A physical model of tropical cyclone central pressure filling at landfall

We derive a simple physically based analytic model which describes the pressure filling of a tropical cyclone (TC) over land. Starting from the axisymmetric mass continuity equation in cylindrical coordinates we derive that the half-life decay of the pressure deficit between the environment and TC centre is proportional to the initial radius of maximum surface wind speed. The initial pressure deficit and column-mean radial inflow speed into the core are the other key variables. The assumptions made in deriving the model are validated against idealised numerical simulations of TC decay over land. Decay half-lives predicted from a range of initial TC states are tested against the idealized simulations and are in good agreement. Dry idealised TC decay simulations show that without latent convective heating, the boundary layer decouples from the vortex above leading to a fast decay of surface winds while a mid-level vortex persists.


Introduction
During TC decay, in the vicinity of the inner core, V is negative (directed toward the centre), 71 filling the pressure deficit. Since the surface pressure, , at a given radius is given by, we can write the tendency of the average pressure within a cylinder of radius as, The tendency of the central pressure, , is then, and since V must tend to zero as tends to zero to avoid infinite central pressure tendency, and 75 invoking L'Hôpitals's rule, giving, We now make the assumption that the gradient of V , evaluated at the central limit, may be 78 linearised: where 0 is the radius of maximum wind speed at = 0 and is the density-weighted column 80 mean radial wind speed at 0 ( = V ( 0 )) which we refer to as the 'column speed' and 81 will typically have a negative sign (directed towards the centre). It is not immediately obvious 82 that this linear gradient approximation is appropriate and this assumption will be examined later 83 in section 2. It is important to note that we do not require to be constant throughout the 84 decay, as is defined at the initial , 0 . We only require that V is linear in out 85 to 0 throughout the decay. It is also worth highlighting that this is the point at which the 86 size of the TC core is introduced explicitly via 0 . The fundamental quantity controlling 87 the central pressure tendency in Equation 8 is the radial gradient of column speed but here we 88 are effectively parameterising this single quantity in terms of two independent terms, the column 89 speed and the initial size of the core. This separation is motivated by the fact that during TC decay 90 the column speed must tend to zero as the TC fills while the core size may remain finite. This sensitive to this choice as long as the above linearity assumption remains valid because is defined 96 as V at 0 and always appears in a ratio with 0 . This is also true of the precise definition 97 of itself. We validate the model here using defined as the radius of maximum 10-m which can be integrated to give an analytical model for the decay of˜as a function of time, It should be noted that for the special case of = 2, Equation 15 simplifies to an algebraic decay 118 form and in the limit = 1, the decay is exponential, Equation 15 then yields an estimate of the central pressure deficit half-life in terms of and the 120 initial state parameters, The half-life therefore depends on three TC state variables: the core size, the initial pressure deficit 123 and the column speed. Interestingly, although the form of the decay for = 1 is exponential, the 124 initial condition,˜0, also appears in the decay constant, and therefore the half-life. We note that 125 the value of will affect the magnitude of the half-life but not its dependence on the three TC state 126 variables.

127
The above model makes two primary assumptions: i) the column-integrated inflow speed is 128 linear in from the centre until , and ii) the core column-integrated speed decays as˜.

129
These assumptions allow us to express the time dependent pressure deficit fraction as a function V is evaluated as above.

206
The parameter is evaluated by regression from˜and the inferred values of using Equation 207 12. Since this cannot be observed or calculated from initial conditions it is also a non-predictive 208 parameter.

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Given that the model depends on both observable ( 0 ,˜0) and non-observable (    We focus our analysis on the first 12 hours of the Control and Dry sets of land-induced decay 227 experiments as in most cases this is long enough for˜to decay to half its initial value. During this 228 period both˜and decay in an exponential-like manner. In the Dry cases tends to 229 decrease to half its initial value over this period whereas is more stable in the Control cases.

230
In the Dry experiments, after eight hours tends to increase noisily.

231
Analysis of the decay of˜reveals that the behaviour during the first hour of the experiments 232 is qualitatively different to the hours following. During the first hour many cases experience 233 a "shock" of transient deepening pressure deficit but then all cases decay monotonically and 234 relatively smoothly. This phenomenon occurs in both Control and Dry sets of experiments but is 235 more prominent in the Dry cases. We do not further examine this "shock" response here and in 236 all following analysis 'Initial' values of TC parameters (˜0, 0 , and 0 ) refer to their values 237 evaluated at 1.5 h after land is imposed in the numerical experiments. We note that the "shock" 238 response may be shorter than one hour, but allowing for one hour was sufficient in all cases to 239 ensure subsequent decay was monotonic and smooth. Initial shock responses in have been .

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The second assumption parameterises the core column speed as a function of pressure deficit.

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The parameter was estimated by using least squares regression and Equation 12

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The values of and 2 for each case are listed in Table 1. as a function of˜with fitted decay 334 curves are shown in Figure 6c. We now examine the structure of the azimuthal average tangential wind speed, , near the start 342 of the decay and 10 hours into the decay for a control case (Figure 7a,b). We choose this case as Dry case the 7 km maximum wind speed decays at a much slower rate that the 10 m wind speed.

365
The mid-level wind speeds is decoupled from the surface level wind speed. This phenomenon is 366 apparent from immediately after the onset of decay until at least = 24 h when the maximum wind 367 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 368 to the mid-level wind in the Control case but decoupled in the Dry case. of the important physical processes which determine the rates at which they then decay. Therefore 378 the simulations are useful to validate the assumptions of the analytic model. 379 We find a strong dependence of simulated decay half-life of central pressure deficit on the initial 380 radius of maximum wind speed, 0 , in both the Control and Dry cases (Figures 3 and 5). This that "smaller storms would tend to decay more rapidly than larger storms since a relatively larger 446 portion of the core of the storm is removed from the energy source more rapidly than in the case 447 of a larger storm". Our simulations confirm the observations, but our model shows that the decay 448 rate is a more fundamental property of the TC vortex geometry and that for the same initial central 449 pressure deficit and core column speed, a large storm will decay slower than a smaller storm. inflow speeds and will therefore depend on many physical processes governing inflow and outflow 472 during the decay. Partly because of this difference of large numbers, the initial core column speed 473 is not easily observable and even calculating it from instantaneous simulation output is challenging.

474
This means that at present we could not evaluateˆ1 /2 directly given standard real-world TC metrics.

475
However, given that we expect it to be governed by a combination of near-surface inflow related suggests the assumption is robust as the cases cover a wide range of initial conditions and decay 506 environments. However, the assumption does not hold during the initial shock response caused by 507 our simulation experimental design, with large non-linear fluctuations near the centre (not shown).

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The second assumption, that decays as˜(Equation 12), also seems well-founded. It is plausible that this pressure filling framework has wider applications as a zero order model.

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The decay of tropical cyclones over the ocean can also be considered in this framework. The be robust as they are derived from mass continuity. More research is required to confirm this.

536
Here we find that the pressure filling half-life depends on the TC state itself, in particular the 537 initial pressure and size. There will therefore be a strong dependence on initial conditions, the state 538 of the TC at the time of landfall. This is quite different from a simple exponential decay where the 539 time constant would just be dependent on environmental conditions and independent of the initial 540 condition. It is not sufficient for forecasts just to know just the environmental conditions, which 541 will affect, for example, the column speed. is the maximum wind speed at 10 m in m s −1 , is the radius of maximum wind speed at 10 m in km, is the central pressure deficit fraction, is the column radial wind speed in m s −1 with subscript denoting value at time land is imposed in simulation ( = 0 h) and subscript 0 denoting 'initial' values used in model taken at = 1.5 h. is the estimated decay exponent (Equation 12) and 2 is the quality of fit parameter for the estimation of .