our decay coefficients with their values in the Gulf Coast and Florida Peninsular regions because they have a large range of Rmax_{0}, which give better estimates of the dependence of decay coefficient on Rmax_{0} (Vickery 2005). Our simulated decay coefficients are similar to the weighted average of their Gulf Coast and Florida Peninsular values and show a similar dependence on Rma*x*_{0} divided by Δ*P*_{0}. We also show the weighted average of all regions which deviated further from our simulations.

Central pressure deficit exponential decay coefficient, *P*_{0} divided by initial radius of maximum wind speed *R*max_{0} for the decay experiments (case numbers in red). Solid line is linear fit to our experiments (*R*^{2} = 0.76). Dashed line is fit from Vickery (2005) Gulf Coast and Florida Peninsula region; dotted line is all regions.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

Central pressure deficit exponential decay coefficient, *P*_{0} divided by initial radius of maximum wind speed *R*max_{0} for the decay experiments (case numbers in red). Solid line is linear fit to our experiments (*R*^{2} = 0.76). Dashed line is fit from Vickery (2005) Gulf Coast and Florida Peninsula region; dotted line is all regions.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

Central pressure deficit exponential decay coefficient, *P*_{0} divided by initial radius of maximum wind speed *R*max_{0} for the decay experiments (case numbers in red). Solid line is linear fit to our experiments (*R*^{2} = 0.76). Dashed line is fit from Vickery (2005) Gulf Coast and Florida Peninsula region; dotted line is all regions.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

The decay half-life is shown as a function of initial TC parameters in Fig. 3 for the Control cases. The half-life is strongly correlated with Rmax_{0} with an *R*^{2} value of 0.76 (*p* values < 0.001). *R*^{2} = 0.27, *p* value = 0.07). Excluding case 2 results in *χ*_{0} is uncorrelated with half-life in the Control cases.

Simulated TC half-life of the Control cases against (a) initial Rmax, (b) initial *χ*, and (d)–(f) model estimated half-lives with subscripts

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

Simulated TC half-life of the Control cases against (a) initial Rmax, (b) initial *χ*, and (d)–(f) model estimated half-lives with subscripts

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

Simulated TC half-life of the Control cases against (a) initial Rmax, (b) initial *χ*, and (d)–(f) model estimated half-lives with subscripts

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

#### 1) Model performance

The physical pressure filling framework developed in section 2 permits an estimation of decay half-life, _{0}, *χ*_{0}, and *k*. We first use values of *χ*_{0}, and *k* estimated for each experiment to calculate the model half-life and compare these to the simulation half-life (Fig. 3d). The model and simulation values are very highly correlated with an *R*^{2} value of 0.96 and RMSE of 1.1 h. There is a mean positive bias in the model estimates of 0.8 h. We then use a constant *k* equal to the mean value of *k* for the Control experiments (*R*^{2} value of 0.82. The RMSE is 1.8 h and the bias is slightly larger at 0.94 h. We can even make a further simplification by assuming both a constant (mean across cases) *k* and *χ*_{0} (*R*^{2} = 0.69, RMSE = 3.0 h and a bias of 0.97 h (Fig. 3f). This simplification increases the error but still appears useful in the Control cases.

As noted above, “initial” values were taken at *t* = 1.5 h after landfall in the simulations because of the apparent early shock response. If we use values of Rmax and *t* = 0) with constant mean *k* and *χ*_{0} as above this slightly increases the correlation of the model estimates with simulation half-lives (*R*^{2} = 0.74) and increases the RMSE (3.06 h) and bias (2.08 h).

#### 2) Model validation

The initial density-weighted column radial wind speed *r* for the Control cases in Fig. 4a. There is an approximately linear increase of _{0}. Goodness-of-fit *R*^{2} values for modeling *r* during the decay are shown in Fig. 4b. These remain high and in nearly all cases stay above 0.98 throughout the decay. That *r* from Rmax_{0} to near *r* = 0 throughout the decay means _{0}, *χ*, divided by Rmax_{0} is an appropriate approximation of the radial gradient of

(a) The density-weighted column-mean radial wind speed *r* at *t* = 1.5 h from the TC center to the radius of maximum wind speed. (b) Goodness-of-fit *R*^{2} of *r* as a function of time. (c) Decay of simulated TC column radial wind speed at radius of maximum wind speed, *χ*, as a function of *k* and goodness-of-fit parameter *R*^{2}. *k* and (f) with both constant mean *k* and *χ*_{0}.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

(a) The density-weighted column-mean radial wind speed *r* at *t* = 1.5 h from the TC center to the radius of maximum wind speed. (b) Goodness-of-fit *R*^{2} of *r* as a function of time. (c) Decay of simulated TC column radial wind speed at radius of maximum wind speed, *χ*, as a function of *k* and goodness-of-fit parameter *R*^{2}. *k* and (f) with both constant mean *k* and *χ*_{0}.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

(a) The density-weighted column-mean radial wind speed *r* at *t* = 1.5 h from the TC center to the radius of maximum wind speed. (b) Goodness-of-fit *R*^{2} of *r* as a function of time. (c) Decay of simulated TC column radial wind speed at radius of maximum wind speed, *χ*, as a function of *k* and goodness-of-fit parameter *R*^{2}. *k* and (f) with both constant mean *k* and *χ*_{0}.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

The second assumption parameterizes the core column speed as a function of pressure deficit. The parameter *k* was estimated by using least squares regression and Eq. (12) for each decay case on values from *t* = 1.5 to 11.5 h after the imposition of land. The mean value of *k* was 1.30 in the Control cases with a standard deviation of 0.35. The mean goodness-of-fit parameter *R*^{2} in the Control cases was 0.89 suggesting the assumption that the core column speed *χ* decays as *k* and *R*^{2} for each case are listed in Table 1. *χ* as a function of

Decay of *k* and *χ*_{0} evaluated per experiment is shown in Fig. 4d. The mean RMSE across the Control experiments was 0.0012. When a constant mean *k* is used the RMSE increases to 0.0020 (Fig. 4e). Finally, when using both constant mean *k* and *χ*_{0}, the mean RMSE increases to 0.0028 (Fig. 4f).

### c. Dry experiments

The Dry runs decayed at twice the rate with a mean half-life of 4.8 h. The half-life for each decay case is shown in Table 1. The decay half-life is shown as a function of initial TC parameters in Fig. 5 for the Dry cases. The half-life is strongly correlated with Rmax_{0} with an *R*^{2} value of 0.97 (*p* values < 0.001). *R*^{2} = 0.12, *p* value = 0.25). Again, excluding case 2, which may appear to be an outlier (Fig. 5b), results in *χ*_{0} is moderately correlated with the half-life in the Dry cases (*R*^{2} = 0.54).

As in Fig. 3, but for the Dry cases with

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

As in Fig. 3, but for the Dry cases with

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

As in Fig. 3, but for the Dry cases with

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

#### 1) Model performance

We use Eq. (18) using *k* and *χ*_{0} evaluated per experiment to calculate model estimate half-lives and compare these to the simulation half-lives in Fig. 5d. The quantities are very highly correlated with *R*^{2} = 099 and an RMSE of 0.75 h. The model half-life has a mean positive bias of 0.65 h. We then use a constant *k* equal to the mean value of *k* for the Dry experiments (*R*^{2} value of 0.97. The RMSE is 0.68 h and the bias is at 0.61 h. We can again make the simplification of using a both a constant *k* and *χ*_{0} (mean across cases) for the Dry experiments (*R*^{2} = 0.93, RMSE = 1.57 h, and a bias of 0.81 h. This simplification work remarkably well for the Dry cases.

#### 2) Model validation

As for the Control cases we present model validation analysis in Fig. 6. The initial column radial wind speed *r*, in both the Control and Dry cases in Fig. 6a. There is an approximately linear increase of _{0}. Goodness-of-fit *R*^{2} values for modeling *r* during the decay remain high and in nearly all cases stay above 0.98.

As in Fig. 4, but for the Dry experiments.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

As in Fig. 4, but for the Dry experiments.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

As in Fig. 4, but for the Dry experiments.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

The mean value of *k* was 1.63 with a standard deviation of 0.09. The mean goodness-of-fit parameter *R*^{2} in the Dry cases and 0.99 suggesting the assumption that the core column speed *χ* decays as *k* and *R*^{2} for each case are listed in Table 1. We show *χ* as a function of

Decay of *k* and *χ*_{0} evaluated per experiment is shown in Fig. 6d. The mean RMSE across the Control experiments was 0.0029. When a constant mean *k* is used the RMSE increases to 0.0030 (Fig. 6e). Finally, when using both constant mean *k* and *χ*_{0} the mean RMSE increases to 0.0040 (Fig. 6f).

### d. Decay structure

We now examine the structure of the azimuthal average tangential wind speed *υ _{t}* near the start of the decay and 10 h into the decay for a control case (Figs. 7a,b). We choose this case as a representative sample and find similar behavior across all cases. In the Control case at

*t*= 1 h we find a well-defined maximum

*υ*at a height of approximately 1.5 km at approximately 20 km radius with speeds exceeding 90 m s

_{t}^{−1}. By

*t*= 10 h, maximum speeds have reduced to around 40 m s

^{−1}but there are three local maxima: one at the top of the inflow layer close to Rmax and two broad midlevel maxima at approximately 20 and 80 km. Similar analysis of azimuthal average radial wind speed

*υ*shows that maximum boundary layer inflow speeds decay from over 40 to 14 m s

_{t}^{−1}over the period

*t*= 1 to 10 h (Figs. 7c,d). At

*t*= 1 h, a substantial outflow channel is apparent beginning at around 1 km near Rmax and rising outwards to the upper levels. Maximum outflow speeds reach 28 m s

^{−1}. By

*t*= 10 h, the maximum outflow speed has reduced to 5 m s

^{−1}.

Azimuthally averaged tangential wind speed *υ _{t}* and radial wind speed

*υ*as a function of radius

_{r}*r*and height

*z*for the Control case 5 at (a),(c) 1 and (b),(d) 10 h after simulated landfall. Dotted lines are zero contours; dashed contours show intervals of 20 m s

^{−1}.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

Azimuthally averaged tangential wind speed *υ _{t}* and radial wind speed

*υ*as a function of radius

_{r}*r*and height

*z*for the Control case 5 at (a),(c) 1 and (b),(d) 10 h after simulated landfall. Dotted lines are zero contours; dashed contours show intervals of 20 m s

^{−1}.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

Azimuthally averaged tangential wind speed *υ _{t}* and radial wind speed

*υ*as a function of radius

_{r}*r*and height

*z*for the Control case 5 at (a),(c) 1 and (b),(d) 10 h after simulated landfall. Dotted lines are zero contours; dashed contours show intervals of 20 m s

^{−1}.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

In the Dry case, the situation at *t* = 1 h is similar to the Control with a low-level maximum *υ _{t}* albeit with attenuated speeds (Fig. 8a). However, by

*t*= 10 h the picture is dramatically different, low-level speeds have largely been reduced to below 10 m s

^{−1}. However, a broad midlevel wind speed maximum centered at a height of about 7 km and at a radius of 40 km has maintained with speeds above 50 m s

^{−1}(Fig. 8b). Analysis of

*υ*shows that at 1 h the maximum inflow speed is 26 m s

_{t}^{−1}, and outflow speeds are below 10 m s

^{−1}, substantially reduced compared to the Control case. By 10 h, all radial wind speeds have reduced to only a few meters per second even though a substantial vortex is still present in the midlevels (Figs. 8c,d).

As in Fig. 7, but for the Dry case 5.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

As in Fig. 7, but for the Dry case 5.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

As in Fig. 7, but for the Dry case 5.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

The maximum wind speeds at a height 10 m and 7 km is shown as a function of time for the Control and Dry case 5 in Fig. 9. The 10 m and 7 km maximum wind speeds decay at a similar rate in the Control case falling to around half their initial values by around *t* = 10 h. However, in the Dry case the 7 km maximum wind speed decays at a much slower rate that the 10 m wind speed. The midlevel wind speeds is decoupled from the surface level wind speed. This phenomenon is apparent from immediately after the onset of decay until at least *t* = 24 h when the maximum wind speed is 44 m s^{−1} at 7 km but only 6 m s^{−1} at 10 m. In summary, the surface wind speed is coupled to the midlevel wind in the Control case but decoupled in the Dry case.

Maximum wind speed Vma*x* as a function of time after simulated landfall at heights of 10 m and 7 km for the Control and Dry case 5.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

Maximum wind speed Vma*x* as a function of time after simulated landfall at heights of 10 m and 7 km for the Control and Dry case 5.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

Maximum wind speed Vma*x* as a function of time after simulated landfall at heights of 10 m and 7 km for the Control and Dry case 5.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0196.1

## 5. Discussion

### a. Simulations

Our set of idealized land-induced decay numerical experiments cover a wide range of initial conditions with a range of Δ*P*_{0}/Rmax_{0} similar to those observations reported by Vickery (2005). The decay constants of the simulations also have a similar dependence on Δ*P*_{0}/Rmax_{0} as reported by Vickery (2005). This close match is perhaps surprising given the idealized nature of our experiments (instantaneous land, no translation, no shear) but gives confidence that our set of experiments does represent the range of TCs making landfall in nature and capture at least some of the important physical processes which determine the rates at which they then decay. Therefore, the simulations are useful to validate the assumptions of the analytic model.

We find a strong dependence of simulated decay half-life of central pressure deficit on the initial radius of maximum wind speed Rmax_{0} in both the Control and Dry cases (Figs. 3 and 5). This is especially clear in the Dry cases (*R*^{2} = 0.97). In the more realistic Control cases the presence of convection fundamentally changes the decay process (see section 1) leading to a noisier dependence of the half-life on Rmax_{0} (*R*^{2} = 0.76). This dependence of pressure deficit decay rate increasing inversely with the radius can be understood in terms of the dominant role of mass convergence during the decay.

Control and Dry decay experiments initialized during the early stage of the Spinup TC (case number < 7) have similar half-lives with very different *χ*_{0}, so variation in Rmax_{0} dominates the changes in decay. In late stage experiments, when _{0} is tempered by an increasing *χ*_{0}. Overall this leads to the very strong correlation of *R*max_{0} with the simulated half-life in our experiments. The relatively weak correlation between half-life and *R*^{2} = 0.27) and Dry (*R*^{2} = 0.12) simulations is certainly related to the limited variability of

While the analytic decay model predicts a linear dependence of half-life on the Rmax and *χ*_{0}, most of the “skill” in the half-life prediction in our sets of simulations comes from the variability of Rmax_{0} (see correlations of half-life with Rmax_{0}, *χ*_{0} in Fig. 3). This can be partly explained through analysis of the coefficient of variation (CV) of the parameters. The CV of Rmax_{0} is approximately double that of both *χ*_{0} in both the Control and Dry cases. Note this is not necessarily reflective of the statistics of real world landfalling TCs. Furthermore, Rmax_{0} and *χ*_{0} are not significantly correlated in the Control experiments (*R*^{2} = 0.15, *p* = 0.18) and have very different relationships with half-life. This supports our choice to parameterize the radial gradient of column-integrated radial wind speed *dV*/*dr* in terms of Rmax_{0} and *χ* in Eq. (9) allowing Rmax_{0} and *χ*_{0} to be treated as independent variables controlling the rate of pressure filling and ultimately the half-life of the decay.

*k*is not directly observable and here is estimated from the decay simulation data. Hence estimating the decay half-life using the model requires knowledge of quantities not observable at the time of landfall. It is worth noting however that the model can still produce useful estimates of decay half-life even if constant estimates of

*k*and

*χ*

_{0}are used. The model with

*χ*

_{0}= 0.026 m s

^{−1}and

*k*= 1.30 may therefore be a practical zero-order estimate of real decay half-lives without knowledge of

*χ*

_{0}and

*k*. This half-life equation then depends only on the observable initial size and pressure deficit:

_{0}is the initial radius of maximum surface wind speed in kilometers and the dimensions of

*χ*have been absorbed into the constant giving it units of hours per kilometer. This is the simplest form of the decay within our framework.

#### Role of moisture

We found that on average the Dry experiments half-life (4.8 h) decayed at approximately twice the rate of the Control experiments (10.5 h). Figure 8 shows that in a Dry case, a strong midlevel vortex persists after 10 h by which time surface friction has destroyed boundary layer wind speeds. The boundary layer has become decoupled from the vortex above it and the frictional braking effects from the surface roughness are felt only within this shallow layer. However, in the Control case 5 the boundary layer is still coupled to the layer above with similar wind speeds seen in both after 10 h. The Dry decoupling results from the lack of convection and much reduced secondary circulation. The Control case transmits the frictional braking felt in the boundary layer into the vortex above though the vertical ascent of lower angular momentum. Thus the height of the vortex experiencing frictional braking is much larger in the Control case. Indeed, we can use the spindown half-life formula proposed by Eliassen and Lystad (1977) and Phillipson and Toumi (2021) to infer vortex height of 5.0 km for the Dry and 12.1 km in the Control cases.

### b. Analytic model

The new analytic physically based framework allows estimates of decay half-life using three TC initial state parameters and a fitted parameter *k*. It differs from other decay models in that it starts with considerations of mass flux and pressure rather than making only empirical assumptions. The model shows that the initial size of the inner core, represented as Rmax, is a key parameter controlling the rate of land-induced decay, with the rate of decay inversely proportional to size. The choice of Rmax itself is somewhat arbitrary and any quantity reflecting the spatial scale of the inner core would give similar results. Rmax was chosen as it is a well-known, frequently used TC parameter and has been recorded for decades in historical TC databases. The linear dependence of the decay rate on Rmax predicted by the model offers a theoretical explanation for the empirical result of Vickery (2005), who found the same linear dependence in the historical record of Atlantic hurricanes. Following Malkin (1959) and DeMaria et al. (2006) they suggest that “smaller storms would tend to decay more rapidly than larger storms since a relatively larger portion of the core of the storm is removed from the energy source more rapidly than in the case of a larger storm.” Our simulations confirm the observations, but our model shows that the decay rate is a more fundamental property of the TC vortex geometry and that for the same initial central pressure deficit and core column speed, a large storm will decay slower than a smaller storm.

Existing theoretical wind speed decay models for TCs are framed in terms of the frictional interaction of the land surface with the TC, and how this leads to a reduction in surface wind speed over time. A theoretical model for turbulent surface drag–driven spindown of cyclones (Eliassen 1971; Eliassen and Lystad 1977) is also successful in predicting the decay rate of hurricane strength Vmax over the ocean (Montgomery et al. 2001) and land (Phillipson and Toumi 2021). That model predicts an algebraic form for the decay of angular momentum (or tangential speed) and a half-life which is proportional to the vortex height and inversely proportional to the drag coefficient *C _{D}* and the initial tangential speed. There is no explicit dependence of Vmax decay on size and this appears fundamentally different to our pressure model. Our model considers the mass in the column and the Eliassen model aims to explain the boundary layer wind at the surface.

The surface pressure model also introduces a new concept, the core column speed *χ*—a radial wind speed representative of the whole column at Rmax_{0}. The column theoretically captures all mass fluxes which make significant contributions to surface pressure tendency. We note that as shown in Fig. 1 the core size as measured by Rmax tends to shrink during the first hours of the land-induced decay experiments. While this phenomenon is interesting in its own right and may prove important in a developing a full understanding of postlandfall dynamics, it is not directly relevant to the decay model we present because *χ* is defined at the initial radius of maximum winds. We only require that the linearity assumption made in Eq. (9) remains valid out to Rmax_{0} throughout the decay process we are modeling.

*χ*_{0} is clearly a simplification of the many factors that affect radial flow at all heights. In practice *χ*_{0} is a small residual of the difference between the large outflow speeds at upper levels and large inflow speeds and will therefore depend on many physical processes governing inflow and outflow during the decay. Partly because of this difference of large numbers, the initial core column speed is not easily observable and even calculating it from instantaneous simulation output is challenging. This means that at present we could not evaluate *χ*_{0} in terms of other, more readily available physical quantities. For example, we observe that *χ*_{0} is dependent on the TC moisture through latent heating. In the Dry experiments, where convective heating is suppressed, the initial core column speed is almost double that in the Control cases. Finally, even just assuming a constant *χ*_{0} value still enables useful predictions of the half-life.

Equation (8) shows that the central pressure tendency is proportional to the central pressure. This leads to the result that the decay half-life is proportional to the initial pressure deficit [Eq. (18)]. That is, for a given initial radius of maximum wind speed and core column speed, TCs with a larger central pressure deficit, therefore smaller central pressure, decay more slowly. Larger pressure deficits take more time to fill and this makes intuitive sense. We can only expect to see this relationship in observations of landfalling TCs if we control for initial radius of maximum wind speed and core column speed (not observed). Even in our own small set of landfall simulations initial central pressure deficit is weakly anticorrelated with half-life. In observational studies Vickery and Twisdale (1995) and Vickery (2005) report that the decay rate increases with initial central pressure deficit. However, these studies did not clearly separate out the simultaneous effect of size. At least some of the strongest storms could also be small. We note that the simulations (and thus our model) agree with the observations of the decay constants as function of the ratio of the pressure deficit and radius reported by Vickery (2005) (see Fig. 2). Kaplan and Demaria (1995) show that storms with stronger surface winds (Vmax) decay more rapidly in agreement with the decay models of Phillipson and Toumi (2021) and Chen and Chavas (2021). Neither of these studies account for the effect of TC size. Furthermore, the decay in surface winds and surface pressure are at least partially decoupled, hence traditional wind–pressure relationships are not applicable during landfall decay. We may therefore expect decay rates of surface wind (Vmax) and pressure deficit to have different controlling factors.

#### 1) Model assumptions

The derivation of our decay model makes two main assumptions. The first is that the column speed varies linearly with radius [Eq. (9)] from the center to Rmax_{0}. This assumption appears justified by the results of the simulations. High *R*^{2} values throughout the decay in the Control and the Dry cases show that *χ* and *r* are highly linear throughout the decay within the core. This suggests the assumption is robust as the cases cover a wide range of initial conditions and decay environments. However, the assumption does not hold during the initial shock response caused by our simulation experimental design, with large nonlinear fluctuations near the center (not shown).

The second assumption, that *χ* decays as *k* parameter was 1.30, reasonably close to 1, at which the decay of central pressure deficit would appear exponential. Vickery (2005) among others assume exponential decays in the historical record of landfalling TCs. The mean *k* in the Dry cases was significantly higher at 1.63. This difference in *k* only leads to a 15% increase in *β* from 0.77 to 0.87 from the Control to the Dry simulations due to the form of Eq. (19). This change actually acts to increase the estimated half-life by 13%. The large reduction in half-lives in the Dry simulations is due to much larger *χ*_{0} acting in opposition to the slightly larger values of *k*.

A larger *k* in the Dry case reflects the faster radial inflow reduction for a similar pressure deficit reduction during the decay compared to the Control decay. In the Dry case almost all the filling happens in the boundary layer and this column speed is rapidly diminished as the inflow collapses. However, the estimated half-life is relatively insensitive to *k*. We find using a mean value of *k* for each of the Control and Dry experiment sets can be sufficient to predict the half-life (as in Figs. 3d and 5d).

The approximation made in obtaining Eq. (14) depends on the central pressure deficit fraction being small (

#### 2) Wider implications

It is plausible that this pressure filling framework has wider applications as a zero-order model. The decay of tropical cyclones over the ocean can also be considered in this framework. The detailed physical causes may be lower sea surface temperatures, wind shear and/or interactions with midlatitude weather. In the model the effect of these physical causes is ultimately captured and collapsed into the column speed. Similarly, in the case of extratropical cyclones we suspect the model is also applicable. The physical cause of the column speed may vary but the half-life linear dependence on initial pressure deficit and size (due to the critical role of convergence) should be robust as they are derived from mass continuity. More research is required to confirm this.

Here we find that the pressure filling half-life depends on the TC state itself, in particular the initial pressure and size. There will therefore be a strong dependence on initial conditions, the state of the TC at the time of landfall. This is quite different from a simple exponential decay where the time constant would just be dependent on environmental conditions and independent of the initial condition. It is not sufficient for forecasts just to know just the environmental conditions, which will affect, for example, the column speed. The strong dependence of the actual time evolution on the initial TC conditions makes it clear that forecasting land decay is very challenging.

Given the limitations of the set of experiments examined here (idealized, limited sampling of initial states) a natural extension to this work would be to test the performance of this new model on a comprehensive set of realistic numerical simulations of landfalling TCs.

## 6. Conclusions

We propose a simple physically based analytical model of pressure filling after landfall and test it against simulations. The simulated half-life of the TC pressure deficit decay increases with TC size in a set of idealized land numerical simulations. When these simulations are simplified, by removing all moisture and therefore latent heating, the decay rate is doubled and the dependence on the size is made even clearer. Despite its simplifications the analytic model does give useful insight. A fundamental insight is that the central pressure deficit half-life is proportional to the initial core size. The initial pressure deficit and the initial core column speed are also important. This framework is quite different from a simple exponential decay.

## Acknowledgments.

The research was supported by the UK Centre for Greening of Finance and Investment (UKRI-NE/V017756/1), the Singapore Green Finance Centre, and the Leverhulme Trust. This work was supported by the Natural Environment Research Council/UKRI (NE/V017756/1) and the U.K.–China Research and Innovation Partnership Fund through the Met Office Climate Science for Service Partnership (CSSP) China as part of the Newton Fund.

## Data availability statement.

Numerical simulation data are available from the authors upon request.

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