The statistical mechanics of meandering R. Bruinsma

To cite this version:

R. Bruinsma. The statistical mechanics of meandering. Journal de Physique, 1990, 51 (9), pp.829-845. ￿10.1051/jphys:01990005109082900￿. ￿jpa-00212412￿

HAL Id: jpa-00212412 https://hal.archives-ouvertes.fr/jpa-00212412 Submitted on 1 Jan 1990

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Classification Physics Abstracts

05.40 - 46.30 - 68.42

The statistical mechanics of meandering

R. Bruinsma

Physics Department & Solid State Science Center, University of California, Los Angeles, Los Angeles, CA, 90024, U.S.A.

(Reçu le 17 juillet 1989, accepté sous forme définitive le 1 b novembre 1989)

Résumé. 2014 On présente un modèle simple décrivant la statistique du serpentement d’écoulements étroits sur un substrat propre et lisse. Ce modèle présente trois régimes différents suivant le débit: (i) pour des faibles débits, le trajet est une marche aléatoire stationnaire ; (ii) pour des débits intermédiaires, des méandres stationnaires apparaissent. L’apparition de ces méandres est très similaire à la physique des transitions de phase continues ; (iii) pour des débits élevés, les méandres commencent à glisser. Le problème du décrochage des méandres est équivalent au problème du décrochage des parois dans les systèmes magnétiques désordonnés. Cette correspondance permet de calculer le débit critique où apparaît le glissement des méandres.

Abstract. 2014 We present a simple mathematical model to describe the statistical properties of the meandering of narrow streams on clean and smooth substrates. The model is shown to contain three different regimes, depending on the flow rate : (i) at low rates, the stream path is a time- independent random walk ; (ii) for intermediate flow rates, static meanders appear. The onset of meandering is found to be closely analogous to the physics of continuous phase transitions ; (iii) at high flow rates, the meanders start to slide downhill. The problem of the depinning of meanders can be mapped onto the problem of domain-wall depinning in disordered magnets. Using this correspondence, we can compute the critical flow rate for the onset of meander sliding.

1. Introduction.

Under non-equilibrium conditions, the surfaces and interphases encountered in solid-state physics can exhibit fascinating patterns. Well known examples are dendrites, diffusion-limited aggregation and ballistic deposition. Fluid interfaces can also show pattern formation as demonstrated by viscous fingering. We will discuss in this paper a familiar hydrodynamic instability - stream meandering [1] - which shows pattern formation. This instability exhibits a number of features reminiscent of critical phenomena in condensed matter physics - a similarity which we will exploit later on. A well known example of the instability is the meandering of rivers. River meandering has been a long-standing fascinating problem with a considerable literature [2] to which even Einstein contributed. It involves a complex interplay between soil erosion and hydrodynamics. In the present paper, we will consider a closely related but simpler problem namely the question of finding the morphology of the stream-path of a narrow stream flowing down a

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01990005109082900 830

(rigid) inclined plane. The fluid is assumed to be non-wetting. There are a number of simplifications in this case : (i) for sufficiently narrow streams and sufficiently low flow rates, one may use the Poiseuille approximation ; (ii) erosion plays no role for a rigid substrate ; (iii) for narrow streams, surface tension provides an important stabilizing action which simplifies the analysis. The problem was investigated experimentally by Nakagawa and Scott [3] and by Walker [4]. We will briefly review their results for different values of the volume flow rate I. (i) For large inclinations of the plane (> 30° ), the stream forms stable meanders. The meanders consist of relatively straight diagonals connected by sharp bends. At low I, the meanders are less prominent while the stream-path is strongly correlated with the path taken by the stream when the flow was turned on. It also depends on height irregularities and chemical contamination. The shape of the meanders of narrow streams differs from that of the meanders of rivers (which are « sine-generated » curves [2]) but for convenience we will retain the name. (ii) With increasing I, the stream-path is reorganized and meandering becomes stronger. The appearance of the meanders appears to be triggered by turbulence and/or deformation of the stream cross-section. For larger I, it may take a long time before the stream-path stabilizes into a static pattern. (iii) Above a critical flow rate, 1,,2, the stream-path is unstable. Meanders constantly break up, reform, and slide downwards. Streams also may bifurcate. (iv) For low inclinations of the plane ( 30° ), there is a second critical current, ici. For I less then Ici the stream breaks up into droplets sliding individually down the plane. The physical origin of the destabilization of straight stream profiles is the centrifugal force f k exerted by a flowing fluid on a curved boundary surface (see Fig. 1). If the boundary forces a narrow stream of fluid to flow along a curve, then the change in momentum of the fluid

Fig. 1. - Forces exerted on a curve in the stream. The centrifugal force f k tries to increase curvature while the surface tension force tries to reduce curvature. The normal of the fS component f g gravitational force tries to slide the stream down the plane. 831 elements, as they move through the curve, must be absorded by the boundary. The resulting force tries to increase the curvature of the stream profile and to deform the cross-section of the stream. The stream becomes longer as a result. The ratio S (L ) of the stream length and geometrical distance L between the initial and final points of the stream is called the sinusuosity. Mandelbrot [5] noted that for rivers, S (L ) has a power-law dependence on L. The instability may be triggered by small initial deformations in the stream cross-section, as was emphasized by Nakagawa and Scott. The increase in length of the stream path is opposed by the surface tension force f which tries to minimize the surface area of the fluid (Fig. 1). At low flow rates, surface tension wins so the stream should be relatively straight. With increased flow rates, surface tension is overcome by the centrifugal force and meandering starts. The third important force is gravity. The component of the gravitational force parallel to the stream-path is reduced in the diagonal sections as compared with that of a stream-path flowing in the direction of steepest descent. This means that the flow velocity also is reduced. Under steady state conditions, the volume flow rate I should be a fixed quantity so the stream cross-section of the diagonal sections must have increased. This indeed is seen experimentally. The diagonal sections become unstable due to the component f g of the gravitational force in the direction normal to the stream-path (Fig.1). This is also the force which is responsible for the sliding. If we rotate a diagonal section towards the horizontal, then the parallel force decreases in magnitude while the normal force increases. The reduction of the parallel force leads, as we saw, to a reduced flow velocity and a concomitant increase in the mass per unit length. The normal force then tries to slide the heavier sections downhill. The aim of this paper is to construct a mathematical model for narrow-stream meandering which is sufficiently simple so that it can be treated either analytically or numerically. Eventhough we are dealing with a simpler problem then river meandering, our model still involves a number of simplifications. The model certainly does not attempt to provide an exact description of the hydrodynamics of the stream. It has however been the experience in growth problems that the large distance geometrical properties of a surface or interface are relatively insensitive to details as long as the basic physical mechanisms are properly included. The hope is thus that the model presented in section 2 is useful for computing large scale properties. Whether this is indeed the case would of course would have to be confirmed experimentally. Some of the limitations of the model are discussed in the final section. In section 3, we first develop an analogy with the theory of continuous phase transitions and discuss the associated « phase-diagram ». In section 4 we compute the relationship between the threshold flow rate 1 c for the onset of meandering and the treshold flow rate Ic2 for the onset of meander sliding. In section 5, we compare with experiment, briefly discuss the dynamical aspects of the problem, such as the sliding velocity, and we finish by reexamining the « Landau » description of section 2.

2. The meander model.

2.1 FORMULATION. - We start by defining the inclined plane down which the stream is flowing. Let r = (x, y ) be a horizontal surface with x e [0, L ] and let y be unbounded. The inclined plane is assumed to have an average height ho (r ) = - ax, with a the tangent of the angle between the horizontal plane and the inclined surface. We already noted that the stream-path appears to be sensitive to height irregularities and chemical contamination. From the theory of wetting [6], we know that chemical contamination has a similar effect as height irregularities and we will include it as such. JOURNAL DE PHYSIQUE. - T. 51, N* 9, lcr MAI 1990 832

We will distinguish two types of height irregularities : (i) « Microscopic » irregularities. These are fluctuations in the height on length-scales less then the stream width 2 R. The stream cannot adjust its course to directly respond to these fluctuations. We will only be concerned with the variation of the average of these microscopic irregularities over area’s of size R2. (ii) « Macroscopic » irregularities. These are irregularities to which the stream-path can respond. There are two contributions : true height irregularities on length scales in excess of 2 R and the statistical fluctuations in the average of the microscopic irregularities over area’s of size R2. We will assume that the surface is flat on large length-scales and ignore the first contribution. The height randomness will be included by adding a (small), uncorrelated, stochastic term to ho. The height profile h (r) is then

Hère, E (r) is a Gaussian random variable with zero average and with a correlation function :

The dimensionless parameter à measures the magnitude of the height disorder. It is assumed to be a small number. According to equation (2.2), there are no correlations in the height irregularities among different parts of the surface. Yet, it should be kept in mind that e only describes the macroscopic fluctuations of the topography of the inclined plane so £ (r ) is smooth on length-scales less then R. The delta-function in equation (2.2) should thus be considered to be rounded on length scales less then R. We will treat R as the short distance « cut-off » : no quantity is allowed to vary on length scales less then R. Next, we turn to the mathematical description of the stream-path. Let r (s ) be the projection of the centre-line of the stream onto the horizontal plane with s the arc-length. The tangent unit vector Î(s) to the projected centre-line is

The curvature K (s) of the projected centre-line is then given by

with û the unit normal. Positive curvature is assigned to bends in r (s ) curved towards the positive y axis and negative curvature in the opposite case (see Fig. 2).

Fig. 2. - Geometry of the inclined plane and the stream-path. 833

There are two parts in determining the equation of motion of r (s ). First we must, for a given r (s ), determine the flow velocity and the stream cross-section. Next, we must use this result to calculate the various forces on r (s ) mentioned in the introduction. To determine flow velocity and cross-section, we assume that the equilibrium contact angle is TT /2 so the cross-section is semi-circular. Let v (s ) be the flow velocity averaged across the stream. It is related to the total flow rate I by

with p the density. The flow rate is a constant under steady-state conditions, so a low flow velocity implies a large cross-section, as seen experimentally. The flow velocity must be determined from the Navier-Stokes equation. We will assume Poiseuille flow. If v (s ) and R (s ) are slowly varying (i.e. if dv/ds v IR and dR/ds 1) then this means that

with n the viscosity and C’ a numerical constant. The quantity in square brackets is the drop in height per unit length along the flow direction. We assume in equation (2.6) that the curvature radius of the stream-path is large compared to R. After using equation (2.5) to eliminate R(s), equation (2.6) reduces to

with a value for R (s ) given by

and C = (2 C’ / TT )1/2. This expresses v(s) in terms of r(s), as desired. Our next problem is to compute the force per unit length, f (s ) n (s ), on r (s ). It is the sum of a centrifugal force fk - resulting from absorption of momentum from the fluid - a gravitational force fg, and a surface tension force f, (see Fig. 1). To compute fk, we note that the momentum dp of a section of fluid of length ds is given by

After a time dt, the fluid element has moved a distance v(s) dt. By Newton’s law, the rate of chance in momentum, (dp (s )/ds ) v (s ), of the fluid element is equal to the force exerted on the boundary by the fluid element. The centrifugal force f k per unit length is then after using equation (2.4). The gravitational force per unit length is

The quantity in brackets is the height drop per unit length in the normal direction. This is the force responsible for the sliding of the meanders. Finally, the surface tension force f S. It can be found as follows. Let y be the interfacial energy between the liquid and air, yls the interfacial energy between the liquid and the 834

substrate, and 7sv the interfacial energy between substrate and air. The surface energy Es is then

From Young’s law it follows that 1’ls = y, if the contact-angle is 7r/2. Equation (2.11 ) can be interpreted as the energy of an elastic string with a line tension 7r yR (s). The restoring force per unit length of an elastic string is the line tension times the curvature so

The minus sign is due to the fact that f, is a restoring force. The total force/unit length is the sum of the three terms. Using equation (2.7) to eliminate R (s ) and v (s ) gives :

with

the « effective » line tension and with

We used everywhere equation (2.7) to eliminate R (s ) and v (s). From the forces exerted on the stream-path, we should now be able to find the equation of motion. The condition for r (s ) to be a stationary stream path is f (s ) = 0 or

For non-stationary stream-paths, we will assume that the normal velocity Vn of the stream- paths is proportional to f (s ) :

with r a dynamic friction coefficient.

2.2 ENERGY CONSERVATION. - Since the stream-path is determined by a combination of dissipative and conservative forces, we cannot (in general) derive equation (2.15) from an energy minimization principle. Energy conservation does give us however a relationship between the average flow velocity and the sinusuosity (ratio of arc-length and L). This is easily derived by noting that the work per second W done by gravity in moving fluid elements from x = 0 to x = L should equal the power P dissipated by viscous losses. The quantity W is equal to

since every fluid element traveling from one end of the stream to another has dropped a vertical distance aL. The dissipation rate is 835

because (n/ C’) v (s)IR2(s) is the average dissipative force/unit volume (see Eq. (2.6)). It follows that if (v2) is the average of v2(s) along the stream-path then

with S(L) the ratio of arc-length and L. Increasing the length of the stream-path reduces the flow velocity of the stream. For a height topography with strong randomness, S (L ) may be a power of L (as for real rivers). In that case, (u2) goes to zero in the large L limit. In our case, where the randomness is weak, we expect S (L ) to go to a constant for large L.

3. The meandering transition.

3.1 LANDAU THEORY. - In this section we will discuss equation (2.15) first from a qualitative « Landau » viewpoint [7] as a guide for the quantitative treatment. The morphology of the stream-path is largely controlled by the effective line tension T(s). For low flow rates the first (positive) contribution to T (s) (oc I1/4) is larger then the second (negative) contribution (oc I3/2) so T(s) is positive (see Eq. (2.14a)). As I increases, T (s ) first increases, reaches a maximum, decreases and then becomes negative. The threshold flow rate I, where T (s ) changes sign is

Note that the threshold depends on the angle of the stream-path. If T (s) is positive, then the line tension tries to straighten the stream-path. The gravitational force favors, on average, a stream profile directed along x We thus expect that Î is close to:k, for positive r (s), since that would satisfy both requirements. The threshold flow rate le for t = i is

For negative T (s), the first term in equation (2.15) tries to increase the length of the stream- path. The most obvious way of increasing the length is by introducing bends and buckles. A natural guess would be that I=I c marks the threshold current I c2 for a dynamic instability. In the following we will investigate what happens to the stream-path for I right around Ic. This means that the tangent Î is always close to k. Note however that even if 1 = k, the cumulative effect of a large number of small displacements in the stream-path could, for large distances L, still lead to a large net displacement along the y direction. This is expected to become particularly noticeable for I close to 1 c where the line tension is small. For I close to Ic, there are considerable simplifications in the equation of motion : since Î = k, the gravitational force is predominantly along the stream direction and the flow velocity, stream cross-section and line tension are to lowest order independent of x. They are given by, respectively, 836

When t = î, it is also convenient to use the following representation for the stream profile :

Since t = î, the derivative dg/dx is small compared to one. By expanding equation (2.15) in ~powers of dg /dx, we get where

We will discuss equation (3.5a) in more detail below. First, we will write it in a different form to exhibit the relation between stream meandering and continuous phase transitions. Neglecting the random contribution in equation (3.5a) we can write it as

with 0 = dg /dx. To solve equation (3.6), think of it as the equation of motion of a « particle » with « position » 0 and « time » x. The particle dissipatively relaxes in a « potential » F (~ ) given by

* For large « time » x, qb must go to a minimum 0 of F (0 ) where aF (0 * )/a 0 = 0. For positive T, F ( ~ ) has a unique minimum at ~ * = 0. By expanding F around 0 = 0, and solving equation (3.6) we find

with 0 (0) the slope at x = 0 and with

The slope 0 of the stream-path thus goes exponentially to zero with a decay length e’. By analogy with the physics of polymers and membranes, we could think of e ’ as a « persistence » length : the stream-path is straight on length-scales small compared to e ’. The « angle-angle » correlation function should decay with e ’ as decay length. Note that ç + goes to zero at 1 = le according to equation (3.9). In reality, we know from the physical meaning of e’ that it cannot be less then the microscopic cutoff R. For negative T, F ( cp ) has two (infinitely deep) minima at

Using equation (2.14b) and equation (3.3),

For large x the slope must assume one of these two values. Apparently, the stream-path spontaneously tilts away from the x axis in either of the two possible directions. 837

In the language of condensed-matter physics we would say that I=Ic is the critical point for a continuous phase-transition. The spontaneously broken symmetry for I> le would be the mirror reflection y - - y while 0 would play the role of order parameter. The corresponding Landau free energy [7] for ~ is F (0 ). As expected from a Landau free energy, F (0 ) has one minimum in the symmetric phase (0 = 0 ) and two degenerate minima in the broken-symmetry phase. The dependence of the order parameter on the control parameter of the transition (the volume flow rate I) is also what is expected from a (mean-field) Landau theory for continuous transitions. However, the persistence length §+ does not diverge at 1 c as would be expected for a correlation length in Landau description. We shall see in section 5 that the Landau picture is not quite valid.

3.2 SYMMETRIC PHASE (I le). - We now will investigate the effect of the randomness (i.e. of e (r)) on the Landau description of the preceding section. We will mostly use methods borrowed from statistical mechanics and consider, wherever possible, equation (2.15) as being derived from a Hamiltonian H. Statistical mechanics problems which involve « quenched-in » randomness, frequently lead to hysteresis, i. e. , with the behaviour depending on the preparation history. For the present purposes, we restrict ourselves to elementary methods which do not take account of the hysteresis but which do help to give intuitive insight. We start with the case I le where the « bare » line tension T has a stabilizing effect on the stream-path. The term T’ (dg /dx )2 is stabilizing as well. If dg ldx « 1, as we have been assuming, we may neglect this term in comparison with T. Neglecting this term, equation (3.5a) reduces to

We can obtain intuitive insight into the long distance behaviour of equation (3.11) by assuming a parabolic stream-path : g (x ) = W(L) (xlL)2 with W(L) « L. The first term in equation (3.11) is of order W (L )/L2 while the second term is if order W(L)/03BE+ L. For L > 03BE +, the first term is neglible compared to the second term. We can estimate the magnitude of last term from its RMS average. The total random force on a straight stream-path of length L is proportional to L 1/2 using the central limit theorem. The force per unit length then scales as 1/L1/2. This random force tries to displace the stream- path away from W(L ) = 0. By balancing it with the restoring force/unit length, proportional to W(L)/L, we find that W (L ) is proportional to L 1/2 which would mean that the stream-path behaves as a random walk. We will investigate the limits 03BE+ -> 0 and § + -.> oo to find the proportionality constant between W (L ) and L 1/2. First consider the critical point I = le where e’ = 0. The resulting equation,

states that the tangent t = (1, dg/dx) is parallel to Vh. The stream-path is entirely determined by the topography of the surface. More precisely, the stream flows perpendicular to constant height contour lines. For our particular choice of the surface topography, equation (2.1), lines of steepest descent are random walks with a bias towards the negative x axis (directed random walk). It is easy to estimate the displacement along the y direction for equation (3.12). The stream- path can only undergo of order L/R sidewise displacement steps (a more rapid variation is unphysical). The typical value of aEl8g for such a step is of order a. According to equation (3.12), the sidewise displacement per step is then + ARla so the RMS displacement 838

W(L) after L/R steps is of order using again the central-limit theorem. Next we consider the limit of large g+ and, consequently, large T. In this case, g (x ) can vary only slowly. Let (g ) be the average of g (x ). For large T, g (x ) must remain close to (g ) . We will use a perturbation expansion in powers of d and of the correction g’ = g - (g ) . To lowest order :

This equation can be treated by once more applying an analogy with mechanics. Let g stand for particle position and x for time. The equation is then of the form of a Langevin equation for a particle with unit mass and friction constant (03BE+ )-1 exposed to a noise source. The correlation function of the noise is using equation (2.2). The effective noise « température » T in this « Langevin » equation is given by kb T = L12 R/ a 2 03BE + , using the fluctuation-dissipation theorem [8]. The statistical properties of the solutions of the Langevin equation are well known. For L > § + , the RMS displacement W(L) = (g(L)2)1/2 is

where the diffusion constant D is given by Einstein’s law D = g + kb T or

This result for W(L) for 1 - Ic reproduces equation (3.13) for 1 = Ic. For large L, W(L) is apparently independent of I which is somewhat surprising. It is also known from the theory of the Langevin equation that the correlation function of dg’ /dx is given by

The « step size » of the random walk W(L ) oc L1/2 is, according to equation (3.18), the persistence length § +. As discussed in the conclusion, this is a fairly large length-scale away from the « critical point » I = 1 C. We should expect the random walk behaviour to break down for length scales less then e’. Finally, note that since the sidewise displacement W (L ) oc L l/2 is small compared to L (for large L) the arc-length scales as L in the large L limit. The sinusuosity S (L ) thus should go to a constant for large L.

3.3 BROKEN SYMMETRY PHASE (I > Ic). - In this regime, we know from experiment that the stream-path makes an angle with the direction of steepest descent. We will thus look for solutions to equation (3.5a) consisting of diagonal sections of non-zero average slope, say dgo/dx. We thus assume 839 with the average fluctuation of the slope, (dg’/dx) , equal to zero. To lowest order in g’, equation (3.5a) becomes where and where e(x, y) has the same statistical properties as E (x, y ). For equation (3.20) to be meaningful, 03BE - should be positive eventhough T 0. Equation (3.20) for I > Ic looks very similar to equation (3.11) for I I,,. There is however an important difference. A finite « DC bias », (03BE- )-1 dgo/dx, has entered. If we would use the Langevin method of the preceding subsection we would find that g’ is always either increasing or decreasing with x. This is inconsistent with our original assumption that (dg’ /dx ) = 0. According to perturbation theory, there are thus no static solutions of equation (3.5a) above Ic. If true, this would mean that we would have to identify Ic with the upper critical current 1 C2 for dynamic instability. It is however easy to see that we actually can construct static solutions by allowing sharp bends in the stream-path. Assume we have a diagonal section of length L. The typical value of the random term in equation (3.20) is then of order =+= t1/ (cr 03BE -). For a diagonal of length L, the average value of the random force, f (L ), is of order

This random term must exceed the DC bias (03BE-1)-1 dgo/dx in order for the diagonal section to be stable. We conclude that equation (3.20) can be satisfied only for diagonal sections of a length no greater then À (1 ) where f (À (I )) = (e- )-1 dgo/dr or

We now construct a stream-path by joining the diagonal sections (see Fig. 1). Each section has a length L less then À with the slopes of the sections alternating between + dgo/dx. The sections are joined by sharp bends. Let the curvature radius of such a bend be p o (with p o > R). The total outwards force on the bend due to the centrifugal force is of order p o (- t/po). This force must be compensated by the line tension ? + T’ (dg /dxo )2 in the two diagonals joined at the bend (see Fig. 1). Under steady-state conditions, the total force on the bend must be zero. By the principle of virtual work we find

Comparing equations (3.21) and (3.24), we see that equation (3.21) guarantees that e’ is positive. For small T, equation (3.24) reduces to

which satisfies our requirement that dgo/dx exceeds cp +. The persistence length e - and the maximum meander size À are then (see Eqs. (3.21) and (3.23)) 840

Note that e - has a power-law dependence different from e ’ and that dgo/dx, unlike 0+, does not behave as the order parameter of a Landau theory. Like §+ , §- does not diverge at 1 C. We have left an important question unanswered. The largest value of the length of the diagonal sections is A (I ). Yet, we could as well construct zig-zag paths of much shorter sections, i.e. of size R. We will discuss in the conclusion why it seems likely that A (I ) indeed is the characteristic length scale of the meanders. In general, there can be many possible stream trajectories above 1 c and we indeed should expect extensive hysteresis.

4. Upper critical current. We have found that for I greater than I c, there are two characteristic length scales in the problem : the meander size À and the persistence length e-. At Ic, À is infinite and ç - zero (or rather R). From equation (3.26b) we see that with increasing flow rate À drops rapidly (for small à) while e - increases with I (Eq. (3.26a)). If À drops to a value less then R, then our zig-zag construction clearly is unphysical. One thus would guess that a (Ic2 ) = R marks the threshold for the dynamic instability. This would however disagree with the experimental observations : the characteristic length scale of the meanders does not shrink to R at Icz. In this section we will compute the actual value of Ic 2’ The predicted relation between 1 c and 1 Cz should be an important test of the model. Assume that I is far enough above Ic for the condition R À « to be valid. In that case, we could neglect (ç - 1)- 1dg’/dx with respect to d2g’/dx2. If we wanted to compute the scaling properties of g’ (x ) then we would in fact never be allowed to neglect this term since, as we saw, it always overwhelms the term d2g’/dx2 in the large L limit. However, the upper critical current does not depend on the large L scaling properties as we shall shortly see. Neglecting dg’ /dx in equation (3.20) gives

This equation has been studied in detail in a different context. A domain wall in a random- bond ferromagnet in the presence of an applied magnetic field obeys a stability conditions which has the same mathematical form as equation (4.1). The relationship between the two problems is as follows : the interfacial energy of the magnetic domain wall corresponds to 03BE - , the applied magnetic field corresponds to dgo/dx and ê’ (r) corresponds to the randomness in the exchange energy. We now can directly translate the known results from the magnetic analog to our problem. For dgo/dx = 0 (i.e. I le)’ the transverse displacement W(L) of an « equilibrium » [9] domain wall is known to depend on L as

where the « roughening » exponent [10] C = 2/3 and where the amplitude A is [11]

These results are of course only valid up to L =-z e - since for larger L we could not neglect dg’ /dx. We thus expect that W (L ) oc L2J3 for L less then e - - For finite dgo/dx (i.e.I > le)’ the random force can prevent sliding if dgo/dx is less then some critical value. The stream-path adjusts itself to the local randomness to make optimal use of the available the random force in The DC - pinning strength supplied by equation (4.1). bias can only trigger sliding if it can dislodge the stream-path out of these energy minima. 841

In the magnetic analog, this critical value corresponds to the coercive field. Translating the known expression [12] for the coercive field to our problem leads to the requirement that dgo/dx must be less then 0,(L) where

Since ~Ce (L ) increases with decreasing L, the critical depinning strength is dominated by the short distance behaviour of g’ (x ). This justifies a posteriori our neglect of dg’ / dx in computing Ie2. The shortest allowed L is the value Le for which W(Le) == R. From equation (4.2) it follows that

or, using equation (4.3),

The physical meaning of Lc is that of a roughening length. The stream-path is smooth on length-scales less then L, and rough on length-scales greater then Lc. Obviously, LC must exceed the persistence length e-. The critical value for depinning 0,(Lc) is given by

Note that the largest allowed value of 0 c corresponds to the smallest value of 03BE - . Since e- is of order R near 1=IC, the critical 0,, value is of order (4/ a )4/3 near Ic. With increasing I, e - increases so Oc decreases. At the same time, dgo/dx increases with 7 (Eq. (3.25)). When dgo/dx exceeds Oc, we lose the static solutions. The critical current to the From IC22 corresponds point dgo/dx = Oc. equations (3.25), (3.26a) and (4.6) it follows that

For this result to be consistent, we must require that the length k (I) exceeds R at I = 1 Cz. From equations (3.26b) and (4.7)

The validity condition for equation (4.7) is then

so the randomness cannot be too weak. If the opposite case, A -- apqRlIc, holds then we should use our earlier estimate À (12 ) = R. In that case it follows from equation (3.26b) that

The theory thus predicts that for very smooth and very clean surfaces, i.e. surfaces for which the inequality of equation (4.9) does not hold, the size of the meanders indeed are of the order of R at the onset of meander sliding, as we guessed earlier on. As we saw, roughness or contamination can lead to larger meanders. 842

5. Discussion and conclusion.

5.1 COMPARISON WITH EXPERIMENT. - How do the predictions of the model compare with the experimental results ? We found that below a critical current Ic the stream-path is a random walk. For low I, the stream-path is relatively straight while near IC the side-wise wandering becomes more pronounced. We also found that above 1C, there is a regime where we still have stationary stream-paths but where they now exhibit zig-zag patterns. The angle of the zig-zags goes continuously to zero at IC. With increasing I, we reached a second threshold such that above there are no static The existence of the I C2 I C22 stream-paths. intermediate regime of static meandering is due to the presence of the height irregularities. For a perfectly smooth substrate we found 1 c = Ic 2. These results do appear to qualitatively reproduce a number of the observations cited in the introduction. To make a quantitative comparison, we will take a stream of water with p = 1 grlcm3, q = 10-2 cm2/s and y = 70 erg/cM2 . For a slope a = 0.1, the critical current 7c is of order 0. 1 gr/s, using equation (3.2). From equation (3.3a), it follows that v is of order 10 cm/s and R of order 1 mm at lC. These values compare reasonably with experiment. Starting with the case of a rough substrate, we could have height irregularities of order 100 U in which case the parameter àla would be of order 1. For L = 1 m, W(L) is then of order 1 cm (Eq. (3.13)). Near Ic, where ç+ goes to zero, one should see the random-walk behaviour most easily. Away from Ic, ç+ is of order 10 cm (Eq. (3.9)). To observe randomwalk behaviour below 1 c would require that the system size L is large compared to 10 cm. To find the upper critical current, we note that the inequality equation (4.9) is satisfied for A = 0.1 so we should use equation (4.7). The resulting upper critical current Ic2 is rather close to Ic : I,, 2/IC - 1 --- 0.06. The length À of the diagonal sections at 1 = 1 C2 is of order 1 cm (Eq. (4.8)). The critical angle oc for the zig-zags (Eq. (4.6)) is of order 0.1. For a smoother substrate, with A/a of order 10- 1, the intermediate phase has a negligible range in I and the meander size at threshold should be of order 0.1 cm, while for very smooth substrates, with 11/ a less then 10- 2, the meanders at threshold have a size of order R because the inequality equation (4.9) is violated. The range of the intermediate phase with static meanders appears to be too small compared with experiment. The most likely source of the problem is the neglect by the model of contact angle hysteresis. The real contact angle can deviate from Young’s law and in fact varies between a maximum and a minimum value [13]. For a small applied force f, a stream line can then absorb the force by deforming the stream cross-section [3]. On one side of the stream (leading edge), the contact angle approaches its maximum value and on the other side (trailing edge) its minimum value. Our stability condition f = 0 for the stream-path is thus not valid in the presence of contact-angle hysteresis. One could phenomenologically include contact-angle hysteresis through a static friction term. More precisely, the absolute value of f must exceed a threshold force f C before it can move the stream profile. The corrected equation of motion would be

It is easy to see that for large f C, 1 Cz indeed becomes large as well. 843

How about the lower critical current 1C1 ? The lower critical current is related to the Rayleigh instability [14]. For I = 0, a semi-circular cylinder of fluid is unstable against droplet formation as was shown by Plateau. If we investigate the stability of equation (2.7), then the Rayleigh instability is encountered for any finite I. Experimentally, free-falling streams of water are indeed always subject to the instability [15]. For streams flowing down an inclined plane, the Rayleigh instability is only seen for small inclinations and then only for low volume flow rates. The reason of this stabilization effect of the substrate must be that the total dissipation rate by viscous loss of individual droplets rolling down the plane is in excess of that of a continuous stream. A proper stability analysis would require us to go beyond the Poiseuille approximation. We simply assumed stability on the basis of the experimental observations. The use of the Poiseuille approximation could lead to other problems. Experimentally, some turbulence is observed [4]. It is seen in particular if the flow rate is suddenly increased for an initially straight stream-path. However, meanders making an angle with the direction of steepest descent have a reduced flow rate and thus a reduced Reynolds number. Although turbulence is likely to be important for time-dependent, transient stream-paths, we do not expect it to be a very significant effect for stable meanders.

5.2 DYNAMICS. - We did not address the formation dynamics of the meanders. The dynamics could however be important in establishing the length scale of the meanders above IC. Recall that in section II we only established an upperbound k (1 ) for the size of the meanders. The observed meander size could be the one which dominates during the growth stage of the meanders. Since the regime I Ic 2is hysteretic the actual meander size may not be a unique. If we use a linear stability analysis in the equation of motion (Eq. (2.16)) then one finds that at early times, the most rapidly growing modes have a wavelength of order R, the smallest allowed value. At later times however, the opposite happens : the largest amplitude mode has the largest allowed wavelength, i.e. A (I ). For this reason, we have assumed that À (1 ) is the typical size of a meander. The dominant stream-path is then the one with the minimum number of sharp bends. Another important dynamical question is the velocity V of the sliding meanders above IC2. Here another analog is of use. We need to solve the dynamic equation of motion equation (2.16). Using the same approximations as in section 3, one arrives at

This equation has been studied for a special class of random potentials, namely with F periodic in y (I.e. é(x, y ) = h (x ) cos (y - gi (x ) ) with h and gi random). For this random potential, equation (5.2) can be mapped onto the equation of motion of a Charge Density Wave (CDW). The position g’ plays the role of the CDW phase and ~ + that of the DC electrical field bias. Within mean-field theory it can be shown [16] that the sliding velocity goes to zero as a powerlaw in the DC bias. For the present problem, this means that we expect that

The exponent £ will be different from that of a mean-field CDW (e = 3/2) because the random potential is not periodic in y and because of corrections to mean-field theory. If the analogy is valid then the depinning at I C2 is a « dynamical critical phenomenon » i. e. we should expect critical dynamical slowing down near IC2. Very long relaxation times were indeed 844

observed by the author for I near IC 2when he tried to reproduce the experiments. A further discussion of dynamical critical transitions is given in reference [16].

5.3 LANDAU THEORY REVISITED. - How well does the simple Landau description of section 2.1 stand up ? The correlation function for the order parameter 0 = dg /dx shows exponential decay for 7 I C (Eq. (3.18)). This agrees with a Landau theory of the symmetric phase. However, for I > 1C2 there is a severe problem. The symmetry y - - y is not really broken. We were forced to introduce sharp bends which connect the two minima of f (~ ). In a Landau theory such objects correspond to domain walls - metastable defects of the broken symmetry phase. In our case, an array of sharp bends is always present. They are an intrinsic feature of the regime I > 1 C. Assume we expand the winding stream-path above Ic in a Fourier series. If the Fourier amplitudes are dominated by a narrow range of wavevectors then we could consider the associated amplitude as a more appropriate order-parameter. In that case, the broken symmetry would be translation-invariance instead of mirror-reflection. Determining whether or not this would be the more appropriate description would depend on further numerical or experimental work. The uncertainty concerning the real order-parameter and the nature of the regime I > IC raises the question whether or not there is a true phase-transition at IC. It coult also be a crossover from a relatively straight stream-path with small Fourier amplitudes to a more winding one with larger amplitudes. Not that this issue does not affect the existence of the depinning transition at 1 Cz. Despite these questions, the analogies between stream-meandering and interfaces in random media in condensed-matter problems are clearly useful on the descriptive level. To what extent these analogies are also quantitatively reliable for computing, say, 1 Cz is a question which must be addressed experimentally.

References

[1] The word meander derives from M~03B103BD03B403C1o03C3, the classical name for a river in present day Turkey which prominently shows this effect. [2] Fluvial Processes in Geomorphology, Eds. L. B. Leopold, M. Gordon Wolman and J. Miller (W. H. Freeman) 1964. [3] NAKAGAWA T. and SCOTT J. C., J. Fluid Mech. 149 (1984) 89. [4] WALKER J., Sci. Am. 253 (1985) 138. [5] MANDELBROT B., The Fractal Geometry of nature, Ch. IV (Freeman) 1982. [6] DE GENNES P. G., Rev. Mod. Phys. 57 (1985) 827. [7] For a discussion of continuous phase transitions, see LANDAU L. and LIFSHITZ E., Statistical Physics, Ch. XIV (Pergamon Press, Oxford) 1970. [8] FORSTER D., Hydrodynamic Fluctuations, Broken symmetry and Correlation Functions, Ch. 6 (Benjamin, Reading) 1975. [9] By equilibrium stream path we mean here the following. For 03A6 + = 0, equation (3.27) is of Hamiltonian form and can be derived from a variational energy H. We can then average over all stream paths with the Boltzmann factor e-H. This leads to a W(L ) proportional to L03B6. See for instance, KARDAR M., J. Appl. Phys. 61 (1987) 3601. Whether this averaging procedure is correct is a rather delicate question for surfaces in a quenched random environment. In our case, if the flow source contains a sufficiently large white noise component then the method could be valid. Equation (4.4) is however believed to be valid under non-equilibrium conditions. 845

[10] HUSE D. A. and HENLEY C. L., Phys. Rev. Lett. 54 (1985) 2708 ; HUSE D. A., HENLEY C. L. and FISHER D. S., Phys. Rev. Lett. 55 (1985) 2924. [11] NATTERMAN T. and RENZ W., Phys. Rev. B 38 (1988) 5184. [12] NATTERMAN T. and VILFAN I., Phys. Rev. Lett. 61 (1988) 223. We are assuming that the « weak- pinning » case applies for small 0394. [13] BLAKE T. D. and HAYNES J. M., Prog. Surf. Memb. Sc. 6 (1973) 125. [14] BOUASSE H., Capillarité (Paris, Librairie Delagrave) 1924, Ch. VII ; CHANDRASEKHAR S., Hydrodynamics and Hydrodynamic Instability (Clarendon, Oxford) 1961. [15] Lord RAYLEIGH, Scientific Papers (Cambridge) 1899, 361. [16] FISHER D. S., Phys. Rev. B 31 (1985) 1396.