The statistical mechanics of meandering

HAL Id: jpa-00212412

https://hal.archives-ouvertes.fr/jpa-00212412

Submitted on 1 Jan 1990

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

The statistical mechanics of meandering

R. Bruinsma

To cite this version:

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.

Classification

Physics

Abstracts 05.40 - 46.30 -

68.42

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

(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. -

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

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. -

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

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

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

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

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

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

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

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

_{pinning strength}

_supplied

_by

the random force in

_{equation (4.1).}

The DC

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

_IC2

2

corresponds

to the

point

dgo/dx

= Oc.

From

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

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

_{I C2}

such that above

_{I C2}

2 there are no static

stream-paths.

The existence of the 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

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

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