Detecting Regime Transitions of the Nocturnal and Polar Near-Surface Temperature Inversion

a. Model description and linear stability analysis

A conceptual model was introduced by van de Wiel et al. (2017) to study regime transitions of near-surface temperature inversions in the nocturnal and polar atmospheric boundary layer. The authors were able to determine a connection between the dynamical stability of the temperature inversion and the ambient wind speed U through their model and measurements. Mathematically speaking, the model is a dynamical system represented by a first-order ordinary differential equation (ODE), which describes the time evolution of the difference between the temperature at a reference height T_r and the surface temperature T_s. Although the equilibrium properties of the system and the dynamical stability properties (i.e., the resilience to perturbations) of all equilibria states were thoroughly discussed in van de Wiel et al. (2017), for the sake of completeness we briefly introduce the model and summarize the linear stability analysis of equilibrium points of the resulting ODE for different values of a bifurcation parameter. The bifurcation parameter is related to the ambient wind speed.

Assuming that the wind speed and temperature are constant at a given height z_r, the following equation describes the evolution of the near-surface inversion strength, based on a simple energy balance at the ground surface:

In this energy balance model, c_υ is the heat capacity of the soil, ΔT = T_r − T_s is the inversion strength between the temperature at height z_r and at the surface z_s, Q_n is the net long wave radiative flux (an energy loss at the surface that will be set as a constant), G is the soil heat flux (an energy storage term that will be parameterized as a linear term), and H is the turbulent sensible heat flux (a nonlinear energy transport term that will be parameterized in the following).

After parameterizing the fluxes, the model has the form

in which Q_i is the isothermal net radiation, λ is a lumped parameter representing all feedbacks from soil heat conduction and radiative cooling as a net linear effect, ρ is the density of air at constant pressure, c_p is the heat capacity of air at constant pressure, c_D = [κ/ln(z_r/z₀)]² is the neutral drag coefficient, with κ ≈ 0.4 the von Kármán constant, z₀ the roughness length, and z_r the reference height, U is the wind speed at height z_r, R_b = z_r(g/T_r)(ΔT/U²) is the bulk Richardson number, and f(R_b) is the stability function used in Monin–Obukhov similarity theory.

The lumped parameter λ corresponds to a linear term in the model as the soil is assumed to respond linearly to the temperature inversion. Moreover, ΔTf(R_b) is a nonlinear term due to the nonlinear dependence of turbulent diffusion on the vertical temperature gradient.

Following van de Wiel et al. (2017), instead of analyzing the dynamical stability of the energy-balance model (2) itself, we will present the linear stability analysis of a simplified system that has a similar mathematical structure but is mathematically convenient to analyze. Using a cutoff, linear form for the stability function, i.e., f(x) = 1 − x and f(x) = 0 for x > 1, the simplified model is

and x(t₀) = x₀. Here, up to dimensional constants, x represents ΔT. The parameter C will be treated as a bifurcation parameter for this simplified system. Similar types of stability functions are typically used in numerical weather prediction tools, and the cutoff form facilitates the mathematical analysis of the model. Note that to be consistent with the original model, the stability function should include a dependence on both the temperature and the wind speed via R_b. Removing this dependence as it is done here changes some of the nonlinearity; however, it makes the mathematical analysis very simple and the qualitative behavior of the system is similar to the original system (see van de Wiel et al. 2017, their Figs. 8 and 10). In that sense, the model loses some physical significance for mathematical convenience, but the qualitative nonlinear feedback processes are maintained. This simplification also has for implication that while C is related to the wind speed, it cannot be directly interpreted as such in the context of the energy balance model. For a deeper discussion of the model, its simplifications and the model parameters, the reader is referred to its thorough presentation by van de Wiel et al. (2017).

For fixed and physically meaningful values of Q_i and λ, Eq. (3) can have either one, two, or three possible equilibrium solutions depending on the fixed values [see illustration in van de Wiel et al. (2017), Figs. 10–12, and related discussion for more details]. The equilibrium solutions will be functions of the parameter C, which we will consider as a bifurcation parameter in the following. Physically, the case of strong thermal coupling between the surface and the atmosphere, corresponding to a large value of λ, results in one unique equilibrium solution whose value depends on C. In van de Wiel et al. (2017), it is hypothesized that such a case is representative of a grass site such as Cabauw, the Netherlands. The solution is linearly stable to perturbations, i.e., linear stability analysis shows that perturbed solutions are attracted back to the equilibrium. The case of no coupling (λ = 0) leads to two equilibrium solutions, one of which is linearly stable and the other unstable to perturbations (i.e., perturbed solutions are repelled by the equilibrium). A weak coupling strength, with an intermediate value of λ that could be representative of a snow surface or another thermally insulated ground surface, results in three possible equilibrium solutions. The two extreme solutions are stable to perturbations, while the middle equilibrium solution is unstable. Perturbed solutions around the middle equilibrium will thus be attracted by either the upper or the lower equilibrium. Plotting those three equilibrium solutions as a function of the bifurcation parameter C results in a back-folded curve, which is qualitatively similar to observations of the temperature inversion shown as a function of wind speed at Dome C (see Vignon et al. 2017). The bifurcation diagram is shown in Fig. 1 for parameter values such that λ > 0 and Q_i > λ, resulting in the case with three possible equilibrium solutions. By convention, the unstable equilibrium branch is denoted by a dashed line. In the following, we will analyze transitions between the two stable equilibria. If the system undergoes random perturbations in this bistable context, a perturbation could drive the system sufficiently far from a stable equilibrium state so that it comes near the unstable equilibrium and finally gets attracted by the second equilibrium. The three possible equilibria are denoted as $x_{e_1}$, $x_{e_2}$, and $x_{e_3}$. To study such regime transitions induced by random perturbations, the conceptual model is extended with a noise term in the following section.

View Full Size

Bifurcation plot for simplified model. The lowest, middle, and upper branches correspond, respectively, to the equilibria $x_{e_1}$, $x_{e_2}$, and $x_{e_3}$.

Citation: Journal of the Atmospheric Sciences 77, 8; 10.1175/JAS-D-19-0287.1

• Download Figure
• Download figure as PowerPoint slide

b. Extending the conceptual model by randomization

The conceptual model (3) can be equivalently written in terms of a gradient system, in which the temperature inversion represented here by x evolves according to the influence of an underlying potential V(x). The randomized model to be introduced will be based on this gradient structure. Specifically, the initial-value problem (3) can be written as

where it is easy to see that the potential is given by

The linear stability analysis discussed in the previous section can thus be understood in the sense that the temperature inversion x equilibrates at a local minimum of a potential V. That is, an equilibrium point x_e satisfies V′(x_e) = 0. Figure 2 sketches the form of the potential with the exemplary parameter values λ = 2, Q_i = 2.5, and C = 6.4. Note that V(x) is a double-well potential in that case where each local minimum corresponds to one of the stable equilibrium points $x_{e_1}$ and $x_{e_3}$, while the local maximum corresponds to the system’s unstable equilibrium $x_{e_2}$.

View Full Size

Example of a potential V(x) [Eq. (4)] and its local approximation through the quadratic potential $\tilde{V}\left(x\right)$ (see text for details).

Citation: Journal of the Atmospheric Sciences 77, 8; 10.1175/JAS-D-19-0287.1

• Download Figure
• Download figure as PowerPoint slide

While the conceptual model (3) has proven very insightful to explain observed sharp transitions in temperature inversions, it only allows for regime transitions when drastic changes in the model parameters (i.e., bifurcations) occur. That is, the model is overly idealized and in reality one can expect regime transitions to also take place due to small natural fluctuations of the temperature inversion itself in certain cases, e.g., when the potential barrier separating the two local minima and corresponding stable equilibria is shallow. Therefore, we will consider an appropriate randomized variant of the model. Specifically, we consider the stochastic differential equation (SDE) model

to account for small random perturbations to the temperature inversion’s rate of change. Here, B denotes a standard Brownian motion (i.e., a stochastic process) and σ > 0 scales the intensity of the fluctuations, while the potential V is as in (4). As the randomized dynamics is characterized by the same potential, also the equilibrium points of the nonrandom model (3) will describe the dominant effects of the randomized model’s dynamics. However, due to the presence of the noise, the stable equilibria of the nonrandom model (3) are not limiting points for the stochastic counterpart in (5), in the sense that the temperature inversion may still fluctuate after reaching a stable equilibrium. The reason is that in a context of two stable equilibria (i.e., for parameter values such that the model (3) exhibits two stable equilibria, denoted earlier as the thermally weakly coupled state), the random perturbations can trigger transitions from one stable equilibrium to another one. We will therefore refer to the formerly stable states as: metastable. Note that depending on the coupling strength and noise intensity, the likelihood of regime transitions can change drastically and the system may or may not exhibit metastable states. The type of noise (additive or multiplicative for example, or noise with a Levy distribution) will also affect regime transitions. In our subsequent simulations and analyses, we will focus on the case of two metastable states with additive noise and leave other cases for future research.

The effect of these random perturbations to a metastable equilibrium point x_e can be understood through a localized approximation of the original dynamics. More precisely, consider a second-order Taylor approximation of the potential around an equilibrium point x_e, yielding the quadratic approximate potential $\tilde{V}$:

For the same parameter values that were used to plot the original potential in Fig. 2, the red line in the same figure shows the approximate quadratic potential around the equilibrium value $x_{e_1}$. Using the locally quadratic potential, we can thus define a locally approximate dynamics for the temperature inversion by replacing V in (5) by $\tilde{V}$, resulting in

where $k\text{:}=d^{2}V/d{x}^2{\left(x\right)|}_{x=x_e}\in \mathbb{R}$ and X is introduced to describe the approximate dynamics of the former x. This approximate dynamics is an example of the well-studied Ornstein–Uhlenbeck process and it provides an accurate description of the full dynamics in the neighborhood to the equilibrium point x_e. Discretizing the Ornstein–Uhlenbeck process X using the Euler–Maruyama scheme with a step-size Δt:= T/L for some positive integer L we furthermore find that the process at discrete times t ∈ {1, …, L} approximately satisfies the difference equation

in the sense that X_t ≈ X(tΔt). By defining $\mu \text{:}=k{x}_e\in \mathbb{R}$, ϕ:= (1 − kΔt) and w_t:= σ{B(tΔt) − B[(t − 1)Δt]} this can be written as

which is a so-called autoregressive model of order 1, denoted AR(1), thanks to the properties of the (scaled) Brownian increments w_t. Consequently, we see that the discretized Ornstein–Uhlenbeck process can be accurately approximated by an AR(1) process. This derivation can also be found in Thomson (1987). Combining this with the observation that the Ornstein–Uhlenbeck process offered an accurate approximation to the original dynamics in the vicinity of a stable equilibrium, we can thus conclude that the local dynamics in the neighborhood of a metastable state can be approximately described by an AR(1) process.

c. Statistical indicator for the dynamical stability of time series

In section 2a we discussed the simplified model by van de Wiel et al. (2017) that was developed to understand regime transitions in near-surface temperature inversions. This model provides a hypothesis that explains the existence of two possible equilibria of the temperature inversion for a given wind speed. In agreement with the randomized conceptual model introduced in the previous section, we say that a system exhibiting at least two metastable equilibria is called metastable. In this section the goal is to describe a methodology for statistically detecting critical transitions based on time series data. For the detection we apply an indicator for the dynamical stability (i.e., the resilience to perturbations) of time series, which was defined by Faranda et al. (2014) and applied to SBL turbulence data in Nevo et al. (2017). The indicator uses a combination of methods from dynamical systems and from statistical modeling. In its definition, deviations from AR(1) processes in the space of so-called autoregressive-moving-average (ARMA) models are used to quantify the dynamical stability of a time series. A time series x_t, $t\in \mathbb{Z}$, is an ARMA(p, q) process if it is stationary and can be written as

with constant ν, coefficients ϕ_p, θ_q,and {w_t} being white noise with positive variance σ². The coefficients ϕ_p and θ_q additionally have to satisfy some constraints (see Brockwell and Davis 2016). Notice that AR(1) = ARMA(1, 0). Intuitively the parameters p and q are related to the memory lag of the process. The longer the system takes to return to the equilibrium after a perturbation, the more memory we expect to observe in the process. Examples of simple systems along with their ARMA(p, q) characteristics can be found in Faranda et al. (2014).

In section 2b, it was shown that the dynamics when the system is close to a stable equilibrium can be approximated by an AR(1) process. We will assume that far from the transition from one dynamical regime to another, the time series of a generic physical observable can be described by an ARMA(p, q) model with a reasonably low number of p, q parameters and coefficients. Indeed, far from a transition, the system will tend to remain around an equilibrium despite random perturbations, and excursions from the equilibrium are short. The idea behind the modeling assumption is that ARMA processes are an important parametric family of stationary time series (Brockwell and Davis 2016). Their importance is due to their flexibility and their capacity to describe many features of stationary time series. Thereby, choosing ARMA(p, q) processes for modeling the dynamics away from a stable state is a reasonable Ansatz. Close to a transition, the resilience of the system to perturbations is weak and longer excursions from the equilibrium occur. The statistical properties (such as the shape and/or the persistence of the autocorrelation function) of the system change drastically, leading to an expected increase of the value p + q (Faranda et al. 2014). Based on this, we use ARMA(p, q) models in the following to analyze the stability of a dynamical system. The dynamical stability indicator that will be defined next will be used to obtain indicators for detecting the system’s proximity to a regime transition.

To quantify the local stability of a time series, we first slice the time series x_t with a moving time window of fixed length τ. In other words, we obtain subsequences {x₁, …, x_τ}, {x₂, …, x_τ+1}, …, {x_t−τ+1, …, x_t} of the original time series that overlap. By slicing the original time series, we obtain a sequence of shorter time series for which it is reasonable to suppose that they are amenable to ARMA modeling. In detail, we assume that the subsequences are realizations of linear processes with Gaussian white noise, which then implies that the process is stationary. We then fit an ARMA(p, q) model for every possible value of (p, q), with p ≤ p_max and q ≤ q_max, to these subsequences, where p_max and q_max are predefined thresholds. Afterward we choose the best fitting ARMA(p, q) model by choosing the one with the minimal Bayesian information criterion (BIC) (Schwarz 1978), which is a commonly used and well-studied tool in statistical model selection. Assuming that we have the maximum likelihood estimator $\widehat{\beta }\text{:}=\left(\widehat{\nu },{\widehat{\varphi }}_1,\dots ,{\widehat{\varphi }}_p,{\widehat{\theta }}_1,\dots ,{\widehat{\theta }}_q\right)$ of the fitted ARMA(p, q) model (which can be obtained using a so-called innovation algorithm, as it is, for example, implemented in the “forecast” R package (Hyndman et al. 2019), which is used for the analyses), the BIC is formally defined as

where $L\left(\widehat{\beta }\right)$ denotes the associated likelihood function evaluated at the maximum likelihood estimator $\widehat{\beta }$. The second term introduces a penalty for high-order models (i.e., those that contain more parameters) to avoid overfitting.

We reiterate that when the system is close to a metastable state, its dynamics can be well approximated by an AR(1) process. When the system is approaching an unstable point separating multiple basins of attraction, the approximation no longer holds as the underlying potential cannot be approximated by a quadratic potential anymore. The change in the shape of the potential introduces new correlations in the time series, resulting in higher-order ARMA terms when fitting such a model to data.

The definition of the stability indicator is based on this observation, in the sense that it assumes that the dynamics near a metastable state can be modeled by an ARMA(1, 0) or AR(1) process. Specifically, the stability indicator is defined as

For a stable state, the most likely statistical model is an AR(1) process and one expects that $\mathrm{\Upsilon }=0$. The indicator $\mathrm{\Upsilon }$ gives a normalized distance between the stable state $\left(\mathrm{\Upsilon }=0\right)$ and the state in which the system is. The limit $\mathrm{\Upsilon }\to 1$ corresponds to a most likely statistical model of high order and probably to a nearing transition. While a formal proof of this statement is still missing, tests performed for systems of increasing complexity in Nevo et al. (2017) showed promising results where the indicator correctly identified changes in the stability of metastable states. Note that the character of the noise present in the physical system (additive noise, multiplicative, Levy process, etc.) will affect the ARMA model approximation and impact the values of $\mathrm{\Upsilon }$. To simplify the notation, we drop the dependence of $\mathrm{\Upsilon }$ on p, q, and τ in the following discussion.

The reliability of $\mathrm{\Upsilon }$ highly depends on the choice of τ, the window length (which we will consider in number of discrete observations in the following), and it relates to the characteristic time scales of the dynamics. Intuitively, the window length, when converted to its equivalent physical duration (i.e., the number of discrete observations multiplied by the discrete sampling time), has to be shorter than the residence time scale in one basin of attraction (i.e., the time spent in the neighborhood of an equilibrium before transitioning to another one) in order to satisfy the local stationarity, but large enough so that statistical model estimation is reliable. In winter at Dome C where the polar winter results in a near absence of daily cycle, no preferred time scale of residence around an equilibrium of the temperature inversion was observed (an equilibrium can remain for several days); however, the transition between two equilibria was observed to take place over a time scale on the order of 10 h (Baas et al. 2019). For nocturnal flows, the residence time scale is tightly connected to the daily cycle and could be of a few hours during the night, or the entire night. The transition between two equilibria typically takes place over a duration of about a half hour. For reliable statistical estimation, multiple tests showed that a minimum window length of 20 discrete points is needed. With a sampling time of 1 min, that means that a moving window of approximately 20 or 30 min may be appropriate.

In addition, the sampling frequency has to be fine enough to sample typical fluctuations of the dynamics. In the following analyses, we find a sampling frequency of 1 min to be appropriate for that purpose. The characteristic time scale here is given by the time scale at which the system recovers from perturbations (which is estimated by linear stability analysis in the case where the model is known; see, e.g., van de Wiel et al. 2017), and the time interval between two observations should be close to or smaller than this quantity so that (small-scale) local fluctuations can be resolved. Since the characteristic time scales of the system cannot be known analytically in many situations, for example when analyzing time series from atmospheric models or from field data, we suggest two data-driven approaches to select a window length:

• In the first approach, the mean residence time around each metastable state as well as the mean transition time between the two states will be estimated based on a data clustering approach. The observations will be clustered in the metastable regimes and an intermediate, transition regime. From the clustered data, the mean residence time in each cluster will be evaluated. This approach will provide an upper bound to select the window length.

• The second approach is motivated by the fact that the indicator $\mathrm{\Upsilon }$ is obtained through a statistical inference procedure through the definition of the BIC, which involves fitting suitable ARMA processes to data. Specifically, a maximum likelihood approach is used, which assumes that subsequences are sampled from a normal distribution. To assess the validity of this statistical approach, a normality test will be implemented as a criterion to select a window length for which the normality assumption is justified and ARMA model estimation is reliable.

Both approaches are applicable when the data-generating model is unknown. This is important in cases where data showing signs of metastability are available, but an underlying model is unknown. A summary of the full algorithmic procedure used to calculate the statistical indicator is given in appendix C.

a. Simulated time series

To generate the simulated data, we use the conceptual randomized model (5), which we recall here for the reader’s convenience:

where V(x) is the energy potential defined in (4). That is, the data-generating model reads

The SDE model (9) is approximated path-wise (i.e., for each realization of the driving Brownian path) using the Euler–Maruyama scheme.

For the purpose of testing the accuracy of the $\mathrm{\Upsilon }$ indicator and its potential to detect nearing regime transitions, one realization {x_t} of the stochastic process is used for each fixed value of the bifurcation parameters C. Multiple fixed values of C are used, resulting in one time series per value of C. The initial parameters are set to t₀ = 0 and $x\left(t_0\right)=\min \left\{x_{e_i}|i=1,\hspace{0.17em}2,\hspace{0.17em}3\right\}$ where $x_{e_i}$ are the three equilibria of the system. To generate the controlled dataset the model parameters are set to λ = 2, Q_i = 2.5, and σ = 0.35. The value of C is varied between C = 5.3 and C = 7.2 with discrete increments of 0.1 and one simulation is done per value of C. The simulations are run for n = 2000 time steps of size Δt = 0.01. The amplitude of the noise, or diffusion coefficient, σ = 0.35 is chosen as it resulted in trajectories for which regime transition could be observed on the time interval [0, T = nΔt = 20]. The range for C is chosen because for these values the time series shows frequent transitions from one metastable state to another. To choose the window length τ we apply both the K-means algorithm [section 2c(1)] and the Anderson–Darling test [section 2c(2)]. The length τ is determined individually for each simulation, i.e., for each fixed value of C. The K-means algorithm can be used to estimate the amount of discrete observation points covering the transition time. We set the cluster number to three as we expect three equilibria. The results of the clustering algorithm are exemplary shown in Fig. 3 for C = 6.4. Note that t ∈ [0, nΔt] = [0, 20], whereas we will express our window length τ in number of discrete points in the following. In this case the equilibria are $x_{e_1}=0.46\hspace{0.17em}\left(\text{metastable}\right)$, $x_{e_2}=0.97\hspace{0.17em}\left(\text{unstable}\right)$, and $x_{e_3}=1.25\hspace{0.17em}\left(\text{metastable}\right)$. The cluster centers, estimated by the K-means algorithm, are 0.46, 0.97, and 1.31, which are a close approximation of the equilibria. Therefore, we expect a good estimation for the amount of points covering the transition. The average time spend in each cluster are (for C = 6.4): mean(T₁) = 112.2, mean(T₂) = 94, and mean(T₃) = 286.67, where mean(T_i) is the average time spent without observed transitions in cluster i ∈ {1, 2, 3} expressed in number of discrete points. The minimal mean residence time is thus mean(T₂) = 94 and provides an upper bound to select a window length that respects the time scales of the system. The window length τ is thus chosen such that it is smaller than the minimal average time spent in one cluster, i.e., for C = 6.4 we choose τ < τ_Kmeans: = min {mean(T_i)|i = 1, 2, 3} = 94. For all tested C, we choose τ = min {mean(T_i)|i = 1, 2, 3} − 5 in order to give room for some uncertainty in the evaluation of the time spent in each cluster, due to potential overlaps of the clusters (we recall that the minimal mean residence time should be understood as an upper bound to select τ). By applying $\mathrm{\Upsilon }$ to the data generated by the simplified model with C = 5.3, C = 6.4, and C = 7.2 we get the results shown in Fig. 4. The solid red lines correspond to the stable equilibria and the dotted red line to the unstable one. The colors ranging from dark blue to yellow represent the stability of the points measured by $\mathrm{\Upsilon }$ and we always color the last point of the subsequence. The simulation is initialized around the stable equilibrium $x_{e_1}$, where short memory of the random perturbations should prevail. As expected, the values of $\mathrm{\Upsilon }$ remains close to 0 [corresponding to a most likely AR(1) model] as long as the simulation oscillates around the equilibrium. The time series eventually approaches the neighborhood of the unstable equilibrium where long memory properties are to be expected and thus higher-order ARMA(p, q) models, hence larger values of $\mathrm{\Upsilon }$, are more likely. This first transition through the unstable equilibrium is well recognized with higher values of $\mathrm{\Upsilon }$ (green dots after the dotted red line).

View Full Size

Clustered simulated time series for C = 6.4 with K-means algorithm. All points with the same color correspond to the same cluster. The window length estimated by the K-means algorithm is τ_Kmeans = 94 time steps, with a discrete time step Δt = 0.01. Hence the window duration corresponds to 0.94 time units.

Citation: Journal of the Atmospheric Sciences 77, 8; 10.1175/JAS-D-19-0287.1

• Download Figure
• Download figure as PowerPoint slide

View Full Size

$\mathrm{\Upsilon }$ for simplified model with (a) C = 5.3, (b) C = 6.4, and (c) C = 7.2 and added noise, estimated for a window length (a) τ = 54, (b) τ = 89, and (c) τ = 143 discrete time steps. The gray dots correspond to missing values, which occur because we only color the last point of the modeled subsequence. Moreover, in rare cases the auto.arima(⋅) R function is not able to find an appropriate model. Full red lines mark the stable equilibria, while the dotted red line marks the unstable equilibrium.

Citation: Journal of the Atmospheric Sciences 77, 8; 10.1175/JAS-D-19-0287.1

• Download Figure
• Download figure as PowerPoint slide

The Anderson–Darling test can be used to find the biggest τ for which we can assume that most of the subsequences are sampled from a normal distribution and hence trust the ARMA model selection and fitting. As shown in Fig. 5, for C = 6.4 the Anderson–Darling test yields that for τ = 67 the median of the p-values for all subsequences is greater than the significance level 0.05. The solid line in the gray boxes is the median of the p-values for a fixed τ while the upper and lower border of the gray boxes refer to the upper and lower quartile of the p-values. The dotted horizontal line is the significance level. We report that the values of τ_AD given by the Anderson–Darling test are ranging from 60 to 70 discrete points for all values for C.

View Full Size

Boxplot of the p-values from the Anderson–Darling test for simulated time series with C = 6.4. The window length estimated by the Anderson–Darling test is τ_AD = 67 discrete time steps.

Citation: Journal of the Atmospheric Sciences 77, 8; 10.1175/JAS-D-19-0287.1

• Download Figure
• Download figure as PowerPoint slide

Figure 6 summarizes the $\mathrm{\Upsilon }$ values obtained for different choices of τ and different values of C in a bifurcation diagram. In the figure, the equilibrium solutions of the deterministic Eq. (3) are shown by a red line for the considered range of values of C. This is the same diagram as shown in Fig. 1, where the upper and lower branches of the equilibrium solution correspond to the two stable equilibria, while the middle one is the unstable equilibrium separating the two basins of attraction of the stable equilibria. A discontinuity in the solution is visible between the upper and middle solution branches, which is due to the discontinuity introduced by the cutoff form of the stability function. The $\mathrm{\Upsilon }$ values obtained for the simulations of the stochastic system [Eq. (9)] for all considered values of C are then shown as a scatterplot along with the equilibrium solution, and the darker color corresponds to higher values of $\mathrm{\Upsilon }$. As the initial condition for all simulations is taken at the lowest equilibrium values, the transitions are expected to occur between the lower and upper equilibrium branches when the system transitions from the basin of attraction of the lowest equilibrium value to that of the highest value. High values of $\mathrm{\Upsilon }$ are indeed mainly found in this region of the diagram.

View Full Size

Bifurcation diagram of the deterministic system (red) and $\mathrm{\Upsilon }$ calculated for simulated data (gray dots): (a) for τ = min{mean(T_i)|i = 1, 2, 3} − 5 (K means), (b) for τ = τ_AD (Anderson–Darling test), and (c) τ = τ_AD if τ_AD < τ_Kmeans − 5, otherwise the subseries are discarded. The red full and dotted lines show, respectively, the stable and unstable branches of the bifurcation diagram.

Citation: Journal of the Atmospheric Sciences 77, 8; 10.1175/JAS-D-19-0287.1

• Download Figure
• Download figure as PowerPoint slide

Three methods are used to select the value of τ and the results associated with these window sizes are shown in Fig. 6. In Fig. 6a, τ = τ_Kmeans − 5 is used. Around the stable branches, values of $\mathrm{\Upsilon }$ are small, denoting that stable states are detected as such. Large values of $\mathrm{\Upsilon }$ are found between the unstable branch and the upper stable branch of the bifurcation diagram, indicating that transitions from the lower to the upper stable branches are detected by the indicator. The fact that the high values are not exactly located around the unstable branch is due to the use of a finite window size for the calculation: in the diagram, the color is always assigned to the last point of the subsequence. For small values of C, e.g., for C = 5.4, large values of $\mathrm{\Upsilon }$ are occasionally inappropriately found around the upper stable branch. For small C, the potential well will be relatively steep and the system rapidly approaches the second equilibrium, so that the detection can be too slow. Figure 6b shows the results for $\mathrm{\Upsilon }$ when choosing τ according to the Anderson–Darling test, denoted as τ_AD. The figure is very similar to the one using τ_Kmeans except that for C ≤ 5.6 there are more high values of $\mathrm{\Upsilon }$ located around the stable branch. This is due to the fact that for these C’s the τ values chosen by the Anderson–Darling test are larger than the ones estimated by the K-means algorithm. Consequently, the local stationarity assumption may break down. For C ≥ 5.9 the values of τ given by the AD test are smaller than those of the K-means algorithm. In these cases $\mathrm{\Upsilon }$ gives a good indication for the stability. Figure 6c is a bifurcation plot showing only the time series for which τ can be chosen to satisfy both the K-means and the Anderson–Darling condition, i.e., τ_AD < τ_Kmeans − 5. Here τ:= τ_AD is used for the analysis and time series that do not satisfy the condition are discarded from the analysis. In this case, we see that large values of $\mathrm{\Upsilon }$ always occur between the unstable branch and the upper stable branch; thus, $\mathrm{\Upsilon }$ is capable of recognizing the location of unstable equilibria for all C and stable equilibria are never assigned a large value of $\mathrm{\Upsilon }$. A note of caution should however be given regarding the reverse transitions from the upper stable branch to the lower stable branch in those numerical examples. Indeed, those are poorly identified. This is probably related to the asymmetry of the underlying potential, which induces different characteristic time scales in the system and hence a need to adapt the value of τ locally, and not just globally for all types of transitions as it is done here. To overcome this difficulty, an adaptive tuning of τ would be required, which will be left for future research. With these possible limitations in mind, $\mathrm{\Upsilon }$ will next be applied to observational data.

As a final remark, the values of τ considered here can be compared to analytical results in this numerical example. Indeed, for this simple bistable example system, analytical results can provide the expected time taken by the system to transition from one of the local equilibria to the bifurcation point (Krumscheid et al. 2015) and can serve as a comparison to the statistical estimations of τ obtained here. As a matter of fact, for C close to the bifurcation point, the results given by the Anderson–Darling test are similar to those given by analytical calculations.

b. Analysis of regime transition in observed nocturnal and polar temperature inversions

In this section we apply the stability indicator $\mathrm{\Upsilon }$ on observational data obtained from one site near Dumosa for which nocturnal data are selected, and from Dome C for which we consider the polar winter. When we plot ΔT over U for both sites (see Fig. 7) we see a clear sign of two distinct states: one when the wind is weak and ΔT is large and one for strong wind where ΔT is small.

View Full Size

Temperature inversion as a function of wind speed as observed at (a) Dumosa and (b) Dome C. The color in (b) corresponds to lower and higher incoming radiation.

Citation: Journal of the Atmospheric Sciences 77, 8; 10.1175/JAS-D-19-0287.1

• Download Figure
• Download figure as PowerPoint slide

1) Dumosa

The first observational dataset consists of temperature measurements from a site near Dumosa. The site was located in a large area with mostly homogeneous and flat terrain, covered by wheat crops, and measurements were taken during the crop season. The temperature measurements were made on the main tower at heights of 3 and 6 m and the wind measurements at 6 m. The frequency of measurements is 1 min. Further details about the observational site can be found in Lang et al. (2018). As we want to use data where we can expect temperature inversions to take place, we exclusively use evening and nighttime data from March until June 2013 (89 days). Each night of data results in a time series of 1020 discrete observations. Similar to the simulated data, we use the K-means algorithm and the Anderson–Darling test to choose the window length τ. The results are shown for all nights considered together in Fig. 8. According to these tests the maximal τ for which we can assume normality (τ_AD) is 19 discrete observations (hence 19 min with the sampling frequency of 1 min) and the τ, which corresponds to the minimal mean residence time in one of three clusters (τ_Kmeans) is 22. Hence, we have τ_AD < τ_Kmeans and choosing τ_AD should be appropriate. The results for $\mathrm{\Upsilon }$ applied to all 89 nights with both choices of window lengths are given in Fig. 9. The results highlight a lower branch with low values of $\mathrm{\Upsilon }$, or dynamics identified as stable, and an upper branch with high values of $\mathrm{\Upsilon }$, or dynamics identified as unstable. In some cases, a proper ARMA(p, q) model cannot be fitted by the statistical methods, resulting in absence of results for some windows. Generally, a reliable ARMA(p, q) fit becomes difficult for a time series with less than 20 observations, and the estimated window lengths are on the lowest end to obtain the statistical estimations. Figure 10 shows the time evolution of ΔT when conditionally averaged for all nights with the wind speed (wsp) being in a given category. The corresponding time evolution of $\mathrm{\Upsilon }$ is shown for the same conditional averages. The window length here is chosen as the most restrictive criterion τ = τ_AD < τ_Kmeans. For low wind speeds, the $\mathrm{\Upsilon }$ values are high (on average), which implies that in this case we have an unstable system. Note that in this dataset, a leveling off of the temperature inversion for low wind speeds [which could correspond to the stable equilibrium of a strong inversion according to the model of van de Wiel et al. (2017)] is not very evident from Fig. 9. It could be that the temperature inversion does not have time to reach the stable equilibrium during the night, or that other instabilities that are not considered in the simplified model arise in strong stability conditions. For example, flow instabilities such as submesomotions, which are favored in strongly stratified situations, could make the system dynamically unstable.

View Full Size

(a) Boxplot of the p-values from the Anderson–Darling test and (b) clustered data with K means for the Dumosa data, for all 89 nights.

Citation: Journal of the Atmospheric Sciences 77, 8; 10.1175/JAS-D-19-0287.1

• Download Figure
• Download figure as PowerPoint slide

View Full Size

(a),(c) Observed temperature inversion vs wind speed relation for the Dumosa data, colored according to $\mathrm{\Upsilon }$ with different window lengths τ: (a) τ = τ_AD and (c) τ = τ_Kmeans. The gray dots correspond to missing values, which occur because we only color the last point of the modeled subsequence. Moreover, in some cases the auto.arima(⋅) R function is not able find an appropriate model. (b),(d) Histograms of $\mathrm{\Upsilon }$ for four different bins of data. The red numbers in (a) and (c) are the bin numbers and the bin separations are shown by the red lines.

Citation: Journal of the Atmospheric Sciences 77, 8; 10.1175/JAS-D-19-0287.1

• Download Figure
• Download figure as PowerPoint slide

View Full Size

Time series of (a) mean ΔT and (b) mean $\mathrm{\Upsilon }$ for different wind speed (wsp) categories for Dumosa data, calculated with τ_Kmeans. The shaded area is the standard deviation.

Citation: Journal of the Atmospheric Sciences 77, 8; 10.1175/JAS-D-19-0287.1

• Download Figure
• Download figure as PowerPoint slide

2) Antarctica

The Dome C data were measured at the Concordia Research Station, which is located on the Antarctica Plateau. It is a French–Italian research facility that was built 3233 m above sea level. It is extensively described for example in Genthon et al. (2010). The Dome C dataset contains 10-min-averaged meteorological data from 2017. Regimes and their transitions were analyzed by Vignon et al. (2017) and Baas et al. (2019). Important for our analysis are measurements of the temperature at height 9.4 m and surface, the wind speed (m s⁻¹) at height 8 m and the radiation made in the polar night, which is from March to September. We focus on the polar night during which multiple regime transitions take place. Following van de Wiel et al. (2017) the data are classified into two subcategories of radiative forcing being the sum of net shortwave and incoming longwave radiation: R₊ = K^↓ − K^↑ + L^↓. Strong cooling is favored in cases of low incoming radiation and when plotting ΔT = T_9.4m − T_s over the wind speed U_8m a back-folding of the points becomes apparent when R⁺ < 80 W m⁻² (van de Wiel et al. 2017, their Fig. 6; and less clearly in our Fig. 7). Therefore, we focus on the case when R⁺ < 80 W m⁻². We apply $\mathrm{\Upsilon }$ to the longest consecutive time series with R⁺ < 80 W m⁻², which is from 1050 LT 3 August to 2150 LT 24 August 2017, i.e., 3091 data points. Our analyzed time series is thus shorter than the one visualized in van de Wiel et al. (2017), which explains the differences in the scatterplot. Again we choose τ with the Anderson–Darling test and the K-means algorithm. The value for τ given by the Anderson–Darling test τ_AD = 43 observations (with a corresponding window duration of 430 min) is much larger than the one given by the K-means algorithm (τ_Kmeans = 10 observations, corresponding to a window duration of 100 min), which is in fact too few points to expect a good fit for the ARMA(p, q) model. In Fig. 11 we see that no transitions are recognized by $\mathrm{\Upsilon }$ with τ = τ_AD (left) but with τ = τ_Kmeans (right) some transitions are noted. This indicates that the data frequency of this dataset is not high enough to give reliable results for the stability indicator.

View Full Size

Temperature inversion between 9.4 m and the surface, as a function of wind speed at 8 m as observed at Dome C, colored according to $\mathrm{\Upsilon }$ with different window lengths τ, expressed in number of discrete observations: (a) τ = 10 (K means) and (b) τ = 43 (AD test).

Citation: Journal of the Atmospheric Sciences 77, 8; 10.1175/JAS-D-19-0287.1

• Download Figure
• Download figure as PowerPoint slide

c. Sensitivity analysis on averaged data

As discussed when applying the indicator $\mathrm{\Upsilon }$ on the Dome C dataset, there is strong indication for the data frequency being crucial for the reliability of the results when applying the stability indicator. Observational data are often stored in block averages, e.g., measurements over 5-min time window are averaged into 1 data point. The issue with this can be that the data frequency can be too low to sample typical fluctuations during the observed transition. In more detail, if the time taken by the system to transition from one metastable state to the next is less than approximately 20 discrete measurement points (the minimum needed to have relevant statistical results according to our tests), then the approach may not be applicable. Therefore, data frequency needs to be high enough to give reliable results for $\mathrm{\Upsilon }$. As a comparative study to illustrate this point, we block average the temperature measurements for the Dumosa data, such that we repeat the analysis based on 5-min-averaged data instead of 1-min averages. Thereby, we reduce the length of the time series for each individual night from 1020 to 204 data points. Again we choose τ with the Anderson–Darling test and the K-means algorithm. There is a clear distinction between the τ estimated by the Anderson–Darling test and the one given by the K-means algorithm for the 5 min data, contrary to the 1 min data where both methods suggested comparable window lengths. Indeed, τ_AD = 31 and τ_Kmeans = 7 for the 5 min averaged data whereas τ_AD = 19 and τ_Kmeans = 22 for the 1 min data. The small value for τ given by the K-means algorithm for the 5 min averaged data suggests that there is only a small amount of points covering the transition time and we cannot fit an ARMA(p, q) model properly to subsequences this short. Moreover, as the value for τ given by the Anderson–Darling test is much bigger than the amount of points covering the transition time we do not expect reliable results for $\mathrm{\Upsilon }$ with this τ. Figure 12 confirms this hypothesis. The left plot is with τ = τ_AD and the right plot is with τ = τ_Kmeans.

View Full Size

Observed temperature inversion vs wind speed relation for the 5-min-averaged Dumosa data, colored according to $\mathrm{\Upsilon }$ with different window lengths τ, expressed in number of discrete observations: (a) τ = τ_AD and (b) τ = τ_Kmeans.

Citation: Journal of the Atmospheric Sciences 77, 8; 10.1175/JAS-D-19-0287.1

• Download Figure
• Download figure as PowerPoint slide