Thermal history models, historically used to understand Earth's geologic history, are being coupled to climate models to map conditions that allow planets to maintain life. However, the lack of structural uncertainty assessment has blurred guidelines for how thermal history models can be used toward this end. Structural uncertainty is intrinsic to the modeling process. Model structure refers to the cause and effect relations that define a model and are assumed to adequately represent a particular real world system. Intrinsic/structural uncertainty is different from input and parameter uncertainties (which are often evaluated for thermal history models). A full uncertainty assessment requires that input/parametric and intrinsic/structural uncertainty be evaluated (one is not a substitute for the other). We quantify the intrinsic uncertainty for several parameterized thermal history models (a subclass of planetary models). We use single perturbation analysis to determine the reactance time of different models. This provides a metric for how long it takes low-amplitude, unmodeled effects to decay or grow. Reactance time is shown to scale inversely with the strength of the dominant model feedback (negative or positive). A perturbed physics analysis is then used to determine uncertainty shadows for model outputs. This provides probability distributions for model predictions. It also tests the structural stability of a model (do model predictions remain qualitatively similar, and within assumed model limits, in the face of intrinsic uncertainty?). Once intrinsic uncertainty is accounted for, model outputs/predictions and comparisons to observational data should be treated in a probabilistic way.

Key Points

The magnitude of intrinsic model uncertainty varies for different thermal history models
Model uncertainty scales with the strength of dominant model feedbacks
Parameterized thermal history results should be viewed probabilistically in light of model uncertainty

Plain Language Summary

The Earth's internal energy is the source of tectonics, volcanism, and geologic activity, all of which effect the surface conditions of our planet. Thermal history models have been used to help understand how this energy, and associated tectonic-volcanic-geologic activity, has evolved over the Earth's history. These models are now being adopted to map conditions that allow for life on planets beyond our own. However, the uncertainty of these models has not been fully assessed which casts a cloud over how their predictions should be viewed. Here we investigate the uncertainty of thermal history models using several techniques. Our results indicate that model predictions should be viewed in a probabilistic sense which breaks from the way they have traditionally been used for Earth application. These results can also be extended to the exoplanet community. They suggest that efforts to delineate conditions that allow for planetary life will need to proceed under a probabilistic umbrella.

1 Introduction

The discovery of terrestrial exoplanets has rejuvenated interest in determining the factors that maximize the life potential of a planet (e.g., Lenardic, 2018b; Meadows & Barnes, 2018; Walker et al., 2018). Modeling the evolution of terrestrial planets, with a focus on mapping scenarios that maintain livable surface conditions, falls under this research umbrella. Consideration of solar energy feeds into this exercise, as does a consideration of a planet's internal energy. Internal energy drives volcanic and tectonic activity. This, in turn, cycles planetary volatiles between surface and interior envelopes, and volatile cycling may be critical for maintaining clement conditions over timescales that allow life to develop and evolve (Kasting et al., 1993; Walker et al., 1981). The internal energy of terrestrial planets decays over time and the interior of a planet, in the long term, cools until it becomes geologically inactive. Determining/mapping particular and/or potential planetary cooling paths is the goal of thermal history modeling (Davies, 1980; Schubert et al., 1980). Thermal history models developed for the Earth have been, and continue to be, adopted and adapted for exoplanet modeling (Kite et al., 2009; Schaefer & Sasselov, 2015). They are also being linked to climate models so as to model the surface conditions of terrestrial planets over geological time scales (Foley, 2015; Foley & Smye, 2018; Jellinek & Jackson, 2015; Lenardic et al., 2016; Rushby et al., 2018).

Thermal history models, like all models, have uncertainties. Informed application and/or evaluation of models requires assessment of all the potential sources of uncertainty (Beck, 1987; Bradley, 2011; Draper & Draper, 1995; Draper et al., 1987; Petersen, 2012; Walker et al., 2003). One source of uncertainty is associated with the fact that planetary models involve input parameters that are not perfectly known (e.g., material properties, initial conditions). This is referred to as input and/or parameter uncertainty. It can be assessed by varying input conditions to determine how model outputs respond (e.g., Loucks & van Peek, 2017; Saltelli et al., 2008, chap. 9). Another source of uncertainty comes from the fact that a model is, at some level, simpler than the phenomenon it seeks to model. This introduces an intrinsic uncertainty associated with the effects of unmodeled factors (both known and unknown). This type of uncertainty is also referred to as epistemic, systemic, and structural uncertainty (Strong & Oakley, 2014; Wieder et al., 2015). Model structure refers to the specific cause and effect relationships that define a model. We use the term intrinsic to highlight that this type of uncertainty is connected to the very definition of a model. Methods for assessing intrinsic uncertainty are model dependent, but a general assumption is that the effects of model simplifications do not alter the ability to make useful comparisons of model outputs to existing and/or forthcoming observations (i.e., the intrinsic uncertainty is assumed to be small relative to the uncertainty associated with observational data). Testing the validity of this assumption requires evaluating the intrinsic uncertainty of a model.

Input/parameter uncertainty is generally evaluated in thermal history studies (e.g., Christensen, 1984; Davies, 1980; Korenaga, 2003) and in studies that couple thermal history models to climate models (e.g., Driscoll & Bercovici, 2013; Foley & Driscoll, 2016; Rushby et al., 2018). Intrinsic uncertainty has not, to the best of out knowledge, been evaluated for thermal history models. It is not correct to assume that an evaluation of input/parameter uncertainty alone is adequate to assess the confidence level of model outputs/predictions (Curry & Webster, 2011; Draper & Draper, 1995; Hosack et al., 2008; Taleb, 2011; Frigg et al., 2013). Thermal history models were originally developed, and still widely used, to model the Earth. As such, there has been a focus on models that can match observations and observational constraints are often built directly into thermal history models that focus on the Earth (Christensen, 1984; Korenaga, 2003). When observational data are available, the proof of a model's worth is often seen exclusively as its ability to account for the data. If a model cannot match the data, then it is considered incorrect in terms of assumptions and/or incomplete in the sense that unmodeled factors are not negligible. The latter can be seen as an indirect assessment of intrinsic uncertainty. However, that type of approach can mix intrinsic and parameter uncertainty in a manner that masks the quantitative aspects of the former. Uncertainty in the observational data itself can add to the screening of intrinsic model uncertainty if the only assessment used is how well model results match data. If the level of intrinsic uncertainty is comparable to input/parameter uncertainty and/or data uncertainty, and the intrinsic uncertainty of a model is not directly assessed, then the confidence given to the model can be inflated (e.g., Frigg et al., 2013).

Intrinsic uncertainty assessments can take on a different level of importance when thermal history models are moved from Earth application to exoplanet modeling. Thermal history models applied to Earth data are postdictive. That is, they seek to provide explanations for existing data. Moving models to applications were observations are forthcoming, such that model predictions are now being used to guide efforts at acquiring observations and/or guide methods to evaluate future observations, changes the way model uncertainty assessments are used (e.g., Hoffman & Hammonds, 1994). In general, this enhances the need for a complete model uncertainty analysis as model results are now being used in a predictive mode to help guide decision making (e.g., Pindyck, 2017).

The primary goal of this work is to quantify the intrinsic uncertainty for a class of thermal history models. We restrict ourselves to providing methods to assess intrinsic uncertainty. In application to any particular planetary modeling problem, intrinsic uncertainty would be coupled to input uncertainty assessment (and potentially uncertainties due to numerical solution methods) to provide a layered uncertainty measure (Taleb, 2011). An associated goal of this work is to assess the structural stability of thermal history models. The outputs from a structurally stable model do not change qualitatively if the model is perturbed. If, on the other hand, small-amplitude perturbations cause the qualitative nature of model solutions to change (e.g., attractors in model solution space appear or disappear), then the model is structurally unstable (George & Oxley, 1985; Guckenheimer & Holmes, 1983). The remainder of this paper develops and assesses intrinsic uncertainty measures for thermal history models that have been applied to the Earth. All of the models assessed assume that the geological manifestation of mantle convection is in the form of plate tectonics. However, the methods we employ can be applied to models that assume other forms of planetary tectonics (Lenardic, 2018a).

2 Methods

2.1 Parameterized Thermal History Models

Thermal history modeling seeks to map the cooling paths of terrestrial planets and/or moons. Subsolidus thermal convection in the rocky interior (the mantle) of terrestrial planets is a key cooling mechanism in their evolutions (Schubert et al., 1979). Modeling mantle convection over the lifetime of a terrestrial planet is, in principal, possible using 3-D numerical models. In practice, this approach comes with high computational and time costs that can be restrictive for mapping large regions of parameter space. For this reason, thermal evolution models that track spatially averaged quantities remain popular for Earth-focused studies (Conrad & Hager, 1999; Foley et al., 2012; Höink & Lenardic, 2013; Korenaga, 2003; Sandu et al., 2011) and for exoplanet studies (Kite et al., 2009; Komacek & Abbot, 2016; Schaefer & Sasselov, 2015). The reduced models are referred to as parameterized thermal history models since they do not solve the full convection equations but use, instead, an empirical formulation that parameterizes variations in convective heat flux as a function of the physical factors that drive and resist convective motions (Schubert et al., 1979, 1980). These models are efficient for traversing vast regions of parameter space (McNamara & Van Keken, 2000). They also allow layers of complexity to be added to base level models in relatively simple and efficient ways, for example, deep water cycling (McGovern & Schubert, 1989; Sandu et al., 2011), planetary carbon cycling (Abbot et al., 2012; Franck & Bounama, 1995; Tajika & Matsui, 1990, 1992; Sleep & Zahnle, 2001; Sleep et al., 2001), coupled thermal evolution, and climate modeling (Foley, 2015; Foley & Driscoll, 2016; Jellinek & Jackson, 2015; Lenardic et al., 2016). They can also be scaled to different planetary mass and/or volume in ways that maintain model efficiency (Valencia et al., 2007).

Parameterized thermal history models are based on a balance between the heat generated within the interior of a terrestrial planet's mantle and the heat flow through the planet's surface according to

$urn:x-wiley:jgre:media:jgre21201:jgre21201-math-0001$ (1)

where C is the heat capacity of the mantle, $urn:x-wiley:jgre:media:jgre21201:jgre21201-math-0002$ is the change in average mantle temperature with time, H is the total amount of heat produced internally, and Q is the total surface heat flow.

For most of the Earth's history, heat is produced principally by the radiogenic decay ²³⁸U, ²³⁵U, ²³²Th, and ⁴⁰K isotopes. The total amount of heat produced at any time is modeled as

$urn:x-wiley:jgre:media:jgre21201:jgre21201-math-0003$ (2)

where H₀ is total present-day heat generation, h_n is the amount of heat produced by a given isotope, λ_n is the decay constant for a given isotope, and t is time. Present-day fractional concentrations and power production for each isotope are represented by c_n and p_n, respectively. The values used for each parameter are given in Table 1. We assume present-day proportions of U:Th:K=1:4:(1.27×10⁴) and normalize by total U to calculate relative isotope concentrations (Turcotte & Schubert, 2002). Although we will not include them here, short-lived isotopes of Al and Fe can also make significant contributions to internal heat early in the history of rocky planets (Macpherson et al., 1995).

Table 1. Radiogenic Heat Production

Isotope	P_n (W/kg)	c_n	h_n	λ_n (1/Ga)
²³⁸U	9.37×10⁻⁵	0.9927	0.372	0.155
²³⁵U	5.69×10⁻⁴	0.0072	0.0164	0.985
²³²Th	2.69×10⁻⁵	4.0	0.430	0.0495
⁴⁰K	2.79×10⁻⁵	1.6256	0.181	0.555

The heat flux through the surface, which depends on convective vigor in the mantle, is typically parameterized using a scaling equation given by Schubert et al. (1979, 1980)

$urn:x-wiley:jgre:media:jgre21201:jgre21201-math-0004$ (3)

where a is a constant, Nu is the Nusselt number defined as the convective heat (Q) flux normalized by the amount of heat that would be conducted over the entire layer, $urn:x-wiley:jgre:media:jgre21201:jgre21201-math-0005$ where k, ΔT, and D are the thermal conductivity, difference between surface and interior temperature,and thickness of the convecting layer, respectively. The Rayleigh number (Ra) is defined as

$urn:x-wiley:jgre:media:jgre21201:jgre21201-math-0006$ (4)

where ρ, g, α, κ, and η(T_m) are density, gravity, thermal expansivity, thermal diffusivity and viscosity, respectively. The scaling parameter, β, varies for different classes of thermal history models. We will return to this issue shortly.

The temperature-dependent viscosity of the mantle is defined as

$urn:x-wiley:jgre:media:jgre21201:jgre21201-math-0007$ (5)

where A is the activation energy, R the universal gas constant, and η₀ a constant (Karato & Wu, 2013). As temperature increases, the viscosity will decrease, leading to an increase in Ra. If we assume that all values in equation 4 are constant except T_m and η(T_m), then combining equations 3 and 4 and the definition of Nu leads to

$urn:x-wiley:jgre:media:jgre21201:jgre21201-math-0008$ (6)

$urn:x-wiley:jgre:media:jgre21201:jgre21201-math-0009$ (7)

where all constants have now been combined into a^′.

Classic thermal history models (CTM) , developed for Earth, set β to a value of 0.33 based on laboratory experiments and boundary layer theory (e.g., Davies, 1980; Schubert et al., 1980). The constant, a^′, in equation 3, is determined based on experimental results or numerical simulations. The viscosity constant, η₀, can be based on experiments or set such that the viscosity at a reference present-day mantle temperature matches constraints on present-day mantle viscosity. With initial conditions, the model is then closed and equation 1 can be integrated forward in time.

A β value of 0.33 suggests that internal mantle viscosity is the dominant resistance to the motion of tectonic plates (e.g., Davies, 1980; Schubert et al., 1980). If plate strength offers significant resistance, then the scaling constant has been argued to be 0.15 or less (Christensen, 1984, 1985; Conrad & Hager, 1999). Korenaga (2003) has argued that β is negative as a result of dehydration during plate formation (plates become stronger in the Earth's past and, as a result, plate velocity and associated mantle cooling decreases even though Ra increases). To account for these different assumptions, we will examine cases with β values of 0.33, 0.15, 0.0, and −0.15. It is worth noting that variations in β represent different assumptions regarding physical processes that are critical to planetary cooling. This distinguishes β variations from uncertainties in material parameters (e.g., viscosity function parameters) and initial conditions. Stated another way, variable β is associated with competing hypotheses regarding planetary cooling.

There is a further break between models that have assumed negative or very low β values and CTM. As noted, CTM have traditionally integrated the energy balance equation forward in time. Workers who argued for low or negative β have taken a different approach and used the present day as the starting point and integrated backward in time (Christensen, 1985; Korenaga, 2003). A key thought behind that approach is the idea that present-day observations, that is, data constraints, have the lowest data uncertainty (i.e., lower than data constraints for past conditions). The constant a^′ in equation 6 is set such that present-day heat flux (Q₀) is achieved for assumed present-day mantle temperature and viscosity values (T₀ and η(T₀), respectively). The present-day ratio of heat produced within the mantle to that which is transferred through the surface, termed the Urey ratio (Ur), can then used to set present-day radiogenic heat production according to H₀=Ur*Q₀. The thermal history model then combines equations 1-2, and 6 resulting in

$urn:x-wiley:jgre:media:jgre21201:jgre21201-math-0010$ (8)

Equation 8 is then integrated backward in time to produce the Earth's thermal history constrained by present-day values of mantle temperature, heat flux, and Urey ratio (we will refer to this as an Earth-Scaled Model [ESM] formulation). Note that, unlike CTM that integrate forward in time and only specify initial mantle temperature, approaches that integrate backward in time often fix the initial temperature and its derivative to present-day values. This leads to a strongly determined system that restricts the model's solution domain.

To allow for an apples to apples comparison of uncertainty accumulation for a given β model, we use the ESM formulation solved forward in time. To accommodate for this and to test low and negative β models in a manner that holds to the methods of the workers who argued for these models, we first solved the energy balance equation backward by prescribing T₀, Q₀, and present-day Ur. Integrating backward to 4,500 Ma, using a fourth-order Runge-Kutta scheme, provided the mantle temperature (T_i) shortly after accretion and differentiation. The value of T_i was then applied as the initial condition for forward integration for a period of 4,500 Myr. Following this procedure, forward integration recovered the initial conditions (i.e., present-day values) of the ESM backward integration approach. This then allows for an apples to apples comparison to evaluate how different models respond to perturbations applied forward in time.

2.2 Perturbations

A useful metric, for dynamic systems models, is their time response—the time it takes a perturbation to decay (Close et al., 2001; Seely, 1964). This metric is referred to as the system's reactance (Cannon & Luecke, 1978). The reactance for variable thermal history models can be gauged by applying a single perturbation to mantle temperature and tracking the time it takes the thermal history path to damp the change in temperature. In addition to being a useful metric on its own, this will also provide insight into potential model uncertainty associated with unmodeled effects that are not singular in time. If the unmodeled effects are associated with variations that would occur on a timescale longer than the system's reactance time, then the system has the potential to damp the variations. If the variations occur on a timescale that is shorter than the systems reactance time, then the variations could be amplified over time and a model has the potential to lose structural stability.

We will perform single-perturbation analysis on full thermal history models and on stripped down versions that exclude the effect of decaying heat sources. The full model analysis will give metrics for full system reactance, while the latter will give metrics for convective reactance for different models. To isolate the influence of the convective reactance, equation 8 was modified by dropping the heat source term from the right-hand side. Using the modified equation, a reference solution was generated by defining an initial temperature and then allowing the mantle to evolve toward colder temperatures. The initial temperature was then perturbed by 15 °C to generate a perturbed solution. The difference between these two cooling paths was used to evaluate the convective reactance. If the two paths were converging, the perturbation was decaying and an e-folding time was defined as the amount of time it took for the perturbation to decrease by 1/e. A divergence between the reference and perturbed solutions indicated perturbation growth. For perturbation growth, the e-folding time was defined as the amount of time it took the perturbation to grow by a factor of e.

This singular in time perturbation concept can be extended by adding a low-amplitude noise term to the governing equation(s) of a model. This is often referred to as a “perturbed physics” analysis. After assessing the reactance times of different thermal history models, we perform uncertainty and stability tests by adding randomized, low-amplitude perturbations to each model. Physically, this additional term will mimic the chaotic nature of mantle convection—a factor that is not included in parameterized thermal history models. As well as assessing structural stability, these tests will give metrics for comparing the intrinsic uncertainty between models; that is, models whose outputs change by a small percentage, due to low-amplitude noise, will have a lower intrinsic uncertainty relative to models whose outputs change by a larger percentage.

For the randomized perturbations, the total distribution, for each model integration, was assumed to be normally distributed according to

$urn:x-wiley:jgre:media:jgre21201:jgre21201-math-0011$ (9)

where x is the percentage by which the current mantle temperature is perturbed and μ and σ the mean and standard deviation characterizing the distribution of perturbations, respectively. During each integration, a perturbation was randomly drawn from the probability density function where μ=0 and 2σ=A. The parameter A is the prescribed amplitude that defines the positive and negative limits between which 95% of perturbation percentages are located. Therefore, the density of large impulses will be much less than the density of smaller impulses. The impulses are applied at a fixed time interval (the interval itself can be varied).

For the case of randomized perturbations, different perturbation time series are possible. For those cases, multiple integrations were performed, for each model, to simulate the manner in which differing random draws affected the uncertainty and stability of different models. For each specific thermal history model, that is, models with different β values, the mean and standard deviation of T_m were calculated at each time step to test the reproducibility of the average mantle temperature that the thermal history models track and to assign associated model uncertainty windows.

2.3 Coupled Thermal History and Deep Water Cycling Models

An advantage of parameterized thermal history models is that added complexity can be incorporated into them relatively efficiently to build models that couple a range of planetary processes (e.g., models that couple the geologic history of a planet to its climate history). Adding complexity has the potential to transform a structurally stable model into an unstable one (or an unstable model into a stable one). Whether this will or will not be the case is not always easy to assess a priori as added complexity in planetary models can introduce multiple new feedback loops. Some of the loops may be negative (which would favor stability), but some may be positive (which could allow for instability). As an example, we will assess the intrinsic uncertainty of a model that couples deep water cycling to thermal history.

The full description of the deep water cycling model can be found in Sandu et al. (2011). A conceptual sketch will be useful at this stage (Figure 1). Warm mantle rises beneath mid-ocean ridges and is subjected to decompressional melting if it passes the mantle solidus, which is depressed by the presence of water. In the volume of melt produced, a fraction is hydrated resulting in a net dehydration of the mantle and therefore an increase in mantle viscosity. This can lower the vigor of mantle convection which can lead to a hotter mantle and, hence, increased melting and dehydration. This allows for the potential of a positive feedback. As the melt carrying the water migrates to the surface, some of the water is locked into the lithosphere whereas the remaining water escapes to the hydrosphere. As the lithosphere is advected away from the mid-ocean ridge, it cools, thickens, and gains water via metamorphic processes. At subduction zones, the downwelling slab is heated and releases some of its bound water back to the mantle. The reinjection of water into the mantle lowers its viscosity. This enhances cooling, and cooler conditions favor enhanced mantle rehydration at subduction zones. This allows for a feedback that works against the previously noted feedback. The degree to which the feedbacks are or are not balanced, at any time in model evolution, will determine a model's uncertainty metrics. This can be assessed following the procedures outlined in the previous section. An advantage of running an analysis on the base level thermal history model and the water cycling version is that the affects of different model components on the final intrinsic uncertainty can be assessed.

Details are in the caption following the image — **Figure 1**
Open in figure viewer PowerPoint

Accounting for mantle melting, associated water loss, and the effect of dehydration on mantle viscosity complicates the feedback structures of thermal-tectonic evolution models. Different feedbacks can dominate the model's overall feedback causing it to shift from positive or negative over evolution time. In the particular model diagrammed, the competition between temperature (negative feedback) and water effects (positive or negative feedback) on the mantle viscosity can lead to evolving model feedback dominance.

2.4 Exoplanet-Scaled Models

The discovery of large terrestrial exoplanets has lead to Earth-based thermal history models being scaled for larger mass and volume planets (Schaefer & Sasselov, 2015; Valencia et al., 2007). Scaling a model has the potential to alter its uncertainty metrics. As an example, we will scale two models of the previous subsections (a water cycling and a negative β model) and assess the intrinsic uncertainty of the scaled models. We will follow the scaling approach of Valencia et al. (2006). The scaling holds core mass fraction constant at a value of 0.3259. Planet and core radius scale as $urn:x-wiley:jgre:media:jgre21201:jgre21201-math-0012$ , where ⊕ indicates the value of Earth. If the index i represents planet radius, then a_p = 0.27. If the index i represents core radius, then a_c = 0.247. From these constraints, mantle volume and therefore average mantle density, ⟨ρ_m⟩ can be calculated. The acceleration due to gravity for each planet is defined by the relationship $urn:x-wiley:jgre:media:jgre21201:jgre21201-math-0013$ . For a scaled water cycling model, ridge length (L) is assumed to scale as 1.5 times the planetary radius. The concentration of mantle water is held constant. This means that total water in the mantle will not be constant. We hold all other parameters fixed and compute thermal evolutions for 1 M_⊕ and 2 M_⊕.

3 Analysis

In this section, we first assess the structural stability and intrinsic uncertainty of several thermal history models using singular in time perturbations and then low-amplitude random perturbations that mimic the chaotic nature of mantle convection. Next, we assess the uncertainty of a model that adds a layer of complexity to a particular base level thermal model to evaluate the influence of competing feedbacks on the evolution of model uncertainty. At the end of this section is an uncertainty analysis of two models, both argued to be applicable to Earth, scaled for large terrestrial exoplanets.

3.1 Impulse Response and Reactance Time Analysis

Figure 2 shows the response of several thermal history models to a singular in time perturbation. Model parameters can be found in Table 2 and were chosen to match present-day constraints. Perturbations decay most rapidly in the CTM (β=0.3) signifying a short reactance time. These models assume that the primary resistance to plate tectonic motions is mantle viscosity, which has an inversely exponential dependence on mantle temperature (e.g., Kohlstedt et al., 1995). This dependence gives rise to the model's strong negative feedback (Tozer, 1972; e.g., if mantle temperature increases, then mantle viscosity decreases which enhances mantle cooling leading to a drop in mantle temperature that works counter to the initial temperature rise). This negative feedback leads to a relatively rapid decay of model perturbations (Figure 2a).

Table 2. Convection Parameters

Parameter	Model	Parameter	Value	Units
Fixed model parameters
		η₀	2.21×10⁹	Pa·s
		A	300	kJ/mol
		R	8.314	J/(Kmol)
		Q₀	36	TW
		T₀	1623	K
Model-dependent parameters
	β=−0.15
		Ur	0.15	—
		T_i	1956	K
	β=0.00
		Ur	0.35	—
		T_i	1939	K
	β=0.15
		Ur	0.55	—
		T_i	2182	K
	β=−0.15
		Ur	0.75	—
		T_i	2016	K

As β was decreased, reactance time increased. Models with decreasing β assume that plate motion is resisted by a combination of plate strength and mantle viscosity (Conrad & Hager, 1999). This weakens the negative feedback discussed above, and, as a result, model reactance time increases (Figure 2b). The assumption for β=0 models is that resistance to plate motion is dominated by plate strength which has no dependence on mantle temperature (Christensen, 1984, 1985). This removes a negative (buffering) feedback from the model system, and the effects of perturbations survive to present day (Figure 2c). The slow convergence of perturbed model paths in Figure 2c is due to the decay of mantle heat sources—over a long evolution time, heat sources will be tapped and all the paths must approach a final nonconvecting state (the model analog for a geologically dead planet). Models with negative β assume that resistance to plate motion is dominated by plate strength and further assume that plate strength increases with mantle temperature (Korenaga, 2003, 2008). This leads to a positive feedback within the model system (Moore & Lenardic, 2015). As a result, evolution paths become highly sensitive to perturbations.

The differences in model reactance times can be highlighted by tracking the percent error between the perturbed and reference model paths (Figure 3). The percent error decayed quickly for models with shorter reactance times. Increasing reactance times, caused by shifting β toward zero, led to a slower decay. For β less than zero, the positive system feedback led to error growth. For any given β, the error decay or growth rate was a function of perturbation sign. Warmer perturbations decayed faster than colder ones for all models with β ≥ 0.

The asymmetry between warm and cold perturbations indicates that reactance time is a function of β and of mantle temperature. To assess the influence of these two effects, we isolated the convective reactance time (τ_c), defined as an e-folding timescale for perturbation growth or decay (this metric removes the effects of decaying heat sources on model reactance times). The value of τ_c was calculated for different initial temperatures and choices of β (Figure 4). For β>0, a positive perturbation increases convective vigor and heat transport leading to a rapid perturbation decay. A negative perturbation decreases convective vigor relative to the unperturbed state. The overall feedback, however, remains negative which still leads to a perturbation decay but at a slower rate than for a positive perturbation. As β was decreased, τ_c increased. The lengthening of τ_c continued until β took on negative values.

When β is reduced past zero, the overall model feedback changes from negative to positive. For negative β, τ_c is the amount of time it takes a perturbation to grow by a value of e. As β became more negative, perturbations grew more rapidly (Figure 4). For a positive feedback, a perturbation that increases mantle temperature decreases convective heat transfer. Decreasing mantle temperature leads to the opposite effect which resulted in a greater divergence between the perturbed and the reference cooling path. The fastest perturbation growth occurred at the combination of coldest temperatures and most negative β values (Figure 4).

3.2 Stochastic Fluctuations and Perturbed Physics Analysis

The response of thermal history models to random fluctuations, over model evolution time, is shown in Figure 5. Each model was subjected to 500 random perturbation sequences. During each sequence, the model was perturbed at a 10 Myr interval. We chose this timescale to assess how short time scale fluctuations, associated with the chaotic nature of mantle convection, can influence model behavior (associated with chaotic convection are potential changes in convective wavelength which can also cause fluctuations in mantle heat loss; Grigné et al., 2005). Each perturbation was drawn at random from a distribution defined by a mean of zero and a standard deviation of 7.5 °C. The maximum single perturbation was then approximately 1% of the total temperature, which is in line with the natural variation due to the chaotic nature of vigorous convection (e.g., Figure 4 of Weller et al., 2016). The random draw meant that the amplitude and sign of the perturbations were stochastic, and the full perturbation time series was nonperiodic. Plotting all of the perturbed paths generates an uncertainty shadow (top row of Figure 5). The uncertainty shadow is plotted along with the mean trend from the 500 perturbed models (blue lines) and the trend from the unperturbed models (black lines). A comparison of the mean trends and the unperturbed model trends shows that all the models maintain structural stability. All the models are not, however, associated with the same uncertainty structure. For the CTM, model uncertainty tended to saturate tightly around the reference solution. As β was decreased, model uncertainty increased.

For any specific model evolution time, a probability distribution of mantle temperatures can be compiled. Distributions for the present day, 1,000 and 2,000 Ma are shown in the bottom row of Figure 5. Each distribution is normally distributed about the unperturbed path with a time-dependent standard deviation. The smaller the standard deviation, the lower the model uncertainty.

Rather than limit the analysis to three specific model times, a continuous uncertainty window can be tracked (Figure 6). For positive β,uncertainty saturates over time. The uncertainty saturation limit is proportional to τ_c. A small τ_c indicates that the system reacts out perturbations quickly, resulting in a smaller accumulation of uncertainty. As β approached zero, but remained positive, τ_c increased, which enhanced uncertainty accumulation, though it remained bounded.

Unlike the β ≥ 0 models, there is no uncertainty saturation limit for the β<0 model. Instead, as the model evolved to lower temperatures, there was an increase of model uncertainty and its accumulation rate. As the mantle secularly cooled, any positive perturbation took the perturbed model further from the reference model, reducing the rate at which it cooled, creating a greater divergence between the two models. Alternatively, any negative perturbation created more rapid cooling in the perturbed model, causing it to also diverge from the reference model.

3.3 Thermal-Volatile Evolution Model

Introducing new layers of complexity to a model has the potential to change its stability and uncertainty properties. As an example, we consider adding deep water cycling to a thermal history model (see Sandu et al., 2011, and the description of Figure 1 for details). Mantle dehydration and rehydration couple with temperature to determine mantle viscosity which, in turn, feeds into convective vigor. As a result, model reactance times, structural stability, and intrinsic uncertainty can all be altered.

Uncertainty was evaluated for the coupled model using the procedures of the previous subsection. The uncertainty shadow from the coupled model is plotted in Figure 7 along with the β=0.3 reference model. Uncertainty tripled relative to the reference case at the final model evolution time. The addition of a positive feedback associated with the deep water cycle decreased the overall negative feedback of the model. Although the uncertainty is not fully bounded over the model run time shown in Figure 7, it is approaching a limit asymptotically (this will be made clear in the next subsection). Enhanced uncertainty in the coupled model is due to the continuous competition between thermal and water cycle effects on mantle viscosity. The competition alters the overall system feedback over model evolution time, which affects the accumulation of uncertainty. This points to the value of performing feedback analysis to map possible model trends consistent with internal feedbacks (e.g., Astrom & Murray, 2008; Crowley et al., 2011) . Feedback analysis can provide qualitative insights into uncertainty potential (e.g., does the model allow for unbounded uncertainty) without having to run the full model. For models of the type explored here, the computational time to run models is not a restriction but for more complex models it can be which adds to the value of feedback analysis as a first step in evaluating uncertainty potential.

An example of how model results, accounting for intrinsic uncertainty, can be compared with observational data is shown in Figure 8 (the data come from Condie et al., 2016). In Figure 8a, the typical method of presenting a model solution with data constraints is shown. Adding an uncertainty shadow (Figure 8b) allows for a more complete comparison of the model prediction with observational data. For the chosen initial conditions and parameters, the coupled model could satisfy observational constraints at a model uncertainty level comparable to that of the observational data itself. Two of the model solutions that determined the model uncertainty shadow are shown along with the mean solution in Figure 8c. Either path, within model uncertainty, is a viable model result. Plotting them shows the level of model deviations that can occur relative to the mean model trend (i.e., the trend from numerous models that account for stochastic fluctuations). The individual paths in Figure 8c evolved similarly in the initial and final stages of model evolution. However, a peaked divergence of ∼150 °C occurred between them at around 2500 Myr of model evolution. Whether this level of model uncertainty is large enough to alter the results from more complex models that, for example, couple climate evolution to thermal evolution will depend on the dynamic properties of the climate models used. Given that many climate models allow for bistable behavior, that is, multiple solutions under equivalent parameter conditions (e.g., Scheffer, 2009), this possibility cannot be ruled out.

3.3.1 Exoplanet Forecasting: Age Dependence

Thermal history models are being projected in time to understand how terrestrial planets evolve outside our own solar system. Doing so introduces new forms of uncertainty. Even for the Earth, the planet with the best observational data, there is debate regarding which thermal history model best represents the Earth's thermal evolution (e.g., Conrad & Hager, 1999; Grigné & Labrosse, 2001; Grigné et al., 2005; Korenaga, 2003; 2008; Moore & Webb, 2013; Sandu et al., 2011; Silver & Behn, 2008). Model selection uncertainty is a form of structural uncertainty: competing models for the Earth's thermal history have different mathematical structures to represent what different researchers consider to be the essential factors that have determined the Earth's thermal evolution.

Figure 9 shows thermal evolution paths from two models that have been applied to the Earth. One model is the deep water cycling model of the previous section (Sandu et al., 2011). The other is an ESM with β<0 value (Korenaga, 2003, 2008). Petrological data constraints are plotted with each model (Condie et al., 2016; Herzberg et al., 2010). The models, as originally presented, did not include uncertainty shadows, which we have determined following the procedures of the previous sections. We have also extended model time beyond the age of the Earth. Although the Earth provides the best observational constraints on a thermal history model, there is still uncertainty in the data. This includes uncertainty in observational constraints on the Earth's past thermal state (e.g., Condie et al., 2016; Herzberg et al., 2010) and uncertainty in observational data that can constrain the Earth's present thermal state (average surface heat flux, present-day radiogenic heat production, the average internal temperature of the Earth's mantle; e.g., Jaupart et al., 2015; Sarafian et al., 2017). Given data and model uncertainties, different models can satisfy observational constraints. Evaluating intrinsic uncertainty with other modeling uncertainties—initial condition and parametric—can broaden the viable model space. Different techniques (e.g., grid search and Monte Carlo) can be used to determine which models can match observational constraints over a larger portion of potential parameter space in an effort to gauge models that are statistically preferred (e.g., Höink & Lenardic, 2013; McNamara & Van Keken, 2000). Such approaches can rule out some classes of models, but multiple competing models can remain viable. In short, models based on different assumptions can fit Earth-based data constraints within allowable uncertainty bands. The critical point for modeling exoplanets is that there are competing hypothesis for the Earth's thermal evolution which brings with it an uncertainty associated with model selection.

The model paths plotted in Figure 9 are for particular parameter and initial conditions that can match data trends within uncertainty. The differences in model structure, and associated uncertainty, become clear when the models are projected forward in time for 10 billion years (Gyr). The intrinsic uncertainty for the deep water cycling model flattens over time. For the ESM model, uncertainty increases over time and after 5 Gyr of evolution the temperatures on the cold side of the uncertainty shadow become so low that the cold paths rapidly decay. This leads to the mean of multiple perturbed models deviating from the unperturbed model trend (for this reason, the uncertainty shadow is cut off at this point—beyond this point, probability distributions of model outputs deviate from Gaussian and can become fat tailed on the cold side of model evolution). The ESM model predicts that volcanic-tectonic activity should end after 5–6 Gyr (mantle temperatures become too cold to allow for continued melt generation). The deep water cycling model predicts a longer volcanic-tectonic lifetime. If model uncertainty is not taken into account this could lead to some discordant claims about the geologically active lifetime of our planet and, by association, about the habitability potential of exoplanets.

3.3.2 Exoplanet Forecasting: Size Dependence

As well as being extended in time, thermal history models can be scaled for larger planets. There is debate about which convective regime is likely to occur on larger planets. Some suggest that plate tectonics is likely (Van Heck & Tackley, 2011; Valencia & O'Connell, 2009; Valencia et al., 2007; Tackley et al., 2013), whereas others suggest that need not be the case (O'Neill & Lenardic, 2007). Keeping in line with the theme of this paper, we only evaluate the uncertainty associated with a plate tectonic mode but do not dismiss that other modes are possible. The simplest method for scaling a planetary model is to increase size and mass of the planet while holding all else equal—an assumption that is used herein only for demonstration. Parameters used for the scaled models can be found in Table 3. For the deep water cycling models, it is assumed that each planet begins with the same mantle water concentration and the same mantle potential temperature, which is consistent with previous studies that have scaled deep water cycle models (e.g., Schaefer & Sasselov, 2015). Similar to previous studies, we report average mantle temperature (T_m) rather than potential temperatures. This means that we have extrapolated the potential temperature along the adiabat to midmantle depth. We use this convention for consistency with previous exoplanet studies. In Figure 10a we plot the reference solutions for 1M⊕ and 2M⊕ planets together with uncertainty shadows. Model differences that arise in the early stages of evolution result from the competition between thermal and deep water cycle effects on mantle viscosity. For the choice of parameters, the Earth-sized model experienced a time window of little to no cooling between 1 and 2 Gyr. During this time, the mantle was dominantly dewatering which tended to increase mantle viscosity, lower convective vigor, and favor mantle heating. That heating tendency, due to more sluggish convection, was balanced by the tendency of temperatures to drop due to decaying heat sources. This lead to a flat line cooling trend. Following this period, water cycled back into the mantle from the surface and cooling was enhanced. Doubling the size of the planet increased the negative thermal feedback such that it outweighed the positive feedback due to mantle dewatering (that positive feedback was critical for the flat line cooling phase of the Earth-sized model). This decreased τ_c for the scaled up model, relative to the Earth model, which lowered its relative uncertainty. This implies that, for this particular model suite, the uncertainty of a reference Earth model will not underestimate the uncertainty of a scaled up model.

Table 3. Deep Water Cycle Model Parameters

Model	Parameter	Description	Value	Units
Convective model
	T_s	Surface temperature	300	K
	H(0)	Initial radiogenic heat	4.51	J/(m³/year)
	Rm	Mantle radius	6,271	km
	Rc	Core radius	3,471	km
	ρ_m	Mantle density	3,000	kg/m³
	k_m	Thermal conductivity	4.2	W/(mK)
	cp	Specific heat	1400	J/(kgK)
	α	Thermal expansivity	3.00×10⁻⁵	K⁻¹
	β	Convective exponent	0.33	—
	λ	Decay constant	3.4×10⁻¹⁰	year⁻¹
	Ra_cr	Critical Rayleigh number	1100	—
Water cycling
	η₀	Viscosity constant	1.7×10¹⁷	Pa·s
	A_cre	Material constant	90	MPa^−r/s
	r	Fugacity exponent	1.2	—
	Q_a	Creep activation energy	4.8×10⁵	J/mol
	χ_d	Degassing efficiency factor	0.03	—
	χ_r	Regassing efficiency factor	0.015	—
	OM	Mass of 1 Earth ocean	1.39×10²¹	kg
	OM(0)	Ocean masses initially in mantle	2	—

The ESM model with β=−0.15 was also scaled for a larger planet (Figure 10b). After 5 Gyr of model evolution, the reference model runs away to cooler temperatures. This signals that the evolution has moved outside of the conditions that the model was designed for from the start (models, in general, assume limits of validity and, as such, what they predict should only be physically interpreted within those limits). In this particular case, the model relies on the assumption that melting can occur within the mantle to generate a chemical lithosphere. Once mantle temperature drops so low that melting cannot occur, the assumed limits of model validity have been crossed and a different formulation would likely need to be implemented. This could be in the form of another plate tectonics model or a transition to a different convective regime. Doubling the mass of the planet increased the resilience against runaway by lessening the positive feedback inherent to this model (cf. Figure 4). As a result, uncertainty grew more slowly than for the Earth model. However, several of perturbed models, that evolved toward the colder side of the uncertainty envelope, did run away at 8 Gyr of evolution (this is why the uncertainty envelope is shown truncated at 8 Gyr).

4 Implications for Planetary Modeling

The goal of our uncertainty analysis, for parameterized planetary thermal history models, was to provide insight into their utility, not to advocate for the correctness or use of one model over another. Uncertainty analysis cannot determine if the assumptions of one model are more or less physically valid than those of a competing model. It can provide guidelines on how to interpret and apply model results.

When intrinsic uncertainty is accounted for, model results are presented as probability distributions (Figure 5). This introduces a probabilistic element to model testing. A mean model path that may have been cast out as unsuccessful, because it did match observations, can remain viable if its uncertainty cloud overlaps data uncertainty. The degree of overlap allows a nonbinary confidence measure to be determined. Model testing can still proceed in a probabilistically manner with confidence cutoffs being stated (e.g., “we consider initial condition and parameter combinations that can match data within two sigma confidence as viable”). If a model uncertainty cloud dwarfs data uncertainty, then it could become difficult to use the data itself to rule out particular model paths. In effect, a model could become irrefutable given the data. An irrefutable model is not necessarily based on invalid assumptions, but it does lose a level of utility.

Though preliminary, we can provide some implications for using models to interpret the Earth's thermal history in light of uncertainty. The larger the spread of model probability distributions, the lower the confidence one can place in using the model to make predictions and/or postdictions. The thermal history models with the highest uncertainty are associated with a positive feedback, which is already known to introduce a strong initial condition dependence to model outputs (Korenaga, 2016). It also brings the possibility that any unmodeled effects (e.g., post formation impacts O'Neill et al., 2017) could have as large an effect on model outputs as do the assumed critical factors that define the model (the tighter the model uncertainty probability distribution, the lower the effect of unmodeled factors relative to the effect of the particular heat flux scaling that the model is based on). For illustration purposes we assumed models that operated in a plate tectonic mode. However, other convective regimes may be possible over Earth's lifetime (e.g., Lenardic, 2018a). The mean evolution path of a plate tectonic model may not lead it to cross a regime boundary that initiates a different mode of tectonics. If uncertainty is considered, then it is possible that the model allows for a tectonic transition. The greater the uncertainty cloud, the greater this potential. A final implication relates to how nonmonotonic thermal-tectonic signals in the rock record are interpreted. Independent of intrinsic uncertainty analysis, certain classes of thermal history models predict monotonically decaying trends for mantle temperature, and by association plate velocities, over the bulk of an evolution path (e.g., CTM with a β of 0.33). Considering intrinsic uncertainty can prevent the incorrect inference that evidence for mantle heating in the past and/or an increase of plate velocities is necessarily arguing against the validity of these models (Figure 8c). It is possible that the magnitude of inferred nonmonotonic trends could favor one model over another but, in order to determine that, the magnitude of allowed fluctuations would need to be quantified. That is to say, an intrinsic/structural uncertainty analysis would need to be preformed.

An implication for exoplanet modeling, in light of intrinsic model uncertainty, relates to planetary habitability. A prominent issue in planetary modeling is mapping conditions that allow a planet to maintain liquid water over geological time scales, a feature considered critical for life as we know it (e.g., Meadows & Barnes, 2018). This has lead to the coupling of thermal history and climate models (e.g., Foley, 2015; Foley & Driscoll, 2016; Rushby et al., 2018). These coupled models track volatile cycling between a planet's interior and its surface envelopes. Coupling a thermal history model to a volatile cycling model can compound model uncertainty in a nonlinear way (Figure 7). Climate models will be associated with their own uncertainties (e.g., Mahadevan & Deutch, 2010; Roe & Baker, 2007). Propagating uncertainties in coupled solid planet and climate models has not been studied to date. If the coupling amplifies intrinsic uncertainty, then the predictions from a coupled model may become highly uncertain, even for a unique combination of initial condition and parameter values. A situation can arise where a unique combination of model inputs (e.g., solar distance, planet mass, and initial water content) leads to an uninhabitable planet based on models that do not account for intrinsic uncertainty. However, if intrinsic uncertainty is accounted for, then this conclusion may no longer hold. Accounting for uncertainty associated with model selection can compound this potential. Coupled climate and thermotectonic evolution modeling studies have dominantly used a single thermal history model, a single β value. However, within uncertainty, different β models can be consistent with Earth constraints (Figure 9). These models can produce very different predictions for planets older than the Earth. Collectively, this allows for the potential that the mapping of habitable conditions to date may be associated with an inflated level of confidence. To determine the degree to which this is the case, future studies will be required that account for layered uncertainty analysis (coupling of structural/intrinsic uncertainty, input uncertainty, and model selection uncertainty). This motivates future work that extends beyond the scope of this paper.

The points raised herein are not a call for less modeling but more, specifically, a layer of modeling that is not geared at making predictions that can be compared to data or that can be used to guide future exploration to gather new observations but, instead, is designed to quantify model uncertainties before the models are put into application modes. With this comes the value of a multimodel approach. Different base level thermal history models can be used collectively to determine which conclusions remain robust in light of model selection uncertainty together with the intrinsic uncertainties of particular models. This adds a complication to using planetary models to guide our thinking about conditions that allow for planetary life beyond Earth, but the problem is not intractable. The value of parameterized thermal history models, and the coupling of such models to simplified climate models, is that they remain simple enough to run millions of models given current computational power. This can provide uncertainty measures which, together with a multimodel approach, can be used to put confidence limits on planetary model predictions—something that will be appreciated by observationally focused colleagues who strive to provide uncertainty measures on observational data. The search for planets that allow for life beyond Earth involves a synergy between modeling and observations (e.g., Kasting, 2012). Both are associated with uncertainty. This does not stand in the way of moving the joint venture forward. Uncertainties can be evaluated and compared. If probability distributions are determined for observations and for models, this opens the path to use, for example, Bayesian analysis to provide well-defined confidence levels for planetary life potential under a range of potential scenarios (e.g., Walker et al., 2018). This approach becomes most effective if exoplanet search strategies are designed with statistical analysis in mind (e.g., Lenardic & Seales, 2019).

5 Conclusions

Thermal history models are associated with uncertainty that is independent of imperfectly known initial conditions and/or input parameter values. This intrinsic uncertainty can be assessed. Isolating dominant model feedbacks can give qualitative insight as to whether uncertainty can be amplified or damped. Single perturbation analysis can bring a more quantitative assessment by determining the reactance time of a model (the time for perturbations to grow or decay by some amount). A perturbed physics approach can provide a further metric by determining an uncertainty shadow for a particular model evolution over time. It can also determine the structural stability of a model and provide model output probability distributions that account for uncertainty. Once intrinsic uncertainty is accounted for in thermal history models, model outputs/predictions and comparisons to observational data should be treated in a statistical/probabilistic way.

Acknowledgments

This work has been supported by NASA Grants 80NSSC18K0828 and UWSC10435. The authors declare no conflict of interest, financial, or otherwise. All numerical data supporting the conclusions of this paper can be seen online (https://github.com/jds16/Data_IntrinsicUncertaintyAndStructuralStability).

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Citing Literature

Volume124, Issue8

August 2019

Pages 2213-2232

Assessing the Intrinsic Uncertainty and Structural Stability of Planetary Models: 1. Parameterized Thermal-Tectonic History Models

Abstract

Key Points

Plain Language Summary

1 Introduction