Length Scales for Continental Deformation in Convergent, Divergent ...

JOURNAL

OF GEOPHYSICAL

RESEARCH,

VOL. 90, NO. B5, PAGES 3551-3557, APRIL

10, 1985

Length Scalesfor Continental Deformation in Convergent, Divergent, and Strike-Slip Environments' Analytical and Approximate Solutions for a Thin Viscous Sheet Model PHILIPENGLAND,GREGORY HOUSEMAN, ! AND LESLIESONDER HoffmanLaboratory,Departmentof GeologicalSciences,Harvard University,Cambridge,Massachusetts

The deformation of a thin viscouslayerthat hasa moving•boundary is investigated for comparison with zones of deformation in the continental lithosphere.Exact analytical soluti6ns, for the case of a Newtonian fluid, and approximatesolutions,for the caseof fluid with power law rhedlogy,show that: When the imposedvelocity vector is normal to the boundary (com.pressionalor extensionalregime) the deformation field decays away from the boundary with a characteristiclength scale 1/3 to 1/10 the wavelength of the imposed boundary velocity distribution for n between 1 and 10, where n is the stress-strainexponent in the rheology; in contrast, when the imposed velocity vector is parallel to the boundary (transcurrentregime), the length scale of the deformation field is approximately 4 times

smaller.In eachcasetheselengthscalesdecrease approximately as n-x/2. The differencein lengthscales arises even in the absenceof any buoyancy forces acting on thickened or thinned crust; such forces would modify the ratio of length scales,but not sufficientlyto affectthis result.

1.

INTRODUCTION

is the pressure.Thus in the horizontal (x, y) plane the force balancesmay be expressedby

Recently,severalauthorshave presentedmodelsfor the deformation of the continentsthat regard the continental lithosphere as a continuum deforming in response to boundary

C•Zxx c•z•, x C•Zzx c•p

o-5+ -Tf + Oz - Ox

conditions imposed by plate interactions [Tapponnier and

Molnar, 1976; Bird and Piper, 1980; Englandand McKenzie, 1982; Vilotte et al., 1982; G. A. Houseman and P. C. England, unpublishedmanuscript, 1984]. These solutions-havemainly been obtained by numerical techniques;the purpose of this paper is to present analytical and approximate solutions to the equations governing the deformation of a thin viscous

(4a)

and

C•Z x•, c•z y•, C•Z zy c•p

+T/y+

=

(4b)

For a Newtonian fluid,

'r0 = 2r/gij (5) sheet [England and McKenzie, 1982; G. A. Houseman and P. C. England, unpublishedmanuscript, 1984] that bring out the wherer/is the viscosity and the strainrate g•jis definedas length scalesinvolved in the deformation. In particular, this model predictsconsiderablyshorteracross-strikelength scales for deformation under strike-slip boundary conditions than for deformationunder compressionalor extensionalboundary conditions.Analytic solutionsare obtained only for the caseof in termsof the componentsof the velocityvectoru. The velocity satisfiesthe incompressibility condition a Newtonian fluid, and approximate solutions are given for

•'•=• •c•xj +c•x,/

the non-Newtonian

case.

v. u =0

2. SOLUTIONSFOR FLOW IN A SEMI-INFINITE, CONSTANTVISCOSITY,HORIZONTAL LAYER WITH

FIXED BOUNDARY

VELOCITIES

The force balancefor creepingflow is

63aij/r3x j = pgai

(1)

(6)

(7)

In section 3 we obtain approximate solutions for nonNewtonian fluids, but in this section we assumethat r/ is a constant. We seek solutions for the velocity in the layer -L < z < 0, representingthe continentallithosphere,and in the semi-infiniteregion of the x-y plane y > 0, subjectto two different sets of boundary conditionson the boundary y- 0

(Figure 1). To model compression and extensionin the y didensity,and aij is the (i, j)th componentof the stresstensor. rection, where a = (0, 0, 1), g is the accelerationdue to gravity, p is The deviatoric

stress tensor is

u(x, o) = o

'rij= aij + (•ijP

(2)

p = --«akk

(3)

where

(8a)

v(x, O) = Vocos (2r•x/it)

(see Figure la) or, alternatively,for a tangentialshear condition,

u(x, O) = Uo sin (2r•x/2)

• Now at ResearchSchoolof Earth Sciences,AustralianNational

(8b)

v(x, O) = 0

University, Canberra.

Copyright 1985by the AmericanGeophysicalUnion.

(seeFigure lb).

In treating the continentallithosphereas a continuumthe incompressibilitycondition (equation (7)) has usually been

Paper number 4B5081. 0148-0227/8 5/004B-5081$05.00 3551

3552

ENGLAND ET AL.' LENGTH SCALES FOR CONTINENTAL DEFORMATION

v½=0

Y= Yrnax or y--•

Y = Ymex or B

• TY,V

ly,v •

Equation (11) may be solvedby means of Fourier transform techniques,with the boundary conditions(equation (8a)) and the constraint that the velocitiesand nonhydrostaticpressure approach zero as y• cc [e.g., Turcotte and Schubert,1982]; this gives

2

x,u

½ = Vosin(2r•x/2)e-2'•y/XE2/2r• + y] u = Vosin(2r•x/2)e-2'•y/XE2r•y/2]

x

x

x=

x=• x=O

x:•

v = Vocos(2r•x/2)e-2'•Y/x[1 + 2•y/2] v(x,O):Vocos 2'n'x X

u(x,O): Uosin 2'trx

u(x,O) = 0

v(x,O)= 0

x

Y = ymax

p = 2r/• cos(2r•x/2)e-2'•y/XE2r•/2]

l• = Vox/•(2•/2)2e - 2•y/XEy]

Y =Ymax

(12a) (12b)

(12c) (12d)

(12e)

where /•, the secondinvariant of the strain rate tensor,is

C

definedin (17). The velocitycomponentsof this solution are illustrated in Figure 2a.

The correspondingsolutionswhen the specifiedvelocityis parallelto the boundaryy = 0 (equation(8b))are

y,v

:- x,u

½ = - Uo sin(2•x/2)e-2•y/X[y] x

x:-• x:•

x:O

•: Vo•O• -•

v:0

u:0

u (x,O) = 0

u = Uo sin(2rcx/2)e-2•Y/x[1 -- 2•y/2]

x=X

2'n'x

u:Uos•n•

(13a) (13b)

v = - Uo cos(2rcx/2)e2'Y/x[2rcy/2]

(13c)

p = -2r/Uo cos(2rcx/2)e-2"Y/x[2rc/2]

(13d)

u:0

v (x,O) = 0

1•= Uov/•(2•/i)e-2'y/Xl(2ny/2)II (13e)

Fig. 1. Schematic boundary conditions for continental defor- , mation zones. In each case the continent is viewed from above; verti-

cal averagesof the stressand velocity fields in horizontal directions are calculated.In Figures la and lc, influx boundary conditionsare imposedon the x axis with y velocity v specifiedas shown and x velocity u fixed at zero. In Figures lb and ld, tangentialvelocity conditions(strike slip) are imposedon the x axis. All other boundaries are kept rigid except where reflectingboundary conditions,as shownby the arrow pairson eachsideboundaryof Figuresla and lb and the left-hand side of Figure l c, allow slip parallel to the boundary. Note that the x and y scalesdiffer in thesefigures.A finite Ymax appliesto solutionsobtainednumerically;the analyticalsolutionsare

The velocity componentsof this solution are illustrated in Figure 2b. 2.2.

Thin ViscousSheet Approximation

The equationsgoverningthe deformationof a thin viscous

sheetare givenby BirdandPiper[1980]andEngland and

McKenzie [1982]. In contrastto the previoussection,where the verticalstrainrate gzzwas zero everywherein the layer, it is assumedhere that the vertical stressazz on the top of the obtained in the half spacey > 0. thin layer (z = z•) is zero. We considerverticalaverages(decombined with the force balance equation (equation (1)) by noted by a bar over the variable)of the rheologyand of the assumingfor the deformationeither a plane strain geometry stresses within the deformingsheetand assumethat the gradi[e.g., Tapponnierand Molnar, 1976] or a thin sheetgeometry entsof topographyand crustalthicknessare very small.Thus [Bird and Piper, 1980; Englandand McKenzie, 1982; G. A. from the verticalcomponentof (1), Housemanand P. C. England,unpublishedmanuscript,1984]. In each of these models the horizontal velocitiesare indepenp = rzz-pg dz' (14) dent of depth,but in the first casethe additionalassumptionis made that the vertical velocityis zero. We investigateboth of If the lithosphereconsistsof a crustof thicknessS and density

z'

these cases below.

Pcoverlyingmantleof densityPmand if we assumelocalcompensationat the baseof the lithosphere, z = -L, and take the

2.1. Plane Strain Deformation Plane strain deformation

of the horizontal

sheet occurs if

vertical average of (14),

the verticalvelocityis zero everywherewithin the layer and if

t,c

all variables are independent of the vertical coordinate z. From (7),

au/ax = -av/ay

(9)

where u and v are the x and y componentsof u, and we may introduce a stream function ½, defined by

u= -a½/ay v = a½/ax

/t,m)S + t'mœ 2

Taking the vertical averageof (4) and assumingzero shear stressat the top and bottom of the layer reducesthe 7z,•and

qzytermsto zero.Weassume thata constitutive relationofthe form

?u= B/•(X/nx)•U

(16)

(10)is validfor theverticalaverages(?u)of thehorizontalcompo-

Differentiating (4a) with respectto y, (4b) with respectto x, subtractingthe latter from the former, and using(5), (6), and (10) givesthe biharmonicequation

nents(zu) of the deviatoric stresstensor[seeEnglandand McKenzie, 1982,AppendixA; England,1983]. B is a constant which here includesthe depth-averagedtemperaturedependenceof the lithosphererheology,n is a constantgreaterthan

ENGLAND ET AL.' LENGTH SCALES FOR CONTINENTAL DEFORMATION

3553

y : ;•/2

!

/

...

_

i

I

/

_

I

i//_ \

(a)

I

x=O

x = X__}_ x=O 2

x- 2

Fig. 2. Velocityvectorsfor planestraindeformation of the semi-infinite layer,calculated from (12) and (13).(a) Deformation in theregion0 < x < )¾2,0 < y < )¾2,withtheboundary condition of Figurela. Vectors havetheirorigins at a gridof pointsthatarespaced at )¾16in thex andy directions. Thex andy scales arethesame.(b)LikeFigure2a except fortheboundary conditions of Figurelb. Linesat thebaseofeachfigurehavelengthsVoand Uo.

or equalto one (for n = 1, B = 2r/),and /• is the secondin-

To obtain the velocity components,we substitute(22) into (20),with Ar = 0. Usingthe boundaryconditionsto determine the constantTx givesfor the indentingboundaryconditions (equation(8a)),with linear theology,

variant of the strain rate tensor'

l• = (gijdi) •/2

(17)

Substituting (16) and (17) into (4) and nondimensionalizing givesan equationfor the horizontalcomponents of the velocity:

V2a= -3V(V. a) + 2(1- 1/n)t•-•

contributions

(23b)

This velocityfield is illustratedin Figure 3a. The corresponding solutionswhen the specifiedvelocityis parallelto the boundaryy = 0 (equation(8b))are

(19) of the forces

arising from variations of crustal thicknessand the forces re-

quiredto deformthe fluid at the referencestrain rate of uo/L. The last term in (18) representsthe contribution to the force

(23c)

(23d)

gPc(1- Pc/Pm)L

of the relative

cos(2r•x/2)e-2•y/it[1 + }(2•y/2)]

l• = Vox•(2r•/2)e-2"Y/x{[« + •(2•y/2)] 2+ • cos 2(2r•x/2)}•/2

[V/•. i + (V. fi)V/•]+ 2Arl•(•- 1/n)svs (18)

Ar is a measure

(23a)

'•zz= •r/Vocos(2•:x/2)e-2"Y/x[2/r/2]

[Englandand McKenzie, 1982,equation (17)], where the operators include only the horizontal derivatives. Lengths have been made dimensionlessby L, velocitiesby u0, and time by L/u o. Ar is the Argand number [England and McKenzie, 1982], given by

Ar= B(uo/L),/.

a = V0sin(2r•x/2)e-2ny/A[}(2•y/2)]

a = Uo sin(2r•x/2)e-2.y/•[1- •2•y/2]

(24a)

t5= -- Uo cos(2r•x/2)e-2.y/St[ •2•zy/2]

(24b)

'•z•= -- -}r/U0cos(2r•x/2)e-2"Y/x[2•:/2]

(24c)

l• = Uov/•(2r•/2)e-2"Y/x[(•(2r•y/2)•)2+ • cos 2(2r•x/2)]•/2 (24d)

balance from variations in crustal thickness $. For a Newton-

Thesevelocitycomponentsare illustratedin Figure 3b. In Figure 4 we comparethe y coordinatedependenceof the principal velocity component(v for indentingboundary conV2fi= -3V(V. fi) + 2Ar SVS (20) ditions, and u for strike-slipconditions)for the plane strain and the thin layer solutions.The velocitysolutionsfor the two Differentiating the x equation with respectto x and the y modelsdiffer only by the factorof-3 s that appearsin the thin equationwith respectto y and adding gives layer solutions.The minor velocity components(equations V2'•zz '-- -- ¬Ar V2S2 (21) (23a) and (24b)) and the pressure(rzz in (23c) and (24c))differ from their counterpartsin the plane strain case by constant where, for a Newtonian fluid, we have made stressesdimen- multiplicativefactorsof 3/5 and 2/5, respectively. sionlessby dividingby flUoiL. 3. THIN VISCOUS SHEET WITH NON-NEWTONIAN We shall neglectthe contributionsdue to crustal thickness RHEOLOGY in what follows, which is equivalent to assumingthat the The equation for the non-Newtonianviscoussheet(equalithospherehasvery high viscosityor to consideringa stagein tion (18)) is strongly nonlinear, so only approximate solutions the deformation at which no appreciablecontrastsin crustal can be obtained. We assumethat the velocity component thicknesshave been achieved[Englandand McKenzie, 1982]. normal to the imposed velocity boundary condition can be (Seesection4 for further discussion.) neglected;for example,the y componentof (18), for the inThe dimensionlessvertical stressthen satisfiesLaplace's denting boundary condition, assumingthat u - 0 and Ar = O, equation,and it followsfrom the form of the boundary con- is ian fluid (n - 1), (18) reducesto

dition

that

7'zz= r•e - 2ny/x cos(2•x/2)

(22)

V2•5 =-3•+2(1-1/n)/•-' a2•5 a•5 a/• ---•xx (25) OY 2•yy•yy +2ax

3554

ENGLAND ET AL.' LENGTH SCALESFOR CONTINENTAL DEFORMATION

I / • '- x \ •

/ /

-" \ I

II/-xXl

. ' ! I

x=0

x=

• , ß- - '

' ....

(b)......

' '

.

x=0

x- 2

Fig.3. LikeFigure2 except thatsolutions areforthedeformation of a Newtonian thinviscous sheet (equations (23) and(24))'(a)velocity vectors fortheboundary condition of Figurela and(b)velocity vectors fortheboundary condition of Figure lb. where

4.

/• -- x/•[(&y/0y) 2+ ¬(&5/Ox)2] •/2

ACCURACY OF APPROXIMATE SOLUTIONS

Numerical solutionsto the governingequation are obtained by the techniquesdescribedby England and McKenzie [1982, Equation (25) may only be further simplifiedif we assumethat Appendix B]. The analytical solutions are obtained for the (&Vc•x) 2