Vinayagame Radjasouria
email: radjasouria.vinayagame@eviden.com

Eviden

Van Man Nguyen
email: van-man.nguyen@eviden.com

Marc Sergent
email: marc.sergent@eviden.com

Samuel Thibault
email: samuel.thibault@u-bordeaux.fr

Emmanuelle Saillard
email: emmanuelle.saillard@inria.fr

Marc Nguyen

Samuel Sergent

Emmanuelle Saillard Thibault

Rethinking Data Race Detection in MPI-RMA Programs

Keywords: Computing methodologies → Parallel programming languages;, Software and its engineering → Software testing and debugging HPC, MPI-RMA, Verification, Data Race, Dynamic Analysis Radjasouria Vinayagame, Van Man

Supercomputers are capable of increasingly more computations, and nodes forming them need to communicate even more efficiently with each other. The Message Passing Interface (MPI) proposes a communication model based on one-sided communications called the MPI Remote Memory Access (MPI-RMA). Thanks to these operations, applications can improve the overlap of communications with computations. However, one-sided communications are complex to write since they are subject to data races. This paper rethinks an existing on-the-fly data race detection algorithm for MPI-RMA programs by improving the storage of memory accesses in a Binary Search Tree using a new insertion algorithm based on fragmentation and merging algorithms. Thus, experimental results on real-life applications show that this new insertion algorithm improves the accuracy of the data race detection and can reduce the overhead of the analysis at runtime by a factor up to two.

exposing the hardware at the software level (network with RMA capabilities) with the Partitioned Global Address Space (PGAS) model. This model is based on one-sided communications which decouple data movement from synchronization. The Message Passing Interface (MPI) standard proposes a similar communication model via MPI Remote Memory Access (MPI-RMA). With MPI-RMA, each MPI process makes a part of its memory available to other MPI processes so the latter can remotely read and write on this "distributed shared memory" with fewer synchronizations than two-sided MPI communications. Consequently, applications migrating from two-sided MPI to MPI-RMA operations should get significant speedup. Nevertheless, most MPI programs still use twosided communications because MPI-RMA programs are challenging to write and error prone. Indeed, as in shared memory models, developers must ensure memory consistency in MPI-RMA programs to avoid data races. Thus, some tools aim at helping developers write correct and productive MPI-RMA applications to promote communication/computation overlap and asynchrony. However, these tools imply a noteworthy overhead at runtime induced by the analysis of the program and thus have scalability issues. This paper enhances an existing on-the-fly data race detection tool for MPI-RMA programs called RMA-Analyzer [START_REF] Tassadit | Dynamic Data Race Detection for MPI-RMA Programs[END_REF]. We propose a new algorithm that enhances its scalability and its accuracy.

The paper is organized as follows: Section 2 provides some background elements and the key concepts this work relies on. Section 3 lists the existing tools that can detect data races in MPI-RMA programs. Section 4 proposes a new algorithm that enhances the memory accesses management in RMA-Analyzer in order to reduce the overhead of the analysis. In Section 5, we compare our contribution against MUST-RMA, a state-of-the-art solution for the detection of data races in MPI-RMA programs. We highlight the improvements we describe in the paper and show a performance analysis. In Section 6, we discuss several subtleties of the MPI-RMA library that may impact the correctness of data race detection tools. Finally, Section 7 concludes this work.

BACKGROUND

This section describes how MPI-RMA works and the errors that can occur due to the specifics of the one-sided communications.

MPI-RMA

In MPI-RMA, each process has a "distributed shared memory" that can be remotely accessed by other MPI processes. In MPI-RMA, these memory regions are called windows and remote memory accesses to these windows are possible during an epoch. Within an epoch, MPI-RMA proposes several communication operations which involve two processes: the 𝑜𝑟𝑖𝑔𝑖𝑛 process which issues the MPI-RMA communication, and the 𝑡𝑎𝑟𝑔𝑒𝑡 process whose window is accessed via the communication. In this paper, we focus on the Passive Target synchronization mode where only the 𝑜𝑟𝑖𝑔𝑖𝑛 process is involved in the synchronization. We consider the two major onesided communication operations: MPI_Put which allows to write a value owned by the 𝑜𝑟𝑖𝑔𝑖𝑛 process to the window of a 𝑡𝑎𝑟𝑔𝑒𝑡 process, and MPI_Get which allows the 𝑜𝑟𝑖𝑔𝑖𝑛 process to locally retrieve a value from the window of a 𝑡𝑎𝑟𝑔𝑒𝑡 process. An example of all possible accesses within an epoch is shown in Figure 1. In the figure, gray parts represent the window. 𝑃 𝑖 → 𝑃 𝑗 means a communication from 𝑃 𝑖 to 𝑃 𝑗 . 𝑜𝑢𝑡𝑊 𝑖𝑛 and 𝑖𝑛𝑊 𝑖𝑛 respectively mean out of the window and in the window.

When using MPI-RMA, four types of memory accesses should be considered [START_REF] Tassadit | Dynamic Data Race Detection for MPI-RMA Programs[END_REF], depending on if the operation is local to the process (𝐿𝑜𝑐𝑎𝑙_ * ) or if it is a remote memory access (𝑅𝑀𝐴_ * ), and on if the operation is a 𝑊 𝑅𝐼𝑇 𝐸 ( * _𝑊 𝑟𝑖𝑡𝑒) or a 𝑅𝐸𝐴𝐷 ( * _𝑅𝑒𝑎𝑑) operation. For instance, an MPI_Put operation is an 𝑅𝑀𝐴_𝑊 𝑟𝑖𝑡𝑒 for the 𝑡𝑎𝑟𝑔𝑒𝑡 process and an 𝑅𝑀𝐴_𝑅𝑒𝑎𝑑 for the 𝑜𝑟𝑖𝑔𝑖𝑛 process. Inversely, an MPI_Get operation is an 𝑅𝑀𝐴_𝑅𝑒𝑎𝑑 for the 𝑡𝑎𝑟𝑔𝑒𝑡 process and an 𝑅𝑀𝐴_𝑊 𝑟𝑖𝑡𝑒 for the 𝑜𝑟𝑖𝑔𝑖𝑛 process. A Store operation is a 𝐿𝑜𝑐𝑎𝑙_𝑊 𝑟𝑖𝑡𝑒 while a Load operation is a 𝐿𝑜𝑐𝑎𝑙_𝑅𝑒𝑎𝑑.

MPI-RMA has shown performance improvement on applications that have migrated from MPI two-sided to MPI-RMA. As an example, Mizan-RMA [START_REF] Li | Mizan-RMA: Accelerating Mizan Graph Processing Framework with MPI RMA[END_REF], a graph processing framework, got a speedup up to 280% when using MPI-RMA compared to a previous version that uses MPI_Send and MPI_Recv operations. Similarly, the Graph500 [START_REF] Li | Scalable Graph500 design with MPI-3 RMA[END_REF] data intensive benchmark got a speedup of 200%. The authors claim that these speedups are achieved through a better overlapping of communications and computations. However, writing efficient and correct MPI-RMA programs can be challenging. Especially because of the following three properties that are crucial to ensure an efficient overlap of communications with computations:

(

Data Races in MPI-RMA Programs

MPI-RMA programs expose memory through an abstraction that allows to read from and write to distant memory at any time, which can lead to data races. A data race occurs if two operations access the same memory range with at least one of them being an RMA access, and one of them being a 𝑊 𝑅𝐼𝑇 𝐸 access (𝑅𝑀𝐴_𝑊 𝑟𝑖𝑡𝑒 or 𝐿𝑜𝑐𝑎𝑙_𝑊 𝑟𝑖𝑡𝑒). In such a scenario, the value in that memory space may change depending on the order in which the operations are executed. A more precise description of data races is presented by Hoefler et al. [START_REF] Hoefler | Remote Memory Access Programming in MPI-3[END_REF] and two examples of data races are presented in Figure 2. In Figure 2a, the 𝑜𝑟𝑖𝑔𝑖𝑛 process issues an MPI_Get operation to read the value of 𝑋 at 𝑡𝑎𝑟𝑔𝑒𝑡 side (RMA_Read) and update buf (RMA_Write). We suppose this operation is followed by a Load operation from buf to loc. Because of the completion property, a data race can occur at the 𝑜𝑟𝑖𝑔𝑖𝑛 side and buf is either equal to 𝑋 or 𝑙𝑜𝑐. Figure 2b shows a data race between two processes. In this example, both 𝑜𝑟𝑖𝑔𝑖𝑛 and 𝑡𝑎𝑟𝑔𝑒𝑡 processes are reading and writing in their own window, and on overlapping address ranges.

Data races can also occur with more than two processes, for instance when multiple processes issue an MPI_Put operation to the same address space of the same target. All possible cases of data races are reported in Figure 3. Possible consistency errors within a process are represented by the first part of the table. We consider three processes: two that can initiate a communication (ORIGIN 1 and ORIGIN 2) and one target process (TARGET). Rows represent the first operation issued. The columns represent another operation issued either by the same process (ORIGIN 1) or another process (TARGET of the first operation or a new process issuing a communication, referred as ORIGIN 2). In each cell, the right bit refers to an error at origin side while the left bit refers to an error at target side. The example Figure 2a is represented by the cell (O1-GET, ORIGIN1-LOAD). "01" means that an error can occur only at origin side. Figure 2b is represented by the cell (O1-GET, TARGET-GET). Depending on if the value is read and written in or out of the window, an error can or cannot occur.

RELATED WORK

Many tools detect data races in multithreaded programs (e.g., Valgrind DRD [START_REF] Müehlenfeld | Fault Detection in Multi-Threaded C++ Server Applications[END_REF], Helgrind [START_REF] Jannesari | Helgrind+: An efficient dynamic race detector[END_REF] or ThreadSanitizer [START_REF] Serebryany | ThreadSanitizer: Data Race Detection in Practice[END_REF]). However, these tools cannot be directly applied to MPI-RMA programs because they are too general, so they struggle with the scale of MPI-RMA programs. On the other hand, tools that are aware of the semantics of MPI-RMA communications can significantly reduce the analysis workload.

Very few approaches exist to detect data races in MPI-RMA programs. Park et al. [START_REF] Park | Detecting Race Conditions in One-Sided Communication of MPI Programs[END_REF] presents an approach that creates a mirror window each time a window is created. Then, each time a new MPI-RMA communication accesses a memory space in the window, a check for data races is performed in the corresponding mirror window containing all previous accesses to that window. This approach does not consider local Load and Store accesses, thus leading to false negative results. Moreover, the implementation of this approach is only compatible with the MPI-2 standard and thus does not support the new MPI-RMA features such as the MPI_Win_lock_all/MPI_Win_unlock_all epoch creation functions. MC-Checker [START_REF] Chen | MC-Checker: Detecting Memory Consistency Errors in MPI One-Sided Applications[END_REF] uses a post-mortem analysis based on a Directed Acyclic Graph (DAG) to detect concurrent regions (in respect to the happens-before relation [START_REF] Lamport | Time, Clocks, and the Ordering of Events in a Distributed System[END_REF]). MC-Checker has been enhanced with MC-CChecker [START_REF] Diep | MC-CChecker: A Clock-Based Approach to Detect Memory Consistency Errors in MPI One-Sided Applications[END_REF]. This new tool reduces the number of false positives and improves the scalability of MC-Checker through a clock-based approach based on the encoded vector clock. Like the work of Park et al., MC-CChecker is only compatible with the MPI-2 standard. MUST-RMA [START_REF] Schwitanski | On-the-Fly Data Race Detection for MPI RMA Programs with MUST[END_REF] is an on-the-fly data race detector for MPI-RMA programs which combines MUST [START_REF] Hilbrich | MUST: A Scalable Approach to Runtime Error Detection in MPI Programs[END_REF] and ThreadSanitizer, a shared-memory data race detector [START_REF] Serebryany | Dynamic Race Detection with LLVM Compiler[END_REF]. MUST-RMA constructs concurrent regions based on the happens-before relation and forwards them to ThreadSanitizer which then checks for data races. A static approach detecting data races in MPI-RMA programs has been proposed by Saillard et al. [START_REF] Saillard | Static Local Concurrency Errors Detection in MPI-RMA Programs[END_REF]. The analysis performs a Breadth First Search (BFS) on the Control Flow Graph to detect data races at compile time. This method enables early detection of errors but is limited to errors occurring at the origin side only. To deal with performance, MC-Checker, MC-CChecker implementations have a lightweight static support that reduces the number of Load/ Store instrumentations. RMA-Analyzer [START_REF] Tassadit | Dynamic Data Race Detection for MPI-RMA Programs[END_REF] captures memory accesses during the execution of a program and stores them in a Binary Search Tree (BST). The implementation considers that a memory access includes information about the exact interval of addresses that are accessed (we are only considering consecutive accesses which means that all the addresses in the interval are accessed), the type of the access, and debug information (e.g., the location of the access in the source code). Thus, as soon as a data race is detected, RMA-Analyzer stops the program and returns an error message including debug information to facilitate the correction of the program. When an MPI window is created, each MPI process creates a BST. The BST is then filled with all memory locations the owner process or other processes accesses to (both remote accesses in the window and local accesses). When a new memory location is accessed, a first traversal of the BST is done in order to check for data races with the previous memory accesses. If there is no error, another traversal is done to insert the new access. We argue that this implementation of the approach may have performance bottlenecks, especially because of the size of the BST that is equal to the number of accesses in a program. Moreover, false negatives are possible because of the approximation made by only considering the lower bound of the interval of addresses when comparing two accesses. These limitations are due to the fact that accesses present in the BST are not disjoint, which leads to false negatives, and are not merged, which leads to slowdown. To solve these issues, we introduce a new insertion algorithm of memory accesses in the BST that increases the accuracy of the analysis, and reduces the memory footprint and overhead induced by the data race detection proposed by RMA-Analyzer.

NEW DATA RACE DETECTION

Our new data race algorithm improves the insertion of memory accesses in the BST. The algorithm is presented in Algorithm 1.

Given a new memory access 𝑛𝑒𝑤𝐴𝑐𝑐, the new insertion algorithm first checks for data races (line 2). If no data race is detected, the algorithm retrieves all the accesses present in 𝐵𝑆𝑇 that are intersecting with 𝑛𝑒𝑤𝐴𝑐𝑐 (line 5). Then, the fragmentation algorithm presented in Section 4.1 is called over all these accesses (line 6). Afterwards, the merging algorithm presented in Section 4.2 is called on 𝑓 𝑟𝑎𝑔𝐴𝑐𝑐 to merge fragmented accesses if possible (line 7). Finally, finish_insertion is called to replace the "old" accesses by the "new" ones (line 8). An example of an insertion is presented in Figure 4.

Step 1 ○ shows the state of the BST at the beginning of the algorithm. The colors correspond to different type of accesses. In step 2 ○, only memory locations that intersect with 𝑛𝑒𝑤𝐴𝑐𝑐 are represented as the rest of the algorithm will only consider these accesses. The following subsections explain the fragmentation and finish_insertion(𝑖𝑛𝑡𝑒𝑟𝐴𝑐𝑐, 𝑚𝑒𝑟𝑔𝑒𝑑𝐴𝑐𝑐, 𝐵𝑆𝑇 ) merging algorithms in details (steps 3 ○ and 4 ○). Finally, step 5 ○ updates the BST.

Improving the Accuracy of the Memory Accesses Insertion Algorithm

RMA-Analyzer can have false negatives caused by non-disjoint accesses. Indeed, when searching for accesses intersecting with a new access in the BST, RMA-Analyzer may miss some of these intersections and potentially miss a data race. This is due to the approximation made by only considering the lower bound of the interval of addresses to compare the accesses. An example of a code leading to a false negative is presented in Figure 8a. The corresponding BST created by RMA-Analyzer is presented in Figure 5a where a data race may be detected if the second memory access is from another process. The third access 𝑟 _𝑓 𝑟𝑎𝑔 represents the remaining subsection. An illustration of the fragmentation algorithm on two accesses is shown in Figure 6. In this figure, we consider that accesses with the same Type have the same access type and the same debug information. Using this new algorithm, the BST constructed by RMA-Analyzer presented in Figure 5a becomes the BST presented in Figure 5b. This fragmentation algorithm has been implemented and tested on the program shown in Figure 8a. It successfully detected the data race between the bold statements, whereas RMA-Analyzer failed to raise the error. However, it can lead to a drastic increase of nodes in the BST. Indeed, each new access possibly increases the nodes in the BST by two : one node is removed and three nodes are added. This may lead to an explosion of the memory usage and slow down the analysis of the BST.

Table 1:

Resulting access type and debug information given two accesses. Rows represent an access already inserted in the BST and columns represent the new access that is about to be inserted. *_R: *_Read, *_Write: *_Write, *-1/*-2: debug information of the 1 𝑠𝑡 /2 𝑛𝑑 access, x: data race.

Local_R-2 Local_W-2 RMA_R-2 RMA_W-2 Local_R-1 Local_R-2 Local_W-2 RMA_R-2 RMA_W-2 Local_W-1 Local_W-1 Local_W-2 RMA_R-2 RMA_W-2 RMA_R-1 RMA_R-1 x RMA_R-2 x RMA_W-1 x x x x

Optimizing the Number of Nodes in the BST

In order to avoid the explosion of the number of nodes in the BST, which could negatively impact the scalability of the data race detection algorithm, we propose a node-merging algorithm.

To merge two nodes, two conditions need to be ensured: (1) the two accesses to be merged must be adjacent, and (2) they must have the same access type and debug information. Two accesses of the same access type cannot be merged if they have different debug information as they will not refer to the same instruction and will not be fixed in the same way. The merging algorithm goes through all the accesses created by the fragmentation algorithm and merges them if possible. Figure 7 illustrates an execution of the merging algorithm. In this figure, since the intervals of Type B are adjacent and are all of the same Type, meaning that they have the same access type and debug information, they can be merged. Figure 8b has multiple adjacent memory accesses : 5,002 memory accesses are made in this program. RMA-Analyzer creates a number of nodes in the BST that is linear in the number of iterations (1,000 in this case). Thus, the BST created by RMA-Analyzer has 5,002 nodes, each iteration adding five new nodes to the BST -variable 𝑖 is read or written 4 times and 𝑏𝑢 𝑓 is read once. The remaining two nodes are the first 𝐿𝑂𝐶𝐴𝐿_𝑅𝐸𝐴𝐷 from i=0 and the last 𝑅𝑀𝐴_𝑊 𝑅𝐼𝑇 𝐸 from the MPI_Get(buf[0],1,X) instruction. The merging algorithm merges all the nodes induced by the MPI_Get communications into only one node in the BST. Indeed, these nodes are accessing adjacent memory spaces and have the same access type and debug information since they are called at the same line in the code. Thus, the merging algorithm updates the BST which is of size two: one node for the variable 𝑖 and one node for all the MPI_Get accesses.

Regarding the complexity of the algorithm, since the most computationally intensive operations used in the new insertion algorithm are searches, insertions and deletions which are logarithmic in time as we use a (balanced) BST, our new insertion algorithm is also logarithmic in time.

EXPERIMENTAL RESULTS

This section first presents an overview of the RMA-Analyzer framework in which our approach has been implemented (Subsection 5.1). Then, Subsection 5.2 presents validation results on a microbenchmark suite we developed. In Subsection 5.3, we compare the overhead induced by our approach with MUST-RMA on two "real-life" applications: MiniVite and CFD-Proxy.

Our experiments were performed on an Eviden cluster that belongs to the Eviden R&D department, located at Echirolles, France. Each node has 2 x AMD rome 24 core (AMD EPYC 7402) with 128GB of RAM. The nodes are connected using the InfiniBand HDR interconnection. All the nodes have an RHEL 8.5 system. Our software stack is built with LLVM-15 and we used an Eviden OpenMPI implementation of MPI, built in its 4.1.5.2 version. For the experiments, we use RMA-Analyzer integrated in PARCOACH [7], a tool dedicated to the detection of deadlocks caused by an improper use of collective calls in parallel applications. The performance analysis is done with MUST-RMA v1.9.0, available on github 1 .

Implementation Details

Our contribution has been implemented in RMA-Analyzer, recently integrated in the tool PARCOACH, based on the LLVM compiler. An overview of the RMA-Analyzer framework is presented in [START_REF] Tassadit | Dynamic Data Race Detection for MPI-RMA Programs[END_REF]. It first collects all memory accesses that are contained within each epoch. To do so, relevant instructions such as memory accesses within an epoch, window creation and destruction (MPI_Win_create/ MPI_Win_free), epoch creation and destruction (MPI_Win_lock_all/ MPI_Win_unlock_all), and synchronization (MPI_Win_flush_all) functions are instrumented during the compilation phase. The instrumentation of the MPI functions are made using the PMPI (Profiling for MPI) interface. The LLVM alias analysis is used to reduce the number of Load/Store instrumentations.

During the execution, the MPI processes store each instrumented memory access in a BST and check for data races. If a data race is detected, RMA-Analyzer stops the execution of the program and reports the error with precise debug information (i.e., line of the conflicting instructions in the source code). More specifically, each time a remote access is initiated by a process (with MPI_Put or MPI_Get), an MPI_Send is called by RMA-Analyzer to inform the target process about this remote access. For each window, a thread is created to receive all the MPI_Send. At the end of the epoch (e.g., MPI_Win_unlock_all), all processes call MPI_Reduce in order to compute the number of remote accesses issued during the epoch towards its window and wait for the pending communications. The BST is implemented using the multiset containers provided by the C++ standard.

Method Validation

We developed a microbenchmark suite containing small programs written in C with correct and incorrect uses of MPI-RMA communications. This suite contains every combination of two one-sided operations by varying the order of the operations, the callers of the operations, and the location that will be accessed twice. The suite contains 154 codes in total and is composed of 47 codes containing a data race and 107 safe codes. Table 2 shows four programs from the suite. The names of the codes correspond to the combination used. For instance, the data race shown in Figure 2a is represented by the program named 𝑙𝑙_𝑔𝑒𝑡_𝑙𝑜𝑎𝑑_𝑜𝑢𝑡𝑤𝑖𝑛𝑑𝑜𝑤_𝑜𝑟𝑖𝑔𝑖𝑛_𝑟𝑎𝑐𝑒, in which an MPI_Get operation followed by a Load operation are issued by the same process (𝑙𝑙 = 𝑙𝑜𝑐𝑎𝑙𝑙𝑜𝑐𝑎𝑙), and both operations are accessing the same address space at 𝑜𝑟𝑖𝑔𝑖𝑛 side, outside its window (𝑜𝑟𝑖𝑔𝑖𝑛_𝑜𝑢𝑡𝑤𝑖𝑛𝑑𝑜𝑤). As shown in the table, all the tools are getting the right result for 𝑙𝑙_𝑔𝑒𝑡_𝑙𝑜𝑎𝑑_𝑜𝑢𝑡𝑤𝑖𝑛𝑑𝑜𝑤_𝑜𝑟𝑖𝑔𝑖𝑛_𝑟𝑎𝑐𝑒 and 𝑙𝑙_𝑔𝑒𝑡_𝑔𝑒𝑡_𝑖𝑛𝑤𝑖𝑛𝑑𝑜𝑤_𝑜𝑟𝑖𝑔𝑖𝑛_𝑠𝑎𝑓 𝑒. However, a false negative is produced by MUST-RMA for 𝑙𝑙_𝑔𝑒𝑡_𝑙𝑜𝑎𝑑_𝑖𝑛𝑤𝑖𝑛𝑑𝑜𝑤_𝑜𝑟𝑖𝑔𝑖𝑛_𝑟𝑎𝑐𝑒. This is because ThreadSanitizer does not instrument stack arrays. Thus, MUST-RMA cannot consider them in its analysis. When using heap arrays, the error is detected by MUST-RMA. For 𝑙𝑙_𝑙𝑜𝑎𝑑_𝑔𝑒𝑡_𝑖𝑛𝑤𝑖𝑛𝑑𝑜𝑤_𝑜𝑟𝑖𝑔𝑖𝑛_𝑠𝑎𝑓 𝑒, a data race is detected by RMA-Analyzer because the implementation does not consider the order of instructions within a process. When a process makes a local 1 https://github.com/RWTH-HPC/must-rma-correctness22-supplemental memory access and then calls an RMA operation, no data race can occur. On the contrary, if the process first calls an RMA operation and then makes a local memory access, a data race can occur if both operations are accessing the same address and one of them is a write operation. Thus, RMA-Analyzer does not distinguish cases such as Load-MPI_Get and MPI_Get-Load and raises an error even though the first case does not contain any error. We fixed this problem in our errors detection algorithm.

Table 3 summarizes the results obtained with RMA-Analyzer, MUST-RMA and our contribution on our microbenchmark suite. All the false positives detected by RMA-Analyzer and the false negatives produced by MUST-RMA are caused by a lack of analysis precision and ThreadSanitizer instrumentation. Our contribution outperforms these tools with no false positive and no false negative.

To ensure our method is able to detect errors in bigger codes, we manually inserted a data race in MiniVite. We duplicated an MPI_Put operation as shown in Figure 9a. Then a data race may occur at target side since both MPI_Put operations are writting in the same address space of the target process. Both RMA-Analyzer and our contribution detect the data race. The output returned by our contribution is depicted in Figure 9b. The error message includes debug information like the name of the file and the line in the source code where the data race is detected. $ mpiexec -n2./miniVitel -n100

for ( int i = 0; i < num_comm_procs ; i ++) { int target_rank = comm_proc [i ]; MPI_Put( scdata . data () + comm_proc_buf_disp [i], ssizes [ target_rank ], MPI_GRAPH_TYPE , target_rank, disp[target_rank] , ssizes [ target_rank ], MPI_GRAPH_TYPE , commwin ) MPI_Put( scdata
Error when inserting memory access of type RMA_WRITE from file ./ dspl . hpp :614 with already inserted interval of type RMA_WRITE from file ./ dspl . hpp :612. The program will be exiting now with MPI_Abort . Error when inserting memory access of type RMA_WRITE from file ./ dspl . hpp :614 with already inserted interval of type RMA_WRITE from file ./ dspl . hpp :612. The program will be exiting now with MPI_Abort .

(b) Report returned by our contribution on code 3.

Figure 9: Data race manually inserted in MiniVite and the output we returned to the developers.

Performance Analysis

In order to evaluate the overhead at runtime induced by the different approaches, we produced a performance analysis on two "real-life" applications: CFD-Proxy [START_REF] Simmendinger | PGAS Community Benchmarks CFD-Proxy version 1.0.1[END_REF] and MiniVite [4]. As MPI-RMA programs are challenging to write, few MPI-RMA applications are available. CFD-Proxy is a proxy-application for computational fluid dynamics. The application uses the Passive Target synchronization mode -using MPI_Win_lock_all and MPI_Win_unlock_all to create an epoch -and contains two windows per MPI process, and two epochs are called in the program: one per window. Figure 10 presents a comparison of the cumulative time spent in the epochs for CFD-Proxy. This experiment has been executed on one node with 12 MPI processes. As illustrated, the overhead is greatly reduced with our contribution compared to RMA-Analyzer. In fact, thanks to the merging algorithm, the number of nodes in the BST is drastically reduced from a BST of size 90,004 with RMA-Analyzer to a BST of size 54 with our method (i.e a reduction of 99.94% of the number of nodes in the BST). This reduction of nodes is possible since the window allocated by a process is actually divided into the number of processes so all the other processes have a dedicated space in the window they can access. Thus, every remote accesses made by a process access the same address space: these accesses can then be merged into one node in the BST. As the execution time of the insertion and deletion operations in the BST depends on the size of the BST, a reduction of its size would also reduce the execution time of the analysis. MUST-RMA has a significant slowdown as ThreadSanitizer instruments all memory accesses in the program while RMA-Analyzer and our contribution both use an alias analysis to filter out useless memory accesses for the error detection. Nonetheless, for our experiments, we only consider the execution time of the epoch. MiniVite is a proxy-application that implements a single phase of Louvain method in distributed memory for graph community detection. The application uses the Passive Target synchronization mode, and contains only one epoch. Figure 11 presents a comparison of the time spent in the epoch when running MiniVite from 32 on 2 nodes to 256 processes on 16 nodes When comparing RMA-Analyzer with our contribution, the performance is substantially the same since our contribution does not merge many nodes in the BST. As shown in Table 4, the number of nodes in the BST is reduced by less than 4%. The cost induced by the new insertion algorithm is thus not compensated by the reduction of time spent in the operations on the BST (insertion, remove and search). This lowlevel of node merging is caused by memory accesses on attributes of adjacent objects. However, the memory space of these attributes are not adjacent to one another. Thus, the nodes associated to these attributes memory accesses cannot be merged. It should be mentioned that when the number of processes is increased, the number of communications between processes increases while the workload per process is reduced. Thus, the overlap between communications and computations is less important with more processes. This explains why the gains in the execution time is not really visible between 128 and 256 processes even for the baseline execution. Thus, when increasing the problem size as in Figure 12 which shows the execution time with 1,280,000 vertices as input, a gain can be seen when running with 256 processes compared to a run with 128 processes because each process has more work to do and a better communication/computation overlap is possible. Nevertheless, the overhead implied by MUST-RMA is even more important when running on a large number of processes. Indeed, in addition to the slowdown introduced by the over-instrumentation of ThreadSanitizer as for CFD-Proxy, when the number of processes greatly increases, the size of the vector clock that is sent to other processes also increases. Thus, sending larger messages also adds overhead at runtime.

DISCUSSION

In this section, we first discuss some synchronization solutions within an epoch in order to avoid a data race by using the MPI_Barrier and MPI_Win_flush_all functions (1). We then discuss the problem of MPI_Win_flush instrumentation in RMA-Analyzer [START_REF] Chen | MC-Checker: Detecting Memory Consistency Errors in MPI One-Sided Applications[END_REF]. Finally, we address the lack of performance with MiniVite due to non-adjacent accesses (3).

(1). According to the MPI standard, an MPI_Barrier which waits for all the processes of the communicator actually does not terminate the communications. However, in practice, MPI_Barrier terminates communications in most MPI implementations. In our approach, we decided to meet the standard. Using only MPI_Win_-flush_all instead does not fix the problem since it does not ensure all processes have finished the communications. The solution consisting in calling MPI_Barrier after MPI_Win_flush_all should be preferred to synchronize the communications within an epoch.

(2). MPI_Win_flush is a synchronization function within an epoch that "completes all outstanding RMA operations initiated by the calling process to the target rank on the specified window. The operations are completed both at the origin and at the target" [START_REF]Message Passing Interface Forum[END_REF]. However, instrumenting MPI_Win_flush is not trivial since the latter only guarantees the ordering of communications of the calling process. Thus, the target of MPI_Win_flush is not aware of the synchronization call issued by the origin process and cannot know in which order remote accesses from several other processes toward its window will complete. That is why simply cleaning the BST of the process calling MPI_Win_flush may lead to false negatives. For instance, a false positive was detected by RMA-Analyzer and MUST-RMA when running them on CFD-Proxy because MPI_Win_flush is not well instrumented by the tools. As a consequence, we cannot support this synchronization function yet. We let for a future work a more in-depth study on the handling of the instrumentation of MPI_Win_flush.

(3). As described in Section 5.3, when runnning on MiniVite, there is no significant gain using our contribution compared to RMA-Analyzer because memory accesses are not adjacent. Some studies address this problem by using polyhedra to abstract memory regions and thus enable the compression of memory accesses even if they are not adjacent (e.g., [START_REF] Ketterlin | Prediction and trace compression of data access addresses through nested loop recognition[END_REF]). We thus assume that using these concepts, the merging algorithm can be extended to non-adjacent accesses when we can ensure that no accesses will be done between the accesses.

CONCLUSION

This paper proposes a better data race detection technique for the tool RMA-Analyzer [START_REF] Tassadit | Dynamic Data Race Detection for MPI-RMA Programs[END_REF]. We present a way to better manage the BST that is used to store all the accesses to the address space of an MPI process. We introduced a new insertion algorithm so the resulting BST detects more errors than the previous version of RMA-Analyzer. For future works, we plan to enhance the static analysis proposed by Saillard et al. [START_REF] Saillard | Static Local Concurrency Errors Detection in MPI-RMA Programs[END_REF] to detect more errors at compile time. We also plan to combine this static analysis to RMA-Analyzer in order to reduce the overhead at runtime.

1 )Figure 1 :

 11 Figure 1: Possible memory access operations within an epoch.

  Data race at the origin process. The asynchronism of the MPI_Get operation makes the value of buf unpredictable. Data race occurring at origin and target sides. Dark grey hatching (resp. light grey) represents the remote read initiated by P1 (resp. P0) that stores the value of Y in X (resp. X in Y).

Figure 2 :

 2 Figure 2: Examples of a data race within a process (left) and between processes (right).

Figure 3 :

 3 Figure 3: Data races situations with 3 processes. Rows represent the first operation, issued by a process origin 1 (O1). Columns represent another operation issued either by the same process (ORIGIN 1) or another process (TARGET or ORIGIN 2).

Figure 4 :

 4 Figure 4: Illustration of the insertion of a new memory access in the BST.

Figure 5 :Figure 6 :Figure 7 :

 567 Figure 5: BST of code 1 with RMA-Analyzer (left) and using our fragmentation algorithm (right). A node contains the memory interval and the access type. Bold nodes are conflicting.

Figure 8 :

 8 Figure 8: Examples of codes to illustrate the fragmentation and merging algorithms.

  . data () + comm_proc_buf_disp [i], ssizes [ target_rank ], MPI_GRAPH_TYPE , target_rank, disp[target_rank] , ssizes [ target_rank ], MPI_GRAPH_TYPE , commwin ) } (a) Code 3, extracted from MiniVite.

Figure 10 :

 10 Figure 10: Cumulative time spent in the epochs in CFD-Proxy for each method (1 node, 12 ranks, 50 Iterations).

Figure 11 :

 11 Figure 11: Execution-time of MiniVite with 32 to 256 MPI processes (2 to 16 nodes) and 640,000 vertices as input.

Figure 12 :

 12 Figure 12: Execution-time of MiniVite with 32 to 256 MPI processes (2 to 16 nodes) and 1,280,000 vertices as input.

  . In the figure, a node in the BST is noted (𝑚𝑒𝑚𝑜𝑟𝑦𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙, 𝑎𝑐𝑐𝑒𝑠𝑠𝑡𝑦𝑝𝑒) and instructions are given in top left. In this example, the node corresponding to the MPI_Put, ([2...12], 𝑅𝑀𝐴_𝑅𝑒𝑎𝑑) is inserted in the left subtree of the node ([4], 𝐿𝑜𝑐𝑎𝑙_𝑅𝑒𝑎𝑑). Then, when searching for the intersecting accesses with new access ([7], 𝐿𝑜𝑐𝑎𝑙_𝑊 𝑟𝑖𝑡𝑒), RMA-Analyzer does not notice the intersection between this new access and ([2...12], 𝑅𝑀𝐴_𝑅𝑒𝑎𝑑). Finally, the new access is inserted in the BST in the right subtree of ([4], 𝐿𝑜𝑐𝑎𝑙_𝑅𝑒𝑎𝑑), without noticing the data race with ([2...12], 𝑅𝑀𝐴_𝑅𝑒𝑎𝑑). This false negative is a consequence of the fact that ([2...12], 𝑅𝑀𝐴_𝑅𝑒𝑎𝑑) and ([4], 𝐿𝑜𝑐𝑎𝑙_𝑅𝑒𝑎𝑑) are not disjoint. To make the accesses disjoint in the BST, we propose a fragmentation algorithm that is called each time a new access is inserted in the BST. The algorithm works as follow: when a new access 𝑛𝑒𝑤_𝑎𝑐𝑐 is inserted in the BST, if this 𝑛𝑒𝑤_𝑎𝑐𝑐 is intersecting with another access 𝐵𝑆𝑇 _𝑎𝑐𝑐 already present in the BST, three new accesses representing the parts of the intersection of the two intervals are created and inserted in the BST if they are not empty. The first access represents the leftmost subsection, that is not part of the intersection between 𝑛𝑒𝑤_𝑎𝑐𝑐 and 𝐵𝑆𝑇 _𝑎𝑐𝑐. The second access represents the intersection of 𝑛𝑒𝑤_𝑎𝑐𝑐 and 𝐵𝑆𝑇 _𝑎𝑐𝑐. Given the access type of 𝑛𝑒𝑤_𝑎𝑐𝑐 and 𝐵𝑆𝑇 _𝑎𝑐𝑐, the access type and the debug information of the resulting access 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛_𝑓 𝑟𝑎𝑔 is given in Table1. Basically, RMA accesses prevail on local accesses and 𝑊 𝑅𝐼𝑇 𝐸 accesses prevail on 𝑅𝐸𝐴𝐷 accesses. Therefore, if both accesses have the same access type, the debug information of the most recent access is kept. In the table, red cells represent cases

Table 2 :

 2 Comparison of RMA-Analyzer, MUST-RMA, and our contribution feedback on four codes from our microbenchmark suite (✓: error detected, x: no error found).

					RMA-Analyzer MUST-RMA Our Contribution
				ll_get_load_outwindow_origin_race	✓	✓	✓
				ll_get_get_inwindow_origin_safe	x	x	x
				ll_get_load_inwindow_origin_race	✓	x	✓
				ll_load_get_inwindow_origin_safe	✓	x	x
	Table 3: Number of False Positive (FP), False Negative (FN),
	True Positive (TP) and True Negative (TN) reported by RMA-
	Analyzer, MUST-RMA, and our contribution on our mi-
	crobenchmark suite.
			RMA-Analyzer MUST-RMA Our Contribution
	FP		6	0	0
	FN	0	15	0
	TP	41	32	47
	TN	107	107	107
		2.0
	Time spent in epoch (s)	1.0 1.5
		0.5
		0.0
			Baseline	RMA-Analyzer	MUST-RMA	Our Contribution

Table 4 :

 4 Number of Nodes in the BST for MiniVite for 32 to 256 MPI processes.

		RMA-Analyzer (640,000/1,280,000) Our Contribution (640,000/1,280,000) Reduction of Nodes
	32	88,528/177,223	88,493/176,916	0.04%/0.17%
	64	48,180/97,347	47,913/97,020	0.55%/0.34%
	128	26,383/52,105	25,916/51,672	1.77%/0.83%
	256	15,544/29,129	14,566/28,127	6.29%/3.44%

SC-W 2023, November 12-17, 2023, Denver, CO, USA Vinayagame et al.


Jeoffrey Dehez

Sandrine Lyser

Perceptions of the risk of COVID-19 infection outdoors: a comparative study of green and blue spaces

Introduction

For many people who have experienced lockdowns and restrictions on indoor activities during the COVID-19 pandemic, outdoor recreation has provided one of the few opportunities to get out of the house, exercise and relax. On May 27th 2020, the European Commission announced its long-term recovery strategy (NextGenerationEU), which will guide and support EU member states as they recover from the impacts of the pandemic. The fundamental policy framework for the recovery is a combination of the EU Green Deal, a strengthening of the EU Single Market and adaptation to the digital age, and an inclusive recovery. One of the objectives of the programme NextGenerationEU is: "Working with nature to protect our planet and health". It is therefore important to understand how people think about and connect with natural environments in order to further increase the impact of this policy. While many studies have documented the changes in outdoor activities that have taken place during this pandemic, few have examined people's perceptions of the associated risks.

It is now widely recognised that risk perception has a major influence on an individual's behaviour when exposed to a hazard, whatever its nature (financial, professional, natural, etc.), and that this perception is dependent on multiple parameters [START_REF] Sjöberg | Risk Perception by the Public and by Experts: A Dilemna in Risk Management[END_REF]. The context in the first year of the COVID-19 pandemic proved to be a breeding ground for diverse representations and beliefs regarding the risk of infection when outdoors. The scientific knowledge base was still under construction during this period, the permitted activities (walking, relaxing, running) were generally not considered 'risky' (unlike other sports such as climbing or shooting), and the potential for infection by such viruses had, from a European perspective at least, only previously been associated with travel to 'exotic' destinations and certainly not with the familiar places frequented during lockdowns.

In addition, there has been an assumption that people have considered outdoor environments taken as a whole [START_REF] Landry | How Has the COVID-19 Pandemic Affected Outdoor Recreation in the U.S.? A Revealded Preference Approah[END_REF]) to be less dangerous than enclosed environments taken as a whole [START_REF] Kim | Perceived Crowding and Risk Perception According to Leisure Activity during COVID-19 Using Spatial Proximity[END_REF]. However, we believe this assumption to be too generalised as some previous works have shown that many people do not seem to perceive naturals environments in the same way [START_REF] Nutsford | Exposure to visible blue space (but not green space) associated with lower psychological distress in a capital city[END_REF].

In this presentation, we propose to examine this question by analysing recreational users' perceptions of the risk of COVID-19 infection in several natural environments in France.

Methods

Our study was conducted in late 2020 (December 7 th -15 th ) and comprised two telephone surveys carried out on a sample of 500 individuals living in the southwest of France each. A professional polling company conducted the surveys. Interviews lasted between 13 and 14 minutes on average. The surveys sought to develop a retrospective overview of 2020 by identifying the respondents' behaviours, attitudes and expectations concerning the options open to them with regard to spending time outdoors in local natural spaces. Each survey focused on a setting in particular, namely forests and Atlantic Ocean beaches. Questions cover activities, motives and trips conditions (time and place), among other things. The respondents were also asked to give their assessment of the risk of infection in each of these two target spaces relative to other destinations, in other words relative to the other target space (i.e. forest or beach) as well as to mountains, urban parks/green spaces and pedestrian zones. To test the possible influence of exposure on perception, the same questions were put to individuals who had and had not visited these places in 2020. Several Socio-demographics were finally included. The statistical analyses were based on Chi-square pairwise comparisons.

Results

Our findings confirm not everyone had the same access to nature during the pandemic. There are important socio-demographic differences between individuals who visited one type of natural environment, both types of natural environments or neither.

The results also showed differences in several variables that characterised the respondents' exposure to the two natural environments, namely times and frequencies of visits, means of travel, composition of groups on the outing and activities engaged in.

Motivations were very varied in both cases. For example, 'contact with nature' was more frequently cited in relation to forest visits, while the need to 'see people' was more frequently reported in relation to beach visits. Compared to the beaches, the restorative function of the forest seems to have received greater recognition therefore.

Finally, forest seemed to be viewed much more positively than the beach in terms of risk perception (Table 1). Neither residential environment (size of hometown, housing type, presence of a garden) nor individual characteristics (gender, age) was statistically significant in explaining these differences, but exposure (i.e. frequentation of one or more natural spaces) was found to be a significant factor.

Discussions and conclusion

Our findings suggest that people enjoying the outdoors make significant differences between potential benefits provided by "green" and "blue" spaces respectively. In the short term, planners can use these results to adapt the conditions of access to both type of natural spaces, favoring those for which there is the greatest demand. In the long term, promotion about the benefits of blue health should be enhanced further.

Although access to nature in France is mostly free, social inequalities persist. Under these conditions, it is necessary to target disadvantaged populations that accumulate inequalities. Generally speaking, it seems important to continue the research on this subject in order to better understand how individual representation and risk perceptions are constructed with respect to new risks and to anticipate associated behaviours.

As the number of studies on this subject grows, international comparisons should also be made given that, on the one hand, cultural specificities mean populations do not react in the same way to epidemics (de Zwart et al. 2009) and, on the other, different health measures (some more restrictive than others) are imposed on different populations.

Table 1 Perception of risks of infection associated with visiting forests and beaches

 1 3% of respondents answered that forests were less risky than urban parks/green spaces from the point of view of COVID-19 infection. Chi-square tests of independence significance levels: ns = not significant; * p < 0.10; ** p < 0.05; *** p < 0.01. Source and field: Frequentation of natural spaces in Aquitaine -Forest 501 respondents & Beach 500 respondents (INRAE 2020). Zwart O., Veldhuijzen I., Elam G., Aro A. R., Abraham T., Bishop G. D., Voeten H. A. C. M., Hendrik Richardus J., Brug J. (2009), Preceived Threat, Risk Perception, and Efficacy Beliefs Related to SARS and Other (Emerging) Infectious Diseases: Results of an International Survey, International Journal of Behavioral Medecine, 2009(16), p.30-40.

	Variable	Forest	Beach	p-value
	Urban parks and gardens			***
	Less risky	87.03	39.00
	Same	7.58	39.40
	More risky	0.80	11.80
	Don't know	4.59	9.80
	Pedestrian zones			***
	Less risky	91.62	57.20
	Same	3.59	23.80
	More risky	1.00	12.00
	Don't know	3.79	7.00
	Forests
	Less risky	NA	18.60
	Same	NA	37.60
	More risky	NA	38.20
	Don't know	NA	5.60
	Beaches
	Less risky	65.27	NA
	Same	24.15	NA
	More risky	2.59	NA
	Don't know	7.98	NA
	Mountains			***
	Less risky	29.14	17.00
	Same	56.09	44.20
	More risky	5.79	30.60
	Don't know	8.98	8.20
	Place of work			***
	Less risky	47.70	33.00
	Same	8.18	13.60
	More risky	0.60	12.80
	Don't know	43.51	40.60
	Note: 87.Selected references

de

Corresponding author : jeoffrey.dehez@inrae.fr


Roland Faure

Jean-François Flot

Dominique Lavenier

HairSplitter: separating strains in metagenome assemblies with long reads

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Jeoffrey Dehez

Sandrine Lyser

The influence of the recreational quality of beaches on risky swimming behaviours

Keywords: beach safety, swimming behaviour, drownings, recreational quality

Introduction

Drowning is the third leading cause of accidental death globally [START_REF] Who | Global Report on Drowning. Preventing a leading killer[END_REF]. Many of these incidents occur in natural environments such as oceans, lakes and rivers, each of which presents its own specific hazards. These drowning fatalities very often occur in the outdoor recreation context.

Preventive measures are put in place to try to combat this problem, including safe bathing areas, which are generally demarcated with flags and patrolled by trained lifeguards. This measure is unfortunately not systematically applied because it requires human, material and financial resources, which are not always available (WHO 2014). Where it is in place, it significantly reduces the risk of incidents (Surf Life Saving Australia 2019). Nevertheless, individuals can still very often be observed swimming outside the patrolled areas [START_REF] Hamilton | Targets to promote swimming between the flags among Australian beachgoers[END_REF]). If we are to increase the effectiveness of these patrol measures, we need to better understand the reasons behind individual 'avoidance' behaviours.

Studies in this area have often highlighted the determining role of knowledge and risk perception, with people frequently underestimating the risks and then unwittingly swimming in dangerous conditions. Age and gender have also been shown to be significant factors, linked to the fact that young males are a priori more attracted than any other group to 'risky' activities [START_REF] Moran | Young) Men behaving badly: Dangerous masculinities and the risk of drowning in aquatic leisure activities[END_REF]. Some studies have approached the subject from a geographical angle. For example, [START_REF] Williamson | A comparison of attitudes and knowledge of beach safety in Australia for beachgoers rural residents and international tourists[END_REF] showed that in Sydney, Australia, coastal area inhabitants swam in patrolled zones less often than visitors from inland areas. Finally, [START_REF] Hamilton | Targets to promote swimming between the flags among Australian beachgoers[END_REF] shed new light in this context by identifying the role of behavioural beliefs, such as the feeling of being constrained in one's choices (if forced to swim in the patrolled areas) and the presence of attractions (such as good wave quality) outside the safe bathing areas (thus inviting one to venture beyond them). This broadening of the reflection is interesting because it reminds us that swimming and other aquatic activities remain recreational activities engaged in first and foremost for pleasure and that any risk element is incidental.

In this study, we therefore analysed the influence of factors associated with the recreational quality of beaches (facilities provided, natural setting, activities available) on people's reported willingness to adopt risky behaviours.

Methods

The data were collected using a telephone survey conducted on a representative sample (n=500) of the population of the Aquitaine region in the southwest of France. The coastline in this area is prone to sea swells, which are generated by depressions crossing the North Atlantic and forming waves that are popular with surfers. However, underwater sandbanks created by a combination of sediment dynamics and marine energy produce dangerous rip currents [START_REF] Castelle | Spatial and temporal patterns of shoreline change of a 280-km high energy disrupted sandy coast from 1950 to 2014[END_REF]. Despite the presence of patrol measures along most of the coastline, there are still a high number of incidents (Castelle et al. 2018a). The survey addressed two types of risky behaviour, namely swimming outside of patrolled areas and swimming on wild beaches. In addition to sociodemographic questions, the survey contained questions on the types of activities the respondents engaged in, the features they considered attractive or unattractive and the information they looked up before going to the beach. The analysis was carried out in two stages. In the first step, we applied a dimension reduction method [START_REF] Chavent | ClustOfVar: An R Package for the Clustering of Variables[END_REF] to the recreational characteristics associated with a trip to the beach. In the second, we used logistic regression models to analyse the influence of these recreational characteristics on reported willingness to adopt risky behaviours.

Results

Our initial analyses have revealed that:

-48% of the respondents reported they had visited the beach in the past year (i.e. 2020). Of these, 51% said they had swum outside a patrolled area at least once, and 36% had gone to a wild beach.

-the primary reason cited by those who had never swum outside a patrolled area was that they liked to 'feel safe' (64%, n=76). The primary reason cited by those who had swum outside a patrolled area was that they had wanted 'to avoid the crowds' (68%, n=83). It appears that COVID-19 social distancing measures did not prompt individuals to move outside the patrolled areas any more than usual.

-the dimension reduction method identified eight synthetic variables that clearly segmented the recreational uses of the beach: 3 variables characterised the activity type, 4 the perceived quality and 1 the sources of information consulted before going to the beach. Among other things, the results showed differences between what characterized the land and sea parts of the beach as well as between the appeal of natural aspects versus facilities provided.

-several of these synthetic variables had a significant influence on the likelihood of avoiding patrolled areas. For example, active pursuits (surfing, fishing, other sports) were associated with venturing outside the areas, while fears related to wave size were associated with cautious behaviour.

-sociodemographic characteristics had relatively little influence. In particular, gender and geographical origin were not statistically significant factors.

Conclusions and Discussions

Managers needs to view the environment also from the perspective of the visitors. Beach lifeguards traditionally focus on avoiding costly liability claims, whilst visitors are seeking to have a positive outdoor experience.

Traditional awareness-raising policies based on age, gender and geographical origin seem to be ill-suited here. In our sample, the type of activity beachcombers are engaged in had a stronger influence on risky behaviours (i.e. going outside supervised areas) than sociodemographic.

Corresponding author : jeoffrey.dehez@inrae.fr


Matteo Mancinelli
email: matteo.mancinelli@univ-poitiers.fr

Eduardo Martini

Vincent Jaunet

Peter Jordan

Including acoustic modes in the

come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

The Vortex-Sheet (V-S) model is a simplified inviscid idealisation of a jet where an infinitely thin vortex sheet separates the interior flow from the outer quiescent fluid. The V-S model was first derived by Landau (1944) and has been largely used since then to: model the jet dynamics and study its stability properties [START_REF] Lessen | On the inviscid stability of the laminar mixing of two parallel streams of a compressible fluid[END_REF][START_REF] Michalke | A note on the spatial jet-lnstability of the compressible cylindrical vortex sheet[END_REF]; describe guided jet modes [START_REF] Tam | On the three families of instability waves of highspeed jets[END_REF][START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF] and the weak, forced resonance they can underpin in free, subsonic jets for jet Mach numbers 0.82 ≤ M j < 1 [START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF]; model resonance generated in subsonic and supersonic impinging jets [START_REF] Bogey | Feedback loop and upwind-propagating waves in ideally expanded supersonic impinging round jets[END_REF][START_REF] Tam | Theoretical model of discrete tone generation by impinging jets[END_REF], in a jet-flap interaction configuration [START_REF] Jordan | Jet-flap interaction tones[END_REF] as well as in free, supersonic screeching jets (Mancinelli et al., 2019a[START_REF] Mancinelli | A complex-valued resonance model for axisymmetric screech tones in supersonic jets[END_REF][START_REF] Mancinelli | Reflection coefficients and screech-tone prediction in supersonic jets[END_REF]. The spatial stability properties of a supersonic vortex sheet inside a circular duct in a co-flow configuration were studied as a function of flow and geometric parameters by [START_REF] Chang | Instability of a supersonic vortex sheet inside a circular duct[END_REF]. A double vortex-sheet model was used by [START_REF] Martini | Acoustic modes in jet and wake stability[END_REF] to further explore the dynamics of guided modes in jets/wakes and the role they play in the onset of absolute instability and their relation to supersonic unstable modes [START_REF] Tam | On the three families of instability waves of highspeed jets[END_REF]. The double V-S model was also used to study the effect of confinement on the spatio-temporal stability properties of planar and non-swirling round jets/wakes [START_REF] Juniper | The effect of confinement on the stability of two-dimensional shear flows[END_REF][START_REF] Juniper | The effect of confinement on the stability of non-swirling round jet/wake flows[END_REF], showing that confinement, under given conditions of density and velocity ratios between the flows and shear-layer thicknesses, may lead to absolute instability of the flow.

The stability properties of confined jets reproduce those of free jets for sufficiently large wall distances for waves whose radial support do not vanish slowly in the cross-stream direction [START_REF] Juniper | The full impulse response of two-dimensional jet/wake flows and implications for confinement[END_REF].

Its wide-spread use shows that the V-S model is very useful for the analysis and modelling of a variety of shear flows. However, where the free jet is concerned, the V-S eigenspectrum does not include free-stream acoustics. Given that free-stream acoustics play an important role in jet noise, e.g., in the dynamics of the jet underpinning the flapping modes in supersonic screech [START_REF] Edgington-Mitchell | Aeroacoustic resonance and self-excitation in screeching and impinging supersonic jets-a review[END_REF][START_REF] Shen | Three-dimensional numerical simulation of the jet screech phenomenon[END_REF] or resonance in impinging jets [START_REF] Edgington-Mitchell | Aeroacoustic resonance and self-excitation in screeching and impinging supersonic jets-a review[END_REF][START_REF] Jaunet | Dynamics of round jet impingement[END_REF], this absence of free-stream acoustics in the eigenbasis of the free jet is a considerable limitation.

The absence of a subset of eigensolutions in the stability problem was already reported by [START_REF] Gustavsson | Initial-value problem for boundary layer flows[END_REF] and [START_REF] Ashpis | The vibrating ribbon problem revisited[END_REF] in the case of boundary-layer flows, showing that waves not forming eigenmodes contribute to the solutions of the associated initial-value problem in the form of a continuous spectrum. This contribution comes out of the deformation of the integration contour in the vicinity of the branch cut in the complex plane. A similar behaviour was also reported by Case (1960) for an inviscid plane Couette flow for waves not satisfying the eigenvalue problem plus boundary conditions.

We here show that boundedness condition at infinity, typically imposed in free-jet problems, leads to the absence of free-stream acoustic waves in the eigensolutions of the unconfined V-S model. We then present a strategy that allows free-stream acoustics to be included in an eigen-description of the free jet by avoiding the bounded condition at infinity. Specifically, we propose a surrogate problem involving a distantly-confined jet. By distantly confined we mean that the boundary is located sufficiently far from the vortex sheet such that its influence on the system dynamics is negligible. We show that, besides discrete freestream acoustic eigenmodes, the discrete eigenmodes of the free jet are recovered. A model is thereby obtained in which both the usual free-jet eigenmodes are obtained along with a subspace associated with free-stream acoustics.

The paper is organised as follows. In §II we recall the vortex-sheet model usually adopted in the literature for free jets and present the derivation of the V-S dispersion relation for confined jets which includes free-stream acoustic eigenmodes. Results are presented in terms of eigenspectra, eigenfunctions and errors between the free-jet modes and their corresponding modes in the confined jet. Conclusions are finally discussed in §III.

II. VORTEX-SHEET MODEL

The jet is modelled as a cylindrical vortex sheet. All variables are normalised by the cylinder diameter D and ambient density and speed of sound, ρ ∞ and c ∞ , respectively. Assuming a parallel base flow q, the Reynolds decomposition,

q (x, r, θ, t) = q (r) + q ′ (x, r, θ, t) , (1)
is applied to decompose the flow-state vector q into its mean and fluctuating component q and q ′ , respectively. The compressible linearised Euler equations reduce to two convected wave equations for the pressure inside and outside the jet,

∂ ∂t + M a ∂ ∂x 2 -T ∇ 2 p ′ i = 0, (2a) ∂ 2 ∂t 2 -∇ 2 p ′ o = 0, ( 2b
)
where the subscripts i and o denote the interior and outer flows, respectively, T = T j /T ∞ is the jet-to-ambient temperature ratio and M a = U j /c ∞ the acoustic Mach number, with U j the jet velocity. The relation between the acoustic and jet Mach numbers is

M j = U j /c j = M a / √ T .
We assume the normal-mode ansatz,

q ′ (x, r, θ, t) = q (r) e i(kx+mθ-ωt) , (3)
where k is the streamwise wavenumber, m the azimuthal mode and ω = 2πStM a a nondimensional frequency, with the Strouhal number defined as St = f D/U j based on the nozzle diameter. Equations (2a) and (2b), written in cylindrical coordinates, reduce to the modified Bessel equation,

∂ 2 ∂r 2 + 1 r ∂ ∂r -γ 2 i,o - m 2 r 2 pi,o = 0, (4)
with

γ i = k 2 - 1 T (ω -M a k) 2 , ( 5a
)
γ o = √ k 2 -ω 2 , ( 5b
)
where the branch cut of the square roots in (5a) and ( 5b) is chosen such that -π/2 ≤ arg (γ i,o ) < π/2. The solutions of (4) in each domain are given by,

     pi (r) = A i I m (γ i r) + B i K m (γ i r) r ≤ 0.5, po (r) = A o I m (γ o r) + B o K m (γ o r) r > 0.5. ( 6a
) (6b)
These solutions have to be additionally constrained using boundary and matching conditions. Next, we investigate the dispersion relations obtained for free and confined jets and show that the imposition of wave-reflection condition at the wall in the confined case allows to describe free-stream acoustic eigenmodes.

A. Dispersion relation for free jets

We here recall the derivation of the dispersion relation for free jets. To enforce solution amplitudes to be bounded B i = 0 and A o = 0 are typically set. This is motivated by the fact that for γ i,o ∈ C K m and I m tend to infinity when r → 0 and r → ∞, respectively. The solutions in (6) become,

     pi (r) = A i I m (γ i r) r ≤ 0.5, po (r) = B o K m (γ o r) r > 0.5. ( 7a
) (7b)
Note that γ i and γ o play the role of radial wavenumbers in the inner and outer flows, respectively, and the functions I m and K m represent incoming and outgoing waves, respectively.

At the vortex sheet, continuity of pressure and vortex-sheet displacement,

         pi 1 2 = po 1 2 , ∂ pi ∂r 1 2 = (ω -kM a ) 2 T ω 2 ∂ po ∂r 1 2 , ( 8a
) (8b) leads to the dispersion relation D f (k, ω; M a , T, m) = 0: 1 1 -kMa ω 2 + 1 T I m γ i 2 γo 2 K m-1 γo 2 + mK m γo 2 K m γo 2 γ i 2 I m-1 γ i 2 -mI m γ i 2 = 0. ( 9
)
Frequency/wavenumber pairs k and ω define vortex-sheet eigenmodes for given values of azimuthal mode m and flow conditions M a and T . To find these modes, we specify either a real or complex frequency ω and compute the associated eigenvalue k which solves (9). The sign of the group velocity of each mode, which reveals whether a mode travels upstream or downstream, is obtained using the Briggs-Bers criterion [START_REF] Bers | Space-time evolution of plasma instabilities-absolute and convective[END_REF][START_REF] Briggs | Electron-stream interaction with plasmas[END_REF] and looking at the asymptotic behaviour of k (ω) for ω i → ∞ [START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF]. The wave is downstream-travelling if lim

ω i →∞ k i = +∞ (10)
and upstream-travelling if lim

ω i →∞ k i = -∞. (11)
Downstream-and upstream-travelling modes are hereafter indicated by a superscript + and -, respectively. To calculate the eigenvalues, we first do a coarse grid search for local minima of D f (k, ω; M a , T, m) in the k r -k i plane, that is later refined using a rootfinder algorithm based on the Levenberg-Marquardt method [START_REF] Levenberg | A method for the solution of certain non-linear problems in least squares[END_REF][START_REF] Marquardt | An algorithm for least-squares estimation of nonlinear parameters[END_REF] to find zeros of

D f (k, ω; M a , T, m).
In what follows we consider the following flow parameters: jet Mach number M j = 1.1, azimuthal mode m = 0, Strouhal number St = 0.68 and jet-to-ambient temperature ratio T ≈ 0.81. We show in appendices A and B that the trends and conclusions presented herein are independent of the jet Mach number and frequency considered, respectively. The absolute value of 1 for the flow conditions mentioned above.

D f (k, ω; M a , T, m) in the k r -k i plane for ω ∈ R is shown in figure

Free-jet eigensolutions

Figure 2 shows the eigenspectrum in the k r -k i plane for the flow parameters listed above.

Lines with phase speed equal to ±c ∞ are shown to identify modes that are supersonic with respect to the free stream. Eigenvalues are calculated for both real and complex values of ω. Distinct families of modes can be identified. The V-S model supports one convectively unstable mode, that is the Kelvin-Helmholtz (K-H) mode, which is hereafter denoted k + KH , and its complex conjugate. The vortex sheet also supports guided modes, which are hereafter denoted k p . These modes belong to a hierarchical family of modes identified by the azimuthal and radial orders m and n, respectively, and are propagative only in a well-defined St-number range which is delimited by a branch point and a saddle point [START_REF] Tam | On the three families of instability waves of highspeed jets[END_REF]. Following [START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF], these modes can be completely trapped inside the jet, experiencing it as a soft-walled duct, depending on the St and radial order n considered. [START_REF] Martini | Acoustic modes in jet and wake stability[END_REF] showed that this behaviour can be explained by the effective impedance of the VS, which for these conditions is close to zero, thus emulating a pressure-release condition. For the flow conditions we consider, we note: the propagative k - p with n = 2, which is characterised by a subsonic phase speed and lies immediately next to the sonic line; the evanescent k ± p for n = 1, which have supersonic phase speed (mode with |k i | > 0 is downstream-travelling, whereas the mode with |k i | < 0 is upstream-travelling); and the propagative k + p with n = 2, which lies on the real axis and has a subsonic phase speed. As outlined in §I, free-stream acoustic modes are not found in the eigenspectrum. We remind the reader that this does not imply that free-stream acoustic waves are not solutions of the unconfined vortex sheet. Blue ⋄ refer to k + KH and its complex conjugate, black △ to evanescent k ± p for n = 1, red • to propagative k - p for n = 2, green 2 to k + p for n = 2.

They are. And they come along as solutions in the impulse-response or initial-value problem in the form of a continuous spectrum. It is their presence in the eigenbasis description which appears to be missing.

Figure 3 shows an example of the pressure eigenfunctions, which are normalised in order to have unitary maximum amplitude. For the sake of brevity, we here report only the eigenfunctions of the K-H mode, the upstream-travelling guided modes of first and second radial orders and the downstream-travelling guided mode of second radial order. As expected, the K-H wave shows an eigenfunction with maximum amplitude at the vortex sheet and a fast radial decay. Concerning the guided jet modes, as outlined above, the k - p modes for n = 1 and 2 show an eigenfunction with a radial support both inside and outside the jet with a slow radial decay, whereas the k + p mode for n = 2 is almost completely trapped in the inner region for the flow conditions considered.

B. Dispersion relation for confined vortex sheet

Considering a confined jet with boundary located at r = r M AX , enforcing to the general solution in ( 6) bounded conditions at r = 0 and soft-wall boundary conditions at r = r M AX , i.e., p (r M AX ) = 0, we obtain,

B i = 0, (12a)
A o = -B o K m (γ o r M AX ) I m (γ o r M AX ) = -B o z (r M AX ) , (12b)
where for notational simplicity we define z (r

M AX ) = K m (γ o r M AX ) /I m (γ o r M AX ). The
Dirichlet boundary condition used at r = r M AX implies a pressure release condition, i.e., a soft-walled duct, and an in-phase wave reflection. The solution for the pressure in the inner and outer flow becomes, respectively, is found in the unconfined case. The contribution of the free acoustics to an impulse response or initial-value problem would appear in the form of a continuous spectrum due to the deformation of the contour integral around the branch cut [START_REF] Ashpis | The vibrating ribbon problem revisited[END_REF]Case, 1960;[START_REF] Gustavsson | Initial-value problem for boundary layer flows[END_REF]. On the contrary, for the confined case, the imposition of the wave-reflection condition at infinity appears to remove the discontinuities along the real and imaginary axes where zeros of the dispersion relation associated with free-stream acoustic eigenmodes are found.

     pi (r) = A i I m (γ i r) r ≤ 0.5 po (r) = B o (-zI m (γ o r) + K m (γ o r)) r > 0.5. ( 13a

Confined-jet eigensolutions

Figure 6 shows the eigenspectrum obtained using the dispersion relation for the confined vortex sheet ( 14). We here consider a maximum radial distance r M AX = 100, the flow may thus be considered as distantly confined. Eigenvalues for both real and complex frequencies are represented. We note that, in addition to the modes observed in figure 2 using the unconfined vortex sheet, propagative and evanescent free-stream acoustic eigenmodes, that hereafter are denoted k ± a , are also present. We note that all the modes found using the dispersion relation ( 9) in unconfined conditions are reproduced by the dispersion relation for the distantly confined vortex sheet ( 14), as shown in figure 7 (for the sake of brevity we hereafter show only results for ω ∈ R). The distantly-confined problem, thus, accurately describes the dynamics of the free vortex sheet while also exhibiting an additional subspace associated with free-stream acoustics.

Figure 8 shows the eigenfunctions obtained using the pressure solution in ( 13) and the dispersion relation in ( 14) for the eigenvalue computation. For the sake of brevity, we only show eigenfunctions for the same modes considered in the unconfined case in figure 3 and additionally one of the propagative and evanescent k - a modes. We note that the eigenfunctions for the K-H and the guided jet modes are identical to those of figure 3 obtained using the unconfined vortex-sheet model. The eigenfunctions of the propagative and evanescent k - a modes show the expected shape for acoustic modes with a spatial support both inside and outside the jet and an algebraic amplitude decay for increasing r. 13) and the dispersion relation for confined jets in ( 14) to find the eigenvalues. Colour lines are the same used in figure 3 for free jets. Additionally, dash-dotted magenta line refers to propagative k - a mode, bold magenta line to evanescent k - a mode.

Asymptotic behaviour for large wall distances

We now study the effect of the domain size on the eigensolution computation in the limit of large wall distances. As outlined by [START_REF] Juniper | The effect of confinement on the stability of non-swirling round jet/wake flows[END_REF], the stability properties of a confined jet are equivalent to those of a free jet for waves whose spatial support decays rapidly in the radial direction if the boundary is located sufficiently far from the flow. In this work we explore values of r M AX → ∞ and thus the stability of the confined and unconfined systems are equivalent. Non-acoustic modes of the confined problem converge to similar solutions for the unconfined problem. We here quantify this convergence by looking at the discrepancy between eigensolutions computed in the unconfined case and eigensolutions computed in the distantly-confined case for different values of r M AX . Specifically, we select the K-H mode, k ± p modes for n = 2 and k - p mode for n = 1 and we compute the error between the eigenvalues and the eigenfunctions calculated in the free and distantly-confined cases for increasing values of r M AX . The error value for the eigenvalues and the eigenfunctions is defined as follows,

ϵ ev = |k f -k c |, (15a)
ϵ ef = 1 - ⟨p f , pc ⟩ ⟨p f , pf ⟩⟨p c , pc ⟩ , (15b)
where the subscripts f and c indicate solutions from the free-and confined-jet cases, respectively, and the inner product ⟨•, •⟩ represents an euclidean dot product. Figure 9 shows the trend of the eigensolution error between the confined and unconfined vortex sheet as a function of r M AX , which we let vary in the range [1,250]. For both eigenvalues and eigenfunctions the confined-jet solution matches the free-jet solution (up to machine precision) as r M AX becomes sufficiently large. We observe that the error trend is different depending on the mode considered. Specifically, the convergence trend is very fast for the K-H wave and the downstream-travelling guided mode of second radial order for which the error value reaches machine precision for r M AX = 5. We point out that the small error increase for the K-H eigenvalue observed for 40 ≤ r M AX ≤ 70 is most likely related to numerical inaccuracies in the root-finder algorithm when searching for zeros of the dispersion relation and, anyway, never exceeds the value of 10 -12 . On the contrary, the convergence speed is slower for the k - p mode with n = 1 and, above all, for the k - p mode with n = 2. For this mode the error value becomes lower than 10 -9 for r M AX ≥ 100, distance at which the confined-jet solution may be considered equal to the free-jet one. We point out that the different convergence rate with r M AX observed for the modes is consistent with their radial support shown in figure 8 and the conclusions reported by [START_REF] Juniper | The full impulse response of two-dimensional jet/wake flows and implications for confinement[END_REF]. Specifically, for modes with eigenfunctions concentrated either in the core or at the vortex sheet of the jet, such as the k + p mode with n = 2 and the k + KH wave, the confined-jet solution equals the free-jet one for low values of r M AX . On the contrary, modes with larger eigenfunction support outside the jet, such as the k - p modes with n = 1 and 2, require a larger r M AX value in order for the confined-jet solution to match the free-jet one.

To better understand the convergence trend reported above, we here analyse the behaviour of the above-defined parameter z (r

M AX ) = K m (γ o r M AX ) /I m (γ o r M AX ) when r M AX → ∞.
For this aim, we consider the zero-order asymptotic expansion at infinity of the Bessel functions I and K, which, following [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF], can be expressed as, lim

r M AX →∞ I m (γ o r M AX ) ≈ e γor M AX √ 2πγ o r M AX , ( 16a
) lim r M AX →∞ K m (γ o r M AX ) ≈ π 2γ o r M AX e -γor M AX , (16b)
such that z for r M AX → ∞ can be evaluated as follows,

z (r M AX → ∞) = lim r M AX →∞ K m (γ o r M AX ) I m (γ o r M AX ) ≈ πe -2γor M AX . ( 17
)
Assuming that the exponent is not purely imaginary, equation ( 17) shows that the convergence of the confined-jet solution to the free-jet one should have an exponential trend with the wall distance r M AX . This result is confirmed by the comparison of the errors between the eigensolutions obtained in the confined and unconfined cases with the exponential fit dictated by ( 17) in figure 9. The convergence rate of all the modes is well approximated by the exponential fit until the error reaches a plateau at machine precision.

As mentioned above, the limit in ( 17), and hence the convergence rate of the errors ϵ ev and ϵ ef , depend on whether γ o (5b) is complex or purely imaginary. Specifically, when γ o ∈ C the limit in ( 17 focus on the effect of the wall distance on the acoustic modes, we here plot a zoom on the free-stream supersonic region. A different discretisation of the continuous branch of the freestream acoustic modes is obtained for each wall distance. We point out that the grid search in the complex plane used to find zeros of the dispersion relation has to be increasingly refined to capture propagative acoustic waves close to the sonic line for increasing r M AX values. Despite this refinement, near-sonic, downstream-travelling, propagative acoustic eigenmodes are found only for few r M AX values and complex frequencies ω. This behaviour is ascribed to numerical issues when searching for zeros of the dispersion relation.

III. CONCLUSIONS

In this paper we dealt with the issue of describing the free-stream acoustic eigenmodes of a jet using a vortex-sheet model. Despite the widespread use of the vortex-sheet model for describing jet dynamics, the absence of eigenmodes associated with free-stream acoustics due to the imposition of bounded solution at infinity, seems to have been overlooked. To describe free-stream acoustics we propose a distantly-confined jet as a surrogate problem for a free jet. The surrogate problem asymptotically exhibits all of the modes supported by the free jet in addition to a discrete family of free-stream acoustic eigenmodes. When the wall is taken sufficiently far from the vortex sheet, the eigensolutions of the confined jet match those of the free jet, such that free-jet stability properties are fully recovered.

Specifically, the discrepancy between the eigensolutions of the confined-and free-jet cases decays exponentially with the wall distance. We show that the convergence rate of the which is consequently less influenced by the wall, and thus a faster convergence to the free-jet eigensolution.

  FIG. 1. (colour online) Absolute value of the free-jet dispersion relation D f (k, ω; M a , T, m) in the

  FIG. 2. (colour online) Eigenspectrum for M j = 1.1, T ≈ 0.81, m = 0 and St = 0.68 obtained

  FIG. 3. (colour online) Normalised pressure eigenfunctions obtained using the unconfined vortex

  match at the vortex-sheet location (8), we obtain the dispersion relation D c (k, ω; M a , T, m, r M AX ) = 0, which differs from the dispersion relation used in the unconfined vortex sheet by the additional terms in z (r M AX ). The absolute value of D c (k, ω; M a , T, m, r M AX ) for r M AX = 100 in the k r -k i plane for a real frequency ω is shown in figure 4. In addition to the minima already observed in figure 1 for the free-jet dispersion relation, a series of local minima on the real and imaginary axes are found within the free-stream supersonic region. The different behaviour of the dispersion relation between unconfined and confined cases is better highlighted in figure 5 which shows a zoomed portrait of the absolute value of D f and D c in the k r -k i plane around the origin. A branch cut inducing a discontinuity and a discontinuous derivative of the dispersion relation on the imaginary and real axes,

  FIG. 4.(colour online) Absolute value of the distantly-confined-jet dispersion relation

  FIG. 5. (colour online) Absolute value of the dispersion relations in the k r -k i plane around the

  FIG. 6. (colour online) Eigenspectrum for M j = 1.1, T ≈ 0.81, m = 0 and St = 0.68 obtained

  FIG. 9. (colour online) Trend of the eigensolution error between the confined-and free-jet cases

  figure 11, which shows the eigenspectra in the case of a confined jet for different values of

  FIG. 10. (colour online) Contour map of γ o (5b) in the k r -k i plane: (a) real part of γ o with

  FIG. 12. (colour online) Eigenspectra for M j = 0.91, T ≈ 0.85, St = 0.68 and m = 0 obtained

  FIG. 13. (colour online) Eigenspectra for M j = 1.1, T ≈ 0.81 and m = 0 in the obtained from

ACKNOWLEDGMENTS M.M. acknowledges the financial support of CNES (Centre National d'Etudes Spatiales) under a post-doctoral grant at the time of the realisation of the work. E.M. acknowledges the support of CEA-CESTA (Commissariat à l' Énergie Atomique -Centre d'Etudes Scientifiques et Techniques d'Aquitaine) under a post-doctoral grant.

APPENDIX A:

We here show that the efficiency of the distantly-confined vortex sheet in describing freestream acoustics along with eigensolutions of the unconfined vortex sheet is independent of the jet-flow condition considered. Figure 12 shows the eigenspectra of both unconfined and distantly-confined vortex-sheet models (r M AX = 100) for the same frequency considered above, i.e., St = 0.68, but for subsonic flow conditions, that is, M j = 0.91 and T ≈ 0.85. As outlined above for supersonic flow conditions, the eigenspectrum of the unconfined vortex sheet does not include free-stream acoustic modes, which, on the contrary, do appear in the distantly-confined case. We further note that all the free-jet modes are recovered in the eigenspectrum of the distantly-confined case.

APPENDIX B:

We here show that the capability of the distantly-confined vortex sheet to describe free unconfined eigensolutions is independent of the frequency considered. For this aim, we report in figure 13 the eigenspectra in the k r -k i and k r -St planes for the same supersonic flow conditions considered above obtained from both the unconfined and distantly-confined vortexsheet models. For the sake of brevity, we only show the K-H mode and the downstream-and upstream-travelling guided modes of first and second radial orders and the Strouhal number is varied in the range [0.02, 0.8]. No differences between the eigenspectra obtained using the unconfined and the distantly-confined vortex-sheet models can be appreciated in figure 13.


Roland Faure
email: roland.faure@irisa.fr

Baptiste Hilaire

Dominique Lavenier

Mapping-friendly Sequence Reductions to process compressed genomic data

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