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DPD Simulation of a Membrane

Description: This project will work toward the simulation of a cell membrane using a type of

molecular dynamics known as Dissipative Particle Dynamics (DPD). In the first week, we will

implement a basic molecular dynamics simulator in C++, probably in 2D, to learn about the

inner workings of particle-based simulators. In the second week, we will learn about the DPD

method and how to implement it in LAMMPS, a DOE-funded software package for molecular

dynamics. We will use LAMMPS to simulate a cell membrane and get some nice 3D visuals.

Time permitting, we can explore some more involved things like how the membrane moves

when subject to a force from polymerization of a rigid polymer network, or how it flexes if the

membrane is pinned at certain points. You can see what these sorts of simulations look like in

this YouTube video.

Prerequisites: While we will make our basic MD simulator in C++, basic programming back-

ground in any language and the ability to Google things are probably su�cient. LAMMPS uses

its own scripting language which is very easy to learn and no prior background with LAMMPS

is assumed. We will not be focusing on the high performance computing aspects like how to

optimize and parallelize code (LAMMPS is already highly optimized), so this project is suited

to beginner or intermediate programmers who are more interested in learning how to approach

the simulation of some biophysical problems.

1

Density Functional Theory Calculations of Molecular

Orbital Evolution During Chemical Reactions

Description: In this project we will focus on the calculation and visualization of the trans-

formation of molecular orbitals following chemical reaction paths. A brief introduction will

be given on background knowledge in chemistry, followed by hands-on tutorials on using the

Atomic Simulation Environment (ASE) Python package to perform the calculations. Methods

that use ensemble parallelization will then be explored to acceleration the turnaround time and

throughput of the calculation.

Prerequisites: There are no formal prerequistes. Python proficiency is a plus but not required.

1

been observed but its existence has been speculated [18].This speculation is supported by the recent discovery oftwo localized exact solutions in PCF by Schneider, Marinc,and Eckhardt [19] which qualitatively resemble localizedstates in the SHE.

The aim of this Letter is to elucidate the origin of theselocalized solutions in PCF.We show that the Navier-Stokesequations in this geometry indeed exhibit homoclinic snak-ing, giving rise to localized counterparts of well-knownspatially periodic equilibria.

In PCF the velocity field uðx; tÞ ¼ ½u; v; w%ðx; y; z; tÞevolves under the incompressible Navier-Stokes equations,

@u

@tþ u 'ru ¼ (rpþ 1

Rer2u; r ' u ¼ 0; (1)

in the domain ! ¼ Lx ) Ly ) Lz where x, y, z are thestreamwise, wall-normal, and spanwise directions, respec-tively. The boundary conditions are periodic in x and z andno-slip at the walls, uðy ¼ *1Þ ¼ *x. The Reynoldsnumber is Re ¼ Uh=!, where U is half the relative veloc-ity of the walls, h half the wall separation, and ! thekinematic viscosity. We treat Re as the control parameterand use as a solution measure the dissipation rate D ¼ðLxLyLzÞ(1

R!ðjr) uj2Þd!. The laminar profile has

D ¼ 1 while solutions such as those shown in Fig. 1have D> 1.

Figure 1 shows two exact solutions of (1) at Re ¼ 400and ! ¼ 4") 2) 16", originally identified in [19] for! ¼ 4") 2) 8". The solutions are localized in the

spanwise z direction and consist of two to three promi-nent pairs of alternating wavy roll-streak structures em-bedded in a laminar background flow. Figures 1(a) and 1(b)are a traveling-wave solution uTW of (1) satisfying½u; v; w%ðx; y; z; tÞ ¼ ½u; v; w%ðx( cxt; y; z; 0Þ, where cx ¼0:028 is the streamwise wave speed. Figures 1(c) and 1(d)are a stationary, time-independent solution uEQ. Theequilibrium uEQ is symmetric under inversion½u; v; w%ðx; y; z; tÞ ¼ ½(u;(v;(w%ð(x;(y;(z; tÞ, andthe traveling-wave uTW has a shift-reflect symmetry,½u; v; w%ðx; y; z; tÞ ¼ ½u; v;(w%ðxþ Lx=2; y;(z; tÞ. Thesesymmetries ensure that neither uEQ nor uTW drifts in thelocalization direction z.To continue these solutions in Re, we combine a

Newton-Krylov hookstep algorithm [20] with quadraticextrapolation in pseudoarclength along the solutionbranch. The Navier-Stokes equations are discretized witha Fourier-Chebyshev-tau scheme in primitive variables and3rd-order semi-implicit backwards differentiation timestepping. Bifurcations along the solution branches arecharacterized by linearized eigenvalues computed withArnoldi iteration. The computations were performed with32) 33) 256 collocation points and 2=3-style dealiasing,resulting in approximately 2) 105 free variables, andvalidated by recomputing with ð3=2Þ3 more grid points ata number of locations along each solution curve [21].The bifurcation diagram in Fig. 2 shows the uTW and

uEQ solutions from Fig. 1 under continuation in Reynoldsnumber. As Re decreases below 180, the solution branches

(a)

x

(b) |

|

y

(c)

x

(d)|

|

y

z

FIG. 1 (color online). Localized traveling-wave uTW (a),(b)and equilibrium uEQ (c),(d) solutions of plane Couette flow atRe ¼ 400, from [19]. The velocity fields are shown in the y ¼ 0midplane in (a),(c), with arrows indicating in-plane velocity andthe color scale indicating streamwise velocity u: dark, light, dark(blue, green, red) correspond to u ¼ (1, 0, þ1. The x-averagedstreamwise velocity is shown in (b),(d), with y expanded by afactor of 3.

130 140 150 160 170 180 1901.2

1.4

1.6

1.8

2

2.2

Re

Du

P

uTW

uEQ

a

b

c

d

α

βγ

FIG. 2 (color online). Snaking of the localized uTW, uEQ

solutions of plane Couette flow in (Re, D) plane. The spatiallyperiodic Nagata solution uP is shown as well; the uTW solutionconnects with it near (131, 1.75). Velocity fields of the localizedsolutions at the saddle-node bifurcations labeled a; b; c; d areshown in Fig. 3. The rung branches are shown with solid linesconnecting the uEQ and uTW in the snaking region; velocityfields for the points marked #, $, % are shown in Fig. 4. Opendots on the uTW traveling-wave branch mark points at which thewave speed passes through zero.

PRL 104, 104501 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending

12 MARCH 2010

104501-2

Cellular Buckling in Long Structures 17

Figure 14. The bifurcation diagram of the homoclinic starting at P = 2, for nonlinearity f (u) = u−u3+3/10 u5.

There is a strong resemblance between this evolution and the form of the solutions thatare found along the bifurcation diagram uncovered in Section 2, for example Figure 5. InFigure 14 we draw a bifurcation diagram for the nonlinearity f2. Although the two figuresare similar in appearance, there is a significant difference. In Figure 5 both solution curvesconsist of even solutions; for the nonlinearity f2, with the additional symmetry u "→ −u,these two sets of solutions are identical (up to a reflection u "→ −u) and we draw them as onecurve in Figure 14 (continuous line). Because of the additional symmetry, there is also a newreversibility in the problem:

R : (u, u′′) → (−u,−u′′) and x → −x,

(compare with Equation (3)). This leads to a second curve of solutions, bifurcating fromP = 2, which are odd (broken line). Further numerical results have found that the bifurc-ation sequence for f2 is the qualitatively similar to that for f1 with the equivalent of the kinktransition at b = 2/9 corresponding to α = 3/16. The degenerate Hamiltonian Hopf whichoccurs for f1 at b = 38/27 has no analogue for f2 other than formally as α → ∞.We believe (but have as yet no proof) that the minimizers of Equation (16) all lie on the

bifurcation diagram in Figure 14. Every horizontal line in this figure intersects the diagramat least twice, and for large values of λ, by the sloping nature of the curves, more than twice.The oscillations in the graph appear to be centred about a mean value P which is close to theMaxwell load which will be described and computed in the next section. At every value ofλ there are therefore several candidates for the global minimizer. In the following section wefirst explore global minimization issues via a simplified caricature, before investigating moreclosely the global minimizer for the strut model and its relation to the above diagram.

4. Maxwell Criterion and Global Stability

Let us now turn to the question of which solutions under conditions of controlled end-shortening may be stable. For an environment rich with underlying disturbance, interest

- 0.04 - 0.02 0�

<10> front

<11> front

1

2

L2 n

orm

Figure 9: Bifurcation diagram for both the �10�- and �11�-hexagon fronts with A = 1, �2 = 0.1.

To describe di↵erent directions and interfaces on the hexagon lattice we use the Bravais-Miller index notation;

see [29]. On a hexagonal lattice there are two principal directions �10�- and �11�-directions, that are at �/3

radians apart.

Setting A = 1, �2 = 0.1, we show the bifurcation diagram for both the principal �10�- and �11�-hexagon fronts

in Figure 9. Here we observe the same type of snaking behaviour seen in the Swift-Hohenberg equation [29]

where the �10�-front (label (1) in Figure 9) snakes over a larger region of parameter space than the �10�-front

(label (2) in Figure 9). As one proceeds up the snake, entire rows of hexagon cells are added to both ends of

the interface. We also expect there to be almost hexagon fronts where single cells are grown along the edge

of the interface; see Lloyd et al. [29, Figure 21].

- 0.1 - 0.05 0

12

3

4

�

L2 n

orm

1

2

3

4

Figure 10: Hexagon �10� front isolas with A = 1, �2 = 0.04. The labelled solutions are for the solid isola branch.The

domain covering hexagon fold occurs at � = �0.092 while the radial spot fold occurs at � = �0.103 and lines up with

the folds of the �10� fronts.

As we decrease �, we find that the bifurcation diagram is made up of isolas of hexagon fronts that go beyond

the saddle-node point for the domain covering hexagons; see Figure 10. These parameter values are the same

as those used by Short et al. [38, Figure 7]. As one transverses the isolas, we see that the localised hexagon

pattern passes to a multi-pulse state involving the hexagon cells; see panel (2) Figure 10. In particular, we

see that the left most folds of the �10�-fronts occur at the fold of the radial spot strongly suggesting that the

localised structure is made-up of radial spots. This explains why the fronts in Figure 10 can exist beyond

the fold of domain covering hexagons. However, it is clear that near the bottom right folds, the interior of

the front does look like domain covering hexagons. We also note that decay to the background state changes

from oscillatory to monotonic as one transverses the bifurcation diagram. This change in the type of decay

18

E0 at fixed !!"1.2, it can be seen in Fig. 2 that cavitysoliton branches bifurcate subcritically at the modulationalinstability threshold.The existence of multipeaked CS structures is shown in

Fig. 2, in which their integral (" !A! dx) is plotted as a func-tion of !E0!2. CS exist on two distinct yet similar brancheswhich correspond to structures with, respectively, odd andeven numbers of peaks. Both bifurcate from the homoge-neous state at the point of modulational instability. Eachbranch, although continuous, is composed of numerous posi-tive slope #upper$ and negative slope #lower$ sections, whichwe will denote by % and L superscripts, respectively. We alsospecify the ‘‘number of peaks’’ #N$ as the number which

have amplitude at least equivalent to that of the lower-branchsolitary cavity soliton CS1

L at given input parameters. A se-quence of these solutions is presented in Fig. 3. Note that theN peaks are ‘‘close packed.’’ As might be guessed, there arenumerous other branches corresponding to structures with atleast one ‘‘gap’’ between adjacent large-amplitude peaks. Ifwe denote such a peak by ‘‘1,’’ and a minimal ‘‘gap’’ by ‘‘0,’’our close-packed CS structures are all of type‘‘ . . . .00011 . . . .111000 . . . . ,’’ which excludes e.g.‘‘ . . . .0001101000 . . . . .’’ We will not examine such ‘‘openstructures’’ in detail, although we note that their existenceand stability is important in connection with the use of CSarrays as pixel or memory arrays &7,22,36'.As N increases, the solutions get broader, and so are even-

tually limited by the computational domain. In the absenceof such constraints, they become very similar to the roll pat-terns described in Ref. &1'. Since a continuum of patterns ofdifferent wave vector are stable in this parameter region, theissue of the limiting peak separation of the multipeaked CSis an interesting question. Another issue arises when we con-sider that additional peaks do not have to be added sym-metrically. By adding peaks on only one side one limits to‘‘ . . . .00000111111 . . . . ,’’ which is not a roll pattern, butcoexistent roll and homogeneous patterns, with a front at theborder between them. These issues will be explored below.Turning now to the dynamical properties of these CS so-

lutions, we have tested their stability by diagonalizing theirJacobian, using the numerical methods mentioned above.Discounting the neutral mode #see below$ possessed by allCS solutions, the stability results are rather simple, in that allpositive-slope branches in Fig. 2 are stable, and all negative-slope branches unstable. More precisely, all nonzero eigen-values of the Jacobian of a positive-slope N-peak CS solutionare negative, so that it is an attractor, self-organizing fromany sufficiently-similar structure into the unique #at givenparameters$ CS solution on its branch.All negative-slope CS are unstable, they in fact have only

FIG. 2. Integral of one-dimensional CS structures against theintracavity field !E0!2. Solid, dotted, and dashed lines, respectively,denote: stable CSN

% , unstable CSoddL , and unstable CSeven

L solutions.Parameters are !!"1.2 and C!5.4.

FIG. 3. Sequences of profiles for odd #left$ and even #right$ CS branches shown in Fig. 2. Dash-dotted, solid, and dashed lines correspondto solutions at !E0!2!1.22, !E0!2!1.33, and !E0!2!1.44. Other parameters are !!"1.2 and C!5.4.

COMPUTATIONALLY DETERMINED . . . . II. . . . PHYSICAL REVIEW E 66, 046606 #2002$

046606-3

Author's personal copy

74 E. Meron / Ecological Modelling 234 (2012) 70– 82

Fig. 5. Mixed patterns predicted by the Gilad et al. model. Shown are numerical solutions of the model equations in bistability ranges of bare soil and spots (a), spots andstripes (b), stripes and gaps (c) and gaps and uniform vegetation (d). Darker gray shades denote higher biomass.From Kletter et al. (2011).

Fig. 6. Mixed patterns in nature: an isolated shrub patch in the northern Negev, Israel (A), mixture of spots and stripes of woody vegetation in Niger (B), mixture of stripesand gaps of woody vegetation in Niger (C), and isolated gaps in the pro-Namib zone of the west coast of southern Africa (D).From Rietkerk et al. (2002) (B and C) and Tlidi et al. (2008) (D).

||u||L2

r(a)

(b)

(c) −0.3 00

(b) (c)

rP1 rP2

rM

u=0

uP

−40 0 40

0

1

2 u(x)

−40 0 40

0

1

2 u(x)

Fig. 7. Homoclinic snaking in the Swift–Hohenberg model. A bifurcation diagram showing intermediate solutions in a bistability range of uniform, u = 0, and pattern statesu = up (a). The intermediate solutions describe localized structures with even (b) and odd (c) numbers of humps. Thick (thin) lines denote stable (unstable) solutions. Theparameter range rp1 < r < rp2 is called the homoclinic snaking range.Courtesy of John Burke.

can form a variety of irregular stable patterns (Meron et al., 2004)as Fig. 5 illustrates. Fig. 6 shows similar types of mixed patterns innature.

The mathematical theory of spatially mixed patterns in bistablesystems is far from being complete. However, significant progresshas been made recently in the case of bistability of uniform and spa-tially periodic states. Fig. 7 shows a bifurcation diagram for a simplepattern-formation model, the Swift–Hohenberg equation,2 that hasa bistability range of a uniform zero state and a periodic pattern.

2 The Swift–Hohenberg equation reads ut = ru + bu2 − u3 − (∂2x + k2

0)2u, where r,

b and k0 are parameters. It can be regarded as the simplest model that captures a

Apart of the zero solution and the periodic solution there are manyintermediate solutions representing spatial mixtures thereof, someof them are shown in the figure (the blue lines). (For interpretationof the references to color in the text, the reader is referred to the webversion of the article.) They correspond to localized structures con-sisting of confined domains of the periodic pattern in a backgroundof the zero state. There are two families of such localized solutions,one with an even number of humps and one with an odd numberof humps. The solution families “snake” upward, giving rise to a

stationary non-uniform instability. In this model the instability destabilizes the zerostate, u = 0 to a stationary periodic pattern with wave number k0.

(a) Plane Couette flow (b) Cellular buckling

(c) Crime hotspots (d) Optical cavity solitons (e) Vegetation patches

Figure 1: Examples of bifurcation diagrams corresponding to widely disparate physical

systems with similar underlying mathematical structure. Figures are reproduced from

published works as follows: (a) plane Couette flow [8]; (b) cellular buckling [2]; (c) crime

hotspots [3]; (d) optical cavity solitons [6]; (e) vegetation patterns [7].

Uncovering the Patterns Behind Patterns

Project description: In this project we will explore localized patterns on the plane.

Such patterns appear in a wide variety of physical contexts, which include – but are not

limited to! – fluid flows, crime hot spots, buckling problems, vegetation growth and optical

systems. Bifurcation diagrams for widely disparate systems have proved to be remarkably

similar; see Figure 1.

Various types of patterns which are periodic in one direction and localized in the other,

including those termed “rolls”, “spots and stripes”, and “squares,” have been investigated

(see, for example, [1], and refer to Figure 2 for visualizations of particular patterns). We

1

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

722 D. AVITABILE, D. J. B. LLOYD, J. BURKE, E. KNOBLOCH, AND B. SANDSTEDE

0.6 0.64 0.68

1 2 3 4

0.6 0.62 0.64 0.66 0.68 0.7

1

2

3

4

Figure 20. The left and right panels contain isolas of localized square patterns for � = 2. In the left panel,stable and unstable solutions along the isola are indicated by solid and dashed curves, respectively; the isolafrom the left panel is drawn in blue in the right panel. Algorithm 2 was used with y � [0, 200] and ntst = 400.

in the upper and lower halves of the domain against each other in the x-direction, the resultingpattern is still an approximate solution. Among this one-parameter family of approximatepatterns, which is parametrized by the relative shift in the x-direction, there are two exactsolutions, one of which is R-symmetric, while the other has �R-symmetry. We expect thatone of these two symmetric patterns is stable under periodic boundary conditions, while theother one is unstable in the direction of the shift x.

4.3. From localized stripe to square patterns. For the almost planar stripe patterns thatwe discussed in the previous section, we observed the growth of square cells along the interfacebetween rolls and the trivial state that then merged to form new rolls. This observation led usto examine localized square patterns on the cylinder. Figure 20 contains continuation resultsfor localized square patterns that we found near the Maxwell point µs = 0.609 of domain-fillingsquare patterns. The pattern profiles from panels 1–4 show that new squares are grown at theinterface with the trivial state. The bifurcation curve is an isola, though, and snaking doesnot occur. It appears as if the snaking of the pattern is inhibited by the relative proximity ofthe roll structures: panel 4 indicates that the localized square pattern tries to grow verticallyoriented rolls. Note that the localized square patterns shown in Figure 20 seem to be stablealong part of the bifurcation curve, which indicates that domain-covering square patterns arestable in this parameter region. This is surprising since domain-covering square patterns areknown to be unstable at onset in the cubic-quintic Swift–Hohenberg equation [14].

To further explore the interaction between stripe and square patterns, we lowered thevalue of � to 1.06247 and again continued localized almost planar stripe patterns in µ. Theresults, shown in Figure 21, indicate that the bifurcation structure is qualitatively similar tothat shown in Figure 10 for � = 2. Starting from the pattern shown in panel 1 of Figure 21,one side of the branch approaches a vertical asymptote given by the Maxwell point of 1D rolls,while the other side of the branch snakes. Along the nonsnaking branch, the pattern growsfour rows of squares, as in panels 2–5, before eventually growing vertically oriented rolls, as inpanel I. Along the snaking branch, an interior plateau of horizontally oriented rolls develops,as shown in panel II. There are a few interesting di↵erences, both in terms of the shape of

(c) square patterns(b) spot and stripe patterns(a) roll patterns


SNAKING OF PLANAR PATTERNS 715

1

0.5

0

-0.5

-1

0.6 0.65 0.7 0.75

1 2

1

2

Figure 8. The left panel contains the bifurcation diagram for snaking planar localized rolls for � = 2.The associated spatial-dynamics interpretation for the y-evolution is shown in the right panel. The roll patternshown in the right panel corresponds to a periodic orbit in the y-dynamics: throughout this paper, we indicateequilibria of the y-dynamics by filled circles, and periodic orbits (in an appropriate Poincare section) by circledcrosses.

0 0.1 0.2 0.3 0.4 0.5 0.7

2

1

1 2

-0.68

0

0.68

Figure 9. Nonsnaking localized rolls for � = 2 are shown in the left panel, while the associated spatial-dynamics interpretation for their y-evolution is shown in the right panel. The roll pattern shown in the rightpanel corresponds to an equilibrium of the y-dynamics. (Recall that equilibria are indicated by filled circles.)

4.1. Planar localized rolls. The 1D localized structures found in Figure 1 can be viewed asplanar localized rolls. Figure 8 shows their bifurcation diagram (which is, of course, identicalto those of localized 1D roll patterns) and their interpretation in terms of the spatial dynamicalsystem (2.2) in the evolution variable y. The rolls in the interior of the localized structure areperiodic in the vertical y-variable, and localized planar rolls can therefore be viewed as R-reversible homoclinic orbits that arise near a heteroclinic cycle from the equilibrium U = 0 toa periodic solution that corresponds to the y-periodic roll pattern: this explains why snakingoccurs. We remark that the localized rolls shown in Figure 8 are found to be alternately stableand unstable.

On the other hand, instead of orienting rolls parallel to the interface with the trivial



1

0.5

0

-0.5

-1

0.6 0.65 0.7 0.75

1 2

1

2


0 0.1 0.2 0.3 0.4 0.5 0.7

2

1

1 2

-0.68

0

0.68






1

0.5

0

-0.5

-1

0.6 0.65 0.7 0.75

1 2

1

2


0 0.1 0.2 0.3 0.4 0.5 0.7

2

1

1 2

-0.68

0

0.68




Figure 2: Examples of patterns observed in the Swift–Hohenberg equation on an infinite

cylinder. See [1] and [5].

would like to use this time and computing resources to better understand the relationships

between these patterns, and, in particular, their connections in parameter space.

As time and interest allow, we may also explore the formation of large localized hexagon

patches. Computing resources were specifically identified as a limiting factor in a 2008

study [4]. However, since this time the underlying package capabilities have progressed

substantially, and revisiting this problem with new computing resources may enable sub-

stantial progress.

Prerequisites: While there are no formal prerequisites for participation in this project,

some familiarity with di↵erential equations and dynamical systems theory would be helpful.

In particular, a basic understanding of bifurcation theory underlies most of the work. I

would be happy to review with any interested students lacking this background; Strogatz

[9] also provides an accessible and useful introduction. Programming for this project will

be conducted in C/C++. Current software is written in Matlab, so our first task will be

converting this software. Depending on our progress we may also use Auto07p, a Fortran

based program for continuation, but no knowledge of Fortran is expected.

2

References

[1] D. Avitabile, D. J. B. Lloyd, J. Burke, E. Knobloch and B. Sandstede. To snake or not

to snake in the planar Swift-Hohenberg equation. SIAM J. Appl. Dyn. Syst. 9 (2010)

704–733.

[2] G. W. Hunt, M. A. Peletier, A. R. Champneys, P. D. Woods, M. A. Wadee, C. J. Budd

and G. J. Lord. Cellular buckling in long structures. Nonlin. Dynam. 21 (2000) 3–29.

[3] D. J. B. Lloyd and H. O’Farrell. On localised hotspots of an urban crime model. Physica

D 253 (2013) 23–39.

[4] D. J. B. Lloyd, B. Sandstede, D. Avitabile and A. R. Champneys. Localized hexagon

patterns of the planar Swift–Hohenberg equation. SIAM J. Appl. Dyn. Syst. 7 (2008)

1049–1100.

[5] E. Makrides and B. Sandstede. Predicting the bifurcation structure of localized snaking

patterns. Physica D 269 (2014).

[6] J. M. McSloy, W. J. Firth, G. K. Harkness and G.-L. Oppo. Computationally deter-

mined existence and stability of transverse structures II. Multipeaked cavity solitons.

Phys. Rev. E 66 (2002) 046606.

[7] E. Meron. Pattern-formation approach to modelling spatially extended ecosystems.

Ecological Modelling 234 (2012) 70–82.

[8] T. M. Schneider, J. F. Gibson and J. Burke. Snakes and ladders: localized solutions of

plane Couette flow. Phys. Rev. Lett. 104 (2010) 104501.

[9] S. H. Strogatz. Nonlinear Dynamics And Chaos: With Applications To Physics, Biol-

ogy, Chemistry, And Engineering. Studies in Nonlinearity, Westview Press, 2001.

3

Computer Simulations on Rocket-Plasma Interactions

! Research Background

Near-Earth space is filled with a plasma, consisting of a huge number of free moving electrons and

ions. The plasma interacts with spacecraft and rockets in space and sometimes causes anomalies

on their systems. The interaction of spacecraft or rocket with the plasma environment is a very

crucial issue for future space development. The computer simulation based on the Particle-In-Cell

method is a very powerful tool to solve the problem.

! Project

The project focuses on a sounding rocket which is launched to investigate the Earth’s ionospheric

region. As a result of interactions of the rocket with the ionospheric plasma, a number of

interesting phenomena will take place around the rocket such as rocket charging, plasma

sheath/wake formation, and non-uniform potential structures. You may use/improve a pre-existing

plasma particle simulator called EMSES to simulate these phenomena. After determining physical

and numerical parameters for the EMSES simulations, you will run the program on the FX10

supercomputer owned by Kobe University. Post-processing is also an important aspect of this

project; the output data should be processed and visualized with ParaView and a virtual reality

system for better understanding of physical phenomena reproduced by the simulations.

! Programming and Computational Skills:

Basic knowledge about Fortran90, parallelization with Message Passing Interface (MPI), and

visualization with ParaView will be helpful, but the project is open also for those who are not

familiar with the subjects. Some materials and short courses will be given before/during the school.

! Project Leader: Yohei Miyake (Kobe University, [email protected])

Simulations of Geophysical Fluids and Planetary Atmospheres

Brief Introduction:

Atmospheric circulation is one of the important applications of computer simulation. In

fact, atmospheric models are used for daily weather prediction and climate prediction of

the Earth’s atmosphere. In addition, atmospheric circulation models are used for

research of atmospheres of planets, such as Mars.

Project:

The project will work on the simulations of atmospheres of the Earth or Mars by use of a

pre-existing codes, DCPAM*, or some geophysical fluids in simple systems by making

codes. Some lectures and tutorials will be given during a first week. Followings are

plausible topics which participants can select based on one’s experiences and interests.

z Perform simulations of Earth’s or Mars’ atmospheres by use of the DCPAM,

z Make a code for geophysical fluids, such as a two-dimensional turbulence, a shallow

water system, and a three-dimensional fluid, and perform simulations,

z Make a tracer transport codes which calculate advection by a meteorological fields,

and perform simulations.

Programing language:

Fortran: Experiences on coding Fortran90 program are helpful, but those who are not

familiar to Fortran90 are welcome.

Project Leader:

Yoshiyuki O. Takahashi (Department of Planetology, Kobe University, Japan)

* DCPAM: Planetary atmosphere general circulation model developed by members of

GFD** Dennou Club. See http://www.gfd-dennou.org/library/dcpam/index.htm.en

for more information of the model.

** GFD: Geophysical Fluid Dynamics

Figure. An example of water vapor distribution in

the Earth’s atmosphere simulated by the DCPAM

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