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CODE 112037
SEMESTER 2° Semester
MODULES Questo insegnamento è un modulo di:



The educational objectives and learning outcomes of the module specifically focus on mixing and dispersion processes. The course aims to provide students with a deep and applied understanding of mixing and dispersion mechanisms in the atmosphere and oceans, through an approach that integrates theory, computer simulations, and data analysis. Below, the educational objectives and expected learning outcomes are outlined.

Educational Objectives

Understanding Mixing and Dispersion Principles: Students will gain a solid theoretical understanding of the mixing and dispersion mechanisms operating in the atmosphere and marine environment, including fundamental concepts such as turbulence, eddy viscosity, the Ekman boundary layer, and the role of atmospheric stability.

Application of Numerical Models and Simulations: Students will learn to use computational tools and specialized software to model and simulate mixing and dispersion processes, acquiring practical skills in managing and analyzing georeferenced data and configuring models of marine circulation and dispersion.

Numerical Analysis and Scientific Programming: Fundamental concepts of numerical analysis and scientific programming will be introduced, with a particular emphasis on using the Python programming language and Jupyter software for developing numerical schemes and analyzing trajectories of particles in the sea.

Integration of Data and Models: Students will be trained in the use of georeferenced data and the NetCDF format, applying data analysis techniques to interpret velocities measured by high-frequency coastal marine radars and to study marine dispersion using advanced models.

Learning Outcomes

By the end of the course, students will be able to:

Analyze and Interpret Mixing Processes: Understand and explain the main mechanisms of mixing and dispersion in the atmosphere and marine environment, identifying the forces and factors that influence these processes.

Develop and Apply Simulations: Use technical skills to develop and implement simulations of mixing and dispersion processes, correctly interpreting the results of the simulations and applying these concepts to real environmental problems.

Manage Complex Environmental Data: Acquire, process, and analyze complex georeferenced data, applying numerical models and analysis techniques to study the dispersion of substances in the atmosphere and marine environment.

Integrate Theory and Practice: Demonstrate the ability to integrate theoretical knowledge with practical applications, developing model-based solutions to address engineering and environmental issues related to mixing and dispersion.

Communicate Scientific Results: Present and discuss the results of simulations and data analyses clearly and effectively, both through written reports and oral presentations, demonstrating a comprehensive understanding of the studied mixing and dispersion processes.

In summary, the course aims to train students capable of applying advanced theoretical knowledge and technical skills in the analysis, modeling, and simulation of mixing and dispersion processes, preparing them to effectively contribute to research and professional practice in the field of environmental engineering.


The module is structured in a balanced educational framework that aims to provide students with comprehensive training in the field of dispersion in air and sea, combining theoretical insights with significant practical applications. The adopted methodology is designed not only to stimulate the learning of fundamental concepts but also to develop technical skills through the use of advanced computational tools. Below are the teaching methods planned for the course detailed.

Theoretical Part

The theoretical component of the course is delivered through lectures that aim to introduce and explain the fundamental principles of mixing processes in air and marine environments. Students will be guided through the analysis of fundamental theorems, the derivation of key equations, and the discussion of illustrative examples that demonstrate the application of concepts in real scenarios.

Practical Computer Exercises

Alongside the theory, the course includes a series of practical computer exercises, where students will have the opportunity to directly apply the learned concepts to solve concrete problems. This component is essential for integrating theory with practice and for developing skills in the use of specialized software for the numerical modeling of mixing processes in geophysical flows. The exercises will cover the implementation of equations governing mixing processes in programming languages such as Python, the simulation of atmospheric and oceanic phenomena, and the analysis of data and results obtained from the simulations.

Final Exercise and Report

The culmination of the practical experience of the course is represented by a final exercise that requires students to tackle a complex problem of dispersion in geophysical flows, applying an integrated approach of theoretical knowledge and technical skills acquired. The final exercise involves developing a project on a specific theme, which could include modeling a particular dispersion phenomenon in air or sea.

At the end of the exercise, students are required to prepare a detailed report documenting the work performed, the methods used, the results obtained, and the conclusions reached. The report represents a fundamental element of course evaluation, as it allows verifying the student's ability to conduct a complete scientific analysis, to critically apply theoretical knowledge to practical problems, and to effectively communicate the results of their work.

Integrated Teaching Approach

The teaching approach of the course is designed to be highly integrated, with constant reference between theory and practice. Theoretical discussions in the classroom are regularly linked to practical applications in computer exercises, encouraging students to develop a deep and applied understanding of the subject. This method enables students to acquire not only a solid theoretical foundation but also the ability to use this knowledge in real and professional contexts, effectively preparing them for future challenges in the field of research and profession.


Mixing in the atmosphere

  • Introduction to the course and review of elementary fluid dynamics
  • Turbulence and closures
  • The concept of eddy viscosity
  • Ekman boundary layer
  • Discussion on the effect of crossing isobars
  • Definition of Air Parcel
  • Dry case and dry-adiabatic lapse rate
  • Moist Adiabatic Lapse Rate deduced from the First Law of Thermodynamics
  • Static stability of the atmosphere
  • Potential temperature
  • Mechanisms of stability
  • Mechanisms of instability
  • Equations of passive scalar transport and 'energetic' balance equation
  • Comments on the balance equations of the passive scalar
  • Energy spectrum and fluctuation flows
  • The Peclet number and proliferation of degrees of freedom
  • Need for a RANS-type description
  • RANS equations and the closure problem
  • Seminar on weather models.
  • Yaglom's law and the flow of "energy" of scalar fluctuations
  • RANS equation for the large-scale square
  • RANS equation for the small-scale square
  • Closure based on phenomenology and eddy diffusivity dependence on the Richardson number.
  • Lagrangian description of the transport problem
  • Evolution of heavy particles and the role of inertia
  • Stokes time and the limit of small Stokes times
  • The Brownian contribution
  • Python exercise on Kolmogorov parallel flow and cellular flows


Mixing in the sea

-) the introduction of some general notions of physical oceanography; 
-) the introduction of the concepts of scientific programming and numerical analysis;
-) the processing of large amounts of georeferenced data in formats commonly used in engineering and environmental physics;
-) the implementation of a marine circulation and dispersion model for simulations of engineering and environmental interest.

The teaching activity in this second part is organized into three sections. In the first it is taught how to configure virtual machines and use the LINUX operating system, together with text editors such as EMACS. Next, the PYTHON programming language and JUPYTER software are introduced. 
The second section introduces the basic concepts of numerical analysis and some numerical schemes for solving ordinary differential equations, specifically studying the equations for inertial oscillations and using PYTHON functions to calculate the trajectories of different particles moving at sea. After having presented autoregressive models and the NetCDF data format, the previously introduced concepts are applied using surface marine velocities measured by high-frequency coastal radars. If time permits, layered ocean models and reduced gravity models are also introduced.
In the last section, the concepts behind different marine circulation models, the basics of FORTRAN and CPP are introduced. Inertial oscillations are then simulated also with the ROMS ocean model, showing the students how to read and visualize its numerical outputs. Finally, the circulation and dispersion of substances in an idealized tidal inlet are simulated always with ROMS.
Students can access the final exam only after taking all the exercises provided during this second part of the course. For the final exam, students may also choose to take a practical test resulting in a corresponding report on the second part of the course where they implement numerical schemes and configure the ROMS model similarly to what it was shown in class.





Fluid Mechanics, 6th Edition - June 4, 2015, Authors: Pijush K. Kundu, Ira M. Cohen, David R Dowling, Language: English, Hardback ISBN: 9780124059351, eBook ISBN: 9780124071513



Exam Board







Class schedule

The timetable for this course is available here: Portale EasyAcademy



The final evaluation of the module includes a balanced approach aimed at assessing both the practical skills and the theoretical understanding of the students. This approach is structured in two main phases: a practical computer exercise and an oral examination.

Practical Computer Exercise

The first phase of the exam consists of a practical exercise conducted using the computer, during which students are called upon to demonstrate their abilities in applying the theoretical concepts learned to concrete environmental interest problems. This part of the exam is designed to evaluate the student's ability to use advanced computational tools, such as specific programming languages like Python, to model the dynamics of geophysical fluids in realistic scenarios.

During the exercise, students will need to solve a set of selected problems that will require data processing, the implementation of numerical models, and the interpretation of the results obtained. The goal is to assess not only the accuracy of the proposed solutions but also the ability to critically and creatively apply the knowledge acquired, as well as clarity in presenting the results.

Oral Examination

The second phase of the exam consists of an oral test, during which the theoretical understanding of the topics covered in the course will be verified. The oral examination aims to evaluate the depth of the student's knowledge on the fundamental principles of geophysical fluid dynamics and the theoretical implications of the phenomena studied. During the oral, students may be asked specific questions on course themes, requiring an in-depth discussion that highlights a solid understanding of the concepts.

In addition to the theoretical part, the oral examination will also include the discussion of a report prepared by the student. This report must be based on the practical computer exercise and aims to evaluate the student's ability to synthesize and effectively communicate the results of the work done, integrating theoretical and applied aspects. The report represents an opportunity for the student to demonstrate their critical and analytical approach to problems, as well as the skills acquired in data interpretation and the formulation of valid conclusions.

Overall Assessment

The final evaluation of the course will be based on the combination of performances obtained in the practical computer exercise and the oral examination, taking into account both the quality of the proposed solutions and the ability to argue and present theoretical concepts. This multidimensional approach to evaluation ensures a comprehensive view of the student's competencies, rewarding not only theoretical knowledge but also practical and analytical abilities, essential for a career in the field of environmental engineering.


The method of assessing student preparedness for the course has been carefully designed to ensure a fair and comprehensive evaluation of the skills acquired, both from a theoretical and practical standpoint. This evaluation process aims to give students the opportunity to fully demonstrate their understanding and abilities, through various phases that include:

Practical Computer Exercise

Students will first be assessed through a practical computer exercise, which will focus on applying the theoretical concepts studied to real problems of geophysical fluid dynamics. This phase of the assessment will verify the student's ability to use specific software and computational tools, such as Python, to model complex phenomena and analyze scientific data. The accuracy of the proposed solutions, clarity in presenting results, and a critical and creative approach to problem-solving will be key elements in evaluating the practical skills acquired.

Oral Examination

Subsequently, students will undergo an oral examination aimed at ascertaining the depth of their theoretical understanding and the ability to link the principles of mixing processes in real environmental contexts. During the oral, the topics covered in the course will be discussed, with targeted questions to explore the students' understanding of fundamental concepts and practical applications. Particular importance will be given to the discussion of a report written by the student, based on the practical exercise, to evaluate synthesis ability, the capacity to effectively communicate scientific results, and the integration between theory and practice.

Report Evaluation

The written report represents a fundamental aspect of the assessment, as it allows evaluating the student's ability to conduct independent analysis, process the data collected during the practical exercise, and present the results in a clear and coherent manner. The quality of the report, in terms of content, structure, and presentation, will be carefully considered in the overall evaluation.

Evaluation Criteria

The final evaluation will consider various criteria, including the technical and theoretical correctness of the answers provided during the oral examination, the originality and analytical approach demonstrated in the practical exercise and the report, as well as the ability to argue, critique, and communicate. The goal is to ensure a holistic evaluation that fairly and accurately reflects the competencies and knowledge acquired by the student during the course.

In summary, the assessment method has been structured to provide a comprehensive and detailed measurement of the student's preparation, emphasizing the importance of integrating theoretical knowledge and practical skills, essential in the field of studying mixing processes.