Issue |
A&A
Volume 504, Number 3, September IV 2009
|
|
---|---|---|
Page(s) | 1041 - 1055 | |
Section | The Sun | |
DOI | https://doi.org/10.1051/0004-6361/200811200 | |
Published online | 02 July 2009 |
Online Material
Appendix A: Model construction
A.1 Data structure
Each vertex in the model has a unique index N and six arrays assigned to it: a position array [X,Y], vertex age array, link array (containing indices of vertices connected to the vertex), two link-type arrays (one for each spatial dimension), determining the type of connections of vertex N to vertices in the link array, and a connection age array, containing the age of connections to vertices in the link array. The values in the link-type and connection age arrays describe connections with vertices whose index occupies a corresponding location in the link array. The values in the link-type array can be either 0, -1 or 1. Zero means that the connection is within the domain, while 1 is a cross-boundary connection directed from vertex Nthrough the right/upper boundary and -1 means a connection through the left/lower boundary. It follows that vertices on the opposite sides of a cross-boundary connection have opposite corresponding link-type values (1and -1).
A.2 Time evolution
Movement of vertices in the domain is restricted to motion towards one of the neighbouring vertices, that is along triangle sides. When a vertex crosses a domain boundary it reappears on the opposite side, with its link type arrays and link type arrays of vertices connected to it updated accordingly. We apply two kinds of motion schemes: random motion and cell-competition. Moreover, each scheme has two different versions of vertex movement: constant velocity and constant acceleration. We label them shortly CA, CV, RAand RV for cell competition constant acceleration, cell competition constant velocity, random walk constant acceleration and random walk constant velocity, respectively. The RV algorithm is as follows: for each vertex a neighbour is chosen randomly and the vertex is moved towards that neighbour by a random displacement



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Figure A.1: Illustration of the procedure preventing cell-flipping. See text for details. |
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A.3 Cell vanishing - vertex merging
In order to allow for vanishing of cells, we have to define a procedure and criteria for the merging of vertices. After a vertex has moved towards one of its neighbours, the distance between them is measured. If it is below a critical value


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Figure A.2: 'Pathological' case of a vertex merger. For details see text. |
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As the age of vertices and lanes (connections) in the model is used to detect
mesogranulation in a way similar to the cork method applied in observations
and simulations, it is important to define the age inheritance rules
properly. Corks are artificial particles which are advected passively on a
horizontal plane by the velocity field. They tend to accumulate in downflow
regions i.e. downflow lanes and vertices separating granules (Cattaneo et al.
2001; Rieutord et al. 2000; Roudier et al. 2003; Ploner et al. 2000). We
postulate a relation between the number of corks accumulated in a downflow
structure and its age: the older the structure the more corks it is likely to
attract. It follows that when two lanes or vertices merge, the corks
naturally remain in the merged lane or vertex. Hence, when two vertices Aand B merge (Fig. A.2), we keep the older age of the connections
A-E and B-E as the age of the new connection of the merged vertex
with E, and similarly for C. The age of the merged vertex is also the
older age of vertices A and B. The merged vertex inherits the neighbours
of both A and B (with link and age arrays updated accordingly). We do not
allow vertices to have multiple references to another vertex (a link array
has to have unique entries); therefore, the domain has to be bigger than
vertices. This also leads to errors in case of ``domain collapse'',
when merging of vertices produces a few giant cells and vertices from
opposing ends of domain become connected also through the domain. Such case
is neither interesting nor sought for, and with proper cell splitting rules
it never occurs.
A.4 Cell splitting - vertex appearance
Since the construction of the two-dimensional model allows for many different schemes for cell splitting, it is reasonable to investigate what differences those schemes produce, both in the granulation characteristics, as well as in the emerging mesogranulation features. Hence, we apply four different cell splitting schemes: critical cell side length, critical cell area, critical cell area plus the longest side, and random splitting. We label them L, A, AL and R, respectively. Thus, a cell-competition constant acceleration model with random splitting is labelled CA/R etc.A.4.1 Critical cell side length (L)
When a connection between two vertices A and B (see Fig. A.3) exceeds a critical length (an uniformly distributed random number between 2and 3.5, evaluated individually for each connection at each timestep; the box size is




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Figure A.3: Illustration of the splitting process, with new vertex X appearing between vertices A and B. Dashed lines are the new connections appearing. |
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A.4.2 Critical cell area (A)
In this scheme the cell is split when it exceeds a critical area value (a
uniformly distributed random number between 1.5 and 2.5; the box size is
). The splitting partner cell is chosen to be the neighbour with
the largest area. The rules of new vertex position and age inheritance of new
structures are like in the ``critical side length'' splitting case.
A.4.3 Critical cell area plus the longest side (AL)
The cell is split when it exceeds a critical area value, and the splitting occurs through the longest side of the cell (in Fig. A.3 the ``splitting connection'' A-B is chosen to be the longest side of the cell), regardless of the area of the neighbour sharing the side with the cell. The rules of new vertex position and age inheritance of new structures are like in the `critical side length' splitting case.
A.4.4 Random splitting (R)
In this scheme the splitting is not based on any cell characteristics. Therefore, to keep the number of cells present in the domain constant throughout the simulation, the number of the splitting events in each timestep is equal to the number of vanishing events that took place in this timestep. The cell that is split is chosen randomly (each cell having the same probability of being chosen), as well as the side through which the splitting occurs. The rules of new vertex position and age inheritance of new structures are like in the ``critical side length'' splitting case.
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