# Modeling Mono Lake's Water Balance

#### Summary

In this activity students build a hydrologic model of Mono Lake in MATLAB and then use the model to evaluate the California State Water Board's 1994 decision regulating diversions from the watershed and design their own water management plan for the lake.

In 1941 the natural water balance of Mono Lake was altered when the Los Angeles Department of Water and Power (LADWP) began diverting water from the watershed. The lake level dropped 45 feet by 1982 threatening the local environment. After a long legal battle the California State Water Board issued an order (D. 1631) limiting water diversions by LADWP in order to return the lake to a desirable level. As of June 2019 the lake surface has yet to reach the target elevation.

In developing a water-balance model of Mono Lake students learn to idealize a hydrologic system as stocks and flows, translate their stock and flow diagram into a water balance equation and solve this equation over time using the forward Euler method. Once students have a working lake model they use MATLAB built-in functions to explore the variance and co-variance of hydrologic data and use these to constrain a probabilistic model of the Lake.

## Context

#### Audience

This activity is intended to be used in 1-2 weeks of class time (3-hour lab, three 50-minute or two 75-minute classes). We use it in an upper-level undergraduate course in earth and environmental modeling, but it could be adapted to an undergraduate hydrology or upper level earth systems course.

#### Skills and concepts that students must have mastered

Students should come to class having completed the background reading on Mono Lake and with basic MATLAB coding skills such as those explained in the MATLAB Onramp tutorial.

#### How the activity is situated in the course

This is the second activity or module in the course. It follows a simple box model of a draining carboy experiment that the students conduct in the first week of class.

## Goals

#### Content/concepts goals for this activity

At the end of this activity students will be able to:
1. Define the components (stocks and flows) and balance of a hydrologic system from published descriptions
2. Implement a numerical simulation of a water balance equation in MATLAB using the forward Euler method
3. Describe the distribution of a single variable within a data set (univariate analysis) and the correlation between variables (multivariate analysis)
4. Estimate model uncertainty using Monte Carlo methods in MATLAB

#### Higher order thinking skills goals for this activity

At the end of this activity students will be able to:
1. Design and construct a dynamic model of a natural hydrologic system using MATLAB scripting
2. Use a probabilistic model to evaluate the efficacy of water diversion policies

#### Other skills goals for this activity

At the end of this activity students will be able to:
1. Present model results and recommendations in concise oral presentations and written reports

## Description and Teaching Materials

In preparation for this activity students should read pages 1-43 of Mono Basin Ecosystem Study Committee (1987). Mono Lake Background slides.pptx (PowerPoint 2007 (.pptx) 174.5MB Jul5 19) can be used to introduce the activity in class when the reading is assigned.

In class the activity can be broken into three blocks:

First, have students use the reading to create a conceptual model of the system (30-50 minutes). Working in groups of 2-3 they draw a concept sketch of the hydrologic system and then idealize the system with a stock and flow diagram (see Menking et al. (2017) or Chapter 3 of Ford (2010) for examples). End with a class discussion of the water balance and determination of the balance equation. These activities are summarized in the student worksheet conceptual model development student.docx (Microsoft Word 2007 (.docx) 20kB Jul1 19).

Optional (50-60 minutes): have students construct a first model using systems dynamics software such as Stella (see https://www.iseesystems.com/store/products/ for purchasing options or to download a trial) or Insight Maker (a free browser-based tool https://insightmaker.com/). The workflow for Insight Maker is described in Systems Dynamics Model.docx (Microsoft Word 2007 (.docx) 409kB Jul12 19).

Second, students construct a numerical model of the system in MATLAB (50-60 minutes). The MATLAB LiveScript document Student_Handout_Mono_Lake water_balance_model.mlx (MATLAB Live Script 6kB Jul2 19) provides an outline, instructions for students to construct their model, guidelines for evaluating and experimenting with their finished model. The students use lake survey data Mono Lake Survey.xlsx (Excel 2007 (.xlsx) 37kB Jul1 19) and water flow data from 1937-1983 Mono Lake Flows 1937-1983.xlsx (Excel 2007 (.xlsx) 23kB Jul1 19) compiled by Vorster (1985) to build their model and lake elevation data from 1850-2017 compiled by the Mono Lake Committee MLC_Lake Levels 1850-2017.xlsx (Excel 2007 (.xlsx) 11kB Jul1 19) to evaluate the results. Note that updated lake elevation data may be available at Mono Basin Clearinghouse.

Third, students modify their predictive model (developed as an experiment at the end of part 2) using Monte Carlo methods to incorporate variance in their model parameters (50-60 minutes). The MATLAB LiveScript document Student_handout_Mono_Lake_Data_Exploration.mlx (MATLAB Live Script 6kB Jul2 19) leads students through a brief statistical exploration of the Vorster data and tools for generating synthetic data sets with the same statistical properties. The MATLAB LiveScript document Student_Handout_Mono_Lake_Monte_Carlo_Model.mlx (MATLAB Live Script 7kB Jul2 19) leads students through the process of modifying their initial predictive water balance model to add Monte Carlo simulation.

## Teaching Notes and Tips

This activity uses a case study approach for teaching hydrologic and modeling content. The Case studies On-Ramp provides a quick overview along with links to additional resources for those who are new to teaching with case studies.

The activity can be taught in 1-2 weeks of class time.

If taught in a single week, the activity will need to be accelerated to make sure that students have time to experiment with their models. This can be done by assigning the conceptual model sheet and/or some of the coding as homework, walking students through some of the coding step-by-step or by providing students the solution to one of the two model scripts (Student_Handout_Mono_Lake water_balance_model.mlx (MATLAB Live Script 6kB Jul2 19) or Student_Handout_Mono_Lake_Monte_Carlo_Model.mlx (MATLAB Live Script 7kB Jul2 19)). The best approach to accelerating the activity will depend on which aspects of the assignment best align with your course goals.

If taught over two weeks, students can work through the first two parts (conceptual model and water balance model) in the first week (possibly including the optional systems dynamics model) and then complete the probabilistic model, experimentation, and assessment in the second week.

Have students hand in a draft model to make sure that they have a working script prior to implementing the Monte Carlo simulation.

## Assessment

Students make group presentations in class and submit individual reports that summarize their recommendations for future LAPWD water diversions from the Monoo Basin. The assignment details are presented at the end of Student_Handout_Mono_Lake_Monte_Carlo_Model.mlx (MATLAB Live Script 7kB Jul2 19). In the written report students are asked to hand in their recommendations and justifications as well as an outline of the modeling process and completed MATLAB scripts. The Model process and rubric.docx (Microsoft Word 2007 (.docx) 19kB Jul2 19) also acts as a grading rubric.

The zipped folder

contains MATLAB solutions and the data files for instructor use.

## References and Resources

Ford, Andrew (2010). Modeling the Environment, Second Edition. Washington, D.C.: Island Press and companion site http://modeling-the-environment.com. An overview of systems dynamics modeling, including a chapter on Mono Lake that was the inspiration for this activity.

Mono Basin Ecosystem Study Committee (1987). The Mono Basin Ecosystem: Effects of Changing Lake Level. Washington: National Academies Press, 288 p. https://www.nap.edu/catalog/1007/the-mono-basin-ecosystem-effects-of-changing-lake-level A thorough report by the National Academies. The introduction and hydrology chapters provide a nice overview for students.

Vorster (1985). A Water Balance Forecast Model for Mono Lake, California (Master's thesis, Calif. State University, Hayward). Available at https://www.monobasinresearch.org/onlinereports/waterbalance.php. Inspiration and data source for model.

https://serc.carleton.edu/microbelife/topics/monolake/index.html. A biological perspective on Mono Lake with a number of useful links.

https://www.monolake.org/. An extensive background on Mono Lake, including human and natural history.

http://www.monobasinresearch.org/index.php A compilation of scientific information and legal documents.

https://ca.water.usgs.gov/california-drought/ An overview of drought in California that puts recent events in context. Students may want to explore this page when putting their models in context.