SIAM News Blog

Numerical Simulation of Permafrost Thermal Dynamics in Alaskan Lake-Soil Systems

By Lina Sorg

40 percent of the Arctic Lowlands—the coastal plain of Alaska and northwest Canada—is covered by lakes. These lakes serve as important heat reservoirs that affect the thaw of permafrost: a permanently frozen layer on or under the Earth's surface. For instance, lakes with a mean annual bed temperature above 0 degrees Celsius do not freeze to their beds during the winter. As the Earth’s climate continues to warm, increases in air temperature and heightened instances of coastal flooding are impacting lake temperatures and driving permafrost thaw. Thawing permafrost dramatically affects the surrounding area and its inhabitants because the soggy water and soil mixture often cannot support the weight of the above vegetation and soil layer. As a consequence, roads, buildings, and even entire villages may incur significant damage.

Figure 1. Arctic lakes on the North Slope (right) in northern Alaska (left). Images courtesy of Google Earth.
During a minisymposium presentation at the 2023 SIAM Conference on Computational Science and Engineering, which is currently taking place in Amsterdam, the Netherlands, Svetlana Tokareva of Los Alamos National Laboratory employed a numerical model to simulate the meteorological conditions that affect lake temperatures and trigger permafrost thermal dynamics. She focused her efforts on the Arctic lakes in northern Alaska (see Figure 1). 

Tokareva began with an overview of an open-source code called Amanzi-ATS, which comprised her computational model. ATS—which stands for Advanced Terrestrial Simulator—is a community platform that helps users test and integrate process-based models to solve problems in ecosystem-based, integrated distributed hydrology. It is freely available on GitHub,  supports multiple spatial domains, and has a flexible mesh infrastructure and an application-centric, high-level design.

The basic code concept of Amanzi-ATS is a process kernel (PK): a partial differential equation (PDE) on a single domain that often represents conservation of a single quantity. PK implements an interface related to time integration and accounts for factors like subsurface flow, surface transport, and snow. Multi-process couplers (MPCs) couple several PKs together. 

Figure 2. Advanced Terrestrial Simulator (ATS) coupled model of heat transfer in a lake-soil system. Figure courtesy of Svetlana Tokareva.
Tokareva then debuted her lake model, which comprises four different layers: snow, ice, water, and soil. Although every lake is a three-dimensional (3D) geometry, Tokareva is only interested in the processes that occur in the vertical direction. For the sake of practical application, she thus chose to model her lake as a one-dimensional vertical column with variable water body depth (see Figure 2).

After sharing a diagram for the coupled model that illustrates the PKs and strong and weak MPCs (see Figure 3), Tokareva introduced the PDEs for temperature distribution in the lake. First, she examined the water layer. Because actual lake depth varies based on factors like precipitation and surface runoff, she used normalized coordinates to keep the domain fixed. Scaling the vertical coordinates by the depth of the lake yielded an atypical formulation of the temperature diffusion equation, but all of the other terms in the PDE are standard.

Tokareva next discussed the PDEs for temperature distribution in the soil, which include a heat equation, Richards equation, Darcy velocity, soil heat capacity, and soil thermal conductivity. The boundary conditions on the lake’s top layer can take the form of either snow, ice, or water depending on the season. The temperature at the bottom of the soil is assumed to be constant.

One advantage of ATS is that users can utilize PKs that other researchers previously implemented for related problems. Tokareva therefore employed an existing PK from a different permafrost project for the snow layer. This model computes the snow surface temperature via the surface energy balance equation and determines heat conduction through snowpack. The heat flux boundary condition allows for two-way coupling between the water and snow.

Figure 3. Building blocks of the Advanced Terrestrial Simulator (ATS). Figure courtesy of Svetlana Tokareva.
To test her model and predict the effect of increasing air temperatures on lake temperature, Tokareva performed validation studies for two locations: Atqasuk Lake in northern Alaska and Fox Den Lake in western Alaska. Atqasuk Lake has a maximum depth of 2.54 meters, while Fox Den Lake has a maximum depth of 1.6 meters. “The surrounding area is classified as continuous permafrost, but the presence and depth of permafrost under the lake is not confirmed,” Tokareva said. The two-year simulation period for Atqasuk Lake spanned August 2013 to August 2015. For this lake, Tokareva pre-processed meteorological data from the South Meade USGS meteorological station and the Atqasuk Lake meteorological station, then inputted data that pertained to factors like air temperature, humidity, precipitation rate, and wind speed. Although her simplified model does not account for all possible physical processes, the ATS performed fairly accurately (with a few differences) when compared to an existing LAKE2.0 model.

Because meteorological data was not locally available at Fox Den, Tokareva used data from the National Weather Service Station in Kotzebue, Ala., as well as short- and longwave radiation from NASA’s Clouds and the Earth’s Radiant Energy System project. The four-year simulation period for Fox Den encompassed June 2009 to June 2013, and the ATS simulation yielded significantly better results than for Atqasuk Lake; the first and second cycles exhibited good consistency between the observed and simulated temperatures.

Tokareva concluded her presentation by reminding the audience that her lake model is still very much a work in progress. She and her team are working to add new influential elements to the model, such as water salinity, turbulent mixing, and biochemistry. They also plan to account for a lake’s actual 3D geometry and intend to eventually examine systems of several lakes that are located close together.

Lina Sorg is the managing editor of SIAM News.
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