Editor's Note: This article originally appeared in SIAM News on October 17, 2023 (https://sinews.siam.org/Details-Page/stratocumulus-clouds-and-predator-prey-dynamics).
The rising global mean temperature within the last century is an alarming symptom of the planet’s changing climate. Earth’s energy budget—the balance of solar radiation that it absorbs from the sun and reemits back into space—controls the variability in global mean temperature. The planet heats up if the amount of energy that it absorbs is larger than the amount of energy that it emits. Clouds play a major role in Earth’s energy budget because they reflect sunlight and therefore reduce the amount of absorbed energy. Stratocumulus clouds are particularly important because they cover large sections of the subtropical oceans and thus reflect a great deal of energy.
Figure 1. Different stratocumulus cloud systems where white regions
represent thick, “puffy” clouds and dark regions represent the absence
of clouds. 1a. Satellite image of stratocumulus clouds. 1b. Contour plot
of cloud depth in a large eddy simulation. Figure 1a adapted from [2],
and Figure 1b adapted from [4].
Figure 1a provides a satellite image of a large stratocumulus cloud system. There are two configurations within the system: thick “puffy” clouds (circled in orange) that reflect a lot of sunlight and energy, and honeycomb-like structures of thin cloud filament (circled in blue) that surround large areas in which clouds are absent. These two configurations are a key area of study because of the impact that stratocumulus clouds exert on Earth’s energy balance.
Our research explores the use of mathematics, computation, and inverse modeling to better understand stratocumulus clouds. Specifically, we utilize inverse modeling to build bridges between two very different mathematical models of stratocumulus clouds.
Large Eddy Simulations
We can construct a very detailed model of Earth’s atmosphere in the form of a large eddy simulation (LES). In Figure 1b, white patches in the LES correspond to thick, “puffy” clouds and dark patches correspond to regions without clouds. The similarity between the simulation and images of Earth’s atmosphere is striking. Because of this accuracy, a single LES requires a great deal of computational power; simulating complex processes, such as Earth’s atmosphere, is complicated and necessitates the use of large supercomputers. The cloud system in Figure 1a covers an area that is roughly 1,000 kilometers (km) \(\times\) 1,000 km, but the LES in Figure 1b covers a much smaller area of about 48 km \(\times\) 48 km. An LES at a global scale and on time scales that are relevant to Earth’s climate is—and will continue to be—infeasible due to computational constraints. The use of lower-dimensional models is one way to resolve clouds and other atmospheric processes in global climate simulations.
Predator-prey Dynamics
We can construct a low-dimensional, phenomenological model for stratocumulus clouds by borrowing ideas from mathematical biology. In a predator-prey model, the prey’s population increases if the predator’s population is low, and vice versa. For example, if there is a high number of bobcats who like to eat rabbits, the rabbit population will decrease. But once the rabbit population is low, the bobcats will have nothing left to eat and their population will also decrease. With fewer bobcats around, the rabbit population will grow and the cycle repeats. We can employ predator-prey dynamics in the context of clouds by thinking of rain as a predator of clouds. A thickening cloud generates rain that then falls from the cloud, causing the cloud to shrink in size. The process subsequently starts all over again.
Describing cloud processes via predator-prey dynamics is the primary motivation behind a phenomenological model for stratocumulus clouds called the nonlinear cloud and rain equation (KTF17) [3]. The KTF17 model is a delay differential equation that describes cloud thickness as a function of time via limit cycles: nonlinear oscillations that are more complex than trigonometric functions (see Figure 2b). The oscillations represent the predator-prey behavior of cloud growth and decay due to rain. Because of its simplicity, we can solve the KTF17 model on a laptop computer. However, this simplicity implies that the model is less detailed than an LES (which requires a supercomputer). For example, KTF17 has no intrinsic spatial scale, while the LES resolves many temporal and spatial scales.
Even though these stratocumulus cloud models are quite different, the LES contains cycles of cloud growth and decay that are similar to the KTF17’s limit cycles. We define what we call an LES feature (see Figure 2a) by averaging all the cloud cycles that are detected in an LES. Our aim is to match KTF17 limit cycles to the LES feature to ultimately improve our understanding of KTF17 as a quantitatively descriptive model for stratocumulus clouds.
Figure 2. Cycles of growth and decay from different stratocumulus cloud models. 2a. Cloud cycles (light blue lines) of the large eddy simulation (LES) and the LES “feature” (dark blue line). The yellow, purple, coral, and brown lines highlight four examples of cloud cycles to illustrate the variation of cycles within an LES. 2b. A cloud cycle (limit cycle) of the nonlinear cloud and rain equation (KTF17) model. The purple line marks the time evolution of the KTF17 model’s cloud depth, and the orange line indicates one limit cycle. Figure courtesy of Rebecca Gjini.
Building Bridges Between Cloud Models
Inverse modeling is an interdisciplinary area of research that combines computational science, mathematics, and Earth science. The goal of inverse modeling is to fit mathematical/computational models to data by determining model parameters that generate compatible model outputs to the available data. For example, inverse modeling derives tomorrow’s weather forecast. First, we estimate today’s atmospheric state from current and past measurements of the weather — this is the inverse modeling, or inversion, step. We then launch a weather forecast from the atmospheric state that we obtained via inverse modeling.
Numerically, inverse modeling relies on optimization, Bayesian statistics, and Markov chain Monte Carlo (MCMC). At their core, all inversions rely on a Bayesian posterior probability distribution
\[p(\theta | y) \propto p_0(\theta)p_l(y | \theta),\]
which describes how data \((y)\) inform model parameters \((\theta)\). More specifically, the posterior is defined by the product of a prior \(p_0(\theta)\) and likelihood \(p_l(y\vert \theta)\). The prior encodes knowledge about model parameters that we already have, while the likelihood directly connects the model, its parameters \((\theta)\), and the data \((y)\). The posterior thus encapsulates everything that we know about the model and its parameters, given prior knowledge and data. The numerical solution of an inverse problem involves finding statistics of the posterior via optimization or approximating the posterior via MCMC.
We now apply these ideas to connect the KTF17 and LES models. The LES feature serves as “data,” and we attempt to calibrate the KTF17 model to the LES via an inversion. More specifically, we use inverse modeling to obtain the parameters of the KTF model \((\theta)\) by matching KTF17 limit cycles to the LES feature. This type of inversion falls under the umbrella of feature-based inversion [5], and a previous study applied it to KTF17 and stratocumulus clouds [4]. The numerical solution of a feature-based inversion relies on MCMC, and particularly on a very robust affine invariant ensemble sampler that is often called the emcee hammer [1].
Figure 3. Feature-based inversion results. 3a. Triangle plot of the posterior distribution. Warmer colors correspond to a higher posterior probability. 3b. Large eddy simulation (LES) feature (in dark blue), the error model (in light blue), the posterior mode of the nonlinear cloud and rain equation (KTF17) model (in green), and 2,000 limit cycles of KTF17 (in orange). Figure courtesy of Rebecca Gjini.
Results and Discussion
Figure 3 depicts feature-based inversion implemented on an LES. Figure 3a is a triangle plot that illustrates the posterior distribution of the KTF17 parameters, allowing us to qualitatively assess KTF17 parameter values that lead to similar cloud cycles to those in the LES. Figure 3b illustrates the way in which the KTF17 model (in green and orange) leads to cycles of cloud growth and decay that are similar to those in an LES (in blue and dark blue). Here, we use an error model to represent variation within the cycles of cloud growth and decay in the LES (see also Figure 2).
More recently, we have started to not only invert a single LES [4] but rather a whole “suite” or collection of LES, wherein each LES represents different meteorological conditions. We wish to identify potential relationships between the KTF17 parameters and meteorological conditions of stratocumulus clouds; such relationships can help link the various models that stem from the cloud and climate communities. Ultimately, these connections will improve our understanding of KTF17 as a quantitatively descriptive model for stratocumulus clouds.
Rebecca Gjini delivered a minisymposium presentation on this research at the 2023 SIAM Conference on Computational Science and Engineering (CSE23), which took place in Amsterdam, the Netherlands, earlier this year. She received funding to attend CSE23 through a SIAM Student Travel Award. To learn more about Student Travel Awards and submit an application, visit the online page.
SIAM Student Travel Awards are made possible in part by the generous support of our community. To make a gift to the Student Travel Fund, visit the SIAM website.
Acknowledgments: I thank Matthias Morzfeld of the Scripps Institution of Oceanography at the University of California, San Diego for his expertise and feedback. I also thank Franziska Glassmeier of Delft University of Technology and Graham Feingold of the National Oceanic and Atmospheric Administration for engaging in interesting and important discussions, sharing their deep knowledge of cloud physics and phenomenological modeling, and providing me with a large suite of LES. Additionally, I thank Spence Lunderman (private sector) for helping me to understand the code that he wrote for his Ph.D. dissertation at the University of Arizona. Last but not least, I thank the U.S. Office of Naval Research for its support via grant N00014-21-1-2309.
References
[1] Goodman, J., & Weare, J. (2010). Ensemble samplers with affine invariance. Comm. App. Math. Comp. Sci., 5(1) 65-80.
[2] Koren, I., & Feingold, G. (2011). Aerosol-cloud-precipitation system as a predator-prey problem. Proc. Natl. Acad. Sci., 108(30), 12227-12232.
[3] Koren, I., Tziperman, E., & Feingold, G. (2017). Exploring the nonlinear cloud and rain equation. Chaos, 27(1), 013107.
[4] Lunderman, S., Morzfeld, M., Glassmeier, F., & Feingold, G. (2020). Estimating parameters of the nonlinear cloud and rain equation from a large-eddy simulation. Phys. D: Nonlin. Phenom., 410, 132500.
[5] Morzfeld, M., Adams, J., Lunderman, S., & Orozco, R. (2018). Feature-based data assimilation in geophysics. Nonlin. Process. Geophys., 25(2), 355-374.
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Rebecca Gjini is a Ph.D. student in the Institute of Geophysics and Planetary Physics at Scripps Institution of Oceanography at the University of California, San Diego. Her main research interests include data assimilation, derivative-free optimization, and cloud microphysics. She hopes to continue using mathematics to improve understanding of the climate system. |