a. Phase-space reconstruction

In a strict mathematical sense, the governing equations for TCs are not closed due to our incomplete understanding of TC dynamics and thermodynamics, especially at the convective and smaller scales. As a consequence, all current representations of TC processes in numerical models must employ some empirical parameterizations that only approximate the true but unknown TC physics. These physical parameterizations generally contain many uncertainties and simplifications, which prevent one from fully understanding TC development.

Early works by Takens and many others (Takens 1981; Brock 1986; Theiler 1987; Sugihara and May 1990; Sugihara 1994; Casdagli 1992) have shown, however, that the dynamics of a nonlinear system can be reconstructed from a single time series of a state variable under some specific conditions, even in the absence of complete governing equations for the system. Assuming that a nonlinear system possesses low-dimensional chaos at its statistically stationary state, it is in fact possible to examine multidimensional phase portraits of a chaotic attractor in the system by reconstructing the attractor in the phase space of time-lagged coordinates. With this phase-space reconstruction, different invariants of the original chaotic attractor can be effectively obtained once the embedding dimension and time delay are properly chosen (Kantz and Schreiber 2003).

There are a range of techniques that have been proposed to find a proper embedding dimension and time delay for phase-space reconstruction such as the averaged mutual information, autoregression, or false nearest neighborhood (Fraser and Swinney 1986; Sugihara 1994; Kantz and Schreiber 2003; Wallot and Monster 2018). These methods all share a common principle that basic invariants of a chaotic attractor must be intrinsic, regardless of the reconstruction methods if the embedding dimension and time lag are sufficiently good. Among several approaches to detect chaos in a phase space reconstructed from a time series, we will present in this study three measures that can characterize deterministic chaos, which include the largest Lyapunov exponent (LLE), the Sugihara and May’s (1990) correlation (SMC) curve, and the correlation dimension.

For the LLE measure, an early algorithm for computing LLE from a given time series was first proposed by Wolf et al. (1985), which has been later improved in many subsequent studies (Rosenstein et al. 1993; Kantz 1994; Balcerzak et al. 2018; Awrejcewicz et al. 2018). For our implementation of the LLE algorithm, a modified version of Wolf’s algorithm presented in Brock (1986) was chosen because of its efficiency. The basic steps of Brock’s scheme are summarized below [see the full proof of the LLE convergence in Brock (1986)]:

• Step 1: From a given time series {a_i}, i = 1, …, N, where N is the number of data sampling, generate a set of m-history $a_t^m\equiv \{a_t,a_{t+\tau },\dots ,a_{t+(m-1)\tau }\},t=1\dots N_m\equiv N-(m-1)\tau$ for the phase-space reconstruction, with a given time delay τ and an embedding dimension m.

• Step 2: Initialize an error growth cycle by finding the nearest neighborhood $a_{t_1}^m$ of the first m-history $a_1^m$ such that $a_{t_1}^m\ne a_1^m$.

• Step 3: Choose a prescribed evolution window q and compute g₁(q) = d₂(1)/d₁(1), where $d_1^{(1)}=\Vert a_{t_1}^m-a_1^m\Vert$ and $d_2^{(1)}=\Vert a_{t_1+q}^m-a_{1+q}^m\Vert$ are distances in the reconstructed phase space with a given metric $\Vert \cdot \Vert$.

• Step 4: Perform a loop from k = 2 to K = max{k|1 + kq ≤ N_m} that repeatedly does the following two main tasks:

1. Find an index t_k of t to minimize a penalty function $p\left(a_t^m-a_{1+(k-1)q}^m,a_{t_{k-1}+q}^m-a_{1+(k-1)q}^m\right)$ defined as follows:

   $\begin{array}{l}p\left(a_t^m-a_{1+(k-1)q}^m,a_{t_{k-1}+q}^m-a_{1+(k-1)q}^m\right)=\Vert a_t^m-a_{1+(k-1)q}^m\Vert \\ +w\left|\theta \left(a_t^m-a_{1+(k-1)q}^m,a_{t_{k-1}+q}^m-a_{1+(k-1)q}^m\right)\right|,\end{array}$

   where w is a weighted parameter for the deviation angle θ.

2. Compute and store the divergence rate of the kth loop defined as g_k(q) = d₂(k)/d₁(k), where $d_1(k)=\Vert a_{t_k}^m-a_{1+(k-1)q}^m\Vert$ and $d_2(k)=\Vert a_{t_k+q}^m-a_{1+kq}^m\Vert$.

• Step 5: Finally, compute LLE λ_q by averaging all g_k(q) as ${\lambda }_q=(1/K){\displaystyle {\sum }_{k=1}^K\left\{\ln [g_k(q)]/q\right\}}$.

We should mention that all LLE algorithms assume a priori the values of the embedding dimension m and the time delay τ. These values are generally not known in advance, given a time series of a state output. While one can always search for (m, τ) using existing algorithms such as the false nearest neighbor or mutual information method (Fraser and Swinney 1986; Sugihara 1994; Rhodes and Morari 1997; Wallot and Monster 2018), it should be noted that the above LLE’s algorithm must converge to a correct LLE of a chaotic attractor with a fractal dimension n, if it exists, for m > 2n + 1 as proven in Brock (1986). As such, one can plot λ_q as a function of (m, τ) and search for the values of (m, τ) for which LLE becomes stabilized. This approach of searching for a LLE in the parameter space (m, τ) is chosen in this study, because it can help reduce various prescribed thresholds for (m, τ) in the current LLE algorithms as also discussed in Kantz and Schreiber (2003).

Along with LLE, Sugihara and May (1990) proposed another measure to detect chaos that is also of interest because of its simplicity and effectiveness. The main idea behind Sugihara and May’s (1990) approach is that a chaotic time series should possess limited predictability, whereas true stochastic variation would have no predictability. Practically, this important property of chaotic time series implies that the correlation between the model forecast and observations must decay with time in a chaotic system.

For the sake of completeness, we summarize here the main step to obtain the Sugihara–May correlation as a function of forecast lead time (T) from a given time series. Detailed discussion of this method as well as its variation can be found in Sugihara and May (1990), and Sugihara (1994), and so will not be duplicated here.

• Step 1: Given a time series {a_i}, i = 1, …, N, one first divides it into an “atlas” (or training) set $\mathcal{A}$ and a test set $\mathcal{T}$.

• Step 2: Reconstruct a phase space with a given embedding dimension m by generating the m histories obtained from lagged time series as $a_t^m=\left(a_t,a_{t+\tau },\dots ,a_{t+(m-1)\tau }\right)$ for both sets $\mathcal{A}$ and $\mathcal{T}$.

• Step 3: For each history $a_i^m\in \mathcal{T}$ (the so-called predictee in Sugihara and May) in the m-dimensional space, search for n_b neighboring points in $\mathcal{A}$ with the minimal distance to $a_i^m$ such that the predictee are within a smallest simplex spanned by these n_b neighboring points.

• Step 4: Choose a lead time T, and a prediction for $a_i^m$ at the lead time T can be then obtained by projecting the entire simplex into the future at leading time T, denoted $a_{i(j)}^f(T)$, where j = 1 … n_b. The prediction value at lead time T for $a_i^m$, denoted by ${\overline{a}}_i^f(T)$, is then computed by taking an ensemble average of n_b values of $a_{i(j)}^f(T)$.

• Step 5: Construct a pair between the prediction $a_i^f(T)$ and the actual value of $a_i^m$ evolution after T steps forward that is obtained directly from the training set $\mathcal{T}$, i.e., $a_{i+T}^m\in \mathcal{T}$.

• Step 6: Repeat steps 3–5 for all data points $a_i\in \mathcal{T}$ and obtain the correlation ρ(T) between ${\overline{a}}_i^f(T)$ and $a_{i+T}^m$ for each lead time T.

• Step 7: Repeat steps 3–6 for different values of T to obtain the curve ρ(T) as a function of T.

Note that in step 4 of the above SMC algorithm, there are several different ways to obtain ${\overline{a}}_i^f(T)$ (also known as “the prediction model”) such as weighted average, regression combination, ensemble average, or neural network. Regardless of the prediction model, the key property of any chaotic time series is that ρ(T) must decay with lead time T in the presence of low-dimensional chaos. In this regard, the SMC curve ρ(T) comprises a criterion for detecting chaotic time series; a deterioration of SMC with the leading time indicates the existence of chaos, whereas a purely stochastic time series would have a constant SMC regardless of how far into the future. More verification and applications of SMC for different systems can be found in Sugihara and May (1990) and Sugihara (1994).

Similar to the LLE algorithm, both the embedding dimension m and the delay time τ have to be given before computing SMC. Our proposed approach to this freedom in choosing these parameters is to again generate an SMC curve ρ(T) for a range of values of (m, τ) as for the LLE analyses. The convergence of the SMC curve for some domain in the (m, τ) parameter space will then indicate the existence of a low-dimensional chaotic attractor in the embedding phase space. By comparing the values of (m, τ) obtained from the convergence of the SMC curves to the values of (m, τ) obtained from the convergence of LLE, one can then further estimate a proper range for (m, τ) that represents the chaotic regime of TC intensity. More in-depth discussion about other methods for choosing optimal parameters (m, τ) can be found in Grassberger et al. (1991).

b. Idealized TC simulations

Given our approaches of searching for chaos from time series described in the previous section, the next step is to generate a time series of TC intensity for the phase-space reconstruction analysis. In principle, one could obtain this time series directly from observation such as flight data or satellite imagery. However, the requirement of a stationary time series for the phase-space reconstruction imposes a strong constraint on possible choices of time series, as real TC intensity contain various stages of TC development in different environments instead of just the mature stage. As such, using a TC model to produce the intensity time series in a fixed environment is the most apparent approach for our purpose. Ideally, one should use full-physics three-dimensional models that are as much realistic as possible such as the Hurricane Weather Research and Forecasting (HWRF) Model. These types of limited-area models are, nevertheless, designed on a rectangle domain with strong constraints by lateral boundary conditions that prevent one from running for a very long time to generate a stationary time series. Because of this, we choose herein an idealized model that allows for a long integration without the issue of lateral boundary asymmetries.

In this regard, the axisymmetric configuration of the cloud model (CM1; Bryan and Fritsch 2002) was used to generate different intensity time series for our phase-space analyses. The model was configured with 359 grid points on a stretching grid in the radial direction with the highest resolution of 2 km in the storm central region and stretched to 6 km outside 1000 km radius. In the vertical direction, a setting of 61 levels with a fixed resolution of 0.5 km was chosen. The model was initialized from the tropical Jordan sounding on an f plane, with fixed sea surface temperature (SST) = 302.15 K.

Because of the requirement of a quasi-stationary time series at the maximum intensity equilibrium, the model was configured for 100-day simulations. A stable maximum intensity equilibrium for this 100-day integration could be obtained by using a suite of physical parameterizations including the YSU boundary layer scheme, the TKE subgrid turbulence scheme, and explicit moisture Kessler scheme with no cumulus parameterization. For the radiative parameterization, an idealized option with the Newtonian cooling relaxation of 2 K day⁻¹ was applied, similar to what used in Kieu and Moon (2016). This choice of the radiative cooling parameterization is sufficient to allow for a stable maximum intensity equilibrium during the entire 100-day simulations as shown in Fig. 1. Given this stable configuration of TC intensity, the time series of the maximum boundary layer inflow (U_MAX), the maximum wind speed (V_MAX) at the model lowest level, the maximum vertical motion in the eyewall region (W_MAX), and the minimum central pressure (P_MIN) were then output at an ultrahigh sampling frequency of 36 s to maximize our time series analyses.

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(a) Time series of the CM1 maximum tangential wind V_MAX (black; unit: m s⁻¹) and the minimum surface pressure deficit P_MIN (blue; unit: hPa) from a 100-day simulations using CM1; (b) a close-up window of the V_MAX time series during the maximum intensity equilibrium from days 57 to 81 of the CM1 simulation; (c)–(e) as in (b), but for the maximum radial inflow U_MAX, the maximum vertical motion in the eyewall region W_MAX, and P_MIN, respectively.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0115.1

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As a step to further verify the effects of random noise on our analyses, a set of sensitivity experiments were also conducted for which random white noise with a given variance was added to CM1 forcing at every time step. This implementation of additive random noise turns CM1 into a stochastic system whose output now contains random fluctuations with an amplitude proportional to the magnitude of random forcing. As discussed in Nguyen et al. (2020), this additive random noise in terms of the Wiener process results in a first-order accuracy for the CM1 finite difference scheme, similar to the Euler–Maruyama method. By choosing a sufficiently small time step (∼20 s at 2-km resolution), the model is able to maintain its numerical stability for a range of stochastic forcing. Note that random noise was applied only to wind components at all CM1 grid points, with a variance in the range of [10⁻³–10⁻¹ m s⁻¹]. Beyond this range, we notice that the model violates the CFL conditions and quickly loses its stability after just a few steps of integration. The main rationale for applying random noise only to the wind field in these sensitivity experiments is because wind components generally most fluctuate with time at any grid point. Adding random noises to the model temperature, pressure, and moisture fields does not change the outcomes, yet these extra noises would cause the model to become more unstable and limit the range of random noise amplitude that we can implement for the wind components. Thus, all stochastic simulations were carried out only for the wind perturbations in this study.

a. Existence of maximum intensity equilibrium

Since the phase-space reconstruction method requires a stationary time series, it is necessary to examine first if the maximum intensity equilibrium exists during TC development. In this regard, Fig. 1a shows the time series of V_MAX as obtained from a 100-day simulation, using CM1. One notices in Fig. 1a that the model vortex experiences a brief rapid intensification during the first 3–5 days and quickly settles down to a mature state after 9–10 days into the model integration. These behaviors are typical in TC development under idealized conditions as shown in various studies (see, e.g., Rotunno and Emanuel 1987; Wang 2001; Bryan and Rotunno 2009; Hakim 2011, 2013; Davis 2015; Kieu and Moon 2016). Although the quasi-stationary equilibrium at the maximum intensity is evident in our simulation as seen in Fig. 1, we note that the existence of such a stable equilibrium is still an open question from the practical standpoint due to the sensitivity of this equilibrium to model configurations as well as the assumptions of a fixed environment in all idealized studies (Smith et al. 2014; Hakim 2011; Kieu and Moon 2016). With the experiment settings described in section 2, the stable equilibrium of the model maximum intensity (MMI)¹ can be at least captured and well maintained during the entire 100-day period, which suffices for examining the phase-space reconstruction for TC intensity. Thus, we will assume that such a maximum intensity equilibrium exists so that the traditional framework of predictability can be applied.

Given the MMI equilibrium, it is apparent that the maximum intensity does not take one single value but highly fluctuates with time, similar to what obtained in previous studies (Hakim 2011, 2013; Kieu and Moon 2016). As shown in Fig. 1, temporal fluctuations at the MMI equilibrium are observed not only for V_MAX but also for other variables including P_MIN, the maximum boundary layer inflow (U_MAX), and the maximum vertical motion in the eyewall region (W_MAX). From the statistical standpoint, these fluctuations show no obvious difference between chaotic and stochastic variability, thus highlighting an important question in TC dynamics: Do these fluctuations reflect the low-dimensional deterministic chaos of TC intensity, model random truncation errors, or a manifestation of high-dimensional nonlinearity projection [the so-called process or stochastic noise in Sugihara (1994) and Casdagli (1992)]?

From the time series output, it should be noted that all numerical models appear to be stochastic (Kantz and Schreiber 2003; Nguyen et al. 2020). This is because numerical truncation errors can be amplified by nonlinearity and projected onto the time series, resulting in an unexplained noise in the model output (Brock 1986; Casdagli 1992; Sugihara 1994; Kantz and Schreiber 2003). This stochastic nature of model time series is especially true for modern modeling systems, which employ also various stochastic parameterization schemes or random switches such as convective triggering mechanism (Palmer 2001; Christensen et al. 2015; Dorrestijn et al. 2015; Zhang et al. 2015). As such, the strong fluctuation of TC intensity as shown in Fig. 1 is always present for any model output.

There are several different techniques in nonlinear time series analyses that can address the distinction between deterministic or stochastic variability. In this study, with a large sample size of the TC intensity state at the MMI equilibrium, we can directly examine the nonlinear chaotic invariants by dividing the long dataset into many smaller overlapped patches, the so-called a sliding window detector method in data analysis, to increase the reliability of our estimation. Specifically in this study, three key measures of deterministic chaos to be examined are (i) the largest Lyapunov exponent, (ii) the Sugihara–May correlation, and (iii) the correlation dimension for TC intensity. These are the main invariants of any chaotic attractor, which can help answer the main question of the potential existence of low-dimensional chaos for TC intensity in CM1 that we wish to explore in this study.

b. Largest Lyapunov exponent

To examine the nature of the variability in the V_MAX, U_MAX, W_MAX, and P_MIN time series during the equilibrium period (days 20–100 of simulation), Fig. 2 shows the LLE λ as a function of embedding dimension m for a range of delay time (τ) between 10 and 60 min. This range of τ is based on the nature of TC dynamics process, which is strongly governed by convective activities at a time scale of minutes to hours. As discussed in Kantz and Schreiber (2003), the choice of τ should have minimum effects on the attractor invariants if the phase-space reconstruction is effective. Thus, it is important to see how sensitive the LLE estimations are to different delay times. Of course, a positive LLE is necessary but not sufficient to conclude whether the variability in a time series is a result of low-dimensional chaos or not. However, the existence of such a positive LLE is a required condition that any chaotic system must possess and so we need to examine it first (Wolf et al. 1985; Fraser and Swinney 1986; Theiler 1987; Brock 1986; Sugihara 1994).

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Dependence of the largest Lyapunov exponent (LLE; unit: 10⁻⁴ s⁻¹) on the embedding dimension m for V_MAX (black), U_MAX (cyan), W_MAX (green), and P_MIN (red; right axis) for (a) delay time τ = 10 min, (b) τ = 30 min, and (c) τ = 45 min. Error bars denote the 95% confidence intervals obtained during the maximum intensity equilibrium. Thin solid lines indicate the plateau limit that LLEs approach when m increases.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0115.1

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One notices two important features from Fig. 2. First, the LLEs derived from all time series display a consistent behavior for all τ between 10 and 60 min, with a decrease of LLE for larger embedding dimension (m) and subsequent leveling off in the range of 0.5–1.4 × 10⁻⁴ s⁻¹ for m ≥ 10. Note that an LLE of 1 × 10⁻⁴ s⁻¹ is equivalent to a doubling time of ∼3 h in the full physical dimension. Thus, the range of LLEs shown in Fig. 2 suggests that an initial error would be doubled every 1–5 h at the maximum intensity equilibrium. While this is relatively broad range, it is important that all LLEs are positive and convergent toward a stable range when m increases. Specifically, the decaying of LLEs with m as seen in Fig. 2 suggests that small embedding dimension m < 10 would not properly capture TC intensity chaotic attractor. As m increases, attractor invariants such as LLEs must converge toward a more stable value, if a low-dimensional chaotic attractor truly exists. In this regard, the decay of LLEs with m in Fig. 2 provides some initial indication about possible existence of intensity chaos that we wish to quantify next.

Second, Fig. 2 shows further that all LLEs converge toward a stable value for the embedding dimension m ≥ 10, regardless of the variables or time delay values used to reconstruct the phase space. Although the value of the stable LLE cannot be precisely pinpointed due to wide range between 0.5 and 1.4 × 10⁻⁴ s⁻¹, the fact that such a stable value for LLE exists for m ≥ 10 is important here. Namely, this convergence of LLEs implies that a low-dimensional chaotic attractor of TC intensity has an intrinsic dimension n ≈ 4–5, according to the Takens embedding theorem.² Of course, finding the exact embedding dimension m from a given time series that can ensure the Takens theorem is difficult, because this embedding dimension is often ad hoc and dependent on choices of parameters such as time delay, sampling frequency, or sample size (Kantz and Schreiber 2003). Nevertheless, our sensitivity estimation of m using different methods such as the false nearest neighbor (FNN) method (Fraser and Swinney 1986; Sugihara 1994; Wallot and Monster 2018) captures a similar minimum range for m ∈ [10–14]. Thus, it is expected that the intensity chaotic attractor would require a minimum embedding dimension m ∼ 10 for the TC intensity phase-space reconstruction as shown in Fig. 2.

It is of interest to note however a distinct behavior of the stable values of LLE estimation from Fig. 2 that the LLE appears to be quite different between the wind (i.e., U_MAX, V_MAX, and W_MAX) and the pressure (i.e., P_MIN) time series. Specifically for the CM1 simulations herein, LLE is ∼0.5–1.4 × 10⁻⁴ s⁻¹ for V_MAX, U_MAX, or W_MAX, but it is noticeably smaller (∼0.1–0.5 × 10⁻⁴ s⁻¹) for P_MIN. In addition, the convergence of LLE for the P_MIN time series occurs for m ≥ 16 as compared to m ≥ 10 for the wind time series. Such difference between LLEs obtained from the wind and the pressure variables may reflect different predictability for different state variables in a multiscale system with the coexistence of fast and slow-varying processes (Shukla 1981; Goswami et al. 1997; Lorenz 1992; DelSole et al. 2017). Much like the predictability of rainfall or convective systems is different from that of temperature or 500-hPa geopotential for general weather systems (e.g., Islam et al. 1993; Weisman et al. 2015; Tan et al. 2004), it is possible that the TC pressure and wind fields possess inherently different predictability ranges as captured in our LLE analyses.

From the physical perspective, the longer predictability range of P_MIN as compared to V_MAX is likely related to the fact that V_MAX is highly sensitive to convective-scale processes (see, e.g., Krishnamurti et al. 2005; Vukicevic et al. 2014), which tend to be very fast evolving in the eyewall region. Note also that the central pressure is an integral measure, whereas wind is driven by pressure gradient. Given that higher-order derivatives tend to be less smooth and peak within the eyewall region, it is expected that V_MAX fluctuations are much higher as reported, e.g., by Zhang et al. (2021). This can help explain why recent studies have proposed to use P_MIN as a measure for TC intensity in operational forecast instead of V_MAX, because it potentially allows for more reliable intensity forecast in the long run (see, e.g., Magnusson et al. 2019; Klotzbach et al. 2020).

While our search for the minimum embedding dimension based on the convergence of LLEs differs from other approaches such as the box counting or the correlation dimension method (Nicolis and Nicolis 1984; Brock 1986; Casdagli 1992), we note that all phase-space reconstruction methods are somewhat subjective and similarly ad hoc due to the wide range of nonlinear dynamical systems and time series characteristics (Kantz and Schreiber 2003). Thus, there is always some uncertainty in determining a proper minimum dimension for embedding phase space, which explains why m has a range of 10–16 or LLEs ∼0.5–1 × 10⁻⁴ s⁻¹ as seen in Fig. 2. Regardless of this uncertainty, the emergence of low-dimensional chaos for TC intensity with a relatively small value of m is still noteworthy, since a large embedding dimension would imply that our time series analysis is insufficient to capture chaotic dynamics.³ From this perspective, the LLE analyses herein could provide some evidence of low-dimensional chaos for TC intensity, at least from the standpoint of error growth on an attractor at the quasi-stationary equilibrium.

c. Sugihara–May correlation

As discussed in Sugihara (1994), detecting chaos based on the existence of a positive LLE in any time series must be cautioned. This is because any fluctuation in a time series could be manifestation of high-dimension nonlinearity or random noise. One could indeed have a nonchaotic system with a positive LLE if there is sufficiently large random noise in the time series (Brock 1986; Casdagli 1992; Sugihara 1994). As such, a positive LLE as shown in Fig. 2 may not be insufficient to guarantee the existence of low-dimensional chaos.

To further examine the potential low-dimensional chaos in TC intensity time series, Fig. 3 shows the SMC as a function of forecast lead time T for all four variables. Again, SMC is obtained by using a modified version of Sugihara and May’s original algorithm, in which the forecast scheme is based on an ensemble average instead of a weighted sum (Sugihara and May 1990) or regression method (Casdagli 1992) as described in the method section. Note also that a fixed embedding dimension m = 10 and the delay time τ = 30 min are chosen for this SCM calculation, based on the results from the LLE analyses in the previous section.

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Dependence of the Sugihara–May correlation (SMC) on the forecast lead time T for V_MAX (black), U_MAX (cyan), W_MAX (green), and P_MIN (red). Error bars denote the 95% confidence intervals obtained during the maximum intensity equilibrium.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0115.1

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One notices in Fig. 3 that SMCs from all four different variables show rapid decay with forecast lead time. As discussed in Sugihara and May (1990), this type of decaying correlation is a characteristic of chaotic dynamics, which is distinct from the pure random noise variability whose SMC is statistically constant. Of further interest in Fig. 3 is the consistency of such decaying SMC among all time series, which confirms the limited predictability for TC intensity due to the low-dimensional chaos, irrespective of model output. Specifically for our CM1 simulation, we observe that SMC decreases from 1.0 to about 0.1 after reaching a limit $T^{*}\approx 3\text{–}5\mathrm{h}$ for the wind variables and 12–18 h for the pressure variable. Such a chaotic decorrelation time is also consistent with the predictability range obtained from the TC energy spectral analyses in Kieu and Rotunno (2022) at the maximum intensity equilibrium.

Similar to the LLE analyses, the time series for the wind components (U_MAX, V_MAX, W_MAX) display a consistent range of predictability among themselves ($T^{*}\approx 3\text{–}5\mathrm{h}$), while P_MIN tends to capture a longer decorrelation time (${T^{\prime }}^{*}\approx 12\text{–}18\mathrm{h}$) as shown in Fig. 3. This difference in SMC between the pressure and the wind time series is robust for a range of embedding dimension, delay time, model physical options, stochastic forcings, or initial conditions in our analyses, so long as the phase space is properly reconstructed. Such a longer decorrelation time in the P_MIN time series again suggests that the pressure field may contain different dynamics, which may allow for more reliable intensity forecast at longer lead times. The fact that both LLE and SMC analyses provide such a consistently different behavior between the wind and pressure variables highlights the possible different predictability for TC intensity when using V_MAX or P_MIN as suggested in the previous studies.

Our additional analyses with different delay time τ or embedding dimension m confirm that the SMC curves display consistent decay and level off only when m ≥ 10, which is comparable with the embedding dimension obtained from the LLE convergence for the wind field or FNN method (not shown). For smaller values of m, the SMC curve does not possess a monotonic decay but highly fluctuates with forecast lead time. These analyses reiterate the results from the LLE analyses that the embedding dimension for TC intensity phase space must be sufficiently large before one can attain consistent characteristics of SMC.

d. Correlation dimension

The consistent convergences of the SMC curve and the LLE for m ≥ 10 is noteworthy, because it suggests the existence of a low-dimensional attractor with dimension n ∼ 4–5 from the Takens theorem. To directly verify this intrinsic dimension of the TC intensity chaotic attractor, the Grassberger–Procaccia (GP) correlation dimension algorithm (Theiler 1987) is used to estimate the dimension of the TC intensity attractor directly from the CM1 time series (Fig. 4a).⁴ While this correlation dimension algorithm has some degree of subjectivity in choosing the best linear fit for correlation integral, these correlation integral curves do show a saturated slope for m ≥ 10 in the scaling region, which corresponds to a correlation dimension of a chaotic attractor n ≈ 5–7 (Fig. 4b). Note that GP correlation dimension is an invariant of any chaotic attractor. Therefore, the consistent slopes of the correlation integrals in Fig. 4 when m increases supports the existence of a chaotic attractor with dimension n ≈ 5–7, slightly larger than what obtained from the LLE and SMC analyses but still within the same range of uncertainty.

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(a) Dependence of the V_MAX correlation integral on the neighborhood radius for a range of embedding dimension m from 2 to 20; (b) dependence of the slope of the correlation integral-radius curve in (a) on the embedding dimension m as obtained from V_MAX (black), U_MAX (cyan), W_MAX (green), and P_MIN (red) time series. The black line denotes the saturated slope of the correlation integral curves at the scaling region.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0115.1

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In the search for the intrinsic dimension of TC chaotic attractor, we should recall a common underlying assumption that possible contributions from random noise must be sufficiently small. This is because noise could strongly interfere with the phase-space reconstruction and result in, e.g., an artificially positive LLE or incorrect correlation dimension estimation (Brock 1986; Sugihara 1994; Casdagli 1992; Kantz and Schreiber 2003). The existence of noise in any model output is natural even for deterministic systems because of the discretization or numeric errors in any model. How random noise impacts our phase-space reconstruction analyses of TC intensity is therefore elusive.

To address the robustness of our correlation dimension estimation in the presence of noise, one could employ different statistical testing methods or noise reduction algorithms that could distinguish the difference between chaotic and stochastic time series (Brock 1986; Baek and Brock 1992; Kantz and Schreiber 2003). Within the model simulation framework, we can however approach this problem differently by carrying out additional experiments in which random processes in the form of stochastic forcing are included in CM1 as described in section 2. Any difference in the estimations of attractor invariants such as LLE, SMC, or correlation dimension between the stochastic and deterministic time series could then reveal the role of random noise in the TC intensity phase-space reconstruction. In this regard, Fig. 5 shows the time series of V_MAX and P_MIN as well as the correlation dimension n as a function of the embedding dimension m that are obtained from the CM1 simulation with stochastic forcing implementation. Despite the existence of noise in CM1 that cause larger intensity fluctuation seen in Fig. 5a, the correlation dimension shown in Fig. 5b still displays a consistent behavior among all time series, similar to that obtained from the CM1 deterministic simulation in Fig. 4. That is, n increases at first and but levels off for m ≥ 10. This saturation of n with increasing embedding dimension m in the presence of noise is significant, because it indicates that the deterministic signals are more dominant, at least in the scaling region. As discussed in Kantz and Schreiber (2003), the random noise generally introduces extra dimensions to any deterministic chaos. As such, the result obtained in the CM1 stochastic simulation is required to establish the existence of TC intensity deterministic chaos, albeit the exact value of the TC intensity chaotic attractor is still not known.

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(a) As in Fig. 1a, but for the V_MAX and P_MIN from the stochastic CM1 integration in which additive random noises are added to model wind fields at every time step of the model integration over the entire model domain, and (b) the correlation integral curves as in Fig. 4b, but obtained from the box in (a).

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0115.1

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We wish to note that the consistent behavior of the correlation dimension n between deterministic and stochastic simulations is only held for a certain range of noise amplitude. For a large value of stochastic forcing, CM1 would crash due to the violation of the model numerical stability, thus preventing us from examining to what extent random processes would dominate chaotic variability. Similar analyses of LLE or SMC for these stochastic simulations capture every similar results as shown in Figs. 2 and 3, thus all together supporting the existence of low-dimensional chaos for TC intensity, even in the presence of random noise.