Wasif Afzal
email: wasif.afzal@mdh.se&hugo.bruneliere@imt-atlantique.fr

Hugo Bruneliere

Davide Di Ruscio
email: davide.diruscio@univaq.it&andrey.sadovykh@softeam.fr

Andrey Sadovykh

Silvia Mazzini
email: silvia.mazzini@intecs.it

Eric Cariou
email: eric.cariou@univ-pau.fr

Dragos Truscan
email: dragos.truscan@abo.fi&jordi.cabot@icrea.cat&agomezlla@uoc.edu

Jordi Cabot

Abel Gómez

Jesús Gorroñogoitia
email: jesus.gorronogoitia@atos.net

Yosu Gorroñogoitia

Luigi Pomante
email: luigi.pomante@univaq.it&smrz@fit.vutbr.cz

Pavel Smrz

The MegaM@Rt2 ECSEL project: MegaModelling at Runtime -Scalable model-based framework for continuous development and runtime validation of complex systems

Keywords: Model-Driven Engineering, Design Time, Runtime, Megamodelling

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In the global context, the European electronic industry faces stiff competition. Electronic systems are becoming more and more complex and software intensive [START_REF] Wallin | Issues Related to Development of E/E Product Line Architectures in Heavy Vehicles[END_REF], which calls for novel engineering practices to tackle advances in productivity and quality of these, now, cyber-physical systems [START_REF] Khaitan | Design Techniques and Applications of Cyberphysical Systems: A Survey[END_REF].

5

Model-Driven Engineering (MDE) refers to a system development methodology where abstractions-or models-are systematically used along the process [START_REF] Kleppe | MDA Explained: The Model Driven Architecture: Practice and Promise[END_REF]. MDE promises many potential benefits (e.g., gains in productivity, portability, maintainability or interoperability) and several studies have been conducted to support these claims with empirical data [START_REF] Anda | Experiences from Introducing UML-based Development in a Large Safety-critical Project[END_REF][START_REF] Baker | Model-Driven Engineering in a Large Industrial Context -Motorola Case Study[END_REF][START_REF] Tomassetti | Maturity of Software Modelling and Model Driven Engineering: A Survey in the Italian Industry[END_REF][START_REF] Petre | No Shit" or "Oh, Shit!": Responses to Observations on the Use of UML in Professional Practice[END_REF][START_REF] Gorschek | On the Use of Software Design Models in Software Development Practice: An Empirical Investigation[END_REF][START_REF] Di Ruscio | Guest Editorial to the Special Issue on Success Stories in Model Driven Engineering[END_REF]. Moreover, in the last years, the technological ecosystem around MDE has flourished, providing developers with a plethora of tools to support modeling tasks, ranging from model management solutions to model transformation and code-generation engines. However, these technologies need to be further developed to scale for real-life industrial projects and provide advantages at runtime. The ultimate objective of enhancing productivity by ensuring quality of development, integration and maintenance while reducing the associated costs can be achieved by the use of techniques that integrate design and runtime aspects within system engineering methods incorporating existing engineering practices [START_REF] Baresi | The Disappearing Boundary Between Developmenttime and Run-time[END_REF]. Industrial scale models, which are usually multi-disciplinary, multi-teams, combine several product lines and typically include strong system quality requirements, can be exploited at runtime by advanced tracing and monitoring. Thus, achieving a continuous system engineering cycle between design and runtime, ensuring the quality of the running system and getting valuable feedback from it that can be used to boost the productivity and provide lessons-learnt for future generations of products [START_REF] Fitzgerald | Continuous Software Engineering: A Roadmap and Agenda[END_REF].

A major challenge in the Model-Driven Engineering of critical software systems is the integration of design and runtime aspects. The system behavior at runtime has to be matched with the design in order to fully understand critical situations, failures in design, and deviations from requirements. Many methods and tools exist for tracing the execution and performing measurements of runtime properties (see e.g. [START_REF] Brahneborg | Round-Trip Time Anomaly Detection[END_REF] [START_REF] Brahneborg | A Black-Box Approach to Latency and Throughput Analysis[END_REF]). However, most of these methods do not allow the integration with system models -the most suitable level for system engineers for analysis and decision-making.

The MegaM@Rt2 (MegaModelling at Runtime) proposal was submitted to the ECSEL in 2015. It received good evaluation scoring: 4.3 in Excellence, 4.6 in Impact and 4 in Implementation. The overly positive and instructive remarks motivated us to continue with MegaM@Rt2 in 2016, and a proposal was submitted for the research and innovation action in the call H2020-ECSEL-2016-RIA, by reinforcing the consortium and clearing the project details. The project officially started on April 1, 2017 and runs for 3 years.

The vision of MegaM@Rt2 is to create a scalable framework for model-based continuous development and validation of large and complex industrial systems by exploiting important features of:

• MARTE, SysML, and others, to express both system functional and nonfunctional properties;

• model-based verification and validation methods at design time and runtime;

• methods for model management/megamodelling;

• methods for traceability over large multi-disciplinary models;

• methods for inference of system deviations from expected behavior and affected design elements. This article is an extension of our previous conference paper [START_REF] Afzal | The MegaM@Rt2 EC-SEL Project: MegaModelling at Runtime -Scalable Model-Based Framework for Continuous Development and Runtime Validation of Complex Systems[END_REF]. Compared to it, we have added many details on the current status of the project as well as the already produced deliverables. At the time of this submission, the project is about to enter its second year of activity. So far, industrial case study requirements and the baseline methodologies provided by the project partners have been collected and analyzed in the context of Work Package 1 (WP1). In addition, a detailed study of the state-of-the art has been performed and corresponding needs for innovation have been identified as part of Work Packages 2, 3 & 4 (WP2-4). In the upcoming phase of the project, we will perform gap analysis between the industrial needs and the baseline methodologies & tools.

Moreover, we will suggest the main features of the MegaM@Rt2 framework as well as a related road map presenting how the different components will be further developed.

Section 2 outlines the mission and the objectives of MegaM@Rt2. Section 3 discusses the main concepts of the development approach proposed by the project. The potential industrial impact of MegaM@Rt2 is summarized in Section 4. The partners of the consortium are presented in Section 5. The work packages (WPs) aiming at achieving the objectives of MegaM@Rt2 are presented in Section 6, whereas Section 7 presents a brief description of each WP along with briefly describing some already produced deliverables. Section 8 concludes the paper.

Project Mission and Objectives

The mission of MegaM@Rt2 is to create a framework incorporating methods and tools for continuous system engineering and validation leveraging the advantages of scalable model-based methods. This will provide benefits in significantly improved productivity, quality and predictability of large and complex industrial systems. Such a mission is realized through the following specific objectives:

• Objective 1. MegaM@Rt2 continuous system engineering: to develop scalable methods and tools for the integration of design artifacts resulting from heterogeneous engineering practices, including the modelling of functional and non-functional properties (e.g. performance, energy consumption, security and safety) based on requirements.

• Objective 2. MegaM@Rt2 runtime analysis: to develop integrated methods and tools for trace analysis based on probes injection to runtime artifacts, as well as improved monitoring in order to validate the system-level requirements.

• Objective 3. MegaM@Rt2 (global) model management: to develop scalable infrastructure for efficient handling and management of numerous, heterogeneous, and large models potentially covering several functional and non-functional aspects.

• Objective 4. MegaM@Rt2 unified traceability management: to develop holistic traceability methods and tools 1) able to link and manage models and their elements from different tools as well as 2) suitable for large distributed cross-functional working teams and 3) allowing to integrate the feedback to the system level models.

• Objective 5. MegaM@Rt2 demonstrators validation: to develop specific demonstrators and validate MegaM@Rt2 technologies through 9 complementary industrial case studies.

• Objective 6. MegaM@Rt2 market uptake: to prepare exploitation of the MegaM@Rt2 technology through open source and commercial tools.

Concept and Approach

In the past, MDE principles and techniques have already shown promising capabilities that have been experimented in a context having software components relying on hardware configurations and their interactions e.g., with their underlying environment, being very often numerous, complex, heterogeneous and strongly interrelated. However, they have generally failed in terms of 1) scalability to support real-world scenarios implied by the full deployment and use of complex electronic components and systems (ECS) and 2) maintaining efficient traceability, integration and communication between two fundamental system life-time phases which are design time and runtime, notably as far as nonfunctional properties and their verification & validation aspects (see e.g. [START_REF] Afzal | A Systematic Review of Search-based Testing for Non-functional System Properties[END_REF][16]) are concerned.

As a consequence, the overall idea of MegaM@Rt2 is to scale up the use of model-based techniques by offering proper methods and related tooling, interacting with both design time and runtime, as well as to validate the designed and developed approach in concrete industrial cases involving complex ECS. To this intent, MegaM@Rt2 proposes an overall model-based approach combining existing and novel techniques. A fundamental challenge notably resides in providing efficient traceability support between the two levels i.e., from design models to runtime ones and back. Moreover, modern large-scale industrial software engineering processes require thorough configuration and model governance to provide the promised productivity gains. Thus, a scalable megamodelling approach is required to manage all the involved artifacts e.g., the many different models, corresponding work flows, configurations, etc. and to better tackle their large diversity in terms of nature, number, size, complexity, etc. Verification and validation of highly configurable systems thus also takes importance (see e.g. [START_REF] Ahmed | Constrained Interaction Testing: A Systematic Literature Study[END_REF]).

To cover all these topics and deal with the complete value chain, MegaM@Rt2 brings together prominent tool vendors and research organisations with state-ofthe-art methods and tools to be validated in highly relevant European industrial case studies. The end users from the space, naval, railway, smart grid, smart warehouse and telecom industry domains aim to drive the project by providing real-world requirements and case studies as well as by validating and endorsing the MegaM@Rt2 results.

Figure 1 provides an overview of the MegaM@Rt2 global approach and emphasizes its key principles and concepts, relating them to the corresponding work packages (described in detail in Section 6). A set of current engineering practices based on SysML, AADL, EAST ADL, but also Matlab/Simulink, AUTOSAR and Method B or Modelica, each one producing as set of specific design models, requirement specifications and resulting software (and sometimes also hardware) artefacts, are integrated into a global system model providing a complete view of the cyber-physical system, and detailing the components, behaviour and desired quality properties of the system. These properties are then object of exhaustive continuous testing and monitoring in the runtime environment (thanks to the configuration of the target platform and the injection of probes in the software or also in the hardware [START_REF] Muttillo | A Design Methodology for Soft-core Platforms on FPGA with SMP Linux, OpenMP Support, and Distributed Hardware Profiling System[END_REF][19] [START_REF] Moro | Hardware Performance Sniffers for Embedded Systems Profiling[END_REF]) to detect deviations in real-time. These deviations, plus all the traces information collected in the process, are analyzed to detect the impacted components in the integrated view of system models. When possible, automatic repairing recommendations will be provided to correct the identified issues and reconfigure or redeploy the system to start the next iteration of the continuous integration process.

Industrial Impact

The ECSEL2 program seeks to invest in projects that strengthen the industrial competitiveness, enable economic growth and improve sustainability.

Europe has a reasonably strong position in the world embedded market (30%), but this is falling as other geographies grow -some at the vanguard and others catching up. The MegaM@Rt2 consortium argues that investment in capability of the software development tools market, although only a fraction, has a very large pay-off. We have seen that the software component of the systems is increasingly more growing in importance. As the hardware becomes commoditized, the added value will rapidly shift to the software. Achieving technological and competitive superiority in software development tools will allow European firms to participate with greater dominance in the overall software market.

Specifically, MegaM@Rt2 achieves this in part through reducing development and exploitation costs and in part by allowing mastery of more complex systems. Reducing development costs and time-to-market is a competitive ad-vantage, allowing on, one hand, greater innovation in each product and allowing faster reaction to hardware changes or new usage scenarios on the other. As the Cyber-Physical Systems' world evolves, the agility to react rapidly to new opportunities is a critical success factor for businesses. Mastering ever more complex systems allows new usage scenarios to emerge, based on optimization of greater problems or more optimized solutions for existing ones.

Improved software will allow the bigger players to better position their overall solutions and engender small businesses fulfilling niche needs for high end bespoke software. Investment in this area is timely and appropriate. The small scale and the under-developed capacity of this market segment can lead to large pay-offs in the related fields, whereas the overall embedded systems are of such a magnitude that it requires vast research investment for significant progress.

The MegaM@Rt2 objectives address several market trends in Cyber-Physical Systems:

• Increasing inclusion of advanced techniques like model-based design, development and validation.

-MegaM@Rt2 supports this trend in the technologies provided through industrial case studies.

• Technology availability and support during extended period (e.g., up to 30 years in the railways).

-MegaM@Rt2 open source solutions support this requirement.

• Convergent combination of multi-domains industrial practices.

-MegaM@Rt2 supports this challenge with multi-domain case studies.

• More and more complex (structure/behaviour) connected systems.

-With a clear support for megamodelling and system analysis at runtime, MegaM@Rt2 supports this trend.

Consortium

The MegaM@Rt2 consortium is large and is composed of partners having This also increases the visibility of the project and attracts more new users who can work collaboratively to extend the project results. In the meantime, the individual exploitation should maximise the MegaM@Rt potential benefits for partners and demonstrate their motivation for the project.

All of these will be discussed in detail in section 2.2.

Figure 2.1: Market & technology Value Chains -In order to deliver excellent results the MegaM@Rt project involves partners covering the market and technology value chains.

The joint exploitation is reinforced by the individual plans of MegaM@Rt partners. The Consortium will exploit the results of the project according to different exploitation profiles:

• End users, solution and service providers, will adopt the MegaM@Rt results in their central methodology groups and disseminate them internally to their operational divisions, leading to products and services produced with higher productivity and less risks. • Tool vendors, engineering tools and technology providers, will integrate their existing products with the MegaM@Rt technologies and will build new products based on the open source results from MegaM@Rt. They will equally improve their consultancy offering with the MegaM@Rt methodology. • Research organisations will integrate MegaM@Rt results in their software/service engineering courses, which will place the MDD approach as a well-established method for development, maintenance and evolution of large complex industrial systems in the education of Europe's next generation of software engineers, and raise their profiles in the areas of software and service modelling. This will impact the subjects of work in the basic research, contributions in standards communities and technology transfer to industrial partners via research cooperation and publications. (Sweden), which also has an extensive experience in both, participating and managing, EU projects. A suitable management strategy has been evolved by bringing together partners that know each other and have already collaborated in the past [START_REF] Cross | A Systems Engineering Approach to Systematic Innovation in an Industry-University Collaboration[END_REF]. To setup the consortium, a complete value-chain has been taken into account by selecting case study owners, technology providers, and research partners (Figure 3):

B2.1.5 Socially important impacts

• Case study owners and end-user partners. Providing knowledge of both end-users needs and development scenarios for complex industrial systems.

• Technology and service providers partners. Providing knowledge and tools in MDE, hardware and software synthesis, collaborative modelling and standardization.

• Research partners. Providing knowledge in megamodelling, MDE, code generation, Verification & Validation and logs analysis.

In the remaining of the section, all the members of the consortium are briefly described with respect to their role in the project.

Case study owners and end-user partners

Nine industrial partners will play the role of case study providers and endusers as described below.

Thalès Research & Technology -TRT (FR) provides a case study in avionics domain and will lead the validation scenarios definition. TRT has an extensive experience with MDE.

ClearSy System Engineering -CSY (FR) provides a case study in safety critical railway systems.

IKERLAN S. Coop -IKER (ES) provides a case study in smart warehouse domain and will lead the experiments with baseline technologies.

Tekne -TEK (IT) provides a case study in short-range communications domain and will lead the requirements analysis activities.

Nokia -NOK (FI) provides a case study in the telecommunications domain and will lead the case studies development activities.

Bombardier Transportation Sweden AB -BT (SE) provides a case study of their train control and management system (train/railway domain).

Volvo Construction Equipment AB -VCE (SE) provides a case study in the vehicular domain (VCE's electrical and electronic system technology platform).

Camea -CAM (CZ) provides the case study in vision-based intelligence.

AinaCom Oy -AINA (FI) will provide a case study in the communication gateway domain.

The technological domains and the applicationa areas of the case studies in the project are summarized in Table 1. Space Systems Finland Ltd. -SSF (FI) will contribute to the MegaM@Rt2 framework with the LIME toolset for runtime monitoring of the implementations and automatic test generation, which was partially funded by SSF.

SSF will work on integrating the toolset to other MegaM@Rt2 tools. SSF will also participate in the application of the tools to the case studies provided by other Finnish partners. Additionally, SSF will share its extensive knowledge of verification and validation methods for safety-critical systems.

Conformiq Software Oy -CON (FI) will contribute to model-based functional test generation in all stages of software process, and to model-based test validation by functional coverage and test correctness analysis with respect to system models. Conformiqs focus is in behavioural models in contrast to e.g. purely architectural models. Conformiq will work on integrating the technology platform to other MegaM@Rt2 tools. In addition, Conformiq will participate in the deployment and application of the platform to the case studies.

Research partners 295

Ten partners will drive the research activities of the consortium. Their names and contributions in the project are summarized in Table 2.

Work Packages

The main expected result of MegaM@Rt2 is a practical framework incorporating methods and tools for continuous system engineering and validation.

As introduced earlier, its overall goal is to leverage the advantages of scalable model-based methods to provide significantly improved productivity, quality and predictability of large and complex industrial systems. This framework will be composed of three main tool sets for 1) system engineering/design & continuous development, 2) related runtime analysis, and 3) global model & traceability management (respectively). As a consequence, we have organized the project around the research work and realization of these tool sets. Their integration and actual application onto a set of concrete use cases, covering different industrial domains, is also a central aspect of the project.

To reflect these principles, the project has been organized in 7 complementary work packages (WPs):

• WP1. Case Study Requirements Analysis & Architecture Specification;

• WP2. MegaM@Rt2 System Engineering;

• WP3. MegaM@Rt2 Runtime Analysis;

• WP4. MegaM@Rt2 Global Model & Traceability Management;

• WP5. Integration, Case Study Development & Evaluation;

• WP6. Dissemination and Exploitation;

• WP7. Management.

The work to be realized in the project is strongly requirements-driven. These requirements are extracted from the use cases as part of WP1, by exploiting the collaboration among the use case providers (mainly large industrial companies) and the technical providers (composed of both service/product companies and experienced researchers from academia). WP1 is also in charge of defining the overall architecture (conceptual and technical) of the MegaM@Rt2 solution.

Most of the research and development effort is concentrated in WP2, WP3 and WP4, which aim at providing the three tool sets previously mentioned. Within WP5, these technical results will be then integrated together, applied on the use cases and finally evaluated for further improvement. The work in the project will follow an iterative and incremental approach divided into three consecutive phases. In the first phase, we will specify the requirements, validation scenarios, global architecture and roadmap. In addition, case study partners will experiment with baseline technologies while technology providers will develop the first set of prototypes. In the second phase, we will consolidate these prototypes, integrate them in a first release of the MegaM@Rt2 framework and run an initial set of validation scenarios. Based on the obtained results, in the third phase, we will integrate and validate the technical solutions, provide final validation and experience reports from the use cases (as well as a final management report).

In parallel, the dissemination (academic or industrial, including the relation with the standardization organizations such as the Object Management Group, OMG) and exploitation (e.g., consortium and individual business plans) activities will be conducted in WP6. The general project management and reporting activities will be performed under the umbrella of WP7.

Work Package Descriptions and Deliverables

We will now present a brief description of each WP along with briefly describing some already produced deliverables.

WP1 -Case Study Requirements Analysis and Architecture Specification

This WP gathers the work on the case studies definition and requirements analysis (by end-users) with the global architecture and road map specification (by technology providers). The industrial partners will set real requirements for research and technology providers. They will closely collaborate and be integrated in the development teams, providing regular feedback on the elaborated technologies. This WP also concentrates on the validation scenarios, i.e. end-toend demonstrators for the MegaM@Rt2 solutions in varied industrial contexts.

End-users will develop methods for gathering the data needed for qualitative and quantitative verification of MegaM@Rt2 achievements. They will run the related experiments in a cost-efficient manner, and will provide representative evaluation of the technologies for large scale usage. From their side, the technology partners will define the architecture and a detailed road map for the technical developments.

Examples of already produced deliverables in WP1

Industry Requirements Specification: This delierable marks the first step of activities for the MegaM@RT2 case study providers. They have defined the case studies to be developed during the project, exposed their current practices, organized the capabilities of the MDE framework they plan to use according to development scenarios, and mapped such scenarios on the time line of the project execution.

On the basis of all above, the case study providers have expressed their enduser requirements for the improved MDE framework that MegaM@RT2 aims to provide. This deliverable is also the first step of the collaboration between the case study providers and the technology providers. This deliverable is also an input to the roadmap development of the MegaM@Rt2 framework, where the capabilities of the baseline tools will be matched to the requirements from the case study providers.

A total of 9 case study providers have given their concrete requirements and their expectations from the MegaM@Rt2 framework (Table 1).

Architecture Specification and Roadmap: This deliverable defines the initial vision of the global architecture of the MegaM@Rt2 framework. As a starting point, we have described the conceptual tools as well as the individual tools by partners. The initial version of the deliverable has concentrated on the following aspects:

• High-level requirements to identify the features, goals and objectives of each tool component. These technology requirements will serve as the starting point for the ongoing refinement, elicitation and traceability work that links the tools and methods development with case study development.

• Functional interfaces that define the high-level services of the tools as well as the integration points. We took attention to extract common interfaces with the goal to match the tools that may easily collaborate.

• Subordinates that are high-level parts of the tool components that help to All in all, in future versions we intend to provide traceability linking Use Case requirements -Conceptual tools of the framework -Individual Tools by partners.

The MegaM@Rt Framework is the main technical result of the project as described in the FFP. The Framework regroups several interconnected tool sets including tool sets for Holistic System Engineering, Model and Traceability Management as well as for Runtime Analysis. Those tool sets are highly interconnected to achieve the goal of linking system models with the runtime analysis of large scale industrial systems. System Engineering deals with integrating the existing industrial practices, verification and validation on system level. The runtime analysis is conducted with monitoring, online testing and verification as well as models@runtime technics. The system models, trace models, runtime models are interconnected the the Model&Traceability Management level. • Deployment to refer the deployment platforms by paying attention to extract the commonalities that would help to identify the facility for an integrated solution.

We have selected Modelio as the common platform for architecture modelling, primarily because SOFTEAM (MegaM@Rt2 technical leader) is an active contributor to Modelio development and has all the technical and support means to help partners to model in a productive way. Figure 4 shows a high-level architecture of MegaM@Rt2 framework in Modelio. The modeled framework regroups several interconnected tool sets including tool sets for Holistic System Engineering, Model and Traceability Management as well as for Runtime Analysis.

WP2 -MegaM@Rt2 System Engineering

This WP gathers the activities related to the definition of the required Domain Specific Languages (DSLs) to support model-based system design, and of the methods and tools to develop integrated system models. One of the strongest points of model-based approaches lies in the support for separation of concerns and definition of specific architectural views. Specific views focus on specific areas of the development from system to software level, including the system functional, logical and physical decomposition, identification of software and hardware components, definition of functional and non-functional properties, software architecture, data, behavior and algorithmic modeling. This WP concentrates on all the modelling and tooling aspects of MegaM@Rt2. The goal is first to provide the foundations for WP3 and WP4, and later, to design, develop and support the MegaM@Rt2 system engineering tool set to be used by industrial partners in WP5.

Example of already produced deliverable in WP2

Foundations for Model-driven Design Methods: This deliverable provides the foundations for the design of the MegaM@Rt2 tool chain. Its objective is to analyse the state-of-the-art in terms of both research approaches and existing modelling solutions and tools in the context of model-based continuous development. Within this task, relevant existing DSLs and modelling technologies have been identified and presented, and the possibilities for their utilization, extension and/or integration within MegaM@Rt2 have been analysed. The objective is to provide an overview of the current state of practice, and define the concepts, features and principles that will be the basis for the development of the MegaM@Rt2 design solutions. In particular, the foundation for models, DSLs and their semantics have been addressed. The content of the deliverable has been organized around three main topics: (i) Systems Modelling, (ii) Verifi- The deliverable ends with a comprehensive catalog of the solutions offered by the different tool providers to all the other members of the consortium. For further information, interested readers can access the full text of the deliverable from the project website 3 .

WP3 -MegaM@Rt2 Runtime Analysis

This WP focuses on the usage and definition of models at runtime level, and on the associated techniques or methods. Models at runtime can be designed or obtained from the system itself. For instance, logging or monitoring the system under the form of models can be performed jointly with the system execution and can help in ensuring a correct system execution. Afterwards, such models can also be analyzed to enhance design models from WP2 and are thus entries of the tools and methods of WP4. Verification and validation issues can be managed directly at runtime, enabling the detection of problems that can be solved at runtime or propagated back to design level. This can be achieved by checking the expected behavior according to functional and nonfunctional properties embedded in the design models, or by analyzing jointly runtime models with the actual system execution to determine if the system fulfills its specifications. To this intent, this WP will notably provide on-line testing and verification techniques.

Example of already produced deliverable in WP3

Foundations for Model-Based Runtime Methods: This deliverable provides a succinct overview of the foundations of model-based runtime methods and technologies in order to support the innovation tasks of WP3 of the MegaM@Rt2 project. The deliverable discusses how runtime artifacts are obtained from design artefacts and from execution logs. In the first category, we overview approaches for generating run time code and models from design models via code generation and, respectively, model transformations. In the second category, we discuss approaches for creating or improving the runtime artifacts by analyzing the runtime execution logs of the system via methods like machine learning and data analytics.

We also discuss how runtime artifacts are used at runtime either by executing them as part of the system at runtime or by using them to generate tests or monitor the system during its operation.

Throughout this deliverable, we have scrutinized the state-of-the art, the state-of-practice, and the baseline technologies which are available for the project participants. To this extent, the deliverable has investigated current methods and tools for their benefits and existing limitations. The results of this deliverable are meant to lay the basis for defining new concepts, methods and tools for coping with these limitations and successfully deploying runtime methods to industrial settings. This deliverable also provides input for the specification of the MegaM@Rt2 runtime tools to support automated code generation and model execution, log analysis, runtime verification and testing activities. The deliverable also includes a collection of relevant solutions and tools provided by the MegaM@Rt2 consortium members as baseline technologies in the project. For further information, interested readers can access the full text of the deliverable from the project website4 .

WP4 -MegaM@Rt2 Global Model and Traceability Management

This WP focuses on megamodelling, also called global model management, in which models for design time (WP2) and models for runtime time (WP3) are to be managed and aligned all together. This relies on the base notion of a megamodel [START_REF] Favre | Towards a Megamodel to Model Software Evolution Through Transformations[END_REF], a model that intends to describe the metadata on the different models involved in a given engineering process, as well as the related interrelationships and corresponding artifacts (transformations, generators, etc.).

Such a (mega)model can be navigated and queried at any time in order to retrieve or compute the required information, notably as far as traceability between models is concerned.

In the context of MegaM@Rt2, a particular focus is put on various scalability topics: not only the size of the models is larger, but there is also a larger number of model users with different roles; there are various kinds of languages (DSLs) involved for different needs, including e.g. user interface (UI) related languages, and various transformations related to them. The second (and directly related)

focus of this WP is on traceability between design time and runtime, as not in all cases the same model can be used for both purposes. WP4 also provides implementation of the tooling for scalable megamodeling/traceability and guidelines for their deployment and practical use in case studies. WP4 is designed to deliver its results incrementally, notably by collecting progressively feedback on the developed features from their application to the project use cases. Figure 5 summarizes the focus of WP4.

Example of already produced deliverable in WP4

Foundations for Model Management and Traceability:

The main goal of WP4 is to elaborate on the required glue between the artifacts produced in WP2 (e.g., design models) and the ones produced in WP3 (e.g., runtime models). As a result, it is expected to provide a so-called global MegaM@Rt2 Model and Traceability Management framework to be a core part of the MegaM@Rt2 overall solution and to be notably deployed on the projects use cases (among possibly others). As the initial step in WP4, this deliverable thus provides an overall state-of-the-art in terms of existing model management and traceability solutions. It presents the main common principles and approaches related to model storage, querying, handling and linking with others models and modeling artifacts, notably via model views [START_REF] Bruneliere | A Feature-based Survey of Model View Approaches[END_REF] and/or so-called megamodels [START_REF] Hebig | On the Unification of Megamodels[END_REF]. It also describes the available traceability and interoperabil- ity solutions [START_REF] Aizenbud-Reshef | Model Traceability[END_REF]. It describes both existing research approaches as well as some more business-oriented tools or environments which are relevant in this given context. Finally, it ends with a list of technical solutions provided by the projects partners. All along the deliverable, a particular importance has been given to aspects related to the scalability of the available solutions.

The main purpose of this deliverable is to prepare the work for specifying the Model and Traceability Management framework to be developed and further used in MegaM@Rt2. Its goal is also to help selecting some of the key problems to be addressed while implementing this framework in the future. Among others, the following big challenges have been identified as important in their respective research areas: scalable model storage and querying, well-synchronized and verified model views, performant and decentralized global model management, efficient integration of inter-model traceability and interoperability support. For further information, interested readers can access the full text of the deliverable from the project website5 .

WP5 -Integration, Case Study Development and Evaluation

This WP provides specific industrial case studies from different domains such as aeronautics, railway, construction and telecommunication. The main goal of WP5 will be to integrate the different technical developments realized in WP2, WP3 and WP4. It will also be in charge of conducting controlled experiments on the case study partner premises, as defined in WP1. Partners in WP5 will perform a preliminary evaluation as feedback for WP2-3-4, and a strong interaction between technology and use case providers is expected. Finally, WP5 will perform the final integration and consolidation of the MegaM@Rt2 solution, as well as the overall validation the obtained results.

WP6 -Dissemination and Exploitation

This WP concentrates on the project impact and community building activities. These activities will provide a solid base to identify the key stakeholders for sustainable exploitation, dissemination, communication and standardization.

Examples of already produced deliverables in WP6

Public Website and Social Media Presence: A twitter account for the project has been created. Twitter handle is: @megamart2 ecsel. The project website URL is: https://megamart2-ecsel.eu/.

Communication Plan: In the initial version of the communication plan, we have identified a preliminary list of stakeholders who would be especially interested in the project and would thus serve as a specific target for our communication and dissemination plan. For further information, interested readers can access the full text of the deliverable from the project website6 .

WP7 -Management

This work package gathers all the activities related to the management of the MegaM@Rt2 project and its consortium. This mostly includes the mandatory official monitoring and reporting tasks (to the ECSEL Joint Unit and the European Commission). The overall objective is to ensure a smooth running of the project and efficient collaborations between all the involved partners. As fundamental to the success of the project, this WP will notably coordinate the establishment of a proper quality plan to be applied to all MegaM@Rt2 results.

It will also deal with the important risk management and Intellectual Property (IP) issues that may appear during the course of the project.

Example of already produced deliverable in WP7

Project Management Guide and Quality Plan: The purpose of this deliverable is to present and describe quality standards and procedures to be applied in the internal management and execution of the project. This document is based on the terms and conditions established in the Grant Agreement signed by the ECSEL-JU. This deliverable describes the management roles and functions, the decision and control procedures, the processes and resources for ensuring the quality of project deliverables.

This deliverable is intended to be used by the project management team and the work package leaders, as well as people who are directly responsible for producing the deliverables, to ensure the quality assurance of project processes and outputs and to avoid eventual deviations from the project work plan.

Conclusion

This paper presented the MegaM@Rt2 ECSEL project. It notably provided the global context and motivation for this project, introduced its mission and targeted objectives, described its general organization in terms of work packages and detailed the composition of its large supporting consortium. As explained in this paper, MegaM@Rt2 mainly intends to create a scalable model-based framework for dealing with the continuous development and validation of the software parts of large and complex industrial CPSs. This framework will notably focus on relating together the actual executions of these systems (i.e., runtime) with the way they are currently specified, developed and maintained (i.e., design time). While there is already quite a lot of support for these two dimensions separately, there is currently no real support for an efficient integration and feedback loop between design time and runtime. We plan to practically realize this by providing the required management and traceability support between all the involved models (both at design time and runtime). The obtained results will be experimented on 9 different use cases covering different industrial domains such as aerospace, railway, telecommunication, networks and construction equipments. In addition to scientific progress in the CPSs and modeling/MDE domains, industrial partners are expected to gain concrete benefits in terms of improvements to their system reliability and decrease in development and maintenance costs.

Figure 1 :

 1 Figure 1: The MegaM@Rt2 Overall Approach.

Figure 2 :

 2 Figure 2: The MegaM@Rt2 Consortium.

  different complementary profiles. It brings together 27 partners coming from 6 European countries, each of which constitutes a national consortium (France, Spain, Italy, Sweden, Finland and Czech Republic). See Figure 2 (the abbrevi-200 ations used in partner names are described in Sections 5.1, 5.2, 5.3). The project consortium is strongly industry-led and consists of 7 Large Enterprises (LE) and 9 Small and Medium Enterprises (SME) accompanied by 11 universities or research and technology transfer organizations (R). An adequate MegaM@Rt 2 68/185

Figure 3 :

 3 Figure 3: The MegaM@Rt2 project involves partners covering the market and technology value chains.

ATOS

  Spain -ATOS (ES) will lead the MegaM@Rt2 framework integration and the exploitation work package. ATOS will contribute to model simulation task force and code generation by providing development for Foundational UML (fUML) and AspectJ. Fent Innovative Software Solutions -FTS (ES) expertise is focused on the development of execution platforms for mixed criticality systems. It is specialized in: (1) Design and development of hypervisor technology; (2) Design and development of real-time operating systems; (3) Adaptation of operating systems to be executed as a partition on top of XtratuM hypervisor. FTS will mainly be involved in Runtime work package and will provide its expertise in execution platforms. Intecs -INT (IT) contributes to the MegaM@Rt2 framework with the CHESS model-driven, component-based methodology and tool chain for the development of high-integrity systems for different domains. INT participates in the development of the CHESS open source project delivered under Eclipse/Polasys. CHESS relies on MARTE, with focus on non functional properties modelling, analysis and correct-by-construction code generation. Ro Technology -RO (IT) will provide advanced design, development and V&V Techniques.

  -ARMINES (FR) Leads activities on scalable model management & traceability. Université de Pau et des Pays de l'Adour -UPPA (FR) Leads activities on models' execution techniques development and contribute with the PauWare library. Universidad de Cantabria -UCAN (ES) Leads development of the design level verification and validation methods tools. Contributes with eSSYN tool suite featuring software synthesis technology. Universitat Oberta de Catalunya -UOC (ES) Leads development of scalable model-based techniques. Contributes with EMFtoCSP verification tool suite. Universit degli Studi dell'Aquila -UAQ (IT) Leads the traceability and provenance task force. Åbo Akademi University -ABO (FI) Leads the runtime verification task and contributes to all the work packages providing expertise in Aspects Oriented Modelling. Further contributes with UPAAL TRON tool suite. Teknologian tutkimuskeskus VTT Oy -VTT (FI) Leads development activities in logs analysis with machine learning and data mining technologies. RISE SICS Västerås AB Västeras -SICS (SE) Contributes in runtime verification and validation methods, their implications and required support from higher modelling levels. Mälardalen University -MDH (SE) Contributes in verification and validation at design-time, verification and testing at run-time, integration of megamodelling and traceability within the overall tool chain. Brno University of Technology -BUT (CZ) Contributes in runtime model optimization and validation through classification and scheduling methods from historical performance data.

Page 8 of 105 -

 105 for the individual tools we clarify their relation to the MegaM@Framework conceptual tools.

Figure 2

 2 Figure 2 MegaM@Rt Framework Architecture Overview

Figure 4 :

 4 Figure 4: MegaM@Rt2 Framework Architecture Overview

  cation and Validation and (iii) Modelling Methodologies. The first one focuses on standard modelling languages and DSLs, state-of-the-art modelling tools and environments, and methodologies towards the participatory development of DSLs. The second topic covers automatic or semi-automatic solutions for the verification and validation of MDE artefacts (e.g., models, transformations). Finally, the third topic covers different state-of-the-art modelling methodologies.

3

  http://hdl.handle.net/20.500.12004/1/P/MMART2/D2.1

Figure 5 :

 5 Figure 5: WP4 -Model and Traceability Management approach.

Table 1 :

 1 Case studies from MegaM@Rt2 Partners.

	No. Technological Domain	Application specific
	1	Avionics	Flight Management System
	2	Railway	Platform Screen Doors Control
	3	Smart warehouse	Deployment and Supervision of Agents
	4	Short range communications Indoor Positioning
	5	Telecommunications	Base Transceiver Station
	6	Transportation	Train Control and Management System
	7	Automotive	Engine Control
	8	ICT Services	SMS Gateway
	9	Traffic monitoring	Intelligent Traffic Surveillance System

5.2. Technology and service providers partners

Eight industrial partners will play the role of technology and service providers as described below.

Softeam -SOFT (FR) will contribute with its expertise in MDE as a tool vendor for Modelio work bench and as an active member of the Object Management Group. SOFT's technical contribution will include the work on user interface generation from Interaction Flow Modeling Language (IFML) specification, code generation with "MDD+aspects" approach and scalable model management with model fragments infrastructure.

Smartesting Solutions & Services -SMA (FR) will lead the work package on runtime methods and tools. SMA's main contribution will be in online testing techniques development. SMA will contribute to baseline technologies with SmartTesting CertifyIt technology.

Table 2 :

 2 MegaM@Rt2 Research Partners.

http://www.ecsel-ju.eu/web/index.php

http://www.ecsel-ju.eu/web/index.php

http://hdl.handle.net/20.500.12004/1/P/MMART2/D3.1

http://hdl.handle.net/20.500.12004/1/P/MMART2/D4.1

http://hdl.handle.net/20.500.12004/1/P/MMART2/D6.2

Acknowledgment This project has received funding from the Electronic Component Systems for European Leadership Joint Undertaking under grant agreement No. 737494. This Joint Undertaking receives support from the European Union's Horizon 2020 research and innovation program and from Sweden, France, Spain, Italy, Finland & Czech Republic.


Masoud Rostami

Vladimir Zeitlin
email: zeitlin@lmd.ens.fr

Luca Montabone

On the role of spatially inhomogeneous diabatic effects upon the evolution of Mars' annular polar vortex

Keywords: Mars' polar vortex, Barotropic Instability, Rotating Shallow Water Model, Radiative relaxation, CO 2 deposition

The structure of Mars' annular polar vortex is not similar to that of its counterpart on Earth and is characterised by a potential vorticity (PV) low in the vicinity of the winter pole, rather than PV monotonically increasing towards the pole. A number of persistent asymmetric high-PV patches around the central low are also typical for the Martian polar vortex. The simplest vertically averaged model of

Introduction

A surprising property of the Martian atmosphere is the latitudinal distribution of zonally averaged potential vorticity (PV), which has a minimum PV at the North winter pole surrounded by a circumpolar annulus of higher PV. The South pole exhibits similar characteristics of smaller intensity [START_REF] Mitchell | Polar vortices on Earth and Mars: A comparative study of the climatology and variability from reanalyses[END_REF]. Such configuration is unstable in the absence of external forcing, and is expected to evolve towards monotonous PV distribution, like e.g. the North polar vortex on

Earth. Yet, this does not happen and the annular PV distribution is persistent on Mars. It should be stressed that the Martian polar vortex is annular only on average, as reanalysis datasets [START_REF] Greybush | Ensemble kalman filter data assimilation of thermal emission spectrometer temperature retrievals into a mars gcm[END_REF][START_REF] Montabone | The mars analysis correction data assimilation (MACDA) dataset v1.0[END_REF] reveal the existence of patches of high PV [START_REF] Mitchell | Polar vortices on Earth and Mars: A comparative study of the climatology and variability from reanalyses[END_REF][START_REF] Waugh | Martian polar vortices: Comparison of reanalyses[END_REF], with time-scale of the order of a sol (a Martian mean solar day) between about 20 and 50 km altitude around the northern hemisphere (NH) winter solstice (L S = 270 • . L S is the solar longitude, the angular position of Mars in its orbit with respect to the position at NH spring equinox). Figure 1 shows the evolution of Ertel potential vorticity (EPV) patches during half a sol just before NH winter solstice of Martian year (MY) 24, from the Mars Analysis Correction Data Assimilation (MACDA) v1.0 reanalysis [START_REF] Montabone | The mars analysis correction data assimilation (MACDA) dataset v1.0[END_REF]. The coherence of such patches throughout local times, and their displacement with the mean flow, are remarkable in this case. Furthermore, maps of 30 sol mean PV show a quasi-continuous belt of high PV with an elliptical shape rather than a circular one [START_REF] Mitchell | Polar vortices on Earth and Mars: A comparative study of the climatology and variability from reanalyses[END_REF][START_REF] Waugh | Martian polar vortices: Comparison of reanalyses[END_REF], likely forced by a stationary wavenumber-2 wave aligned with the hemisphere-scale topography, as shown by [START_REF] Barnes | Mars atmospheric dynamics as simulated by the nasa ames general circulation model: 3. winter quasi-stationary eddies[END_REF]; Hollingsworth and Barnes (1996). It should be noted that there are no satellite measurements of winds in the Martian atmosphere so far, and reanalysis datasets have been obtained with the support of Martian general circulation models (MGCM) (see [START_REF] Lewis | Modelling the martian atmosphere[END_REF] for a general review of Mars GCMs).

It is natural to look for an explanation of the existence and longevity of the Mars' annular polar vortex in terms of diabatic effects, and indeed such attempts exist in the literature [START_REF] Waugh | Martian polar vortices: Comparison of reanalyses[END_REF][START_REF] Seviour | The stability of mars' annular polar vortex[END_REF][START_REF] Toigo | What causes mars' annular polar vortices[END_REF].

For example, [START_REF] Toigo | What causes mars' annular polar vortices[END_REF] showed that the latent heat released by CO 2 "condensation" in a MGCM model tends to produce an annular polar vortex.

Yet the complexity of general circulation models, including a plethora of various physical phenomena and their parameterisations, makes it difficult to isolate the particular dynamical role played by each of them. This is why simplified shallowwater models are often used for this purpose. Thus, recently [START_REF] Seviour | The stability of mars' annular polar vortex[END_REF] obtained an annular PV structure in a shallow-water model by adding thermal relaxation towards an equilibrium state. In all aforementioned studies the CO 2 deposition process is considered homogeneous, without incorporating the effects of aerosols, which may serve as "condensation" nuclei, and thus affect the spatial pattern of deposition (Although the term "condensation" is frequently used in the literature, we use the formally correct "deposition" term to avoid confusion with gas -liquid phase transition).

In this work, we use the simplest column-averaged model of the Martian atmosphere in order to understand the role of two major diabatic effects in the winter season: the radiative relaxation, and the spatially inhomogeneous CO 2 deposition through the vapour-solid phase transition. The model is the rotating shallow water (RSW) model, which is not introduced in an ad hoc manner, but is obtained by vertical averaging of the primitive equations in pressure coordinates [START_REF] Zeitlin | Chapter 1 introduction: Fundamentals of rotating shallow water model in the geophysical fluid dynamics perspective[END_REF]. By the same procedure, the model allows to incorporate the profile of Mars' polar vortex as obtained from the MACDA v1.0 dataset. Using the zonally averaged profile of the polar vortex from the reanalysis, we are able to study its stability and identify the unstable modes that shape the vortex at the nonlinear stage of evolution. A major new feature and improvement of the model is that, while radiative relaxation is incorporated in the standard way, we use recipes similar to those applied for the construction of so-called moist-convective RSW models [START_REF] Bouchut | Fronts and nonlinear waves in a simplified shallow-water model of the atmosphere with moisture and convection[END_REF] in order to include convective fluxes produced by the latent heat release due to CO 2 deposition. Moreover, we extend this approach to allow for spatially inhomogeneous deposition with aerosols serving as deposition nuclei, as it is known that homogeneous ice nucleation is negligible in Mars' atmosphere [START_REF] Wood | Nucleation and growth of co 2 ice crystals in the martian atmosphere[END_REF][START_REF] Colaprete | Carbon dioxide clouds in an early dense martian atmosphere[END_REF][START_REF] Määttänen | Nucleation studies in the Martian atmosphere[END_REF]. Aerosol nuclei could be any particles, such as mineral dust, water ice crystals, or micrometeorites, which have potential for triggering CO 2 ice nucleation. They are considered to be advected by the flow. We identify the dynamical role of both radiative cooling/heating and CO 2 ice deposition, and show that, even in this simple model, their combined influence allows to explain how the evolution of the most unstable mode of the zonally and time-averaged profile of Mars' polar vortex produces a structure similar to that observed in MACDA, including high PV patches and elliptic shape.

We should stress that in the present study, unlike what is frequently done with GCMs, we do not seek to reproduce the formation of the polar vortex ex nihilo -a process in which the diabatic effects are surely already important. On the contrary, once the vortex is formed, we show how the adiabatic evolution towards the monotonous axisymmetric distribution is modified by each of the two diabatic effects mentioned above, and how their combination produces the characteristics observed in the reanalyses.

The paper is organised as follows. In section 2 we describe the model and introduce the new parametrisation of CO 2 deposition. We present in section 3 the background state and sketch the results of its linear stability analysis. We also present the unstable modes, which are used later for initialisation of numerical simulations of nonlinear evolution. Section 4 contains comparison of numerical simulations of the evolution of the most unstable mode of the zonally averaged polar vortex profile with and without diabatic effects, analysis of the results, and comparisons with MACDA results. Section 5 is devoted to discussion and conclusions.

The model

Rotating shallow water on the polar tangent plane

We write down the RSW equations that are obtained by vertical averaging of the adiabatic atmospheric primitive equations, with pseudo-height [START_REF] Hoskins | Atmospheric frontogenesis models: Mathematical formulation and solution[END_REF] as vertical coordinate, between a pair of material surfaces in the absence of forcing and dissipation. Anticipating application to a zonally averaged [START_REF] Montabone | The mars analysis correction data assimilation (MACDA) dataset v1.0[END_REF] in MY 24 at L S ≈ 268 • (sol-of-year=512). Each panel is separated by two hours, for a total evolution time of half a sol. The order of the sequence is from panel (a) to (f), with panel (a) corresponding to 02:00 a.m. at 0 • longitude. The longitude-latitude grid (dotted lines) is 30 • × 30 • . The projection is orthographic, centred on the North pole. The prime meridian is pointing downward. The grey area around the North pole is where the calculation of the latitudinal derivative of the zonal wind is not carried out (we use a conservative approach to avoid possible artifacts in the calculation of EPV from MACDA v1.0 dataset at very high latitudes).

Note that we use c p = 820 JKg -1 K -1 to define the isentropic levels, consistently with the study by [START_REF] Mitchell | Polar vortices on Earth and Mars: A comparative study of the climatology and variability from reanalyses[END_REF]. If one uses c p = 745 JKg -1 K -1 as in other studies, these figures correspond to an isentropic level of 287 K.

polar vortex, we rewrite the equations in cylindrical coordinates. We will not work on the whole rotating sphere, but rather limit ourselves to the polar region, which will be considered in the so-called gamma-plane approximation, i.e. the tangent plane approximation with the curvature correction to the Coriolis parameter.

Dv Dt + (f + v r )(ẑ × v ) = -g ∇ H h, (2.1a) Dh Dt + h∇ H .v = 0.
(2.1b) Equation (2.1a) expresses momentum conservation in the fluid layer in the presence of the Coriolis force, and equation (2.1b) expresses conservation of mass. In these equations (r, θ) are polar coordinates in the tangent plane, ẑ is the unit vector normal to the plane, v = ur + v θ, u and v are radial and azimuthal components of the horizontal velocity v , respectively. g = g(θ/θ 0 ) is effective gravity defined by the mean potential temperature of the atmospheric layer θ normalised by a reference value θ 0 , and f is the Coriolis parameter, h is the thickness of the layer defined as pseudo-height (note that the atmospheric shallow water model is "upside-down", in a sense that the free surface is on the ground and the fixed pseudo-height (pressure) level is on the top, while geopotential is constant on the ground, cf. [START_REF] Bouchut | Fronts and nonlinear waves in a simplified shallow-water model of the atmosphere with moisture and convection[END_REF][START_REF] Rostami | Improved moist-convective rotating shallow water model and its application to instabilities of hurricane-like vortices[END_REF], D/Dt = ∂ t + v .∇ H is the material derivative, and ∇ H = r∂ r + θ(1/r)∂ θ is the two-dimensional gradient in cylindrical coordinates. The Coriolis parameter in the gamma-plane approximation has the form f = f 0 -γr 2 , where f 0 = 2Ω, and Ω is the angular velocity of the planet's rotation. In fact, at latitude φ the Coriolis parameter is f = 2Ω sin φ, and the gamma-plane formula retains only the first non-trivial term in the Taylor expansion of sin φ at φ = π/2. The gamma-plane approximation can be "improved" by including subsequent terms of the Taylor expansion. We will be using sponges at the boundaries of the calculational domain in numerical simulations below, which, roughly, correspond to radiative boundary conditions allowing to dump inertiagravity waves produced by the evolution of the vortex. Although topography can be straightforwardly introduced in the model, we will not include it in the present study.

Inclusion of diabatic effects

Radiative relaxation

The first diabatic effect to be included in equations (2.1a) -(2.1b) is the radiative relaxation of pressure field towards an equilibrium configuration of thickness in pseudo-height coordinates h = H(r, θ), with some relaxation time τ r . Thus, to include this effect, the equation (2.1b) is modified following the well-known equation (common for all atmospheres):

Dh Dt + h∇ H .v = - h -H τ r . (2.2)
This parametrisation has been applied to Mars' polar vortex in [START_REF] Seviour | The stability of mars' annular polar vortex[END_REF], and is also used for Earth's polar vortex, e.g. [START_REF] Rong | Vacillations in a shallow-water model of the stratosphere[END_REF],

where a relaxation toward a monotonic equilibrium profile has been considered. It is known that the radiative relaxation rate on Mars is much faster than that on Earth [START_REF] Goody | A discussion of martian atmospheric dynamics[END_REF][START_REF] Imamura | Radiative damping of gravity waves in the terrestrial planetary atmospheres[END_REF][START_REF] Eckermann | Scale-dependent infrared radiative damping rates on mars and their role in the deposition of gravity-wave momentum flux[END_REF]. [START_REF] Seviour | The stability of mars' annular polar vortex[END_REF] tested weak and strong relaxation with τ r = 10, 2, 0.5 sols. The simplest parameterisation is that with radiative heating and cooling rates having the same scale for all altitudes. Yet, parametrisations of radiative heating and cooling rates in MGCMs suggest departures from the local thermodynamic equilibrium above ≈ 80 km altitude [START_REF] López-Puertas | Radiative energy balance of co2 non-lte infrared emissions in the martian atmosphere[END_REF][START_REF] López-Valverde | Nonlocal thermodynamic equilibrium in general circulation models of the martian atmosphere 1. effects of the local thermodynamic equilibrium approximation on thermal cooling and solar heating[END_REF][START_REF] González-Galindo | A ground-to-exosphere martian general circulation model: 1. seasonal, diurnal, and solar cycle variation of thermospheric temperatures[END_REF], although we will neglect them.

CO 2 deposition

The second diabatic effect is specific for Mars and consists of the latent heat release due to deposition of CO 2 , which is the main component of the Martian atmosphere (about 96% based on the latest measurements by Curiosity rover [START_REF] Mahaffy | Abundance and isotopic composition of gases in the martian atmosphere from the curiosity rover[END_REF]), and formation of CO 2 ice clouds. Martian CO 2 ice clouds have been observed mostly during the polar nights in the lower troposphere [START_REF] Pettengill | Winter clouds over the north martian polar cap[END_REF][START_REF] Hayne | Carbon dioxide snow clouds on mars: South polar winter observations by the mars climate sounder[END_REF][START_REF] Hu | Mars atmospheric co2 condensation above the north and south poles as revealed by radio occultation, climate sounder, and laser ranging observations[END_REF], but also in the equatorial and mid-latitude mesosphere [START_REF] González-Galindo | The martian mesosphere as revealed by co2 cloud observations and general circulation modeling[END_REF]. The primary mechanism of CO 2 ice-cloud formation in the Martian atmosphere is heterogeneous nucleation on the dust particles [START_REF] Määttänen | Nucleation studies in the Martian atmosphere[END_REF][START_REF] Listowski | Modeling the microphysics of co2 ice clouds within wave-induced cold pockets in the martian mesosphere[END_REF]. It is worth noting that by dust in this article we mean any type of deposition nuclei, which can be of different origins: mineral dust elevated from the ground by winds, water ice crystals, or any type of micrometeorites. A sufficient local concentration of nuclei is assumed to be a necessary condition for CO 2 deposition in this study.

Deposition in Mars GCM models has not been coupled to aerosols so far, and only depends on the temperature, while here the inhomogeneity introduced by the aerosol spatial distribution will play a decisive role in both CO 2 deposition and radiative relaxation.

In the simplest version of the model we are using (2.1a) -(2.1b), the potential temperature θ is considered to be horizontally uniform. So the deposition phase transition in the model is controlled uniquely by the remaining thermodynamical variable, the pressure h, and starts once the pressure is higher than some threshold pressure, h th . The deposition then leads to pressure relaxation towards the threshold value with a characteristic time τ c . The deposition threshold pressure is taken to be above the equilibrium pressure H, by a tunable amount * . A necessary condition to trigger the deposition is that the bulk amount of nuclei in the air column, N , also exceeds some threshold value, N th . The deposition of (2009). We follow here the recipe of this paper, although the origin of the convective flux is different. Like in the moist-convective RSW, the momentum equation is unchanged, while the mass conservation acquires sources/sinks due to radiative relaxation and CO 2 deposition, and takes the following form:

Dh Dt + h∇ H .v = - h -H τ r -C. (2.3) C = α * h -h th τ c H(h -h th )(N -N th )H(N -N th ), (2.4)
where H denotes Heaviside (step-) function, the constant α * can be absorbed in the relaxation time τ c without loss of generality, and the deposition threshold is restricted by a threshold pressure given by

h th = H(r) + * . (2.5)
As follows from (2.4), we consider that deposition can not take place if the bulk amount of nuclei in the air column N is below N th , and stops once N drops below this threshold. The nuclei conservation equation is given by

DN Dt + N ∇ H .v = -δC + S, (2.6)
where the first term in the r.h.s. corresponds to depletion of nuclei due to deposition, and the second one is a nuclei source. We will not dwell into discussion of parameterisations of S (for example, the one proportional to the wind v could be used, like what is used for surface evaporation, see e.g. Rostami In what follows we will first use the adiabatic version of the model (2.1a), (2.1b) in order to identify the unstable modes of the zonally and time-averaged Mars' polar vortex, and then use these latter to initialise numerical simulations of the nonlinear evolution of instabilities in adiabatic and diabatic versions of the model.

We will diagnose nonlinear evolution of the instabilities with the help of potential vorticity (PV). Indeed, as is well-known, the adiabatic version of the model (2.1a),

(2.1b) possesses a Lagrangian invariant, the potential vorticity q = (ζ + f )/h:

D Dt ζ + f h = 0, (2.7)
where ζ is the relative vorticity in the tangent plane:

ζ = ẑ • ∇ × v .
In the presence of diabatic effects PV acquires sources/sinks

D Dt ζ + f h = ζ + f h 2 h -H τ r + C . (2.8)
which modify its evolution.

3 Background vortex and its instabilities

A reminder on vortex solutions and linear stability analysis

The first stage of our analysis is identification of unstable modes of the zonally and time-averaged Mars' polar vortex. Azimuthally-symmetric stationary state with a given velocity profile V (r), which we retrieve from MACDA data below, is a solution of the equations (2.1a), (2.1b) provided V (r) and the pressure (thickness) H(r) are in cyclo-geostrophic (or gradient wind) equilibrium:

( V r + f)V = g∂ r H. (3.1)
Linear stability of any solution of this kind is analysed by the standard means.

The equations (2.1a), (2.1b) are linearised, considering small perturbations of the axisymmetric background flow, denoted by primes. It is convenient to work with non-dimensional equations, and we will be using the following scaling, with non-dimensional variables marked with an asterisk:

(u , v , V )(r, θ, t) = L d f 0 (u * , v * , V * )(r * , θ * , t * ), (3.2a) r = L d r * , θ = θ * , t = 1 f 0 t * , H(r) = H 0 H * (r * ), h = H 0 η * . (3.2b)
Here H 0 is a characteristic thickness, and L d is the Rossby deformation radius

L d = √ gH 0 /f 0 .
The advantage of this non-dimensionalisation is that it is based on a single adjustable parameter, L d , while f 0 is fixed for a given planet.

We are looking for the normal-mode solutions with harmonic dependence on time and polar angle

(u * , v * , η * )(r * , θ * , t * ) = Re[(iũ, ṽ, η)(r * )e i(lθ * -ωt * ) ], (3.3)
where l and ω are the azimuthal wavenumber and the frequency, respectively. The corresponding eigenproblem can be written as:

                  lV * r * sin(φ) + 2V * r * -D r * sin(φ) + V * r * + D r * V * lV * r * l r * H * D r * + 1 r * D r * (r * H * ) lH * r * lV * r *                   ×               ũ ṽ η               = ω               ũ ṽ η               , (3.4)
with D r * denoting differentiation with respect to r * . Complex eigen-frequencies ω = ω r +iω I with positive imaginary part (ω I > 0), correspond to instabilities with linear growth rate σ = ω I . In the formulas above sin φ in the Coriolis parameter should be understood in the sense of Taylor expansion in r, where the first two terms correspond to the gamma-plane approximation, but higher terms could be also retained, heuristically.

Numerical analysis of the eigenproblem (3.4) is performed by the pseudospectral collocation method [START_REF] Trefethen | Spectral Methods in MATLAB[END_REF]. The system is discretised over an N -point grid in the radial direction, and D r * becomes the Chebyshev differentiation operator. To avoid the Runge phenomenon, Chebyshev collocation points are used, with a stretching in the radial direction allowing to accumulate the collocation points in the most dynamically interesting region near the centre of the vortex.

The eigenproblem (3.4) is singular, possessing so-called critical levels, where the angular velocity of the eigenmodes ω/l is equal to the local angular velocity of the flow V * /r * . Numerical solutions of (3.4) can produce pseudo-modes which are not smooth at critical levels, and not converge to smooth solutions with increasing resolution. Such modes were carefully analysed and eliminated. We should say that stability analyses of atmospheric jets are known in literature for a long time, e.g. [START_REF] Hartmann | Barotropic instability of the polar night jet stream[END_REF], including jets on Mars [START_REF] Michelangeli | Barotropic instability of midlatitude zonal jets on mars, earth and venus[END_REF]. However, they are usually done 1) with ad hoc analytic velocity profiles, and 2)within the quasi-geostrophic model which, in our case, is a small Rossby number limit of the system (2.1a), (2.1b), with the velocity field given in terms of the geostrophic stream-function, and thus having only very weak divergence. We perform the stability analysis of the velocity profile fitted from the data, and find the unstable modes, which we use later for initialisations of numerical simulations in the "full" model, as even a small divergence is crucial in determining the transport properties of the flow.

Stationary barotropic Mars' polar vortex and its instabilities

To construct a stationary solution of our shallow water model corresponding to the Martian winter polar vortex we use the MACDA v1.0 dataset [START_REF] Montabone | The mars analysis correction data assimilation (MACDA) dataset v1.0[END_REF]. As we are mostly interested in the atmospheric configuration that occurs during middle winter, when deposition and related latent heat release are expected [START_REF] Hu | Mars atmospheric co2 condensation above the north and south poles as revealed by radio occultation, climate sounder, and laser ranging observations[END_REF], we consider data about the NH winter solstice (L S ≈ 270 • ) of MY 24. We take 14 layers up from the surface in the MACDA dataset, which corresponds to the pressure range 0.9995P s -0.2P s , where P s is the surface Instead of numerical interpolation of the velocity profile data obtained by the above-described procedure, it is convenient to use its analytic fit, which can be achieved by a simple parametrisation with a small number of parameters. The fit for azimuthal velocity of the stationary vortex we are using is as follows:

V (r) =                          (r -r 0 ) α e -m(r-r 0 ) β max (r -r 0 ) α e -m(r-r 0 ) β , r ≥ r 00 . m 0 r, r < r 00 (3.5)
In this formula r is the non-dimensional distance from the vortex center, measures the intensity of the velocity field in the circumpolar jet, r 0 gives the nondimensional radius of maximum wind (RMW), and other parameters allow to fit the shape of the distribution. As the center of the Martian winter polar vortex is relatively calm, with low-speed winds, we suppose a slow solid rotation in this region, i.e. linear V (r) with a gentle slope m 0 ≈ 0.02 . The parameter r 00 gives the size of this domain. A cubic Hermite interpolation between the two regions is done to prevent jump appearance in vorticity profile. An advantage of this parametrisation is the possibility to tune the ascending or descending trend of the wind near and far from the peak. The velocity profile is normalised by its maximum, so the maximum value of velocity is equal to . We list in Table 1 the values of the parameters we have chosen, together with parameters of diabatic corrections which are used in nonlinear simulations below. The upper left panel of Figure 2 provides a comparison of the MACDA data with the fit (3.5) using the values of parameters given in Table 1.

Results of the linear stability analysis of the zonally averaged vortex profile

We omit the details of computations following the procedure sketched in Section 3, and present only the main results and their analysis. For the barotropic velocity profile V (r), described in subsection 3.2, the most unstable modes have azimuthal wavenumbers l = 2 and l = 3, with a slightly higher growth rate for the latter. The azimuthal and radial structure of the unstable mode with l = 3 is presented in Figure 2, together with the background vortex profile. Nevertheless, the growth rates of modes with azimuthal wave-numbers l = 1, 4 are not negligible. This result is compatible with those of [START_REF] Barnes | Mars atmospheric dynamics as simulated by the nasa ames general circulation model: 2. transient baroclinic eddies[END_REF] at L d ≈ 1700 -1800km. Figure 3 represents a variation of the linear growth rate with azimuthal wavenumber for different values of L d . Linear stability analysis of the profile (3.5) demonstrates that when the RMW of the background velocity profile is closer to the North Pole with higher curvature at the RMW, the most unstable mode shifts to lower azimuthal wavenumbers and higher growth rates, and vice versa. On this basis we can expect strong wavenumber two and wavenumber three instabilities with growth rates larger than 1 sol -1 in the Martian upper atmosphere during winter, as velocity peak is higher and closer to the pole in the upper layers. This is com-patible with findings of [START_REF] Conrath | Planetary-scale wave structure in the martian atmosphere[END_REF], where a stationary wavenumber 2 pattern in the upper atmosphere was revealed by using Mariner 9 Infrared Interferometer Spectrometer (IRIS) data. The gradient of PV in MACDA data in the lower layers is much smaller than its counterpart in the upper layers, and the RMW is larger than in the upper ones. So if the upper layers are excluded from vertical averaging, the growth rates of corresponding unstable modes become smaller and the most unstable azimuthal wavenumber shifts to higher values, and vice versa.

Increasing the amplitude of the initial velocity, i.e. the Rossby number, leads to an increase of growth rates, but the azimuthal wavenumber of the most unstable mode remains the same. This relationship is almost linear, similarly to what was found in (Rostami and Zeitlin, 2017;Rostami et al., 2017). Although we did not introduce them, adding Ekman friction and viscosity would lead to lower values of phase speeds and growth rates, cf. [START_REF] Barnes | Linear baroclinic instability in the martian atmosphere[END_REF].

Nonlinear evolution of instability

Once the unstable modes of stationary polar vortex are identified, the question arises about nonlinear evolution of the instability. In the present section we compare the adiabatic evolution of the instability with diabatic evolution in the configurations including 1) only classical radiative relaxation (R), with deposition term switched off in (2.3), but including the passive tracer N , 2) only spatially inhomogeneous CO 2 deposition (D), due to the coupling of the pressure field with the tracer, with deposition term switched on but radiative relaxation term switched off in (2.3), and 3) both radiative relaxation and CO 2 deposition (RD). We per-form fully nonlinear numerical simulations using a finite-volume scheme designed for moist-convective RSW [START_REF] Bouchut | Fronts and nonlinear waves in a simplified shallow-water model of the atmosphere with moisture and convection[END_REF]. Inclusion of the radiative relaxation in the scheme is straightforward, and the deposition corrections discussed in section 2 are incorporated along the same lines as condensation in the moistconvective RSW. In order to analyse the development of the barotropic instability, the variables u, v, h are initialised by superimposing a small perturbation corresponding to the most unstable mode with l = 3 onto the background vortex profile obtained by vertical averaging from MACDA data. The initial distribution of N , which is N 0 , is uniform (0.005 below N th ). It should be noted that changes in the scale of N and N th can be absorbed by rescaling of the relaxation time, and it is the relative difference between the two which is the most important factor: the smaller the difference the stronger the effect.

Comparisons of adiabatic and diabatic (in different configurations) nonlinear evolution of the unstable mode as seen in the PV field are presented in Figure 4. In all environments, the growing initial l = 3 perturbation leads to formation of the triangular eye-wall at early stages, which is gradually transformed into a quasiannular belt of higher PV around the eye. Yet, re-distribution of PV in the belt, leading to spatial non-uniformity, occurs rather early (t ≈ 150f -1 ) in the simulations involving deposition (the second and the fourth rows of Fig. 4), while this is not the case in the adiabatic simulation nor in the one with radiative relaxation only. Due to space limitations we skip the snapshots between t = 200f -1 0 and t = 300f -1 0 in all simulations. They show an evolution towards the last displayed state without qualitative disruptions. The annular PV structure is clearly visible in all cases at initial steps. As time goes on the form of the PV belt is modu-lated by other azimuthal modes emerging due to nonlinear interactions in course of evolution. We show behavior of the normalised amplitudes of azimuthal modes of the perturbation, extracted from the simulations of nonlinear evolution of the instability in Figure 5. As follows from the Figure, the l = 1 and l = 2 modes are growing faster than modes with l = 4 and l = 5 during instability saturation.

However, with the choice of parameters made, the evolution in configurations with deposition has a tendency to saturate the l = 2 mode, which is mainly responsible for the ellipticity, faster then in purely radiative configuration. Differences among four configurations are manifest in Figure 4. Yet, although patches of higher PV appear in simulations with deposition at later stages (the two last columns of the Figure), they are less pronounced than in the reanalysis data. In order to track the patchiness of the PV field we continued simulation for even longer times. The results are presented in Figure 6, and clearly show that only the combination of radiative relaxation and deposition gives a PV field qualitatively similar to the reanalysis, while relaxation or deposition alone do not. The form of the PV belt in different configurations is consistent with Figure 5 and is corroborated by further evolution of the azimuthal modes presented in Figure 7.

In order to better quantify the influence of both diabatic effects upon the annularity of the resulting distributions of potential vorticity and velocity, we present radial distributions of zonally and time averaged PV and velocity at the late stages of the evolution of the instability in the left and right panels of Figure 8, respectively. As follows from all the displayed figures, the influence of diabatic effects upon the developing instability is drastic. While the adiabatic evolution follows the standard scenario of growth and breaking of the unstable mode, which leads to filling up the PV minimum in the centre of the vortex and forming a monotonous PV profile, this process is arrested in the presence of radiative relaxation, which leaves the PV distribution hollow. Deposition alone, D, leads to non-uniform filling of the centre of the PV distribution, with formation of high PV patches, as compared with the adiabatic evolution. Nevertheless, in this case the ring of high PV collapses at later times and gives a centred distribution of PV, like the diabatic one. Together, radiative relaxation and deposition lead to hollow PV distribution with high PV patches in the PV belt around the PV minimum at the centre. We checked the sensitivity of the just described scenarios to the values of parameters.

The results are presented in Figure 9, and show that big qualitative differences between D, R, and RD scenarios do not change with the changes of N 0 , * and τ r , although some quantitative differences, e.g. in the height of the PV maximum, do exist. The presence of patches is clearly visible at all stages of the depositionrelaxation scenario, and is qualitatively similar to what is observed in MACDA data, cf. Figure 1. Such patches were also detected in the Ensemble Mars Atmosphere Reanalysis System (EMARS) dataset and GCM simulations, cf. [START_REF] Waugh | Martian polar vortices: Comparison of reanalyses[END_REF]. In order to quantify the patchiness of each case in a robust way, we apply a spatio-temporal diagnostics of patchiness based on its definition, which is the variance of PV about the mean distribution of PV. The results are presented in Figure 10, and show that, as the time goes on, the patchiness of both D and R falls down, while RD is enhanced similarly to results from reanalyses. The presence of these zones of high PV explains the average enhancement of PV at the RM W observed in Figure 8. Yet the ellipticity of the annulus is better accounted for in the purely radiative simulation, cf. Figures 6 and7. It should be emphasised that the initialisation was made with solely the most unstable mode with l = 3, and the system keeps memory of this for long times.

In order to understand the origin of enhanced zones of PV in simulations with deposition, we trace in Figures 11 and12 the evolution of the dust field N in all simulations. We should recall that the convective sink of any nature enhances PV, as follows from (2.8). The related change in h leads to additional radiative relaxation and the two source/sink terms in (2.3) thus influence each other. The deposition sink is strongly inhomogeneous in space as it is operational only when a high concentration of N occurs in the region of strong enough pressure. As follows from Figures 11 and12, the zones of high N are much more intermittent and have more complex structure in the configurations with deposition, as compared to the purely radiative configuration, which explains the patchiness of enhanced PV. Let us recall that we take the initial distribution of dust as uniform in all configurations. Its evolution is well correlated with that of the divergence field.

The dust is rapidly expelled out of the center of the vortex, and evolves around the RM W and beyond. Although this has no direct relation to the goal of this paper, we should emphasise that the evolution of the divergence field displays bursts of emission of inertia-gravity waves related to the convective events due to deposition (not shown here). 1). Vertical dotted line represents the radius of maximum wind of the non-perturbed vortex.

] to [300f -1 0 ] in different environ- ments (see Table
consequences of radiative relaxation and CO 2 deposition, and reproduce patches of high potential vorticity, which are observed in atmospheric reanalysis datasets.

We show that it is the barotropic instability of the azimuthally averaged Mars' po- 1).

datasets. At large relaxation times the hollow PV tower cannot resist the instability. At low Rossby deformation radii, the hollow PV tower is maintained even at larger relaxation times. On the other hand, combination of thermal relaxation and the inhomogeneous CO 2 deposition triggered by nucleation is shown to lead the appearance of patches of high-PV inside the annulus, similar to what is observed in MACDA and EMARS datasets. This is the key new finding of the present work demonstrating dynamical influence of the deposition which is conditioned by the presence of deposition nuclei in a sufficient amount. We should emphasize that linking radiative properties to the presence of tracers (e.g. dust) can be done in the model along the same lines as in section 2.2.2. We also showed that qualitative differences among the relaxation-deposition scenario and others are not sensitive to the particular choice of values of the parameters of deposition and relaxation, within a reasonable range.

The key advantage of the simple rotating shallow water model we are using is We should emphasize that in the present study we were attempting to gain a gen-eral understanding of the respective roles of deposition and radiative relaxation, without making an effort to adjust parameters and initializations to fit the observations in the best possible way. Nevertheless, we sketched general dependencies of the results on the values of the parameters. Neither we introduce topography which, as already mentioned in Introduction, could account for the ellipticity of the polar vortex. Data assimilation techniques can be straightforwardly applied to the model with realistic topography, if necessary, but this was not our goal here.

Figure 1 .

 1 Figure 1. This figure shows the evolution of the Ertel potential vorticity on the 275K isentropic level (about 25km altitude in the NW winter polar region) from the MACDA v1.0 dataset (Montabone et al., 2014) in MY 24 at L S ≈ 268 • (sol-of-year=512). Each

  CO 2 is accompanied by a latent heat release, which creates a vertical convective flux resulting in a sink in the mass conservation equation. Additional convective fluxes are easily introduced in the derivation of RSW model by vertical averaging the primitive equations, as was shown for the moist-convective RSW in Bouchut et al.

  pressure. Thus H 0 ≈ 16.5km, and L d ≈ 1750km. The background velocity profile is obtained by vertical, zonal, and solar longitude (i.e. time) averaging within 270 • ≤ L S ≤ 271 • of corresponding MACDA data. The corresponding pseudoheight profile follows from the cyclo-geostrophic balance (3.1). We thus obtain a full stationary solution of the equations of the model whose linear stability can be analysed by the above-described method. The results both on linear stability and on nonlinear saturation of the instability are sensitive to the value of the Rossby deformation radius L d .

Figure 2 .

 2 Figure 2. Upper left panel: radial profile of barotropic azimuthal velocity of Mars' north polar vortex superimposed on vertically and time-averaged MACDA data, and the corresponding profile of relative vorticity. Upper right panel: thickness H(r) of the background vortex. Lower left panel: 2D structure of the most unstable mode with l = 3. Colours: pressure (thickness) anomaly, arrows: velocity field. Lower right panel: radial structure of the most unstable mode. Vertical lines indicate the critical levels.

Figure 3 .

 3 Figure 3. Linear growth rate of the barotropic structure as a function of azimuthal wavenumber at different values of the Rossby deformation radius. L d ≈ 1200, 1740, 2200km correspond to vertical averaging over 12, 14, 16 layers, respectively.

Figure 4 .Figure 5 .

 45 Figure 4. Nonlinear evolution of the l = 3 instability in different environments, as seen in PV field measured in units of H 0 /f 0 . From top to bottom: adiabatic evolution, evolution with CO 2 deposition only (D), evolution with radiative relaxation only (R), evolution with both radiative relaxation and deposition (RD). White dashed lines indicate initial radius of maximum wind (RM W ). 12.5 [f 0 -1 ] ≈ 1[ sol].

Figure 6 .Figure 7 .

 67 Figure 6. Late -time evolution of the PV field in D, R, and RD configurations.

Figure 8 .

 8 Figure 8. Radial distribution of zonal means of PV (left panel) and zonal velocity (right panel) averaged over a period of time from [250f -1 0 ] to [300f -1 0 ] in different environ-

Figure 9 .Figure 10 .

 910 Figure 9. Sensitivity of radial distributions of zonally and time averaged PV to different values of the initial uniformly distributed concentration of dust N 0 , radiative relaxation time (τ r ), and * in different environments (see Table 1). The vertical dotted line represents the radius of maximum wind of the non-perturbed vortex. Time interval: from [0f -1 0 ] to [900f -1 0 ].

Figure 11 .

 11 Figure 11. Evolution of the tracer (dust) field during nonlinear saturation of the l = 3 instability in different environments. From top to bottom: adiabatic evolution, evolution with deposition only (D), evolution with radiative relaxation only (R), evolution with both radiative relaxation and deposition (RD). The thick dashed lines indicate the initial radius of maximum wind (RM W ). 12.5 [f 0 -1 ] ≈ 1[ sol]. Only a part of the calculational domain is presented. The colorbar in the RD case has been changed for clarity.

Figure 12 .

 12 Figure 12. Same as figure 11 for later times.that it is based on momentum and mass conservation with only a couple of basic parametrisations. It is thus robust and does not rely on numerous fine-tuned parametrisations of physical processes. The model captures only the most basic features of the dynamics, which however turns to be sufficient to understand the salient properties of the Mars' polar vortex. The model can be improved in many ways by including topographic effects, nuclei (dust) sources and sinks, but also by passing from its barotropic version to a two-or multi-layer baroclinic version, which would allow to include the effects of stratification and wind shear.

  The values of parameters, τ r , τ c , N th , * used in numerical simulations presented in this paper are given in Table1. For simplicity N is initialised with a uniform value close to N th . Deposition time scale τ c , which is the key parameter, is chosen Initial conditions and parametrisations of the numerical experiments. Adiabatic: diabatic effects switched off, R: environment including just the radiative relaxation, D: environment including only the CO 2 deposition, RD: environment including both radiative relaxation and deposition, N th : threshold value of bulk concentration of nuclei above which deposition occurs. ∆t: the numerical time step. Fixed parameters of all simulations: = 0.2489, α = 4.1, β = 2.3, m = 3.2, L d = 1740 km, r 0 = 0.35, r 00 = 0.4.

	Table 1
	Parameter Adiabatic	R	D	RD
	τ r	-	0.5 sol ∞ sol 0.5 sol
	τ c /α *	-	-	5∆t	5∆t
	*	-	-	0.001	0.001
	N th	-	-	0.9	0.9

and Zeitlin (2017)

), and will consider for simplicity that production and depletion of nuclei are either negligible, or compensate each other, i.e. that the r.h.s. of (2.6) is zero, and deposition nuclei (i.e the tracer field) are just transported, but not produced, nor eliminated. to be fast: just few numerical time steps. As in simulations with moist-convective RSW

[START_REF] Bouchut | Fronts and nonlinear waves in a simplified shallow-water model of the atmosphere with moisture and convection[END_REF]

, this poses no numerical problems. Lower values of τ c result in stronger deposition. The results of linear stability analysis depend on L d , see below.

Acknowledgements

The authors thank A. Spiga for discussions and valuable suggestions and comments on the manuscript. V.Z is grateful to P. Read for a discussion that stimulated the present study. L.M. acknowledges funding from NASA's Mars Data Analysis program (MDAP) to study the variability of the Martian polar vortices (grant N. NNX13AK02G). We furthermore thank the German Climate Computing Center (Deutsches Klimarechenzentrum, DKRZ) for providing computing time on the high performance computing system, as well as support.