Two dimensional patterns in bacterial veils arise from

self-generated, three-dimensional fluid flows

N. G. Cogan ∗

and C. W. Wolgemuth †

∗N.G. Cogan. Address: Department of Mathematics, Florida State University, 208 Love Building, Talla-hassee, Fl 32317, U.S.A., Tel.: (850)644-7196, Fax: (850)644-7153

†C. W. Wolgemuth. Address: University of Connecticut Health Center, 263 Farmington Avenue, Farm-ington, CT 06030 Tel: (860) 679-1655

1

Abstract

The behavior of collections of oceanic bacteria is controlled by metabolic (chemo-taxis) and physical (fluid motion) processes. Some sulfur-oxidizing bacteria, such asThiovulum majus, unite these two processes via a material interface produced by thebacteria and upon which the bacteria are transiently attached. This interface, termed abacterial veil, is formed by exo-polymeric substances (EPS) produced by the bacteria.By adhering to the veil while continuing to rotate their flagella, the bacteria are ableto exert force on the fluid surroundings. This behavior induces a fluid flow that, inturn, causes the bacteria to aggregate leading to the formation of a physical patternin the veil. These striking patterns are very similar in flavor to the classic convectioninstability observed when a shallow fluid is heated from below. However, the physicsare very different since the flow around the veil is mediated by the bacteria and affectsthe bacterial densities.

In this study, we extend a model of a one-dimensional veil in a two-dimensionalfluid to the more realistic two-dimensional veil in a three-dimensional fluid. The linearstability analysis indicates that the Peclet number serves as a bifurcation parameter,which is consistent with experimental observations. We also solve the nonlinear prob-lem numerically and are able to obtain patterns that are similar to those observed inthe experiments.

Keywords: Fluid/structure, biofilm, pattern formation, bacterial veils, instability, Bound-ary integral method

1 Introduction

Mathematical analysis has proven to be a valuable tool for understanding how the coupledmotion of microorganisms and fluid flow creates striking patterns. For example, recentanalysis of densely packed, motile bacteria indicates that complex flows with vortices andjets can be produced by the interactions between the bacteria and the fluid [1, 2]. Anotherexample is the formation of convection cells by dense populations of bacteria that respondto differences in gravity or acceleration (gravitaxes) [3, 4]. In each of these examples, thegroup behavior was modulated by the interaction between the motion of the bacteria andthe external fluid flow that is driven by the bacterial motion.

In this paper, we focus on another pattern that is formed by swimming bacteria. Here themotion of the bacteria drives the fluid which, in turn, drives the aggragation of the bacteria.However, this behavior is interupted by the presence of a semi-permeable material (veil) thatis produced by the bacteria and on which the bacteria may be transiently attached. The veilmaterial undergoes a transition from a uniform distribution into either stripes or spottedpatterns that are similar to convection rolls and cells in appearance, although the physicalmechanism is quite different [5, 6, 7, 8]. In this paper, we we extend the analysis of a onedimensional veil in a two dimensional fluid to a two dimensional veil in a three dimensionalfluid. We find that the model captures the relevant experimental observations.

Polymer veils are formed by certain marine bacteria, such as Thiovulum majus and Thio-

turbo danicus [5]. These species are relatively fast swimmers with T. danicus reaching speedsaround 100 µm/s and T. majus going up to 600 µm/s [8]. Both species aggregate by chemo-taxis to a thin region (<100 µm in thickness). Once there, the bacteria produce a mucus

2

layer to which the bacteria can transiently attach. The bacteria typically attach to the oxicside of the veil by a stalk, which allows the cell body to rotate when the flagella are turned;the attachment also holds the bacteria in place in the ambient fluid. The rotation of thebacterial flagella produces the thrust force that drives the swimming of the bacteria; how-ever, when the bacteria are attached to the veil, flagellar rotation induces a fluid flow thatdraws oxygen-rich fluid from above the veil across the veil layer. Meanwhile, the bacteriaconstantly detach and reattach to the veil. The coupled chemotactic and advective motionof the bacteria reinforces perturbations leading to spatially-organized aggregation of the bac-teria. After approximately three days, the attached bacteria surround voids in the veil andthe observed fluid dynamics resembles that of a convective instability where the fluid motionin various regions is dominated by vertical motion [9, 6]. Depending on the experimentalset-up the voids can form either rolls or regularly spaced patterns.

The material interface on which the bacteria are attached is composed of a polymericgel, which is a viscoelastic material whose material properties are affected by the ionicenvironment as well as the current and past position. In turn, the motion of the veil affectsthe fluid velocity (and hence the bacterial distribution) since the veil acts as a semi-permeablelayer that interacts with the fluid through the inter-phase (fluid/polymer) drag.

A model for the coupled fluid and bacterial motion was introduced in [10] where theone-dimensional model was studied. Near hole in the veil the fluid dynamics tends to pushfree bacteria back onto the veil away from the hole. This feedback causes the populationto aggregate in regularly spaced regions. Presumably the gel deteriorates between thesepopulation peaks, reinforcing the formation of holes. This heuristic argument is confirmedby the analysis of the one-dimensional veil immersed in a two-dimensional fluid [10]. However,this analysis did not address the formation of two-dimensional patterns. Therefore, in thepresent investigation, we extend the analysis of the model to a two-dimensional veil immersedin a three-dimensional fluid. We show that aggregation can form both striped and spottedpatterns. The Peclet number acts as a bifurcation parameter with veils higher above thesurface tending to display honeycombed patterns. We note that the mechanism for patternformation given here differs from other models that assume a radially symmetric velocityprofile and local attraction of free bacteria to bound bacteria [11] since we develop the flowfield as a dynamic variable.

2 The model

We begin with a review of the development of the model in [10]. The model is posed intwo dimensions, where the veil is a one-dimensional interface aligned with the x-axis andimmersed in the two-dimensional fluid. The model does not account for the production ofthe veil, but assumes that the veil is located at a fixed height, h, above the ocean floor. Thebacteria are considered to be either adhered to the veil (bound) or swimming above the veil(free). To simplify the analysis, it is assumed that there is no interaction between the fluidand the veil. Rather, the fluid motion is created by forces from the bound bacteria, and thisfluid motion affects the swim path of the free bacteria. Free bacteria bind to the veil at a rateτf . As bacteria swimming near the veil are observed to swim in parabolic trajectories thatbring the bacteria back to the veil [8], it is presumed that this rate represents the average

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time between release from the veil and return to the veil. It is also likely that this ratedepends on the details of the fluid flow; however, we treat this rate as a constant. Boundbacteria break free of the veil at a rate τb.

The free bacteria swim, chemotact and are advected by the fluid. Free swimming ismodeled as a diffusive term, with diffusion coefficient, Df . Instead of treating chemotaxisexplicitly, we assume that the chemotaxis of the bacteria tends to move the bacteria backto the veil. Therefore, free cells swim an average height, δ, above the veil. This treatmenteffectively averages over the vertical motion of the bacteria, and we only consider advectionparallel to the veil, simplifying the equations substantially. Denoting the density of free andbound bacteria as nf and nb, respectively, we find that the densities are determined by

∂nb

∂t= − 1

τbnb +

1

τfnf (1)

∂nf

∂t= Df∆nf − ∂nfvx,δ

∂x+

1

τbnb −

1

τfnf . (2)

Here, vx,δ denotes the x-component of the fluid velocity evaluated at the height δ abovethe veil. In these equations, the time rate of change of the concentration of bound bacteriais balanced by detachment and attachment, whereas the time rate of change of the freebacteria is balanced by swimming (e.g., diffusion), advection by the fluid parallel to the veil,and detachment/attachment.

Scaling the fluid motion using a typical veil height and the bacterial velocity scales leadsto a Reynolds number on the order of 10−1, indicating that the flow is well approximated bythe incompressible Stokes equation:

µ∆v −∇p = K (3)

∇ · v = 0. (4)

The force, K on the right hand side represents the force that the bound bacteria exerts onthe fluid as they rotate. We assume that this force is down and proportional to the numberof bound bacteria, K = −αnby. Using the boundary integral method [12], we represent thefluid motion as an integral over the entire veil,

vi =1

4πµ

∫

dx0Gij(x,x0)K, (5)

where Gij is the Green’s function (or Stokeslet) relating the flow at x generated by forces atlocation x0.

We now consider a three-dimensional fluid with a two-dimensional veil, where the veil isparallel to the x-y plane and located a height h above the z-axis. This leads to an analogousset of equations, where the dynamic equations governing the bacterial populations are,

∂nb

∂t= − 1

τbnb +

1

τfnf (6)

∂nf

∂t= Df∆nf − (

∂nfvx,δ

∂x+

∂nfvy,δ

∂y) +

1

τbnb −

1

τfnf . (7)

4

Where vx,δ and vy,δ are the x and y-components of the fluid velocity at a height δ above theveil. The fluid motion is governed by the three-dimensional form of Eqs. 3- 4.

Following [10], we rescale the equations by x = x/h, t = Df t/h2, and the equilibrium

concentration of bound bacteria n0, which leads to a dimensionless set of equations withfour control parameters, the two rate constants, a Peclet number Pc, and the dimension-less height that the bacteria swim above the veil. We denote the scaled rate constants ask−

= τD/τb and k+ = τD/τf , where the characteristic time scale is τd = h2/Df . The dimen-

sionless swimming height is denoted δ = δ/h and Pc = αh2n0/8πDfµ defines the relativeimportance of advection to diffusion. Changing the dependent and independent variables tothe dimensionless variables (and dropping the hats) we obtain

∂nb

∂t= −k

−nb + k+nf (8)

∂nf

∂t= ∆nf − Pc(

∂nfvx,δ

∂x+

∂nfvy,δ

∂y) + k

−nb − k+nf . (9)

The dimensionless velocity can be calculated analytically using the boundary integralmethod [12],

v (x) = −∫

∞

−∞

∫

∞

−∞

nb (x0) [Gxz (x,x0) x + Gyz (x,x0) y] dx0dy0 , (10)

where Gxz and Gyz are the components of the Green’s function tensor that give the velocityin x and y when a force acts on the fluid in the z direction. These functions can be writtenin terms of the Stokeslet, the doublet tensor, and the dipole tensor. Specifically,

Gxz =(x − x0) (z − z0)

[

(x − x0)2 + (y − y0)

2 + (z − z0)2]3/2

− (x − xI) (z − zI)[

(x − x0)2 + (y − y0)

2 + (z − zI)2]3/2

+

(

6 (x − x0) (z − zI)[

(x − x0)2 + (y − y0)

2 + (z − zI)2]5/2

)

−2 (2 + δ)

(

3 (x − x0) (z − zI)[

(x − x0)2 + (y − y0)

2 + (z − zI)2]5/2

)

+2 (x − x0)

[

(x − x0)2 + (y − y0)

2 + (z − zI)2]3/2

, (11)

where the subscript I denotes the image charge points, and we have used that xI = x0 andyI = y0. δ is the average distance that the free bacteria swim above the veil. Collectingsimilar terms, Eq. 11 simplifies to

Gxz =δ (x − x0)

[

(x − x0)2 + (y − y0)

2 + (z − z0)2]3/2

− δ (x − x0)[

(x − x0)2 + (y − y0)

2 + (z − zI)2]3/2

− 6 (1 + δ) (x − x0) (z − zI)[

(x − x0)2 + (y − y0)

2 + (z − zI)2]5/2

, (12)

5

which can be re-written in terms of derivatives of x and z as

Gxz = − ∂

∂x

(

δ[

(x − x0)2 + (y − y0)

2 + (z − z0)2]1/2

)

+

(

δ∂

∂x− 2 (1 + δ)

∂2

∂x∂z

)

(

1[

(x − x0)2 + (y − y0)

2 + (z − zI)2]1/2

)

. (13)

Likewise, the yz component is

Gyz = − ∂

∂y

(

δ[

(x − x0)2 + (y − y0)

2 + (z − z0)2]1/2

)

+

(

δ∂

∂y− 2 (1 + δ)

∂2

∂y∂z

)

(

1[

(x − x0)2 + (y − y0)

2 + (z − zI)2]1/2

)

. (14)

Equations 8, 9, and 10 give a closed set of nonlinear equations for the concentrations offree and bound bacteria, and the fluid velocity. We are interested in how the parameterseffect the development of patterns on the veil. Because we are not treating the polymerconcentration explicitly, we assume that if there are regions without bound bacteria, thepolymer degrades leaving holes.

In the next two sections we consider the stability of a uniform distribution of boundbacteria to perturbations. We first linearize the equations about the uniform distributionand show that one can obtain a dispersion relation that predicts the wavelengths of themaximally unstable modes. We then confirm the linear analysis with numerical simulationsof the fully nonlinear system. We note that the dispersion relation that we obtain dependsonly on the magnitude of the perturbation modes. Therefore the linear analysis cannotdifferentiate between parameters that yield stripes or cells. To break the symmetry in thesimulations we also show that if the domain is longer in one direction, it is possible to observestripes as well as cells.

3 Linear Stability

In this section we consider the linear stability of the two-dimensional veil immersed in athree dimensional fluid. To analyze the stability of the homogeneous distribution of bacteriawe expand the free and bound bacterial concentrations as,

nb = n0

b +∑

qx,qy

Aqxqyeγt cos (qxx0 + qyy0) , (15)

where n0

b is a constant and Aqxqyis the amplitude of the different modes.

Expanding Equations 7, 6, 18 and 19, neglecting to first order in ǫ leads to a linear systemof linear equations for the perturbations n

(1)f and n

(1)b ,

(γn1b + k

−n1

b − k+n1f )e

γt cos (qxx0 + qyy0) = 0 (16)

(

(γ + q21 + q2

2 + k+)n1f − k

−n1

b

)

eγt cos (qxx0 + qyy0) = Pcn0f

(

∂

∂xv

(1)x,δ +

∂

∂yv

(1)y,δ

)

. (17)

6

The terms v(1)x,δ and v

(1)y,δ denote the order ǫ terms in the x and y components of the velocities.

The velocities are zero up to leading order (by symmetry), so the advection term dependson the initial free density and the order ǫ velocities.

To calculate the order ǫ velocities, we need to calculate

vx,δ = − α

8πµ

∫

∞

−∞

dx0nb(x0)Gx,y (18)

vy,δ = − α

8πµ

∫

∞

−∞

dx0nb(x0)Gz,y, (19)

where n0

b is a constant and Aqxqyis the amplitude of the different modes.

To calculate the velocity, we will consider just one mode. The total velocity will just bea sum over all modes. The velocity in the x direction is

vx,δ =αAqxqy

δ

8πµ

∂

∂x

∫

∞

−∞

∫

∞

−∞

cos (qxx0 + qyy0)[

(x − x0)2 + (y − y0)

2 + (z − z0)2]1/2

dx0dy0

−(

δ∂

∂x− 2h (h + δ)

∂2

∂x∂z

)

αAqxqy

8πµ

∫

∞

−∞

∫

∞

−∞

cos (qxx0 + qyy0)[

(x − x0)2 + (y − y0)

2 + (z − zI)2]1/2

dx0dy0.

Using the results from the Section 4, we can calculate the velocity in the x direction,

vx,δ = −αqxAqxqy

8πµsin (qxx + qyy)

(

δ

√

c1

|q|K−1

2

(|q|c1) −(

δ − 2h (h + δ)∂

∂z

)√

c2

|q|K−1

2

(|q|c2)

)

,

where |q| = q2x + q2

y, c1 = z − z0 = δ, and c2 = z − zI = 2h + δ. The y component of thevelocity is calculated in a similar fashion,

vy,δ = −αqyAqxqy

8πµsin (qxx + qyy)

(

δ

√

c1

|q|K−1

2

(|q|c1) −(

δ − 2h (h + δ)∂

∂z

)√

c2

|q|K−1

2

(|q|c2)

)

.

Note that ∂/∂z = ∂/∂c2. Therefore, the z derivatives of the Bessel function can becalculated using the recurrence relations,

∂

∂c2

(√c2K−

1

2

(|q|c2))

=1

2√

c2

K−

1

2

(|q|c2) +√

c2∂

∂c2

K−

1

2

(|q|c2)

=1

2√

c2K

−1

2

(|q|c2) −√

|q|c2

2

(

K−

3

2

(|q|c2) + K 1

2

(|q|c2))

(20)

Using the velocities obtained above in the linearized equations 16 and 17, yields a rela-tionship between γ and the modes of the perturbations. Arbitrary perturbations with modesqx in the x direction and qy in the y direction are unstable if the real part of γ is positive.Moreover, the spatial pattern of the unstable mode is determined by qx and qx. Rolls occurwhen only perturbations in one direction lead to positive growth rates while regularly spacedtarget patterns occur when neither mode is zero. However, it is clear that the perturbedvelocities depend only |q| which implies that the linear analysis cannot distinguish between

7

Figure 1: Example of the dispersion curve for the linearized system. The parameters arek+ = 10, k

−= 3 while the Peclet number is varied. We see that as the Peclet number

increases from zero the uniform steady-state moves from linearly stable to unstable, with amaximally unstable wave number at approximately 3 for Pc = 200.

rolls and hexagonal patterns. The numerical solution of the fully nonlinear system doescapture both regimes.

From the linearized equation, it is possible to obtain the dispersion curve, relating thegrowth rate, given by the real part of γ, to the mode, q. This relationship is quite compli-cated, rather than show the exact dispersion relationship, we show a representative plot inFigure 1.

In the following section, we describe the dynamic behavior of the full system. In partic-ular, we investigate the formation of rolls and cells in various parameter regimes. We note,that all the results agree with the linear analysis; namely, regimes where there are unstablemodels (Peclet number larger than about 100), yield non-uniform solutions. We are able todistinguish between rolls and cells in the nonlinear regime.

4 Nonlinear Behavior

To investigate the nonlinear behavior, we solve the full equations 7, 6, and 10. We note thatthere are difficulties in solving these equations on an infinite domain. For practical results,these equations are typically solved on a periodic domain; however to implement this, weneed the periodic Green’s function. There are several methods available to calculate this

8

in a one dimensional domain ([12]), but the methods do not translate simply into higherdimensions. To avoid this issue, we first embed the domain, D = (x1, x2) × (y1, y2) in aperiodic collection of N2 copies of D. We then approximate the behavior on a single domainwith periodic boundary conditions to the behavior on the center grid.

On the full grid domain we use standard methods to solve the equations. The Green’sfunction is well defined and analytic, so that the integrals that yield the velocities can becalculated explicitly on the domain (as in Section 6.1). The calculated velocity satisfies Eq.10, so we are left with solving Equations 6, 7, with known velocities. To solve the equations,we use Crank-Nicolson which gives second order accuracy in time and space. In general, weuse a 70 × 70 grid for the square sub-domain (−.5, .5) × (−.5, .5) and 70 × 210 grid for therectangular sub-domain (−.5, .5)× (−1.5, 1.5). By computing both a full and half time-stepof Crank-Nicolson, provides an error estimate. In general, we use a time step of dt = 0.01.We set the tolerance of the error to 10−4 and reduce the time step by half if this toleranceis not met.

4.1 Spots

We first study the development of spatial patterns using the Peclet number as the controlparameter. Recalling that the dispersion curve (see Figure 1) indicates that for small Pecletnumbers, the uniform state is stable, while higher values can yield unstable modes. We alsocomment that increasing Peclet number can be thought of as an increase in the height ofthe veil above the ocean floor (h) or as an increase in the force each bacteria induces on thefluid (α). This interpretation concurs with experimental observations that veils formed overdepressions or formed by stronger swimmers tend to be spatially heterogeneous [6, 8].

We explore the nonlinear dynamics of the system for varying Peclet numbers. In Figures2, 3 and 4, we show snapshots of the system at various times. We clearly see how quicklythe system moves from the uniform state when it is perturbed by a small random amount.

We can also visualize the flow associated with the patterns. In Figure 5, we show cross-sections of the flow in three dimensions. The stream lines are indicated, with the colorindicating the velocity. We see representative flows that are essentially radially symmetricnear the aggregated attache bacteria agreeing with the imposed flow in [11]. Near the veilthe flow is transvers and above the veil we see the characteristic reversal in the streamlinesbringing the free bacteria back to the veil.

4.2 Stripes

We now investigate the development of stripes. Because of the similarity of the systemand analysis to that of the well known convection instability, it seems clear that there areparameter regimes which tend toward a stable striped configuration. However, the lineartheory cannot address this since the dispersion relation depends only on |q|. One way toforce the development of stripes is to break the symmetry of the domain. By elongating they-direction and adjusting the Peclet number, we can find parameter regimes for which thex-direction is too small to allow an instability in that direction while the y component of thedomain is large enough to allow for an instability. In this domain and for the appropriatePeclet numbers, the stripe steady-state is found.

9

Figure 2: The dynamics of the system for Pc = 1, the uniform steady-state is apparentlystable, which agrees with the linear analysis. The parameters are k+ = 10, k

−= 3.

Figure 3: The dynamics of the system for Pc = 100. Considering the color scale, we see thatthe uniform steady-state is stable. The parameters are k+ = 10, k

−= 3

10

Figure 4: The dynamics of the system for Pc = 200. Here the uniform state is unstable andthe maximally unstable modes yields the development of spatial patterns. The parametersare k+ = 10, k

−= 3. The Peclet number large enough that the linearized system is unstable,

agreeing with the nonlinear computations.

11

Figure 5: A snapshot of the three dimensional flow patterns. The lines show the streamlinesof the flow and the magnitude is shown by the colormap, with white being fastest flow. Thisfigure shows 1/4 of the full domain.

As before, the uniform state is apparently stable for low Peclet numbers (Figure 6, 7).As the Peclet number increases, some modes become unstable. Figure 8, show snapshots ofthe attached bacteria for larger Peclet number (Pc = 200). Here the system is clearly notmoving towards the uniform state, but to a striped pattern.

5 Discussion

Here we have derived the equations that describe the dynamics and pattern formation of atwo-dimensional bacterial veil immersed in a three-dimensional fluid. Linear stability anal-ysis of this system shows that a uniformly dense colony of bacteria will go unstable whenthe Peclet number for the system exceeds a certain value. This Peclet number is primarilydependent on the height of the veil off the ocean floor and the thrust force that is generatedby an individual swimming bacteria. The height of the veil also strongly determines themaximally-unstable mode of the system, and therefore patterns have a characteristic length-scale that is comparable to the height of the veil. Simulations show that rolls and hexagonalpatterns are possible.

As mentioned before, the sulfur-oxidizing bacteria are fairly fast swimmers, with velocitiesof at least 50µm/s and as high as 600µm/s [6, 5], whereas E. coli has a maximum velocity ofonly 30µm/s. We speculate that one benefit of such high swimming speeds is the generationof the instability that leads to the convective patterns described here. As we noted, a uniform

12

Figure 6: The dynamics of the solution of the fully nonlinear system for Pc = 1. Theuniform state is stable and the initial perturbation decays. All parameters are the same asthe symmetric case,

layer of bacteria will not produce any flow. Therefore, to achieve the downward flows throughthe bacterial colony that bring in oxygen-rich fluid, the bacterial system must have a Pecletnumber large enough to produce the instability. One way to achieve this large Peclet numberis to increase the swimming speed, as fast swimming translates to a larger force exerted onthe fluid by the attached bacteria.

On a broader scale, understanding the underlying processes which influence our globalclimate requires understanding the global carbon cycle, as atmospheric carbon dioxide is themajor player in global warming. The metabolism of microorganisms found in the oceans is asignificant contributor to the carbon cycling. Especially, many coastal sediments in shallowwater show a microbial activity which sometimes even surpasses the metabolic activity ofrain forests (e.g. if measuring respiration rates per unit area). As marine water contains largeamounts of sulfate, a major part of sediment microorganisms are specialized in metabolizingsulfur compounds like sulfide, elemental sulfur, or sulfate [13]. These bacteria link the globalcarbon cycle with the global sulfur cycle. Thus, it is of interest to understand the processesof the sulfur cycle and their influence on the carbon cycle.

The sulfur cycle is coupled to the metabolism of the organisms. In this study, we focuson a novel behavior displayed by bacteria that presumably increases their metabolic rates.In particular, it seems that the bacteria optimize their metabolism by fixing their locationwithin the ocean. This leads to a new pattern formation problem that is mathematicallystriking and biologically explanatory.

13

Figure 7: The dynamics of the solution the fully nonlinear system for Pc = 100. The uniformstate is stable, although there is a weak striped pattern that is transient apparent in thelower right.

6 Appendix

6.1 Calculating the integrals

For the 3D veil linear stability analysis, we need to calculate the velocity field that is gen-erated by a doubly-periodic perturbation in the bound bacterial density, nb = n0

b cos(qxx0 +qyy0), where qx and qy are the wave numbers for the perturbation in the x and y directions,respectively. Since the Greens function for bounded fluid motion can be defined using agenerating function, it is possible to calculate the double integrals of the generating functionand then take derivatives with respect to x, y, and z to get the velocity that is created bythe infinite distribution of bound bacteria. Therefore, we need to calculate the integral,

∫

∞

−∞

dy0

∫

∞

−∞

dx0

n0

b cos(qxx0 + qyy0)(

(x − x0)2 + (y − y0)

2 + (z − z0)2)1/2

(21)

Making the substitutions u = (x0 −x) and c2 = (y−y0)2 +(z−z0)

2 and using trig identities,we rewrite this integral as

n0

b

∫

∞

−∞

dy0

∫

∞

−∞

ducos(qxu) cos(qxx + qyy0) − sin(qxu) sin(qxx + qyy0)

(u2 + c2)1/2(22)

14

Figure 8: The dynamics of the solution the fully nonlinear system for Pc = 200. ThePeclet number is sufficiently high so that there is an unstable mode, that is dominated bydifferentiation in the x-direction.

The integral containing cos(qxu) is a Bessel function, and the term containing sin(qxu) inte-grates to zero:

2n0

b

∫

∞

−∞

dy0 cos(qxx + qyy0)K0

[

qx

(

(y − y0)2 + (z − z0)

2)1/2]

(23)

To solve this integral, we make the substitutions v = (y0 − y) and c = z − z0. Using trigidentities, we get the integral into the form

2n0

b

∫

∞

−∞

dv (cos(qyv) cos(qxx + qyy) − sin(qyv) sin(qxx + qyy))K0

[

qx

(

v2 + c2)1/2]

(24)

As in the previous case, the term containing sin(qyv) integrates to zero. The other term canbe integrated using 6.726 from “Table of Integrals, Series, and Products” by Gradshteyn andRyzhik. I get that Eq. 1 becomes,

n0

b

√

8πc

|q| K−

1

2

(|q|c) cos(qxx + qyy) (25)

where |q|2 = q2x + q2

y .

15

References

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